(* ========================================================================= *) (* Lebesgue measure, measurable functions (defined via the gauge integral). *) (* *) (* (c) Copyright, John Harrison 1998-2008 *) (* ========================================================================= *) needs "Library/card.ml";; needs "Library/permutations.ml";; needs "Multivariate/integration.ml";; needs "Multivariate/determinants.ml";; prioritize_real();; (* ------------------------------------------------------------------------- *) (* Lebesgue measure in the case where the measure is finite. This is our *) (* default notion of "measurable", but we also define "lebesgue_measurable" *) (* further down. Note that in neither case do we assume the set is bounded. *) (* ------------------------------------------------------------------------- *) parse_as_infix("has_measure",(12,"right"));; let has_measure = new_definition `s has_measure m <=> ((\x. vec 1) has_integral (lift m)) s`;; let measurable = new_definition `measurable s <=> ?m. s has_measure m`;; let measure = new_definition `measure s = @m. s has_measure m`;; let HAS_MEASURE_MEASURE = prove (`!s. measurable s <=> s has_measure (measure s)`, REWRITE_TAC[measure; measurable] THEN MESON_TAC[]);; let HAS_MEASURE_UNIQUE = prove (`!s m1 m2. s has_measure m1 /\ s has_measure m2 ==> m1 = m2`, REWRITE_TAC[has_measure; GSYM LIFT_EQ] THEN MESON_TAC[HAS_INTEGRAL_UNIQUE]);; let MEASURE_UNIQUE = prove (`!s m. s has_measure m ==> measure s = m`, MESON_TAC[HAS_MEASURE_UNIQUE; HAS_MEASURE_MEASURE; measurable]);; let HAS_MEASURE_MEASURABLE_MEASURE = prove (`!s m. s has_measure m <=> measurable s /\ measure s = m`, REWRITE_TAC[HAS_MEASURE_MEASURE] THEN MESON_TAC[MEASURE_UNIQUE]);; let HAS_MEASURE_IMP_MEASURABLE = prove (`!s m. s has_measure m ==> measurable s`, REWRITE_TAC[measurable] THEN MESON_TAC[]);; let HAS_MEASURE = prove (`!s m. s has_measure m <=> ((\x. if x IN s then vec 1 else vec 0) has_integral (lift m)) (:real^N)`, SIMP_TAC[HAS_INTEGRAL_RESTRICT_UNIV; has_measure]);; let MEASURABLE = prove (`!s. measurable s <=> (\x. vec 1:real^1) integrable_on s`, REWRITE_TAC[measurable; integrable_on; has_measure; EXISTS_DROP; LIFT_DROP]);; let MEASURABLE_INTEGRABLE = prove (`measurable s <=> (\x. if x IN s then vec 1 else vec 0:real^1) integrable_on UNIV`, REWRITE_TAC[measurable; integrable_on; HAS_MEASURE; EXISTS_DROP; LIFT_DROP]);; let MEASURE_INTEGRAL = prove (`!s. measurable s ==> measure s = drop (integral s (\x. vec 1))`, REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP] THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_REWRITE_TAC[GSYM has_measure; GSYM HAS_MEASURE_MEASURE]);; let MEASURE_INTEGRAL_UNIV = prove (`!s. measurable s ==> measure s = drop(integral UNIV (\x. if x IN s then vec 1 else vec 0))`, REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP] THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_REWRITE_TAC[GSYM HAS_MEASURE; GSYM HAS_MEASURE_MEASURE]);; let INTEGRAL_MEASURE = prove (`!s. measurable s ==> integral s (\x. vec 1) = lift(measure s)`, SIMP_TAC[GSYM DROP_EQ; LIFT_DROP; MEASURE_INTEGRAL]);; let INTEGRAL_MEASURE_UNIV = prove (`!s. measurable s ==> integral UNIV (\x. if x IN s then vec 1 else vec 0) = lift(measure s)`, SIMP_TAC[GSYM DROP_EQ; LIFT_DROP; MEASURE_INTEGRAL_UNIV]);; let INTEGRABLE_ON_INDICATOR = prove (`!s t:real^N->bool. indicator s integrable_on t <=> measurable(s INTER t)`, ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN REWRITE_TAC[MEASURABLE_INTEGRABLE; GSYM indicator] THEN REPEAT GEN_TAC THEN REWRITE_TAC[indicator; IN_INTER] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[VEC_EQ; ARITH_EQ]));; let ABSOLUTELY_INTEGRABLE_ON_INDICATOR = prove (`!s t:real^N->bool. indicator s absolutely_integrable_on t <=> measurable(s INTER t)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM INTEGRABLE_ON_INDICATOR] THEN REWRITE_TAC[absolutely_integrable_on; indicator; COND_RAND] THEN REWRITE_TAC[NORM_1; DROP_VEC; REAL_ABS_NUM; LIFT_NUM]);; let INTEGRAL_INDICATOR = prove (`!s t:real^M->bool. measurable(s INTER t) ==> integral t (indicator s) = lift(measure(s INTER t))`, SIMP_TAC[MEASURE_INTEGRAL; LIFT_DROP; indicator; INTEGRAL_RESTRICT_INTER]);; let HAS_MEASURE_INTERVAL = prove (`(!a b:real^N. interval[a,b] has_measure content(interval[a,b])) /\ (!a b:real^N. interval(a,b) has_measure content(interval[a,b]))`, MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [REWRITE_TAC[has_measure] THEN ONCE_REWRITE_TAC[LIFT_EQ_CMUL] THEN REWRITE_TAC[HAS_INTEGRAL_CONST]; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN SIMP_TAC[HAS_MEASURE] THEN MATCH_MP_TAC(REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> a /\ b ==> c ==> d`] HAS_INTEGRAL_SPIKE) THEN EXISTS_TAC `interval[a:real^N,b] DIFF interval(a,b)` THEN REWRITE_TAC[NEGLIGIBLE_FRONTIER_INTERVAL] THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] INTERVAL_OPEN_SUBSET_CLOSED) THEN SET_TAC[]);; let MEASURABLE_INTERVAL = prove (`(!a b:real^N. measurable (interval[a,b])) /\ (!a b:real^N. measurable (interval(a,b)))`, REWRITE_TAC[measurable] THEN MESON_TAC[HAS_MEASURE_INTERVAL]);; let MEASURE_INTERVAL = prove (`(!a b:real^N. measure(interval[a,b]) = content(interval[a,b])) /\ (!a b:real^N. measure(interval(a,b)) = content(interval[a,b]))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN REWRITE_TAC[HAS_MEASURE_INTERVAL]);; let MEASURE_INTERVAL_1 = prove (`(!a b:real^1. measure(interval[a,b]) = if drop a <= drop b then drop b - drop a else &0) /\ (!a b:real^1. measure(interval(a,b)) = if drop a <= drop b then drop b - drop a else &0)`, REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; PRODUCT_1; drop]);; let MEASURE_INTERVAL_1_ALT = prove (`(!a b:real^1. measure(interval[a,b]) = if drop a < drop b then drop b - drop a else &0) /\ (!a b:real^1. measure(interval(a,b)) = if drop a < drop b then drop b - drop a else &0)`, REWRITE_TAC[MEASURE_INTERVAL_1] THEN REAL_ARITH_TAC);; let MEASURE_INTERVAL_2 = prove (`(!a b:real^2. measure(interval[a,b]) = if a$1 <= b$1 /\ a$2 <= b$2 then (b$1 - a$1) * (b$2 - a$2) else &0) /\ (!a b:real^2. measure(interval(a,b)) = if a$1 <= b$1 /\ a$2 <= b$2 then (b$1 - a$1) * (b$2 - a$2) else &0)`, REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[DIMINDEX_2; FORALL_2; PRODUCT_2]);; let MEASURE_INTERVAL_2_ALT = prove (`(!a b:real^2. measure(interval[a,b]) = if a$1 < b$1 /\ a$2 < b$2 then (b$1 - a$1) * (b$2 - a$2) else &0) /\ (!a b:real^2. measure(interval(a,b)) = if a$1 < b$1 /\ a$2 < b$2 then (b$1 - a$1) * (b$2 - a$2) else &0)`, REWRITE_TAC[MEASURE_INTERVAL_2] THEN REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC [`(a:real^2)$1 = (b:real^2)$1`; `(a:real^2)$2 = (b:real^2)$2`] THEN ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_SUB_REFL; REAL_LE_REFL; REAL_ABS_NUM; COND_ID] THEN ASM_REWRITE_TAC[REAL_LT_LE]);; let MEASURE_INTERVAL_3 = prove (`(!a b:real^3. measure(interval[a,b]) = if a$1 <= b$1 /\ a$2 <= b$2 /\ a$3 <= b$3 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) else &0) /\ (!a b:real^3. measure(interval(a,b)) = if a$1 <= b$1 /\ a$2 <= b$2 /\ a$3 <= b$3 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) else &0)`, REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[DIMINDEX_3; FORALL_3; PRODUCT_3]);; let MEASURE_INTERVAL_3_ALT = prove (`(!a b:real^3. measure(interval[a,b]) = if a$1 < b$1 /\ a$2 < b$2 /\ a$3 < b$3 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) else &0) /\ (!a b:real^3. measure(interval(a,b)) = if a$1 < b$1 /\ a$2 < b$2 /\ a$3 < b$3 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) else &0)`, REWRITE_TAC[MEASURE_INTERVAL_3] THEN REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC [`(a:real^3)$1 = (b:real^3)$1`; `(a:real^3)$2 = (b:real^3)$2`; `(a:real^3)$3 = (b:real^3)$3`] THEN ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_SUB_REFL; REAL_LE_REFL; REAL_ABS_NUM; COND_ID] THEN ASM_REWRITE_TAC[REAL_LT_LE]);; let MEASURE_INTERVAL_4 = prove (`(!a b:real^4. measure(interval[a,b]) = if a$1 <= b$1 /\ a$2 <= b$2 /\ a$3 <= b$3 /\ a$4 <= b$4 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) * (b$4 - a$4) else &0) /\ (!a b:real^4. measure(interval(a,b)) = if a$1 <= b$1 /\ a$2 <= b$2 /\ a$3 <= b$3 /\ a$4 <= b$4 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) * (b$4 - a$4) else &0)`, REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[DIMINDEX_4; FORALL_4; PRODUCT_4]);; let MEASURE_INTERVAL_4_ALT = prove (`(!a b:real^4. measure(interval[a,b]) = if a$1 < b$1 /\ a$2 < b$2 /\ a$3 < b$3 /\ a$4 < b$4 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) * (b$4 - a$4) else &0) /\ (!a b:real^4. measure(interval(a,b)) = if a$1 < b$1 /\ a$2 < b$2 /\ a$3 < b$3 /\ a$4 < b$4 then (b$1 - a$1) * (b$2 - a$2) * (b$3 - a$3) * (b$4 - a$4) else &0)`, REWRITE_TAC[MEASURE_INTERVAL_4] THEN REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC [`(a:real^4)$1 = (b:real^4)$1`; `(a:real^4)$2 = (b:real^4)$2`; `(a:real^4)$3 = (b:real^4)$3`; `(a:real^4)$4 = (b:real^4)$4`] THEN ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_SUB_REFL; REAL_LE_REFL; REAL_ABS_NUM; COND_ID] THEN ASM_REWRITE_TAC[REAL_LT_LE]);; let MEASURABLE_INTER = prove (`!s t:real^N->bool. measurable s /\ measurable t ==> measurable (s INTER t)`, REWRITE_TAC[MEASURABLE_INTEGRABLE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN SUBGOAL_THEN `(\x. if x IN s INTER t then vec 1 else vec 0):real^N->real^1 = (\x. lambda i. min (((if x IN s then vec 1 else vec 0):real^1)$i) (((if x IN t then vec 1 else vec 0):real^1)$i))` SUBST1_TAC THENL [SIMP_TAC[FUN_EQ_THM; CART_EQ; LAMBDA_BETA] THEN X_GEN_TAC `x:real^N` THEN REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC [`(x:real^N) IN s`; `(x:real^N) IN t`] THEN ASM_SIMP_TAC[IN_INTER; VEC_COMPONENT] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_MIN THEN CONJ_TAC THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[VEC_COMPONENT; REAL_POS]);; let MEASURABLE_UNION = prove (`!s t:real^N->bool. measurable s /\ measurable t ==> measurable (s UNION t)`, REWRITE_TAC[MEASURABLE_INTEGRABLE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN SUBGOAL_THEN `(\x. if x IN s UNION t then vec 1 else vec 0):real^N->real^1 = (\x. lambda i. max (((if x IN s then vec 1 else vec 0):real^1)$i) (((if x IN t then vec 1 else vec 0):real^1)$i))` SUBST1_TAC THENL [SIMP_TAC[FUN_EQ_THM; CART_EQ; LAMBDA_BETA] THEN X_GEN_TAC `x:real^N` THEN REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC [`(x:real^N) IN s`; `(x:real^N) IN t`] THEN ASM_SIMP_TAC[IN_UNION; VEC_COMPONENT] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_MAX THEN CONJ_TAC THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[VEC_COMPONENT; REAL_POS]);; let HAS_MEASURE_DISJOINT_UNION = prove (`!s1 s2 m1 m2. s1 has_measure m1 /\ s2 has_measure m2 /\ DISJOINT s1 s2 ==> (s1 UNION s2) has_measure (m1 + m2)`, REPEAT GEN_TAC THEN REWRITE_TAC[HAS_MEASURE; CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_INTEGRAL_ADD) THEN REWRITE_TAC[LIFT_ADD] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN GEN_TAC THEN REPEAT(COND_CASES_TAC THEN REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID]) THEN ASM SET_TAC[]);; let MEASURE_DISJOINT_UNION = prove (`!s t. measurable s /\ measurable t /\ DISJOINT s t ==> measure(s UNION t) = measure s + measure t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_DISJOINT_UNION; GSYM HAS_MEASURE_MEASURE]);; let MEASURE_DISJOINT_UNION_EQ = prove (`!s t u. measurable s /\ measurable t /\ s UNION t = u /\ DISJOINT s t ==> measure s + measure t = measure u`, MESON_TAC[MEASURE_DISJOINT_UNION]);; let HAS_MEASURE_POS_LE = prove (`!m s:real^N->bool. s has_measure m ==> &0 <= m`, REWRITE_TAC[HAS_MEASURE] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM(CONJUNCT2 LIFT_DROP)] THEN REWRITE_TAC[drop] THEN MATCH_MP_TAC(ISPEC `(\x. if x IN s then vec 1 else vec 0):real^N->real^1` HAS_INTEGRAL_COMPONENT_POS) THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[DIMINDEX_1; ARITH; IN_UNIV] THEN GEN_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[GSYM drop; DROP_VEC; REAL_POS]);; let MEASURE_POS_LE = prove (`!s. measurable s ==> &0 <= measure s`, REWRITE_TAC[HAS_MEASURE_MEASURE; HAS_MEASURE_POS_LE]);; let HAS_MEASURE_SUBSET = prove (`!s1 s2:real^N->bool m1 m2. s1 has_measure m1 /\ s2 has_measure m2 /\ s1 SUBSET s2 ==> m1 <= m2`, REPEAT GEN_TAC THEN REWRITE_TAC[has_measure] THEN STRIP_TAC THEN GEN_REWRITE_TAC BINOP_CONV [GSYM(CONJUNCT2 LIFT_DROP)] THEN MATCH_MP_TAC(ISPEC `(\x. vec 1):real^N->real^1` HAS_INTEGRAL_SUBSET_DROP_LE) THEN MAP_EVERY EXISTS_TAC [`s1:real^N->bool`; `s2:real^N->bool`] THEN ASM_REWRITE_TAC[DROP_VEC; REAL_POS]);; let MEASURE_SUBSET = prove (`!s t. measurable s /\ measurable t /\ s SUBSET t ==> measure s <= measure t`, REWRITE_TAC[HAS_MEASURE_MEASURE] THEN MESON_TAC[HAS_MEASURE_SUBSET]);; let HAS_MEASURE_0 = prove (`!s:real^N->bool. s has_measure &0 <=> negligible s`, GEN_TAC THEN EQ_TAC THENL [ALL_TAC; REWRITE_TAC[NEGLIGIBLE; has_measure] THEN DISCH_THEN(MP_TAC o SPEC `(:real^N)`) THEN ONCE_REWRITE_TAC[GSYM HAS_INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[IN_UNIV; indicator; LIFT_NUM]] THEN REWRITE_TAC[negligible] THEN REWRITE_TAC[has_measure] THEN ONCE_REWRITE_TAC[GSYM HAS_INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[LIFT_NUM] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [HAS_INTEGRAL_ALT]) THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN REWRITE_TAC[integrable_on; IN_UNIV] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [GSYM HAS_INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[indicator] THEN DISCH_THEN(X_CHOOSE_TAC `y:real^1`) THEN SUBGOAL_THEN `y:real^1 = vec 0` (fun th -> ASM_MESON_TAC[th]) THEN REWRITE_TAC[GSYM DROP_EQ; GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL [MATCH_MP_TAC(ISPEC `(\x. if x IN interval [a,b] then if x IN s then vec 1 else vec 0 else vec 0):real^N->real^1` HAS_INTEGRAL_DROP_LE) THEN EXISTS_TAC `(\x. if x IN s then vec 1 else vec 0):real^N->real^1`; REWRITE_TAC[DROP_VEC] THEN MATCH_MP_TAC(ISPEC `(\x. if x IN interval [a,b] then if x IN s then vec 1 else vec 0 else vec 0):real^N->real^1` HAS_INTEGRAL_DROP_POS)] THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN REWRITE_TAC[DROP_VEC; REAL_POS; REAL_LE_REFL]);; let MEASURE_EQ_0 = prove (`!s. negligible s ==> measure s = &0`, MESON_TAC[MEASURE_UNIQUE; HAS_MEASURE_0]);; let NEGLIGIBLE_IMP_MEASURABLE = prove (`!s:real^N->bool. negligible s ==> measurable s`, MESON_TAC[HAS_MEASURE_0; measurable]);; let HAS_MEASURE_EMPTY = prove (`{} has_measure &0`, REWRITE_TAC[HAS_MEASURE_0; NEGLIGIBLE_EMPTY]);; let MEASURE_EMPTY = prove (`measure {} = &0`, SIMP_TAC[MEASURE_EQ_0; NEGLIGIBLE_EMPTY]);; let MEASURABLE_EMPTY = prove (`measurable {}`, REWRITE_TAC[measurable] THEN MESON_TAC[HAS_MEASURE_EMPTY]);; let MEASURABLE_SING = prove (`!a:real^N. measurable {a}`, MESON_TAC[NEGLIGIBLE_IMP_MEASURABLE; NEGLIGIBLE_SING]);; let MEASURABLE_MEASURE_EQ_0 = prove (`!s. measurable s ==> (measure s = &0 <=> negligible s)`, REWRITE_TAC[HAS_MEASURE_MEASURE; GSYM HAS_MEASURE_0] THEN MESON_TAC[MEASURE_UNIQUE]);; let NEGLIGIBLE_EQ_MEASURE_0 = prove (`!s:real^N->bool. negligible s <=> measurable s /\ measure s = &0`, MESON_TAC[NEGLIGIBLE_IMP_MEASURABLE; MEASURABLE_MEASURE_EQ_0]);; let MEASURE_SING = prove (`!a:real^N. measure {a} = &0`, MESON_TAC[NEGLIGIBLE_EQ_MEASURE_0; NEGLIGIBLE_SING]);; let MEASURABLE_MEASURE_POS_LT = prove (`!s. measurable s ==> (&0 < measure s <=> ~negligible s)`, SIMP_TAC[REAL_LT_LE; MEASURE_POS_LE; GSYM MEASURABLE_MEASURE_EQ_0] THEN REWRITE_TAC[EQ_SYM_EQ]);; let NEGLIGIBLE_INTERVAL = prove (`(!a b. negligible(interval[a,b]) <=> interval(a,b) = {}) /\ (!a b. negligible(interval(a,b)) <=> interval(a,b) = {})`, REWRITE_TAC[GSYM HAS_MEASURE_0] THEN MESON_TAC[HAS_MEASURE_INTERVAL; CONTENT_EQ_0_INTERIOR; INTERIOR_CLOSED_INTERVAL; HAS_MEASURE_UNIQUE]);; let MEASURABLE_UNIONS = prove (`!f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> measurable s) ==> measurable (UNIONS f)`, REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[UNIONS_0; UNIONS_INSERT; MEASURABLE_EMPTY] THEN REWRITE_TAC[IN_INSERT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_UNION THEN ASM_SIMP_TAC[]);; let HAS_MEASURE_DIFF_SUBSET = prove (`!s1 s2 m1 m2. s1 has_measure m1 /\ s2 has_measure m2 /\ s2 SUBSET s1 ==> (s1 DIFF s2) has_measure (m1 - m2)`, REPEAT GEN_TAC THEN REWRITE_TAC[HAS_MEASURE; CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_INTEGRAL_SUB) THEN REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN REWRITE_TAC[VECTOR_SUB_REFL; VECTOR_SUB_RZERO] THEN ASM SET_TAC[]);; let MEASURABLE_DIFF = prove (`!s t:real^N->bool. measurable s /\ measurable t ==> measurable (s DIFF t)`, SUBGOAL_THEN `!s t:real^N->bool. measurable s /\ measurable t /\ t SUBSET s ==> measurable (s DIFF t)` ASSUME_TAC THENL [REWRITE_TAC[measurable] THEN MESON_TAC[HAS_MEASURE_DIFF_SUBSET]; ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s DIFF (s INTER t)`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[MEASURABLE_INTER] THEN SET_TAC[]]);; let MEASURE_DIFF_SUBSET = prove (`!s t. measurable s /\ measurable t /\ t SUBSET s ==> measure(s DIFF t) = measure s - measure t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_DIFF_SUBSET; GSYM HAS_MEASURE_MEASURE]);; let HAS_MEASURE_UNION_NEGLIGIBLE = prove (`!s t:real^N->bool m. s has_measure m /\ negligible t ==> (s UNION t) has_measure m`, REWRITE_TAC[HAS_MEASURE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE THEN MAP_EVERY EXISTS_TAC [`(\x. if x IN s then vec 1 else vec 0):real^N->real^1`; `t:real^N->bool`] THEN ASM_SIMP_TAC[IN_DIFF; IN_UNIV; IN_UNION]);; let HAS_MEASURE_DIFF_NEGLIGIBLE = prove (`!s t:real^N->bool m. s has_measure m /\ negligible t ==> (s DIFF t) has_measure m`, REWRITE_TAC[HAS_MEASURE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE THEN MAP_EVERY EXISTS_TAC [`(\x. if x IN s then vec 1 else vec 0):real^N->real^1`; `t:real^N->bool`] THEN ASM_SIMP_TAC[IN_DIFF; IN_UNIV; IN_UNION]);; let HAS_MEASURE_UNION_NEGLIGIBLE_EQ = prove (`!s t:real^N->bool m. negligible t ==> ((s UNION t) has_measure m <=> s has_measure m)`, REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[HAS_MEASURE_UNION_NEGLIGIBLE] THEN SUBST1_TAC(SET_RULE `s:real^N->bool = (s UNION t) DIFF (t DIFF s)`) THEN MATCH_MP_TAC HAS_MEASURE_DIFF_NEGLIGIBLE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_DIFF THEN ASM_REWRITE_TAC[]);; let HAS_MEASURE_DIFF_NEGLIGIBLE_EQ = prove (`!s t:real^N->bool m. negligible t ==> ((s DIFF t) has_measure m <=> s has_measure m)`, REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[HAS_MEASURE_DIFF_NEGLIGIBLE] THEN SUBST1_TAC(SET_RULE `s:real^N->bool = (s DIFF t) UNION (t INTER s)`) THEN MATCH_MP_TAC HAS_MEASURE_UNION_NEGLIGIBLE THEN ASM_SIMP_TAC[NEGLIGIBLE_INTER]);; let HAS_MEASURE_ALMOST = prove (`!s s' t m. s has_measure m /\ negligible t /\ s UNION t = s' UNION t ==> s' has_measure m`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE `s UNION t = s' UNION t ==> s' = (s UNION t) DIFF (t DIFF s')`)) THEN ASM_SIMP_TAC[HAS_MEASURE_DIFF_NEGLIGIBLE; HAS_MEASURE_UNION_NEGLIGIBLE; NEGLIGIBLE_DIFF]);; let HAS_MEASURE_ALMOST_EQ = prove (`!s s' t. negligible t /\ s UNION t = s' UNION t ==> (s has_measure m <=> s' has_measure m)`, MESON_TAC[HAS_MEASURE_ALMOST]);; let MEASURABLE_ALMOST = prove (`!s s' t. measurable s /\ negligible t /\ s UNION t = s' UNION t ==> measurable s'`, REWRITE_TAC[measurable] THEN MESON_TAC[HAS_MEASURE_ALMOST]);; let HAS_MEASURE_NEGLIGIBLE_UNION = prove (`!s1 s2:real^N->bool m1 m2. s1 has_measure m1 /\ s2 has_measure m2 /\ negligible(s1 INTER s2) ==> (s1 UNION s2) has_measure (m1 + m2)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_ALMOST THEN MAP_EVERY EXISTS_TAC [`(s1 DIFF (s1 INTER s2)) UNION (s2 DIFF (s1 INTER s2)):real^N->bool`; `s1 INTER s2:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC HAS_MEASURE_DISJOINT_UNION THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC HAS_MEASURE_ALMOST THEN EXISTS_TAC `s1:real^N->bool`; MATCH_MP_TAC HAS_MEASURE_ALMOST THEN EXISTS_TAC `s2:real^N->bool`; SET_TAC[]] THEN EXISTS_TAC `s1 INTER s2:real^N->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]);; let MEASURE_NEGLIGIBLE_UNION = prove (`!s t. measurable s /\ measurable t /\ negligible(s INTER t) ==> measure(s UNION t) = measure s + measure t`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_NEGLIGIBLE_UNION; GSYM HAS_MEASURE_MEASURE]);; let MEASURE_NEGLIGIBLE_UNION_EQ = prove (`!s t u. measurable s /\ measurable t /\ s UNION t = u /\ negligible(s INTER t) ==> measure s + measure t = measure u`, MESON_TAC[MEASURE_NEGLIGIBLE_UNION]);; let HAS_MEASURE_NEGLIGIBLE_SYMDIFF = prove (`!s t:real^N->bool m. s has_measure m /\ negligible((s DIFF t) UNION (t DIFF s)) ==> t has_measure m`, REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_ALMOST THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `(s DIFF t) UNION (t DIFF s):real^N->bool`] THEN ASM_REWRITE_TAC[] THEN SET_TAC[]);; let MEASURABLE_NEGLIGIBLE_SYMDIFF = prove (`!s t:real^N->bool. measurable s /\ negligible((s DIFF t) UNION (t DIFF s)) ==> measurable t`, REWRITE_TAC[measurable] THEN MESON_TAC[HAS_MEASURE_NEGLIGIBLE_SYMDIFF]);; let MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ = prove (`!s t:real^N->bool. negligible(s DIFF t UNION t DIFF s) ==> (measurable s <=> measurable t)`, MESON_TAC[MEASURABLE_NEGLIGIBLE_SYMDIFF; UNION_COMM]);; let MEASURE_NEGLIGIBLE_SYMDIFF = prove (`!s t:real^N->bool. negligible(s DIFF t UNION t DIFF s) ==> measure s = measure t`, REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC [`measurable(s:real^N->bool)`; `measurable(t:real^N->bool)`] THENL [ASM_MESON_TAC[HAS_MEASURE_NEGLIGIBLE_SYMDIFF; MEASURE_UNIQUE; HAS_MEASURE_MEASURE]; ASM_MESON_TAC[MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ]; ASM_MESON_TAC[MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ]; REWRITE_TAC[measure] THEN AP_TERM_TAC THEN ABS_TAC THEN ASM_MESON_TAC[measurable]]);; let NEGLIGIBLE_SYMDIFF_EQ = prove (`!s t:real^N->bool. negligible (s DIFF t UNION t DIFF s) ==> (negligible s <=> negligible t)`, REPEAT STRIP_TAC THEN EQ_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP; GSYM NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN SET_TAC[]);; let NEGLIGIBLE_DELETE = prove (`!a:real^N. negligible(s DELETE a) <=> negligible s`, GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{a:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]);; let HAS_MEASURE_NEGLIGIBLE_UNIONS = prove (`!m f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> s has_measure (m s)) /\ (!s t. s IN f /\ t IN f /\ ~(s = t) ==> negligible(s INTER t)) ==> (UNIONS f) has_measure (sum f m)`, GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[SUM_CLAUSES; UNIONS_0; UNIONS_INSERT; HAS_MEASURE_EMPTY] THEN REWRITE_TAC[IN_INSERT] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN STRIP_TAC THEN STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_UNION THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN REWRITE_TAC[INTER_UNIONS] THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[]);; let MEASURE_NEGLIGIBLE_UNIONS = prove (`!m f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> s has_measure (m s)) /\ (!s t. s IN f /\ t IN f /\ ~(s = t) ==> negligible(s INTER t)) ==> measure(UNIONS f) = sum f m`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_NEGLIGIBLE_UNIONS]);; let HAS_MEASURE_DISJOINT_UNIONS = prove (`!m f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> s has_measure (m s)) /\ (!s t. s IN f /\ t IN f /\ ~(s = t) ==> DISJOINT s t) ==> (UNIONS f) has_measure (sum f m)`, REWRITE_TAC[DISJOINT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_UNIONS THEN ASM_SIMP_TAC[NEGLIGIBLE_EMPTY]);; let MEASURE_DISJOINT_UNIONS = prove (`!m f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> s has_measure (m s)) /\ (!s t. s IN f /\ t IN f /\ ~(s = t) ==> DISJOINT s t) ==> measure(UNIONS f) = sum f m`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_DISJOINT_UNIONS]);; let HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE = prove (`!f:A->real^N->bool s. FINITE s /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> negligible((f x) INTER (f y))) ==> (UNIONS (IMAGE f s)) has_measure (sum s (\x. measure(f x)))`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `sum s (\x. measure(f x)) = sum (IMAGE (f:A->real^N->bool) s) measure` SUBST1_TAC THENL [CONV_TAC SYM_CONV THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC SUM_IMAGE_NONZERO THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:A`; `y:A`]) THEN ASM_SIMP_TAC[INTER_ACI; MEASURABLE_MEASURE_EQ_0]; MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_UNIONS THEN ASM_SIMP_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[FINITE_IMAGE; HAS_MEASURE_MEASURE]]);; let MEASURE_NEGLIGIBLE_UNIONS_IMAGE = prove (`!f:A->real^N->bool s. FINITE s /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> negligible((f x) INTER (f y))) ==> measure(UNIONS (IMAGE f s)) = sum s (\x. measure(f x))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE]);; let HAS_MEASURE_DISJOINT_UNIONS_IMAGE = prove (`!f:A->real^N->bool s. FINITE s /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> DISJOINT (f x) (f y)) ==> (UNIONS (IMAGE f s)) has_measure (sum s (\x. measure(f x)))`, REWRITE_TAC[DISJOINT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE THEN ASM_SIMP_TAC[NEGLIGIBLE_EMPTY]);; let MEASURE_DISJOINT_UNIONS_IMAGE = prove (`!f:A->real^N->bool s. FINITE s /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> DISJOINT (f x) (f y)) ==> measure(UNIONS (IMAGE f s)) = sum s (\x. measure(f x))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_DISJOINT_UNIONS_IMAGE]);; let HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE_STRONG = prove (`!f:A->real^N->bool s. FINITE {x | x IN s /\ ~(f x = {})} /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> negligible((f x) INTER (f y))) ==> (UNIONS (IMAGE f s)) has_measure (sum s (\x. measure(f x)))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:A->real^N->bool`; `{x | x IN s /\ ~((f:A->real^N->bool) x = {})}`] HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE) THEN ASM_SIMP_TAC[IN_ELIM_THM; FINITE_RESTRICT] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL [GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_UNIONS; IN_IMAGE; IN_ELIM_THM] THEN MESON_TAC[NOT_IN_EMPTY]; CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN SIMP_TAC[SUBSET; IN_ELIM_THM; TAUT `a /\ ~(a /\ b) <=> a /\ ~b`] THEN REWRITE_TAC[MEASURE_EMPTY]]);; let MEASURE_NEGLIGIBLE_UNIONS_IMAGE_STRONG = prove (`!f:A->real^N->bool s. FINITE {x | x IN s /\ ~(f x = {})} /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> negligible((f x) INTER (f y))) ==> measure(UNIONS (IMAGE f s)) = sum s (\x. measure(f x))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE_STRONG]);; let HAS_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG = prove (`!f:A->real^N->bool s. FINITE {x | x IN s /\ ~(f x = {})} /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> DISJOINT (f x) (f y)) ==> (UNIONS (IMAGE f s)) has_measure (sum s (\x. measure(f x)))`, REWRITE_TAC[DISJOINT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE_STRONG THEN ASM_SIMP_TAC[NEGLIGIBLE_EMPTY]);; let MEASURE_DISJOINT_UNIONS_IMAGE_STRONG = prove (`!f:A->real^N->bool s. FINITE {x | x IN s /\ ~(f x = {})} /\ (!x. x IN s ==> measurable(f x)) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> DISJOINT (f x) (f y)) ==> measure(UNIONS (IMAGE f s)) = sum s (\x. measure(f x))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG]);; let MEASURE_UNION = prove (`!s t:real^N->bool. measurable s /\ measurable t ==> measure(s UNION t) = measure(s) + measure(t) - measure(s INTER t)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s UNION t = (s INTER t) UNION (s DIFF t) UNION (t DIFF s)`] THEN ONCE_REWRITE_TAC[REAL_ARITH `a + b - c:real = c + (a - c) + (b - c)`] THEN MP_TAC(ISPECL [`s DIFF t:real^N->bool`; `t DIFF s:real^N->bool`] MEASURE_DISJOINT_UNION) THEN ASM_SIMP_TAC[MEASURABLE_DIFF] THEN ANTS_TAC THENL [SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`s INTER t:real^N->bool`; `(s DIFF t) UNION (t DIFF s):real^N->bool`] MEASURE_DISJOINT_UNION) THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_UNION; MEASURABLE_INTER] THEN ANTS_TAC THENL [SET_TAC[]; ALL_TAC] THEN REPEAT(DISCH_THEN SUBST1_TAC) THEN AP_TERM_TAC THEN BINOP_TAC THEN REWRITE_TAC[REAL_EQ_SUB_LADD] THEN MATCH_MP_TAC EQ_TRANS THENL [EXISTS_TAC `measure((s DIFF t) UNION (s INTER t):real^N->bool)`; EXISTS_TAC `measure((t DIFF s) UNION (s INTER t):real^N->bool)`] THEN (CONJ_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_DISJOINT_UNION THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTER]; AP_TERM_TAC] THEN SET_TAC[]));; let MEASURE_UNION_LE = prove (`!s t:real^N->bool. measurable s /\ measurable t ==> measure(s UNION t) <= measure s + measure t`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[MEASURE_UNION] THEN REWRITE_TAC[REAL_ARITH `a + b - c <= a + b <=> &0 <= c`] THEN MATCH_MP_TAC MEASURE_POS_LE THEN ASM_SIMP_TAC[MEASURABLE_INTER]);; let MEASURE_UNIONS_LE = prove (`!f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> measurable s) ==> measure(UNIONS f) <= sum f (\s. measure s)`, REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[UNIONS_0; UNIONS_INSERT; SUM_CLAUSES] THEN REWRITE_TAC[MEASURE_EMPTY; REAL_LE_REFL] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN REWRITE_TAC[IN_INSERT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(s:real^N->bool) + measure(UNIONS f:real^N->bool)` THEN ASM_SIMP_TAC[MEASURE_UNION_LE; MEASURABLE_UNIONS] THEN REWRITE_TAC[REAL_LE_LADD] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[]);; let MEASURABLE_INSERT = prove (`!x s:real^N->bool. measurable(x INSERT s) <=> measurable s`, REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]);; let MEASURABLE_DELETE = prove (`!x s:real^N->bool. measurable(s DELETE x) <=> measurable s`, REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]);; let MEASURE_INSERT = prove (`!x s:real^N->bool. measure(x INSERT s) = measure s`, REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]);; let MEASURE_DELETE = prove (`!x s:real^N->bool. measure(s DELETE x) = measure s`, REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]);; let MEASURE_UNIONS_LE_IMAGE = prove (`!f:A->bool s:A->(real^N->bool). FINITE f /\ (!a. a IN f ==> measurable(s a)) ==> measure(UNIONS (IMAGE s f)) <= sum f (\a. measure(s a))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum (IMAGE s (f:A->bool)) (\k:real^N->bool. measure k)` THEN ASM_SIMP_TAC[MEASURE_UNIONS_LE; FORALL_IN_IMAGE; FINITE_IMAGE] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC SUM_IMAGE_LE THEN ASM_SIMP_TAC[MEASURE_POS_LE]);; let MEASURE_SUB_LE_MEASURE_DIFF = prove (`!s t:real^N->bool. measurable s /\ measurable t ==> measure s - measure t <= measure(s DIFF t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_LE_SUB_RADD] THEN TRANS_TAC REAL_LE_TRANS `measure((s DIFF t) UNION t:real^N->bool)` THEN ASM_SIMP_TAC[MEASURE_UNION_LE; MEASURABLE_DIFF] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNION; MEASURABLE_DIFF] THEN SET_TAC[]);; let MEASURE_SUB_LE_MEASURE_SYMDIFF = prove (`!s t:real^N->bool. measurable s /\ measurable t ==> abs(measure s - measure t) <= measure((s DIFF t) UNION (t DIFF s))`, REWRITE_TAC[REAL_ARITH `abs(s - t) <= e <=> s - t <= e /\ t - s <= e`] THEN REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_SUB_LE_MEASURE_DIFF o lhand o snd) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNION; MEASURABLE_DIFF] THEN SET_TAC[]);; let MEASURABLE_INNER_OUTER = prove (`!s:real^N->bool. measurable s <=> !e. &0 < e ==> ?t u. t SUBSET s /\ s SUBSET u /\ measurable t /\ measurable u /\ abs(measure t - measure u) < e`, GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [GEN_TAC THEN DISCH_TAC THEN REPEAT(EXISTS_TAC `s:real^N->bool`) THEN ASM_REWRITE_TAC[SUBSET_REFL; REAL_SUB_REFL; REAL_ABS_NUM]; ALL_TAC] THEN REWRITE_TAC[MEASURABLE_INTEGRABLE] THEN MATCH_MP_TAC INTEGRABLE_STRADDLE THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->bool`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(\x. if x IN t then vec 1 else vec 0):real^N->real^1`; `(\x. if x IN u then vec 1 else vec 0):real^N->real^1`; `lift(measure(t:real^N->bool))`; `lift(measure(u:real^N->bool))`] THEN ASM_REWRITE_TAC[GSYM HAS_MEASURE; GSYM HAS_MEASURE_MEASURE] THEN ASM_REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[DROP_VEC; REAL_POS; REAL_LE_REFL]) THEN ASM SET_TAC[]);; let HAS_MEASURE_INNER_OUTER = prove (`!s:real^N->bool m. s has_measure m <=> (!e. &0 < e ==> ?t. t SUBSET s /\ measurable t /\ m - e < measure t) /\ (!e. &0 < e ==> ?u. s SUBSET u /\ measurable u /\ measure u < m + e)`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [HAS_MEASURE_MEASURABLE_MEASURE] THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "t") (LABEL_TAC "u")) THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [GEN_REWRITE_TAC I [MEASURABLE_INNER_OUTER] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN REMOVE_THEN "u" (MP_TAC o SPEC `e / &2`) THEN REMOVE_THEN "t" (MP_TAC o SPEC `e / &2`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN REWRITE_TAC[IMP_IMP; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ t <= u /\ m - e / &2 < t /\ u < m + e / &2 ==> abs(t - u) < e`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `~(&0 < x - y) /\ ~(&0 < y - x) ==> x = y`) THEN CONJ_TAC THEN DISCH_TAC THENL [REMOVE_THEN "u" (MP_TAC o SPEC `measure(s:real^N->bool) - m`) THEN ASM_REWRITE_TAC[REAL_SUB_ADD2; GSYM REAL_NOT_LE]; REMOVE_THEN "t" (MP_TAC o SPEC `m - measure(s:real^N->bool)`) THEN ASM_REWRITE_TAC[REAL_SUB_SUB2; GSYM REAL_NOT_LE]] THEN ASM_MESON_TAC[MEASURE_SUBSET]]);; let HAS_MEASURE_INNER_OUTER_LE = prove (`!s:real^N->bool m. s has_measure m <=> (!e. &0 < e ==> ?t. t SUBSET s /\ measurable t /\ m - e <= measure t) /\ (!e. &0 < e ==> ?u. s SUBSET u /\ measurable u /\ measure u <= m + e)`, REWRITE_TAC[HAS_MEASURE_INNER_OUTER] THEN MESON_TAC[REAL_ARITH `&0 < e /\ m - e / &2 <= t ==> m - e < t`; REAL_ARITH `&0 < e /\ u <= m + e / &2 ==> u < m + e`; REAL_ARITH `&0 < e <=> &0 < e / &2`; REAL_LT_IMP_LE]);; let NEGLIGIBLE_OUTER = prove (`!s:real^N->bool. negligible s <=> !e. &0 < e ==> ?t. s SUBSET t /\ measurable t /\ measure t < e`, GEN_TAC THEN REWRITE_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_INNER_OUTER] THEN REWRITE_TAC[REAL_ADD_LID] THEN MATCH_MP_TAC(TAUT `a ==> (a /\ b <=> b)`) THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN EXISTS_TAC `{}:real^N->bool` THEN REWRITE_TAC[EMPTY_SUBSET; MEASURABLE_EMPTY; MEASURE_EMPTY] THEN ASM_REAL_ARITH_TAC);; let NEGLIGIBLE_OUTER_LE = prove (`!s:real^N->bool. negligible s <=> !e. &0 < e ==> ?t. s SUBSET t /\ measurable t /\ measure t <= e`, REWRITE_TAC[NEGLIGIBLE_OUTER] THEN MESON_TAC[REAL_LT_IMP_LE; REAL_ARITH `&0 < e ==> &0 < e / &2 /\ (x <= e / &2 ==> x < e)`]);; let HAS_MEASURE_LIMIT = prove (`!s. s has_measure m <=> !e. &0 < e ==> ?B. &0 < B /\ !a b. ball(vec 0,B) SUBSET interval[a,b] ==> ?z. (s INTER interval[a,b]) has_measure z /\ abs(z - m) < e`, GEN_TAC THEN REWRITE_TAC[HAS_MEASURE] THEN GEN_REWRITE_TAC LAND_CONV [HAS_INTEGRAL] THEN REWRITE_TAC[IN_UNIV] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM HAS_INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[MESON[IN_INTER] `(if x IN k INTER s then a else b) = (if x IN s then if x IN k then a else b else b)`] THEN REWRITE_TAC[EXISTS_LIFT; GSYM LIFT_SUB; NORM_LIFT]);; let MEASURE_LIMIT = prove (`!s:real^N->bool e. measurable s /\ &0 < e ==> ?B. &0 < B /\ !a b. ball(vec 0,B) SUBSET interval[a,b] ==> abs(measure(s INTER interval[a,b]) - measure s) < e`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_MEASURE_MEASURE]) THEN GEN_REWRITE_TAC LAND_CONV [HAS_MEASURE_LIMIT] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[MEASURE_UNIQUE]);; let INTEGRABLE_ON_CONST = prove (`!c:real^N. (\x:real^M. c) integrable_on s <=> c = vec 0 \/ measurable s`, GEN_TAC THEN ASM_CASES_TAC `c:real^N = vec 0` THEN ASM_REWRITE_TAC[INTEGRABLE_0; MEASURABLE] THEN EQ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; VEC_COMPONENT] THEN DISCH_THEN(X_CHOOSE_THEN `k:num` STRIP_ASSUME_TAC) THEN DISCH_THEN(MP_TAC o ISPEC `(\y. lambda i. y$k / (c:real^N)$k):real^N->real^1` o MATCH_MP(REWRITE_RULE[IMP_CONJ] INTEGRABLE_LINEAR)) THEN ASM_SIMP_TAC[vec; o_DEF; REAL_DIV_REFL] THEN DISCH_THEN MATCH_MP_TAC THEN SIMP_TAC[linear; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA] THEN REAL_ARITH_TAC; DISCH_THEN(MP_TAC o ISPEC `(\y. lambda i. (c:real^N)$i * y$i):real^1->real^N` o MATCH_MP(REWRITE_RULE[IMP_CONJ] INTEGRABLE_LINEAR)) THEN ANTS_TAC THENL [SIMP_TAC[linear; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA] THEN REAL_ARITH_TAC; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[FUN_EQ_THM; CART_EQ; o_THM; LAMBDA_BETA; VEC_COMPONENT] THEN REWRITE_TAC[REAL_MUL_RID]]]);; let ABSOLUTELY_INTEGRABLE_ON_CONST = prove (`!c. (\x. c) absolutely_integrable_on s <=> c = vec 0 \/ measurable s`, REWRITE_TAC[absolutely_integrable_on; INTEGRABLE_ON_CONST] THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; DROP_VEC; NORM_EQ_0]);; let HAS_INTEGRAL_CONST_GEN = prove (`!s c. measurable s ==> (((\x. c):real^M->real^N) has_integral (measure s % c)) s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(\x. vec 1):real^M->real^1`; `integral s ((\x. vec 1):real^M->real^1)`; `s:real^M->bool`; `\v. drop v % (c:real^N)`] HAS_INTEGRAL_LINEAR) THEN ASM_SIMP_TAC[o_DEF; DROP_VEC; VECTOR_MUL_LID; GSYM MEASURE_INTEGRAL] THEN DISCH_THEN MATCH_MP_TAC THEN SIMP_TAC[LINEAR_VMUL_DROP; LINEAR_ID; GSYM HAS_INTEGRAL_INTEGRAL] THEN ASM_REWRITE_TAC[INTEGRABLE_ON_CONST]);; let INTEGRAL_CONST_GEN = prove (`!s c. measurable s ==> integral s ((\x. c):real^M->real^N) = measure s % c`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC HAS_INTEGRAL_CONST_GEN THEN ASM_REWRITE_TAC[]);; let OPEN_NOT_NEGLIGIBLE = prove (`!s:real^N->bool. open s /\ ~(s = {}) ==> ~(negligible s)`, GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; RIGHT_AND_EXISTS_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN DISCH_TAC THEN SUBGOAL_THEN `negligible(interval[a - e / (&(dimindex(:N))) % vec 1:real^N, a + e / (&(dimindex(:N))) % vec 1])` MP_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `cball(a:real^N,e)` THEN CONJ_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET]; ALL_TAC] THEN REWRITE_TAC[SUBSET; IN_INTERVAL; IN_CBALL; VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID; REAL_ARITH `a - e <= x /\ x <= a + e <=> abs(x - a) <= e`; dist] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC SUM_BOUND_GEN THEN REWRITE_TAC[FINITE_NUMSEG; CARD_NUMSEG_1; NUMSEG_EMPTY; NOT_LT] THEN REWRITE_TAC[IN_NUMSEG; VECTOR_SUB_COMPONENT; DIMINDEX_GE_1] THEN ASM_MESON_TAC[REAL_ABS_SUB]; REWRITE_TAC[NEGLIGIBLE_INTERVAL; INTERVAL_NE_EMPTY] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; REAL_MUL_RID; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `a - e < a + e <=> &0 < e`] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1]]);; let NOT_NEGLIGIBLE_UNIV = prove (`~negligible(:real^N)`, SIMP_TAC[OPEN_NOT_NEGLIGIBLE; OPEN_UNIV; UNIV_NOT_EMPTY]);; let NEGLIGIBLE_EMPTY_INTERIOR = prove (`!s:real^N->bool. negligible s ==> interior s = {}`, MESON_TAC[OPEN_NOT_NEGLIGIBLE; INTERIOR_SUBSET; OPEN_INTERIOR; NEGLIGIBLE_SUBSET]);; let HAS_INTEGRAL_NEGLIGIBLE_EQ_AE = prove (`!f:real^M->real^N s t. negligible t /\ (!x i. x IN s DIFF t /\ 1 <= i /\ i <= dimindex (:N) ==> &0 <= f x$i) ==> ((f has_integral vec 0) s <=> negligible {x | x IN s /\ ~(f x = vec 0)})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x. if x IN t then vec 0 else (f:real^M->real^N) x`; `s:real^M->bool`] HAS_INTEGRAL_NEGLIGIBLE_EQ) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[VEC_COMPONENT; IN_DIFF; REAL_LE_REFL]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL [MATCH_MP_TAC HAS_INTEGRAL_SPIKE_EQ; MATCH_MP_TAC NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET] THEN EXISTS_TAC `t:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Properties of measure under simple affine transformations. *) (* ------------------------------------------------------------------------- *) let HAS_MEASURE_AFFINITY = prove (`!s m c y. s has_measure y ==> (IMAGE (\x:real^N. m % x + c) s) has_measure abs(m) pow (dimindex(:N)) * y`, REPEAT GEN_TAC THEN ASM_CASES_TAC `m = &0` THENL [ASM_REWRITE_TAC[REAL_ABS_NUM; VECTOR_ADD_LID; VECTOR_MUL_LZERO] THEN ONCE_REWRITE_TAC[MATCH_MP (ARITH_RULE `~(x = 0) ==> x = SUC(x - 1)`) (SPEC_ALL DIMINDEX_NONZERO)] THEN DISCH_TAC THEN REWRITE_TAC[real_pow; REAL_MUL_LZERO; HAS_MEASURE_0] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{c:real^N}` THEN SIMP_TAC[NEGLIGIBLE_FINITE; FINITE_RULES] THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[HAS_MEASURE] THEN ONCE_REWRITE_TAC[HAS_INTEGRAL] THEN REWRITE_TAC[IN_UNIV] THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real / abs(m) pow dimindex(:N)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_POW_LT; GSYM REAL_ABS_NZ; REAL_POW_LT] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `abs(m) * B + norm(c:real^N)` THEN ASM_SIMP_TAC[REAL_ARITH `&0 < B /\ &0 <= x ==> &0 < B + x`; NORM_POS_LE; REAL_LT_MUL; GSYM REAL_ABS_NZ; REAL_POW_LT] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN ASM_SIMP_TAC[VECTOR_EQ_AFFINITY; UNWIND_THM1] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`if &0 <= m then inv m % u + --(inv m % c):real^N else inv m % v + --(inv m % c)`; `if &0 <= m then inv m % v + --(inv m % c):real^N else inv m % u + --(inv m % c)`]) THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b ==> c) ==> (a ==> b) ==> c`) THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN DISCH_THEN(MP_TAC o SPEC `m % x + c:real^N`) THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[IN_BALL; IN_INTERVAL] THEN CONJ_TAC THENL [REWRITE_TAC[NORM_ARITH `dist(vec 0,x) = norm(x:real^N)`] THEN DISCH_TAC THEN MATCH_MP_TAC(NORM_ARITH `norm(x:real^N) < a ==> norm(x + y) < a + norm(y)`) THEN ASM_SIMP_TAC[NORM_MUL; REAL_LT_LMUL; GSYM REAL_ABS_NZ]; ALL_TAC] THEN SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_NEG_COMPONENT; COND_COMPONENT] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[REAL_ARITH `m * u + --(m * c):real = (u - c) * m`] THEN SUBST1_TAC(REAL_ARITH `inv(m) = if &0 <= inv(m) then abs(inv m) else --(abs(inv m))`) THEN SIMP_TAC[REAL_LE_INV_EQ] THEN REWRITE_TAC[REAL_ARITH `(x - y:real) * --z = (y - x) * z`] THEN REWRITE_TAC[REAL_ABS_INV; GSYM real_div] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; GSYM REAL_ABS_NZ] THEN ASM_REWRITE_TAC[real_abs] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN DISCH_TAC THEN DISCH_THEN(X_CHOOSE_THEN `z:real^1` (fun th -> EXISTS_TAC `(abs m pow dimindex (:N)) % z:real^1` THEN MP_TAC th)) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP(REAL_FIELD `~(x = &0) ==> ~(inv x = &0)`)) THEN REWRITE_TAC[TAUT `a ==> b ==> c <=> b /\ a ==> c`] THEN DISCH_THEN(MP_TAC o SPEC `--(inv m % c):real^N` o MATCH_MP HAS_INTEGRAL_AFFINITY) THEN ASM_REWRITE_TAC[IMAGE_AFFINITY_INTERVAL; REAL_INV_INV] THEN SIMP_TAC[COND_ID] THEN COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; VECTOR_MUL_LNEG; VECTOR_MUL_RNEG] THEN ASM_SIMP_TAC[REAL_MUL_RINV; VECTOR_MUL_LID; VECTOR_NEG_NEG] THEN REWRITE_TAC[VECTOR_ARITH `(u + --c) + c:real^N = u`] THEN REWRITE_TAC[REAL_ABS_INV; REAL_INV_INV; GSYM REAL_POW_INV] THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN REWRITE_TAC[LIFT_CMUL; GSYM VECTOR_SUB_LDISTRIB] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_POW; REAL_ABS_ABS] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_POW_LT; GSYM REAL_ABS_NZ]);; let STRETCH_GALOIS = prove (`!x:real^N y:real^N m. (!k. 1 <= k /\ k <= dimindex(:N) ==> ~(m k = &0)) ==> ((y = (lambda k. m k * x$k)) <=> (lambda k. inv(m k) * y$k) = x)`, REPEAT GEN_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN MATCH_MP_TAC(MESON[] `(!x. p x ==> (q x <=> r x)) ==> (!x. p x) ==> ((!x. q x) <=> (!x. r x))`) THEN GEN_TAC THEN ASM_CASES_TAC `1 <= k /\ k <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_FIELD);; let HAS_MEASURE_STRETCH = prove (`!s m y. s has_measure y ==> (IMAGE (\x:real^N. lambda k. m k * x$k) s :real^N->bool) has_measure abs(product (1..dimindex(:N)) m) * y`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `!k. 1 <= k /\ k <= dimindex(:N) ==> ~(m k = &0)` THENL [ALL_TAC; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN SUBGOAL_THEN `product(1..dimindex (:N)) m = &0` SUBST1_TAC THENL [ASM_MESON_TAC[PRODUCT_EQ_0_NUMSEG]; ALL_TAC] THEN REWRITE_TAC[REAL_ABS_NUM; REAL_MUL_LZERO; HAS_MEASURE_0] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | x$k = &0}` THEN ASM_SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE; SUBSET; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[IN_ELIM_THM; LAMBDA_BETA; REAL_MUL_LZERO]] THEN UNDISCH_TAC `(s:real^N->bool) has_measure y` THEN REWRITE_TAC[HAS_MEASURE] THEN ONCE_REWRITE_TAC[HAS_INTEGRAL] THEN REWRITE_TAC[IN_UNIV] THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN SUBGOAL_THEN `&0 < abs(product(1..dimindex(:N)) m)` ASSUME_TAC THENL [ASM_MESON_TAC[REAL_ABS_NZ; REAL_LT_DIV; PRODUCT_EQ_0_NUMSEG]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real / abs(product(1..dimindex(:N)) m)`) THEN ASM_SIMP_TAC[REAL_LT_DIV] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `sup(IMAGE (\k. abs(m k) * B) (1..dimindex(:N)))` THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[REAL_LT_SUP_FINITE; FINITE_IMAGE; NUMSEG_EMPTY; FINITE_NUMSEG; IN_NUMSEG; GSYM NOT_LE; DIMINDEX_GE_1; IMAGE_EQ_EMPTY; EXISTS_IN_IMAGE] THEN ASM_MESON_TAC[IN_NUMSEG; DIMINDEX_GE_1; LE_REFL; REAL_LT_MUL; REAL_ABS_NZ]; DISCH_TAC] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN DISCH_TAC THEN ASM_SIMP_TAC[IN_IMAGE; STRETCH_GALOIS; UNWIND_THM1] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(lambda k. min (inv(m k) * (u:real^N)$k) (inv(m k) * (v:real^N)$k)):real^N`; `(lambda k. max (inv(m k) * (u:real^N)$k) (inv(m k) * (v:real^N)$k)):real^N`]) THEN MATCH_MP_TAC(TAUT `a /\ (b ==> a ==> c) ==> (a ==> b) ==> c`) THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^1` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN SUBGOAL_THEN `!k. 1 <= k /\ k <= dimindex (:N) ==> ~(inv(m k) = &0)` MP_TAC THENL [ASM_SIMP_TAC[REAL_INV_EQ_0]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_INTEGRAL_STRETCH)] THEN (MP_TAC(ISPECL [`u:real^N`; `v:real^N`; `\i:num. inv(m i:real)`] IMAGE_STRETCH_INTERVAL) THEN SUBGOAL_THEN `~(interval[u:real^N,v] = {})` ASSUME_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> ~(s = {}) ==> ~(t = {})`)) THEN ASM_REWRITE_TAC[BALL_EQ_EMPTY; GSYM REAL_NOT_LT]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM)) THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `b SUBSET s ==> b' SUBSET IMAGE f b ==> b' SUBSET IMAGE f s`)) THEN REWRITE_TAC[IN_BALL; SUBSET; NORM_ARITH `dist(vec 0:real^N,x) = norm x`; IN_IMAGE] THEN ASM_SIMP_TAC[STRETCH_GALOIS; REAL_INV_EQ_0; UNWIND_THM1; REAL_INV_INV] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `norm(sup(IMAGE(\k. abs(m k)) (1..dimindex(:N))) % x:real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN SIMP_TAC[LAMBDA_BETA; VECTOR_MUL_COMPONENT; REAL_ABS_MUL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= abs y`) THEN ASM_SIMP_TAC[REAL_LE_SUP_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; FINITE_NUMSEG; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[EXISTS_IN_IMAGE; IN_NUMSEG] THEN ASM_MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN REWRITE_TAC[NORM_MUL] THEN MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `abs(sup(IMAGE(\k. abs(m k)) (1..dimindex(:N)))) * B` THEN SUBGOAL_THEN `&0 < sup(IMAGE(\k. abs(m k)) (1..dimindex(:N)))` ASSUME_TAC THENL [ASM_SIMP_TAC[REAL_LT_SUP_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; FINITE_NUMSEG; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[EXISTS_IN_IMAGE; GSYM REAL_ABS_NZ; IN_NUMSEG] THEN ASM_MESON_TAC[DIMINDEX_GE_1; LE_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ; REAL_ARITH `&0 < x ==> &0 < abs x`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sup(IMAGE(\k. abs(m k)) (1..dimindex(:N))) * B` THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ; REAL_ARITH `&0 < x ==> abs x <= x`] THEN ASM_SIMP_TAC[REAL_LE_SUP_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; FINITE_NUMSEG; GSYM NOT_LE; DIMINDEX_GE_1] THEN ASM_SIMP_TAC[EXISTS_IN_IMAGE; REAL_LE_RMUL_EQ] THEN ASM_SIMP_TAC[REAL_SUP_LE_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; FINITE_NUMSEG; GSYM NOT_LE; DIMINDEX_GE_1] THEN MP_TAC(ISPEC `IMAGE (\k. abs (m k)) (1..dimindex(:N))` SUP_FINITE) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[]; MATCH_MP_TAC(MESON[] `s = t /\ P z ==> (f has_integral z) s ==> Q ==> ?w. (f has_integral w) t /\ P w`) THEN SIMP_TAC[GSYM PRODUCT_INV; FINITE_NUMSEG; GSYM REAL_ABS_INV] THEN REWRITE_TAC[REAL_INV_INV] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC(SET_RULE `(!x. f x = x) ==> IMAGE f s = s`) THEN SIMP_TAC[o_THM; LAMBDA_BETA; CART_EQ] THEN ASM_SIMP_TAC[REAL_MUL_ASSOC; REAL_MUL_RINV; REAL_MUL_LID]; REWRITE_TAC[NORM_1; DROP_SUB; LIFT_DROP; DROP_CMUL] THEN REWRITE_TAC[GSYM REAL_SUB_LDISTRIB; ETA_AX] THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_ABS] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN ASM_MESON_TAC[NORM_1; DROP_SUB; LIFT_DROP]]]);; let HAS_MEASURE_TRANSLATION = prove (`!s m a. s has_measure m ==> (IMAGE (\x:real^N. a + x) s) has_measure m`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `&1`; `a:real^N`; `m:real`] HAS_MEASURE_AFFINITY) THEN REWRITE_TAC[VECTOR_MUL_LID; REAL_ABS_NUM; REAL_POW_ONE; REAL_MUL_LID] THEN REWRITE_TAC[VECTOR_ADD_SYM]);; let NEGLIGIBLE_TRANSLATION = prove (`!s a. negligible s ==> negligible (IMAGE (\x:real^N. a + x) s)`, SIMP_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_TRANSLATION]);; let HAS_MEASURE_TRANSLATION_EQ = prove (`!a s m. (IMAGE (\x:real^N. a + x) s) has_measure m <=> s has_measure m`, REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[HAS_MEASURE_TRANSLATION] THEN DISCH_THEN(MP_TAC o SPEC `--a:real^N` o MATCH_MP HAS_MEASURE_TRANSLATION) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ARITH `--a + a + b:real^N = b`] THEN SET_TAC[]);; add_translation_invariants [HAS_MEASURE_TRANSLATION_EQ];; let MEASURE_TRANSLATION = prove (`!a s. measure(IMAGE (\x:real^N. a + x) s) = measure s`, REWRITE_TAC[measure; HAS_MEASURE_TRANSLATION_EQ]);; add_translation_invariants [MEASURE_TRANSLATION];; let NEGLIGIBLE_TRANSLATION_REV = prove (`!s a. negligible (IMAGE (\x:real^N. a + x) s) ==> negligible s`, SIMP_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_TRANSLATION_EQ]);; let NEGLIGIBLE_TRANSLATION_EQ = prove (`!a s. negligible (IMAGE (\x:real^N. a + x) s) <=> negligible s`, SIMP_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_TRANSLATION_EQ]);; add_translation_invariants [NEGLIGIBLE_TRANSLATION_EQ];; let MEASURABLE_TRANSLATION_EQ = prove (`!a:real^N s. measurable (IMAGE (\x. a + x) s) <=> measurable s`, REWRITE_TAC[measurable; HAS_MEASURE_TRANSLATION_EQ]);; add_translation_invariants [MEASURABLE_TRANSLATION_EQ];; let MEASURABLE_TRANSLATION = prove (`!s a:real^N. measurable s ==> measurable (IMAGE (\x. a + x) s)`, REWRITE_TAC[MEASURABLE_TRANSLATION_EQ]);; let HAS_MEASURE_SCALING = prove (`!s m c. s has_measure m ==> (IMAGE (\x:real^N. c % x) s) has_measure (abs(c) pow dimindex(:N)) * m`, REPEAT GEN_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `c:real`; `vec 0:real^N`; `m:real`] HAS_MEASURE_AFFINITY) THEN REWRITE_TAC[VECTOR_ADD_RID]);; let HAS_MEASURE_SCALING_EQ = prove (`!s m c. ~(c = &0) ==> (IMAGE (\x:real^N. c % x) s has_measure (abs(c) pow dimindex(:N)) * m <=> s has_measure m)`, REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[HAS_MEASURE_SCALING] THEN DISCH_THEN(MP_TAC o SPEC `inv(c):real` o MATCH_MP HAS_MEASURE_SCALING) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; GSYM REAL_ABS_MUL] THEN REWRITE_TAC[GSYM REAL_POW_MUL; VECTOR_MUL_ASSOC; REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[GSYM REAL_ABS_MUL; REAL_MUL_LINV] THEN REWRITE_TAC[REAL_POW_ONE; REAL_ABS_NUM; REAL_MUL_LID; VECTOR_MUL_LID] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN SET_TAC[]);; let MEASURABLE_SCALING = prove (`!s c. measurable s ==> measurable (IMAGE (\x:real^N. c % x) s)`, REWRITE_TAC[measurable] THEN MESON_TAC[HAS_MEASURE_SCALING]);; let MEASURABLE_SCALING_EQ = prove (`!s:real^N->bool c. measurable (IMAGE (\x. c % x) s) <=> c = &0 \/ measurable s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[IMAGE_CONST; VECTOR_MUL_LZERO] THEN MESON_TAC[MEASURABLE_SING; MEASURABLE_EMPTY]; EQ_TAC THEN REWRITE_TAC[MEASURABLE_SCALING] THEN DISCH_THEN(MP_TAC o SPEC `inv(c):real` o MATCH_MP MEASURABLE_SCALING) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_LID; IMAGE_ID]]);; let MEASURABLE_AFFINITY_EQ = prove (`!s m c:real^N. measurable (IMAGE (\x. m % x + c) s) <=> m = &0 \/ measurable s`, REWRITE_TAC[AFFINITY_SCALING_TRANSLATION; MEASURABLE_TRANSLATION_EQ; MEASURABLE_SCALING_EQ; IMAGE_o]);; let MEASURABLE_AFFINITY = prove (`!s m c:real^N. measurable s ==> measurable (IMAGE (\x. m % x + c) s)`, SIMP_TAC[MEASURABLE_AFFINITY_EQ]);; let MEASURE_SCALING = prove (`!s c. measurable s ==> measure(IMAGE (\x:real^N. c % x) s) = (abs(c) pow dimindex(:N)) * measure s`, REWRITE_TAC[HAS_MEASURE_MEASURE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN ASM_SIMP_TAC[HAS_MEASURE_SCALING]);; let MEASURE_AFFINITY = prove (`!s m c:real^N. measurable s ==> measure(IMAGE (\x. m % x + c) s) = abs m pow dimindex (:N) * measure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_UNIQUE THEN MATCH_MP_TAC HAS_MEASURE_AFFINITY THEN ASM_REWRITE_TAC[GSYM HAS_MEASURE_MEASURE]);; let NEGLIGIBLE_SCALING = prove (`!s c. negligible s ==> negligible (IMAGE (\x:real^N. c % x) s)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM HAS_MEASURE_0] THEN DISCH_THEN(MP_TAC o SPEC `c:real` o MATCH_MP HAS_MEASURE_SCALING) THEN REWRITE_TAC[REAL_MUL_RZERO]);; let NEGLIGIBLE_SCALING_EQ = prove (`!s:real^N->bool c. negligible (IMAGE (\x. c % x) s) <=> c = &0 \/ negligible s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[IMAGE_CONST; VECTOR_MUL_LZERO] THEN MESON_TAC[NEGLIGIBLE_SING; NEGLIGIBLE_EMPTY]; EQ_TAC THEN REWRITE_TAC[NEGLIGIBLE_SCALING] THEN DISCH_THEN(MP_TAC o SPEC `inv(c):real` o MATCH_MP NEGLIGIBLE_SCALING) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_LID; IMAGE_ID]]);; let NEGLIGIBLE_AFFINITY_EQ = prove (`!s m c:real^N. negligible (IMAGE (\x. m % x + c) s) <=> m = &0 \/ negligible s`, REWRITE_TAC[AFFINITY_SCALING_TRANSLATION; NEGLIGIBLE_TRANSLATION_EQ; NEGLIGIBLE_SCALING_EQ; IMAGE_o]);; let NEGLIGIBLE_AFFINITY = prove (`!s m c:real^N. negligible s ==> negligible (IMAGE (\x. m % x + c) s)`, SIMP_TAC[NEGLIGIBLE_AFFINITY_EQ]);; let NOT_MEASURABLE_UNIV = prove (`~measurable(:real^N)`, DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `&2` o MATCH_MP MEASURE_SCALING) THEN MATCH_MP_TAC(REAL_RING `a = b /\ ~(b = &0) /\ ~(c = &1) ==> a = c * b ==> F`) THEN REWRITE_TAC[REAL_POW_EQ_1] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[MEASURABLE_MEASURE_EQ_0; NOT_NEGLIGIBLE_UNIV] THEN SIMP_TAC[DIMINDEX_GE_1; LE_1] THEN AP_TERM_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN EXISTS_TAC `inv(&2) % x:real^N` THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_LID]);; (* ------------------------------------------------------------------------- *) (* Measurability of countable unions and intersections of various kinds. *) (* ------------------------------------------------------------------------- *) let HAS_MEASURE_NESTED_UNIONS = prove (`!s:num->real^N->bool B. (!n. measurable(s n)) /\ (!n. measure(s n) <= B) /\ (!n. s(n) SUBSET s(SUC n)) ==> measurable(UNIONS { s(n) | n IN (:num) }) /\ ((\n. lift(measure(s n))) --> lift(measure(UNIONS { s(n) | n IN (:num) }))) sequentially`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b /\ (b ==> c))`] THEN SIMP_TAC[MEASURE_INTEGRAL_UNIV; LIFT_DROP] THEN REWRITE_TAC[MEASURABLE_INTEGRABLE] THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `b /\ c ==> b /\ (b ==> c)`) THEN MATCH_MP_TAC MONOTONE_CONVERGENCE_INCREASING THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [REPEAT GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN REWRITE_TAC[DROP_VEC; REAL_POS; REAL_LE_REFL] THEN ASM SET_TAC[]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN COND_CASES_TAC THENL [MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o PART_MATCH (rand o rand) TRANSITIVE_STEPWISE_LE_EQ o concl) THEN ASM_REWRITE_TAC[SUBSET_TRANS; SUBSET_REFL] THEN ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_UNIONS]) THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN SIMP_TAC[NOT_EXISTS_THM; IN_UNIV; LIM_CONST]]; RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEASURABLE_INTEGRABLE]) THEN ASM_SIMP_TAC[INTEGRAL_MEASURE_UNIV] THEN REWRITE_TAC[bounded; SIMPLE_IMAGE; FORALL_IN_IMAGE] THEN EXISTS_TAC `B:real` THEN REWRITE_TAC[IN_UNIV; NORM_LIFT] THEN REWRITE_TAC[real_abs] THEN ASM_MESON_TAC[MEASURE_POS_LE]]);; let MEASURABLE_NESTED_UNIONS = prove (`!s:num->real^N->bool B. (!n. measurable(s n)) /\ (!n. measure(s n) <= B) /\ (!n. s(n) SUBSET s(SUC n)) ==> measurable(UNIONS { s(n) | n IN (:num) })`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_NESTED_UNIONS) THEN SIMP_TAC[]);; let HAS_MEASURE_COUNTABLE_NEGLIGIBLE_UNIONS = prove (`!s:num->real^N->bool B. (!n. measurable(s n)) /\ (!m n. ~(m = n) ==> negligible(s m INTER s n)) /\ (!n. sum (0..n) (\k. measure(s k)) <= B) ==> measurable(UNIONS { s(n) | n IN (:num) }) /\ ((\n. lift(measure(s n))) sums lift(measure(UNIONS { s(n) | n IN (:num) }))) (from 0)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`\n. UNIONS (IMAGE s (0..n)):real^N->bool`; `B:real`] HAS_MEASURE_NESTED_UNIONS) THEN REWRITE_TAC[sums; FROM_0; INTER_UNIV] THEN SUBGOAL_THEN `!n. (UNIONS (IMAGE s (0..n)):real^N->bool) has_measure (sum(0..n) (\k. measure(s k)))` MP_TAC THENL [GEN_TAC THEN MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE THEN ASM_SIMP_TAC[FINITE_NUMSEG]; ALL_TAC] THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN ASSUME_TAC(GEN `n:num` (MATCH_MP MEASURE_UNIQUE (SPEC `n:num` th)))) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[measurable]; ALL_TAC] THEN GEN_TAC THEN MATCH_MP_TAC SUBSET_UNIONS THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; IN_NUMSEG] THEN ARITH_TAC; ALL_TAC] THEN SIMP_TAC[LIFT_SUM; FINITE_NUMSEG; o_DEF] THEN SUBGOAL_THEN `UNIONS {UNIONS (IMAGE s (0..n)) | n IN (:num)}:real^N->bool = UNIONS (IMAGE s (:num))` (fun th -> REWRITE_TAC[th] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[]) THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNIONS] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[EXISTS_IN_IMAGE; EXISTS_IN_UNIONS; IN_UNIV] THEN REWRITE_TAC[IN_UNIONS; EXISTS_IN_IMAGE] THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN MESON_TAC[LE_REFL]);; let NEGLIGIBLE_COUNTABLE_UNIONS_GEN = prove (`!f. COUNTABLE f /\ (!s:real^N->bool. s IN f ==> negligible s) ==> negligible(UNIONS f)`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN ASM_REWRITE_TAC[UNIONS_0; NEGLIGIBLE_EMPTY] THEN MP_TAC(ISPEC `f:(real^N->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE; IN_UNIV] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN ASM_REWRITE_TAC[]);; let HAS_MEASURE_COUNTABLE_NEGLIGIBLE_UNIONS_BOUNDED = prove (`!s:num->real^N->bool. (!n. measurable(s n)) /\ (!m n. ~(m = n) ==> negligible(s m INTER s n)) /\ bounded(UNIONS { s(n) | n IN (:num) }) ==> measurable(UNIONS { s(n) | n IN (:num) }) /\ ((\n. lift(measure(s n))) sums lift(measure(UNIONS { s(n) | n IN (:num) }))) (from 0)`, REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN MATCH_MP_TAC HAS_MEASURE_COUNTABLE_NEGLIGIBLE_UNIONS THEN EXISTS_TAC `measure(interval[a:real^N,b])` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(UNIONS (IMAGE (s:num->real^N->bool) (0..n)))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS_IMAGE THEN ASM_SIMP_TAC[FINITE_NUMSEG]; MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE]; ASM SET_TAC[]]]);; let MEASURABLE_COUNTABLE_UNIONS_BOUNDED = prove (`!s:num->real^N->bool. (!n. measurable(s n)) /\ bounded(UNIONS { s(n) | n IN (:num) }) ==> measurable(UNIONS { s(n) | n IN (:num) })`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `UNIONS { s(n):real^N->bool | n IN (:num) } = UNIONS { UNIONS {s(m) | m IN 0..n} | n IN (:num)}` SUBST1_TAC THENL [GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[GSYM CONJ_ASSOC; UNWIND_THM2; IN_UNIONS; IN_ELIM_THM] THEN REWRITE_TAC[IN_NUMSEG; IN_UNIV; LE_0] THEN MESON_TAC[LE_REFL]; MATCH_MP_TAC MEASURABLE_NESTED_UNIONS THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN EXISTS_TAC `measure(interval[a:real^N,b])` THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG]; DISCH_TAC] THEN CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[MEASURABLE_INTERVAL] THEN ASM SET_TAC[]; GEN_TAC THEN REWRITE_TAC[NUMSEG_CLAUSES; LE_0] THEN SET_TAC[]]]);; let MEASURE_COUNTABLE_UNIONS_LE_STRONG = prove (`!d:num->(real^N->bool) B. (!n. measurable(d n)) /\ (!n. measure(UNIONS {d k | k <= n}) <= B) ==> measurable(UNIONS {d n | n IN (:num)}) /\ measure(UNIONS {d n | n IN (:num)}) <= B`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`\n. UNIONS {(d:num->(real^N->bool)) k | k IN (0..n)}`; `B:real`] HAS_MEASURE_NESTED_UNIONS) THEN REWRITE_TAC[] THEN SUBGOAL_THEN `UNIONS {UNIONS {d k | k IN (0..n)} | n IN (:num)} = UNIONS {d n:real^N->bool | n IN (:num)}` SUBST1_TAC THENL [GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_UNIONS; EXISTS_IN_GSPEC; IN_UNIV; IN_NUMSEG; LE_0] THEN MESON_TAC[LE_REFL]; ALL_TAC] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE]; ASM_REWRITE_TAC[IN_NUMSEG; LE_0]; GEN_TAC THEN REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC SUBSET_UNIONS THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET_NUMSEG] THEN ARITH_TAC]; STRIP_TAC THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM(CONJUNCT2 LIFT_DROP)] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_UBOUND) THEN EXISTS_TAC `\n. lift(measure(UNIONS {d k | k IN 0..n} :real^N->bool))` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `0` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN ASM_REWRITE_TAC[LIFT_DROP; IN_NUMSEG; LE_0]]);; let MEASURE_COUNTABLE_UNIONS_LE = prove (`!d:num->(real^N->bool) B. (!n. measurable(d n)) /\ (!n. sum(0..n) (\k. measure(d k)) <= B) ==> measurable(UNIONS {d n | n IN (:num)}) /\ measure(UNIONS {d n | n IN (:num)}) <= B`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN MP_TAC(ISPECL [`0..n`;`d:num->real^N->bool`] MEASURE_UNIONS_LE_IMAGE) THEN ASM_REWRITE_TAC[FINITE_NUMSEG] THEN REPEAT(FIRST_X_ASSUM (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; numseg; LE_0; IN_ELIM_THM] THEN MESON_TAC[REAL_LE_TRANS]);; let MEASURABLE_COUNTABLE_UNIONS_STRONG = prove (`!s:num->real^N->bool B. (!n. measurable(s n)) /\ (!n. measure(UNIONS {s k | k <= n}) <= B) ==> measurable(UNIONS { s(n) | n IN (:num) })`, MESON_TAC[MEASURE_COUNTABLE_UNIONS_LE_STRONG; REAL_LE_REFL]);; let MEASURABLE_COUNTABLE_UNIONS = prove (`!s:num->real^N->bool B. (!n. measurable(s n)) /\ (!n. sum (0..n) (\k. measure(s k)) <= B) ==> measurable(UNIONS { s(n) | n IN (:num) })`, MESON_TAC[MEASURE_COUNTABLE_UNIONS_LE; REAL_LE_REFL]);; let MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN = prove (`!D B. COUNTABLE D /\ (!d:real^N->bool. d IN D ==> measurable d) /\ (!D'. D' SUBSET D /\ FINITE D' ==> measure(UNIONS D') <= B) ==> measurable(UNIONS D) /\ measure(UNIONS D) <= B`, REPEAT GEN_TAC THEN ASM_CASES_TAC `D:(real^N->bool)->bool = {}` THENL [ASM_SIMP_TAC[UNIONS_0; MEASURABLE_EMPTY; SUBSET_EMPTY] THEN MESON_TAC[FINITE_EMPTY]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MP_TAC(ISPEC `D:(real^N->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:num->real^N->bool` SUBST1_TAC) THEN REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; FORALL_SUBSET_IMAGE] THEN REWRITE_TAC[IN_UNIV; SUBSET_UNIV] THEN REPEAT DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o SPEC `{k:num | k <= n}`) THEN SIMP_TAC[FINITE_NUMSEG_LE; FINITE_IMAGE] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN REPLICATE_TAC 3 AP_TERM_TAC THEN SET_TAC[]]);; let MEASURE_COUNTABLE_UNIONS_LE_GEN = prove (`!D B. COUNTABLE D /\ (!d:real^N->bool. d IN D ==> measurable d) /\ (!D'. D' SUBSET D /\ FINITE D' ==> sum D' (\d. measure d) <= B) ==> measurable(UNIONS D) /\ measure(UNIONS D) <= B`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `D':(real^N->bool)->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `D':(real^N->bool)->bool`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC MEASURE_UNIONS_LE THEN ASM SET_TAC[]);; let MEASURABLE_COUNTABLE_INTERS = prove (`!s:num->real^N->bool. (!n. measurable(s n)) ==> measurable(INTERS { s(n) | n IN (:num) })`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `INTERS { s(n):real^N->bool | n IN (:num) } = s 0 DIFF (UNIONS {s 0 DIFF s n | n IN (:num)})` SUBST1_TAC THENL [GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_INTERS; IN_DIFF; IN_UNIONS] THEN REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_COUNTABLE_UNIONS_STRONG THEN EXISTS_TAC `measure(s 0:real^N->bool)` THEN ASM_SIMP_TAC[MEASURABLE_DIFF; LE_0] THEN GEN_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; IN_ELIM_THM; IN_DIFF] THEN MESON_TAC[IN_DIFF]] THEN ONCE_REWRITE_TAC[GSYM IN_NUMSEG_0] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; MEASURABLE_DIFF; MEASURABLE_UNIONS]);; let MEASURABLE_COUNTABLE_INTERS_GEN = prove (`!D. COUNTABLE D /\ ~(D = {}) /\ (!d:real^N->bool. d IN D ==> measurable d) ==> measurable(INTERS D)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `D:(real^N->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE; IN_UNIV] THEN GEN_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN MATCH_MP_TAC MEASURABLE_COUNTABLE_INTERS THEN ASM SET_TAC[]);; let MEASURE_COUNTABLE_UNIONS_APPROACHABLE = prove (`!D B e. COUNTABLE D /\ (!d. d IN D ==> measurable d) /\ (!D'. D' SUBSET D /\ FINITE D' ==> measure(UNIONS D') <= B) /\ &0 < e ==> ?D'. D' SUBSET D /\ FINITE D' /\ measure(UNIONS D) - e < measure(UNIONS D':real^N->bool)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `D:(real^N->bool)->bool = {}` THENL [DISCH_TAC THEN EXISTS_TAC `{}:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[EMPTY_SUBSET; FINITE_EMPTY; UNIONS_0; MEASURE_EMPTY] THEN ASM_REAL_ARITH_TAC; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MP_TAC(ISPEC `D:(real^N->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:num->real^N->bool` SUBST1_TAC) THEN REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; EXISTS_SUBSET_IMAGE; FORALL_SUBSET_IMAGE] THEN REWRITE_TAC[IN_UNIV; SUBSET_UNIV] THEN REPEAT DISCH_TAC THEN MP_TAC(ISPECL [`\n. UNIONS(IMAGE (d:num->real^N->bool) {k | k <= n})`; `B:real`] HAS_MEASURE_NESTED_UNIONS) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[MEASURABLE_UNIONS; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG_LE; IN_ELIM_THM] THEN GEN_TAC THEN MATCH_MP_TAC SUBSET_UNIONS THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `UNIONS {UNIONS (IMAGE d {k | k <= n}) | n IN (:num)}:real^N->bool = UNIONS (IMAGE d (:num))` SUBST1_TAC THENL [REWRITE_TAC[UNIONS_IMAGE] THEN REWRITE_TAC[UNIONS_GSPEC] THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM; EXTENSION] THEN MESON_TAC[LE_REFL]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[LIM_SEQUENTIALLY; DIST_REAL; GSYM drop; LIFT_DROP] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[LE_REFL] THEN DISCH_TAC THEN EXISTS_TAC `{k:num | k <= n}` THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LE] THEN ASM_SIMP_TAC[REAL_ARITH `abs(x - u) < e /\ &0 < e ==> u - e < x`]]);; let HAS_MEASURE_NESTED_INTERS = prove (`!s:num->real^N->bool. (!n. measurable(s n)) /\ (!n. s(SUC n) SUBSET s(n)) ==> measurable(INTERS {s n | n IN (:num)}) /\ ((\n. lift(measure (s n))) --> lift(measure (INTERS {s n | n IN (:num)}))) sequentially`, GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`\n. (s:num->real^N->bool) 0 DIFF s n`; `measure(s 0:real^N->bool)`] HAS_MEASURE_NESTED_UNIONS) THEN ASM_SIMP_TAC[MEASURABLE_DIFF] THEN ANTS_TAC THENL [CONJ_TAC THEN X_GEN_TAC `n:num` THENL [MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_DIFF; SUBSET_DIFF] THEN SET_TAC[]; REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `n:num`)) THEN SET_TAC[]]; SUBGOAL_THEN `UNIONS {s 0 DIFF s n | n IN (:num)} = s 0 DIFF INTERS {s n :real^N->bool | n IN (:num)}` (fun th -> REWRITE_TAC[th]) THENL [REWRITE_TAC[DIFF_INTERS] THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [DISCH_TAC THEN SUBGOAL_THEN `measurable(s 0 DIFF (s 0 DIFF INTERS {s n | n IN (:num)}) :real^N->bool)` MP_TAC THENL [ASM_SIMP_TAC[MEASURABLE_DIFF]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `t SUBSET s ==> s DIFF (s DIFF t) = t`) THEN REWRITE_TAC[SUBSET; INTERS_GSPEC; IN_ELIM_THM] THEN SET_TAC[]; MP_TAC(ISPECL [`sequentially`; `lift(measure(s 0:real^N->bool))`] LIM_CONST) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_SUB) THEN REWRITE_TAC[GSYM LIFT_SUB] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN BINOP_TAC THEN REWRITE_TAC[LIFT_EQ; FUN_EQ_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[REAL_ARITH `s - m:real = n <=> m = s - n`] THEN MATCH_MP_TAC MEASURE_DIFF_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_COUNTABLE_INTERS] THENL [ALL_TAC; SET_TAC[]] THEN MP_TAC(ISPEC `\m n:num. (s n :real^N->bool) SUBSET (s m)` TRANSITIVE_STEPWISE_LE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [SET_TAC[]; MESON_TAC[LE_0]]]]);; let INTEGRAL_ZERO_ON_SUBINTERVALS_IMP_ZERO_AE_ALT = prove (`!f:real^M->real^N. (!a b. (f has_integral (vec 0)) (interval[a,b])) ==> negligible{x | ~(f x = vec 0)}`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `(:real^M)` OPEN_COUNTABLE_UNION_CLOSED_INTERVALS) THEN REWRITE_TAC[OPEN_UNIV; SUBSET_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | x IN d /\ ~((f:real^M->real^N) x = vec 0)} | d IN D}` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `d:real^M->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:real^M->bool`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC INTEGRAL_ZERO_ON_SUBINTERVALS_IMP_ZERO_AE THEN ASM_REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* Measurability of compact and bounded open sets. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_COMPACT = prove (`!s:real^N->bool. compact s ==> measurable s`, let lemma = prove (`!f s:real^N->bool. (!n. FINITE(f n)) /\ (!n. s SUBSET UNIONS(f n)) /\ (!x. ~(x IN s) ==> ?n. ~(x IN UNIONS(f n))) /\ (!n a. a IN f(SUC n) ==> ?b. b IN f(n) /\ a SUBSET b) /\ (!n a. a IN f(n) ==> measurable a) ==> measurable s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!n. UNIONS(f(SUC n):(real^N->bool)->bool) SUBSET UNIONS(f n)` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `s = INTERS { UNIONS(f n) | n IN (:num) }:real^N->bool` SUBST1_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE; IN_UNIV] THEN REWRITE_TAC[IN_IMAGE] THEN ASM SET_TAC[]; MATCH_MP_TAC MEASURABLE_COUNTABLE_INTERS THEN ASM_REWRITE_TAC[] THEN GEN_TAC THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_MESON_TAC[]]) in REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN EXISTS_TAC `\n. { k | ?u:real^N. (!i. 1 <= i /\ i <= dimindex(:N) ==> integer(u$i)) /\ k = { x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> u$i / &2 pow n <= x$i /\ x$i < (u$i + &1) / &2 pow n } /\ ~(s INTER k = {})}` THEN REWRITE_TAC[IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN SUBGOAL_THEN `?N. !x:real^N i. x IN s /\ 1 <= i /\ i <= dimindex(:N) ==> abs(x$i * &2 pow n) < &N` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN REWRITE_TAC[BOUNDED_POS; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN MP_TAC(SPEC `B * &2 pow n` (MATCH_MP REAL_ARCH REAL_LT_01)) THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[REAL_MUL_RID] THEN X_GEN_TAC `N:num` THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_POW; REAL_ABS_NUM] THEN SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS; REAL_LET_TRANS]; ALL_TAC] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `IMAGE (\u. {x | !i. 1 <= i /\ i <= dimindex(:N) ==> (u:real^N)$i <= (x:real^N)$i * &2 pow n /\ x$i * &2 pow n < u$i + &1}) {u | !i. 1 <= i /\ i <= dimindex(:N) ==> integer (u$i) /\ abs(u$i) <= &N}` THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_IMAGE THEN MATCH_MP_TAC FINITE_CART THEN REWRITE_TAC[GSYM REAL_BOUNDS_LE; FINITE_INTSEG]; REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_IMAGE] THEN X_GEN_TAC `l:real^N->bool` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_REVERSE_INTEGERS THEN ASM_SIMP_TAC[INTEGER_CLOSED] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_THEN `x:real^N` MP_TAC) THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `k:num`)) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `k:num`]) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]; X_GEN_TAC `n:num` THEN REWRITE_TAC[SUBSET; IN_UNIONS; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `(lambda i. floor(&2 pow n * (x:real^N)$i)):real^N` THEN ONCE_REWRITE_TAC[TAUT `(a /\ b /\ c) /\ d <=> b /\ a /\ c /\ d`] THEN REWRITE_TAC[UNWIND_THM2] THEN SIMP_TAC[LAMBDA_BETA; FLOOR] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_MUL_SYM; FLOOR]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_CLOSED) THEN REWRITE_TAC[closed; open_def] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPECL [`inv(&2)`; `e / &(dimindex(:N))`] REAL_ARCH_POW_INV) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; DIMINDEX_GE_1; ARITH_RULE `0 < x <=> 1 <= x`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(a /\ b /\ c) /\ d <=> b /\ a /\ c /\ d`] THEN REWRITE_TAC[UNWIND_THM2] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `u:real^N` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC o CONJUNCT2) THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `d < e ==> x <= d ==> x < e`)) THEN REWRITE_TAC[dist] THEN W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN MATCH_MP_TAC(REAL_ARITH `a <= b ==> x <= a ==> x <= b`) THEN GEN_REWRITE_TAC (funpow 3 RAND_CONV) [GSYM CARD_NUMSEG_1] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN MATCH_MP_TAC SUM_BOUND THEN SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; VECTOR_SUB_COMPONENT] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `k:num`)) THEN ASM_REWRITE_TAC[real_div; REAL_ADD_RDISTRIB] THEN REWRITE_TAC[REAL_MUL_LID; GSYM REAL_POW_INV] THEN REAL_ARITH_TAC; MAP_EVERY X_GEN_TAC [`n:num`; `a:real^N->bool`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o SYM) ASSUME_TAC) THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(a /\ b /\ c) /\ d <=> b /\ a /\ c /\ d`] THEN REWRITE_TAC[UNWIND_THM2] THEN EXISTS_TAC `(lambda i. floor((u:real^N)$i / &2)):real^N` THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; LAMBDA_BETA; FLOOR] THEN MATCH_MP_TAC(SET_RULE `~(s INTER a = {}) /\ a SUBSET b ==> ~(s INTER b = {}) /\ a SUBSET b`) THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "a" THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[real_pow; real_div; REAL_INV_MUL; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN MP_TAC(SPEC `(u:real^N)$k / &2` FLOOR) THEN REWRITE_TAC[REAL_ARITH `u / &2 < floor(u / &2) + &1 <=> u < &2 * floor(u / &2) + &2`] THEN ASM_SIMP_TAC[REAL_LT_INTEGERS; INTEGER_CLOSED; FLOOR_FRAC] THEN REAL_ARITH_TAC; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`n:num`; `a:real^N->bool`; `u:real^N`] THEN DISCH_THEN(SUBST1_TAC o CONJUNCT1 o CONJUNCT2) THEN ONCE_REWRITE_TAC[MEASURABLE_INNER_OUTER] THEN GEN_TAC THEN DISCH_TAC THEN EXISTS_TAC `interval(inv(&2 pow n) % u:real^N, inv(&2 pow n) % (u + vec 1))` THEN EXISTS_TAC `interval[inv(&2 pow n) % u:real^N, inv(&2 pow n) % (u + vec 1)]` THEN REWRITE_TAC[MEASURABLE_INTERVAL; MEASURE_INTERVAL] THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_ABS_0] THEN REWRITE_TAC[SUBSET; IN_INTERVAL; IN_ELIM_THM] THEN CONJ_TAC THEN X_GEN_TAC `y:real^N` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT; VEC_COMPONENT] THEN REAL_ARITH_TAC]);; let MEASURABLE_OPEN = prove (`!s:real^N->bool. bounded s /\ open s ==> measurable s`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> s = t DIFF (t DIFF s)`)) THEN MATCH_MP_TAC MEASURABLE_DIFF THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_DIFF; BOUNDED_INTERVAL] THEN MATCH_MP_TAC CLOSED_DIFF THEN ASM_REWRITE_TAC[CLOSED_INTERVAL]);; let MEASURE_OPEN_POS_LT = prove (`!s. open s /\ bounded s /\ ~(s = {}) ==> &0 < measure s`, MESON_TAC[OPEN_NOT_NEGLIGIBLE; MEASURABLE_MEASURE_POS_LT; MEASURABLE_OPEN]);; let MEASURE_OPEN_POS_LT_EQ = prove (`!s. open s /\ bounded s ==> (&0 < measure s <=> ~(s = {}))`, MESON_TAC[MEASURE_OPEN_POS_LT; MEASURE_EMPTY; REAL_LT_REFL]);; let MEASURABLE_CLOSURE = prove (`!s. bounded s ==> measurable(closure s)`, SIMP_TAC[MEASURABLE_COMPACT; COMPACT_EQ_BOUNDED_CLOSED; CLOSED_CLOSURE; BOUNDED_CLOSURE]);; let MEASURABLE_INTERIOR = prove (`!s. bounded s ==> measurable(interior s)`, SIMP_TAC[MEASURABLE_OPEN; OPEN_INTERIOR; BOUNDED_INTERIOR]);; let MEASURABLE_FRONTIER = prove (`!s:real^N->bool. bounded s ==> measurable(frontier s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_SIMP_TAC[MEASURABLE_CLOSURE; MEASURABLE_INTERIOR] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN REWRITE_TAC[INTERIOR_SUBSET; CLOSURE_SUBSET]);; let MEASURE_FRONTIER = prove (`!s:real^N->bool. bounded s ==> measure(frontier s) = measure(closure s) - measure(interior s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN MATCH_MP_TAC MEASURE_DIFF_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_CLOSURE; MEASURABLE_INTERIOR] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `s:real^N->bool` THEN REWRITE_TAC[INTERIOR_SUBSET; CLOSURE_SUBSET]);; let MEASURE_CLOSURE = prove (`!s:real^N->bool. bounded s /\ negligible(frontier s) ==> measure(closure s) = measure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN ASM_SIMP_TAC[MEASURABLE_CLOSURE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN REWRITE_TAC[frontier] THEN SET_TAC[]);; let MEASURE_INTERIOR = prove (`!s:real^N->bool. bounded s /\ negligible(frontier s) ==> measure(interior s) = measure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN ASM_SIMP_TAC[MEASURABLE_INTERIOR] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN REWRITE_TAC[frontier] THEN SET_TAC[]);; let MEASURABLE_JORDAN = prove (`!s:real^N->bool. bounded s /\ negligible(frontier s) ==> measurable s`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[MEASURABLE_INNER_OUTER] THEN GEN_TAC THEN DISCH_TAC THEN EXISTS_TAC `interior(s):real^N->bool` THEN EXISTS_TAC `closure(s):real^N->bool` THEN ASM_SIMP_TAC[MEASURABLE_INTERIOR; MEASURABLE_CLOSURE] THEN REWRITE_TAC[INTERIOR_SUBSET; CLOSURE_SUBSET] THEN ONCE_REWRITE_TAC[REAL_ABS_SUB] THEN ASM_SIMP_TAC[GSYM MEASURE_FRONTIER; REAL_ABS_NUM; MEASURE_EQ_0]);; let HAS_MEASURE_ELEMENTARY = prove (`!d s. d division_of s ==> s has_measure (sum d content)`, REPEAT STRIP_TAC THEN REWRITE_TAC[has_measure] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP DIVISION_OF_FINITE) THEN ASM_SIMP_TAC[LIFT_SUM] THEN MATCH_MP_TAC HAS_INTEGRAL_COMBINE_DIVISION THEN ASM_REWRITE_TAC[o_THM] THEN REWRITE_TAC[GSYM has_measure] THEN ASM_MESON_TAC[HAS_MEASURE_INTERVAL; division_of]);; let MEASURABLE_ELEMENTARY = prove (`!d s. d division_of s ==> measurable s`, REWRITE_TAC[measurable] THEN MESON_TAC[HAS_MEASURE_ELEMENTARY]);; let MEASURE_ELEMENTARY = prove (`!d s. d division_of s ==> measure s = sum d content`, MESON_TAC[HAS_MEASURE_ELEMENTARY; MEASURE_UNIQUE]);; let MEASURABLE_INTER_INTERVAL = prove (`!s a b:real^N. measurable s ==> measurable (s INTER interval[a,b])`, SIMP_TAC[MEASURABLE_INTER; MEASURABLE_INTERVAL]);; let MEASURABLE_INSIDE = prove (`!s:real^N->bool. compact s ==> measurable(inside s)`, SIMP_TAC[MEASURABLE_OPEN; BOUNDED_INSIDE; COMPACT_IMP_CLOSED; OPEN_INSIDE; COMPACT_IMP_BOUNDED]);; (* ------------------------------------------------------------------------- *) (* We can split off part of a measurable set of chosen size. *) (* ------------------------------------------------------------------------- *) let PART_MEASURES = prove (`!s:real^N->bool m. measurable s /\ &0 <= m /\ m <= measure s ==> ?t u. DISJOINT t u /\ t UNION u = s /\ measurable t /\ measure t = m /\ measurable u /\ measure u = measure s - m`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP MEASURE_POS_LE) THEN ASM_CASES_TAC `measure(s:real^N->bool) = m` THENL [MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `{}:real^N->bool`] THEN ASM_REWRITE_TAC[UNION_EMPTY; DISJOINT_EMPTY; MEASURE_EMPTY] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[MEASURABLE_EMPTY] THEN REWRITE_TAC[REAL_SUB_REFL]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `measure(s:real^N->bool) - m`] MEASURE_LIMIT) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`\x. lift(measure(s INTER interval[--(lambda i. drop x):real^N,(lambda i. drop x)]))`; `vec 0:real^1`; `lift B`; `m:real`; `1`] IVT_INCREASING_COMPONENT_ON_1) THEN ASM_REWRITE_TAC[GSYM drop; LIFT_DROP; DIMINDEX_1; LE_REFL; DROP_VEC] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC; GSYM EXISTS_DROP] THEN DISCH_THEN(X_CHOOSE_THEN `b:real` MP_TAC) THEN ABBREV_TAC `c:real^N = lambda i. b` THEN STRIP_TAC THEN EXISTS_TAC `s INTER interval[--c:real^N,c]` THEN EXISTS_TAC `s DIFF interval[--c:real^N,c]` THEN REPLICATE_TAC 2 (MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [SET_TAC[]; DISCH_TAC]) THEN MP_TAC(ISPECL [`s INTER interval[--c:real^N,c]`; `s DIFF interval[--c:real^N,c]`] MEASURE_DISJOINT_UNION) THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTER; MEASURABLE_INTERVAL] THEN REAL_ARITH_TAC] THEN REPEAT CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[INTERVAL_SING; GSYM vec; VECTOR_NEG_0] THEN TRANS_TAC REAL_LE_TRANS `measure {vec 0:real^N}` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_SING; MEASURE_SING] THEN REWRITE_TAC[INTER_SUBSET]; REWRITE_TAC[MEASURE_SING] THEN ASM_REAL_ARITH_TAC]; MATCH_MP_TAC(REAL_ARITH `abs(measure (s INTER i) - measure s) < measure s - m ==> m <= measure(s INTER i:real^N->bool)`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN SIMP_TAC[SUBSET; IN_BALL_0; IN_INTERVAL; LAMBDA_BETA; VECTOR_NEG_COMPONENT; GSYM REAL_ABS_BOUNDS] THEN MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS; REAL_LT_IMP_LE]] THEN MATCH_MP_TAC(INST_TYPE [`:N`,`:P`] CONTINUOUS_ON_COMPARISON) THEN EXISTS_TAC `\x. lift(measure(interval[--(lambda i. drop x):real^N, (lambda i. drop x)]))` THEN CONJ_TAC THENL [REWRITE_TAC[MEASURE_INTERVAL; continuous_on] THEN REWRITE_TAC[CONTENT_CLOSED_INTERVAL_CASES] THEN SIMP_TAC[LAMBDA_BETA; VECTOR_NEG_COMPONENT] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN SIMP_TAC[REAL_ARITH `--x <= x <=> &0 <= x`; LIFT_DROP] THEN SIMP_TAC[PRODUCT_CONST; CARD_NUMSEG_1; REAL_ARITH `x - --x = &2 * x`; FINITE_NUMSEG] THEN SUBGOAL_THEN `(\x. lift((&2 * drop x) pow (dimindex(:N)))) continuous_on (:real^1)` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_LIFT_POW THEN SIMP_TAC[LIFT_CMUL; LIFT_DROP; CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID]; REWRITE_TAC[continuous_on; IN_UNIV] THEN MESON_TAC[]]; REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[DIST_SYM]; ALL_TAC] THEN REWRITE_TAC[LIFT_DROP; IN_INTERVAL_1; DROP_VEC; DIST_LIFT] THEN MAP_EVERY X_GEN_TAC [`x:real`; `y:real`] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ABS_SUB] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DIFF_SUBSET o rand o lhand o snd) THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DIFF_SUBSET o funpow 3 rand o snd) THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_INTERVAL] THEN ASM_SIMP_TAC[SUBSET_INTERVAL; VECTOR_NEG_COMPONENT; LAMBDA_BETA; REAL_LE_NEG2; SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`] THEN REPEAT(DISCH_THEN(SUBST1_TAC o SYM)) THEN ASM_SIMP_TAC[real_abs; MEASURE_POS_LE; MEASURABLE_DIFF; MEASURABLE_INTER; MEASURABLE_INTERVAL] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTER; MEASURABLE_INTERVAL] THEN SET_TAC[]]);; let HALF_MEASURES = prove (`!s:real^N->bool. measurable s ==> ?t u. DISJOINT t u /\ t UNION u = s /\ measurable t /\ measure t = measure s / &2 /\ measurable u /\ measure u = measure s / &2`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `measure(s:real^N->bool) / &2`] PART_MEASURES) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP MEASURE_POS_LE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let MULTIPART_MEASURES = prove (`!s:real^N->bool n. measurable s /\ ~(n = 0) ==> ?f. FINITE f /\ CARD f <= n /\ pairwise DISJOINT f /\ UNIONS f = s /\ !t. t IN f ==> t SUBSET s /\ measurable t /\ measure t = measure s / &n`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?k:num->real^N->bool. !m. k m = @t. t SUBSET s DIFF UNIONS {k i | i < m} /\ measurable t /\ measure t = measure s / &n` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(MATCH_MP WF_REC WF_num) THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN REWRITE_TAC[UNIONS_GSPEC; MESON[] `(?m:num. m < n /\ a IN f m) <=> ~(!m. m < n ==> ~(a IN f m))`] THEN ASM_SIMP_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!m. m < n - 1 ==> k m SUBSET s DIFF UNIONS {k i | i < m} /\ measurable(k m) /\ measure((k:num->real^N->bool) m) = measure(s:real^N->bool) / &n` MP_TAC THENL [MATCH_MP_TAC num_WF THEN X_GEN_TAC `m:num` THEN FIRST_X_ASSUM(SUBST1_TAC o SPEC `m:num`) THEN REWRITE_TAC[IMP_IMP] THEN STRIP_TAC THEN CONV_TAC SELECT_CONV THEN MP_TAC(ISPECL [`s DIFF UNIONS {(k:num->real^N->bool) i | i < m}`; `measure(s:real^N->bool) / &n`] PART_MEASURES) THEN ANTS_TAC THENL [ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; MEASURE_POS_LE]; MATCH_MP_TAC MONO_EXISTS THEN STRIP_TAC THEN ASM SET_TAC[]] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LT; FORALL_IN_IMAGE] THEN X_GEN_TAC `p:num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `p:num`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ASM SET_TAC[]]; W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_DIFF_SUBSET o rand o snd) THEN ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LT; FORALL_IN_IMAGE]; REWRITE_TAC[FORALL_IN_GSPEC]] THEN X_GEN_TAC `p:num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `p:num`) THEN (ANTS_TAC THENL [ASM_ARITH_TAC; ASM SET_TAC[]]); DISCH_THEN SUBST1_TAC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN W(MP_TAC o PART_MATCH(lhand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o rand o rand o snd) THEN REWRITE_TAC[FINITE_NUMSEG_LT; FORALL_IN_GSPEC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (MESON[LT_TRANS] `(!i. i < m /\ i < n - 1 ==> P i) ==> m < n - 1 ==> (!i. i < m ==> P i)`)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN ANTS_TAC THENL [REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `j:num`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN SIMP_TAC[SUM_CONST; FINITE_NUMSEG_LT; CARD_NUMSEG_LT] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1; REAL_SUB_RDISTRIB; REAL_DIV_RMUL; REAL_LT_IMP_NZ; GSYM REAL_MUL_ASSOC] THEN REWRITE_TAC[REAL_ARITH `s <= s * a - b * s <=> &0 <= s * (a - b - &1)`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[MEASURE_POS_LE] THEN REWRITE_TAC[REAL_LE_SUB_LADD] THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC]; FIRST_X_ASSUM(K ALL_TAC o SPEC `0`) THEN DISCH_TAC THEN EXISTS_TAC `(s DIFF UNIONS(IMAGE k {m | m < n - 1})) INSERT (IMAGE (k:num->real^N->bool) {m | m < n - 1})` THEN REWRITE_TAC[FORALL_IN_INSERT; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SIMP_TAC[FINITE_INSERT; FINITE_IMAGE; FINITE_NUMSEG_LT] THEN REPEAT CONJ_TAC THENL [SIMP_TAC[CARD_CLAUSES; FINITE_IMAGE; FINITE_NUMSEG_LT] THEN MATCH_MP_TAC(ARITH_RULE `~(n = 0) /\ m <= n - 1 ==> (if p then m else SUC m) <= n`) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC RAND_CONV [GSYM CARD_NUMSEG_LT] THEN MATCH_MP_TAC CARD_IMAGE_LE THEN REWRITE_TAC[FINITE_NUMSEG_LT]; REWRITE_TAC[PAIRWISE_INSERT] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[PAIRWISE_IMAGE] THEN REWRITE_TAC[pairwise; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `j:num`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]; REWRITE_TAC[UNIONS_INSERT] THEN MATCH_MP_TAC(SET_RULE `t SUBSET s ==> (s DIFF t) UNION t = s`) THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET_DIFF]; MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LT; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; W(MP_TAC o PART_MATCH(lhand o rand) MEASURE_DIFF_SUBSET o lhand o snd) THEN ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LT; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[REAL_ARITH `s - t = s / a <=> t = (&1 - inv a) * s`] THEN W(MP_TAC o PART_MATCH(lhand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o lhand o snd) THEN ASM_SIMP_TAC[FINITE_NUMSEG_LT; IN_ELIM_THM] THEN ANTS_TAC THENL [MATCH_MP_TAC WLOG_LT THEN RULE_ASSUM_TAC(REWRITE_RULE[UNIONS_GSPEC]) THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN SIMP_TAC[SUM_CONST; CARD_NUMSEG_LT; FINITE_NUMSEG_LT] THEN ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; LE_1] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM REAL_OF_NUM_EQ]) THEN CONV_TAC REAL_FIELD; ASM SET_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* A nice lemma for negligibility proofs. *) (* ------------------------------------------------------------------------- *) let STARLIKE_NEGLIGIBLE_BOUNDED_MEASURABLE = prove (`!s. measurable s /\ bounded s /\ (!c x:real^N. &0 <= c /\ x IN s /\ (c % x) IN s ==> c = &1) ==> negligible s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(&0 < measure(s:real^N->bool))` (fun th -> ASM_MESON_TAC[th; MEASURABLE_MEASURE_POS_LT]) THEN DISCH_TAC THEN MP_TAC(SPEC `(vec 0:real^N) INSERT s` BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC) THEN ASM_SIMP_TAC[BOUNDED_INSERT; COMPACT_IMP_BOUNDED; NOT_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN REWRITE_TAC[INSERT_SUBSET] THEN STRIP_TAC THEN SUBGOAL_THEN `?N. EVEN N /\ &0 < &N /\ measure(interval[--a:real^N,a]) < (&N * measure(s:real^N->bool)) / &4 pow dimindex (:N)` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o SPEC `measure(interval[--a:real^N,a]) * &4 pow (dimindex(:N))` o MATCH_MP REAL_ARCH) THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN SIMP_TAC[GSYM REAL_LT_LDIV_EQ; ASSUME `&0 < measure(s:real^N->bool)`] THEN DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `2 * (N DIV 2 + 1)` THEN REWRITE_TAC[EVEN_MULT; ARITH] THEN CONJ_TAC THENL [ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < a ==> a <= b ==> x < b`)) THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`UNIONS (IMAGE (\m. IMAGE (\x:real^N. (&m / &N) % x) s) (1..N))`; `interval[--a:real^N,a]`] MEASURE_SUBSET) THEN MP_TAC(ISPECL [`measure:(real^N->bool)->real`; `IMAGE (\m. IMAGE (\x:real^N. (&m / &N) % x) s) (1..N)`] HAS_MEASURE_DISJOINT_UNIONS) THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM HAS_MEASURE_MEASURE] THEN MATCH_MP_TAC MEASURABLE_SCALING THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ ~c ==> d <=> a /\ b /\ ~d ==> c`] THEN SUBGOAL_THEN `!m n. m IN 1..N /\ n IN 1..N /\ ~(DISJOINT (IMAGE (\x:real^N. &m / &N % x) s) (IMAGE (\x. &n / &N % x) s)) ==> m = n` ASSUME_TAC THENL [MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[DISJOINT; GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[EXISTS_IN_IMAGE; IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN DISCH_THEN(MP_TAC o AP_TERM `(%) (&N / &m) :real^N->real^N`) THEN SUBGOAL_THEN `~(&N = &0) /\ ~(&m = &0)` STRIP_ASSUME_TAC THENL [REWRITE_TAC[REAL_OF_NUM_EQ] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_NUMSEG])) THEN ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE (BINDER_CONV o BINDER_CONV) [GSYM CONTRAPOS_THM]) THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_FIELD `~(x = &0) /\ ~(y = &0) ==> x / y * y / x = &1`] THEN ASM_SIMP_TAC[REAL_FIELD `~(x = &0) /\ ~(y = &0) ==> x / y * z / x = z / y`] THEN REWRITE_TAC[VECTOR_MUL_LID] THEN DISCH_THEN SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`&n / &m`; `y:real^N`]) THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; REAL_FIELD `~(y = &0) ==> (x / y = &1 <=> x = y)`] THEN REWRITE_TAC[REAL_OF_NUM_EQ; EQ_SYM_EQ]; ALL_TAC] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_TAC] THEN REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[measurable] THEN ASM_MESON_TAC[]; REWRITE_TAC[MEASURABLE_INTERVAL]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`--a:real^N`; `a:real^N`] CONVEX_INTERVAL) THEN DISCH_THEN(MP_TAC o REWRITE_RULE[CONVEX_ALT] o CONJUNCT1) THEN DISCH_THEN(MP_TAC o SPECL [`vec 0:real^N`; `x:real^N`; `&n / &N`]) THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN DISCH_THEN MATCH_MP_TAC THEN SIMP_TAC[REAL_LE_DIV; REAL_POS] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_NUMSEG]) THEN DISCH_THEN(MP_TAC o MATCH_MP (ARITH_RULE `1 <= n /\ n <= N ==> 0 < N /\ n <= N`)) THEN SIMP_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_LT; REAL_LE_LDIV_EQ] THEN SIMP_TAC[REAL_MUL_LID]; ALL_TAC] THEN FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP MEASURE_UNIQUE) THEN ASM_SIMP_TAC[MEASURE_SCALING; REAL_NOT_LE] THEN FIRST_X_ASSUM(K ALL_TAC o SPEC `&0`) THEN MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `sum (1..N) (measure o (\m. IMAGE (\x:real^N. &m / &N % x) s))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC REAL_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_IMAGE THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[SET_RULE `DISJOINT s s <=> s = {}`; IMAGE_EQ_EMPTY] THEN DISCH_THEN SUBST_ALL_TAC THEN ASM_MESON_TAC[REAL_LT_REFL; MEASURE_EMPTY]] THEN FIRST_X_ASSUM(K ALL_TAC o SPEC `0`) THEN ASM_SIMP_TAC[o_DEF; MEASURE_SCALING; SUM_RMUL] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < a ==> a <= b ==> x < b`)) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN ONCE_REWRITE_TAC[REAL_ARITH `(a * b) * c:real = (a * c) * b`] THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ] THEN REWRITE_TAC[GSYM SUM_RMUL] THEN REWRITE_TAC[GSYM REAL_POW_MUL] THEN REWRITE_TAC[REAL_ABS_DIV; REAL_ABS_NUM] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `M:num` SUBST_ALL_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN REWRITE_TAC[GSYM REAL_OF_NUM_MUL] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_MUL]) THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_ARITH `&0 < &2 * x <=> &0 < x`]) THEN ASM_SIMP_TAC[REAL_FIELD `&0 < y ==> x / (&2 * y) * &4 = x * &2 / y`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(M..(2*M)) (\i. (&i * &2 / &M) pow dimindex (:N))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN SIMP_TAC[REAL_POW_LE; REAL_LE_MUL; REAL_LE_DIV; REAL_POS] THEN REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG; SUBSET] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_OF_NUM_LT]) THEN ARITH_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(M..(2*M)) (\i. &2)` THEN CONJ_TAC THENL [REWRITE_TAC[SUM_CONST_NUMSEG] THEN REWRITE_TAC[ARITH_RULE `(2 * M + 1) - M = M + 1`] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC SUM_LE THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&2 pow (dimindex(:N))` THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM REAL_POW_1] THEN MATCH_MP_TAC REAL_POW_MONO THEN REWRITE_TAC[DIMINDEX_GE_1] THEN ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC REAL_POW_LE2 THEN REWRITE_TAC[REAL_POS; ARITH; real_div; REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LE_RDIV_EQ] THEN REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_LE] THEN UNDISCH_TAC `M:num <= n` THEN ARITH_TAC);; let STARLIKE_NEGLIGIBLE_LEMMA = prove (`!s. compact s /\ (!c x:real^N. &0 <= c /\ x IN s /\ (c % x) IN s ==> c = &1) ==> negligible s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC STARLIKE_NEGLIGIBLE_BOUNDED_MEASURABLE THEN ASM_MESON_TAC[MEASURABLE_COMPACT; COMPACT_IMP_BOUNDED]);; let STARLIKE_NEGLIGIBLE = prove (`!s a. closed s /\ (!c x:real^N. &0 <= c /\ (a + x) IN s /\ (a + c % x) IN s ==> c = &1) ==> negligible s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_TRANSLATION_REV THEN EXISTS_TAC `--a:real^N` THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN MATCH_MP_TAC STARLIKE_NEGLIGIBLE_LEMMA THEN CONJ_TAC THENL [MATCH_MP_TAC CLOSED_INTER_COMPACT THEN REWRITE_TAC[COMPACT_INTERVAL] THEN ASM_SIMP_TAC[CLOSED_TRANSLATION]; REWRITE_TAC[IN_IMAGE; IN_INTER] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `x:real^N = --a + y <=> y = a + x`] THEN REWRITE_TAC[UNWIND_THM2] THEN ASM MESON_TAC[]]);; let STARLIKE_NEGLIGIBLE_STRONG = prove (`!s a. closed s /\ (!c x:real^N. &0 <= c /\ c < &1 /\ (a + x) IN s ==> ~((a + c % x) IN s)) ==> negligible s`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC STARLIKE_NEGLIGIBLE THEN EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `~(x < y) /\ ~(y < x) ==> x = y`) THEN STRIP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`inv c:real`; `c % x:real^N`]) THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_ARITH `&1 < c ==> ~(c = &0)`] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_1] THEN MATCH_MP_TAC REAL_LT_INV2 THEN ASM_REWRITE_TAC[REAL_LT_01]);; (* ------------------------------------------------------------------------- *) (* In particular. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_HYPERPLANE = prove (`!a b. ~(a = vec 0 /\ b = &0) ==> negligible {x:real^N | a dot x = b}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_SIMP_TAC[DOT_LZERO; SET_RULE `{x | F} = {}`; NEGLIGIBLE_EMPTY] THEN MATCH_MP_TAC STARLIKE_NEGLIGIBLE THEN SUBGOAL_THEN `?x:real^N. ~(a dot x = b)` MP_TAC THENL [MATCH_MP_TAC(MESON[] `!a:real^N. P a \/ P(--a) ==> ?x. P x`) THEN EXISTS_TAC `a:real^N` THEN REWRITE_TAC[DOT_RNEG] THEN MATCH_MP_TAC(REAL_ARITH `~(a = &0) ==> ~(a = b) \/ ~(--a = b)`) THEN ASM_REWRITE_TAC[DOT_EQ_0]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN REWRITE_TAC[CLOSED_HYPERPLANE; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN MAP_EVERY X_GEN_TAC [`t:real`; `y:real^N`] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `&0 <= t /\ ac + ay = b /\ ac + t * ay = b ==> ((ay = &0 ==> ac = b) /\ (t - &1) * ay = &0)`)) THEN ASM_SIMP_TAC[REAL_ENTIRE; REAL_SUB_0] THEN CONV_TAC TAUT);; let NEGLIGIBLE_LOWDIM = prove (`!s:real^N->bool. dim(s) < dimindex(:N) ==> negligible s`, GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `span(s):real^N->bool` THEN REWRITE_TAC[SPAN_INC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | a dot x = &0}` THEN ASM_SIMP_TAC[NEGLIGIBLE_HYPERPLANE]);; let NEGLIGIBLE_AFFINE_HULL = prove (`!s:real^N->bool. FINITE s /\ CARD(s) <= dimindex(:N) ==> negligible(affine hull s)`, REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[AFFINE_HULL_EMPTY; NEGLIGIBLE_EMPTY] THEN SUBGOAL_THEN `!x s:real^N->bool n. ~(x IN s) /\ (x INSERT s) HAS_SIZE n /\ n <= dimindex(:N) ==> negligible(affine hull(x INSERT s))` (fun th -> MESON_TAC[th; HAS_SIZE; FINITE_INSERT]) THEN X_GEN_TAC `orig:real^N` THEN GEOM_ORIGIN_TAC `orig:real^N` THEN SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; SPAN_INSERT_0; HULL_INC] THEN REWRITE_TAC[HAS_SIZE; FINITE_INSERT; IMP_CONJ] THEN SIMP_TAC[CARD_CLAUSES] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_LOWDIM THEN MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `CARD(s:real^N->bool)` THEN ASM_SIMP_TAC[DIM_LE_CARD; DIM_SPAN] THEN ASM_ARITH_TAC);; let NEGLIGIBLE_AFFINE_HULL_1 = prove (`!a:real^1. negligible (affine hull {a})`, REPEAT GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_AFFINE_HULL THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; FINITE_EMPTY; DIMINDEX_1] THEN ARITH_TAC);; let NEGLIGIBLE_AFFINE_HULL_2 = prove (`!a b:real^2. negligible (affine hull {a,b})`, REPEAT GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_AFFINE_HULL THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; FINITE_EMPTY; DIMINDEX_2] THEN ARITH_TAC);; let NEGLIGIBLE_AFFINE_HULL_3 = prove (`!a b c:real^3. negligible (affine hull {a,b,c})`, REPEAT GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_AFFINE_HULL THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; FINITE_EMPTY; DIMINDEX_3] THEN ARITH_TAC);; let NEGLIGIBLE_CONVEX_HULL = prove (`!s:real^N->bool. FINITE s /\ CARD(s) <= dimindex(:N) ==> negligible(convex hull s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP NEGLIGIBLE_AFFINE_HULL) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]);; let NEGLIGIBLE_CONVEX_HULL_1 = prove (`!a:real^1. negligible (convex hull {a})`, REPEAT GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_CONVEX_HULL THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; FINITE_EMPTY; DIMINDEX_1] THEN ARITH_TAC);; let NEGLIGIBLE_CONVEX_HULL_2 = prove (`!a b:real^2. negligible (convex hull {a,b})`, REPEAT GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_CONVEX_HULL THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; FINITE_EMPTY; DIMINDEX_2] THEN ARITH_TAC);; let NEGLIGIBLE_CONVEX_HULL_3 = prove (`!a b c:real^3. negligible (convex hull {a,b,c})`, REPEAT GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_CONVEX_HULL THEN SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; FINITE_EMPTY; DIMINDEX_3] THEN ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Measurability of bounded convex sets. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_CONVEX_FRONTIER = prove (`!s:real^N->bool. convex s ==> negligible(frontier s)`, SUBGOAL_THEN `!s:real^N->bool. convex s /\ (vec 0) IN s ==> negligible(frontier s)` ASSUME_TAC THENL [ALL_TAC; X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[FRONTIER_EMPTY; NEGLIGIBLE_EMPTY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\x:real^N. --a + x) s`) THEN ASM_SIMP_TAC[CONVEX_TRANSLATION; IN_IMAGE] THEN ASM_REWRITE_TAC[UNWIND_THM2; VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN REWRITE_TAC[FRONTIER_TRANSLATION; NEGLIGIBLE_TRANSLATION_EQ]] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` DIM_SUBSET_UNIV) THEN REWRITE_TAC[ARITH_RULE `d:num <= e <=> d < e \/ d = e`] THEN STRIP_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `closure s:real^N->bool` THEN REWRITE_TAC[frontier; SUBSET_DIFF] THEN MATCH_MP_TAC NEGLIGIBLE_LOWDIM THEN ASM_REWRITE_TAC[DIM_CLOSURE]; ALL_TAC] THEN SUBGOAL_THEN `?a:real^N. a IN interior s` CHOOSE_TAC THENL [X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC (ISPEC `s:real^N->bool` BASIS_EXISTS) THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN MP_TAC(ISPEC `b:real^N->bool` INTERIOR_SIMPLEX_NONEMPTY) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[INSERT_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC STARLIKE_NEGLIGIBLE_STRONG THEN EXISTS_TAC `a:real^N` THEN REWRITE_TAC[FRONTIER_CLOSED] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[frontier; IN_DIFF; DE_MORGAN_THM] THEN DISJ2_TAC THEN SIMP_TAC[VECTOR_ARITH `a + c % x:real^N = (a + x) - (&1 - c) % ((a + x) - a)`] THEN MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN RULE_ASSUM_TAC(REWRITE_RULE[frontier; IN_DIFF]) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let MEASURABLE_CONVEX = prove (`!s:real^N->bool. convex s /\ bounded s ==> measurable s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_JORDAN THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_FRONTIER]);; let NEGLIGIBLE_CONVEX_INTERIOR = prove (`!s:real^N->bool. convex s ==> (negligible s <=> interior s = {})`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[OPEN_NOT_NEGLIGIBLE; INTERIOR_SUBSET; OPEN_INTERIOR; NEGLIGIBLE_SUBSET]; DISCH_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `frontier s:real^N->bool` THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_FRONTIER] THEN ASM_REWRITE_TAC[frontier; DIFF_EMPTY; CLOSURE_SUBSET]]);; (* ------------------------------------------------------------------------- *) (* Various special cases. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_SPHERE = prove (`!a:real^N r. negligible (sphere(a,r))`, REWRITE_TAC[GSYM FRONTIER_CBALL] THEN SIMP_TAC[NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_CBALL]);; let MEASURABLE_BALL = prove (`!a r. measurable(ball(a,r))`, SIMP_TAC[MEASURABLE_OPEN; BOUNDED_BALL; OPEN_BALL]);; let MEASURABLE_CBALL = prove (`!a r. measurable(cball(a,r))`, SIMP_TAC[MEASURABLE_COMPACT; COMPACT_CBALL]);; let MEASURE_BALL_POS = prove (`!x:real^N e. &0 < measure(ball(x,e)) <=> &0 < e`, SIMP_TAC[MEASURE_OPEN_POS_LT_EQ; OPEN_BALL; BOUNDED_BALL; BALL_EQ_EMPTY] THEN REAL_ARITH_TAC);; let MEASURE_CBALL_POS = prove (`!x:real^N e. &0 < measure(cball(x,e)) <=> &0 < e`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM MEASURE_BALL_POS] THEN AP_TERM_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `sphere(x:real^N,e)` THEN REWRITE_TAC[GSYM SPHERE_UNION_BALL; NEGLIGIBLE_SPHERE] THEN SET_TAC[]);; let HAS_INTEGRAL_OPEN_INTERVAL = prove (`!f a b y. (f has_integral y) (interval(a,b)) <=> (f has_integral y) (interval[a,b])`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM INTERIOR_CLOSED_INTERVAL] THEN MATCH_MP_TAC HAS_INTEGRAL_INTERIOR THEN MATCH_MP_TAC NEGLIGIBLE_CONVEX_FRONTIER THEN REWRITE_TAC[CONVEX_INTERVAL]);; let INTEGRABLE_ON_OPEN_INTERVAL = prove (`!f a b. f integrable_on interval(a,b) <=> f integrable_on interval[a,b]`, REWRITE_TAC[integrable_on; HAS_INTEGRAL_OPEN_INTERVAL]);; let INTEGRAL_OPEN_INTERVAL = prove (`!f a b. integral(interval(a,b)) f = integral(interval[a,b]) f`, REWRITE_TAC[integral; HAS_INTEGRAL_OPEN_INTERVAL]);; let ABSOLUTELY_INTEGRABLE_ON_OPEN_INTERVAL = prove (`!f:real^M->real^N a b. f absolutely_integrable_on interval(a,b) <=> f absolutely_integrable_on interval[a,b]`, REWRITE_TAC[absolutely_integrable_on; INTEGRABLE_ON_OPEN_INTERVAL]);; let MEASURABLE_SEGMENT = prove (`(!a b:real^N. measurable(segment[a,b])) /\ (!a b:real^N. measurable(segment(a,b)))`, SIMP_TAC[MEASURABLE_CONVEX; CONVEX_SEGMENT; BOUNDED_SEGMENT]);; let MEASURE_SEGMENT_1 = prove (`(!a b:real^1. measure(segment[a,b]) = norm(b - a)) /\ (!a b:real^1. measure(segment(a,b)) = norm(b - a))`, REWRITE_TAC[SEGMENT_1] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[MEASURE_INTERVAL_1] THEN REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB] THEN ASM_REAL_ARITH_TAC);; let NEGLIGIBLE_SEGMENT = prove (`(!a b:real^N. negligible(segment[a,b]) <=> 2 <= dimindex(:N) \/ a = b) /\ (!a b:real^N. negligible(segment(a,b)) <=> 2 <= dimindex(:N) \/ a = b)`, SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR; CONVEX_SEGMENT] THEN REWRITE_TAC[INTERIOR_SEGMENT] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[SEGMENT_EQ_EMPTY]);; let MEASURE_BALL_SCALING = prove (`!a:real^N c r. &0 <= c ==> measure(ball(a,c * r)) = c pow dimindex(:N) * measure(ball(a,r))`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `c = &0` THEN ASM_SIMP_TAC[REAL_MUL_LZERO; BALL_EMPTY; REAL_LE_REFL; MEASURE_EMPTY; REAL_POW_ZERO; DIMINDEX_NONZERO] THEN SUBGOAL_THEN `&0 < c` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^N = c % vec 0`) THEN ASM_SIMP_TAC[BALL_SCALING; MEASURE_SCALING; MEASURABLE_BALL] THEN ASM_REWRITE_TAC[real_abs; VECTOR_MUL_RZERO]);; let MEASURE_CBALL_SCALING = prove (`!a:real^N c r. &0 <= c ==> measure(cball(a,c * r)) = c pow dimindex(:N) * measure(cball(a,r))`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `c = &0` THEN ASM_SIMP_TAC[REAL_MUL_LZERO; CBALL_SING; REAL_LE_REFL; MEASURE_SING; REAL_POW_ZERO; DIMINDEX_NONZERO] THEN SUBGOAL_THEN `&0 < c` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^N = c % vec 0`) THEN ASM_SIMP_TAC[CBALL_SCALING; MEASURE_SCALING; MEASURABLE_CBALL] THEN ASM_REWRITE_TAC[real_abs; VECTOR_MUL_RZERO]);; (* ------------------------------------------------------------------------- *) (* An existence theorem for "improper" integrals. Hake's theorem implies *) (* that if the integrals over subintervals have a limit then the integral *) (* exists. This is incomparable: we only need a priori to assume that *) (* the integrals are bounded, and we get absolute integrability, but we *) (* also need a (rather weak) bound assumption on the function. *) (* ------------------------------------------------------------------------- *) let ABSOLUTELY_INTEGRABLE_IMPROPER = prove (`!f:real^M->real^N a b. (!c d. interval[c,d] SUBSET interval(a,b) ==> f integrable_on interval[c,d]) /\ bounded { integral (interval[c,d]) f | interval[c,d] SUBSET interval(a,b)} /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> ?g. g absolutely_integrable_on interval[a,b] /\ ((!x. x IN interval[a,b] ==> (f x)$i <= drop(g x)) \/ (!x. x IN interval[a,b] ==> (f x)$i >= drop(g x)))) ==> f absolutely_integrable_on interval[a,b]`, REPEAT GEN_TAC THEN ASM_CASES_TAC `content(interval[a:real^M,b]) = &0` THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_ON_NULL] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM CONTENT_LT_NZ; CONTENT_POS_LT_EQ]) THEN STRIP_TAC THEN ONCE_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_COMPONENTWISE] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[real_ge] THEN SUBGOAL_THEN `(!n. interval[a + inv(&n + &1) % (b - a),b - inv(&n + &1) % (b - a)] SUBSET interval(a:real^M,b)) /\ (!n. interval[a + inv(&n + &1) % (b - a),b - inv(&n + &1) % (b - a)] SUBSET interval[a:real^M,b])` STRIP_ASSUME_TAC THENL [REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ t SUBSET u ==> s SUBSET t /\ s SUBSET u`) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED] THEN REWRITE_TAC[SUBSET_INTERVAL] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_SUB_COMPONENT] THEN ASM_SIMP_TAC[REAL_ARITH `a < a + x <=> &0 < x`; REAL_ARITH `b - x < b <=> &0 < x`; REAL_LT_MUL; REAL_SUB_LT; REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`]; ALL_TAC] THEN SUBGOAL_THEN `!n. interval[a + inv(&n + &1) % (b - a),b - inv(&n + &1) % (b - a)] SUBSET interval[a + inv(&(SUC n) + &1) % (b - a):real^M, b - inv(&(SUC n) + &1) % (b - a)]` ASSUME_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL] THEN GEN_TAC THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `a + x * y <= a + w * y <=> &0 <= (w - x) * y`; REAL_ARITH `b - w * y <= b - x * y <=> &0 <= (w - x) * y`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_SUB_LE; REAL_LT_IMP_LE; GSYM REAL_OF_NUM_SUC] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^1` STRIP_ASSUME_TAC) THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_COMPONENT_UBOUND THEN EXISTS_TAC `g:real^M->real^1` THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; DIMINDEX_1] THEN ASM_REWRITE_TAC[IMP_IMP; FORALL_1; GSYM drop; LIFT_DROP] THEN SUBGOAL_THEN `(\x. lift((f:real^M->real^N) x$i)) = (\x. g x - (g x - lift(f x$i)))` SUBST1_TAC THENL [ABS_TAC THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN MATCH_MP_TAC INTEGRABLE_SUB THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN MP_TAC(ISPECL [`\n x. if x IN interval[a + inv(&n + &1) % (b - a), b - inv(&n + &1) % (b - a)] then g x - lift((f:real^M->real^N) x $i) else vec 0`; `\x. g x - lift((f:real^M->real^N) x$i)`; `interval(a:real^M,b)`] MONOTONE_CONVERGENCE_INCREASING) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[INTEGRABLE_ON_OPEN_INTERVAL]] THEN REWRITE_TAC[INTEGRABLE_RESTRICT_INTER; INTEGRAL_RESTRICT_INTER] THEN ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`] THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC INTEGRABLE_SUB THEN CONJ_TAC THENL [ASM_MESON_TAC[INTEGRABLE_SUBINTERVAL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; RULE_ASSUM_TAC(ONCE_REWRITE_RULE[INTEGRABLE_COMPONENTWISE]) THEN ASM_MESON_TAC[]]; ALL_TAC]; MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_COMPONENT_LBOUND THEN EXISTS_TAC `g:real^M->real^1` THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; DIMINDEX_1] THEN ASM_REWRITE_TAC[IMP_IMP; FORALL_1; GSYM drop; LIFT_DROP] THEN SUBGOAL_THEN `(\x. lift((f:real^M->real^N) x$i)) = (\x. (lift(f x$i) - g x) + g x)` SUBST1_TAC THENL [ABS_TAC THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN MATCH_MP_TAC INTEGRABLE_ADD THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN MP_TAC(ISPECL [`\n x. if x IN interval[a + inv(&n + &1) % (b - a), b - inv(&n + &1) % (b - a)] then lift((f:real^M->real^N) x $i) - g x else vec 0`; `\x. lift((f:real^M->real^N) x$i) - g x`; `interval(a:real^M,b)`] MONOTONE_CONVERGENCE_INCREASING) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[INTEGRABLE_ON_OPEN_INTERVAL]] THEN REWRITE_TAC[INTEGRABLE_RESTRICT_INTER; INTEGRAL_RESTRICT_INTER] THEN ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`] THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC INTEGRABLE_SUB THEN CONJ_TAC THENL [RULE_ASSUM_TAC(ONCE_REWRITE_RULE[INTEGRABLE_COMPONENTWISE]) THEN ASM_MESON_TAC[]; ASM_MESON_TAC[INTEGRABLE_SUBINTERVAL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]]; ALL_TAC]] THEN (REPEAT CONJ_TAC THENL [REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL]) THEN ASM_SIMP_TAC[DROP_SUB; DROP_VEC; REAL_SUB_LE; LIFT_DROP] THEN ASM SET_TAC[]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTERVAL] THEN DISCH_TAC THEN MATCH_MP_TAC LIM_EVENTUALLY THEN MP_TAC(SPEC `inf({(x - a:real^M)$i / (b - a)$i | i IN 1..dimindex(:M)} UNION {(b - x:real^M)$i / (b - a)$i | i IN 1..dimindex(:M)})` REAL_ARCH_INV) THEN SIMP_TAC[REAL_LT_INF_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1; FINITE_UNION; IMAGE_UNION; EMPTY_UNION] THEN REWRITE_TAC[FORALL_IN_UNION; FORALL_IN_IMAGE] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; IN_NUMSEG; EVENTUALLY_SEQUENTIALLY] THEN ASM_SIMP_TAC[REAL_SUB_LT; REAL_LT_RDIV_EQ; REAL_MUL_LZERO] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN MATCH_MP_TAC(MESON[] `(!x. ~P x) ==> (?x. P x) ==> Q`) THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[] THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; REAL_ARITH `a + y <= x /\ x <= b - y <=> y <= x - a /\ y <= b - x`] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `inv(&N)` THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_SUB_LT; REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT; LE_1] THEN ASM_ARITH_TAC; FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_INTEGRALS_OVER_SUBINTERVALS) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[BOUNDED_POS; FORALL_IN_GSPEC; IN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `B + C:real` THEN ASM_SIMP_TAC[REAL_LT_ADD] THEN RULE_ASSUM_TAC(ONCE_REWRITE_RULE[INTEGRABLE_COMPONENTWISE]) THEN GEN_TAC THEN W(MP_TAC o PART_MATCH (lhs o rand) INTEGRAL_SUB o rand o lhand o snd) THEN ASM_SIMP_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[INTEGRABLE_SUBINTERVAL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; DISCH_THEN SUBST1_TAC]]) THENL [MATCH_MP_TAC(NORM_ARITH `norm(x:real^N) <= c /\ norm(y) <= b ==> norm(x - y) <= b + c`); MATCH_MP_TAC(NORM_ARITH `norm(x:real^N) <= c /\ norm(y) <= b ==> norm(x - y) <= c + b`)] THEN ASM_SIMP_TAC[] THEN IMP_REWRITE_TAC[GSYM LIFT_INTEGRAL_COMPONENT] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM INTEGRABLE_COMPONENTWISE]) THEN ASM_SIMP_TAC[NORM_LIFT] THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN ASM_SIMP_TAC[]);; (* ------------------------------------------------------------------------- *) (* Crude upper bounds for measure of balls. *) (* ------------------------------------------------------------------------- *) let MEASURE_CBALL_BOUND = prove (`!x:real^N d. &0 <= d ==> measure(cball(x,d)) <= (&2 * d) pow (dimindex(:N))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(interval[x - d % vec 1:real^N,x + d % vec 1])` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_CBALL; MEASURABLE_INTERVAL] THEN REWRITE_TAC[SUBSET; IN_CBALL; IN_INTERVAL] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; dist] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; VEC_COMPONENT] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN MP_TAC(ISPECL [`x - y:real^N`; `i:num`] COMPONENT_LE_NORM) THEN ASM_REWRITE_TAC[VECTOR_SUB_COMPONENT] THEN ASM_REAL_ARITH_TAC; SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_POW_LE; REAL_LE_MUL; REAL_POS] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `(x + a) - (x - a):real = &2 * a`] THEN REWRITE_TAC[PRODUCT_CONST_NUMSEG; VECTOR_MUL_COMPONENT; VEC_COMPONENT] THEN REWRITE_TAC[REAL_MUL_RID; ADD_SUB; REAL_LE_REFL]]);; let MEASURE_BALL_BOUND = prove (`!x:real^N d. &0 <= d ==> measure(ball(x,d)) <= (&2 * d) pow (dimindex(:N))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(cball(x:real^N,d))` THEN ASM_SIMP_TAC[MEASURE_CBALL_BOUND] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[BALL_SUBSET_CBALL; MEASURABLE_BALL; MEASURABLE_CBALL]);; (* ------------------------------------------------------------------------- *) (* Negligibility of image under non-injective linear map. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_LINEAR_SINGULAR_IMAGE = prove (`!f:real^N->real^N s. linear f /\ ~(!x y. f(x) = f(y) ==> x = y) ==> negligible(IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP LINEAR_SINGULAR_IMAGE_HYPERPLANE) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | a dot x = &0}` THEN ASM_SIMP_TAC[NEGLIGIBLE_HYPERPLANE]);; (* ------------------------------------------------------------------------- *) (* Some technical lemmas used in the approximation results that follow. *) (* Proof of the covering lemma is an obvious multidimensional generalization *) (* of Lemma 3, p65 of Swartz's "Introduction to Gauge Integrals". *) (* ------------------------------------------------------------------------- *) let COVERING_LEMMA = prove (`!a b:real^N s g. s SUBSET interval[a,b] /\ ~(interval(a,b) = {}) /\ gauge g ==> ?d. COUNTABLE d /\ (!k. k IN d ==> k SUBSET interval[a,b] /\ ~(interior k = {}) /\ (?c d. k = interval[c,d])) /\ (!k1 k2. k1 IN d /\ k2 IN d /\ ~(k1 = k2) ==> interior k1 INTER interior k2 = {}) /\ (!k. k IN d ==> ?x. x IN (s INTER k) /\ k SUBSET g(x)) /\ (!u v. interval[u,v] IN d ==> ?n. !i. 1 <= i /\ i <= dimindex(:N) ==> v$i - u$i = (b$i - a$i) / &2 pow n) /\ s SUBSET UNIONS d`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?d. COUNTABLE d /\ (!k. k IN d ==> k SUBSET interval[a,b] /\ ~(interior k = {}) /\ (?c d:real^N. k = interval[c,d])) /\ (!k1 k2. k1 IN d /\ k2 IN d ==> k1 SUBSET k2 \/ k2 SUBSET k1 \/ interior k1 INTER interior k2 = {}) /\ (!x. x IN s ==> ?k. k IN d /\ x IN k /\ k SUBSET g(x)) /\ (!u v. interval[u,v] IN d ==> ?n. !i. 1 <= i /\ i <= dimindex(:N) ==> v$i - u$i = (b$i - a$i) / &2 pow n) /\ (!k. k IN d ==> FINITE {l | l IN d /\ k SUBSET l})` ASSUME_TAC THENL [EXISTS_TAC `IMAGE (\(n,v). interval[(lambda i. a$i + &(v$i) / &2 pow n * ((b:real^N)$i - (a:real^N)$i)):real^N, (lambda i. a$i + (&(v$i) + &1) / &2 pow n * (b$i - a$i))]) {n,v | n IN (:num) /\ v IN {v:num^N | !i. 1 <= i /\ i <= dimindex(:N) ==> v$i < 2 EXP n}}` THEN CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_IMAGE THEN MATCH_MP_TAC COUNTABLE_PRODUCT_DEPENDENT THEN REWRITE_TAC[NUM_COUNTABLE; IN_UNIV] THEN GEN_TAC THEN MATCH_MP_TAC FINITE_IMP_COUNTABLE THEN MATCH_MP_TAC FINITE_CART THEN REWRITE_TAC[FINITE_NUMSEG_LT]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`n:num`; `v:num^N`] THEN REWRITE_TAC[IN_ELIM_PAIR_THM] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN DISCH_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REWRITE_TAC[INTERIOR_CLOSED_INTERVAL] THEN SIMP_TAC[INTERVAL_NE_EMPTY; SUBSET_INTERVAL; LAMBDA_BETA] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN ASM_SIMP_TAC[REAL_LE_LADD; REAL_LE_RMUL_EQ; REAL_SUB_LT; REAL_LE_MUL_EQ; REAL_LT_LADD; REAL_LT_RMUL_EQ; REAL_LE_ADDR; REAL_ARITH `a + x * (b - a) <= b <=> &0 <= (&1 - x) * (b - a)`] THEN SIMP_TAC[REAL_LE_DIV2_EQ; REAL_LT_DIV2_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_ARITH `x <= x + &1 /\ x < x + &1`] THEN REWRITE_TAC[REAL_SUB_LE] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_POS; REAL_MUL_LID] THEN SIMP_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN ASM_SIMP_TAC[ARITH_RULE `x + 1 <= y <=> x < y`; REAL_LT_IMP_LE]; ALL_TAC] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IN_ELIM_PAIR_THM; IN_UNIV] THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN GEN_REWRITE_TAC BINDER_CONV [SWAP_FORALL_THM] THEN MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [SWAP_FORALL_THM] THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN SET_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`v:num^N`; `w:num^N`] THEN REPEAT DISCH_TAC THEN REWRITE_TAC[INTERIOR_CLOSED_INTERVAL; SUBSET_INTERVAL] THEN SIMP_TAC[DISJOINT_INTERVAL; LAMBDA_BETA] THEN MATCH_MP_TAC(TAUT `p \/ q \/ r ==> (a ==> p) \/ (b ==> q) \/ r`) THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(a /\ b ==> ~c)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN ASM_SIMP_TAC[REAL_LE_LADD; REAL_LE_RMUL_EQ; REAL_SUB_LT; LAMBDA_BETA] THEN REWRITE_TAC[NOT_IMP; REAL_LE_LADD] THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_ARITH `~(x + &1 <= x)`] THEN DISJ2_TAC THEN MATCH_MP_TAC(MESON[] `(!i. ~P i ==> Q i) ==> (!i. Q i) \/ (?i. P i)`) THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[DE_MORGAN_THM; REAL_NOT_LE] THEN UNDISCH_TAC `m:num <= n` THEN REWRITE_TAC[LE_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN `p:num` SUBST1_TAC) THEN ONCE_REWRITE_TAC[ADD_SYM] THEN REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_MUL] THEN REWRITE_TAC[REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_LT_POW2; REAL_LT_DIV2_EQ] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_LT_POW2; REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ] THEN SIMP_TAC[REAL_LT_INTEGERS; INTEGER_CLOSED] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?e. &0 < e /\ !y. (!i. 1 <= i /\ i <= dimindex(:N) ==> abs((x:real^N)$i - (y:real^N)$i) <= e) ==> y IN g(x)` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [gauge]) THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `e / &2 / &(dimindex(:N))` THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1; ARITH] THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ x IN s ==> x IN t`) THEN EXISTS_TAC `ball(x:real^N,e)` THEN ASM_REWRITE_TAC[IN_BALL] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN ASM_REWRITE_TAC[dist] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(1..dimindex(:N)) (\i. abs((x - y:real^N)$i))` THEN REWRITE_TAC[NORM_LE_L1] THEN MATCH_MP_TAC SUM_BOUND_GEN THEN ASM_SIMP_TAC[IN_NUMSEG; FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1; VECTOR_SUB_COMPONENT; CARD_NUMSEG_1]; ALL_TAC] THEN REWRITE_TAC[EXISTS_IN_IMAGE; EXISTS_PAIR_THM; IN_ELIM_PAIR_THM] THEN MP_TAC(SPECL [`&1 / &2`; `e / norm(b - a:real^N)`] REAL_ARCH_POW_INV) THEN SUBGOAL_THEN `&0 < norm(b - a:real^N)` ASSUME_TAC THENL [ASM_MESON_TAC[VECTOR_SUB_EQ; NORM_POS_LT; INTERVAL_SING]; ALL_TAC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[REAL_LT_DIV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[real_div; REAL_MUL_LID; REAL_POW_INV] THEN DISCH_TAC THEN SIMP_TAC[IN_ELIM_THM; IN_INTERVAL; SUBSET; LAMBDA_BETA] THEN MATCH_MP_TAC(MESON[] `(!x. Q x ==> R x) /\ (?x. P x /\ Q x) ==> ?x. P x /\ Q x /\ R x`) THEN CONJ_TAC THENL [REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`w:num^N`; `y:real^N`] THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `(a + n <= x /\ x <= a + m) /\ (a + n <= y /\ y <= a + m) ==> abs(x - y) <= m - n`)) THEN MATCH_MP_TAC(REAL_ARITH `y * z <= e ==> a <= ((x + &1) * y) * z - ((x * y) * z) ==> a <= e`) THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `n < e * x ==> &0 <= e * (inv y - x) ==> n <= e / y`)) THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_SUB_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[REAL_SUB_LT] THEN MP_TAC(SPECL [`b - a:real^N`; `i:num`] COMPONENT_LE_NORM) THEN ASM_SIMP_TAC[VECTOR_SUB_COMPONENT] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IN_UNIV; AND_FORALL_THM] THEN REWRITE_TAC[TAUT `(a ==> c) /\ (a ==> b) <=> a ==> b /\ c`] THEN REWRITE_TAC[GSYM LAMBDA_SKOLEM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN SUBGOAL_THEN `(x:real^N) IN interval[a,b]` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN STRIP_TAC THEN DISJ_CASES_TAC(MATCH_MP (REAL_ARITH `x <= y ==> x = y \/ x < y`) (ASSUME `(x:real^N)$i <= (b:real^N)$i`)) THENL [EXISTS_TAC `2 EXP n - 1` THEN SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_LT; EXP_LT_0; LE_1; ARITH] THEN ASM_REWRITE_TAC[REAL_SUB_ADD; REAL_ARITH `a - &1 < a`] THEN MATCH_MP_TAC(REAL_ARITH `&1 * (b - a) = x /\ y <= x ==> a + y <= b /\ b <= a + x`) THEN ASM_SIMP_TAC[REAL_EQ_MUL_RCANCEL; REAL_LT_IMP_NZ; REAL_LE_RMUL_EQ; REAL_SUB_LT; REAL_LT_INV_EQ; REAL_LT_POW2] THEN SIMP_TAC[GSYM REAL_OF_NUM_POW; REAL_MUL_RINV; REAL_POW_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(SPEC `&2 pow n * ((x:real^N)$i - (a:real^N)$i) / ((b:real^N)$i - (a:real^N)$i)` FLOOR_POS) THEN ANTS_TAC THENL [ASM_MESON_TAC[REAL_LE_MUL; REAL_LE_MUL; REAL_POW_LE; REAL_POS; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_LE_DIV]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[GSYM REAL_OF_NUM_LT; GSYM REAL_OF_NUM_POW] THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN REWRITE_TAC[REAL_ARITH `a + b * c <= x /\ x <= a + b' * c <=> b * c <= x - a /\ x - a <= b' * c`] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; GSYM REAL_LE_RDIV_EQ; REAL_SUB_LT; GSYM real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_LT_POW2] THEN SIMP_TAC[FLOOR; REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `((x:real^N)$i - (a:real^N)$i) / ((b:real^N)$i - (a:real^N)$i) * &2 pow n` THEN REWRITE_TAC[FLOOR] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN ASM_SIMP_TAC[REAL_LT_RMUL_EQ; REAL_LT_POW2] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_MUL_LID; REAL_SUB_LT] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[IN_IMAGE; EXISTS_PAIR_THM] THEN REWRITE_TAC[EQ_INTERVAL; IN_ELIM_PAIR_THM] THEN REWRITE_TAC[INTERVAL_EQ_EMPTY; IN_UNIV; IN_ELIM_THM] THEN SIMP_TAC[TAUT `a /\ b /\ c <=> ~(a /\ b ==> ~c)`; LAMBDA_BETA] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN ASM_SIMP_TAC[REAL_LT_LADD; REAL_LT_RMUL_EQ; REAL_SUB_LT; REAL_LT_DIV2_EQ; REAL_LT_POW2; REAL_ARITH `~(v + &1 < v)`] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`n:num`; `v:num^N`] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN DISCH_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `IMAGE (\(n,v). interval[(lambda i. a$i + &(v$i) / &2 pow n * ((b:real^N)$i - (a:real^N)$i)):real^N, (lambda i. a$i + (&(v$i) + &1) / &2 pow n * (b$i - a$i))]) {m,v | m IN 0..n /\ v IN {v:num^N | !i. 1 <= i /\ i <= dimindex(:N) ==> v$i < 2 EXP m}}` THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_IMAGE THEN MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN REWRITE_TAC[FINITE_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_CART THEN REWRITE_TAC[FINITE_NUMSEG_LT]; ALL_TAC] THEN GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`m:num`; `w:num^N`] THEN DISCH_TAC THEN DISCH_TAC THEN SIMP_TAC[IN_IMAGE; EXISTS_PAIR_THM; IN_ELIM_PAIR_THM] THEN MAP_EVERY EXISTS_TAC [`m:num`; `w:num^N`] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_NUMSEG; GSYM NOT_LT; LT] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET_INTERVAL]) THEN SIMP_TAC[NOT_IMP; LAMBDA_BETA] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN ASM_SIMP_TAC[REAL_LE_LADD; REAL_LE_RMUL_EQ; REAL_SUB_LT] THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_ARITH `x <= x + &1`] THEN DISCH_THEN(MP_TAC o SPEC `1`) THEN REWRITE_TAC[LE_REFL; DIMINDEX_GE_1] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `w / m <= v / n /\ (v + &1) / n <= (w + &1) / m ==> inv n <= inv m`)) THEN REWRITE_TAC[REAL_NOT_LE] THEN MATCH_MP_TAC REAL_LT_INV2 THEN ASM_REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO_LT THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN SUBGOAL_THEN `?d. COUNTABLE d /\ (!k. k IN d ==> k SUBSET interval[a,b] /\ ~(interior k = {}) /\ (?c d:real^N. k = interval[c,d])) /\ (!k1 k2. k1 IN d /\ k2 IN d ==> k1 SUBSET k2 \/ k2 SUBSET k1 \/ interior k1 INTER interior k2 = {}) /\ (!k. k IN d ==> (?x. x IN s INTER k /\ k SUBSET g x)) /\ (!u v. interval[u,v] IN d ==> ?n. !i. 1 <= i /\ i <= dimindex(:N) ==> v$i - u$i = (b$i - a$i) / &2 pow n) /\ (!k. k IN d ==> FINITE {l | l IN d /\ k SUBSET l}) /\ s SUBSET UNIONS d` MP_TAC THENL [FIRST_X_ASSUM(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{k:real^N->bool | k IN d /\ ?x. x IN (s INTER k) /\ k SUBSET g x}` THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `d:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; X_GEN_TAC `k:real^N->bool` THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{l:real^N->bool | l IN d /\ k SUBSET l}` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ASM SET_TAC[]]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{k:real^N->bool | k IN d /\ !k'. k' IN d /\ ~(k = k') ==> ~(k SUBSET k')}` THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `d:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[FORALL_IN_UNIONS] THEN MAP_EVERY X_GEN_TAC [`k:real^N->bool`; `x:real^N`] THEN DISCH_TAC THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN MP_TAC(ISPEC `\k l:real^N->bool. k IN d /\ l IN d /\ l SUBSET k /\ ~(k = l)` WF_FINITE) THEN REWRITE_TAC[WF] THEN ANTS_TAC THENL [CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `l:real^N->bool` THEN ASM_CASES_TAC `(l:real^N->bool) IN d` THEN ASM_REWRITE_TAC[EMPTY_GSPEC; FINITE_RULES] THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{m:real^N->bool | m IN d /\ l SUBSET m}` THEN ASM_SIMP_TAC[] THEN SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `\l:real^N->bool. l IN d /\ x IN l`) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]);; let COUNTABLE_ELEMENTARY_DIVISION = prove (`!d. COUNTABLE d /\ (!k. k IN d ==> ?a b:real^N. k = interval[a,b]) ==> ?d'. COUNTABLE d' /\ (!k. k IN d' ==> ~(k = {}) /\ ?a b. k = interval[a,b]) /\ (!k l. k IN d' /\ l IN d' /\ ~(k = l) ==> interior k INTER interior l = {}) /\ UNIONS d' = UNIONS d`, let lemma = prove (`!s. UNIONS(s DELETE {}) = UNIONS s`, REWRITE_TAC[EXTENSION; IN_UNIONS; IN_DELETE] THEN MESON_TAC[NOT_IN_EMPTY]) in REWRITE_TAC[IMP_CONJ; FORALL_COUNTABLE_AS_IMAGE] THEN REWRITE_TAC[UNIONS_0; EMPTY_UNIONS] THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN EXISTS_TAC `{}:(real^N->bool)->bool` THEN REWRITE_TAC[NOT_IN_EMPTY; COUNTABLE_EMPTY]; ALL_TAC] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`d:num->real^N->bool`; `a:num->real^N`; `b:num->real^N`] THEN DISCH_TAC THEN (CHOOSE_THEN MP_TAC o prove_recursive_functions_exist num_RECURSION) `x 0 = ({}:(real^N->bool)->bool) /\ (!n. x(SUC n) = @q. (x n) SUBSET q /\ q division_of (d n) UNION UNIONS(x n))` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN SUBGOAL_THEN `!n:num. (x n) division_of UNIONS {d k:real^N->bool | k < n}` ASSUME_TAC THENL [INDUCT_TAC THEN ASM_REWRITE_TAC[LT; SET_RULE `UNIONS {f x |x| F} = {}`; DIVISION_OF_TRIVIAL] THEN FIRST_ASSUM(MP_TAC o SPECL [`(a:num->real^N) n`; `(b:num->real^N) n`] o MATCH_MP ELEMENTARY_UNION_INTERVAL_STRONG o MATCH_MP DIVISION_OF_UNION_SELF) THEN DISCH_THEN(ASSUME_TAC o SELECT_RULE) THEN REWRITE_TAC[SET_RULE `{f x | x = a \/ q x} = f a INSERT {f x | q x}`] THEN REWRITE_TAC[UNIONS_INSERT] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM o last o CONJUNCTS) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!m n. m <= n ==> (x:num->(real^N->bool)->bool) m SUBSET x n` ASSUME_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN REPEAT(CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(a:num->real^N) n`; `(b:num->real^N) n`] o MATCH_MP ELEMENTARY_UNION_INTERVAL_STRONG o MATCH_MP DIVISION_OF_UNION_SELF o SPEC `n:num`) THEN DISCH_THEN(ASSUME_TAC o SELECT_RULE) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN EXISTS_TAC `UNIONS(IMAGE x (:num)) DELETE ({}:real^N->bool)` THEN REWRITE_TAC[COUNTABLE_DELETE; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_UNIONS; FORALL_IN_IMAGE; IN_DELETE; IN_UNIV] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_UNIONS THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; FORALL_IN_IMAGE; IN_UNIV] THEN GEN_TAC THEN MATCH_MP_TAC FINITE_IMP_COUNTABLE THEN ASM_MESON_TAC[DIVISION_OF_FINITE]; MAP_EVERY X_GEN_TAC [`n:num`; `k:real^N->bool`] THEN ASM_MESON_TAC[division_of]; REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN GEN_REWRITE_TAC BINDER_CONV [SWAP_FORALL_THM] THEN MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[INTER_COMM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`k:real^N->bool`; `l:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`m:num`; `n:num`]) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of] o SPEC `n:num`) THEN ASM SET_TAC[]; REWRITE_TAC[lemma] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_UNIV; FORALL_IN_UNIONS; SUBSET; IN_UNIONS; EXISTS_IN_IMAGE] THENL [X_GEN_TAC `k:real^N->bool` THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of] o SPEC `n:num`) THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN ASM SET_TAC[]; MAP_EVERY X_GEN_TAC [`n:num`; `y:real^N`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of] o SPEC `SUC n`) THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN REWRITE_TAC[EXTENSION; IN_UNIONS; EXISTS_IN_GSPEC] THEN DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN ASM_MESON_TAC[ARITH_RULE `n < SUC n`]]]);; let EXPAND_CLOSED_OPEN_INTERVAL = prove (`!a b:real^N e. &0 < e ==> ?c d. interval[a,b] SUBSET interval(c,d) /\ measure(interval(c,d)) <= measure(interval[a,b]) + e`, let lemma = prove (`!f n. (\x. lift(product(1..n) (\i. f i + drop x))) continuous at (vec 0)`, GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[PRODUCT_CLAUSES_NUMSEG; ARITH_EQ; CONTINUOUS_CONST] THEN REWRITE_TAC[ARITH_RULE `1 <= SUC n`] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[o_DEF; LIFT_ADD; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_ADD; CONTINUOUS_AT_ID; CONTINUOUS_CONST]) in REPEAT GEN_TAC THEN ABBREV_TAC `m:real^N = midpoint(a,b)` THEN POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `m:real^N` THEN REWRITE_TAC[midpoint; VECTOR_ARITH `inv(&2) % (a + b):real^N = vec 0 <=> a = --b`] THEN REPEAT GEN_TAC THEN DISCH_THEN SUBST1_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `interval[--b:real^N,b] = {}` THENL [MAP_EVERY EXISTS_TAC [`--b:real^N`; `b:real^N`] THEN REWRITE_TAC[MEASURE_INTERVAL] THEN ASM_REWRITE_TAC[CONTENT_EMPTY; EMPTY_SUBSET] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTERVAL_NE_EMPTY]) THEN REWRITE_TAC[VECTOR_NEG_COMPONENT; REAL_ARITH `--x <= x <=> &0 <= x`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`\i. &2 * (b:real^N)$i`; `dimindex(:N)`] lemma) THEN REWRITE_TAC[continuous_at; DIST_LIFT; FORALL_LIFT; DIST_0; DROP_VEC] THEN REWRITE_TAC[NORM_LIFT; LIFT_DROP; REAL_ADD_RID] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN MAP_EVERY EXISTS_TAC [`--(b + k / &4 % vec 1:real^N)`; `b + k / &4 % vec 1:real^N`] THEN REWRITE_TAC[MEASURE_INTERVAL; SUBSET_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_SIMP_TAC[REAL_ARITH `--x <= x <=> &0 <= x`; REAL_LT_ADDR; REAL_ARITH `&0 < k / &4 <=> &0 < k`; REAL_ARITH `&0 <= b /\ &0 < k ==> --(b + k) < b`; REAL_ARITH `&0 <= b /\ &0 < k ==> --(b + k) < --b`; REAL_ARITH `&0 <= b /\ &0 < k ==> &0 <= b + k`] THEN REWRITE_TAC[REAL_ARITH `b - --b = &2 * b`; REAL_ADD_LDISTRIB] THEN MATCH_MP_TAC(REAL_ARITH `abs(a - b) < e ==> a <= b + e`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Outer and inner approximation of measurable set by well-behaved sets. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_OUTER_INTERVALS_BOUNDED = prove (`!s a b:real^N e. measurable s /\ s SUBSET interval[a,b] /\ &0 < e ==> ?d. COUNTABLE d /\ (!k. k IN d ==> k SUBSET interval[a,b] /\ ~(k = {}) /\ (?c d. k = interval[c,d])) /\ (!k1 k2. k1 IN d /\ k2 IN d /\ ~(k1 = k2) ==> interior k1 INTER interior k2 = {}) /\ (!u v. interval[u,v] IN d ==> ?n. !i. 1 <= i /\ i <= dimindex(:N) ==> v$i - u$i = (b$i - a$i) / &2 pow n) /\ (!k. k IN d /\ ~(interval(a,b) = {}) ==> ~(interior k = {})) /\ s SUBSET UNIONS d /\ measurable (UNIONS d) /\ measure (UNIONS d) <= measure s + e`, let lemma = prove (`(!x y. (x,y) IN IMAGE (\z. f z,g z) s ==> P x y) <=> (!z. z IN s ==> P (f z) (g z))`, REWRITE_TAC[IN_IMAGE; PAIR_EQ] THEN MESON_TAC[]) in REPEAT GEN_TAC THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THENL [ASM_REWRITE_TAC[SUBSET_EMPTY] THEN STRIP_TAC THEN EXISTS_TAC `{}:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; UNIONS_0; MEASURE_EMPTY; REAL_ADD_LID; SUBSET_REFL; COUNTABLE_EMPTY; MEASURABLE_EMPTY] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN STRIP_TAC THEN ASM_CASES_TAC `interval(a:real^N,b) = {}` THEN ASM_REWRITE_TAC[] THENL [EXISTS_TAC `{interval[a:real^N,b]}` THEN REWRITE_TAC[UNIONS_1; COUNTABLE_SING] THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_INSERT; NOT_IN_EMPTY; SUBSET_REFL; MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IN_SING; EQ_INTERVAL] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `0` THEN ASM_REWRITE_TAC[real_pow; REAL_DIV_1]; SUBGOAL_THEN `measure(interval[a:real^N,b]) = &0 /\ measure(s:real^N->bool) = &0` (fun th -> ASM_SIMP_TAC[th; REAL_LT_IMP_LE; REAL_ADD_LID]) THEN SUBGOAL_THEN `interval[a:real^N,b] has_measure &0 /\ (s:real^N->bool) has_measure &0` (fun th -> MESON_TAC[th; MEASURE_UNIQUE]) THEN REWRITE_TAC[HAS_MEASURE_0] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[NEGLIGIBLE_INTERVAL]; ASM_MESON_TAC[NEGLIGIBLE_SUBSET]]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [measurable]) THEN DISCH_THEN(X_CHOOSE_TAC `m:real`) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP MEASURE_UNIQUE) THEN SUBGOAL_THEN `((\x:real^N. if x IN s then vec 1 else vec 0) has_integral (lift m)) (interval[a,b])` ASSUME_TAC THENL [ONCE_REWRITE_TAC[GSYM HAS_INTEGRAL_RESTRICT_UNIV] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_MEASURE]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_EQ) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP HAS_INTEGRAL_INTEGRABLE) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [has_integral]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(SPECL [`a:real^N`; `b:real^N`; `s:real^N->bool`; `g:real^N->real^N->bool`] COVERING_LEMMA) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:(real^N->bool)->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[INTERIOR_EMPTY]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`(\x. if x IN s then vec 1 else vec 0):real^N->real^1`; `a:real^N`; `b:real^N`; `g:real^N->real^N->bool`; `e:real`] HENSTOCK_LEMMA_PART1) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC "*") THEN SUBGOAL_THEN `!k l:real^N->bool. k IN d /\ l IN d /\ ~(k = l) ==> negligible(k INTER l)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`k:real^N->bool`; `l:real^N->bool`]) THEN ASM_SIMP_TAC[] THEN SUBGOAL_THEN `?x y:real^N u v:real^N. k = interval[x,y] /\ l = interval[u,v]` MP_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN DISCH_THEN(REPEAT_TCL CHOOSE_THEN (CONJUNCTS_THEN SUBST_ALL_TAC)) THEN REWRITE_TAC[INTERIOR_CLOSED_INTERVAL] THEN DISCH_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `(interval[x:real^N,y] DIFF interval(x,y)) UNION (interval[u:real^N,v] DIFF interval(u,v)) UNION (interval (x,y) INTER interval (u,v))` THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN ASM_REWRITE_TAC[UNION_EMPTY] THEN SIMP_TAC[NEGLIGIBLE_UNION; NEGLIGIBLE_FRONTIER_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `!D. FINITE D /\ D SUBSET d ==> measurable(UNIONS D :real^N->bool) /\ measure(UNIONS D) <= m + e` ASSUME_TAC THENL [GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `?t:(real^N->bool)->real^N. !k. k IN D ==> t(k) IN (s INTER k) /\ k SUBSET (g(t k))` (CHOOSE_THEN (LABEL_TAC "+")) THENL [REWRITE_TAC[GSYM SKOLEM_THM] THEN ASM SET_TAC[]; ALL_TAC] THEN REMOVE_THEN "*" (MP_TAC o SPEC `IMAGE (\k. (t:(real^N->bool)->real^N) k,k) D`) THEN ASM_SIMP_TAC[VSUM_IMAGE; PAIR_EQ] THEN REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL [REWRITE_TAC[tagged_partial_division_of; fine] THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN REWRITE_TAC[lemma; RIGHT_FORALL_IMP_THM; IMP_CONJ; PAIR_EQ] THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[SUBSET]]; ALL_TAC] THEN USE_THEN "+" (MP_TAC o REWRITE_RULE[IN_INTER]) THEN SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN ASM_SIMP_TAC[VSUM_SUB] THEN SUBGOAL_THEN `D division_of (UNIONS D:real^N->bool)` ASSUME_TAC THENL [REWRITE_TAC[division_of] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP MEASURABLE_ELEMENTARY) THEN SUBGOAL_THEN `vsum D (\k:real^N->bool. content k % vec 1) = lift(measure(UNIONS D))` SUBST1_TAC THENL [ONCE_REWRITE_TAC[GSYM DROP_EQ] THEN ASM_SIMP_TAC[LIFT_DROP; DROP_VSUM; o_DEF; DROP_CMUL; DROP_VEC] THEN SIMP_TAC[REAL_MUL_RID; ETA_AX] THEN ASM_MESON_TAC[MEASURE_ELEMENTARY]; ALL_TAC] THEN SUBGOAL_THEN `vsum D (\k. integral k (\x:real^N. if x IN s then vec 1 else vec 0)) = lift(sum D (\k. measure(k INTER s)))` SUBST1_TAC THENL [ASM_SIMP_TAC[LIFT_SUM; o_DEF] THEN MATCH_MP_TAC VSUM_EQ THEN X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[] THEN SUBGOAL_THEN `measurable(k:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; ALL_TAC] THEN ASM_SIMP_TAC[GSYM INTEGRAL_MEASURE_UNIV; MEASURABLE_INTER] THEN REWRITE_TAC[MESON[IN_INTER] `(if x IN k INTER s then a else b) = (if x IN k then if x IN s then a else b else b)`] THEN REWRITE_TAC[INTEGRAL_RESTRICT_UNIV]; ALL_TAC] THEN ASM_REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `y <= m ==> abs(x - y) <= e ==> x <= m + e`) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(UNIONS D INTER s:real^N->bool)` THEN CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "m" THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_INTER THEN ASM_REWRITE_TAC[]] THEN REWRITE_TAC[INTER_UNIONS] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS_IMAGE_STRONG THEN ASM_SIMP_TAC[FINITE_RESTRICT] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL; MEASURABLE_INTER]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`k:real^N->bool`; `l:real^N->bool`] THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k INTER l:real^N->bool` THEN ASM_SIMP_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `FINITE(d:(real^N->bool)->bool)` THENL [ASM_MESON_TAC[SUBSET_REFL]; ALL_TAC] THEN MP_TAC(ISPEC `d:(real^N->bool)->bool` COUNTABLE_AS_INJECTIVE_IMAGE) THEN ASM_REWRITE_TAC[INFINITE] THEN DISCH_THEN(X_CHOOSE_THEN `s:num->real^N->bool` (CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC)) THEN MP_TAC(ISPECL [`s:num->real^N->bool`; `m + e:real`] HAS_MEASURE_COUNTABLE_NEGLIGIBLE_UNIONS) THEN MATCH_MP_TAC(TAUT `a /\ (a /\ b ==> c) ==> (a ==> b) ==> c`) THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN RULE_ASSUM_TAC(REWRITE_RULE[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_UNIV]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM]) THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL; MEASURABLE_INTER]; ASM_MESON_TAC[]; X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (s:num->real^N->bool) (0..n)`) THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; IMAGE_SUBSET; SUBSET_UNIV] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> x <= e ==> y <= e`) THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS_IMAGE THEN ASM_MESON_TAC[FINITE_NUMSEG; MEASURABLE_INTERVAL]; ALL_TAC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM(CONJUNCT2 LIFT_DROP)] THEN REWRITE_TAC[drop] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_COMPONENT_UBOUND) THEN EXISTS_TAC `\n. vsum(from 0 INTER (0..n)) (\n. lift(measure(s n:real^N->bool)))` THEN ASM_REWRITE_TAC[GSYM sums; TRIVIAL_LIMIT_SEQUENTIALLY] THEN REWRITE_TAC[DIMINDEX_1; ARITH; EVENTUALLY_SEQUENTIALLY] THEN SIMP_TAC[VSUM_COMPONENT; ARITH; DIMINDEX_1] THEN ASM_REWRITE_TAC[GSYM drop; LIFT_DROP; FROM_INTER_NUMSEG]);; let MEASURABLE_OUTER_CLOSED_INTERVALS = prove (`!s:real^N->bool e. measurable s /\ &0 < e ==> ?d. COUNTABLE d /\ (!k. k IN d ==> ~(k = {}) /\ (?a b. k = interval[a,b])) /\ (!k l. k IN d /\ l IN d /\ ~(k = l) ==> interior k INTER interior l = {}) /\ s SUBSET UNIONS d /\ measurable (UNIONS d) /\ measure (UNIONS d) <= measure s + e`, let lemma = prove (`UNIONS (UNIONS {d n | n IN (:num)}) = UNIONS {UNIONS(d n) | n IN (:num)}`, REWRITE_TAC[SIMPLE_IMAGE; UNIONS_IMAGE] THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM; IN_UNIV] THEN MESON_TAC[]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `?d. COUNTABLE d /\ (!k. k IN d ==> ?a b:real^N. k = interval[a,b]) /\ s SUBSET UNIONS d /\ measurable (UNIONS d) /\ measure (UNIONS d) <= measure s + e` MP_TAC THENL [ALL_TAC; DISCH_THEN(X_CHOOSE_THEN `d1:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `d1:(real^N->bool)->bool` COUNTABLE_ELEMENTARY_DIVISION) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:(real^N->bool)->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN ASM_REWRITE_TAC[]] THEN MP_TAC(ISPECL [`\n. s INTER (ball(vec 0:real^N,&n + &1) DIFF ball(vec 0,&n))`; `measure(s:real^N->bool)`] HAS_MEASURE_COUNTABLE_NEGLIGIBLE_UNIONS) THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_DIFF; MEASURABLE_BALL] THEN SUBGOAL_THEN `!m n. ~(m = n) ==> (s INTER (ball(vec 0,&m + &1) DIFF ball(vec 0,&m))) INTER (s INTER (ball(vec 0,&n + &1) DIFF ball(vec 0,&n))) = ({}:real^N->bool)` ASSUME_TAC THENL [MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[INTER_COMM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `m1 SUBSET n ==> (s INTER (m1 DIFF m)) INTER (s INTER (n1 DIFF n)) = {}`) THEN MATCH_MP_TAC SUBSET_BALL THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; ALL_TAC] THEN ANTS_TAC THENL [ASM_SIMP_TAC[NEGLIGIBLE_EMPTY] THEN X_GEN_TAC `n:num` THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o lhand o snd) THEN ASM_SIMP_TAC[FINITE_NUMSEG; DISJOINT] THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_DIFF; MEASURABLE_BALL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC MEASURE_SUBSET THEN SIMP_TAC[SUBSET; FORALL_IN_UNIONS; IMP_CONJ; FORALL_IN_IMAGE; RIGHT_FORALL_IMP_THM; IN_INTER] THEN ASM_SIMP_TAC[MEASURABLE_UNIONS; FINITE_NUMSEG; FORALL_IN_IMAGE; FINITE_IMAGE; MEASURABLE_INTER; MEASURABLE_DIFF; MEASURABLE_BALL]; ALL_TAC] THEN SUBGOAL_THEN `UNIONS {s INTER (ball(vec 0,&n + &1) DIFF ball(vec 0,&n)) | n IN (:num)} = (s:real^N->bool)` ASSUME_TAC THENL [REWRITE_TAC[EXTENSION; IN_UNIONS; EXISTS_IN_GSPEC; IN_UNIV; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `?n. (x:real^N) IN ball(vec 0,&n)` MP_TAC THENL [REWRITE_TAC[IN_BALL_0; REAL_ARCH_LT]; GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN ASM_CASES_TAC `n = 0` THENL [ASM_REWRITE_TAC[IN_BALL_0; GSYM REAL_NOT_LE; NORM_POS_LE]; STRIP_TAC THEN EXISTS_TAC `n - 1` THEN REWRITE_TAC[IN_DIFF] THEN ASM_SIMP_TAC[REAL_OF_NUM_ADD; SUB_ADD; LE_1] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; ASM_REWRITE_TAC[] THEN DISCH_TAC] THEN MP_TAC(MATCH_MP MONO_FORALL (GEN `n:num` (ISPECL [`s INTER (ball(vec 0:real^N,&n + &1) DIFF ball(vec 0,&n))`; `--(vec(n + 1)):real^N`; `vec(n + 1):real^N`; `e / &2 / &2 pow n`] MEASURABLE_OUTER_INTERVALS_BOUNDED))) THEN ANTS_TAC THENL [ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; REAL_LT_POW2] THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_DIFF; MEASURABLE_BALL] THEN REWRITE_TAC[SUBSET; IN_INTER; IN_INTERVAL; IN_BALL_0; IN_DIFF; REAL_NOT_LT; REAL_OF_NUM_ADD; VECTOR_NEG_COMPONENT; VEC_COMPONENT; REAL_BOUNDS_LE] THEN MESON_TAC[COMPONENT_LE_NORM; REAL_LET_TRANS; REAL_LT_IMP_LE]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM]] THEN X_GEN_TAC `d:num->(real^N->bool)->bool` THEN STRIP_TAC THEN EXISTS_TAC `UNIONS {d n | n IN (:num)} :(real^N->bool)->bool` THEN REWRITE_TAC[lemma] THEN CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_UNIONS THEN ASM_REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE] THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_UNIONS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN ASM_MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; IN_UNIV; IN_UNIONS] THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(0..n) (\k. measure(s INTER (ball(vec 0:real^N,&k + &1) DIFF ball(vec 0,&k))) + e / &2 / &2 pow k)` THEN ASM_SIMP_TAC[SUM_LE_NUMSEG] THEN REWRITE_TAC[SUM_ADD_NUMSEG] THEN MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THENL [W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o lhand o snd) THEN ASM_SIMP_TAC[DISJOINT; FINITE_NUMSEG; MEASURABLE_DIFF; MEASURABLE_INTER; MEASURABLE_BALL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNIONS; FORALL_IN_IMAGE; FINITE_NUMSEG; FINITE_IMAGE; MEASURABLE_DIFF; MEASURABLE_INTER; MEASURABLE_BALL] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN MATCH_MP_TAC SUBSET_UNIONS THEN REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET_UNIV]; REWRITE_TAC[real_div; SUM_LMUL; REAL_INV_POW; SUM_GP; LT] THEN REWRITE_TAC[GSYM real_div] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `e / &2 * (&1 - x) / (&1 / &2) <= e <=> &0 <= e * x`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_POW_LE THEN CONV_TAC REAL_RAT_REDUCE_CONV]);; let MEASURABLE_OUTER_OPEN_INTERVALS = prove (`!s:real^N->bool e. measurable s /\ &0 < e ==> ?d. COUNTABLE d /\ (!k. k IN d ==> ~(k = {}) /\ (?a b. k = interval(a,b))) /\ s SUBSET UNIONS d /\ measurable (UNIONS d) /\ measure (UNIONS d) <= measure s + e`, let lemma = prove (`!s. UNIONS(s DELETE {}) = UNIONS s`, REWRITE_TAC[EXTENSION; IN_UNIONS; IN_DELETE] THEN MESON_TAC[NOT_IN_EMPTY]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `e / &2`] MEASURABLE_OUTER_CLOSED_INTERVALS) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `dset:(real^N->bool)->bool` THEN ASM_CASES_TAC `dset:(real^N->bool)->bool = {}` THENL [ASM_REWRITE_TAC[UNIONS_0; SUBSET_EMPTY] THEN STRIP_TAC THEN EXISTS_TAC `{}:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[UNIONS_0; NOT_IN_EMPTY; MEASURE_EMPTY; SUBSET_REFL] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN SUBGOAL_THEN `?f. dset = IMAGE (f:num->(real^N->bool)) (:num) DELETE {} /\ (!m n. f m = f n ==> m = n \/ f n = {})` MP_TAC THENL [ASM_CASES_TAC `FINITE(dset:(real^N->bool)->bool)` THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FINITE_HAS_SIZE]) THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_SIZE_INDEX) THEN ABBREV_TAC `m = CARD(dset:(real^N->bool)->bool)` THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\i. if i < m then (f:num->real^N->bool) i else {}` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_DELETE; IN_IMAGE; IN_UNIV] THEN ASM_MESON_TAC[]; MP_TAC(ISPEC `dset:(real^N->bool)->bool` COUNTABLE_AS_INJECTIVE_IMAGE) THEN ASM_REWRITE_TAC[INFINITE] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN ASM_REWRITE_TAC[SET_RULE `s = s DELETE a <=> ~(a IN s)`] THEN ASM_MESON_TAC[]]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:num->real^N->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC) THEN FIRST_X_ASSUM(MP_TAC o check (is_forall o concl)) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV; FORALL_AND_THM; SKOLEM_THM; IMP_CONJ; RIGHT_FORALL_IMP_THM; IN_DELETE; lemma] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!x. ~(P x) ==> ~(P x) /\ Q x) ==> (!x. P x ==> Q x) ==> !x. Q x`)) THEN ANTS_TAC THENL [MESON_TAC[EMPTY_AS_INTERVAL]; ALL_TAC] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:num->real^N`; `b:num->real^N`] THEN DISCH_TAC THEN DISCH_TAC THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN GEN_REWRITE_TAC I [IMP_CONJ] THEN DISCH_THEN(MP_TAC o MATCH_MP(MESON[] `(!x y. ~(P x) /\ ~(P y) /\ ~(f x = f y) ==> Q x y) ==> (!x y. P x ==> Q x y) /\ (!x y. P y ==> Q x y) ==> (!x y. ~(f x = f y) ==> Q x y)`)) THEN SIMP_TAC[INTERIOR_EMPTY; INTER_EMPTY] THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?d. COUNTABLE d /\ (!k. k IN d ==> ?a b:real^N. k = interval(a,b)) /\ s SUBSET UNIONS d /\ measurable (UNIONS d) /\ measure (UNIONS d) <= measure s + e` MP_TAC THENL [ALL_TAC; DISCH_THEN(X_CHOOSE_TAC `d:(real^N->bool)->bool`) THEN EXISTS_TAC `d DELETE ({}:real^N->bool)` THEN ASM_SIMP_TAC[lemma; COUNTABLE_DELETE; IN_DELETE]] THEN MP_TAC(GEN `n:num` (ISPECL [`(a:num->real^N) n`; `(b:num->real^N) n`; `e / &2 pow (n + 2)`] EXPAND_CLOSED_OPEN_INTERVAL)) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2; SKOLEM_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN MAP_EVERY X_GEN_TAC [`A:num->real^N`; `B:num->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `IMAGE (\n. interval(A n:real^N,B n)) (:num)` THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; FORALL_IN_IMAGE; IN_UNIV] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_UNIONS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`n:num`; `x:real^N`] THEN ASM_REWRITE_TAC[IN_UNIONS; EXISTS_IN_IMAGE; IN_UNIV] THEN ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(0..n) (\i. measure(interval[a i:real^N,b i]) + e / &2 pow (i + 2))` THEN ASM_SIMP_TAC[SUM_LE_NUMSEG] THEN REWRITE_TAC[SUM_ADD_NUMSEG] THEN REWRITE_TAC[real_div; REAL_INV_MUL; SUM_LMUL; REAL_POW_ADD; SUM_RMUL] THEN REWRITE_TAC[REAL_INV_POW; SUM_GP; LT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC(REAL_ARITH `s <= m + e / &2 /\ &0 <= e * x ==> s + e * (&1 - x) / (&1 / &2) * &1 / &4 <= m + e`) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_POW_LE; REAL_LT_IMP_LE; REAL_LE_DIV; REAL_POS] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN W(MP_TAC o PART_MATCH (rhs o rand) MEASURE_NEGLIGIBLE_UNIONS_IMAGE o lhand o snd) THEN REWRITE_TAC[FINITE_NUMSEG; MEASURABLE_INTERVAL] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN ASM_CASES_TAC `interval[(a:num->real^N) i,b i] = interval[a j,b j]` THENL [UNDISCH_TAC `!m n. (d:num->real^N->bool) m = d n ==> m = n \/ d n = {}` THEN DISCH_THEN(MP_TAC o SPECL [`i:num`; `j:num`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[INTER_EMPTY; NEGLIGIBLE_EMPTY]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (BINDER_CONV o BINDER_CONV o RAND_CONV o LAND_CONV) [GSYM INTERIOR_INTER]) THEN DISCH_THEN(MP_TAC o SPECL [`i:num`; `j:num`]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_MEASURABLE_MEASURE] THEN SIMP_TAC[MEASURABLE_INTER; MEASURABLE_INTERVAL] THEN MATCH_MP_TAC(MESON[MEASURE_EMPTY] `measure(interior s) = measure s ==> interior s = {} ==> measure s = &0`) THEN MATCH_MP_TAC MEASURE_INTERIOR THEN SIMP_TAC[BOUNDED_INTER; BOUNDED_INTERVAL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_INTER; CONVEX_INTERVAL]]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURE_SUBSET THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS THEN SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE; MEASURABLE_INTERVAL; FINITE_NUMSEG]; ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_UNIONS THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[IN_IMAGE; IN_UNIV] THEN MESON_TAC[]]);; let MEASURABLE_OUTER_OPEN = prove (`!s:real^N->bool e. measurable s /\ &0 < e ==> ?t. open t /\ s SUBSET t /\ measurable t /\ measure t < measure s + e`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `e / &2`] MEASURABLE_OUTER_OPEN_INTERVALS) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:(real^N->bool)->bool` THEN STRIP_TAC THEN EXISTS_TAC `UNIONS d :real^N->bool` THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e /\ m <= s + e / &2 ==> m < s + e`] THEN MATCH_MP_TAC OPEN_UNIONS THEN ASM_MESON_TAC[OPEN_INTERVAL]);; let MEASURABLE_INNER_COMPACT = prove (`!s:real^N->bool e. measurable s /\ &0 < e ==> ?t. compact t /\ t SUBSET s /\ measurable t /\ measure s < measure t + e`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_MEASURE_MEASURE]) THEN GEN_REWRITE_TAC LAND_CONV [HAS_MEASURE_LIMIT] THEN DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> &0 < e / &4`] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MP_TAC(ISPEC `ball(vec 0:real^N,B)` BOUNDED_SUBSET_CLOSED_INTERVAL) THEN REWRITE_TAC[BOUNDED_BALL; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real` THEN STRIP_TAC THEN MP_TAC(ISPECL [`interval[a:real^N,b] DIFF s`; `e/ &4`] MEASURABLE_OUTER_OPEN) THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTERVAL; REAL_ARITH `&0 < e ==> &0 < e / &4`] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `interval[a:real^N,b] DIFF t` THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN ASM_SIMP_TAC[CLOSED_DIFF; CLOSED_INTERVAL; BOUNDED_DIFF; BOUNDED_INTERVAL]; ASM SET_TAC[]; ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTERVAL]; MATCH_MP_TAC(REAL_ARITH `&0 < e /\ measure(s) < measure(interval[a,b] INTER s) + e / &4 /\ measure(t) < measure(interval[a,b] DIFF s) + e / &4 /\ measure(interval[a,b] INTER s) + measure(interval[a,b] DIFF s) = measure(interval[a,b]) /\ measure(interval[a,b] INTER t) + measure(interval[a,b] DIFF t) = measure(interval[a,b]) /\ measure(interval[a,b] INTER t) <= measure t ==> measure s < measure(interval[a,b] DIFF t) + e`) THEN ASM_SIMP_TAC[MEASURE_SUBSET; INTER_SUBSET; MEASURABLE_INTER; MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [FIRST_ASSUM(SUBST_ALL_TAC o SYM o MATCH_MP MEASURE_UNIQUE) THEN ONCE_REWRITE_TAC[INTER_COMM] THEN ASM_REAL_ARITH_TAC; CONJ_TAC THEN MATCH_MP_TAC MEASURE_DISJOINT_UNION_EQ THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_DIFF; MEASURABLE_INTERVAL] THEN SET_TAC[]]]);; let OPEN_MEASURABLE_INNER_DIVISION = prove (`!s:real^N->bool e. open s /\ measurable s /\ &0 < e ==> ?D. D division_of UNIONS D /\ UNIONS D SUBSET s /\ measure s < measure(UNIONS D) + e`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `e / &2`] MEASURE_LIMIT) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN MP_TAC(ISPEC `ball(vec 0:real^N,B)` BOUNDED_SUBSET_CLOSED_INTERVAL) THEN ASM_REWRITE_TAC[BOUNDED_BALL; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPEC `s INTER interval(a - vec 1:real^N,b + vec 1)` OPEN_COUNTABLE_UNION_CLOSED_INTERVALS) THEN ASM_SIMP_TAC[OPEN_INTER; OPEN_INTERVAL; SUBSET_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`D:(real^N->bool)->bool`; `measure(s:real^N->bool)`; `e / &2`] MEASURE_COUNTABLE_UNIONS_APPROACHABLE) THEN ASM_REWRITE_TAC[REAL_HALF] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(UNIONS D :real^N->bool)` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_INTERVAL]; ASM_SIMP_TAC[SUBSET_UNIONS]]; ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_INTERVAL; INTER_SUBSET]]; DISCH_THEN(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `d:(real^N->bool)->bool` ELEMENTARY_UNIONS_INTERVALS) THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `p:(real^N->bool)->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `UNIONS p :real^N->bool = UNIONS d` SUBST1_TAC THENL [ASM_MESON_TAC[division_of]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `UNIONS D :real^N->bool` THEN ASM_SIMP_TAC[SUBSET_UNIONS; INTER_SUBSET]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `ms' - e / &2 < mud ==> ms < ms' + e / &2 ==> ms < mud + e`)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `abs(sc - s) < e / &2 ==> sc <= so /\ sc <= s ==> s < so + e / &2`)) THEN CONJ_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_INTERVAL; INTER_SUBSET] THEN MATCH_MP_TAC(SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`) THEN REWRITE_TAC[SUBSET_INTERVAL; VECTOR_SUB_COMPONENT; VEC_COMPONENT; VECTOR_ADD_COMPONENT] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN REAL_ARITH_TAC]]);; let OUTER_MEASURE = prove (`!s:real^N->bool. bounded s ==> ?t. s SUBSET t /\ measurable t /\ !t'. s SUBSET t' /\ measurable t' ==> negligible(t DIFF t')`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `IMAGE measure {u:real^N->bool | s SUBSET u /\ measurable u /\ open u}` INF) THEN REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_GSPEC] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN ASM_MESON_TAC[BOUNDED_SUBSET_BALL; OPEN_BALL; BOUNDED_BALL; MEASURABLE_OPEN; MEASURE_POS_LE]; ALL_TAC] THEN ABBREV_TAC `b = inf(IMAGE measure { u:real^N->bool | s SUBSET u /\ measurable u /\ open u})` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(fun th -> MP_TAC(GEN `n:num` (SPEC `b + inv(&n + &1)` th)) THEN MP_TAC(SPEC `&0` th)) THEN SIMP_TAC[MEASURE_POS_LE; MEASURABLE_OPEN] THEN DISCH_TAC THEN REWRITE_TAC[REAL_ARITH `b + x <= b <=> ~(&0 < x)`] THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LE] THEN REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:num->real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `INTERS {u n | n IN (:num)} : real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_INTERS; FORALL_IN_GSPEC] THEN ASM_SIMP_TAC[MEASURABLE_COUNTABLE_INTERS; MEASURABLE_OPEN] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (funpow 3 rand) MEASURABLE_MEASURE_POS_LT o snd) THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_COUNTABLE_INTERS; MEASURABLE_OPEN] THEN MATCH_MP_TAC(TAUT `~p ==> (p <=> ~q) ==> q`) THEN DISCH_TAC THEN MP_TAC(ISPECL [`INTERS {u n | n IN (:num)} INTER t:real^N->bool`; `measure(INTERS {u n | n IN (:num)} DIFF t:real^N->bool)`] MEASURABLE_OUTER_OPEN) THEN ASM_SIMP_TAC[MEASURABLE_COUNTABLE_INTERS; MEASURABLE_INTER] THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `v:real^N->bool` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[REAL_NOT_LT] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNION o lhand o snd) THEN ASM_SIMP_TAC[MEASURABLE_COUNTABLE_INTERS; MEASURABLE_INTER; MEASURABLE_DIFF; SET_RULE `DISJOINT (s INTER t) (s DIFF t)`] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[SET_RULE `(s INTER t) UNION (s DIFF t) = s`] THEN TRANS_TAC REAL_LE_TRANS `b:real` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[REAL_ARITH `x <= y <=> x - y <= &0`] THEN ONCE_REWRITE_TAC[REAL_LE_TRANS_LTE] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [REAL_ARITH_TAC; X_GEN_TAC `n:num`] THEN REWRITE_TAC[REAL_ARITH `x - y <= z <=> x <= y + z`] THEN TRANS_TAC REAL_LE_TRANS `measure((u:num->real^N->bool) n)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_COUNTABLE_INTERS] THEN SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Transformation of measure by linear maps. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_LINEAR_IMAGE_INTERVAL = prove (`!f a b. linear f ==> measurable(IMAGE f (interval[a,b]))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_CONVEX THEN CONJ_TAC THENL [MATCH_MP_TAC CONVEX_LINEAR_IMAGE THEN ASM_MESON_TAC[CONVEX_INTERVAL]; MATCH_MP_TAC BOUNDED_LINEAR_IMAGE THEN ASM_MESON_TAC[BOUNDED_INTERVAL]]);; let HAS_MEASURE_LINEAR_SUFFICIENT = prove (`!f:real^N->real^N m. linear f /\ (!a b. IMAGE f (interval[a,b]) has_measure (m * measure(interval[a,b]))) ==> !s. measurable s ==> (IMAGE f s) has_measure (m * measure s)`, REPEAT GEN_TAC THEN STRIP_TAC THEN DISJ_CASES_TAC(REAL_ARITH `m < &0 \/ &0 <= m`) THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`vec 0:real^N`; `vec 1:real^N`]) THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_POS_LE) THEN MATCH_MP_TAC(TAUT `~a ==> a ==> b`) THEN MATCH_MP_TAC(REAL_ARITH `&0 < --m * x ==> ~(&0 <= m * x)`) THEN MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[REAL_NEG_GT0] THEN REWRITE_TAC[MEASURE_INTERVAL] THEN MATCH_MP_TAC CONTENT_POS_LT THEN SIMP_TAC[VEC_COMPONENT; REAL_LT_01]; ALL_TAC] THEN ASM_CASES_TAC `!x y. (f:real^N->real^N) x = f y ==> x = y` THENL [ALL_TAC; SUBGOAL_THEN `!s. negligible(IMAGE (f:real^N->real^N) s)` ASSUME_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_LINEAR_SINGULAR_IMAGE]; ALL_TAC] THEN SUBGOAL_THEN `m * measure(interval[vec 0:real^N,vec 1]) = &0` MP_TAC THENL [MATCH_MP_TAC(ISPEC `IMAGE (f:real^N->real^N) (interval[vec 0,vec 1])` HAS_MEASURE_UNIQUE) THEN ASM_REWRITE_TAC[HAS_MEASURE_0]; REWRITE_TAC[REAL_ENTIRE; MEASURE_INTERVAL] THEN MATCH_MP_TAC(TAUT `~b /\ (a ==> c) ==> a \/ b ==> c`) THEN CONJ_TAC THENL [SIMP_TAC[CONTENT_EQ_0_INTERIOR; INTERIOR_CLOSED_INTERVAL; INTERVAL_NE_EMPTY; VEC_COMPONENT; REAL_LT_01]; ASM_SIMP_TAC[REAL_MUL_LZERO; HAS_MEASURE_0]]]] THEN MP_TAC(ISPEC `f:real^N->real^N` LINEAR_INJECTIVE_ISOMORPHISM) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^N->real^N` STRIP_ASSUME_TAC) THEN UNDISCH_THEN `!x y. (f:real^N->real^N) x = f y ==> x = y` (K ALL_TAC) THEN SUBGOAL_THEN `!s. bounded s /\ measurable s ==> (IMAGE (f:real^N->real^N) s) has_measure (m * measure s)` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN SUBGOAL_THEN `!d. COUNTABLE d /\ (!k. k IN d ==> k SUBSET interval[a,b] /\ ~(k = {}) /\ (?c d. k = interval[c,d])) /\ (!k1 k2. k1 IN d /\ k2 IN d /\ ~(k1 = k2) ==> interior k1 INTER interior k2 = {}) ==> IMAGE (f:real^N->real^N) (UNIONS d) has_measure (m * measure(UNIONS d))` ASSUME_TAC THENL [REWRITE_TAC[IMAGE_UNIONS] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `!g:real^N->real^N. linear g ==> !k l. k IN d /\ l IN d /\ ~(k = l) ==> negligible((IMAGE g k) INTER (IMAGE g l))` MP_TAC THENL [REPEAT STRIP_TAC THEN ASM_CASES_TAC `!x y. (g:real^N->real^N) x = g y ==> x = y` THENL [ALL_TAC; ASM_MESON_TAC[NEGLIGIBLE_LINEAR_SINGULAR_IMAGE; NEGLIGIBLE_INTER]] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `frontier(IMAGE (g:real^N->real^N) k INTER IMAGE g l) UNION interior(IMAGE g k INTER IMAGE g l)` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s SUBSET (t DIFF u) UNION u`) THEN REWRITE_TAC[CLOSURE_SUBSET]] THEN MATCH_MP_TAC NEGLIGIBLE_UNION THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_CONVEX_FRONTIER THEN MATCH_MP_TAC CONVEX_INTER THEN CONJ_TAC THEN MATCH_MP_TAC CONVEX_LINEAR_IMAGE THEN ASM_MESON_TAC[CONVEX_INTERVAL]; ALL_TAC] THEN REWRITE_TAC[INTERIOR_INTER] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^N->real^N) (interior k) INTER IMAGE g (interior l)` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^N->real^N) (interior k INTER interior l)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[IMAGE_CLAUSES; NEGLIGIBLE_EMPTY]; ASM SET_TAC[]]; MATCH_MP_TAC(SET_RULE `s SUBSET u /\ t SUBSET v ==> (s INTER t) SUBSET (u INTER v)`) THEN CONJ_TAC THEN MATCH_MP_TAC INTERIOR_IMAGE_SUBSET THEN ASM_MESON_TAC[LINEAR_CONTINUOUS_AT]]; ALL_TAC] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `f:real^N->real^N` th) THEN MP_TAC(SPEC `\x:real^N. x` th)) THEN ASM_REWRITE_TAC[LINEAR_ID; SET_RULE `IMAGE (\x. x) s = s`] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `FINITE(d:(real^N->bool)->bool)` THENL [MP_TAC(ISPECL [`IMAGE (f:real^N->real^N)`; `d:(real^N->bool)->bool`] HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE) THEN ANTS_TAC THENL [ASM_MESON_TAC[measurable]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum d (\k:real^N->bool. m * measure k)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[MEASURE_UNIQUE]; ALL_TAC] THEN REWRITE_TAC[SUM_LMUL] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS THEN ASM_REWRITE_TAC[GSYM HAS_MEASURE_MEASURE] THEN ASM_MESON_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN MP_TAC(ISPEC `d:(real^N->bool)->bool` COUNTABLE_AS_INJECTIVE_IMAGE) THEN ASM_REWRITE_TAC[INFINITE] THEN DISCH_THEN(X_CHOOSE_THEN `s:num->real^N->bool` (CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC)) THEN MP_TAC(ISPEC `s:num->real^N->bool` HAS_MEASURE_COUNTABLE_NEGLIGIBLE_UNIONS_BOUNDED) THEN MP_TAC(ISPEC `\n:num. IMAGE (f:real^N->real^N) (s n)` HAS_MEASURE_COUNTABLE_NEGLIGIBLE_UNIONS_BOUNDED) THEN RULE_ASSUM_TAC(REWRITE_RULE[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_UNIV]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM]) THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_LINEAR_IMAGE_INTERVAL]; ASM_MESON_TAC[]; ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[GSYM IMAGE_UNIONS; IMAGE_o] THEN MATCH_MP_TAC BOUNDED_LINEAR_IMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN REWRITE_TAC[UNIONS_SUBSET] THEN EXISTS_TAC `interval[a:real^N,b]` THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN ASM SET_TAC[]]; ALL_TAC] THEN STRIP_TAC THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL]; ASM_MESON_TAC[]; MATCH_MP_TAC BOUNDED_SUBSET THEN REWRITE_TAC[UNIONS_SUBSET] THEN EXISTS_TAC `interval[a:real^N,b]` THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN ASM SET_TAC[]]; ALL_TAC] THEN STRIP_TAC THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN SUBGOAL_THEN `m * measure (UNIONS (IMAGE s (:num)):real^N->bool) = measure(UNIONS (IMAGE (\x. IMAGE f (s x)) (:num)):real^N->bool)` (fun th -> ASM_REWRITE_TAC[GSYM HAS_MEASURE_MEASURE; th]) THEN ONCE_REWRITE_TAC[GSYM LIFT_EQ] THEN MATCH_MP_TAC SERIES_UNIQUE THEN EXISTS_TAC `\n:num. lift(measure(IMAGE (f:real^N->real^N) (s n)))` THEN EXISTS_TAC `from 0` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUMS_EQ THEN EXISTS_TAC `\n:num. m % lift(measure(s n:real^N->bool))` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM LIFT_CMUL; LIFT_EQ] THEN ASM_MESON_TAC[MEASURE_UNIQUE]; REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC SERIES_CMUL THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN REWRITE_TAC[HAS_MEASURE_INNER_OUTER_LE] THEN CONJ_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THENL [MP_TAC(ISPECL [`interval[a,b] DIFF s:real^N->bool`; `a:real^N`; `b:real^N`; `e / (&1 + abs m)`] MEASURABLE_OUTER_INTERVALS_BOUNDED) THEN ANTS_TAC THENL [ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTERVAL] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < &1 + abs x`; REAL_LT_DIV] THEN SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE f (interval[a,b]) DIFF IMAGE (f:real^N->real^N) (UNIONS d)` THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:(real^N->bool)->bool`) THEN ASM_SIMP_TAC[IMAGE_SUBSET] THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_DIFF; measurable]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(IMAGE f (interval[a,b])) - measure(IMAGE (f:real^N->real^N) (UNIONS d))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC REAL_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_DIFF_SUBSET THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[measurable]; ALL_TAC]) THEN MATCH_MP_TAC IMAGE_SUBSET THEN ASM_SIMP_TAC[UNIONS_SUBSET]] THEN UNDISCH_TAC `!a b. IMAGE (f:real^N->real^N) (interval [a,b]) has_measure m * measure (interval [a,b])` THEN DISCH_THEN(ASSUME_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN REPEAT(FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP MEASURE_UNIQUE)) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `m * measure(s:real^N->bool) - m * e / (&1 + abs m)` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `a - x <= a - y <=> y <= x`] THEN REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &1 + abs x`] THEN GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM REAL_SUB_LDISTRIB] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `d <= a + e ==> a = i - s ==> s - e <= i - d`)) THEN MATCH_MP_TAC MEASURE_DIFF_SUBSET THEN ASM_REWRITE_TAC[MEASURABLE_INTERVAL]; MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`; `e / (&1 + abs m)`] MEASURABLE_OUTER_INTERVALS_BOUNDED) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &1 + abs x`] THEN DISCH_THEN(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE (f:real^N->real^N) (UNIONS d)` THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:(real^N->bool)->bool`) THEN ASM_SIMP_TAC[IMAGE_SUBSET] THEN SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `m * measure(s:real^N->bool) + m * e / (&1 + abs m)` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM REAL_ADD_LDISTRIB] THEN ASM_SIMP_TAC[REAL_LE_LMUL]; REWRITE_TAC[REAL_LE_LADD] THEN REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &1 + abs x`] THEN GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ] THEN REAL_ARITH_TAC]]; ALL_TAC] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[HAS_MEASURE_LIMIT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_MEASURE_MEASURE]) THEN GEN_REWRITE_TAC LAND_CONV [HAS_MEASURE_LIMIT] THEN DISCH_THEN(MP_TAC o SPEC `e / (&1 + abs m)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &1 + abs x`] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*"))) THEN MP_TAC(ISPEC `ball(vec 0:real^N,B)` BOUNDED_SUBSET_CLOSED_INTERVAL) THEN REWRITE_TAC[BOUNDED_BALL; LEFT_IMP_EXISTS_THM] THEN REMOVE_THEN "*" MP_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `c:real^N` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `d:real^N` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `z:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`interval[c:real^N,d]`; `vec 0:real^N`] BOUNDED_SUBSET_BALL) THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN DISCH_THEN(X_CHOOSE_THEN `D:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `f:real^N->real^N` LINEAR_BOUNDED_POS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `D * C:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `s INTER (IMAGE (h:real^N->real^N) (interval[a,b]))`) THEN SUBGOAL_THEN `IMAGE (f:real^N->real^N) (s INTER IMAGE h (interval [a,b])) = (IMAGE f s) INTER interval[a,b]` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ANTS_TAC THENL [ASM_SIMP_TAC[BOUNDED_INTER; BOUNDED_LINEAR_IMAGE; BOUNDED_INTERVAL] THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_LINEAR_IMAGE_INTERVAL]; ALL_TAC] THEN DISCH_TAC THEN EXISTS_TAC `m * measure(s INTER (IMAGE (h:real^N->real^N) (interval[a,b])))` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `m * e / (&1 + abs m)` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_ARITH `&0 < &1 + abs x`] THEN GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN ASM_SIMP_TAC[REAL_LT_RMUL_EQ] THEN REAL_ARITH_TAC] THEN REWRITE_TAC[GSYM REAL_SUB_LDISTRIB; REAL_ABS_MUL] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [real_abs] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `abs(z - m) < e ==> z <= w /\ w <= m ==> abs(w - m) <= e`)) THEN SUBST1_TAC(SYM(MATCH_MP MEASURE_UNIQUE (ASSUME `s INTER interval [c:real^N,d] has_measure z`))) THEN CONJ_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_LINEAR_IMAGE_INTERVAL; MEASURABLE_INTERVAL; INTER_SUBSET] THEN MATCH_MP_TAC(SET_RULE `!v. t SUBSET v /\ v SUBSET u ==> s INTER t SUBSET s INTER u`) THEN EXISTS_TAC `ball(vec 0:real^N,D)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `!f. (!x. h(f x) = x) /\ IMAGE f s SUBSET t ==> s SUBSET IMAGE h t`) THEN EXISTS_TAC `f:real^N->real^N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(vec 0:real^N,D * C)` THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_BALL_0] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `C * norm(x:real^N)` THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ]);; let INDUCT_MATRIX_ROW_OPERATIONS = prove (`!P:real^N^N->bool. (!A i. 1 <= i /\ i <= dimindex(:N) /\ row i A = vec 0 ==> P A) /\ (!A. (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> A$i$j = &0) ==> P A) /\ (!A m n. P A /\ 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(lambda i j. A$i$(swap(m,n) j))) /\ (!A m n c. P A /\ 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(lambda i. if i = m then row m A + c % row n A else row i A)) ==> !A. P A`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "zero_row") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "diagonal") MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "swap_cols") (LABEL_TAC "row_op")) THEN SUBGOAL_THEN `!k A:real^N^N. (!i j. 1 <= i /\ i <= dimindex(:N) /\ k <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> A$i$j = &0) ==> P A` (fun th -> GEN_TAC THEN MATCH_MP_TAC th THEN EXISTS_TAC `dimindex(:N) + 1` THEN ARITH_TAC) THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN USE_THEN "diagonal" MATCH_MP_TAC THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[LE_0]; ALL_TAC] THEN X_GEN_TAC `k:num` THEN DISCH_THEN(LABEL_TAC "ind_hyp") THEN DISJ_CASES_THEN2 SUBST1_TAC ASSUME_TAC (ARITH_RULE `k = 0 \/ 1 <= k`) THEN ASM_REWRITE_TAC[ARITH] THEN ASM_CASES_TAC `k <= dimindex(:N)` THENL [ALL_TAC; REPEAT STRIP_TAC THEN REMOVE_THEN "ind_hyp" MATCH_MP_TAC THEN ASM_ARITH_TAC] THEN SUBGOAL_THEN `!A:real^N^N. ~(A$k$k = &0) /\ (!i j. 1 <= i /\ i <= dimindex (:N) /\ SUC k <= j /\ j <= dimindex (:N) /\ ~(i = j) ==> A$i$j = &0) ==> P A` (LABEL_TAC "nonzero_hyp") THENL [ALL_TAC; X_GEN_TAC `A:real^N^N` THEN DISCH_TAC THEN ASM_CASES_TAC `row k (A:real^N^N) = vec 0` THENL [REMOVE_THEN "zero_row" MATCH_MP_TAC THEN ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN SIMP_TAC[VEC_COMPONENT; row; LAMBDA_BETA] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `l:num` THEN STRIP_TAC THEN ASM_CASES_TAC `l:num = k` THENL [REMOVE_THEN "nonzero_hyp" MATCH_MP_TAC THEN ASM_MESON_TAC[]; ALL_TAC] THEN REMOVE_THEN "swap_cols" (MP_TAC o SPECL [`(lambda i j. (A:real^N^N)$i$swap(k,l) j):real^N^N`; `k:num`; `l:num`]) THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[swap] THEN REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA])] THEN REMOVE_THEN "nonzero_hyp" MATCH_MP_TAC THEN ONCE_REWRITE_TAC[ARITH_RULE `SUC k <= i <=> 1 <= i /\ SUC k <= i`] THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN ASM_REWRITE_TAC[swap] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `l:num <= k` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`k:num`; `l:num`]) THEN ASM_REWRITE_TAC[] THEN ARITH_TAC; ALL_TAC] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC] THEN SUBGOAL_THEN `!l A:real^N^N. ~(A$k$k = &0) /\ (!i j. 1 <= i /\ i <= dimindex (:N) /\ SUC k <= j /\ j <= dimindex (:N) /\ ~(i = j) ==> A$i$j = &0) /\ (!i. l <= i /\ i <= dimindex(:N) /\ ~(i = k) ==> A$i$k = &0) ==> P A` MP_TAC THENL [ALL_TAC; DISCH_THEN(MP_TAC o SPEC `dimindex(:N) + 1`) THEN REWRITE_TAC[CONJ_ASSOC; ARITH_RULE `~(n + 1 <= i /\ i <= n)`]] THEN MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "main") (LABEL_TAC "aux")) THEN USE_THEN "ind_hyp" MATCH_MP_TAC THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN ASM_CASES_TAC `j:num = k` THENL [ASM_REWRITE_TAC[] THEN USE_THEN "aux" MATCH_MP_TAC THEN ASM_ARITH_TAC; REMOVE_THEN "main" MATCH_MP_TAC THEN ASM_ARITH_TAC]; ALL_TAC] THEN X_GEN_TAC `l:num` THEN DISCH_THEN(LABEL_TAC "inner_hyp") THEN GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "main") (LABEL_TAC "aux")) THEN ASM_CASES_TAC `l:num = k` THENL [REMOVE_THEN "inner_hyp" MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REMOVE_THEN "aux" MATCH_MP_TAC THEN ASM_ARITH_TAC; ALL_TAC] THEN DISJ_CASES_TAC(ARITH_RULE `l = 0 \/ 1 <= l`) THENL [REMOVE_THEN "ind_hyp" MATCH_MP_TAC THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN ASM_CASES_TAC `j:num = k` THENL [ASM_REWRITE_TAC[] THEN REMOVE_THEN "aux" MATCH_MP_TAC THEN ASM_ARITH_TAC; REMOVE_THEN "main" MATCH_MP_TAC THEN ASM_ARITH_TAC]; ALL_TAC] THEN ASM_CASES_TAC `l <= dimindex(:N)` THENL [ALL_TAC; REMOVE_THEN "inner_hyp" MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC] THEN REMOVE_THEN "inner_hyp" (MP_TAC o SPECL [`(lambda i. if i = l then row l (A:real^N^N) + --(A$l$k/A$k$k) % row k A else row i A):real^N^N`]) THEN ANTS_TAC THENL [SUBGOAL_THEN `!i. l <= i ==> 1 <= i` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[ARITH_RULE `SUC k <= j <=> 1 <= j /\ SUC k <= j`] THEN ASM_SIMP_TAC[LAMBDA_BETA; row; COND_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN ASM_SIMP_TAC[REAL_FIELD `~(y = &0) ==> x + --(x / y) * y = &0`] THEN REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num = l` THEN ASM_REWRITE_TAC[] THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_RING `x = &0 /\ y = &0 ==> x + z * y = &0`) THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; REPEAT STRIP_TAC THEN REMOVE_THEN "aux" MATCH_MP_TAC THEN ASM_ARITH_TAC]; ALL_TAC] THEN DISCH_TAC THEN REMOVE_THEN "row_op" (MP_TAC o SPECL [`(lambda i. if i = l then row l A + --(A$l$k / A$k$k) % row k A else row i (A:real^N^N)):real^N^N`; `l:num`; `k:num`; `(A:real^N^N)$l$k / A$k$k`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; row; COND_COMPONENT] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);; let INDUCT_MATRIX_ELEMENTARY = prove (`!P:real^N^N->bool. (!A B. P A /\ P B ==> P(A ** B)) /\ (!A i. 1 <= i /\ i <= dimindex(:N) /\ row i A = vec 0 ==> P A) /\ (!A. (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> A$i$j = &0) ==> P A) /\ (!m n. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(lambda i j. (mat 1:real^N^N)$i$(swap(m,n) j))) /\ (!m n c. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(lambda i j. if i = m /\ j = n then c else if i = j then &1 else &0)) ==> !A. P A`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(fun th -> MATCH_MP_TAC INDUCT_MATRIX_ROW_OPERATIONS THEN MP_TAC th) THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THEN REWRITE_TAC[] THEN DISCH_THEN(fun th -> X_GEN_TAC `A:real^N^N` THEN MP_TAC th) THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `(P:real^N^N->bool) A` THENL [REWRITE_TAC[GSYM IMP_CONJ]; REWRITE_TAC[GSYM IMP_CONJ_ALT]] THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[CART_EQ] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA; matrix_mul; row] THENL [ASM_SIMP_TAC[mat; IN_DIMINDEX_SWAP; LAMBDA_BETA] THEN ONCE_REWRITE_TAC[COND_RAND] THEN SIMP_TAC[SUM_DELTA; REAL_MUL_RZERO; REAL_MUL_RID] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[swap; IN_NUMSEG]) THEN ASM_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `i:num = m` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[REAL_MUL_LZERO] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN ASM_SIMP_TAC[SUM_DELTA; LAMBDA_BETA; IN_NUMSEG; REAL_MUL_LID]] THEN ASM_SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum {m,n} (\k. (if k = n then c else if m = k then &1 else &0) * (A:real^N^N)$k$j)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_SUPERSET THEN ASM_SIMP_TAC[SUBSET; IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM; IN_NUMSEG; REAL_MUL_LZERO] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[SUM_CLAUSES; FINITE_RULES; IN_INSERT; NOT_IN_EMPTY] THEN REAL_ARITH_TAC]);; let INDUCT_MATRIX_ELEMENTARY_ALT = prove (`!P:real^N^N->bool. (!A B. P A /\ P B ==> P(A ** B)) /\ (!A i. 1 <= i /\ i <= dimindex(:N) /\ row i A = vec 0 ==> P A) /\ (!A. (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> A$i$j = &0) ==> P A) /\ (!m n. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(lambda i j. (mat 1:real^N^N)$i$(swap(m,n) j))) /\ (!m n. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(lambda i j. if i = m /\ j = n \/ i = j then &1 else &0)) ==> !A. P A`, GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC INDUCT_MATRIX_ELEMENTARY THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `c = &0` THENL [FIRST_X_ASSUM(fun th -> MATCH_MP_TAC th THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`]) THEN ASM_SIMP_TAC[LAMBDA_BETA; COND_ID]; ALL_TAC] THEN SUBGOAL_THEN `(lambda i j. if i = m /\ j = n then c else if i = j then &1 else &0) = ((lambda i j. if i = j then if j = n then inv c else &1 else &0):real^N^N) ** ((lambda i j. if i = m /\ j = n \/ i = j then &1 else &0):real^N^N) ** ((lambda i j. if i = j then if j = n then c else &1 else &0):real^N^N)` SUBST1_TAC THENL [ALL_TAC; REPEAT(MATCH_MP_TAC(ASSUME `!A B:real^N^N. P A /\ P B ==> P(A ** B)`) THEN CONJ_TAC) THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(fun th -> MATCH_MP_TAC th THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`]) THEN ASM_SIMP_TAC[LAMBDA_BETA]] THEN SIMP_TAC[CART_EQ; matrix_mul; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[SUM_DELTA; IN_NUMSEG; REAL_ARITH `(if p then &1 else &0) * (if q then c else &0) = if q then if p then c else &0 else &0`] THEN REWRITE_TAC[REAL_ARITH `(if p then x else &0) * y = (if p then x * y else &0)`] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN ASM_SIMP_TAC[SUM_DELTA; IN_NUMSEG] THEN ASM_CASES_TAC `i:num = m` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `j:num = n` THEN ASM_REWRITE_TAC[REAL_MUL_LID; EQ_SYM_EQ] THEN ASM_CASES_TAC `i:num = n` THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_MUL_LID; REAL_MUL_RZERO]);; let INDUCT_LINEAR_ELEMENTARY = prove (`!P. (!f g. linear f /\ linear g /\ P f /\ P g ==> P(f o g)) /\ (!f i. linear f /\ 1 <= i /\ i <= dimindex(:N) /\ (!x. (f x)$i = &0) ==> P f) /\ (!c. P(\x. lambda i. c i * x$i)) /\ (!m n. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(\x. lambda i. x$swap(m,n) i)) /\ (!m n. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> P(\x. lambda i. if i = m then x$m + x$n else x$i)) ==> !f:real^N->real^N. linear f ==> P f`, GEN_TAC THEN MP_TAC(ISPEC `\A:real^N^N. P(\x:real^N. A ** x):bool` INDUCT_MATRIX_ELEMENTARY_ALT) THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_IMP THEN CONJ_TAC THENL [ALL_TAC; DISCH_TAC THEN X_GEN_TAC `f:real^N->real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `matrix(f:real^N->real^N)`) THEN ASM_SIMP_TAC[MATRIX_WORKS] THEN REWRITE_TAC[ETA_AX]] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`A:real^N^N`; `B:real^N^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`\x:real^N. (A:real^N^N) ** x`; `\x:real^N. (B:real^N^N) ** x`]) THEN ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR; o_DEF] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_ASSOC]; ALL_TAC] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`A:real^N^N`; `m:num`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`\x:real^N. (A:real^N^N) ** x`; `m:num`]) THEN ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN DISCH_THEN MATCH_MP_TAC THEN UNDISCH_TAC `row m (A:real^N^N) = vec 0` THEN ASM_SIMP_TAC[CART_EQ; row; LAMBDA_BETA; VEC_COMPONENT; matrix_vector_mul; REAL_MUL_LZERO; SUM_0]; ALL_TAC] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [DISCH_TAC THEN X_GEN_TAC `A:real^N^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\i. (A:real^N^N)$i$i`) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; matrix_vector_mul; FUN_EQ_THM; LAMBDA_BETA] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `i:num`] THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum(1..dimindex(:N)) (\j. if j = i then (A:real^N^N)$i$j * (x:real^N)$j else &0)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUM_DELTA; IN_NUMSEG]; ALL_TAC] THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_MUL_LZERO]; ALL_TAC] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `m:num` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; matrix_vector_mul; FUN_EQ_THM; LAMBDA_BETA; mat; IN_DIMINDEX_SWAP] THENL [ONCE_REWRITE_TAC[SWAP_GALOIS] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN SIMP_TAC[SUM_DELTA; REAL_MUL_LID; REAL_MUL_LZERO; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[swap] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; MAP_EVERY X_GEN_TAC [`x:real^N`; `i:num`] THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = m` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN ASM_SIMP_TAC[SUM_DELTA; REAL_MUL_LZERO; REAL_MUL_LID; IN_NUMSEG] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum {m,n} (\j. if n = j \/ j = m then (x:real^N)$j else &0)` THEN CONJ_TAC THENL [SIMP_TAC[SUM_CLAUSES; FINITE_RULES; IN_INSERT; NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[REAL_ADD_RID]; CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN ASM_SIMP_TAC[SUBSET; IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM; IN_NUMSEG; REAL_MUL_LZERO] THEN ASM_ARITH_TAC]]);; let LAMBDA_SWAP_GALOIS = prove (`!x:real^N y:real^N. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) ==> (x = (lambda i. y$swap(m,n) i) <=> (lambda i. x$swap(m,n) i) = y)`, SIMP_TAC[CART_EQ; LAMBDA_BETA; IN_DIMINDEX_SWAP] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `swap(m,n) (i:num)`) THEN ASM_SIMP_TAC[IN_DIMINDEX_SWAP] THEN ASM_MESON_TAC[REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM] SWAP_IDEMPOTENT]);; let LAMBDA_ADD_GALOIS = prove (`!x:real^N y:real^N. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) ==> (x = (lambda i. if i = m then y$m + y$n else y$i) <=> (lambda i. if i = m then x$m - x$n else x$i) = y)`, SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `n:num`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);; let HAS_MEASURE_SHEAR_INTERVAL = prove (`!a b:real^N m n. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ ~(m = n) /\ ~(interval[a,b] = {}) /\ &0 <= a$n ==> (IMAGE (\x. (lambda i. if i = m then x$m + x$n else x$i)) (interval[a,b]):real^N->bool) has_measure measure (interval [a,b])`, let lemma = prove (`!s t u v:real^N->bool. measurable s /\ measurable t /\ measurable u /\ negligible(s INTER t) /\ negligible(s INTER u) /\ negligible(t INTER u) /\ s UNION t UNION u = v ==> v has_measure (measure s) + (measure t) + (measure u)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_UNION] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[MEASURE_UNION; MEASURABLE_UNION] THEN ASM_SIMP_TAC[MEASURE_EQ_0; UNION_OVER_INTER; MEASURE_UNION; MEASURABLE_UNION; NEGLIGIBLE_INTER; MEASURABLE_INTER] THEN REAL_ARITH_TAC) and lemma' = prove (`!s t u a:real^N. measurable s /\ measurable t /\ s UNION (IMAGE (\x. a + x) t) = u /\ negligible(s INTER (IMAGE (\x. a + x) t)) ==> measure s + measure t = measure u`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN ASM_SIMP_TAC[MEASURE_NEGLIGIBLE_UNION; MEASURABLE_TRANSLATION_EQ; MEASURE_TRANSLATION]) in REWRITE_TAC[INTERVAL_NE_EMPTY] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `linear((\x. lambda i. if i = m then x$m + x$n else x$i):real^N->real^N)` ASSUME_TAC THENL [ASM_SIMP_TAC[linear; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; CART_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`IMAGE (\x. lambda i. if i = m then x$m + x$n else x$i) (interval[a:real^N,b]):real^N->bool`; `interval[a,(lambda i. if i = m then (b:real^N)$m + b$n else b$i)] INTER {x:real^N | (basis m - basis n) dot x <= a$m}`; `interval[a,(lambda i. if i = m then b$m + b$n else b$i)] INTER {x:real^N | (basis m - basis n) dot x >= (b:real^N)$m}`; `interval[a:real^N, (lambda i. if i = m then (b:real^N)$m + b$n else b$i)]`] lemma) THEN ANTS_TAC THENL [ASM_SIMP_TAC[CONVEX_LINEAR_IMAGE; CONVEX_INTERVAL; CONVEX_HALFSPACE_LE; CONVEX_HALFSPACE_GE; CONVEX_INTER; MEASURABLE_CONVEX; BOUNDED_INTER; BOUNDED_LINEAR_IMAGE; BOUNDED_INTERVAL] THEN REWRITE_TAC[INTER] THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_INTER; IN_IMAGE] THEN ASM_SIMP_TAC[LAMBDA_ADD_GALOIS; UNWIND_THM1] THEN ASM_SIMP_TAC[IN_INTERVAL; IN_ELIM_THM; LAMBDA_BETA; DOT_BASIS; DOT_LSUB] THEN ONCE_REWRITE_TAC[MESON[] `(!i:num. P i) <=> P m /\ (!i. ~(i = m) ==> P i)`] THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[TAUT `(p /\ x) /\ (q /\ x) /\ r <=> x /\ p /\ q /\ r`; TAUT `(p /\ x) /\ q /\ (r /\ x) <=> x /\ p /\ q /\ r`; TAUT `((p /\ x) /\ q) /\ (r /\ x) /\ s <=> x /\ p /\ q /\ r /\ s`; TAUT `(a /\ x \/ (b /\ x) /\ c \/ (d /\ x) /\ e <=> f /\ x) <=> x ==> (a \/ b /\ c \/ d /\ e <=> f)`] THEN ONCE_REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN REPEAT CONJ_TAC THEN MATCH_MP_TAC NEGLIGIBLE_INTER THEN DISJ2_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THENL [EXISTS_TAC `{x:real^N | (basis m - basis n) dot x = (a:real^N)$m}`; EXISTS_TAC `{x:real^N | (basis m - basis n) dot x = (b:real^N)$m}`; EXISTS_TAC `{x:real^N | (basis m - basis n) dot x = (b:real^N)$m}`] THEN (CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_HYPERPLANE THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN ASM_MESON_TAC[BASIS_INJ]; ASM_SIMP_TAC[DOT_LSUB; DOT_BASIS; SUBSET; IN_ELIM_THM; NOT_IN_EMPTY] THEN FIRST_X_ASSUM(MP_TAC o SPEC `m:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]); ALL_TAC] THEN ASM_SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_LINEAR_IMAGE_INTERVAL; MEASURABLE_INTERVAL] THEN MP_TAC(ISPECL [`interval[a,(lambda i. if i = m then (b:real^N)$m + b$n else b$i)] INTER {x:real^N | (basis m - basis n) dot x <= a$m}`; `interval[a,(lambda i. if i = m then b$m + b$n else b$i)] INTER {x:real^N | (basis m - basis n) dot x >= (b:real^N)$m}`; `interval[a:real^N, (lambda i. if i = m then (a:real^N)$m + b$n else (b:real^N)$i)]`; `(lambda i. if i = m then (a:real^N)$m - (b:real^N)$m else &0):real^N`] lemma') THEN ANTS_TAC THENL [ASM_SIMP_TAC[CONVEX_LINEAR_IMAGE; CONVEX_INTERVAL; CONVEX_HALFSPACE_LE; CONVEX_HALFSPACE_GE; CONVEX_INTER; MEASURABLE_CONVEX; BOUNDED_INTER; BOUNDED_LINEAR_IMAGE; BOUNDED_INTERVAL] THEN REWRITE_TAC[INTER] THEN REWRITE_TAC[EXTENSION; IN_UNION; IN_INTER; IN_IMAGE] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `x:real^N = (lambda i. p i) + y <=> x - (lambda i. p i) = y`] THEN ASM_SIMP_TAC[IN_INTERVAL; IN_ELIM_THM; LAMBDA_BETA; DOT_BASIS; DOT_LSUB; UNWIND_THM1; VECTOR_SUB_COMPONENT] THEN ONCE_REWRITE_TAC[MESON[] `(!i:num. P i) <=> P m /\ (!i. ~(i = m) ==> P i)`] THEN ASM_SIMP_TAC[REAL_SUB_RZERO] THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN FIRST_ASSUM(MP_TAC o SPEC `n:num`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `m:num`) THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `!i. ~(i = m) ==> 1 <= i /\ i <= dimindex (:N) ==> (a:real^N)$i <= (x:real^N)$i /\ x$i <= (b:real^N)$i` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ONCE_REWRITE_TAC[TAUT `((a /\ b) /\ c) /\ (d /\ e) /\ f <=> (b /\ e) /\ a /\ c /\ d /\ f`] THEN ONCE_REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN MATCH_MP_TAC NEGLIGIBLE_INTER THEN DISJ2_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | (basis m - basis n) dot x = (a:real^N)$m}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_HYPERPLANE THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN ASM_MESON_TAC[BASIS_INJ]; ASM_SIMP_TAC[DOT_LSUB; DOT_BASIS; SUBSET; IN_ELIM_THM; NOT_IN_EMPTY] THEN FIRST_ASSUM(MP_TAC o SPEC `n:num`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `m:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]; ALL_TAC] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(REAL_ARITH `a:real = b + c ==> a = x + b ==> x = c`) THEN ASM_SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES; LAMBDA_BETA] THEN REPEAT(COND_CASES_TAC THENL [ALL_TAC; FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN MATCH_MP_TAC(TAUT `p ==> ~p ==> q`) THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN FIRST_ASSUM(MP_TAC o SPEC `n:num`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `m:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]) THEN SUBGOAL_THEN `1..dimindex(:N) = m INSERT ((1..dimindex(:N)) DELETE m)` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_INSERT; IN_DELETE; IN_NUMSEG] THEN ASM_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[PRODUCT_CLAUSES; FINITE_DELETE; FINITE_NUMSEG] THEN ASM_SIMP_TAC[IN_DELETE] THEN MATCH_MP_TAC(REAL_RING `s1:real = s3 /\ s2 = s3 ==> ((bm + bn) - am) * s1 = ((am + bn) - am) * s2 + (bm - am) * s3`) THEN CONJ_TAC THEN MATCH_MP_TAC PRODUCT_EQ THEN SIMP_TAC[IN_DELETE] THEN REAL_ARITH_TAC);; let HAS_MEASURE_LINEAR_IMAGE = prove (`!f:real^N->real^N s. linear f /\ measurable s ==> (IMAGE f s) has_measure (abs(det(matrix f)) * measure s)`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC INDUCT_LINEAR_ELEMENTARY THEN REPEAT CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `g:real^N->real^N`] THEN REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `IMAGE (g:real^N->real^N) s`) (MP_TAC o SPEC `s:real^N->bool`)) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [HAS_MEASURE_MEASURABLE_MEASURE] THEN STRIP_TAC THEN ASM_SIMP_TAC[MATRIX_COMPOSE; DET_MUL; REAL_ABS_MUL] THEN REWRITE_TAC[IMAGE_o; GSYM REAL_MUL_ASSOC]; MAP_EVERY X_GEN_TAC [`f:real^N->real^N`; `m:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `~(!x y. (f:real^N->real^N) x = f y ==> x = y)` ASSUME_TAC THENL [ASM_SIMP_TAC[LINEAR_SINGULAR_INTO_HYPERPLANE] THEN EXISTS_TAC `basis m:real^N` THEN ASM_SIMP_TAC[BASIS_NONZERO; DOT_BASIS]; ALL_TAC] THEN MP_TAC(ISPEC `matrix f:real^N^N` INVERTIBLE_DET_NZ) THEN ASM_SIMP_TAC[INVERTIBLE_LEFT_INVERSE; MATRIX_LEFT_INVERTIBLE_INJECTIVE; MATRIX_WORKS; REAL_ABS_NUM; REAL_MUL_LZERO] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[HAS_MEASURE_0] THEN ASM_SIMP_TAC[NEGLIGIBLE_LINEAR_SINGULAR_IMAGE]; MAP_EVERY X_GEN_TAC [`c:num->real`; `s:real^N->bool`] THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o REWRITE_RULE[HAS_MEASURE_MEASURE]) THEN FIRST_ASSUM(MP_TAC o SPEC `c:num->real` o MATCH_MP HAS_MEASURE_STRETCH) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[matrix; LAMBDA_BETA] THEN W(MP_TAC o PART_MATCH (lhs o rand) DET_DIAGONAL o rand o snd) THEN REWRITE_TAC[diagonal_matrix] THEN SIMP_TAC[LAMBDA_BETA; BASIS_COMPONENT; REAL_MUL_RZERO] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC PRODUCT_EQ_NUMSEG THEN REWRITE_TAC[REAL_MUL_RID]; MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_LINEAR_SUFFICIENT THEN ASM_SIMP_TAC[linear; LAMBDA_BETA; IN_DIMINDEX_SWAP; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; CART_EQ] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN SUBGOAL_THEN `matrix (\x:real^N. lambda i. x$swap (m,n) i):real^N^N = transp(lambda i j. (mat 1:real^N^N)$i$swap (m,n) j)` SUBST1_TAC THENL [ASM_SIMP_TAC[MATRIX_EQ; LAMBDA_BETA; IN_DIMINDEX_SWAP; matrix_vector_mul; CART_EQ; matrix; mat; basis; COND_COMPONENT; transp] THEN REWRITE_TAC[EQ_SYM_EQ]; ALL_TAC] THEN REWRITE_TAC[DET_TRANSP] THEN W(MP_TAC o PART_MATCH (lhs o rand) DET_PERMUTE_COLUMNS o rand o lhand o rand o snd) THEN ASM_SIMP_TAC[PERMUTES_SWAP; IN_NUMSEG; ETA_AX] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[DET_I; REAL_ABS_SIGN; REAL_MUL_RID; REAL_MUL_LID] THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THENL [ASM_SIMP_TAC[IMAGE_CLAUSES; HAS_MEASURE_EMPTY; MEASURE_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `~(IMAGE (\x:real^N. lambda i. x$swap (m,n) i) (interval[a,b]):real^N->bool = {})` MP_TAC THENL [ASM_REWRITE_TAC[IMAGE_EQ_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE (\x:real^N. lambda i. x$swap (m,n) i) (interval[a,b]):real^N->bool = interval[(lambda i. a$swap (m,n) i), (lambda i. b$swap (m,n) i)]` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_INTERVAL; IN_IMAGE] THEN ASM_SIMP_TAC[LAMBDA_SWAP_GALOIS; UNWIND_THM1] THEN SIMP_TAC[LAMBDA_BETA] THEN GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `swap(m,n) (i:num)`) THEN ASM_SIMP_TAC[IN_DIMINDEX_SWAP] THEN ASM_MESON_TAC[REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM] SWAP_IDEMPOTENT]; ALL_TAC] THEN REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_INTERVAL] THEN REWRITE_TAC[MEASURE_INTERVAL] THEN ASM_SIMP_TAC[CONTENT_CLOSED_INTERVAL; GSYM INTERVAL_NE_EMPTY] THEN DISCH_THEN(K ALL_TAC) THEN SIMP_TAC[LAMBDA_BETA] THEN ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; IN_DIMINDEX_SWAP] THEN MP_TAC(ISPECL [`\i. (b - a:real^N)$i`; `swap(m:num,n)`; `1..dimindex(:N)`] (GSYM PRODUCT_PERMUTE)) THEN REWRITE_TAC[o_DEF] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[PERMUTES_SWAP; IN_NUMSEG]; MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_LINEAR_SUFFICIENT THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[linear; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; CART_EQ] THEN REAL_ARITH_TAC; DISCH_TAC] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN SUBGOAL_THEN `det(matrix(\x. lambda i. if i = m then (x:real^N)$m + x$n else x$i):real^N^N) = &1` SUBST1_TAC THENL [ASM_SIMP_TAC[matrix; basis; COND_COMPONENT; LAMBDA_BETA] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE `~(m:num = n) ==> m < n \/ n < m`)) THENL [W(MP_TAC o PART_MATCH (lhs o rand) DET_UPPERTRIANGULAR o lhs o snd); W(MP_TAC o PART_MATCH (lhs o rand) DET_LOWERTRIANGULAR o lhs o snd)] THEN ASM_SIMP_TAC[LAMBDA_BETA; BASIS_COMPONENT; COND_COMPONENT; matrix; REAL_ADD_RID; COND_ID; PRODUCT_CONST_NUMSEG; REAL_POW_ONE] THEN DISCH_THEN MATCH_MP_TAC THEN REPEAT GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[REAL_ABS_NUM; REAL_MUL_LID] THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THENL [ASM_SIMP_TAC[IMAGE_CLAUSES; HAS_MEASURE_EMPTY; MEASURE_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `IMAGE (\x. lambda i. if i = m then x$m + x$n else x$i) (interval [a,b]) = IMAGE (\x:real^N. (lambda i. if i = m \/ i = n then a$n else &0) + x) (IMAGE (\x:real^N. lambda i. if i = m then x$m + x$n else x$i) (IMAGE (\x. (lambda i. if i = n then --(a$n) else &0) + x) (interval[a,b])))` SUBST1_TAC THENL [REWRITE_TAC[GSYM IMAGE_o] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[FUN_EQ_THM; o_THM; VECTOR_ADD_COMPONENT; LAMBDA_BETA; CART_EQ] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `i:num`] THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = m` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `i:num = n` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC HAS_MEASURE_TRANSLATION THEN SUBGOAL_THEN `measure(interval[a,b]) = measure(IMAGE (\x:real^N. (lambda i. if i = n then --(a$n) else &0) + x) (interval[a,b]):real^N->bool)` SUBST1_TAC THENL [REWRITE_TAC[MEASURE_TRANSLATION]; ALL_TAC] THEN SUBGOAL_THEN `~(IMAGE (\x:real^N. (lambda i. if i = n then --(a$n) else &0) + x) (interval[a,b]):real^N->bool = {})` MP_TAC THENL [ASM_SIMP_TAC[IMAGE_EQ_EMPTY]; ALL_TAC] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `c + x:real^N = &1 % x + c`] THEN ASM_REWRITE_TAC[IMAGE_AFFINITY_INTERVAL; REAL_POS] THEN DISCH_TAC THEN MATCH_MP_TAC HAS_MEASURE_SHEAR_INTERVAL THEN ASM_SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN REAL_ARITH_TAC]);; let MEASURABLE_LINEAR_IMAGE = prove (`!f:real^N->real^N s. linear f /\ measurable s ==> measurable(IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_LINEAR_IMAGE) THEN SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE]);; let MEASURE_LINEAR_IMAGE = prove (`!f:real^N->real^N s. linear f /\ measurable s ==> measure(IMAGE f s) = abs(det(matrix f)) * measure s`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_LINEAR_IMAGE) THEN SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE]);; let HAS_MEASURE_LINEAR_IMAGE_ALT = prove (`!f:real^N->real^N s m. linear f /\ s has_measure m ==> (IMAGE f s) has_measure (abs(det(matrix f)) * m)`, MESON_TAC[MEASURE_UNIQUE; measurable; HAS_MEASURE_LINEAR_IMAGE]);; let HAS_MEASURE_LINEAR_IMAGE_SAME = prove (`!f s. linear f /\ measurable s /\ abs(det(matrix f)) = &1 ==> (IMAGE f s) has_measure (measure s)`, MESON_TAC[HAS_MEASURE_LINEAR_IMAGE; REAL_MUL_LID]);; let MEASURE_LINEAR_IMAGE_SAME = prove (`!f:real^N->real^N s. linear f /\ measurable s /\ abs(det(matrix f)) = &1 ==> measure(IMAGE f s) = measure s`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_LINEAR_IMAGE_SAME) THEN SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE]);; let MEASURABLE_LINEAR_IMAGE_EQ = prove (`!f:real^N->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (measurable (IMAGE f s) <=> measurable s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE MEASURABLE_LINEAR_IMAGE));; add_linear_invariants [MEASURABLE_LINEAR_IMAGE_EQ];; let NEGLIGIBLE_LINEAR_IMAGE = prove (`!f:real^N->real^N s. linear f /\ negligible s ==> negligible(IMAGE f s)`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM HAS_MEASURE_0] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HAS_MEASURE_LINEAR_IMAGE_ALT) THEN REWRITE_TAC[REAL_MUL_RZERO]);; let NEGLIGIBLE_LINEAR_IMAGE_EQ = prove (`!f:real^N->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (negligible (IMAGE f s) <=> negligible s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE NEGLIGIBLE_LINEAR_IMAGE));; add_linear_invariants [NEGLIGIBLE_LINEAR_IMAGE_EQ];; let HAS_MEASURE_ORTHOGONAL_IMAGE = prove (`!f:real^N->real^N s m. orthogonal_transformation f /\ s has_measure m ==> (IMAGE f s) has_measure m`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_LINEAR_IMAGE_ALT) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC(REAL_RING `x = &1 ==> x * m = m`) THEN REWRITE_TAC[REAL_ARITH `abs x = &1 <=> x = &1 \/ x = -- &1`] THEN MATCH_MP_TAC DET_ORTHOGONAL_MATRIX THEN ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX]);; let HAS_MEASURE_ORTHOGONAL_IMAGE_EQ = prove (`!f:real^N->real^N s m. orthogonal_transformation f ==> ((IMAGE f s) has_measure m <=> s has_measure m)`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[HAS_MEASURE_ORTHOGONAL_IMAGE] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_INVERSE_o) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_ORTHOGONAL_IMAGE) THEN ASM_SIMP_TAC[GSYM IMAGE_o; IMAGE_I]);; add_linear_invariants [REWRITE_RULE[ORTHOGONAL_TRANSFORMATION] HAS_MEASURE_ORTHOGONAL_IMAGE_EQ];; let MEASURE_ORTHOGONAL_IMAGE_EQ = prove (`!f:real^N->real^N s. orthogonal_transformation f ==> measure(IMAGE f s) = measure s`, SIMP_TAC[measure; HAS_MEASURE_ORTHOGONAL_IMAGE_EQ]);; add_linear_invariants [REWRITE_RULE[ORTHOGONAL_TRANSFORMATION] MEASURE_ORTHOGONAL_IMAGE_EQ];; let HAS_MEASURE_ISOMETRY = prove (`!f:real^M->real^N s m. dimindex(:M) = dimindex(:N) /\ linear f /\ (!x. norm(f x) = norm x) ==> (IMAGE f s has_measure m <=> s has_measure m)`, REPEAT STRIP_TAC THEN TRANS_TAC EQ_TRANS `IMAGE ((\x. lambda i. x$i):real^N->real^M) (IMAGE (f:real^M->real^N) s) has_measure m` THEN CONJ_TAC THENL [SPEC_TAC(`IMAGE (f:real^M->real^N) s`,`s:real^N->bool`) THEN GEN_TAC THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[has_measure] THEN W(MP_TAC o PART_MATCH (lhand o rand) HAS_INTEGRAL_TWIZZLE_EQ o lhand o snd) THEN REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ONCE_ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM I_DEF; PERMUTES_I]; REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC HAS_MEASURE_ORTHOGONAL_IMAGE_EQ THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; o_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC LINEAR_COMPOSE THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT]; X_GEN_TAC `x:real^M` THEN TRANS_TAC EQ_TRANS `norm((f:real^M->real^N) x)` THEN CONJ_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[]] THEN SIMP_TAC[NORM_EQ; dot; LAMBDA_BETA] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN SIMP_TAC[LAMBDA_BETA]]]);; let MEASURABLE_LINEAR_IMAGE_EQ_GEN = prove (`!f:real^M->real^N s. dimindex(:M) = dimindex(:N) /\ linear f /\ (!x y. f x = f y ==> x = y) ==> (measurable(IMAGE f s) <=> measurable s)`, REPEAT STRIP_TAC THEN TRANS_TAC EQ_TRANS `measurable(IMAGE ((\x. lambda i. x$i):real^N->real^M) (IMAGE (f:real^M->real^N) s))` THEN CONJ_TAC THENL [CONV_TAC SYM_CONV THEN REWRITE_TAC[measurable] THEN AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_MEASURE_ISOMETRY THEN ONCE_ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT]; SIMP_TAC[NORM_EQ; dot; LAMBDA_BETA] THEN ASM_MESON_TAC[]]; REWRITE_TAC[GSYM IMAGE_o] THEN MATCH_MP_TAC MEASURABLE_LINEAR_IMAGE_EQ THEN CONJ_TAC THENL [MATCH_MP_TAC LINEAR_COMPOSE THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT]; SIMP_TAC[CART_EQ; LAMBDA_BETA; o_DEF] THEN ASM_MESON_TAC[CART_EQ]]]);; let MEASURE_ISOMETRY = prove (`!f:real^M->real^N s. dimindex(:M) = dimindex(:N) /\ linear f /\ (!x. norm(f x) = norm x) ==> measure (IMAGE f s) = measure s`, REPEAT GEN_TAC THEN REWRITE_TAC[measure] THEN DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP HAS_MEASURE_ISOMETRY th]));; let MEASURABLE_CONVEX_EQ = prove (`!s:real^N->bool. convex s ==> (measurable s <=> bounded s \/ interior s = {})`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `interior s:real^N->bool = {}` THENL [ASM_MESON_TAC[NEGLIGIBLE_CONVEX_INTERIOR; NEGLIGIBLE_IMP_MEASURABLE]; EQ_TAC THEN ASM_SIMP_TAC[MEASURABLE_CONVEX]] THEN DISCH_TAC THEN GEN_REWRITE_TAC I [TAUT `p <=> ~ ~ p`] THEN DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAYS) THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM; RELATIVE_INTERIOR_NONEMPTY_INTERIOR] THEN DISCH_THEN(X_CHOOSE_THEN `l:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REPEAT(POP_ASSUM MP_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `l:real^N` THEN X_GEN_TAC `l:real` THEN ASM_CASES_TAC `l = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; DE_MORGAN_THM] THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `z:real^N`) THEN MP_TAC(ISPEC `interior s:real^N->bool` OPEN_CONTAINS_INTERVAL) THEN REWRITE_TAC[OPEN_INTERIOR] THEN DISCH_THEN(MP_TAC o SPEC `z:real^N`) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN STRIP_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN GEN_GEOM_ORIGIN_TAC `u:real^N` ["l"] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `!t. &0 <= t ==> measure(interval[vec 0:real^N, (lambda i. if i = 1 then t else (v:real^N)$i)]) <= measure(s:real^N->bool)` MP_TAC THENL [X_GEN_TAC `t:real` THEN STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[MEASURABLE_INTERVAL] THEN TRANS_TAC SUBSET_TRANS `interior(s:real^N->bool)` THEN REWRITE_TAC[INTERIOR_SUBSET] THEN SIMP_TAC[SUBSET; IN_INTERVAL; LAMBDA_BETA; VEC_COMPONENT] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(lambda i. if i = 1 then &0 else (x:real^N)$i):real^N`; `(x:real^N)$1 / l:real`]) THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE; DIMINDEX_GE_1; LE_REFL] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN SIMP_TAC[IN_INTERVAL; LAMBDA_BETA; VEC_COMPONENT] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL]) THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[VEC_COMPONENT] THEN REAL_ARITH_TAC; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT] THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT; REAL_FIELD `&0 < l ==> x / l * l * y = x * y`] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]; DISCH_THEN(MP_TAC o SPEC `v$1 * (measure(s:real^N->bool) + &1) / measure(interval[vec 0:real^N,v])`) THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N) ==> &0 < (v:real^N)$i` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL]) THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[VEC_COMPONENT] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN SIMP_TAC[LAMBDA_BETA; VEC_COMPONENT] THEN REWRITE_TAC[MESON[] `(&0 <= if p then x else y) <=> if p then &0 <= x else &0 <= y`] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [REAL_LE_DIV; REAL_LE_ADD; REAL_POS; MEASURE_POS_LE; REAL_LE_MUL; REAL_LE_MUL; MEASURABLE_INTERVAL; DIMINDEX_GE_1; LE_REFL; LAMBDA_BETA; VEC_COMPONENT; REAL_LT_IMP_LE; PRODUCT_POS_LE_NUMSEG; REAL_SUB_RZERO] THEN SIMP_TAC[PRODUCT_CLAUSES_LEFT; DIMINDEX_GE_1; COND_ID] THEN SIMP_TAC[ARITH; ARITH_RULE `2 <= n ==> ~(n = 1)`] THEN MATCH_MP_TAC(REAL_ARITH `x = y + &1 ==> ~(x <= y)`) THEN MATCH_MP_TAC(REAL_FIELD `&0 < v /\ &0 < p ==> (v * m / (v * p)) * p = m`) THEN ASM_SIMP_TAC[DIMINDEX_GE_1; LE_REFL] THEN MATCH_MP_TAC PRODUCT_POS_LT_NUMSEG THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Measure of a standard simplex. *) (* ------------------------------------------------------------------------- *) let CONGRUENT_IMAGE_STD_SIMPLEX = prove (`!p. p permutes 1..dimindex(:N) ==> {x:real^N | &0 <= x$(p 1) /\ x$(p(dimindex(:N))) <= &1 /\ (!i. 1 <= i /\ i < dimindex(:N) ==> x$(p i) <= x$(p(i + 1)))} = IMAGE (\x:real^N. lambda i. sum(1..inverse p(i)) (\j. x$j)) {x | (!i. 1 <= i /\ i <= dimindex (:N) ==> &0 <= x$i) /\ sum (1..dimindex (:N)) (\i. x$i) <= &1}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN ASM_SIMP_TAC[IN_ELIM_THM; LAMBDA_BETA; LAMBDA_BETA_PERM; LE_REFL; ARITH_RULE `i < n ==> i <= n /\ i + 1 <= n`; ARITH_RULE `1 <= n + 1`; DIMINDEX_GE_1] THEN STRIP_TAC THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP PERMUTES_INVERSES th]) THEN ASM_SIMP_TAC[SUM_SING_NUMSEG; DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[GSYM ADD1; SUM_CLAUSES_NUMSEG; ARITH_RULE `1 <= SUC n`] THEN ASM_SIMP_TAC[REAL_LE_ADDR] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC] THEN REWRITE_TAC[SUBSET; IN_IMAGE; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN EXISTS_TAC `(lambda i. if i = 1 then x$(p 1) else (x:real^N)$p(i) - x$p(i - 1)):real^N` THEN ASM_SIMP_TAC[IN_ELIM_THM; LAMBDA_BETA; LAMBDA_BETA_PERM; LE_REFL; ARITH_RULE `i < n ==> i <= n /\ i + 1 <= n`; ARITH_RULE `1 <= n + 1`; DIMINDEX_GE_1; CART_EQ] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `i:num` THEN STRIP_TAC THEN SUBGOAL_THEN `1 <= inverse (p:num->num) i /\ !x. x <= inverse p i ==> x <= dimindex(:N)` ASSUME_TAC THENL [ASM_MESON_TAC[PERMUTES_INVERSE; IN_NUMSEG; LE_TRANS; PERMUTES_IN_IMAGE]; ASM_SIMP_TAC[LAMBDA_BETA] THEN ASM_SIMP_TAC[SUM_CLAUSES_LEFT; ARITH]] THEN SIMP_TAC[ARITH_RULE `2 <= n ==> ~(n = 1)`] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o BINDER_CONV) [GSYM REAL_MUL_LID] THEN ONCE_REWRITE_TAC[SUM_PARTIAL_PRE] THEN REWRITE_TAC[REAL_SUB_REFL; REAL_MUL_RZERO; SUM_0; COND_ID] THEN REWRITE_TAC[REAL_MUL_LID; ARITH; REAL_SUB_RZERO] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE `1 <= p ==> p = 1 \/ 2 <= p`) o CONJUNCT1) THEN ASM_SIMP_TAC[ARITH] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP PERMUTES_INVERSES th]) THEN REWRITE_TAC[REAL_ADD_RID] THEN TRY REAL_ARITH_TAC THEN ASM_MESON_TAC[PERMUTES_INVERSE_EQ; PERMUTES_INVERSE]; X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_SUB_LE] THEN FIRST_X_ASSUM(MP_TAC o SPEC `i - 1`) THEN ASM_SIMP_TAC[SUB_ADD] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_ARITH_TAC; SIMP_TAC[SUM_CLAUSES_LEFT; DIMINDEX_GE_1; ARITH; ARITH_RULE `2 <= n ==> ~(n = 1)`] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o RAND_CONV o BINDER_CONV) [GSYM REAL_MUL_LID] THEN ONCE_REWRITE_TAC[SUM_PARTIAL_PRE] THEN REWRITE_TAC[REAL_SUB_REFL; REAL_MUL_RZERO; SUM_0; COND_ID] THEN REWRITE_TAC[REAL_MUL_LID; ARITH; REAL_SUB_RZERO] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_ADD_RID] THEN ASM_REWRITE_TAC[REAL_ARITH `x + y - x:real = y`] THEN ASM_MESON_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n /\ ~(2 <= n) ==> n = 1`]]);; let HAS_MEASURE_IMAGE_STD_SIMPLEX = prove (`!p. p permutes 1..dimindex(:N) ==> {x:real^N | &0 <= x$(p 1) /\ x$(p(dimindex(:N))) <= &1 /\ (!i. 1 <= i /\ i < dimindex(:N) ==> x$(p i) <= x$(p(i + 1)))} has_measure (measure (convex hull (vec 0 INSERT {basis i:real^N | 1 <= i /\ i <= dimindex(:N)})))`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONGRUENT_IMAGE_STD_SIMPLEX] THEN ASM_SIMP_TAC[GSYM STD_SIMPLEX] THEN MATCH_MP_TAC HAS_MEASURE_LINEAR_IMAGE_SAME THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[linear; CART_EQ] THEN ASM_SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; GSYM SUM_ADD_NUMSEG; GSYM SUM_LMUL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN REPEAT STRIP_TAC THEN REWRITE_TAC[]; MATCH_MP_TAC MEASURABLE_CONVEX THEN REWRITE_TAC[CONVEX_CONVEX_HULL] THEN MATCH_MP_TAC BOUNDED_CONVEX_HULL THEN REWRITE_TAC[BOUNDED_INSERT] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN MATCH_MP_TAC FINITE_IMAGE THEN REWRITE_TAC[GSYM numseg; FINITE_NUMSEG]; MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `abs(det ((lambda i. ((lambda i j. if j <= i then &1 else &0):real^N^N) $inverse p i) :real^N^N))` THEN CONJ_TAC THENL [AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[CART_EQ] THEN ASM_SIMP_TAC[matrix; LAMBDA_BETA; BASIS_COMPONENT; COND_COMPONENT; LAMBDA_BETA_PERM; PERMUTES_INVERSE] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum (1..inverse (p:num->num) i) (\k. if k = j then &1 else &0)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_EQ THEN ASM_SIMP_TAC[IN_NUMSEG; PERMUTES_IN_IMAGE; basis] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LAMBDA_BETA THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG; LE_TRANS; PERMUTES_INVERSE]; ASM_SIMP_TAC[SUM_DELTA; IN_NUMSEG]]; ALL_TAC] THEN ASM_SIMP_TAC[PERMUTES_INVERSE; DET_PERMUTE_ROWS; ETA_AX] THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_SIGN; REAL_MUL_LID] THEN MATCH_MP_TAC(REAL_ARITH `x = &1 ==> abs x = &1`) THEN ASM_SIMP_TAC[DET_LOWERTRIANGULAR; GSYM NOT_LT; LAMBDA_BETA] THEN REWRITE_TAC[LT_REFL; PRODUCT_CONST_NUMSEG; REAL_POW_ONE]]);; let HAS_MEASURE_STD_SIMPLEX = prove (`(convex hull (vec 0:real^N INSERT {basis i | 1 <= i /\ i <= dimindex(:N)})) has_measure inv(&(FACT(dimindex(:N))))`, let lemma = prove (`!f:num->real. (!i. 1 <= i /\ i < n ==> f i <= f(i + 1)) <=> (!i j. 1 <= i /\ i <= j /\ j <= n ==> f i <= f j)`, GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [GEN_TAC THEN INDUCT_TAC THEN SIMP_TAC[LE; REAL_LE_REFL] THEN STRIP_TAC THEN ASM_SIMP_TAC[REAL_LE_REFL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(f:num->real) j` THEN ASM_SIMP_TAC[ARITH_RULE `SUC x <= y ==> x <= y`] THEN REWRITE_TAC[ADD1] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]) in MP_TAC(ISPECL [`\p. {x:real^N | &0 <= x$(p 1) /\ x$(p(dimindex(:N))) <= &1 /\ (!i. 1 <= i /\ i < dimindex(:N) ==> x$(p i) <= x$(p(i + 1)))}`; `{p | p permutes 1..dimindex(:N)}`] HAS_MEASURE_NEGLIGIBLE_UNIONS_IMAGE) THEN ASM_SIMP_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE] HAS_MEASURE_IMAGE_STD_SIMPLEX; IN_ELIM_THM] THEN ASM_SIMP_TAC[SUM_CONST; FINITE_PERMUTATIONS; FINITE_NUMSEG; CARD_PERMUTATIONS; CARD_NUMSEG_1] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`p:num->num`; `q:num->num`] THEN STRIP_TAC THEN SUBGOAL_THEN `?i. i IN 1..dimindex(:N) /\ ~(p i:num = q i)` MP_TAC THENL [ASM_MESON_TAC[permutes; FUN_EQ_THM]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN REWRITE_TAC[TAUT `a ==> ~(b /\ ~c) <=> a /\ b ==> c`] THEN REWRITE_TAC[IN_NUMSEG] THEN DISCH_THEN(X_CHOOSE_THEN `k:num` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | (basis(p(k:num)) - basis(q k)) dot x = &0}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_HYPERPLANE THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN MATCH_MP_TAC BASIS_NE THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]; ALL_TAC] THEN REWRITE_TAC[SUBSET; IN_INTER; IN_ELIM_THM; DOT_LSUB; VECTOR_SUB_EQ] THEN ASM_SIMP_TAC[DOT_BASIS; GSYM IN_NUMSEG; PERMUTES_IN_IMAGE] THEN SUBGOAL_THEN `?l. (q:num->num) l = p(k:num)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[permutes]; ALL_TAC] THEN SUBGOAL_THEN `1 <= l /\ l <= dimindex(:N)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]; ALL_TAC] THEN SUBGOAL_THEN `k:num < l` ASSUME_TAC THENL [REWRITE_TAC[GSYM NOT_LE] THEN REWRITE_TAC[LE_LT] THEN ASM_MESON_TAC[PERMUTES_INJECTIVE; IN_NUMSEG]; ALL_TAC] THEN SUBGOAL_THEN `?m. (p:num->num) m = q(k:num)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[permutes]; ALL_TAC] THEN SUBGOAL_THEN `1 <= m /\ m <= dimindex(:N)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]; ALL_TAC] THEN SUBGOAL_THEN `k:num < m` ASSUME_TAC THENL [REWRITE_TAC[GSYM NOT_LE] THEN REWRITE_TAC[LE_LT] THEN ASM_MESON_TAC[PERMUTES_INJECTIVE; IN_NUMSEG]; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[lemma] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`k:num`; `l:num`]) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`k:num`; `m:num`]) THEN ASM_SIMP_TAC[LT_IMP_LE; IMP_IMP; REAL_LE_ANTISYM; REAL_SUB_0] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG; DOT_BASIS]; ALL_TAC] THEN REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN DISCH_THEN(ASSUME_TAC o CONJUNCT2) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_CONVEX THEN REWRITE_TAC[CONVEX_CONVEX_HULL] THEN MATCH_MP_TAC BOUNDED_CONVEX_HULL THEN REWRITE_TAC[BOUNDED_INSERT] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN MATCH_MP_TAC FINITE_IMAGE THEN REWRITE_TAC[GSYM numseg; FINITE_NUMSEG]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_FIELD `~(y = &0) ==> (x = inv y <=> y * x = &1)`; REAL_OF_NUM_EQ; FACT_NZ] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `measure(interval[vec 0:real^N,vec 1])` THEN CONJ_TAC THENL [AP_TERM_TAC; REWRITE_TAC[MEASURE_INTERVAL; CONTENT_UNIT]] THEN REWRITE_TAC[lemma] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM; IN_ELIM_THM] THEN SIMP_TAC[IMP_IMP; IN_INTERVAL; LAMBDA_BETA; VEC_COMPONENT] THEN X_GEN_TAC `p:num->num` THEN STRIP_TAC THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN X_GEN_TAC `i:num` THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THENL [EXISTS_TAC `(x:real^N)$(p 1)`; EXISTS_TAC `(x:real^N)$(p(dimindex(:N)))`] THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o SPEC `i:num` o MATCH_MP PERMUTES_SURJECTIVE) THEN ASM_MESON_TAC[LE_REFL; PERMUTES_IN_IMAGE; IN_NUMSEG]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `s SUBSET UNIONS(IMAGE f t) <=> !x. x IN s ==> ?y. y IN t /\ x IN f y`] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTERVAL; IN_ELIM_THM] THEN SIMP_TAC[VEC_COMPONENT] THEN DISCH_TAC THEN MP_TAC(ISPEC `\i j. ~((x:real^N)$j <= x$i)` TOPOLOGICAL_SORT) THEN REWRITE_TAC[REAL_NOT_LE; REAL_NOT_LT] THEN ANTS_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPECL [`dimindex(:N)`; `1..dimindex(:N)`]) THEN REWRITE_TAC[HAS_SIZE_NUMSEG_1; EXTENSION; IN_IMAGE; IN_NUMSEG] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->num` (CONJUNCTS_THEN2 (ASSUME_TAC o GSYM) ASSUME_TAC)) THEN EXISTS_TAC `\i. if i IN 1..dimindex(:N) then f(i) else i` THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[ARITH_RULE `1 <= i /\ i <= j /\ j <= n <=> 1 <= i /\ 1 <= j /\ i <= n /\ j <= n /\ i <= j`] THEN ASM_SIMP_TAC[IN_NUMSEG; LE_REFL; DIMINDEX_GE_1] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[LE_REFL; DIMINDEX_GE_1; LE_LT; REAL_LE_LT]] THEN SIMP_TAC[PERMUTES_FINITE_SURJECTIVE; FINITE_NUMSEG] THEN SIMP_TAC[IN_NUMSEG] THEN ASM_MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Hence the measure of a general simplex. *) (* ------------------------------------------------------------------------- *) let HAS_MEASURE_SIMPLEX_0 = prove (`!l:(real^N)list. LENGTH l = dimindex(:N) ==> (convex hull (vec 0 INSERT set_of_list l)) has_measure abs(det(vector l)) / &(FACT(dimindex(:N)))`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `vec 0 INSERT (set_of_list l) = IMAGE (\x:real^N. transp(vector l:real^N^N) ** x) (vec 0 INSERT {basis i:real^N | 1 <= i /\ i <= dimindex(:N)})` SUBST1_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[IMAGE_CLAUSES; GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_RZERO] THEN AP_TERM_TAC THEN SIMP_TAC[matrix_vector_mul; vector; transp; LAMBDA_BETA; basis] THEN ONCE_REWRITE_TAC[COND_RAND] THEN SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA] THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; IN_NUMSEG] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(b /\ c ==> ~a)`] THEN X_GEN_TAC `y:real^N` THEN SIMP_TAC[LAMBDA_BETA; REAL_MUL_RID] THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REWRITE_TAC[NOT_IMP; REAL_MUL_RID; GSYM CART_EQ] THEN ASM_REWRITE_TAC[IN_SET_OF_LIST; MEM_EXISTS_EL] THEN EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC) THENL [EXISTS_TAC `SUC i`; EXISTS_TAC `i - 1`] THEN ASM_REWRITE_TAC[SUC_SUB1] THEN ASM_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[CONVEX_HULL_LINEAR_IMAGE; MATRIX_VECTOR_MUL_LINEAR] THEN SUBGOAL_THEN `det(vector l:real^N^N) = det(matrix(\x:real^N. transp(vector l) ** x))` SUBST1_TAC THENL [REWRITE_TAC[MATRIX_OF_MATRIX_VECTOR_MUL; DET_TRANSP]; ALL_TAC] THEN REWRITE_TAC[real_div] THEN ASM_SIMP_TAC[GSYM(REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE] HAS_MEASURE_STD_SIMPLEX)] THEN MATCH_MP_TAC HAS_MEASURE_LINEAR_IMAGE THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN MATCH_MP_TAC MEASURABLE_CONVEX THEN REWRITE_TAC[CONVEX_CONVEX_HULL] THEN MATCH_MP_TAC BOUNDED_CONVEX_HULL THEN REWRITE_TAC[BOUNDED_INSERT] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN MATCH_MP_TAC FINITE_IMAGE THEN REWRITE_TAC[GSYM numseg; FINITE_NUMSEG]);; let HAS_MEASURE_SIMPLEX = prove (`!a l:(real^N)list. LENGTH l = dimindex(:N) ==> (convex hull (set_of_list(CONS a l))) has_measure abs(det(vector(MAP (\x. x - a) l))) / &(FACT(dimindex(:N)))`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `MAP (\x:real^N. x - a) l` HAS_MEASURE_SIMPLEX_0) THEN ASM_REWRITE_TAC[LENGTH_MAP; set_of_list] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N` o MATCH_MP HAS_MEASURE_TRANSLATION) THEN REWRITE_TAC[GSYM CONVEX_HULL_TRANSLATION] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[IMAGE_CLAUSES; VECTOR_ADD_RID; SET_OF_LIST_MAP] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ARITH `a + x - a:real^N = x`; SET_RULE `IMAGE (\x. x) s = s`]);; let MEASURABLE_CONVEX_HULL = prove (`!s. bounded s ==> measurable(convex hull s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_CONVEX THEN ASM_SIMP_TAC[CONVEX_CONVEX_HULL; BOUNDED_CONVEX_HULL]);; let MEASURABLE_SIMPLEX = prove (`!l. measurable(convex hull (set_of_list l))`, GEN_TAC THEN MATCH_MP_TAC MEASURABLE_CONVEX_HULL THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN REWRITE_TAC[FINITE_SET_OF_LIST]);; let MEASURE_SIMPLEX = prove (`!a l:(real^N)list. LENGTH l = dimindex(:N) ==> measure(convex hull (set_of_list(CONS a l))) = abs(det(vector(MAP (\x. x - a) l))) / &(FACT(dimindex(:N)))`, MESON_TAC[HAS_MEASURE_SIMPLEX; HAS_MEASURE_MEASURABLE_MEASURE]);; (* ------------------------------------------------------------------------- *) (* Area of a triangle. *) (* ------------------------------------------------------------------------- *) let HAS_MEASURE_TRIANGLE = prove (`!a b c:real^2. convex hull {a,b,c} has_measure abs((b$1 - a$1) * (c$2 - a$2) - (b$2 - a$2) * (c$1 - a$1)) / &2`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`a:real^2`; `[b;c]:(real^2)list`] HAS_MEASURE_SIMPLEX) THEN REWRITE_TAC[LENGTH; DIMINDEX_2; ARITH; set_of_list; MAP] THEN CONV_TAC NUM_REDUCE_CONV THEN SIMP_TAC[DET_2; VECTOR_2] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; DIMINDEX_2; ARITH]);; let MEASURABLE_TRIANGLE = prove (`!a b c:real^N. measurable(convex hull {a,b,c})`, REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_CONVEX THEN REWRITE_TAC[CONVEX_CONVEX_HULL] THEN MATCH_MP_TAC BOUNDED_CONVEX_HULL THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN REWRITE_TAC[FINITE_INSERT; FINITE_RULES]);; let MEASURE_TRIANGLE = prove (`!a b c:real^2. measure(convex hull {a,b,c}) = abs((b$1 - a$1) * (c$2 - a$2) - (b$2 - a$2) * (c$1 - a$1)) / &2`, REWRITE_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE] HAS_MEASURE_TRIANGLE]);; (* ------------------------------------------------------------------------- *) (* Volume of a tetrahedron. *) (* ------------------------------------------------------------------------- *) let HAS_MEASURE_TETRAHEDRON = prove (`!a b c d:real^3. convex hull {a,b,c,d} has_measure abs((b$1 - a$1) * (c$2 - a$2) * (d$3 - a$3) + (b$2 - a$2) * (c$3 - a$3) * (d$1 - a$1) + (b$3 - a$3) * (c$1 - a$1) * (d$2 - a$2) - (b$1 - a$1) * (c$3 - a$3) * (d$2 - a$2) - (b$2 - a$2) * (c$1 - a$1) * (d$3 - a$3) - (b$3 - a$3) * (c$2 - a$2) * (d$1 - a$1)) / &6`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`a:real^3`; `[b;c;d]:(real^3)list`] HAS_MEASURE_SIMPLEX) THEN REWRITE_TAC[LENGTH; DIMINDEX_3; ARITH; set_of_list; MAP] THEN CONV_TAC NUM_REDUCE_CONV THEN SIMP_TAC[DET_3; VECTOR_3] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; DIMINDEX_3; ARITH]);; let MEASURABLE_TETRAHEDRON = prove (`!a b c d:real^N. measurable(convex hull {a,b,c,d})`, REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_CONVEX THEN REWRITE_TAC[CONVEX_CONVEX_HULL] THEN MATCH_MP_TAC BOUNDED_CONVEX_HULL THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN REWRITE_TAC[FINITE_INSERT; FINITE_RULES]);; let MEASURE_TETRAHEDRON = prove (`!a b c d:real^3. measure(convex hull {a,b,c,d}) = abs((b$1 - a$1) * (c$2 - a$2) * (d$3 - a$3) + (b$2 - a$2) * (c$3 - a$3) * (d$1 - a$1) + (b$3 - a$3) * (c$1 - a$1) * (d$2 - a$2) - (b$1 - a$1) * (c$3 - a$3) * (d$2 - a$2) - (b$2 - a$2) * (c$1 - a$1) * (d$3 - a$3) - (b$3 - a$3) * (c$2 - a$2) * (d$1 - a$1)) / &6`, REWRITE_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE] HAS_MEASURE_TETRAHEDRON]);; (* ------------------------------------------------------------------------- *) (* Measure is continuous with Hausdorff distance: several formulations. *) (* ------------------------------------------------------------------------- *) let MEASURE_CONTINUOUS_WITH_HAUSDIST = prove (`!s:real^N->bool e. bounded s /\ convex s /\ ~(s = {}) /\ &0 < e ==> ?d. &0 < d /\ !t. bounded t /\ convex t /\ ~(t = {}) /\ hausdist(s,t) < d ==> abs(measure(t) - measure(s)) < e`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`frontier s:real^N->bool`; `e:real`] MEASURABLE_OUTER_OPEN) THEN FIRST_ASSUM(MP_TAC o MATCH_MP NEGLIGIBLE_CONVEX_FRONTIER) THEN REWRITE_TAC[NEGLIGIBLE_EQ_MEASURE_0] THEN STRIP_TAC THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_ADD_LID] THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `setdist(frontier s,(:real^N) DIFF u)` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN ASM_SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED; COMPACT_FRONTIER_BOUNDED; GSYM OPEN_CLOSED; SETDIST_POS_LE; FRONTIER_EQ_EMPTY] THEN REWRITE_TAC[SET_RULE `UNIV DIFF s = {} <=> s = UNIV`; DE_MORGAN_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[NOT_BOUNDED_UNIV]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[NOT_MEASURABLE_UNIV]; ASM SET_TAC[]]; X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `u < e ==> a <= u ==> a < e`)) THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_SUB_LE_MEASURE_SYMDIFF o lhand o snd) THEN ASM_SIMP_TAC[MEASURABLE_CONVEX] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNION; MEASURABLE_DIFF; MEASURABLE_CONVEX] THEN ONCE_REWRITE_TAC[UNION_COMM] THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`; `setdist(frontier s,(:real^N) DIFF u)`] CONVEX_SYMDIFF_CLOSE_TO_FRONTIER) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_BALL_0] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[NORM_ARITH `~(norm(y:real^N) < e) = e <= dist(x,x + y)`] THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]]);; let MEASURE_CONTINUOUS_WITH_HAUSDIST_EXPLICIT = prove (`!s:real^N->bool e. bounded s /\ convex s /\ &0 < e ==> ?d. &0 < d /\ !t. convex t /\ (!y. y IN s ==> ?x. x IN t /\ dist(x,y) < d) /\ (!y. y IN t ==> ?x. x IN s /\ dist(x,y) < d) ==> abs(measure(t) - measure(s)) < e`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY; GSYM NOT_EXISTS_THM; MEMBER_NOT_EMPTY] THEN EXISTS_TAC `&1` THEN SIMP_TAC[MEASURE_EMPTY] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `e:real`] MEASURE_CONTINUOUS_WITH_HAUSDIST) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN EXISTS_TAC `d / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN X_GEN_TAC `t:real^N->bool` THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY; GSYM NOT_EXISTS_THM; MEMBER_NOT_EMPTY] THEN EXISTS_TAC `&1` THEN SIMP_TAC[MEASURE_EMPTY] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `{a + b:real^N | a IN s /\ b IN ball(vec 0,d / &2)}` THEN ASM_SIMP_TAC[BOUNDED_SUMS; BOUNDED_BALL] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_BALL_0; VECTOR_ARITH `x:real^N = a + b <=> b = x - a`] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM2] THEN ASM_MESON_TAC[dist; DIST_SYM]; DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 < d /\ x <= d / &2 ==> x < d`) THEN ASM_SIMP_TAC[REAL_HAUSDIST_LE_EQ] THEN ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS; SETDIST_LE_DIST; IN_SING; DIST_SYM; SETDIST_SYM]]);; let MEASURE_SEMICONTINUOUS_WITH_HAUSDIST_EXPLICIT = prove (`!s:real^N->bool e. bounded s /\ negligible(frontier s) /\ &0 < e ==> ?d. &0 < d /\ !s'. measurable s' /\ (!y. y IN s' ==> ?x. x IN s /\ dist(x,y) < d) ==> measure(s') < measure(s) + e`, REWRITE_TAC[NEGLIGIBLE_EQ_MEASURE_0] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM_REWRITE_TAC[NOT_IN_EMPTY; GSYM NOT_EXISTS_THM; MEMBER_NOT_EMPTY] THEN EXISTS_TAC `&1` THEN SIMP_TAC[MEASURE_EMPTY] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`frontier s:real^N->bool`; `e:real`] MEASURABLE_OUTER_OPEN) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_ADD_LID] THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `setdist(frontier s,(:real^N) DIFF u)` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN ASM_SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED; COMPACT_FRONTIER_BOUNDED; GSYM OPEN_CLOSED; SETDIST_POS_LE; FRONTIER_EQ_EMPTY] THEN REWRITE_TAC[SET_RULE `UNIV DIFF s = {} <=> s = UNIV`; DE_MORGAN_THM] THEN CONJ_TAC THENL [ASM_MESON_TAC[NOT_BOUNDED_UNIV]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[NOT_MEASURABLE_UNIV]; ASM SET_TAC[]]; X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `u < e ==> t - s <= u ==> t < s + e`)) THEN TRANS_TAC REAL_LE_TRANS `measure(t DIFF s:real^N->bool)` THEN ASM_SIMP_TAC[MEASURE_SUB_LE_MEASURE_DIFF; MEASURABLE_JORDAN; NEGLIGIBLE_EQ_MEASURE_0] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_JORDAN; MEASURABLE_DIFF; NEGLIGIBLE_EQ_MEASURE_0] THEN REWRITE_TAC[SUBSET; IN_DIFF] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`segment[x:real^N,y]`; `s:real^N->bool`] CONNECTED_INTER_FRONTIER) THEN REWRITE_TAC[CONNECTED_SEGMENT] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_DIFF] THEN ANTS_TAC THENL [CONJ_TAC THENL [EXISTS_TAC `y:real^N`; EXISTS_TAC `x:real^N`] THEN ASM_REWRITE_TAC[ENDS_IN_SEGMENT]; DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `dist(x:real^N,z) < setdist(frontier s,(:real^N) DIFF u)` MP_TAC THENL [ASM_MESON_TAC[REAL_LET_TRANS; DIST_IN_CLOSED_SEGMENT; DIST_SYM]; GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[REAL_NOT_LT] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]]]);; let MEASURE_SEMICONTINUOUS_WITH_HAUSDIST_BOUND = prove (`!s s' r e a:real^N. bounded s /\ convex s /\ ball(a,r) SUBSET s /\ &0 < r /\ bounded s' /\ measurable s' /\ hausdist(s,s') <= e * r /\ &0 < e ==> measure(s') <= measure(s) * (&1 + e) pow dimindex(:N)`, REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s':real^N->bool`; `closure s:real^N->bool`] HAUSDIST_COMPACT_SUMS) THEN ASM_REWRITE_TAC[HAUSDIST_CLOSURE; COMPACT_CLOSURE; CLOSURE_EQ_EMPTY] THEN ANTS_TAC THENL [ASM_MESON_TAC[REAL_NOT_LE; BALL_EQ_EMPTY; SUBSET_EMPTY]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] MEASURE_SUBSET))) THEN ASM_SIMP_TAC[MEASURABLE_COMPACT; COMPACT_SUMS; COMPACT_CLOSURE; COMPACT_CBALL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN TRANS_TAC REAL_LE_TRANS `measure {&1 % y + e % z:real^N | y IN closure s /\ z IN closure s}` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_COMPACT; COMPACT_SUMS; COMPACT_CLOSURE; COMPACT_CBALL] THEN CONJ_TAC THENL [REWRITE_TAC[SET_RULE `{a % x + b % y:real^N | x IN s /\ y IN t} = {x + y | x IN IMAGE (\x. a % x) s /\ y IN IMAGE (\x. b % x) t}`] THEN ASM_SIMP_TAC[MEASURABLE_COMPACT; COMPACT_SUMS; COMPACT_CLOSURE; COMPACT_CBALL; COMPACT_SCALING]; REWRITE_TAC[VECTOR_MUL_LID] THEN MATCH_MP_TAC(SET_RULE `t SUBSET IMAGE (\x. e % x) u ==> {y + z:real^N | y IN s /\ z IN t} SUBSET {y + e % z | y IN s /\ z IN u}`) THEN TRANS_TAC SUBSET_TRANS `IMAGE (\x:real^N. e % x) (cball(vec 0,r))` THEN CONJ_TAC THENL [ASM_SIMP_TAC[GSYM CBALL_SCALING; VECTOR_MUL_RZERO] THEN MATCH_MP_TAC SUBSET_CBALL THEN ASM_MESON_TAC[HAUSDIST_SYM]; MATCH_MP_TAC IMAGE_SUBSET THEN ASM_SIMP_TAC[GSYM CLOSURE_BALL; SUBSET_CLOSURE]]]; ASM_SIMP_TAC[CONVEX_SUMS_MULTIPLES; REAL_POS; REAL_LT_IMP_LE; CONVEX_CLOSURE] THEN ASM_SIMP_TAC[MEASURE_SCALING; COMPACT_CLOSURE; MEASURABLE_COMPACT] THEN ASM_SIMP_TAC[MEASURE_CLOSURE; NEGLIGIBLE_CONVEX_FRONTIER] THEN MATCH_MP_TAC(REAL_ARITH `y * x = a ==> x * y <= a`) THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_REAL_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Steinhaus's theorem. (Stromberg's proof as given on Wikipedia.) *) (* ------------------------------------------------------------------------- *) let STEINHAUS = prove (`!s:real^N->bool. measurable s /\ &0 < measure s ==> ?d. &0 < d /\ ball(vec 0,d) SUBSET {x - y | x IN s /\ y IN s}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `measure(s:real^N->bool) / &3`] MEASURABLE_INNER_COMPACT) THEN MP_TAC(ISPECL [`s:real^N->bool`; `measure(s:real^N->bool) / &3`] MEASURABLE_OUTER_OPEN) THEN ASM_REWRITE_TAC[REAL_ARITH `&0 < x / &3 <=> &0 < x`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`k:real^N->bool`; `(:real^N) DIFF u`] SEPARATE_COMPACT_CLOSED) THEN ASM_REWRITE_TAC[GSYM OPEN_CLOSED] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_BALL_0; IN_ELIM_THM] THEN X_GEN_TAC `v:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `~((IMAGE (\x:real^N. v + x) k) INTER k = {})` MP_TAC THENL [DISCH_TAC THEN MP_TAC(ISPECL [`IMAGE (\x:real^N. v + x) k`; `k:real^N->bool`] MEASURE_UNION) THEN ASM_REWRITE_TAC[MEASURABLE_TRANSLATION_EQ; MEASURE_EMPTY] THEN REWRITE_TAC[MEASURE_TRANSLATION; REAL_SUB_RZERO] THEN MATCH_MP_TAC(REAL_ARITH `!s:real^N->bool u:real^N->bool. measure u < measure s + measure s / &3 /\ measure s < measure k + measure s / &3 /\ measure x <= measure u ==> ~(measure x = measure k + measure k)`) THEN MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `u:real^N->bool`] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_TRANSLATION_EQ; MEASURABLE_UNION] THEN ASM_REWRITE_TAC[UNION_SUBSET] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `v + x:real^N`]) THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; NORM_ARITH `d <= dist(x:real^N,v + x) <=> ~(norm v < d)`]; REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY; IN_IMAGE] THEN REWRITE_TAC[VECTOR_ARITH `v:real^N = x - y <=> x = v + y`] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* A measurable set with cardinality less than c is negligible. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_NONNEGLIGIBLE_IMP_LARGE = prove (`!s:real^N->bool. measurable s /\ &0 < measure s ==> s =_c (:real)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `FINITE(s:real^N->bool)` THENL [ASM_MESON_TAC[NEGLIGIBLE_FINITE; MEASURABLE_MEASURE_POS_LT]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP STEINHAUS) THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN REWRITE_TAC[CARD_LE_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN REWRITE_TAC[CARD_EQ_EUCLIDEAN]; ALL_TAC] THEN TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN CONJ_TAC THENL [MESON_TAC[CARD_EQ_EUCLIDEAN; CARD_EQ_SYM; CARD_EQ_IMP_LE]; ALL_TAC] THEN TRANS_TAC CARD_LE_TRANS `interval(vec 0:real^N,vec 1)` THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_EQ_IMP_LE THEN ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN MATCH_MP_TAC HOMEOMORPHIC_IMP_CARD_EQ THEN MATCH_MP_TAC HOMEOMORPHIC_OPEN_INTERVAL_UNIV THEN REWRITE_TAC[UNIT_INTERVAL_NONEMPTY]; ALL_TAC] THEN TRANS_TAC CARD_LE_TRANS `interval[vec 0:real^N,vec 1]` THEN SIMP_TAC[INTERVAL_OPEN_SUBSET_CLOSED; CARD_LE_SUBSET] THEN TRANS_TAC CARD_LE_TRANS `cball(vec 0:real^N,d / &2)` THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_EQ_IMP_LE THEN MATCH_MP_TAC HOMEOMORPHIC_IMP_CARD_EQ THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT THEN REWRITE_TAC[CONVEX_INTERVAL; COMPACT_INTERVAL; INTERIOR_CLOSED_INTERVAL; CONVEX_CBALL; COMPACT_CBALL; UNIT_INTERVAL_NONEMPTY; INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN TRANS_TAC CARD_LE_TRANS `ball(vec 0:real^N,d)` THEN CONJ_TAC THENL [MATCH_MP_TAC CARD_LE_SUBSET THEN REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN TRANS_TAC CARD_LE_TRANS `IMAGE (\(x:real^N,y). x - y) (s *_c s)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[mul_c; CARD_LE_SUBSET; SET_RULE `IMAGE f {g x y | P x /\ Q y} = {f(g x y) | P x /\ Q y}`]; ALL_TAC] THEN TRANS_TAC CARD_LE_TRANS `((s:real^N->bool) *_c s)` THEN REWRITE_TAC[CARD_LE_IMAGE] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN MATCH_MP_TAC CARD_SQUARE_INFINITE THEN ASM_REWRITE_TAC[INFINITE]);; let MEASURABLE_SMALL_IMP_NEGLIGIBLE = prove (`!s:real^N->bool. measurable s /\ s <_c (:real) ==> negligible s`, GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `a /\ b ==> c <=> a ==> ~c ==> ~b`] THEN SIMP_TAC[GSYM MEASURABLE_MEASURE_POS_LT] THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_NONNEGLIGIBLE_IMP_LARGE) THEN REWRITE_TAC[lt_c] THEN MESON_TAC[CARD_EQ_IMP_LE; CARD_EQ_SYM]);; (* ------------------------------------------------------------------------- *) (* Austin's Lemma. *) (* ------------------------------------------------------------------------- *) let AUSTIN_LEMMA = prove (`!D. FINITE D /\ (!d. d IN D ==> ?k a b. d = interval[a:real^N,b] /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> b$i - a$i = k)) ==> ?D'. D' SUBSET D /\ pairwise DISJOINT D' /\ measure(UNIONS D') >= measure(UNIONS D) / &3 pow (dimindex(:N))`, GEN_TAC THEN WF_INDUCT_TAC `CARD(D:(real^N->bool)->bool)` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*")) THEN ASM_CASES_TAC `D:(real^N->bool)->bool = {}` THENL [ASM_REWRITE_TAC[SUBSET_EMPTY; UNWIND_THM2; PAIRWISE_EMPTY] THEN REWRITE_TAC[UNIONS_0; real_ge; MEASURE_EMPTY; NOT_IN_EMPTY] THEN REWRITE_TAC[REAL_ARITH `&0 / x = &0`; REAL_LE_REFL]; ALL_TAC] THEN SUBGOAL_THEN `?d:real^N->bool. d IN D /\ !d'. d' IN D ==> measure d' <= measure d` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `IMAGE measure (D:(real^N->bool)->bool)` SUP_FINITE) THEN ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `{c:real^N->bool | c IN (D DELETE d) /\ c INTER d = {}}`) THEN ANTS_TAC THENL [MATCH_MP_TAC CARD_PSUBSET THEN ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[FINITE_DELETE; FINITE_RESTRICT; IN_ELIM_THM; real_ge] THEN ANTS_TAC THENL [ASM_SIMP_TAC[IN_DELETE]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `D':(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(d:real^N->bool) INSERT D'` THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[pairwise; IN_INSERT] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?a3 b3:real^N. measure(interval[a3,b3]) = &3 pow dimindex(:N) * measure d /\ !c. c IN D /\ ~(c INTER d = {}) ==> c SUBSET interval[a3,b3]` STRIP_ASSUME_TAC THENL [USE_THEN "*" (MP_TAC o SPEC `d:real^N->bool`) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:real`; `a:real^N`; `b:real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN EXISTS_TAC `inv(&2) % (a + b) - &3 / &2 % (b - a):real^N` THEN EXISTS_TAC `inv(&2) % (a + b) + &3 / &2 % (b - a):real^N` THEN CONJ_TAC THENL [REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `(x + &3 / &2 * a) - (x - &3 / &2 * a) = &3 * a`; REAL_ARITH `x - a <= x + a <=> &0 <= a`] THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= &3 / &2 * x - &0 <=> &0 <= x`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO] THEN SIMP_TAC[PRODUCT_CONST; FINITE_NUMSEG; CARD_NUMSEG_1; REAL_POW_MUL]; X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `c:real^N->bool`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k':real`; `a':real^N`; `b':real^N`] THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [DISJOINT_INTERVAL]) THEN REWRITE_TAC[NOT_EXISTS_THM; SUBSET_INTERVAL] THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `i <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `interval[a':real^N,b']`) THEN ASM_REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[DE_MORGAN_THM; REAL_NOT_LT] THEN REWRITE_TAC[REAL_ARITH `a$k <= b$k <=> &0 <= b$k - a$k`] THEN ASM_SIMP_TAC[IN_NUMSEG] THEN ASM_CASES_TAC `&0 <= k` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `&0 <= k'` THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(fun th -> SIMP_TAC[th] THEN MP_TAC(ISPEC `i:num` th))) THEN ASM_SIMP_TAC[PRODUCT_CONST; CARD_NUMSEG_1; FINITE_NUMSEG] THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] REAL_POW_LE2_REV)) THEN ASM_SIMP_TAC[DIMINDEX_GE_1; LE_1] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN REWRITE_TAC[UNIONS_INSERT] THEN SUBGOAL_THEN `!d:real^N->bool. d IN D ==> measurable d` ASSUME_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_DISJOINT_UNION o rand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET)) THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_DELETE]; DISCH_THEN SUBST1_TAC] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(interval[a3:real^N,b3]) + measure(UNIONS D DIFF interval[a3,b3])` THEN CONJ_TAC THENL [W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNION o rand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[MEASURABLE_UNIONS; MEASURABLE_DIFF; MEASURABLE_INTERVAL] THEN SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[MEASURABLE_UNIONS]; ASM_MESON_TAC[MEASURABLE_UNIONS; MEASURABLE_DIFF; MEASURABLE_INTERVAL; MEASURABLE_UNION]; SET_TAC[]]]; ASM_REWRITE_TAC[REAL_ARITH `a * x + y <= (x + z) * a <=> y <= z * a`] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `y <= a ==> x <= y ==> x <= a`)) THEN SIMP_TAC[REAL_LE_DIV2_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_UNIONS; MEASURABLE_INTERVAL; IN_ELIM_THM; IN_DELETE; FINITE_DELETE; FINITE_RESTRICT] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Some differentiability-like properties of the indefinite integral. *) (* The first two proofs are minor variants of each other, but it was more *) (* work to derive one from the other. *) (* ------------------------------------------------------------------------- *) let INTEGRABLE_CCONTINUOUS_EXPLICIT = prove (`!f:real^M->real^N. (!a b. f integrable_on interval[a,b]) ==> ?k. negligible k /\ !x e. ~(x IN k) /\ &0 < e ==> ?d. &0 < d /\ !h. &0 < h /\ h < d ==> norm(inv(content(interval[x,x + h % vec 1])) % integral (interval[x,x + h % vec 1]) f - f(x)) < e`, REPEAT STRIP_TAC THEN REWRITE_TAC[IN_UNIV] THEN MAP_EVERY ABBREV_TAC [`box = \h x. interval[x:real^M,x + h % vec 1]`; `box2 = \h x. interval[x:real^M - h % vec 1,x + h % vec 1]`; `i = \h:real x:real^M. inv(content(box h x)) % integral (box h x) (f:real^M->real^N)`] THEN SUBGOAL_THEN `?k. negligible k /\ !x e. ~(x IN k) /\ &0 < e ==> ?d. &0 < d /\ !h. &0 < h /\ h < d ==> norm(i h x - (f:real^M->real^N) x) < e` MP_TAC THENL [ALL_TAC; MAP_EVERY EXPAND_TAC ["i"; "box"] THEN REWRITE_TAC[]] THEN EXISTS_TAC `{x | ~(!e. &0 < e ==> ?d. &0 < d /\ !h. &0 < h /\ h < d ==> norm(i h x - (f:real^M->real^N) x) < e)}` THEN SIMP_TAC[IN_ELIM_THM] THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | !d. &0 < d ==> ?h. &0 < h /\ h < d /\ inv(&k + &1) <= dist(i h x,(f:real^M->real^N) x)} | k IN (:num)}` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`y:real^M`; `e:real`] THEN STRIP_TAC THEN REWRITE_TAC[SIMPLE_IMAGE; UNIONS_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:num` THEN DISCH_TAC THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LT] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real` THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[dist] THEN MATCH_MP_TAC (REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `inv(&k)` THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `jj:num` THEN SUBGOAL_THEN `&0 < inv(&jj + &1)` MP_TAC THENL [REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; SPEC_TAC(`inv(&jj + &1)`,`mu:real`) THEN GEN_TAC THEN DISCH_TAC] THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN ASM_CASES_TAC `negligible(interval[a:real^M,b])` THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET; INTER_SUBSET]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[NEGLIGIBLE_INTERVAL]) THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `a - vec 1:real^M`; `b + vec 1:real^M`] HENSTOCK_LEMMA) THEN ANTS_TAC THENL [ASM_MESON_TAC[INTEGRABLE_ON_SUBINTERVAL; SUBSET_UNIV]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `(e * mu) / &2 / &6 pow (dimindex(:M))`) THEN ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; REAL_LT_MUL; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[SET_RULE `{x | P x} INTER s = {x | x IN s /\ P x}`] THEN ABBREV_TAC `E = {x | x IN interval[a,b] /\ !d. &0 < d ==> ?h. &0 < h /\ h < d /\ mu <= dist(i h x,(f:real^M->real^N) x)}` THEN SUBGOAL_THEN `!x. x IN E ==> ?h. &0 < h /\ (box h x:real^M->bool) SUBSET (g x) /\ (box h x:real^M->bool) SUBSET interval[a - vec 1,b + vec 1] /\ mu <= dist(i h x,(f:real^M->real^N) x)` MP_TAC THENL [X_GEN_TAC `x:real^M` THEN EXPAND_TAC "E" THEN REWRITE_TAC[IN_ELIM_THM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [gauge]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o SPEC `x:real^M`) THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `min (&1) (d / &(dimindex(:M)))`)) THEN REWRITE_TAC[REAL_LT_MIN; REAL_LT_01; GSYM CONJ_ASSOC] THEN ASM_SIMP_TAC[REAL_LT_DIV; DIMINDEX_GE_1; LE_1; REAL_OF_NUM_LT] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(x:real^M,d)` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "box" THEN REWRITE_TAC[SUBSET; IN_INTERVAL; IN_BALL] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `sum(1..dimindex(:M)) (\i. abs((x - y:real^M)$i))` THEN REWRITE_TAC[NORM_LE_L1] THEN MATCH_MP_TAC SUM_BOUND_LT_GEN THEN REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; IN_NUMSEG] THEN SIMP_TAC[NOT_LT; DIMINDEX_GE_1; CARD_NUMSEG_1; VECTOR_SUB_COMPONENT] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; UNDISCH_TAC `(x:real^M) IN interval[a,b]` THEN EXPAND_TAC "box" THEN REWRITE_TAC[SUBSET; IN_INTERVAL] THEN DISCH_THEN(fun th -> X_GEN_TAC `y:real^M` THEN MP_TAC th) THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `uv:real^M->real` THEN REWRITE_TAC[TAUT `(a ==> b /\ c) <=> (a ==> b) /\ (a ==> c)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPECL [`a:real^M`; `b:real^M`; `E:real^M->bool`; `\x:real^M. if x IN E then ball(x,uv x) else g(x)`] COVERING_LEMMA) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[INTERVAL_NE_EMPTY] THEN CONJ_TAC THENL [EXPAND_TAC "E" THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[gauge] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN RULE_ASSUM_TAC(REWRITE_RULE[gauge]) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_TAC `D:(real^M->bool)->bool`) THEN EXISTS_TAC `UNIONS D:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `measurable(UNIONS D:real^M->bool) /\ measure(UNIONS D) <= measure(interval[a:real^M,b])` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `?d. d SUBSET D /\ FINITE d /\ measure(UNIONS D:real^M->bool) <= &2 * measure(UNIONS d)` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `measure(UNIONS D:real^M->bool) = &0` THENL [EXISTS_TAC `{}:(real^M->bool)->bool` THEN ASM_REWRITE_TAC[FINITE_EMPTY; EMPTY_SUBSET; MEASURE_EMPTY; UNIONS_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV; MP_TAC(ISPECL [`D:(real^M->bool)->bool`; `measure(interval[a:real^M,b])`; `measure(UNIONS D:real^M->bool) / &2`] MEASURE_COUNTABLE_UNIONS_APPROACHABLE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; REAL_HALF] THEN ASM_SIMP_TAC[GSYM MEASURABLE_MEASURE_EQ_0] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL; MEASURABLE_UNIONS]; ALL_TAC]) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN REAL_ARITH_TAC]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o el 3 o CONJUNCTS) THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN ONCE_REWRITE_TAC[TAUT `a /\ b <=> ~(a ==> ~b)`] THEN SIMP_TAC[IN_INTER] THEN REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN DISCH_THEN(X_CHOOSE_TAC `tag:(real^M->bool)->real^M`) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `D <= &2 * d ==> d <= e / &2 ==> D <= e`)) THEN MP_TAC(ISPEC `IMAGE (\k:real^M->bool. (box2:real->real^M->real^M->bool) (uv(tag k):real) ((tag k:real^M))) d` AUSTIN_LEMMA) THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN EXPAND_TAC "box2" THEN EXISTS_TAC `&2 * uv((tag:(real^M->bool)->real^M) k):real` THEN EXISTS_TAC `(tag:(real^M->bool)->real^M) k - uv(tag k) % vec 1:real^M` THEN EXISTS_TAC `(tag:(real^M->bool)->real^M) k + uv(tag k) % vec 1:real^M` THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[EXISTS_SUBSET_IMAGE; real_ge] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN DISCH_THEN(X_CHOOSE_THEN `p:(real^M->bool)->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REAL_ARITH `d <= d' /\ p <= e ==> d' <= p ==> d <= e`) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN EXPAND_TAC "box2" THEN REWRITE_TAC[MEASURABLE_INTERVAL]; REWRITE_TAC[SUBSET; IN_UNIONS; EXISTS_IN_IMAGE] THEN X_GEN_TAC `z:real^M` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `(z:real^M) IN k` THEN SPEC_TAC(`z:real^M`,`z:real^M`) THEN REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(tag k:real^M,uv(tag(k:real^M->bool)))` THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN EXPAND_TAC "box2" THEN REWRITE_TAC[SUBSET; IN_BALL; IN_INTERVAL] THEN X_GEN_TAC `z:real^M` THEN REWRITE_TAC[dist] THEN DISCH_TAC THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN SIMP_TAC[REAL_ARITH `x - h <= y /\ y <= x + h <=> abs(x - y) <= h`] THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LT_IMP_LE; REAL_LE_TRANS]]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(UNIONS (IMAGE (\k:real^M->bool. (box:real->real^M->real^M->bool) (uv(tag k):real) ((tag k:real^M))) p)) * &6 pow dimindex (:M)` THEN CONJ_TAC THENL [SUBGOAL_THEN `!box. IMAGE (\k:real^M->bool. (box:real->real^M->real^M->bool) (uv(tag k):real) ((tag k:real^M))) p = IMAGE (\t. box (uv t) t) (IMAGE tag p)` (fun th -> REWRITE_TAC[th]) THENL [REWRITE_TAC[GSYM IMAGE_o; o_DEF]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhs o rand) MEASURE_NEGLIGIBLE_UNIONS_IMAGE o lhand o rand o snd) THEN W(MP_TAC o PART_MATCH (lhs o rand) MEASURE_NEGLIGIBLE_UNIONS_IMAGE o lhand o lhand o rand o snd) THEN MATCH_MP_TAC(TAUT `fp /\ (mb /\ mb') /\ (db /\ db') /\ (m1 /\ m2 ==> p) ==> (fp /\ mb /\ db ==> m1) ==> (fp /\ mb' /\ db' ==> m2) ==> p`) THEN SUBGOAL_THEN `FINITE(p:(real^M->bool)->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ASM_SIMP_TAC[FINITE_IMAGE]] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MAP_EVERY EXPAND_TAC ["box"; "box2"] THEN REWRITE_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[IMP_IMP; RIGHT_IMP_FORALL_THM; AND_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`k1:real^M->bool`; `k2:real^M->bool`] THEN MATCH_MP_TAC(TAUT `(q ==> r) /\ (p ==> q) ==> (p ==> q) /\ (p ==> r)`) THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN MATCH_MP_TAC(SET_RULE `s SUBSET s' /\ t SUBSET t' ==> (s INTER t) SUBSET (s' INTER t')`) THEN CONJ_TAC THEN MAP_EVERY EXPAND_TAC ["box"; "box2"] THEN REWRITE_TAC[SUBSET_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN STRIP_TAC THEN MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s = {} ==> negligible s`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `k1:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `k2:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [EXPAND_TAC "box2" THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `x - e <= x + e <=> &0 <= e`] THEN SUBGOAL_THEN `&0 <= uv((tag:(real^M->bool)->real^M) k1) /\ &0 <= uv((tag:(real^M->bool)->real^M) k2)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; REAL_LT_IMP_LE]; ASM_REWRITE_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN MATCH_MP_TAC MONO_NOT THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; SET_TAC[]]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN SUBST1_TAC) THEN REWRITE_TAC[GSYM SUM_RMUL] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `t:real^M` THEN DISCH_THEN(K ALL_TAC) THEN SUBST1_TAC(REAL_ARITH `&6 = &2 * &3`) THEN REWRITE_TAC[REAL_POW_MUL; REAL_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN MAP_EVERY EXPAND_TAC ["box"; "box2"] THEN REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `a <= a + x <=> &0 <= x`; REAL_ARITH `a - x <= a + x <=> &0 <= x`] THEN COND_CASES_TAC THEN REWRITE_TAC[REAL_MUL_LZERO] THEN REWRITE_TAC[REAL_ARITH `(t + h) - (t - h):real = &2 * h`; REAL_ARITH `(t + h) - t:real = h`] THEN REWRITE_TAC[PRODUCT_MUL_NUMSEG; PRODUCT_CONST_NUMSEG] THEN REWRITE_TAC[ADD_SUB; REAL_MUL_AC]; ALL_TAC] THEN SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN SUBGOAL_THEN `FINITE(p:(real^M->bool)->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_LCANCEL_IMP THEN EXISTS_TAC `mu:real` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\k. (tag:(real^M->bool)->real^M) k, (box(uv(tag k):real) (tag k):real^M->bool)) p`) THEN ANTS_TAC THENL [REWRITE_TAC[tagged_partial_division_of; fine] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IN_IMAGE; PAIR_EQ] THEN REWRITE_TAC[MESON[] `(!x j. (?k. (x = tag k /\ j = g k) /\ k IN d) ==> P x j) <=> (!k. k IN d ==> P (tag k) (g k))`] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN REPEAT CONJ_TAC THENL [EXPAND_TAC "box" THEN REWRITE_TAC[IN_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 < u ==> x <= x /\ x <= x + u`) THEN ASM_MESON_TAC[SUBSET]; ASM_MESON_TAC[SUBSET]; EXPAND_TAC "box" THEN MESON_TAC[]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k1:real^M->bool` THEN ASM_CASES_TAC `(k1:real^M->bool) IN p` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k2:real^M->bool` THEN ASM_CASES_TAC `(k2:real^M->bool) IN p` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(tag:(real^M->bool)->real^M) k1 = tag k2` THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [EXPAND_TAC "box2" THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `x - e <= x + e <=> &0 <= e`] THEN SUBGOAL_THEN `&0 <= uv((tag:(real^M->bool)->real^M) k1) /\ &0 <= uv((tag:(real^M->bool)->real^M) k2)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; REAL_LT_IMP_LE]; ASM_REWRITE_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN MATCH_MP_TAC MONO_NOT THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; MATCH_MP_TAC(SET_RULE `i1 SUBSET s1 /\ i2 SUBSET s2 ==> DISJOINT s1 s2 ==> i1 INTER i2 = {}`) THEN CONJ_TAC THEN MATCH_MP_TAC(MESON[INTERIOR_SUBSET; SUBSET_TRANS] `s SUBSET t ==> interior s SUBSET t`) THEN MAP_EVERY EXPAND_TAC ["box"; "box2"] THEN REWRITE_TAC[SUBSET_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]; ASM_MESON_TAC[SUBSET]]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `e = e' /\ y <= x ==> x < e ==> y <= e'`) THEN CONJ_TAC THENL [REWRITE_TAC[real_div; REAL_MUL_AC]; ALL_TAC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE o lhand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN EXPAND_TAC "box" THEN REWRITE_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `a' <= e ==> a <= a' ==> a <= e`) THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; GSYM SUM_RMUL] THEN MATCH_MP_TAC SUM_LE_INCLUDED THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; RIGHT_EXISTS_AND_THM; FINITE_IMAGE] THEN REWRITE_TAC[NORM_POS_LE; EXISTS_IN_IMAGE] THEN EXISTS_TAC `SND:real^M#(real^M->bool)->real^M->bool` THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `&0 < uv(tag(k:real^M->bool):real^M):real` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `&0 < measure(box(uv(tag(k:real^M->bool):real^M):real) (tag k):real^M->bool)` MP_TAC THENL [EXPAND_TAC "box" THEN REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < x ==> a <= a + x`] THEN MATCH_MP_TAC PRODUCT_POS_LT_NUMSEG THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (funpow 2 RAND_CONV) [MATCH_MP(REAL_ARITH `&0 < x ==> x = abs x`) th] THEN ASSUME_TAC th) THEN REWRITE_TAC[real_div; GSYM REAL_ABS_INV] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM NORM_MUL] THEN SUBGOAL_THEN `mu <= dist(i (uv(tag(k:real^M->bool):real^M):real) (tag k):real^N, f(tag k))` MP_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> m <= x ==> m <= y`) THEN ONCE_REWRITE_TAC[DIST_SYM] THEN EXPAND_TAC "i" THEN REWRITE_TAC[dist; VECTOR_SUB_LDISTRIB] THEN UNDISCH_TAC `&0 < measure(box(uv(tag(k:real^M->bool):real^M):real) (tag k):real^M->bool)` THEN EXPAND_TAC "box" THEN REWRITE_TAC[MEASURE_INTERVAL] THEN SIMP_TAC[VECTOR_MUL_ASSOC; REAL_LT_IMP_NZ; REAL_MUL_LINV] THEN REWRITE_TAC[VECTOR_MUL_LID]);; let INTEGRABLE_CCONTINUOUS_EXPLICIT_SYMMETRIC = prove (`!f:real^M->real^N. (!a b. f integrable_on interval[a,b]) ==> ?k. negligible k /\ !x e. ~(x IN k) /\ &0 < e ==> ?d. &0 < d /\ !h. &0 < h /\ h < d ==> norm(inv(content(interval[x - h % vec 1,x + h % vec 1])) % integral (interval[x - h % vec 1,x + h % vec 1]) f - f(x)) < e`, REPEAT STRIP_TAC THEN MAP_EVERY ABBREV_TAC [`box = \h x. interval[x - h % vec 1:real^M,x + h % vec 1]`; `i = \h:real x:real^M. inv(content(box h x)) % integral (box h x) (f:real^M->real^N)`] THEN SUBGOAL_THEN `?k. negligible k /\ !x e. ~(x IN k) /\ &0 < e ==> ?d. &0 < d /\ !h. &0 < h /\ h < d ==> norm(i h x - (f:real^M->real^N) x) < e` MP_TAC THENL [ALL_TAC; MAP_EVERY EXPAND_TAC ["i"; "box"] THEN REWRITE_TAC[]] THEN EXISTS_TAC `{x | ~(!e. &0 < e ==> ?d. &0 < d /\ !h. &0 < h /\ h < d ==> norm(i h x - (f:real^M->real^N) x) < e)}` THEN SIMP_TAC[IN_ELIM_THM] THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | !d. &0 < d ==> ?h. &0 < h /\ h < d /\ inv(&k + &1) <= dist(i h x,(f:real^M->real^N) x)} | k IN (:num)}` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`y:real^M`; `e:real`] THEN STRIP_TAC THEN REWRITE_TAC[SIMPLE_IMAGE; UNIONS_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:num` THEN DISCH_TAC THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LT] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real` THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[dist] THEN MATCH_MP_TAC (REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `inv(&k)` THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `jj:num` THEN SUBGOAL_THEN `&0 < inv(&jj + &1)` MP_TAC THENL [REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; SPEC_TAC(`inv(&jj + &1)`,`mu:real`) THEN GEN_TAC THEN DISCH_TAC] THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN ASM_CASES_TAC `negligible(interval[a:real^M,b])` THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET; INTER_SUBSET]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[NEGLIGIBLE_INTERVAL]) THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `a - vec 1:real^M`; `b + vec 1:real^M`] HENSTOCK_LEMMA) THEN ANTS_TAC THENL [ASM_MESON_TAC[INTEGRABLE_ON_SUBINTERVAL; SUBSET_UNIV]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `(e * mu) / &2 / &3 pow (dimindex(:M))`) THEN ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; REAL_LT_MUL; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN REWRITE_TAC[SET_RULE `{x | P x} INTER s = {x | x IN s /\ P x}`] THEN ABBREV_TAC `E = {x | x IN interval[a,b] /\ !d. &0 < d ==> ?h. &0 < h /\ h < d /\ mu <= dist(i h x,(f:real^M->real^N) x)}` THEN SUBGOAL_THEN `!x. x IN E ==> ?h. &0 < h /\ (box h x:real^M->bool) SUBSET (g x) /\ (box h x:real^M->bool) SUBSET interval[a - vec 1,b + vec 1] /\ mu <= dist(i h x,(f:real^M->real^N) x)` MP_TAC THENL [X_GEN_TAC `x:real^M` THEN EXPAND_TAC "E" THEN REWRITE_TAC[IN_ELIM_THM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [gauge]) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o SPEC `x:real^M`) THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `min (&1) (d / &(dimindex(:M)))`)) THEN REWRITE_TAC[REAL_LT_MIN; REAL_LT_01; GSYM CONJ_ASSOC] THEN ASM_SIMP_TAC[REAL_LT_DIV; DIMINDEX_GE_1; LE_1; REAL_OF_NUM_LT] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(x:real^M,d)` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "box" THEN REWRITE_TAC[SUBSET; IN_INTERVAL; IN_BALL] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN SIMP_TAC[REAL_ARITH `x - h <= y /\ y <= x + h <=> abs(x - y) <= h`] THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `sum(1..dimindex(:M)) (\i. abs((x - y:real^M)$i))` THEN REWRITE_TAC[NORM_LE_L1] THEN MATCH_MP_TAC SUM_BOUND_LT_GEN THEN REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; IN_NUMSEG] THEN SIMP_TAC[NOT_LT; DIMINDEX_GE_1; CARD_NUMSEG_1; VECTOR_SUB_COMPONENT] THEN ASM_MESON_TAC[REAL_LET_TRANS]; UNDISCH_TAC `(x:real^M) IN interval[a,b]` THEN EXPAND_TAC "box" THEN REWRITE_TAC[SUBSET; IN_INTERVAL] THEN DISCH_THEN(fun th -> X_GEN_TAC `y:real^M` THEN MP_TAC th) THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `uv:real^M->real` THEN REWRITE_TAC[TAUT `(a ==> b /\ c) <=> (a ==> b) /\ (a ==> c)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPECL [`a:real^M`; `b:real^M`; `E:real^M->bool`; `\x:real^M. if x IN E then ball(x,uv x) else g(x)`] COVERING_LEMMA) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[INTERVAL_NE_EMPTY] THEN CONJ_TAC THENL [EXPAND_TAC "E" THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[gauge] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN RULE_ASSUM_TAC(REWRITE_RULE[gauge]) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_TAC `D:(real^M->bool)->bool`) THEN EXISTS_TAC `UNIONS D:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `measurable(UNIONS D:real^M->bool) /\ measure(UNIONS D) <= measure(interval[a:real^M,b])` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `?d. d SUBSET D /\ FINITE d /\ measure(UNIONS D:real^M->bool) <= &2 * measure(UNIONS d)` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `measure(UNIONS D:real^M->bool) = &0` THENL [EXISTS_TAC `{}:(real^M->bool)->bool` THEN ASM_REWRITE_TAC[FINITE_EMPTY; EMPTY_SUBSET; MEASURE_EMPTY; UNIONS_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV; MP_TAC(ISPECL [`D:(real^M->bool)->bool`; `measure(interval[a:real^M,b])`; `measure(UNIONS D:real^M->bool) / &2`] MEASURE_COUNTABLE_UNIONS_APPROACHABLE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; REAL_HALF] THEN ASM_SIMP_TAC[GSYM MEASURABLE_MEASURE_EQ_0] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL; MEASURABLE_UNIONS]; ALL_TAC]) THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN REAL_ARITH_TAC]]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o el 3 o CONJUNCTS) THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN ONCE_REWRITE_TAC[TAUT `a /\ b <=> ~(a ==> ~b)`] THEN SIMP_TAC[IN_INTER] THEN REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN DISCH_THEN(X_CHOOSE_TAC `tag:(real^M->bool)->real^M`) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `D <= &2 * d ==> d <= e / &2 ==> D <= e`)) THEN MP_TAC(ISPEC `IMAGE (\k:real^M->bool. (box:real->real^M->real^M->bool) (uv(tag k):real) ((tag k:real^M))) d` AUSTIN_LEMMA) THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN EXPAND_TAC "box" THEN EXISTS_TAC `&2 * uv((tag:(real^M->bool)->real^M) k):real` THEN EXISTS_TAC `(tag:(real^M->bool)->real^M) k - uv(tag k) % vec 1:real^M` THEN EXISTS_TAC `(tag:(real^M->bool)->real^M) k + uv(tag k) % vec 1:real^M` THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[EXISTS_SUBSET_IMAGE; real_ge] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN DISCH_THEN(X_CHOOSE_THEN `p:(real^M->bool)->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REAL_ARITH `d <= d' /\ p <= e ==> d' <= p ==> d <= e`) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN EXPAND_TAC "box" THEN REWRITE_TAC[MEASURABLE_INTERVAL]; REWRITE_TAC[SUBSET; IN_UNIONS; EXISTS_IN_IMAGE] THEN X_GEN_TAC `z:real^M` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `(z:real^M) IN k` THEN SPEC_TAC(`z:real^M`,`z:real^M`) THEN REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(tag k:real^M,uv(tag(k:real^M->bool)))` THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN EXPAND_TAC "box" THEN REWRITE_TAC[SUBSET; IN_BALL; IN_INTERVAL] THEN X_GEN_TAC `z:real^M` THEN REWRITE_TAC[dist] THEN DISCH_TAC THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN SIMP_TAC[REAL_ARITH `x - h <= y /\ y <= x + h <=> abs(x - y) <= h`] THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LT_IMP_LE; REAL_LE_TRANS]]; ALL_TAC] THEN SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN SUBGOAL_THEN `FINITE(p:(real^M->bool)->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_LCANCEL_IMP THEN EXISTS_TAC `mu:real` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\k. (tag:(real^M->bool)->real^M) k, (box(uv(tag k):real) (tag k):real^M->bool)) p`) THEN ANTS_TAC THENL [REWRITE_TAC[tagged_partial_division_of; fine] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IN_IMAGE; PAIR_EQ] THEN REWRITE_TAC[MESON[] `(!x j. (?k. (x = tag k /\ j = g k) /\ k IN d) ==> P x j) <=> (!k. k IN d ==> P (tag k) (g k))`] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN REPEAT CONJ_TAC THENL [EXPAND_TAC "box" THEN REWRITE_TAC[IN_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 < u ==> x - u <= x /\ x <= x + u`) THEN ASM_MESON_TAC[SUBSET]; ASM_MESON_TAC[SUBSET]; EXPAND_TAC "box" THEN MESON_TAC[]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k1:real^M->bool` THEN ASM_CASES_TAC `(k1:real^M->bool) IN p` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k2:real^M->bool` THEN ASM_CASES_TAC `(k2:real^M->bool) IN p` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(tag:(real^M->bool)->real^M) k1 = tag k2` THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [EXPAND_TAC "box" THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `x - e <= x + e <=> &0 <= e`] THEN SUBGOAL_THEN `&0 <= uv((tag:(real^M->bool)->real^M) k1) /\ &0 <= uv((tag:(real^M->bool)->real^M) k2)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; REAL_LT_IMP_LE]; ASM_REWRITE_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN MATCH_MP_TAC MONO_NOT THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; MATCH_MP_TAC(SET_RULE `i1 SUBSET s1 /\ i2 SUBSET s2 ==> DISJOINT s1 s2 ==> i1 INTER i2 = {}`) THEN REWRITE_TAC[INTERIOR_SUBSET]]; ASM_MESON_TAC[SUBSET]]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `e = e' /\ y <= x ==> x < e ==> y <= e'`) THEN CONJ_TAC THENL [REWRITE_TAC[real_div; REAL_MUL_AC]; ALL_TAC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE o lhand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN EXPAND_TAC "box" THEN REWRITE_TAC[MEASURABLE_INTERVAL]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `a' <= e ==> a <= a' ==> a <= e`) THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; GSYM SUM_RMUL] THEN MATCH_MP_TAC SUM_LE_INCLUDED THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; RIGHT_EXISTS_AND_THM; FINITE_IMAGE] THEN REWRITE_TAC[NORM_POS_LE; EXISTS_IN_IMAGE] THEN EXISTS_TAC `SND:real^M#(real^M->bool)->real^M->bool` THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `&0 < uv(tag(k:real^M->bool):real^M):real` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `&0 < measure(box(uv(tag(k:real^M->bool):real^M):real) (tag k):real^M->bool)` MP_TAC THENL [EXPAND_TAC "box" THEN REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < x ==> a - x <= a + x`] THEN MATCH_MP_TAC PRODUCT_POS_LT_NUMSEG THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (funpow 2 RAND_CONV) [MATCH_MP(REAL_ARITH `&0 < x ==> x = abs x`) th] THEN ASSUME_TAC th) THEN REWRITE_TAC[real_div; GSYM REAL_ABS_INV] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM NORM_MUL] THEN SUBGOAL_THEN `mu <= dist(i (uv(tag(k:real^M->bool):real^M):real) (tag k):real^N, f(tag k))` MP_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> m <= x ==> m <= y`) THEN ONCE_REWRITE_TAC[DIST_SYM] THEN EXPAND_TAC "i" THEN REWRITE_TAC[dist; VECTOR_SUB_LDISTRIB] THEN UNDISCH_TAC `&0 < measure(box(uv(tag(k:real^M->bool):real^M):real) (tag k):real^M->bool)` THEN EXPAND_TAC "box" THEN REWRITE_TAC[MEASURE_INTERVAL] THEN SIMP_TAC[VECTOR_MUL_ASSOC; REAL_LT_IMP_NZ; REAL_MUL_LINV] THEN REWRITE_TAC[VECTOR_MUL_LID]);; let HAS_VECTOR_DERIVATIVE_INDEFINITE_INTEGRAL = prove (`!f:real^1->real^N a b. f integrable_on interval[a,b] ==> ?k. negligible k /\ !x. x IN interval[a,b] DIFF k ==> ((\x. integral(interval[a,x]) f) has_vector_derivative f(x)) (at x within interval[a,b])`, SUBGOAL_THEN `!f:real^1->real^N a b. f integrable_on interval[a,b] ==> ?k. negligible k /\ !x e. x IN interval[a,b] DIFF k /\ &0 < e ==> ?d. &0 < d /\ !x'. x' IN interval[a,b] /\ drop x < drop x' /\ drop x' < drop x + d ==> norm(integral(interval[x,x']) f - drop(x' - x) % f x) / norm(x' - x) < e` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPEC `(\x. if x IN interval[a,b] then f x else vec 0):real^1->real^N` INTEGRABLE_CCONTINUOUS_EXPLICIT) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT GEN_TAC THEN MATCH_MP_TAC INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `(:real^1)` THEN ASM_REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV; SUBSET_UNIV]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^1->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `e:real`] THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^1`; `e:real`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^1` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `drop y - drop x`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `x + (drop y - drop x) % vec 1 = y` SUBST1_TAC THENL [REWRITE_TAC[GSYM DROP_EQ; DROP_ADD; DROP_CMUL; DROP_VEC] THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[CONTENT_1; REAL_LT_IMP_LE] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> x < e ==> y < e`) THEN ASM_SIMP_TAC[REAL_EQ_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ; GSYM DROP_EQ; REAL_LT_IMP_NE] THEN SUBGOAL_THEN `norm(y - x) = abs(drop y - drop x)` SUBST1_TAC THENL [REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB]; ALL_TAC] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM NORM_MUL)] THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_FIELD `x < y ==> (y - x) * inv(y - x) = &1`] THEN AP_TERM_TAC THEN REWRITE_TAC[DROP_SUB; VECTOR_MUL_LID] THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_EQ THEN X_GEN_TAC `z:real^1` THEN REWRITE_TAC[DIFF_EMPTY] THEN DISCH_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(ISPECL [`f:real^1->real^N`; `a:real^1`; `b:real^1`] th) THEN MP_TAC(ISPECL [`\x. (f:real^1->real^N) (--x)`; `--b:real^1`; `--a:real^1`] th)) THEN ASM_REWRITE_TAC[INTEGRABLE_REFLECT] THEN DISCH_THEN(X_CHOOSE_THEN `k2:real^1->bool` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "2"))) THEN DISCH_THEN(X_CHOOSE_THEN `k1:real^1->bool` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "1"))) THEN EXISTS_TAC `k1 UNION IMAGE (--) k2:real^1->bool` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_UNION THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_LINEAR_IMAGE THEN ASM_REWRITE_TAC[linear] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_DIFF; IN_UNION; DE_MORGAN_THM] THEN REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `x:real^1 = --x' <=> --x = x'`] THEN REWRITE_TAC[UNWIND_THM1] THEN STRIP_TAC THEN REWRITE_TAC[has_vector_derivative; HAS_DERIVATIVE_WITHIN] THEN CONJ_TAC THENL [REWRITE_TAC[linear; DROP_ADD; DROP_CMUL] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN REMOVE_THEN "2" (MP_TAC o SPECL [`--x:real^1`; `e:real`]) THEN REMOVE_THEN "1" (MP_TAC o SPECL [`x:real^1`; `e:real`]) THEN ASM_REWRITE_TAC[IN_DIFF; IN_INTERVAL_REFLECT] THEN DISCH_THEN(X_CHOOSE_THEN `d1:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "1"))) THEN DISCH_THEN(X_CHOOSE_THEN `d2:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "2"))) THEN EXISTS_TAC `min d1 d2:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_INTERVAL_1] THEN REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN STRIP_TAC THEN SUBGOAL_THEN `drop x < drop y \/ drop y < drop x` DISJ_CASES_TAC THENL [ASM_REAL_ARITH_TAC; REMOVE_THEN "1" (MP_TAC o SPEC `y:real^1`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> x < e ==> y < e`) THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC(VECTOR_ARITH `c + a:real^N = b ==> a = b - c`) THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN MATCH_MP_TAC INTEGRABLE_SUBINTERVAL THEN MAP_EVERY EXISTS_TAC [`a:real^1`; `b:real^1`] THEN ASM_REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; REMOVE_THEN "2" (MP_TAC o SPEC `--y:real^1`) THEN ANTS_TAC THENL [SIMP_TAC[DROP_NEG] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `norm(--y - --x) = abs(drop y - drop x)` SUBST1_TAC THENL [REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB; DROP_NEG] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> x < e ==> y < e`) THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[INTEGRAL_REFLECT] THEN REWRITE_TAC[VECTOR_NEG_NEG; DROP_SUB; DROP_NEG] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `x - (--a - --b) % y:real^N = --(--x - (a - b) % y)`] THEN REWRITE_TAC[NORM_NEG] THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC(VECTOR_ARITH `b + a = c ==> --a:real^N = b - c`) THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN MATCH_MP_TAC INTEGRABLE_SUBINTERVAL THEN MAP_EVERY EXISTS_TAC [`a:real^1`; `b:real^1`] THEN ASM_REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]);; let ABSOLUTELY_INTEGRABLE_LEBESGUE_POINTS = prove (`!f:real^M->real^N. (!a b. f absolutely_integrable_on interval[a,b]) ==> ?k. negligible k /\ !x e. ~(x IN k) /\ &0 < e ==> ?d. &0 < d /\ !h. &0 < h /\ h < d ==> norm(inv(content(interval[x - h % vec 1, x + h % vec 1])) % integral (interval[x - h % vec 1, x + h % vec 1]) (\t. lift(norm(f t - f x)))) < e`, REPEAT STRIP_TAC THEN MP_TAC(GEN `r:real^N` (ISPEC `\t. lift(norm((f:real^M->real^N) t - r))` INTEGRABLE_CCONTINUOUS_EXPLICIT_SYMMETRIC)) THEN REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP MONO_FORALL) THEN ANTS_TAC THENL [REPEAT GEN_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_CONST]; ALL_TAC] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `k:real^N->real^M->bool` THEN STRIP_TAC THEN EXISTS_TAC `UNIONS (IMAGE (k:real^N->real^M->bool) {x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)})` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_RATIONAL_COORDINATES] THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `e:real`] THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; NOT_EXISTS_THM] THEN REWRITE_TAC[TAUT `~(p /\ q) <=> p ==> ~q`] THEN STRIP_TAC THEN MP_TAC(SET_RULE `(f:real^M->real^N) x IN (:real^N)`) THEN REWRITE_TAC[GSYM CLOSURE_RATIONAL_COORDINATES] THEN REWRITE_TAC[CLOSURE_APPROACHABLE; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &3 <=> &0 < e`] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`r:real^N`; `x:real^M`; `e / &3`]) THEN ASM_SIMP_TAC[REAL_ARITH `&0 < e / &3 <=> &0 < e`] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `h:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `h:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `norm(y1:real^N) < e / &3 /\ norm(i1 - i2) <= e / &3 ==> norm(i1 - y1) < e / &3 ==> norm(i2) < e`) THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_NORM] THEN CONJ_TAC THENL [ASM_MESON_TAC[dist; DIST_SYM]; ALL_TAC] THEN REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `abs(inv(content(interval[x - h % vec 1,x + h % vec 1]))) * drop(integral (interval[x - h % vec 1,x + h % vec 1]) (\x:real^M. lift(e / &3)))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_SUB o rand o lhand o snd) THEN ANTS_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_CONST]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[INTEGRABLE_CONST] THEN CONJ_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_CONST]; X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_NORM; LIFT_DROP; GSYM LIFT_SUB] THEN ASM_MESON_TAC[NORM_ARITH `dist(r,x) < e / &3 ==> abs(norm(y - r:real^N) - norm(y - x)) <= e / &3`]]]; ASM_CASES_TAC `content(interval[x - h % vec 1:real^M,x + h % vec 1]) = &0` THENL [ASM_REWRITE_TAC[REAL_INV_0; REAL_ABS_NUM; REAL_MUL_LZERO] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_ABS_INV] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; GSYM REAL_ABS_NZ] THEN REWRITE_TAC[INTEGRAL_CONST; DROP_CMUL; LIFT_DROP] THEN SIMP_TAC[real_abs; CONTENT_POS_LE; REAL_MUL_SYM; REAL_LE_REFL]]]);; (* ------------------------------------------------------------------------- *) (* Measurability of a function on a set (not necessarily itself measurable). *) (* ------------------------------------------------------------------------- *) parse_as_infix("measurable_on",(12,"right"));; let measurable_on = new_definition `(f:real^M->real^N) measurable_on s <=> ?k g. negligible k /\ (!n. (g n) continuous_on (:real^M)) /\ (!x. ~(x IN k) ==> ((\n. g n x) --> if x IN s then f(x) else vec 0) sequentially)`;; let MEASURABLE_ON_UNIV = prove (`(\x. if x IN s then f(x) else vec 0) measurable_on (:real^M) <=> f measurable_on s`, REWRITE_TAC[measurable_on; IN_UNIV; ETA_AX]);; (* ------------------------------------------------------------------------- *) (* Lebesgue measurability (like "measurable" but allowing infinite measure) *) (* ------------------------------------------------------------------------- *) let lebesgue_measurable = new_definition `lebesgue_measurable s <=> (indicator s) measurable_on (:real^N)`;; (* ------------------------------------------------------------------------- *) (* Relation between measurability and integrability. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE = prove (`!f:real^M->real^N g s. f measurable_on s /\ g integrable_on s /\ (!x. x IN s ==> norm(f x) <= drop(g x)) ==> f integrable_on s`, let lemma = prove (`!f:real^M->real^N g a b. f measurable_on (:real^M) /\ g integrable_on interval[a,b] /\ (!x. x IN interval[a,b] ==> norm(f x) <= drop(g x)) ==> f integrable_on interval[a,b]`, REPEAT GEN_TAC THEN REWRITE_TAC[measurable_on; IN_UNIV] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `h:num->real^M->real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `interval[a:real^M,b] DIFF k` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC DOMINATED_CONVERGENCE_INTEGRABLE THEN MAP_EVERY EXISTS_TAC [`h:num->real^M->real^N`; `g:real^M->real^1`] THEN ASM_SIMP_TAC[IN_DIFF] THEN REWRITE_TAC[LEFT_AND_FORALL_THM] THEN X_GEN_TAC `n:num` THEN UNDISCH_TAC `(g:real^M->real^1) integrable_on interval [a,b]` THEN SUBGOAL_THEN `(h:num->real^M->real^N) n absolutely_integrable_on interval[a,b]` MP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_CONTINUOUS THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; REWRITE_TAC[IMP_IMP; absolutely_integrable_on; GSYM CONJ_ASSOC] THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THEN MATCH_MP_TAC INTEGRABLE_SPIKE_SET THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]]) in ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRABLE_ON_ALL_INTERVALS_INTEGRABLE_BOUND THEN EXISTS_TAC `g:real^M->real^1` THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN MATCH_MP_TAC lemma THEN EXISTS_TAC `(\x. if x IN s then g x else vec 0):real^M->real^1` THEN RULE_ASSUM_TAC(ONCE_REWRITE_RULE[INTEGRABLE_ALT]) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[NORM_0; DROP_VEC; REAL_POS]);; let MEASURABLE_BOUNDED_AE_BY_INTEGRABLE_IMP_INTEGRABLE = prove (`!f:real^M->real^N g s k. f measurable_on s /\ g integrable_on s /\ negligible k /\ (!x. x IN s DIFF k ==> norm(f x) <= drop(g x)) ==> f integrable_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `\x. if x IN k then lift(norm((f:real^M->real^N) x)) else g x` THEN ASM_SIMP_TAC[COND_RAND; IN_DIFF; LIFT_DROP; REAL_LE_REFL; COND_ID] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE) THEN MAP_EVERY EXISTS_TAC [`g:real^M->real^1`; `k:real^M->bool`] THEN ASM_SIMP_TAC[IN_DIFF]);; let MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE = prove (`!f:real^M->real^N g s. f measurable_on s /\ g integrable_on s /\ (!x. x IN s ==> norm(f x) <= drop(g x)) ==> f absolutely_integrable_on s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^M->real^1`] ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_BOUND) THEN DISCH_THEN MATCH_MP_TAC THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[NORM_REAL; GSYM drop] THEN ASM_MESON_TAC[REAL_ABS_LE; REAL_LE_TRANS]; ASM_MESON_TAC[MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE]; MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ASM_REWRITE_TAC[IMP_IMP; DIMINDEX_1; FORALL_1; GSYM drop] THEN ASM_MESON_TAC[NORM_ARITH `norm(x) <= a ==> &0 <= a`]]);; let INTEGRAL_DROP_LE_MEASURABLE = prove (`!f g s:real^N->bool. f measurable_on s /\ g integrable_on s /\ (!x. x IN s ==> &0 <= drop(f x) /\ drop(f x) <= drop(g x)) ==> drop(integral s f) <= drop(integral s g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `g:real^N->real^1` THEN ASM_SIMP_TAC[NORM_REAL; GSYM drop; real_abs]);; let INTEGRABLE_SUBINTERVALS_IMP_MEASURABLE = prove (`!f:real^M->real^N. (!a b. f integrable_on interval[a,b]) ==> f measurable_on (:real^M)`, REPEAT STRIP_TAC THEN REWRITE_TAC[measurable_on; IN_UNIV] THEN MAP_EVERY ABBREV_TAC [`box = \h x. interval[x:real^M,x + h % vec 1]`; `i = \h:real x:real^M. inv(content(box h x)) % integral (box h x) (f:real^M->real^N)`] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `(\n x. i (inv(&n + &1)) x):num->real^M->real^N` THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN CONJ_TAC THENL [REWRITE_TAC[continuous_on; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`n:num`; `x:real^M`; `e:real`] THEN DISCH_TAC THEN EXPAND_TAC "i" THEN EXPAND_TAC "box" THEN MP_TAC(ISPECL [`f:real^M->real^N`; `x - &2 % vec 1:real^M`; `x + &2 % vec 1:real^M`; `x:real^M`; `x + inv(&n + &1) % vec 1:real^M`; `e * (&1 / (&n + &1)) pow dimindex(:M)`] INDEFINITE_INTEGRAL_CONTINUOUS) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[IN_INTERVAL; VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; REAL_MUL_RID; VECTOR_MUL_COMPONENT; VEC_COMPONENT] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [SUBGOAL_THEN `&0 <= inv(&n + &1) /\ inv(&n + &1) <= &1` MP_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ; REAL_ARITH `&0 <= &n + &1`] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN REAL_ARITH_TAC; MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_POW_LT THEN MATCH_MP_TAC REAL_LT_DIV THEN REAL_ARITH_TAC]; DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min k (&1)` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_01] THEN ASM_REWRITE_TAC[dist] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN REWRITE_TAC[CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `a <= a + x <=> &0 <= x`] THEN REWRITE_TAC[REAL_LE_INV_EQ; REAL_ARITH `&0 <= &n + &1`] THEN REWRITE_TAC[REAL_ARITH `(x + inv y) - x = &1 / y`] THEN REWRITE_TAC[PRODUCT_CONST_NUMSEG; ADD_SUB] THEN REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_POW; REAL_ABS_DIV] THEN REWRITE_TAC[REAL_ABS_NUM; REAL_ARITH `abs(&n + &1) = &n + &1`] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_DIV; REAL_POW_LT; REAL_ARITH `&0 < &1 /\ &0 < &n + &1`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[VECTOR_ARITH `(y + i) - (x + i):real^N = y - x`; VECTOR_ARITH `(y - i) - (x - i):real^N = y - x`] THEN ASM_SIMP_TAC[IN_INTERVAL; REAL_LT_IMP_LE] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; dist; VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i /\ i <= dimindex(:M)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= i /\ i <= &1 /\ abs(x - y) <= &1 ==> (x - &2 <= y /\ y <= x + &2) /\ (x - &2 <= y + i /\ y + i <= x + &2)`) THEN ASM_SIMP_TAC[REAL_LE_INV_EQ; REAL_INV_LE_1; REAL_ARITH `&0 <= &n + &1 /\ &1 <= &n + &1`] THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LT_IMP_LE; NORM_SUB; REAL_LE_TRANS]]; FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_CCONTINUOUS_EXPLICIT) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN ASM_CASES_TAC `negligible(k:real^M->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^M` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN MP_TAC(SPEC `d:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN MAP_EVERY EXPAND_TAC ["i"; "box"] THEN REWRITE_TAC[dist] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]);; let INTEGRABLE_IMP_MEASURABLE = prove (`!f:real^M->real^N s. f integrable_on s ==> f measurable_on s`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV; GSYM MEASURABLE_ON_UNIV] THEN SPEC_TAC(`\x. if x IN s then (f:real^M->real^N) x else vec 0`, `f:real^M->real^N`) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRABLE_SUBINTERVALS_IMP_MEASURABLE THEN REPEAT GEN_TAC THEN MATCH_MP_TAC INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[SUBSET_UNIV]);; let ABSOLUTELY_INTEGRABLE_MEASURABLE = prove (`!f:real^M->real^N s. f absolutely_integrable_on s <=> f measurable_on s /\ (\x. lift(norm(f x))) integrable_on s`, REPEAT GEN_TAC THEN REWRITE_TAC[absolutely_integrable_on] THEN MATCH_MP_TAC(TAUT `(a ==> b) /\ (b /\ c ==> a) ==> (a /\ c <=> b /\ c)`) THEN REWRITE_TAC[INTEGRABLE_IMP_MEASURABLE] THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `\x. lift(norm((f:real^M->real^N) x))` THEN ASM_REWRITE_TAC[LIFT_DROP; REAL_LE_REFL]);; (* ------------------------------------------------------------------------- *) (* Composing continuous and measurable functions; a few variants. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_ON_COMPOSE_CONTINUOUS = prove (`!f:real^M->real^N g:real^N->real^P. f measurable_on (:real^M) /\ g continuous_on (:real^N) ==> (g o f) measurable_on (:real^M)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[measurable_on; IN_UNIV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `h:num->real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\n x. (g:real^N->real^P) ((h:num->real^M->real^N) n x)` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[ETA_AX] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_SEQUENTIALLY]) THEN ASM_SIMP_TAC[o_DEF; IN_UNIV]]);; let MEASURABLE_ON_COMPOSE_CONTINUOUS_0 = prove (`!f:real^M->real^N g:real^N->real^P s. f measurable_on s /\ g continuous_on (:real^N) /\ g(vec 0) = vec 0 ==> (g o f) measurable_on s`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_COMPOSE_CONTINUOUS) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; o_DEF] THEN ASM_MESON_TAC[]);; let MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL = prove (`!f:real^M->real^N g:real^N->real^P a b. f measurable_on (:real^M) /\ (!x. f(x) IN interval(a,b)) /\ g continuous_on interval(a,b) ==> (g o f) measurable_on (:real^M)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[measurable_on; IN_UNIV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `h:num->real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(\n x. (g:real^N->real^P) (lambda i. max ((a:real^N)$i + (b$i - a$i) / (&n + &2)) (min ((h n x:real^N)$i) ((b:real^N)$i - (b$i - a$i) / (&n + &2))))) :num->real^M->real^P` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL [MP_TAC(ISPECL [`(:real^M)`; `(lambda i. (b:real^N)$i - (b$i - (a:real^N)$i) / (&n + &2)):real^N`] CONTINUOUS_ON_CONST) THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP CONTINUOUS_ON_MIN) THEN MP_TAC(ISPECL [`(:real^M)`; `(lambda i. (a:real^N)$i + ((b:real^N)$i - a$i) / (&n + &2)):real^N`] CONTINUOUS_ON_CONST) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP CONTINUOUS_ON_MAX) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[FUN_EQ_THM; CART_EQ; LAMBDA_BETA]; MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval(a:real^N,b)` THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M` o CONJUNCT1) THEN SIMP_TAC[IN_INTERVAL; LAMBDA_BETA] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `&0 < ((b:real^N)$i - (a:real^N)$i) / (&n + &2) /\ ((b:real^N)$i - (a:real^N)$i) / (&n + &2) <= (b$i - a$i) / &2` MP_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_LDIV_EQ; REAL_ARITH `&0 < &n + &2`] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[real_div]] THEN MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC REAL_LE_INV2 THEN REAL_ARITH_TAC]]; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC LIM_CONTINUOUS_FUNCTION THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `((\n. (lambda i. ((a:real^N)$i + ((b:real^N)$i - a$i) / (&n + &2)))) --> a) sequentially /\ ((\n. (lambda i. ((b:real^N)$i - ((b:real^N)$i - a$i) / (&n + &2)))) --> b) sequentially` MP_TAC THENL [ONCE_REWRITE_TAC[LIM_COMPONENTWISE_LIFT] THEN SIMP_TAC[LAMBDA_BETA] THEN CONJ_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN REWRITE_TAC[real_sub] THEN GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN REWRITE_TAC[LIFT_ADD] THEN MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST; LIFT_NEG; real_div; LIFT_CMUL] THEN GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_NEG_0] THEN TRY(MATCH_MP_TAC LIM_NEG) THEN REWRITE_TAC[VECTOR_NEG_0] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^1 = ((b:real^N)$j + --((a:real^N)$j)) % vec 0`) THEN MATCH_MP_TAC LIM_CMUL THEN REWRITE_TAC[LIM_SEQUENTIALLY; DIST_0; NORM_LIFT] THEN X_GEN_TAC `e:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN REWRITE_TAC[REAL_ABS_INV] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LT; LE_1; REAL_OF_NUM_LE; REAL_ABS_NUM] THEN ASM_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[TAUT `a ==> b /\ c ==> d <=> a /\ c ==> b ==> d`] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_MIN) THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_MAX) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN BINOP_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; FUN_EQ_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL]) THEN ASM_MESON_TAC[REAL_ARITH `a < x /\ x < b ==> max a (min x b) = x`]]);; let MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET = prove (`!f:real^M->real^N g:real^N->real^P s. closed s /\ f measurable_on (:real^M) /\ (!x. f(x) IN s) /\ g continuous_on s ==> (g o f) measurable_on (:real^M)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`g:real^N->real^P`; `(:real^N)`; `s:real^N->bool`] TIETZE_UNBOUNDED) THEN ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN] THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^N->real^P` THEN DISCH_TAC THEN SUBGOAL_THEN `(g:real^N->real^P) o (f:real^M->real^N) = h o f` SUBST1_TAC THENL [ASM_SIMP_TAC[FUN_EQ_THM; o_THM]; ALL_TAC] THEN MATCH_MP_TAC MEASURABLE_ON_COMPOSE_CONTINUOUS THEN ASM_REWRITE_TAC[]);; let MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET_0 = prove (`!f:real^M->real^N g:real^N->real^P s t. closed s /\ f measurable_on t /\ (!x. f(x) IN s) /\ g continuous_on s /\ vec 0 IN s /\ g(vec 0) = vec 0 ==> (g o f) measurable_on t`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN MP_TAC(ISPECL [`(\x. if x IN t then f x else vec 0):real^M->real^N`; `g:real^N->real^P`; `s:real^N->bool`] MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[MEASURABLE_ON_UNIV] THEN ASM_MESON_TAC[]; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; o_THM] THEN ASM_MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Basic closure properties of measurable functions. *) (* ------------------------------------------------------------------------- *) let CONTINUOUS_IMP_MEASURABLE_ON = prove (`!f:real^M->real^N. f continuous_on (:real^M) ==> f measurable_on (:real^M)`, REPEAT STRIP_TAC THEN REWRITE_TAC[measurable_on; IN_UNIV] THEN EXISTS_TAC `{}:real^M->bool` THEN REWRITE_TAC[NEGLIGIBLE_EMPTY] THEN EXISTS_TAC `\n:num. (f:real^M->real^N)` THEN ASM_REWRITE_TAC[LIM_CONST]);; let MEASURABLE_ON_CONST = prove (`!k:real^N. (\x. k) measurable_on (:real^M)`, SIMP_TAC[CONTINUOUS_IMP_MEASURABLE_ON; CONTINUOUS_ON_CONST]);; let MEASURABLE_ON_0 = prove (`!s. (\x. vec 0) measurable_on s`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_CONST; COND_ID]);; let MEASURABLE_ON_CMUL = prove (`!c f:real^M->real^N s. f measurable_on s ==> (\x. c % f x) measurable_on s`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC MEASURABLE_ON_COMPOSE_CONTINUOUS_0 THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID]);; let MEASURABLE_ON_CMUL_EQ = prove (`!f:real^M->real^N s c. (\x. c % f x) measurable_on s <=> c = &0 \/ f measurable_on s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; MEASURABLE_ON_0] THEN EQ_TAC THEN REWRITE_TAC[MEASURABLE_ON_CMUL] THEN DISCH_THEN(MP_TAC o SPEC `inv(c):real` o MATCH_MP MEASURABLE_ON_CMUL) THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV] THEN REWRITE_TAC[VECTOR_MUL_LID; ETA_AX]);; let MEASURABLE_ON_NEG = prove (`!f:real^M->real^N s. f measurable_on s ==> (\x. --(f x)) measurable_on s`, REWRITE_TAC[VECTOR_ARITH `--x:real^N = --(&1) % x`; MEASURABLE_ON_CMUL]);; let MEASURABLE_ON_NEG_EQ = prove (`!f:real^M->real^N s. (\x. --(f x)) measurable_on s <=> f measurable_on s`, REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_NEG) THEN REWRITE_TAC[VECTOR_NEG_NEG; ETA_AX]);; let MEASURABLE_ON_NORM = prove (`!f:real^M->real^N s. f measurable_on s ==> (\x. lift(norm(f x))) measurable_on s`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o ISPEC `\x:real^N. lift(norm x)` o MATCH_MP (REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_COMPOSE_CONTINUOUS_0)) THEN REWRITE_TAC[o_DEF; NORM_0; LIFT_NUM] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[continuous_on; IN_UNIV; DIST_LIFT] THEN GEN_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[] THEN NORM_ARITH_TAC);; let MEASURABLE_ON_LIFT_ABS = prove (`!f:real^N->real s. (\x. lift(f x)) measurable_on s ==> (\x. lift(abs(f x))) measurable_on s`, REWRITE_TAC[GSYM NORM_LIFT; MEASURABLE_ON_NORM]);; let MEASURABLE_ON_PASTECART = prove (`!f:real^M->real^N g:real^M->real^P s. f measurable_on s /\ g measurable_on s ==> (\x. pastecart (f x) (g x)) measurable_on s`, REPEAT GEN_TAC THEN REWRITE_TAC[measurable_on] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `k1:real^M->bool` MP_TAC) (X_CHOOSE_THEN `k2:real^M->bool` MP_TAC)) THEN DISCH_THEN(X_CHOOSE_THEN `g2:num->real^M->real^P` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `g1:num->real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `k1 UNION k2:real^M->bool` THEN ASM_SIMP_TAC[NEGLIGIBLE_UNION] THEN EXISTS_TAC `(\n x. pastecart (g1 n x) (g2 n x)) :num->real^M->real^(N,P)finite_sum` THEN ASM_SIMP_TAC[CONTINUOUS_ON_PASTECART; ETA_AX; IN_UNION; DE_MORGAN_THM] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_CASES_TAC `(x:real^M) IN s` THEN REWRITE_TAC[GSYM PASTECART_VEC] THEN ASM_SIMP_TAC[LIM_PASTECART]);; let MEASURABLE_ON_COMBINE = prove (`!h:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P s. f measurable_on s /\ g measurable_on s /\ (\x. h (fstcart x) (sndcart x)) continuous_on UNIV /\ h (vec 0) (vec 0) = vec 0 ==> (\x. h (f x) (g x)) measurable_on s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(\x:real^M. (h:real^N->real^P->real^Q) (f x) (g x)) = (\x. h (fstcart x) (sndcart x)) o (\x. pastecart (f x) (g x))` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; FSTCART_PASTECART; SNDCART_PASTECART; o_THM]; MATCH_MP_TAC MEASURABLE_ON_COMPOSE_CONTINUOUS_0 THEN ASM_SIMP_TAC[MEASURABLE_ON_PASTECART; FSTCART_VEC; SNDCART_VEC]]);; let MEASURABLE_ON_ADD = prove (`!f:real^M->real^N g:real^M->real^N s. f measurable_on s /\ g measurable_on s ==> (\x. f x + g x) measurable_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_COMBINE THEN ASM_REWRITE_TAC[VECTOR_ADD_LID] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]);; let MEASURABLE_ON_SUB = prove (`!f:real^M->real^N g:real^M->real^N s. f measurable_on s /\ g measurable_on s ==> (\x. f x - g x) measurable_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_COMBINE THEN ASM_REWRITE_TAC[VECTOR_SUB_RZERO] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]);; let MEASURABLE_ON_MAX = prove (`!f:real^M->real^N g:real^M->real^N s. f measurable_on s /\ g measurable_on s ==> (\x. (lambda i. max ((f x)$i) ((g x)$i)):real^N) measurable_on s`, let lemma = REWRITE_RULE[] (ISPEC `(\x y. lambda i. max (x$i) (y$i)):real^N->real^N->real^N` MEASURABLE_ON_COMBINE) in REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT] THEN REWRITE_TAC[REAL_ARITH `max x x = x`; LAMBDA_ETA] THEN SIMP_TAC[continuous_on; LAMBDA_BETA; IN_UNIV; DIST_LIFT] THEN GEN_TAC THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^(N,N)finite_sum`; `e:real`] THEN DISCH_TAC THEN EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[dist] THEN X_GEN_TAC `y:real^(N,N)finite_sum` THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `abs(x - y) < e /\ abs(x' - y') < e ==> abs(max x x' - max y y') < e`) THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN CONJ_TAC THEN MATCH_MP_TAC(REAL_ARITH `norm(x) < e /\ abs(x$i) <= norm x ==> abs(x$i) < e`) THEN ASM_SIMP_TAC[COMPONENT_LE_NORM; GSYM FSTCART_SUB; GSYM SNDCART_SUB] THEN ASM_MESON_TAC[REAL_LET_TRANS; NORM_FSTCART; NORM_SNDCART]);; let MEASURABLE_ON_MIN = prove (`!f:real^M->real^N g:real^M->real^N s. f measurable_on s /\ g measurable_on s ==> (\x. (lambda i. min ((f x)$i) ((g x)$i)):real^N) measurable_on s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_NEG)) THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_MAX) THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_NEG) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN SIMP_TAC[CART_EQ; VECTOR_NEG_COMPONENT; LAMBDA_BETA] THEN REAL_ARITH_TAC);; let MEASURABLE_ON_DROP_MUL = prove (`!f g:real^M->real^N s. f measurable_on s /\ g measurable_on s ==> (\x. drop(f x) % g x) measurable_on s`, let lemma = REWRITE_RULE[] (ISPEC `\x y. drop x % y :real^N` MEASURABLE_ON_COMBINE) in REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; ETA_AX; LIFT_DROP] THEN CONJ_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]);; let MEASURABLE_ON_LIFT_MUL = prove (`!f g s. (\x. lift(f x)) measurable_on s /\ (\x. lift(g x)) measurable_on s ==> (\x. lift(f x * g x)) measurable_on s`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_DROP_MUL) THEN REWRITE_TAC[LIFT_CMUL; LIFT_DROP]);; let MEASURABLE_ON_MUL = prove (`!f g s. (\x. lift(f x)) measurable_on s /\ g measurable_on s ==> (\x. f x % g x) measurable_on s`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_DROP_MUL) THEN REWRITE_TAC[LIFT_DROP]);; let MEASURABLE_ON_VSUM = prove (`!f:A->real^M->real^N s t. FINITE t /\ (!i. i IN t ==> (f i) measurable_on s) ==> (\x. vsum t (\i. f i x)) measurable_on s`, GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[VSUM_CLAUSES; MEASURABLE_ON_0; MEASURABLE_ON_ADD; IN_INSERT; ETA_AX]);; let MEASURABLE_ON_COMPONENTWISE = prove (`!f:real^M->real^N s. f measurable_on s <=> (!i. 1 <= i /\ i <= dimindex (:N) ==> (\x. lift (f x$i)) measurable_on s)`, let lemma = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> (!i. 1 <= i /\ i <= dimindex(:N) ==> (\x. lift(f x$i)) measurable_on (:real^M))`, REPEAT GEN_TAC THEN EQ_TAC THENL [REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPEC `\x:real^N. lift(x$i)` o MATCH_MP (REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_COMPOSE_CONTINUOUS)) THEN ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT; o_DEF]; ALL_TAC] THEN REWRITE_TAC[measurable_on; IN_UNIV] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:num->real^M->bool`; `g:num->num->real^M->real^1`] THEN DISCH_TAC THEN EXISTS_TAC `UNIONS(IMAGE k (1..dimindex(:N))):real^M->bool` THEN EXISTS_TAC `(\n x. lambda i. drop(g i n x)):num->real^M->real^N` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN ASM_SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; FORALL_IN_IMAGE; FINITE_IMAGE]; GEN_TAC THEN ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT] THEN ASM_SIMP_TAC[LAMBDA_BETA; LIFT_DROP; ETA_AX]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_UNIONS; EXISTS_IN_IMAGE] THEN REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(a /\ b) <=> a ==> ~b`] THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[LIM_COMPONENTWISE_LIFT] THEN ASM_SIMP_TAC[LAMBDA_BETA; LIFT_DROP; ETA_AX]]) in REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN GEN_REWRITE_TAC LAND_CONV [lemma] THEN REWRITE_TAC[COND_RAND; COND_RATOR; VEC_COMPONENT; LIFT_NUM]);; let MEASURABLE_ON_CONST_EQ = prove (`!s:real^M->bool c:real^N. (\x. c) measurable_on s <=> c = vec 0 \/ lebesgue_measurable s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `c:real^N = vec 0` THEN ASM_REWRITE_TAC[MEASURABLE_ON_0] THEN ONCE_REWRITE_TAC[MEASURABLE_ON_COMPONENTWISE] THEN EQ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[VEC_COMPONENT] THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `inv((c:real^N)$i)` o MATCH_MP MEASURABLE_ON_CMUL) THEN ASM_SIMP_TAC[GSYM LIFT_CMUL; REAL_MUL_LINV; LIFT_NUM] THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[lebesgue_measurable; indicator]; REWRITE_TAC[lebesgue_measurable; indicator; MEASURABLE_ON_UNIV] THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(c:real^N)$i` o MATCH_MP MEASURABLE_ON_CMUL) THEN REWRITE_TAC[GSYM LIFT_NUM; GSYM LIFT_CMUL; REAL_MUL_RID]]);; let MEASURABLE_ON_LIFT_POW = prove (`!f:real^M->real s n. (\x. lift(f x)) measurable_on s /\ (n = 0 ==> lebesgue_measurable s) ==> (\x. lift(f x pow n)) measurable_on s`, REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC num_INDUCTION THEN SIMP_TAC[MEASURABLE_ON_CONST_EQ; CONJUNCT1 real_pow] THEN X_GEN_TAC `n:num` THEN ASM_CASES_TAC `n = 0` THEN ASM_CASES_TAC `lebesgue_measurable(s:real^M->bool)` THEN ASM_REWRITE_TAC[NOT_SUC; real_pow; REAL_MUL_RID] THEN ASM_SIMP_TAC[MEASURABLE_ON_LIFT_MUL]);; let MEASURABLE_ON_LIFT_PRODUCT = prove (`!f:A->real^N->real s t. FINITE t /\ (t = {} ==> lebesgue_measurable s) /\ (!i. i IN t ==> (\x. lift(f i x)) measurable_on s) ==> (\x. lift(product t (\i. f i x))) measurable_on s`, GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[PRODUCT_CLAUSES; MEASURABLE_ON_CONST_EQ] THEN REWRITE_TAC[FORALL_IN_INSERT; NOT_INSERT_EMPTY] THEN MAP_EVERY X_GEN_TAC [`a:A`; `k:A->bool`] THEN REWRITE_TAC[IMP_IMP] THEN STRIP_TAC THEN ASM_CASES_TAC `k:A->bool = {}` THEN ASM_SIMP_TAC[PRODUCT_CLAUSES; REAL_MUL_RID; ETA_AX] THEN MATCH_MP_TAC MEASURABLE_ON_LIFT_MUL THEN ASM_REWRITE_TAC[ETA_AX] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]);; let MEASURABLE_ON_SPIKE = prove (`!f:real^M->real^N g s t. negligible s /\ (!x. x IN t DIFF s ==> g x = f x) ==> f measurable_on t ==> g measurable_on t`, REPEAT GEN_TAC THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN REWRITE_TAC[measurable_on] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_THEN(X_CHOOSE_THEN `k:real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `s UNION k:real^M->bool` THEN ASM_SIMP_TAC[DE_MORGAN_THM; IN_UNION; NEGLIGIBLE_UNION]);; let MEASURABLE_ON_SPIKE_SET = prove (`!f:real^M->real^N s t. negligible (s DIFF t UNION t DIFF s) ==> f measurable_on s ==> f measurable_on t`, REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[measurable_on] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:num->real^M->real^N` THEN DISCH_THEN(X_CHOOSE_THEN `k:real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `k UNION (s DIFF t UNION t DIFF s):real^M->bool` THEN ASM_SIMP_TAC[NEGLIGIBLE_UNION; IN_UNION; DE_MORGAN_THM] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(x:real^M) IN t`] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let MEASURABLE_ON_SPIKE_SET_EQ = prove (`!f:real^M->real^N s t. negligible (s DIFF t UNION t DIFF s) ==> (f measurable_on s <=> f measurable_on t)`, REPEAT STRIP_TAC THEN EQ_TAC THEN MATCH_MP_TAC MEASURABLE_ON_SPIKE_SET THEN POP_ASSUM MP_TAC THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]);; let MEASURABLE_ON_EQ = prove (`!f g:real^M->real^N s. (!x. x IN s ==> f x = g x) /\ f measurable_on s ==> g measurable_on s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC MEASURABLE_ON_SPIKE THEN EXISTS_TAC `{}:real^M->bool` THEN REWRITE_TAC[NEGLIGIBLE_EMPTY] THEN ASM SET_TAC[]);; let MEASURABLE_ON_RESTRICT = prove (`!f:real^M->real^N s. f measurable_on (:real^M) /\ lebesgue_measurable s ==> (\x. if x IN s then f(x) else vec 0) measurable_on (:real^M)`, REPEAT GEN_TAC THEN REWRITE_TAC[lebesgue_measurable; indicator] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_DROP_MUL) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DROP_VEC] THEN VECTOR_ARITH_TAC);; let MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET = prove (`!f s t. s SUBSET t /\ f measurable_on t /\ lebesgue_measurable s ==> f measurable_on s`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[IN_UNIV] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_RESTRICT) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN ASM SET_TAC[]);; let MEASURABLE_ON_OPEN_INTERVAL = prove (`!f:real^M->real^N a b. f measurable_on interval(a,b) <=> f measurable_on interval[a,b]`, REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_ON_SPIKE_SET_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `interval[a:real^M,b] DIFF interval(a,b)` THEN REWRITE_TAC[NEGLIGIBLE_FRONTIER_INTERVAL] THEN MP_TAC(ISPECL [`a:real^M`; `b:real^M`] INTERVAL_OPEN_SUBSET_CLOSED) THEN SET_TAC[]);; let MEASURABLE_ON_CASES = prove (`!P f g:real^M->real^N s. lebesgue_measurable {x | P x} /\ f measurable_on s /\ g measurable_on s ==> (\x. if P x then f x else g x) measurable_on s`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `!x. (if x IN s then if P x then (f:real^M->real^N) x else g x else vec 0) = (if x IN {x | P x} then if x IN s then f x else vec 0 else vec 0) + (if x IN (:real^M) DIFF {x | P x} then if x IN s then g x else vec 0 else vec 0)` (fun th -> REWRITE_TAC[th]) THENL [GEN_TAC THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM; IN_DIFF] THEN MESON_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID]; MATCH_MP_TAC MEASURABLE_ON_ADD THEN CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_ON_RESTRICT THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [lebesgue_measurable]) THEN REWRITE_TAC[lebesgue_measurable] THEN SUBGOAL_THEN `((\x. vec 1):real^M->real^1) measurable_on (:real^M)` MP_TAC THENL [REWRITE_TAC[MEASURABLE_ON_CONST]; REWRITE_TAC[IMP_IMP]] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_SUB) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[indicator; IN_DIFF; IN_UNIV; IN_ELIM_THM; FUN_EQ_THM] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(P:real^M->bool) x` THEN ASM_REWRITE_TAC[] THEN CONV_TAC VECTOR_ARITH]);; let MEASURABLE_ON_LIMIT = prove (`!f:num->real^M->real^N g s k. (!n. (f n) measurable_on s) /\ negligible k /\ (!x. x IN s DIFF k ==> ((\n. f n x) --> g x) sequentially) ==> g measurable_on s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`vec 0:real^N`; `vec 1:real^N`] HOMEOMORPHIC_OPEN_INTERVAL_UNIV) THEN REWRITE_TAC[INTERVAL_NE_EMPTY; VEC_COMPONENT; REAL_LT_01] THEN REWRITE_TAC[homeomorphic; homeomorphism; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h':real^N->real^N`; `h:real^N->real^N`] THEN REWRITE_TAC[IN_UNIV] THEN STRIP_TAC THEN SUBGOAL_THEN `((h':real^N->real^N) o (h:real^N->real^N) o (\x. if x IN s then g x else vec 0)) measurable_on (:real^M)` MP_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[o_DEF; MEASURABLE_ON_UNIV]] THEN SUBGOAL_THEN `!y:real^N. norm(h y:real^N) <= &(dimindex(:N))` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE h UNIV = s ==> (!z. z IN s ==> P z) ==> !y. P(h y)`)) THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_INTERVAL] THEN REWRITE_TAC[VEC_COMPONENT] THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(1..dimindex(:N)) (\i. abs((y:real^N)$i))` THEN REWRITE_TAC[NORM_LE_L1] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN MATCH_MP_TAC SUM_BOUND THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < x /\ x < &1 ==> abs(x) <= &1`]; ALL_TAC] THEN MATCH_MP_TAC MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL THEN MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `vec 1:real^N`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_SUBINTERVALS_IMP_MEASURABLE THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `interval[a:real^M,b] DIFF k` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC DOMINATED_CONVERGENCE_INTEGRABLE THEN MAP_EVERY EXISTS_TAC [`(\n x. h(if x IN s then f n x else vec 0:real^N)):num->real^M->real^N`; `(\x. vec(dimindex(:N))):real^M->real^1`] THEN REWRITE_TAC[o_DEF; INTEGRABLE_CONST] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `(\x. vec(dimindex(:N))):real^M->real^1` THEN ASM_REWRITE_TAC[ETA_AX; INTEGRABLE_CONST] THEN ASM_SIMP_TAC[DROP_VEC] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] MEASURABLE_ON_SPIKE_SET) THEN EXISTS_TAC `interval[a:real^M,b:real^M]` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN MATCH_MP_TAC(REWRITE_RULE[indicator; lebesgue_measurable] MEASURABLE_ON_RESTRICT) THEN REWRITE_TAC[MEASURABLE_ON_UNIV] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`(\x. if x IN s then f (n:num) x else vec 0):real^M->real^N`; `h:real^N->real^N`] MEASURABLE_ON_COMPOSE_CONTINUOUS) THEN ASM_REWRITE_TAC[o_DEF] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[MEASURABLE_ON_UNIV; ETA_AX]; MATCH_MP_TAC INTEGRABLE_IMP_MEASURABLE THEN REWRITE_TAC[INTEGRABLE_CONST]]; MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `interval[a:real^M,b:real^M]` THEN REWRITE_TAC[INTEGRABLE_CONST] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]]; MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `interval[a:real^M,b:real^M]` THEN REWRITE_TAC[INTEGRABLE_CONST] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; ASM_SIMP_TAC[DROP_VEC]; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[LIM_CONST] THEN MATCH_MP_TAC LIM_CONTINUOUS_FUNCTION THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_UNIV; IN_UNIV]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]]]; REWRITE_TAC[o_THM] THEN ASM SET_TAC[]]);; let MEASURABLE_ON_BILINEAR = prove (`!op:real^N->real^P->real^Q f g s:real^M->bool. bilinear op /\ f measurable_on s /\ g measurable_on s ==> (\x. op (f x) (g x)) measurable_on s`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[measurable_on; LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `ff:num->real^M->real^N`] THEN REPLICATE_TAC 3 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`k':real^M->bool`; `gg:num->real^M->real^P`] THEN REPLICATE_TAC 3 DISCH_TAC THEN EXISTS_TAC `k UNION k':real^M->bool` THEN EXISTS_TAC `\n:num x:real^M. (op:real^N->real^P->real^Q) (ff n x) (gg n x)` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN CONJ_TAC THENL [GEN_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`] BILINEAR_CONTINUOUS_ON_COMPOSE)) THEN ASM_REWRITE_TAC[ETA_AX]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `(if x IN s then (op:real^N->real^P->real^Q) (f x) (g x) else vec 0) = op (if x IN s then f(x:real^M) else vec 0) (if x IN s then g(x:real^M) else vec 0)` SUBST1_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bilinear]) THEN DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o GEN `y:real^N` o MATCH_MP LINEAR_0 o SPEC `y:real^N`) (MP_TAC o GEN `z:real^P` o MATCH_MP LINEAR_0 o SPEC `z:real^P`)) THEN MESON_TAC[]; REPEAT STRIP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`] LIM_BILINEAR)) THEN ASM_SIMP_TAC[]]]);; let ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT = prove (`!op:real^N->real^P->real^Q f g s:real^M->bool. bilinear op /\ f measurable_on s /\ bounded (IMAGE f s) /\ g absolutely_integrable_on s ==> (\x. op (f x) (g x)) absolutely_integrable_on s`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\x:real^M. lift(B * C * norm((g:real^M->real^P) x))` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_BILINEAR)) THEN ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_MEASURABLE]; REWRITE_TAC[LIFT_CMUL] THEN REPEAT(MATCH_MP_TAC INTEGRABLE_CMUL) THEN RULE_ASSUM_TAC(REWRITE_RULE[absolutely_integrable_on]) THEN ASM_REWRITE_TAC[]; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[LIFT_DROP] THEN TRANS_TAC REAL_LE_TRANS `B * norm((f:real^M->real^N) x) * norm(g x:real^P)` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[NORM_POS_LE]]);; let MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE_AE = prove (`!f:real^M->real^N g s t. f measurable_on s /\ g integrable_on s /\ negligible t /\ (!x. x IN s DIFF t ==> norm(f x) <= drop(g x)) ==> f absolutely_integrable_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] ABSOLUTELY_INTEGRABLE_SPIKE) THEN MAP_EVERY EXISTS_TAC [`\x. if x IN s DIFF t then (f:real^M->real^N) x else vec 0`; `t:real^M->bool`] THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\x. if x IN s DIFF t then (g:real^M->real^1) x else vec 0` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p ==> q ==> r <=> q ==> p ==> r`] MEASURABLE_ON_SPIKE)); FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p ==> q ==> r <=> q ==> p ==> r`] INTEGRABLE_SPIKE)); ASM_MESON_TAC[REAL_LE_REFL; NORM_0; DROP_VEC]] THEN EXISTS_TAC `t:real^M->bool` THEN ASM_SIMP_TAC[]);; (* ------------------------------------------------------------------------- *) (* Some handy lemmas about square integrable functions. These exist in *) (* far more generality in "Multivariate/lpspaces.ml" *) (* ------------------------------------------------------------------------- *) let ABSOLUTELY_INTEGRABLE_SQUARE_INTEGRABLE_PRODUCT = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P s. bilinear bop /\ f measurable_on s /\ (\x. lift(norm(f x) pow 2)) integrable_on s /\ g measurable_on s /\ (\x. lift(norm(g x) pow 2)) integrable_on s ==> (\x. bop (f x) (g x)) absolutely_integrable_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN ASM_SIMP_TAC[MEASURABLE_ON_BILINEAR] THEN FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. B / &2 % lift(norm((f:real^M->real^N) x) pow 2 + norm((g:real^M->real^P) x) pow 2)` THEN ASM_SIMP_TAC[LIFT_ADD; INTEGRABLE_ADD; INTEGRABLE_CMUL] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH lhand th o lhand o snd)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[DROP_CMUL; LIFT_DROP; DROP_ADD; REAL_ARITH `B * x * y <= B / &2 * (x pow 2 + y pow 2) <=> &0 <= B * (x - y) pow 2`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LE_POW_2; REAL_LT_IMP_LE]);; let SQUARE_INTEGRAL_SQUARE_INTEGRABLE_PRODUCT_LE = prove (`!f:real^M->real^N g:real^M->real^P s. f measurable_on s /\ (\x. lift(norm(f x) pow 2)) integrable_on s /\ g measurable_on s /\ (\x. lift(norm(g x) pow 2)) integrable_on s ==> drop(integral s (\x. lift(norm(f x) * norm(g x)))) pow 2 <= drop(integral s (\x. lift(norm(f x) pow 2))) * drop(integral s (\x. lift(norm(g x) pow 2)))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x y. lift(drop x * drop y)`; `\x. lift(norm((f:real^M->real^N) x))`; `\x. lift(norm((g:real^M->real^P) x))`; `s:real^M->bool`] ABSOLUTELY_INTEGRABLE_SQUARE_INTEGRABLE_PRODUCT) THEN ASM_SIMP_TAC[LIFT_DROP; NORM_LIFT; REAL_ABS_NORM; MEASURABLE_ON_NORM] THEN REWRITE_TAC[BILINEAR_LIFT_MUL] THEN DISCH_THEN(ASSUME_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN MAP_EVERY ABBREV_TAC [`a = sqrt(drop(integral s (\x. lift(norm((f:real^M->real^N) x) pow 2))))`; `b = sqrt(drop(integral s (\x. lift(norm((g:real^M->real^P) x) pow 2))))`] THEN ASM_CASES_TAC `a = &0` THENL [UNDISCH_TAC `a = &0` THEN EXPAND_TAC "a" THEN REWRITE_TAC[SQRT_EQ_0] THEN DISCH_TAC THEN MP_TAC(ISPECL [`\x. lift(norm((f:real^M->real^N) x) pow 2)`; `s:real^M->bool`] HAS_INTEGRAL_NEGLIGIBLE_EQ) THEN REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p ==> q /\ r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_1; DIMINDEX_1] THEN REWRITE_TAC[GSYM drop; LIFT_DROP; REAL_LE_POW_2] THEN ASM_SIMP_TAC[GSYM INTEGRAL_EQ_HAS_INTEGRAL] THEN ASM_REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; DROP_VEC; NORM_EQ_0; REAL_RING `x pow 2 = &0 <=> x = &0`] THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `x = &0 ==> x <= &0 * y`) THEN REWRITE_TAC[REAL_RING `x pow 2 = &0 <=> x = &0`] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC HAS_INTEGRAL_NEGLIGIBLE THEN EXISTS_TAC `{x | x IN s /\ ~((f:real^M->real^N) x = vec 0)}` THEN ASM_SIMP_TAC[NORM_0; REAL_MUL_LZERO; LIFT_NUM; SET_RULE `x IN s DIFF {x | x IN s /\ ~P x} <=> x IN s /\ P x`]; ALL_TAC] THEN ASM_CASES_TAC `b = &0` THENL [UNDISCH_TAC `b = &0` THEN EXPAND_TAC "b" THEN REWRITE_TAC[SQRT_EQ_0] THEN DISCH_TAC THEN MP_TAC(ISPECL [`\x. lift(norm((g:real^M->real^P) x) pow 2)`; `s:real^M->bool`] HAS_INTEGRAL_NEGLIGIBLE_EQ) THEN REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p ==> q /\ r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_1; DIMINDEX_1] THEN REWRITE_TAC[GSYM drop; LIFT_DROP; REAL_LE_POW_2] THEN ASM_SIMP_TAC[GSYM INTEGRAL_EQ_HAS_INTEGRAL] THEN ASM_REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; DROP_VEC; NORM_EQ_0; REAL_RING `x pow 2 = &0 <=> x = &0`] THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `x = &0 ==> x <= y * &0`) THEN REWRITE_TAC[REAL_RING `x pow 2 = &0 <=> x = &0`] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC HAS_INTEGRAL_NEGLIGIBLE THEN EXISTS_TAC `{x | x IN s /\ ~((g:real^M->real^P) x = vec 0)}` THEN ASM_SIMP_TAC[NORM_0; REAL_MUL_RZERO; LIFT_NUM; SET_RULE `x IN s DIFF {x | x IN s /\ ~P x} <=> x IN s /\ P x`]; ALL_TAC] THEN MP_TAC(ISPECL [`\x. lift((a * norm((g:real^M->real^P) x) - b * norm((f:real^M->real^N) x)) pow 2)`; `s:real^M->bool`] INTEGRAL_DROP_POS) THEN REWRITE_TAC[LIFT_DROP; REAL_LE_POW_2] THEN REWRITE_TAC[REAL_ARITH `(a * g - b * f:real) pow 2 = (a pow 2 * g pow 2 + b pow 2 * f pow 2) - (&2 * a * b) * (f * g)`] THEN REWRITE_TAC[LIFT_ADD; LIFT_SUB] THEN ONCE_REWRITE_TAC[LIFT_CMUL] THEN ASM_SIMP_TAC[INTEGRABLE_ADD; INTEGRABLE_SUB; INTEGRABLE_CMUL; INTEGRAL_ADD; INTEGRAL_SUB; INTEGRAL_CMUL] THEN REWRITE_TAC[DROP_SUB; DROP_ADD; DROP_CMUL] THEN SUBGOAL_THEN `drop(integral s (\x. lift(norm((f:real^M->real^N) x) pow 2))) = a pow 2 /\ drop(integral s (\x. lift(norm((g:real^M->real^P) x) pow 2))) = b pow 2` (CONJUNCTS_THEN SUBST1_TAC) THENL [MAP_EVERY EXPAND_TAC ["a"; "b"] THEN REWRITE_TAC[REAL_SQRT_POW_2; REAL_ARITH `x = abs x <=> &0 <= x`] THEN ASM_SIMP_TAC[INTEGRAL_DROP_POS; LIFT_DROP; REAL_LE_POW_2]; REWRITE_TAC[REAL_ARITH `&0 <= (a pow 2 * b pow 2 + b pow 2 * a pow 2) - (&2 * a * b) * i <=> (a * b) * i <= (a * b) pow 2`]] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `x <= y ==> &0 <= x ==> abs x <= abs y`)) THEN SUBGOAL_THEN `&0 <= a /\ &0 <= b` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["a"; "b"] THEN CONJ_TAC THEN MATCH_MP_TAC SQRT_POS_LE THEN ASM_SIMP_TAC[INTEGRAL_DROP_POS; LIFT_DROP; REAL_LE_POW_2]; ASM_SIMP_TAC[INTEGRAL_DROP_POS; LIFT_DROP; REAL_LE_MUL; NORM_POS_LE]] THEN REWRITE_TAC[REAL_LE_SQUARE_ABS; REAL_ARITH `((a * b) * i) pow 2 <= (a * b) pow 2 pow 2 <=> &0 <= (a * b) pow 2 * ((a pow 2 * b pow 2) - i pow 2)`] THEN SUBGOAL_THEN `&0 < a * b` ASSUME_TAC THENL [MATCH_MP_TAC REAL_LT_MUL THEN ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_POW_LT]] THEN REWRITE_TAC[LIFT_CMUL] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Natural closure properties of measurable functions; the intersection *) (* one is actually quite tedious since we end up reinventing cube roots *) (* before they actually get introduced in transcendentals.ml *) (* ------------------------------------------------------------------------- *) let MEASURABLE_ON_EMPTY = prove (`!f:real^M->real^N. f measurable_on {}`, ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[NOT_IN_EMPTY; MEASURABLE_ON_CONST]);; let MEASURABLE_ON_INTER = prove (`!f:real^M->real^N s t. f measurable_on s /\ f measurable_on t ==> f measurable_on (s INTER t)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN ONCE_REWRITE_TAC[MEASURABLE_ON_COMPONENTWISE] THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `p /\ q ==> r <=> p /\ p ==> q ==> r`] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_LIFT_MUL) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_LIFT_MUL) THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[VEC_COMPONENT; REAL_ARITH `(if p then x else &0) * (if q then y else &0) = if p /\ q then x * y else &0`] THEN SUBGOAL_THEN `!s. (\x. lift (drop x pow 3)) continuous_on s` ASSUME_TAC THENL [GEN_TAC THEN REWRITE_TAC[REAL_ARITH `(x:real) pow 3 = x * x * x`] THEN REWRITE_TAC[LIFT_CMUL] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_ON_MUL THEN ASM_REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID]); ALL_TAC] THEN SUBGOAL_THEN `?r. !x. lift(drop(r x) pow 3) = x` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM SKOLEM_THM; FORALL_LIFT; GSYM EXISTS_DROP; LIFT_EQ] THEN X_GEN_TAC `x:real` THEN MP_TAC(ISPECL [`\x. lift (drop x pow 3)`; `lift(--(abs x + &1))`; `lift(abs x + &1)`;`x:real`; `1`] IVT_INCREASING_COMPONENT_1) THEN REWRITE_TAC[GSYM drop; LIFT_DROP; EXISTS_DROP] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REWRITE_TAC[DIMINDEX_1; LE_REFL] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `(:real^1)`) THEN ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_UNIV; IN_UNIV]; REWRITE_TAC[REAL_BOUNDS_LE; REAL_POW_NEG; ARITH] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ &0 <= x pow 2 /\ &0 <= x pow 3 ==> x <= (x + &1) pow 3`) THEN SIMP_TAC[REAL_POW_LE; REAL_ABS_POS]]; ALL_TAC] THEN SUBGOAL_THEN `!x. r(lift(x pow 3)) = lift x` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN GEN_TAC THEN MATCH_MP_TAC REAL_POW_EQ_ODD THEN EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH; GSYM LIFT_EQ; LIFT_DROP]; ALL_TAC] THEN SUBGOAL_THEN `(r:real^1->real^1) continuous_on (:real^1)` ASSUME_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_INVERSE_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`\x. lift(drop x pow 3)`; `(:real^1)`] THEN ASM_REWRITE_TAC[LIFT_DROP] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN SUBST1_TAC(SYM th)) THEN MATCH_MP_TAC INJECTIVE_INTO_1D_IMP_OPEN_MAP THEN ASM_REWRITE_TAC[PATH_CONNECTED_UNIV; LIFT_EQ] THEN SIMP_TAC[REAL_POW_EQ_ODD_EQ; ARITH; DROP_EQ]; ONCE_REWRITE_TAC[REAL_ARITH `&0 = &0 pow 3`] THEN REWRITE_TAC[REAL_ARITH `(x * x) * x:real = x pow 3`; IN_INTER] THEN REWRITE_TAC[MESON[] `(if p then x pow 3 else y pow 3) = (if p then x else y:real) pow 3`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(MP_TAC o ISPEC `r:real^1->real^1` o MATCH_MP (REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_COMPOSE_CONTINUOUS)) THEN ASM_REWRITE_TAC[o_DEF]]);; let MEASURABLE_ON_DIFF = prove (`!f:real^M->real^N s t. f measurable_on s /\ f measurable_on t ==> f measurable_on (s DIFF t)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP MEASURABLE_ON_INTER) THEN FIRST_ASSUM(MP_TAC o CONJUNCT1) THEN REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_SUB) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_DIFF; IN_INTER] THEN X_GEN_TAC `x:real^M` THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(x:real^M) IN t`] THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);; let MEASURABLE_ON_UNION = prove (`!f:real^M->real^N s t. f measurable_on s /\ f measurable_on t ==> f measurable_on (s UNION t)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP MEASURABLE_ON_INTER) THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_ADD) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP MEASURABLE_ON_SUB) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_UNION; IN_INTER] THEN X_GEN_TAC `x:real^M` THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(x:real^M) IN t`] THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);; let MEASURABLE_ON_UNIONS = prove (`!f:real^M->real^N k. FINITE k /\ (!s. s IN k ==> f measurable_on s) ==> f measurable_on (UNIONS k)`, GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[UNIONS_0; MEASURABLE_ON_EMPTY; UNIONS_INSERT] THEN SIMP_TAC[FORALL_IN_INSERT; MEASURABLE_ON_UNION]);; let MEASURABLE_ON_COUNTABLE_UNIONS = prove (`!f:real^M->real^N k. COUNTABLE k /\ (!s. s IN k ==> f measurable_on s) ==> f measurable_on (UNIONS k)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `k:(real^M->bool)->bool = {}` THEN ASM_REWRITE_TAC[UNIONS_0; MEASURABLE_ON_EMPTY] THEN MP_TAC(ISPEC `k:(real^M->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:num->real^M->bool` THEN DISCH_THEN SUBST_ALL_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN MATCH_MP_TAC MEASURABLE_ON_LIMIT THEN EXISTS_TAC `(\n x. if x IN UNIONS (IMAGE d (0..n)) then f x else vec 0): num->real^M->real^N` THEN EXISTS_TAC `{}:real^M->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY; MEASURABLE_ON_UNIV] THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC MEASURABLE_ON_UNIONS THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FORALL_IN_IMAGE]) THEN SIMP_TAC[FORALL_IN_IMAGE; IN_UNIV; FINITE_IMAGE; FINITE_NUMSEG]; X_GEN_TAC `x:real^M` THEN DISCH_THEN(K ALL_TAC) THEN ASM_CASES_TAC `(x:real^M) IN UNIONS (IMAGE d (:num))` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LIM_EVENTUALLY THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_IMAGE; IN_UNIV; EVENTUALLY_SEQUENTIALLY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_UNIONS]) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[EXISTS_IN_IMAGE; IN_NUMSEG; LE_0] THEN ASM_MESON_TAC[]; MATCH_MP_TAC ALWAYS_EVENTUALLY THEN ASM SET_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* Negligibility of a Lipschitz image of a negligible set. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE = prove (`!f:real^M->real^N s. dimindex(:M) <= dimindex(:N) /\ negligible s /\ (!x. x IN s ==> ?t b. open t /\ x IN t /\ !y. y IN s INTER t ==> norm(f y - f x) <= b * norm(y - x)) ==> negligible(IMAGE f s)`, let lemma = prove (`!f:real^M->real^N s B. dimindex(:M) <= dimindex(:N) /\ bounded s /\ negligible s /\ &0 < B /\ (!x. x IN s ==> ?t. open t /\ x IN t /\ !y. y IN s INTER t ==> norm(f y - f x) <= B * norm(y - x)) ==> negligible(IMAGE f s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[NEGLIGIBLE_OUTER] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `e / &2 / (&2 * B * &(dimindex(:M))) pow (dimindex(:N))`] MEASURABLE_OUTER_OPEN) THEN ANTS_TAC THENL [ASM_SIMP_TAC[NEGLIGIBLE_IMP_MEASURABLE] THEN MATCH_MP_TAC REAL_LT_DIV THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC REAL_POW_LT THEN REPEAT(MATCH_MP_TAC REAL_LT_MUL THEN CONJ_TAC) THEN ASM_SIMP_TAC[DIMINDEX_GE_1; REAL_OF_NUM_LT; ARITH; LE_1]; ALL_TAC] THEN ASM_SIMP_TAC[NEGLIGIBLE_IMP_MEASURABLE; REAL_HALF; MEASURE_EQ_0] THEN REWRITE_TAC[REAL_ADD_LID] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!x. ?r. &0 < r /\ r <= &1 / &2 /\ (x IN s ==> !y. norm(y - x:real^M) < r ==> y IN t /\ (y IN s ==> norm(f y - f x:real^N) <= B * norm(y - x)))` MP_TAC THENL [X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; EXISTS_TAC `&1 / &4` THEN REAL_ARITH_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `t INTER u :real^M->bool` open_def) THEN ASM_SIMP_TAC[OPEN_INTER; OPEN_BALL] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTER; dist]] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min (&1 / &2) r` THEN ASM_REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_MESON_TAC[]; FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl)) THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN X_GEN_TAC `r:real^M->real` THEN STRIP_TAC] THEN SUBGOAL_THEN `?c. s SUBSET interval[--(vec c):real^M,vec c] /\ ~(interval(--(vec c):real^M,vec c) = {})` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN DISCH_THEN(X_CHOOSE_THEN `c:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `abs c + &1` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET; INTERVAL_NE_EMPTY] THEN REWRITE_TAC[IN_INTERVAL; VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[REAL_BOUNDS_LE] THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`--(vec c):real^M`; `(vec c):real^M`; `s:real^M->bool`; `\x:real^M. ball(x,r x)`] COVERING_LEMMA) THEN ASM_REWRITE_TAC[gauge; OPEN_BALL; CENTRE_IN_BALL] THEN REWRITE_TAC[VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!k. k IN D ==> ?u v z. k = interval[u,v] /\ ~(interval(u,v) = {}) /\ z IN s /\ z IN interval[u,v] /\ interval[u:real^M,v] SUBSET ball(z,r z)` MP_TAC THENL [X_GEN_TAC `d:real^M->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `?u v:real^M. d = interval[u,v]` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^M` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^M` THEN DISCH_THEN SUBST_ALL_TAC THEN ASM_MESON_TAC[SUBSET; INTERIOR_CLOSED_INTERVAL; IN_INTER]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:(real^M->bool)->real^M`; `v:(real^M->bool)->real^M`; `z:(real^M->bool)->real^M`] THEN DISCH_THEN(LABEL_TAC "*") THEN EXISTS_TAC `UNIONS(IMAGE (\d:real^M->bool. interval[(f:real^M->real^N)(z d) - (B * &(dimindex(:M)) * ((v(d):real^M)$1 - (u(d):real^M)$1)) % vec 1:real^N, f(z d) + (B * &(dimindex(:M)) * (v(d)$1 - u(d)$1)) % vec 1]) D)` THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN SUBGOAL_THEN `(y:real^M) IN UNIONS D` MP_TAC THENL [ASM_MESON_TAC[SUBSET]; REWRITE_TAC[UNIONS_IMAGE]] THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(y:real^M) IN ball(z(d:real^M->bool),r(z d))` MP_TAC THENL [ASM_MESON_TAC[SUBSET]; REWRITE_TAC[IN_BALL; dist]] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN DISCH_TAC THEN SUBGOAL_THEN `y IN t /\ norm((f:real^M->real^N) y - f(z d)) <= B * norm(y - z(d:real^M->bool))` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[IN_INTERVAL] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `z - b <= y /\ y <= z + b <=> abs(y - z) <= b`] THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS)) THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN SIMP_TAC[GSYM SUM_CONST; FINITE_NUMSEG] THEN MATCH_MP_TAC SUM_LE_NUMSEG THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `((v:(real^M->bool)->real^M) d - u d)$j` THEN REWRITE_TAC[VECTOR_SUB_COMPONENT] THEN CONJ_TAC THENL [SUBGOAL_THEN `y IN interval[(u:(real^M->bool)->real^M) d,v d] /\ (z d) IN interval[u d,v d]` MP_TAC THENL [ASM_MESON_TAC[]; REWRITE_TAC[IN_INTERVAL]] THEN DISCH_THEN(CONJUNCTS_THEN (MP_TAC o SPEC `j:num`)) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; MATCH_MP_TAC REAL_EQ_IMP_LE THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(u:(real^M->bool)->real^M) d`; `(v:(real^M->bool)->real^M) d`]) THEN ASM_MESON_TAC[DIMINDEX_GE_1; LE_REFL]]; ALL_TAC] THEN MATCH_MP_TAC(MESON[] `(x <= e / &2 ==> x < e) /\ P /\ x <= e / &2 ==> P /\ x < e`) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE; MEASURABLE_INTERVAL] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `D':(real^M->bool)->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_IMAGE_LE o lhand o snd) THEN ASM_SIMP_TAC[MEASURE_POS_LE; MEASURABLE_INTERVAL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&2 * B * &(dimindex(:M))) pow (dimindex(:N)) * sum D' (\d:real^M->bool. measure d)` THEN SUBGOAL_THEN `FINITE(D':(real^M->bool)->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[MEASURE_INTERVAL] THEN X_GEN_TAC `d:real^M->bool` THEN DISCH_TAC THEN REWRITE_TAC[CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; REAL_ARITH `(a - x <= a + x <=> &0 <= x) /\ (a + x) - (a - x) = &2 * x`] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RID] THEN ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1] THEN SUBGOAL_THEN `d = interval[u d:real^M,v d]` (fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [th]) THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:M) ==> ((u:(real^M->bool)->real^M) d)$i <= (v d:real^M)$i` MP_TAC THENL [ASM_MESON_TAC[SUBSET; INTERVAL_NE_EMPTY; REAL_LT_IMP_LE]; ALL_TAC] THEN SIMP_TAC[REAL_SUB_LE; DIMINDEX_GE_1; LE_REFL] THEN DISCH_TAC THEN REWRITE_TAC[PRODUCT_CONST_NUMSEG; REAL_POW_MUL] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH; GSYM REAL_MUL_ASSOC; ADD_SUB; DIMINDEX_GE_1; LE_1] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `((v d:real^M)$1 - ((u:(real^M->bool)->real^M) d)$1) pow (dimindex(:M))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_POW_MONO_INV THEN ASM_SIMP_TAC[REAL_SUB_LE; DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN MATCH_MP_TAC(REAL_ARITH `abs x <= a ==> x <= a`) THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN REWRITE_TAC[DIMINDEX_GE_1; LE_REFL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC(NORM_ARITH `!z r. norm(z - u) < r /\ norm(z - v) < r /\ r <= &1 / &2 ==> norm(v - u:real^M) <= &1`) THEN MAP_EVERY EXISTS_TAC [`(z:(real^M->bool)->real^M) d`; `r((z:(real^M->bool)->real^M) d):real`] THEN ASM_REWRITE_TAC[GSYM dist; GSYM IN_BALL] THEN SUBGOAL_THEN `(u:(real^M->bool)->real^M) d IN interval[u d,v d] /\ (v:(real^M->bool)->real^M) d IN interval[u d,v d]` MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET]] THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY]; GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN SIMP_TAC[GSYM PRODUCT_CONST; FINITE_NUMSEG] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC PRODUCT_EQ_NUMSEG THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(u:(real^M->bool)->real^M) d`; `(v:(real^M->bool)->real^M) d`]) THEN ASM_MESON_TAC[DIMINDEX_GE_1; LE_REFL; SUBSET]]; MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&2 * B * &(dimindex(:M))) pow dimindex(:N) * measure(t:real^M->bool)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LT_IMP_LE; ALL_TAC]; MATCH_MP_TAC REAL_LT_IMP_LE THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN W(MP_TAC o PART_MATCH (rand o rand) REAL_LT_RDIV_EQ o snd)] THEN ASM_SIMP_TAC[REAL_POW_LT; REAL_LT_MUL; LE_1; DIMINDEX_GE_1; REAL_ARITH `&0 < &2 * B <=> &0 < B`; REAL_OF_NUM_LT] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(UNIONS D':real^M->bool)` THEN CONJ_TAC THENL [MP_TAC(ISPECL [`D':(real^M->bool)->bool`; `UNIONS D':real^M->bool`] MEASURE_ELEMENTARY) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[division_of] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET]] THEN GEN_TAC THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[SUBSET; INTERIOR_EMPTY]]; DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[MEASURE_INTERVAL; SUBSET]]; MATCH_MP_TAC MEASURE_SUBSET THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_MESON_TAC[MEASURABLE_INTERVAL; SUBSET]; ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `UNIONS D:real^M->bool` THEN ASM_SIMP_TAC[SUBSET_UNIONS] THEN REWRITE_TAC[SUBSET; FORALL_IN_UNIONS] THEN X_GEN_TAC `d:real^M->bool` THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN DISCH_TAC THEN REWRITE_TAC[GSYM SUBSET] THEN SUBGOAL_THEN `d SUBSET ball(z d:real^M,r(z d))` MP_TAC THENL [ASM_MESON_TAC[]; REWRITE_TAC[SUBSET; IN_BALL; dist] THEN ASM_MESON_TAC[NORM_SUB]]]]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `s = UNIONS {{x | x IN s /\ norm(x:real^M) <= &n /\ ?t. open t /\ x IN t /\ !y. y IN s INTER t ==> norm(f y - f x:real^N) <= (&n + &1) * norm(y - x)} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `max (norm(x:real^M)) b` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[REAL_MAX_LE] THEN X_GEN_TAC `n:num` THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `b * norm(y - x:real^M)` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IMAGE_UNIONS] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[GSYM IMAGE_o; COUNTABLE_IMAGE; NUM_COUNTABLE] THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[IN_UNIV] THEN MATCH_MP_TAC lemma THEN EXISTS_TAC `&n + &1` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(vec 0:real^M,&n)` THEN SIMP_TAC[BOUNDED_CBALL; SUBSET; IN_CBALL_0; IN_ELIM_THM]; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; REAL_ARITH_TAC; REWRITE_TAC[IN_ELIM_THM; IN_INTER] THEN MESON_TAC[]]]);; let NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE_LOWDIM = prove (`!f:real^M->real^N s. dimindex(:M) < dimindex(:N) /\ (!x. x IN s ==> ?t b. open t /\ x IN t /\ !y. y IN s INTER t ==> norm(f y - f x) <= b * norm(y - x)) ==> negligible(IMAGE f s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(f:real^M->real^N) o (sndcart:real^(1,M)finite_sum->real^M)`; `IMAGE (pastecart (vec 0:real^1)) (s:real^M->bool)`] NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; SNDCART_PASTECART; ETA_AX] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_REWRITE_TAC[ARITH_RULE `1 + m <= n <=> m < n`] THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^(1,M)finite_sum | x$1 = &0}` THEN SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE; DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; IN_ELIM_THM] THEN SIMP_TAC[pastecart; LAMBDA_BETA; DIMINDEX_1; LE_REFL; VEC_COMPONENT; DIMINDEX_FINITE_SUM; ARITH_RULE `1 <= 1 + n`]; REWRITE_TAC[IN_INTER; IMP_CONJ; FORALL_IN_IMAGE] THEN REWRITE_TAC[GSYM dist; DIST_PASTECART_CANCEL; SNDCART_PASTECART] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_INTER] THEN MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `b:real`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(:real^1) PCROSS (u:real^M->bool)`; `b:real`] THEN ASM_SIMP_TAC[OPEN_PCROSS; OPEN_UNIV; PASTECART_IN_PCROSS; IN_UNIV] THEN ASM_SIMP_TAC[dist]]);; let NEGLIGIBLE_LIPSCHITZ_IMAGE_UNIV = prove (`!f:real^N->real^N s B. negligible s /\ (!x y. norm(f x - f y) <= B * norm(x - y)) ==> negligible(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM_REWRITE_TAC[LE_REFL] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`interval(a - vec 1:real^N,a + vec 1)`; `B:real`] THEN ASM_REWRITE_TAC[OPEN_INTERVAL; IN_INTERVAL] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; VEC_COMPONENT] THEN REAL_ARITH_TAC);; let NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE = prove (`!f:real^M->real^N s. dimindex(:M) <= dimindex(:N) /\ negligible s /\ f differentiable_on s ==> negligible(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM_REWRITE_TAC[IN_INTER] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [differentiable_on]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[differentiable; HAS_DERIVATIVE_WITHIN_ALT] THEN DISCH_THEN(X_CHOOSE_THEN `f':real^M->real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[REAL_LT_01; REAL_MUL_RID] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP LINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `ball(x:real^M,d)` THEN EXISTS_TAC `B + &1` THEN ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN REWRITE_TAC[IN_BALL; dist; REAL_ADD_RDISTRIB] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(NORM_ARITH `!d. norm(y - x - d:real^N) <= z /\ norm(d) <= b ==> norm(y - x) <= b + z`) THEN EXISTS_TAC `(f':real^M->real^N)(y - x)` THEN ASM_MESON_TAC[NORM_SUB]);; let NEGLIGIBLE_DIFFERENTIABLE_IMAGE_LOWDIM = prove (`!f:real^M->real^N s. dimindex(:M) < dimindex(:N) /\ f differentiable_on s ==> negligible(IMAGE f s)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP (ARITH_RULE `m < n ==> !x:num. x <= m ==> x <= n`)) THEN SUBGOAL_THEN `(f:real^M->real^N) = (f o ((\x. lambda i. x$i):real^N->real^M)) o ((\x. lambda i. if i <= dimindex(:M) then x$i else &0):real^M->real^N)` SUBST1_TAC THENL [SIMP_TAC[FUN_EQ_THM; o_THM] THEN GEN_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA]; ONCE_REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN REWRITE_TAC[LE_REFL] THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{y:real^N | y$(dimindex(:N)) = &0}` THEN SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE; LE_REFL; DIMINDEX_GE_1] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN SIMP_TAC[LAMBDA_BETA; LE_REFL; DIMINDEX_GE_1] THEN ASM_REWRITE_TAC[GSYM NOT_LT]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [differentiable_on]) THEN REWRITE_TAC[differentiable_on; FORALL_IN_IMAGE] THEN STRIP_TAC THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC DIFFERENTIABLE_CHAIN_WITHIN THEN CONJ_TAC THENL [MATCH_MP_TAC DIFFERENTIABLE_LINEAR THEN SIMP_TAC[linear; LAMBDA_BETA; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT]; FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN BINOP_TAC THENL [AP_TERM_TAC; MATCH_MP_TAC(SET_RULE `(!x. f(g x) = x) ==> s = IMAGE f (IMAGE g s)`)] THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA]]]]);; (* ------------------------------------------------------------------------- *) (* Simplest case of Sard's theorem (we don't need continuity of derivative). *) (* ------------------------------------------------------------------------- *) let BABY_SARD = prove (`!f:real^M->real^N f' s. dimindex(:M) <= dimindex(:N) /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s) /\ rank(matrix(f' x)) < dimindex(:N)) ==> negligible(IMAGE f s)`, let lemma = prove (`!p w e m. dim p < dimindex(:N) /\ &0 <= m /\ &0 <= e ==> ?s. measurable s /\ {z:real^N | norm(z - w) <= m /\ ?t. t IN p /\ norm(z - w - t) <= e} SUBSET s /\ measure s <= (&2 * e) * (&2 * m) pow (dimindex(:N) - 1)`, REPEAT GEN_TAC THEN GEN_GEOM_ORIGIN_TAC `w:real^N` ["t"; "p"] THEN REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN REWRITE_TAC[VECTOR_SUB_RZERO; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^N` THEN GEOM_BASIS_MULTIPLE_TAC 1 `a:real^N` THEN X_GEN_TAC `a:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN ASM_CASES_TAC `a = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `interval[--(lambda i. if i = 1 then e else m):real^N, (lambda i. if i = 1 then e else m)]` THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL] THEN SIMP_TAC[VECTOR_NEG_COMPONENT; LAMBDA_BETA] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN REWRITE_TAC[REAL_BOUNDS_LE] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THENL [ALL_TAC; ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS]] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN ASM_SIMP_TAC[SPAN_SUPERSET; IN_ELIM_THM; DOT_BASIS; DOT_LMUL; DIMINDEX_GE_1; LE_REFL; REAL_ENTIRE; REAL_LT_IMP_NZ] THEN MP_TAC(ISPECL [`x - y:real^N`; `1`] COMPONENT_LE_NORM) THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; ARITH; DIMINDEX_GE_1] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN SIMP_TAC[VECTOR_NEG_COMPONENT; LAMBDA_BETA] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_POW_LE; REAL_POS] THEN REWRITE_TAC[REAL_ARITH `x - --x = &2 * x`] THEN SIMP_TAC[PRODUCT_CLAUSES_LEFT; DIMINDEX_GE_1] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS] THEN SIMP_TAC[ARITH; ARITH_RULE `2 <= n ==> ~(n = 1)`] THEN SIMP_TAC[PRODUCT_CONST_NUMSEG; DIMINDEX_GE_1; REAL_LE_REFL; ARITH_RULE `1 <= n ==> (n + 1) - 2 = n - 1`]]) in let semma = prove (`!f:real^M->real^N f' s B. dimindex(:M) <= dimindex(:N) /\ &0 < B /\ bounded s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s) /\ rank(matrix(f' x)) < dimindex(:N) /\ onorm(f' x) <= B) ==> negligible(IMAGE f s)`, REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `!x. x IN s ==> linear((f':real^M->real^M->real^N) x)` ASSUME_TAC THENL [ASM_MESON_TAC[has_derivative]; ALL_TAC] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN SUBGOAL_THEN `?c. s SUBSET interval(--(vec c):real^M,vec c) /\ ~(interval(--(vec c):real^M,vec c) = {})` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN DISCH_THEN(X_CHOOSE_THEN `c:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `abs c + &1` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[SUBSET; INTERVAL_NE_EMPTY] THEN REWRITE_TAC[IN_INTERVAL; VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[REAL_BOUNDS_LT] THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTERVAL_NE_EMPTY]) THEN DISCH_THEN(MP_TAC o SPEC `1`) THEN REWRITE_TAC[VEC_COMPONENT; DIMINDEX_GE_1; LE_REFL; VECTOR_NEG_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `--x < x <=> &0 < &2 * x`; REAL_OF_NUM_MUL] THEN DISCH_TAC THEN SUBGOAL_THEN `?d. &0 < d /\ d <= B /\ (d * &2) * (&4 * B) pow (dimindex(:N) - 1) <= e / &(2 * c) pow dimindex(:M) / &(dimindex(:M)) pow dimindex(:M)` STRIP_ASSUME_TAC THENL [EXISTS_TAC `min B (e / &(2 * c) pow dimindex(:M) / &(dimindex(:M)) pow dimindex(:M) / (&4 * B) pow (dimindex(:N) - 1) / &2)` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_ARITH `min x y <= x`] THEN CONJ_TAC THENL [REPEAT(MATCH_MP_TAC REAL_LT_DIV THEN CONJ_TAC) THEN ASM_SIMP_TAC[REAL_POW_LT; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1; REAL_ARITH `&0 < &4 * B <=> &0 < B`; ARITH]; ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_POW_LT; REAL_ARITH `&0 < &4 * B <=> &0 < B`; ARITH] THEN REAL_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `!x. ?r. &0 < r /\ r <= &1 / &2 /\ (x IN s ==> !y. y IN s /\ norm(y - x) < r ==> norm((f:real^M->real^N) y - f x - f' x (y - x)) <= d * norm(y - x))` MP_TAC THENL [X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; EXISTS_TAC `&1 / &4` THEN REAL_ARITH_TAC] THEN UNDISCH_THEN `!x. x IN s ==> ((f:real^M->real^N) has_derivative f' x) (at x within s)` (MP_TAC o REWRITE_RULE[HAS_DERIVATIVE_WITHIN_ALT]) THEN ASM_SIMP_TAC[RIGHT_IMP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `d:real`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min r (&1 / &2)` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_MIN_LE; REAL_LE_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_MESON_TAC[]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `r:real^M->real` THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(LABEL_TAC "*")] THEN MP_TAC(ISPECL [`--(vec c):real^M`; `(vec c):real^M`; `s:real^M->bool`; `\x:real^M. ball(x,r x)`] COVERING_LEMMA) THEN ASM_REWRITE_TAC[gauge; OPEN_BALL; CENTRE_IN_BALL] THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET_TRANS; INTERVAL_OPEN_SUBSET_CLOSED]; ALL_TAC] THEN REWRITE_TAC[VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!k:real^M->bool. k IN D ==> ?t. measurable(t) /\ IMAGE (f:real^M->real^N) (k INTER s) SUBSET t /\ measure t <= e / &(2 * c) pow (dimindex(:M)) * measure(k)` MP_TAC THENL [X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `?u v:real^M. k = interval[u,v]` (REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC) THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?x:real^M. x IN (s INTER interval[u,v]) /\ interval[u,v] SUBSET ball(x,r x)` MP_TAC THENL [ASM_MESON_TAC[]; REWRITE_TAC[IN_INTER]] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^M` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`IMAGE ((f':real^M->real^M->real^N) x) (:real^M)`; `(f:real^M->real^N) x`; `d * norm(v - u:real^M)`; `(&2 * B) * norm(v - u:real^M)`] lemma) THEN ANTS_TAC THENL [ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS; NORM_POS_LE; REAL_LT_IMP_LE] THEN MP_TAC(ISPEC `matrix ((f':real^M->real^M->real^N) x)` RANK_DIM_IM) THEN ASM_SIMP_TAC[MATRIX_WORKS] THEN REWRITE_TAC[ETA_AX] THEN ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN REPEAT(MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[]) THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] SUBSET_TRANS) THEN REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[IN_INTER; EXISTS_IN_IMAGE; IN_UNIV] THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPECL [`x:real^M`; `y:real^M`]) THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_BALL; SUBSET; NORM_SUB; dist]; ALL_TAC] THEN DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THENL [REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC(NORM_ARITH `norm(z) <= B /\ d <= B ==> norm(y - x - z:real^N) <= d ==> norm(y - x) <= &2 * B`) THEN CONJ_TAC THENL [MP_TAC(ISPEC `(f':real^M->real^M->real^N) x` ONORM) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `y - x:real^M` o CONJUNCT1) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_SIMP_TAC[ONORM_POS_LE; NORM_POS_LE]; MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; NORM_POS_LE]]; DISCH_THEN(fun th -> EXISTS_TAC `y - x:real^M` THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ]] THEN MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL])) THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[VECTOR_SUB_COMPONENT] THEN REAL_ARITH_TAC; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[REAL_ARITH `&2 * (&2 * B) * n = (&4 * B) * n`] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [REAL_POW_MUL] THEN SIMP_TAC[REAL_ARITH `(&2 * d * n) * a * b = d * &2 * a * (n * b)`] THEN REWRITE_TAC[GSYM(CONJUNCT2 real_pow)] THEN SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> SUC(n - 1) = n`] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `e / &(2 * c) pow (dimindex(:M)) / (&(dimindex(:M)) pow dimindex(:M)) * norm(v - u:real^M) pow dimindex(:N)` THEN CONJ_TAC THENL [REWRITE_TAC[REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[NORM_POS_LE; REAL_POW_LE]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [real_div] THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_POW_LE; REAL_LT_IMP_LE] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `norm(v - u:real^M) pow dimindex(:M)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_POW_MONO_INV THEN ASM_REWRITE_TAC[NORM_POS_LE] THEN SUBGOAL_THEN `u IN ball(x:real^M,r x) /\ v IN ball(x,r x)` MP_TAC THENL [ASM_MESON_TAC[SUBSET; ENDS_IN_INTERVAL; INTERIOR_EMPTY]; REWRITE_TAC[IN_BALL] THEN SUBGOAL_THEN `(r:real^M->real) x <= &1 / &2` MP_TAC THENL [ASM_REWRITE_TAC[]; CONV_TAC NORM_ARITH]]; REMOVE_THEN "*" (K ALL_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^M`; `v:real^M`]) THEN ASM_REWRITE_TAC[REAL_ARITH `x - --x = &2 * x`] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_OF_NUM_MUL] THEN X_GEN_TAC `p:num` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(sum(1..dimindex(:M)) (\i. abs((v - u:real^M)$i))) pow (dimindex(:M))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_POW_LE2 THEN SIMP_TAC[NORM_POS_LE; NORM_LE_L1]; REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM REAL_SUB_LE] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_DIV; REAL_LT_POW2] THEN ASM_SIMP_TAC[SUM_CONST_NUMSEG; PRODUCT_CONST_NUMSEG; VECTOR_SUB_COMPONENT; ADD_SUB] THEN REWRITE_TAC[REAL_POW_MUL; REAL_MUL_SYM] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN BINOP_TAC THEN REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[REAL_ABS_REFL] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_DIV; REAL_LT_POW2]]]]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:(real^M->bool)->(real^N->bool)` THEN DISCH_TAC THEN EXISTS_TAC `UNIONS (IMAGE (g:(real^M->bool)->(real^N->bool)) D)` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `D':(real^M->bool)->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum D' (\k:real^M->bool. e / &(2 * c) pow (dimindex(:M)) * measure k)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN REWRITE_TAC[SUM_LMUL] THEN REWRITE_TAC[REAL_ARITH `e / b * x:real = (e * x) / b`] THEN ASM_SIMP_TAC[REAL_POW_LT; REAL_LE_LDIV_EQ; REAL_LE_LMUL_EQ] THEN MP_TAC(ISPECL [`D':(real^M->bool)->bool`; `UNIONS D':real^M->bool`] MEASURE_ELEMENTARY) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[division_of] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET]] THEN GEN_TAC THEN DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[SUBSET; INTERIOR_EMPTY]]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `y = z /\ x <= e ==> x = y ==> z <= e`) THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[MEASURE_INTERVAL; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(interval[--(vec c):real^M,vec c])` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS; ASM SET_TAC[]] THEN ASM_MESON_TAC[SUBSET; MEASURABLE_INTERVAL]; SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VEC_COMPONENT; VECTOR_NEG_COMPONENT; REAL_ARITH `x - --x = &2 * x /\ (--x <= x <=> &0 <= &2 * x)`] THEN ASM_SIMP_TAC[REAL_OF_NUM_MUL; REAL_LT_IMP_LE] THEN REWRITE_TAC[PRODUCT_CONST_NUMSEG; ADD_SUB; REAL_LE_REFL]]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `s = UNIONS {{x | x IN s /\ norm(x:real^M) <= &n /\ onorm((f':real^M->real^M->real^N) x) <= &n} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_MAX_LE; REAL_ARCH_SIMPLE]; REWRITE_TAC[IMAGE_UNIONS] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[GSYM IMAGE_o; COUNTABLE_IMAGE; NUM_COUNTABLE] THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[IN_UNIV] THEN MATCH_MP_TAC semma THEN MAP_EVERY EXISTS_TAC [`f':real^M->real^M->real^N`; `&n + &1:real`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(vec 0:real^M,&n)` THEN SIMP_TAC[BOUNDED_CBALL; SUBSET; IN_CBALL_0; IN_ELIM_THM]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN ASM_SIMP_TAC[REAL_ARITH `x <= n ==> x <= n + &1`] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_DERIVATIVE_WITHIN_SUBSET)) THEN SET_TAC[]]]);; let BABY_SARD_ALT = prove (`!f:real^M->real^N s. dimindex(:M) <= dimindex(:N) /\ (!x. x IN s ==> ?f'. (f has_derivative f') (at x within s) /\ rank(matrix f') < dimindex (:N)) ==> negligible(IMAGE f s)`, REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC BABY_SARD THEN ASM_MESON_TAC[]);; let NEGLIGIBLE_INFINITE_PREIMAGES_MOSTLY_DIFFERENTIABLE_GEN = prove (`!f:real^N->real^N s. (!y. compact {x | x IN s /\ f x = y}) /\ negligible (IMAGE f {x | x IN s /\ ~(f differentiable (at x within s))}) ==> negligible {y | INFINITE {x | x IN s /\ f x = y}}`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MP_TAC(ISPECL [`f:real^N->real^N`; `{x | x IN s /\ ?f'. (f has_derivative f') (at x within s) /\ ~invertible(matrix f':real^N^N)}`] BABY_SARD_ALT) THEN REWRITE_TAC[IN_ELIM_THM; LE_REFL] THEN ANTS_TAC THENL [X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[GSYM DET_EQ_0_RANK; INVERTIBLE_DET_NZ] THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HAS_DERIVATIVE_WITHIN_SUBSET) THEN REWRITE_TAC[SUBSET_RESTRICT]; REWRITE_TAC[IMP_IMP; GSYM NEGLIGIBLE_UNION_EQ]] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; INFINITE] THEN X_GEN_TAC `y:real^N` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[DE_MORGAN_THM; IN_UNION; SET_RULE `~(y IN IMAGE f s) <=> (!x. f x = y ==> ~(x IN s))`] THEN REWRITE_TAC[IN_ELIM_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN REWRITE_TAC[GSYM IMP_CONJ_ALT; differentiable] THEN REWRITE_TAC[AND_FORALL_THM; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN REWRITE_TAC[MESON[] `~(?x. P x /\ Q x) /\ (?x. P x) <=> (?x. P x /\ ~Q x) /\ (!x. P x ==> ~Q x)`] THEN REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM; RIGHT_IMP_EXISTS_THM] THEN DISCH_THEN(MP_TAC o CONJUNCT1) THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f':real^N->real^N->real^N` THEN STRIP_TAC THEN MATCH_MP_TAC DIFFERENTIABLE_FINITE_PREIMAGES_GEN THEN EXISTS_TAC `f':real^N->real^N->real^N` THEN ASM_SIMP_TAC[GSYM INVERTIBLE_DET_NZ]);; let NEGLIGIBLE_INFINITE_PREIMAGES_MOSTLY_DIFFERENTIABLE = prove (`!f:real^N->real^N s. f continuous_on s /\ compact s /\ negligible (IMAGE f {x | x IN s /\ ~(f differentiable (at x within s))}) ==> negligible {y | INFINITE {x | x IN s /\ f x = y}}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_INFINITE_PREIMAGES_MOSTLY_DIFFERENTIABLE_GEN THEN ASM_REWRITE_TAC[] THEN GEN_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CLOSED_IN_COMPACT)) THEN ONCE_REWRITE_TAC[GSYM IN_SING] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN ASM_REWRITE_TAC[CLOSED_SING]);; let NEGLIGIBLE_INFINITE_PREIMAGES_DIFFERENTIABLE = prove (`!f:real^N->real^N s. compact s /\ f differentiable_on s ==> negligible {y | INFINITE {x | x IN s /\ f(x) = y}}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_INFINITE_PREIMAGES_MOSTLY_DIFFERENTIABLE THEN ASM_SIMP_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_ON] THEN RULE_ASSUM_TAC(REWRITE_RULE[differentiable_on]) THEN REWRITE_TAC[TAUT `p /\ ~q <=> ~(p ==> q)`] THEN ASM_REWRITE_TAC[EMPTY_GSPEC; IMAGE_CLAUSES; NEGLIGIBLE_EMPTY]);; (* ------------------------------------------------------------------------- *) (* Also negligibility of BV low-dimensional image. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_IMAGE_BOUNDED_VARIATION_INTERVAL = prove (`!f:real^1->real^N s. 2 <= dimindex(:N) /\ f has_bounded_variation_on s /\ is_interval s ==> negligible(IMAGE f s)`, let lemma = prove (`!f:real^1->real^N a b. 2 <= dimindex(:N) /\ f has_bounded_variation_on interval[a,b] ==> negligible(IMAGE f (interval[a,b]))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FACTOR_THROUGH_VARIATION) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` (STRIP_ASSUME_TAC o GSYM)) THEN SUBGOAL_THEN `IMAGE (f:real^1->real^N) (interval[a,b]) = IMAGE g { lift(vector_variation (interval[a,u]) f) | u | u IN interval[a,b]}` SUBST1_TAC THENL [REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o; o_DEF] THEN ASM SET_TAC[]; MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE_LOWDIM THEN ASM_REWRITE_TAC[DIMINDEX_1; ARITH_RULE `1 < n <=> 2 <= n`] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`(:real^1)`; `&1`] THEN ASM_SIMP_TAC[GSYM dist; REAL_MUL_LID; IN_INTER; IN_UNIV; OPEN_UNIV]]) in REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^1->real^N) (closure s INTER UNIONS {interval[--vec n,vec n] | n IN (:num)})` THEN CONJ_TAC THENL [REWRITE_TAC[INTER_UNIONS; IMAGE_UNIONS] THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; FORALL_IN_IMAGE; o_THM] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(K ALL_TAC) THEN MP_TAC(ISPEC `closure s INTER interval[--vec n:real^1,vec n]` IS_INTERVAL_COMPACT) THEN SIMP_TAC[CLOSED_INTER_COMPACT; COMPACT_INTERVAL; CLOSED_CLOSURE] THEN RULE_ASSUM_TAC(REWRITE_RULE[IS_INTERVAL_CONVEX_1]) THEN ASM_SIMP_TAC[IS_INTERVAL_CONVEX_1; CONVEX_CLOSURE; CONVEX_INTER; CONVEX_INTERVAL; LEFT_IMP_EXISTS_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HAS_BOUNDED_VARIATION_ON_SUBSET THEN EXISTS_TAC `closure s:real^1->bool` THEN ASM_SIMP_TAC[HAS_BOUNDED_VARIATION_ON_CLOSURE; IS_INTERVAL_CONVEX_1] THEN ASM SET_TAC[]; MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET_INTER; CLOSURE_SUBSET] THEN MATCH_MP_TAC(SET_RULE `(!x. ?n. x IN f n) ==> s SUBSET UNIONS {f n | n IN UNIV}`) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; GSYM REAL_ABS_BOUNDS] THEN REWRITE_TAC[GSYM FORALL_DROP; DROP_VEC; REAL_ARCH_SIMPLE]]);; let NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE_LOWDIM = prove (`!f:real^1->real^N s. 2 <= dimindex(:N) /\ f absolutely_continuous_on s /\ is_interval s ==> negligible(IMAGE f s)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_EXTENDS_TO_CLOSURE)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^1->real^N) (closure s INTER UNIONS {interval[--vec n,vec n] | n IN (:num)})` THEN CONJ_TAC THENL [REWRITE_TAC[INTER_UNIONS; IMAGE_UNIONS] THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; FORALL_IN_IMAGE; o_THM] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(K ALL_TAC) THEN MP_TAC(ISPEC `closure s INTER interval[--vec n:real^1,vec n]` IS_INTERVAL_COMPACT) THEN SIMP_TAC[CLOSED_INTER_COMPACT; COMPACT_INTERVAL; CLOSED_CLOSURE] THEN RULE_ASSUM_TAC(REWRITE_RULE[IS_INTERVAL_CONVEX_1]) THEN ASM_SIMP_TAC[IS_INTERVAL_CONVEX_1; CONVEX_CLOSURE; CONVEX_INTER; CONVEX_INTERVAL; LEFT_IMP_EXISTS_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_IMAGE_BOUNDED_VARIATION_INTERVAL THEN ASM_REWRITE_TAC[IS_INTERVAL_INTERVAL] THEN MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_ON_IMP_HAS_BOUNDED_VARIATION_ON THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `closure s:real^1->bool` THEN ASM SET_TAC[]; MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> g x = f x) /\ s SUBSET c /\ u = UNIV ==> IMAGE f s SUBSET IMAGE g (c INTER u)`) THEN ASM_REWRITE_TAC[CLOSURE_SUBSET; UNIONS_GSPEC; EXTENSION] THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM] THEN REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; GSYM REAL_ABS_BOUNDS] THEN REWRITE_TAC[GSYM FORALL_DROP; DROP_VEC; REAL_ARCH_SIMPLE]]);; let NEGLIGIBLE_RECTIFIABLE_PATH_IMAGE = prove (`!g:real^1->real^N. 2 <= dimindex(:N) /\ rectifiable_path g ==> negligible(path_image g)`, REWRITE_TAC[rectifiable_path; path_image] THEN SIMP_TAC[NEGLIGIBLE_IMAGE_BOUNDED_VARIATION_INTERVAL; IS_INTERVAL_INTERVAL]);; let INTERIOR_RECTIFIABLE_PATH_IMAGE = prove (`!g:real^1->real^N. 2 <= dimindex(:N) /\ rectifiable_path g ==> interior(path_image g) = {}`, MESON_TAC[NEGLIGIBLE_RECTIFIABLE_PATH_IMAGE; NEGLIGIBLE_EMPTY_INTERIOR]);; (* ------------------------------------------------------------------------- *) (* Properties of Lebesgue measurable sets. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_ON_INDICATOR = prove (`!s t. indicator t measurable_on s <=> lebesgue_measurable(s INTER t)`, REPEAT GEN_TAC THEN REWRITE_TAC[lebesgue_measurable] THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[indicator; IN_UNIV; IN_INTER] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN ASM_MESON_TAC[]);; let MEASURABLE_ON_INDICATOR_SUBSET = prove (`!s t:real^N->bool. t SUBSET s ==> (indicator t measurable_on s <=> lebesgue_measurable t)`, SIMP_TAC[MEASURABLE_ON_INDICATOR; SET_RULE `t SUBSET s ==> s INTER t = t`]);; let MEASURABLE_IMP_LEBESGUE_MEASURABLE = prove (`!s:real^N->bool. measurable s ==> lebesgue_measurable s`, REPEAT STRIP_TAC THEN REWRITE_TAC[lebesgue_measurable] THEN MATCH_MP_TAC INTEGRABLE_IMP_MEASURABLE THEN ASM_REWRITE_TAC[indicator; GSYM MEASURABLE_INTEGRABLE]);; let NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE = prove (`!s:real^N->bool. negligible s ==> lebesgue_measurable s`, SIMP_TAC[NEGLIGIBLE_IMP_MEASURABLE; MEASURABLE_IMP_LEBESGUE_MEASURABLE]);; let LEBESGUE_MEASURABLE_EMPTY = prove (`lebesgue_measurable {}`, SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_EMPTY]);; let LEBESGUE_MEASURABLE_UNIV = prove (`lebesgue_measurable (:real^N)`, REWRITE_TAC[lebesgue_measurable; indicator; IN_UNIV; MEASURABLE_ON_CONST]);; let LEBESGUE_MEASURABLE_COMPACT = prove (`!s:real^N->bool. compact s ==> lebesgue_measurable s`, SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_COMPACT]);; let LEBESGUE_MEASURABLE_BALL = prove (`!a:real^N r. lebesgue_measurable(ball(a,r))`, SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_BALL]);; let LEBESGUE_MEASURABLE_CBALL = prove (`!a:real^N r. lebesgue_measurable(cball(a,r))`, SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_CBALL]);; let LEBESGUE_MEASURABLE_INTERVAL = prove (`(!a b:real^N. lebesgue_measurable(interval[a,b])) /\ (!a b:real^N. lebesgue_measurable(interval(a,b)))`, SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_INTERVAL]);; let LEBESGUE_MEASURABLE_INTER = prove (`!s t:real^N->bool. lebesgue_measurable s /\ lebesgue_measurable t ==> lebesgue_measurable(s INTER t)`, REWRITE_TAC[indicator; lebesgue_measurable; MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_INTER]);; let LEBESGUE_MEASURABLE_UNION = prove (`!s t:real^N->bool. lebesgue_measurable s /\ lebesgue_measurable t ==> lebesgue_measurable(s UNION t)`, REWRITE_TAC[indicator; lebesgue_measurable; MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_UNION]);; let LEBESGUE_MEASURABLE_DIFF = prove (`!s t:real^N->bool. lebesgue_measurable s /\ lebesgue_measurable t ==> lebesgue_measurable(s DIFF t)`, REWRITE_TAC[indicator; lebesgue_measurable; MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_DIFF]);; let LEBESGUE_MEASURABLE_COMPL = prove (`!s. lebesgue_measurable((:real^N) DIFF s) <=> lebesgue_measurable s`, MESON_TAC[LEBESGUE_MEASURABLE_DIFF; LEBESGUE_MEASURABLE_UNIV; COMPL_COMPL]);; let LEBESGUE_MEASURABLE_ON_SUBINTERVALS = prove (`!s. lebesgue_measurable s <=> !a b:real^N. lebesgue_measurable(s INTER interval[a,b])`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[LEBESGUE_MEASURABLE_INTERVAL; LEBESGUE_MEASURABLE_INTER] THEN REWRITE_TAC[lebesgue_measurable] THEN DISCH_TAC THEN MATCH_MP_TAC INTEGRABLE_SUBINTERVALS_IMP_MEASURABLE THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `(\x. vec 1):real^N->real^1` THEN REWRITE_TAC[INTEGRABLE_CONST] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; indicator; IN_INTER] THEN MESON_TAC[]; REPEAT STRIP_TAC THEN REWRITE_TAC[indicator] THEN COND_CASES_TAC THEN REWRITE_TAC[DROP_VEC; NORM_REAL; GSYM drop] THEN REAL_ARITH_TAC]);; let LEBESGUE_MEASURABLE_CLOSED = prove (`!s:real^N->bool. closed s ==> lebesgue_measurable s`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[LEBESGUE_MEASURABLE_ON_SUBINTERVALS] THEN ASM_SIMP_TAC[CLOSED_INTER_COMPACT; LEBESGUE_MEASURABLE_COMPACT; COMPACT_INTERVAL]);; let LEBESGUE_MEASURABLE_OPEN = prove (`!s:real^N->bool. open s ==> lebesgue_measurable s`, REWRITE_TAC[OPEN_CLOSED] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM LEBESGUE_MEASURABLE_COMPL] THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_CLOSED]);; let LEBESGUE_MEASURABLE_OPEN_IN = prove (`!s t:real^N->bool. open_in (subtopology euclidean s) t /\ lebesgue_measurable s ==> lebesgue_measurable t`, MESON_TAC[OPEN_IN_OPEN; LEBESGUE_MEASURABLE_OPEN; LEBESGUE_MEASURABLE_INTER]);; let LEBESGUE_MEASURABLE_CLOSED_IN = prove (`!s t:real^N->bool. closed_in (subtopology euclidean s) t /\ lebesgue_measurable s ==> lebesgue_measurable t`, MESON_TAC[CLOSED_IN_CLOSED; LEBESGUE_MEASURABLE_CLOSED; LEBESGUE_MEASURABLE_INTER]);; let LEBESGUE_MEASURABLE_UNIONS = prove (`!f. FINITE f /\ (!s. s IN f ==> lebesgue_measurable s) ==> lebesgue_measurable (UNIONS f)`, REWRITE_TAC[indicator; lebesgue_measurable; MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_UNIONS]);; let LEBESGUE_MEASURABLE_COUNTABLE_UNIONS = prove (`!f:(real^N->bool)->bool. COUNTABLE f /\ (!s. s IN f ==> lebesgue_measurable s) ==> lebesgue_measurable (UNIONS f)`, REWRITE_TAC[indicator; lebesgue_measurable; MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_COUNTABLE_UNIONS]);; let LEBESGUE_MEASURABLE_COUNTABLE_UNIONS_EXPLICIT = prove (`!s:num->real^N->bool. (!n. lebesgue_measurable(s n)) ==> lebesgue_measurable(UNIONS {s n | n IN (:num)})`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE; IN_UNIV; NUM_COUNTABLE]);; let LEBESGUE_MEASURABLE_COUNTABLE_INTERS = prove (`!f:(real^N->bool)->bool. COUNTABLE f /\ (!s. s IN f ==> lebesgue_measurable s) ==> lebesgue_measurable (INTERS f)`, REPEAT STRIP_TAC THEN REWRITE_TAC[INTERS_UNIONS; LEBESGUE_MEASURABLE_COMPL] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; COUNTABLE_IMAGE; LEBESGUE_MEASURABLE_COMPL]);; let LEBESGUE_MEASURABLE_COUNTABLE_INTERS_EXPLICIT = prove (`!s:num->real^N->bool. (!n. lebesgue_measurable(s n)) ==> lebesgue_measurable(INTERS {s n | n IN (:num)})`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; COUNTABLE_IMAGE; NUM_COUNTABLE]);; let LEBESGUE_MEASURABLE_INTERS = prove (`!f:(real^N->bool)->bool. FINITE f /\ (!s. s IN f ==> lebesgue_measurable s) ==> lebesgue_measurable (INTERS f)`, SIMP_TAC[LEBESGUE_MEASURABLE_COUNTABLE_INTERS; FINITE_IMP_COUNTABLE]);; let GDELTA_IMP_LEBESGUE_MEASURABLE = prove (`!s:real^N->bool. gdelta s ==> lebesgue_measurable s`, GEN_TAC THEN REWRITE_TAC[gdelta; INTERSECTION_OF] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_OPEN]);; let FSIGMA_IMP_LEBESGUE_MEASURABLE = prove (`!s:real^N->bool. fsigma s ==> lebesgue_measurable s`, GEN_TAC THEN REWRITE_TAC[fsigma; UNION_OF] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_CLOSED]);; let BOREL_IMP_LEBESGUE_MEASURABLE = prove (`!s:real^N->bool. borel s ==> lebesgue_measurable s`, MATCH_MP_TAC borel_INDUCT THEN REWRITE_TAC[LEBESGUE_MEASURABLE_OPEN; LEBESGUE_MEASURABLE_COMPL] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COUNTABLE_UNIONS]);; let LEBESGUE_MEASURABLE_IFF_MEASURABLE = prove (`!s:real^N->bool. bounded s ==> (lebesgue_measurable s <=> measurable s)`, REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN REWRITE_TAC[lebesgue_measurable; indicator; MEASURABLE_INTEGRABLE] THEN SUBGOAL_THEN `?a b:real^N. s = s INTER interval[a,b]` (REPEAT_TCL CHOOSE_THEN SUBST1_TAC) THENL [REWRITE_TAC[SET_RULE `s = s INTER t <=> s SUBSET t`] THEN ASM_MESON_TAC[BOUNDED_SUBSET_CLOSED_INTERVAL]; ALL_TAC] THEN REWRITE_TAC[IN_INTER; MESON[] `(if P x /\ Q x then a else b) = (if Q x then if P x then a else b else b)`] THEN REWRITE_TAC[MEASURABLE_ON_UNIV; INTEGRABLE_RESTRICT_UNIV] THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `(\x. vec 1):real^N->real^1` THEN ASM_REWRITE_TAC[INTEGRABLE_CONST; NORM_REAL; DROP_VEC; GSYM drop] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN SIMP_TAC[DROP_VEC] THEN REAL_ARITH_TAC);; let LEBESGUE_MEASURABLE_MEASURABLE_ON_SUBINTERVALS = prove (`!s:real^N->bool. lebesgue_measurable s <=> (!a b. measurable(s INTER interval[a,b]))`, MESON_TAC[LEBESGUE_MEASURABLE_ON_SUBINTERVALS; LEBESGUE_MEASURABLE_IFF_MEASURABLE; BOUNDED_INTER; BOUNDED_INTERVAL]);; let LEBESGUE_MEASURABLE_MEASURABLE_ON_COUNTABLE_SUBINTERVALS = prove (`!s:real^N->bool. lebesgue_measurable s <=> (!n. measurable(s INTER interval[--vec n,vec n]))`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [LEBESGUE_MEASURABLE_MEASURABLE_ON_SUBINTERVALS] THEN EQ_TAC THEN SIMP_TAC[] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `!a b:real^N. ?n. s INTER interval[a,b] = ((s INTER interval[--vec n,vec n]) INTER interval[a,b])` (fun th -> ASM_MESON_TAC[th; MEASURABLE_INTERVAL; MEASURABLE_INTER]) THEN REPEAT GEN_TAC THEN MP_TAC(ISPECL [`interval[a:real^N,b]`; `vec 0:real^N`] BOUNDED_SUBSET_CBALL) THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `r:real` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `i SUBSET b ==> b SUBSET n ==> s INTER i = (s INTER n) INTER i`)) THEN REWRITE_TAC[SUBSET; IN_CBALL_0; IN_INTERVAL; VEC_COMPONENT; VECTOR_NEG_COMPONENT; GSYM REAL_ABS_BOUNDS] THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS]);; let MEASURABLE_ON_MEASURABLE_SUBSET = prove (`!f s t. s SUBSET t /\ f measurable_on t /\ measurable s ==> f measurable_on s`, MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET; MEASURABLE_IMP_LEBESGUE_MEASURABLE]);; let LEBESGUE_MEASURABLE_JORDAN = prove (`!s:real^N->bool. negligible(frontier s) ==> lebesgue_measurable s`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[LEBESGUE_MEASURABLE_ON_SUBINTERVALS] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN MATCH_MP_TAC MEASURABLE_IMP_LEBESGUE_MEASURABLE THEN MATCH_MP_TAC MEASURABLE_JORDAN THEN SIMP_TAC[BOUNDED_INTER; BOUNDED_INTERVAL] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `frontier s UNION frontier(interval[a:real^N,b])` THEN ASM_REWRITE_TAC[FRONTIER_INTER_SUBSET; NEGLIGIBLE_UNION_EQ] THEN SIMP_TAC[NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_INTERVAL]);; let LEBESGUE_MEASURABLE_CONVEX = prove (`!s:real^N->bool. convex s ==> lebesgue_measurable s`, SIMP_TAC[LEBESGUE_MEASURABLE_JORDAN; NEGLIGIBLE_CONVEX_FRONTIER]);; let LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF = prove (`!s t:real^N->bool. lebesgue_measurable s /\ negligible((s DIFF t) UNION (t DIFF s)) ==> lebesgue_measurable t`, REPEAT STRIP_TAC THEN SUBST1_TAC(SET_RULE `t:real^N->bool = (s DIFF (s DIFF t)) UNION (t DIFF s)`) THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN CONJ_TAC THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFF THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]);; let LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ = prove (`!s t:real^N->bool. negligible(s DIFF t UNION t DIFF s) ==> (lebesgue_measurable s <=> lebesgue_measurable t)`, MESON_TAC[LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF; UNION_COMM]);; let LEBESGUE_MEASURABLE_INSERT = prove (`!s a:real^N. lebesgue_measurable(a INSERT s) <=> lebesgue_measurable s`, REPEAT GEN_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{a:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]);; let LEBESGUE_MEASURABLE_DELETE = prove (`!s a:real^N. lebesgue_measurable(s DELETE a) <=> lebesgue_measurable s`, REPEAT GEN_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{a:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]);; let ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_INTER = prove (`!f:real^M->real^N s t. f absolutely_integrable_on s /\ lebesgue_measurable t ==> f absolutely_integrable_on (s INTER t)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\x. lift(norm(if x IN s then (f:real^M->real^N) x else vec 0))` THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; IN_UNIV; IN_INTER; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REWRITE_TAC[MESON[] `(if p /\ q then x else y) = if q then if p then x else y else y`] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_CASES THEN ASM_REWRITE_TAC[SET_RULE `{x | x IN s} = s`; MEASURABLE_ON_0] THEN ASM_SIMP_TAC[INTEGRABLE_IMP_MEASURABLE; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN t` THEN ASM_REWRITE_TAC[REAL_LE_REFL; LIFT_DROP; NORM_0; NORM_POS_LE]]);; let ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET = prove (`!f:real^M->real^N s t. f absolutely_integrable_on s /\ t SUBSET s /\ lebesgue_measurable t ==> f absolutely_integrable_on t`, MESON_TAC[ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_INTER; SET_RULE `s SUBSET t ==> s = t INTER s`]);; (* ------------------------------------------------------------------------- *) (* Invariance theorems for Lebesgue measurability. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_ON_TRANSLATION = prove (`!f:real^M->real^N s a. f measurable_on (IMAGE (\x. a + x) s) ==> (\x. f(a + x)) measurable_on s`, REPEAT GEN_TAC THEN REWRITE_TAC[measurable_on; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `g:num->real^M->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `IMAGE (\x:real^M. --a + x) k` THEN EXISTS_TAC `\n. (g:num->real^M->real^N) n o (\x. a + x)` THEN ASM_REWRITE_TAC[NEGLIGIBLE_TRANSLATION_EQ] THEN CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; X_GEN_TAC `x:real^M` THEN FIRST_X_ASSUM(MP_TAC o SPEC `a + x:real^M`) THEN REWRITE_TAC[o_DEF; IN_IMAGE] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `x:real^N = --a + y <=> a + x = y`] THEN REWRITE_TAC[UNWIND_THM1; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]]);; let MEASURABLE_ON_TRANSLATION_EQ = prove (`!f:real^M->real^N s a. (\x. f(a + x)) measurable_on s <=> f measurable_on (IMAGE (\x. a + x) s)`, REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[MEASURABLE_ON_TRANSLATION] THEN MP_TAC(ISPECL [`\x. (f:real^M->real^N) (a + x)`; `IMAGE (\x:real^M. a + x) s`; `--a:real^M`] MEASURABLE_ON_TRANSLATION) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF; ETA_AX; IMAGE_ID; VECTOR_ARITH `--a + a + x:real^N = x /\ a + --a + x = x`]);; let NEGLIGIBLE_LINEAR_IMAGE_GEN = prove (`!f:real^M->real^N s. linear f /\ negligible s /\ dimindex(:M) <= dimindex(:N) ==> negligible (IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_LINEAR]);; let MEASURABLE_ON_LINEAR_IMAGE_EQ_GEN = prove (`!f:real^M->real^N h:real^N->real^P s. dimindex(:M) = dimindex(:N) /\ linear f /\ (!x y. f x = f y ==> x = y) ==> ((h o f) measurable_on s <=> h measurable_on (IMAGE f s))`, let lemma = prove (`!f:real^N->real^P g:real^M->real^N h s. dimindex(:M) = dimindex(:N) /\ linear g /\ linear h /\ (!x. h(g x) = x) /\ (!x. g(h x) = x) ==> (f o g) measurable_on s ==> f measurable_on (IMAGE g s)`, REPEAT GEN_TAC THEN REWRITE_TAC[measurable_on] THEN STRIP_TAC THEN DISCH_THEN(X_CHOOSE_THEN `k:real^M->bool` (X_CHOOSE_THEN `G:num->real^M->real^P` STRIP_ASSUME_TAC)) THEN EXISTS_TAC `IMAGE (g:real^M->real^N) k` THEN EXISTS_TAC `\n x. (G:num->real^M->real^P) n ((h:real^N->real^M) x)` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_LINEAR_IMAGE_GEN THEN ASM_MESON_TAC[LE_REFL]; GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_MESON_TAC[LINEAR_CONTINUOUS_ON; CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(h:real^N->real^M) y`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_REWRITE_TAC[o_THM] THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]]) in REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; FUN_EQ_THM; o_THM; I_THM] THEN X_GEN_TAC `g:real^N->real^M` THEN STRIP_TAC THEN SUBGOAL_THEN `!y:real^N. (f:real^M->real^N) ((g:real^N->real^M) y) = y` ASSUME_TAC THENL [SUBGOAL_THEN `IMAGE (f:real^M->real^N) UNIV = UNIV` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[EXTENSION; IN_UNIV; IN_IMAGE] THEN ASM_MESON_TAC[LINEAR_SURJECTIVE_IFF_INJECTIVE_GEN]; ALL_TAC] THEN EQ_TAC THENL [ASM_MESON_TAC[lemma]; DISCH_TAC] THEN MP_TAC(ISPECL [`(h:real^N->real^P) o (f:real^M->real^N)`; `g:real^N->real^M`; `f:real^M->real^N`; `IMAGE (f:real^M->real^N) s`] lemma) THEN ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; ETA_AX] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[]);; let MEASURABLE_ON_LINEAR_IMAGE_EQ = prove (`!f:real^N->real^N h:real^N->real^P s. linear f /\ (!x y. f x = f y ==> x = y) ==> ((h o f) measurable_on s <=> h measurable_on (IMAGE f s))`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_LINEAR_IMAGE_EQ_GEN THEN ASM_MESON_TAC[]);; let LEBESGUE_MEASURABLE_TRANSLATION = prove (`!a:real^N s. lebesgue_measurable (IMAGE (\x. a + x) s) <=> lebesgue_measurable s`, ONCE_REWRITE_TAC[LEBESGUE_MEASURABLE_ON_SUBINTERVALS] THEN SIMP_TAC[LEBESGUE_MEASURABLE_IFF_MEASURABLE; BOUNDED_INTER; BOUNDED_INTERVAL] THEN GEOM_TRANSLATE_TAC[]);; add_translation_invariants [LEBESGUE_MEASURABLE_TRANSLATION];; let LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ = prove (`!f:real^N->real^N s. linear f /\ (!x y. f x = f y ==> x = y) ==> (lebesgue_measurable (IMAGE f s) <=> lebesgue_measurable s)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LINEAR_INJECTIVE_IMP_SURJECTIVE) THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN DISCH_TAC THEN FIRST_ASSUM(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC o MATCH_MP LINEAR_BIJECTIVE_LEFT_RIGHT_INVERSE) THEN REWRITE_TAC[lebesgue_measurable] THEN MP_TAC(ISPECL [`g:real^N->real^N`; `indicator(s:real^N->bool)`; `(:real^N)`] MEASURABLE_ON_LINEAR_IMAGE_EQ) THEN ASM_REWRITE_TAC[indicator; o_DEF] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; MATCH_MP_TAC EQ_IMP] THEN BINOP_TAC THENL [AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]; AP_TERM_TAC THEN ASM SET_TAC[]]);; add_linear_invariants [LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ];; let MEASURABLE_ON_REFLECT = prove (`!f:real^M->real^N s. (\x. f(--x)) measurable_on s <=> f measurable_on (IMAGE (--) s)`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN MATCH_MP_TAC MEASURABLE_ON_LINEAR_IMAGE_EQ THEN REWRITE_TAC[linear] THEN CONV_TAC VECTOR_ARITH);; (* ------------------------------------------------------------------------- *) (* Various common equivalent forms of function measurability. *) (* ------------------------------------------------------------------------- *) let (MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT, MEASURABLE_ON_SIMPLE_FUNCTION_LIMIT) = (CONJ_PAIR o prove) (`(!f:real^M->real^N. f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | f(x)$k < a}) /\ (!f:real^M->real^N. f measurable_on (:real^M) <=> ?g. (!n. (g n) measurable_on (:real^M)) /\ (!n. FINITE(IMAGE (g n) (:real^M))) /\ (!x. ((\n. g n x) --> f x) sequentially))`, let lemma0 = prove (`!f:real^M->real^1 n m. integer m /\ m / &2 pow n <= drop(f x) /\ drop(f x) < (m + &1) / &2 pow n /\ abs(m) <= &2 pow (2 * n) ==> vsum {k | integer k /\ abs(k) <= &2 pow (2 * n)} (\k. k / &2 pow n % indicator {y:real^M | k / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1) / &2 pow n} x) = lift(m / &2 pow n)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `vsum {m} (\k. k / &2 pow n % indicator {y:real^M | k / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1) / &2 pow n} x)` THEN CONJ_TAC THENL [MATCH_MP_TAC VSUM_SUPERSET THEN ASM_REWRITE_TAC[SING_SUBSET; IN_ELIM_THM; IN_SING] THEN X_GEN_TAC `k:real` THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISJ2_TAC THEN ASM_REWRITE_TAC[indicator; IN_ELIM_THM] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `F ==> p`) THEN UNDISCH_TAC `~(k:real = m)` THEN ASM_SIMP_TAC[REAL_EQ_INTEGERS] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN REAL_ARITH_TAC; ASM_REWRITE_TAC[VSUM_SING; indicator; IN_ELIM_THM; LIFT_EQ_CMUL]]) in let lemma1 = prove (`!f:real^M->real^1. (!a b. lebesgue_measurable {x | a <= drop(f x) /\ drop(f x) < b}) ==> ?g. (!n. (g n) measurable_on (:real^M)) /\ (!n. FINITE(IMAGE (g n) (:real^M))) /\ (!x. ((\n. g n x) --> f x) sequentially)`, REPEAT STRIP_TAC THEN EXISTS_TAC `\n x. vsum {k | integer k /\ abs(k) <= &2 pow (2 * n)} (\k. k / &2 pow n % indicator {y:real^M | k / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1) / &2 pow n} x)` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC MEASURABLE_ON_VSUM THEN REWRITE_TAC[REAL_ABS_BOUNDS; FINITE_INTSEG; IN_ELIM_THM] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_CMUL THEN ASM_REWRITE_TAC[GSYM lebesgue_measurable; ETA_AX]; X_GEN_TAC `n:num` THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `IMAGE (\k. lift(k / &2 pow n)) {k | integer k /\ abs(k) <= &2 pow (2 * n)}` THEN CONJ_TAC THENL [SIMP_TAC[REAL_ABS_BOUNDS; FINITE_INTSEG; FINITE_IMAGE]; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_IMAGE] THEN ASM_CASES_TAC `?k. integer k /\ abs k <= &2 pow (2 * n) /\ k / &2 pow n <= drop(f(x:real^M)) /\ drop(f x) < (k + &1) / &2 pow n` THENL [FIRST_X_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS) THEN X_GEN_TAC `m:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC lemma0 THEN ASM_REWRITE_TAC[]; EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[IN_ELIM_THM; INTEGER_CLOSED; REAL_ABS_NUM] THEN SIMP_TAC[REAL_POW_LE; REAL_POS; real_div; REAL_MUL_LZERO] THEN REWRITE_TAC[LIFT_NUM; GSYM real_div] THEN MATCH_MP_TAC VSUM_EQ_0 THEN X_GEN_TAC `k:real` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISJ2_TAC THEN REWRITE_TAC[indicator; IN_ELIM_THM] THEN ASM_MESON_TAC[]]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN MP_TAC(ISPECL [`&2`; `abs(drop((f:real^M->real^1) x))`] REAL_ARCH_POW) THEN ANTS_TAC THENL [REAL_ARITH_TAC; DISCH_THEN(X_CHOOSE_TAC `N1:num`)] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN REWRITE_TAC[REAL_POW_INV] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `N2:num` MP_TAC) THEN SUBST1_TAC(REAL_ARITH `inv(&2 pow N2) = &1 / &2 pow N2`) THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_POW2] THEN DISCH_TAC THEN EXISTS_TAC `MAX N1 N2` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN ABBREV_TAC `m = floor(&2 pow n * drop(f(x:real^M)))` THEN SUBGOAL_THEN `dist(lift(m / &2 pow n),(f:real^M->real^1) x) < e` MP_TAC THENL [REWRITE_TAC[DIST_REAL; GSYM drop; LIFT_DROP] THEN MATCH_MP_TAC REAL_LT_LCANCEL_IMP THEN EXISTS_TAC `abs(&2 pow n)` THEN REWRITE_TAC[GSYM REAL_ABS_MUL; REAL_SUB_LDISTRIB] THEN SIMP_TAC[REAL_DIV_LMUL; REAL_POW_EQ_0; GSYM REAL_ABS_NZ; REAL_OF_NUM_EQ; ARITH] THEN MATCH_MP_TAC(REAL_ARITH `x <= y /\ y < x + &1 /\ &1 <= z ==> abs(x - y) < z`) THEN EXPAND_TAC "m" THEN REWRITE_TAC[FLOOR] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `e * &2 pow N2` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_ABS_POW; REAL_ABS_NUM] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; MATCH_MP_TAC(NORM_ARITH `x:real^1 = y ==> dist(y,z) < e ==> dist(x,z) < e`) THEN MATCH_MP_TAC lemma0 THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN EXPAND_TAC "m" THEN REWRITE_TAC[FLOOR] THEN SIMP_TAC[REAL_ABS_BOUNDS; REAL_LE_FLOOR; REAL_FLOOR_LE; INTEGER_CLOSED] THEN MATCH_MP_TAC(REAL_ARITH `abs(x) <= e ==> --e <= x /\ x - &1 < e`) THEN REWRITE_TAC[MULT_2; REAL_POW_ADD; REAL_ABS_MUL; REAL_ABS_POW; REAL_ABS_NUM] THEN MATCH_MP_TAC REAL_LE_LMUL THEN SIMP_TAC[REAL_POW_LE; REAL_POS] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e ==> e <= d ==> x <= d`))] THEN MATCH_MP_TAC REAL_POW_MONO THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC]) in MATCH_MP_TAC(MESON[] `(!f. P f ==> Q f) /\ (!f. Q f ==> R f) /\ (!f. R f ==> P f) ==> (!f. P f <=> Q f) /\ (!f. P f <=> R f)`) THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `g:real^M->real^N` THEN DISCH_TAC THEN ABBREV_TAC `f:real^M->real^N = \x. --(g x)` THEN SUBGOAL_THEN `(f:real^M->real^N) measurable_on (:real^M)` ASSUME_TAC THENL [EXPAND_TAC "f" THEN MATCH_MP_TAC MEASURABLE_ON_NEG THEN ASM_SIMP_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM REAL_LT_NEG2] THEN X_GEN_TAC `a:real` THEN SPEC_TAC(`--a:real`,`a:real`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FUN_EQ_THM]) THEN SIMP_TAC[GSYM VECTOR_NEG_COMPONENT] THEN DISCH_THEN(K ALL_TAC) THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `k:num` o GEN_REWRITE_RULE I [MEASURABLE_ON_COMPONENTWISE]) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MP_TAC(GEN `d:real` (ISPECL [`\x. lift ((f:real^M->real^N) x$k)`; `(\x. lift a + (lambda i. d)):real^M->real^1`; `(:real^M)`] MEASURABLE_ON_MIN)) THEN ASM_REWRITE_TAC[MEASURABLE_ON_CONST] THEN DISCH_THEN(fun th -> MP_TAC(GEN `n:num` (ISPEC `&n + &1` (MATCH_MP MEASURABLE_ON_CMUL (MATCH_MP MEASURABLE_ON_SUB (CONJ (SPEC `inv(&n + &1)` th) (SPEC `&0` th))))))) THEN REWRITE_TAC[lebesgue_measurable; indicator] THEN DISCH_THEN(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_LIMIT)) THEN EXISTS_TAC `{}:real^M->bool` THEN REWRITE_TAC[NEGLIGIBLE_EMPTY; IN_DIFF; IN_UNIV; NOT_IN_EMPTY] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN SIMP_TAC[LIM_SEQUENTIALLY; DIST_REAL; VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; LAMBDA_BETA; DIMINDEX_1; ARITH] THEN REWRITE_TAC[GSYM drop; LIFT_DROP; REAL_ADD_RID] THEN SIMP_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`; REAL_ARITH `&0 < d ==> (min x (a + d) - min x a = if x <= a then &0 else if x <= a + d then x - a else d)`] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN ASM_CASES_TAC `a < (f:real^M->real^N) x $k` THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[REAL_ARITH `(x:real^N)$k <= a <=> ~(a < x$k)`] THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; DROP_VEC; REAL_SUB_REFL; REAL_ABS_NUM] THEN MP_TAC(SPEC `((f:real^M->real^N) x)$k - a` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN SUBGOAL_THEN `a + inv(&n + &1) < ((f:real^M->real^N) x)$k` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `N < f - a ==> n <= N ==> a + n < f`)) THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[REAL_MUL_RINV; REAL_ARITH `~(&n + &1 = &0)`] THEN ASM_REAL_ARITH_TAC]; REPEAT STRIP_TAC THEN SUBGOAL_THEN `!k. 1 <= k /\ k <= dimindex(:N) ==> ?g. (!n. (g n) measurable_on (:real^M)) /\ (!n. FINITE(IMAGE (g n) (:real^M))) /\ (!x. ((\n. g n x) --> lift((f x:real^N)$k)) sequentially)` MP_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma1 THEN ASM_SIMP_TAC[LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | Q x} DIFF {x | ~P x}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFF THEN ASM_SIMP_TAC[REAL_NOT_LE]; GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM]] THEN REWRITE_TAC[SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `g:num->num->real^M->real^1` MP_TAC) THEN REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN EXISTS_TAC `\n x. (lambda k. drop((g:num->num->real^M->real^1) k n x)):real^N` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN ONCE_REWRITE_TAC[MEASURABLE_ON_COMPONENTWISE] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA; LIFT_DROP; ETA_AX]; X_GEN_TAC `n:num` THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> lift(x$i) IN IMAGE (g i (n:num)) (:real^M)}` THEN ASM_SIMP_TAC[GSYM IN_IMAGE_LIFT_DROP; SET_RULE `{x | x IN s} = s`; FINITE_IMAGE; FINITE_CART] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; IN_UNIV] THEN SIMP_TAC[IN_IMAGE; IN_UNIV; LAMBDA_BETA; DROP_EQ] THEN MESON_TAC[]; X_GEN_TAC `x:real^M` THEN ONCE_REWRITE_TAC[LIM_COMPONENTWISE_LIFT] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA; LIFT_DROP; ETA_AX]]; X_GEN_TAC `f:real^M->real^N` THEN DISCH_THEN(X_CHOOSE_THEN `g:num->real^M->real^N` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC MEASURABLE_ON_LIMIT THEN MAP_EVERY EXISTS_TAC [`g:num->real^M->real^N`; `{}:real^M->bool`] THEN ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]]);; let MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GE = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | f(x)$k >= a}`, GEN_TAC THEN REWRITE_TAC[REAL_ARITH `x >= a <=> ~(x < a)`] THEN REWRITE_TAC[SET_RULE `{x | ~P x} = UNIV DIFF {x | P x}`] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT]);; let MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | f(x)$k > a}`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEASURABLE_ON_NEG_EQ] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT] THEN GEN_REWRITE_TAC LAND_CONV [MESON[REAL_NEG_NEG] `(!x. P x) <=> (!x:real. P(--x))`] THEN REWRITE_TAC[real_gt; VECTOR_NEG_COMPONENT; REAL_LT_NEG2]);; let MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | f(x)$k <= a}`, GEN_TAC THEN REWRITE_TAC[REAL_ARITH `x <= a <=> ~(x > a)`] THEN REWRITE_TAC[SET_RULE `{x | ~P x} = UNIV DIFF {x | P x}`] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]);; let (MEASURABLE_ON_PREIMAGE_OPEN_INTERVAL, MEASURABLE_ON_PREIMAGE_OPEN) = (CONJ_PAIR o prove) (`(!f:real^M->real^N. f measurable_on (:real^M) <=> !a b. lebesgue_measurable {x | f(x) IN interval(a,b)}) /\ (!f:real^M->real^N. f measurable_on (:real^M) <=> !t. open t ==> lebesgue_measurable {x | f(x) IN t})`, let ulemma = prove (`{x | f x IN UNIONS D} = UNIONS {{x | f(x) IN s} | s IN D}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in MATCH_MP_TAC(MESON[] `(!f. P f ==> Q f) /\ (!f. Q f ==> R f) /\ (!f. R f ==> P f) ==> (!f. P f <=> Q f) /\ (!f. P f <=> R f)`) THEN REPEAT CONJ_TAC THENL [REPEAT STRIP_TAC THEN SUBGOAL_THEN `{x | (f:real^M->real^N) x IN interval(a,b)} = INTERS {{x | a$k < f(x)$k} | k IN 1..dimindex(:N)} INTER INTERS {{x | (--b)$k < --(f(x))$k} | k IN 1..dimindex(:N)}` SUBST1_TAC THENL [REWRITE_TAC[IN_INTERVAL; GSYM IN_NUMSEG] THEN REWRITE_TAC[VECTOR_NEG_COMPONENT; REAL_LT_NEG2] THEN REWRITE_TAC[INTERS_GSPEC] THEN SET_TAC[]; MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN CONJ_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_INTERS THEN SIMP_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]); FIRST_X_ASSUM(MP_TAC o MATCH_MP MEASURABLE_ON_NEG) THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]] THEN ASM_SIMP_TAC[real_gt]]; REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_COUNTABLE_UNION_OPEN_INTERVALS) THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN ONCE_REWRITE_TAC[ulemma] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `i:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM]; REPEAT STRIP_TAC THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{x:real^M | (f x)$k < a} = {x | f x IN {y:real^N | y$k < a}}`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[OPEN_HALFSPACE_COMPONENT_LT]]);; let MEASURABLE_ON_PREIMAGE_CLOSED = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !t. closed t ==> lebesgue_measurable {x | f(x) IN t}`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM LEBESGUE_MEASURABLE_COMPL; closed] THEN REWRITE_TAC[SET_RULE `UNIV DIFF {x | f x IN t} = {x | f x IN (UNIV DIFF t)}`] THEN REWRITE_TAC[MESON[COMPL_COMPL] `(!s. P(UNIV DIFF s)) <=> (!s. P s)`] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN]);; let MEASURABLE_ON_PREIMAGE_CLOSED_INTERVAL = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a b. lebesgue_measurable {x | f(x) IN interval[a,b]}`, let ulemma = prove (`{x | f x IN UNIONS D} = UNIONS {{x | f(x) IN s} | s IN D}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in GEN_TAC THEN EQ_TAC THENL [SIMP_TAC[MEASURABLE_ON_PREIMAGE_CLOSED; CLOSED_INTERVAL]; DISCH_TAC] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_COUNTABLE_UNION_CLOSED_INTERVALS) THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN ONCE_REWRITE_TAC[ulemma] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `i:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM]);; let MEASURABLE_ON_PREIMAGE_BOREL = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !t. borel t ==> lebesgue_measurable {x | f x IN t}`, let lemma = prove (`{x | f x IN UNIONS u} = UNIONS {{x | f x IN t} | t IN u}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in GEN_TAC THEN EQ_TAC THENL [DISCH_TAC; MESON_TAC[MEASURABLE_ON_PREIMAGE_OPEN; OPEN_IMP_BOREL]] THEN MATCH_MP_TAC borel_INDUCT THEN ASM_REWRITE_TAC[GSYM MEASURABLE_ON_PREIMAGE_OPEN] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL; lemma; SET_RULE `{x | f x IN UNIV DIFF t} = UNIV DIFF {x | f x IN t}`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE]);; let LEBESGUE_MEASURABLE_PREIMAGE_OPEN = prove (`!f:real^M->real^N t. f measurable_on (:real^M) /\ open t ==> lebesgue_measurable {x | f(x) IN t}`, SIMP_TAC[MEASURABLE_ON_PREIMAGE_OPEN]);; let LEBESGUE_MEASURABLE_PREIMAGE_CLOSED = prove (`!f:real^M->real^N t. f measurable_on (:real^M) /\ closed t ==> lebesgue_measurable {x | f(x) IN t}`, SIMP_TAC[MEASURABLE_ON_PREIMAGE_CLOSED]);; let LEBESGUE_MEASURABLE_PREIMAGE_BOREL = prove (`!f:real^M->real^N t. f measurable_on (:real^M) /\ borel t ==> lebesgue_measurable {x | f(x) IN t}`, SIMP_TAC[MEASURABLE_ON_PREIMAGE_BOREL]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_LE = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a. lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k <= (a:real^N)$k}`, GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [GEN_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{x | !k. P k ==> f x$k <= a k} = {x | f(x) IN {y | !k. P k ==> y$k <= a k}}`] THEN FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_CLOSED]) THEN REWRITE_TAC[CLOSED_INTERVAL_LEFT]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE] THEN MAP_EVERY X_GEN_TAC [`a:real`; `k:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `{x | (f:real^M->real^N) x$k <= a} = UNIONS {{x | !j. 1 <= j /\ j <= dimindex(:N) ==> f x$j <= ((lambda i. if i = k then a else &n):real^N)$j} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN SIMP_TAC[LAMBDA_BETA] THEN SPEC_TAC(`(f:real^M->real^N) x`,`y:real^N`) THEN GEN_TAC THEN ASM_CASES_TAC `(y:real^N)$k <= a` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `sup {(y:real^N)$j | j IN 1..dimindex(:N)}` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[REAL_SUP_LE_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN ASM_MESON_TAC[]; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; NUM_COUNTABLE; FORALL_IN_IMAGE]]]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_GE = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a. lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k >= (a:real^N)$k}`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEASURABLE_ON_NEG_EQ] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_ORTHANT_LE] THEN GEN_REWRITE_TAC LAND_CONV [MESON[VECTOR_NEG_NEG] `(!x:real^N. P x) <=> (!x. P(--x))`] THEN REWRITE_TAC[REAL_ARITH `--x <= --y <=> x >= y`; VECTOR_NEG_COMPONENT]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_LT = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a. lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k < (a:real^N)$k}`, GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [GEN_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{x | !k. P k ==> f x$k < a k} = {x | f(x) IN {y | !k. P k ==> y$k < a k}}`] THEN FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_OPEN]) THEN REWRITE_TAC[OPEN_INTERVAL_LEFT]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT] THEN MAP_EVERY X_GEN_TAC [`a:real`; `k:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `{x | (f:real^M->real^N) x$k < a} = UNIONS {{x | !j. 1 <= j /\ j <= dimindex(:N) ==> f x$j < ((lambda i. if i = k then a else &n):real^N)$j} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN SIMP_TAC[LAMBDA_BETA] THEN SPEC_TAC(`(f:real^M->real^N) x`,`y:real^N`) THEN GEN_TAC THEN ASM_CASES_TAC `(y:real^N)$k < a` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN ASM_REWRITE_TAC[] THEN MP_TAC(SPEC `&1 + sup {(y:real^N)$j | j IN 1..dimindex(:N)}` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[REAL_ARITH `&1 + x <= y <=> x <= y - &1`] THEN SIMP_TAC[REAL_SUP_LE_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN ASM_MESON_TAC[REAL_ARITH `x <= y - &1 ==> x < y`]; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; NUM_COUNTABLE; FORALL_IN_IMAGE]]]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_GT = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !a. lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k > (a:real^N)$k}`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEASURABLE_ON_NEG_EQ] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_ORTHANT_LT] THEN GEN_REWRITE_TAC LAND_CONV [MESON[VECTOR_NEG_NEG] `(!x:real^N. P x) <=> (!x. P(--x))`] THEN REWRITE_TAC[REAL_ARITH `--x < --y <=> x > y`; VECTOR_NEG_COMPONENT]);; let MEASURABLE_ON_SIMPLE_FUNCTION_LIMIT_INCREASING = prove (`!f:real^N->real^1. f measurable_on (:real^N) /\ (!x. &0 <= drop(f x)) <=> ?g. (!n x. &0 <= drop(g n x) /\ drop(g n x) <= drop(f x)) /\ (!n x. drop(g n x) <= drop(g(SUC n) x)) /\ (!n. (g n) measurable_on (:real^N)) /\ (!n. FINITE(IMAGE (g n) (:real^N))) /\ (!x. ((\n. g n x) --> f x) sequentially)`, let lemma = prove (`!f:real^M->real^1 n m. integer m /\ m / &2 pow n <= drop(f x) /\ drop(f x) < (m + &1) / &2 pow n /\ abs(m) <= &2 pow (2 * n) ==> vsum {k | integer k /\ abs(k) <= &2 pow (2 * n)} (\k. k / &2 pow n % indicator {y:real^M | k / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1) / &2 pow n} x) = lift(m / &2 pow n)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `vsum {m} (\k. k / &2 pow n % indicator {y:real^M | k / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1) / &2 pow n} x)` THEN CONJ_TAC THENL [MATCH_MP_TAC VSUM_SUPERSET THEN ASM_REWRITE_TAC[SING_SUBSET; IN_ELIM_THM; IN_SING] THEN X_GEN_TAC `k:real` THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISJ2_TAC THEN ASM_REWRITE_TAC[indicator; IN_ELIM_THM] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `F ==> p`) THEN UNDISCH_TAC `~(k:real = m)` THEN ASM_SIMP_TAC[REAL_EQ_INTEGERS] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN REAL_ARITH_TAC; ASM_REWRITE_TAC[VSUM_SING; indicator; IN_ELIM_THM; LIFT_EQ_CMUL]]) in REPEAT STRIP_TAC THEN EQ_TAC THENL [STRIP_TAC; DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THENL [GEN_REWRITE_TAC RAND_CONV [MEASURABLE_ON_SIMPLE_FUNCTION_LIMIT] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]; MESON_TAC[REAL_LE_TRANS]]] THEN SUBGOAL_THEN `!a b. lebesgue_measurable {x:real^N | a <= drop(f x) /\ drop(f x) < b}` ASSUME_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | Q x} DIFF {x | ~P x}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFF THEN REWRITE_TAC[REAL_NOT_LE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT]) THEN SIMP_TAC[drop; FORALL_1; DIMINDEX_1]; FIRST_X_ASSUM(K ALL_TAC o GEN_REWRITE_RULE I [measurable_on])] THEN REWRITE_TAC[FORALL_AND_THM; GSYM CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[] `(!x. P x /\ R x ==> Q x) /\ (?x. P x /\ R x) ==> (?x. P x /\ Q x /\ R x)`) THEN CONJ_TAC THENL [X_GEN_TAC `g:num->real^N->real^1` THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`n:num`; `x:real^N`] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY] o SPEC `x:real^N`) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o SPEC `drop((g:num->real^N->real^1) n x - f x)`) THEN ASM_REWRITE_TAC[DROP_SUB; REAL_SUB_LT; NOT_EXISTS_THM] THEN X_GEN_TAC `N:num` THEN DISCH_THEN(MP_TAC o SPEC `N + n:num`) THEN REWRITE_TAC[LE_ADD; DIST_REAL; GSYM drop] THEN MATCH_MP_TAC(REAL_ARITH `f < g /\ g <= g' ==> ~(abs(g' - f) < g - f)`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ARITH_RULE `n:num <= N + n`) THEN SPEC_TAC(`N + n:num`,`m:num`) THEN SPEC_TAC(`n:num`,`n:num`) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ALL_TAC] THEN EXISTS_TAC `\n x. vsum {k | integer k /\ abs(k) <= &2 pow (2 * n)} (\k. k / &2 pow n % indicator {y:real^N | k / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1) / &2 pow n} x)` THEN REWRITE_TAC[] THEN SUBGOAL_THEN `!n. FINITE {k | integer k /\ abs k <= &2 pow (2 * n)}` ASSUME_TAC THENL [SIMP_TAC[REAL_ABS_BOUNDS; FINITE_INTSEG; FINITE_IMAGE]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[VSUM_REAL; LIFT_DROP; o_DEF] THEN MATCH_MP_TAC SUM_POS_LE THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `k:real` THEN STRIP_TAC THEN REWRITE_TAC[DROP_CMUL] THEN ASM_CASES_TAC `&0 <= k` THENL [MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_POW_LE; REAL_POS] THEN REWRITE_TAC[DROP_INDICATOR_POS_LE]; MATCH_MP_TAC(REAL_ARITH `x = &0 ==> &0 <= x`) THEN REWRITE_TAC[REAL_ENTIRE] THEN DISJ2_TAC THEN REWRITE_TAC[indicator] THEN COND_CASES_TAC THEN REWRITE_TAC[DROP_VEC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_ELIM_THM]) THEN MATCH_MP_TAC(TAUT `~b ==> a /\ b ==> c`) THEN REWRITE_TAC[REAL_NOT_LT] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&0` THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_POW2] THEN ASM_SIMP_TAC[GSYM REAL_LT_INTEGERS; REAL_MUL_LZERO; INTEGER_CLOSED] THEN ASM_REAL_ARITH_TAC]; REPEAT GEN_TAC THEN SIMP_TAC[VSUM_REAL; LIFT_DROP; o_DEF; DROP_CMUL] THEN TRANS_TAC REAL_LE_TRANS `sum {k | integer k /\ abs(k) <= &2 pow (2 * n)} (\k. k / &2 pow n * (drop(indicator {y:real^N | k / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1 / &2) / &2 pow n} x) + drop(indicator {y:real^N | (k + &1 / &2) / &2 pow n <= drop(f y) /\ drop(f y) < (k + &1) / &2 pow n} x)))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN X_GEN_TAC `k:real` THEN STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[indicator; IN_ELIM_THM] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[DROP_VEC]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_ARITH `x / y = (&2 * x) * inv(&2) * inv(y)`] THEN REWRITE_TAC[GSYM REAL_INV_MUL] THEN REWRITE_TAC[GSYM real_div] THEN REWRITE_TAC[GSYM(CONJUNCT2 real_pow); REAL_ARITH `&2 * (k + &1 / &2) = &2 * k + &1`; REAL_ARITH `&2 * (k + &1) = (&2 * k + &1) + &1`] THEN ASM_SIMP_TAC[REAL_ADD_LDISTRIB; SUM_ADD] THEN MATCH_MP_TAC(REAL_ARITH `!g. sum s f <= sum s g /\ a + sum s g <= b ==> a + sum s f <= b`) THEN EXISTS_TAC `\k. (&2 * k + &1) / &2 pow SUC n * drop (indicator {y | (&2 * k + &1) / &2 pow SUC n <= drop ((f:real^N->real^1) y) /\ drop (f y) < ((&2 * k + &1) + &1) / &2 pow SUC n} x)` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_RMUL THEN SIMP_TAC[DROP_INDICATOR_POS_LE; REAL_LE_DIV2_EQ; REAL_LT_POW2] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPEC `\x. &2 * x` SUM_IMAGE) THEN MP_TAC(ISPEC `\x. &2 * x + &1` SUM_IMAGE) THEN REWRITE_TAC[REAL_EQ_ADD_RCANCEL; REAL_EQ_MUL_LCANCEL] THEN REWRITE_TAC[REAL_OF_NUM_EQ; ARITH; IMP_CONJ; o_DEF] THEN REPEAT(DISCH_THEN(fun th -> ONCE_REWRITE_TAC[GSYM th])) THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_UNION o lhand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_IMAGE; SET_RULE `DISJOINT (IMAGE f s) (IMAGE g s) <=> !x. x IN s ==> !y. y IN s ==> ~(f x = g y)`] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `i:real` THEN STRIP_TAC THEN X_GEN_TAC `j:real` THEN STRIP_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `&2 * x = &2 * y + &1 ==> &2 * abs(x - y) = &1`)) THEN SUBGOAL_THEN `integer(i - j)` MP_TAC THENL [ASM_SIMP_TAC[INTEGER_CLOSED]; REWRITE_TAC[integer]] THEN STRIP_TAC THEN ASM_REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_EQ] THEN DISCH_THEN(MP_TAC o AP_TERM `EVEN`) THEN REWRITE_TAC[EVEN_MULT; ARITH]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC SUM_SUBSET THEN ASM_SIMP_TAC[FINITE_UNION; FINITE_IMAGE] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> x IN u) /\ (!x. x IN t ==> x IN u) ==> !x. x IN (s UNION t) DIFF u ==> P x`) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_ELIM_THM] THEN SIMP_TAC[INTEGER_CLOSED; ARITH_RULE `2 * SUC n = 2 + 2 * n`] THEN REWRITE_TAC[REAL_POW_ADD] THEN CONJ_TAC THENL [REAL_ARITH_TAC; REPEAT STRIP_TAC] THEN MATCH_MP_TAC(REAL_ARITH `abs(x) <= n /\ &1 <= n ==> abs(&2 * x + &1) <= &2 pow 2 * n`) THEN ASM_REWRITE_TAC[REAL_LE_POW2]; X_GEN_TAC `k:real` THEN REWRITE_TAC[IN_ELIM_THM; IN_DIFF] THEN STRIP_TAC THEN REWRITE_TAC[DROP_CMUL] THEN ASM_CASES_TAC `&0 <= k` THENL [MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_POW_LE; REAL_POS] THEN REWRITE_TAC[DROP_INDICATOR_POS_LE]; MATCH_MP_TAC(REAL_ARITH `x = &0 ==> &0 <= x`) THEN REWRITE_TAC[REAL_ENTIRE] THEN DISJ2_TAC THEN REWRITE_TAC[indicator] THEN COND_CASES_TAC THEN REWRITE_TAC[DROP_VEC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_ELIM_THM]) THEN MATCH_MP_TAC(TAUT `~b ==> a /\ b ==> c`) THEN REWRITE_TAC[REAL_NOT_LT] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&0` THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_POW2] THEN ASM_SIMP_TAC[GSYM REAL_LT_INTEGERS; REAL_MUL_LZERO; INTEGER_CLOSED] THEN ASM_REAL_ARITH_TAC]]; X_GEN_TAC `n:num` THEN MATCH_MP_TAC MEASURABLE_ON_VSUM THEN REWRITE_TAC[REAL_ABS_BOUNDS; FINITE_INTSEG; IN_ELIM_THM] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_CMUL THEN ASM_REWRITE_TAC[GSYM lebesgue_measurable; ETA_AX]; X_GEN_TAC `n:num` THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `IMAGE (\k. lift(k / &2 pow n)) {k | integer k /\ abs(k) <= &2 pow (2 * n)}` THEN CONJ_TAC THENL [SIMP_TAC[REAL_ABS_BOUNDS; FINITE_INTSEG; FINITE_IMAGE]; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_IMAGE] THEN ASM_CASES_TAC `?k. integer k /\ abs k <= &2 pow (2 * n) /\ k / &2 pow n <= drop(f(x:real^N)) /\ drop(f x) < (k + &1) / &2 pow n` THENL [FIRST_X_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS) THEN X_GEN_TAC `m:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC lemma THEN ASM_REWRITE_TAC[]; EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[IN_ELIM_THM; INTEGER_CLOSED; REAL_ABS_NUM] THEN SIMP_TAC[REAL_POW_LE; REAL_POS; real_div; REAL_MUL_LZERO] THEN REWRITE_TAC[LIFT_NUM; GSYM real_div] THEN MATCH_MP_TAC VSUM_EQ_0 THEN X_GEN_TAC `k:real` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISJ2_TAC THEN REWRITE_TAC[indicator; IN_ELIM_THM] THEN ASM_MESON_TAC[]]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN MP_TAC(ISPECL [`&2`; `abs(drop((f:real^N->real^1) x))`] REAL_ARCH_POW) THEN ANTS_TAC THENL [REAL_ARITH_TAC; DISCH_THEN(X_CHOOSE_TAC `N1:num`)] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN REWRITE_TAC[REAL_POW_INV] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `N2:num` MP_TAC) THEN SUBST1_TAC(REAL_ARITH `inv(&2 pow N2) = &1 / &2 pow N2`) THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_POW2] THEN DISCH_TAC THEN EXISTS_TAC `MAX N1 N2` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN ABBREV_TAC `m = floor(&2 pow n * drop(f(x:real^N)))` THEN SUBGOAL_THEN `dist(lift(m / &2 pow n),(f:real^N->real^1) x) < e` MP_TAC THENL [REWRITE_TAC[DIST_REAL; GSYM drop; LIFT_DROP] THEN MATCH_MP_TAC REAL_LT_LCANCEL_IMP THEN EXISTS_TAC `abs(&2 pow n)` THEN REWRITE_TAC[GSYM REAL_ABS_MUL; REAL_SUB_LDISTRIB] THEN SIMP_TAC[REAL_DIV_LMUL; REAL_POW_EQ_0; GSYM REAL_ABS_NZ; REAL_OF_NUM_EQ; ARITH] THEN MATCH_MP_TAC(REAL_ARITH `x <= y /\ y < x + &1 /\ &1 <= z ==> abs(x - y) < z`) THEN EXPAND_TAC "m" THEN REWRITE_TAC[FLOOR] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `e * &2 pow N2` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_ABS_POW; REAL_ABS_NUM] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; MATCH_MP_TAC(NORM_ARITH `x:real^1 = y ==> dist(y,z) < e ==> dist(x,z) < e`) THEN MATCH_MP_TAC lemma THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN EXPAND_TAC "m" THEN REWRITE_TAC[FLOOR] THEN SIMP_TAC[REAL_ABS_BOUNDS; REAL_LE_FLOOR; REAL_FLOOR_LE; INTEGER_CLOSED] THEN MATCH_MP_TAC(REAL_ARITH `abs(x) <= e ==> --e <= x /\ x - &1 < e`) THEN REWRITE_TAC[MULT_2; REAL_POW_ADD; REAL_ABS_MUL; REAL_ABS_POW; REAL_ABS_NUM] THEN MATCH_MP_TAC REAL_LE_LMUL THEN SIMP_TAC[REAL_POW_LE; REAL_POS] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e ==> e <= d ==> x <= d`))] THEN MATCH_MP_TAC REAL_POW_MONO THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* Some "iff" variants of integrability on subsets. *) (* ------------------------------------------------------------------------- *) let ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET_EQ, ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT_EQ = (CONJ_PAIR o prove) (`(!f:real^M->real^N s. f absolutely_integrable_on s <=> f measurable_on s /\ !t. t SUBSET s /\ lebesgue_measurable t ==> f integrable_on t) /\ (!f:real^M->real^N s. f absolutely_integrable_on s <=> f measurable_on s /\ !g. g measurable_on s /\ bounded(IMAGE g s) ==> (\x. drop(g x) % f x) integrable_on s)`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC (BINOP_CONV o RAND_CONV o LAND_CONV) [MEASURABLE_ON_COMPONENTWISE] THEN ONCE_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_COMPONENTWISE; INTEGRABLE_COMPONENTWISE] THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC(MESON[] `(!i. P i ==> R i) /\ (!i. R i ==> Q i) /\ (!i. Q i ==> P i) ==> ((!i. P i) <=> (!i. Q i)) /\ ((!i. P i) <=> (!i. R i))`) THEN REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i /\ i <= dimindex(:N)` THEN ASM_REWRITE_TAC[VECTOR_MUL_COMPONENT; LIFT_CMUL] THEN ABBREV_TAC `h x = lift((f:real^M->real^N) x$i)` THEN POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[ETA_AX] THEN SPEC_TAC(`h:real^M->real^1`,`f:real^M->real^1`) THEN GEN_TAC THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_MEASURABLE] THEN ASM_CASES_TAC `(f:real^M->real^1) measurable_on s` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[FORALL_IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `\x. B % lift(norm((f:real^M->real^1) x))` THEN REWRITE_TAC[NORM_MUL; GSYM NORM_1; DROP_CMUL; LIFT_DROP] THEN ASM_SIMP_TAC[REAL_LE_RMUL; NORM_POS_LE] THEN ASM_SIMP_TAC[INTEGRABLE_CMUL; MEASURABLE_ON_MUL; LIFT_DROP; ETA_AX]; FIRST_X_ASSUM(MP_TAC o SPEC `indicator(t:real^M->bool)`) THEN ASM_SIMP_TAC[MEASURABLE_ON_INDICATOR_SUBSET] THEN ANTS_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `{vec 0:real^1,vec 1}` THEN REWRITE_TAC[BOUNDED_INSERT; BOUNDED_EMPTY] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INSERT; indicator] THEN MESON_TAC[]; ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN REWRITE_TAC[indicator] THEN ASM_MESON_TAC[DROP_VEC; VECTOR_MUL_LZERO; VECTOR_MUL_LID; SUBSET]]; FIRST_ASSUM(fun th -> MP_TAC(SPEC `{x:real^M | (if x IN s then f x else vec 0:real^1)$1 > &0}` th) THEN MP_TAC(SPEC `{x:real^M | (if x IN s then f x else vec 0:real^1)$1 < &0}` th)) THEN FIRST_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE I [GSYM MEASURABLE_ON_UNIV]) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT]) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]) THEN SIMP_TAC[DIMINDEX_1; ARITH] THEN REPLICATE_TAC 2 (DISCH_THEN(K ALL_TAC)) THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[real_gt; VEC_COMPONENT; REAL_LT_REFL] THEN REWRITE_TAC[SUBSET_RESTRICT; SET_RULE `{x | if x IN s then Q x else F} = {x | x IN s /\ Q x}`] THEN DISCH_THEN(MP_TAC o MATCH_MP INTEGRABLE_NEG) THEN ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN REWRITE_TAC[IN_ELIM_THM; GSYM drop; IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP INTEGRABLE_ADD) THEN MATCH_MP_TAC EQ_IMP THEN REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[VECTOR_ADD_LID] THEN REWRITE_TAC[NORM_1; GSYM DROP_EQ; LIFT_DROP; DROP_ADD] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[DROP_VEC; DROP_NEG]) THEN ASM_REAL_ARITH_TAC]);; let ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT_EQ_ALT = prove (`!f:real^M->real^N s. lebesgue_measurable s ==> (f absolutely_integrable_on s <=> !g. g measurable_on s /\ bounded(IMAGE g s) ==> (\x. drop(g x) % f x) integrable_on s)`, REPEAT STRIP_TAC THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT_EQ] THEN REWRITE_TAC[TAUT `(p /\ q <=> q) <=> q ==> p`] THEN DISCH_THEN(MP_TAC o SPEC `(\x. vec 1):real^M->real^1`) THEN ASM_REWRITE_TAC[DROP_CMUL; DROP_VEC; MEASURABLE_ON_CONST_EQ] THEN REWRITE_TAC[VECTOR_MUL_LID; ETA_AX] THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[INTEGRABLE_IMP_MEASURABLE]] THEN MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `{vec 1:real^1}` THEN REWRITE_TAC[BOUNDED_SING] THEN SET_TAC[]);; let ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET_EQ_ALT = prove (`!f:real^M->real^N s. lebesgue_measurable s ==> (f absolutely_integrable_on s <=> !t. t SUBSET s /\ lebesgue_measurable t ==> f integrable_on t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET_EQ] THEN ASM_MESON_TAC[SUBSET_REFL; INTEGRABLE_IMP_MEASURABLE]);; (* ------------------------------------------------------------------------- *) (* More connections with measure where Lebesgue measurability is useful. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_LEBESGUE_MEASURABLE_SUBSET = prove (`!s t:real^N->bool. lebesgue_measurable s /\ measurable t /\ s SUBSET t ==> measurable s`, REWRITE_TAC[lebesgue_measurable; MEASURABLE_INTEGRABLE] THEN REWRITE_TAC[indicator] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `(\x. if x IN t then vec 1 else vec 0):real^N->real^1` THEN ASM_REWRITE_TAC[IN_UNIV] THEN GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[DROP_VEC; NORM_REAL; GSYM drop]) THEN REWRITE_TAC[REAL_ABS_NUM; REAL_LE_REFL; REAL_POS] THEN ASM SET_TAC[]);; let MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE = prove (`!s t:real^N->bool. lebesgue_measurable s /\ measurable t ==> measurable(s INTER t)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `t:real^N->bool` THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_INTER; MEASURABLE_IMP_LEBESGUE_MEASURABLE; INTER_SUBSET]);; let MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE = prove (`!s t:real^N->bool. measurable s /\ lebesgue_measurable t ==> measurable(s INTER t)`, MESON_TAC[INTER_COMM; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE]);; let LEBESGUE_MEASURABLE_MEASURABLE_INTER_EQ = prove (`!s:real^N->bool. lebesgue_measurable s <=> !t. measurable t ==> measurable(s INTER t)`, MESON_TAC[LEBESGUE_MEASURABLE_MEASURABLE_ON_SUBINTERVALS; MEASURABLE_INTERVAL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE]);; let MEASURABLE_INTER_HALFSPACE_LE = prove (`!s a i. measurable s ==> measurable(s INTER {x:real^N | x$i <= a})`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !z:real^N. z$i = z$k` CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE THEN ASM_SIMP_TAC[CLOSED_HALFSPACE_COMPONENT_LE; LEBESGUE_MEASURABLE_CLOSED]);; let MEASURABLE_INTER_HALFSPACE_GE = prove (`!s a i. measurable s ==> measurable(s INTER {x:real^N | x$i >= a})`, REPEAT GEN_TAC THEN SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !z:real^N. z$i = z$k` CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE THEN ASM_SIMP_TAC[CLOSED_HALFSPACE_COMPONENT_GE; LEBESGUE_MEASURABLE_CLOSED]);; let MEASURABLE_MEASURABLE_DIFF_LEBESGUE_MEASURABLE = prove (`!s t. measurable s /\ lebesgue_measurable t ==> measurable(s DIFF t)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE; LEBESGUE_MEASURABLE_COMPL]);; let MEASURABLE_OPEN_IN = prove (`!s t:real^N->bool. open_in (subtopology euclidean s) t /\ measurable s ==> measurable t`, MESON_TAC[MEASURABLE_LEBESGUE_MEASURABLE_SUBSET; OPEN_IN_IMP_SUBSET; LEBESGUE_MEASURABLE_OPEN_IN; MEASURABLE_IMP_LEBESGUE_MEASURABLE]);; let MEASURABLE_CLOSED_IN = prove (`!s t:real^N->bool. closed_in (subtopology euclidean s) t /\ measurable s ==> measurable t`, MESON_TAC[MEASURABLE_LEBESGUE_MEASURABLE_SUBSET; CLOSED_IN_IMP_SUBSET; LEBESGUE_MEASURABLE_CLOSED_IN; MEASURABLE_IMP_LEBESGUE_MEASURABLE]);; let MEASURABLE_ON_REAL_SGN = prove (`!f:real^N->real s. (\x. lift(f x)) measurable_on s ==> (\x. lift(real_sgn(f x))) measurable_on s`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[real_sgn; MESON[DROP_VEC; REAL_LT_REFL; LIFT_DROP] `(if P x then lift(if &0 < f x then &1 else if f x < &0 then -- &1 else &0) else vec 0) = if (&0 < (if P x then f x else &0)) then lift(&1) else if ((if P x then f x else &0) < &0) then lift(-- &1) else lift(&0)`] THEN REWRITE_TAC[MESON[LIFT_NUM] `(if P then lift(f x) else vec 0) = lift(if P then f x else &0)`] THEN ABBREV_TAC `h x = if x IN s then (f:real^N->real) x else &0` THEN DISCH_TAC THEN MATCH_MP_TAC MEASURABLE_ON_CASES THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]); REWRITE_TAC[MEASURABLE_ON_CONST] THEN MATCH_MP_TAC MEASURABLE_ON_CASES THEN REWRITE_TAC[MEASURABLE_ON_CONST] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT])] THEN REWRITE_TAC[real_gt; DIMINDEX_1; FORALL_1] THEN SIMP_TAC[GSYM drop; LIFT_DROP]);; let LEBESGUE_MEASURABLE_INNER_COMPACT = prove (`!s:real^N->bool. lebesgue_measurable s /\ ~negligible s ==> ?k. k SUBSET s /\ compact k /\ &0 < measure k`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?a b:real^N. ~(negligible(s INTER interval[a,b]))` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_ON_INTERVALS]; ALL_TAC] THEN MP_TAC(ISPECL [`s INTER interval[a:real^N,b]`; `measure(s INTER interval[a:real^N,b])`] MEASURABLE_INNER_COMPACT) THEN ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_INTERVAL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^N->bool` THEN SIMP_TAC[SUBSET_INTER; REAL_LT_ADDL]);; let CHOOSE_LARGE_MEASURABLE_SUBSET = prove (`!s:real^N->bool B. lebesgue_measurable s /\ ~measurable s ==> ?t. t SUBSET s /\ measurable t /\ B <= measure t`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\n. s INTER interval[--vec n:real^N,vec n]`; `B:real`] MEASURABLE_NESTED_UNIONS) THEN ASM_REWRITE_TAC[GSYM LEBESGUE_MEASURABLE_MEASURABLE_ON_COUNTABLE_SUBINTERVALS] THEN MATCH_MP_TAC(TAUT `(~p ==> s) /\ q /\ ~r ==> (p /\ q ==> r) ==> s`) THEN REWRITE_TAC[NOT_FORALL_THM; REAL_NOT_LE] THEN REPEAT CONJ_TAC THENL [DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN EXISTS_TAC `s INTER interval[--vec n:real^N,vec n]` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; INTER_SUBSET] THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_INTERVAL]; GEN_TAC THEN MATCH_MP_TAC(SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`) THEN REWRITE_TAC[SUBSET_INTERVAL; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `~measurable s ==> t = s ==> ~measurable t`)) THEN ONCE_REWRITE_TAC[SET_RULE `{s INTER f n | n IN (:num)} = {s INTER t | t IN IMAGE f (:num)}`] THEN REWRITE_TAC[GSYM INTER_UNIONS] THEN MATCH_MP_TAC(SET_RULE `u = UNIV ==> s INTER u = s`) THEN REWRITE_TAC[UNIONS_IMAGE; EXTENSION; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN MP_TAC(ISPEC `norm(x:real^N)` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[IN_INTERVAL; VECTOR_NEG_COMPONENT] THEN REWRITE_TAC[GSYM REAL_ABS_BOUNDS; VEC_COMPONENT] THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS]]);; let CHOOSE_LARGE_COMPACT_SUBSET = prove (`!s:real^N->bool B. lebesgue_measurable s /\ ~measurable s ==> ?t. t SUBSET s /\ compact t /\ B <= measure t`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `B + &1`] CHOOSE_LARGE_MEASURABLE_SUBSET) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`u:real^N->bool`; `&1`] MEASURABLE_INNER_COMPACT) THEN ASM_REWRITE_TAC[REAL_LT_01] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; ASM_REAL_ARITH_TAC]);; let CHOOSE_LARGE_COMPACT_SUBSET_ALT = prove (`!(s:real^N->bool) B. lebesgue_measurable s /\ B < measure s ==> ?t. compact t /\ t SUBSET s /\ B < measure t`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `measurable(s:real^N->bool)` THENL [MP_TAC(ISPECL [`s:real^N->bool`; `measure(s:real^N->bool) - B`] MEASURABLE_INNER_COMPACT); MP_TAC(ISPECL [`s:real^N->bool`; `B + &1:real`] CHOOSE_LARGE_COMPACT_SUBSET)] THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN REAL_ARITH_TAC);; let OUTER_LEBESGUE_MEASURE = prove (`!s:real^N->bool. ?t. s SUBSET t /\ lebesgue_measurable t /\ !t'. s SUBSET t' /\ lebesgue_measurable t' ==> negligible(t DIFF t')`, GEN_TAC THEN MP_TAC (GEN `n:num` (SPEC `s INTER cball(vec 0:real^N,&n)` OUTER_MEASURE)) THEN SIMP_TAC[BOUNDED_INTER; BOUNDED_CBALL; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:num->real^N->bool` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN EXISTS_TAC `UNIONS {u n | n IN (:num)}:real^N->bool` THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COUNTABLE_UNIONS_EXPLICIT; MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(SPEC `norm(x:real^N)` REAL_ARCH_SIMPLE) THEN REWRITE_TAC[GSYM IN_CBALL_0] THEN ASM SET_TAC[]; X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN REWRITE_TAC[UNIONS_DIFF; SET_RULE `{f x | x IN {g y | y IN s}} = {f(g x) | x IN s}`] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `n:num` THEN ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s DIFF (s INTER t)`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE] THEN ASM SET_TAC[]]);; let OUTER_MEASURE_GEN = prove (`!s u:real^N->bool. s SUBSET u /\ measurable u ==> ?t. s SUBSET t /\ measurable t /\ !t'. s SUBSET t' /\ lebesgue_measurable t' ==> negligible(t DIFF t')`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` OUTER_LEBESGUE_MEASURE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `u UNION (t DIFF u):real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNION THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_IMP_MEASURABLE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE]);; let OUTER_MEASURE_EQ = prove (`!s:real^N->bool. (?t. s SUBSET t /\ measurable t) <=> (?t. s SUBSET t /\ measurable t /\ !t'. s SUBSET t' /\ lebesgue_measurable t' ==> negligible(t DIFF t'))`, GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; OUTER_MEASURE_GEN]);; (* ------------------------------------------------------------------------- *) (* Continuity of measure with respect to parameters determining region. *) (* ------------------------------------------------------------------------- *) let CONTINUOUS_ON_MEASURE_IN_PORTION = prove (`!(f:real^M->real^N->real) s t. measurable s /\ (!a. a IN t ==> lebesgue_measurable {x | f a x <= &0}) /\ (!a. a IN t ==> negligible {x | f a x = &0}) /\ (!x. x IN s ==> (\a. lift(f a x)) continuous_on t) ==> (\a. lift(measure {x | x IN s /\ f a x <= &0})) continuous_on t`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON_SEQUENTIALLY; IN_UNIV; IN_DELETE] THEN MAP_EVERY X_GEN_TAC [`a:num->real^M`; `c:real^M`] THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN ASM_SIMP_TAC[MEASURE_INTEGRAL_UNIV; LIFT_DROP; o_DEF; MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE] THEN MATCH_MP_TAC(TAUT `g integrable_on s /\ (f --> integral s g) sequentially ==> (f --> integral s g) sequentially`) THEN MATCH_MP_TAC DOMINATED_CONVERGENCE_AE THEN EXISTS_TAC `indicator(s:real^N->bool)` THEN EXISTS_TAC `{x | (f:real^M->real^N->real) c x = &0}` THEN ASM_REWRITE_TAC[GSYM MEASURABLE_INTEGRABLE; indicator] THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE] THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_ELIM_THM; IN_INTER] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`n:num`; `x:real^N`] THEN DISCH_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[NORM_1; DROP_VEC; REAL_ABS_NUM; REAL_LE_REFL] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DROP_VEC; REAL_ABS_NUM; REAL_POS; REAL_LE_REFL]; X_GEN_TAC `x:real^N` THEN DISCH_TAC] THEN MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[EVENTUALLY_TRUE] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CONTINUOUS_ON_SEQUENTIALLY] THEN DISCH_THEN(MP_TAC o SPECL [`a:num->real^M`; `c:real^M`]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[tendsto] THEN DISCH_THEN(MP_TAC o SPEC `abs((f:real^M->real^N->real) c x)`) THEN ASM_REWRITE_TAC[GSYM REAL_ABS_NZ; REAL_SUB_0; o_THM; DIST_LIFT] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `n:num` THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let CONTINUOUS_ON_MEASURE_IN_HALFSPACE = prove (`!s b. measurable s ==> (\a. lift(measure {x | x IN s /\ a dot x <= b})) continuous_on ((:real^N) DELETE vec 0)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `a:real <= b <=> a - b <= &0`] THEN MATCH_MP_TAC CONTINUOUS_ON_MEASURE_IN_PORTION THEN ASM_SIMP_TAC[IN_DELETE; REAL_SUB_0; NEGLIGIBLE_HYPERPLANE] THEN REWRITE_TAC[REAL_ARITH `a - b <= &0 <=> a:real <= b`] THEN SIMP_TAC[LEBESGUE_MEASURABLE_CONVEX; CONVEX_HALFSPACE_LE; LIFT_SUB] THEN SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_LIFT_DOT2; CONTINUOUS_ON_ID]);; (* ------------------------------------------------------------------------- *) (* Negigibility of set with uncountably many disjoint translates. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_DISJOINT_TRANSLATES = prove (`!s:real^N->bool k z. lebesgue_measurable s /\ z limit_point_of k /\ pairwise (\a b. DISJOINT (IMAGE (\x. a + x) s) (IMAGE (\x. b + x) s)) k ==> negligible s`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN ABBREV_TAC `t = s INTER interval[a:real^N,b]` THEN SUBGOAL_THEN `measurable(t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_INTERVAL; INTER_COMM]; ALL_TAC] THEN SUBGOAL_THEN `bounded(t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[BOUNDED_INTER; BOUNDED_INTERVAL]; ALL_TAC] THEN ASM_SIMP_TAC[GSYM MEASURABLE_MEASURE_EQ_0] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ ~(&0 < x) ==> x = &0`) THEN ASM_SIMP_TAC[MEASURE_POS_LE] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `&1` o GEN_REWRITE_RULE I [LIMPT_INFINITE_CBALL]) THEN REWRITE_TAC[REAL_LT_01] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `measure(IMAGE (\x:real^N. z + x) (interval[a - vec 1,b + vec 1]))` o MATCH_MP REAL_ARCH) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN MP_TAC(ISPECL [`n:num`; `k INTER cball(z:real^N,&1)`] CHOOSE_SUBSET_STRONG) THEN ANTS_TAC THENL [ASM_MESON_TAC[INFINITE]; ALL_TAC] THEN REWRITE_TAC[SUBSET_INTER; LEFT_IMP_EXISTS_THM; REAL_NOT_LT] THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN TRANS_TAC REAL_LE_TRANS `measure(UNIONS(IMAGE (\a. IMAGE (\x:real^N. a + x) t) u))` THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN SUBGOAL_THEN `UNIONS(IMAGE (\a. IMAGE (\x:real^N. a + x) t) u) has_measure &n * measure(t:real^N->bool)` MP_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\a. IMAGE (\x:real^N. a + x) t`; `u:real^N->bool`] HAS_MEASURE_DISJOINT_UNIONS_IMAGE) THEN ASM_SIMP_TAC[MEASURABLE_TRANSLATION_EQ; MEASURE_TRANSLATION; SUM_CONST] THEN DISCH_THEN MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN STRIP_TAC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[REAL_LE_REFL]; MATCH_MP_TAC MEASURE_SUBSET] THEN ASM_REWRITE_TAC[MEASURABLE_TRANSLATION_EQ; MEASURABLE_INTERVAL] THEN REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN X_GEN_TAC `e:real^N` THEN DISCH_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE; UNWIND_THM1; VECTOR_ARITH `d + e:real^N = z + y <=> e + d - z = y`] THEN SUBGOAL_THEN `x IN interval[a:real^N,b]` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTERVAL]] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[VEC_COMPONENT] THEN MATCH_MP_TAC(REAL_ARITH `abs(d) <= &1 ==> a <= x /\ x <= b ==> a - &1 <= x + d /\ x + d <= b + &1`) THEN SUBGOAL_THEN `e IN cball(z:real^N,&1)` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_CBALL]] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] dist] THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN MESON_TAC[REAL_LE_TRANS; COMPONENT_LE_NORM]);; (* ------------------------------------------------------------------------- *) (* Sometimes convenient to restrict the sets in "preimage" characterization *) (* of measurable functions to choose points from a dense set. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE_DENSE = prove (`!f:real^M->real^N r. closure (IMAGE lift r) = (:real^1) ==> (f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) /\ a IN r ==> lebesgue_measurable {x | f(x)$k <= a})`, REPEAT STRIP_TAC THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE] THEN EQ_TAC THEN SIMP_TAC[] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real`; `k:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `!n. ?x. x IN r /\ a < x /\ x < a + inv(&n + &1)` MP_TAC THENL [GEN_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN REWRITE_TAC[IN_UNIV; CLOSURE_APPROACHABLE; EXISTS_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPECL [`lift(a + inv(&n + &1) / &2)`; `inv(&n + &1) / &2`]) THEN REWRITE_TAC[REAL_HALF; DIST_LIFT; REAL_LT_INV_EQ] THEN ANTS_TAC THENL [REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN SIMP_TAC[] THEN REAL_ARITH_TAC; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `t:num->real` THEN DISCH_TAC THEN SUBGOAL_THEN `{x | (f:real^M->real^N) x$k <= a} = INTERS {{x | (f x)$k <= t n} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[INTERS_GSPEC; IN_UNIV; EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN EQ_TAC THENL [ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_FORALL_THM; REAL_NOT_LE] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `i < f - a ==> !j. j <= i /\ a < t /\ t < a + j ==> &0 < f - t`)) THEN EXISTS_TAC `inv(&n + &1)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]);; let MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GE_DENSE = prove (`!f:real^M->real^N r. closure (IMAGE lift r) = (:real^1) ==> (f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) /\ a IN r ==> lebesgue_measurable {x | f(x)$k >= a})`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEASURABLE_ON_NEG_EQ] THEN MP_TAC(ISPECL [`(\x. --f x):real^M->real^N`; `IMAGE (--) r:real->bool`] MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE_DENSE) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[GSYM IMAGE_o; o_DEF; LIFT_NEG] THEN ASM_REWRITE_TAC[GSYM o_DEF; IMAGE_o; CLOSURE_NEGATIONS] THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_UNIV] THEN MESON_TAC[VECTOR_NEG_NEG]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[IMP_CONJ] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN REWRITE_TAC[VECTOR_NEG_COMPONENT; REAL_ARITH `--x <= --y <=> x >= y`]]);; let MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT_DENSE = prove (`!f:real^M->real^N r. closure (IMAGE lift r) = (:real^1) ==> (f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) /\ a IN r ==> lebesgue_measurable {x | f(x)$k < a})`, GEN_TAC THEN REWRITE_TAC[REAL_ARITH `x < a <=> ~(x >= a)`] THEN REWRITE_TAC[SET_RULE `{x | ~P x} = UNIV DIFF {x | P x}`] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL] THEN SIMP_TAC[GSYM MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GE_DENSE]);; let MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT_DENSE = prove (`!f:real^M->real^N r. closure (IMAGE lift r) = (:real^1) ==> (f measurable_on (:real^M) <=> !a k. 1 <= k /\ k <= dimindex(:N) /\ a IN r ==> lebesgue_measurable {x | f(x)$k > a})`, GEN_TAC THEN REWRITE_TAC[REAL_ARITH `x > a <=> ~(x <= a)`] THEN REWRITE_TAC[SET_RULE `{x | ~P x} = UNIV DIFF {x | P x}`] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL] THEN SIMP_TAC[GSYM MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE_DENSE]);; let MEASURABLE_ON_PREIMAGE_CLOSED_INTERVAL_DENSE = prove (`!f:real^M->real^N t. closure t = (:real^N) ==> (f measurable_on (:real^M) <=> !a b. a IN t /\ b IN t ==> lebesgue_measurable {x | f(x) IN interval[a,b]})`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [MEASURABLE_ON_PREIMAGE_CLOSED_INTERVAL] THEN EQ_TAC THEN SIMP_TAC[] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN SUBGOAL_THEN `!n. ?u v:real^N. (u IN t /\ u IN interval[(a - lambda i. inv(&n + &1)),a]) /\ (v IN t /\ v IN interval[b,(b + lambda i. inv(&n + &1))])` MP_TAC THENL [GEN_TAC THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; LEFT_EXISTS_AND_THM] THEN CONJ_TAC THEN MATCH_MP_TAC(SET_RULE `~(interior s INTER t = {}) /\ interior s SUBSET s ==> ?x. x IN t /\ x IN s`) THEN REWRITE_TAC[INTERIOR_INTERVAL; INTERVAL_OPEN_SUBSET_CLOSED] THEN W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_EQ_EMPTY o rand o snd) THEN REWRITE_TAC[OPEN_INTERVAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[INTER_UNIV; INTERVAL_NE_EMPTY] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN REWRITE_TAC[REAL_ARITH `a - i < a <=> &0 < i`; REAL_ARITH `b < b + i <=> &0 < i`] THEN REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:num->real^N`; `v:num->real^N`] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN_INTERVAL] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA]] THEN STRIP_TAC THEN SUBGOAL_THEN `{x | (f:real^M->real^N) x IN interval[a,b]} = INTERS {{x | f x IN interval[u n,v n]} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[INTERS_GSPEC; IN_UNIV; EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTERVAL] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN ASM_CASES_TAC `1 <= k /\ k <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN EQ_TAC THENL [ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_FORALL_THM; REAL_NOT_LE] THEN REWRITE_TAC[DE_MORGAN_THM; EXISTS_OR_THM; REAL_NOT_LE] THEN MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN STRIP_TAC THENL [MATCH_MP_TAC(REAL_ARITH `!a i j. i < a - f /\ j <= i /\ a - j <= t /\ t <= a ==> &0 < t - f`) THEN EXISTS_TAC `(a:real^N)$k`; MATCH_MP_TAC(REAL_ARITH `!a i j. i < f - a /\ j <= i /\ a <= t /\ t <= a + j ==> &0 < f - t`) THEN EXISTS_TAC `(b:real^N)$k`] THEN MAP_EVERY EXISTS_TAC [`inv(&n)`; `inv(&n + &1)`] THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]);; let MEASURABLE_ON_PREIMAGE_OPEN_INTERVAL_DENSE = prove (`!f:real^M->real^N t. closure t = (:real^N) ==> (f measurable_on (:real^M) <=> !a b. a IN t /\ b IN t ==> lebesgue_measurable {x | f(x) IN interval(a,b)})`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [MEASURABLE_ON_PREIMAGE_OPEN_INTERVAL] THEN EQ_TAC THEN SIMP_TAC[] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN SUBGOAL_THEN `!n. ?u v:real^N. (u IN t /\ u IN interval[a,(a + lambda i. inv(&n + &1))]) /\ (v IN t /\ v IN interval[(b - lambda i. inv(&n + &1)),b])` MP_TAC THENL [GEN_TAC THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; LEFT_EXISTS_AND_THM] THEN CONJ_TAC THEN MATCH_MP_TAC(SET_RULE `~(interior s INTER t = {}) /\ interior s SUBSET s ==> ?x. x IN t /\ x IN s`) THEN REWRITE_TAC[INTERIOR_INTERVAL; INTERVAL_OPEN_SUBSET_CLOSED] THEN W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_EQ_EMPTY o rand o snd) THEN REWRITE_TAC[OPEN_INTERVAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[INTER_UNIV; INTERVAL_NE_EMPTY] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN REWRITE_TAC[REAL_ARITH `a - i < a <=> &0 < i`; REAL_ARITH `b < b + i <=> &0 < i`] THEN REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:num->real^N`; `v:num->real^N`] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN_INTERVAL] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA]] THEN STRIP_TAC THEN SUBGOAL_THEN `{x | (f:real^M->real^N) x IN interval(a,b)} = UNIONS {{x | f x IN interval(u n,v n)} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[UNIONS_GSPEC; IN_UNIV; EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTERVAL] THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LET_TRANS; REAL_LTE_TRANS]] THEN SPEC_TAC(`(f:real^M->real^N) x`,`y:real^N`) THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `&0 < inf { min ((y - a:real^N)$i) ((b - y:real^N)$i) | i IN 1..dimindex(:N)}` MP_TAC THENL [SIMP_TAC[REAL_LT_INF_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; REAL_LT_MIN; REAL_SUB_LT; VECTOR_SUB_COMPONENT; IN_NUMSEG]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN SIMP_TAC[REAL_LT_INF_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[FORALL_IN_IMAGE; VECTOR_SUB_COMPONENT; IN_NUMSEG] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `k:num`])) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `inv(&n + &1) <= inv(&n)` MP_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_LE_DENSE = prove (`!f:real^M->real^N t. closure t = (:real^N) ==> (f measurable_on (:real^M) <=> !a. a IN t ==> lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k <= (a:real^N)$k})`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [MEASURABLE_ON_PREIMAGE_ORTHANT_LE] THEN EQ_TAC THEN SIMP_TAC[] THEN DISCH_TAC THEN X_GEN_TAC `a:real^N` THEN SUBGOAL_THEN `!n. ?u:real^N. u IN t /\ u IN interval[a,(a + lambda i. inv(&n + &1))]` MP_TAC THENL [GEN_TAC THEN MATCH_MP_TAC(SET_RULE `~(interior s INTER t = {}) /\ interior s SUBSET s ==> ?x. x IN t /\ x IN s`) THEN REWRITE_TAC[INTERIOR_INTERVAL; INTERVAL_OPEN_SUBSET_CLOSED] THEN W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_EQ_EMPTY o rand o snd) THEN REWRITE_TAC[OPEN_INTERVAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[INTER_UNIV; INTERVAL_NE_EMPTY] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN REWRITE_TAC[REAL_ARITH `b < b + i <=> &0 < i`] THEN REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:num->real^N` THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN_INTERVAL] THEN SIMP_TAC[VECTOR_ADD_COMPONENT; LAMBDA_BETA]] THEN STRIP_TAC THEN SUBGOAL_THEN `{x | !i. 1 <= i /\ i <= dimindex(:N) ==> (f:real^M->real^N) x$i <= (a:real^N)$i} = INTERS {{x | !i. 1 <= i /\ i <= dimindex(:N) ==> (f:real^M->real^N) x$i <= (u n:real^N)$i} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[INTERS_GSPEC; IN_UNIV; EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN ASM_CASES_TAC `1 <= k /\ k <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN EQ_TAC THENL [ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_FORALL_THM; REAL_NOT_LE] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `!a i j. i < f - a /\ j <= i /\ a <= t /\ t <= a + j ==> &0 < f - t`) THEN EXISTS_TAC `(a:real^N)$k` THEN MAP_EVERY EXISTS_TAC [`inv(&n)`; `inv(&n + &1)`] THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_GE_DENSE = prove (`!f:real^M->real^N t. closure t = (:real^N) ==> (f measurable_on (:real^M) <=> !a. a IN t ==> lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k >= (a:real^N)$k})`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEASURABLE_ON_NEG_EQ] THEN MP_TAC(ISPECL [`(\x. --f x):real^M->real^N`; `IMAGE (--) t:real^N->bool`] MEASURABLE_ON_PREIMAGE_ORTHANT_LE_DENSE) THEN ASM_REWRITE_TAC[CLOSURE_NEGATIONS; FORALL_IN_IMAGE] THEN REWRITE_TAC[VECTOR_NEG_COMPONENT; REAL_ARITH `--x <= --y <=> x >= y`] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_UNIV] THEN MESON_TAC[VECTOR_NEG_NEG]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_LT_DENSE = prove (`!f:real^M->real^N t. closure t = (:real^N) ==> (f measurable_on (:real^M) <=> !a. a IN t ==> lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k < (a:real^N)$k})`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [MEASURABLE_ON_PREIMAGE_ORTHANT_LT] THEN EQ_TAC THEN SIMP_TAC[] THEN DISCH_TAC THEN X_GEN_TAC `a:real^N` THEN SUBGOAL_THEN `!n. ?u:real^N. u IN t /\ u IN interval[(a - lambda i. inv(&n + &1)):real^N,a]` MP_TAC THENL [GEN_TAC THEN MATCH_MP_TAC(SET_RULE `~(interior s INTER t = {}) /\ interior s SUBSET s ==> ?x. x IN t /\ x IN s`) THEN REWRITE_TAC[INTERIOR_INTERVAL; INTERVAL_OPEN_SUBSET_CLOSED] THEN W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_EQ_EMPTY o rand o snd) THEN REWRITE_TAC[OPEN_INTERVAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[INTER_UNIV; INTERVAL_NE_EMPTY] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN REWRITE_TAC[REAL_ARITH `b - i < b <=> &0 < i`] THEN REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:num->real^N` THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN_INTERVAL] THEN SIMP_TAC[VECTOR_ADD_COMPONENT; LAMBDA_BETA]] THEN STRIP_TAC THEN SUBGOAL_THEN `{x | !i. 1 <= i /\ i <= dimindex(:N) ==> (f:real^M->real^N) x$i < (a:real^N)$i} = UNIONS {{x | !i. 1 <= i /\ i <= dimindex(:N) ==> (f:real^M->real^N) x$i < (u n:real^N)$i} | n IN (:num)}` SUBST1_TAC THENL [REWRITE_TAC[UNIONS_GSPEC; IN_UNIV; EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTERVAL] THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LET_TRANS; REAL_LTE_TRANS]] THEN SPEC_TAC(`(f:real^M->real^N) x`,`y:real^N`) THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `&0 < inf { (a - y:real^N)$i | i IN 1..dimindex(:N)}` MP_TAC THENL [SIMP_TAC[REAL_LT_INF_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; REAL_LT_MIN; REAL_SUB_LT; VECTOR_SUB_COMPONENT; IN_NUMSEG]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN SIMP_TAC[REAL_LT_INF_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN REWRITE_TAC[FORALL_IN_IMAGE; VECTOR_SUB_COMPONENT; IN_NUMSEG] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `k:num`])) THEN ASM_SIMP_TAC[VECTOR_SUB_COMPONENT; LAMBDA_BETA] THEN SUBGOAL_THEN `inv(&n + &1) <= inv(&n)` MP_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE]]);; let MEASURABLE_ON_PREIMAGE_ORTHANT_GT_DENSE = prove (`!f:real^M->real^N t. closure t = (:real^N) ==> (f measurable_on (:real^M) <=> !a. a IN t ==> lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex(:N) ==> f(x)$k > (a:real^N)$k})`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM MEASURABLE_ON_NEG_EQ] THEN MP_TAC(ISPECL [`(\x. --f x):real^M->real^N`; `IMAGE (--) t:real^N->bool`] MEASURABLE_ON_PREIMAGE_ORTHANT_LT_DENSE) THEN ASM_REWRITE_TAC[CLOSURE_NEGATIONS; FORALL_IN_IMAGE] THEN REWRITE_TAC[VECTOR_NEG_COMPONENT; REAL_ARITH `--x < --y <=> x > y`] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_UNIV] THEN MESON_TAC[VECTOR_NEG_NEG]);; (* ------------------------------------------------------------------------- *) (* Localized variants of function measurability equivalents. *) (* ------------------------------------------------------------------------- *) let [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED_INTERVAL; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GE; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LE; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LT; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN_INTERVAL; MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL] = (CONJUNCTS o prove) (`(!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !t. closed t ==> lebesgue_measurable {x | x IN s /\ f x IN t})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !a b. lebesgue_measurable {x | x IN s /\ f x IN interval[a,b]})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !t. open t ==> lebesgue_measurable {x | x IN s /\ f x IN t})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | x IN s /\ (f x)$k >= a})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | x IN s /\ (f x)$k > a})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | x IN s /\ (f x)$k <= a})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {x | x IN s /\ (f x)$k < a})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !a b. lebesgue_measurable {x | x IN s /\ f x IN interval(a,b)})) /\ (!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !t. borel t ==> lebesgue_measurable {x | x IN s /\ f x IN t}))`, let lemma = prove (`!f s P. {x | P(if x IN s then f x else vec 0)} = if P(vec 0) then s INTER {x | P(f x)} UNION ((:real^M) DIFF s) else {x | x IN s /\ P(f x)}`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN SET_TAC[]) in ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[MEASURABLE_ON_PREIMAGE_CLOSED]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_CLOSED_INTERVAL]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GE]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN_INTERVAL]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_BOREL]] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] lemma) THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN TRY(MATCH_MP_TAC(TAUT `(q <=> q') ==> (p ==> q <=> p ==> q')`)) THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN EQ_TAC THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_UNION; LEBESGUE_MEASURABLE_COMPL] THEN UNDISCH_TAC `lebesgue_measurable(s:real^M->bool)` THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_INTER) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]);; let LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_OPEN = prove (`!f:real^M->real^N s t. f measurable_on s /\ lebesgue_measurable s /\ open t ==> lebesgue_measurable {x | x IN s /\ f(x) IN t}`, MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN]);; let LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED = prove (`!f:real^M->real^N s t. f measurable_on s /\ lebesgue_measurable s /\ closed t ==> lebesgue_measurable {x | x IN s /\ f(x) IN t}`, MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED]);; let MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN_EQ = prove (`!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s <=> !t. open t ==> lebesgue_measurable {x | x IN s /\ f(x) IN t}`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_OPEN] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(SPEC `(:real^N)` th)) THEN REWRITE_TAC[OPEN_UNIV; SET_RULE `{x | x IN s /\ f x IN UNIV} = s`] THEN SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN]);; let MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED_EQ = prove (`!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s <=> !t. closed t ==> lebesgue_measurable {x | x IN s /\ f(x) IN t}`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(SPEC `(:real^N)` th)) THEN REWRITE_TAC[CLOSED_UNIV; SET_RULE `{x | x IN s /\ f x IN UNIV} = s`] THEN SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED]);; let [MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED; MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED_INTERVAL; MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN; MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GE; MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT; MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LE; MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LT; MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN_INTERVAL] = (CONJUNCTS o prove) (`(!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !t. closed t ==> measurable {x | x IN s /\ f x IN t})) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !a b. measurable {x | x IN s /\ f x IN interval[a,b]})) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !t. open t ==> measurable {x | x IN s /\ f x IN t})) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> measurable {x | x IN s /\ (f x)$k >= a})) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> measurable {x | x IN s /\ (f x)$k > a})) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> measurable {x | x IN s /\ (f x)$k <= a})) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !a k. 1 <= k /\ k <= dimindex(:N) ==> measurable {x | x IN s /\ (f x)$k < a})) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !a b. measurable {x | x IN s /\ f x IN interval(a,b)}))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP MEASURABLE_IMP_LEBESGUE_MEASURABLE) THENL [ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED]; ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED_INTERVAL]; ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN]; ASM_SIMP_TAC [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GE]; ASM_SIMP_TAC [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT]; ASM_SIMP_TAC [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LE]; ASM_SIMP_TAC [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LT]; ASM_SIMP_TAC [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN_INTERVAL]] THEN EQ_TAC THEN SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[] THEN SET_TAC[]);; let MEASURABLE_MEASURABLE_PREIMAGE_OPEN = prove (`!f:real^M->real^N s t. f measurable_on s /\ measurable s /\ open t ==> measurable {x | x IN s /\ f(x) IN t}`, MESON_TAC[MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN]);; let MEASURABLE_MEASURABLE_PREIMAGE_CLOSED = prove (`!f:real^M->real^N s t. f measurable_on s /\ measurable s /\ closed t ==> measurable {x | x IN s /\ f(x) IN t}`, MESON_TAC[MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED]);; let MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN_EQ = prove (`!f:real^M->real^N s. f measurable_on s /\ measurable s <=> !t. open t ==> measurable {x | x IN s /\ f(x) IN t}`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[MEASURABLE_MEASURABLE_PREIMAGE_OPEN] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(SPEC `(:real^N)` th)) THEN REWRITE_TAC[OPEN_UNIV; SET_RULE `{x | x IN s /\ f x IN UNIV} = s`] THEN SIMP_TAC[MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN]);; let MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED_EQ = prove (`!f:real^M->real^N s. f measurable_on s /\ measurable s <=> !t. closed t ==> measurable {x | x IN s /\ f(x) IN t}`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[MEASURABLE_MEASURABLE_PREIMAGE_CLOSED] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(SPEC `(:real^N)` th)) THEN REWRITE_TAC[CLOSED_UNIV; SET_RULE `{x | x IN s /\ f x IN UNIV} = s`] THEN SIMP_TAC[MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED]);; let LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_BOREL = prove (`!f:real^M->real^N s t. f measurable_on s /\ lebesgue_measurable s /\ borel t ==> lebesgue_measurable {x | x IN s /\ f(x) IN t}`, MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL]);; let MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL_EQ = prove (`!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s <=> !t. borel t ==> lebesgue_measurable {x | x IN s /\ f x IN t}`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_BOREL] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(SPEC `(:real^N)` th)) THEN REWRITE_TAC[BOREL_UNIV; SET_RULE `{x | x IN s /\ f x IN UNIV} = s`] THEN SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL]);; (* ------------------------------------------------------------------------- *) (* Measurability of analytic sets and related results. *) (* ------------------------------------------------------------------------- *) let SUSLIN_LEBESGUE_MEASURABLE = prove (`suslin lebesgue_measurable:(real^N->bool)->bool = lebesgue_measurable`, let lemma = prove (`!l m a b:A. APPEND l [a] = APPEND m [b] <=> l = m /\ a = b`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[] THEN DISCH_THEN(fun th -> MP_TAC(AP_TERM `BUTLAST:A list->A list` th) THEN MP_TAC(AP_TERM `LAST:A list->A` th)) THEN REWRITE_TAC[LAST_APPEND; BUTLAST_APPEND; NOT_CONS_NIL] THEN SIMP_TAC[LAST; BUTLAST; APPEND_NIL]) in REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUSLIN_SUPERSET] THEN REWRITE_TAC[SUBSET; suslin; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN] THEN SUBGOAL_THEN `!e. (!l:num list. lebesgue_measurable(e l)) /\ e [] = (:real^N) ==> lebesgue_measurable(suslin_operation e)` MP_TAC THENL [X_GEN_TAC `e:num list->real^N->bool` THEN STRIP_TAC; DISCH_TAC THEN X_GEN_TAC `e:num list->real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\l:num list. if l = [] then (:real^N) else e l`) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[LEBESGUE_MEASURABLE_UNIV]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SIMP_TAC[suslin_operation] THEN REPEAT(AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. P x ==> f x = g x) ==> {f x | P x} = {g x | P x}`) THEN REPEAT STRIP_TAC) THEN ASM_SIMP_TAC[LIST_OF_SEQ_EQ_NIL; LE_1]] THEN ONCE_REWRITE_TAC[LEBESGUE_MEASURABLE_MEASURABLE_INTER_EQ] THEN X_GEN_TAC `t:real^N->bool` THEN DISCH_TAC THEN ABBREV_TAC `f = \l. UNIONS { INTERS { (e:num list->real^N->bool)(list_of_seq s n) | 1 <= n} | list_of_seq s (LENGTH l) = l}` THEN SUBGOAL_THEN `!l. (f:num list->real^N->bool) l SUBSET e l` ASSUME_TAC THENL [X_GEN_TAC `l:num list` THEN ASM_CASES_TAC `l:num list = []` THEN ASM_REWRITE_TAC[SUBSET_UNIV] THEN EXPAND_TAC "f" THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `s:num->num` THEN DISCH_TAC THEN MATCH_MP_TAC INTERS_SUBSET_STRONG THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN EXISTS_TAC `LENGTH(l:num list)` THEN ASM_SIMP_TAC[LE_1; LENGTH_EQ_NIL; SUBSET_REFL]; ALL_TAC] THEN SUBGOAL_THEN `!l. ?u. f l INTER t SUBSET u /\ u SUBSET e l /\ u SUBSET t /\ measurable u /\ !v. (f:num list->real^N->bool) l INTER t SUBSET v /\ lebesgue_measurable v ==> negligible(u DIFF v)` MP_TAC THENL [X_GEN_TAC `l:num list` THEN MP_TAC(ISPECL [`(f:num list->real^N->bool) l INTER t`; `t:real^N->bool`] OUTER_MEASURE_GEN) THEN ASM_REWRITE_TAC[INTER_SUBSET] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `u INTER t INTER e(l:num list):real^N->bool` THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE; MEASURABLE_INTER] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN X_GEN_TAC `v:real^N->bool` THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN SET_TAC[]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM]] THEN X_GEN_TAC `g:num list->real^N->bool` THEN STRIP_TAC THEN ABBREV_TAC `h = \l. (g l:real^N->bool) DIFF UNIONS {g(APPEND l [i]) | i IN (:num)}` THEN SUBGOAL_THEN `!l. negligible((h:num list->real^N->bool) l)` ASSUME_TAC THENL [X_GEN_TAC `l:num list` THEN EXPAND_TAC "h" THEN REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COUNTABLE_UNIONS_EXPLICIT; FORALL_IN_GSPEC; MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN SUBGOAL_THEN `f l:real^N->bool = UNIONS {f (APPEND l [i]) | i IN (:num)}` SUBST1_TAC THENL [EXPAND_TAC "f" THEN REWRITE_TAC[LENGTH_APPEND; LENGTH; ADD_CLAUSES] THEN REWRITE_TAC[list_of_seq; lemma] THEN REWRITE_TAC[UNIONS_GSPEC; INTERS_GSPEC] THEN SET_TAC[]; REWRITE_TAC[INTER_UNIONS] THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN MATCH_MP_TAC UNIONS_MONO_IMAGE THEN ASM_REWRITE_TAC[o_THM; IN_UNIV]]; ALL_TAC] THEN ABBREV_TAC `H:real^N->bool = UNIONS {h l | l IN (:num list)}` THEN SUBGOAL_THEN `negligible(H:real^N->bool)` ASSUME_TAC THENL [EXPAND_TAC "H" THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[COUNTABLE_LIST; NUM_COUNTABLE; COUNTABLE_IMAGE]; ALL_TAC] THEN MATCH_MP_TAC MEASURABLE_NEGLIGIBLE_SYMDIFF THEN EXISTS_TAC `(g:num list->real^N->bool) []` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `H:real^N->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `e SUBSET g /\ g DIFF h SUBSET e ==> (g DIFF e) UNION (e DIFF g) SUBSET h`) THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `(f:num list->real^N->bool) [] INTER t` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "f" THEN REWRITE_TAC[suslin_operation; LENGTH; LIST_OF_SEQ_EQ_NIL; IN_UNIV] THEN REWRITE_TAC[SUBSET_REFL]; ALL_TAC] THEN EXPAND_TAC "H" THEN REWRITE_TAC[SUBSET; IN_DIFF; UNIONS_GSPEC; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM; NOT_EXISTS_THM] THEN STRIP_TAC THEN REWRITE_TAC[suslin_operation; IN_INTER; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[UNIONS_GSPEC; INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN SUBGOAL_THEN `?l:num->num list. (!n. LENGTH(l n) = n /\ (x:real^N) IN g(l n)) /\ (!n m. m < n ==> EL m (l(SUC n)) = EL m (l n))` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONJ_TAC THENL [ASM_MESON_TAC[LENGTH]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `l:num list`] THEN STRIP_TAC THEN MATCH_MP_TAC(MESON[] `(?t h. P(APPEND t [h])) ==> (?l. P l)`) THEN EXISTS_TAC `l:num list` THEN ASM_SIMP_TAC[LENGTH; LENGTH_APPEND; EL_APPEND; ADD_CLAUSES] THEN SUBGOAL_THEN `~(x IN (h:num list->real^N->bool) l)` MP_TAC THENL [ASM_REWRITE_TAC[]; EXPAND_TAC "h" THEN REWRITE_TAC[UNIONS_GSPEC]] THEN ASM SET_TAC[]; EXISTS_TAC `(\n. EL n (l(SUC n))):num->num` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[] THEN SUBGOAL_THEN `list_of_seq (\n. EL n (l (SUC n))) n :num list = l n` SUBST1_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_SIMP_TAC[LIST_EQ; LENGTH_LIST_OF_SEQ; EL_LIST_OF_SEQ] THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[GSYM LE_SUC_LT] THEN SIMP_TAC[LE_EXISTS; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:num` THEN DISCH_THEN(K ALL_TAC) THEN SPEC_TAC(`d:num`,`d:num`) THEN INDUCT_TAC THEN ASM_SIMP_TAC[ADD_CLAUSES; ARITH_RULE `m < SUC(m + d)`]]);; let ANALYTIC_IMP_LEBESGUE_MEASURABLE = prove (`!s:real^N->bool. analytic s ==> lebesgue_measurable s`, ONCE_REWRITE_TAC[GSYM SUSLIN_LEBESGUE_MEASURABLE] THEN REWRITE_TAC[analytic] THEN GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] SUSLIN_MONO) THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COMPACT]);; let LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC = prove (`!f:real^M->real^N s t. f measurable_on s /\ lebesgue_measurable s /\ analytic t ==> lebesgue_measurable {x | x IN s /\ f x IN t}`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM SUSLIN_LEBESGUE_MEASURABLE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [analytic]) THEN REWRITE_TAC[suslin; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `g:num list->real^N->bool` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN EXISTS_TAC `\l:num list. {x | x IN s /\ (f:real^M->real^N) x IN g l}` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `l:num list` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[IN] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_BOREL THEN ASM_SIMP_TAC[COMPACT_IMP_BOREL]; REWRITE_TAC[suslin_operation; UNIONS_GSPEC; INTERS_GSPEC] THEN MP_TAC LE_REFL THEN SET_TAC[]]);; let LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC = prove (`!f:real^M->real^N t. f measurable_on (:real^M) /\ analytic t ==> lebesgue_measurable {x | f x IN t}`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{x | P x} = {x | x IN UNIV /\ P x}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV]);; let MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC_EQ = prove (`!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s <=> !t. analytic t ==> lebesgue_measurable {x | x IN s /\ f x IN t}`, REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC] THEN REWRITE_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL_EQ] THEN SIMP_TAC[BOREL_IMP_ANALYTIC]);; let MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC = prove (`!f:real^M->real^N s. lebesgue_measurable s ==> (f measurable_on s <=> !t. analytic t ==> lebesgue_measurable {x | x IN s /\ f x IN t})`, SIMP_TAC[GSYM MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC_EQ]);; let MEASURABLE_ON_PREIMAGE_ANALYTIC = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> !t. analytic t ==> lebesgue_measurable {x | f x IN t}`, SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_ANALYTIC; LEBESGUE_MEASURABLE_UNIV; IN_UNIV]);; (* ------------------------------------------------------------------------- *) (* Some additional lemmas about measurability and absolute integrals. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_ON_LIFT_INV = prove (`!f:real^N->real s. (\x. lift(f x)) measurable_on s /\ negligible {x | x IN s /\ f x = &0} ==> (\x. lift(inv(f x))) measurable_on s`, let lemma = prove (`!f:real^N->real s. (\x. lift(f x)) measurable_on s /\ (!x. x IN s ==> ~(f x = &0)) ==> (\x. lift(inv(f x))) measurable_on s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `lebesgue_measurable(s:real^N->bool)` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEASURABLE_ON_UNIV]) THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN] THEN DISCH_THEN(MP_TAC o SPEC `(:real^1) DELETE vec 0`) THEN SIMP_TAC[OPEN_UNIV; OPEN_DELETE; IN_DELETE; IN_UNIV] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_SIMP_TAC[GSYM DROP_EQ; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[MESON[] `~(if p then F else T) <=> p`] THEN REWRITE_TAC[SET_RULE `{x | x IN s} = s`]; UNDISCH_TAC `(\x:real^N. lift(f x)) measurable_on s` THEN ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN] THEN DISCH_TAC THEN X_GEN_TAC `u:real^1->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (lift o inv o drop) (u DELETE vec 0)`) THEN ANTS_TAC THENL [MATCH_MP_TAC OPEN_IN_OPEN_TRANS THEN EXISTS_TAC `(:real^1) DELETE vec 0` THEN SIMP_TAC[OPEN_DELETE; OPEN_UNIV] THEN MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN MAP_EVERY EXISTS_TAC [`lift o inv o drop`; `(:real^1) DELETE vec 0`] THEN CONJ_TAC THENL [REWRITE_TAC[HOMEOMORPHISM; SUBSET; FORALL_IN_IMAGE; IN_DELETE; IN_UNIV; o_THM; FORALL_LIFT; LIFT_DROP] THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[REAL_INV_INV; REAL_INV_EQ_0] THEN MATCH_MP_TAC CONTINUOUS_ON_INV THEN REWRITE_TAC[IN_DELETE; o_DEF; LIFT_DROP; CONTINUOUS_ON_ID] THEN SIMP_TAC[GSYM DROP_EQ; DROP_VEC]; MATCH_MP_TAC OPEN_SUBSET THEN ASM_SIMP_TAC[OPEN_DELETE] THEN SET_TAC[]]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE; o_THM; LIFT_DROP; IN_ELIM_THM] THEN REWRITE_TAC[LIFT_EQ; EXISTS_LIFT; LIFT_DROP; IN_DELETE; GSYM LIFT_NUM] THEN ASM_MESON_TAC[REAL_INV_INV; REAL_INV_EQ_0]]]) in REPEAT STRIP_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] MEASURABLE_ON_SPIKE_SET) THEN EXISTS_TAC `{x:real^N | x IN s /\ ~(f x = &0)}` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC lemma THEN SIMP_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] MEASURABLE_ON_SPIKE_SET) THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]);; let MEASURABLE_ON_LIFT_DIV = prove (`!f:real^N->real g s. (\x. lift(f x)) measurable_on s /\ (\x. lift(g x)) measurable_on s /\ negligible {x | x IN s /\ g x = &0} ==> (\x. lift(f x / g x)) measurable_on s`, REPEAT STRIP_TAC THEN REWRITE_TAC[real_div] THEN MATCH_MP_TAC MEASURABLE_ON_LIFT_MUL THEN ASM_SIMP_TAC[MEASURABLE_ON_LIFT_INV]);; let ABSOLUTELY_INTEGRABLE_UNION = prove (`!f:real^M->real^N s t. f absolutely_integrable_on s /\ f absolutely_integrable_on t ==> f absolutely_integrable_on (s UNION t)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(f:real^M->real^N) absolutely_integrable_on ({x | x IN s /\ ~(f x = vec 0)} INTER {x | x IN t /\ ~(f x = vec 0)})` MP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN CONJ_TAC THENL [UNDISCH_TAC `(f:real^M->real^N) absolutely_integrable_on s`; UNDISCH_TAC `(f:real^M->real^N) absolutely_integrable_on t`] THEN DISCH_THEN(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN] THEN DISCH_THEN(MP_TAC o SPEC `(:real^N) DELETE (vec 0)`) THEN SIMP_TAC[OPEN_DELETE; OPEN_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]; ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV])) THEN ONCE_REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_ADD) THEN ONCE_REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_SUB) THEN REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:real^M` THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(x:real^M) IN t`] THEN ASM_REWRITE_TAC[IN_INTER; IN_UNION; IN_ELIM_THM] THEN ASM_CASES_TAC `(f:real^M->real^N) x = vec 0` THEN ASM_REWRITE_TAC[] THEN CONV_TAC VECTOR_ARITH]);; let ABSOLUTELY_INTEGRABLE_DIFF = prove (`!f:real^M->real^N s t. f absolutely_integrable_on s /\ f absolutely_integrable_on t ==> f absolutely_integrable_on (s DIFF t)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_UNION) THEN FIRST_ASSUM(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_SUB) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:real^M` THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(x:real^M) IN t`] THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNION] THEN CONV_TAC VECTOR_ARITH);; let ABSOLUTELY_INTEGRABLE_INTER = prove (`!f:real^M->real^N s t. f absolutely_integrable_on s /\ f absolutely_integrable_on t ==> f absolutely_integrable_on (s INTER t)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_UNION) THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_ADD) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_SUB) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:real^M` THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(x:real^M) IN t`] THEN ASM_REWRITE_TAC[IN_INTER; IN_UNION] THEN CONV_TAC VECTOR_ARITH);; let INTEGRAL_COUNTABLE_UNIONS_ALT = prove (`!f:real^M->real^N s. f absolutely_integrable_on (UNIONS {s m | m IN (:num)}) /\ (!m. lebesgue_measurable(s m)) ==> (!n. f absolutely_integrable_on UNIONS {s m | m IN 0..n}) /\ ((\n. integral (UNIONS {s m | m IN 0..n}) f) --> integral (UNIONS {s m | m IN (:num)}) f) sequentially`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET)) THEN CONJ_TAC THENL [SET_TAC[]; MATCH_MP_TAC LEBESGUE_MEASURABLE_UNIONS] THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; SIMPLE_IMAGE] THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG]; DISCH_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN MATCH_MP_TAC(TAUT `p /\ q ==> p ==> q`) THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV; GSYM INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC DOMINATED_CONVERGENCE THEN EXISTS_TAC `\x. if x IN UNIONS {s m | m IN (:num)} then lift(norm((f:real^M->real^N) x)) else vec 0` THEN REWRITE_TAC[INTEGRAL_RESTRICT_UNIV; INTEGRABLE_RESTRICT_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN REWRITE_TAC[IN_UNIV] THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL; LIFT_DROP; REAL_LE_REFL; NORM_0; DROP_VEC; NORM_POS_LE]) THEN ASM SET_TAC[]; X_GEN_TAC `x:real^M` THEN COND_CASES_TAC THENL [MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN MESON_TAC[]; MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN ASM SET_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* Regularity properties and Steinhaus, this time for Lebesgue measure. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_MEASURABLE_OUTER_OPEN = prove (`!s:real^N->bool e. lebesgue_measurable s /\ &0 < e ==> ?t. open t /\ s SUBSET t /\ measurable(t DIFF s) /\ measure(t DIFF s) < e`, REPEAT STRIP_TAC THEN MP_TAC(GEN `n:num` (ISPECL [`s INTER ball(vec 0:real^N,&2 pow n)`; `e / &4 / &2 pow n`] MEASURABLE_OUTER_OPEN)) THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; REAL_LT_DIV; MEASURABLE_BALL; REAL_LT_INV_EQ; REAL_LT_POW2; REAL_ARITH `&0 < e / &4 <=> &0 < e`] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `t:num->real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `UNIONS(IMAGE t (:num)):real^N->bool` THEN ASM_SIMP_TAC[OPEN_UNIONS; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; UNIONS_IMAGE; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(ISPEC `norm(x:real^N)` REAL_ARCH_POW2) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[IN_BALL_0; IN_INTER]; REWRITE_TAC[UNIONS_DIFF; SET_RULE `{f x | x IN IMAGE g s} = {f(g(x)) | x IN s}`] THEN MATCH_MP_TAC(MESON[REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`] `&0 < e /\ P /\ x <= e / &2 ==> P /\ x < e`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_DIFF_LEBESGUE_MEASURABLE] THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(0..n) (\i. e / &4 / &2 pow i)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE_NUMSEG THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(t i DIFF (s INTER ball(vec 0:real^N,&2 pow i)))` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_DIFF_LEBESGUE_MEASURABLE; MEASURABLE_INTER; MEASURABLE_BALL; LEBESGUE_MEASURABLE_INTER; MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN SET_TAC[]; ASM_SIMP_TAC[MEASURE_DIFF_SUBSET; MEASURABLE_DIFF; MEASURABLE_BALL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE] THEN ASM_SIMP_TAC[REAL_ARITH `t < s + e ==> t - s <= e`]]; REWRITE_TAC[real_div; SUM_LMUL; REAL_INV_POW; SUM_GP] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[CONJUNCT1 LT] THEN ASM_SIMP_TAC[GSYM REAL_MUL_ASSOC; REAL_LE_LMUL_EQ] THEN REWRITE_TAC[REAL_ARITH `&1 / &4 * (&1 - x) * &2 <= &1 / &2 <=> &0 <= x`] THEN MATCH_MP_TAC REAL_POW_LE THEN CONV_TAC REAL_RAT_REDUCE_CONV]]);; let LEBESGUE_MEASURABLE_INNER_CLOSED = prove (`!s:real^N->bool e. lebesgue_measurable s /\ &0 < e ==> ?t. closed t /\ t SUBSET s /\ measurable(s DIFF t) /\ measure(s DIFF t) < e`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM LEBESGUE_MEASURABLE_COMPL] THEN DISCH_THEN(X_CHOOSE_TAC `t:real^N->bool` o MATCH_MP LEBESGUE_MEASURABLE_OUTER_OPEN) THEN EXISTS_TAC `(:real^N) DIFF t` THEN POP_ASSUM MP_TAC THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THEN REWRITE_TAC[GSYM OPEN_CLOSED] THENL [SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC] THEN SET_TAC[]);; let STEINHAUS_LEBESGUE = prove (`!s:real^N->bool. lebesgue_measurable s /\ ~negligible s ==> ?d. &0 < d /\ ball(vec 0,d) SUBSET {x - y | x IN s /\ y IN s}`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN REWRITE_TAC[NOT_FORALL_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN MP_TAC(ISPEC `s INTER interval[a:real^N,b]` STEINHAUS) THEN ASM_SIMP_TAC[GSYM MEASURABLE_MEASURE_POS_LT; MEASURABLE_INTERVAL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE] THEN SET_TAC[]);; let LEBESGUE_MEASURABLE_REGULAR_OUTER = prove (`!s:real^N->bool. lebesgue_measurable s ==> ?k c. negligible k /\ (!n. open(c n)) /\ s = INTERS {c n | n IN (:num)} DIFF k`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LEBESGUE_MEASURABLE_OUTER_OPEN)) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&2 pow n)`) THEN REWRITE_TAC[REAL_LT_POW2; SKOLEM_THM; REAL_LT_INV_EQ] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `c:num->real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `INTERS {c n | n IN (:num)} DIFF s:real^N->bool` THEN EXISTS_TAC `c:num->real^N->bool` THEN ASM_REWRITE_TAC[SET_RULE `s = t DIFF (t DIFF s) <=> s SUBSET t`] THEN ASM_REWRITE_TAC[SUBSET_INTERS; FORALL_IN_GSPEC] THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_POW_INV]] THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN EXISTS_TAC `(c:num->real^N->bool) n DIFF s` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS]]);; let LEBESGUE_MEASURABLE_REGULAR_INNER = prove (`!s:real^N->bool. lebesgue_measurable s ==> ?k c. negligible k /\ (!n. compact(c n)) /\ s = UNIONS {c n | n IN (:num)} UNION k`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LEBESGUE_MEASURABLE_INNER_CLOSED)) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&2 pow n)`) THEN REWRITE_TAC[REAL_LT_POW2; SKOLEM_THM; REAL_LT_INV_EQ] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `c:num->real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `s DIFF UNIONS {c n | n IN (:num)}:real^N->bool` THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN CONJ_TAC THENL [REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_POW_INV]] THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN EXISTS_TAC `s DIFF (c:num->real^N->bool) n` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS]]; SUBGOAL_THEN `?d. (!n. compact(d n:real^N->bool)) /\ UNIONS {d n | n IN (:num)} = UNIONS {c n | n IN (:num)}` MP_TAC THENL [MP_TAC(GEN `n:num` (ISPEC `(c:num->real^N->bool) n` CLOSED_UNION_COMPACT_SUBSETS)) THEN ASM_REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM] THEN DISCH_THEN (X_CHOOSE_THEN `d:num->num->real^N->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `COUNTABLE {d n m:real^N->bool | n IN (:num) /\ m IN (:num)}` MP_TAC THENL [MATCH_MP_TAC COUNTABLE_PRODUCT_DEPENDENT THEN REWRITE_TAC[NUM_COUNTABLE]; DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COUNTABLE_AS_IMAGE)) THEN ANTS_TAC THENL [SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[SET_RULE `s = t UNION (s DIFF t) <=> t SUBSET s`] THEN ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC]]]);; let LEBESGUE_MEASURABLE_SMALL_IMP_NEGLIGIBLE = prove (`!s:real^N->bool. lebesgue_measurable s /\ s <_c (:real) ==> negligible s`, GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[TAUT `p ==> q <=> ~(p /\ ~q)`] THEN DISCH_THEN(MP_TAC o MATCH_MP STEINHAUS_LEBESGUE) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `s:real^N->bool`; `(:real)`] CARD_MUL_LT_INFINITE) THEN ASM_REWRITE_TAC[real_INFINITE; mul_c; CARD_NOT_LT] THEN TRANS_TAC CARD_LE_TRANS `ball(vec 0:real^N,r)` THEN CONJ_TAC THENL [ASM_MESON_TAC[CARD_EQ_BALL; CARD_EQ_SYM; CARD_EQ_IMP_LE]; ALL_TAC] THEN TRANS_TAC CARD_LE_TRANS `{x - y:real^N | x IN s /\ y IN s}` THEN ASM_SIMP_TAC[CARD_LE_SUBSET] THEN SUBGOAL_THEN `{x - y:real^N | x IN s /\ y IN s} = IMAGE (\z. FST z - SND z) {x,y | x IN s /\ y IN s}` (fun th -> REWRITE_TAC[th; CARD_LE_IMAGE]) THEN REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PAIR_THM; IN_ELIM_PAIR_THM] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* A Lebesgue measurable set is almost an F_sigma or G_delta. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_MEASURABLE_ALMOST_FSIGMA = prove (`!s:real^N->bool. lebesgue_measurable s ==> ?c t. fsigma c /\ negligible t /\ c UNION t = s /\ DISJOINT c t`, REPEAT STRIP_TAC THEN REWRITE_TAC[fsigma; UNION_OF; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?c b a. P a b c)`] THEN REWRITE_TAC[TAUT `(p /\ q /\ r) /\ s /\ t /\ u <=> r /\ t /\ u /\ p /\ q /\ s`] THEN REWRITE_TAC[UNWIND_THM1] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LEBESGUE_MEASURABLE_INNER_CLOSED)) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; SKOLEM_THM; FORALL_AND_THM] THEN X_GEN_TAC `f:num->real^N->bool` THEN STRIP_TAC THEN EXISTS_TAC `IMAGE (f:num->real^N->bool) (:num)` THEN EXISTS_TAC `s DIFF UNIONS (IMAGE (f:num->real^N->bool) (:num))` THEN ASM_SIMP_TAC[SET_RULE `DISJOINT s (u DIFF s)`; COUNTABLE_IMAGE; NUM_COUNTABLE; FORALL_IN_IMAGE; IN_UNIV; UNIONS_SUBSET; SET_RULE `s UNION (u DIFF s) = u <=> s SUBSET u`] THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `s DIFF (f:num->real^N->bool) n` THEN ASM_REWRITE_TAC[UNIONS_IMAGE] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `inv(&n + &1)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN TRANS_TAC REAL_LE_TRANS `inv(&n)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC);; let LEBESGUE_MEASURABLE_ALMOST_GDELTA = prove (`!s:real^N->bool. lebesgue_measurable s ==> ?c t. gdelta c /\ negligible t /\ s UNION t = c /\ DISJOINT t s`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM LEBESGUE_MEASURABLE_COMPL] THEN DISCH_THEN(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_ALMOST_FSIGMA) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:real^N->bool`; `t:real^N->bool`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(:real^N) DIFF c`; `t:real^N->bool`] THEN ASM_REWRITE_TAC[GDELTA_COMPLEMENT] THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Existence of nonmeasurable subsets of any set of positive measure. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_IFF_LEBESGUE_MEASURABLE_SUBSETS = prove (`!s:real^N->bool. negligible s <=> !t. t SUBSET s ==> lebesgue_measurable t`, let lemma = prove (`!s:real^N->bool. lebesgue_measurable s /\ (!x y q. x IN s /\ y IN s /\ rational q /\ y = q % basis 1 + x ==> y = x) ==> negligible s`, SIMP_TAC[VECTOR_ARITH `q + x:real^N = x <=> q = vec 0`; VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_GE_1; ARITH] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN MP_TAC(ISPEC `s:real^N->bool` STEINHAUS_LEBESGUE) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN FIRST_ASSUM(X_CHOOSE_TAC `q:real` o MATCH_MP RATIONAL_BETWEEN) THEN FIRST_X_ASSUM (MP_TAC o SPEC `q % basis 1:real^N` o GEN_REWRITE_RULE I [SUBSET]) THEN SIMP_TAC[IN_BALL_0; NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; ARITH; NOT_IMP] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_ELIM_THM]] THEN ASM_REWRITE_TAC[REAL_MUL_RID; IN_ELIM_THM; NOT_EXISTS_THM; VECTOR_ARITH `q:real^N = x - y <=> x = q + y`] THEN ASM_CASES_TAC `q = &0` THENL [ASM_REAL_ARITH_TAC; ASM_MESON_TAC[]]) in GEN_TAC THEN EQ_TAC THENL [MESON_TAC[NEGLIGIBLE_SUBSET; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]; DISCH_TAC] THEN ABBREV_TAC `(canonize:real^N->real^N) = \x. @y. y IN s /\ ?q. rational q /\ q % basis 1 + y = x` THEN SUBGOAL_THEN `!x:real^N. x IN s ==> canonize x IN s /\ ?q. rational q /\ q % basis 1 + canonize x = x` ASSUME_TAC THENL [GEN_TAC THEN DISCH_TAC THEN EXPAND_TAC "canonize" THEN CONV_TAC SELECT_CONV THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `&0` THEN REWRITE_TAC[RATIONAL_CLOSED] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN ABBREV_TAC `v = IMAGE (canonize:real^N->real^N) s` THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE (\q. IMAGE (\x:real^N. q % basis 1 + x) v) rational)` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_IMAGE; SUBSET; IN_ELIM_THM] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN SIMP_TAC[COUNTABLE_RATIONAL; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[NEGLIGIBLE_TRANSLATION_EQ] THEN GEN_TAC THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC lemma THEN CONJ_TAC THENL [FIRST_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN EXPAND_TAC "v" THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN X_GEN_TAC `q:real` THEN REPEAT DISCH_TAC THEN EXPAND_TAC "canonize" THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `z:real^N` THEN AP_TERM_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `y:real^N` th) THEN MP_TAC(SPEC `x:real^N` th)) THEN ASM_REWRITE_TAC[VECTOR_ARITH `q % b + x:real^N = y <=> x = y - q % b`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH `x - q % b:real^N = y - r % b - s % b <=> y + (q - r - s) % b = x /\ x + (r + s - q) % b = y`] THEN STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (BINDER_CONV o RAND_CONV o RAND_CONV o LAND_CONV) [SYM th]) THEN SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_GE_1; ARITH; VECTOR_ARITH `y - q % b:real^N = (y + r % b) - s % b <=> (q + r - s) % b = vec 0`] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[REAL_ARITH `a + b - c = &0 <=> c = a + b`; UNWIND_THM2] THEN ASM_SIMP_TAC[RATIONAL_CLOSED]);; let NEGLIGIBLE_IFF_MEASURABLE_SUBSETS = prove (`!s:real^N->bool. negligible s <=> !t. t SUBSET s ==> measurable t`, MESON_TAC[NEGLIGIBLE_SUBSET; NEGLIGIBLE_IMP_MEASURABLE; MEASURABLE_IMP_LEBESGUE_MEASURABLE; NEGLIGIBLE_IFF_LEBESGUE_MEASURABLE_SUBSETS]);; let NON_MEASURABLE_SET = prove (`?s:real^N->bool. ~lebesgue_measurable s`, MP_TAC(ISPEC `(:real^N)` NEGLIGIBLE_IFF_LEBESGUE_MEASURABLE_SUBSETS) THEN REWRITE_TAC[NOT_NEGLIGIBLE_UNIV] THEN MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Preserving Lebesgue measurability vs. preserving negligibility. *) (* ------------------------------------------------------------------------- *) let PRESERVES_LEBESGUE_MEASURABLE_IMP_PRESERVES_NEGLIGIBLE = prove (`!f s:real^N->bool. (!t. negligible t /\ t SUBSET s ==> lebesgue_measurable(IMAGE f t)) ==> (!t. negligible t /\ t SUBSET s ==> negligible(IMAGE f t))`, REPEAT STRIP_TAC THEN REWRITE_TAC[NEGLIGIBLE_IFF_LEBESGUE_MEASURABLE_SUBSETS] THEN REWRITE_TAC[FORALL_SUBSET_IMAGE] THEN ASM_MESON_TAC[NEGLIGIBLE_SUBSET; SUBSET_TRANS]);; let PRESERVES_NEGLIGIBLE_IMAGE = prove (`!f:real^M->real^N s. (!t. negligible t /\ t SUBSET s ==> negligible(IMAGE f t)) <=> (!t. negligible t /\ t SUBSET s ==> lebesgue_measurable(IMAGE f t))`, REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[PRESERVES_LEBESGUE_MEASURABLE_IMP_PRESERVES_NEGLIGIBLE] THEN MESON_TAC[NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]);; let PRESERVES_NEGLIGIBLE_IMAGE_UNIV = prove (`!f:real^M->real^N. (!t. negligible t ==> negligible(IMAGE f t)) <=> (!t. negligible t ==> lebesgue_measurable(IMAGE f t))`, MESON_TAC[PRESERVES_NEGLIGIBLE_IMAGE; SUBSET_UNIV]);; let LEBESGUE_MEASURABLE_CONTINUOUS_IMAGE = prove (`!f:real^M->real^N s. f continuous_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible(IMAGE f t)) ==> !t. lebesgue_measurable t /\ t SUBSET s ==> lebesgue_measurable(IMAGE f t)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP LEBESGUE_MEASURABLE_REGULAR_INNER) THEN ASM_REWRITE_TAC[IMAGE_UNION; IMAGE_UNIONS] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN SUBGOAL_THEN `(k:real^M->bool) SUBSET s` ASSUME_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o; FORALL_IN_IMAGE] THEN SIMP_TAC[IN_UNIV; COUNTABLE_IMAGE; NUM_COUNTABLE] THEN GEN_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);; let PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE = prove (`!f:real^M->real^N s. f continuous_on s ==> ((!t. lebesgue_measurable t /\ t SUBSET s ==> lebesgue_measurable (IMAGE f t)) <=> (!t. negligible t /\ t SUBSET s ==> negligible (IMAGE f t)))`, MESON_TAC[LEBESGUE_MEASURABLE_CONTINUOUS_IMAGE; PRESERVES_LEBESGUE_MEASURABLE_IMP_PRESERVES_NEGLIGIBLE; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]);; let PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_ALT = prove (`!f:real^M->real^N. f continuous_on s ==> ((!t. lebesgue_measurable t /\ t SUBSET s ==> lebesgue_measurable (IMAGE f t)) <=> (!t. negligible t /\ t SUBSET s ==> lebesgue_measurable(IMAGE f t)))`, SIMP_TAC[PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE] THEN MESON_TAC[PRESERVES_LEBESGUE_MEASURABLE_IMP_PRESERVES_NEGLIGIBLE; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]);; let LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE = prove (`!f:real^M->real^N s. dimindex(:M) <= dimindex(:N) /\ f differentiable_on s /\ lebesgue_measurable s ==> lebesgue_measurable(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC (REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM] LEBESGUE_MEASURABLE_CONTINUOUS_IMAGE) THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[SUBSET_REFL; DIFFERENTIABLE_IMP_CONTINUOUS_ON] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN ASM_MESON_TAC[DIFFERENTIABLE_ON_SUBSET]);; let LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN = prove (`!f:real^M->real^N s. linear f /\ lebesgue_measurable s /\ dimindex(:M) <= dimindex(:N) ==> lebesgue_measurable(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_LINEAR]);; let MEASURABLE_LINEAR_IMAGE_GEN = prove (`!f:real^M->real^N s. linear f /\ measurable s /\ dimindex(:M) <= dimindex(:N) ==> measurable(IMAGE f s)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE `m:num <= n ==> m < n \/ m = n`)) THENL [MATCH_MP_TAC NEGLIGIBLE_IMP_MEASURABLE THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_LOWDIM THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_LINEAR]; ASM_CASES_TAC `!x y. (f:real^M->real^N) x = f y ==> x = y` THENL [ASM_MESON_TAC[MEASURABLE_LINEAR_IMAGE_EQ_GEN]; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_IMP_MEASURABLE THEN MATCH_MP_TAC NEGLIGIBLE_LOWDIM THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`] DIM_IMAGE_KERNEL_GEN) THEN ASM_REWRITE_TAC[SUBSPACE_UNIV; DIM_UNIV] THEN ONCE_ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC(ARITH_RULE `x <= y /\ ~(d = 0) ==> x < y + d`) THEN SIMP_TAC[DIM_SUBSET; IMAGE_SUBSET; SUBSET_UNIV] THEN REWRITE_TAC[IN_UNIV; DIM_EQ_0] THEN FIRST_ASSUM(MP_TAC o MATCH_MP LINEAR_INJECTIVE_0) THEN ASM SET_TAC[]]);; let LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ_GEN = prove (`!f:real^M->real^N s. dimindex(:M) = dimindex(:N) /\ linear f /\ (!x y. f x = f y ==> x = y) ==> (lebesgue_measurable(IMAGE f s) <=> lebesgue_measurable s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!y. f((g:real^N->real^M) y) = y` ASSUME_TAC THENL [MP_TAC(ISPEC `f:real^M->real^N` LINEAR_SURJECTIVE_IFF_INJECTIVE_GEN) THEN ASM_MESON_TAC[]; ALL_TAC] THEN EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN; LE_REFL]] THEN DISCH_TAC THEN SUBGOAL_THEN `s = IMAGE (g:real^N->real^M) (IMAGE f s)` SUBST1_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN; LE_REFL]]);; (* ------------------------------------------------------------------------- *) (* Lebesgue-Lebesgue measurability (preimage of measurable is measurable). *) (* ------------------------------------------------------------------------- *) let DOUBLE_LEBESGUE_MEASURABLE = prove (`!f:real^M->real^N. (!t. lebesgue_measurable t ==> lebesgue_measurable {x | f x IN t}) <=> f measurable_on (:real^M) /\ (!t. negligible t ==> lebesgue_measurable {x | f x IN t})`, GEN_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[MEASURABLE_ON_PREIMAGE_BOREL; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE; BOREL_IMP_LEBESGUE_MEASURABLE] THEN STRIP_TAC THEN X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_ALMOST_FSIGMA) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`b:real^N->bool`; `n:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN REWRITE_TAC[SET_RULE `{x | f x IN s UNION t} = {x | f x IN s} UNION {x | f x IN t}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN ASM_SIMP_TAC[FSIGMA_IMP_BOREL]);; let DOUBLE_LEBESGUE_MEASURABLE_ON = prove (`!f:real^M->real^N s t. lebesgue_measurable s /\ lebesgue_measurable t /\ IMAGE f s SUBSET t ==> ((!u. lebesgue_measurable u /\ u SUBSET t ==> lebesgue_measurable {x | x IN s /\ f x IN u}) <=> f measurable_on s /\ (!u. negligible u /\ u SUBSET t ==> lebesgue_measurable {x | x IN s /\ f x IN u}))`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL] THEN EQ_TAC THENL [SIMP_TAC[NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE] THEN DISCH_TAC THEN X_GEN_TAC `v:real^N->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t INTER v:real^N->bool`) THEN ASM_SIMP_TAC[INTER_SUBSET; LEBESGUE_MEASURABLE_INTER; BOREL_IMP_LEBESGUE_MEASURABLE] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]; DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "B") (LABEL_TAC "N")) THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_ALMOST_FSIGMA) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`b:real^N->bool`; `n:real^N->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN REWRITE_TAC[SET_RULE `{x | x IN a /\ f x IN s UNION t} = {x | x IN a /\ f x IN s} UNION {x | x IN a /\ f x IN t}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN CONJ_TAC THENL [REMOVE_THEN "B" MATCH_MP_TAC; REMOVE_THEN "N" MATCH_MP_TAC] THEN ASM_SIMP_TAC[FSIGMA_IMP_BOREL] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Measurability of continuous functions. *) (* ------------------------------------------------------------------------- *) let CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET = prove (`!f:real^M->real^N s. f continuous_on s /\ lebesgue_measurable s ==> f measurable_on s`, let lemma = prove (`!s. lebesgue_measurable s ==> ?u:num->real^M->bool. (!n. closed(u n)) /\ (!n. u n SUBSET s) /\ (!n. measurable(s DIFF u n) /\ measure(s DIFF u n) < inv(&n + &1)) /\ (!n. u(n) SUBSET u(SUC n))`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!n t. closed t /\ t SUBSET s ==> ?u:real^M->bool. closed u /\ t SUBSET u /\ u SUBSET s /\ measurable(s DIFF u) /\ measure(s DIFF u) < inv(&n + &1)` MP_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s DIFF t:real^M->bool`; `inv(&n + &1)`] LEBESGUE_MEASURABLE_INNER_CLOSED) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_DIFF; LEBESGUE_MEASURABLE_CLOSED] THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t UNION u:real^M->bool` THEN ASM_SIMP_TAC[CLOSED_UNION] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[SET_RULE `s DIFF (t UNION u) = s DIFF t DIFF u`]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `v:num->(real^M->bool)->(real^M->bool)` THEN DISCH_TAC THEN MP_TAC(prove_recursive_functions_exist num_RECURSION `(u:num->real^M->bool) 0 = v 0 {} /\ (!n. u(SUC n) = v (SUC n) (u n))`) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:num->real^M->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `!n. closed(u n) /\ (u:num->real^M->bool) n SUBSET s` ASSUME_TAC THENL [INDUCT_TAC THEN ASM_SIMP_TAC[CLOSED_EMPTY; EMPTY_SUBSET]; ASM_SIMP_TAC[]] THEN INDUCT_TAC THEN ONCE_ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[CLOSED_EMPTY; EMPTY_SUBSET]]) in REPEAT STRIP_TAC THEN FIRST_ASSUM(X_CHOOSE_THEN `u:num->real^M->bool` STRIP_ASSUME_TAC o MATCH_MP lemma) THEN SUBGOAL_THEN `lebesgue_measurable((:real^M) DIFF s)` MP_TAC THENL [ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `v:num->real^M->bool` STRIP_ASSUME_TAC o MATCH_MP lemma) THEN REWRITE_TAC[measurable_on] THEN EXISTS_TAC `(:real^M) DIFF (UNIONS {u n | n IN (:num)} UNION UNIONS {v n | n IN (:num)})` THEN SUBGOAL_THEN `!n. ?g. g continuous_on (:real^M) /\ (!x. x IN u(n) UNION v(n:num) ==> g x = if x IN s then (f:real^M->real^N)(x) else vec 0)` MP_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC TIETZE_UNBOUNDED THEN ASM_SIMP_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN; CLOSED_UNION] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN ASM_SIMP_TAC[CONTINUOUS_ON_CONST] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]; REWRITE_TAC[SKOLEM_THM] THEN MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `g:num->real^M->real^N` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `(s DIFF UNIONS {u n | n IN (:num)}) UNION ((:real^M) DIFF s DIFF UNIONS {v n | n IN (:num)})` THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC NEGLIGIBLE_UNION THEN CONJ_TAC THEN REWRITE_TAC[NEGLIGIBLE_OUTER] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPEC `e:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THENL [EXISTS_TAC `s DIFF u(n:num):real^M->bool`; EXISTS_TAC `(:real^M) DIFF s DIFF v(n:num):real^M->bool`] THEN (CONJ_TAC THENL [SET_TAC[]; ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC REAL_LT_TRANS THEN EXISTS_TAC `inv(&n + &1)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_TRANS THEN EXISTS_TAC `inv(&n)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_INV2 THEN ASM_REWRITE_TAC[REAL_OF_NUM_LT] THEN CONJ_TAC THENL [ASM_ARITH_TAC; REAL_ARITH_TAC]); X_GEN_TAC `x:real^M` THEN REWRITE_TAC[SET_RULE `~(x IN (UNIV DIFF (s UNION t))) <=> x IN s \/ x IN t`] THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN REWRITE_TAC[OR_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `n:num` THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_UNION] THEN SUBGOAL_THEN `!i j. i <= j ==> (u:num->real^M->bool)(i) SUBSET u(j) /\ (v:num->real^M->bool)(i) SUBSET v(j)` (fun th -> ASM_MESON_TAC[SUBSET; th]) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN SET_TAC[]]);; let CONTINUOUS_IMP_MEASURABLE_ON_CLOSED_SUBSET = prove (`!f:real^M->real^N s. f continuous_on s /\ closed s ==> f measurable_on s`, SIMP_TAC[CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET; LEBESGUE_MEASURABLE_CLOSED]);; let CONTINUOUS_AE_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET = prove (`!f:real^M->real^N s m. f continuous_on (s DIFF m) /\ lebesgue_measurable s /\ negligible m ==> f measurable_on s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(f:real^M->real^N) measurable_on (s DIFF m)` MP_TAC THENL [MATCH_MP_TAC CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_DIFF; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]; MATCH_MP_TAC MEASURABLE_ON_SPIKE_SET THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]]);; let MEASURABLE_CONTINUOUS_COMPOSE = prove (`!f:real^N->real^P g:real^M->real^N. f measurable_on UNIV /\ g continuous_on UNIV /\ (!k. negligible k ==> negligible {x | g x IN k}) ==> (f o g) measurable_on UNIV`, REWRITE_TAC[measurable_on; IN_UNIV] THEN REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`k:real^N->bool`; `h:num->real^N->real^P`] THEN STRIP_TAC THEN EXISTS_TAC `{x | (g:real^M->real^N) x IN k}` THEN EXISTS_TAC `\n x:real^M. (h:num->real^N->real^P) n (g x)` THEN ASM_SIMP_TAC[IN_ELIM_THM; o_DEF] THEN GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]);; let MEASURABLE_ON_COMPOSE_REV = prove (`!f:real^N->real^P g:real^M->real^N s t. lebesgue_measurable s /\ IMAGE g s = t /\ (!k. lebesgue_measurable k /\ k SUBSET s ==> lebesgue_measurable (IMAGE g k)) /\ (f o g) measurable_on s ==> f measurable_on t`, REPEAT GEN_TAC THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SUBGOAL_THEN `lebesgue_measurable(t:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `u:real^P->bool` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[o_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{x | x IN s /\ (f:real^N->real^P) (g(x:real^M)) IN u}`) THEN ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]);; let MEASURABLE_ON_CONTINUOUS_COMPOSE_REV = prove (`!f:real^N->real^P g:real^M->real^N s t. lebesgue_measurable s /\ IMAGE g s = t /\ (!k. negligible k /\ k SUBSET s ==> lebesgue_measurable (IMAGE g k)) /\ g continuous_on s /\ (f o g) measurable_on s ==> f measurable_on t`, REPEAT GEN_TAC THEN REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] MEASURABLE_ON_COMPOSE_REV)) THEN ASM_MESON_TAC[PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_ALT]);; let MEASURABLE_ON_COMPOSE_GEN = prove (`!f:real^N->real^P g:real^M->real^N s t. lebesgue_measurable t /\ IMAGE g s SUBSET t /\ (!k. lebesgue_measurable k /\ k SUBSET t ==> lebesgue_measurable {x | x IN s /\ g x IN k}) /\ f measurable_on t ==> (f o g) measurable_on s`, REPEAT GEN_TAC THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SUBGOAL_THEN `lebesgue_measurable(s:real^M->bool)` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]; ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN]] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `u:real^P->bool` THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[o_THM] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{x | x IN t /\ (f:real^N->real^P) x IN u}`) THEN ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]);; let MEASURABLE_ON_COMPOSE_ALT = prove (`!f:real^N->real^P g:real^M->real^N s t. lebesgue_measurable s /\ lebesgue_measurable t /\ IMAGE g s SUBSET t /\ (!k. negligible k /\ k SUBSET t ==> lebesgue_measurable {x | x IN s /\ g x IN k}) /\ g measurable_on s /\ f measurable_on t ==> (f o g) measurable_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_COMPOSE_GEN THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[DOUBLE_LEBESGUE_MEASURABLE_ON]);; let MEASURABLE_ON_CONTINUOUS_COMPOSE = prove (`!f:real^N->real^P g:real^M->real^N s t. lebesgue_measurable s /\ lebesgue_measurable t /\ IMAGE g s SUBSET t /\ f measurable_on t /\ g continuous_on s /\ (!k. negligible k /\ k SUBSET t ==> lebesgue_measurable {x | x IN s /\ g x IN k}) ==> (f o g) measurable_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_COMPOSE_ALT THEN EXISTS_TAC `t:real^N->bool` THEN ASM_SIMP_TAC[CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET]);; let MEASURABLE_ON_DIFFERENTIABLE_IMAGE = prove (`!f:real^N->real^P g:real^M->real^N s. dimindex(:M) <= dimindex(:N) /\ lebesgue_measurable s /\ g differentiable_on s /\ (f o g) measurable_on s ==> f measurable_on (IMAGE g s)`, REPEAT GEN_TAC THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] MEASURABLE_ON_CONTINUOUS_COMPOSE_REV)) THEN ASM_SIMP_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_ON] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_MESON_TAC[DIFFERENTIABLE_ON_SUBSET; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]);; let BOREL_MEASURABLE_IMP_MEASURABLE_ON = prove (`!f:real^M->real^N s. f borel_measurable_on s /\ lebesgue_measurable s ==> f measurable_on s`, REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC borel_measurable_INDUCT THEN CONJ_TAC THENL [MESON_TAC[CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET]; REPEAT GEN_TAC THEN ASM_CASES_TAC `lebesgue_measurable(s:real^M->bool)` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_LIMIT THEN MAP_EVERY EXISTS_TAC [`f:num->real^M->real^N`; `{}:real^M->bool`] THEN ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY; DIFF_EMPTY]]);; (* ------------------------------------------------------------------------- *) (* Connections between measurability and Baire-ness. *) (* ------------------------------------------------------------------------- *) let BAIRE_IMP_MEASURABLE_ON = prove (`!(f:real^M->real^N) s. baire n s f /\ lebesgue_measurable s ==> f measurable_on s`, MESON_TAC[BOREL_MEASURABLE_IMP_MEASURABLE_ON; BAIRE_IMP_BOREL_MEASURABLE]);; let BAIRE_IMP_MEASURABLE_ON_UNIV = prove (`!(f:real^M->real^N) s. baire n (:real^M) f ==> f measurable_on (:real^M)`, MESON_TAC[BAIRE_IMP_MEASURABLE_ON; LEBESGUE_MEASURABLE_UNIV]);; let MEASURABLE_EQ_ALMOST_BAIRE1 = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> ?s. negligible((:real^M) DIFF s) /\ baire 1 s f`, GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[measurable_on; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`s:real^M->bool`; `g:num->real^M->real^N`] THEN STRIP_TAC THEN EXISTS_TAC `(:real^M) DIFF s` THEN ASM_REWRITE_TAC[SET_RULE `UNIV DIFF (UNIV DIFF s) = s`] THEN REWRITE_TAC[num_CONV `1`; baire] THEN EXISTS_TAC `g:num->real^M->real^N` THEN ASM_SIMP_TAC[IN_DIFF] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; DISCH_THEN(X_CHOOSE_THEN `s:real^M->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP BAIRE_IMP_BOREL_MEASURABLE) THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] BOREL_MEASURABLE_IMP_MEASURABLE_ON)) THEN ONCE_REWRITE_TAC[GSYM LEBESGUE_MEASURABLE_COMPL] THEN ASM_SIMP_TAC[NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE] THEN MATCH_MP_TAC MEASURABLE_ON_SPIKE_SET THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]]);; let MEASURABLE_EQ_ALMOST_BAIRE2 = prove (`!f:real^M->real^N. f measurable_on (:real^M) <=> ?g. baire 2 (:real^M) g /\ negligible {x | ~(g x = f x)}`, GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[measurable_on; IN_UNIV; LEFT_IMP_EXISTS_THM]; DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP BAIRE_IMP_BOREL_MEASURABLE) THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] BOREL_MEASURABLE_IMP_MEASURABLE_ON)) THEN REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV] THEN MATCH_MP_TAC MEASURABLE_ON_SPIKE THEN EXISTS_TAC `{x | ~((g:real^M->real^N) x = f x)}` THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; IN_ELIM_THM] THEN MESON_TAC[]] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `g:num->real^M->real^N`] THEN STRIP_TAC THEN ABBREV_TAC `s = {x | ?l. ((\n. (g:num->real^M->real^N) n x) --> l) sequentially}` THEN SUBGOAL_THEN `!x. ?l. x IN s ==> ((\n. (g:num->real^M->real^N) n x) --> l) sequentially` MP_TAC THENL [EXPAND_TAC "s" THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^M->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`h:real^M->real^N`; `s:real^M->bool`; `2`] BAIRE_EXTENSION) THEN CONV_TAC NUM_REDUCE_CONV THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC BAIRE_MONO THEN EXISTS_TAC `SUC 0` THEN REWRITE_TAC[baire] THEN CONV_TAC NUM_REDUCE_CONV THEN EXISTS_TAC `g:num->real^M->real^N` THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC] THEN SUBGOAL_THEN `s = INTERS {UNIONS {INTERS {{x | dist((g:num->real^M->real^N) m x,g n x) <= e} | m IN {p | N <= p} /\ n IN {p | N <= p}} | N IN (:num)} | e IN {q | q IN rational /\ &0 < q}}` SUBST1_TAC THENL [EXPAND_TAC "s" THEN REWRITE_TAC[CONVERGENT_EQ_CAUCHY] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^M` THEN SIMP_TAC[cauchy; UNIONS_GSPEC; INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN REWRITE_TAC[IN; GE] THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN MP_TAC(SPECL [`&0`; `e:real`] RATIONAL_BETWEEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `q:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `q:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[REAL_LT_TRANS; REAL_LT_IMP_LE; REAL_LET_TRANS]; MATCH_MP_TAC COUNTABLE_INTERSECTION_OF_INTERS THEN SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; COUNTABLE_RESTRICT; COUNTABLE_RATIONAL; FORALL_IN_IMAGE] THEN X_GEN_TAC `e:real` THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC COUNTABLE_INTERSECTION_OF_INC THEN REWRITE_TAC[ARITH_RULE `2 = 1 + 1`; o_THM] THEN MATCH_MP_TAC COUNTABLE_UNION_OF_BAIRE_INDICATOR THEN MATCH_MP_TAC COUNTABLE_UNION_OF_UNIONS THEN SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_SUBSET_NUM; FORALL_IN_IMAGE] THEN X_GEN_TAC `N:num` THEN REWRITE_TAC[IN_UNIV; GSYM FSIGMA_BAIRE] THEN MATCH_MP_TAC CLOSED_IMP_FSIGMA THEN MATCH_MP_TAC CLOSED_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN REWRITE_TAC[dist; GSYM IN_CBALL_0] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN REWRITE_TAC[CLOSED_CBALL] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[ETA_AX] THEN ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_UNIV; IN_UNIV]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `!x. ~(x IN k) ==> (h:real^M->real^N) x = f x` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `\n. (g:num->real^M->real^N) n x` THEN ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Versions of the Lebesgue density theorem, both integral and measure *) (* forms. Later we have cosmetically nicer ones using real limits. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_DENSITY_THEOREM_INTEGRAL_NORM_CBALL = prove (`!f:real^M->real^N. (!a b. f absolutely_integrable_on interval[a,b]) ==> ?k. negligible k /\ !x. ~(x IN k) ==> ((\e. inv(measure(cball(x,drop e))) % integral (cball(x,drop e)) (\y. lift(norm(f y - f x)))) --> vec 0) (at (vec 0) within {t | &0 < drop t})`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_LEBESGUE_POINTS) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[LIM_WITHIN; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `e / &(dimindex(:M)) pow dimindex(:M)`]) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_POW_LT; DIST_0; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[FORALL_LIFT; NORM_LIFT; LIFT_DROP] THEN X_GEN_TAC `h:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `h:real`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS) THEN REWRITE_TAC[REAL_ABS_MUL; NORM_MUL] THEN ONCE_REWRITE_TAC [REAL_ARITH `x <= (abs a * b) * c <=> x <= (abs(a) * c) * b`] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[REAL_ABS_POS] THEN CONJ_TAC THENL [SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1] THEN REWRITE_TAC[REAL_ABS_INV; real_div; GSYM REAL_INV_MUL] THEN MATCH_MP_TAC REAL_LE_INV2 THEN CONJ_TAC THENL [REWRITE_TAC[GSYM REAL_ABS_NZ; CONTENT_EQ_0] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT; VECTOR_SUB_COMPONENT] THEN ASM_REAL_ARITH_TAC; SIMP_TAC[real_abs; CONTENT_POS_LE; MEASURE_POS_LE; MEASURABLE_CBALL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(interval[x - h / &(dimindex(:M)) % vec 1:real^M, x + h / &(dimindex(:M)) % vec 1]) * &(dimindex (:M)) pow dimindex (:M)` THEN CONJ_TAC THENL [REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT; VECTOR_SUB_COMPONENT; REAL_MUL_RID] THEN ASM_SIMP_TAC[REAL_ARITH `x - h <= x + h <=> &0 <= h`; REAL_LE_DIV; REAL_POS; REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_ARITH `(x + h) - (x - h) = &2 * h`; PRODUCT_CONST_NUMSEG_1; REAL_POW_DIV; REAL_POW_MUL] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> y <= x`) THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN AP_TERM_TAC THEN MATCH_MP_TAC REAL_DIV_RMUL THEN REWRITE_TAC[REAL_POW_EQ_0; REAL_OF_NUM_EQ; DIMINDEX_NONZERO]; MATCH_MP_TAC REAL_LE_RMUL THEN SIMP_TAC[REAL_POS; REAL_POW_LE] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL; MEASURABLE_CBALL] THEN REWRITE_TAC[SUBSET; IN_INTERVAL; IN_CBALL] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT; VECTOR_SUB_COMPONENT; REAL_MUL_RID; REAL_ARITH `x - h <= y /\ y <= x + h <=> abs(x - y) <= h`] THEN STRIP_TAC THEN REWRITE_TAC[dist] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(1..dimindex(:M)) (\i. abs((x - y:real^M)$i))` THEN REWRITE_TAC[NORM_LE_L1] THEN MATCH_MP_TAC SUM_BOUND_GEN THEN ASM_REWRITE_TAC[CARD_NUMSEG_1; VECTOR_SUB_COMPONENT; IN_NUMSEG] THEN REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1]]]; REWRITE_TAC[NORM_POS_LE] THEN SUBGOAL_THEN `cball(x:real^M,h) SUBSET interval[x - h % vec 1,x + h % vec 1]` ASSUME_TAC THENL [REWRITE_TAC[SUBSET; IN_CBALL; IN_INTERVAL] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT; VECTOR_SUB_COMPONENT; REAL_MUL_RID; REAL_ARITH `x - h <= y /\ y <= x + h <=> abs(x - y) <= h`] THEN REWRITE_TAC[dist; GSYM VECTOR_SUB_COMPONENT] THEN MESON_TAC[REAL_LE_TRANS; COMPONENT_LE_NORM]; ALL_TAC] THEN SUBGOAL_THEN `(f:real^M->real^N) absolutely_integrable_on cball(x,h)` ASSUME_TAC THENL [ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET; LEBESGUE_MEASURABLE_CBALL]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[NORM_1] THEN MATCH_MP_TAC(REAL_ARITH `a <= b ==> a <= abs b`) THEN REWRITE_TAC[GSYM NORM_1] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN ASM_REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE; ABSOLUTELY_INTEGRABLE_SUB; ABSOLUTELY_INTEGRABLE_ON_CONST; MEASURABLE_CBALL; MEASURABLE_INTERVAL; IN_UNIV] THEN GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[NORM_LIFT; REAL_ABS_NORM; LIFT_DROP; NORM_0; DROP_VEC; NORM_POS_LE; REAL_LE_REFL]) THEN ASM SET_TAC[]]);; let LEBESGUE_DENSITY_THEOREM_INTEGRAL_NORM_BALL = prove (`!f:real^M->real^N. (!a b. f absolutely_integrable_on interval[a,b]) ==> ?k. negligible k /\ !x. ~(x IN k) ==> ((\e. inv(measure(ball(x,drop e))) % integral (ball(x,drop e)) (\y. lift(norm(f y - f x)))) --> vec 0) (at (vec 0) within {t | &0 < drop t})`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_DENSITY_THEOREM_INTEGRAL_NORM_CBALL) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN REWRITE_TAC[EVENTUALLY_WITHIN; IN_ELIM_THM; FORALL_LIFT; LIFT_DROP] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[DIST_0; NORM_LIFT; REAL_LT_01] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_SIMP_TAC[GSYM CLOSURE_BALL; MEASURE_CLOSURE; BOUNDED_BALL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_BALL] THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET THEN ASM_SIMP_TAC[CLOSURE_BALL] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `sphere(x:real^M,e)` THEN REWRITE_TAC[NEGLIGIBLE_SPHERE] THEN REWRITE_TAC[GSYM SPHERE_UNION_BALL] THEN SET_TAC[]);; let LEBESGUE_DENSITY_THEOREM_INTEGRAL_CBALL = prove (`!f:real^M->real^N s. (!a b. f absolutely_integrable_on (s INTER interval[a,b])) ==> ?k. negligible k /\ !x. ~(x IN k) ==> ((\e. inv(measure(cball(x,drop e))) % integral (s INTER cball(x,drop e)) f) --> (if x IN s then f x else vec 0)) (at (vec 0) within {t | &0 < drop t})`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_INTER] THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_INTER] THEN ABBREV_TAC `g:real^M->real^N = \x. if x IN s then f x else vec 0` THEN RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM]) THEN ASM_REWRITE_TAC[] THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`g:real^M->real^N`,`f:real^M->real^N`) THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_DENSITY_THEOREM_INTEGRAL_NORM_CBALL) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [LIM_NULL] THEN GEN_REWRITE_TAC (LAND_CONV o RATOR_CONV o LAND_CONV o ABS_CONV) [GSYM LIFT_DROP] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_NULL_COMPARISON) THEN REWRITE_TAC[EVENTUALLY_WITHIN; IN_ELIM_THM; DIST_0; FORALL_LIFT] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[LIFT_DROP; NORM_LIFT; REAL_LT_01] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN REWRITE_TAC[DROP_CMUL] THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div); REAL_LE_RDIV_EQ; MEASURE_CBALL_POS] THEN MATCH_MP_TAC(REAL_ARITH `abs(x * y) <= z ==> y * x <= z`) THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM; GSYM NORM_MUL] THEN ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ; MEASURE_CBALL_POS] THEN SIMP_TAC[GSYM INTEGRAL_CONST_GEN; MEASURABLE_CBALL; VECTOR_MUL_LID] THEN SUBGOAL_THEN `(f:real^M->real^N) absolutely_integrable_on cball(x,e)` ASSUME_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `interval[x - e % vec 1:real^M,x + e % vec 1]` THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_CBALL] THEN REWRITE_TAC[SUBSET; IN_CBALL; IN_INTERVAL] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT; VECTOR_SUB_COMPONENT; REAL_MUL_RID; REAL_ARITH `x - h <= y /\ y <= x + h <=> abs(x - y) <= h`] THEN REWRITE_TAC[dist; GSYM VECTOR_SUB_COMPONENT] THEN MESON_TAC[REAL_LE_TRANS; COMPONENT_LE_NORM]; ASM_SIMP_TAC[GSYM INTEGRAL_SUB; INTEGRABLE_ON_CONST; MEASURABLE_CBALL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE; ABSOLUTELY_INTEGRABLE_SUB; ABSOLUTELY_INTEGRABLE_ON_CONST; MEASURABLE_CBALL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REWRITE_TAC[LIFT_DROP; REAL_LE_REFL]);; let LEBESGUE_DENSITY_THEOREM_INTEGRAL_BALL = prove (`!f:real^M->real^N s. (!a b. f absolutely_integrable_on (s INTER interval[a,b])) ==> ?k. negligible k /\ !x. ~(x IN k) ==> ((\e. inv(measure(ball(x,drop e))) % integral (s INTER ball(x,drop e)) f) --> (if x IN s then f x else vec 0)) (at (vec 0) within {t | &0 < drop t})`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_DENSITY_THEOREM_INTEGRAL_CBALL) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN REWRITE_TAC[EVENTUALLY_WITHIN; IN_ELIM_THM; FORALL_LIFT; LIFT_DROP] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[DIST_0; NORM_LIFT; REAL_LT_01] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_SIMP_TAC[GSYM CLOSURE_BALL; MEASURE_CLOSURE; BOUNDED_BALL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_BALL] THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET THEN ASM_SIMP_TAC[CLOSURE_BALL] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `sphere(x:real^M,e)` THEN REWRITE_TAC[NEGLIGIBLE_SPHERE] THEN REWRITE_TAC[GSYM SPHERE_UNION_BALL] THEN SET_TAC[]);; let LEBESGUE_DENSITY_THEOREM_LIFT_CBALL = prove (`!s:real^N->bool. lebesgue_measurable s ==> ?k. negligible k /\ !x. ~(x IN k) ==> ((\e. lift(measure(s INTER cball(x,drop e)) / measure(cball(x,drop e)))) --> (if x IN s then vec 1 else vec 0)) (at (vec 0) within {t | &0 < drop t})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`indicator(s:real^N->bool)`; `(:real^N)`] LEBESGUE_DENSITY_THEOREM_INTEGRAL_CBALL) THEN REWRITE_TAC[IN_UNIV; INTER_UNIV] THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_ON_INDICATOR] THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_INTERVAL] THEN REWRITE_TAC[indicator; INTEGRAL_RESTRICT_INTER] THEN ASM_SIMP_TAC[INTEGRAL_CONST_GEN; MEASURABLE_CBALL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE] THEN REWRITE_TAC[GSYM LIFT_EQ_CMUL] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div; LIFT_CMUL]);; let LEBESGUE_DENSITY_THEOREM_LIFT_BALL = prove (`!s:real^N->bool. lebesgue_measurable s ==> ?k. negligible k /\ !x. ~(x IN k) ==> ((\e. lift(measure(s INTER ball(x,drop e)) / measure(ball(x,drop e)))) --> (if x IN s then vec 1 else vec 0)) (at (vec 0) within {t | &0 < drop t})`, GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP LEBESGUE_DENSITY_THEOREM_LIFT_CBALL) THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN REWRITE_TAC[GSYM FORALL_DROP] THEN X_GEN_TAC `e:real` THEN AP_TERM_TAC THEN BINOP_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `sphere(x:real^N,e)` THEN REWRITE_TAC[NEGLIGIBLE_SPHERE] THEN REWRITE_TAC[GSYM SPHERE_UNION_BALL] THEN SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* More refined form of "derivative is zero at a maximum or minimum" *) (* where we only need the derivative to hold w.r.t. a Lebesgue measurable *) (* set based at a point of density 1 (actually this could be further *) (* sharpened to density > 1/2, but it hardly seems worth it). *) (* ------------------------------------------------------------------------- *) let DIFFERENTIAL_ZERO_MAXMIN_DENSITY = prove (`!f f' s a:real^N. (f has_derivative f') (at a within s) /\ (eventually (\x. drop(f a) <= drop(f x)) (at a within s) \/ eventually (\x. drop(f x) <= drop(f a)) (at a within s)) /\ lebesgue_measurable s /\ ((\e. lift(measure(s INTER ball(a,drop e)) / measure(ball(a,drop e)))) --> vec 1) (at (vec 0) within {t | &0 < drop t}) ==> f' = \v. vec 0`, let lemma = prove (`!k a d:real^N. &0 < k /\ k < &1 /\ ~(d = vec 0) ==> ?e. &0 < e /\ !r. &0 < r ==> measurable({x | d dot (x - a) > k * norm d * norm(x - a)} INTER ball(a,r)) /\ e * measure(ball(a,r)) <= measure({x | d dot (x - a) > k * norm d * norm(x - a)} INTER ball(a,r))`, REPEAT GEN_TAC THEN GEN_GEOM_ORIGIN_TAC `a:real^N` ["d"] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_SUB_RZERO] THEN EXISTS_TAC `measure({x:real^N | d dot x > k * norm d * norm x} INTER ball(vec 0,&1)) / measure(ball(vec 0:real^N,&1))` THEN ONCE_REWRITE_TAC[REAL_ARITH `(a / b) * c:real = (a * c) / b`] THEN SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_01; MEASURE_BALL_POS] THEN SUBGOAL_THEN `open({x:real^N | d dot x > k * norm d * norm x} INTER ball(vec 0,&1))` ASSUME_TAC THENL [MATCH_MP_TAC OPEN_INTER THEN REWRITE_TAC[OPEN_BALL; real_gt] THEN ONCE_REWRITE_TAC[MESON[REAL_SUB_LT; LIFT_DROP] `x < y <=> &0 < drop(lift(y - x))`] THEN ONCE_REWRITE_TAC[SET_RULE `{x | &0 < drop(f x)} = {x | f x IN {y | &0 < drop y}}`] THEN MATCH_MP_TAC CONTINUOUS_OPEN_PREIMAGE_UNIV THEN REWRITE_TAC[drop; REWRITE_RULE[real_gt] OPEN_HALFSPACE_COMPONENT_GT] THEN REWRITE_TAC[LIFT_SUB; LIFT_CMUL] THEN GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_DOT] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_CMUL) THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_NORM]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[REAL_MUL_LZERO] THEN MATCH_MP_TAC MEASURE_OPEN_POS_LT THEN ASM_SIMP_TAC[BOUNDED_INTER; BOUNDED_BALL] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `(k + &1) / &2 % inv(norm d) % d:real^N` THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; IN_BALL_0; DOT_RMUL] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0] THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN REWRITE_TAC[REAL_ARITH `e * inv d * p > k * d * q <=> q * k * d < (e * p) / d`] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; NORM_POS_LT; GSYM REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM REAL_POW_2; NORM_POW_2] THEN ASM_SIMP_TAC[REAL_LT_RMUL_EQ; REAL_MUL_ASSOC; DOT_POS_LT] THEN ASM_SIMP_TAC[REAL_MUL_RID; REAL_ARITH `&0 < k ==> abs((k + &1) / &2) = (k + &1) / &2`] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN MATCH_MP_TAC REAL_LT_LMUL THEN ASM_REAL_ARITH_TAC; X_GEN_TAC `r:real` THEN DISCH_TAC THEN SUBGOAL_THEN `ball(vec 0:real^N,r) = ball(r % vec 0,r * &1)` SUBST1_TAC THENL [SIMP_TAC[VECTOR_MUL_RZERO; REAL_MUL_RID]; ALL_TAC] THEN ASM_SIMP_TAC[BALL_SCALING; MEASURE_SCALING; MEASURABLE_BALL] THEN SUBGOAL_THEN `{x:real^N | d dot x > k * norm d * norm x} INTER IMAGE (\x. r % x) (ball (vec 0,&1)) = IMAGE (\x. r % x) ({x:real^N | d dot x > k * norm d * norm x} INTER ball(vec 0,&1))` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x y. f x = f y ==> x = y) /\ (!x. f x IN s <=> x IN s) ==> s INTER IMAGE f t = IMAGE f (s INTER t)`) THEN ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; IN_ELIM_THM] THEN ASM_SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_LT_LMUL_EQ; REAL_ARITH `&0 < r ==> (r * d > g * e * abs r * f <=> r * g * e * f < r * d)`] THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[MEASURABLE_SCALING; MEASURE_SCALING; MEASURABLE_OPEN; BOUNDED_INTER; BOUNDED_BALL] THEN REWRITE_TAC[REAL_MUL_AC; REAL_LE_REFL]]) in SUBGOAL_THEN `!f f' s a:real^N. (f has_derivative f') (at a within s) /\ eventually (\x. drop(f x) <= drop(f a)) (at a within s) /\ lebesgue_measurable s /\ ((\e. lift(measure(s INTER ball(a,drop e)) / measure(ball(a,drop e)))) --> vec 1) (at (vec 0) within {t | &0 < drop t}) ==> f' = \v. vec 0` ASSUME_TAC THENL [ALL_TAC; REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`(--) o (f:real^N->real^1)`; `(--) o (f':real^N->real^1)`]); FIRST_X_ASSUM(MP_TAC o SPECL [`f:real^N->real^1`; `f':real^N->real^1`])] THEN DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`; `a:real^N`]) THEN ASM_SIMP_TAC[HAS_DERIVATIVE_NEG; o_DEF; DROP_NEG; REAL_LE_NEG2] THEN REWRITE_TAC[FUN_EQ_THM; VECTOR_NEG_EQ_0]] THEN REPEAT GEN_TAC THEN REWRITE_TAC[has_derivative_within; GSYM CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN REWRITE_TAC[tendsto] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!e. &0 < e ==> P e) /\ Q ==> !e. &0 < e ==> P e /\ Q`)) THEN REWRITE_TAC[GSYM EVENTUALLY_AND; DIST_0; NORM_MUL; REAL_ABS_INV] THEN REWRITE_TAC[REAL_ABS_NORM] THEN DISCH_TAC THEN SUBGOAL_THEN `!e. &0 < e ==> eventually (\x. drop(f'(x - a:real^N)) <= e * norm(x - a)) (at a within s)` MP_TAC THENL [X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_CASES_TAC `x:real^N = a` THENL [ASM_REWRITE_TAC[VECTOR_SUB_REFL] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_0 th]) THEN REWRITE_TAC[DROP_VEC; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL]; ASM_SIMP_TAC[REAL_LT_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN REWRITE_TAC[NORM_1; DROP_SUB; DROP_ADD] THEN REAL_ARITH_TAC]; FIRST_X_ASSUM(K ALL_TAC o SPEC `&0:real`)] THEN FIRST_ASSUM(MP_TAC o MATCH_MP LINEAR_TO_REALS) THEN ABBREV_TAC `d = row 1 (matrix(f':real^N->real^1))` THEN DISCH_THEN SUBST1_TAC THEN REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (free_in `f':real^N->real^1`) o concl)) THEN REWRITE_TAC[FUN_EQ_THM; GSYM DROP_EQ; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[FORALL_DOT_EQ_0] THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o GEN `e:real` o SPEC `e * norm(d:real^N)`) THEN ASM_SIMP_TAC[REAL_LT_MUL_EQ; NORM_POS_LT] THEN DISCH_THEN(MP_TAC o SPEC `&1 / &2`) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MP_TAC(ISPECL [`&1 / &2`; `a:real^N`; `d:real^N`] lemma) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_ELIM_THM; FORALL_LIFT; LIFT_DROP; DIST_0] THEN REWRITE_TAC[DIST_LIFT; GSYM LIFT_NUM; NORM_LIFT] THEN DISCH_THEN(X_CHOOSE_THEN `R:real` STRIP_ASSUME_TAC) THEN REWRITE_TAC[EVENTUALLY_WITHIN; GSYM DIST_NZ] THEN DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN ABBREV_TAC `r = min R k / &2` THEN SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `r:real`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `m <= &1 - e ==> abs(m - &1) < e ==> F`) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; MEASURE_BALL_POS] THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `y <= m - x ==> e * m <= x ==> y <= (&1 - e) * m`) THEN ASM_SIMP_TAC[GSYM MEASURE_DIFF_SUBSET; MEASURABLE_BALL; INTER_SUBSET] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_INTER; MEASURABLE_BALL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE] THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTER; MEASURABLE_BALL] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s /\ x IN b ==> ~(x IN h)) ==> s INTER b SUBSET b DIFF (h INTER b)`) THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_BALL] THEN STRIP_TAC THEN ASM_CASES_TAC `x:real^N = a` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[VECTOR_SUB_REFL; DOT_RZERO; NORM_0; real_gt] THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_REFL] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC);; let DIFFERENTIAL_ZERO_LEVELSET_DENSITY = prove (`!f:real^M->real^N f' s a c. (f has_derivative f') (at a within s) /\ eventually (\x. f x = c) (at a within s) /\ lebesgue_measurable s /\ ((\e. lift (measure (s INTER ball (a,drop e)) / measure (ball (a,drop e)))) --> vec 1) (at (vec 0) within {t | &0 < drop t}) ==> f' = \h. vec 0`, ONCE_REWRITE_TAC[HAS_DERIVATIVE_COMPONENTWISE_WITHIN] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N) ==> (\x. lift((f':real^M->real^N) x$i)) = (\x. vec 0)` MP_TAC THENL [ALL_TAC; SIMP_TAC[FUN_EQ_THM; GSYM DROP_EQ; LIFT_DROP; DROP_VEC] THEN SIMP_TAC[CART_EQ; VEC_COMPONENT]] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC DIFFERENTIAL_ZERO_MAXMIN_DENSITY THEN MAP_EVERY EXISTS_TAC [`\x. lift((f:real^M->real^N) x$i)`; `s:real^M->bool`; `a:real^M`] THEN ASM_SIMP_TAC[LIFT_DROP] THEN ASM_CASES_TAC `(c:real^N)$i <= (f:real^M->real^N) a$i` THENL [DISJ2_TAC; DISJ1_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_MONO)) THEN ASM_SIMP_TAC[] THEN ASM_REAL_ARITH_TAC);; let NEGLIGIBLE_POINTS_OF_AMBIGUOUS_DERIVATIVE = prove (`!f:real^M->real^N s. lebesgue_measurable s ==> negligible {x | x IN s /\ ?y z. (f has_derivative y) (at x within s) /\ (f has_derivative z) (at x within s) /\ ~(y = z)}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^M | x IN s /\ ~(((\e. lift(measure(s INTER ball(x,drop e)) / measure(ball(x,drop e)))) --> vec 1) (at (vec 0) within {t | &0 < drop t}))}` THEN CONJ_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_DENSITY_THEOREM_LIFT_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `k:real^M->bool` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN ONCE_REWRITE_TAC[TAUT `p /\ ~q ==> r <=> p ==> ~r ==> q`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[]; REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN MATCH_MP_TAC(TAUT `(p ==> r ==> ~q) ==> p /\ q ==> p /\ ~r`) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN REPEAT DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`f':real^M->real^N`; `f'':real^M->real^N`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[FUN_EQ_THM] THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN GEN_REWRITE_TAC I [GSYM FUN_EQ_THM] THEN MATCH_MP_TAC DIFFERENTIAL_ZERO_LEVELSET_DENSITY THEN MAP_EVERY EXISTS_TAC [`\x. (f:real^M->real^N) x - f x`; `s:real^M->bool`; `x:real^M`; `vec 0:real^N`] THEN ASM_SIMP_TAC[HAS_DERIVATIVE_SUB] THEN REWRITE_TAC[VECTOR_SUB_REFL; EVENTUALLY_TRUE]]);; (* ------------------------------------------------------------------------- *) (* Can only have countably many disjoint sets of positive measure. *) (* ------------------------------------------------------------------------- *) let PAIRWISE_DISJOINT_LEBESGUE_MEASURABLE_IMP_COUNTABLE = prove (`!f:(real^N->bool)->bool. pairwise (\s t. negligible (s INTER t)) f /\ (!s. s IN f ==> lebesgue_measurable s /\ ~negligible s) ==> COUNTABLE f`, REPEAT STRIP_TAC THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE (\(m,n). {s | s IN f /\ inv(&m + &1) < measure(s INTER interval[--vec n:real^N,vec n])}) ((:num) CROSS (:num)))` THEN CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_UNIONS THEN SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_CROSS; NUM_COUNTABLE] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_CROSS; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN MATCH_MP_TAC FINITE_IMP_COUNTABLE THEN MATCH_MP_TAC(MESON[] `!n. ~(FINITE s ==> n <= CARD s) ==> FINITE s`) THEN EXISTS_TAC `1 + (m + 1) * (2 * n) EXP dimindex(:N)` THEN MATCH_MP_TAC(ONCE_REWRITE_RULE [GSYM CONTRAPOS_THM] CHOOSE_SUBSET_STRONG) THEN REWRITE_TAC[NOT_EXISTS_THM; SUBSET; HAS_SIZE; IN_ELIM_THM] THEN X_GEN_TAC `t:(real^N->bool)->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`\s. s INTER interval[--vec n:real^N,vec n]`; `t:(real^N->bool)->bool`] MEASURE_NEGLIGIBLE_UNIONS_IMAGE) THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_INTERVAL; NOT_IMP] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN MATCH_MP_TAC MONO_IMP THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN SET_TAC[]; MATCH_MP_TAC(REAL_ARITH `!z. x <= z /\ z < y ==> ~(x = y)`) THEN EXISTS_TAC `measure(interval[--vec n:real^N,vec n])` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_IMAGE] THEN SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_INTERVAL]; TRANS_TAC REAL_LET_TRANS `sum (t:(real^N->bool)->bool) (\s. inv(&m + &1))` THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUM_CONST; MEASURE_INTERVAL; GSYM REAL_OF_NUM_MUL; GSYM REAL_OF_NUM_ADD; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `-- &n <= &n /\ x - --x = &2 * x`] THEN SIMP_TAC[REAL_FIELD `(&1 + (&m + &1) * x) * inv(&m + &1) = x + inv(&m + &1)`] THEN MATCH_MP_TAC(REAL_ARITH `x <= y /\ &0 <= z ==> x <= y + z`) THEN REWRITE_TAC[REAL_LE_INV_EQ; REAL_ARITH `&0 <= &m + &1`] THEN REWRITE_TAC[PRODUCT_CONST_NUMSEG; REAL_OF_NUM_MUL; REAL_OF_NUM_POW; REAL_OF_NUM_LE; ADD_SUB; LE_REFL]; MATCH_MP_TAC SUM_LT_ALL THEN ASM_SIMP_TAC[GSYM CARD_EQ_0] THEN REWRITE_TAC[MULT_EQ_0; EXP_EQ_0; ARITH_EQ; ADD_EQ_0]]]]; REWRITE_TAC[SUBSET; UNIONS_IMAGE] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN REWRITE_TAC[EXISTS_PAIR_THM; IN_CROSS; IN_UNIV; IN_ELIM_THM] THEN FIRST_X_ASSUM(MP_TAC o SPEC `s:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [NEGLIGIBLE_ON_COUNTABLE_INTERVALS]) THEN REWRITE_TAC[NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN ASM_SIMP_TAC[GSYM MEASURABLE_MEASURE_POS_LT; MEASURABLE_INTERVAL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ARCH_EVENTUALLY_INV1] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[LE_REFL]]);; (* ------------------------------------------------------------------------- *) (* Various Vitali-type covering lemmas. *) (* ------------------------------------------------------------------------- *) let WIENER_COVERING_LEMMA_BALLS = prove (`!k a:A->real^N r s. FINITE k /\ s SUBSET UNIONS(IMAGE (\i. ball(a i,r i)) k) ==> ?c. c SUBSET k /\ pairwise (\i j. DISJOINT (ball(a i,r i)) (ball(a j,r j))) c /\ s SUBSET UNIONS(IMAGE (\i. ball(a i,&3 * r i)) c)`, REPLICATE_TAC 3 GEN_TAC THEN WF_INDUCT_TAC `CARD(k:A->bool)` THEN X_GEN_TAC `s:real^N->bool` THEN ASM_CASES_TAC `k:A->bool = {}` THEN ASM_REWRITE_TAC[PAIRWISE_EMPTY; SUBSET_EMPTY; IMAGE_CLAUSES; FINITE_EMPTY; UNWIND_THM2] THEN STRIP_TAC THEN MP_TAC(ISPEC `IMAGE (r:A->real) k` SUP_FINITE) THEN ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `n:A` THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_CASES_TAC `(r:A->real) n <= &0` THENL [EXISTS_TAC `{}:A->bool` THEN ASM_REWRITE_TAC[PAIRWISE_EMPTY; EMPTY_SUBSET] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET u ==> u = {} ==> s SUBSET t`)) THEN REWRITE_TAC[EMPTY_UNIONS; FORALL_IN_IMAGE; BALL_EQ_EMPTY] THEN ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `{i | i IN k /\ DISJOINT (ball((a:A->real^N) i,r i)) (ball(a n,r n))}`) THEN ANTS_TAC THENL [MATCH_MP_TAC CARD_PSUBSET THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `i IN s ==> ~P i ==> {j | j IN s /\ P j} PSUBSET s`)) THEN ASM_REWRITE_TAC[GSYM DISJOINT_EMPTY_REFL; BALL_EQ_EMPTY]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `s DIFF ball((a:A->real^N) n,&3 * r n)`) THEN ASM_SIMP_TAC[FINITE_RESTRICT] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET UNIONS(IMAGE h f) ==> g SUBSET f /\ (!i. i IN f DIFF g ==> h i SUBSET b) ==> s DIFF b SUBSET UNIONS(IMAGE h g)`)) THEN REWRITE_TAC[SUBSET_RESTRICT; SET_RULE `i IN k DIFF {x | x IN k /\ P x} <=> i IN k /\ ~P i`] THEN X_GEN_TAC `m:A` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `m:A`) THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[DISJOINT; EXTENSION; SUBSET; NOT_IN_EMPTY; IN_BALL; IN_INTER] THEN CONV_TAC NORM_ARITH; DISCH_THEN(X_CHOOSE_THEN `c:A->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(n:A) INSERT c` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[PAIRWISE_INSERT; IMAGE_CLAUSES; UNIONS_INSERT] THEN ASM SET_TAC[]]);; let WIENER_COVERING_LEMMA_CBALLS = prove (`!k a:A->real^N r s. FINITE k /\ s SUBSET UNIONS(IMAGE (\i. cball(a i,r i)) k) ==> ?c. c SUBSET k /\ pairwise (\i j. DISJOINT (cball(a i,r i)) (cball(a j,r j))) c /\ s SUBSET UNIONS(IMAGE (\i. cball(a i,&3 * r i)) c)`, REPLICATE_TAC 3 GEN_TAC THEN WF_INDUCT_TAC `CARD(k:A->bool)` THEN X_GEN_TAC `s:real^N->bool` THEN ASM_CASES_TAC `k:A->bool = {}` THEN ASM_REWRITE_TAC[PAIRWISE_EMPTY; SUBSET_EMPTY; IMAGE_CLAUSES; FINITE_EMPTY; UNWIND_THM2] THEN STRIP_TAC THEN MP_TAC(ISPEC `IMAGE (r:A->real) k` SUP_FINITE) THEN ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `n:A` THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_CASES_TAC `(r:A->real) n < &0` THENL [EXISTS_TAC `{}:A->bool` THEN ASM_REWRITE_TAC[PAIRWISE_EMPTY; EMPTY_SUBSET] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET u ==> u = {} ==> s SUBSET t`)) THEN REWRITE_TAC[EMPTY_UNIONS; FORALL_IN_IMAGE; CBALL_EQ_EMPTY] THEN ASM_MESON_TAC[REAL_LET_TRANS]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `{i | i IN k /\ DISJOINT (cball((a:A->real^N) i,r i)) (cball(a n,r n))}`) THEN ANTS_TAC THENL [MATCH_MP_TAC CARD_PSUBSET THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `i IN s ==> ~P i ==> {j | j IN s /\ P j} PSUBSET s`)) THEN ASM_REWRITE_TAC[GSYM DISJOINT_EMPTY_REFL; CBALL_EQ_EMPTY]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `s DIFF cball((a:A->real^N) n,&3 * r n)`) THEN ASM_SIMP_TAC[FINITE_RESTRICT] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET UNIONS(IMAGE h f) ==> g SUBSET f /\ (!i. i IN f DIFF g ==> h i SUBSET b) ==> s DIFF b SUBSET UNIONS(IMAGE h g)`)) THEN REWRITE_TAC[SUBSET_RESTRICT; SET_RULE `i IN k DIFF {x | x IN k /\ P x} <=> i IN k /\ ~P i`] THEN X_GEN_TAC `m:A` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `m:A`) THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[DISJOINT; EXTENSION; SUBSET; NOT_IN_EMPTY; IN_CBALL; IN_INTER] THEN CONV_TAC NORM_ARITH; DISCH_THEN(X_CHOOSE_THEN `c:A->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(n:A) INSERT c` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[PAIRWISE_INSERT; IMAGE_CLAUSES; UNIONS_INSERT] THEN ASM SET_TAC[]]);; let VITALI_COVERING_LEMMA_CBALLS_BALLS = prove (`!a:A->real^N r k B. (!i. i IN k ==> &0 < r i /\ r i <= B) ==> ?c. COUNTABLE c /\ c SUBSET k /\ pairwise (\i j. DISJOINT (cball(a i,r i)) (cball(a j,r j))) c /\ !i. i IN k ==> ?j. j IN c /\ ~DISJOINT (cball(a i,r i)) (cball(a j,r j)) /\ cball(a i,r i) SUBSET ball(a j,&5 * r j)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c. (!n. c n SUBSET k /\ (!i:A. i IN c n ==> B / &2 pow n <= r i) /\ pairwise (\i j. DISJOINT (cball(a i,r i)) (cball(a j,r j))) (c n) /\ !i. i IN k /\ B / &2 pow n < r i ==> ?j. j IN c n /\ ~DISJOINT (cball(a i:real^N,r i)) (cball(a j,r j)) /\ cball(a i,r i) SUBSET ball(a j,&5 * r j)) /\ (!n. c n SUBSET c(SUC n))` MP_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONJ_TAC THENL [EXISTS_TAC `{}:A->bool` THEN REWRITE_TAC[EMPTY_SUBSET] THEN REWRITE_TAC[PAIRWISE_EMPTY; NOT_IN_EMPTY; real_pow; REAL_DIV_1] THEN ASM_MESON_TAC[REAL_LET_ANTISYM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `c:A->bool`] THEN STRIP_TAC THEN ABBREV_TAC `d = {i | i IN k /\ B / &2 pow SUC n < r i /\ cball((a:A->real^N) i,r i) INTER UNIONS (IMAGE (\j. cball(a j,r j)) c) = {}}` THEN MP_TAC(ISPEC `\c. c SUBSET d /\ pairwise (\i j. DISJOINT (cball((a:A->real^N) i,r i)) (cball(a j,r j))) c` ZL_SUBSETS_UNIONS) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [X_GEN_TAC `f:(A->bool)->bool` THEN SIMP_TAC[UNIONS_SUBSET; pairwise] THEN SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `e:A->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `c UNION e:A->bool` THEN ASM_REWRITE_TAC[SUBSET_UNION; PAIRWISE_UNION] THEN REWRITE_TAC[FORALL_IN_UNION] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `i:A` THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `B / &2 pow n` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[real_div] THEN MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [SUBGOAL_THEN `&0 < r(i:A) /\ r i <= B` MP_TAC THENL [ASM SET_TAC[]; REAL_ARITH_TAC]; REWRITE_TAC[REAL_INV_POW] THEN MATCH_MP_TAC REAL_POW_MONO_INV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ARITH_TAC]; REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LT_IMP_LE THEN ASM SET_TAC[]; ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `i:A` THEN STRIP_TAC THEN ASM_CASES_TAC `B / &2 pow n < r(i:A)` THENL [ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT])] THEN MATCH_MP_TAC(MESON[] `(!j. P j ==> Q j ==> R j) /\ (?j. P j /\ Q j) ==> (?j. P j /\ Q j /\ R j)`) THEN CONJ_TAC THENL [X_GEN_TAC `j:A` THEN REWRITE_TAC[IN_UNION] THEN STRIP_TAC THEN REWRITE_TAC[DISJOINT; GSYM MEMBER_NOT_EMPTY; IN_INTER; SUBSET; IN_BALL; IN_CBALL] THEN MATCH_MP_TAC(NORM_ARITH `r < &2 * s ==> (?x. dist(i:real^N,x) <= r /\ dist(j,x) <= s) ==> !z. dist(i,z) <= r ==> dist(j,z) < &5 * s`) THEN TRANS_TAC REAL_LET_TRANS `B / &2 pow n` THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC(REAL_ARITH `&0 < j /\ a <= j ==> a < &2 * j`) THEN ASM SET_TAC[]; REWRITE_TAC[REAL_ARITH `a / b < &2 * c <=> a / &2 / b < c`] THEN REWRITE_TAC[real_div; GSYM REAL_MUL_ASSOC; GSYM REAL_INV_MUL] THEN REWRITE_TAC[GSYM real_div; GSYM(CONJUNCT2 real_pow)] THEN ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `(i:A) IN d` THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o SPEC `(i:A) INSERT e`) THEN ASM_REWRITE_TAC[INSERT_SUBSET; SET_RULE `s SUBSET x INSERT s`] THEN ASM_REWRITE_TAC[PAIRWISE_INSERT; SET_RULE `s = x INSERT s <=> x IN s`] THEN ASM_CASES_TAC `(i:A) IN e` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[UNIONS_UNION; IMAGE_UNION] THEN SUBGOAL_THEN `~(cball((a:A->real^N) i,r i) = {})` MP_TAC THENL [ASM_SIMP_TAC[CBALL_EQ_EMPTY; REAL_NOT_LT]; ASM SET_TAC[]] THEN MATCH_MP_TAC REAL_LT_IMP_LE THEN ASM SET_TAC[]; REWRITE_TAC[FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:num->A->bool` THEN STRIP_TAC THEN EXISTS_TAC `UNIONS (IMAGE (c:num->A->bool) (:num))` THEN MATCH_MP_TAC(TAUT `q /\ (q /\ r ==> p) /\ r /\ s ==> p /\ q /\ r /\ s`) THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; SPEC_TAC(`UNIONS (IMAGE (c:num->A->bool) (:num))`,`q:A->bool`) THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\i. cball((a:A->real^N) i,r i)`; `q:A->bool`] COUNTABLE_IMAGE_INJ_EQ) THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`i:A`; `j:A`] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `~(s = {}) /\ DISJOINT s t ==> ~(s = t)`) THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM_SIMP_TAC[CBALL_EQ_EMPTY] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(x < &0)`) THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC COUNTABLE_DISJOINT_NONEMPTY_INTERIOR_SUBSETS THEN SIMP_TAC[IMP_CONJ; FORALL_IN_IMAGE; INTERIOR_CBALL; BALL_EQ_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN REWRITE_TAC[GSYM REAL_NOT_LT; pairwise] THEN ASM SET_TAC[]; MATCH_MP_TAC PAIRWISE_CHAIN_UNIONS THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_IMAGE_2; IN_UNIV] THEN MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(MESON[] `(!x y. P x y ==> Q x y) ==> (!x y. P x y ==> Q x y \/ R x y)`) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM SET_TAC[]; X_GEN_TAC `i:A` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `(r:A->real) i / B`] REAL_ARCH_POW_INV) THEN SUBGOAL_THEN `&0 < (r:A->real) i /\ r i <= B` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < B` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_MUL_LZERO] THEN REWRITE_TAC[GSYM REAL_INV_POW] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[UNIONS_IMAGE; IN_UNIV; IN_ELIM_THM] THEN ASM SET_TAC[]]]);; let VITALI_COVERING_LEMMA_CBALLS = prove (`!s a:A->real^N r k B. s SUBSET UNIONS (IMAGE (\i. cball(a i,r i)) k) /\ (!i. i IN k ==> &0 < r i /\ r i <= B) ==> ?c. COUNTABLE c /\ c SUBSET k /\ pairwise (\i j. DISJOINT (cball(a i,r i)) (cball(a j,r j))) c /\ s SUBSET UNIONS (IMAGE (\i. cball(a i,&5 * r i)) c)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPEC `a:A->real^N` o MATCH_MP VITALI_COVERING_LEMMA_CBALLS_BALLS) THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN TRANS_TAC SUBSET_TRANS `UNIONS(IMAGE(\i. ball((a:A->real^N) i,&5 * r i)) c)` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC UNIONS_MONO_IMAGE THEN REWRITE_TAC[BALL_SUBSET_CBALL]);; let VITALI_COVERING_LEMMA_BALLS = prove (`!s a:A->real^N r k B. s SUBSET UNIONS (IMAGE (\i. ball(a i,r i)) k) /\ (!i. i IN k ==> &0 < r i /\ r i <= B) ==> ?c. COUNTABLE c /\ c SUBSET k /\ pairwise (\i j. DISJOINT (ball(a i,r i)) (ball(a j,r j))) c /\ s SUBSET UNIONS (IMAGE (\i. ball(a i,&5 * r i)) c)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPEC `a:A->real^N` o MATCH_MP VITALI_COVERING_LEMMA_CBALLS_BALLS) THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_IMP)) THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN ASM_MESON_TAC[BALL_SUBSET_CBALL; SET_RULE `s SUBSET s' /\ t SUBSET t' /\ DISJOINT s' t' ==> DISJOINT s t`]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN TRANS_TAC SUBSET_TRANS `UNIONS(IMAGE(\i. cball((a:A->real^N) i,r i)) k)` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC UNIONS_MONO_IMAGE THEN REWRITE_TAC[BALL_SUBSET_CBALL]]);; let VITALI_COVERING_THEOREM_CBALLS = prove (`!k a:A->real^N r s. (!i. i IN k ==> &0 < r i) /\ (!x d. x IN s /\ &0 < d ==> ?i. i IN k /\ x IN cball(a i,r i) /\ r i < d) ==> ?c. COUNTABLE c /\ c SUBSET k /\ pairwise (\i j. DISJOINT (cball(a i,r i)) (cball(a j,r j))) c /\ negligible(s DIFF UNIONS {cball(a i,r i) | i IN c})`, SUBGOAL_THEN `!k a:A->real^N r s. (!i. i IN k ==> &0 < r i /\ r i <= &1) /\ (!x d. x IN s /\ &0 < d ==> ?i. i IN k /\ x IN cball(a i,r i) /\ r i < d) ==> ?c. COUNTABLE c /\ c SUBSET k /\ pairwise (\i j. DISJOINT (cball(a i,r i)) (cball(a j,r j))) c /\ negligible(s DIFF UNIONS {cball(a i,r i) | i IN c})` MP_TAC THENL [ALL_TAC; REPEAT STRIP_TAC THEN ABBREV_TAC `k' = {i | i IN k /\ (r:A->real) i <= &1}` THEN FIRST_X_ASSUM(MP_TAC o SPECL [`k':A->bool`; `a:A->real^N`; `r:A->real`; `s:real^N->bool`]) THEN ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_ARITH `&0 < d ==> &0 < min d (&1) /\ (x < min d (&1) ==> x < d /\ x <= &1)`]] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`a:A->real^N`; `r:A->real`; `k:A->bool`; `&1`] VITALI_COVERING_LEMMA_CBALLS_BALLS) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:A->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN MAP_EVERY X_GEN_TAC [`l:real^N`; `u:real^N`] THEN ONCE_REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN ABBREV_TAC `d = {i | i IN c /\ ~DISJOINT (ball((a:A->real^N) i,&5 * r i)) (interval[l,u])}` THEN SUBGOAL_THEN `COUNTABLE(d:A->bool)` ASSUME_TAC THENL [EXPAND_TAC "d" THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COUNTABLE_SUBSET)) THEN REWRITE_TAC[SUBSET_RESTRICT]; ALL_TAC] THEN SUBGOAL_THEN `measurable(UNIONS(IMAGE (\i. cball((a:A->real^N) i,r i)) d))` ASSUME_TAC THENL [MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `interval[l - vec 6:real^N,u + vec 6]` THEN REWRITE_TAC[MEASURABLE_INTERVAL] THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COUNTABLE_UNIONS; FORALL_IN_IMAGE; LEBESGUE_MEASURABLE_CBALL; COUNTABLE_IMAGE] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN EXPAND_TAC "d" THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `i:A` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN SIMP_TAC[SUBSET; SET_RULE `~DISJOINT s t <=> ?x. x IN s /\ x IN t`] THEN REWRITE_TAC[IN_CBALL; IN_BALL; IN_INTERVAL] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[VEC_COMPONENT] THEN MATCH_MP_TAC(REAL_ARITH `abs(y - x) <= &6 * &1 ==> l <= x /\ x <= u ==> l - &6 <= y /\ y <= u + &6`) THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&6 * (r:A->real) i` THEN CONJ_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [dist])) THEN CONV_TAC NORM_ARITH; MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_POS] THEN ASM SET_TAC[]]; ALL_TAC] THEN MP_TAC(ISPECL [`IMAGE (\i. cball((a:A->real^N) i,r i)) d`; `measure(UNIONS(IMAGE (\i. cball((a:A->real^N) i,r i)) d))`; `e / &5 pow (dimindex(:N))`] MEASURE_COUNTABLE_UNIONS_APPROACHABLE) THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE; MEASURABLE_CBALL] THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_POW_LT; REAL_ARITH `&0 < &5`; CONJ_ASSOC] THEN ONCE_REWRITE_TAC[TAUT `p /\ FINITE s <=> FINITE s /\ p`] THEN REWRITE_TAC[GSYM CONJ_ASSOC; FORALL_FINITE_SUBSET_IMAGE] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[MEASURABLE_UNIONS; FORALL_IN_IMAGE; MEASURABLE_CBALL; FINITE_IMAGE] THEN ASM SET_TAC[]; REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE]] THEN DISCH_THEN(X_CHOOSE_THEN `d1:A->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `UNIONS(IMAGE (\i. ball((a:A->real^N) i,&5 * r i)) (d DIFF d1))` THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; IN_DIFF; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MP_TAC(ISPEC `(:real^N) DIFF UNIONS (IMAGE (\i:A. cball(a i,r i)) d1)` OPEN_CONTAINS_BALL) THEN ASM_SIMP_TAC[GSYM closed; CLOSED_UNIONS; CLOSED_CBALL; FINITE_IMAGE; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> DISJOINT s t`] THEN DISCH_THEN(X_CHOOSE_THEN `q:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `q / &2`]) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `i:A` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:A`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `j:A` STRIP_ASSUME_TAC) THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN EXISTS_TAC `j:A` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[IN_DIFF] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `DISJOINT s (UNIONS (IMAGE f k)) ==> (?x. x IN s /\ x IN f i) ==> ~(i IN k)`)) THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `~DISJOINT s t ==> ?x. x IN s /\ x IN t`)) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_CBALL]) THEN REWRITE_TAC[IN_CBALL; IN_BALL] THEN UNDISCH_TAC `(r:A->real) i < q / &2` THEN CONV_TAC NORM_ARITH; MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_GEN THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_IMAGE THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] COUNTABLE_SUBSET)) THEN SET_TAC[]; REWRITE_TAC[FORALL_IN_IMAGE; MEASURABLE_BALL]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `d2:A->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_IMAGE_LE o lhand o snd) THEN ASM_SIMP_TAC[MEASURABLE_BALL; MEASURE_POS_LE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[o_DEF] THEN TRANS_TAC REAL_LE_TRANS `sum d2 (\i. &5 pow dimindex(:N) * measure(ball((a:A->real^N) i,r i)))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC MEASURE_BALL_SCALING THEN CONV_TAC REAL_RAT_REDUCE_CONV; REWRITE_TAC[SUM_LMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM]] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_POW_LT; REAL_ARITH `&0 < &5`] THEN REWRITE_TAC[GSYM INTERIOR_CBALL] THEN SIMP_TAC[MEASURE_INTERIOR; BOUNDED_CBALL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_CBALL] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o lhand o snd) THEN ASM_REWRITE_TAC[MEASURABLE_CBALL] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `n - e < m ==> m + p <= n ==> p <= e`)) THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNION o lhand o snd) THEN ASM_SIMP_TAC[MEASURABLE_UNIONS; FORALL_IN_IMAGE; MEASURABLE_CBALL; FINITE_IMAGE] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNION; MEASURABLE_UNIONS; FORALL_IN_IMAGE; MEASURABLE_CBALL; FINITE_IMAGE] THEN ASM SET_TAC[]]);; let VITALI_COVERING_THEOREM_BALLS = prove (`!k a:A->real^N r s. (!x d. x IN s /\ &0 < d ==> ?i. i IN k /\ x IN ball(a i,r i) /\ r i < d) ==> ?c. COUNTABLE c /\ c SUBSET k /\ pairwise (\i j. DISJOINT (ball(a i,r i)) (ball(a j,r j))) c /\ negligible(s DIFF UNIONS {ball(a i,r i) | i IN c})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`{i | i IN k /\ &0 < (r:A->real) i}`; `a:A->real^N`; `r:A->real`; `s:real^N->bool`] VITALI_COVERING_THEOREM_CBALLS) THEN SIMP_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^N`; `d:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `d:real`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `i:A` THEN ASM_CASES_TAC `(r:A->real) i <= &0` THEN ASM_SIMP_TAC[BALL_EMPTY; NOT_IN_EMPTY] THEN SIMP_TAC[REWRITE_RULE[SUBSET] BALL_SUBSET_CBALL] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:A->bool` THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_IMP)) THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN ASM_MESON_TAC[BALL_SUBSET_CBALL; SET_RULE `s SUBSET s' /\ t SUBSET t' /\ DISJOINT s' t' ==> DISJOINT s t`]; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `s DIFF UNIONS {cball((a:A->real^N) i,r i) | i IN c} UNION UNIONS {sphere(a i,r i) | i IN c}` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN REWRITE_TAC[GSYM BALL_UNION_SPHERE; SIMPLE_IMAGE] THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE; NEGLIGIBLE_SPHERE]; REWRITE_TAC[UNIONS_IMAGE] THEN SET_TAC[]]]]);; (* ------------------------------------------------------------------------- *) (* Negligibility is a local property (we actually use the topological *) (* notion, which looks iconoclastic but is perfectly sensible). More *) (* interestingly, it is equivalent, to localized zero density. *) (* ------------------------------------------------------------------------- *) let LOCALLY_NEGLIGIBLE_ALT = prove (`!s:real^N->bool. negligible s <=> !x. x IN s ==> ?u. open_in (subtopology euclidean s) u /\ x IN u /\ negligible u`, GEN_TAC THEN EQ_TAC THENL [MESON_TAC[OPEN_IN_REFL]; ALL_TAC] THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^N->real^N->bool` THEN DISCH_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE (u:real^N->real^N->bool) s)` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MP_TAC(ISPECL[`IMAGE (u:real^N->real^N->bool) s`; `s:real^N->bool`] LINDELOF_OPEN_IN) THEN ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_COUNTABLE_SUBSET_IMAGE] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ASM SET_TAC[]);; let LOCALLY_NEGLIGIBLE = prove (`!s:real^N->bool. locally negligible s <=> negligible s`, GEN_TAC THEN REWRITE_TAC[locally] THEN EQ_TAC THENL [GEN_REWRITE_TAC RAND_CONV [LOCALLY_NEGLIGIBLE_ALT] THEN MESON_TAC[OPEN_IN_REFL; SUBSET_REFL; NEGLIGIBLE_SUBSET]; DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `x:real^N`] THEN STRIP_TAC THEN REPEAT(EXISTS_TAC `w INTER s:real^N->bool`) THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL; INTER_SUBSET; SUBSET_REFL] THEN ASM_MESON_TAC[SUBSET; IN_INTER; OPEN_IN_IMP_SUBSET; NEGLIGIBLE_SUBSET]]);; let LOCALLY_LEBESGUE_MEASURABLE_ALT = prove (`!s:real^N->bool. lebesgue_measurable s <=> !x. x IN s ==> ?u. open_in (subtopology euclidean s) u /\ x IN u /\ lebesgue_measurable u`, GEN_TAC THEN EQ_TAC THENL [MESON_TAC[OPEN_IN_REFL]; ALL_TAC] THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^N->real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `s = UNIONS (IMAGE (u:real^N->real^N->bool) s)` SUBST1_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[open_in]) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL[`IMAGE (u:real^N->real^N->bool) s`; `s:real^N->bool`] LINDELOF_OPEN_IN) THEN ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_COUNTABLE_SUBSET_IMAGE] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ASM SET_TAC[]);; let LOCALLY_LEBESGUE_MEASURABLE = prove (`!s:real^N->bool. locally lebesgue_measurable s <=> lebesgue_measurable s`, GEN_TAC THEN REWRITE_TAC[locally] THEN EQ_TAC THENL [DISCH_TAC THEN ONCE_REWRITE_TAC[LOCALLY_LEBESGUE_MEASURABLE_ALT] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s:real^N->bool`; `x:real^N`]) THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN ASM_MESON_TAC[LEBESGUE_MEASURABLE_OPEN_IN; OPEN_IN_SUBSET_TRANS]; DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`w:real^N->bool`; `x:real^N`] THEN STRIP_TAC THEN REPEAT(EXISTS_TAC `w INTER s:real^N->bool`) THEN ASM_SIMP_TAC[OPEN_IN_INTER; OPEN_IN_REFL; INTER_SUBSET; SUBSET_REFL] THEN ASM_MESON_TAC[SUBSET; IN_INTER; OPEN_IN_IMP_SUBSET; LEBESGUE_MEASURABLE_OPEN_IN; LEBESGUE_MEASURABLE_INTER]]);; let NEGLIGIBLE_EQ_ZERO_DENSITY_ALT = prove (`!s:real^N->bool. negligible s <=> !x e. x IN s /\ &0 < e ==> ?d u. &0 < d /\ d <= e /\ s INTER ball(x,d) SUBSET u /\ measurable u /\ measure u < e * measure(ball(x,d))`, GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [REPEAT STRIP_TAC THEN EXISTS_TAC `e:real` THEN EXISTS_TAC `s INTER ball(x:real^N,e)` THEN ASM_SIMP_TAC[SUBSET_REFL; MEASURABLE_INTER; MEASURABLE_BALL; NEGLIGIBLE_IMP_MEASURABLE; REAL_LE_REFL] THEN MATCH_MP_TAC(REAL_ARITH `x = &0 /\ &0 < y ==> x < y`) THEN ASM_SIMP_TAC[MEASURE_BALL_POS; REAL_LT_MUL] THEN ASM_MESON_TAC[MEASURE_EQ_0; NEGLIGIBLE_SUBSET; INTER_SUBSET]; ALL_TAC] THEN ONCE_REWRITE_TAC[LOCALLY_NEGLIGIBLE_ALT] THEN X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN EXISTS_TAC `s INTER ball(z:real^N,&1)` THEN ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; IN_INTER] THEN REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_01] THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`{(x:real^N,d) | x IN s /\ &0 < d /\ ball(x,d) SUBSET ball(z,&1) /\ ?u. s INTER ball(x,d) SUBSET u /\ measurable u /\ measure u < e / measure(ball(z:real^N,&1)) * measure(ball(x,d))}`; `FST:real^N#real->real^N`; `SND:real^N#real->real`; `s INTER ball(z:real^N,&1)`] VITALI_COVERING_THEOREM_BALLS) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^N`; `d:real`] THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN EXISTS_TAC `x:real^N` THEN MP_TAC(ISPEC `ball(z:real^N,&1)` OPEN_CONTAINS_BALL) THEN REWRITE_TAC[OPEN_BALL] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `k:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `min (e / measure(ball(z:real^N,&1)) / &2) (min (d / &2) k)`]) THEN ASM_SIMP_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_DIV; MEASURE_BALL_POS; REAL_LT_01] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real` THEN REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN REPEAT CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `ball(x:real^N,k)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_BALL THEN ASM_REAL_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_LTE_TRANS)) THEN MATCH_MP_TAC REAL_LE_RMUL THEN SIMP_TAC[MEASURE_POS_LE; MEASURABLE_BALL] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> min (x / &2) y <= x`) THEN ASM_SIMP_TAC[REAL_LT_DIV; MEASURE_BALL_POS; REAL_LT_01]; ASM_REAL_ARITH_TAC]; ALL_TAC] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N#real->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!p. p IN c ==> ?u. s INTER ball p SUBSET u /\ measurable(u:real^N->bool) /\ measure u < e / measure (ball(z:real^N,&1)) * measure(ball p)` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`)) THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MESON_TAC[]; GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM]] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^N#real->real^N->bool` THEN DISCH_TAC THEN EXISTS_TAC `((s INTER ball(z:real^N,&1)) DIFF UNIONS (IMAGE ball c)) UNION UNIONS(IMAGE u c)` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `measurable(UNIONS(IMAGE (u:real^N#real->real^N->bool) c)) /\ measure(UNIONS(IMAGE u c)) <= e` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] MEASURABLE_NEGLIGIBLE_SYMDIFF); MATCH_MP_TAC(REAL_ARITH `x = y ==> x <= e ==> y <= e`) THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `d:real^N#real->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN TRANS_TAC REAL_LE_TRANS `sum d (\p:real^N#real. e / measure(ball(z:real^N,&1)) * measure(ball p))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LT_IMP_LE THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUM_LMUL] THEN ONCE_REWRITE_TAC[REAL_ARITH `a / b * c:real = (a * c) / b`] THEN SIMP_TAC[REAL_LE_LDIV_EQ; MEASURE_BALL_POS; REAL_LT_01] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN TRANS_TAC REAL_LE_TRANS `measure(UNIONS (IMAGE ball d):real^N->bool)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_DISJOINT_UNIONS_IMAGE THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_PAIR_THM; MEASURABLE_BALL]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_BALL; UNIONS_SUBSET; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM; MEASURABLE_BALL; FINITE_IMAGE]]);; let NEGLIGIBLE_EQ_ZERO_DENSITY = prove (`!s:real^N->bool. negligible s <=> !x r e. x IN s /\ &0 < e /\ &0 < r ==> ?d u. &0 < d /\ d <= r /\ s INTER ball(x,d) SUBSET u /\ measurable u /\ measure u < e * measure(ball(x,d))`, GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [NEGLIGIBLE_EQ_ZERO_DENSITY_ALT] THEN EQ_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THENL [MAP_EVERY X_GEN_TAC [`r:real`; `e:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `min r e:real`) THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LE_MIN] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_LTE_TRANS)) THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ; MEASURE_BALL_POS] THEN REAL_ARITH_TAC; X_GEN_TAC `e:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`e:real`; `e:real`]) THEN ASM_MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* A handy but by no means optimal measurability lemma. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_MEASURABLE_POINTS_OF_CONVERGENCE = prove (`!f:real^M->real^N->real^P g s. lebesgue_measurable s /\ (!y. (\x. f x y) continuous_on s) /\ g continuous_on s ==> lebesgue_measurable {x | x IN s /\ ?l. (f x --> l) (at (g x))}`, let lemma = prove (`x IN s /\ x IN INTERS f <=> x IN s /\ x IN INTERS {s INTER t | t IN f}`, REWRITE_TAC[INTERS_GSPEC] THEN SET_TAC[]) in REPEAT STRIP_TAC THEN REWRITE_TAC[CONVERGENT_EQ_CAUCHY_AT] THEN MATCH_MP_TAC(MESON[] `!t. s = t /\ lebesgue_measurable t ==> lebesgue_measurable s`) THEN EXISTS_TAC `INTERS { UNIONS { {z | (z:real^M) IN s /\ z IN INTERS { {z | &0 < dist (x,g z) /\ dist(x,g z) < inv(&n + &1) /\ &0 < dist (y,g z) /\ dist(y,g z) < inv(&n + &1) ==> dist((f:real^M->real^N->real^P) z x,f z y) <= inv(&m + &1) } | x IN UNIV /\ y IN UNIV}} | n IN (:num) } | m IN (:num) }` THEN CONJ_TAC THENL [REWRITE_TAC[UNIONS_GSPEC; INTERS_GSPEC; IN_UNIV] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `z:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_CASES_TAC `(z:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN W(MP_TAC o PART_MATCH (lhand o rand) FORALL_POS_MONO_1_EQ o lhand o snd) THEN ANTS_TAC THENL [MESON_TAC[REAL_LT_TRANS]; DISCH_THEN SUBST1_TAC] THEN MATCH_MP_TAC(MESON[] `(!n. P n ==> Q n) /\ (!n. Q(n + 1) ==> P n) ==> ((!n. P n) <=> (!n. Q n))`) THEN CONJ_TAC THEN X_GEN_TAC `m:num` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THENL [W(MP_TAC o PART_MATCH (lhand o rand) FORALL_POS_MONO_1_EQ o rand o snd); W(MP_TAC o PART_MATCH (lhand o rand) FORALL_POS_MONO_1_EQ o lhand o snd)] THEN (ANTS_TAC THENL [MESON_TAC[REAL_LT_TRANS]; DISCH_THEN SUBST1_TAC]) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[CONTRAPOS_THM] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN MATCH_MP_TAC MONO_IMP THEN SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN MATCH_MP_TAC REAL_LT_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LT] THEN ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; SIMPLE_IMAGE] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN X_GEN_TAC `m:num` THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; NUM_COUNTABLE; SIMPLE_IMAGE] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_CLOSED_IN THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[lemma] THEN ONCE_REWRITE_TAC[GSYM INTER] THEN MATCH_MP_TAC(SET_RULE `(f = {} ==> closed_in top (s INTER INTERS f)) /\ (~(f = {}) ==> closed_in top (s INTER INTERS f)) ==> closed_in top (s INTER INTERS f)`) THEN SIMP_TAC[INTERS_0; INTER_UNIV; CLOSED_IN_REFL] THEN DISCH_TAC THEN MATCH_MP_TAC CLOSED_IN_INTER THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_INTERS THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[SET_RULE `s INTER {z | P z ==> Q z} = s DIFF {z | P z} UNION s INTER {z | Q z}`] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN CONJ_TAC THENL [REWRITE_TAC[GSYM IN_BALL; GSYM DIST_NZ] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN REWRITE_TAC[SET_RULE `~(a = z) /\ z IN s <=> z IN s DELETE a`] THEN REWRITE_TAC[SET_RULE `s DIFF {x | P x /\ Q x} = {x | x IN s /\ ~P x} UNION {x | x IN s /\ ~Q x}`] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN CONJ_TAC THEN REWRITE_TAC[SET_RULE `~(x IN s DELETE a) <=> x IN a INSERT (UNIV DIFF s)`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_INSERT THEN REWRITE_TAC[GSYM OPEN_CLOSED; OPEN_BALL]; REWRITE_TAC[dist; SET_RULE `s INTER {x | P x} = {x | x IN s /\ P x}`] THEN REWRITE_TAC[GSYM IN_CBALL_0] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN ASM_SIMP_TAC[CLOSED_CBALL; CONTINUOUS_ON_SUB]]);; (* ------------------------------------------------------------------------- *) (* Measurability of the set of points of differentiability and of the *) (* partial derivatives and vector derivatives. These proofs are both *) (* similar and have some technicalities to handle the extra generality *) (* of "within s" (we show that the set of points where this makes the *) (* limit ill-defined is negligible). In the unrestricted case, the same *) (* idea works and shows the set of points of differentiability is Borel. *) (* ------------------------------------------------------------------------- *) let BOREL_POINTS_OF_DIFFERENTIABILITY = prove (`!f:real^M->real^N. borel {x | f differentiable at x}`, let lemur = prove (`!f:real^M->real^N x. f differentiable (at x) <=> !e. &0 < e ==> ?d A. &0 < d /\ (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> rational(A$i$j)) /\ !y. norm(y - x) < d ==> norm(f y - f x - A ** (y - x)) <= e * norm(y - x)`, REPEAT GEN_TAC THEN REWRITE_TAC[differentiable; HAS_DERIVATIVE_AT_ALT] THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_THEN `f':real^M->real^N` STRIP_ASSUME_TAC) THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC (ISPEC `matrix(f':real^M->real^N)` MATRIX_RATIONAL_APPROXIMATION) THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `B:real^M^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_ARITH `x / &2 * y = (x * y) / &2`] THEN MATCH_MP_TAC(NORM_ARITH `norm(d' - d:real^N) <= e / &2 ==> norm(y - x - d') <= e / &2 ==> norm(y - x - d) <= e`) THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN TRANS_TAC REAL_LE_TRANS `onorm(\x. (matrix(f':real^M->real^N) - B) ** x) * norm(y - x:real^M)` THEN ASM_SIMP_TAC[ONORM; LINEAR_COMPOSE_SUB; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[REAL_ARITH `(e * x) / &2 = e / &2 * x`] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN MAP_EVERY X_GEN_TAC [`d:num->real`; `A:num->real^M^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> ?a. ((\n. lift((A n:real^M^N)$i$j)) --> a) sequentially` MP_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[CONVERGENT_EQ_CAUCHY; cauchy] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e / &2` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN REWRITE_TAC[GE] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN ABBREV_TAC `y:real^M = x + min ((d:num->real) m) (d n) / &2 % basis j` THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPECL [`m:num`; `y:real^M`] th) THEN MP_TAC(SPECL [`n:num`; `y:real^M`] th)) THEN MATCH_MP_TAC(TAUT `(q /\ p) /\ (r /\ s ==> t) ==> (p ==> r) ==> (q ==> s) ==> t`) THEN CONJ_TAC THENL [REWRITE_TAC[GSYM REAL_LT_MIN] THEN EXPAND_TAC "y" THEN REWRITE_TAC[VECTOR_ADD_SUB; NORM_MUL] THEN ASM_SIMP_TAC[NORM_BASIS; REAL_MUL_RID] THEN MATCH_MP_TAC(REAL_ARITH `&0 < d ==> abs(d / &2) < d`) THEN ASM_REWRITE_TAC[REAL_LT_MIN]; DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH `norm(y - x - d) <= a /\ norm(y - x - e) <= b ==> norm(d - e:real^N) <= a + b`)) THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN EXPAND_TAC "y" THEN REWRITE_TAC[VECTOR_ADD_SUB] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_RMUL; NORM_MUL] THEN ASM_SIMP_TAC[NORM_BASIS; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `a * x <= n * a + m * a <=> a * x <= a * (n + m)`] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_ARITH `&0 < x ==> &0 < abs x`; REAL_HALF; REAL_LT_MIN] THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_BASIS; column] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[COMPONENT_LE_NORM; REAL_LE_TRANS] `norm(x:real^N) <= a ==> !i. abs(x$i) <= a`)) THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_SIMP_TAC[DIST_LIFT; LAMBDA_BETA; MATRIX_SUB_COMPONENT] THEN MATCH_MP_TAC(REAL_ARITH `x < e / &2 /\ y < e / &2 ==> abs(a - b) <= x + y ==> abs(b - a) < e`) THEN CONJ_TAC THEN TRANS_TAC REAL_LET_TRANS `inv(&N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]; ALL_TAC] THEN REWRITE_TAC[EXISTS_LIFT] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; LAMBDA_SKOLEM] THEN DISCH_THEN(X_CHOOSE_THEN `B:real^M^N` (LABEL_TAC "*")) THEN EXISTS_TAC `\x. (B:real^M^N) ** x` THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o REWRITE_RULE[tendsto]) THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!i. P i ==> !j. Q j ==> !e. R i j e) ==> !e i. P i ==> !j. Q j ==> R i j e`)) THEN DISCH_THEN(MP_TAC o SPEC `e / &2 / &(dimindex(:N)) / &(dimindex(:M))`) THEN REWRITE_TAC[GSYM IN_NUMSEG] THEN ASM_SIMP_TAC[CONV_RULE (RAND_CONV SYM_CONV) (SPEC_ALL EVENTUALLY_FORALL); FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1; REAL_LT_DIV; LE_1; REAL_HALF; REAL_OF_NUM_LT] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; LEFT_IMP_EXISTS_THM] THEN MP_TAC(SPEC `e / &2` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_THEN(MP_TAC o SPEC `n + m:num`) THEN REWRITE_TAC[LE_ADD; RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN REWRITE_TAC[IN_NUMSEG] THEN DISCH_TAC THEN EXISTS_TAC `(d:num->real) (n + m)` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n + m:num`; `y:real^M`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `norm(d' - d:real^N) <= e' - e ==> norm(y - x - d') <= e ==> norm(y - x - d) <= e'`) THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN TRANS_TAC REAL_LE_TRANS `onorm(\x. ((A:num->real^M^N) (n + m) - B) ** x) * norm(y - x:real^M)` THEN SIMP_TAC[ONORM; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[GSYM REAL_SUB_RDISTRIB] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `d <= e / &2 /\ x <= e / &2 ==> x <= e - d`) THEN CONJ_TAC THENL [TRANS_TAC REAL_LE_TRANS `inv(&m)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC; TRANS_TAC REAL_LE_TRANS `&(dimindex(:N)) * &(dimindex(:M)) * e / &2 / &(dimindex(:N)) / &(dimindex(:M))` THEN CONJ_TAC THENL [MATCH_MP_TAC ONORM_LE_MATRIX_COMPONENT THEN RULE_ASSUM_TAC(REWRITE_RULE[DIST_LIFT]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; MATRIX_SUB_COMPONENT]; SIMP_TAC[REAL_DIV_LMUL; REAL_OF_NUM_EQ; DIMINDEX_NONZERO] THEN ASM_REAL_ARITH_TAC]]) in GEN_TAC THEN SUBGOAL_THEN `{x | (f:real^M->real^N) differentiable at x} = {x | f continuous at x} INTER INTERS { UNIONS { UNIONS { {x | f continuous at x} INTER INTERS {{x | f continuous at x /\ (norm(y - x) < d ==> norm(f y - f x - A ** (y - x)) <= e * norm(y - x))} |y| y IN UNIV} |d| d IN {d | d IN rational /\ &0 < d}} |A| A IN {A | !i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> rational (A$i$j)}} |e| e IN {e | e IN rational /\ &0 < e}}` SUBST1_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. P x ==> Q x) /\ (!x. Q x ==> (P x <=> x IN s)) ==> {x | P x} = {x | Q x} INTER s`) THEN REWRITE_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_AT] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[lemur] THEN REWRITE_TAC[UNIONS_GSPEC; INTERS_GSPEC; IN_ELIM_THM; IN_INTER] THEN REWRITE_TAC[SET_RULE `x IN rational <=> rational x`] THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP RATIONAL_BETWEEN) THEN DISCH_THEN(X_CHOOSE_THEN `e':real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `e':real`) THEN ASM_REWRITE_TAC[IN_UNIV] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `A:real^M^N` THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP RATIONAL_BETWEEN) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d':real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN (ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN MATCH_MP_TAC BOREL_INTER THEN SIMP_TAC[GDELTA_IMP_BOREL; GDELTA_POINTS_OF_CONTINUITY] THEN MATCH_MP_TAC BOREL_INTERS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_RATIONAL]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `e:real` THEN STRIP_TAC] THEN MATCH_MP_TAC BOREL_UNIONS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC COUNTABLE_CART THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COUNTABLE_CART THEN REWRITE_TAC[COUNTABLE_RATIONAL; SET_RULE `{x | s x} = s`]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `A:real^M^N` THEN STRIP_TAC] THEN MATCH_MP_TAC BOREL_UNIONS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_RATIONAL]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `d:real` THEN STRIP_TAC] THEN MATCH_MP_TAC CLOSED_IN_BOREL THEN EXISTS_TAC `{x | (f:real^M->real^N) continuous at x}` THEN SIMP_TAC[GDELTA_IMP_BOREL; GDELTA_POINTS_OF_CONTINUITY] THEN MATCH_MP_TAC(MESON[SET_RULE `s INTER INTERS {} = s`; CLOSED_IN_REFL] `(~(u = {}) ==> closed_in (subtopology euclidean s) (s INTER INTERS u)) ==> closed_in (subtopology euclidean s) (s INTER INTERS u)`) THEN DISCH_TAC THEN MATCH_MP_TAC CLOSED_IN_INTER THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_INTERS THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[SET_RULE `{x | P x /\ (Q x ==> R x)} = {x | x IN {x | P x} /\ ~Q x} UNION {x | x IN {x | P x} /\ R x}`] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN REWRITE_TAC[REAL_NOT_LT] THEN CONJ_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `&0 <= d <=> &0 <= drop(lift d)`] THEN ONCE_REWRITE_TAC[SET_RULE `&0 <= drop x <=> x IN {x | &0 <= drop x}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN REWRITE_TAC[REWRITE_RULE[real_ge] CLOSED_HALFSPACE_COMPONENT_GE; drop] THEN REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN REWRITE_TAC[LIFT_CMUL] THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_CMUL) THEN REWRITE_TAC[CONTINUOUS_ON_CONST; dist] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN CONJ_TAC THEN REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_LDISTRIB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN SIMP_TAC[MATRIX_VECTOR_MUL_LINEAR; LINEAR_CONTINUOUS_ON] THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN SET_TAC[]);; let LEBESGUE_MEASURABLE_POINTS_OF_DIFFERENTIABILITY_WITHIN = prove (`!f:real^M->real^N s. lebesgue_measurable s ==> lebesgue_measurable {x | x IN s /\ f differentiable (at x within s)}`, let lemma = prove (`!f:real^M->real^N x s. (!n. ~(n = vec 0) ==> ?k. &0 < k /\ !e. &0 < e ==> ?y. y IN s DELETE x /\ dist(x,y) < e /\ k * norm(y - x) <= abs(n dot (y - x))) ==> (f differentiable (at x within s) <=> !e. &0 < e ==> ?d A. &0 < d /\ (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> rational(A$i$j)) /\ !y. y IN s /\ norm(y - x) < d ==> norm(f y - f x - A ** (y - x)) <= e * norm (y - x))`, REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `trivial_limit (at (x:real^M) within s)` THENL [ASM_SIMP_TAC[LIM_TRIVIAL; differentiable; has_derivative] THEN MATCH_MP_TAC(TAUT `p /\ q ==> (p <=> q)`) THEN CONJ_TAC THENL [MESON_TAC[LINEAR_ZERO]; ALL_TAC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TRIVIAL_LIMIT_WITHIN]) THEN REWRITE_TAC[LIMPT_APPROACHABLE; NOT_FORALL_THM; NOT_IMP] THEN REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q /\ r) <=> p /\ r ==> ~q`] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN REWRITE_TAC[dist] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `mat 0:real^M^N` THEN SIMP_TAC[MAT_COMPONENT; MATRIX_VECTOR_MUL_LZERO] THEN REWRITE_TAC[COND_ID; RATIONAL_NUM; VECTOR_SUB_RZERO] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL]; RULE_ASSUM_TAC(REWRITE_RULE[TRIVIAL_LIMIT_WITHIN])] THEN REWRITE_TAC[differentiable; HAS_DERIVATIVE_WITHIN_ALT] THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_THEN `f':real^M->real^N` STRIP_ASSUME_TAC) THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC (ISPEC `matrix(f':real^M->real^N)` MATRIX_RATIONAL_APPROXIMATION) THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `B:real^M^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_ARITH `x / &2 * y = (x * y) / &2`] THEN MATCH_MP_TAC(NORM_ARITH `norm(d' - d:real^N) <= e / &2 ==> norm(y - x - d') <= e / &2 ==> norm(y - x - d) <= e`) THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN TRANS_TAC REAL_LE_TRANS `onorm(\x. (matrix(f':real^M->real^N) - B) ** x) * norm(y - x:real^M)` THEN ASM_SIMP_TAC[ONORM; LINEAR_COMPOSE_SUB; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[REAL_ARITH `(e * x) / &2 = e / &2 * x`] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN MAP_EVERY X_GEN_TAC [`d:num->real`; `A:num->real^M^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> ?a. ((\n. lift((A n:real^M^N)$i$j)) --> a) sequentially` MP_TAC THENL [ALL_TAC; REWRITE_TAC[EXISTS_LIFT] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; LAMBDA_SKOLEM] THEN DISCH_THEN(X_CHOOSE_THEN `B:real^M^N` (LABEL_TAC "*")) THEN EXISTS_TAC `\x. (B:real^M^N) ** x` THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o REWRITE_RULE[tendsto]) THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!i. P i ==> !j. Q j ==> !e. R i j e) ==> !e i. P i ==> !j. Q j ==> R i j e`)) THEN DISCH_THEN(MP_TAC o SPEC `e / &2 / &(dimindex(:N)) / &(dimindex(:M))`) THEN REWRITE_TAC[GSYM IN_NUMSEG] THEN ASM_SIMP_TAC[CONV_RULE (RAND_CONV SYM_CONV) (SPEC_ALL EVENTUALLY_FORALL); FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1; REAL_LT_DIV; LE_1; REAL_HALF; REAL_OF_NUM_LT] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; LEFT_IMP_EXISTS_THM] THEN MP_TAC(SPEC `e / &2` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_THEN(MP_TAC o SPEC `n + m:num`) THEN REWRITE_TAC[LE_ADD; RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN REWRITE_TAC[IN_NUMSEG] THEN DISCH_TAC THEN EXISTS_TAC `(d:num->real) (n + m)` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n + m:num`; `y:real^M`]) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `norm(d' - d:real^N) <= e' - e ==> norm(y - x - d') <= e ==> norm(y - x - d) <= e'`) THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN TRANS_TAC REAL_LE_TRANS `onorm(\x. ((A:num->real^M^N) (n + m) - B) ** x) * norm(y - x:real^M)` THEN SIMP_TAC[ONORM; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[GSYM REAL_SUB_RDISTRIB] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `d <= e / &2 /\ x <= e / &2 ==> x <= e - d`) THEN CONJ_TAC THENL [TRANS_TAC REAL_LE_TRANS `inv(&m)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC; TRANS_TAC REAL_LE_TRANS `&(dimindex(:N)) * &(dimindex(:M)) * e / &2 / &(dimindex(:N)) / &(dimindex(:M))` THEN CONJ_TAC THENL [MATCH_MP_TAC ONORM_LE_MATRIX_COMPONENT THEN RULE_ASSUM_TAC(REWRITE_RULE[DIST_LIFT]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; MATRIX_SUB_COMPONENT]; SIMP_TAC[REAL_DIV_LMUL; REAL_OF_NUM_EQ; DIMINDEX_NONZERO] THEN ASM_REAL_ARITH_TAC]]] THEN SUBGOAL_THEN `{x | cauchy (\n. (A n:real^M^N) ** x)} = (:real^M)` ASSUME_TAC THENL [ALL_TAC; MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `basis j:real^M` o MATCH_MP (SET_RULE `s = UNIV ==> !x. x IN s`)) THEN REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `l:real^N` MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `i:num` o MATCH_MP LIM_COMPONENT) THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_BASIS; column; LAMBDA_BETA] THEN MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `span s = s /\ span s = UNIV ==> s = UNIV`) THEN CONJ_TAC THENL [REWRITE_TAC[SPAN_EQ_SELF; GSYM CONVERGENT_EQ_CAUCHY] THEN REWRITE_TAC[subspace; IN_ELIM_THM] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_ADD_LDISTRIB; MATRIX_VECTOR_MUL_RZERO] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_RMUL] THEN CONJ_TAC THENL [EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[LIM_CONST]; REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN CONJ_TAC THEN REPEAT GEN_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD); DISCH_THEN(MP_TAC o MATCH_MP LIM_CMUL)] THEN REWRITE_TAC[] THEN MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM DIM_EQ_FULL] THEN MATCH_MP_TAC(ARITH_RULE `n:num <= N /\ ~(n < N) ==> n = N`) THEN REWRITE_TAC[DIM_SUBSET_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^M` MP_TAC o MATCH_MP ORTHOGONAL_TO_SUBSPACE_EXISTS) THEN REWRITE_TAC[FORALL_IN_GSPEC; orthogonal] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (ASSUME `~(d:real^M = vec 0)`)) THEN REWRITE_TAC[IN_DELETE; ONCE_REWRITE_RULE[DIST_SYM] dist] THEN DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?y:num->real^M z. (!n. y n IN s) /\ (!n. ~(y n = x)) /\ (!n. k * norm(y n - x) <= abs(d dot (y n - x))) /\ (y --> x) sequentially /\ ((\n. inv(norm(y n - x)) % (y n - x)) --> z) sequentially` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?y:num->real^M. (!n. y n IN s /\ ~(y n = x) /\ k * norm(y n - x) <= abs(d dot (y n - x)) /\ norm(y n - x) < inv(&n + &1)) /\ (!n. norm(y(SUC n) - x) < norm(y n - x))` MP_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_01]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `y:real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `min (norm(y:real^M - x)) (inv(&(SUC n) + &1))`) THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_INV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN ANTS_TAC THENL [REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[]]; REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPEC `sphere(vec 0:real^M,&1)` compact) THEN REWRITE_TAC[COMPACT_SPHERE; IN_SPHERE_0] THEN DISCH_THEN(MP_TAC o SPEC `\n:num. inv(norm(y n - x:real^M)) % (y n - x)`) THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0; VECTOR_SUB_EQ] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^M` THEN REWRITE_TAC[o_DEF] THEN DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(y:num->real^M) o (r:num->num)` THEN ASM_REWRITE_TAC[o_THM] THEN MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[LIM_SEQUENTIALLY; dist] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [MESON_TAC[REAL_LT_TRANS]; ALL_TAC] THEN X_GEN_TAC `n:num` THEN EXISTS_TAC `n:num` THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN TRANS_TAC REAL_LTE_TRANS `inv(&m + &1)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `k <= abs((d:real^M) dot z)` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o ISPEC `\x:real^M. lift(abs(d dot x) - k)` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN ANTS_TAC THENL [REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `abs x = abs(drop(lift x))`] THEN REWRITE_TAC[drop] THEN MATCH_MP_TAC CONTINUOUS_LIFT_ABS_COMPONENT THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_DOT]; DISCH_THEN(MP_TAC o SPEC `&0` o MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_DROP_LBOUND)) THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIFT_DROP; REAL_SUB_LE] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[o_DEF] THEN REWRITE_TAC[DOT_RMUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_ABS_MUL] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ]]; MATCH_MP_TAC(REAL_ARITH `&0 < k /\ x = &0 ==> k <= abs x ==> F`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC] THEN REWRITE_TAC[cauchy] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e / &2` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN REWRITE_TAC[dist; GE] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e / &2 ==> a <= &2 * x ==> a < e`)) THEN FIRST_X_ASSUM(MP_TAC o ISPEC `\x:real^M. lift(norm((A:num->real^M^N) m ** x - A n ** x) - &2 / &N)` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST] THEN MATCH_MP_TAC CONTINUOUS_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `&0` o MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_DROP_UBOUND)) THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIFT_DROP; REAL_SUB_LE] THEN ANTS_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [tendsto]) THEN DISCH_THEN(MP_TAC o SPEC `min ((d:num->real) m) (d n)`) THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `p:num` THEN REWRITE_TAC[dist] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(ISPECL [`m:num`; `(y:num->real^M) p`] th) THEN MP_TAC(ISPECL [`n:num`; `(y:num->real^M) p`] th)) THEN ASM_REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH `norm(y - x - a:real^N) <= d /\ norm(y - x - b) <= e ==> norm(b - a) <= d + e`)) THEN REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; MATRIX_VECTOR_MUL_RMUL] THEN REWRITE_TAC[REAL_ARITH `a - b <= &0 <=> a <= b`; NORM_MUL] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NORM; GSYM REAL_ADD_RDISTRIB] THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div); REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `a <= inv c /\ b <= inv c ==> a + b <= &2 / c`) THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC) in REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF THEN EXISTS_TAC `{x | x IN s /\ (f:real^M->real^N) continuous (at x within s)} INTER {x | !e. &0 < e ==> ?d A. &0 < d /\ (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> rational(A$i$j)) /\ !y. y IN s /\ norm(y - x) < d ==> norm(f y - f x - A ** (y - x)) <= e * norm (y - x)}` THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `!s. t = s /\ lebesgue_measurable s ==> lebesgue_measurable t`) THEN EXISTS_TAC `{x | x IN s /\ (f:real^M->real^N) continuous (at x within s)} INTER INTERS { UNIONS { UNIONS { {x | x IN s /\ f continuous (at x within s)} INTER INTERS {{x | x IN s /\ f continuous (at x within s) /\ (norm(y - x) < d ==> norm(f y - f x - A ** (y - x)) <= e * norm(y - x))} |y| y IN s} |d| d IN {d | d IN rational /\ &0 < d}} |A| A IN {A | !i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> rational (A$i$j)}} |e| e IN {e | e IN rational /\ &0 < e}}` THEN CONJ_TAC THENL [GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; INTERS_GSPEC; UNIONS_GSPEC] THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(f:real^M->real^N) continuous (at x within s)` THEN ASM_REWRITE_TAC[IN_UNIV] THEN REWRITE_TAC[UNIONS_GSPEC; INTERS_GSPEC; IN_ELIM_THM; IN_INTER] THEN REWRITE_TAC[SET_RULE `x IN rational <=> rational x`] THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP RATIONAL_BETWEEN) THEN DISCH_THEN(X_CHOOSE_THEN `e':real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `e':real`) THEN ASM_REWRITE_TAC[IN_UNIV] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `A:real^M^N` THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP RATIONAL_BETWEEN) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d':real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[] THEN (ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN CONJ_TAC THENL [MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] GDELTA_POINTS_OF_CONTINUITY_WITHIN) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[GDELTA_IMP_LEBESGUE_MEASURABLE; LEBESGUE_MEASURABLE_INTER]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_RATIONAL]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `e:real` THEN STRIP_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC COUNTABLE_CART THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COUNTABLE_CART THEN REWRITE_TAC[COUNTABLE_RATIONAL; SET_RULE `{x | s x} = s`]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `A:real^M^N` THEN STRIP_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_RATIONAL]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `d:real` THEN STRIP_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_CLOSED_IN THEN EXISTS_TAC `{x | x IN s /\ (f:real^M->real^N) continuous (at x within s)}` THEN CONJ_TAC THENL [ALL_TAC; MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] GDELTA_POINTS_OF_CONTINUITY_WITHIN) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[GDELTA_IMP_LEBESGUE_MEASURABLE; LEBESGUE_MEASURABLE_INTER]] THEN MATCH_MP_TAC(MESON[SET_RULE `s INTER INTERS {} = s`; CLOSED_IN_REFL] `(~(u = {}) ==> closed_in (subtopology euclidean s) (s INTER INTERS u)) ==> closed_in (subtopology euclidean s) (s INTER INTERS u)`) THEN DISCH_TAC THEN MATCH_MP_TAC CLOSED_IN_INTER THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_INTERS THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN REWRITE_TAC[SET_RULE `{x | P x /\ (Q x ==> R x)} = {x | x IN {x | P x} /\ ~Q x} UNION {x | x IN {x | P x} /\ R x}`] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN REWRITE_TAC[REAL_NOT_LT] THEN CONJ_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `&0 <= d <=> &0 <= drop(lift d)`] THEN ONCE_REWRITE_TAC[SET_RULE `&0 <= drop x <=> x IN {x | &0 <= drop x}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN REWRITE_TAC[REWRITE_RULE[real_ge] CLOSED_HALFSPACE_COMPONENT_GE; drop] THEN REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN REWRITE_TAC[LIFT_CMUL] THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_CMUL) THEN REWRITE_TAC[CONTINUOUS_ON_CONST; dist] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN CONJ_TAC THEN REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_LDISTRIB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN SIMP_TAC[MATRIX_VECTOR_MUL_LINEAR; LINEAR_CONTINUOUS_ON] THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_WITHIN_SUBSET) THEN REWRITE_TAC[SUBSET_RESTRICT]; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x | x IN s /\ ~(!n. ~(n:real^M = vec 0) ==> (?k. &0 < k /\ (!e. &0 < e ==> (?y. y IN s DELETE x /\ dist (x,y) < e /\ k * norm (y - x) <= abs (n dot (y - x))))))}` THEN CONJ_TAC THENL [ALL_TAC; GEN_REWRITE_TAC I [SUBSET] THEN X_GEN_TAC `x:real^M` THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[IN_ELIM_THM; IN_INTER; IN_UNION; IN_DIFF] THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(fun th -> ONCE_REWRITE_TAC[GSYM (MATCH_MP lemma th)]) THEN MESON_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN]] THEN GEN_REWRITE_TAC I [NEGLIGIBLE_EQ_ZERO_DENSITY] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `r:real`; `e:real`] THEN REWRITE_TAC[IN_ELIM_THM; SUBSET; NOT_FORALL_THM; NOT_IMP; IN_INTER; IN_DELETE; GSYM CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `n:real^M` MP_TAC) MP_TAC) THEN POP_ASSUM MP_TAC THEN GEOM_BASIS_MULTIPLE_TAC 1 `n:real^M` THEN REPEAT GEN_TAC THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN ASM_CASES_TAC `n = &0` THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN DISCH_TAC THEN STRIP_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[NOT_EXISTS_THM; MESON[] `(!k. ~(&0 < k /\ !e. &0 < e ==> P e k)) <=> (!k. &0 < k ==> ?e. &0 < e /\ ~P e k)`] THEN SIMP_TAC[DOT_LMUL; DOT_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_ABS_MUL] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < n ==> abs n * a = a * n`] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN REWRITE_TAC[REAL_ARITH `(a * b) / c:real = a / c * b`] THEN REWRITE_TAC[TAUT `~(p /\ q) <=> p ==> ~q`] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; REAL_NOT_LE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(n * e / &2) / &(dimindex(:M)) pow dimindex(:M)`) THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; REAL_POW_LT; REAL_HALF; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1] THEN REWRITE_TAC[REAL_ARITH `((n * e) / x) / n * p:real = n * (e * p) / x / n`] THEN ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d r:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN REWRITE_TAC[REAL_ARITH `min d r:real <= r`; BALL_MIN_INTER; IN_INTER] THEN EXISTS_TAC `ball(x:real^M,min d r) INTER {y | abs ((y - x)$1) < (e / &2 * min d r) / &(dimindex(:M)) pow dimindex(:M) }` THEN SUBGOAL_THEN `!b a:real^M. open {x | abs((x - a)$1) < b}` ASSUME_TAC THENL [POP_ASSUM_LIST(K ALL_TAC) THEN REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^M` THEN REWRITE_TAC[VECTOR_SUB_RZERO; OPEN_STRIP_COMPONENT_LT]; ALL_TAC] THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE; MEASURABLE_BALL; LEBESGUE_MEASURABLE_OPEN] THEN CONJ_TAC THENL [X_GEN_TAC `y:real^M` THEN SIMP_TAC[IN_INTER; BALL_MIN_INTER] THEN REWRITE_TAC[IN_BALL; IN_ELIM_THM] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN ASM_CASES_TAC `y:real^M = x` THEN ASM_SIMP_TAC[VECTOR_SUB_REFL; VEC_COMPONENT; REAL_ABS_NUM; REAL_LT_MUL; REAL_LT_DIV; REAL_POW_LT; REAL_HALF; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1; REAL_LT_MIN] THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LTE_TRANS) THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_LE_LMUL_EQ; REAL_POW_LT; REAL_HALF; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e * y /\ x <= e / &2 * y ==> x < e * y`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[REAL_LT_MUL; MEASURE_BALL_POS; GSYM BALL_MIN_INTER; REAL_LT_MIN]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `measure(interval [(x - lambda i. if i = 1 then (e / &2 * min d r) / &(dimindex(:M)) pow dimindex(:M) else min d r):real^M, (x + lambda i. if i = 1 then (e / &2 * min d r) / &(dimindex(:M)) pow dimindex(:M) else min d r)])` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE; MEASURABLE_BALL; LEBESGUE_MEASURABLE_OPEN; MEASURABLE_INTERVAL] THEN REWRITE_TAC[SUBSET; IN_INTER; IN_BALL; IN_INTERVAL; IN_ELIM_THM] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `x - e <= y /\ y <= x + e <=> abs(y - x) <= e`] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; REAL_LT_IMP_LE] THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `e / &2 * measure(interval[(x - lambda i. min d r / &(dimindex(:M))):real^M, (x + lambda i. min d r / &(dimindex(:M)))])` THEN CONJ_TAC THENL [SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES; VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN REWRITE_TAC[REAL_ARITH `x - e <= x + e <=> &0 <= e`; REAL_ARITH `(x + e) - (x - e) = &2 * e`] THEN REWRITE_TAC[MESON[] `&0 <= (if p then x else y) <=> if p then &0 <= x else &0 <= y`] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_MIN; REAL_LE_DIV; REAL_POW_LE; REAL_POS; REAL_HALF; REAL_LT_IMP_LE; COND_ID] THEN SIMP_TAC[PRODUCT_CLAUSES_LEFT; DIMINDEX_GE_1; ARITH] THEN SIMP_TAC[ARITH_RULE `2 <= i ==> ~(i = 1)`] THEN REWRITE_TAC[PRODUCT_CONST_NUMSEG] THEN REWRITE_TAC[ARITH_RULE `(n + 1) - 2 = n - 1`] THEN REWRITE_TAC[REAL_ARITH `(&2 * (e * m) / p) * x = e * ((&2 * m) * x) / p`] THEN REWRITE_TAC[GSYM(CONJUNCT2 real_pow)] THEN SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> SUC(n - 1) = n`] THEN REWRITE_TAC[REAL_ARITH `&2 * x / y = (&2 * x) / y`] THEN REWRITE_TAC[REAL_POW_DIV; REAL_LE_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_HALF; GSYM BALL_MIN_INTER] THEN SIMP_TAC[GSYM INTERIOR_CBALL; MEASURE_INTERIOR; BOUNDED_CBALL; FRONTIER_CBALL; NEGLIGIBLE_SPHERE] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL; MEASURABLE_CBALL] THEN REWRITE_TAC[SUBSET; IN_INTERVAL; IN_CBALL] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN REWRITE_TAC[dist] THEN W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC SUM_BOUND_GEN THEN REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; CARD_NUMSEG_1] THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);; let LEBESGUE_MEASURABLE_POINTS_OF_DIFFERENTIABILITY_AT = prove (`!f:real^M->real^N. lebesgue_measurable {x | f differentiable (at x)}`, GEN_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`] LEBESGUE_MEASURABLE_POINTS_OF_DIFFERENTIABILITY_WITHIN) THEN REWRITE_TAC[IN_UNIV; LEBESGUE_MEASURABLE_UNIV; WITHIN_UNIV]);; let MEASURABLE_ON_PARTIAL_DERIVATIVES = prove (`!f:real^M->real^N f' s i j. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f'(x)) (at x within s)) /\ 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> (\x. lift(matrix(f' x)$i$j)) measurable_on s`, let lemma = prove (`!f:real^M->real^N s a. linear f /\ ((\x. inv(norm(x - a)) % f(x - a)) --> vec 0) (at a within s) /\ (!n. ~(n = vec 0) ==> ?k. &0 < k /\ !e. &0 < e ==> ?x. x IN s DELETE a /\ dist(a,x) < e /\ k * norm(x - a) <= abs(n dot (x - a))) ==> f = \x. vec 0`, REPEAT GEN_TAC THEN REWRITE_TAC[LIM_WITHIN] THEN GEN_GEOM_ORIGIN_TAC `a:real^M` ["n"] THEN REWRITE_TAC[GSYM LIM_WITHIN] THEN REWRITE_TAC[DIST_0; VECTOR_SUB_RZERO; IN_DELETE] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[FUN_EQ_THM; VEC_COMPONENT] THEN MATCH_MP_TAC(SET_RULE `span {x | P x} = {x | P x} /\ span {x | P x} = UNIV ==> !x. P x`) THEN REWRITE_TAC[GSYM DIM_EQ_FULL; SPAN_EQ_SELF] THEN ASM_SIMP_TAC[SUBSPACE_KERNEL] THEN MATCH_MP_TAC(ARITH_RULE `n:num <= N /\ ~(n < N) ==> n = N`) THEN REWRITE_TAC[DIM_SUBSET_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^M` MP_TAC o MATCH_MP ORTHOGONAL_TO_SUBSPACE_EXISTS) THEN REWRITE_TAC[FORALL_IN_GSPEC; orthogonal] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (ASSUME `~(d:real^M = vec 0)`)) THEN DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?y:num->real^M. (!n. y n IN s /\ ~(y n = vec 0) /\ k * norm(y n) <= abs(d dot y n) /\ norm(y n) < inv(&n + &1)) /\ (!n. norm(y(SUC n)) < norm(y n))` MP_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_01]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `y:real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `min (norm(y:real^M)) (inv(&(SUC n) + &1))`) THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_INV_EQ; NORM_POS_LT] THEN ANTS_TAC THENL [REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[]]; REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN ABBREV_TAC `z:num->real^M = \n. inv(norm(y n)) % y n` THEN MP_TAC(ISPEC `sphere(vec 0:real^M,&1)` compact) THEN REWRITE_TAC[COMPACT_SPHERE; IN_SPHERE_0] THEN DISCH_THEN(MP_TAC o SPEC `z:num->real^M`) THEN ANTS_TAC THENL [EXPAND_TAC "z" THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0]; REWRITE_TAC[NOT_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`l:real^M`; `r:num->num`] THEN STRIP_TAC THEN SUBGOAL_THEN `k <= abs((d:real^M) dot l)` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o ISPEC `\x:real^M. lift(abs(d dot x) - k)` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN ANTS_TAC THENL [REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `abs x = abs(drop(lift x))`] THEN REWRITE_TAC[drop] THEN MATCH_MP_TAC CONTINUOUS_LIFT_ABS_COMPONENT THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_DOT]; DISCH_THEN(MP_TAC o SPEC `&0` o MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_DROP_LBOUND)) THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIFT_DROP; REAL_SUB_LE] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `k:num` THEN EXPAND_TAC "z" THEN REWRITE_TAC[o_DEF] THEN REWRITE_TAC[DOT_RMUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_ABS_MUL] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT]]; MATCH_MP_TAC(REAL_ARITH `&0 < k /\ x = &0 ==> k <= abs x ==> F`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `(f:real^M->real^N) o z o (r:num->num)` THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL [UNDISCH_TAC `((z:num->real^M) o (r:num->num) --> l) sequentially` THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_CONTINUOUS_FUNCTION) THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_AT]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_WITHIN_SEQUENTIALLY]) THEN DISCH_THEN(MP_TAC o SPEC `(y:num->real^M) o (r:num->num)`) THEN ASM_REWRITE_TAC[IN_DELETE; o_THM] THEN ANTS_TAC THENL [MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[LIM_SEQUENTIALLY; DIST_0] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [MESON_TAC[REAL_LT_TRANS]; ALL_TAC] THEN X_GEN_TAC `n:num` THEN EXISTS_TAC `n:num` THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN TRANS_TAC REAL_LTE_TRANS `inv(&m + &1)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN EXPAND_TAC "z" THEN REWRITE_TAC[o_DEF] THEN ASM_MESON_TAC[LINEAR_CMUL]]]) in REPLICATE_TAC 3 GEN_TAC THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LE] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop; LIFT_DROP] THEN X_GEN_TAC `b:real` THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF THEN EXISTS_TAC `{x | x IN s /\ !e. &0 < e ==> ?d A. &0 < d /\ A$m$n < b /\ (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> rational(A$i$j)) /\ !y. y IN s /\ norm(y - x) < d ==> norm((f:real^M->real^N) y - f x - A ** (y - x)) <= e * norm(y - x)}` THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `!s. t = s /\ lebesgue_measurable s ==> lebesgue_measurable t`) THEN EXISTS_TAC `s INTER INTERS { UNIONS { UNIONS { s INTER INTERS {{x | x IN s /\ (norm(y - x) < d ==> norm((f:real^M->real^N) y - f x - A ** (y - x)) <= e * norm(y - x))} |y| y IN s} |d| d IN {d | d IN rational /\ &0 < d}} |A| A IN {A | A$m$n < b /\ !i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> rational (A$i$j)}} |e| e IN {e | e IN rational /\ &0 < e}}` THEN CONJ_TAC THENL [GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTER; IN_ELIM_THM; INTERS_GSPEC; UNIONS_GSPEC] THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[IN_UNIV] THEN REWRITE_TAC[UNIONS_GSPEC; INTERS_GSPEC; IN_ELIM_THM; IN_INTER] THEN REWRITE_TAC[SET_RULE `x IN rational <=> rational x`] THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP RATIONAL_BETWEEN) THEN DISCH_THEN(X_CHOOSE_THEN `e':real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `e':real`) THEN ASM_REWRITE_TAC[IN_UNIV] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `A:real^M^N` THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `&0 < d` THEN DISCH_THEN(MP_TAC o MATCH_MP RATIONAL_BETWEEN) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d':real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[] THEN (ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_RATIONAL]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `e:real` THEN STRIP_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN MATCH_MP_TAC COUNTABLE_INTER THEN DISJ2_TAC THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC COUNTABLE_CART THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COUNTABLE_CART THEN REWRITE_TAC[COUNTABLE_RATIONAL; SET_RULE `{x | s x} = s`]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `A:real^M^N` THEN STRIP_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN SIMP_TAC[COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_RATIONAL]; REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `d:real` THEN STRIP_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_CLOSED_IN THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[SET_RULE `s INTER INTERS {} = s`; CLOSED_IN_REFL] `(~(u = {}) ==> closed_in (subtopology euclidean s) (s INTER INTERS u)) ==> closed_in (subtopology euclidean s) (s INTER INTERS u)`) THEN DISCH_TAC THEN MATCH_MP_TAC CLOSED_IN_INTER THEN REWRITE_TAC[CLOSED_IN_REFL] THEN MATCH_MP_TAC CLOSED_IN_INTERS THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN REWRITE_TAC[SET_RULE `{x | P x /\ (Q x ==> R x)} = {x | P x /\ ~Q x} UNION {x | P x /\ R x}`] THEN MATCH_MP_TAC CLOSED_IN_UNION THEN REWRITE_TAC[REAL_NOT_LT] THEN CONJ_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `&0 <= d <=> &0 <= drop(lift d)`] THEN ONCE_REWRITE_TAC[SET_RULE `&0 <= drop x <=> x IN {x | &0 <= drop x}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN REWRITE_TAC[REWRITE_RULE[real_ge] CLOSED_HALFSPACE_COMPONENT_GE; drop] THEN REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN REWRITE_TAC[LIFT_CMUL] THEN TRY(MATCH_MP_TAC CONTINUOUS_ON_CMUL) THEN REWRITE_TAC[CONTINUOUS_ON_CONST; dist] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN CONJ_TAC THEN REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_LDISTRIB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN SIMP_TAC[MATRIX_VECTOR_MUL_LINEAR; LINEAR_CONTINUOUS_ON] THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN ASM_MESON_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN; differentiable]; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x | x IN s /\ ~(!n. ~(n:real^M = vec 0) ==> (?k. &0 < k /\ (!e. &0 < e ==> (?y. y IN s DELETE x /\ dist (x,y) < e /\ k * norm (y - x) <= abs (n dot (y - x))))))}` THEN CONJ_TAC THENL [UNDISCH_TAC `lebesgue_measurable(s:real^M->bool)` THEN POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_TAC THEN GEN_REWRITE_TAC I [NEGLIGIBLE_EQ_ZERO_DENSITY] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `r:real`; `e:real`] THEN REWRITE_TAC[IN_ELIM_THM; SUBSET; NOT_FORALL_THM; NOT_IMP; IN_INTER; IN_DELETE; GSYM CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `n:real^M` MP_TAC) MP_TAC) THEN POP_ASSUM MP_TAC THEN GEOM_BASIS_MULTIPLE_TAC 1 `n:real^M` THEN REPEAT GEN_TAC THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN ASM_CASES_TAC `n = &0` THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN DISCH_TAC THEN STRIP_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[NOT_EXISTS_THM; MESON[] `(!k. ~(&0 < k /\ !e. &0 < e ==> P e k)) <=> (!k. &0 < k ==> ?e. &0 < e /\ ~P e k)`] THEN SIMP_TAC[DOT_LMUL; DOT_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_ABS_MUL] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < n ==> abs n * a = a * n`] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN REWRITE_TAC[REAL_ARITH `(a * b) / c:real = a / c * b`] THEN REWRITE_TAC[TAUT `~(p /\ q) <=> p ==> ~q`] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; REAL_NOT_LE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(n * e / &2) / &(dimindex(:M)) pow dimindex(:M)`) THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; REAL_POW_LT; REAL_HALF; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1] THEN REWRITE_TAC[REAL_ARITH `((n * e) / x) / n * p:real = n * (e * p) / x / n`] THEN ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d r:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN REWRITE_TAC[REAL_ARITH `min d r:real <= r`; BALL_MIN_INTER; IN_INTER] THEN EXISTS_TAC `ball(x:real^M,min d r) INTER {y | abs ((y - x)$1) < (e / &2 * min d r) / &(dimindex(:M)) pow dimindex(:M) }` THEN SUBGOAL_THEN `!b a:real^M. open {x | abs((x - a)$1) < b}` ASSUME_TAC THENL [POP_ASSUM_LIST(K ALL_TAC) THEN REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^M` THEN REWRITE_TAC[VECTOR_SUB_RZERO; OPEN_STRIP_COMPONENT_LT]; ALL_TAC] THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE; MEASURABLE_BALL; LEBESGUE_MEASURABLE_OPEN] THEN CONJ_TAC THENL [X_GEN_TAC `y:real^M` THEN SIMP_TAC[IN_INTER; BALL_MIN_INTER] THEN REWRITE_TAC[IN_BALL; IN_ELIM_THM] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN ASM_CASES_TAC `y:real^M = x` THEN ASM_SIMP_TAC[VECTOR_SUB_REFL; VEC_COMPONENT; REAL_ABS_NUM; REAL_LT_MUL; REAL_LT_DIV; REAL_POW_LT; REAL_HALF; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1; REAL_LT_MIN] THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LTE_TRANS) THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_LE_LMUL_EQ; REAL_POW_LT; REAL_HALF; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e * y /\ x <= e / &2 * y ==> x < e * y`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[REAL_LT_MUL; MEASURE_BALL_POS; GSYM BALL_MIN_INTER; REAL_LT_MIN]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `measure(interval [(x - lambda i. if i = 1 then (e / &2 * min d r) / &(dimindex(:M)) pow dimindex(:M) else min d r):real^M, (x + lambda i. if i = 1 then (e / &2 * min d r) / &(dimindex(:M)) pow dimindex(:M) else min d r)])` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE; MEASURABLE_BALL; LEBESGUE_MEASURABLE_OPEN; MEASURABLE_INTERVAL] THEN REWRITE_TAC[SUBSET; IN_INTER; IN_BALL; IN_INTERVAL; IN_ELIM_THM] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `x - e <= y /\ y <= x + e <=> abs(y - x) <= e`] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; REAL_LT_IMP_LE] THEN W(MP_TAC o PART_MATCH lhand COMPONENT_LE_NORM o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `e / &2 * measure(interval[(x - lambda i. min d r / &(dimindex(:M))):real^M, (x + lambda i. min d r / &(dimindex(:M)))])` THEN CONJ_TAC THENL [SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES; VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN REWRITE_TAC[REAL_ARITH `x - e <= x + e <=> &0 <= e`; REAL_ARITH `(x + e) - (x - e) = &2 * e`] THEN REWRITE_TAC[MESON[] `&0 <= (if p then x else y) <=> if p then &0 <= x else &0 <= y`] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_MIN; REAL_LE_DIV; REAL_POW_LE; REAL_POS; REAL_HALF; REAL_LT_IMP_LE; COND_ID] THEN SIMP_TAC[PRODUCT_CLAUSES_LEFT; DIMINDEX_GE_1; ARITH] THEN SIMP_TAC[ARITH_RULE `2 <= i ==> ~(i = 1)`] THEN REWRITE_TAC[PRODUCT_CONST_NUMSEG] THEN REWRITE_TAC[ARITH_RULE `(n + 1) - 2 = n - 1`] THEN REWRITE_TAC[REAL_ARITH `(&2 * (e * m) / p) * x = e * ((&2 * m) * x) / p`] THEN REWRITE_TAC[GSYM(CONJUNCT2 real_pow)] THEN SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> SUC(n - 1) = n`] THEN REWRITE_TAC[REAL_ARITH `&2 * x / y = (&2 * x) / y`] THEN REWRITE_TAC[REAL_POW_DIV; REAL_LE_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_HALF; GSYM BALL_MIN_INTER] THEN SIMP_TAC[GSYM INTERIOR_CBALL; MEASURE_INTERIOR; BOUNDED_CBALL; FRONTIER_CBALL; NEGLIGIBLE_SPHERE] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_INTERVAL; MEASURABLE_CBALL] THEN REWRITE_TAC[SUBSET; IN_INTERVAL; IN_CBALL] THEN SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN REWRITE_TAC[dist] THEN W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC SUM_BOUND_GEN THEN REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; CARD_NUMSEG_1] THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `(!x. ~(x IN s) ==> (x IN t <=> x IN u)) ==> (t DIFF u) UNION (u DIFF t) SUBSET s`) THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN EQ_TAC THENL [ALL_TAC; DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [has_derivative_within] o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; DIST_0] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC (ISPEC `matrix((f':real^M->real^M->real^N) x) - lambda i j. if i = m /\ j = n then e / &4 else &0` MATRIX_RATIONAL_APPROXIMATION) THEN DISCH_THEN(MP_TAC o SPEC `e / &6`) THEN ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &6 <=> &0 < e`] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `B:real^M^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MP_TAC(GEN `f:real^M->real^N` (ISPECL [`f:real^M->real^N`; `m:num`; `n:num`] COMPONENT_LE_ONORM)) THEN DISCH_THEN(fun th -> FIRST_ASSUM(fun th' -> MP_TAC(PART_MATCH (rand o rand) th (lhand(concl th'))))) THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR; MATRIX_OF_MATRIX_VECTOR_MUL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `n < e ==> (p < e ==> u < v) ==> p <= n ==> u < v`)) THEN ASM_SIMP_TAC[MATRIX_SUB_COMPONENT; LAMBDA_BETA] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN ASM_CASES_TAC `y:real^M = x` THEN ASM_SIMP_TAC[VECTOR_SUB_REFL; LINEAR_0; MATRIX_VECTOR_MUL_LINEAR; MATRIX_VECTOR_MUL_RZERO; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL] THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[GSYM DIST_NZ] THEN ASM_REWRITE_TAC[dist] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN REWRITE_TAC[REAL_ARITH `x / &2 * y = (x * y) / &2`] THEN MATCH_MP_TAC(NORM_ARITH `norm(d' - d:real^N) <= e / &2 ==> norm(y - (x + d')) < e / &2 ==> norm(y - x - d) <= e`) THEN ASM_SIMP_TAC[GSYM MATRIX_WORKS] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN MATCH_MP_TAC(NORM_ARITH `!y. norm(y) <= e / &6 /\ norm(x - y) <= e / &4 ==> norm(x:real^N) <= e / &2`) THEN EXISTS_TAC `((matrix((f':real^M->real^M->real^N) x) - lambda i j. if i = m /\ j = n then e / &4 else &0) - B) ** (y - x)` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN REWRITE_TAC[REAL_ARITH `(x * y) / &6 = x / &6 * y`] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e ==> y <= x ==> y <= e`)) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN SIMP_TAC[ONORM; MATRIX_VECTOR_MUL_LINEAR]; ALL_TAC] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_RDISTRIB; VECTOR_ARITH `m - b - (m - e - b):real^M = e`] THEN TRANS_TAC REAL_LE_TRANS `norm(e / &4 % (y - x:real^M)$n % basis m:real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; matrix_vector_mul; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = m` THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT; REAL_MUL_LZERO; REAL_MUL_RZERO; SUM_0] THEN REWRITE_TAC[COND_RAND; REAL_MUL_RID; COND_RATOR; REAL_MUL_LZERO] THEN SIMP_TAC[SUM_DELTA; IN_NUMSEG] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[NORM_MUL; NORM_BASIS] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_ARITH `&0 < e ==> (abs(e / &4) * x * &1 <= (e * y) / &4 <=> e * x <= e * y)`] THEN REWRITE_TAC[COMPONENT_LE_NORM]] THEN DISCH_THEN(MP_TAC o GEN `i:num` o SPEC `inv(&i + &1)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN MAP_EVERY X_GEN_TAC [`d:num->real`; `A:num->real^M^N`] THEN STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM LIFT_DROP] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_UBOUND) THEN EXISTS_TAC `\i. lift((A:num->real^M^N) i$m$n)` THEN ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY; LIFT_DROP; REAL_LT_IMP_LE] THEN SUBGOAL_THEN `!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:M) ==> ?a. ((\n. lift((A n:real^M^N)$i$j)) --> a) sequentially` MP_TAC THENL [SUBGOAL_THEN `{x | cauchy (\n. (A n:real^M^N) ** x)} = (:real^M)` ASSUME_TAC THENL [ALL_TAC; MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `basis j:real^M` o MATCH_MP (SET_RULE `s = UNIV ==> !x. x IN s`)) THEN REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `l:real^N` MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `i:num` o MATCH_MP LIM_COMPONENT) THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_BASIS; column; LAMBDA_BETA] THEN MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `span s = s /\ span s = UNIV ==> s = UNIV`) THEN CONJ_TAC THENL [REWRITE_TAC[SPAN_EQ_SELF; GSYM CONVERGENT_EQ_CAUCHY] THEN REWRITE_TAC[subspace; IN_ELIM_THM] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_ADD_LDISTRIB; MATRIX_VECTOR_MUL_RZERO] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_RMUL] THEN CONJ_TAC THENL [EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[LIM_CONST]; REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN CONJ_TAC THEN REPEAT GEN_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD); DISCH_THEN(MP_TAC o MATCH_MP LIM_CMUL)] THEN REWRITE_TAC[] THEN MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM DIM_EQ_FULL] THEN MATCH_MP_TAC(ARITH_RULE `n:num <= N /\ ~(n < N) ==> n = N`) THEN REWRITE_TAC[DIM_SUBSET_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `d:real^M` MP_TAC o MATCH_MP ORTHOGONAL_TO_SUBSPACE_EXISTS) THEN REWRITE_TAC[FORALL_IN_GSPEC; orthogonal] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (ASSUME `~(d:real^M = vec 0)`)) THEN REWRITE_TAC[IN_DELETE; ONCE_REWRITE_RULE[DIST_SYM] dist] THEN DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?y:num->real^M z. (!n. y n IN s) /\ (!n. ~(y n = x)) /\ (!n. k * norm(y n - x) <= abs(d dot (y n - x))) /\ (y --> x) sequentially /\ ((\n. inv(norm(y n - x)) % (y n - x)) --> z) sequentially` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?y:num->real^M. (!i. y i IN s /\ ~(y i = x) /\ k * norm(y i - x) <= abs(d dot (y i - x)) /\ norm(y i - x) < inv(&i + &1)) /\ (!i. norm(y(SUC i) - x) < norm(y i - x))` MP_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_01]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `y:real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `min (norm(y:real^M - x)) (inv(&(SUC i) + &1))`) THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_INV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN ANTS_TAC THENL [REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[]]; REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN MP_TAC(ISPEC `sphere(vec 0:real^M,&1)` compact) THEN REWRITE_TAC[COMPACT_SPHERE; IN_SPHERE_0] THEN DISCH_THEN(MP_TAC o SPEC `\i:num. inv(norm(y i - x:real^M)) % (y i - x)`) THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0; VECTOR_SUB_EQ] THEN GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^M` THEN REWRITE_TAC[o_DEF] THEN DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(y:num->real^M) o (r:num->num)` THEN ASM_REWRITE_TAC[o_THM] THEN MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[LIM_SEQUENTIALLY; dist] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [MESON_TAC[REAL_LT_TRANS]; ALL_TAC] THEN X_GEN_TAC `i:num` THEN EXISTS_TAC `i:num` THEN X_GEN_TAC `p:num` THEN DISCH_TAC THEN TRANS_TAC REAL_LTE_TRANS `inv(&p + &1)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `k <= abs((d:real^M) dot z)` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o ISPEC `\x:real^M. lift(abs(d dot x) - k)` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN ANTS_TAC THENL [REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `abs x = abs(drop(lift x))`] THEN REWRITE_TAC[drop] THEN MATCH_MP_TAC CONTINUOUS_LIFT_ABS_COMPONENT THEN REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_DOT]; DISCH_THEN(MP_TAC o SPEC `&0` o MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_DROP_LBOUND)) THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIFT_DROP; REAL_SUB_LE] THEN DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[o_DEF] THEN REWRITE_TAC[DOT_RMUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_ABS_MUL] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ]]; MATCH_MP_TAC(REAL_ARITH `&0 < k /\ x = &0 ==> k <= abs x ==> F`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC] THEN REWRITE_TAC[cauchy] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e / &2` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN REWRITE_TAC[dist; GE] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e / &2 ==> a <= &2 * x ==> a < e`)) THEN FIRST_X_ASSUM(MP_TAC o ISPEC `\x:real^M. lift(norm((A:num->real^M^N) i ** x - A j ** x) - &2 / &N)` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST] THEN MATCH_MP_TAC CONTINUOUS_LIFT_NORM_COMPOSE THEN MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `&0` o MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_DROP_UBOUND)) THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIFT_DROP; REAL_SUB_LE] THEN ANTS_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [tendsto]) THEN DISCH_THEN(MP_TAC o SPEC `min ((d:num->real) i) (d j)`) THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `p:num` THEN REWRITE_TAC[dist] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(ISPECL [`i:num`; `(y:num->real^M) p`] th) THEN MP_TAC(ISPECL [`j:num`; `(y:num->real^M) p`] th)) THEN ASM_REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH `norm(y - x - a:real^N) <= d /\ norm(y - x - b) <= e ==> norm(b - a) <= d + e`)) THEN REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; MATRIX_VECTOR_MUL_RMUL] THEN REWRITE_TAC[REAL_ARITH `a - b <= &0 <=> a <= b`; NORM_MUL] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NORM; GSYM REAL_ADD_RDISTRIB] THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div); REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `a <= inv c /\ b <= inv c ==> a + b <= &2 / c`) THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; REWRITE_TAC[EXISTS_LIFT] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; LAMBDA_SKOLEM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; RIGHT_IMP_FORALL_THM] THEN X_GEN_TAC `B:real^M^N` THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN DISCH_TAC] THEN SUBGOAL_THEN `(f':real^M->real^M->real^N) x = \y. B ** y` (fun th -> ASM_SIMP_TAC[MATRIX_OF_MATRIX_VECTOR_MUL; th]) THEN REWRITE_TAC[FUN_EQ_THM] THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN RULE_ASSUM_TAC(REWRITE_RULE[has_derivative_within]) THEN ASM_SIMP_TAC[GSYM MATRIX_WORKS] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN MATCH_MP_TAC lemma THEN MAP_EVERY EXISTS_TAC [`s:real^M->bool`; `x:real^M`] THEN ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o C MATCH_MP (ASSUME `(x:real^M) IN s`)) THEN GEN_REWRITE_TAC LAND_CONV [GSYM LIM_NEG_EQ] THEN REWRITE_TAC[VECTOR_NEG_0] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM) THEN ASM_SIMP_TAC[MATRIX_WORKS; MATRIX_VECTOR_MUL_SUB_RDISTRIB; VECTOR_ARITH `--(a % (y - (x + d))) - a % (d - b):real^N = --(a % (y - x - b))`] THEN ONCE_REWRITE_TAC[GSYM LIM_NEG_EQ] THEN REWRITE_TAC[VECTOR_NEG_NEG; VECTOR_NEG_0] THEN REWRITE_TAC[LIM_WITHIN] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e / &2` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `q:num` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `((\p. lift(onorm(\y. (A:num->real^M^N) p ** y - B ** y))) --> vec 0) sequentially` MP_TAC THENL [MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\p:num. sum (1..dimindex(:N)) (\i. sum (1..dimindex(:M)) (\j. abs((A p - B:real^M^N)$i$j)))` THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB; NORM_LIFT] THEN SIMP_TAC[real_abs; ONORM_POS_LE; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[GSYM real_abs; ONORM_LE_MATRIX_COMPONENT_SUM] THEN REWRITE_TAC[LIFT_SUM; o_DEF] THEN REPEAT(MATCH_MP_TAC LIM_NULL_VSUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN REPEAT STRIP_TAC) THEN REWRITE_TAC[GSYM NORM_LIFT; GSYM LIM_NULL_NORM] THEN REWRITE_TAC[LIFT_SUB; MATRIX_SUB_COMPONENT; GSYM LIM_NULL] THEN ASM_SIMP_TAC[]; ALL_TAC] THEN REWRITE_TAC[LIM_SEQUENTIALLY; DIST_0] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `p:num` THEN DISCH_THEN(MP_TAC o SPEC `p + q:num`) THEN REWRITE_TAC[LE_ADD] THEN REWRITE_TAC[NORM_LIFT] THEN DISCH_TAC THEN EXISTS_TAC `(d:num->real) (p + q)` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[dist] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`p + q:num`; `y:real^M`]) THEN ASM_REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(NORM_ARITH `norm(b - c:real^N) < e - d ==> norm(y - x - b) <= d ==> norm(y - x - c) < e`) THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN TRANS_TAC REAL_LET_TRANS `onorm(\x. ((A:num->real^M^N)(p + q) - B) ** x) * norm(y - x:real^M)` THEN SIMP_TAC[ONORM; MATRIX_VECTOR_MUL_LINEAR] THEN TRANS_TAC REAL_LTE_TRANS `e / &2 * norm(y - x:real^M)` THEN ASM_SIMP_TAC[REAL_LT_RMUL_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_SUB_RDISTRIB; REAL_ARITH `abs x < e ==> x < e`] THEN REWRITE_TAC[REAL_ARITH `d * n <= e * n - n / q <=> n * (d + inv q) <= n * e`] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC(REAL_ARITH `x < e / &2 ==> e / &2 + x <= e`) THEN TRANS_TAC REAL_LET_TRANS `inv(&q)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC);; let MEASURABLE_ON_VECTOR_DERIVATIVE_GEN = prove (`!f:real^1->real^N f' s. lebesgue_measurable s /\ (!x. x IN s ==> (f has_vector_derivative f'(x)) (at x within s)) ==> f' measurable_on s`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[MEASURABLE_ON_COMPONENTWISE] THEN REWRITE_TAC[has_vector_derivative] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> p /\ q ==> r ==> s`] MEASURABLE_ON_PARTIAL_DERIVATIVES)) THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[DIMINDEX_1; FORALL_1] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_EQ) THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REWRITE_TAC[GSYM drop; LIFT_DROP; matrix] THEN ASM_SIMP_TAC[LAMBDA_BETA; VECTOR_MUL_COMPONENT; CART_EQ; FORALL_1; DIMINDEX_1; DROP_BASIS] THEN REWRITE_TAC[GSYM drop; LIFT_DROP; REAL_MUL_LID]);; let MEASURABLE_ON_VECTOR_DERIVATIVE = prove (`!f:real^1->real^N f' s k. negligible k /\ lebesgue_measurable s /\ (!x. x IN (s DIFF k) ==> (f has_vector_derivative f'(x)) (at x)) ==> f' measurable_on s`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(f':real^1->real^N) measurable_on (s DIFF k)` MP_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_VECTOR_DERIVATIVE_GEN THEN EXISTS_TAC `f:real^1->real^N` THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_DIFF; HAS_VECTOR_DERIVATIVE_AT_WITHIN; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]; MATCH_MP_TAC MEASURABLE_ON_SPIKE_SET THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]]);; let MEASURABLE_ON_DET_JACOBIAN = prove (`!f:real^N->real^N f' s. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s)) ==> (\x. lift(det(matrix(f' x)))) measurable_on s`, REPEAT STRIP_TAC THEN REWRITE_TAC[det; LIFT_SUM; o_DEF] THEN MATCH_MP_TAC MEASURABLE_ON_VSUM THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG; FORALL_IN_GSPEC] THEN X_GEN_TAC `p:num->num` THEN DISCH_TAC THEN REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC MEASURABLE_ON_CMUL THEN MATCH_MP_TAC MEASURABLE_ON_LIFT_PRODUCT THEN ASM_REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_PARTIAL_DERIVATIVES THEN EXISTS_TAC `f:real^N->real^N` THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]);; (* ------------------------------------------------------------------------- *) (* Luzin's theorem (Talvila and Loeb's proof from Marius Junge's notes). *) (* ------------------------------------------------------------------------- *) let LUZIN = prove (`!f:real^M->real^N s e. measurable s /\ f measurable_on s /\ &0 < e ==> ?k. compact k /\ k SUBSET s /\ measure(s DIFF k) < e /\ f continuous_on k`, REPEAT STRIP_TAC THEN X_CHOOSE_THEN `v:num->real^N->bool` STRIP_ASSUME_TAC UNIV_SECOND_COUNTABLE_SEQUENCE THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN SUBGOAL_THEN `!n. ?k k'. compact k /\ k SUBSET {x | x IN s /\ (f:real^M->real^N) x IN v n} /\ compact k' /\ k' SUBSET {x | x IN s /\ f x IN ((:real^N) DIFF v n)} /\ measure(s DIFF (k UNION k')) < e / &4 / &2 pow n` MP_TAC THENL [GEN_TAC THEN MP_TAC(ISPECL [`{x:real^M | x IN s /\ f(x) IN (v:num->real^N->bool) n}`; `e / &4 / &2 / &2 pow n`] MEASURABLE_INNER_COMPACT) THEN ASM_SIMP_TAC[REAL_OF_NUM_LT; ARITH; REAL_LT_DIV; REAL_LT_POW2] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`{x:real^M | x IN s /\ f(x) IN (:real^N) DIFF v(n:num)}`; `e / &4 / &2 / &2 pow n`] MEASURABLE_INNER_COMPACT) THEN ASM_SIMP_TAC[GSYM OPEN_CLOSED; REAL_LT_DIV; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k':real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `measure(({x | x IN s /\ (f:real^M->real^N) x IN v n} DIFF k) UNION ({x | x IN s /\ f x IN ((:real^N) DIFF v(n:num))} DIFF k'))` THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_UNION; MEASURABLE_COMPACT; GSYM OPEN_CLOSED] THEN SET_TAC[]; ASM_SIMP_TAC[MEASURE_UNION; MEASURABLE_DIFF; MEASURABLE_COMPACT; GSYM OPEN_CLOSED; MEASURE_DIFF_SUBSET] THEN MATCH_MP_TAC(REAL_ARITH `s < k + e / &4 / &2 / d /\ s' < k' + e / &4 / &2 / d /\ m = &0 ==> (s - k) + (s' - k') - m < e / &4 / d`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[MEASURE_EMPTY] `s = {} ==> measure s = &0`) THEN SET_TAC[]]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; IN_DIFF; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`k:num->real^M->bool`; `k':num->real^M->bool`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN EXISTS_TAC `INTERS {k n UNION k' n | n IN (:num)} :real^M->bool` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC COMPACT_INTERS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; COMPACT_UNION] THEN SET_TAC[]; REWRITE_TAC[INTERS_GSPEC] THEN ASM SET_TAC[]; REWRITE_TAC[DIFF_INTERS; SET_RULE `{f y | y IN {g x | x IN s}} = {f(g x) | x IN s}`] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC (MESON[] `measurable s /\ measure s <= b ==> measure s <= b`) THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_UNION; MEASURABLE_COMPACT] THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(0..n) (\i. e / &4 / &2 pow i)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUM_LE_NUMSEG; REAL_LT_IMP_LE]; ALL_TAC] THEN ASM_SIMP_TAC[real_div; SUM_LMUL; REAL_LE_LMUL_EQ; REAL_ARITH `(e * inv(&4)) * s <= e * inv(&2) <=> e * s <= e * &2`] THEN REWRITE_TAC[REAL_INV_POW; SUM_GP; LT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `(&1 - s) / (&1 / &2) <= &2 <=> &0 <= s`] THEN MATCH_MP_TAC REAL_POW_LE THEN CONV_TAC REAL_RAT_REDUCE_CONV; REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN REWRITE_TAC[INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[CONTINUOUS_WITHIN_OPEN; IN_ELIM_THM] THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `?n:num. (f:real^M->real^N)(x) IN v(n) /\ v(n) SUBSET t` STRIP_ASSUME_TAC THENL [UNDISCH_THEN `!s. open s ==> (?k. s:real^N->bool = UNIONS {v(n:num) | n IN k})` (MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; UNIONS_GSPEC] THEN ASM SET_TAC[]; EXISTS_TAC `(:real^M) DIFF k'(n:num)` THEN ASM_SIMP_TAC[GSYM closed; COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]]]);; let LUZIN_EQ,LUZIN_EQ_ALT = (CONJ_PAIR o prove) (`(!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !e. &0 < e ==> ?k. compact k /\ k SUBSET s /\ measure(s DIFF k) < e /\ f continuous_on k)) /\ (!f:real^M->real^N s. measurable s ==> (f measurable_on s <=> !e. &0 < e ==> ?k g. compact k /\ k SUBSET s /\ measure(s DIFF k) < e /\ g continuous_on (:real^M) /\ (!x. x IN k ==> g x = f x)))`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `measurable(s:real^M->bool)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ (q ==> r) /\ (r ==> p) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[LUZIN]; MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC TIETZE_UNBOUNDED THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; SUBTOPOLOGY_UNIV; GSYM CLOSED_IN]; DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&2 pow n)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN MAP_EVERY X_GEN_TAC [`k:num->real^M->bool`; `g:num->real^M->real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_ON_LIMIT THEN MAP_EVERY EXISTS_TAC [`g:num->real^M->real^N`; `s DIFF UNIONS {INTERS {k m | n <= m} | n IN (:num)}:real^M->bool`] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN ASM_MESON_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; SIMP_TAC[DIFF_UNIONS_NONEMPTY; SET_RULE `~({f x | x IN UNIV} = {})`] THEN REWRITE_TAC[NEGLIGIBLE_OUTER] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPECL [`inv(&2)`; `e / &4`] REAL_ARCH_POW_INV) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_POW_INV]] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `s DIFF INTERS {k m | n:num <= m}:real^M->bool` THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[INTERS_GSPEC; FORALL_IN_GSPEC] THEN ASM SET_TAC[]; MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_COUNTABLE_INTERS_GEN THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; MEASURABLE_COMPACT] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[LE_REFL]] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN MESON_TAC[NUM_COUNTABLE; COUNTABLE_SUBSET; SUBSET_UNIV]; REWRITE_TAC[DIFF_INTERS] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC (MESON[] `measurable s /\ measure s <= b ==> measure s <= b`) THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_GEN THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; MEASURABLE_COMPACT; MEASURABLE_DIFF] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN REWRITE_TAC[SET_RULE `{x | x IN s} = s`] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN MESON_TAC[NUM_COUNTABLE; COUNTABLE_SUBSET; SUBSET_UNIV]; REWRITE_TAC[SIMPLE_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `ns:num->bool` THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN STRIP_TAC THEN REWRITE_TAC[GSYM IMAGE_o] THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_IMAGE_LE o lhand o snd) THEN ASM_SIMP_TAC[o_DEF; MEASURE_POS_LE; MEASURABLE_DIFF; MEASURABLE_COMPACT] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN FIRST_ASSUM(MP_TAC o SPEC `\x:num. x` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum (n..m) (\i. measure(s DIFF k i:real^M->bool))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN ASM_SIMP_TAC[MEASURE_POS_LE; MEASURABLE_DIFF; MEASURABLE_COMPACT; FINITE_NUMSEG; SUBSET; IN_NUMSEG]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum (n..m) (\i. inv(&2 pow i))` THEN ASM_SIMP_TAC[SUM_LE_NUMSEG; REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_INV_POW; SUM_GP; LT] THEN COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC(REAL_ARITH `a <= e / &4 /\ &0 <= b ==> (a - b) / (&1 / &2) <= e / &2`) THEN REWRITE_TAC[real_div; REAL_MUL_LID; REAL_POW_INV] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LT_IMP_LE; REAL_LE_INV_EQ; REAL_LT_POW2]]]; REWRITE_TAC[SET_RULE `s DIFF (s DIFF t) = s INTER t`] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[UNIONS_GSPEC; IN_INTER] THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM; INTERS_GSPEC] THEN STRIP_TAC THEN MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN ASM_MESON_TAC[]]]);; let LUZIN_SIGMA = prove (`!f:real^M->real^N s. lebesgue_measurable s /\ f measurable_on s ==> ?u. COUNTABLE u /\ pairwise DISJOINT u /\ (!k. k IN u ==> compact k /\ k SUBSET s /\ f continuous_on k) /\ negligible(s DIFF UNIONS u)`, let lemma = prove (`!f:real^M->real^N s. measurable s /\ f measurable_on s ==> ?u. COUNTABLE u /\ pairwise DISJOINT u /\ (!k. k IN u ==> compact k /\ k SUBSET s /\ f continuous_on k) /\ negligible(s DIFF UNIONS u)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?g. !n. g n = @k. compact k /\ k SUBSET (s DIFF UNIONS {g(i:num) | i < n}) /\ measure((s DIFF UNIONS {g i | i < n}) DIFF k) < inv(&n + &1) /\ (f:real^M->real^N) continuous_on k` MP_TAC THENL [MATCH_MP_TAC WF_REC_num THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN AP_TERM_TAC THEN BINOP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPLICATE_TAC 2 (AP_THM_TAC THEN AP_TERM_TAC) THEN AP_TERM_TAC THEN ASM SET_TAC[]; DISCH_THEN(X_CHOOSE_TAC `g:num->real^M->bool`)] THEN SUBGOAL_THEN `!n:num. compact (g n) /\ (g n) SUBSET (s DIFF UNIONS {g(i:num) | i < n}) /\ measure (s DIFF UNIONS {g i | i < n} DIFF (g n)) < inv(&n + &1) /\ (f:real^M->real^N) continuous_on (g n)` MP_TAC THENL [MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SPEC `n:num`) THEN CONV_TAC SELECT_CONV THEN MATCH_MP_TAC LUZIN THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC MEASURABLE_ON_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_DIFF]] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; MEASURABLE_COMPACT] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC FINITE_IMAGE THEN REWRITE_TAC[FINITE_NUMSEG_LT]; FIRST_X_ASSUM(K ALL_TAC o SPEC `0`) THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN EXISTS_TAC `IMAGE (g:num->real^M->bool) (:num)` THEN ASM_SIMP_TAC[NUM_COUNTABLE; COUNTABLE_IMAGE; FORALL_IN_IMAGE; IN_UNIV] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[PAIRWISE_IMAGE] THEN REWRITE_TAC[pairwise; IN_UNIV] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN ASM SET_TAC[]; ASM SET_TAC[]; REWRITE_TAC[NEGLIGIBLE_OUTER] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [MESON_TAC[REAL_LT_TRANS]; ALL_TAC] THEN X_GEN_TAC `n:num` THEN EXISTS_TAC `s DIFF UNIONS {(g:num->real^M->bool) i | i < n} DIFF g n` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_SIMP_TAC[MEASURABLE_COMPACT] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_SIMP_TAC[MEASURABLE_COMPACT] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; MEASURABLE_COMPACT] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC FINITE_IMAGE THEN REWRITE_TAC[FINITE_NUMSEG_LT]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `!a. ?u. COUNTABLE u /\ pairwise DISJOINT u /\ (!k. k IN u ==> compact k /\ k SUBSET s INTER {x | !i. 1 <= i /\ i <= dimindex(:M) ==> a$i <= x$i /\ x$i < a$i + &1} /\ (f:real^M->real^N) continuous_on k) /\ negligible(s INTER {x | !i. 1 <= i /\ i <= dimindex(:M) ==> (a:real^M)$i <= x$i /\ x$i < a$i + &1} DIFF UNIONS u)` MP_TAC THENL [GEN_TAC THEN MATCH_MP_TAC lemma THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[INTER_SUBSET] THEN MATCH_MP_TAC MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE THEN ASM_REWRITE_TAC[] THEN (MATCH_MP_TAC MEASURABLE_CONVEX THEN CONJ_TAC THENL [MATCH_MP_TAC IS_INTERVAL_CONVEX THEN REWRITE_TAC[is_interval; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `interval[a:real^M,a + vec 1]` THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN SIMP_TAC[SUBSET; IN_INTERVAL; VECTOR_ADD_COMPONENT; VEC_COMPONENT] THEN SIMP_TAC[REAL_LT_IMP_LE; IN_ELIM_THM]]); REWRITE_TAC[SKOLEM_THM; IN_ELIM_THM; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `u:real^M->(real^M->bool)->bool` THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN EXISTS_TAC `UNIONS (IMAGE (u:real^M->(real^M->bool)->bool) {x | !i. 1 <= i /\ i <= dimindex(:M) ==> integer(x$i)})` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_UNIONS THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SIMP_TAC[COUNTABLE_INTEGER_COORDINATES; COUNTABLE_IMAGE]; REWRITE_TAC[pairwise] THEN REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_IN_UNIONS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN X_GEN_TAC `a:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN X_GEN_TAC `b:real^M` THEN DISCH_TAC THEN X_GEN_TAC `l:real^M->bool` THEN REPEAT DISCH_TAC THEN ASM_CASES_TAC `a:real^M = b` THENL [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN DISCH_THEN(X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `(a:real^M)$i - (b:real^M)$i` REAL_ABS_INTEGER_LEMMA) THEN ASM_SIMP_TAC[INTEGER_CLOSED; REAL_SUB_0] THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(ISPECL [`a:real^M`; `k:real^M->bool`] th) THEN MP_TAC(ISPECL [`b:real^M`; `l:real^M->bool`] th)) THEN ASM_REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(CONJUNCTS_THEN (MP_TAC o el 1 o CONJUNCTS)) THEN MATCH_MP_TAC(SET_RULE `DISJOINT i j ==> k SUBSET s INTER i ==> l SUBSET s INTER j ==> DISJOINT k l`) THEN REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s /\ x IN t ==> F`] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[FORALL_IN_UNIONS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN ASM SET_TAC[]; FIRST_ASSUM(MP_TAC o ISPEC `{x:real^M | !i. 1 <= i /\ i <= dimindex(:M) ==> integer(x$i)}` o MATCH_MP (MESON[FORALL_IN_IMAGE; COUNTABLE_IMAGE; NEGLIGIBLE_COUNTABLE_UNIONS_GEN] `(!a. negligible(f a)) ==> !s. COUNTABLE s ==> negligible(UNIONS (IMAGE f s))`)) THEN REWRITE_TAC[COUNTABLE_INTEGER_COORDINATES] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; SUBSET; IN_DIFF; IN_INTER] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN STRIP_TAC THEN EXISTS_TAC `(lambda i. floor((x:real^M)$i)):real^M` THEN ASM_SIMP_TAC[LAMBDA_BETA; FLOOR] THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `(lambda i. floor((x:real^M)$i)):real^M` THEN ASM_SIMP_TAC[LAMBDA_BETA; FLOOR]]);; let LUZIN_SIGMA_EXPLICIT = prove (`!f:real^M->real^N s. lebesgue_measurable s /\ f measurable_on s ==> ?k. (!n. compact(k n)) /\ (!n. k n SUBSET s) /\ (!n. f continuous_on k n) /\ pairwise (\m n. DISJOINT (k m) (k n)) (:num) /\ negligible(s DIFF UNIONS {k n | n IN (:num)})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:(real^M->bool)->bool` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COUNTABLE_AS_INJECTIVE_IMAGE_SUBSET]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; INJECTIVE_ON_ALT] THEN MAP_EVERY X_GEN_TAC [`k:num->real^M->bool`; `t:num->bool`] THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN EXISTS_TAC `\n. if n IN t then (k:num->real^M->bool) n else {}` THEN RULE_ASSUM_TAC(REWRITE_RULE[FORALL_IN_IMAGE]) THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[COMPACT_EMPTY]; ASM_MESON_TAC[EMPTY_SUBSET]; ASM_MESON_TAC[CONTINUOUS_ON_EMPTY]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [pairwise]) THEN REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[GSYM UNIONS_INSERT_EMPTY] THEN AP_TERM_TAC THEN SET_TAC[]]);; let LUZIN_SIGMA_NESTED = prove (`!f:real^M->real^N s. lebesgue_measurable s /\ f measurable_on s ==> ?k. (!n. compact(k n)) /\ (!n. k n SUBSET s) /\ (!n. f continuous_on k n) /\ (!n. k n SUBSET k(SUC n)) /\ negligible(s DIFF UNIONS {k n | n IN (:num)})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA_EXPLICIT) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `c:num->real^M->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\n. UNIONS {(c:num->real^M->bool) m | m <= n}` THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC COMPACT_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; FINITE_NUMSEG_LE; FINITE_IMAGE]; ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC]; X_GEN_TAC `k:num` THEN MP_TAC(ISPECL [`subtopology euclidean (UNIONS {c m | m:num <= k}:real^M->bool)`; `euclidean:(real^N)topology`; `\n:num. (f:real^M->real^N)`] PASTING_LEMMA_LOCALLY_FINITE) THEN REWRITE_TAC[CONTINUOUS_MAP_EUCLIDEAN; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBTOPOLOGY_SUBTOPOLOGY] THEN ONCE_REWRITE_TAC[TAUT`closed_in a b /\ c <=> ~(closed_in a b ==> ~c)`] THEN SIMP_TAC[ISPEC `euclidean` CLOSED_IN_IMP_SUBSET; SET_RULE `s SUBSET u ==> u INTER s = s`] THEN REWRITE_TAC[NOT_IMP] THEN DISCH_THEN MATCH_MP_TAC THEN MAP_EVERY EXISTS_TAC [`c:num->real^M->bool`; `{n:num | n <= k}`] THEN REWRITE_TAC[FORALL_IN_UNIONS] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM_SIMP_TAC[CLOSED_SUBSET_EQ; COMPACT_IMP_CLOSED; UNIONS_GSPEC] THEN SIMP_TAC[FINITE_RESTRICT; FINITE_NUMSEG_LE] THEN REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN X_GEN_TAC `j:num` THEN DISCH_TAC THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC(MESON[OPEN_IN_REFL] `P(s:real^N->bool) ==> ?t. open_in (subtopology euclidean s) t /\ P t`) THEN ASM SET_TAC[]; REWRITE_TAC[LE; UNIONS_GSPEC] THEN SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN REWRITE_TAC[SUBSET; IN_DIFF; UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN MESON_TAC[LE_REFL]]);; let PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_GEN = prove (`!f:real^M->real^N s. f measurable_on s ==> ((!t. lebesgue_measurable t /\ t SUBSET s ==> lebesgue_measurable (IMAGE f t)) <=> (!t. negligible t /\ t SUBSET s ==> negligible (IMAGE f t)))`, REPEAT STRIP_TAC THEN EQ_TAC THENL [MESON_TAC[PRESERVES_LEBESGUE_MEASURABLE_IMP_PRESERVES_NEGLIGIBLE; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]; DISCH_TAC THEN X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `t:real^M->bool`] LUZIN_SIGMA) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `IMAGE (f:real^M->real^N) t = IMAGE f (UNIONS u) UNION IMAGE f (t DIFF UNIONS u)` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN CONJ_TAC THENL [REWRITE_TAC[IMAGE_UNIONS] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[COMPACT_CONTINUOUS_IMAGE; LEBESGUE_MEASURABLE_COMPACT]; MATCH_MP_TAC NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Several variants of measurability of the Banach indicatrix. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_MEASURABLE_PREIMAGE_CARD_LE = prove (`!f:real^M->real^N s n. f measurable_on s /\ lebesgue_measurable s /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) ==> lebesgue_measurable {y | FINITE {x | x IN s /\ f x = y} /\ CARD {x | x IN s /\ f x = y} <= n}`, REPEAT STRIP_TAC THEN MP_TAC(SPECL[`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA_NESTED) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:num->real^M->bool` THEN STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF THEN EXISTS_TAC `{y | FINITE(UNIONS {{x | x IN c k /\ f x = (y:real^N)} | k IN (:num)}) /\ CARD(UNIONS {{x:real^M | x IN c k /\ f x = y} | k IN (:num)}) <= n}` THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `!s:real^M->bool. lebesgue_measurable s /\ s = t ==> lebesgue_measurable t`) THEN EXISTS_TAC `INTERS {{y | FINITE {x | x IN c k /\ (f:real^M->real^N) x = y} /\ CARD {x | x IN c k /\ (f:real^M->real^N) x = y} <= n} | k IN (:num)}` THEN CONJ_TAC THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS_EXPLICIT THEN X_GEN_TAC `k:num` THEN MATCH_MP_TAC GDELTA_IMP_LEBESGUE_MEASURABLE THEN MATCH_MP_TAC GDELTA_PREIMAGE_CARD_LE THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_IMP_FSIGMA]; REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CARD_LE_UNIONS_CHAIN THEN ASM_REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_GSPEC] THEN MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!m n. m <= n ==> (c:num->real^M->bool) m SUBSET c n` MP_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LE; SET_TAC[]] THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; REWRITE_TAC[SUBSET_INTERS; FORALL_IN_GSPEC] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNIV] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(MESON[FINITE_SUBSET; CARD_SUBSET; LE_TRANS] `s SUBSET t ==> FINITE t /\ CARD t <= n ==> FINITE s /\ CARD s <= n`) THEN REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]]]; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (s DIFF UNIONS {c n | n IN (:num)} )` THEN ASM_SIMP_TAC[SUBSET_DIFF] THEN MATCH_MP_TAC(SET_RULE `(!y. ~(P y <=> Q y) ==> y IN t) ==> ({y | P y} DIFF {y | Q y}) UNION ({y | Q y} DIFF {y | P y}) SUBSET t`) THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `~(FINITE(f y) /\ CARD(f y) <= n <=> FINITE(g y) /\ CARD(g y) <= n) ==> ~(f y = g y)`)) THEN REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]]);; let LEBESGUE_MEASURABLE_PREIMAGE_HAS_SIZE = prove (`!f:real^M->real^N s n. f measurable_on s /\ lebesgue_measurable s /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) ==> lebesgue_measurable {y | {x | x IN s /\ f x = y} HAS_SIZE n}`, REPEAT STRIP_TAC THEN REWRITE_TAC[HAS_SIZE] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LEBESGUE_MEASURABLE_PREIMAGE_CARD_LE) THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `n = 0` THENL [DISCH_THEN(MP_TAC o SPEC `0`) THEN ASM_REWRITE_TAC[LE]; ALL_TAC] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `n - 1` th) THEN MP_TAC(SPEC `n:num` th)) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_DIFF) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[HAS_SIZE] THEN REWRITE_TAC[EXTENSION; IN_DIFF; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN ASM_CASES_TAC `FINITE {x | x IN s /\ (f:real^M->real^N) x = y}` THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC);; let LEBESGUE_MEASURABLE_PREIMAGE_FINITE = prove (`!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) ==> lebesgue_measurable {y | FINITE {x | x IN s /\ f x = y}}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LEBESGUE_MEASURABLE_PREIMAGE_HAS_SIZE) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_COUNTABLE_UNIONS_EXPLICIT) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[HAS_SIZE; UNIONS_GSPEC; IN_UNIV; EXTENSION; IN_ELIM_THM] THEN MESON_TAC[]);; let LEBESGUE_MEASURABLE_PREIMAGE_INFINITE = prove (`!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) ==> lebesgue_measurable {y | INFINITE {x | x IN s /\ f x = y}}`, REWRITE_TAC[INFINITE; SET_RULE `{x | ~P x} = UNIV DIFF {x | P x}`] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL; LEBESGUE_MEASURABLE_PREIMAGE_FINITE]);; let MEASURABLE_ON_BANACH_INDICATRIX = prove (`!f:real^M->real^N s c. f measurable_on s /\ lebesgue_measurable s /\ (!t. t SUBSET s /\ negligible t ==> negligible (IMAGE f t)) ==> (\y. if FINITE {x | x IN s /\ f x = y} then lift(&(CARD{x | x IN s /\ f x = y})) else c) measurable_on (:real^N)`, REPEAT STRIP_TAC THEN SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LE; LEBESGUE_MEASURABLE_UNIV] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop; LIFT_DROP; IN_UNIV] THEN X_GEN_TAC `a:real` THEN REPLICATE_TAC 2 (ONCE_REWRITE_TAC[COND_RAND]) THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[LIFT_DROP] THEN ASM_CASES_TAC `drop c <= a` THEN ASM_REWRITE_TAC[MESON[] `(if p then x else F) <=> p /\ x`; MESON[] `(if p then x else T) <=> ~p \/ p /\ x`] THENL [ONCE_REWRITE_TAC[SET_RULE `{x | ~P x \/ Q x} = (UNIV DIFF {x | P x}) UNION {x | Q x}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPL; LEBESGUE_MEASURABLE_PREIMAGE_FINITE]; ALL_TAC] THEN (ASM_CASES_TAC `a < &0` THEN ASM_SIMP_TAC[REAL_ARITH `a < &0 ==> ~(&n <= a)`] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_EMPTY; EMPTY_GSPEC] THEN RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT]) THEN SUBGOAL_THEN `!n. &n <= a <=> &n <= floor a` (fun th -> ONCE_REWRITE_TAC[th]) THENL [ASM_MESON_TAC[REAL_LE_FLOOR; INTEGER_CLOSED]; ALL_TAC] THEN FIRST_ASSUM(X_CHOOSE_THEN `n:num` SUBST1_TAC o MATCH_MP FLOOR_POS) THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_PREIMAGE_CARD_LE THEN ASM_REWRITE_TAC[]));; (* ------------------------------------------------------------------------- *) (* A measurable subset on which a function is injective, the simple version *) (* first and then some approximate variants with a "better" subsets. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_MEASURABLE_DOMAIN_OF_INJECTIVITY = prove (`!f:real^M->real^N s. f measurable_on s ==> ?t. lebesgue_measurable t /\ t SUBSET s /\ IMAGE f t = IMAGE f s /\ !x y. x IN t /\ y IN t /\ f x = f y ==> x = y`, SUBGOAL_THEN `!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s ==> ?t. lebesgue_measurable t /\ t SUBSET s /\ IMAGE f t = IMAGE f s /\ !x y. x IN t /\ y IN t /\ f x = f y ==> x = y` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `u:(real^M->bool)->bool`] BOREL_DOMAIN_OF_INJECTIVITY_CONTINUOUS_GEN) THEN ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^M->bool` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!y. ?x. y IN IMAGE f s DIFF IMAGE f (UNIONS u) ==> x IN s DIFF UNIONS u /\ (f:real^M->real^N) x = y` MP_TAC THENL [SET_TAC[]; REWRITE_TAC[SKOLEM_THM]] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `b UNION IMAGE (g:real^N->real^M) (IMAGE (f:real^M->real^N) s DIFF IMAGE f (UNIONS u))` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF THEN EXISTS_TAC `b:real^M->bool` THEN ASM_SIMP_TAC[BOREL_IMP_LEBESGUE_MEASURABLE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN ASM SET_TAC[]; ASM SET_TAC[]; ABBREV_TAC `v:real^M->bool = UNIONS u` THEN ASM_REWRITE_TAC[IMAGE_UNION] THEN ASM SET_TAC[]; ABBREV_TAC `v:real^M->bool = UNIONS u` THEN ASM SET_TAC[]]; REPEAT STRIP_TAC THEN SUBGOAL_THEN `lebesgue_measurable {x | x IN s /\ ~((f:real^M->real^N) x = vec 0)}` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o SPEC `(:real^N) DELETE (vec 0)` o GEN_REWRITE_RULE I [MEASURABLE_ON_PREIMAGE_OPEN] o GEN_REWRITE_RULE I [GSYM MEASURABLE_ON_UNIV]) THEN SIMP_TAC[OPEN_DELETE; OPEN_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`f:real^M->real^N`; `{x | x IN s /\ ~((f:real^M->real^N) x = vec 0)}`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[SUBSET_RESTRICT]; REWRITE_TAC[SET_RULE `IMAGE f {x | x IN s /\ ~(f x = a)} = IMAGE f s DELETE a`]] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC) THEN ASM_CASES_TAC `vec 0 IN IMAGE (f:real^M->real^N) s` THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_IMAGE]) THEN DISCH_THEN(X_CHOOSE_THEN `a:real^M` (STRIP_ASSUME_TAC o GSYM)) THEN EXISTS_TAC `(a:real^M) INSERT t` THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_INSERT] THEN ASM SET_TAC[]; EXISTS_TAC `t:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]]);; let BOREL_DOMAIN_OF_INJECTIVITY = prove (`!f:real^M->real^N s. f measurable_on s /\ lebesgue_measurable s /\ (!n. n SUBSET s /\ negligible n ==> negligible(IMAGE f n)) ==> ?t. borel t /\ t SUBSET s /\ negligible(IMAGE f s DIFF IMAGE f t) /\ !x y. x IN t /\ y IN t /\ f x = f y ==> x = y`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `u:(real^M->bool)->bool`] BOREL_DOMAIN_OF_INJECTIVITY_CONTINUOUS_GEN) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^M->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (s DIFF UNIONS u)` THEN ASM_SIMP_TAC[SUBSET_DIFF] THEN ASM SET_TAC[]);; let GDELTA_DOMAIN_OF_INJECTIVITY_MEASURABLE = prove (`!f:real^M->real^N s u e. f measurable_on s /\ lebesgue_measurable s /\ &0 < e /\ IMAGE f s SUBSET u /\ measurable u /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) ==> ?t. t SUBSET s /\ gdelta t /\ bounded t /\ measurable t /\ measurable(IMAGE f t) /\ measure(IMAGE f s DIFF IMAGE f t) < e /\ (!x y. x IN t /\ y IN t /\ f x = f y ==> x = y)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA_NESTED) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `k:num->real^M->bool` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_GEN) THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPEC `\n:num. IMAGE (f:real^M->real^N) s DIFF IMAGE f (k n)` HAS_MEASURE_NESTED_INTERS) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `n:num`; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFF THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[SUBSET_REFL; LEBESGUE_MEASURABLE_COMPACT]; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN SUBGOAL_THEN `measure(INTERS {IMAGE (f:real^M->real^N) s DIFF IMAGE f (k n) | n IN (:num)}) = &0` SUBST1_TAC THENL [MATCH_MP_TAC MEASURE_EQ_0 THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (s DIFF UNIONS {k n | n IN (:num)})` THEN ASM_SIMP_TAC[SUBSET_DIFF] THEN REWRITE_TAC[INTERS_GSPEC; UNIONS_GSPEC] THEN ASM SET_TAC[]; REWRITE_TAC[LIM_SEQUENTIALLY; DIST_LIFT]] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[REAL_SUB_RZERO; LE_REFL] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(k:num->real^M->bool) n`] GDELTA_DOMAIN_OF_INJECTIVITY_CONTINUOUS) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[INJECTIVE_ON_ALT] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[BOUNDED_SUBSET; COMPACT_IMP_BOUNDED]; MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `(k:num->real^M->bool) n` THEN ASM_SIMP_TAC[GDELTA_IMP_LEBESGUE_MEASURABLE; MEASURABLE_COMPACT]; ASM_SIMP_TAC[MEASURABLE_COMPACT; COMPACT_CONTINUOUS_IMAGE]; ASM_SIMP_TAC[REAL_ARITH `abs x < e ==> x < e`]; ASM_SIMP_TAC[]; ASM_MESON_TAC[SUBSET_TRANS]]);; (* ------------------------------------------------------------------------- *) (* Relations between image measure sizes and preimage cardinality. These *) (* are natural generalizations (from continuous functions on intervals) of *) (* two of the "Theorems of Banach" in Saks, "Theory of the Integral" IX.7. *) (* ------------------------------------------------------------------------- *) let LUZIN_NPROPERTY_IMP_COUNTABLE_PREIMAGES = prove (`!f:real^M->real^N s. lebesgue_measurable s /\ f measurable_on s /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) ==> negligible {y | ~COUNTABLE {x | x IN s /\ f x = y}}`, let lemma = prove (`!f:real^M->real^N s. lebesgue_measurable s /\ f measurable_on s /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) ==> ?t. t SUBSET s /\ lebesgue_measurable t /\ negligible(IMAGE f s DIFF IMAGE f t) /\ !y. COUNTABLE {x | x IN t /\ f x = y}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:(real^M->bool)->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `!k. k IN u ==> ?k'. measurable k' /\ k' SUBSET k /\ IMAGE f k' = IMAGE f k /\ !y. FINITE {x | x IN k' /\ (f:real^M->real^N) x = y}` MP_TAC THENL [X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `k:real^M->bool`] GDELTA_DOMAIN_OF_INJECTIVITY_CONTINUOUS) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN REWRITE_TAC[SUBSET_RESTRICT] THEN ASM_SIMP_TAC[MEASURABLE_COMPACT] THEN ASM_SIMP_TAC[GDELTA_IMP_LEBESGUE_MEASURABLE]; MATCH_MP_TAC(MESON[FINITE_SUBSET; FINITE_SING] `(?a. s SUBSET {a}) ==> FINITE s`) THEN ASM SET_TAC[]]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM]] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:(real^M->bool)->(real^M->bool)` THEN DISCH_TAC THEN EXISTS_TAC `UNIONS(IMAGE (c:(real^M->bool)->(real^M->bool)) u)` THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE]; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (s DIFF UNIONS u)` THEN ASM_SIMP_TAC[SUBSET_DIFF] THEN MATCH_MP_TAC (SET_RULE `IMAGE f u = IMAGE f t ==> (IMAGE f s DIFF IMAGE f u) SUBSET IMAGE f (s DIFF t)`) THEN ASM SET_TAC[]; X_GEN_TAC `y:real^N` THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | x IN c k /\ (f:real^M->real^N) x = y} | (k:real^M->bool) IN u}` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC COUNTABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMP_COUNTABLE]]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LUZIN_SIGMA) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:(real^M->bool)->bool` THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS{{y | ~COUNTABLE {x | x IN k /\ f x = y}} | k IN u} UNION IMAGE (f:real^M->real^N) (s DIFF UNIONS u)` THEN REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNION; UNIONS_GSPEC; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (MESON[COUNTABLE_EMPTY] `~COUNTABLE s ==> ~(s = {})`)) THEN REWRITE_TAC[SET_RULE `~({x | x IN s /\ f x = y} = {}) <=> y IN IMAGE f s`] THEN DISCH_TAC THEN REWRITE_TAC[TAUT `p \/ q <=> ~q ==> p`] THEN DISCH_TAC THEN REWRITE_TAC[MESON[] `(?x. P x /\ ~Q x) <=> ~(!x. P x ==> Q x)`] THEN DISCH_TAC THEN UNDISCH_TAC `~COUNTABLE {x | x IN s /\ (f:real^M->real^N) x = y}` THEN REWRITE_TAC[] THEN SUBGOAL_THEN `{x | x IN s /\ (f:real^M->real^N) x = y} = {x | x IN UNIONS u /\ f x = y}` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | x IN k /\ (f:real^M->real^N) x = y} | k IN u}` THEN ASM_SIMP_TAC[COUNTABLE_UNIONS; FORALL_IN_IMAGE; SIMPLE_IMAGE; COUNTABLE_IMAGE] THEN REWRITE_TAC[UNIONS_IMAGE] THEN ASM SET_TAC[]] THEN SUBGOAL_THEN `compact k /\ (f:real^M->real^N) continuous_on k /\ (!t. t SUBSET k /\ negligible t ==> negligible(IMAGE f t))` MP_TAC THENL [ASM_MESON_TAC[SUBSET_TRANS]; ALL_TAC] THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`k:real^M->bool`,`s:real^M->bool`) THEN REPEAT STRIP_TAC THEN ABBREV_TAC `m = sup { measure t | t SUBSET s /\ lebesgue_measurable t /\ negligible(IMAGE f s DIFF IMAGE f t) /\ !y. COUNTABLE {x | x IN t /\ (f:real^M->real^N) x = y}}` THEN FIRST_ASSUM(MP_TAC o C SPEC SUP o rand o lhs o concl) THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(?x. P x) ==> ~({f x | P x} = {})`) THEN MATCH_MP_TAC lemma THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT; CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET]; ASM_MESON_TAC[MEASURE_SUBSET; MEASURABLE_LEBESGUE_MEASURABLE_SUBSET; MEASURABLE_COMPACT]]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `m - inv(&n + &1)`) THEN REWRITE_TAC[REAL_ARITH `m <= m - a <=> ~(&0 < a)`] THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[NOT_FORALL_THM; SKOLEM_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:num->real^M->bool` THEN REWRITE_TAC[FORALL_AND_THM] THEN REWRITE_TAC[REAL_NOT_LE] THEN STRIP_TAC THEN ABBREV_TAC `t:real^M->bool = UNIONS {h n | n IN (:num)}` THEN SUBGOAL_THEN `lebesgue_measurable(t:real^M->bool)` ASSUME_TAC THENL [EXPAND_TAC "t" THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS_EXPLICIT THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s DIFF t:real^M->bool`] lemma) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_DIFF; LEBESGUE_MEASURABLE_COMPACT] THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET; SUBSET_TRANS; SUBSET_DIFF; LEBESGUE_MEASURABLE_DIFF; LEBESGUE_MEASURABLE_COMPACT; CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET]; DISCH_THEN(X_CHOOSE_THEN `u:real^M->bool` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `measurable(t:real^M->bool) /\ measurable(u:real^M->bool) /\ (!n. measurable((h:num->real^M->bool) n))` STRIP_ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[MEASURABLE_COMPACT] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!y. COUNTABLE {x | x IN t /\ (f:real^M->real^N) x = y}` ASSUME_TAC THENL [GEN_TAC THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | x IN h n /\ (f:real^M->real^N) x = y} | n IN (:num)}` THEN ASM_SIMP_TAC[COUNTABLE_UNIONS; FORALL_IN_IMAGE; SIMPLE_IMAGE; COUNTABLE_IMAGE; NUM_COUNTABLE] THEN EXPAND_TAC "t" THEN REWRITE_TAC[UNIONS_IMAGE] THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `negligible(IMAGE (f:real^M->real^N) s DIFF IMAGE f t)` ASSUME_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) s DIFF IMAGE f (h 0)` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "t" THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `measure(t UNION u:real^M->bool) <= m` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_UNION] THEN REWRITE_TAC[SET_RULE `{x | x IN s UNION t /\ P x} = {x | x IN s /\ P x} UNION {x | x IN t /\ P x}`] THEN ASM_REWRITE_TAC[COUNTABLE_UNION] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `(IMAGE f (s DIFF t) DIFF IMAGE f u) UNION (IMAGE (f:real^M->real^N) s DIFF IMAGE f t)` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN SET_TAC[]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_DISJOINT_UNION o lhand o lhand o snd) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `t + u <= m ==> &0 <= u ==> ~(&0 < m - t) ==> u = &0`)) THEN ASM_SIMP_TAC[MEASURE_POS_LE] THEN ANTS_TAC THENL [GEN_REWRITE_TAC RAND_CONV [GSYM ARCH_EVENTUALLY_INV1] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[LE_REFL; REAL_NOT_LT] THEN ONCE_REWRITE_TAC[REAL_ARITH `a - b <= c <=> a - c <= b`] THEN TRANS_TAC REAL_LE_TRANS `measure((h:num->real^M->bool) n)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "t" THEN SET_TAC[]; ASM_SIMP_TAC[MEASURABLE_MEASURE_EQ_0] THEN DISCH_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (s DIFF (t UNION u))` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) u UNION (IMAGE f s DIFF IMAGE f t) UNION (IMAGE f (s DIFF t) DIFF IMAGE f u)` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ]; SET_TAC[]] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]; REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `{x | x IN t /\ (f:real^M->real^N) x = y} UNION {x | x IN u /\ (f:real^M->real^N) x = y}` THEN ASM_REWRITE_TAC[COUNTABLE_UNION] THEN ASM SET_TAC[]]);; let BANACH_SPROPERTY_IMP_LUZIN_NPROPERTY_OUTER = prove (`!f:real^M->real^N s. (!e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> ?u. IMAGE f t SUBSET u /\ measurable u /\ measure u < e) ==> (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t))`, REPEAT GEN_TAC THEN DISCH_THEN(LABEL_TAC "*") THEN REPEAT STRIP_TAC THEN REWRITE_TAC[NEGLIGIBLE_OUTER] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[MEASURE_EQ_0; NEGLIGIBLE_IMP_MEASURABLE]);; let BANACH_SPROPERTY_IMP_LUZIN_NPROPERTY = prove (`!f:real^M->real^N s. (!e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable (IMAGE f t) /\ measure (IMAGE f t) < e) ==> (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t))`, REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC BANACH_SPROPERTY_IMP_LUZIN_NPROPERTY_OUTER THEN ASM_MESON_TAC[SUBSET_REFL]);; let BANACH_SPROPERTY_OUTER = prove (`!f:real^M->real^N s. f measurable_on s /\ (!e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> ?u. IMAGE f t SUBSET u /\ measurable u /\ measure u < e) ==> (!e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable (IMAGE f t) /\ measure (IMAGE f t) < e)`, REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o snd o EQ_IMP_RULE o MATCH_MP PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_GEN) THEN ANTS_TAC THENL [ONCE_REWRITE_TAC[CONJ_SYM] THEN MATCH_MP_TAC BANACH_SPROPERTY_IMP_LUZIN_NPROPERTY_OUTER THEN ASM_REWRITE_TAC[]; DISCH_TAC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`)) THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN ASM_MESON_TAC[MEASURABLE_LEBESGUE_MEASURABLE_SUBSET; MEASURE_SUBSET; REAL_LET_TRANS]);; let BANACH_SPROPERTY_IMP_PRESERVES_MEASURABLE = prove (`!f:real^M->real^N s. (!e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable (IMAGE f t) /\ measure (IMAGE f t) < e) ==> (!t. t SUBSET s /\ measurable t ==> measurable(IMAGE f t))`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_LT_01] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC THEN MP_TAC(SPEC `measure(t:real^M->bool) / d` REAL_ARCH_LT) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN ASM_CASES_TAC `n = 0` THEN ASM_SIMP_TAC[GSYM REAL_NOT_LE; REAL_LE_DIV; REAL_LT_IMP_LE; MEASURE_POS_LE] THEN REWRITE_TAC[REAL_NOT_LE] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_HALF] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN DISCH_TAC THEN MP_TAC(ISPECL [`t:real^M->bool`; `n:num`] MULTIPART_MEASURES) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN EXPAND_TAC "t" THEN REWRITE_TAC[IMAGE_UNIONS] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `c:real^M->bool` THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `measurable s /\ measure s < &1 ==> measurable(s:real^M->bool)`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);; let BANACH_SPROPERTY_IMP_FINITE_PREIMAGES = prove (`!f:real^M->real^N s. f measurable_on s /\ measurable s /\ (!e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable (IMAGE f t) /\ measure (IMAGE f t) < e) ==> negligible {y | INFINITE {x | x IN s /\ f x = y}}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP MEASURABLE_IMP_LEBESGUE_MEASURABLE) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP BANACH_SPROPERTY_IMP_PRESERVES_MEASURABLE) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP BANACH_SPROPERTY_IMP_LUZIN_NPROPERTY) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LEBESGUE_MEASURABLE_PREIMAGE_INFINITE) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LEBESGUE_MEASURABLE_INNER_COMPACT)) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `k:real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `?c. (!n. c n SUBSET s DIFF UNIONS {c m | (m:num) < n}) /\ (!n. measurable(c n)) /\ (!n. measurable(IMAGE (f:real^M->real^N) (c n))) /\ (!n. measure(k:real^N->bool) / &2 <= measure(IMAGE f (c n))) /\ (!n x y. x IN c n /\ y IN c n ==> (f x = f y <=> x = y))` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `?c. !n. c n = @t. t SUBSET s DIFF UNIONS {c m | (m:num) < n} /\ measurable t /\ measurable(IMAGE (f:real^M->real^N) t) /\ measure(k:real^N->bool) / &2 <= measure(IMAGE f t) /\ (!x y. x IN t /\ y IN t ==> (f x = f y <=> x = y))` MP_TAC THENL [MATCH_MP_TAC(MATCH_MP WF_REC WF_num) THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN REWRITE_TAC[UNIONS_GSPEC; MESON[] `(?m:num. m < n /\ a IN f m) <=> ~(!m. m < n ==> ~(a IN f m))`] THEN ASM_SIMP_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `c:num->real^M->bool` THEN REWRITE_TAC[AND_FORALL_THM] THEN DISCH_THEN(fun th -> MATCH_MP_TAC num_WF THEN MP_TAC th) THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN DISCH_THEN SUBST1_TAC THEN DISCH_TAC THEN CONV_TAC SELECT_CONV THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s DIFF UNIONS {(c:num->real^M->bool) m | m < n}`; `IMAGE (f:real^M->real^N) s`; `measure(k:real^N->bool) / &2`] GDELTA_DOMAIN_OF_INJECTIVITY_MEASURABLE) THEN ASM_REWRITE_TAC[REAL_HALF] THEN ANTS_TAC THENL [MATCH_MP_TAC(TAUT `r /\ s /\ (q ==> p) /\ q ==> p /\ q /\ r /\ s`) THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[SUBSET_REFL]] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET_TRANS; SUBSET_DIFF]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET; SUBSET_DIFF]; MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[FINITE_NUMSEG_LT; FINITE_IMAGE]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^M->bool`] THEN REWRITE_TAC[INJECTIVE_ON_ALT] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < k / &2 ==> k <= x + y ==> k / &2 <= y`)) THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_UNION_LE o rand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[SUBSET_REFL] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THENL [REWRITE_TAC[SUBSET_DIFF]; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[FINITE_NUMSEG_LT; FINITE_IMAGE]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS)] THEN ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> (IMAGE f s DIFF IMAGE f t) UNION IMAGE f t = IMAGE f s`] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_COMPACT] THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[SUBSET_DIFF] THEN MATCH_MP_TAC MEASURABLE_DIFF THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[FINITE_NUMSEG_LT; FINITE_IMAGE]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `{y | INFINITE {x | x IN s /\ (f:real^M->real^N) x = y}}` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[SET_RULE `y IN IMAGE f s <=> ~({x | x IN s /\ f x = y} = {})`] THEN MATCH_MP_TAC(MESON[FINITE_EMPTY] `~FINITE s ==> ~(s = {})`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INFINITE]) THEN REWRITE_TAC[CONTRAPOS_THM] THEN SUBGOAL_THEN `FINITE {x | x IN UNIONS {c m | (m:num) < n} /\ (f:real^M->real^N) x = y}` MP_TAC THENL [REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN REWRITE_TAC[INTER_UNIONS; FINITE_UNIONS; SIMPLE_IMAGE] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[GSYM IMAGE_o; FINITE_IMAGE; FINITE_NUMSEG_LT] THEN REWRITE_TAC[FORALL_IN_IMAGE; o_DEF; FORALL_IN_GSPEC; IMAGE_ID] THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN MATCH_MP_TAC(MESON[FINITE_SUBSET; FINITE_SING] `(?a. s SUBSET {a}) ==> FINITE s`) THEN ASM SET_TAC[]; REWRITE_TAC[IMP_IMP; GSYM FINITE_UNION] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN SET_TAC[]]; FIRST_X_ASSUM(MP_TAC o SPEC `measure(k:real^N->bool) / &2`) THEN ASM_SIMP_TAC[REAL_HALF; NOT_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `(c:num->real^M->bool) n`) THEN ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LT] THEN MATCH_MP_TAC(MESON[] `(!n. P n) /\ ~(!n. ~Q n) ==> ?n. P n /\ Q n`) THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[REAL_NOT_LT]] THEN DISCH_TAC THEN MP_TAC(ISPEC `measure(s:real^M->bool) / d` REAL_ARCH_LT) THEN REWRITE_TAC[NOT_EXISTS_THM; REAL_NOT_LT] THEN X_GEN_TAC `n:num` THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ] THEN TRANS_TAC REAL_LE_TRANS `measure(UNIONS {(c:num->real^M->bool) m | m < n})` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o rand o snd) THEN ASM_REWRITE_TAC[FINITE_NUMSEG_LT; FORALL_IN_GSPEC] THEN ANTS_TAC THENL [MATCH_MP_TAC WLOG_LT THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_LT] THEN SIMP_TAC[GSYM SUM_CONST; FINITE_NUMSEG_LT] THEN MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC; FINITE_NUMSEG_LT]; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[UNIONS_SUBSET] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNIONS; ASM SET_TAC[]] THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LT]]]);; let ABSOLUTELY_CONTINUOUS_MEASURE_IMAGE = prove (`!f:real^M->real^N s u. f measurable_on s /\ IMAGE f s SUBSET u /\ measurable u /\ (!t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t)) /\ negligible {y | INFINITE {x | x IN s /\ f x = y}} ==> (!e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable (IMAGE f t) /\ measure (IMAGE f t) < e)`, REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MATCH_MP_TAC(MESON[] `((!d. ~P d) ==> F) ==> ?d. P d`) THEN DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&2 pow n)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_LT_POW2] THEN GEN_REWRITE_TAC (RAND_CONV o BINDER_CONV) [NOT_FORALL_THM] THEN REWRITE_TAC[SKOLEM_THM; NOT_EXISTS_THM; NOT_IMP] THEN X_GEN_TAC `t:num->real^M->bool` THEN REWRITE_TAC[FORALL_AND_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN SUBGOAL_THEN `!n:num. measurable(IMAGE (f:real^M->real^N) (t n))` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN FIRST_ASSUM(MP_TAC o MATCH_MP PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_GEN) THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE]; ASM_REWRITE_TAC[REAL_NOT_LT] THEN DISCH_TAC] THEN MAP_EVERY ABBREV_TAC [`c:real^M->bool = INTERS {UNIONS {t k | n <= k} | n IN (:num)}`; `d = INTERS {UNIONS {IMAGE (f:real^M->real^N) (t k) | n <= k} | n IN (:num)}`] THEN MP_TAC(ISPEC `\n:num. UNIONS {IMAGE (f:real^M->real^N) (t k) | n <= k}` HAS_MEASURE_NESTED_INTERS) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [REWRITE_TAC[ARITH_RULE `SUC n <= k <=> n <= k /\ ~(n = k)`] THEN CONJ_TAC THENL [X_GEN_TAC `n:num`; SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_SUBSET_NUM; FORALL_IN_IMAGE] THEN FIRST_ASSUM(MP_TAC o MATCH_MP PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_GEN) THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)] THEN DISCH_THEN(MP_TAC o SPEC `e:real` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] LIM_DROP_LBOUND)) THEN REWRITE_TAC[LIFT_DROP; TRIVIAL_LIMIT_SEQUENTIALLY; NOT_IMP; REAL_NOT_LE] THEN CONJ_TAC THENL [MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[] THEN TRANS_TAC REAL_LE_TRANS `measure(IMAGE (f:real^M->real^N) (t(n:num)))` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_SUBSET_NUM; FORALL_IN_IMAGE] THEN FIRST_ASSUM(MP_TAC o MATCH_MP PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_GEN) THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE]; MP_TAC(SPEC `n:num` LE_REFL) THEN REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `&0 < e ==> x = &0 ==> x < e`)) THEN ASM_SIMP_TAC[MEASURABLE_MEASURE_EQ_0]] THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN ANTS_TAC THENL [EXPAND_TAC "c" THEN CONJ_TAC THENL [MATCH_MP_TAC INTERS_SUBSET THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[EMPTY_UNIONS; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN ASM SET_TAC[]; REWRITE_TAC[NEGLIGIBLE_OUTER] THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `d:real`] REAL_ARCH_POW_INV) THEN ASM_REWRITE_TAC[REAL_POW_INV] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `UNIONS {t k | SUC N <= k}:real^M->bool` THEN CONJ_TAC THENL [EXPAND_TAC "c" THEN REWRITE_TAC[SUBSET; INTERS_GSPEC; UNIONS_GSPEC] THEN SET_TAC[]; ALL_TAC] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[REAL_LET_TRANS] `a < d ==> measurable(s:real^N->bool) /\ measure s <= a ==> measurable s /\ measure s < d`)) THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_GEN THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_SUBSET_NUM; FORALL_IN_IMAGE] THEN ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `k:num->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_IMAGE_LE o lhand o snd) THEN ASM_SIMP_TAC[o_DEF; MEASURE_POS_LE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN FIRST_ASSUM(X_CHOOSE_THEN `M:num` MP_TAC o ISPEC `\n:num. n` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN REWRITE_TAC[] THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `sum(SUC N..M) (\n. measure(t n:real^M->bool))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN ASM_SIMP_TAC[FINITE_NUMSEG; SUBSET; IN_NUMSEG; MEASURE_POS_LE] THEN ASM SET_TAC[]; TRANS_TAC REAL_LE_TRANS `sum(SUC N..M) (\n. inv(&2 pow n))` THEN ASM_SIMP_TAC[SUM_LE_NUMSEG; REAL_LT_IMP_LE] THEN REWRITE_TAC[SUM_GP; REAL_INV_POW] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN COND_CASES_TAC THEN SIMP_TAC[REAL_POW_LE; REAL_ARITH `&0 <= &1 / &2`; real_pow; REAL_ARITH `(&1 / &2 * x - &1 / &2 * y) / (&1 / &2) <= x <=> &0 <= y`]]]; UNDISCH_TAC `negligible {y | INFINITE {x | x IN s /\ (f:real^M->real^N) x = y}}` THEN REWRITE_TAC[IMP_IMP; GSYM NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET)] THEN EXPAND_TAC "d" THEN GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[INTERS_GSPEC; IN_UNIV; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM] THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM; IN_UNION] THEN ONCE_REWRITE_TAC[TAUT `p \/ q <=> ~q ==> p`] THEN EXPAND_TAC "c" THEN REWRITE_TAC[IN_IMAGE; INTERS_GSPEC; IN_ELIM_THM; IN_UNIV; UNIONS_GSPEC] THEN REWRITE_TAC[NOT_EXISTS_THM; NOT_FORALL_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[INFINITE; RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN DISCH_TAC THEN DISCH_THEN(X_CHOOSE_TAC `r:real^M->num`) THEN FIRST_ASSUM(X_CHOOSE_THEN `M:num` MP_TAC o ISPEC `r:real^M->num` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `k:num` MP_TAC o SPEC `M:num`) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^M` STRIP_ASSUME_TAC) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN REPEAT(ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC]) THEN ASM_MESON_TAC[LE_TRANS]);; let ABSOLUTELY_CONTINUOUS_MEASURE_DIFFERENTIABLE_IMAGE_GEN = prove (`!f:real^N->real^N s. compact s /\ f continuous_on s /\ negligible (IMAGE f {x | x IN s /\ ~(f differentiable (at x within s))}) ==> !e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable(IMAGE f t) /\ measure(IMAGE f t) < e`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_MEASURE_IMAGE THEN EXISTS_TAC `IMAGE (f:real^N->real^N) s` THEN REWRITE_TAC[SUBSET_REFL] THEN ASM_SIMP_TAC[CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET; COMPACT_CONTINUOUS_IMAGE; LEBESGUE_MEASURABLE_COMPACT; MEASURABLE_COMPACT; NEGLIGIBLE_INFINITE_PREIMAGES_MOSTLY_DIFFERENTIABLE] THEN X_GEN_TAC `n:real^N->bool` THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE f {x | x IN s /\ ~(f differentiable at x within s)} UNION IMAGE (f:real^N->real^N) {x | x IN n /\ f differentiable at x within s}` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN REWRITE_TAC[LE_REFL] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN REWRITE_TAC[SUBSET_RESTRICT]; REWRITE_TAC[differentiable_on; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] DIFFERENTIABLE_WITHIN_SUBSET) THEN ASM SET_TAC[]]);; let ABSOLUTELY_CONTINUOUS_MEASURE_DIFFERENTIABLE_IMAGE = prove (`!f:real^N->real^N s. compact s /\ f differentiable_on s ==> !e. &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable(IMAGE f t) /\ measure(IMAGE f t) < e`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_MEASURE_DIFFERENTIABLE_IMAGE_GEN THEN ASM_SIMP_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_ON] THEN RULE_ASSUM_TAC(REWRITE_RULE[differentiable_on]) THEN ASM_REWRITE_TAC[TAUT `p /\ ~q <=> ~(p ==> q)`] THEN REWRITE_TAC[EMPTY_GSPEC; IMAGE_CLAUSES; NEGLIGIBLE_EMPTY]);; (* ------------------------------------------------------------------------- *) (* More refined measurability bounds for Lipschitz or differentiable images. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_MEASURABLE_LOCALLY_LIPSCHITZ_IMAGE = prove (`!f:real^M->real^N s. dimindex(:M) <= dimindex(:N) /\ lebesgue_measurable s /\ (!x. x IN s ==> ?t b. open t /\ x IN t /\ !y. y IN s INTER t ==> norm(f y - f x) <= b * norm(y - x)) ==> lebesgue_measurable(IMAGE f s)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(f:real^M->real^N) continuous_on s` MP_TAC THENL [REWRITE_TAC[continuous_on] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; dist] THEN MAP_EVERY X_GEN_TAC [`t:real^M->bool`; `B:real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [open_def]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d (e / (abs B + &1))` THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MIN; REAL_ARITH `&0 < abs B + &1`] THEN X_GEN_TAC `y:real^M` THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_ARITH `&0 < abs B + &1`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS) THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN TRANS_TAC REAL_LE_TRANS `B * norm(y - x:real^M)` THEN ASM_SIMP_TAC[REAL_LE_RMUL; NORM_POS_LE; REAL_ARITH `B <= abs B + &1`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTER] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[dist]; DISCH_THEN(MP_TAC o MATCH_MP PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE) THEN DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_REFL]] THEN X_GEN_TAC `u:real^M->bool` THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);; let MEASURABLE_LOCALLY_LIPSCHITZ_IMAGE,MEASURE_LOCALLY_LIPSCHITZ_IMAGE = (CONJ_PAIR o prove) (`(!f:real^M->real^N s B. dimindex(:M) <= dimindex(:N) /\ measurable s /\ (!x. x IN s ==> ?t. open t /\ x IN t /\ (!y. y IN s INTER t ==> norm(f y - f x) <= B * norm(y - x))) ==> measurable(IMAGE f s)) /\ (!f:real^M->real^N s B. dimindex(:M) <= dimindex(:N) /\ measurable s /\ (!x. x IN s ==> ?t. open t /\ x IN t /\ (!y. y IN s INTER t ==> norm(f y - f x) <= B * norm(y - x))) ==> measure(IMAGE f s) <= B pow dimindex(:N) * measure s)`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC I [TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. ?e. x IN s ==> &0 < e /\ !y. y IN s INTER ball(x,e) ==> norm((f:real^M->real^N) y - f x) <= B * norm(y - x)` MP_TAC THENL [X_GEN_TAC `x:real^M` THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[OPEN_CONTAINS_BALL; SUBSET; IN_INTER] THEN MESON_TAC[]; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN X_GEN_TAC `r:real^M->real` THEN DISCH_THEN(LABEL_TAC "*") THEN ASM_CASES_TAC `B < &0` THENL [SUBGOAL_THEN `negligible(s:real^M->bool)` ASSUME_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE THEN MATCH_MP_TAC DISCRETE_IMP_COUNTABLE THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN EXISTS_TAC `(r:real^M->real) x` THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M` o CONJUNCT2) THEN ASM_REWRITE_TAC[IN_INTER; IN_BALL] THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN ANTS_TAC THENL [ALL_TAC; CONV_TAC NORM_ARITH] THEN MATCH_MP_TAC(NORM_ARITH `&0 < --x ==> ~(norm(a:real^M) <= x)`) THEN REWRITE_TAC[GSYM REAL_MUL_LNEG] THEN MATCH_MP_TAC REAL_LT_MUL THEN ASM_SIMP_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC; SUBGOAL_THEN `negligible(IMAGE (f:real^M->real^N) s)` ASSUME_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM SET_TAC[]; RULE_ASSUM_TAC(REWRITE_RULE[NEGLIGIBLE_EQ_MEASURE_0]) THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_LE_REFL]]]; RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT])] THEN ASM_CASES_TAC `dimindex(:M) < dimindex(:N)` THENL [SUBGOAL_THEN `negligible(IMAGE (f:real^M->real^N) s)` MP_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE_LOWDIM THEN ASM SET_TAC[]; ASM_SIMP_TAC[NEGLIGIBLE_EQ_MEASURE_0; REAL_POS; REAL_POW_LE; REAL_LE_MUL; MEASURE_POS_LE]]; ALL_TAC] THEN ASM_CASES_TAC `B = &0` THENL [SUBGOAL_THEN `negligible(IMAGE (f:real^M->real^N) s)` MP_TAC THENL [ALL_TAC; ASM_SIMP_TAC[NEGLIGIBLE_EQ_MEASURE_0; REAL_POS; REAL_POW_LE; REAL_LE_MUL; MEASURE_POS_LE]] THEN MP_TAC(ISPEC `{ball(x:real^M,r x) | x IN s}` LINDELOF) THEN REWRITE_TAC[FORALL_IN_IMAGE; SIMPLE_IMAGE; OPEN_BALL] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_COUNTABLE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^M->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (UNIONS (IMAGE (\x. s INTER ball(x,r x)) c))` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE (\x. {(f:real^M->real^N) x}) c)` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE; NEGLIGIBLE_SING]; REWRITE_TAC[IMAGE_UNIONS; GSYM IMAGE_o; o_DEF] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x SUBSET g x) ==> UNIONS (IMAGE f s) SUBSET UNIONS (IMAGE g s)`) THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN SUBGOAL_THEN `(x:real^M) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`)) THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[REAL_MUL_LZERO; NORM_ARITH `norm(x - y:real^M) <= &0 <=> x = y`] THEN SET_TAC[]]; TRANS_TAC SUBSET_TRANS `IMAGE (f:real^M->real^N) (s INTER UNIONS(IMAGE (\x. ball (x,r x)) c))` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[]; REWRITE_TAC[INTER_UNIONS] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; UNIONS_IMAGE; IN_ELIM_THM; IN_INTER] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `x:real^M` THEN ASM_SIMP_TAC[CENTRE_IN_BALL]]; SUBGOAL_THEN `&0 < B` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]] THEN SUBGOAL_THEN `measurable(IMAGE (f:real^M->real^N) s) /\ !e. &0 < e ==> measure(IMAGE f s) <= B pow dimindex(:N) * (measure s + e)` MP_TAC THENL [MATCH_MP_TAC(MESON[REAL_LT_01] `(!e. &0 < e ==> P /\ Q e) ==> P /\ (!e. &0 < e ==> Q e)`); DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `(measure(IMAGE (f:real^M->real^N) s) / B pow dimindex(:N) - measure s) / &2`)) THEN ASM_SIMP_TAC[REAL_SUB_LT; REAL_LT_RDIV_EQ; REAL_ADD_LDISTRIB; REAL_HALF; REAL_LT_IMP_NZ; REAL_DIV_LMUL; REAL_POW_LT; REAL_ARITH `B * (x - y) / &2 = (B * x - B * y) / &2`] THEN REAL_ARITH_TAC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `e:real`] MEASURABLE_OUTER_OPEN) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`{(x,t) | x IN s /\ t <= (r:real^M->real) x /\ ball(x,t) SUBSET u}`; `FST:real^M#real->real^M`; `SND:real^M#real->real`; `s:real^M->bool`] VITALI_COVERING_THEOREM_BALLS) THEN ASM_REWRITE_TAC[EXISTS_IN_GSPEC] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^M`; `d:real`] THEN STRIP_TAC THEN EXISTS_TAC `x:real^M` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d':real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min (d / &2) (min d' (r(x:real^M)))` THEN ASM_SIMP_TAC[CENTRE_IN_BALL; REAL_LT_MIN; REAL_MIN_LE; REAL_LE_REFL] THEN CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN TRANS_TAC SUBSET_TRANS `ball(x:real^M,d')` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_BALL THEN ASM_REAL_ARITH_TAC; DISCH_THEN(X_CHOOSE_THEN `c:real^M#real->bool` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `measurable(IMAGE (f:real^M->real^N) (s INTER UNIONS {ball i | i IN c})) /\ measure(IMAGE (f:real^M->real^N) (s INTER UNIONS {ball i | i IN c})) <= B pow dimindex(:N) * (measure(s:real^M->bool) + e)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] MEASURABLE_NEGLIGIBLE_SYMDIFF); MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^M->real^N) (s DIFF UNIONS {ball i | i IN c})` THEN (CONJ_TAC THENL [ALL_TAC; SET_TAC[]]) THEN MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]] THEN MATCH_MP_TAC(MESON[MEASURABLE_LEBESGUE_MEASURABLE_SUBSET; REAL_LE_TRANS; MEASURE_SUBSET] `!t. lebesgue_measurable s /\ s SUBSET t /\ measurable t /\ measure t <= b ==> measurable s /\ measure s <= b`) THEN EXISTS_TAC `UNIONS {ball((f:real^M->real^N) (FST x),B * SND x) | (x:real^M#real) IN c}` THEN CONJ_TAC THENL [REWRITE_TAC[INTER_UNIONS; SIMPLE_IMAGE; GSYM IMAGE_o; IMAGE_UNIONS] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`)) THEN REWRITE_TAC[FORALL_IN_GSPEC; o_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_INTER; MEASURABLE_BALL] THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[INTER_UNIONS; IMAGE_UNIONS] THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN MATCH_MP_TAC UNIONS_MONO_IMAGE THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`)) THEN REWRITE_TAC[FORALL_IN_GSPEC; o_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `t:real`] THEN STRIP_TAC THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[IN_INTER; IN_BALL] THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `y:real^M`)) THEN ASM_REWRITE_TAC[IN_INTER; GSYM dist; IN_BALL] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [DIST_SYM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN REWRITE_TAC[MEASURABLE_BALL] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `k:real^M#real->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ASM_REWRITE_TAC[MEASURABLE_BALL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN TRANS_TAC REAL_LE_TRANS `sum k (\a:real^M#real. abs B pow dimindex(:N) * measure(ball a))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `t:real`] THEN DISCH_TAC THEN SUBGOAL_THEN `(f:real^M->real^N) x = B % inv(B) % f x` SUBST1_TAC THENL [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]; ALL_TAC] THEN ASM_SIMP_TAC[BALL_SCALING; MEASURE_SCALING; MEASURABLE_BALL] THEN SUBGOAL_THEN `dimindex(:N) = dimindex(:M)` SUBST1_TAC THENL [ASM_ARITH_TAC; MATCH_MP_TAC REAL_LE_LMUL] THEN ASM_SIMP_TAC[REAL_POW_LE; REAL_ABS_POS] THEN ONCE_REWRITE_TAC[GSYM VECTOR_ADD_RID] THEN REWRITE_TAC[BALL_TRANSLATION; MEASURE_TRANSLATION] THEN SUBGOAL_THEN `ball(vec 0:real^N,t) = IMAGE (\x. lambda i. x$i) (ball(vec 0:real^M,t))` SUBST1_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_BALL_0; NORM_LT_SQUARE] THEN SIMP_TAC[dot; LAMBDA_BETA] THEN SUBGOAL_THEN `dimindex(:M) = dimindex(:N)` SUBST1_TAC THENL [ASM_ARITH_TAC; REWRITE_TAC[]] THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN EXISTS_TAC `(lambda i. (y:real^N)$i):real^M` THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN SUBGOAL_THEN `dimindex(:N) = dimindex(:M)` SUBST1_TAC THENL [ASM_ARITH_TAC; SIMP_TAC[LAMBDA_BETA]]; MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC MEASURE_ISOMETRY THEN CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[linear; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA; CART_EQ] THEN X_GEN_TAC `z:real^M` THEN SIMP_TAC[NORM_EQ; dot; LAMBDA_BETA] THEN SUBGOAL_THEN `dimindex(:M) = dimindex(:N)` SUBST1_TAC THENL [ASM_ARITH_TAC; REWRITE_TAC[]]]; REWRITE_TAC[SUM_LMUL] THEN ASM_REWRITE_TAC[real_abs] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_POW_LE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `u < s + e ==> x <= u ==> x <= s + e`)) THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o lhand o snd) THEN ASM_REWRITE_TAC[FORALL_PAIR_THM; MEASURABLE_BALL] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNIONS; FORALL_IN_IMAGE; FORALL_PAIR_THM; MEASURABLE_BALL; FINITE_IMAGE] THEN ASM SET_TAC[]]);; let LEBESGUE_MEASURABLE_LIPSCHITZ_IMAGE = prove (`!f:real^M->real^N s B. dimindex(:M) <= dimindex(:N) /\ lebesgue_measurable s /\ (!x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> lebesgue_measurable(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`(:real^M)`; `B:real`] THEN ASM_SIMP_TAC[OPEN_UNIV; IN_UNIV; IN_INTER]);; let MEASURABLE_LIPSCHITZ_IMAGE = prove (`!f:real^M->real^N s B. dimindex(:M) <= dimindex(:N) /\ measurable s /\ (!x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> measurable(IMAGE f s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_LOCALLY_LIPSCHITZ_IMAGE THEN EXISTS_TAC `B:real` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `(:real^M)` THEN ASM_SIMP_TAC[OPEN_UNIV; IN_UNIV; IN_INTER]);; let MEASURE_LIPSCHITZ_IMAGE = prove (`!f:real^M->real^N s B. dimindex(:M) <= dimindex(:N) /\ measurable s /\ (!x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> measure(IMAGE f s) <= B pow dimindex(:N) * measure s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_LOCALLY_LIPSCHITZ_IMAGE THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `(:real^M)` THEN ASM_SIMP_TAC[OPEN_UNIV; IN_UNIV; IN_INTER]);; let MEASURABLE_BOUNDED_DIFFERENTIABLE_IMAGE, MEASURE_BOUNDED_DIFFERENTIABLE_IMAGE = (CONJ_PAIR o prove) (`(!f:real^N->real^N f' s B. measurable s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s) /\ abs(det(matrix(f' x))) <= B) ==> measurable(IMAGE f s)) /\ (!f:real^N->real^N f' s B. measurable s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s) /\ abs(det(matrix(f' x))) <= B) ==> measure(IMAGE f s) <= B * measure s)`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN ASM_CASES_TAC `B < &0` THENL [ASM_SIMP_TAC[REAL_ARITH `B < &0 ==> ~(abs x <= B)`] THEN SIMP_TAC[GSYM NOT_EXISTS_THM; MEMBER_NOT_EMPTY] THEN REWRITE_TAC[IMAGE_CLAUSES; MEASURE_EMPTY; MEASURABLE_EMPTY] THEN REAL_ARITH_TAC; RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT]) THEN STRIP_TAC] THEN SUBGOAL_THEN `measurable(IMAGE (f:real^N->real^N) s) /\ !e. &0 < e ==> measure(IMAGE f s) <= (B + e) * measure s` MP_TAC THENL [MATCH_MP_TAC(MESON[REAL_LT_01] `(!e. &0 < e ==> P /\ Q e) ==> P /\ (!e. &0 < e ==> Q e)`); FIRST_X_ASSUM(MP_TAC o MATCH_MP MEASURE_POS_LE) THEN REWRITE_TAC[REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[REAL_MUL_RZERO; REAL_LT_01]; FIRST_X_ASSUM(MP_TAC o SPEC `(measure(IMAGE (f:real^N->real^N) s) / measure s - B) / &2`) THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN ASM_REAL_ARITH_TAC]] THEN SUBGOAL_THEN `!d e. &0 < d /\ &0 < e ==> measurable(IMAGE (f:real^N->real^N) s) /\ measure(IMAGE f s) <= (B + e) * (measure s + d)` MP_TAC THENL [ALL_TAC; GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[REAL_LT_01]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `(measure(IMAGE (f:real^N->real^N) s) / (B + e) - measure s) / &2`) THEN SUBGOAL_THEN `&0 < B + e` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[]] THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM REAL_LE_LDIV_EQ)] THEN ASM_REAL_ARITH_TAC] THEN MAP_EVERY X_GEN_TAC [`m:real`; `e:real`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `m:real`] MEASURABLE_OUTER_OPEN) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `(f:real^N->real^N) differentiable_on s` ASSUME_TAC THENL [REWRITE_TAC[differentiable_on; differentiable] THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x d. x IN s /\ &0 < d ==> ?r. &0 < r /\ r < d /\ ball(x,r) SUBSET u /\ measurable(IMAGE (f:real^N->real^N) (s INTER ball(x,r))) /\ measure(IMAGE f (s INTER ball(x,r))) <= (B + e) * measure(ball(x,r))` MP_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE ((f':real^N->real^N->real^N) x) (ball(vec 0,&1))`; `e * measure(ball(vec 0:real^N,&1))`] MEASURE_SEMICONTINUOUS_WITH_HAUSDIST_EXPLICIT) THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [has_derivative_within]) THEN ASM_SIMP_TAC[CONVEX_LINEAR_IMAGE; CONVEX_BALL; NEGLIGIBLE_CONVEX_FRONTIER] THEN ASM_SIMP_TAC[BOUNDED_LINEAR_IMAGE; BOUNDED_BALL] THEN ASM_SIMP_TAC[REAL_LT_MUL; MEASURE_BALL_POS; REAL_LT_01] THEN DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `l:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_WITHIN]) THEN DISCH_THEN(MP_TAC o SPEC `k:real`) THEN ASM_REWRITE_TAC[DIST_0] THEN DISCH_THEN(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN ABBREV_TAC `r = min (min l (n / &2)) (min (&1) (d / &2))` THEN EXISTS_TAC `r:real` THEN REPLICATE_TAC 2 (MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_TAC]) THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `ball(x:real^N,l)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_BALL THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\x. inv(r) % x) (IMAGE (\y. --(f x) + y) (IMAGE (f:real^N->real^N) (s INTER ball(x,r))))`) THEN ASM_SIMP_TAC[MEASURE_LINEAR_IMAGE; MEASURABLE_BALL; MEASURE_TRANSLATION; MEASURABLE_TRANSLATION] THEN MATCH_MP_TAC(TAUT `q /\ (q ==> p) /\ (p ==> r ==> s) ==> (p /\ q ==> r) ==> s`) THEN CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_IMAGE; EXISTS_IN_IMAGE; IN_INTER; IN_BALL] THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN ASM_CASES_TAC `y:real^N = x` THENL [EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[DIST_REFL; REAL_LT_01] THEN FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP LINEAR_0) THEN REWRITE_TAC[VECTOR_ARITH `r % (--x + x):real^N = vec 0`] THEN ASM_REWRITE_TAC[DIST_REFL]; EXISTS_TAC `inv(r) % (y - x):real^N` THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]) THEN ASM_REWRITE_TAC[DIST_0; NORM_MUL; REAL_ABS_INV; DIST_MUL] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_ARITH `&0 < x ==> &0 < abs x`] THEN REWRITE_TAC[NORM_ARITH `norm(y - x:real^N) < &1 * r <=> dist(x,y) < r`] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[GSYM DIST_NZ] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; GSYM dist; GSYM DIST_NZ] THEN MATCH_MP_TAC(NORM_ARITH `a <= b ==> dist(y:real^N,x + f) < a ==> dist(--x + y,f) < b`) THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN CONJ_TAC THENL [DISCH_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) LEBESGUE_MEASURABLE_IFF_MEASURABLE o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `{a + b | a IN IMAGE ((f':real^N->real^N->real^N) x) (ball(vec 0,&1)) /\ b IN ball(vec 0,k)}` THEN ASM_SIMP_TAC[BOUNDED_SUMS; BOUNDED_BALL; BOUNDED_LINEAR_IMAGE] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> r /\ p /\ q`] THEN REWRITE_TAC[UNWIND_THM2; VECTOR_ARITH `x:real^N = a + b <=> b = x - a`] THEN REWRITE_TAC[IN_BALL_0; GSYM dist] THEN ASM_MESON_TAC[DIST_SYM]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN THEN REWRITE_TAC[LE_REFL; LINEAR_SCALING] THEN REWRITE_TAC[LEBESGUE_MEASURABLE_TRANSLATION] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_SIMP_TAC[LE_REFL; MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_INTER; MEASURABLE_BALL] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]; ALL_TAC] THEN DISCH_THEN(MP_TAC o ISPEC `\x:real^N. r % x` o MATCH_MP(ONCE_REWRITE_RULE[IMP_CONJ_ALT] MEASURABLE_LINEAR_IMAGE)) THEN REWRITE_TAC[LINEAR_SCALING] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM IMAGE_o] THEN ASM_SIMP_TAC[o_DEF; VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN REWRITE_TAC[VECTOR_MUL_LID; IMAGE_ID] THEN DISCH_THEN(MP_TAC o SPEC `(f:real^N->real^N) x` o MATCH_MP MEASURABLE_TRANSLATION) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM IMAGE_o] THEN REWRITE_TAC[o_DEF; VECTOR_ARITH `x + --x + y:real^N = y`] THEN REWRITE_TAC[IMAGE_ID] THEN DISCH_TAC THEN ASM_SIMP_TAC[MEASURE_SCALING; MEASURABLE_TRANSLATION] THEN REWRITE_TAC[MEASURE_TRANSLATION; REAL_ABS_INV; REAL_POW_INV] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_POW_LT; REAL_ARITH `&0 < r ==> &0 < abs r`] THEN MATCH_MP_TAC(REAL_ARITH `y <= z ==> x < y ==> x <= z`) THEN ONCE_REWRITE_TAC[REAL_ARITH `(a * b + c) * r:real = a * r * b + c * r`] THEN SIMP_TAC[GSYM MEASURE_SCALING; MEASURABLE_BALL] THEN ASM_SIMP_TAC[GSYM BALL_SCALING] THEN MATCH_MP_TAC(REAL_ARITH `x <= a * m /\ w * z * y <= b * m ==> x + (w * y) * z <= (a + b) * m`) THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REWRITE_TAC[REAL_ABS_POS] THEN SIMP_TAC[MEASURE_POS_LE; MEASURABLE_BALL] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; GSYM REAL_MUL_ASSOC]; ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN SIMP_TAC[GSYM MEASURE_SCALING; MEASURABLE_BALL] THEN ASM_SIMP_TAC[GSYM BALL_SCALING]] THEN REWRITE_TAC[VECTOR_MUL_RZERO; REAL_MUL_RID] THEN SUBST1_TAC(VECTOR_ARITH `x:real^N = x + vec 0`) THEN REWRITE_TAC[BALL_TRANSLATION; MEASURE_TRANSLATION; REAL_LE_REFL]; ALL_TAC] THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; LEFT_IMP_EXISTS_THM; SKOLEM_THM] THEN X_GEN_TAC `r:real^N->real->real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`{(x:real^N,(r:real^N->real->real) x t) | x IN s /\ &0 < t}`; `FST:real^N#real->real^N`; `SND:real^N#real->real`; `s:real^N->bool`] VITALI_COVERING_THEOREM_BALLS) THEN ASM_REWRITE_TAC[EXISTS_IN_GSPEC] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^N`; `d:real`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `d:real`] THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ASM_MESON_TAC[]; DISCH_THEN(X_CHOOSE_THEN `c:real^N#real->bool` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `measurable(IMAGE (f:real^N->real^N) (s INTER UNIONS {ball i | i IN c})) /\ measure(IMAGE (f:real^N->real^N) (s INTER UNIONS {ball i | i IN c})) <= (B + e) * (measure s + m)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] MEASURABLE_NEGLIGIBLE_SYMDIFF); MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^N->real^N) (s DIFF UNIONS {ball i | i IN c})` THEN (CONJ_TAC THENL [ALL_TAC; SET_TAC[]]) THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN ASM_REWRITE_TAC[LE_REFL] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN REWRITE_TAC[SUBSET_DIFF; differentiable_on; differentiable] THEN ASM_MESON_TAC[]] THEN REWRITE_TAC[INTER_UNIONS; SIMPLE_IMAGE; GSYM IMAGE_o; IMAGE_UNIONS] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN REWRITE_TAC[o_DEF] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`)) THEN REWRITE_TAC[FORALL_IN_GSPEC; o_THM] THEN ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `k:real^N#real->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ASM_REWRITE_TAC[MEASURABLE_BALL] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`)) THEN REWRITE_TAC[FORALL_IN_GSPEC; o_THM] THEN ASM SET_TAC[]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)] THEN TRANS_TAC REAL_LE_TRANS `sum k (\a:real^N#real. (B + e) * measure(ball a))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN REPEAT(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> (!x. x IN t ==> P x) ==> (!x. x IN s ==> P x)`))) THEN REWRITE_TAC[FORALL_IN_GSPEC; o_THM] THEN ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUM_LMUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `u < s + e ==> x <= u ==> x <= s + e`)) THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNIONS_IMAGE o lhand o snd) THEN ASM_REWRITE_TAC[FORALL_PAIR_THM; MEASURABLE_BALL] THEN ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNIONS; FORALL_IN_IMAGE; FORALL_PAIR_THM; MEASURABLE_BALL; FINITE_IMAGE] THEN ASM SET_TAC[]);; let MEASURABLE_DIFFERENTIABLE_IMAGE,MEASURE_DIFFERENTIABLE_IMAGE = (CONJ_PAIR o prove) (`(!f:real^N->real^N f' s. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f'(x)) (at x within s)) /\ (\x. lift(abs(det(matrix(f' x))))) integrable_on s ==> measurable(IMAGE f s)) /\ (!f:real^N->real^N f' s b. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f'(x)) (at x within s)) /\ (\x. lift(abs(det(matrix(f' x))))) integrable_on s /\ drop(integral s (\x. lift(abs(det(matrix(f' x)))))) <= b ==> measure(IMAGE f s) <= b)`, let lemma = prove (`!f:real^N->real^N f' s. measurable s /\ (!x. x IN s ==> (f has_derivative f'(x)) (at x within s)) /\ (\x. lift(abs(det(matrix(f' x))))) integrable_on s ==> measurable(IMAGE f s) /\ measure(IMAGE f s) <= drop(integral s (\x. lift(abs(det(matrix(f' x))))))`, REPEAT GEN_TAC THEN STRIP_TAC THEN ABBREV_TAC `m = integral s (\x:real^N. lift(abs(det(matrix(f' x):real^N^N))))` THEN SUBGOAL_THEN `measurable(IMAGE (f:real^N->real^N) s) /\ !e. &0 < e ==> measure(IMAGE (f:real^N->real^N) s) <= drop m + e * measure s` MP_TAC THENL [MATCH_MP_TAC(MESON[REAL_LT_01] `(!e. &0 < e ==> P /\ Q e) ==> P /\ (!e. &0 < e ==> Q e)`); MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP MEASURE_POS_LE) THEN REWRITE_TAC[REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_RID] THENL [MESON_TAC[REAL_LT_01]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN REWRITE_TAC[REAL_ARITH `(m + e) - f:real = (m - f) + e`] THEN ABBREV_TAC `x = drop m - measure (IMAGE (f:real^N->real^N) s)` THEN DISCH_THEN(MP_TAC o SPEC `--x / &2 / measure(s:real^N->bool)`) THEN ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ; REAL_LT_RDIV_EQ] THEN REAL_ARITH_TAC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN ABBREV_TAC `t = \n. {x | x IN s /\ &n * e <= abs(det(matrix(f'(x:real^N)):real^N^N)) /\ abs(det(matrix(f' x))) < (&n + &1) * e}` THEN SUBGOAL_THEN `!n. measurable((t:num->real^N->bool) n)` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "t" THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x /\ Q x} = {x | x IN s /\ P x} INTER {x | x IN s /\ Q x}`] THEN MATCH_MP_TAC MEASURABLE_INTER THEN CONJ_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE) THENL [ASM_SIMP_TAC[MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GE]; ASM_SIMP_TAC[MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LT]] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; real_ge] THEN SIMP_TAC[GSYM drop; LIFT_DROP]; ALL_TAC] THEN SUBGOAL_THEN `s:real^N->bool = UNIONS {t n | n IN (:num)}` (fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [th] THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [th]) THENL [EXPAND_TAC "t" THEN REWRITE_TAC[UNIONS_GSPEC; IN_UNIV; IN_ELIM_THM] THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; GSYM REAL_LT_LDIV_EQ] THEN MP_TAC(ISPEC `abs(det(matrix(f'(x:real^N)):real^N^N)) / e` FLOOR_POS) THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE; REAL_ABS_POS] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[FLOOR]; ALL_TAC] THEN REWRITE_TAC[IMAGE_UNIONS] THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE THEN REWRITE_TAC[o_THM] THEN CONJ_TAC THEN X_GEN_TAC `n:num` THENL [MATCH_MP_TAC MEASURABLE_BOUNDED_DIFFERENTIABLE_IMAGE THEN MAP_EVERY EXISTS_TAC [`f':real^N->real^N->real^N`; `(&n + &1) * e`] THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "t" THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HAS_DERIVATIVE_WITHIN_SUBSET) THEN SET_TAC[]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `sum(0..n) (\k. ((&k + &1) * e) * measure(t k:real^N->bool))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE_NUMSEG THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC MEASURE_BOUNDED_DIFFERENTIABLE_IMAGE THEN EXISTS_TAC `f':real^N->real^N->real^N` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "t" THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HAS_DERIVATIVE_WITHIN_SUBSET) THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[REAL_ADD_RDISTRIB; GSYM REAL_MUL_ASSOC; SUM_ADD_NUMSEG] THEN REWRITE_TAC[REAL_MUL_LID] THEN ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THENL [ASM_SIMP_TAC[SUM_LMUL; REAL_LE_LMUL_EQ] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_DISJOINT_UNIONS_IMAGE_STRONG o lhand o snd) THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_NUMSEG] THEN ANTS_TAC THENL [MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[ARITH_RULE `k < n <=> k + 1 <= n`] THEN EXPAND_TAC "t" THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_ADD; SET_RULE `DISJOINT {x | P x} {x | Q x} <=> !x. P x /\ Q x ==> F`] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; GSYM REAL_LT_LDIV_EQ] THEN REAL_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNIONS; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; UNIONS_SUBSET] THEN EXPAND_TAC "t" THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUM_VSUM; o_DEF; LIFT_CMUL] THEN ASM_SIMP_TAC[GSYM INTEGRAL_MEASURE] THEN ASM_SIMP_TAC[GSYM INTEGRAL_CMUL; INTEGRABLE_ON_CONST] THEN TRANS_TAC REAL_LE_TRANS `drop(vsum (0..n) (\k. integral (t k) (\x:real^N. lift(abs(det(matrix(f' x):real^N^N))))))` THEN CONJ_TAC THENL [REWRITE_TAC[DROP_VSUM; o_DEF] THEN MATCH_MP_TAC SUM_LE_NUMSEG THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[INTEGRABLE_ON_CONST; DROP_CMUL] THEN REWRITE_TAC[DROP_VEC; LIFT_DROP; REAL_MUL_RID] THEN CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "t" THEN SET_TAC[]] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "t" THEN SET_TAC[]] THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN ASM_REWRITE_TAC[DIMINDEX_1; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; FORALL_1; GSYM drop; LIFT_DROP; REAL_ABS_POS]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `drop(vsum {t k | k IN 0..n} (\t. integral t (\x:real^N. lift(abs(det(matrix(f' x):real^N^N))))))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN REWRITE_TAC[SIMPLE_IMAGE] THEN W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE_NONZERO o rand o snd) THEN REWRITE_TAC[FINITE_NUMSEG] THEN ANTS_TAC THENL [MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[ARITH_RULE `k < n <=> k + 1 <= n`] THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_ADD] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `i = j ==> DISJOINT i j ==> i = {}`)) THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[INTEGRAL_EMPTY]] THEN EXPAND_TAC "t" THEN REWRITE_TAC[SET_RULE `DISJOINT {x | P x} {x | Q x} <=> !x. P x /\ Q x ==> F`] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; GSYM REAL_LT_LDIV_EQ] THEN ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF]]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `drop(integral (UNIONS {t k | k IN 0..n}) (\x:real^N. lift(abs(det(matrix(f' x):real^N^N)))))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC HAS_INTEGRAL_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE; FINITE_NUMSEG] THEN CONJ_TAC THENL [X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "t" THEN SET_TAC[]] THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN ASM_REWRITE_TAC[DIMINDEX_1; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; FORALL_1; GSYM drop; LIFT_DROP; REAL_ABS_POS]; REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[FORALL_IN_IMAGE_2] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[INTER_COMM]; ALL_TAC] THEN REWRITE_TAC[ARITH_RULE `k < n <=> k + 1 <= n`] THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_ADD] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s = {} ==> negligible s`) THEN EXPAND_TAC "t" THEN REWRITE_TAC[SET_RULE `{x | P x} INTER {x | Q x} = {} <=> !x. P x /\ Q x ==> F`] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; GSYM REAL_LT_LDIV_EQ] THEN ASM_REAL_ARITH_TAC]; EXPAND_TAC "m" THEN MATCH_MP_TAC INTEGRAL_SUBSET_DROP_LE THEN ASM_REWRITE_TAC[LIFT_DROP; REAL_ABS_POS; UNIONS_SUBSET] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN CONJ_TAC THENL [EXPAND_TAC "t" THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN ASM_REWRITE_TAC[DIMINDEX_1; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; FORALL_1; GSYM drop; LIFT_DROP; REAL_ABS_POS]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN EXPAND_TAC "t" THEN SET_TAC[]; MATCH_MP_TAC LEBESGUE_MEASURABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; MEASURABLE_IMP_LEBESGUE_MEASURABLE]]]) in REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; MESON[REAL_LE_REFL; REAL_LE_TRANS] `(!a. x <= a ==> y <= a) <=> y <= x`] THEN REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN STRIP_TAC THEN SUBGOAL_THEN `s = UNIONS {interval[--vec n:real^N,vec n] INTER s | n IN (:num)}` (fun th -> SUBST1_TAC th THEN REWRITE_TAC[IMAGE_UNIONS] THEN REWRITE_TAC[SET_RULE `IMAGE f {g x | x IN s} = {f(g x) | x IN s}`] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG THEN SUBST1_TAC(SYM th)) THENL [REWRITE_TAC[UNIONS_GSPEC] THEN SUBGOAL_THEN `!x:real^N. ?n. x IN interval[--vec n,vec n]` MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN X_GEN_TAC `x:real^N` THEN MP_TAC(ISPEC `norm(x:real^N)` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[IN_INTERVAL; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN ASM_MESON_TAC[REAL_ABS_BOUNDS; COMPONENT_LE_NORM; REAL_LE_TRANS]; ALL_TAC] THEN REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `n:num` THEN SUBGOAL_THEN `UNIONS {IMAGE f (interval[--vec k,vec k] INTER s) | k <= n} = IMAGE (f:real^N->real^N) (interval [--vec n,vec n] INTER s)` SUBST1_TAC THENL [REWRITE_TAC[UNIONS_GSPEC] THEN SUBGOAL_THEN `!x:real^N n. x IN interval[--vec n,vec n] <=> ?k. k <= n /\ x IN interval[--vec k,vec k]` MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN REPEAT GEN_TAC THEN EQ_TAC THENL [MESON_TAC[LE_REFL]; ALL_TAC] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `k:num` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN REWRITE_TAC[SUBSET_INTERVAL; VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN ASM_REWRITE_TAC[REAL_OF_NUM_LE; REAL_LE_NEG2]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `interval[--vec n:real^N,vec n] INTER s`] lemma) THEN ASM_SIMP_TAC[MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE; MEASURABLE_INTERVAL] THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_WITHIN_SUBSET; INTER_SUBSET; IN_INTER]; ALL_TAC]; MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC INTEGRAL_SUBSET_DROP_LE THEN ASM_REWRITE_TAC[LIFT_DROP; REAL_ABS_POS; INTER_SUBSET]] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_INTER] THEN MATCH_MP_TAC INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[SUBSET_UNIV; INTEGRABLE_RESTRICT_UNIV]);; let NEGLIGIBLE_DIFFERENTIABLE_PREIMAGE = prove (`!f:real^M->real^N f' s t. dimindex(:M) = dimindex(:N) /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s) /\ rank(matrix(f' x)) = dimindex(:N)) /\ negligible t ==> negligible {x | x IN s /\ f x IN t}`, REPEAT STRIP_TAC THEN ABBREV_TAC `u = \n. {x | x IN s /\ !y. y IN s /\ norm(y - x) < inv(&n + &1) ==> norm(y - x:real^M) <= (&n + &1) * norm(f y - f x:real^N)}` THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | x IN u n /\ (f:real^M->real^N) x IN t} | n IN (:num)}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `n:num` THEN ONCE_REWRITE_TAC[LOCALLY_NEGLIGIBLE_ALT] THEN X_GEN_TAC `a:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN ABBREV_TAC `v = {x | x IN u n /\ (f:real^M->real^N) x IN t} INTER ball(a,inv(&n + &1) / &2)` THEN EXISTS_TAC `v:real^M->bool` THEN EXPAND_TAC "v" THEN SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN ASM_REWRITE_TAC[IN_ELIM_THM; IN_INTER; CENTRE_IN_BALL; REAL_HALF] THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN SUBGOAL_THEN `!x y:real^M. x IN v /\ y IN v ==> norm(x - y) <= (&n + &1) * norm(f x - f y:real^N)` ASSUME_TAC THENL [EXPAND_TAC "v" THEN REWRITE_TAC[IN_ELIM_THM; IN_INTER; IN_BALL] THEN EXPAND_TAC "u" THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[NORM_ARITH `dist(a:real^N,x) < e / &2 /\ dist(a,y) < e / &2 ==> norm(x - y) < e`]; ALL_TAC] THEN SUBGOAL_THEN `?g. !x. x IN v ==> g((f:real^M->real^N) x) = x` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM INJECTIVE_ON_LEFT_INVERSE] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `y:real^M`]) THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; real_div; NORM_0; REAL_MUL_LZERO] THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^N->real^M) (IMAGE (f:real^M->real^N) v INTER t)` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC NEGLIGIBLE_LOCALLY_LIPSCHITZ_IMAGE THEN CONJ_TAC THENL [ASM_MESON_TAC[LE_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[IN_INTER; IMP_CONJ; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET; INTER_SUBSET]; ALL_TAC] THEN X_GEN_TAC `x:real^M` THEN REPEAT DISCH_TAC THEN EXISTS_TAC `ball((f:real^M->real^N) x,&1)` THEN REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_01] THEN ASM_MESON_TAC[]; REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNIV; UNIONS_GSPEC] THEN X_GEN_TAC `x:real^M` THEN SIMP_TAC[] THEN DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[HAS_DERIVATIVE_WITHIN_ALT] THEN STRIP_TAC THEN MP_TAC(ISPEC `\h:real^M. matrix((f':real^M->real^M->real^N) x) ** h` LINEAR_INJECTIVE_LEFT_INVERSE) THEN ASM_REWRITE_TAC[GSYM FULL_RANK_INJECTIVE] THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_LINEAR; MATRIX_WORKS] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; REWRITE_TAC[ETA_AX]] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] LINEAR_INVERTIBLE_BOUNDED_BELOW_POS)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN MP_TAC(SPEC `B:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC; NOT_SUC] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(K ALL_TAC) THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `B - inv(&n + &1)`) THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `d:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC; NOT_SUC] THEN X_GEN_TAC `m:num` THEN DISCH_THEN(K ALL_TAC) THEN STRIP_TAC THEN EXISTS_TAC `MAX m n` THEN EXPAND_TAC "u" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LT_TRANS `inv(&m + &1)` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LTE_TRANS `inv(&(MAX m n) + &1)` THEN ASM_SIMP_TAC[]; REWRITE_TAC[REAL_SUB_RDISTRIB] THEN MATCH_MP_TAC(NORM_ARITH `b <= norm(y) /\ d <= c ==> norm(f - f' - y:real^N) <= b - c ==> d <= norm(f - f')`) THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE]] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* A one-way version of change-of-variables not assuming injectivity. *) (* ------------------------------------------------------------------------- *) let ABSOLUTELY_INTEGRABLE_ON_IMAGE,INTEGRAL_ON_IMAGE_DROP_UBOUND_LE = let lemma = prove (`!a:real. {x | x IN s /\ f x <= a} = UNIONS {{x | x IN s /\ f x = b} |b| b IN IMAGE f s /\ b <= a}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in let version0 = prove (`!f:real^N->real^1 g:real^N->real^N g' s b. lebesgue_measurable s /\ f measurable_on (IMAGE g s) /\ (!x. x IN s ==> &0 <= drop(f(g x))) /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!t. t SUBSET IMAGE g s /\ lebesgue_measurable t ==> lebesgue_measurable {x | x IN s /\ g x IN t}) /\ (\x. abs(det(matrix(g' x))) % f(g x)) integrable_on s /\ drop (integral s (\x. abs(det(matrix (g' x))) % f(g x))) <= b ==> f integrable_on (IMAGE g s) /\ drop(integral (IMAGE g s) f) <= b`, REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `?h:num->real^N->real^1. (!n x. &0 <= drop(h n x)) /\ (!n x. x IN IMAGE (g:real^N->real^N) s ==> drop(h n x) <= drop(f x)) /\ (!n x. drop(h n x) <= drop(h (SUC n) x)) /\ (!n. (h n) measurable_on (:real^N)) /\ (!n. FINITE(IMAGE (h n) (:real^N))) /\ (!x. x IN IMAGE g s ==> ((\n. h n x) --> f x) sequentially)` STRIP_ASSUME_TAC THENL [MP_TAC(fst(EQ_IMP_RULE(ISPEC `\x. if x IN IMAGE (g:real^N->real^N) s then (f:real^N->real^1) x else vec 0` MEASURABLE_ON_SIMPLE_FUNCTION_LIMIT_INCREASING))) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[MEASURABLE_ON_UNIV] THEN GEN_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[DROP_VEC; REAL_POS] THEN ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `!n y. lebesgue_measurable {x:real^N | x IN s /\ g x IN {u | (h:num->real^N->real^1) n u = y}}` MP_TAC THENL [REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{x | x IN s /\ g x IN {x | P x}} = {x | x IN s /\ g x IN {y | y IN IMAGE g s /\ P y}}`] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[SUBSET_RESTRICT] THEN ONCE_REWRITE_TAC[GSYM IN_SING] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED THEN REWRITE_TAC[CLOSED_SING; ETA_AX] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[SUBSET_UNIV]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_REWRITE_TAC[LE_REFL; differentiable_on; differentiable] THEN ASM_MESON_TAC[]; REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC] THEN SUBGOAL_THEN `!k. (h:num->real^N->real^1) k integrable_on IMAGE (g:real^N->real^N) s /\ drop(integral (IMAGE g s) (h k)) <= drop(integral s (\x. abs(det(matrix(g' x):real^N^N)) % h k (g x)))` MP_TAC THENL [X_GEN_TAC `n:num` THEN ABBREV_TAC `r = IMAGE ((h:num->real^N->real^1) n) (:real^N)` THEN SUBGOAL_THEN `FINITE(r:real^1->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(h:num->real^N->real^1) n = \x. vsum r (\y. drop y % indicator {x | h n x = y} x)` (fun th -> ONCE_REWRITE_TAC[th]) THENL [REWRITE_TAC[indicator; COND_RAND; VECTOR_MUL_RZERO; IN_ELIM_THM] THEN REWRITE_TAC[GSYM VSUM_RESTRICT_SET; FUN_EQ_THM] THEN X_GEN_TAC `x:real^N` THEN SUBGOAL_THEN `(h:num->real^N->real^1) n x IN r` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[SET_RULE `a IN s ==> {y | y IN s /\ a = y} = {a}`] THEN REWRITE_TAC[VSUM_SING] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; DROP_VEC; REAL_MUL_RID]; ALL_TAC] THEN SUBGOAL_THEN `!y. y IN r ==> (\t. drop y % indicator{x | (h:num->real^N->real^1) n x = y} t) integrable_on (IMAGE (g:real^N->real^N) s) /\ drop(integral (IMAGE g s) (\t. drop y % indicator {x | h n x = y} t)) <= drop(integral s (\t. abs(det(matrix(g' t):real^N^N)) % drop y % (indicator {x | h n x = y} (g t))))` ASSUME_TAC THENL [X_GEN_TAC `y:real^1` THEN DISCH_TAC THEN ASM_CASES_TAC `y:real^1 = vec 0` THEN ASM_REWRITE_TAC[DROP_VEC; VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; INTEGRABLE_0; INTEGRAL_0; REAL_LE_REFL] THEN ASM_REWRITE_TAC[INTEGRABLE_CMUL_EQ; GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM; ETA_AX] THEN ONCE_REWRITE_TAC[TAUT `p /\ q <=> ~(p ==> ~q)`] THEN SIMP_TAC[INTEGRAL_CMUL; ETA_AX] THEN REWRITE_TAC[INTEGRABLE_ON_INDICATOR] THEN SIMP_TAC[INTEGRAL_INDICATOR; DROP_CMUL; LIFT_DROP; ETA_AX] THEN REWRITE_TAC[indicator; COND_RAND; VECTOR_MUL_RZERO] THEN REWRITE_TAC[REWRITE_RULE[IN] INTEGRAL_RESTRICT_INTER; NOT_IMP] THEN REWRITE_TAC[SET_RULE `(\x. g x IN {x | h n x = y}) INTER s = {x | x IN s /\ h n (g x) = y}`] THEN REWRITE_TAC[SET_RULE `{x | h n x = y} INTER IMAGE g s = IMAGE g {x | x IN s /\ h n (g x) = y}`] THEN REWRITE_TAC[GSYM LIFT_NUM; GSYM LIFT_CMUL; REAL_MUL_RID] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL; LIFT_DROP] THEN SUBGOAL_THEN `(\t:real^N. lift(abs(det(matrix(g' t):real^N^N)))) integrable_on {t | t IN s /\ (h:num->real^N->real^1) n (g t) = y}` ASSUME_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = {x | P x} INTER s`] THEN REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_INTER] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `\x. inv(drop y) % abs(det(matrix(g' x):real^N^N)) % (f:real^N->real^1) (g(x:real^N))` THEN ASM_SIMP_TAC[INTEGRABLE_CMUL] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MESON[] `(if p then if q then x else vec 0 else vec 0) = (if p /\ q then x else vec 0)`] THEN REWRITE_TAC[SET_RULE `x IN s /\ x IN t <=> x IN (s INTER t)`] THEN REWRITE_TAC[MEASURABLE_ON_UNIV] THEN MATCH_MP_TAC MEASURABLE_ON_LIFT_ABS THEN MATCH_MP_TAC MEASURABLE_ON_DET_JACOBIAN THEN EXISTS_TAC `g:real^N->real^N` THEN ASM_REWRITE_TAC[SET_RULE `s INTER {x | P x} = {x | x IN s /\ P x}`] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[SUBSET_RESTRICT]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `&0 <= drop y` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[DROP_CMUL; NORM_0; REAL_LE_MUL; REAL_ABS_POS; REAL_LE_INV_EQ] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN SUBGOAL_THEN `&0 < drop y` ASSUME_TAC THENL [ASM_REWRITE_TAC[REAL_LT_LE; GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM]; ASM_SIMP_TAC[REAL_LE_RDIV_EQ]] THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_ABS] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN ASM SET_TAC[]]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_DIFFERENTIABLE_IMAGE; ASM_SIMP_TAC[INTEGRAL_CMUL] THEN REWRITE_TAC[DROP_CMUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC MEASURE_DIFFERENTIABLE_IMAGE] THEN EXISTS_TAC `g':real^N->real^N->real^N` THEN ASM_SIMP_TAC[REAL_LE_REFL] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[SUBSET_RESTRICT]; ASM_SIMP_TAC[INTEGRABLE_VSUM; INTEGRABLE_CMUL; ETA_AX] THEN REWRITE_TAC[GSYM VSUM_LMUL] THEN W(MP_TAC o PART_MATCH (lhand o rand) INTEGRAL_VSUM o rand o lhand o snd) THEN ASM_SIMP_TAC[INTEGRABLE_CMUL; ETA_AX] THEN DISCH_THEN SUBST1_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) INTEGRAL_VSUM o rand o rand o snd) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [X_GEN_TAC `y:real^1` THEN DISCH_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `\x:real^N. abs(det(matrix(g' x):real^N^N)) % (f:real^N->real^1) (g x)` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_MUL THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_LIFT_ABS THEN MATCH_MP_TAC MEASURABLE_ON_DET_JACOBIAN THEN EXISTS_TAC `g:real^N->real^N` THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC MEASURABLE_ON_CMUL THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `y:real^1`]) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[lebesgue_measurable; indicator; IN_ELIM_THM] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN MESON_TAC[]]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_ABS; DROP_CMUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN REWRITE_TAC[GSYM NORM_MUL] THEN TRANS_TAC REAL_LE_TRANS `drop((h:num->real^N->real^1) n (g(x:real^N)))` THEN ASM_SIMP_TAC[FUN_IN_IMAGE] THEN REWRITE_TAC[indicator; IN_ELIM_THM] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; NORM_0] THEN REWRITE_TAC[NORM_1; DROP_CMUL; DROP_VEC; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `abs x <= x <=> &0 <= x`] THEN ASM SET_TAC[]]; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[DROP_VSUM; o_DEF] THEN MATCH_MP_TAC SUM_LE THEN ASM_SIMP_TAC[]]]; REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN SUBGOAL_THEN `!n. drop (integral s (\x. abs(det(matrix(g' x):real^N^N)) % (h:num->real^N->real^1) n (g(x:real^N)))) <= b` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_SIMP_TAC[DROP_CMUL; REAL_LE_LMUL; REAL_ABS_POS; FUN_IN_IMAGE] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `\x:real^N. abs(det(matrix(g' x):real^N^N)) % (f:real^N->real^1) (g x)` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_MUL THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_LIFT_ABS THEN MATCH_MP_TAC MEASURABLE_ON_DET_JACOBIAN THEN EXISTS_TAC `g:real^N->real^N` THEN ASM_REWRITE_TAC[]; ASM_SIMP_TAC [MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LE] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop] THEN X_GEN_TAC `a:real` THEN REWRITE_TAC[lemma] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNIONS THEN ASM_SIMP_TAC[FORALL_IN_GSPEC; GSYM LIFT_EQ; LIFT_DROP] THEN MATCH_MP_TAC(MESON[] `(FINITE {f x | x IN s} ==> FINITE {f x | x IN s /\ P x}) /\ FINITE {f x | x IN s} ==> FINITE {f x | x IN s /\ P x}`) THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN SET_TAC[]; ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC FINITE_IMAGE THEN MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `IMAGE drop (IMAGE ((h:num->real^N->real^1) n) (:real^N))` THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN SET_TAC[]]]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_ABS; DROP_CMUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN ASM_SIMP_TAC[NORM_1; real_abs; FUN_IN_IMAGE]]; ALL_TAC] THEN MP_TAC(ISPECL [`h:num->real^N->real^1`; `f:real^N->real^1`; `IMAGE (g:real^N->real^N) s`] MONOTONE_CONVERGENCE_INCREASING) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[bounded; FORALL_IN_GSPEC; IN_UNIV; RIGHT_AND_EXISTS_THM] THEN EXISTS_TAC `b:real` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[NORM_1; real_abs] THEN ASM_SIMP_TAC[INTEGRAL_DROP_POS] THEN ASM_MESON_TAC[REAL_LE_TRANS]; STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_UBOUND) THEN EXISTS_TAC `\k. integral (IMAGE (g:real^N->real^N) s) ((h:num->real^N->real^1) k)` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN ASM_MESON_TAC[REAL_LE_TRANS]]) in let version1 = prove (`!f:real^N->real^1 g:real^N->real^N g' s b. (!x. x IN s ==> &0 <= drop(f(g x))) /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (\x. abs(det(matrix(g' x))) % f(g x)) integrable_on s /\ drop(integral s (\x. abs(det(matrix (g' x))) % f(g x))) <= b ==> f integrable_on (IMAGE g s) /\ drop(integral (IMAGE g s) f) <= b`, REPEAT GEN_TAC THEN STRIP_TAC THEN ABBREV_TAC `s' = {x:real^N | x IN s /\ ~(abs(det(matrix(g' x):real^N^N)) % (f:real^N->real^1)(g x) = vec 0)}` THEN SUBGOAL_THEN `lebesgue_measurable(s':real^N->bool)` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN] THEN DISCH_THEN(MP_TAC o SPEC `(:real^1) DELETE (vec 0)`) THEN SIMP_TAC[OPEN_DELETE; OPEN_UNIV] THEN EXPAND_TAC "s'" THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `lebesgue_measurable(IMAGE (g:real^N->real^N) s')` ASSUME_TAC THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_REWRITE_TAC[LE_REFL; differentiable_on; differentiable] THEN X_GEN_TAC `x:real^N` THEN EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN EXISTS_TAC `(g':real^N->real^N->real^N) x` THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN REWRITE_TAC[SUBSET_RESTRICT] THEN ASM_SIMP_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(f:real^N->real^1) measurable_on (IMAGE (g:real^N->real^N) s')` ASSUME_TAC THENL [ASM_SIMP_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN] THEN X_GEN_TAC `t:real^1->bool` THEN DISCH_TAC THEN REWRITE_TAC[SET_RULE `{x | x IN IMAGE g s /\ f x IN t} = IMAGE g {x | x IN s /\ f(g x) IN t}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_REWRITE_TAC[LE_REFL; differentiable_on; differentiable] THEN CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN EXISTS_TAC `(g':real^N->real^N->real^N) x` THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(\x:real^N. (f:real^N->real^1)(g x)) measurable_on s'` MP_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_EQ THEN EXISTS_TAC `\x:real^N. inv(abs(det(matrix(g' x):real^N^N))) % abs(det(matrix(g' x):real^N^N)) % (f:real^N->real^1)(g x)` THEN CONJ_TAC THENL [EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_ASSOC] THEN SIMP_TAC[VECTOR_MUL_EQ_0; DE_MORGAN_THM; REAL_MUL_LINV] THEN REWRITE_TAC[VECTOR_MUL_LID]; MATCH_MP_TAC MEASURABLE_ON_MUL THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_LIFT_INV THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_LIFT_ABS THEN MATCH_MP_TAC MEASURABLE_ON_DET_JACOBIAN THEN EXISTS_TAC `g:real^N->real^N` THEN ASM_REWRITE_TAC[SET_RULE `s INTER {x | P x} = {x | x IN s /\ P x}`] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]; MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s = {} ==> negligible s`) THEN EXPAND_TAC "s'" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REWRITE_TAC[VECTOR_MUL_EQ_0; NOT_IN_EMPTY] THEN MESON_TAC[]]; FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN EXPAND_TAC "s'" THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN MESON_TAC[VECTOR_MUL_EQ_0]]]; GEN_REWRITE_TAC LAND_CONV [GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN] THEN DISCH_THEN(MP_TAC o SPEC `t DELETE (vec 0:real^1)`) THEN ASM_SIMP_TAC[OPEN_DELETE; IN_DELETE] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN EXPAND_TAC "s'" THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN MESON_TAC[VECTOR_MUL_EQ_0]]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^1`; `g:real^N->real^N`; `g':real^N->real^N->real^N`; `s':real^N->bool`; `b:real`] version0) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_SIMP_TAC[]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]; GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV) [CONJ_SYM] THEN W(MP_TAC o PART_MATCH (lhand o rand) DOUBLE_LEBESGUE_MEASURABLE_ON o snd) THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN DISCH_THEN SUBST1_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN ASM_REWRITE_TAC[LE_REFL; differentiable_on; differentiable] THEN X_GEN_TAC `x:real^N` THEN EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN EXISTS_TAC `(g':real^N->real^N->real^N) x` THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]; X_GEN_TAC `n:real^N->bool` THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_PREIMAGE THEN EXISTS_TAC `g':real^N->real^N->real^N` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_EQ_0; DE_MORGAN_THM] THEN REWRITE_TAC[REAL_ABS_ZERO] THEN X_GEN_TAC `x:real^N` THEN SIMP_TAC[RANK_EQ_FULL_DET] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]]; ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM INTEGRABLE_RESTRICT_UNIV]) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN AP_TERM_TAC] THEN REWRITE_TAC[FUN_EQ_THM] THEN EXPAND_TAC "s'" THEN REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[VECTOR_MUL_EQ_0]; EXPAND_TAC "s'" THEN REWRITE_TAC[VECTOR_MUL_EQ_0; DE_MORGAN_THM] THEN REWRITE_TAC[SET_RULE `IMAGE g {x | x IN s /\ ~(P x) /\ ~(Q(g x))} = IMAGE g {x | x IN s /\ ~P x} INTER {y | ~Q y}`] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_INTER; GSYM INTEGRAL_RESTRICT_INTER] THEN REWRITE_TAC[IN_ELIM_THM; MESON[] `(if ~(x = a) then x else a) = x`] THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_SPIKE_SET; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET] THEN REWRITE_TAC[REAL_ABS_ZERO] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^N->real^N) {x | x IN s /\ det(matrix(g' x):real^N^N) = &0}` THEN (CONJ_TAC THENL [ALL_TAC; SET_TAC[]]) THEN MATCH_MP_TAC BABY_SARD THEN EXISTS_TAC `g':real^N->real^N->real^N` THEN SIMP_TAC[IN_ELIM_THM; LE_REFL; DET_EQ_0_RANK] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]]) in let version2 = prove (`!f:real^N->real^1 g:real^N->real^N g' s. (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s ==> f absolutely_integrable_on (IMAGE g s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x. --((f:real^N->real^1) x)`; `g:real^N->real^N`; `g':real^N->real^N->real^N`; `{x | x IN s /\ drop(f((g:real^N->real^N) x)) < &0}`; `drop(integral {x | x IN s /\ drop(f((g:real^N->real^N) x)) < &0} (\x. abs(det(matrix(g' x):real^N^N)) % --(f(g x))))`] version1) THEN MP_TAC(ISPECL [`f:real^N->real^1`; `g:real^N->real^N`; `g':real^N->real^N->real^N`; `{x | x IN s /\ &0 < drop(f((g:real^N->real^N) x))}`; `drop(integral {x | x IN s /\ &0 < drop(f((g:real^N->real^N) x))} (\x. abs(det(matrix(g' x):real^N^N)) % f(g x)))`] version1) THEN REWRITE_TAC[REAL_LE_REFL; SET_RULE `IMAGE g {x | x IN s /\ P(g x)} = {y | y IN IMAGE g s /\ P y}`] THEN REWRITE_TAC[INTEGRABLE_NEG_EQ; ETA_AX] THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE; REAL_NEG_GE0; DROP_NEG] THEN REWRITE_TAC[VECTOR_MUL_RNEG; INTEGRABLE_NEG_EQ] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(\x. abs(det(matrix(g' x))) % f((g:real^N->real^N) x)) integrable_on {x | x IN s /\ &0 < drop(abs(det(matrix(g' x):real^N^N)) % f(g x))}` MP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET)) THEN REWRITE_TAC[SUBSET_RESTRICT] THEN FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT] THEN DISCH_THEN(MP_TAC o SPEC `&0`) THEN REWRITE_TAC[FORALL_1; DIMINDEX_1; real_gt; GSYM drop] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[DROP_VEC; REAL_LT_REFL]; ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `det(matrix(g'(x:real^N)):real^N^N) = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; COND_ID; REAL_ABS_NUM; DROP_CMUL] THEN ASM_SIMP_TAC[REAL_LT_MUL_EQ; GSYM REAL_ABS_NZ]]; DISCH_THEN(MP_TAC o ISPEC `(\x. vec 0):real^N->real^1` o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`] ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_LBOUND) o CONJUNCT1) THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE; DROP_VEC] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0] THEN MATCH_MP_TAC(TAUT `(q ==> p ==> r) ==> p ==> q ==> r`)] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(\x. abs(det(matrix(g' x))) % f((g:real^N->real^N) x)) integrable_on {x | x IN s /\ drop(abs(det(matrix(g' x):real^N^N)) % f(g x)) < &0}` MP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET)) THEN REWRITE_TAC[SUBSET_RESTRICT] THEN FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT] THEN DISCH_THEN(MP_TAC o SPEC `&0`) THEN REWRITE_TAC[FORALL_1; DIMINDEX_1; real_gt; GSYM drop] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[DROP_VEC; REAL_LT_REFL]; ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `det(matrix(g'(x:real^N)):real^N^N) = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; COND_ID; REAL_ABS_NUM] THEN REWRITE_TAC[DROP_CMUL; REAL_ARITH `a * b < &0 <=> &0 < a * --b`] THEN ASM_SIMP_TAC[REAL_LT_MUL_EQ; GSYM REAL_ABS_NZ; REAL_NEG_GT0]]; DISCH_THEN(MP_TAC o ISPEC `(\x. vec 0):real^N->real^1` o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> q ==> p /\ r ==> s`] ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_UBOUND) o CONJUNCT1) THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE; DROP_VEC] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0; IMP_IMP]] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_UNION) THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM; IN_UNION] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(f:real^N->real^1) x = vec 0` THEN ASM_REWRITE_TAC[COND_ID] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM DROP_EQ]) THEN REWRITE_TAC[DROP_VEC; REAL_ARITH `~(x = &0) <=> x < &0 \/ &0 < x`] THEN MESON_TAC[]) in let ABSOLUTELY_INTEGRABLE_ON_IMAGE = prove (`!f:real^M->real^N g:real^M->real^M g' s. (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s ==> f absolutely_integrable_on (IMAGE g s)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_COMPONENTWISE] THEN REWRITE_TAC[RIGHT_AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i /\ i <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; LIFT_CMUL; version2]) in let INTEGRAL_ON_IMAGE_DROP_UBOUND_LE = prove (`!f:real^N->real^1 g:real^N->real^N g' s b. (!x. x IN s ==> &0 <= drop(f(g x))) /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (\x. abs(det(matrix(g' x))) % f(g x)) integrable_on s /\ drop(integral s (\x. abs(det(matrix (g' x))) % f(g x))) <= b ==> drop(integral (IMAGE g s) f) <= b`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP version1) THEN SIMP_TAC[]) in ABSOLUTELY_INTEGRABLE_ON_IMAGE,INTEGRAL_ON_IMAGE_DROP_UBOUND_LE;; (* ------------------------------------------------------------------------- *) (* The classic change-of-variables theorem. We have two versions with quite *) (* general hypotheses, the first that the transforming function has a *) (* continuous inverse, the second that the base set is Lebesgue measurable. *) (* I am not sure if one can eliminate both of these hypotheses, but anyway *) (* it's hard to imagine a useful application to a non-measurable set. *) (* ------------------------------------------------------------------------- *) let HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_INVERTIBLE = prove (`!f:real^M->real^N g:real^M->real^M h g' s b. (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!x. x IN s ==> h(g x) = x) /\ h continuous_on IMAGE g s ==> ((\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s /\ integral s (\x. abs(det(matrix(g' x))) % f(g x)) = b <=> f absolutely_integrable_on (IMAGE g s) /\ integral (IMAGE g s) f = b)`, let version0 = prove (`!f:real^N->real^1 g:real^N->real^N h g' h' s t b. (!x. x IN s ==> g(x) IN t /\ h(g x) = x) /\ (!y. y IN t ==> h(y) IN s /\ g(h y) = y) /\ (!y. y IN t ==> &0 <= drop(f y)) /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!y. y IN t ==> (h has_derivative h' y) (at y within t)) /\ (!y. y IN t ==> h' y o g'(h y) = I) ==> (f integrable_on t /\ drop(integral t f) <= b <=> (\x. abs(det(matrix(g' x))) % f(g x)) integrable_on s /\ drop (integral s (\x. abs(det(matrix (g' x))) % f(g x))) <= b)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `IMAGE (g:real^N->real^N) s = t /\ IMAGE h t = s` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN EQ_TAC THEN STRIP_TAC THENL [EXPAND_TAC "s"; EXPAND_TAC "t"] THEN (CONJ_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_IMAGE; MATCH_MP_TAC INTEGRAL_ON_IMAGE_DROP_UBOUND_LE]) THENL [EXISTS_TAC `h':real^N->real^N->real^N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_LBOUND THEN EXISTS_TAC `(\x. vec 0):real^N->real^1` THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0; DROP_VEC; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[GSYM REAL_ABS_MUL; REAL_LE_MUL; DROP_CMUL; REAL_ABS_POS] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] INTEGRABLE_EQ)); EXISTS_TAC `h':real^N->real^N->real^N` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "t" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_SIMP_TAC[DROP_CMUL; REAL_LE_MUL; REAL_ABS_POS] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] INTEGRABLE_EQ)); FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_EQ THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV)]; EXISTS_TAC `g':real^N->real^N->real^N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_LBOUND THEN EXISTS_TAC `(\x. vec 0):real^N->real^1` THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0; DROP_VEC] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_ABS_POS; DROP_CMUL]; EXISTS_TAC `g':real^N->real^N->real^N` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]] THEN ASM_SIMP_TAC[GSYM DET_MUL; VECTOR_MUL_ASSOC; GSYM REAL_ABS_MUL] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN CONV_TAC SYM_CONV THEN GEN_REWRITE_TAC RAND_CONV [GSYM VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_MUL_RCANCEL] THEN DISJ1_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[has_derivative]) THEN ASM_SIMP_TAC[GSYM MATRIX_COMPOSE] THEN REWRITE_TAC[MATRIX_I; DET_I; REAL_ABS_NUM]) in let version1 = prove (`!f:real^N->real^1 g:real^N->real^N h g' h' s t b. (!x. x IN s ==> g(x) IN t /\ h(g x) = x) /\ (!y. y IN t ==> h(y) IN s /\ g(h y) = y) /\ (!y. y IN t ==> &0 <= drop(f y)) /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!y. y IN t ==> (h has_derivative h' y) (at y within t)) /\ (!y. y IN t ==> h' y o g'(h y) = I) ==> (((\x. abs(det(matrix(g' x))) % f(g x)) has_integral b) s <=> (f has_integral b) t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN SPEC_TAC(`b:real^1`,`b:real^1`) THEN REWRITE_TAC[FORALL_LIFT; GSYM DROP_EQ; LIFT_DROP] THEN REWRITE_TAC[MESON[REAL_LE_TRANS; REAL_LE_ANTISYM] `(!b. P x /\ f x = b <=> Q x /\ g x = b) <=> (!b. P x /\ f x <= b <=> Q x /\ g x <= b)`] THEN GEN_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC version0 THEN MAP_EVERY EXISTS_TAC [`h:real^N->real^N`; `h':real^N->real^N->real^N`] THEN ASM_REWRITE_TAC[]) in let version2 = prove (`!f:real^N->real^1 g:real^N->real^N h g' h' s t b. (!x. x IN s ==> g(x) IN t /\ h(g x) = x) /\ (!y. y IN t ==> h(y) IN s /\ g(h y) = y) /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!y. y IN t ==> (h has_derivative h' y) (at y within t)) /\ (!y. y IN t ==> h' y o g'(h y) = I) ==> ((\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s /\ integral s (\x. abs(det(matrix(g' x))) % f(g x)) = b <=> f absolutely_integrable_on t /\ integral t f = b)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `IMAGE (g:real^N->real^N) s = t /\ IMAGE h t = s` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^1`; `g:real^N->real^N`; `h:real^N->real^N`; `g':real^N->real^N->real^N`; `h':real^N->real^N->real^N`; `{x | x IN s /\ &0 < drop(f((g:real^N->real^N) x))}`; `{y:real^N | y IN t /\ &0 < drop(f y)}`] version1) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IN_ELIM_THM] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THENL [EXISTS_TAC `s:real^N->bool`; EXISTS_TAC `t:real^N->bool`] THEN ASM_REWRITE_TAC[differentiable_on; differentiable] THEN ASM SET_TAC[]; REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN DISCH_THEN(LABEL_TAC "+")] THEN MP_TAC(ISPECL [`\x. --((f:real^N->real^1) x)`; `g:real^N->real^N`; `h:real^N->real^N`; `g':real^N->real^N->real^N`; `h':real^N->real^N->real^N`; `{x | x IN s /\ drop(f((g:real^N->real^N) x)) < &0}`; `{y:real^N | y IN t /\ drop(f y) < &0}`] version1) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; IN_ELIM_THM] THEN ASM_SIMP_TAC[DROP_NEG; REAL_NEG_GT0; REAL_LT_IMP_LE] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THENL [EXISTS_TAC `s:real^N->bool`; EXISTS_TAC `t:real^N->bool`] THEN ASM_REWRITE_TAC[differentiable_on; differentiable] THEN ASM SET_TAC[]; REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN DISCH_THEN(LABEL_TAC "-")] THEN EQ_TAC THEN STRIP_TAC THENL [SUBGOAL_THEN `(\x. abs(det(matrix(g' x):real^N^N)) % f(g x)) absolutely_integrable_on {x | x IN s /\ &0 < drop((f:real^N->real^1)(g x))} /\ (\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on {x:real^N | x IN s /\ drop(f(g x)) < &0}` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THENL [REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT]] THEN DISCH_THEN(MP_TAC o SPEC `&0`) THEN REWRITE_TAC[FORALL_1; DIMINDEX_1; real_gt; GSYM drop] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[DROP_VEC; REAL_LT_REFL; DROP_CMUL] THEN REWRITE_TAC[REAL_ARITH `(a * b < &0 <=> b < &0) <=> (&0 < a * --b <=> &0 < --b)`] THEN MATCH_MP_TAC(CONJUNCT1 REAL_LT_MUL_EQ) THEN REWRITE_TAC[GSYM REAL_ABS_NZ] THEN FIRST_X_ASSUM(MP_TAC o SPEC `(g:real^N->real^N) x`) THEN (SUBGOAL_THEN `(g:real^N->real^N) x IN t` ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]]) THEN DISCH_THEN(MP_TAC o AP_TERM `(det o matrix):(real^N->real^N)->real`) THEN RULE_ASSUM_TAC(REWRITE_RULE[has_derivative]) THEN ASM_SIMP_TAC[o_THM; MATRIX_COMPOSE; DET_MUL; MATRIX_I; DET_I] THEN CONV_TAC REAL_RING; ALL_TAC] THEN REMOVE_THEN "-" (MP_TAC o SPEC `integral {x | x IN s /\ drop(f((g:real^N->real^N) x)) < &0} (\x:real^N. abs(det(matrix(g' x):real^N^N)) % --f(g x))`) THEN ASM_SIMP_TAC[VECTOR_MUL_RNEG; INTEGRABLE_NEG_EQ; INTEGRAL_NEG; IMP_CONJ; VECTOR_EQ_NEG2; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC th) THEN DISCH_THEN(MP_TAC o ISPEC `(\x. vec 0):real^N->real^1` o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> q ==> p /\ r ==> s`] ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_UBOUND)) THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE; DROP_VEC] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0; IMP_IMP; GSYM CONJ_ASSOC] THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN REMOVE_THEN "+" (MP_TAC o SPEC `integral {x | x IN s /\ &0 < drop (f((g:real^N->real^N) x))} (\x:real^N. abs(det(matrix(g' x):real^N^N)) % f(g x))`) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; IMP_CONJ] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC th) THEN DISCH_THEN(MP_TAC o ISPEC `(\x. vec 0):real^N->real^1` o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`] ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_LBOUND)) THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE; DROP_VEC] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0] THEN REPLICATE_TAC 2 (GEN_REWRITE_TAC I [IMP_IMP]) THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRABLE_INTEGRAL; GSYM CONJ_ASSOC] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q ==> p' ==> q' ==> r) <=> p /\ p' ==> q /\ q' ==> r`] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_UNION) THEN REWRITE_TAC[IMP_IMP] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_UNION; IN_ELIM_THM] THEN REWRITE_TAC[GSYM LEFT_OR_DISTRIB; GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM; REAL_ARITH `&0 < x \/ x < &0 <=> ~(x = &0)`] THEN MESON_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HAS_INTEGRAL_UNION))) THEN ANTS_TAC THENL [REWRITE_TAC[REAL_LT_ANTISYM; EMPTY_GSPEC; NEGLIGIBLE_EMPTY; SET_RULE `{x | x IN s /\ P x} INTER {x | x IN s /\ Q x} = {x | x IN s /\ P x /\ Q x}`]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_UNION o rand o rator o lhand o snd) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN ANTS_TAC THENL [REWRITE_TAC[REAL_LT_ANTISYM; EMPTY_GSPEC; NEGLIGIBLE_EMPTY; SET_RULE `{x | x IN s /\ P x} INTER {x | x IN s /\ Q x} = {x | x IN s /\ P x /\ Q x}`]; DISCH_THEN(SUBST1_TAC o SYM)] THEN DISCH_THEN(MP_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN EXPAND_TAC "b" THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM; IN_UNIV; SET_RULE `{x | x IN s /\ P x} UNION {x | x IN s /\ Q x} = {x | x IN s /\ (P x \/ Q x)}`] THEN REWRITE_TAC[REAL_ARITH `&0 < x \/ x < &0 <=> ~(x = &0)`] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM] THEN MESON_TAC[VECTOR_MUL_EQ_0]]; SUBGOAL_THEN `(f:real^N->real^1) absolutely_integrable_on {y | y IN t /\ &0 < drop(f y)} /\ (f:real^N->real^1) absolutely_integrable_on {y | y IN t /\ drop(f y) < &0}` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_IMP_MEASURABLE o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THENL [REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT]; REWRITE_TAC[MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LT]] THEN DISCH_THEN(MP_TAC o SPEC `&0`) THEN REWRITE_TAC[FORALL_1; DIMINDEX_1; real_gt; GSYM drop] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN t` THEN ASM_REWRITE_TAC[DROP_VEC; REAL_LT_REFL; DROP_CMUL] THEN REWRITE_TAC[REAL_ARITH `(a * b < &0 <=> b < &0) <=> (&0 < a * --b <=> &0 < --b)`] THEN MATCH_MP_TAC(CONJUNCT1 REAL_LT_MUL_EQ) THEN REWRITE_TAC[GSYM REAL_ABS_NZ] THEN FIRST_X_ASSUM(MP_TAC o SPEC `(g:real^N->real^N) x`) THEN (SUBGOAL_THEN `(g:real^N->real^N) x IN t` ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]]) THEN DISCH_THEN(MP_TAC o AP_TERM `(det o matrix):(real^N->real^N)->real`) THEN RULE_ASSUM_TAC(REWRITE_RULE[has_derivative]) THEN ASM_SIMP_TAC[o_THM; MATRIX_COMPOSE; DET_MUL; MATRIX_I; DET_I] THEN CONV_TAC REAL_RING; ALL_TAC] THEN REMOVE_THEN "-" (MP_TAC o SPEC `integral {y | y IN t /\ drop(f y) < &0} (\x. --((f:real^N->real^1) x))`) THEN ASM_SIMP_TAC[VECTOR_MUL_RNEG; INTEGRABLE_NEG_EQ; INTEGRAL_NEG; IMP_CONJ; VECTOR_EQ_NEG2; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC th) THEN DISCH_THEN(MP_TAC o ISPEC `(\x. vec 0):real^N->real^1` o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> q ==> p /\ r ==> s`] ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_UBOUND)) THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE; DROP_VEC] THEN ANTS_TAC THENL [REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0; DROP_CMUL; REAL_ARITH `x * y <= &0 <=> &0 <= x * --y`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRABLE_INTEGRAL]] THEN REMOVE_THEN "+" (MP_TAC o SPEC `integral {y | y IN t /\ &0 < drop(f y)} (f:real^N->real^1)`) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; IMP_CONJ] THEN DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC th) THEN DISCH_THEN(MP_TAC o ISPEC `(\x. vec 0):real^N->real^1` o MATCH_MP (ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`] ABSOLUTELY_INTEGRABLE_ABSOLUTELY_INTEGRABLE_DROP_LBOUND)) THEN SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_LE; DROP_VEC; DROP_CMUL; REAL_LE_MUL; REAL_LT_IMP_LE; REAL_ABS_POS] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0] THEN REPLICATE_TAC 2 (GEN_REWRITE_TAC I [IMP_IMP]) THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRABLE_INTEGRAL; GSYM CONJ_ASSOC] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q ==> p' ==> q' ==> r) <=> p /\ p' ==> q /\ q' ==> r`] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_UNION) THEN REWRITE_TAC[IMP_IMP] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_UNION; IN_ELIM_THM] THEN REWRITE_TAC[GSYM LEFT_OR_DISTRIB; GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM; REAL_ARITH `&0 < x \/ x < &0 <=> ~(x = &0)`] THEN MESON_TAC[VECTOR_MUL_EQ_0]; DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HAS_INTEGRAL_UNION))) THEN ANTS_TAC THENL [REWRITE_TAC[REAL_LT_ANTISYM; EMPTY_GSPEC; NEGLIGIBLE_EMPTY; SET_RULE `{x | x IN s /\ P x} INTER {x | x IN s /\ Q x} = {x | x IN s /\ P x /\ Q x}`]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_UNION o rand o rator o lhand o snd) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN ANTS_TAC THENL [REWRITE_TAC[REAL_LT_ANTISYM; EMPTY_GSPEC; NEGLIGIBLE_EMPTY; SET_RULE `{x | x IN s /\ P x} INTER {x | x IN s /\ Q x} = {x | x IN s /\ P x /\ Q x}`]; DISCH_THEN(SUBST1_TAC o SYM)] THEN DISCH_THEN(MP_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN EXPAND_TAC "b" THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM; IN_UNIV; SET_RULE `{x | x IN s /\ P x} UNION {x | x IN s /\ Q x} = {x | x IN s /\ (P x \/ Q x)}`] THEN REWRITE_TAC[REAL_ARITH `&0 < x \/ x < &0 <=> ~(x = &0)`] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM] THEN MESON_TAC[VECTOR_MUL_EQ_0]]]) in let version3 = prove (`!f:real^M->real^N g:real^M->real^M h g' h' s t b. (!x. x IN s ==> g(x) IN t /\ h(g x) = x) /\ (!y. y IN t ==> h(y) IN s /\ g(h y) = y) /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!y. y IN t ==> (h has_derivative h' y) (at y within t)) /\ (!y. y IN t ==> h' y o g'(h y) = I) ==> ((\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s /\ integral s (\x. abs(det(matrix(g' x))) % f(g x)) = b <=> f absolutely_integrable_on t /\ integral t f = b)`, REPEAT STRIP_TAC THEN REWRITE_TAC[MESON[HAS_INTEGRAL_INTEGRABLE_INTEGRAL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] `f absolutely_integrable_on s /\ integral s f = b <=> f absolutely_integrable_on s /\ (f has_integral b) s`] THEN ONCE_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_COMPONENTWISE; HAS_INTEGRAL_COMPONENTWISE] THEN REWRITE_TAC[AND_FORALL_THM] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i /\ i <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[MESON[HAS_INTEGRAL_INTEGRABLE_INTEGRAL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] `f absolutely_integrable_on s /\ (f has_integral b) s <=> f absolutely_integrable_on s /\ integral s f = b`] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; LIFT_CMUL] THEN MP_TAC(ISPEC `\x. (lift((f:real^M->real^N) x$i))` version2) THEN REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN MAP_EVERY EXISTS_TAC [`h:real^M->real^M`; `h':real^M->real^M->real^M`] THEN ASM_REWRITE_TAC[]) in let version4 = prove (`!f:real^M->real^N g:real^M->real^M h g' s b. (!x. x IN s ==> (g has_derivative g' x) (at x within s) /\ invertible(matrix(g' x))) /\ (!x. x IN s ==> h(g x) = x) /\ h continuous_on IMAGE g s ==> ((\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s /\ integral s (\x. abs(det(matrix(g' x))) % f(g x)) = b <=> f absolutely_integrable_on (IMAGE g s) /\ integral (IMAGE g s) f = b)`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [TAUT `p /\ q <=> ~(p ==> ~q)`] THEN SIMP_TAC[has_derivative; MATRIX_INVERTIBLE] THEN REWRITE_TAC[GSYM has_derivative] THEN REWRITE_TAC[NOT_IMP; RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_AND_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `h':real^M->real^M->real^M` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC version3 THEN EXISTS_TAC `h:real^M->real^M` THEN EXISTS_TAC `\x:real^M d. (h':real^M->real^M->real^M) (h x) d` THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; FUN_IN_IMAGE; ETA_AX] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_INVERSE_WITHIN THEN EXISTS_TAC `(g':real^M->real^M->real^M) x` THEN ASM_MESON_TAC[FUN_IN_IMAGE; CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^M->real^M`; `h:real^M->real^M`; `g':real^M->real^M->real^M`; `{x | x IN s /\ invertible(matrix((g':real^M->real^M->real^M) x))}`; `b:real^N`] version4) THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[SUBSET_RESTRICT]; MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `IMAGE (g:real^M->real^M) s` THEN ASM_SIMP_TAC[] THEN SET_TAC[]]; MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL [BINOP_TAC THENL [ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN AP_THM_TAC THEN AP_TERM_TAC; ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC] THEN REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM; INVERTIBLE_DET_NZ] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COND_SWAP] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_ABS_NUM; VECTOR_MUL_LZERO]; BINOP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SPIKE_SET_EQ; AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^M->real^M) {x | x IN s /\ ~invertible(matrix(g' x):real^M^M)}` THEN (CONJ_TAC THENL [ALL_TAC; SET_TAC[]]) THEN MATCH_MP_TAC BABY_SARD THEN EXISTS_TAC `g':real^M->real^M->real^M` THEN ASM_SIMP_TAC[LE_REFL; IN_ELIM_THM; GSYM DET_EQ_0_RANK; GSYM DET_EQ_0] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[SUBSET_RESTRICT]]]);; let HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES = prove (`!f:real^M->real^N g:real^M->real^M g' s b. lebesgue_measurable s /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ g x = g y ==> x = y) ==> ((\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s /\ integral s (\x. abs(det(matrix(g' x))) % f(g x)) = b <=> f absolutely_integrable_on (IMAGE g s) /\ integral (IMAGE g s) f = b)`, let lemma = prove (`UNIONS {IMAGE f (g x) | P x} = IMAGE f (UNIONS {g x | P x})`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in let version6 = prove (`!f:real^M->real^N g:real^M->real^M g' s b. compact s /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ g x = g y ==> x = y) ==> ((\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s /\ integral s (\x. abs(det(matrix(g' x))) % f(g x)) = b <=> f absolutely_integrable_on (IMAGE g s) /\ integral (IMAGE g s) f = b)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN DISCH_THEN(X_CHOOSE_TAC `h:real^M->real^M`) THEN MATCH_MP_TAC HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_INVERTIBLE THEN EXISTS_TAC `h:real^M->real^M` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN ASM_REWRITE_TAC[differentiable_on; differentiable] THEN ASM_MESON_TAC[]) in let version7 = prove (`!f:real^M->real^N g:real^M->real^M g' u b. COUNTABLE u /\ (!s. s IN u ==> compact s) /\ (!x. x IN UNIONS u ==> (g has_derivative g' x) (at x within UNIONS u)) /\ (!x y. x IN UNIONS u /\ y IN UNIONS u /\ g x = g y ==> x = y) ==> ((\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on (UNIONS u) /\ integral (UNIONS u) (\x. abs(det(matrix(g' x))) % f(g x)) = b <=> f absolutely_integrable_on (IMAGE g (UNIONS u)) /\ integral (IMAGE g (UNIONS u)) f = b)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_CASES_TAC `u:(real^M->bool)->bool = {}` THENL [ASM_REWRITE_TAC[UNIONS_0; INTEGRAL_EMPTY; ABSOLUTELY_INTEGRABLE_ON_EMPTY; IMAGE_CLAUSES] THEN MESON_TAC[]; MP_TAC(ISPEC `u:(real^M->bool)->bool` COUNTABLE_AS_IMAGE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN POP_ASSUM_LIST(K ALL_TAC)] THEN X_GEN_TAC `s:num->real^M->bool` THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN STRIP_TAC THEN REWRITE_TAC[IMAGE_UNIONS; GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[SET_RULE `IMAGE s (:num) = {s n | n IN (:num)}`] THEN EQ_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] INTEGRAL_COUNTABLE_UNIONS_ALT)) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT] THENL [ALL_TAC; ANTS_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT; LE_REFL] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_SUBSET THEN EXISTS_TAC `UNIONS(IMAGE s (:num)):real^M->bool` THEN REWRITE_TAC[differentiable_on; differentiable] THEN ASM SET_TAC[]; ALL_TAC]] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN (SUBGOAL_THEN `(!n. (f:real^M->real^N) absolutely_integrable_on UNIONS {IMAGE (g:real^M->real^M)(s m) | m IN 0..n} <=> (\x. abs(det(matrix(g' x):real^M^M)) % f(g x)) absolutely_integrable_on UNIONS {s m | m IN 0..n}) /\ (!n. integral (UNIONS {IMAGE (g:real^M->real^M)(s m) | m IN 0..n}) f = integral (UNIONS {s m | m IN 0..n}) (\x. abs(det(matrix(g' x))) % f(g x)))` MP_TAC THENL [REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(MESON[] `(p \/ q) /\ (!b. p /\ x = b <=> q /\ y = b) ==> (p <=> q) /\ x = y`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[]; GEN_TAC] THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[lemma] THEN MATCH_MP_TAC version6 THEN ASM_SIMP_TAC[COMPACT_UNIONS; SIMPLE_IMAGE; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE s (:num)):real^M->bool` THEN ASM SET_TAC[]; ALL_TAC]) THENL [DISCH_THEN(MP_TAC o GSYM); ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN2 (fun th -> RULE_ASSUM_TAC(REWRITE_RULE[th])) (fun th -> REWRITE_TAC[th])) THEN DISCH_TAC THENL [CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC DOMINATED_CONVERGENCE_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\n x. if x IN UNIONS {IMAGE g (s m:real^M->bool) | m IN 0..n} then (f:real^M->real^N) x else vec 0`; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[TRIVIAL_LIMIT_SEQUENTIALLY] (ISPEC `sequentially` LIM_UNIQUE)))) THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN MATCH_MP_TAC(TAUT `g integrable_on s /\ (f --> integral s g) net ==> (f --> integral s g) net`) THEN MATCH_MP_TAC DOMINATED_CONVERGENCE] THEN REWRITE_TAC[] THEN EXISTS_TAC `\x. if x IN UNIONS {IMAGE g (s m:real^M->bool) | m IN (:num)} then lift(norm((f:real^M->real^N) x)) else vec 0` THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV; INTEGRABLE_RESTRICT_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN (REPEAT CONJ_TAC THENL [ALL_TAC; REPEAT GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[LIFT_DROP; DROP_VEC; NORM_0; REAL_LE_REFL; NORM_POS_LE]) THEN ASM SET_TAC[]; GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN COND_CASES_TAC THENL [MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN MESON_TAC[]; MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN ASM SET_TAC[]]] THEN ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC(MESON[MONOTONE_CONVERGENCE_INCREASING] `!f. (!k. f k integrable_on s) /\ (!k x. x IN s ==> drop (f k x) <= drop (f (SUC k) x)) /\ (!x. x IN s ==> ((\k. f k x) --> g x) sequentially) /\ bounded {integral s (f k) | k IN (:num)} ==> (g:real^M->real^1) integrable_on s`) THEN EXISTS_TAC `\n x. if x IN UNIONS {IMAGE g (s m:real^M->bool) | m IN 0..n} then lift(norm((f:real^M->real^N) x)) else vec 0` THEN REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV; INTEGRAL_RESTRICT_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN REPEAT CONJ_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[LIFT_DROP; DROP_VEC; NORM_0; REAL_LE_REFL; NORM_POS_LE]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_NUMSEG; LE]) THEN ASM SET_TAC[]; GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN COND_CASES_TAC THENL [MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN MESON_TAC[]; MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN ASM SET_TAC[]]; ALL_TAC] THEN MP_TAC(GEN `n:num` (ISPECL [`\x. lift(norm((f:real^M->real^N) x))`; `g:real^M->real^M`; `g':real^M->real^M->real^M`; `UNIONS {s m | m IN 0..n}:real^M->bool`; `integral (UNIONS {s m | m IN 0..n}) (\x:real^M. abs(det(matrix(g' x):real^M^M)) % lift(norm((f:real^M->real^N) (g x))))`] version6)) THEN DISCH_THEN(MP_TAC o MATCH_MP MONO_FORALL) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[COMPACT_UNIONS; SIMPLE_IMAGE; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN GEN_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE s (:num)):real^M->bool` THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!n. P n <=> Q n) ==> (!n. P n) ==> !n. Q n`)) THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q ==> r) ==> (p ==> q) ==> r`) THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[GSYM REAL_ABS_ABS] THEN REWRITE_TAC[GSYM NORM_MUL] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET)) THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COMPACT THEN ASM_SIMP_TAC[COMPACT_UNIONS; SIMPLE_IMAGE; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG]; DISCH_TAC THEN REWRITE_TAC[lemma; FORALL_AND_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN EXISTS_TAC `drop(integral (UNIONS {s m | m IN (:num)}) (\x:real^M. abs(det(matrix(g' x):real^M^M)) % lift(norm((f:real^M->real^N) (g x)))))` THEN X_GEN_TAC `m:num` THEN DISCH_THEN(K ALL_TAC) THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[GSYM REAL_ABS_ABS] THEN REWRITE_TAC[GSYM NORM_MUL] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE]; REPEAT GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[LIFT_DROP; DROP_VEC; NORM_0; NORM_MUL; DROP_CMUL; REAL_ABS_ABS; NORM_LIFT; REAL_ABS_NORM; NORM_POS_LE; REAL_LE_REFL; REAL_LE_MUL; NORM_POS_LE; REAL_ABS_POS]) THEN ASM SET_TAC[]]]); CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC DOMINATED_CONVERGENCE_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\n x. if (x:real^M) IN UNIONS {s m | m IN 0..n} then abs(det(matrix(g' x):real^M^M)) % (f:real^M->real^N) (g x) else vec 0`; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[TRIVIAL_LIMIT_SEQUENTIALLY] (ISPEC `sequentially` LIM_UNIQUE)))) THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN MATCH_MP_TAC(TAUT `g integrable_on s /\ (f --> integral s g) net ==> (f --> integral s g) net`) THEN MATCH_MP_TAC DOMINATED_CONVERGENCE] THEN REWRITE_TAC[] THEN EXISTS_TAC `\x. if (x:real^M) IN UNIONS {s m | m IN (:num)} then lift(norm(abs(det(matrix(g' x):real^M^M)) % (f:real^M->real^N) (g x))) else vec 0` THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV; INTEGRABLE_RESTRICT_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN (REPEAT CONJ_TAC THENL [ALL_TAC; REPEAT GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[LIFT_DROP; DROP_VEC; NORM_0; REAL_LE_REFL; NORM_POS_LE]) THEN ASM SET_TAC[]; GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN COND_CASES_TAC THENL [MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN MESON_TAC[]; MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN ASM SET_TAC[]]] THEN ONCE_REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC(MESON[MONOTONE_CONVERGENCE_INCREASING] `!f. (!k. f k integrable_on s) /\ (!k x. x IN s ==> drop (f k x) <= drop (f (SUC k) x)) /\ (!x. x IN s ==> ((\k. f k x) --> g x) sequentially) /\ bounded {integral s (f k) | k IN (:num)} ==> (g:real^M->real^1) integrable_on s`) THEN EXISTS_TAC `\n x. if (x:real^M) IN UNIONS {s m | m IN 0..n} then lift(norm(abs(det(matrix(g' x):real^M^M)) % (f:real^M->real^N) (g x))) else vec 0` THEN REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV; INTEGRAL_RESTRICT_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN REPEAT CONJ_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[LIFT_DROP; DROP_VEC; NORM_0; REAL_LE_REFL; NORM_POS_LE]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_NUMSEG; LE]) THEN ASM SET_TAC[]; GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN COND_CASES_TAC THENL [MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN MESON_TAC[]; MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN REWRITE_TAC[UNIONS_GSPEC; IN_NUMSEG; LE_0; IN_ELIM_THM] THEN GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_UNIONS]) THEN REWRITE_TAC[EXISTS_IN_GSPEC] THEN ASM SET_TAC[]]; ALL_TAC] THEN MP_TAC(GEN `n:num` (ISPECL [`\x. lift(norm((f:real^M->real^N) x))`; `g:real^M->real^M`; `g':real^M->real^M->real^M`; `UNIONS {s m | m IN 0..n}:real^M->bool`; `integral (IMAGE (g:real^M->real^M) (UNIONS {s m | m IN 0..n})) (\x:real^M. lift(norm((f:real^M->real^N) x)))`] version6)) THEN DISCH_THEN(MP_TAC o MATCH_MP MONO_FORALL) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[COMPACT_UNIONS; SIMPLE_IMAGE; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN GEN_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE s (:num)):real^M->bool` THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `(!n. P n <=> Q n) ==> (!n. Q n) ==> !n. P n`)) THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q ==> r) ==> (p ==> q) ==> r`) THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET)) THEN CONJ_TAC THENL [REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[COMPACT_UNIONS; SIMPLE_IMAGE; FORALL_IN_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN REWRITE_TAC[differentiable_on; differentiable] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN EXISTS_TAC `(g':real^M->real^M->real^M) x` THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `UNIONS (IMAGE s (:num)):real^M->bool` THEN ASM SET_TAC[]; DISCH_TAC THEN REWRITE_TAC[lemma; FORALL_AND_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[GSYM REAL_ABS_ABS] THEN REWRITE_TAC[GSYM NORM_MUL] THEN REWRITE_TAC[REAL_ABS_ABS] THEN DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN EXISTS_TAC `drop(integral (IMAGE (g:real^M->real^M) (UNIONS {s m | m IN (:num)})) (\x. lift(norm((f:real^M->real^N) x))))` THEN X_GEN_TAC `m:num` THEN DISCH_THEN(K ALL_TAC) THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REWRITE_TAC[GSYM lemma] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN REPEAT GEN_TAC THEN REWRITE_TAC[IN_UNIV] THEN REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[LIFT_DROP; DROP_VEC; NORM_0; NORM_MUL; DROP_CMUL; REAL_ABS_ABS; NORM_LIFT; REAL_ABS_NORM; NORM_POS_LE; REAL_LE_REFL; REAL_LE_MUL; NORM_POS_LE; REAL_ABS_POS]) THEN ASM SET_TAC[]])]) in REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_ALMOST_FSIGMA) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:real^M->bool`; `n:real^M->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `(\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on c /\ integral c (\x. abs(det(matrix(g' x):real^M^M)) % f(g x)) = b <=> (f:real^M->real^N) absolutely_integrable_on (IMAGE g c) /\ integral (IMAGE (g:real^M->real^M) c) f = b` MP_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FSIGMA_UNIONS_COMPACT]) THEN REWRITE_TAC[UNION_OF] THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^M->bool)->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN MATCH_MP_TAC version7 THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL [BINOP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SPIKE_SET_EQ; AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `n:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; BINOP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SPIKE_SET_EQ; AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^M->real^M) n` THEN (CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN ASM_REWRITE_TAC[LE_REFL; differentiable_on; differentiable] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN EXISTS_TAC `(g':real^M->real^M->real^M) x` THEN MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM SET_TAC[]]]);; let ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES = prove (`!f:real^M->real^N g:real^M->real^M g' s. lebesgue_measurable s /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ g x = g y ==> x = y) ==> (f absolutely_integrable_on (IMAGE g s) <=> (\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES) THEN MESON_TAC[]);; let INTEGRAL_CHANGE_OF_VARIABLES = prove (`!f:real^M->real^N g:real^M->real^M g' s. lebesgue_measurable s /\ (!x. x IN s ==> (g has_derivative g' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ g x = g y ==> x = y) /\ (f absolutely_integrable_on (IMAGE g s) \/ (\x. abs(det(matrix(g' x))) % f(g x)) absolutely_integrable_on s) ==> integral (IMAGE g s) f = integral s (\x. abs(det(matrix(g' x))) % f(g x))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (p /\ q /\ r) /\ s`] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES) THEN ASM_MESON_TAC[]);; let HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_1 = prove (`!f:real^1->real^N g:real^1->real^1 g' s b. lebesgue_measurable s /\ (!x y. x IN s /\ y IN s /\ g x = g y ==> x = y) /\ (!x. x IN s ==> (g has_vector_derivative lift (g' (drop x))) (at x within s)) ==> ((\x. abs (g' (drop x)) % f (g x)) absolutely_integrable_on s /\ integral s (\x. abs (g' (drop x)) % f (g x)) = b <=> f absolutely_integrable_on IMAGE g s /\ integral (IMAGE g s) f = b)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[MESON[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; HAS_INTEGRAL_INTEGRABLE_INTEGRAL] `f absolutely_integrable_on s /\ integral s f = b <=> f absolutely_integrable_on s /\ (f has_integral b) s`] THEN ONCE_REWRITE_TAC[HAS_INTEGRAL_COMPONENTWISE; ABSOLUTELY_INTEGRABLE_COMPONENTWISE] THEN REWRITE_TAC[AND_FORALL_THM] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i /\ i <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[MESON[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; HAS_INTEGRAL_INTEGRABLE_INTEGRAL] `f absolutely_integrable_on s /\ (f has_integral b) s <=> f absolutely_integrable_on s /\ integral s f = b`] THEN MP_TAC(ISPECL [`\x. lift((f:real^1->real^N) x$i)`; `g:real^1->real^1`; `(\x h. g'(drop x) % h) :real^1->real^1->real^1`; `s:real^1->bool`; `lift((b:real^N)$i)`] HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [has_vector_derivative] o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL; LIFT_DROP]; DISCH_THEN(SUBST1_TAC o SYM)] THEN SIMP_TAC[MATRIX_CMUL; LINEAR_ID; MATRIX_ID; DET_CMUL] THEN REWRITE_TAC[DET_I; DIMINDEX_1; REAL_POW_1; REAL_MUL_RID] THEN REWRITE_TAC[LIFT_CMUL; VECTOR_MUL_COMPONENT]);; let ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES_1 = prove (`!f:real^1->real^N g:real^1->real^1 g' s. lebesgue_measurable s /\ (!x y. x IN s /\ y IN s /\ g x = g y ==> x = y) /\ (!x. x IN s ==> (g has_vector_derivative lift (g' (drop x))) (at x within s)) ==> (f absolutely_integrable_on IMAGE g s <=> (\x. abs(g'(drop x)) % f(g x)) absolutely_integrable_on s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_1) THEN MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Change of variable for measure. *) (* ------------------------------------------------------------------------- *) let HAS_MEASURE_DIFFERENTIABLE_IMAGE = prove (`!f:real^N->real^N f' s m. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> (IMAGE f s has_measure m <=> ((\x. lift(abs(det(matrix(f' x))))) has_integral (lift m)) s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(\x. vec 1):real^N->real^1`; `f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`; `lift m`] HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM LIFT_NUM; GSYM LIFT_CMUL; REAL_MUL_RID] THEN SIMP_TAC[ABSOLUTELY_INTEGRABLE_EQ_INTEGRABLE_POS; LIFT_DROP; REAL_ABS_POS; REAL_POS] THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[LIFT_NUM; has_measure]);; let MEASURABLE_DIFFERENTIABLE_IMAGE_EQ = prove (`!f:real^N->real^N f' s. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> (measurable (IMAGE f s) <=> (\x. lift(abs(det(matrix(f' x))))) integrable_on s)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_DIFFERENTIABLE_IMAGE) THEN SIMP_TAC[measurable; integrable_on] THEN REWRITE_TAC[GSYM EXISTS_LIFT]);; let MEASURABLE_DIFFERENTIABLE_IMAGE_ALT = prove (`!f:real^N->real^N f' s. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> (measurable (IMAGE f s) <=> (\x. lift(abs(det(matrix(f' x))))) absolutely_integrable_on s)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`] MEASURABLE_DIFFERENTIABLE_IMAGE_EQ) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_EQ_INTEGRABLE_POS THEN REWRITE_TAC[LIFT_DROP; REAL_ABS_POS]);; let MEASURE_DIFFERENTIABLE_IMAGE_EQ = prove (`!f:real^N->real^N f' s. lebesgue_measurable s /\ (!x. x IN s ==> (f has_derivative f' x) (at x within s)) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\ (\x. lift(abs(det(matrix(f' x))))) integrable_on s ==> measure (IMAGE f s) = drop(integral s (\x. lift(abs(det(matrix(f' x))))))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`] MEASURABLE_DIFFERENTIABLE_IMAGE_EQ) THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[MEASURE_INTEGRAL] THEN DISCH_THEN(K ALL_TAC) THEN AP_TERM_TAC THEN REWRITE_TAC[LIFT_EQ_CMUL] THEN MATCH_MP_TAC INTEGRAL_CHANGE_OF_VARIABLES THEN ASM_REWRITE_TAC[GSYM LIFT_EQ_CMUL] THEN DISJ2_TAC THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_EQ_INTEGRABLE_POS; LIFT_DROP; REAL_ABS_POS]);; (* ------------------------------------------------------------------------- *) (* Change of variables for integrals again: special case of linear function. *) (* ------------------------------------------------------------------------- *) let HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_LINEAR = prove (`!f:real^M->real^N g:real^M->real^M s b. linear g ==> ((\x. abs(det(matrix g)) % f(g x)) absolutely_integrable_on s /\ integral s (\x. abs(det(matrix g)) % f(g x)) = b <=> f absolutely_integrable_on (IMAGE g s) /\ integral (IMAGE g s) f = b)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `det(matrix g:real^M^M) = &0` THENL [ASM_REWRITE_TAC[REAL_ABS_NUM; VECTOR_MUL_LZERO] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0; INTEGRAL_0] THEN SUBGOAL_THEN `negligible(IMAGE (g:real^M->real^M) s)` ASSUME_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_LINEAR_SINGULAR_IMAGE THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LINEAR_INJECTIVE_LEFT_INVERSE)) THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `h:real^M->real^M` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o AP_TERM `(det o matrix):(real^M->real^M)->real`) THEN ASM_SIMP_TAC[o_THM; MATRIX_COMPOSE; DET_MUL; MATRIX_I; DET_I] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_ON_NEGLIGIBLE; INTEGRAL_ON_NEGLIGIBLE]]; MATCH_MP_TAC HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_INVERTIBLE THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [DET_EQ_0]) THEN ASM_SIMP_TAC[MATRIX_INVERTIBLE; FUN_EQ_THM; o_THM; I_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^M->real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; HAS_DERIVATIVE_LINEAR]]);; let ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES_LINEAR = prove (`!f:real^M->real^N g:real^M->real^M s. linear g ==> ((\x. abs(det(matrix g)) % f(g x)) absolutely_integrable_on s <=> f absolutely_integrable_on (IMAGE g s))`, MESON_TAC[HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_LINEAR]);; let ABSOLUTELY_INTEGRABLE_ON_LINEAR_IMAGE = prove (`!f:real^M->real^N g:real^M->real^M s. linear g ==> (f absolutely_integrable_on (IMAGE g s) <=> (f o g) absolutely_integrable_on s \/ det(matrix g) = &0)`, ASM_SIMP_TAC[GSYM ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES_LINEAR] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_CMUL_EQ; o_DEF; REAL_ABS_ZERO] THEN CONV_TAC TAUT);; let INTEGRAL_CHANGE_OF_VARIABLES_LINEAR = prove (`!f:real^M->real^N g:real^M->real^M s. linear g /\ (f absolutely_integrable_on (IMAGE g s) \/ (f o g) absolutely_integrable_on s) ==> integral (IMAGE g s) f = abs(det(matrix g)) % integral s (f o g)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^M->real^M`; `s:real^M->bool`] ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES_LINEAR) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `(p \/ q) /\ (p /\ q ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(DISJ_CASES_THEN MP_TAC) THEN ASM_SIMP_TAC[o_DEF; ABSOLUTELY_INTEGRABLE_CMUL]; DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^M->real^M`; `s:real^M->bool`] HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES_LINEAR) THEN ASM_REWRITE_TAC[o_DEF] THEN DISCH_THEN(fun th -> REWRITE_TAC[GSYM th]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ABSOLUTELY_INTEGRABLE_CMUL_EQ] o CONJUNCT1) THEN STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INTEGRAL_0] THEN ASM_SIMP_TAC[INTEGRAL_CMUL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]]);; (* ------------------------------------------------------------------------- *) (* Approximation of L_1 functions by bounded continuous ones. *) (* Note that 100/fourier.ml has some generalizations to L_p spaces. *) (* ------------------------------------------------------------------------- *) let ABSOLUTELY_INTEGRABLE_APPROXIMATE_CONTINUOUS = prove (`!f:real^M->real^N s e. lebesgue_measurable s /\ f absolutely_integrable_on s /\ &0 < e ==> ?g. g absolutely_integrable_on s /\ g continuous_on (:real^M) /\ bounded (IMAGE g (:real^M)) /\ norm(integral s (\x. lift(norm(f x - g x)))) < e`, let lemma = prove (`!f:real^M->real^N s e. measurable s /\ f absolutely_integrable_on s /\ &0 < e ==> ?g. g absolutely_integrable_on s /\ g continuous_on (:real^M) /\ bounded (IMAGE g (:real^M)) /\ norm(integral s (\x. lift(norm(f x - g x)))) < e`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?h. h absolutely_integrable_on s /\ bounded (IMAGE h (:real^M)) /\ norm(integral s (\x. lift(norm(f x - h x:real^N)))) < e / &2` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`\n x. lift(norm (f x - (lambda i. max (--(&n)) (min (&n) ((f:real^M->real^N)(x)$i)))))`; `(\x. vec 0):real^M->real^1`; `\x. lift(norm((f:real^M->real^N)(x)))`; `s:real^M->bool`] DOMINATED_CONVERGENCE) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `!n. ((\x. lambda i. max (--(&n)) (min (&n) ((f x:real^N)$i))) :real^M->real^N) absolutely_integrable_on s` ASSUME_TAC THENL [GEN_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `(\x. lambda i. &n):real^M->real^N` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] ABSOLUTELY_INTEGRABLE_MIN)) THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_ON_CONST] THEN DISCH_THEN(MP_TAC o SPEC `(\x. lambda i. --(&n)):real^M->real^N` o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] ABSOLUTELY_INTEGRABLE_MAX)) THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_ON_CONST] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA]; ALL_TAC] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_SUB]; ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; MAP_EVERY X_GEN_TAC [`n:num`; `x:real^M`] THEN DISCH_TAC THEN REWRITE_TAC[LIFT_DROP; NORM_LIFT; REAL_ABS_NORM] THEN MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN SIMP_TAC[LAMBDA_BETA; VECTOR_SUB_COMPONENT] THEN REAL_ARITH_TAC; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN MP_TAC(SPEC `norm((f:real^M->real^N) x)` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[DIST_0; NORM_LIFT; REAL_ABS_NORM; GSYM LIFT_SUB] THEN MATCH_MP_TAC(NORM_ARITH `&0 < d /\ x = y ==> norm(x:real^N - y) < d`) THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `abs(x) <= n ==> x = max (--n) (min n x)`) THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS; REAL_OF_NUM_LE]]; DISCH_THEN(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[INTEGRAL_0; DIST_0; LE_REFL] THEN DISCH_TAC THEN EXISTS_TAC `(\x. lambda i. max (--(&n)) (min (&n) ((f:real^M->real^N)(x)$i))):real^M->real^N` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[BOUNDED_COMPONENTWISE] THEN REWRITE_TAC[bounded; FORALL_IN_IMAGE] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN EXISTS_TAC `&n` THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN ASM_SIMP_TAC[NORM_LIFT; LAMBDA_BETA] THEN REAL_ARITH_TAC]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?k g. negligible k /\ (!n. g n continuous_on (:real^M)) /\ (!n x. norm(g n x:real^N) <= norm(B % vec 1:real^N)) /\ (!x. x IN (s DIFF k) ==> ((\n. g n x) --> h x) sequentially)` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `(h:real^M->real^N) measurable_on s` MP_TAC THENL [ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_MEASURABLE]; ALL_TAC] THEN REWRITE_TAC[measurable_on] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_THEN(X_CHOOSE_THEN `g:num->real^M->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(\n x. lambda i. max (--B) (min B (((g n x):real^N)$i))): num->real^M->real^N` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN MP_TAC(ISPECL [`(:real^M)`; `(lambda i. B):real^N`] CONTINUOUS_ON_CONST) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP CONTINUOUS_ON_MIN) THEN MP_TAC(ISPECL [`(:real^M)`; `(lambda i. --B):real^N`] CONTINUOUS_ON_CONST) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP CONTINUOUS_ON_MAX) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[FUN_EQ_THM; CART_EQ; LAMBDA_BETA]; REPEAT STRIP_TAC THEN MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN SIMP_TAC[LAMBDA_BETA; VEC_COMPONENT; VECTOR_MUL_COMPONENT] THEN REAL_ARITH_TAC; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `ee:real` THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `norm(c - a:real^N) <= norm(b - a) ==> dist(b,a) < ee ==> dist(c,a) < ee`) THEN MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN SIMP_TAC[LAMBDA_BETA; VECTOR_SUB_COMPONENT] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP NORM_BOUND_COMPONENT_LE) THEN DISCH_THEN(MP_TAC o SPEC `k:num`) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]; ALL_TAC] THEN SUBGOAL_THEN `!n. (g:num->real^M->real^N) n absolutely_integrable_on s` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `(\x. lift(norm(B % vec 1:real^N))):real^M->real^1` THEN ASM_REWRITE_TAC[LIFT_DROP; INTEGRABLE_ON_CONST] THEN ONCE_REWRITE_TAC[GSYM MEASURABLE_ON_UNIV] THEN MATCH_MP_TAC(REWRITE_RULE[lebesgue_measurable; indicator] MEASURABLE_ON_RESTRICT) THEN ASM_SIMP_TAC[CONTINUOUS_IMP_MEASURABLE_ON; ETA_AX] THEN MATCH_MP_TAC INTEGRABLE_IMP_MEASURABLE THEN ASM_REWRITE_TAC[GSYM MEASURABLE_INTEGRABLE]; ALL_TAC] THEN MP_TAC(ISPECL [`\n x. lift(norm((g:num->real^M->real^N) n x - h x))`; `(\x. vec 0):real^M->real^1`; `(\x. lift(B + norm(B % vec 1:real^N))):real^M->real^1`; `s DIFF k:real^M->bool`] DOMINATED_CONVERGENCE) THEN ASM_SIMP_TAC[INTEGRAL_0; INTEGRABLE_ON_CONST; MEASURABLE_DIFF; NEGLIGIBLE_IMP_MEASURABLE] THEN ANTS_TAC THENL [REWRITE_TAC[NORM_LIFT; REAL_ABS_NORM] THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_SUB; ETA_AX] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; REPEAT STRIP_TAC THEN REWRITE_TAC[LIFT_DROP] THEN MATCH_MP_TAC(NORM_ARITH `norm(g:real^N) <= b /\ norm(h) <= a ==> norm(g - h) <= a + b`) THEN ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[GSYM LIM_NULL_NORM; GSYM LIM_NULL]]; REWRITE_TAC[LIM_SEQUENTIALLY] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[LE_REFL; DIST_0] THEN DISCH_TAC THEN EXISTS_TAC `(g:num->real^M->real^N) n` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[bounded; FORALL_IN_IMAGE; IN_UNIV] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `norm(integral s (\x. lift(norm(f x - h x)))) + norm(integral s (\x. lift(norm ((g:num->real^M->real^N) n x - h x))))` THEN CONJ_TAC THENL [MATCH_MP_TAC(NORM_ARITH `norm(x:real^N) <= norm(y + z:real^N) ==> norm(x) <= norm(y) + norm(z)`) THEN W(MP_TAC o PART_MATCH (lhs o rand) (GSYM INTEGRAL_ADD) o rand o rand o snd) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_SUB; ETA_AX] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(MESON[] `norm x = drop x /\ norm(a:real^N) <= drop x ==> norm a <= norm x`) THEN CONJ_TAC THENL [MATCH_MP_TAC NORM_1_POS THEN MATCH_MP_TAC INTEGRAL_DROP_POS THEN SIMP_TAC[DROP_ADD; LIFT_DROP; NORM_POS_LE; REAL_LE_ADD] THEN MATCH_MP_TAC INTEGRABLE_ADD THEN CONJ_TAC; MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[DROP_ADD; LIFT_DROP; NORM_LIFT; REAL_ABS_NORM] THEN REWRITE_TAC[NORM_ARITH `norm(f - g:real^N) <= norm(f - h) + norm(g - h)`] THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC INTEGRABLE_ADD THEN CONJ_TAC]] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_SUB; ETA_AX]; MATCH_MP_TAC(REAL_ARITH `a < e / &2 /\ b < e / &2 ==> a + b < e`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e ==> x = y ==> y < e`)) THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `(!u v. f absolutely_integrable_on (s INTER interval[u,v])) /\ (!u v. (f:real^M->real^N) absolutely_integrable_on (s DIFF interval[u,v]))` STRIP_ASSUME_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_INTER; LEBESGUE_MEASURABLE_INTERVAL; LEBESGUE_MEASURABLE_DIFF; LEBESGUE_MEASURABLE_UNIV]; ALL_TAC] THEN SUBGOAL_THEN `?a b. norm(integral (s INTER interval[a,b]) (\x. lift(norm(f x))) - integral s (\x. lift(norm((f:real^M->real^N) x)))) < e / &3` STRIP_ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [absolutely_integrable_on]) THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRAL] THEN REWRITE_TAC[HAS_INTEGRAL_ALT; INTEGRAL_RESTRICT_INTER] THEN DISCH_THEN(MP_TAC o SPEC `e / &3` o CONJUNCT2) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MESON_TAC[BOUNDED_SUBSET_CLOSED_INTERVAL; BOUNDED_BALL]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s INTER interval[a:real^M,b]`; `e / &3`] lemma) THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_INTERVAL; REAL_ARITH `&0 < e / &3 <=> &0 < e`] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?c d. interval[a:real^M,b] SUBSET interval(c,d) /\ measure(interval(c,d)) - measure(interval[a,b]) < e / &3 / B` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`a:real^M`; `b:real^M`; `e / &3 / B / &2`] EXPAND_CLOSED_OPEN_INTERVAL) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; REAL_ARITH `&0 < &3`] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `&0 < e ==> x <= y + e / &2 ==> x - y < e`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &3`]; ALL_TAC] THEN MP_TAC(ISPECL [`\x. if x IN interval[a,b] then (g:real^M->real^N) x else vec 0`; `(:real^M)`; `interval[a,b] UNION ((:real^M) DIFF interval(c,d))`; `B:real`] TIETZE) THEN REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN; IN_UNIV] THEN ANTS_TAC THENL [ASM_SIMP_TAC[REAL_LT_IMP_LE; FORALL_IN_UNION] THEN SIMP_TAC[CLOSED_UNION; CLOSED_INTERVAL; GSYM OPEN_CLOSED; OPEN_INTERVAL; IN_DIFF; IN_UNIV] THEN ASM_SIMP_TAC[COND_RAND; NORM_0; COND_RATOR; REAL_LT_IMP_LE; COND_ID] THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN SIMP_TAC[CLOSED_INTERVAL; GSYM OPEN_CLOSED; OPEN_INTERVAL] THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ASM SET_TAC[]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^M->real^N`] THEN REWRITE_TAC[FORALL_IN_UNION; bounded; FORALL_IN_IMAGE; IN_UNIV] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_EQ THEN EXISTS_TAC `\x. if x IN s INTER interval(c,d) then (h:real^M->real^N) x else vec 0` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_INTER] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `(\x. lift B):real^M->real^1` THEN ASM_REWRITE_TAC[INTEGRABLE_CONST; LIFT_DROP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_CASES THEN ASM_REWRITE_TAC[SET_RULE `{x | x IN s} = s`; MEASURABLE_ON_0] THEN MATCH_MP_TAC CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN REWRITE_TAC[LEBESGUE_MEASURABLE_INTERVAL] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; REWRITE_TAC[INTEGRABLE_ON_OPEN_INTERVAL; INTEGRABLE_CONST]; GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[NORM_0; REAL_LT_IMP_LE]]; DISCH_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(!u v. h absolutely_integrable_on (s INTER interval[u,v])) /\ (!u v. (h:real^M->real^N) absolutely_integrable_on (s DIFF interval[u,v]))` STRIP_ASSUME_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_INTER; LEBESGUE_MEASURABLE_INTERVAL; LEBESGUE_MEASURABLE_DIFF; LEBESGUE_MEASURABLE_UNIV]; ALL_TAC] THEN TRANS_TAC REAL_LET_TRANS `norm(integral (s INTER interval[a,b]) (\x. lift(norm((f:real^M->real^N) x - h x)))) + norm(integral (s DIFF interval[a,b]) (\x. lift(norm(f x - h x))))` THEN CONJ_TAC THENL [MATCH_MP_TAC(NORM_ARITH `a + b:real^N = c ==> norm(c) <= norm(a) + norm(b)`) THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_UNION o lhand o snd) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_SUB; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REWRITE_TAC[NEGLIGIBLE_EMPTY; SET_RULE `(s INTER t) INTER (s DIFF t) = {} /\ (s INTER t) UNION (s DIFF t) = s`] THEN DISCH_THEN SUBST1_TAC THEN REFL_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH `norm(integral s f) < e / &3 ==> integral s f = integral s g /\ y < &2 / &3 * e ==> norm(integral s g) + y < e`)) THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_EQ THEN ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC REAL_LET_TRANS `drop(integral (s DIFF interval[a,b]) (\x. lift(norm((f:real^M->real^N) x)) + lift(norm(h x:real^N))))` THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_SUB; ABSOLUTELY_INTEGRABLE_ADD; LIFT_DROP; DROP_ADD; NORM_LIFT; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN CONV_TAC NORM_ARITH; ASM_SIMP_TAC[INTEGRAL_ADD; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_NORM; DROP_ADD]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e / &3 ==> z = x /\ y <= e / &3 ==> z + y < &2 / &3 * e`)) THEN CONJ_TAC THENL [REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB] THEN MATCH_MP_TAC(REAL_ARITH `z + y = x /\ &0 <= y ==> y = abs(z - x)`) THEN ASM_SIMP_TAC[INTEGRAL_DROP_POS; LIFT_DROP; NORM_POS_LE; ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REWRITE_TAC[GSYM DROP_ADD; DROP_EQ] THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_UNION o lhand o snd) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REWRITE_TAC[NEGLIGIBLE_EMPTY; SET_RULE `(s INTER t) INTER (s DIFF t) = {} /\ (s INTER t) UNION (s DIFF t) = s`] THEN DISCH_THEN SUBST1_TAC THEN REFL_TAC; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `drop(integral (interval(c,d) DIFF interval[a,b]) (\x:real^M. lift B))` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV; IN_UNIV] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; INTEGRABLE_ON_CONST; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_INTERVAL] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_DIFF] THEN ASM_CASES_TAC `x IN interval(c:real^M,d)` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `x IN interval[a:real^M,b]` THEN ASM_REWRITE_TAC[] THEN REPEAT COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_LE_REFL; LIFT_DROP; NORM_0; REAL_LT_IMP_LE; DROP_VEC] THEN ASM_MESON_TAC[IN_DIFF; IN_UNIV; NORM_0; REAL_LE_REFL]; SIMP_TAC[LIFT_EQ_CMUL; INTEGRAL_CMUL; INTEGRABLE_ON_CONST; MEASURABLE_DIFF; MEASURABLE_INTERVAL; INTEGRAL_MEASURE] THEN REWRITE_TAC[DROP_CMUL; DROP_VEC; REAL_MUL_RID] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x < e ==> y = x ==> y <= e`)) THEN MATCH_MP_TAC MEASURE_DIFF_SUBSET THEN ASM_REWRITE_TAC[MEASURABLE_INTERVAL]]);; (* ------------------------------------------------------------------------- *) (* A kind of continuity of the (absolute) integral under translation. *) (* ------------------------------------------------------------------------- *) let CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_NORM = prove (`!f:real^M->real^N. f absolutely_integrable_on (:real^M) ==> ((\a. integral (:real^M) (\x. lift(norm(f(a + x) - f x)))) --> vec 0) (at (vec 0))`, let lemma = prove (`!f:real^M->real^N. lift(norm(f x)) - (if p x then lift(norm(f x)) else vec 0) = lift(norm(f x - (if p x then f x else vec 0)))`, REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_SUB_REFL] THEN REWRITE_TAC[NORM_0; LIFT_NUM]) in SUBGOAL_THEN `!f:real^M->real^N. f absolutely_integrable_on (:real^M) /\ f continuous_on (:real^M) ==> ((\a. integral (:real^M) (\x. lift(norm(f(a + x) - f x)))) --> vec 0) (at (vec 0))` ASSUME_TAC THENL [REPEAT STRIP_TAC; X_GEN_TAC `f:real^M->real^N` THEN STRIP_TAC THEN REWRITE_TAC[LIM_AT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`; `e / &3`] ABSOLUTELY_INTEGRABLE_APPROXIMATE_CONTINUOUS) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `g:real^M->real^N`) THEN ASM_REWRITE_TAC[LIM_AT] THEN DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[DIST_0] THEN UNDISCH_THEN `norm(integral(:real^M) (\x. lift(norm(f x - g x:real^N)))) < e / &3` (fun th -> MP_TAC th THEN MP_TAC th) THEN MP_TAC(ISPEC `a:real^M` TRANSLATION_UNIV) THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[GSYM INTEGRAL_TRANSLATION] THEN MATCH_MP_TAC(NORM_ARITH `norm(w) <= norm(x + y + z) ==> norm(x:real^N) < e / &3 ==> norm(y:real^N) < e / &3 ==> norm(z:real^N) < e / &3 ==> norm(w:real^N) < e`) THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_ADD o funpow 3 rand o snd) THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_ADD o funpow 2 rand o snd) THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN GEN_REWRITE_TAC RAND_CONV [NORM_1] THEN MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= abs y`) THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[IN_UNIV; DROP_ADD; DROP_SUB; LIFT_DROP] THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_NORM] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; CONV_TAC NORM_ARITH]]] THEN REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN REPEAT(MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ADD THEN CONJ_TAC) THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THEN ASM_REWRITE_TAC[ETA_AX] THEN SUBST1_TAC(SYM(ISPEC `--a:real^M` TRANSLATION_UNIV)) THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN REWRITE_TAC[VECTOR_ARITH `a + --a + x:real^N = x`] THEN ASM_REWRITE_TAC[ETA_AX]] THEN REWRITE_TAC[LIM_AT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`\x. lift(norm((f:real^M->real^N) x))`; `(:real^M)`; `integral (:real^M) (\x. lift(norm((f:real^M->real^N) x)))`] HAS_INTEGRAL_ALT) THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL; IN_UNIV; ETA_AX] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [absolutely_integrable_on]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `ball(vec 0:real^M,B + &1)` BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC) THEN REWRITE_TAC[BOUNDED_BALL; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:real^M` THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `interval[--(c + vec 1):real^M,c + vec 1]`] COMPACT_UNIFORMLY_CONTINUOUS) THEN REWRITE_TAC[COMPACT_INTERVAL] THEN ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC] THEN REWRITE_TAC[uniformly_continuous_on] THEN SUBGOAL_THEN `&0 < content(interval[--(c + vec 1):real^M,c + vec 1])` ASSUME_TAC THENL [REWRITE_TAC[CONTENT_LT_NZ; CONTENT_EQ_0_INTERIOR] THEN MATCH_MP_TAC(SET_RULE `!s. ~(s = {}) /\ s SUBSET t ==> ~(t = {})`) THEN EXISTS_TAC `interval[--c:real^M,c]` THEN REWRITE_TAC[INTERIOR_INTERVAL; SUBSET_INTERVAL] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> ~(s = {}) ==> ~(t = {})`)) THEN REWRITE_TAC[BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `e / &3 / content(interval[--(c + vec 1):real^M,c + vec 1])`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &3`; DIST_0] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d (&1)` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_01] THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN SUBGOAL_THEN `norm(integral(interval[--c,c]) (\x. lift(norm((f:real^M->real^N)(a + x) - f x)))) <= e / &3` MP_TAC THENL [REWRITE_TAC[NORM_1] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x <= a ==> abs x <= a`) THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_DROP_POS THEN REWRITE_TAC[LIFT_DROP; NORM_POS_LE] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN REWRITE_TAC[GSYM INTERVAL_TRANSLATION] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[SUBSET_UNIV]; ALL_TAC] THEN REWRITE_TAC[drop] THEN TRANS_TAC REAL_LE_TRANS `e / &3 / content (interval [--(c + vec 1),c + vec 1]) * content(interval[--c:real^M,c])` THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_COMPONENT_UBOUND THEN REWRITE_TAC[DIMINDEX_1; LE_REFL] THEN CONJ_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN REWRITE_TAC[GSYM INTERVAL_TRANSLATION] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[SUBSET_UNIV]; ALL_TAC] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTERVAL; VECTOR_NEG_COMPONENT] THEN REWRITE_TAC[GSYM REAL_ABS_BOUNDS] THEN DISCH_TAC THEN REWRITE_TAC[GSYM drop; LIFT_DROP; GSYM dist] THEN MATCH_MP_TAC REAL_LT_IMP_LE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[NORM_ARITH `dist(a + x:real^M,x) = norm a`] THEN REWRITE_TAC[IN_INTERVAL; VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; VEC_COMPONENT; GSYM REAL_ABS_BOUNDS] THEN CONJ_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`a:real^M`; `i:num`] COMPONENT_LE_NORM) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_ARITH `e / &3 / x * y <= e / &3 <=> (e * y) / x <= e`] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_LMUL_EQ] THEN MATCH_MP_TAC CONTENT_SUBSET THEN REWRITE_TAC[SUBSET_INTERVAL; VECTOR_NEG_COMPONENT; VECTOR_ADD_COMPONENT; VEC_COMPONENT] THEN REAL_ARITH_TAC]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[IN_UNIV; ETA_AX] THEN MATCH_MP_TAC(NORM_ARITH `norm(y - x:real^M) < &2 * e / &3 ==> norm(x) <= e / &3 ==> norm y < e`) THEN SUBGOAL_THEN `ball(vec 0,B) SUBSET interval[a + --c,a + c] /\ ball(vec 0:real^M,B) SUBSET interval[--c,c]` MP_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[INTERVAL_TRANSLATION; TRANSLATION_SUBSET_GALOIS_RIGHT]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `ball(vec 0:real^M,B + &1)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_BALL_0; FORALL_IN_IMAGE] THEN UNDISCH_TAC `norm(a:real^M) < &1` THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN(fun th -> FIRST_ASSUM(MP_TAC o C MATCH_MP th))) THEN MP_TAC(ISPEC `a:real^M` TRANSLATION_UNIV) THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[INTERVAL_TRANSLATION; GSYM INTEGRAL_TRANSLATION; IMP_IMP] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [NORM_SUB] THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[IN_UNIV; ETA_AX] THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_SUB o rand o funpow 3 lhand o snd) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV] THEN RULE_ASSUM_TAC(REWRITE_RULE[absolutely_integrable_on]) THEN ASM_REWRITE_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_SUB o rand o lhand o rand o lhand o snd) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV] THEN MP_TAC(ISPEC `\x. lift(norm((f:real^M->real^N) x))` INTEGRABLE_TRANSLATION) THEN ASM_SIMP_TAC[GSYM INTERVAL_TRANSLATION; TRANSLATION_UNIV]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[lemma] THEN MATCH_MP_TAC(NORM_ARITH `norm(z:real^M) <= norm(x + y:real^M) ==> norm x < e / &3 /\ norm y < e / &3 ==> norm z < &2 * e / &3`) THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_SUB o rand o lhand o snd) THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_ADD o rand o rand o snd) THEN ANTS_TAC THENL [ALL_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[NORM_1] THEN MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs a`) THEN REWRITE_TAC[GSYM NORM_1] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_UNIV] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[NORM_LIFT; GSYM LIFT_SUB; LIFT_DROP; GSYM LIFT_ADD; VECTOR_SUB_RZERO] THEN CONV_TAC NORM_ARITH]]] THEN REPEAT CONJ_TAC THEN REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN REPEAT(MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC) THEN REPEAT(MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ADD THEN CONJ_TAC) THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN ASM_REWRITE_TAC[TRANSLATION_UNIV; GSYM INTERVAL_TRANSLATION] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[SUBSET_UNIV]);; let CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_NORM_GEN = prove (`!f:real^M->real^N s t. f absolutely_integrable_on s /\ lebesgue_measurable t /\ t SUBSET s ==> ((\a. integral t (\x. lift(norm(f(a + x) - f x)))) --> vec 0) (at (vec 0) within {a | IMAGE (\x. a + x) t SUBSET s})`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV]) THEN DISCH_THEN(MP_TAC o MATCH_MP CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_NORM) THEN MATCH_MP_TAC(MESON[LIM_AT_WITHIN] `((f --> l) (at x within s) ==> (g --> m) (at x within s)) ==> ((f --> l) (at x) ==> (g --> m) (at x within s))`) THEN REWRITE_TAC[] THEN GEN_REWRITE_TAC (LAND_CONV o RATOR_CONV o LAND_CONV o ABS_CONV) [GSYM LIFT_DROP] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_NULL_COMPARISON) THEN REWRITE_TAC[EVENTUALLY_WITHIN] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; IN_ELIM_THM; DIST_0] THEN X_GEN_TAC `a:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[IN_UNIV; CONJ_ASSOC] THEN CONJ_TAC THENL [REWRITE_TAC[INTEGRABLE_RESTRICT_UNIV] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN REWRITE_TAC[REWRITE_RULE[IN] ABSOLUTELY_INTEGRABLE_RESTRICT_INTER] THEN REWRITE_TAC[SET_RULE `(\x. P x) INTER UNIV = {x | P x}`] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN REPEAT CONJ_TAC THEN REWRITE_TAC[IN_TRANSLATION_GALOIS_ALT; SET_RULE `{x | x IN s} = s`] THEN ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; VECTOR_ARITH `a + --a + x:real^N = x`] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_TRANSLATION]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[LIFT_DROP] THEN ASM_CASES_TAC `(x:real^M) IN t` THEN ASM_REWRITE_TAC[NORM_0; NORM_POS_LE; NORM_LIFT; REAL_ABS_NORM] THEN REPEAT(COND_CASES_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN REWRITE_TAC[REAL_LE_REFL]]);; let CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_GEN = prove (`!f:real^M->real^N s t. f absolutely_integrable_on s /\ lebesgue_measurable t ==> (\a. integral t (\x. f(a + x))) continuous_on {a | IMAGE (\x. a + x) t SUBSET s}`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON] THEN X_GEN_TAC `a:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[LIM_NULL] THEN ONCE_REWRITE_TAC[LIM_WITHIN_ZERO] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `IMAGE (\x:real^M. a + x) t`] CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_NORM_GEN) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_TRANSLATION] THEN REWRITE_TAC[GSYM INTEGRAL_TRANSLATION] THEN MATCH_MP_TAC(MESON[LIM_WITHIN_SUBSET] `t SUBSET s /\ ((x --> l) (at a within s) ==> (y --> m) (at a within s)) ==> (x --> l) (at a within s) ==> (y --> m) (at a within t)`) THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC; GSYM IMAGE_o] THEN REWRITE_TAC[IN_ELIM_THM; VECTOR_ARITH `b - a + a + x:real^N = b + x`]; ALL_TAC] THEN REWRITE_TAC[] THEN GEN_REWRITE_TAC (LAND_CONV o RATOR_CONV o LAND_CONV o ABS_CONV) [GSYM LIFT_DROP] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_NULL_COMPARISON) THEN REWRITE_TAC[EVENTUALLY_WITHIN] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; IN_ELIM_THM] THEN X_GEN_TAC `b:real^M` THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[VECTOR_ARITH `b + a + x:real^N = (a + b) + x`] THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_SUB o rand o lhand o snd) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[INTEGRABLE_TRANSLATION] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_TRANSLATION]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[LIFT_DROP; REAL_LE_REFL] THEN CONJ_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_TRANSLATION]);; let CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION = prove (`!f:real^M->real^N. f absolutely_integrable_on (:real^M) ==> (\a. integral (:real^M) (\x. f(a + x))) continuous_on (:real^M)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`; `(:real^M)`] CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_GEN) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV; SUBSET_UNIV; UNIV_GSPEC]);; let CONTINUOUS_MEASURE_TRANSLATION_SYMDIFF = prove (`!s:real^N->bool. measurable s ==> ((\a. lift(measure(((IMAGE (\x. a + x) s) DIFF s) UNION (s DIFF IMAGE (\x. a + x) s)))) --> vec 0) (at (vec 0))`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [GSYM LIM_AT_REFLECT] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [SET_RULE `s = s INTER UNIV`]) THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_ON_INDICATOR] THEN REWRITE_TAC[VECTOR_NEG_0] THEN DISCH_THEN(MP_TAC o MATCH_MP CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_NORM) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `a:real^N` THEN ASM_SIMP_TAC[MEASURE_INTEGRAL; MEASURABLE_UNION; MEASURABLE_DIFF; MEASURABLE_TRANSLATION_EQ; LIFT_DROP] THEN GEN_REWRITE_TAC RAND_CONV [GSYM INTEGRAL_RESTRICT_UNIV] THEN AP_TERM_TAC THEN ABS_TAC THEN ASM_REWRITE_TAC[indicator; IN_DIFF; IN_UNION] THEN REWRITE_TAC[IN_TRANSLATION_GALOIS] THEN REWRITE_TAC[VECTOR_ARITH `x - --a:real^N = a + x`] THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; NORM_1] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[DROP_SUB; DROP_VEC]) THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let CONTINUOUS_MEASURE_TRANSLATION_DIFF = prove (`!s:real^N->bool. measurable s ==> ((\a. lift(measure((IMAGE (\x. a + x) s) DIFF s))) --> vec 0) (at (vec 0))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_MEASURE_TRANSLATION_SYMDIFF) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_NULL_COMPARISON) THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `a:real^N` THEN ASM_SIMP_TAC[NORM_LIFT; real_abs; MEASURE_POS_LE; MEASURABLE_DIFF; MEASURABLE_TRANSLATION_EQ] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNION; MEASURABLE_DIFF; MEASURABLE_TRANSLATION_EQ] THEN SET_TAC[]);; let CONTINUOUS_MEASURE_DIFFERENTIABLE_IMAGE_TRANSLATION = prove (`!f:real^N->real^N s k. open s /\ f differentiable_on s /\ compact k ==> (\a. lift(measure(IMAGE f (IMAGE (\x. a + x) k)))) continuous_on {a | IMAGE (\x. a + x) k SUBSET s}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`k:real^N->bool`; `s:real^N->bool`] OPEN_TRANSLATION_SUBSET_PREIMAGE) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN MP_TAC(ISPEC `{a:real^N | IMAGE (\x. a + x) k SUBSET s}` OPEN_CONTAINS_CBALL) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `b:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN REWRITE_TAC[continuous_at] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `{x + d:real^N | x IN k /\ d IN cball(b,r)}`] ABSOLUTELY_CONTINUOUS_MEASURE_DIFFERENTIABLE_IMAGE) THEN ASM_SIMP_TAC[COMPACT_SUMS; COMPACT_CBALL] THEN SUBGOAL_THEN `{x + d:real^N | x IN k /\ d IN cball (b,r)} SUBSET s` ASSUME_TAC THENL [REWRITE_TAC[FORALL_IN_GSPEC; SUBSET] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `d:real^N`] THEN STRIP_TAC THEN UNDISCH_TAC `cball(b:real^N,r) SUBSET {a | IMAGE (\x. a + x) k SUBSET s}` THEN GEN_REWRITE_TAC LAND_CONV [SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `d:real^N`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_ELIM_THM; SUBSET; FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[VECTOR_ADD_SYM]; ALL_TAC] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] DIFFERENTIABLE_ON_SUBSET)) THEN ASM_REWRITE_TAC[]; DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[]] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `IMAGE (\x:real^N. b + x) k` CONTINUOUS_MEASURE_TRANSLATION_SYMDIFF) THEN ASM_SIMP_TAC[MEASURABLE_COMPACT; COMPACT_TRANSLATION] THEN REWRITE_TAC[LIM_AT; DIST_LIFT; DIST_0; NORM_LIFT] THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min t r:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c - b:real^N`) THEN ASM_REWRITE_TAC[GSYM dist; GSYM DIST_NZ] THEN ASM_CASES_TAC `c:real^N = b` THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_ABS_NUM] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[VECTOR_ARITH `c - b + b + x:real^N = c + x`] THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN FIRST_X_ASSUM(MP_TAC o SPEC (rand(rand(lhand(concl th)))))) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[REAL_ARITH `abs x < d ==> x < d`] THEN ASM_SIMP_TAC[MEASURABLE_UNION; MEASURABLE_DIFF; MEASURABLE_COMPACT; COMPACT_TRANSLATION_EQ] THEN MATCH_MP_TAC(SET_RULE `s UNION t SUBSET u ==> (s DIFF t UNION t DIFF s) SUBSET u`) THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [VECTOR_ADD_SYM] THEN SUBGOAL_THEN `b IN cball(b:real^N,r) /\ c IN cball(b,r)` MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[IN_CBALL; DIST_REFL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)] THEN SUBGOAL_THEN `IMAGE (\x. b + x) k SUBSET {x + d:real^N | x IN k /\ d IN cball(b,r)} /\ IMAGE (\x. c + x) k SUBSET {x + d:real^N | x IN k /\ d IN cball(b,r)}` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN REWRITE_TAC[FORALL_IN_IMAGE; SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `x:real^N` THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[VECTOR_ARITH `b + x:real^N = x + c <=> b = c`] THEN ASM_REWRITE_TAC[UNWIND_THM1; IN_CBALL; DIST_REFL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN MATCH_MP_TAC(REAL_ARITH `x <= y + d /\ y <= x + d ==> d < e ==> abs(x - y) < e`) THEN CONJ_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_UNION_LE o rand o snd) THEN (ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]]]) THEN CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^N->real^N) {x + d | x IN k /\ d IN cball(b,r)}` THEN (REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL [REPEAT(MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN CONJ_TAC) THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_TRANSLATION; LEBESGUE_MEASURABLE_COMPACT; LEBESGUE_MEASURABLE_UNION; LEBESGUE_MEASURABLE_DIFF] THEN REWRITE_TAC[LE_REFL] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_SIMP_TAC[COMPACT_SUMS; COMPACT_CBALL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_ON]]));; (* ------------------------------------------------------------------------- *) (* A kind of mean-value theorem for integrals w.r.t. arbitrarily small *) (* subintervals, similar to the one in Saint-Raymond's "Local Inversion..." *) (* ------------------------------------------------------------------------- *) let SUBINTERVAL_MEAN_VALUE_THEOREM = prove (`!f:real^N->real^1 a b n. ~(interval[a,b] = {}) /\ ~(n = 0) /\ f absolutely_integrable_on interval[a,b] ==> ?c d. c + inv(&n) % (b - a) = d /\ interval[c,d] SUBSET interval[a,b] /\ measure(interval[a,b]) % integral (interval[c,d]) f = measure(interval[c,d]) % integral (interval[a,b]) f`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `&0 < &n /\ ~(&n = &0)` STRIP_ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_EQ]) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[UNWIND_THM1]] THEN ASM_CASES_TAC `content(interval[a:real^N,b]) = &0` THENL [EXISTS_TAC `a:real^N` THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN CONJ_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTENT_0_SUBSET)) THEN ASM_SIMP_TAC[INTEGRAL_NULL; VECTOR_MUL_RZERO]; ASM_REWRITE_TAC[SUBSET_INTERVAL] THEN ASM_SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT; REAL_LE_REFL; REAL_LE_ADDR; REAL_LE_MUL_EQ; REAL_LT_INV_EQ] THEN REWRITE_TAC[REAL_SUB_LE; REAL_ARITH `a + n * (b - a) <= b <=> &0 <= (&1 - n) * (b - a)`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN ASM_SIMP_TAC[LE_1; REAL_OF_NUM_LE]]; RULE_ASSUM_TAC(REWRITE_RULE[CONTENT_POS_LT_EQ; GSYM CONTENT_LT_NZ])] THEN SUBGOAL_THEN `!c. measure(interval[c:real^N,c + inv(&n) % (b - a)]) = measure(interval[a,b]) / &n pow dimindex(:N)` (fun th -> REWRITE_TAC[th]) THENL [GEN_TAC THEN REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; REAL_ADD_SUB] THEN REWRITE_TAC[REAL_ARITH `c <= c + inv x * y <=> &0 <= y / x`] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_MUL_LZERO; VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[REAL_SUB_LE; PRODUCT_MUL_NUMSEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN SIMP_TAC[PRODUCT_CONST; FINITE_NUMSEG; CARD_NUMSEG_1; REAL_POW_INV] THEN REWRITE_TAC[REAL_MUL_AC]; REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC; VECTOR_MUL_LCANCEL] THEN REWRITE_TAC[LEFT_OR_DISTRIB; EXISTS_OR_THM] THEN DISJ2_TAC] THEN MATCH_MP_TAC(SET_RULE `(!a b. segment[a,b] SUBSET {x | P x} ==> segment [f a,f b] SUBSET IMAGE f (segment [a,b])) /\ (!a b. a IN {x | P x} /\ b IN {x | P x} ==> segment[a,b] SUBSET {x | P x}) /\ (?a b. P a /\ P b /\ c IN segment[f a,f b]) ==> ?x. P x /\ f x = c`) THEN REPEAT CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`c:real^N`; `d:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC SUBSET_CONTINUOUS_IMAGE_SEGMENT_1 THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET)) THEN ONCE_REWRITE_TAC[MESON[VECTOR_ADD_RID] `interval[c:real^N,c + d] = interval[c + vec 0,c + d]`] THEN REWRITE_TAC[INTERVAL_TRANSLATION; GSYM INTEGRAL_TRANSLATION] THEN MATCH_MP_TAC CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_GEN THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_INTERVAL]; REWRITE_TAC[GSYM CONVEX_CONTAINS_SEGMENT] THEN ONCE_REWRITE_TAC[MESON[VECTOR_ADD_RID] `interval[c:real^N,c + d] = interval[c + vec 0,c + d]`] THEN REWRITE_TAC[INTERVAL_TRANSLATION] THEN MATCH_MP_TAC CONVEX_TRANSLATION_SUBSET_PREIMAGE THEN REWRITE_TAC[CONVEX_INTERVAL]; MATCH_MP_TAC(MESON[INTERVAL_SUBSET_SEGMENT_1; SUBSET] `(?c d. P c /\ P d /\ x IN interval[(f:real^N->real^1) c,f d]) ==> ?c d. P c /\ P d /\ x IN segment[f c,f d]`) THEN REWRITE_TAC[IN_INTERVAL_1; MESON[] `(?a b. P a /\ P b /\ Q a /\ R b) <=> (?c. P c /\ Q c) /\ (?d. P d /\ R d)`] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN SPEC_TAC(`f:real^N->real^1`,`f:real^N->real^1`) THEN MATCH_MP_TAC(MESON[] `!g. (!f. P f ==> P(g f)) /\ (!f c. P f ==> R (g f) c ==> Q f c) /\ (!f. P f ==> ?d. R f d) ==> !f. P f ==> (?c. Q f c) /\ (?d. R f d)`) THEN EXISTS_TAC `\f:real^N->real^1. \x. --(f x)` THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_NEG_EQ] THEN CONJ_TAC THENL [SIMP_TAC[INTEGRAL_NEG; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_RNEG; DROP_NEG] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `x = --y ==> --f <= x ==> y <= f`) THEN REWRITE_TAC[GSYM DROP_NEG] THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_NEG THEN ASM_MESON_TAC[INTEGRABLE_SUBINTERVAL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; X_GEN_TAC `f:real^N->real^1` THEN STRIP_TAC]] THEN MP_TAC(ISPECL [`f:real^N->real^1`; `{interval[(lambda i. a$i + &((m:num^N)$i) / &n * (b$i - a$i)):real^N, lambda i. (a:real^N)$i + &(m$i + 1) / &n * ((b:real^N)$i - a$i)] |m| !i. 1 <= i /\ i <= dimindex(:N) ==> m$i < n}`; `interval[a:real^N,b]`] INTEGRAL_COMBINE_DIVISION_TOPDOWN) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN MATCH_MP_TAC(TAUT `p /\ (p /\ q ==> r) ==> (p ==> q) ==> r`) THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN REWRITE_TAC[division_of; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM; EXISTS_IN_GSPEC; SET_RULE `UNIONS f = s <=> (!t. t IN f ==> t SUBSET s) /\ (!x. x IN s ==> ?t. t IN f /\ x IN t)`] THEN REWRITE_TAC[IMP_IMP] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC FINITE_IMAGE THEN MATCH_MP_TAC FINITE_CART THEN REWRITE_TAC[FINITE_NUMSEG_LT]; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> (p ==> r) /\ p /\ q /\ s`] THEN CONJ_TAC THENL [SIMP_TAC[]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `m:num^N` THEN DISCH_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN CONJ_TAC THENL [SIMP_TAC[SUBSET_INTERVAL; LAMBDA_BETA] THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[REAL_LE_ADDR; REAL_ARITH `a + x * (b - a) <= b <=> &0 <= (&1 - x) * (b - a)`] THEN X_GEN_TAC `i:num` THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_SUB_LE; REAL_LE_DIV; REAL_POS; REAL_LT_IMP_LE; REAL_LE_LDIV_EQ] THEN REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN ASM_SIMP_TAC[ARITH_RULE `m < n ==> m + 1 <= n`]; SIMP_TAC[INTERVAL_NE_EMPTY; LAMBDA_BETA; REAL_LE_LADD] THEN X_GEN_TAC `i:num` THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_OF_NUM_LE] THEN ARITH_TAC]; GEN_REWRITE_TAC BINDER_CONV [RIGHT_IMP_FORALL_THM] THEN REWRITE_TAC[IMP_IMP] THEN MATCH_MP_TAC(MESON[] `(!x y. ~(x = y) ==> R x y) ==> !x y. P x /\ P y /\ ~(f x = f y) ==> R x y`) THEN REWRITE_TAC[DISJOINT_INTERVAL; INTERIOR_INTERVAL] THEN REWRITE_TAC[MESON[] `(?i. P i /\ Q i /\ R i) <=> ~(!i. P i /\ Q i ==> ~R i)`] THEN SIMP_TAC[LAMBDA_BETA; REAL_LE_LADD] THEN REPEAT GEN_TAC THEN REWRITE_TAC[CART_EQ; NOT_FORALL_THM; NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN DISJ2_TAC THEN DISJ2_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE `~(m:num = n) ==> m + 1 <= n \/ n + 1 <= m`)) THEN MATCH_MP_TAC MONO_OR THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_OF_NUM_LE]; X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTERVAL] THEN DISCH_TAC THEN SIMP_TAC[LAMBDA_BETA] THEN REWRITE_TAC[AND_FORALL_THM; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN REWRITE_TAC[GSYM LAMBDA_SKOLEM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `(x:real^N)$i = (b:real^N)$i` THENL [EXISTS_TAC `n - 1` THEN ASM_SIMP_TAC[EXP_EQ_0; ARITH_EQ; ARITH_RULE `~(n = 0) ==> n - 1 < n /\ n - 1 + 1 = n`] THEN ASM_SIMP_TAC[REAL_DIV_REFL] THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN REWRITE_TAC[REAL_ARITH `a + x * (b - a) <= b <=> &0 <= (&1 - x) * (b - a)`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_MUL_LID; REAL_OF_NUM_LE] THEN ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `(a:real^N)$i < (b:real^N)$i` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(SPEC `&n * ((x:real^N)$i - (a:real^N)$i) / ((b:real^N)$i - a$i)` FLOOR_POS) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_DIV; REAL_SUB_LE; REAL_LT_IMP_LE] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN CONJ_TAC THENL [REWRITE_TAC[GSYM REAL_OF_NUM_LT] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN SIMP_TAC[GSYM REAL_NOT_LE; REAL_LE_FLOOR; INTEGER_CLOSED] THEN REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN REWRITE_TAC[REAL_NOT_LE; GSYM REAL_OF_NUM_POW] THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ] THEN ASM_REAL_ARITH_TAC; FIRST_ASSUM(MP_TAC o MATCH_MP (MESON[FLOOR; REAL_LT_IMP_LE] `floor x = n ==> n <= x /\ x <= n + &1`)) THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM REAL_LE_LDIV_EQ); ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM REAL_LE_RDIV_EQ)] THEN UNDISCH_TAC `(a:real^N)$i < (b:real^N)$i` THEN SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_SUB_LT] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC]]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)] THEN FIRST_ASSUM(MP_TAC o MATCH_MP DIVISION_OF_FINITE) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `FINITE s ==> ~(s = {}) ==> FINITE s /\ ~(s = {})`)) THEN ANTS_TAC THENL [REWRITE_TAC[SET_RULE `{f x | P x} = {} <=> !n. ~(P n)`] THEN REWRITE_TAC[NOT_FORALL_THM; GSYM LAMBDA_SKOLEM] THEN ASM_MESON_TAC[LE_1]; STRIP_TAC] THEN DISCH_THEN(MP_TAC o AP_TERM `drop`) THEN REWRITE_TAC[DROP_VSUM; o_DEF] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `a = b ==> ~(b < a)`)) THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] SUM_BOUND_LT_GEN)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LT] THEN DISCH_THEN(X_CHOOSE_THEN `m:num^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN W(MP_TAC o PART_MATCH (lhand o rand) CARD_IMAGE_INJ o rand o rand o lhand o lhand o snd) THEN SIMP_TAC[CARD_CART; FINITE_NUMSEG_LT; CARD_NUMSEG_LT; FINITE_CART] THEN ANTS_TAC THENL [REWRITE_TAC[IN_ELIM_THM; EQ_INTERVAL; INTERVAL_EQ_EMPTY; CART_EQ] THEN REWRITE_TAC[MESON[] `(?x. P x /\ Q x /\ R x) <=> ~(!x. P x /\ Q x ==> ~R x)`] THEN SIMP_TAC[LAMBDA_BETA; REAL_EQ_ADD_LCANCEL; REAL_LT_LADD] THEN ASM_SIMP_TAC[REAL_LT_RMUL_EQ; REAL_SUB_LT; REAL_LT_DIV2_EQ] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD; REAL_ARITH `~(x + &1 < x)`] THEN ASM_REWRITE_TAC[real_div; REAL_EQ_MUL_RCANCEL] THEN ASM_SIMP_TAC[REAL_SUB_0; REAL_LT_IMP_NZ; REAL_INV_EQ_0] THEN ASM_SIMP_TAC[REAL_EQ_ADD_RCANCEL; REAL_OF_NUM_EQ; REAL_LT_IMP_NE]; DISCH_THEN SUBST1_TAC] THEN SIMP_TAC[NPRODUCT_CONST; FINITE_NUMSEG; CARD_NUMSEG_1] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [REAL_ARITH `x / y:real = inv y * x`] THEN REWRITE_TAC[GSYM REAL_OF_NUM_POW; REAL_INV_POW; GSYM DROP_CMUL] THEN MATCH_MP_TAC(MESON[] `p u = v /\ interval[u,v] SUBSET s ==> a <= drop(integral (interval[u,v]) f) ==> ?d. interval[d,p d] SUBSET s /\ a <= drop(integral(interval[d,p d]) f)`) THEN CONJ_TAC THENL [SIMP_TAC[CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC; FIRST_ASSUM(MP_TAC o last o CONJUNCTS o GEN_REWRITE_RULE I [division_of]) THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `UNIONS s = a ==> !i. i IN s ==> i SUBSET a`)) THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[]]);; let SUBINTERVAL_MEAN_VALUE_THEOREM_SEQ = prove (`!f:real^N->real^1 a b. ~(interval[a,b] = {}) /\ f absolutely_integrable_on interval[a,b] ==> ?c d. (!n. ?m. ~(m = 0) /\ c n + inv(&m) % (b - a) = d n) /\ (!n. ~(interval[c n,d n] = {})) /\ ((\n. d n - c n) --> vec 0) sequentially /\ (!n. interval[c n,d n] SUBSET interval[a,b]) /\ (!n. interval[c(SUC n),d(SUC n)] SUBSET interval[c n,d n]) /\ (!n. measure(interval[a,b]) % integral (interval[c n,d n]) f = measure(interval[c n,d n]) % integral (interval[a,b]) f)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?p. ((!n. ?m. ~(m = 0) /\ FST(p n) + inv(&m) % (b - a) = SND(p n)) /\ (!n. dist(FST(p n),SND(p n)) < inv(&n + &1)) /\ (!n. interval[FST(p n),SND(p n)] SUBSET interval[a,b]) /\ (!n. measure(interval[a,b]) % integral (interval[FST(p n),SND(p n)]) f = measure(interval[FST(p n),SND(p n)]) % integral (interval[a,b]) (f:real^N->real^1))) /\ (!n. interval[FST(p(SUC n)),SND(p(SUC n))] SUBSET interval[FST(p n),SND(p n)])` MP_TAC THENL [GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV o TOP_DEPTH_CONV) [AND_FORALL_THM] THEN MATCH_MP_TAC DEPENDENT_CHOICE THEN REWRITE_TAC[EXISTS_PAIR_THM; FORALL_PAIR_THM]; REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `p:num->real^N#real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `FST o (p:num->real^N#real^N)` THEN EXISTS_TAC `SND o (p:num->real^N#real^N)` THEN ASM_REWRITE_TAC[o_THM] THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV o LAND_CONV) [GSYM PAIR] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SKOLEM_THM]) THEN PURE_REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (fun th -> PURE_REWRITE_TAC[GSYM th])) THEN UNDISCH_TAC `~(interval[a:real^N,b] = {})` THEN PURE_REWRITE_TAC[INTERVAL_NE_EMPTY] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT] THEN SIMP_TAC[REAL_LE_ADDR; REAL_LE_MUL; REAL_LE_INV_EQ; REAL_POS; VECTOR_SUB_COMPONENT; REAL_SUB_LE]; MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\n. inv(&n + &1)` THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; EVENTUALLY_TRUE] THEN REWRITE_TAC[SEQ_HARMONIC_OFFSET]]] THEN CONJ_TAC THENL [MP_TAC(ISPEC `dist(a:real^N,b) + &1` REAL_ARCH_SIMPLE) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[NORM_ARITH `~(dist(p:real^N,q) + &1 <= &0)`] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real^1`; `a:real^N`; `b:real^N`; `n:num`] SUBINTERVAL_MEAN_VALUE_THEOREM) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; CONV_TAC REAL_RAT_REDUCE_CONV] THEN EXPAND_TAC "d" THEN REWRITE_TAC[NORM_ARITH `dist(c:real^N,c + x) = norm x`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] NORM_MUL] THEN ASM_SIMP_TAC[REAL_ABS_INV; REAL_ABS_NUM; REAL_LT_LDIV_EQ; GSYM real_div; REAL_OF_NUM_LT; LE_1; REAL_MUL_LID] THEN UNDISCH_TAC `dist(a:real^N,b) + &1 <= &n` THEN CONV_TAC NORM_ARITH; MAP_EVERY X_GEN_TAC [`k:num`; `c:real^N`; `d:real^N`] THEN STRIP_TAC THEN MP_TAC(ISPEC `dist(c:real^N,d) * &(k + 2) + &1` REAL_ARCH_SIMPLE) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN ASM_CASES_TAC `n = 0` THENL [MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> ~(x + &1 <= &0)`) THEN SIMP_TAC[REAL_LE_MUL; REAL_POS; DIST_POS_LE]; DISCH_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^1`; `c:real^N`; `d:real^N`; `n:num`] SUBINTERVAL_MEAN_VALUE_THEOREM) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL]] THEN UNDISCH_TAC `~(interval[a:real^N,b] = {})` THEN EXPAND_TAC "d" THEN REWRITE_TAC[INTERVAL_NE_EMPTY] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT] THEN SIMP_TAC[REAL_LE_ADDR; REAL_LE_MUL; REAL_LE_INV_EQ; REAL_POS; VECTOR_SUB_COMPONENT; REAL_SUB_LE]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [MAP_EVERY EXPAND_TAC ["v"; "d"] THEN REWRITE_TAC[VECTOR_ADD_SUB] THEN EXISTS_TAC `m * n:num` THEN ASM_REWRITE_TAC[MULT_EQ_0; VECTOR_MUL_ASSOC] THEN REWRITE_TAC[REAL_OF_NUM_MUL; GSYM REAL_INV_MUL] THEN REWRITE_TAC[MULT_SYM]; EXPAND_TAC "v" THEN REWRITE_TAC[NORM_ARITH `dist(u:real^N,u + v) = norm v`] THEN REWRITE_TAC[REAL_OF_NUM_SUC; REAL_OF_NUM_ADD] THEN REWRITE_TAC[ARITH_RULE `SUC(SUC n) = n + 2`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] NORM_MUL] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NUM; GSYM real_div] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN SIMP_TAC[REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; ARITH_RULE `0 < n + 2`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[SUBSET_TRANS]; REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM DROP_EQ])) THEN REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL] THEN MATCH_MP_TAC(REAL_RING `((m0 = &0 ==> i0 = &0) /\ (m1 = &0 ==> i1 = &0) /\ (m2 = &0 ==> i2 = &0)) /\ ((m0 = &0 ==> m1 = &0) /\ (m1 = &0 ==> m2 = &0)) ==> m0 * i1 = m1 * i0 ==> m1 * i2 = m2 * i1 ==> m0 * i2 = m2 * i0`) THEN REWRITE_TAC[MESON[LIFT_DROP; LIFT_NUM] `drop x = &0 <=> x = vec 0`] THEN SIMP_TAC[MEASURE_INTERVAL; INTEGRAL_NULL] THEN ASM_MESON_TAC[CONTENT_0_SUBSET; SUBSET_TRANS]]]);; let SUBINTERVAL_MEAN_VALUE_THEOREM_ALT = prove (`!f:real^N->real^1 a b. ~(interval[a,b] = {}) /\ f absolutely_integrable_on interval[a,b] ==> ?x. x IN interval[a,b] /\ !e. &0 < e ==> ?c d n. ~(n = 0) /\ c + inv(&n) % (b - a) = d /\ x IN interval[c,d] /\ interval[c,d] SUBSET interval[a,b] /\ diameter(interval[c,d]) < e /\ measure(interval[a,b]) % integral (interval[c,d]) f = measure(interval[c,d]) % integral (interval[a,b]) f`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^1`; `a:real^N`; `b:real^N`] SUBINTERVAL_MEAN_VALUE_THEOREM_SEQ) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:num->real^N`; `d:num->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPEC `\n. interval[(c:num->real^N) n,d n]` COMPACT_NEST) THEN ASM_REWRITE_TAC[COMPACT_INTERVAL] THEN ANTS_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN SET_TAC[]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_GSPEC; IN_ELIM_THM]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNIV] THEN DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[NORM_ARITH `dist(x - y:real^N,vec 0) = dist(x,y)`] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[LE_REFL] THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(ASSUME_TAC o SPEC `n:num`)) THEN FIRST_X_ASSUM(X_CHOOSE_THEN `m:num` STRIP_ASSUME_TAC) THEN MAP_EVERY EXISTS_TAC [`(c:num->real^N) n`; `(d:num->real^N) n`; `m:num`] THEN ASM_REWRITE_TAC[DIAMETER_INTERVAL; GSYM dist]);; (* ------------------------------------------------------------------------- *) (* A kind of intermediate value result for a function where all points are *) (* a weak kind of Lebesgue point (inspired by Saint-Raymond's paper). *) (* ------------------------------------------------------------------------- *) let WEAK_LEBESGUE_POINTS_IMP_IVT = prove (`!f:real^N->real^1 a b s. open s /\ connected s /\ ~(interval(a,b) = {}) /\ locally ((absolutely_integrable_on) f) s /\ (!c x. x IN s /\ (!n. ?u v. &0 < u /\ c n = IMAGE (\x. u % x + v) (interval[a,b])) /\ eventually (\n. x IN c n) sequentially /\ ((\n. lift(diameter(c n))) --> vec 0) sequentially ==> ((\n. inv(measure(c n)) % integral (c n) f) --> f x) sequentially) ==> connected(IMAGE f s)`, REPLICATE_TAC 3 GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s /\ t <=> s /\ p /\ q /\ r /\ t`] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC(MESON[] `!P. ((!s. P s ==> R s) ==> (!s. Q s ==> R s)) /\ (!s. P s ==> R s) ==> !s. Q s ==> R s`) THEN EXISTS_TAC `\t. (f:real^N->real^1) absolutely_integrable_on t` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [DISCH_THEN(LABEL_TAC "*") THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN STRIP_TAC THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN REWRITE_TAC[FORALL_IN_IMAGE_2] THEN MATCH_MP_TAC CONNECTED_EQUIVALENCE_RELATION THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [locally]) THEN DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`; `x:real^N`]) THEN ASM_SIMP_TAC[OPEN_IN_REFL; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN ASM_SIMP_TAC[OPEN_IN_OPEN_EQ] THEN STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `connected_component u (x:real^N)`) THEN ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN ASM_SIMP_TAC[OPEN_CONNECTED_COMPONENT] THEN ANTS_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `v:real^N->bool` THEN ASM_SIMP_TAC[OPEN_CONNECTED_COMPONENT; LEBESGUE_MEASURABLE_OPEN] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; SUBSET]; DISCH_TAC] THEN EXISTS_TAC `connected_component u (x:real^N)` THEN ASM_SIMP_TAC[OPEN_CONNECTED_COMPONENT] THEN REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; SUBSET]; ALL_TAC] THEN X_GEN_TAC `y:real^N` THEN GEN_REWRITE_TAC LAND_CONV [GSYM IN] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_IFF_CONNECTED_COMPONENT]) THEN REWRITE_TAC[FORALL_IN_IMAGE_2] THEN DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN ANTS_TAC THENL [ASM_MESON_TAC[IN; CONNECTED_COMPONENT_REFL]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s x ==> t x`) THEN REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MONO THEN MATCH_MP_TAC IMAGE_SUBSET THEN ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; SUBSET_TRANS]; REPEAT STRIP_TAC] THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [MESON_TAC[CONNECTED_COMPONENT_SYM]; ALL_TAC] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP] THEN ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`] THEN REWRITE_TAC[FORALL_IN_IMAGE_2] THEN MAP_EVERY X_GEN_TAC [`p:real^N`; `q:real^N`] THEN STRIP_TAC THEN STRIP_TAC THEN REWRITE_TAC[connected_component] THEN EXISTS_TAC `interval[(f:real^N->real^1) p,f q]` THEN REWRITE_TAC[CONNECTED_INTERVAL] THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1] THEN REWRITE_TAC[SUBSET; IN_INTERVAL_1] THEN X_GEN_TAC `y:real^1` THEN STRIP_TAC THEN ASM_CASES_TAC `y = (f:real^N->real^1) p` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `y = (f:real^N->real^1) q` THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `drop(f(p:real^N)) < drop y /\ drop y < drop(f q) /\ drop(f p) < drop(f q)` STRIP_ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[GSYM DROP_EQ]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPEC `s:real^N->bool` path_connected) THEN ASM_SIMP_TAC[PATH_CONNECTED_EQ_CONNECTED] THEN DISCH_THEN(MP_TAC o SPECL [`p:real^N`; `q:real^N`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?d. &0 < d /\ !r:real^N. r IN path_image g ==> ball(r,d) SUBSET s` STRIP_ASSUME_TAC THENL [ASM_CASES_TAC `s = (:real^N)` THENL [ASM_MESON_TAC[SUBSET_UNIV; REAL_LT_01]; ALL_TAC] THEN EXISTS_TAC `setdist(path_image g,(:real^N) DIFF s)` THEN ASM_SIMP_TAC[SETDIST_POS_LT; SETDIST_EQ_0_COMPACT_CLOSED; COMPACT_PATH_IMAGE; GSYM OPEN_CLOSED; PATH_IMAGE_NONEMPTY] THEN CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[SUBSET; IN_BALL]] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[REAL_NOT_LT] THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM_REWRITE_TAC[IN_UNIV; IN_DIFF]; ALL_TAC] THEN MP_TAC(ISPECL [`a:real^N`; `b:real^N`] INTERVAL_OPEN_SUBSET_CLOSED) THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THEN ASM_REWRITE_TAC[SUBSET_EMPTY] THEN DISCH_THEN(K ALL_TAC) THEN FIRST_ASSUM(MP_TAC o SPECL [`\n. IMAGE (\x. inv(&n + &1) % x + (p - inv(&n + &1) % a)) (interval[a:real^N,b])`; `p:real^N`]) THEN FIRST_ASSUM(MP_TAC o SPECL [`\n. IMAGE (\x. inv(&n + &1) % x + (q - inv(&n + &1) % a)) (interval[a:real^N,b])`; `q:real^N`]) THEN MATCH_MP_TAC(TAUT `(p /\ r) /\ (p /\ r ==> q /\ s ==> t) ==> (p ==> q) ==> (r ==> s) ==> t`) THEN CONJ_TAC THENL [CONJ_TAC THEN (ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN EXISTS_TAC `inv(&n + &1)` THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN SET_TAC[]; ASM_REWRITE_TAC[IMAGE_AFFINITY_INTERVAL; DIAMETER_INTERVAL; REAL_LE_INV_EQ; REAL_ARITH `&0 <= &n + &1`] THEN REWRITE_TAC[VECTOR_ARITH `c % a + p - c % b:real^N = p + c % (a - b)`] THEN REWRITE_TAC[VECTOR_SUB_REFL; VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_ADD_SUB; REAL_ARITH `p + inv x * (b - a) < p <=> &0 < (a - b) / x`] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_LT; GSYM INTERVAL_EQ_EMPTY] THEN ASM_REWRITE_TAC[EVENTUALLY_TRUE; NORM_MUL; LIFT_CMUL] THEN MATCH_MP_TAC LIM_NULL_VMUL THEN REWRITE_TAC[REAL_ABS_INV; REAL_ARITH `abs(&n + &1) = &n + &1`] THEN REWRITE_TAC[SEQ_HARMONIC_OFFSET]]); DISCH_THEN(CONJUNCTS_THEN(MP_TAC o last o CONJUNCTS)) THEN GEN_REWRITE_TAC I [IMP_IMP] THEN REWRITE_TAC[tendsto; AND_FORALL_THM] THEN REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`] THEN DISCH_THEN(MP_TAC o SPEC `min (drop(f(q:real^N)) - drop y) (drop y - drop(f p))`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPEC `&0` ARCH_EVENTUALLY_LT) THEN REWRITE_TAC[GSYM EVENTUALLY_AND; IMP_IMP]] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[IMAGE_AFFINITY_INTERVAL; DIAMETER_INTERVAL; LE_REFL; REAL_LE_INV_EQ; REAL_ARITH `&0 <= &n + &1`] THEN REWRITE_TAC[VECTOR_ARITH `c % a + p - c % b:real^N = p + c % (a - b)`] THEN REWRITE_TAC[VECTOR_SUB_REFL; VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_ADD_SUB; REAL_ARITH `p + inv x * (b - a) < p <=> &0 < (a - b) / x`] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_LT; GSYM INTERVAL_EQ_EMPTY] THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[REAL_OF_NUM_LT; LT_REFL] THEN ASM_CASES_TAC `0 < n` THEN ASM_REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[MESON[VECTOR_ADD_RID] `interval[a:real^N,b] = interval[a + vec 0,b]`] THEN REWRITE_TAC[INTERVAL_TRANSLATION; MEASURE_TRANSLATION] THEN REWRITE_TAC[GSYM INTEGRAL_TRANSLATION; DIST_0; NORM_LIFT; REAL_ABS_NORM] THEN ABBREV_TAC `c:real^N = inv(&n + &1) % (b - a)` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[DIST_1] THEN DISCH_THEN(STRIP_ASSUME_TAC o MATCH_MP (REAL_ARITH `abs(z - q) < min (q - y) (y - p) /\ abs(x - p) < min (q - y) (y - p) ==> x < y /\ y < z`)) THEN MP_TAC(ISPECL [`f:real^N->real^1`; `s:real^N->bool`; `interval[vec 0:real^N,c]`] CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_GEN) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `path_image g:real^N->bool` o MATCH_MP(REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SUBGOAL_THEN `!a:real^N. a IN path_image g ==> interval[a,a + c] SUBSET s` ASSUME_TAC THENL [SUBGOAL_THEN `!a:real^N. interval[a,a + c] SUBSET ball(a,d)` MP_TAC THENL [GEN_TAC; ASM SET_TAC[]] THEN ONCE_REWRITE_TAC[MESON[VECTOR_ADD_RID] `interval[a:real^N,b] = interval[a + vec 0,b]`] THEN REWRITE_TAC[INTERVAL_TRANSLATION; TRANSLATION_SUBSET_GALOIS_LEFT] THEN REWRITE_TAC[GSYM BALL_TRANSLATION; VECTOR_ADD_LINV] THEN REWRITE_TAC[SUBSET; IN_BALL_0] THEN X_GEN_TAC `z:real^N` THEN REWRITE_TAC[IN_INTERVAL; VEC_COMPONENT] THEN DISCH_TAC THEN TRANS_TAC REAL_LET_TRANS `norm(c:real^N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN ASM_SIMP_TAC[REAL_ARITH `&0 <= z /\ z <= c ==> abs z <= abs c`]; REWRITE_TAC[GSYM INTERVAL_TRANSLATION; VECTOR_ADD_RID] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC]] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONNECTED_CONTINUOUS_IMAGE)) THEN ASM_SIMP_TAC[CONNECTED_PATH_IMAGE; GSYM IS_INTERVAL_CONNECTED_1] THEN REWRITE_TAC[IS_INTERVAL_1; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `p:real^N`) THEN ANTS_TAC THENL [ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `q:real^N`) THEN ANTS_TAC THENL [ASM_MESON_TAC[PATHFINISH_IN_PATH_IMAGE]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `measure(interval[vec 0:real^N,c]) % y:real^1`) THEN SUBGOAL_THEN `~(interval(vec 0:real^N,c) = {})` ASSUME_TAC THENL [EXPAND_TAC "c" THEN REWRITE_TAC[INTERVAL_NE_EMPTY; VEC_COMPONENT] THEN SIMP_TAC[VECTOR_MUL_COMPONENT; REAL_LT_MUL_EQ; REAL_LT_INV_EQ; VECTOR_SUB_COMPONENT; REAL_SUB_LT; REAL_ARITH `&0 < &n + &1`] THEN ASM_REWRITE_TAC[GSYM INTERVAL_NE_EMPTY]; MP_TAC(ISPECL [`vec 0:real^N`; `c:real^N`] INTERVAL_OPEN_SUBSET_CLOSED) THEN ASM_CASES_TAC `interval[vec 0:real^N,c] = {}` THEN ASM_REWRITE_TAC[SUBSET_EMPTY] THEN DISCH_THEN(K ALL_TAC)] THEN SUBGOAL_THEN `&0 < measure(interval[vec 0:real^N,c])` ASSUME_TAC THENL [SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_INTERVAL] THEN SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR; CONVEX_INTERVAL] THEN ASM_REWRITE_TAC[INTERIOR_INTERVAL]; ALL_TAC] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] DROP_CMUL] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; GSYM REAL_LE_RDIV_EQ] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN ASM_SIMP_TAC[GSYM DROP_CMUL; REAL_LT_IMP_LE; IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `r:real^N` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN REWRITE_TAC[INTEGRAL_TRANSLATION] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `measure i % y:real^N = integral (IMAGE t i) f ==> measure(IMAGE t i) = measure i ==> integral (IMAGE t i) f = measure(IMAGE t i) % y`)) THEN ANTS_TAC THENL [REWRITE_TAC[MEASURE_TRANSLATION]; ALL_TAC] THEN REWRITE_TAC[GSYM INTERVAL_TRANSLATION; VECTOR_ADD_RID] THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real^1`; `r:real^N`; `r + c:real^N`] SUBINTERVAL_MEAN_VALUE_THEOREM_ALT) THEN ANTS_TAC THENL [CONJ_TAC THENL [ONCE_REWRITE_TAC[MESON[VECTOR_ADD_RID] `interval[a:real^N,b] = interval[a + vec 0,b]`] THEN ASM_REWRITE_TAC[INTERVAL_TRANSLATION; IMAGE_EQ_EMPTY]; ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET; LEBESGUE_MEASURABLE_INTERVAL]]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `x:real^N` THEN GEN_REWRITE_TAC RAND_CONV [CONJ_SYM] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o GEN `m:num` o SPEC `inv(&m + &1)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &m + &1`] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN MAP_EVERY X_GEN_TAC [`u:num->real^N`; `v:num->real^N`; `k:num->num`] THEN ASM_REWRITE_TAC[VECTOR_ADD_SUB; VECTOR_MUL_ASSOC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VECTOR_MUL_LCANCEL] THEN ONCE_REWRITE_TAC[MESON[VECTOR_ADD_RID] `interval[a:real^N,b] = interval[a + vec 0,b]`] THEN ASM_REWRITE_TAC[INTERVAL_TRANSLATION; MEASURE_TRANSLATION] THEN REWRITE_TAC[GSYM INTERVAL_TRANSLATION; VECTOR_ADD_RID] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ; VECTOR_ADD_RID] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`\n. interval[(u:num->real^N) n,v n]`; `x:real^N`]) THEN ASM_REWRITE_TAC[EVENTUALLY_TRUE] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `m:num` THEN MAP_EVERY EXISTS_TAC [`inv(&(k(m:num))) * inv(&n + &1)`; `u(m:num) - (inv(&(k(m:num))) * inv(&n + &1)) % a:real^N`] THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_INV_EQ; REAL_OF_NUM_LT; LE_1; REAL_ARITH `&0 < &n + &1`] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV o RAND_CONV) [GSYM th]) THEN EXPAND_TAC "c" THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL] THEN SIMP_TAC[REAL_LE_MUL; REAL_LE_INV_EQ; REAL_POS; REAL_ARITH `&0 <= &n + &1`] THEN COND_CASES_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[VECTOR_SUB_ADD2] THEN REWRITE_TAC[VECTOR_ARITH `k % b + (u - k % a):real^N = u + k % (b - a)`] THEN REWRITE_TAC[VECTOR_MUL_ASSOC]; MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\n. inv(&n + &1)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; EVENTUALLY_TRUE; NORM_LIFT; real_abs; DIAMETER_POS_LE; BOUNDED_INTERVAL] THEN REWRITE_TAC[SEQ_HARMONIC_OFFSET]]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ; TRIVIAL_LIMIT_SEQUENTIALLY] (ISPEC `sequentially` LIM_UNIQUE)) THEN MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[] THEN GEN_REWRITE_TAC RAND_CONV [GSYM VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC REAL_MUL_LINV THEN SIMP_TAC[MEASURABLE_MEASURE_EQ_0; MEASURABLE_INTERVAL] THEN SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR; CONVEX_INTERVAL] THEN REWRITE_TAC[INTERIOR_INTERVAL] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV o RAND_CONV) [GSYM th]) THEN ONCE_REWRITE_TAC[MESON[VECTOR_ADD_RID] `interval(a:real^N,b) = interval(a + vec 0,b)`] THEN REWRITE_TAC[INTERVAL_TRANSLATION; IMAGE_EQ_EMPTY] THEN UNDISCH_TAC `~(interval(vec 0:real^N,c) = {})` THEN REWRITE_TAC[INTERVAL_NE_EMPTY; VEC_COMPONENT; VECTOR_MUL_COMPONENT] THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_INV_EQ; REAL_OF_NUM_LT; LE_1]]);; (* ------------------------------------------------------------------------- *) (* More from Saint Raymond's paper "Local Inversion for Differentiable *) (* Functions and the Darboux Property". The earlier parts are back in *) (* Multivariate/derivatives.ml but this part of the reasoning needs measure. *) (* ------------------------------------------------------------------------- *) let MEASURE_DIFFERENTIABLE_IMAGE_APPROX_GEN = prove (`!f:real^N->real^N s c a. open s /\ f differentiable_on s /\ (!v. v SUBSET s /\ open v ==> open(IMAGE f v)) /\ a IN s /\ ~(det(jacobian f (at a)) = &0) /\ ((\n. lift(diameter (c n))) --> vec 0) sequentially /\ (!n. a IN closure(c n)) /\ (?A. convex A /\ bounded A /\ ~(interior A = {}) /\ !n. ?t z. &0 < t /\ IMAGE (\x. t % x + z) A = c n) ==> eventually (\n. measurable(IMAGE f (c n))) sequentially /\ ((\n. lift(measure(IMAGE f (c n)) / measure(c n))) --> lift(abs(det(jacobian f (at a))))) sequentially`, REPEAT GEN_TAC THEN REWRITE_TAC[jacobian] THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [differentiable_on]) THEN REWRITE_TAC[differentiable; RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f':real^N->real^N->real^N` THEN ASM_SIMP_TAC[HAS_DERIVATIVE_WITHIN_OPEN] THEN DISCH_TAC THEN SUBGOAL_THEN `frechet_derivative (f:real^N->real^N) (at a) = f' a` SUBST_ALL_TAC THENL [MATCH_MP_TAC HAS_FRECHET_DERIVATIVE_UNIQUE_AT THEN ASM_SIMP_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(f:real^N->real^N) continuous_on s` ASSUME_TAC THENL [MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN ASM_MESON_TAC[differentiable]; ALL_TAC] THEN SUBGOAL_THEN `!u v. bounded u /\ open u /\ closure u SUBSET s /\ open v /\ connected v /\ ~(v INTER IMAGE f u = {}) /\ v INTER IMAGE f (frontier u) = {} ==> v SUBSET IMAGE (f:real^N->real^N) u` (LABEL_TAC "L3") THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN MP_TAC(ISPECL [`v:real^N->bool`; `IMAGE (f:real^N->real^N) u`] CONNECTED_INTER_FRONTIER) THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `v INTER s = {} ==> t SUBSET s ==> v INTER t = {}`)) THEN MATCH_MP_TAC FRONTIER_OPEN_MAP_IMAGE_SUBSET THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[OPEN_INTERIOR] THEN ASM_MESON_TAC[INTERIOR_SUBSET; CLOSURE_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN SUBGOAL_THEN `?u:real^N->bool. open u /\ a IN u /\ u SUBSET s` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET_REFL]; ALL_TAC] THEN SUBGOAL_THEN `linear((f':real^N->real^N->real^N) a)` ASSUME_TAC THENL [ASM_MESON_TAC[has_derivative]; ALL_TAC] THEN SUBGOAL_THEN `!n:num. bounded(c n) /\ convex(c n) /\ ~(interior(c n):real^N->bool = {})` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(CHOOSE_THEN MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM o SPEC `n:num`) THEN ASM_SIMP_TAC[CONVEX_AFFINITY; BOUNDED_AFFINITY; INTERIOR_AFFINITY] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ; IMAGE_EQ_EMPTY] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `eventually (\n. closure(c n) SUBSET (u:real^N->bool)) sequentially` MP_TAC THENL [UNDISCH_TAC `open(u:real^N->bool)` THEN REWRITE_TAC[OPEN_CONTAINS_CBALL] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [tendsto]) THEN DISCH_THEN(MP_TAC o SPEC `r:real`) THEN ASM_REWRITE_TAC[DIST_0] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[NORM_LIFT; real_abs] THEN ASM_SIMP_TAC[DIAMETER_POS_LE] THEN DISCH_TAC THEN TRANS_TAC SUBSET_TRANS `cball(a:real^N,r)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_CBALL] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `diameter(closure((c:num->real^N->bool) n))` THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[REAL_LT_IMP_LE; DIAMETER_CLOSURE]] THEN MATCH_MP_TAC DIST_LE_DIAMETER THEN ASM_SIMP_TAC[CLOSURE_INC; BOUNDED_CLOSURE]; REWRITE_TAC[EVENTUALLY_SEQUENTIALLY]] THEN DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC) THEN ONCE_REWRITE_TAC[LE_EXISTS] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[FORALL_UNWIND_THM2] THEN DISCH_TAC THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN EXISTS_TAC `N:num` THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [X_GEN_TAC `n:num` THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^N->real^N) (closure(c(N + n:num)))` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFFERENTIABLE_IMAGE THEN ASM_SIMP_TAC[LE_REFL; LEBESGUE_MEASURABLE_CONVEX]; MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_CLOSURE] THEN MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON; SIMP_TAC[IMAGE_SUBSET; CLOSURE_SUBSET]] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN ASM_MESON_TAC[differentiable; CLOSURE_SUBSET; SUBSET_TRANS]; DISCH_TAC] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[ADD_SYM] SEQ_OFFSET_REV) THEN EXISTS_TAC `N:num` THEN ABBREV_TAC `d:num->real^N->bool = \n. closure(c(N + n))` THEN MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC `\n:num. lift(measure(IMAGE (f:real^N->real^N) (d n)) / measure(d n))` THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN CONJ_TAC THENL [EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN EXPAND_TAC "d" THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN BINOP_TAC THENL [MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (f:real^N->real^N) (frontier(c(N + n:num)))` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN ASM_SIMP_TAC[LE_REFL; NEGLIGIBLE_CONVEX_FRONTIER] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[frontier; DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN ASM_MESON_TAC[SUBSET_TRANS; SUBSET_DIFF; differentiable]; MP_TAC(ISPEC `(c:num->real^N->bool)(N + n)` INTERIOR_SUBSET) THEN MP_TAC(ISPEC `(c:num->real^N->bool)(N + n)` CLOSURE_SUBSET) THEN REWRITE_TAC[frontier] THEN SET_TAC[]]; MATCH_MP_TAC MEASURE_CLOSURE THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_FRONTIER]]; ALL_TAC] THEN SUBGOAL_THEN `!n. a IN (d:num->real^N->bool) n /\ d n SUBSET u /\ compact(d n) /\ convex(d n)` MP_TAC THENL [EXPAND_TAC "d" THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[COMPACT_CLOSURE; CONVEX_CLOSURE]; REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN SUBGOAL_THEN `((\n. lift(diameter(d n:real^N->bool))) --> vec 0) sequentially` ASSUME_TAC THENL [EXPAND_TAC "d" THEN REWRITE_TAC[DIAMETER_CLOSURE] THEN FIRST_ASSUM(MP_TAC o SPEC `N:num` o MATCH_MP (ONCE_REWRITE_RULE[ADD_SYM] SEQ_OFFSET)) THEN REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?A. compact A /\ convex A /\ vec 0 IN interior A /\ !n:num. ?t z. &0 < t /\ IMAGE (\x:real^N. t % x + z) A = d n` MP_TAC THENL [FIRST_X_ASSUM(X_CHOOSE_THEN `A:real^N->bool` MP_TAC) THEN REWRITE_TAC[SKOLEM_THM; GSYM MEMBER_NOT_EMPTY; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`z:real^N`; `t:num->real`; `a:num->real^N`] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN EXISTS_TAC `IMAGE (\x:real^N. --z + x) (closure A)` THEN REWRITE_TAC[GSYM IMAGE_o; INTERIOR_TRANSLATION] THEN REWRITE_TAC[IN_TRANSLATION_GALOIS; COMPACT_TRANSLATION_EQ] THEN REWRITE_TAC[VECTOR_ARITH `vec 0 - --z:real^N = z`] THEN ASM_SIMP_TAC[CONVEX_TRANSLATION_EQ; CONVEX_INTERIOR_CLOSURE] THEN ASM_SIMP_TAC[COMPACT_CLOSURE; CONVEX_CLOSURE] THEN EXPAND_TAC "d" THEN REWRITE_TAC[o_DEF] THEN ASM_REWRITE_TAC[CLOSURE_AFFINITY] THEN EXISTS_TAC `\n. (t:num->real) (N + n)` THEN EXISTS_TAC `\n. (a:num->real^N) (N + n) + t(N + n) % z` THEN X_GEN_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN CONV_TAC VECTOR_ARITH; DISCH_THEN(X_CHOOSE_THEN `A:real^N->bool` MP_TAC)] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:num->real`; `z:num->real^N`] THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (free_in `c:num->real^N->bool` o concl))) THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN SPEC_TAC(`d:num->real^N->bool`,`c:num->real^N->bool`) THEN GEN_TAC THEN REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN SUBGOAL_THEN `bounded(A:real^N->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[COMPACT_IMP_BOUNDED]; ALL_TAC] THEN SUBGOAL_THEN `~(A:real^N->bool = {}) /\ ~(interior A = {})` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `A:real^N->bool` INTERIOR_SUBSET) THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < measure(A:real^N->bool)` ASSUME_TAC THENL [ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_COMPACT] THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR]; ALL_TAC] THEN SUBGOAL_THEN `!n. ~(interior((c:num->real^N->bool) n) = {})` ASSUME_TAC THENL [GEN_TAC THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV) [GSYM th]) THEN REWRITE_TAC[INTERIOR_AFFINITY] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ; IMAGE_EQ_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `!n. &0 < measure((c:num->real^N->bool) n)` ASSUME_TAC THENL [ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_COMPACT] THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR]; ALL_TAC] THEN ONCE_REWRITE_TAC[LIM_NULL] THEN MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC `\n. lift((measure(IMAGE (\x. --(f a) + x) (IMAGE (f:real^N->real^N) (c n))) - measure(IMAGE (f' a) (IMAGE (\x. --a + x) (c n)):real^N->bool)) / measure(c(n:num)))` THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN CONJ_TAC THENL [EXISTS_TAC `0` THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[LE_0] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div; LIFT_CMUL] THEN ASM_SIMP_TAC[MEASURE_LINEAR_IMAGE; MEASURABLE_COMPACT; MEASURABLE_TRANSLATION; MEASURE_TRANSLATION] THEN REWRITE_TAC[LIFT_SUB; LIFT_CMUL; VECTOR_SUB_LDISTRIB] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; LIFT_DROP; DROP_SUB] THEN ASM_SIMP_TAC[REAL_FIELD `&0 < x ==> inv x * y * x = y`] THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC MEASURE_DIFFERENTIABLE_IMAGE_EQ THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT; CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; SUBSET]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURABLE_DIFFERENTIABLE_IMAGE_EQ o snd) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT] THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; SUBSET]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_TRANS]]; ALL_TAC] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN ABBREV_TAC `b = \p:num. f a + (f':real^N->real^N->real^N) a (z p - a)` THEN SUBGOAL_THEN `!e. &0 < e /\ e < &1 ==> eventually (\n. IMAGE (f:real^N->real^N) (c n) SUBSET IMAGE (\x. b n + x) (IMAGE (\x. (t n * (&1 + e)) % x) (IMAGE ((f':real^N->real^N->real^N) a) A)) /\ IMAGE (\x. b n + x) (IMAGE (\x. (t n * (&1 - e)) % x) (IMAGE (f' a) (interior A))) SUBSET IMAGE f (interior(c n))) sequentially` ASSUME_TAC THENL [ALL_TAC; REWRITE_TAC[tendsto; DIST_0] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN SUBGOAL_THEN `((\x. ((&1 + drop x) pow dimindex(:N) - &1) % lift(abs(det(matrix((f':real^N->real^N->real^N) a))))) --> vec 0) (at (vec 0))` MP_TAC THENL [MATCH_MP_TAC LIM_NULL_VMUL THEN REWRITE_TAC[LIFT_SUB; GSYM LIM_NULL] THEN SUBST1_TAC(SYM(SPEC `dimindex(:N)` REAL_POW_ONE)) THEN MATCH_MP_TAC LIM_LIFT_POW THEN REWRITE_TAC[LIFT_ADD; LIFT_NUM; REAL_POW_ONE; LIFT_DROP] THEN GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN SIMP_TAC[LIM_ADD; LIM_CONST; LIM_AT_ID]; REWRITE_TAC[LIM_AT; DIST_0]] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `?k. &0 < k /\ k < d /\ k < &1` STRIP_ASSUME_TAC THENL [REWRITE_TAC[GSYM REAL_LT_MIN; GSYM REAL_LT_BETWEEN] THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_01]; FIRST_X_ASSUM(MP_TAC o SPEC `k:real`)] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[NORM_LIFT] THEN SUBGOAL_THEN `measurable(IMAGE (f:real^N->real^N) (c(n:num))) /\ measurable(IMAGE f (interior (c n))) /\ measure(IMAGE f (interior (c n))) = measure(IMAGE f (c n))` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC(TAUT `p /\ (q <=> p) /\ r ==> p /\ q /\ r`) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_TRANS]; SUBGOAL_THEN `negligible(IMAGE (f:real^N->real^N) (frontier(c(n:num))))` ASSUME_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN ASM_SIMP_TAC[LE_REFL; NEGLIGIBLE_CONVEX_FRONTIER] THEN MATCH_MP_TAC DIFFERENTIABLE_ON_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN CONJ_TAC THENL [REWRITE_TAC[differentiable_on]; ASM SET_TAC[]] THEN ASM_MESON_TAC[differentiable; HAS_DERIVATIVE_AT_WITHIN]; CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ; MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN ASM_SIMP_TAC[frontier; CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN MP_TAC(ISPEC `(c:num->real^N->bool) n` INTERIOR_SUBSET) THEN SET_TAC[]]]; ASM_REWRITE_TAC[]] THEN DISCH_THEN(CONJUNCTS_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE [TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`] MEASURE_SUBSET))) THEN ASM_SIMP_TAC[MEASURABLE_TRANSLATION_EQ; MEASURABLE_SCALING_EQ; MEASURABLE_INTERIOR; MEASURABLE_LINEAR_IMAGE; MEASURABLE_COMPACT] THEN SUBGOAL_THEN `(\x. --f a + f x) = (\y. --f a + y) o (f:real^N->real^N)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN REWRITE_TAC[MEASURE_TRANSLATION; IMAGE_o] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[REAL_ABS_DIV; REAL_LT_LDIV_EQ; REAL_ARITH `&0 < x ==> abs x = x`] THEN SUBGOAL_THEN `IMAGE (\x. (f':real^N->real^N->real^N) a (--a + x)) (c n) = IMAGE (f' a) (IMAGE (\x. (--a + z(n:num)) + x) (IMAGE (\x. t n % x) A))` SUBST1_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN REWRITE_TAC[VECTOR_ADD_AC]; ALL_TAC] THEN FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o RAND_CONV) [GSYM th]) THEN REWRITE_TAC[REAL_ARITH `abs x < a <=> --a < x /\ x < a`] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a <= m ==> b + d < a ==> b < m - d`)); FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `m <= a ==> a < b + d ==> m - d < b`))] THEN REWRITE_TAC[GSYM REAL_MUL_LNEG; GSYM REAL_ADD_RDISTRIB] THEN ASM_SIMP_TAC[MEASURE_LINEAR_IMAGE; MEASURE_TRANSLATION; MEASURE_SCALING; MEASURABLE_COMPACT; MEASURABLE_SCALING; MEASURABLE_TRANSLATION_EQ; MEASURABLE_LINEAR_IMAGE; MEASURE_AFFINITY; MEASURABLE_INTERIOR] THEN REWRITE_TAC[REAL_MUL_ASSOC; GSYM REAL_ADD_RDISTRIB] THEN ASM_SIMP_TAC[MEASURE_INTERIOR; NEGLIGIBLE_CONVEX_FRONTIER] THEN MATCH_MP_TAC REAL_LT_RMUL THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ABS_MUL; REAL_POW_MUL] THEN REWRITE_TAC[REAL_ARITH `(--e + d) * t < (t * k) * d <=> t * (d - e) < t * d * k`] THEN REWRITE_TAC[REAL_ARITH `(t * k) * d < (e + d) * t <=> t * d * k < t * (d + e)`] THEN MATCH_MP_TAC REAL_LT_LMUL THEN ASM_SIMP_TAC[REAL_POW_LT; REAL_ARITH `&0 < x ==> abs x = x`] THENL [REWRITE_TAC[REAL_ARITH `x - a < x * k <=> (&1 - k) * x < a`] THEN FIRST_ASSUM(MP_TAC o SPEC `--lift k`); REWRITE_TAC[REAL_ARITH `x * k < x + e <=> (k - &1) * x < e`] THEN FIRST_ASSUM(MP_TAC o SPEC `lift k`)] THEN REWRITE_TAC[NORM_NEG; NORM_LIFT; DROP_NEG; LIFT_DROP] THEN (ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS) THEN REWRITE_TAC[NORM_MUL; NORM_LIFT; REAL_ABS_ABS] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_ABS_POS] THENL [MATCH_MP_TAC(REAL_ARITH `y - x = z ==> x - y <= abs z`); MATCH_MP_TAC(REAL_ARITH `x - y = z ==> x - y <= abs z`)] THEN ASM_SIMP_TAC[real_abs; REAL_SUB_LE; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[GSYM real_sub; REAL_LE_ADD; REAL_POS; REAL_LT_IMP_LE]] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN ABBREV_TAC `M = diameter(A:real^N->bool)` THEN SUBGOAL_THEN `&0 < M` ASSUME_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 < M <=> &0 <= M /\ ~(M = &0)`] THEN EXPAND_TAC "M" THEN ASM_SIMP_TAC[DIAMETER_POS_LE; DIAMETER_EQ_0] THEN DISCH_THEN(CHOOSE_THEN (MP_TAC o AP_TERM `interior:(real^N->bool)->(real^N->bool)`)) THEN ASM_REWRITE_TAC[INTERIOR_SING]; ALL_TAC] THEN SUBGOAL_THEN `!u:real^N. u IN A ==> norm(u) <= M` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_ARITH `norm(u:real^N) = dist(vec 0,u)`] THEN EXPAND_TAC "M" THEN MATCH_MP_TAC DIST_LE_DIAMETER THEN ASM_MESON_TAC[SUBSET; INTERIOR_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `!p. norm(a - (z:num->real^N) p) <= M * t p` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN TRANS_TAC REAL_LE_TRANS `diameter(IMAGE (\x. t n % x + (z:num->real^N) n) A)` THEN REWRITE_TAC[GSYM dist] THEN CONJ_TAC THENL [MATCH_MP_TAC DIST_LE_DIAMETER THEN REPEAT(CONJ_TAC THENL [ASM_SIMP_TAC[BOUNDED_AFFINITY]; ALL_TAC]) THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN ASM_MESON_TAC[SUBSET; INTERIOR_SUBSET]; ASM_SIMP_TAC[DIAMETER_AFFINITY] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_MUL_AC; REAL_LE_REFL]]; ALL_TAC] THEN SUBGOAL_THEN `!x y. (f':real^N->real^N->real^N) a x = f' a y <=> x = y` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP MATRIX_INVERTIBLE) THEN ASM_SIMP_TAC[INVERTIBLE_DET_NZ; FUN_EQ_THM; I_THM; o_THM] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?d. &0 < d /\ cball(vec 0,d / &2) SUBSET IMAGE (\x. (&1 - e) % x) (IMAGE (f' a) (interior A)) /\ (!x x'. x IN IMAGE ((f':real^N->real^N->real^N) a) A /\ norm(x' - x) <= d ==> x' IN interior(IMAGE (\x. (&1 + e) % x) (IMAGE (f' a) A))) /\ (!x x'. x IN IMAGE ((f':real^N->real^N->real^N) a) (frontier A) /\ norm(x' - x) <= d ==> ~(x' IN IMAGE (\x. (&1 - e) % x) (IMAGE (f' a) A)))` MP_TAC THENL [SUBGOAL_THEN `?d. &0 < d /\ cball(vec 0,d / &2) SUBSET IMAGE (\x. (&1 - e) % x) (IMAGE ((f':real^N->real^N->real^N) a) (interior A))` (X_CHOOSE_THEN `d0:real` STRIP_ASSUME_TAC) THENL [MATCH_MP_TAC(MESON [REAL_ARITH `&0 < d ==> &0 < &2 * d /\ (&2 * d) / &2 = d`] `(?d. &0 < d /\ P d) ==> (?d. &0 < d /\ P(d / &2))`) THEN MATCH_MP_TAC(MESON[OPEN_CONTAINS_CBALL] `open s /\ x IN s ==> ?d. &0 < d /\ cball(x,d) SUBSET s`) THEN ASM_SIMP_TAC[OPEN_SCALING_EQ; REAL_SUB_0; REAL_LT_IMP_NE] THEN ASM_SIMP_TAC[OPEN_INVERTIBLE_LINEAR_IMAGE; OPEN_INTERIOR; INVERTIBLE_DET_NZ; GSYM IMAGE_o] THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[o_DEF] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP LINEAR_0) THEN REWRITE_TAC[VECTOR_MUL_RZERO]; ALL_TAC] THEN MP_TAC(ISPECL [`IMAGE ((f':real^N->real^N->real^N) a) A`; `&1 + e`] CONVEX_NEARBY_IN_SCALING_RELATIVE_INTERIOR) THEN MP_TAC(ISPECL [`IMAGE ((f':real^N->real^N->real^N) a) A`; `&1 - e`] CONVEX_NEARBY_NOT_IN_SCALING) THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX; CONVEX_LINEAR_IMAGE] THEN ASM_SIMP_TAC[RELATIVE_INTERIOR_NONEMPTY_INTERIOR] THEN UNDISCH_THEN `(vec 0:real^N) IN interior A` (MP_TAC o ISPEC `(f':real^N->real^N->real^N) a` o MATCH_MP FUN_IN_IMAGE) THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP LINEAR_0) THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `min d0 (min d1 d2):real` THEN ASM_REWRITE_TAC[REAL_LE_MIN; REAL_LT_MIN] THEN CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `cball(vec 0:real^N,d0 / &2)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_CBALL THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist)] THEN STRIP_TAC THENL [REWRITE_TAC[INTERIOR_SCALING] THEN COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[INTERIOR_INJECTIVE_LINEAR_IMAGE] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `s = UNIV ==> x IN s`) THEN MATCH_MP_TAC AFFINE_HULL_NONEMPTY_INTERIOR THEN ASM_SIMP_TAC[INTERIOR_INJECTIVE_LINEAR_IMAGE; IMAGE_EQ_EMPTY]; FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `x:real^N` THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE] THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR]]; DISCH_THEN(X_CHOOSE_THEN `k:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)] THEN REWRITE_TAC[EVENTUALLY_AND] THEN MATCH_MP_TAC MONO_AND THEN SUBGOAL_THEN `((f:real^N->real^N) has_derivative f'(a)) (at a)` MP_TAC THENL [ASM_SIMP_TAC[]; REWRITE_TAC[HAS_DERIVATIVE_AT_ALT]] THEN DISCH_THEN(MP_TAC o SPEC `k / M / &2` o CONJUNCT2) THEN ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "D"))) THEN SUBGOAL_THEN `eventually (\n. (c n) SUBSET ball(a:real^N,r)) sequentially` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN DISCH_THEN(MP_TAC o SPEC `r:real`) THEN ASM_REWRITE_TAC[DIST_0] THEN REWRITE_TAC[GSYM EVENTUALLY_SEQUENTIALLY; NORM_LIFT] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[SUBSET; IN_BALL] THEN DISCH_THEN(ASSUME_TAC o MATCH_MP (REAL_ARITH `abs x < a ==> x < a`)) THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN TRANS_TAC REAL_LET_TRANS `diameter((c:num->real^N->bool) n)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIST_LE_DIAMETER THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED]; ALL_TAC] THEN DISCH_THEN(fun th -> CONJ_TAC THEN DISCH_THEN(LABEL_TAC "*") THEN MP_TAC th) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[] THEN DISCH_TAC THENL [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `?v. v IN A /\ t(n:num) % v + z n:real^N = y` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `norm(y - a:real^N) <= &2 * M * t(n:num)` ASSUME_TAC THENL [EXPAND_TAC "y" THEN MATCH_MP_TAC(NORM_ARITH `norm(x) <= m /\ norm(z - y) <= m ==> norm((x + y) - z:real^N) <= &2 * m`) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[NORM_MUL; real_abs; REAL_LT_IMP_LE; REAL_LE_LMUL_EQ]; ALL_TAC] THEN SUBGOAL_THEN `(\x:real^N. (t n * (&1 + e)) % x) = (\x. t(n:num) % x) o (\x. (&1 + e) % x)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; VECTOR_MUL_ASSOC; REAL_MUL_SYM]; ALL_TAC] THEN REWRITE_TAC[IMAGE_o] THEN ONCE_REWRITE_TAC[GSYM IMAGE_o] THEN REWRITE_TAC[o_DEF] THEN ONCE_REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `inv(t(n:num)) % ((f:real^N->real^N) y - b n)` THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN CONJ_TAC THENL [CONV_TAC VECTOR_ARITH; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] INTERIOR_SUBSET) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `v:real^N`; FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[OPEN_INTERIOR; OPEN_TRANSLATION_EQ] THEN REWRITE_TAC[CONNECTED_TRANSLATION_EQ] THEN ASM_SIMP_TAC[CONVEX_CLOSURE_INTERIOR] THEN ASM_SIMP_TAC[BOUNDED_INTERIOR; COMPACT_IMP_BOUNDED] THEN ASM_SIMP_TAC[CLOSURE_CLOSED; COMPACT_IMP_CLOSED] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[OPEN_SCALING_EQ; CONNECTED_SCALING_EQ] THEN ASM_SIMP_TAC[CONNECTED_LINEAR_IMAGE; CONVEX_CONNECTED; CONVEX_INTERIOR] THEN ASM_SIMP_TAC[OPEN_INVERTIBLE_LINEAR_IMAGE; INVERTIBLE_DET_NZ; OPEN_INTERIOR] THEN ASM_REWRITE_TAC[REAL_ENTIRE; REAL_SUB_0] THEN ASM_SIMP_TAC[REAL_LT_IMP_NE] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o funpow 3 RAND_CONV) [GSYM th]) THEN REWRITE_TAC[INTERIOR_AFFINITY] THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ] THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN MATCH_MP_TAC(SET_RULE `(?x. x IN s /\ f x IN t) ==> ~(t INTER IMAGE f s = {})`) THEN EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN SUBGOAL_THEN `(\x. b(n:num) + (t n * (&1 - e)) % (f':real^N->real^N->real^N) a x) = (\x. b n + x) o (\x. t n % x) o (\x. (&1 - e) % x) o f' a` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; VECTOR_ADD_SYM; VECTOR_MUL_ASSOC]; REWRITE_TAC[IMAGE_o; IN_TRANSLATION_GALOIS]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `c SUBSET d ==> x IN IMAGE f c ==> x IN IMAGE f d`)) THEN ASM_SIMP_TAC[GSYM CBALL_SCALING; VECTOR_MUL_RZERO] THEN REMOVE_THEN "D" (MP_TAC o SPEC `(z:num->real^N) n`) THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM dist); GSYM IN_BALL] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^N` THEN CONJ_TAC THENL [CONV_TAC VECTOR_ARITH; ALL_TAC] THEN ASM_MESON_TAC[SUBSET; INTERIOR_SUBSET]; EXPAND_TAC "b" THEN REWRITE_TAC[IN_CBALL_0]] THEN MATCH_MP_TAC(REAL_ARITH `x = y /\ a <= b ==> x <= a ==> y <= b`) THEN CONJ_TAC THENL [CONV_TAC NORM_ARITH; REWRITE_TAC[dist]] THEN TRANS_TAC REAL_LE_TRANS `k / M / &2 * M * t(n:num)` THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_DIV; REAL_HALF] THEN ASM_SIMP_TAC[REAL_LE_REFL; REAL_FIELD `&0 < M ==> k / M / &2 * M * t = t * k / &2`]; ALL_TAC] THEN SUBGOAL_THEN `(\x:real^N. (t n * (&1 - e)) % x) = (\x. t(n:num) % x) o (\x. (&1 - e) % x)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; VECTOR_MUL_ASSOC; REAL_MUL_SYM]; ALL_TAC] THEN REWRITE_TAC[IMAGE_o] THEN ONCE_REWRITE_TAC[GSYM IMAGE_o] THEN REWRITE_TAC[SET_RULE `IMAGE f s INTER t = {} <=> !x. f x IN t ==> ~(x IN s)`] THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(MP_TAC o ISPEC `\y:real^N. inv(t(n:num)) % (y - b n)` o MATCH_MP FUN_IN_IMAGE) THEN REWRITE_TAC[o_DEF; VECTOR_ADD_SUB; VECTOR_MUL_ASSOC; GSYM IMAGE_o] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ; VECTOR_MUL_LID] THEN SPEC_TAC(`y:real^N`,`y:real^N`) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV o funpow 3 RAND_CONV) [GSYM th]) THEN ASM_SIMP_TAC[INTERIOR_AFFINITY; REAL_LT_IMP_NZ] THEN ASM_SIMP_TAC[FRONTIER_AFFINITY; REAL_LT_IMP_NZ] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `v:real^N` THEN SUBGOAL_THEN `frontier(interior A):real^N->bool = frontier A` SUBST1_TAC THENL [REWRITE_TAC[frontier; INTERIOR_INTERIOR] THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC CONVEX_CLOSURE_INTERIOR THEN ASM_REWRITE_TAC[]; DISCH_TAC] THEN SUBGOAL_THEN `(\x. (&1 - e) % (f':real^N->real^N->real^N) a x) = (\x. (&1 - e) % x) o (f' a)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; REWRITE_TAC[GSYM IMAGE_o]] THEN MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ ~(x IN IMAGE f t) ==> ~(x IN IMAGE f s)`) THEN EXISTS_TAC `A:real^N->bool` THEN REWRITE_TAC[INTERIOR_SUBSET; IMAGE_o] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `v:real^N` THEN ASM_REWRITE_TAC[] THEN ABBREV_TAC `y = t n % v + (z:num->real^N) n` THEN SUBGOAL_THEN `y IN (c:num->real^N->bool) n` ASSUME_TAC THENL [FIRST_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN EXPAND_TAC "y" THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `v:real^N` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM (MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN REWRITE_TAC[FRONTIER_SUBSET_EQ] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]; ALL_TAC] THEN SUBGOAL_THEN `norm(y - a:real^N) <= &2 * M * t(n:num)` ASSUME_TAC THENL [EXPAND_TAC "y" THEN MATCH_MP_TAC(NORM_ARITH `norm(x) <= m /\ norm(z - y) <= m ==> norm((x + y) - z:real^N) <= &2 * m`) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[NORM_MUL; real_abs; REAL_LT_IMP_LE; REAL_LE_LMUL_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM (MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN REWRITE_TAC[FRONTIER_SUBSET_EQ] THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]; ALL_TAC]] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_LCANCEL_IMP THEN EXISTS_TAC `abs((t:num->real) n)` THEN REWRITE_TAC[GSYM NORM_MUL] THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RINV; real_abs; REAL_LT_IMP_LE; REAL_LT_IMP_NZ] THEN EXPAND_TAC "b" THEN REWRITE_TAC[VECTOR_ARITH `&1 % y - &1 % (x + z) - w:real^N = y - x - (w + z)`] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_SUB th]) THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN REWRITE_TAC[VECTOR_ARITH `a + b - c:real^N = (a + b) - c`] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_ADD th)]) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_SUB th)]) THEN REMOVE_THEN "D" (MP_TAC o SPEC `y:real^N`) THEN GEN_REWRITE_TAC (funpow 3 LAND_CONV) [GSYM dist] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[GSYM IN_BALL] THEN (ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN TRANS_TAC REAL_LE_TRANS `k / M / &2 * &2 * M * t(n:num)` THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_DIV; REAL_HALF] THEN ASM_SIMP_TAC[REAL_FIELD `&0 < M ==> k / M / &2 * &2 * M * t = t * k`] THEN REWRITE_TAC[REAL_LE_REFL]);; let MEASURE_DIFFERENTIABLE_IMAGE_APPROX = prove (`!f:real^N->real^N f' s c a. open s /\ a IN s /\ (!x. x IN s ==> (f has_derivative f' x) (at x) /\ ~(det(matrix (f' x)) = &0)) /\ ((\n. lift(diameter (c n))) --> vec 0) sequentially /\ (!n. a IN closure(c n)) /\ (?A. convex A /\ bounded A /\ ~(interior A = {}) /\ !n. ?t z. &0 < t /\ IMAGE (\x. t % x + z) A = c n) ==> eventually (\n. measurable(IMAGE f (c n))) sequentially /\ ((\n. lift(measure(IMAGE f (c n)) / measure(c n))) --> lift(abs(det(matrix(f' a))))) sequentially`, REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN SUBGOAL_THEN `matrix(f' a) = jacobian (f:real^N->real^N) (at a)` (fun th -> REWRITE_TAC[th] THEN MATCH_MP_TAC MEASURE_DIFFERENTIABLE_IMAGE_APPROX_GEN THEN REWRITE_TAC[GSYM th]) THENL [REWRITE_TAC[jacobian] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC HAS_FRECHET_DERIVATIVE_UNIQUE_AT THEN ASM_SIMP_TAC[]; EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN CONJ_TAC THENL [ASM_MESON_TAC[differentiable]; ALL_TAC] THEN ASM_MESON_TAC[DIFFERENTIABLE_IMP_OPEN_MAP; SUBSET]]);; let CONNECTED_JACOBIAN_RANGE = prove (`!f:real^N->real^N f' s. open s /\ connected s /\ (!x. x IN s ==> (f has_derivative f' x) (at x) /\ ~(det(matrix (f' x)) = &0)) ==> connected (IMAGE (\x. lift(det(matrix(f' x)))) s)`, REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `connected (IMAGE (\x. lift(abs(det(matrix((f':real^N->real^N->real^N) x))))) s)` MP_TAC THENL [ALL_TAC; DISCH_THEN(fun th -> MP_TAC(MATCH_MP CONNECTED_NEGATIONS th) THEN MP_TAC th) THEN MATCH_MP_TAC(MESON[] `t = s \/ u = s ==> connected t ==> connected u ==> connected s`) THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> a x = f x) \/ (!x. x IN s ==> n(a x) = f x) ==> IMAGE a s = IMAGE f s \/ IMAGE n (IMAGE a s) = IMAGE f s`) THEN REWRITE_TAC[LIFT_EQ; GSYM LIFT_NEG] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`] JACOBIAN_SIGN_INVARIANCE) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN REAL_ARITH_TAC] THEN SUBGOAL_THEN `!x. x IN s ==> ?g. linear g /\ (f':real^N->real^N->real^N) x o g = I /\ g o f' x = I` MP_TAC THENL [REPEAT STRIP_TAC THEN W(MP_TAC o PART_MATCH (rand o rand) MATRIX_INVERTIBLE o snd) THEN REWRITE_TAC[INVERTIBLE_DET_NZ] THEN ASM_MESON_TAC[has_derivative]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM]] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; SKOLEM_THM] THEN X_GEN_TAC `g':real^N->real^N->real^N` THEN REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REWRITE_TAC[FORALL_AND_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC WEAK_LEBESGUE_POINTS_IMP_IVT THEN MAP_EVERY EXISTS_TAC [`--vec 1:real^N`; `vec 1:real^N`] THEN SUBGOAL_THEN `~(interval[--vec 1:real^N,vec 1] = {}) /\ ~(interval(--vec 1:real^N,vec 1) = {})` STRIP_ASSUME_TAC THENL [REWRITE_TAC[INTERVAL_NE_EMPTY; VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ASM_REWRITE_TAC[locally]] THEN GEN_REWRITE_TAC BINOP_CONV [SWAP_FORALL_THM] THEN GEN_REWRITE_TAC I [AND_FORALL_THM] THEN X_GEN_TAC `a:real^N` THEN ASM_CASES_TAC `(a:real^N) IN s` THENL [ASM_REWRITE_TAC[]; ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; SUBSET]] THEN SUBGOAL_THEN `(f:real^N->real^N) continuous_on s` ASSUME_TAC THENL [MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN ASM_MESON_TAC[differentiable]; ALL_TAC] THEN SUBGOAL_THEN `?u. open u /\ a IN u /\ u SUBSET s /\ !x y. x IN u /\ y IN u ==> ((f:real^N->real^N) x = f y <=> x = y)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `a:real^N`; `s:real^N->bool`] INVERSE_FUNCTION_THEOREM) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[homeomorphism] THEN MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_SIMP_TAC[OPEN_IN_OPEN_EQ] THEN X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN MP_TAC(ISPEC `u INTER w:real^N->bool` OPEN_IMP_LOCALLY_COMPACT) THEN ASM_SIMP_TAC[OPEN_INTER; LOCALLY_COMPACT; OPEN_IN_OPEN_EQ] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[IN_INTER] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N->bool` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N->bool` THEN REWRITE_TAC[SUBSET_INTER] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[SUBSET_TRANS]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURABLE_DIFFERENTIABLE_IMAGE_ALT o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT] THEN ASM_MESON_TAC[SUBSET; HAS_DERIVATIVE_AT_WITHIN]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN MATCH_MP_TAC(MESON[] `!Q'. ((!x. P x /\ Q' x /\ R x ==> S x) ==> (!x. P x /\ Q x /\ R x ==> S x)) /\ (!x. P x /\ Q' x /\ R x ==> S x) ==> (!x. P x /\ Q x /\ R x ==> S x)`) THEN EXISTS_TAC `\c:num->real^N->bool. (!n. a IN c n) /\ (!n. c n SUBSET u)` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [DISCH_TAC THEN X_GEN_TAC `c:num->real^N->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_MP)) THEN DISCH_THEN(MP_TAC o SPEC `\n:num. (a:real^N) IN c n /\ c n SUBSET u`) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [UNDISCH_TAC `open(u:real^N->bool)` THEN REWRITE_TAC[open_def] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[DIST_0; NORM_LIFT; EVENTUALLY_SEQUENTIALLY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN SUBGOAL_THEN `bounded((c:num->real^N->bool) n)` MP_TAC THENL [FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC o SPEC `n:num`) THEN ASM_SIMP_TAC[BOUNDED_AFFINITY; BOUNDED_INTERVAL]; SIMP_TAC[real_abs; DIAMETER_POS_LE] THEN REPEAT STRIP_TAC] THEN REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN TRANS_TAC REAL_LET_TRANS `diameter((c:num->real^N->bool) n)` THEN ASM_SIMP_TAC[DIST_LE_DIAMETER]; REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; MESON[LE_ADD; LE_EXISTS] `(!n:num. N <= n ==> P n) <=> (!n. P(N + n))`] THEN REWRITE_TAC[FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\n. (c:num->real^N->bool) (N + n)`) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[ADD_SYM] THEN FIRST_X_ASSUM(MP_TAC o SPEC `N:num` o MATCH_MP SEQ_OFFSET) THEN REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC SEQ_OFFSET_REV THEN EXISTS_TAC `N:num` THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `linear((f':real^N->real^N->real^N) a)` ASSUME_TAC THENL [ASM_MESON_TAC[has_derivative]; ALL_TAC] THEN X_GEN_TAC `c:num->real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `!n. compact((c:num->real^N->bool) n)` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC o SPEC `n:num`) THEN ASM_SIMP_TAC[COMPACT_AFFINITY; COMPACT_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `!n. ~(interior((c:num->real^N->bool) n) = {})` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[INTERIOR_AFFINITY; INTERIOR_INTERVAL] THEN ASM_SIMP_TAC[IMAGE_EQ_EMPTY; REAL_LT_IMP_NZ]; ALL_TAC] THEN SUBGOAL_THEN `!n. convex((c:num->real^N->bool) n)` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC o SPEC `n:num`) THEN ASM_SIMP_TAC[CONVEX_AFFINITY; CONVEX_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `!n. &0 < measure((c:num->real^N->bool) n)` ASSUME_TAC THENL [ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_COMPACT] THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR]; ALL_TAC] THEN MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC `\n:num. lift(measure (IMAGE (f:real^N->real^N) (c n)) / measure (c n))` THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN CONJ_TAC THENL [EXISTS_TAC `0` THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[LE_0] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div; LIFT_CMUL] THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN MATCH_MP_TAC MEASURE_DIFFERENTIABLE_IMAGE_EQ THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT; CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; SUBSET]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURABLE_DIFFERENTIABLE_IMAGE_EQ o snd) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT] THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; SUBSET]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_TRANS]]; MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`; `c:num->real^N->bool`; `a:real^N`] MEASURE_DIFFERENTIABLE_IMAGE_APPROX) THEN ASM_SIMP_TAC[CLOSURE_INC] THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[]] THEN EXISTS_TAC `interval[--vec 1:real^N,vec 1]` THEN REWRITE_TAC[CONVEX_INTERVAL; BOUNDED_INTERVAL; INTERIOR_INTERVAL] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN ASM_REWRITE_TAC[]]);; let CONNECTED_JACOBIAN_RANGE_ALT = prove (`!f:real^N->real^N f' s t a b. open s /\ connected s /\ convex t /\ t SUBSET s /\ ~(interior t = {}) /\ (!x. x IN s ==> (f has_derivative f' x) (at x) /\ ~(det(matrix (f' x)) = &0)) /\ a IN t /\ b IN t ==> segment(lift(det(matrix(f' a))),lift(det(matrix(f' b)))) SUBSET IMAGE (\x. lift(det(matrix(f' x)))) (interior t)`, let lemma = prove (`!f:real^N->real^N f' s t. open s /\ connected s /\ (!x. x IN s ==> (f has_derivative f' x) (at x) /\ ~(det(matrix (f' x)) = &0)) /\ t SUBSET s /\ convex t /\ ~(interior t = {}) ==> !e x. &0 < e /\ x IN t ==> (?x'. x' IN interior t /\ det(matrix(f' x')) <= det(matrix(f' x)) + e) /\ (?x'. x' IN interior t /\ det(matrix(f' x)) <= det(matrix(f' x')) + e)`, REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `(f:real^N->real^N) continuous_on s` ASSUME_TAC THENL [MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN ASM_MESON_TAC[differentiable]; ALL_TAC] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `(x:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(?x'. x' IN interior t /\ abs(det(matrix((f':real^N->real^N->real^N) x'))) <= abs(det(matrix(f' x))) + e) /\ (?x'. x' IN interior t /\ abs(det(matrix(f' x))) <= abs(det(matrix(f' x'))) + e)` MP_TAC THENL [ALL_TAC; MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`] JACOBIAN_SIGN_INVARIANCE) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THENL [ALL_TAC; GEN_REWRITE_TAC RAND_CONV [CONJ_SYM]] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN (SUBGOAL_THEN `(y:real^N) IN s` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET; INTERIOR_SUBSET]; ALL_TAC]) THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[REAL_ARITH `x < &0 ==> ~(&0 <= x)`] THEN REAL_ARITH_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `z:real^N`) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN ABBREV_TAC `A = convex hull (x INSERT cball(z:real^N,r))` THEN SUBGOAL_THEN `compact(A:real^N->bool) /\ convex A` STRIP_ASSUME_TAC THENL [EXPAND_TAC "A" THEN REWRITE_TAC[CONVEX_CONVEX_HULL] THEN MATCH_MP_TAC COMPACT_CONVEX_HULL THEN SIMP_TAC[COMPACT_INSERT; COMPACT_CBALL]; ALL_TAC] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN SUBGOAL_THEN `~(interior(A:real^N->bool) = {})` ASSUME_TAC THENL [MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ ~(s = {}) ==> ~(t = {})`) THEN EXISTS_TAC `ball(z:real^N,r)` THEN ASM_REWRITE_TAC[BALL_EQ_EMPTY; REAL_NOT_LE] THEN REWRITE_TAC[GSYM INTERIOR_CBALL] THEN MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC(SET_RULE `!x. x INSERT s SUBSET t ==> s SUBSET t`) THEN EXISTS_TAC `x:real^N` THEN EXPAND_TAC "A" THEN REWRITE_TAC[HULL_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `(A:real^N->bool) SUBSET t` ASSUME_TAC THENL [EXPAND_TAC "A" THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[INSERT_SUBSET]; ALL_TAC] THEN ABBREV_TAC `c = \n. IMAGE (\y:real^N. x + y) (IMAGE (\x. inv(&n + &1) % x) (IMAGE (\y. --x + y) A))` THEN SUBGOAL_THEN `!n. x IN (c:num->real^N->bool) n` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN EXPAND_TAC "c" THEN REWRITE_TAC[IN_TRANSLATION_GALOIS; VECTOR_SUB_REFL] THEN GEN_REWRITE_TAC I [IN_IMAGE] THEN EXISTS_TAC `vec 0:real^N` THEN REWRITE_TAC[VECTOR_MUL_RZERO; IN_TRANSLATION_GALOIS] THEN REWRITE_TAC[VECTOR_ARITH `vec 0 - --x:real^N = x:real^N`] THEN EXPAND_TAC "A" THEN MATCH_MP_TAC HULL_INC THEN REWRITE_TAC[IN_INSERT]; ALL_TAC] THEN SUBGOAL_THEN `!n. convex((c:num->real^N->bool) n) /\ compact(c n) /\ bounded(c n) /\ ~(interior(c n) = {})` MP_TAC THENL [EXPAND_TAC "c" THEN REWRITE_TAC[CONVEX_TRANSLATION_EQ; COMPACT_TRANSLATION_EQ; BOUNDED_TRANSLATION_EQ; INTERIOR_TRANSLATION; CONVEX_SCALING_EQ; COMPACT_SCALING_EQ; IMAGE_EQ_EMPTY; BOUNDED_SCALING_EQ; INTERIOR_SCALING; REAL_INV_EQ_0; REAL_ARITH `~(&n + &1 = &0)`] THEN ASM_REWRITE_TAC[]; REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC] THEN SUBGOAL_THEN `!n. &0 < measure((c:num->real^N->bool) n)` ASSUME_TAC THENL [ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_COMPACT] THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR]; ALL_TAC] THEN SUBGOAL_THEN `!n. (c:num->real^N->bool) n SUBSET t` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN EXPAND_TAC "c" THEN REWRITE_TAC[TRANSLATION_SUBSET_GALOIS_LEFT] THEN MATCH_MP_TAC(SET_RULE `IMAGE g s SUBSET t /\ IMAGE f (IMAGE g s) SUBSET IMAGE g s ==> IMAGE f (IMAGE g s) SUBSET t`) THEN ASM_SIMP_TAC[IMAGE_SUBSET] THEN REWRITE_TAC[SUBSET] THEN ONCE_REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `w:real^N` THEN DISCH_TAC THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a % x:real^N = (&1 - a) % vec 0 + a % x`] THEN MATCH_MP_TAC IN_CONVEX_SET THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ; REAL_ARITH `&0 <= &n + &1`] THEN REWRITE_TAC[CONVEX_TRANSLATION_EQ; IN_TRANSLATION_GALOIS] THEN ASM_REWRITE_TAC[VECTOR_ARITH `vec 0 - --x:real^N = x`] THEN EXPAND_TAC "A" THEN SIMP_TAC[HULL_INC; IN_INSERT] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `((\n. lift(diameter(c n:real^N->bool))) --> vec 0) sequentially` (LABEL_TAC "D") THENL [EXPAND_TAC "c" THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[DIAMETER_SCALING; BOUNDED_SCALING; BOUNDED_TRANSLATION; DIAMETER_TRANSLATION] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] LIFT_CMUL] THEN MATCH_MP_TAC LIM_NULL_CMUL THEN REWRITE_TAC[REAL_ABS_INV; REAL_ARITH `abs(&n + &1) = &n + &1`] THEN REWRITE_TAC[SEQ_HARMONIC_OFFSET]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`; `c:num->real^N->bool`; `x:real^N`] MEASURE_DIFFERENTIABLE_IMAGE_APPROX) THEN ASM_SIMP_TAC[CLOSURE_INC] THEN ANTS_TAC THENL [EXISTS_TAC `A:real^N->bool` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN EXPAND_TAC "c" THEN REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN EXISTS_TAC `inv(&n + &1)` THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN EXISTS_TAC `(&1 - inv(&n + &1)) % x:real^N` THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN CONV_TAC VECTOR_ARITH; STRIP_TAC] THEN SUBGOAL_THEN `?u. open u /\ x IN u /\ u SUBSET s /\ !a b. a IN u /\ b IN u ==> ((f:real^N->real^N) a = f b <=> a = b)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `x:real^N`; `s:real^N->bool`] INVERSE_FUNCTION_THEOREM) THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[homeomorphism] THEN MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `eventually (\n. c n SUBSET (u:real^N->bool)) sequentially` MP_TAC THENL [UNDISCH_TAC `open(u:real^N->bool)` THEN REWRITE_TAC[OPEN_CONTAINS_CBALL] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN REMOVE_THEN "D" (MP_TAC o GEN_REWRITE_RULE I [tendsto]) THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[DIST_0] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[NORM_LIFT; real_abs] THEN ASM_SIMP_TAC[DIAMETER_POS_LE] THEN DISCH_TAC THEN TRANS_TAC SUBSET_TRANS `cball(x:real^N,d)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_CBALL] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `diameter((c:num->real^N->bool) n)` THEN ASM_SIMP_TAC[DIST_LE_DIAMETER; REAL_LT_IMP_LE]; REMOVE_THEN "D" (K ALL_TAC)] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [tendsto]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[DIST_LIFT] THEN REWRITE_TAC[EVENTUALLY_AND; REAL_ARITH `abs(x - y) < e <=> x < y + e /\ y < x + e`] THEN MATCH_MP_TAC(TAUT `(r /\ p ==> s) /\ (r /\ q ==> t) ==> p /\ q ==> r ==> s /\ t`) THEN REWRITE_TAC[GSYM EVENTUALLY_AND] THEN GEN_REWRITE_TAC BINOP_CONV [GSYM CONTRAPOS_THM] THEN REWRITE_TAC[MESON[] `~(?x. P x /\ Q x) <=> !x. P x ==> ~Q x`] THEN REWRITE_TAC[REAL_NOT_LE] THEN CONJ_TAC THEN DISCH_TAC THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN REWRITE_TAC[LE_REFL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[REAL_NOT_LT] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `x + e <= y <=> x <= y - e`] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ONCE_REWRITE_TAC[GSYM LIFT_DROP] THEN REWRITE_TAC[LIFT_CMUL] THEN ASM_SIMP_TAC[GSYM INTEGRAL_CONST_GEN; MEASURABLE_COMPACT] THEN REWRITE_TAC[LIFT_DROP] THENL [W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_DIFFERENTIABLE_IMAGE_EQ o rand o snd); W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_DIFFERENTIABLE_IMAGE_EQ o lhand o snd)] THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT] THEN (ANTS_TAC THENL [REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; SUBSET]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURABLE_DIFFERENTIABLE_IMAGE_EQ o snd) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT] THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; SUBSET]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_TRANS]; DISCH_THEN SUBST1_TAC]) THEN MATCH_MP_TAC INTEGRAL_DROP_LE_AE THEN EXISTS_TAC `frontier((c:num->real^N->bool) n)` THEN ASM_SIMP_TAC[INTEGRABLE_ON_CONST; MEASURABLE_COMPACT; LIFT_DROP] THEN ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_FRONTIER; SET_DIFF_FRONTIER] THEN (CONJ_TAC THENL [W(MP_TAC o PART_MATCH (rand o rand) MEASURABLE_DIFFERENTIABLE_IMAGE_EQ o snd) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_COMPACT] THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; SUBSET]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_TRANS]; REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LT_IMP_LE THEN REWRITE_TAC[REAL_LT_SUB_LADD] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; SUBSET_INTERIOR]])) in REPLICATE_TAC 4 GEN_TAC THEN MATCH_MP_TAC(MESON[REAL_LE_TOTAL] `!f:A->real. (!a b. P a b ==> P b a) /\ (!a b. f a <= f b ==> P a b) ==> !a b. P a b`) THEN EXISTS_TAC `\x. det(matrix((f':real^N->real^N->real^N) x))` THEN CONJ_TAC THENL [REWRITE_TAC[SEGMENT_SYM; CONJ_ACI]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN SIMP_TAC[SEGMENT_1; LIFT_DROP] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET; IN_INTERVAL_1; LIFT_DROP; FORALL_LIFT] THEN X_GEN_TAC `y:real` THEN STRIP_TAC THEN REWRITE_TAC[IN_IMAGE; LIFT_EQ] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`; `t:real^N->bool`] lemma) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(fun th -> MP_TAC(SPECL [`det(matrix((f':real^N->real^N->real^N) b)) - y`; `b:real^N`] th) THEN MP_TAC(SPECL [`y - det(matrix((f':real^N->real^N->real^N) a))`; `a:real^N`] th)) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT1)] THEN REWRITE_TAC[REAL_ARITH `a + y - a:real = y`] THEN DISCH_THEN(X_CHOOSE_THEN `a':real^N` STRIP_ASSUME_TAC) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN REWRITE_TAC[REAL_ARITH `b <= x + b - y <=> y <= x`] THEN DISCH_THEN(X_CHOOSE_THEN `b':real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `interior t:real^N->bool`] CONNECTED_JACOBIAN_RANGE) THEN ASM_SIMP_TAC[OPEN_INTERIOR; CONVEX_CONNECTED; CONVEX_INTERIOR] THEN ANTS_TAC THENL [ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET]; ALL_TAC] THEN REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1] THEN REWRITE_TAC[IS_INTERVAL_1; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN DISCH_THEN(MP_TAC o SPEC `a':real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `b':real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `lift y`) THEN ASM_REWRITE_TAC[LIFT_DROP; IN_IMAGE; LIFT_EQ]);; let CONNECTED_JACOBIAN_RANGE_SUBSET = prove (`!f:real^N->real^N f' s t. open s /\ connected s /\ convex t /\ t SUBSET s /\ ~(interior t = {}) /\ (!x. x IN s ==> (f has_derivative f' x) (at x) /\ ~(det(matrix (f' x)) = &0)) ==> connected (IMAGE (\x. lift(det(matrix(f' x)))) t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1] THEN REWRITE_TAC[IS_INTERVAL_1; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN REWRITE_TAC[FORALL_LIFT; LIFT_DROP] THEN X_GEN_TAC `y:real` THEN REPEAT DISCH_TAC THEN REWRITE_TAC[IN_IMAGE; LIFT_EQ] THEN ASM_CASES_TAC `det(matrix((f':real^N->real^N->real^N) a)) = y` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `det(matrix((f':real^N->real^N->real^N) b)) = y` THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`; `t:real^N->bool`] CONNECTED_JACOBIAN_RANGE_ALT) THEN DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; SEGMENT_1; LIFT_DROP] THEN COND_CASES_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN DISCH_THEN(MP_TAC o SPEC `lift y`) THEN ASM_REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; REAL_LT_LE] THEN ASM_REWRITE_TAC[LIFT_DROP; IN_IMAGE; LIFT_EQ] THEN MESON_TAC[REWRITE_RULE[SUBSET] INTERIOR_SUBSET]);; let CONNECTED_JACOBIAN_GRAPH = prove (`!f:real^N->real^N f' s. open s /\ connected s /\ (!x. x IN s ==> (f has_derivative f' x) (at x) /\ ~(det(matrix (f' x)) = &0)) ==> connected {pastecart x (lift(det(matrix(f' x)))) | x IN s}`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[SET_RULE `{f x | x IN {}} = {}`; CONNECTED_EMPTY] THEN SUBGOAL_THEN `(f:real^N->real^N) continuous_on s` ASSUME_TAC THENL [MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN ASM_SIMP_TAC[DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN ASM_MESON_TAC[differentiable]; ALL_TAC] THEN ABBREV_TAC `G = {pastecart x (lift(det(matrix((f':real^N->real^N->real^N) x)))) | x IN s}` THEN REWRITE_TAC[CONNECTED_CLOPEN] THEN X_GEN_TAC `X:real^(N,1)finite_sum->bool` THEN REWRITE_TAC[TAUT `p ==> q <=> ~(p /\ ~q)`] THEN DISCH_THEN(STRIP_ASSUME_TAC o REWRITE_RULE[DE_MORGAN_THM]) THEN ABBREV_TAC `Y = {x | x IN s /\ pastecart x (lift(det(matrix((f':real^N->real^N->real^N) x)))) IN X}` THEN ABBREV_TAC `Z = {x | x IN s /\ ~(pastecart x (lift(det(matrix((f':real^N->real^N->real^N) x)))) IN X)}` THEN SUBGOAL_THEN `fsigma(Y:real^N->bool) /\ fsigma(Z:real^N->bool)` MP_TAC THENL [CONJ_TAC THENL [EXPAND_TAC "Y" THEN UNDISCH_TAC `open_in (subtopology euclidean G) (X:real^(N,1)finite_sum->bool)`; EXPAND_TAC "Z" THEN UNDISCH_TAC `closed_in (subtopology euclidean G) (X:real^(N,1)finite_sum->bool)` THEN REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ ~(f x IN t)} = {x | x IN s /\ f x IN ({f x | x IN s} DIFF t)}`]] THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "Y" THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN SIMP_TAC[OPEN_IN_OPEN; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u:real^(N,1)finite_sum->bool` THEN STRIP_TAC THEN EXPAND_TAC "G" THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ f x IN {f x | x IN s} INTER u} = {x | x IN s /\ f x IN u}`] THEN MATCH_MP_TAC FSIGMA_BAIRE1_PREIMAGE_OPEN THEN ASM_SIMP_TAC[OPEN_IMP_FSIGMA] THEN MATCH_MP_TAC BAIRE_PASTECART THEN SIMP_TAC[CONTINUOUS_ON_IMP_BAIRE; CONTINUOUS_ON_ID] THEN MATCH_MP_TAC BAIRE1_DET_JACOBIAN THEN ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[fsigma; UNION_OF; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[NOT_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`YY:(real^N->bool)->bool`; `ZZ:(real^N->bool)->bool`] THEN STRIP_TAC THEN MP_TAC (ISPECL [`{frontier Y INTER A:real^N->bool | A IN YY /\ ~(A = {})} UNION {frontier Y INTER B | B IN ZZ /\ ~(B = {})}`; `frontier Y INTER s:real^N->bool`] BAIRE_ALT) THEN REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC LOCALLY_COMPACT_INTER THEN ASM_SIMP_TAC[OPEN_IMP_LOCALLY_COMPACT; CLOSED_IMP_LOCALLY_COMPACT; FRONTIER_CLOSED]; ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC CONNECTED_INTER_FRONTIER THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM SET_TAC[]; REWRITE_TAC[COUNTABLE_UNION] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN CONJ_TAC THEN MATCH_MP_TAC COUNTABLE_IMAGE THEN MATCH_MP_TAC COUNTABLE_RESTRICT THEN ASM_REWRITE_TAC[]; SIMP_TAC[UNIONS_UNION; INTER_EMPTY; SET_RULE `(!x. x = {} ==> f x = {}) ==> UNIONS {f x | P x /\ ~(x = {})} = UNIONS {f x | P x}`] THEN REWRITE_TAC[SIMPLE_IMAGE; GSYM UNIONS_UNION; GSYM IMAGE_UNION] THEN SUBGOAL_THEN `s:real^N->bool = UNIONS YY UNION UNIONS ZZ` SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[UNIONS_IMAGE; IN_INTER; IN_ELIM_THM] THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_IN_UNION; EXISTS_IN_GSPEC] THEN REWRITE_TAC[OR_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `A:real^N->bool` MP_TAC) THEN REWRITE_TAC[GSYM RIGHT_OR_DISTRIB; GSYM DISJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 (CONJUNCTS_THEN ASSUME_TAC) MP_TAC) THEN SUBGOAL_THEN `closure(frontier Y INTER A):real^N->bool = frontier Y INTER A` SUBST1_TAC THENL [ASM_MESON_TAC[CLOSURE_CLOSED; FRONTIER_CLOSED; CLOSED_INTER]; ALL_TAC] THEN REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[UNWIND_THM2] THEN REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `w:real^N->bool` THEN (ASM_CASES_TAC `open(s INTER w:real^N->bool)` THENL [DISCH_THEN(CONJUNCTS_THEN2 (K ALL_TAC) MP_TAC) THEN UNDISCH_TAC `open(s INTER w:real^N->bool)`; ASM_MESON_TAC[OPEN_INTER]] THEN SUBGOAL_THEN `(s INTER w:real^N->bool) SUBSET s` MP_TAC THENL [SET_TAC[]; REWRITE_TAC[INTER_ASSOC]]) THEN SPEC_TAC(`s INTER w:real^N->bool`,`w:real^N->bool`) THEN X_GEN_TAC `W:real^N->bool` THEN REWRITE_TAC[IMP_IMP] THEN UNDISCH_TAC `(A:real^N->bool) IN YY \/ A IN ZZ` THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN ONCE_REWRITE_TAC[TAUT `p ==> q \/ r ==> s ==> t <=> (q ==> p /\ s ==> t) /\ (r ==> p /\ s ==> t)`] THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`X:real^(N,1)finite_sum->bool`; `Y:real^N->bool`; `Z:real^N->bool`; `YY:(real^N->bool)->bool`; `ZZ:(real^N->bool)->bool`] THEN REWRITE_TAC[FORALL_AND_THM] THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `(!x y. P x y ==> Q y x) ==> (!x y. P x y) ==> (!x y. Q x y)`) THEN MAP_EVERY X_GEN_TAC [`ZZ:(real^N->bool)->bool`; `YY:(real^N->bool)->bool`] THEN MATCH_MP_TAC(MESON[] `(!x y. P x y ==> Q y x) ==> (!x y. P x y) ==> (!x y. Q x y)`) THEN MAP_EVERY X_GEN_TAC [`Z:real^N->bool`; `Y:real^N->bool`] THEN DISCH_TAC THEN X_GEN_TAC `X:real^(N,1)finite_sum->bool` THEN DISCH_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `G DIFF X:real^(N,1)finite_sum->bool`) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_SIMP_TAC[OPEN_IN_DIFF; CLOSED_IN_DIFF; OPEN_IN_REFL; CLOSED_IN_REFL] THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM SET_TAC[]; SUBGOAL_THEN `frontier Y INTER s:real^N->bool = frontier Z INTER s` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[FRONTIER_CLOSURES] THEN ONCE_REWRITE_TAC[SET_RULE `(s INTER t) INTER u = s INTER u INTER t`] THEN W(MP_TAC o PART_MATCH (lhand o rand) OPEN_INTER_CLOSURE_EQ o rand o lhand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) OPEN_INTER_CLOSURE_EQ o rand o rand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(SET_RULE `t' = u /\ u' = t ==> u INTER s INTER u' = t INTER s INTER t'`) THEN CONJ_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]]; REPEAT STRIP_TAC] THEN SUBGOAL_THEN `frontier Z INTER s:real^N->bool = frontier Y INTER s` ASSUME_TAC THENL [REWRITE_TAC[FRONTIER_CLOSURES] THEN ONCE_REWRITE_TAC[SET_RULE `(s INTER t) INTER u = s INTER u INTER t`] THEN W(MP_TAC o PART_MATCH (lhand o rand) OPEN_INTER_CLOSURE_EQ o rand o lhand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) OPEN_INTER_CLOSURE_EQ o rand o rand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(SET_RULE `t' = u /\ u' = t ==> u INTER s INTER u' = t INTER s INTER t'`) THEN CONJ_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?a:real^N. a IN frontier Y /\ a IN W /\ a IN A` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?r. &0 < r /\ ball(a,r) SUBSET W /\ (!x y. x IN ball(a,r) /\ y IN ball(a,r) ==> ((f:real^N->real^N) x = f y <=> x = y))` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `a:real^N`; `W:real^N->bool`] INVERSE_FUNCTION_THEOREM) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[homeomorphism]] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `u:real^N->bool` OPEN_CONTAINS_BALL) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?b:real^N. b IN ball(a,r / &3) /\ b IN Z` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `(a:real^N) IN frontier Y` THEN REWRITE_TAC[FRONTIER_STRADDLE] THEN DISCH_THEN(MP_TAC o SPEC `r / &3`) THEN ASM_REWRITE_TAC[REAL_ARITH `&0 < r / &3 <=> &0 < r`] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN REWRITE_TAC[GSYM IN_BALL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`a:real^N`; `r / &3`; `r:real`] SUBSET_BALL) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `b IN ball(a:real^N,r)` ASSUME_TAC THENL [MP_TAC(ISPECL [`a:real^N`; `r / &3`; `r:real`] SUBSET_BALL) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `(b:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `~((b:real^N) IN frontier Y)` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ABBREV_TAC `d = setdist({b:real^N},s INTER frontier Y)` THEN SUBGOAL_THEN `~(s INTER frontier Y:real^N->bool = {})` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&0 < d` ASSUME_TAC THENL [EXPAND_TAC "d" THEN REWRITE_TAC[SETDIST_POS_LT; SETDIST_EQ_0_SING] THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`s:real^N->bool`; `frontier Y:real^N->bool`] OPEN_INTER_CLOSURE_EQ) THEN ASM_SIMP_TAC[CLOSURE_CLOSED; FRONTIER_CLOSED] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `d < r / &3` ASSUME_TAC THENL [TRANS_TAC REAL_LET_TRANS `dist(b:real^N,a)` THEN CONJ_TAC THENL [EXPAND_TAC "d" THEN MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]; ASM_MESON_TAC[IN_BALL; DIST_SYM]]; ALL_TAC] THEN SUBGOAL_THEN `ball(b:real^N,d) SUBSET Z` ASSUME_TAC THENL [MP_TAC(ISPECL [`ball(b:real^N,d)`; `Z:real^N->bool`] CONNECTED_INTER_FRONTIER) THEN REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`; CONNECTED_BALL] THEN ASM_CASES_TAC `ball(b:real^N,d) SUBSET Z` THEN ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `b:real^N` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; IN_INTER]; REWRITE_TAC[EXTENSION; IN_BALL; NOT_IN_EMPTY; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN REWRITE_TAC[REAL_NOT_LT] THEN EXPAND_TAC "d" THEN MATCH_MP_TAC SETDIST_LE_DIST THEN SUBGOAL_THEN `x IN ball(a:real^N,r)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MAP_EVERY UNDISCH_TAC [`b IN ball(a:real^N,r / &3)`; `dist(b:real^N,x) < d`; `d < r / &3`] THEN REWRITE_TAC[IN_BALL] THEN CONV_TAC NORM_ARITH]; ALL_TAC] THEN SUBGOAL_THEN `~(sphere(b:real^N,d) INTER s INTER frontier Y = {})` MP_TAC THENL [MP_TAC(ISPECL [`closure(s INTER frontier Y):real^N->bool`; `{b:real^N}`] SETDIST_CLOSED_COMPACT) THEN ASM_REWRITE_TAC[CLOSED_CLOSURE; CLOSURE_EQ_EMPTY] THEN REWRITE_TAC[COMPACT_SING; NOT_INSERT_EMPTY; RIGHT_EXISTS_AND_THM] THEN REWRITE_TAC[SETDIST_CLOSURE; IN_SING; UNWIND_THM2] THEN ONCE_REWRITE_TAC[SETDIST_SYM] THEN ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; NOT_IN_EMPTY] THEN REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] IN_SPHERE] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(x:real^N) IN s` MP_TAC THENL [SUBGOAL_THEN `x IN ball(a:real^N,r)` MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MAP_EVERY UNDISCH_TAC [`b IN ball(a:real^N,r / &3)`; `dist(x:real^N,b) = d`; `d < r / &3`] THEN REWRITE_TAC[IN_BALL] THEN CONV_TAC NORM_ARITH; REWRITE_TAC[IMP_IMP; GSYM IN_INTER] THEN ASM_SIMP_TAC[GSYM OPEN_INTER_CLOSURE_EQ] THEN SIMP_TAC[CLOSURE_CLOSED; FRONTIER_CLOSED]]; PURE_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN DISCH_THEN(X_CHOOSE_THEN `a':real^N` STRIP_ASSUME_TAC)] THEN SUBGOAL_THEN `(a':real^N) IN ball(a,r)` ASSUME_TAC THENL [MAP_EVERY UNDISCH_TAC [`b IN ball(a:real^N,r / &3)`; `a' IN sphere(b:real^N,d)`; `d < r / &3`] THEN REWRITE_TAC[IN_BALL; IN_SPHERE] THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN SUBGOAL_THEN `(a':real^N) IN W` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(a':real^N) IN Y` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?p k. &0 < p /\ p <= d /\ &0 < k /\ G INTER (ball(a',&2 * p) PCROSS ball(lift(det(matrix((f':real^N->real^N->real^N) a'))),k)) SUBSET X` STRIP_ASSUME_TAC THENL [UNDISCH_TAC `open_in (subtopology euclidean G) (X:real^(N,1)finite_sum->bool)` THEN REWRITE_TAC[OPEN_IN_OPEN; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `U:real^(N,1)finite_sum->bool` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] PASTECART_IN_INTERIOR)) THEN DISCH_THEN(MP_TAC o SPECL [`a':real^N`; `lift(det(matrix((f':real^N->real^N->real^N) a')))`]) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`P:real^N->bool`; `K:real^1->bool`] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s /\ t ==> u <=> t ==> (p /\ q) /\ (r /\ s) ==> u`] THEN DISCH_TAC THEN DISCH_THEN(CONJUNCTS_THEN(MP_TAC o MATCH_MP (MESON[OPEN_CONTAINS_BALL] `open s /\ a IN s ==> ?d. &0 < d /\ ball(a:real^N,d) SUBSET s`))) THEN DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `p:real` STRIP_ASSUME_TAC) THEN MAP_EVERY EXISTS_TAC [`min d (p / &2)`; `k:real`] THEN ASM_REWRITE_TAC[REAL_HALF; REAL_LT_MIN; REAL_MIN_LE; REAL_LE_REFL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET u ==> t SUBSET s ==> g INTER t SUBSET g INTER u`)) THEN ASM_REWRITE_TAC[SUBSET_PCROSS] THEN REPEAT DISJ2_TAC THEN TRANS_TAC SUBSET_TRANS `ball(a':real^N,p)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_BALL THEN REAL_ARITH_TAC; ALL_TAC] THEN ABBREV_TAC `b':real^N = a' + p / d % (b - a')` THEN SUBGOAL_THEN `ball(b':real^N,p) SUBSET ball(b,d)` ASSUME_TAC THENL [EXPAND_TAC "b'" THEN REWRITE_TAC[SUBSET_BALLS; dist] THEN REWRITE_TAC[VECTOR_ARITH `(a + x % (b - a)) - b:real^N = (x - &1) % (b - a)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_SPHERE]) THEN ASM_REWRITE_TAC[NORM_MUL; GSYM dist] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < d ==> x * d = x * abs d`] THEN REWRITE_TAC[GSYM REAL_ABS_MUL] THEN ASM_SIMP_TAC[REAL_FIELD `&0 < d ==> (p / d - &1) * d = p - d`] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`; `s:real^N->bool`; `cball(b':real^N,p)`; `a':real^N`; `b':real^N`] CONNECTED_JACOBIAN_RANGE_ALT) THEN ASM_SIMP_TAC[CONVEX_CBALL; INTERIOR_CBALL; BALL_EQ_EMPTY; REAL_NOT_LE] THEN ASM_SIMP_TAC[CENTRE_IN_CBALL; NOT_IMP; REAL_LT_IMP_LE] THEN SUBGOAL_THEN `dist(a':real^N,b') = p` ASSUME_TAC THENL [EXPAND_TAC "b'" THEN REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN MATCH_MP_TAC(REAL_FIELD `&0 < d /\ n = d ==> p / d * n = p`) THEN ASM_REWRITE_TAC[GSYM dist; GSYM IN_SPHERE]; ALL_TAC] THEN REPEAT CONJ_TAC THENL [TRANS_TAC SUBSET_TRANS `cball(b:real^N,d)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[GSYM CLOSURE_BALL] THEN ASM_SIMP_TAC[SUBSET_CLOSURE]; ALL_TAC] THEN TRANS_TAC SUBSET_TRANS `W:real^N->bool` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC SUBSET_TRANS `ball(a:real^N,r)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET_BALLS] THEN MAP_EVERY UNDISCH_TAC [`b IN ball(a:real^N,r / &3)`; `d < r / &3`] THEN REWRITE_TAC[IN_BALL] THEN CONV_TAC NORM_ARITH; REWRITE_TAC[IN_CBALL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[REAL_LE_REFL]; ALL_TAC] THEN SUBGOAL_THEN `(b':real^N) IN Z` ASSUME_TAC THENL [MP_TAC(ISPECL [`b':real^N`; `p:real`] CENTRE_IN_BALL) THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `ball(lift(det(matrix((f':real^N->real^N->real^N) a'))),k)` o MATCH_MP (SET_RULE `s SUBSET t ==> !u. ~(u INTER s = {}) ==> ?x. x IN t /\ x IN u`)) THEN REWRITE_TAC[EXISTS_IN_IMAGE; NOT_IMP] THEN CONJ_TAC THENL [W(MP_TAC o PART_MATCH (rand o rand) OPEN_INTER_CLOSURE_EQ_EMPTY o rand o snd) THEN REWRITE_TAC[OPEN_BALL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[CLOSURE_SEGMENT] THEN COND_CASES_TAC THENL [MP_TAC(ASSUME `(b':real^N) IN Z`) THEN UNDISCH_THEN `UNIONS ZZ:real^N->bool = Z` (K ALL_TAC) THEN EXPAND_TAC "Z" THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET t ==> x IN s ==> P /\ ~(x IN t) ==> Q`)) THEN EXPAND_TAC "G" THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_ELIM_THM; IN_INTER] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ASM_REWRITE_TAC[IN_BALL; REAL_ARITH `p < &2 * p <=> &0 < p`] THEN EXISTS_TAC `b':real^N` THEN ASM SET_TAC[]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN EXISTS_TAC `lift(det(matrix((f':real^N->real^N->real^N) a')))` THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; ENDS_IN_SEGMENT]]; DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `G INTER ball(a':real^N,&2 * p) PCROSS ball(lift(det(matrix(f' a'):real^N^N)),k) SUBSET X` THEN EXPAND_TAC "G" THEN REWRITE_TAC[SUBSET; IN_INTER; FORALL_IN_GSPEC; IMP_CONJ] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[PASTECART_IN_PCROSS; NOT_IMP] THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; MAP_EVERY UNDISCH_TAC [`dist(a':real^N,b') = p`; `x IN ball(b':real^N,p)`] THEN REWRITE_TAC[IN_BALL] THEN CONV_TAC NORM_ARITH; ASM SET_TAC[]]]);; (* ------------------------------------------------------------------------- *) (* A derivative-free formulation of (absolute) integration by parts. *) (* ------------------------------------------------------------------------- *) let ABSOLUTE_INTEGRATION_BY_PARTS = prove (`!(bop:real^M->real^N->real^P) (f:real^1->real^M) g f' g' a b. bilinear bop /\ drop a <= drop b /\ f' absolutely_integrable_on interval[a,b] /\ g' absolutely_integrable_on interval[a,b] /\ (!x. x IN interval[a,b] ==> (f' has_integral f(x)) (interval[a,x])) /\ (!x. x IN interval[a,b] ==> (g' has_integral g(x)) (interval[a,x])) ==> (\x. bop (f x) (g' x)) absolutely_integrable_on interval[a,b] /\ (\x. bop (f' x) (g x)) absolutely_integrable_on interval[a,b] /\ integral (interval[a,b]) (\x. bop (f x) (g' x)) + integral (interval[a,b]) (\x. bop (f' x) (g x)) = bop (f b) (g b) - bop (f a) (g a)`, REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `(f:real^1->real^M) continuous_on interval[a,b] /\ (g:real^1->real^N) continuous_on interval[a,b]` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THENL [EXISTS_TAC `\x. integral(interval[a,x]) (f':real^1->real^M)`; EXISTS_TAC `\x. integral(interval[a,x]) (g':real^1->real^N)`] THEN ASM_SIMP_TAC[INDEFINITE_INTEGRAL_CONTINUOUS_RIGHT; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_SIMP_TAC[]; ALL_TAC] THEN MP_TAC(GEN `n:num` (ISPECL [`g':real^1->real^N`; `interval[a:real^1,b]`; `inv(&n + &1)`] ABSOLUTELY_INTEGRABLE_APPROXIMATE_CONTINUOUS)) THEN MP_TAC(GEN `n:num` (ISPECL [`f':real^1->real^M`; `interval[a:real^1,b]`; `inv(&n + &1)`] ABSOLUTELY_INTEGRABLE_APPROXIMATE_CONTINUOUS)) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_INTERVAL; REAL_LT_INV_EQ] THEN REWRITE_TAC[REAL_ARITH `&0 < &n + &1`; FORALL_AND_THM; SKOLEM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `ff':num->real^1->real^M` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `gg':num->real^1->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(!n. ((ff':num->real^1->real^M) n) continuous_on interval[a,b]) /\ (!n. ((gg':num->real^1->real^N) n) continuous_on interval[a,b])` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC] THEN MAP_EVERY ABBREV_TAC [`ff = \n x. integral(interval[a,x]) ((ff':num->real^1->real^M) n)`; `gg = \n x. integral(interval[a,x]) ((gg':num->real^1->real^N) n)`] THEN SUBGOAL_THEN `(!n. (ff:num->real^1->real^M) n continuous_on interval[a,b]) /\ (!n. (gg:num->real^1->real^N) n continuous_on interval[a,b])` STRIP_ASSUME_TAC THENL [REPEAT STRIP_TAC THEN MAP_EVERY EXPAND_TAC ["ff"; "gg"] THEN MATCH_MP_TAC INDEFINITE_INTEGRAL_CONTINUOUS_RIGHT THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; ALL_TAC] THEN SUBGOAL_THEN `(f:real^1->real^M) absolutely_integrable_on interval[a,b] /\ (g:real^1->real^N) absolutely_integrable_on interval[a,b] /\ (!n. (ff:num->real^1->real^M) n absolutely_integrable_on interval[a,b]) /\ (!n. (gg:num->real^1->real^N) n absolutely_integrable_on interval[a,b])` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_CONTINUOUS]; ALL_TAC] THEN SUBGOAL_THEN `(!f:real^1->real^M g:real^1->real^N. f absolutely_integrable_on interval[a,b] /\ g continuous_on interval[a,b] ==> (\x. (bop:real^M->real^N->real^P) (f x) (g x)) absolutely_integrable_on interval[a,b]) /\ (!f:real^1->real^M g:real^1->real^N. f continuous_on interval[a,b] /\ g absolutely_integrable_on interval[a,b] ==> (\x. (bop:real^M->real^N->real^P) (f x) (g x)) absolutely_integrable_on interval[a,b])` STRIP_ASSUME_TAC THENL [REPEAT STRIP_TAC THENL [MP_TAC(GEN `g:real^1->real^N` (ISPECL [`\x y. (bop:real^M->real^N->real^P) y x`; `g:real^1->real^N`] ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT)); MP_TAC(GEN `f:real^1->real^M` (ISPECL [`bop:real^M->real^N->real^P`; `f:real^1->real^M`] ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT))] THEN ASM_REWRITE_TAC[BILINEAR_SWAP] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[CONTINUOUS_IMP_MEASURABLE_ON_CLOSED_SUBSET; CLOSED_INTERVAL] THEN MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[COMPACT_INTERVAL]; ALL_TAC] THEN REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[]; STRIP_TAC] THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `\n. (integral (interval [a,b]) (\x. bop (ff n x) (gg' n x)) + integral (interval [a,b]) (\x. bop (ff' n x) (gg n x))) - ((bop:real^M->real^N->real^P) (ff n b) (gg n b) - bop ((ff:num->real^1->real^M) n a) (gg n a))` THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[VECTOR_ARITH `(i + j) - b:real^N = vec 0 <=> j = b - i`] THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC INTEGRATION_BY_PARTS THEN MAP_EVERY EXISTS_TAC [`(ff:num->real^1->real^M) n`; `(gg':num->real^1->real^N) n`; `{}:real^1->bool`] THEN ASM_REWRITE_TAC[COUNTABLE_EMPTY; DIFF_EMPTY] THEN REWRITE_TAC[VECTOR_ARITH `b - a - (b - a - i):real^N = i`] THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL] THEN ASM_SIMP_TAC[BILINEAR_CONTINUOUS_ON_COMPOSE; ETA_AX; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN X_GEN_TAC `x:real^1` THEN REPEAT STRIP_TAC THENL [MP_TAC(ISPECL [`(ff':num->real^1->real^M) n`; `a:real^1`; `b:real^1`; `x:real^1`] INTEGRAL_HAS_VECTOR_DERIVATIVE_POINTWISE) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN ANTS_TAC THENL [ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET; CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]; EXPAND_TAC "ff"]; MP_TAC(ISPECL [`(gg':num->real^1->real^N) n`; `a:real^1`; `b:real^1`; `x:real^1`] INTEGRAL_HAS_VECTOR_DERIVATIVE_POINTWISE) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN ANTS_TAC THENL [ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET; CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]; EXPAND_TAC "gg"]] THEN REWRITE_TAC[has_vector_derivative; has_derivative] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM(CONJUNCT1 INTERIOR_INTERVAL)]) THEN SIMP_TAC[NETLIMIT_WITHIN; NETLIMIT_AT; LIM_WITHIN_INTERIOR]] THEN MATCH_MP_TAC LIM_SUB THEN CONJ_TAC THENL [MATCH_MP_TAC LIM_ADD; MATCH_MP_TAC LIM_SUB THEN CONJ_TAC THEN MP_TAC(ISPECL [`sequentially`; `bop:real^M->real^N->real^P`] LIM_BILINEAR) THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN (SUBGOAL_THEN `!x. x IN interval[a,b] ==> (f:real^1->real^M) x = integral (interval[a,x]) f' /\ (g:real^1->real^N) x = integral (interval[a,x]) g'` MP_TAC THENL [ASM_MESON_TAC[INTEGRAL_UNIQUE]; ALL_TAC]) THEN ASM_SIMP_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1] THEN DISCH_THEN(K ALL_TAC) THEN MAP_EVERY EXPAND_TAC ["ff"; "gg"] THEN REWRITE_TAC[INTEGRAL_REFL; LIM_CONST] THEN ONCE_REWRITE_TAC[LIM_NULL] THEN ASM_SIMP_TAC[GSYM INTEGRAL_SUB; INTEGRABLE_CONTINUOUS; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\n. inv(&n)` THEN REWRITE_TAC[SEQ_HARMONIC] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) ABSOLUTELY_INTEGRABLE_LE o lhand o snd) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_SUB; ETA_AX] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN ONCE_REWRITE_TAC[NORM_SUB] THEN MATCH_MP_TAC(REAL_ARITH `abs x < a ==> x <= a`) THEN REWRITE_TAC[GSYM NORM_1] THEN TRANS_TAC REAL_LTE_TRANS `inv(&n + &1)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN UNDISCH_TAC `1 <= n` THEN ARITH_TAC] THEN FIRST_ASSUM(X_CHOOSE_THEN `M:real` STRIP_ASSUME_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN SUBGOAL_THEN `?B. &0 < B /\ norm (integral(interval[a,b]) (\x:real^1. lift(norm(f' x:real^M)))) <= B /\ norm (integral(interval[a,b]) (\x. lift(norm(g' x:real^N)))) <= B` STRIP_ASSUME_TAC THENL [REWRITE_TAC[NORM_REAL] THEN MESON_TAC[REAL_ARITH `&0 < max (abs a) (abs b) + &1 /\ abs a <= max (abs a) (abs b) + &1 /\ abs b <= max (abs a) (abs b) + &1`]; ALL_TAC] THEN CONJ_TAC THEN ONCE_REWRITE_TAC[LIM_NULL] THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; GSYM INTEGRAL_SUB; ABSOLUTELY_INTEGRABLE_CONTINUOUS; ETA_AX] THEN MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\n. &2 * M * B * inv(&n + &1) + M * inv(&n + &1) pow 2` THEN REWRITE_TAC[] THEN (CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[LIFT_ADD; LIFT_CMUL] THEN MATCH_MP_TAC LIM_NULL_ADD THEN CONJ_TAC THEN REPEAT(MATCH_MP_TAC LIM_NULL_CMUL) THEN REWRITE_TAC[SEQ_HARMONIC_OFFSET] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^1 = drop(vec 0) % vec 0`) THEN REWRITE_TAC[REAL_POW_2; LIFT_CMUL] THEN MATCH_MP_TAC LIM_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP; SEQ_HARMONIC_OFFSET]]) THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THENL [ONCE_REWRITE_TAC[VECTOR_ARITH `bop a b - bop c d:real^N = (bop a b - bop a d) + (bop a d - bop c d)`]; ONCE_REWRITE_TAC[VECTOR_ARITH `bop a b - bop c d:real^N = (bop a b - bop c b) + (bop c b - bop c d)`]] THEN W(MP_TAC o PART_MATCH (lhand o rand) INTEGRAL_ADD o rand o lhand o snd) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; INTEGRABLE_SUB; ETA_AX] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[REAL_ARITH `&2 * M * B * e + M * e pow 2 = M * (B + e) * e + M * B * e`] THEN MATCH_MP_TAC(NORM_ARITH `norm(x:real^N) <= a /\ norm y <= b ==> norm(x + y) <= a + b`) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bilinear]) THEN DISCH_THEN(CONJUNCTS_THEN(MP_TAC o GEN_ALL o GSYM o MATCH_MP LINEAR_SUB o SPEC_ALL)) THEN SIMP_TAC[] THEN REPEAT(DISCH_THEN(K ALL_TAC)) THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THENL [EXISTS_TAC `drop(integral(interval[a,b]) (\x. lift(M * (B + inv(&n + &1)) * norm((gg':num->real^1->real^N) n x - g' x))))`; EXISTS_TAC `drop(integral(interval[a,b]) (\x. lift(M * inv(&n + &1) * norm((g':real^1->real^N) x))))`; EXISTS_TAC `drop(integral(interval[a,b]) (\x. lift(M * (B + inv(&n + &1)) * norm((ff':num->real^1->real^M) n x - f' x))))`; EXISTS_TAC `drop(integral(interval[a,b]) (\x. lift(M * inv(&n + &1) * norm((f':real^1->real^M) x))))`] THEN (CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL; REWRITE_TAC[REAL_MUL_ASSOC] THEN REWRITE_TAC[LIFT_CMUL] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4) [INTEGRAL_CMUL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_SUB; ABSOLUTELY_INTEGRABLE_CONTINUOUS; ETA_AX] THEN REWRITE_TAC[DROP_CMUL; GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_ARITH `inv x * a <= B * inv x <=> inv x * a <= inv x * B`] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_LT_IMP_LE; REAL_LE_INV_EQ; REAL_POS] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN MATCH_MP_TAC(REAL_ARITH `abs x <= a ==> x <= a`) THEN ASM_SIMP_TAC[GSYM NORM_1; REAL_LT_IMP_LE]]) THEN REWRITE_TAC[REAL_MUL_ASSOC] THEN REWRITE_TAC[LIFT_CMUL] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4) [VECTOR_MUL_ASSOC; INTEGRABLE_CMUL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_SUB; CONTINUOUS_ON_SUB; ABSOLUTELY_INTEGRABLE_CONTINUOUS; ETA_AX] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `norm((bop:real^M->real^N->real^P) a b) <= M * norm a * norm b /\ M * norm a * norm b <= c ==> norm(bop a b) <= c`) THEN ASM_REWRITE_TAC[DROP_CMUL; LIFT_DROP] THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_ARITH `a * x <= y * a <=> x * a <= y * a`] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_REWRITE_TAC[NORM_POS_LE] THENL [MATCH_MP_TAC(NORM_ARITH `!f'. norm(f':real^N) <= B /\ norm(f - f') <= i ==> norm(f) <= B + i`) THEN EXISTS_TAC `(f:real^1->real^M) x` THEN CONJ_TAC THENL [SUBGOAL_THEN `(f:real^1->real^M) x = integral(interval[a,x]) f'` SUBST1_TAC THENL [ASM_MESON_TAC[INTEGRAL_UNIQUE]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) ABSOLUTELY_INTEGRABLE_LE o lhand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)] THEN TRANS_TAC REAL_LE_TRANS `drop(integral (interval[a,b]) (\x. lift(norm((f':real^1->real^M) x))))` THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_SUBSET_DROP_LE THEN REWRITE_TAC[LIFT_DROP; NORM_POS_LE] THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL]; MATCH_MP_TAC(REAL_ARITH `abs x <= a ==> x <= a`) THEN ASM_SIMP_TAC[GSYM NORM_1; REAL_LT_IMP_LE]]; ALL_TAC]; ALL_TAC; MATCH_MP_TAC(NORM_ARITH `!g'. norm(g':real^N) <= B /\ norm(g - g') <= i ==> norm(g) <= B + i`) THEN EXISTS_TAC `(g:real^1->real^N) x` THEN CONJ_TAC THENL [SUBGOAL_THEN `(g:real^1->real^N) x = integral(interval[a,x]) g'` SUBST1_TAC THENL [ASM_MESON_TAC[INTEGRAL_UNIQUE]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) ABSOLUTELY_INTEGRABLE_LE o lhand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)] THEN TRANS_TAC REAL_LE_TRANS `drop(integral (interval[a,b]) (\x. lift(norm((g':real^1->real^N) x))))` THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_SUBSET_DROP_LE THEN REWRITE_TAC[LIFT_DROP; NORM_POS_LE] THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL]; MATCH_MP_TAC(REAL_ARITH `abs x <= a ==> x <= a`) THEN ASM_SIMP_TAC[GSYM NORM_1; REAL_LT_IMP_LE]]; ALL_TAC]; ALL_TAC] THEN (SUBGOAL_THEN `(f:real^1->real^M) x = integral(interval[a,x]) f' /\ (g:real^1->real^N) x = integral(interval[a,x]) g'` (CONJUNCTS_THEN SUBST1_TAC) THENL [ASM_MESON_TAC[INTEGRAL_UNIQUE]; MAP_EVERY EXPAND_TAC ["ff"; "gg"]]) THEN (W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_SUB o rand o lhand o snd) THEN ANTS_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN W(MP_TAC o PART_MATCH (lhand o rand) ABSOLUTELY_INTEGRABLE_LE o lhand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL; ETA_AX]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)]) THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[REAL_LE_TRANS; REAL_LT_IMP_LE] `norm(integral i f) < e ==> drop(integral j f) <= norm(integral i f) ==> drop(integral j f) <= e`) o SPEC `n:num`) THEN REWRITE_TAC[NORM_REAL; GSYM drop] THEN MATCH_MP_TAC(REAL_ARITH `a <= x ==> a <= abs x`) THEN MATCH_MP_TAC INTEGRAL_SUBSET_DROP_LE THEN REWRITE_TAC[LIFT_DROP; NORM_POS_LE] THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN REWRITE_TAC[ETA_AX] THEN ASM_SIMP_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_SIMP_TAC[SUBSET_INTERVAL_1; GSYM IN_INTERVAL_1; REAL_LE_REFL; ETA_AX]);; (* ------------------------------------------------------------------------- *) (* Measurability of inverse function (including sections / selections). *) (* ------------------------------------------------------------------------- *) let DOUBLE_LEBESGUE_MEASURABLE_INVERSE_FUNCTION_GEN = prove (`!f:real^M->real^N s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible(IMAGE f t)) ==> (!t. lebesgue_measurable t ==> lebesgue_measurable(IMAGE f (s INTER t)))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE_GEN) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `t INTER {x | x IN s /\ ~((f:real^M->real^N) x = vec 0)}`) THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEASURABLE_ON_UNIV]) THEN REWRITE_TAC[MEASURABLE_ON_PREIMAGE_OPEN] THEN DISCH_THEN(MP_TAC o SPEC `(:real^N) DELETE vec 0`) THEN SIMP_TAC[OPEN_DELETE; OPEN_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{vec 0:real^N}` THEN REWRITE_TAC[NEGLIGIBLE_SING] THEN SET_TAC[]]);; let DOUBLE_LEBESGUE_MEASURABLE_INVERSE_FUNCTION = prove (`!f:real^M->real^N g s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible(IMAGE f t)) /\ (!x. x IN s ==> g(f x) = x) ==> !t. lebesgue_measurable t ==> lebesgue_measurable {y | y IN IMAGE f s /\ g y IN t}`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] DOUBLE_LEBESGUE_MEASURABLE_INVERSE_FUNCTION_GEN) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]);; let LEBESGUE_MEASURABLE_MEASURABLE_IMAGE,MEASURABLE_ON_INVERSE_FUNCTION = (CONJ_PAIR o prove) (`(!f:real^M->real^N s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible(IMAGE f t)) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> lebesgue_measurable(IMAGE f s)) /\ (!f:real^M->real^N g s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible(IMAGE f t)) /\ (!x. x IN s ==> g(f x) = x) ==> g measurable_on (IMAGE f s))`, REWRITE_TAC[INJECTIVE_ON_LEFT_INVERSE; AND_FORALL_THM] THEN GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [SWAP_FORALL_THM] THEN REWRITE_TAC[INJECTIVE_ON_LEFT_INVERSE; AND_FORALL_THM] THEN GEN_TAC THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `g:real^N->real^M` THEN ONCE_REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> r /\ q`] THEN STRIP_TAC THEN REWRITE_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN_EQ] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `g:real^N->real^M`; `s:real^M->bool`] DOUBLE_LEBESGUE_MEASURABLE_INVERSE_FUNCTION) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_OPEN]);; let DOUBLE_LEBESGUE_MEASURABLE_LEFT_INVERSE = prove (`!f:real^M->real^N s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible (IMAGE f t)) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> ?g. (!x. x IN s ==> g(f x) = x) /\ (!t. lebesgue_measurable t ==> lebesgue_measurable {y | y IN IMAGE f s /\ g y IN t})`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[INJECTIVE_ON_LEFT_INVERSE] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN CONJ_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DOUBLE_LEBESGUE_MEASURABLE_INVERSE_FUNCTION THEN ASM_REWRITE_TAC[]);; let MEASURABLE_ON_LEFT_INVERSE = prove (`!f:real^M->real^N s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible (IMAGE f t)) /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) ==> ?g. (!x. x IN s ==> g(f x) = x) /\ g measurable_on IMAGE f s`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[INJECTIVE_ON_LEFT_INVERSE] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN CONJ_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_ON_INVERSE_FUNCTION THEN ASM_REWRITE_TAC[]);; let DOUBLE_LEBESGUE_MEASURABLE_RIGHT_INVERSE = prove (`!f:real^M->real^N s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible (IMAGE f t)) ==> ?g. (!y. y IN IMAGE f s ==> g y IN s /\ f(g y) = y) /\ (!t. lebesgue_measurable t ==> lebesgue_measurable {y | y IN IMAGE f s /\ g y IN t})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LEBESGUE_MEASURABLE_DOMAIN_OF_INJECTIVITY) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `t:real^M->bool`] DOUBLE_LEBESGUE_MEASURABLE_LEFT_INVERSE) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET; SUBSET_TRANS]; MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]]);; let MEASURABLE_ON_RIGHT_INVERSE = prove (`!f:real^M->real^N s. f measurable_on s /\ (!t. negligible t /\ t SUBSET s ==> negligible (IMAGE f t)) ==> ?g. (!y. y IN IMAGE f s ==> g y IN s /\ f(g y) = y) /\ g measurable_on IMAGE f s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`] LEBESGUE_MEASURABLE_DOMAIN_OF_INJECTIVITY) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `t:real^M->bool`] MEASURABLE_ON_LEFT_INVERSE) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET; SUBSET_TRANS]; MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Egorov's thoerem. *) (* ------------------------------------------------------------------------- *) let EGOROV = prove (`!f:num->real^M->real^N g s t. measurable s /\ negligible t /\ (!n. f n measurable_on s) /\ (!x. x IN s DIFF t ==> ((\n. f n x) --> g x) sequentially) ==> !d. &0 < d ==> ?k. k SUBSET s /\ measurable k /\ measure k < d /\ !e. &0 < e ==> ?N. !n x. N <= n /\ x IN s DIFF k ==> dist(f n x,g x) < e`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(g:real^M->real^N) measurable_on s` ASSUME_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_LIMIT THEN ASM_MESON_TAC[]; ALL_TAC] THEN ABBREV_TAC `e = \n m. UNIONS{{x | x IN s /\ dist((f:num->real^M->real^N) k x,g x) >= inv(&m + &1)} | n <= k}` THEN SUBGOAL_THEN `!m n. measurable ((e:num->num->real^M->bool) n m)` ASSUME_TAC THENL [REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "e" THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN SET_TAC[]] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_SUBSET_NUM; FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_ARITH `dist(a:real^M,b) >= e <=> ~(dist(vec 0,a - b) < e)`] THEN REWRITE_TAC[GSYM IN_BALL; SET_RULE `~(x IN s) <=> x IN UNIV DIFF s`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_CLOSED THEN ASM_SIMP_TAC[GSYM OPEN_CLOSED; OPEN_BALL; MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN MATCH_MP_TAC MEASURABLE_ON_SUB THEN CONJ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] MEASURABLE_ON_SPIKE_SET) THEN EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[ETA_AX] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!m. ?k. measure((e:num->num->real^M->bool) k m) < d / &2 pow (m + 2)` MP_TAC THENL [GEN_TAC THEN MP_TAC(ISPEC `\n. (e:num->num->real^M->bool) n m` HAS_MEASURE_NESTED_INTERS) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [GEN_TAC THEN EXPAND_TAC "e" THEN REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ARITH_TAC; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)] THEN SUBGOAL_THEN `measure (INTERS {(e:num->num->real^M->bool) n m | n IN (:num)}) = &0` SUBST1_TAC THENL [MATCH_MP_TAC MEASURE_EQ_0 THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `t:real^M->bool` THEN ASM_REWRITE_TAC[INTERS_GSPEC; SUBSET; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_CASES_TAC `(x:real^M) IN t` THEN ASM_REWRITE_TAC[IN_DIFF] THEN EXPAND_TAC "e" THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; IN_DIFF] THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[LIM_SEQUENTIALLY; NOT_FORALL_THM; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `inv(&m + &1)`) THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &m + &1`] THEN REWRITE_TAC[DE_MORGAN_THM; real_ge; REAL_NOT_LE] THEN MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[LIM_SEQUENTIALLY; LIFT_NUM; DIST_0; NORM_LIFT] THEN DISCH_THEN(MP_TAC o SPEC `d / &2 pow (m + 2)`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_POW2] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_THEN(MP_TAC o SPEC `N:num`) THEN REWRITE_TAC[LE_REFL] THEN REAL_ARITH_TAC; REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `k:num->num` THEN DISCH_TAC] THEN EXISTS_TAC `UNIONS {(e:num->num->real^M->bool) (k m) m | m IN (:num)}` THEN CONJ_TAC THENL [REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN EXPAND_TAC "e" THEN REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`\m. (e:num->num->real^M->bool) (k m) m`; `d / &2`] MEASURE_COUNTABLE_UNIONS_LE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [X_GEN_TAC `n:num`; ASM_MESON_TAC[REAL_ARITH `&0 < d /\ x <= d / &2 ==> x < d`]] THEN TRANS_TAC REAL_LE_TRANS `sum(0..n) (\m. d / &2 pow (m + 2))` THEN ASM_SIMP_TAC[SUM_LE_NUMSEG; REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_POW_ADD; real_div; REAL_INV_POW; REAL_MUL_ASSOC] THEN REWRITE_TAC[SUM_RMUL; SUM_LMUL; SUM_GP; CONJUNCT1 LT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> (&1 - x) / (&1 / &2) * &1 / &4 <= &1 / &2`) THEN MATCH_MP_TAC REAL_POW_LE THEN CONV_TAC REAL_RAT_REDUCE_CONV; X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN EXISTS_TAC `(k:num->num) m` THEN MAP_EVERY X_GEN_TAC [`n:num`; `x:real^M`] THEN EXPAND_TAC "e" THEN REWRITE_TAC[IN_DIFF; UNIONS_GSPEC; IN_ELIM_THM] THEN REWRITE_TAC[NOT_EXISTS_THM; IN_UNIV] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPECL [`m:num`; `n:num`]) THEN ASM_REWRITE_TAC[REAL_NOT_LE; real_ge] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `i < e ==> m <= i ==> d < m ==> d < e`)) THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]);; (* ------------------------------------------------------------------------- *) (* The Lebesgue differentiation theorem. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_DIFFERENTIATION_THEOREM_COMPACT = prove (`!f:real^1->real^N a b. f has_bounded_variation_on interval[a,b] ==> negligible {x | x IN interval[a,b] /\ ~(f differentiable at x)}`, let lemma0 = prove (`k <= y - x ==> &0 < k ==> ?q. rational q /\ k / &3 < q - x /\ k / &3 < y - q`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(x + y) / &2`; `k / &6`] RATIONAL_APPROXIMATION) THEN ANTS_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]] THEN ASM_REAL_ARITH_TAC) in let lemma1 = prove (`!f:real^1->real^1 a b. f has_bounded_variation_on interval[a,b] ==> ?t. negligible t /\ !x. x IN interval[a,b] DIFF t ==> ?B. &0 < B /\ eventually (\y. norm(f y - f x) <= B * norm(y - x)) (at x)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?t. negligible t /\ !x. x IN interval[a,b] DIFF t /\ (f:real^1->real^1) continuous at x ==> ?B. &0 < B /\ eventually (\y. norm(f y - f x) <= B * norm(y - x)) (at x)` STRIP_ASSUME_TAC THENL [ALL_TAC; EXISTS_TAC `t UNION {x | x IN interval[a,b] /\ ~((f:real^1->real^1) continuous at x)}` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; SET_RULE `x IN i DIFF (t UNION {x | x IN i /\ ~P x}) <=> x IN i DIFF t /\ P x`] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE THEN MATCH_MP_TAC HAS_BOUNDED_VARIATION_COUNTABLE_DISCONTINUITIES THEN ASM_REWRITE_TAC[IS_INTERVAL_INTERVAL]] THEN ABBREV_TAC `t = {x | x IN interval(a,b) /\ (f:real^1->real^1) continuous at x /\ ~(?B. &0 < B /\ eventually (\y. norm(f y - f x) <= B * norm (y - x)) (at x))}` THEN EXISTS_TAC `{a:real^1,b} UNION t` THEN CONJ_TAC THENL [REWRITE_TAC[NEGLIGIBLE_UNION_EQ; NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY]; EXPAND_TAC "t" THEN REWRITE_TAC[GSYM OPEN_CLOSED_INTERVAL_1; SET_RULE `s DIFF (t UNION u) = s DIFF t DIFF u`] THEN EXPAND_TAC "t" THEN SET_TAC[]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [has_bounded_variation_on]) THEN REWRITE_TAC[has_bounded_setvariation_on] THEN DISCH_THEN(X_CHOOSE_TAC `B:real`) THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN SUBGOAL_THEN `!x. x IN t ==> ?u v. u IN interval[a,b] /\ v IN interval[a,b] /\ x IN interval(u,v) /\ (&3 * (abs B + &1) / e) * norm(v - u) <= norm((f:real^1->real^1) u - f v)` MP_TAC THENL [ABBREV_TAC `M = &3 * (abs B + &1) / e` THEN SUBGOAL_THEN `&0 < M` ASSUME_TAC THENL [EXPAND_TAC "M" THEN REWRITE_TAC[REAL_ARITH `&0 < &3 * M <=> &0 < M`] THEN MATCH_MP_TAC REAL_LT_DIV THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `x:real^1` THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_ELIM_THM; NOT_EXISTS_THM] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN(MP_TAC o SPEC `&3 * M`) THEN ASM_REWRITE_TAC[REAL_ARITH `&0 < &3 * M <=> &0 < M`] THEN REWRITE_TAC[EVENTUALLY_AT; NOT_EXISTS_THM] THEN MP_TAC(ISPEC `interval(a:real^1,b)` OPEN_CONTAINS_BALL) THEN REWRITE_TAC[OPEN_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; IN_BALL] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LE] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^1` MP_TAC) THEN ASM_CASES_TAC `y:real^1 = x` THEN ASM_REWRITE_TAC[DIST_REFL; REAL_LT_REFL] THEN STRIP_TAC THEN SUBGOAL_THEN `y IN interval(a:real^1,b)` ASSUME_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN SUBGOAL_THEN `x IN interval[a:real^1,b] /\ y IN interval[a:real^1,b]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET]; ALL_TAC] THEN MP_TAC(SPECL [`drop x`; `drop y`] REAL_LT_TOTAL) THEN ASM_REWRITE_TAC[DROP_EQ] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_at]) THEN DISCH_THEN(MP_TAC o SPEC `M * norm(y - x:real^1)`) THEN ASM_SIMP_TAC[REAL_LT_MUL; NORM_POS_LT; VECTOR_SUB_EQ] THEN DISCH_THEN(X_CHOOSE_THEN `d':real` STRIP_ASSUME_TAC) THENL [ABBREV_TAC `u = x - lift(min (norm(y - x)) (min d d') / &2)` THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `u:real^1`)) THEN EXPAND_TAC "u" THEN REWRITE_TAC[NORM_ARITH `dist(x:real^1,x - a) = norm a`; NORM_ARITH `dist(x - a:real^1,x) = norm a`] THEN REWRITE_TAC[NORM_LIFT] THEN ASM_REWRITE_TAC[] THEN REPEAT(ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[dist]) THEN ASM_REAL_ARITH_TAC; DISCH_TAC]) THEN MAP_EVERY EXISTS_TAC [`u:real^1`; `y:real^1`] THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] INTERVAL_OPEN_SUBSET_CLOSED] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN EXPAND_TAC "u" THEN REWRITE_TAC[DROP_SUB; LIFT_DROP] THEN RULE_ASSUM_TAC(REWRITE_RULE[dist]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH `a < norm(y - x:real^N) ==> !b. dist(u,x) < b /\ c <= a - b ==> c <= norm(u - y)`)) THEN EXISTS_TAC `M * norm(y - x:real^1)` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_ARITH `a <= (&3 * M) * y - M * y <=> a <= M * &2 * y`] THEN MATCH_MP_TAC(NORM_ARITH `norm(x - u:real^N) <= norm(y - x) ==> norm(y - u) <= &2 * norm(y - x)`) THEN EXPAND_TAC "u" THEN REWRITE_TAC[NORM_LIFT; VECTOR_ARITH `x - (x - l):real^N = l`] THEN RULE_ASSUM_TAC(REWRITE_RULE[dist]) THEN ASM_REAL_ARITH_TAC; ABBREV_TAC `u = x + lift(min (norm(y - x)) (min d d') / &2)` THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `u:real^1`)) THEN EXPAND_TAC "u" THEN REWRITE_TAC[NORM_ARITH `dist(x:real^1,x + a) = norm a`; NORM_ARITH `dist(x + a:real^1,x) = norm a`] THEN REWRITE_TAC[NORM_LIFT] THEN ASM_REWRITE_TAC[] THEN REPEAT(ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[dist]) THEN ASM_REAL_ARITH_TAC; DISCH_TAC]) THEN MAP_EVERY EXISTS_TAC [`y:real^1`; `u:real^1`] THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] INTERVAL_OPEN_SUBSET_CLOSED] THEN CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN EXPAND_TAC "u" THEN REWRITE_TAC[DROP_ADD; LIFT_DROP] THEN RULE_ASSUM_TAC(REWRITE_RULE[dist]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH `a < norm(y - x:real^N) ==> !b. dist(u,x) < b /\ c <= a - b ==> c <= norm(y - u)`)) THEN EXISTS_TAC `M * norm(y - x:real^1)` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_ARITH `a <= (&3 * M) * y - M * y <=> a <= M * &2 * y`] THEN MATCH_MP_TAC(NORM_ARITH `norm(x - u:real^N) <= norm(y - x) ==> norm(u - y) <= &2 * norm(y - x)`) THEN EXPAND_TAC "u" THEN REWRITE_TAC[NORM_LIFT; NORM_NEG; VECTOR_ARITH `x - (x + l):real^N = --l`] THEN RULE_ASSUM_TAC(REWRITE_RULE[dist]) THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^1->real^1`; `v:real^1->real^1`] THEN DISCH_TAC THEN MP_TAC(ISPEC `IMAGE (\x:real^1. interval(u x:real^1,v x)) t` LINDELOF) THEN REWRITE_TAC[FORALL_IN_IMAGE; OPEN_INTERVAL] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_COUNTABLE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^1->bool` STRIP_ASSUME_TAC) THEN ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x /\ R x) <=> ~(!x. Q x /\ P x ==> ~R x)`] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN SUBGOAL_THEN `?p. FINITE p /\ p SUBSET IMAGE (\x:real^1. interval[u x:real^1,v x]) c /\ e < measure(UNIONS p)` MP_TAC THENL [ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x /\ R x) <=> ~(!x. Q x /\ P x ==> ~R x)`] THEN REWRITE_TAC[REAL_NOT_LT] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN))) THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE; MEASURABLE_INTERVAL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ e < x ==> x <= e ==> F`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `UNIONS(IMAGE f c) = u ==> t SUBSET u /\ (!x. x IN c ==> f x SUBSET g x) ==> t SUBSET UNIONS(IMAGE g c)`)) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED] THEN ASM SET_TAC[]; REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `p:real^1->bool` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPEC `IMAGE (\x:real^1. interval[u x:real^1,v x]) p` AUSTIN_LEMMA) THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE; DIMINDEX_1; FORALL_1] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^1->bool)->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `e < m ==> a < e / &3 ==> a >= m / &3 pow 1 ==> F`)) THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE o lhand o snd) THEN ANTS_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET; FINITE_IMAGE]; ALL_TAC] THEN ASM_MESON_TAC[SUBSET; IN_IMAGE; MEASURABLE_INTERVAL]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)] THEN SUBGOAL_THEN `e / &3 = (abs B + &1) / (&3 * (abs B + &1) / e)` SUBST1_TAC THENL [UNDISCH_TAC `&0 < e` THEN CONV_TAC REAL_FIELD; ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_ARITH `&0 < &3 * M <=> &0 < M`; REAL_LT_DIV; REAL_ARITH `&0 < abs B + &1`]] THEN MATCH_MP_TAC(REAL_ARITH `x <= b ==> x < abs b + &1`) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`D:(real^1->bool)->bool`; `UNIONS D:real^1->bool`]) THEN ANTS_TAC THENL [REWRITE_TAC[division_of; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET; FINITE_IMAGE]; ALL_TAC] THEN SIMP_TAC[SET_RULE `s IN t ==> s SUBSET UNIONS t`] THEN REPEAT CONJ_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> P(f x)) ==> !x. x IN s ==> P x`)) THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY]; MESON_TAC[]] THEN ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET]; REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `interior s SUBSET s /\ interior t SUBSET t /\ DISJOINT s t ==> interior s INTER interior t = {}`) THEN REWRITE_TAC[INTERIOR_SUBSET] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM SET_TAC[]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> f x SUBSET u) ==> UNIONS s SUBSET u`)) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTERVAL_1]] THEN REAL_ARITH_TAC]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN REWRITE_TAC[GSYM SUM_RMUL] THEN MATCH_MP_TAC SUM_LE THEN CONJ_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET; FINITE_IMAGE]; ALL_TAC] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> P(f x)) ==> !x. x IN s ==> P x`)) THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTERVAL_1]] THEN STRIP_TAC THEN SUBGOAL_THEN `drop((u:real^1->real^1) x) <= drop(v x)` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[INTERVAL_LOWERBOUND_1; INTERVAL_UPPERBOUND_1] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH `x <= norm(u - v:real^N) ==> a <= x ==> a <= norm(v - u)`)) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC(REAL_RING `x:real = y ==> x * c = c * y`) THEN ASM_REWRITE_TAC[MEASURE_INTERVAL_1; NORM_REAL] THEN REWRITE_TAC[GSYM drop; DROP_SUB] THEN ASM_REAL_ARITH_TAC]) in let lemma2 = prove (`!f a b k. f has_bounded_variation_on interval[a,b] /\ drop a < drop b /\ &0 < k ==> negligible {x | x IN interval[a,b] /\ !s. open s /\ x IN s ==> (?u v. u IN interval[a,b] /\ u IN s /\ v IN interval[a,b] /\ v IN s /\ x IN interval(u,v) /\ k <= (drop(f v) - drop(f u)) / (drop v - drop u)) /\ (?u v. u IN interval[a,b] /\ u IN s /\ v IN interval[a,b] /\ v IN s /\ x IN interval(u,v) /\ (drop(f v) - drop(f u)) / (drop v - drop u) <= --k)}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC(MESON[] `(!t. s = t ==> negligible t) ==> negligible s`) THEN X_GEN_TAC `t':real^1->bool` THEN DISCH_TAC THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o CONJUNCT2 o MATCH_MP HAS_BOUNDED_VARIATION_WORKS_ON_INTERVAL) THEN DISCH_THEN(MP_TAC o SPEC `vector_variation (interval[a,b]) (f:real^1->real^1) - k * e / &3`) THEN ASM_SIMP_TAC[REAL_LT_MUL; REAL_ARITH `v <= v - e <=> ~(&0 < e)`; REAL_ARITH `&0 < x / &3 <=> &0 < x`] THEN REWRITE_TAC[NOT_FORALL_THM; LEFT_IMP_EXISTS_THM; NOT_IMP; REAL_NOT_LE] THEN X_GEN_TAC `D:(real^1->bool)->bool` THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP DIVISION_OF_FINITE) THEN ABBREV_TAC `t:real^1->bool = t' DIFF UNIONS {frontier i | i IN D}` THEN SUBGOAL_THEN `?c:real^1->bool. t SUBSET c /\ measurable c /\ measure c <= e` MP_TAC THENL [ALL_TAC; DISCH_THEN(X_CHOOSE_THEN `c:real^1->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `c UNION UNIONS {frontier i:real^1->bool | i IN D}` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `negligible(UNIONS {frontier i:real^1->bool | i IN D})` ASSUME_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION_NONEMPTY th]) THEN REPEAT STRIP_TAC THEN REWRITE_TAC[FRONTIER_CLOSED_INTERVAL; OPEN_CLOSED_INTERVAL_1] THEN ASM_SIMP_TAC[ENDS_IN_INTERVAL; INSERT_SUBSET; EMPTY_SUBSET; SET_RULE `t SUBSET s ==> s DIFF (s DIFF t) = t`] THEN REWRITE_TAC[NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY]; ASM_SIMP_TAC[MEASURABLE_UNION; NEGLIGIBLE_IMP_MEASURABLE] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNION_LE o lhand o snd) THEN RULE_ASSUM_TAC(REWRITE_RULE[NEGLIGIBLE_EQ_MEASURE_0]) THEN ASM_SIMP_TAC[MEASURABLE_UNION] THEN ASM_REAL_ARITH_TAC]] THEN SUBGOAL_THEN `!x. x IN t ==> ?c d u v. interval[c,d] IN D /\ x IN interval(c,d) /\ u IN interval(c,d) /\ v IN interval(c,d) /\ x IN interval(u,v) /\ (drop(f c) <= drop(f d) ==> drop(f v) - drop(f u) <= --k * (drop v - drop u)) /\ (drop(f d) < drop(f c) ==> k * (drop v - drop u) <= drop(f v) - drop(f u))` MP_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[IN_DIFF] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o last o CONJUNCTS o GEN_REWRITE_RULE I [division_of]) THEN DISCH_THEN(MP_TAC o SPEC `x:real^1` o MATCH_MP (SET_RULE `s = t ==> !x. x IN t ==> x IN s`)) THEN ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_UNIONS]] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION_NONEMPTY th]) THEN MAP_EVERY X_GEN_TAC [`c:real^1`; `d:real^1`] THEN REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`c:real^1`; `d:real^1`] THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `interval[c:real^1,d]` o MATCH_MP (SET_RULE `~(x IN UNIONS {f y | y IN s}) ==> !a. a IN s ==> ~(x IN f a)`)) THEN ASM_REWRITE_TAC[FRONTIER_CLOSED_INTERVAL; IN_DIFF] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `(x:real^1) IN t'` THEN EXPAND_TAC "t'" THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `interval(c:real^1,d)`)) THEN ASM_REWRITE_TAC[OPEN_INTERVAL] THEN DISJ_CASES_TAC(SPECL [`drop(f(c:real^1))`; `drop(f(d:real^1))`] REAL_LET_TOTAL) THEN ASM_REWRITE_TAC[] THENL [DISCH_THEN(MP_TAC o CONJUNCT2); DISCH_THEN(MP_TAC o CONJUNCT1)] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^1` THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^1` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN TRY(DISCH_TAC THEN ASM_REAL_ARITH_TAC) THEN (SUBGOAL_THEN `drop u < drop v` ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC]) THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_SUB_LT; GSYM REAL_LE_RDIV_EQ]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:real^1->real^1`; `d:real^1->real^1`; `u:real^1->real^1`; `v:real^1->real^1`] THEN DISCH_TAC THEN MP_TAC(ISPEC `IMAGE (\x:real^1. interval(u x:real^1,v x)) t` LINDELOF) THEN REWRITE_TAC[FORALL_IN_IMAGE; OPEN_INTERVAL] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[EXISTS_COUNTABLE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `c:real^1->bool` STRIP_ASSUME_TAC) THEN ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x /\ R x) <=> ~(!x. Q x /\ P x ==> ~R x)`] THEN REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN SUBGOAL_THEN `?p. FINITE p /\ p SUBSET IMAGE (\x:real^1. interval[u x:real^1,v x]) c /\ e < measure(UNIONS p)` MP_TAC THENL [ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x /\ R x) <=> ~(!x. Q x /\ P x ==> ~R x)`] THEN REWRITE_TAC[REAL_NOT_LT] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN))) THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE; MEASURABLE_INTERVAL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ e < x ==> x <= e ==> F`) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `UNIONS(IMAGE f c) = u ==> t SUBSET u /\ (!x. x IN c ==> f x SUBSET g x) ==> t SUBSET UNIONS(IMAGE g c)`)) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED] THEN ASM SET_TAC[]; REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `p:real^1->bool` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPEC `IMAGE (\x:real^1. interval[u x:real^1,v x]) p` AUSTIN_LEMMA) THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE; DIMINDEX_1; FORALL_1] THEN REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `d:(real^1->bool)->bool` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `e < m ==> a < e / &3 ==> a >= m / &3 pow 1 ==> F`)) THEN SUBGOAL_THEN `UNIONS d = UNIONS {UNIONS {i:real^1->bool | i IN d /\ i SUBSET j} | j IN D}` SUBST1_TAC THENL [REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM] THEN REWRITE_TAC[EXTENSION; IN_UNIONS; IN_ELIM_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN SUBGOAL_THEN `!i:real^1->bool. i IN d ==> ?j. j IN D /\ i SUBSET j` MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> P(f x)) ==> !y. y IN s ==> P y`)) THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN STRIP_TAC THEN EXISTS_TAC `interval[(c:real^1->real^1) x,d x]` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC SUBSET_TRANS `interval((c:real^1->real^1) x,d x)` THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE o lhand o snd) THEN ANTS_TAC THENL [ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `i:real^1->bool` THEN DISCH_TAC THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_RESTRICT THEN ASM_MESON_TAC[FINITE_SUBSET; FINITE_IMAGE]; REWRITE_TAC[FORALL_IN_GSPEC; IMP_CONJ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> P(f x)) ==> !y. y IN s ==> P y`)) THEN REWRITE_TAC[MEASURABLE_INTERVAL]]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)] THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_IMAGE_LE o lhand o snd) THEN ASM_REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL [X_GEN_TAC `i:real^1->bool` THEN DISCH_TAC THEN MATCH_MP_TAC MEASURE_POS_LE THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_RESTRICT THEN ASM_MESON_TAC[FINITE_SUBSET; FINITE_IMAGE]; REWRITE_TAC[FORALL_IN_GSPEC; IMP_CONJ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> P(f x)) ==> !y. y IN s ==> P y`)) THEN REWRITE_TAC[MEASURABLE_INTERVAL]]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS)] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `v - e < s ==> v - s < e`)) THEN MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`; `D:(real^1->bool)->bool`] VECTOR_VARIATION_ON_DIVISION) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[GSYM SUM_SUB] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS) THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; GSYM SUM_RMUL] THEN MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION_NONEMPTY th]) THEN SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY] THEN MAP_EVERY X_GEN_TAC [`l:real^1`; `r:real^1`] THEN STRIP_TAC THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN SUBGOAL_THEN `FINITE(d:(real^1->bool)->bool)` ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET; FINITE_IMAGE]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE o lhand o snd) THEN ASM_SIMP_TAC[FINITE_RESTRICT; FORALL_IN_GSPEC] THEN ANTS_TAC THENL [REWRITE_TAC[IMP_CONJ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> P(f x)) ==> !y. y IN s ==> P y`)) THEN REWRITE_TAC[MEASURABLE_INTERVAL]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ; GSYM SUM_RMUL] THEN ABBREV_TAC `d' = {i | i IN d /\ i SUBSET interval[l:real^1,r]}` THEN SUBGOAL_THEN `FINITE(d':(real^1->bool)->bool)` ASSUME_TAC THENL [EXPAND_TAC "d'" THEN ASM_SIMP_TAC[FINITE_RESTRICT]; ALL_TAC] THEN SUBGOAL_THEN `pairwise DISJOINT (d':(real^1->bool)->bool)` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_MONO)) THEN EXPAND_TAC "d'" THEN REWRITE_TAC[SUBSET_RESTRICT]; ALL_TAC] THEN MP_TAC(ISPECL [`d':(real^1->bool)->bool`; `l:real^1`; `r:real^1`] PARTIAL_DIVISION_EXTEND_INTERVAL) THEN ANTS_TAC THENL [EXPAND_TAC "d'" THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[division_of] THEN ASM_SIMP_TAC[FINITE_RESTRICT] THEN REWRITE_TAC[FORALL_AND_THM; TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN REPEAT CONJ_TAC THENL [SET_TAC[]; EXPAND_TAC "d'" THEN REWRITE_TAC[IN_ELIM_THM; IMP_CONJ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET IMAGE f t ==> (!x. x IN t ==> P(f x)) ==> !y. y IN s ==> P y`)) THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN REPEAT DISCH_TAC THEN EXISTS_TAC `x:real^1` THEN MATCH_MP_TAC(REWRITE_RULE[SUBSET] INTERVAL_OPEN_SUBSET_CLOSED) THEN ASM SET_TAC[]; ASM SET_TAC[]; REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `interior s SUBSET s /\ interior t SUBSET t /\ DISJOINT s t ==> interior s INTER interior t = {}`) THEN REWRITE_TAC[INTERIOR_SUBSET] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM_MESON_TAC[]]; DISCH_THEN(X_CHOOSE_THEN `d'':(real^1->bool)->bool` STRIP_ASSUME_TAC)] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP DIVISION_OF_FINITE) THEN REWRITE_TAC[NORM_1; DROP_SUB] THEN REWRITE_TAC[REAL_ARITH `abs(r - l) = if l <= r then r - l else l - r`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [TRANS_TAC REAL_LE_TRANS `sum d'' (\i. abs(drop(f(interval_upperbound i)) - drop(f(interval_lowerbound i))) - (drop((f:real^1->real^1)(interval_upperbound i)) - drop(f(interval_lowerbound i))))` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE_INCLUDED THEN ASM_REWRITE_TAC[REAL_ARITH `&0 <= abs x - x`] THEN EXISTS_TAC `\x:real^1->bool. x` THEN REWRITE_TAC[TAUT `p /\ a = b /\ q <=> a = b /\ p /\ q`] THEN REWRITE_TAC[UNWIND_THM2] THEN MATCH_MP_TAC(SET_RULE `d' SUBSET d'' /\ (!x. x IN d'' ==> x IN d' ==> P x) ==> !x. x IN d' ==> x IN d'' /\ P x`) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION_NONEMPTY th]) THEN SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY] THEN MAP_EVERY X_GEN_TAC [`w:real^1`;`z:real^1`] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?x:real^1. x IN p /\ interval[w:real^1,z] = interval[u x,v x]` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[EQ_INTERVAL] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^1` (CONJUNCTS_THEN2 MP_TAC (CONJUNCTS_THEN SUBST_ALL_TAC))) THEN DISCH_TAC THEN ASM_SIMP_TAC[MEASURE_INTERVAL_1; GSYM INTERVAL_NE_EMPTY_1] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= k * a /\ x <= --k * a ==> a * k <= abs x - x`) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; GSYM INTERVAL_NE_EMPTY_1; REAL_LT_IMP_LE] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC(ASSUME `D division_of interval[a:real^1,b]`) THEN REWRITE_TAC[division_of] THEN DISCH_THEN(MP_TAC o SPECL [`interval[l:real^1,r]`; `interval[(c:real^1->real^1) x,d x]`] o el 2 o CONJUNCTS) THEN ASM_REWRITE_TAC[EQ_INTERVAL] THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN MATCH_MP_TAC (SET_RULE `!u. u SUBSET t /\ ~(u = {}) ==> ~(t = {})`) THEN EXISTS_TAC `interval((u:real^1->real^1) x,v x)` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM(CONJUNCT1 INTERIOR_INTERVAL)] THEN MATCH_MP_TAC SUBSET_INTERIOR THEN ASM SET_TAC[]; REWRITE_TAC[INTERIOR_INTERVAL; SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]; W(MP_TAC o PART_MATCH (lhand o rand) SUM_SUB o lhand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(REAL_ARITH `x <= x' /\ y = y' ==> x - y <= x' - y'`) THEN CONJ_TAC THENL [MP_TAC(ISPECL [`f:real^1->real^1`; `interval[l:real^1,r]`] HAS_BOUNDED_VARIATION_WORKS) THEN REWRITE_TAC[NORM_1; DROP_SUB] THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_REFL]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_ON_SUBSET)) THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM SET_TAC[]; ASM_SIMP_TAC[GSYM DROP_SUB; GSYM(REWRITE_RULE[o_DEF] DROP_VSUM)] THEN AP_TERM_TAC THEN MATCH_MP_TAC ADDITIVE_DIVISION_1 THEN ASM_REWRITE_TAC[GSYM INTERVAL_NE_EMPTY_1]]]; TRANS_TAC REAL_LE_TRANS `sum d'' (\i. abs(drop(f(interval_upperbound i)) - drop(f(interval_lowerbound i))) + (drop((f:real^1->real^1)(interval_upperbound i)) - drop(f(interval_lowerbound i))))` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE_INCLUDED THEN ASM_REWRITE_TAC[REAL_ARITH `&0 <= abs x + x`] THEN EXISTS_TAC `\x:real^1->bool. x` THEN REWRITE_TAC[TAUT `p /\ a = b /\ q <=> a = b /\ p /\ q`] THEN REWRITE_TAC[UNWIND_THM2] THEN MATCH_MP_TAC(SET_RULE `d' SUBSET d'' /\ (!x. x IN d'' ==> x IN d' ==> P x) ==> !x. x IN d' ==> x IN d'' /\ P x`) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION_NONEMPTY th]) THEN SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY] THEN MAP_EVERY X_GEN_TAC [`w:real^1`;`z:real^1`] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?x:real^1. x IN p /\ interval[w:real^1,z] = interval[u x,v x]` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[EQ_INTERVAL] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^1` (CONJUNCTS_THEN2 MP_TAC (CONJUNCTS_THEN SUBST_ALL_TAC))) THEN DISCH_TAC THEN ASM_SIMP_TAC[MEASURE_INTERVAL_1; GSYM INTERVAL_NE_EMPTY_1] THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN MATCH_MP_TAC(REAL_ARITH `k * a <= x ==> a * k <= abs x + x`) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC(ASSUME `D division_of interval[a:real^1,b]`) THEN REWRITE_TAC[division_of] THEN DISCH_THEN(MP_TAC o SPECL [`interval[l:real^1,r]`; `interval[(c:real^1->real^1) x,d x]`] o el 2 o CONJUNCTS) THEN ASM_REWRITE_TAC[EQ_INTERVAL; GSYM REAL_NOT_LE] THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN MATCH_MP_TAC (SET_RULE `!u. u SUBSET t /\ ~(u = {}) ==> ~(t = {})`) THEN EXISTS_TAC `interval((u:real^1->real^1) x,v x)` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM(CONJUNCT1 INTERIOR_INTERVAL)] THEN MATCH_MP_TAC SUBSET_INTERIOR THEN ASM SET_TAC[]; REWRITE_TAC[INTERIOR_INTERVAL; SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]; ASM_SIMP_TAC[SUM_ADD] THEN MATCH_MP_TAC(REAL_ARITH `x <= x' /\ y = --y' ==> x + y <= x' - y'`) THEN CONJ_TAC THENL [MP_TAC(ISPECL [`f:real^1->real^1`; `interval[l:real^1,r]`] HAS_BOUNDED_VARIATION_WORKS) THEN REWRITE_TAC[NORM_1; DROP_SUB] THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_REFL]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_ON_SUBSET)) THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM SET_TAC[]; REWRITE_TAC[REAL_NEG_SUB] THEN ASM_SIMP_TAC[GSYM DROP_SUB; GSYM(REWRITE_RULE[o_DEF] DROP_VSUM)] THEN AP_TERM_TAC THEN MATCH_MP_TAC ADDITIVE_DIVISION_1 THEN ASM_REWRITE_TAC[GSYM INTERVAL_NE_EMPTY_1]]]]) in let lemma3 = prove (`!f a b k. f has_bounded_variation_on interval[a,b] /\ drop a < drop b /\ &0 < k ==> negligible {x | x IN interval[a,b] /\ !n. ?u v. u IN ball(x,inv(&n + &1)) /\ u IN interval[a,b] /\ v IN ball(x,inv(&n + &1)) /\ v IN interval[a,b] /\ ~(u = x) /\ ~(v = x) /\ k <= (drop(f v) - drop(f x)) / (drop v - drop x) /\ (drop(f u) - drop(f x)) / (drop u - drop x) <= --k}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `k / &2` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] lemma2)) THEN ASM_REWRITE_TAC[REAL_HALF] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP NEGLIGIBLE_COUNTABLE o REWRITE_RULE[IS_INTERVAL_INTERVAL] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_COUNTABLE_DISCONTINUITIES)) THEN SUBGOAL_THEN `negligible {a:real^1,b}` MP_TAC THENL [REWRITE_TAC[NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY]; ALL_TAC] THEN REWRITE_TAC[GSYM NEGLIGIBLE_UNION_EQ; IMP_IMP] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNION] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[TAUT `(p \/ ~q) \/ r <=> ~p /\ q ==> r`] THEN REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN X_GEN_TAC `s:real^1->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `?n. ball(x:real^1,inv (&n + &1)) SUBSET s INTER interval(a,b)` STRIP_ASSUME_TAC THENL [MP_TAC(ISPEC `s INTER interval(a:real^1,b)` OPEN_CONTAINS_BALL) THEN ASM_SIMP_TAC[OPEN_INTER; OPEN_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `x:real^1`) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[OPEN_CLOSED_INTERVAL_1; IN_DIFF; IN_INSERT; IN_INTER; NOT_IN_EMPTY]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `e:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN TRANS_TAC SUBSET_TRANS `ball(x:real^1,e)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_BALL THEN TRANS_TAC REAL_LE_TRANS `inv(&n)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC; FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [SUBSET_INTER]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `s SUBSET interval(a:real^1,b) ==> interval(a,b) SUBSET interval[a,b] ==> s SUBSET interval[a,b]`)) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN STRIP_TAC] THEN CONJ_TAC THENL [MP_TAC(SPECL [`drop v`; `drop x`] REAL_LT_TOTAL) THEN ASM_REWRITE_TAC[DROP_EQ] THEN STRIP_TAC THENL [EXISTS_TAC `v:real^1` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(\y. lift((drop(f v) - drop(f y)) / (drop v - drop y))) continuous at x` MP_TAC THENL [REWRITE_TAC[real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN ASM_SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_AT_INV) THEN ASM_SIMP_TAC[REAL_SUB_0; DROP_EQ; LIFT_SUB; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST; CONTINUOUS_AT_ID]; REWRITE_TAC[continuous_at; DIST_LIFT] THEN DISCH_THEN(MP_TAC o SPEC `k / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC] THEN EXISTS_TAC `x + lift(min d (inv(&n + &1)) / &2)` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x:real^N,x + r) = norm r`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < y`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_ADD; REAL_LT_ADDR; LIFT_DROP] THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; ONCE_REWRITE_TAC[REAL_ARITH `x / y:real = --x * --(inv y)`] THEN REWRITE_TAC[GSYM REAL_INV_NEG; REAL_NEG_SUB] THEN REWRITE_TAC[GSYM real_div] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `k <= y ==> abs(y' - y) < k / &2 ==> --k / -- &2 <= y'`)) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x + y:real^N,x) = norm y`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < x`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC]; GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN EXISTS_TAC `v:real^1` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(\y. lift((drop(f v) - drop(f y)) / (drop v - drop y))) continuous at x` MP_TAC THENL [REWRITE_TAC[real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN ASM_SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_AT_INV) THEN ASM_SIMP_TAC[REAL_SUB_0; DROP_EQ; LIFT_SUB; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST; CONTINUOUS_AT_ID]; REWRITE_TAC[continuous_at; DIST_LIFT] THEN DISCH_THEN(MP_TAC o SPEC `k / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC] THEN EXISTS_TAC `x - lift(min d (inv(&n + &1)) / &2)` THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> r /\ (p /\ q) /\ s`] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x:real^N,x - r) = norm r`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < y`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; LIFT_DROP] THEN REWRITE_TAC[REAL_ARITH `x - a < x <=> &0 < a`] THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `k <= y ==> abs(y' - y) < k / &2 ==> k / &2 <= y'`)) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x - y:real^N,x) = norm y`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < x`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC]]; MP_TAC(SPECL [`drop u`; `drop x`] REAL_LT_TOTAL) THEN ASM_REWRITE_TAC[DROP_EQ] THEN STRIP_TAC THENL [EXISTS_TAC `u:real^1` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(\y. lift((drop(f u) - drop(f y)) / (drop u - drop y))) continuous at x` MP_TAC THENL [REWRITE_TAC[real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN ASM_SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_AT_INV) THEN ASM_SIMP_TAC[REAL_SUB_0; DROP_EQ; LIFT_SUB; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST; CONTINUOUS_AT_ID]; REWRITE_TAC[continuous_at; DIST_LIFT] THEN DISCH_THEN(MP_TAC o SPEC `k / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC] THEN EXISTS_TAC `x + lift(min d (inv(&n + &1)) / &2)` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x:real^N,x + r) = norm r`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < y`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_ADD; REAL_LT_ADDR; LIFT_DROP] THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; ONCE_REWRITE_TAC[REAL_ARITH `x / y:real = --x * --(inv y)`] THEN REWRITE_TAC[GSYM REAL_INV_NEG; REAL_NEG_SUB] THEN REWRITE_TAC[GSYM real_div] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `y <= --k ==> abs(y' - y) < k / &2 ==> y' <= --(--k / -- &2)`)) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x + y:real^N,x) = norm y`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < x`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC]; GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN EXISTS_TAC `u:real^1` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(\y. lift((drop(f u) - drop(f y)) / (drop u - drop y))) continuous at x` MP_TAC THENL [REWRITE_TAC[real_div; LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN ASM_SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_AT_INV) THEN ASM_SIMP_TAC[REAL_SUB_0; DROP_EQ; LIFT_SUB; LIFT_DROP] THEN SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST; CONTINUOUS_AT_ID]; REWRITE_TAC[continuous_at; DIST_LIFT] THEN DISCH_THEN(MP_TAC o SPEC `k / &2`) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC] THEN EXISTS_TAC `x - lift(min d (inv(&n + &1)) / &2)` THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s <=> r /\ (p /\ q) /\ s`] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x:real^N,x - r) = norm r`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < y`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_SUB; LIFT_DROP] THEN REWRITE_TAC[REAL_ARITH `x - a < x <=> &0 < a`] THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `y <= --k ==> abs(y' - y) < k / &2 ==> y' <= --(k / &2)`)) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_BALL; NORM_ARITH `dist(x - y:real^N,x) = norm y`] THEN REWRITE_TAC[NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> abs(min x y / &2) < x`) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC]]]) in let lemma4 = prove (`!f a b k. f has_bounded_variation_on interval[a,b] /\ drop a < drop b /\ &0 < k ==> negligible {x | x IN interval[a,b] /\ !n. ?u v. u IN ball(x,inv(&n + &1)) /\ u IN interval[a,b] /\ v IN ball(x,inv(&n + &1)) /\ v IN interval[a,b] /\ ~(u = x) /\ ~(v = x) /\ k <= (drop(f v) - drop(f x)) / (drop v - drop x) - (drop(f u) - drop(f x)) / (drop u - drop x)}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP lemma1) THEN DISCH_THEN(X_CHOOSE_THEN `u:real^1->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `u UNION UNIONS {{x | x IN interval[a,b] /\ !n. ?u v. u IN ball(x,inv(&n + &1)) /\ u IN interval[a,b] /\ v IN ball(x,inv(&n + &1)) /\ v IN interval[a,b] /\ ~(u = x) /\ ~(v = x) /\ k / &3 <= (drop(f v) - drop(f x)) / (drop v - drop x) - q /\ (drop(f u) - drop(f x)) / (drop u - drop x) - q <= --(k / &3)} | q IN rational}` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN SIMP_TAC[COUNTABLE_RATIONAL; COUNTABLE_IMAGE; SIMPLE_IMAGE] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN X_GEN_TAC `q:real` THEN DISCH_THEN(K ALL_TAC) THEN MP_TAC(ISPECL [`\x:real^1. f(x) - q % x`; `a:real^1`; `b:real^1`; `k / &3`] lemma3) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN MATCH_MP_TAC HAS_BOUNDED_VARIATION_ON_SUB THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HAS_BOUNDED_VARIATION_ON_CMUL THEN SIMP_TAC[HAS_BOUNDED_VARIATION_ON_ID; BOUNDED_INTERVAL]; REWRITE_TAC[DROP_SUB; DROP_CMUL; REAL_ARITH `((a - q * a') - (b - q * b')) / i:real = (a - b) / i - q * (a' - b') / i`] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r /\ s /\ t /\ u /\ v <=> ~(t /\ u ==> ~(p /\ q /\ r /\ s /\ v))`] THEN SIMP_TAC[REAL_DIV_REFL; DROP_EQ; REAL_SUB_0; REAL_MUL_RID]]; GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; IN_UNION] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN ASM_CASES_TAC `(x:real^1) IN u` THEN ASM_REWRITE_TAC[]] THEN SUBGOAL_THEN `?N B. &0 < B /\ !n u. N <= n /\ u IN ball(x,inv(&n + &1)) /\ ~(u = x) ==> abs((drop(f u) - drop(f x)) / (drop u - drop x)) <= B` STRIP_ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[IN_DIFF] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `B:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[EVENTUALLY_AT; GSYM DIST_NZ; IN_BALL] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `e:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`n:num`; `u:real^1`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `u:real^1`) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN TRANS_TAC REAL_LT_TRANS `inv(&n + &1)` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LET_TRANS `inv(&N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LT; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC; ASM_SIMP_TAC[REAL_ABS_DIV; REAL_LE_LDIV_EQ; GSYM REAL_ABS_NZ; REAL_SUB_0; DROP_EQ] THEN REWRITE_TAC[GSYM DROP_SUB; GSYM NORM_1]]; ALL_TAC] THEN SUBGOAL_THEN `!n. ~({ u:real^1 | u IN ball(x,inv(&n + &1)) /\ u IN interval[a,b] /\ ~(u = x)} = {})` ASSUME_TAC THENL [GEN_TAC THEN MATCH_MP_TAC(SET_RULE `interval(a,b) SUBSET interval[a,b] /\ ~(s INTER interval(a,b) SUBSET {c}) ==> ~({x | x IN s /\ x IN interval[a,b] /\ ~(x = c)} = {})`) THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED; BALL_1] THEN REWRITE_TAC[GSYM(CONJUNCT1 INTERIOR_INTERVAL)] THEN REWRITE_TAC[GSYM INTERIOR_INTER; INTER_INTERVAL_1] THEN DISCH_THEN(MP_TAC o MATCH_MP SUBSET_CLOSURE) THEN REWRITE_TAC[GSYM INTERVAL_SING; CLOSURE_INTERVAL] THEN MATCH_MP_TAC(MESON[CONVEX_CLOSURE_INTERIOR] `convex s /\ ~(interior s = {}) /\ closure s = s /\ ~(s SUBSET t) ==> ~(closure(interior s) SUBSET t)`) THEN REWRITE_TAC[CONVEX_INTERVAL; CLOSURE_INTERVAL] THEN REWRITE_TAC[INTERIOR_INTERVAL; INTERVAL_EQ_EMPTY_1; SUBSET_INTERVAL_1; LIFT_DROP; DROP_ADD; DROP_SUB] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN MP_TAC(ISPEC `&n + &1` REAL_LT_INV_EQ) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `?l. ((\n. lift(inf {(drop (f u) - drop (f x)) / (drop u - drop x) |u| u IN ball (x,inv (&n + &1)) /\ u IN interval[a,b] /\ ~(u = x)})) --> l) sequentially` (CHOOSE_THEN (LABEL_TAC "l")) THENL [MATCH_MP_TAC(MATCH_MP MONO_EXISTS (GEN `l:real^1` (ISPECL [`f:num->real^1`; `l:real^1`; `N:num`] SEQ_OFFSET_REV))) THEN MATCH_MP_TAC CONVERGENT_BOUNDED_MONOTONE_1 THEN REWRITE_TAC[LIFT_DROP] THEN CONJ_TAC THENL [REWRITE_TAC[bounded; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN EXISTS_TAC `B:real` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_BOUNDS] THEN MATCH_MP_TAC REAL_INF_BOUNDS THEN REWRITE_TAC[GSYM REAL_ABS_BOUNDS; FORALL_IN_GSPEC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_MESON_TAC[ARITH_RULE `N:num <= n + N`]; DISJ1_TAC THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC REAL_LE_INF_SUBSET THEN REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `b SUBSET c ==> {f x | x IN b /\ P x} SUBSET {f x | x IN c /\ P x}`) THEN MATCH_MP_TAC SUBSET_BALL THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ARITH_TAC; EXISTS_TAC `--B:real` THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM_MESON_TAC[ARITH_RULE `N:num <= n + N`; REAL_ARITH `abs x <= B ==> --B <= x`]]]; ALL_TAC] THEN SUBGOAL_THEN `?m. ((\n. lift(sup {(drop (f u) - drop (f x)) / (drop u - drop x) |u| u IN ball (x,inv (&n + &1)) /\ u IN interval[a,b] /\ ~(u = x)})) --> m) sequentially` (CHOOSE_THEN (LABEL_TAC "m")) THENL [MATCH_MP_TAC(MATCH_MP MONO_EXISTS (GEN `l:real^1` (ISPECL [`f:num->real^1`; `l:real^1`; `N:num`] SEQ_OFFSET_REV))) THEN MATCH_MP_TAC CONVERGENT_BOUNDED_MONOTONE_1 THEN REWRITE_TAC[LIFT_DROP] THEN CONJ_TAC THENL [REWRITE_TAC[bounded; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN EXISTS_TAC `B:real` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_BOUNDS] THEN MATCH_MP_TAC REAL_SUP_BOUNDS THEN REWRITE_TAC[GSYM REAL_ABS_BOUNDS; FORALL_IN_GSPEC] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_MESON_TAC[ARITH_RULE `N:num <= n + N`]; DISJ2_TAC THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC REAL_SUP_LE_SUBSET THEN REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `b SUBSET c ==> {f x | x IN b /\ P x} SUBSET {f x | x IN c /\ P x}`) THEN MATCH_MP_TAC SUBSET_BALL THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ARITH_TAC; EXISTS_TAC `B:real` THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM_MESON_TAC[ARITH_RULE `N:num <= n + N`; REAL_ARITH `abs x <= B ==> x <= B`]]]; ALL_TAC] THEN SUBGOAL_THEN `k <= drop m - drop l` MP_TAC THENL [REMOVE_THEN "l" MP_TAC THEN REMOVE_THEN "m" MP_TAC THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_SUB) THEN REWRITE_TAC[GSYM DROP_SUB] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] LIM_DROP_LBOUND) THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; DROP_SUB; LIFT_DROP] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `N:num` THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[REAL_LE_SUB_LADD] THEN DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `u:real^1` MP_TAC o SPEC `n:num`) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^1` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MATCH_MP_TAC(REAL_ARITH `i <= u /\ v <= s ==> k <= v - u ==> k + i <= s`) THEN CONJ_TAC THENL [MATCH_MP_TAC INF_LE_ELEMENT; MATCH_MP_TAC ELEMENT_LE_SUP] THEN (CONJ_TAC THENL [REWRITE_TAC[FORALL_IN_GSPEC]; ASM SET_TAC[]]) THENL [EXISTS_TAC `--B:real`; EXISTS_TAC `B:real`] THEN ASM_MESON_TAC[ARITH_RULE `N:num <= n + N`; REAL_ARITH `abs x <= B ==> --B <= x /\ x <= B`]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP lemma0) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real` THEN STRIP_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[TAUT `p1 /\ q1 /\ p2 /\ q2 /\ r1 /\ r2 /\ s2 /\ s1 <=> (p1 /\ q1 /\ r1 /\ s1) /\ (p2 /\ q2 /\ r2 /\ s2)`] THEN REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM] THEN CONJ_TAC THEN REWRITE_TAC[REAL_LE_LT; LEFT_OR_DISTRIB; EXISTS_OR_THM] THEN DISJ1_TAC THEN REWRITE_TAC[REAL_NOT_LT; SET_RULE `(?u. p u /\ q u /\ r u /\ s u) <=> ~(!u. p u /\ q u /\ r u ==> ~s u)`] THEN REWRITE_TAC[REAL_ARITH `--k <= a - b <=> b - k <= a`; REAL_ARITH `a - b <= k / &3 <=> a <= k / &3 + b`] THENL [REMOVE_THEN "l" MP_TAC THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN DISCH_THEN(MP_TAC o SPEC `(q - drop l) - k / &3`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `M:num` (MP_TAC o SPEC `M + n:num`)) THEN REWRITE_TAC[LE_ADD; DIST_REAL; GSYM drop; LIFT_DROP] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `abs(j - l) < q - l - k ==> ~(q - k <= j)`)) THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] REAL_LE_INF)); REMOVE_THEN "m" MP_TAC THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN DISCH_THEN(MP_TAC o SPEC `(drop m - q) - k / &3`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `M:num` (MP_TAC o SPEC `M + n:num`)) THEN REWRITE_TAC[LE_ADD; DIST_REAL; GSYM drop; LIFT_DROP] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `abs(j - m) < m - q - k ==> ~(j <= k + q)`)) THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] REAL_SUP_LE))] THEN MATCH_MP_TAC (TAUT `p /\ (r ==> q) ==> ~(p /\ q) ==> ~r`) THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN REWRITE_TAC[FORALL_IN_GSPEC; IMP_IMP] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN MATCH_MP_TAC SUBSET_BALL THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ARITH_TAC) in let lemma5 = prove (`!f a b k. f has_bounded_variation_on interval[a,b] /\ drop a < drop b /\ &0 < k ==> negligible {x | x IN interval[a,b] /\ !n. ?u v. u IN ball(x,inv(&n + &1)) /\ u IN interval[a,b] /\ v IN ball(x,inv(&n + &1)) /\ v IN interval[a,b] /\ ~(u = x) /\ ~(v = x) /\ k <= abs((drop(f v) - drop(f x)) / (drop v - drop x) - (drop(f u) - drop(f x)) / (drop u - drop x))}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(--) o (f:real^1->real^1)`; `a:real^1`; `b:real^1`; `k:real`] lemma4) THEN MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`; `k:real`] lemma4) THEN ASM_SIMP_TAC[HAS_BOUNDED_VARIATION_ISOMETRIC_COMPOSE; NORM_ARITH `dist(--x:real^N,--y) = dist(x,y)`] THEN REWRITE_TAC[IMP_IMP; GSYM NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN GEN_REWRITE_TAC I [SUBSET] THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[LEFT_OR_FORALL_THM] THEN REWRITE_TAC[RIGHT_OR_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN FIRST_X_ASSUM(MP_TAC o SPEC `m + n:num`) THEN REWRITE_TAC[REAL_ARITH `a <= abs x <=> a <= x \/ a <= --x`; o_DEF; TAUT `p /\ (q \/ r) <=> p /\ q \/ p /\ r`; EXISTS_OR_THM] THEN MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN REWRITE_TAC[DROP_NEG; REAL_ARITH `--x - --y:real = --(x - y)`; REAL_ARITH `--a / b:real = --(a / b)`] THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THEN REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> x IN s ==> x IN t`) THEN MATCH_MP_TAC SUBSET_BALL THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REAL_ARITH_TAC) in let lemma6 = prove (`!f:real^1->real^1 a b. f has_bounded_variation_on interval[a,b] /\ drop a < drop b ==> negligible {x | x IN interval[a,b] /\ ~(f differentiable (at x within interval[a,b]))}`, REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_DIFFERENTIABLE] THEN REWRITE_TAC[has_vector_derivative; EXISTS_LIFT] THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL; LIFT_DROP; has_derivative_within] THEN REWRITE_TAC[EXISTS_DROP; LIFT_DROP; LINEAR_SCALING] THEN REWRITE_TAC[NORM_1; GSYM REAL_ABS_INV] THEN ONCE_REWRITE_TAC[LIM_NULL_NORM] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_ABS] THEN REWRITE_TAC[GSYM NORM_MUL] THEN REWRITE_TAC[GSYM LIM_NULL_NORM] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a % (y - (x + d)):real^N = a % (y - x) - a % d`] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `a % x = a % lift(drop x)`] THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN SIMP_TAC[DROP_EQ; DROP_SUB; REAL_FIELD `~(x' = x) ==> (inv(x' - x) * h) * (x' - x):real = h`] THEN REWRITE_TAC[LIFT_DROP; ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN REWRITE_TAC[GSYM LIM_NULL] THEN REWRITE_TAC[CONVERGENT_EQ_CAUCHY_WITHIN; DIST_LIFT] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | x IN interval[a,b] /\ !n. ?u v. u IN ball(x,inv(&n + &1)) /\ u IN interval[a,b] /\ v IN ball(x,inv(&n + &1)) /\ v IN interval[a,b] /\ ~(u = x) /\ ~(v = x) /\ inv(&m + &1) <= abs((drop(f v) - drop(f x)) / (drop v - drop x) - (drop(f u) - drop(f x)) / (drop u - drop x))} | m IN (:num)}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN GEN_TAC THEN MATCH_MP_TAC lemma5 THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; GEN_REWRITE_TAC I [SUBSET] THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_TAC THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[REAL_LT_TRANS]; X_GEN_TAC `n:num` THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN REWRITE_TAC[GSYM REAL_NOT_LT; NOT_FORALL_THM; GSYM DIST_NZ] THEN REWRITE_TAC[IN_BALL] THEN DISCH_THEN(X_CHOOSE_TAC `p:num`) THEN EXISTS_TAC `inv(&p + &1)` THEN REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN ASM_MESON_TAC[DIST_SYM]]]) in let lemma7 = prove (`!f:real^1->real^1 a b k. f has_bounded_variation_on interval[a,b] ==> negligible {x | x IN interval[a,b] /\ ~(f differentiable at x)}`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `interval[a:real^1,b] = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; EMPTY_GSPEC; NEGLIGIBLE_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `a <= b ==> a = b \/ a < b`)) THEN REWRITE_TAC[DROP_EQ] THEN STRIP_TAC THENL [ASM_REWRITE_TAC[INTERVAL_SING] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{b:real^1}` THEN REWRITE_TAC[SUBSET_RESTRICT; NEGLIGIBLE_SING]; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `(a:real^1) INSERT b INSERT {x | x IN interval[a,b] /\ ~((f:real^1->real^1) differentiable (at x within interval[a,b]))}` THEN ASM_REWRITE_TAC[NEGLIGIBLE_INSERT] THEN CONJ_TAC THENL [MATCH_MP_TAC lemma6 THEN ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s /\ ~(x = a) /\ ~(x = b) ==> (Q x <=> P x)) ==> {x | x IN s /\ P x} SUBSET a INSERT b INSERT {x | x IN s /\ Q x}`) THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[VECTOR_DIFFERENTIABLE] THEN AP_TERM_TAC THEN ABS_TAC THEN REWRITE_TAC[has_vector_derivative] THEN REWRITE_TAC[has_derivative_at; has_derivative_within] THEN AP_TERM_TAC THEN MATCH_MP_TAC LIM_WITHIN_INTERIOR THEN REWRITE_TAC[OPEN_CLOSED_INTERVAL_1; INTERIOR_INTERVAL] THEN ASM SET_TAC[]]) in REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[DIFFERENTIABLE_COMPONENTWISE_AT] THEN REWRITE_TAC[GSYM IN_NUMSEG; NOT_FORALL_THM; NOT_IMP] THEN REWRITE_TAC[SET_RULE `x IN a /\ (?i. i IN s /\ P i x) <=> ?i. i IN s /\ x IN {x | x IN a /\ P i x}`] THEN REWRITE_TAC[SET_RULE `{x | ?i. i IN k /\ x IN f i} = UNIONS {f i | i IN k}`] THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN MATCH_MP_TAC lemma7 THEN FIRST_ASSUM(MATCH_MP_TAC o ONCE_REWRITE_RULE[HAS_BOUNDED_VARIATION_ON_COMPONENTWISE]) THEN ASM_REWRITE_TAC[]);; let LEBESGUE_DIFFERENTIATION_THEOREM = prove (`!f:real^1->real^N s. is_interval s /\ f has_bounded_variation_on s ==> negligible {x | x IN s /\ ~(f differentiable at x)}`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `negligible {x | x IN frontier s /\ ~(f differentiable at x)} /\ negligible {x | x IN interior s /\ ~((f:real^1->real^N) differentiable at x)}` MP_TAC THENL [ALL_TAC; REWRITE_TAC[GSYM NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[frontier] THEN MP_TAC(ISPEC `s:real^1->bool` CLOSURE_SUBSET) THEN SET_TAC[]] THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_FINITE THEN MATCH_MP_TAC FINITE_RESTRICT THEN ASM_MESON_TAC[CARD_FRONTIER_INTERVAL_1]; SUBGOAL_THEN `(f:real^1->real^N) has_bounded_variation_on interior s` MP_TAC THENL [ASM_MESON_TAC[HAS_BOUNDED_VARIATION_ON_SUBSET; INTERIOR_SUBSET]; ALL_TAC] THEN MP_TAC(ISPEC `s:real^1->bool` OPEN_INTERIOR) THEN POP_ASSUM_LIST(K ALL_TAC)] THEN SPEC_TAC(`interior s:real^1->bool`,`s:real^1->bool`) THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_COUNTABLE_UNION_CLOSED_INTERVALS) THEN DISCH_THEN(X_CHOOSE_THEN `u:(real^1->bool)->bool` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN REWRITE_TAC[INTER_UNIONS] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `i:real^1->bool` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:real^1->bool`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `s INTER {x | P x} = {x | x IN s /\ P x}`] THEN MATCH_MP_TAC LEBESGUE_DIFFERENTIATION_THEOREM_COMPACT THEN ASM_MESON_TAC[HAS_BOUNDED_VARIATION_ON_SUBSET]);; let LEBESGUE_DIFFERENTIATION_THEOREM_ALT = prove (`!f:real^1->real^N s. is_interval s /\ f has_bounded_variation_on s ==> ?t. t SUBSET s /\ negligible t /\ !x. x IN s DIFF t ==> f differentiable at x`, REPEAT STRIP_TAC THEN EXISTS_TAC `{x | x IN s /\ ~((f:real^1->real^N) differentiable at x)}` THEN ASM_SIMP_TAC[LEBESGUE_DIFFERENTIATION_THEOREM; SUBSET_RESTRICT] THEN REWRITE_TAC[IN_DIFF; IN_ELIM_THM] THEN CONV_TAC TAUT);; let LEBESGUE_DIFFERENTIATION_THEOREM_GEN = prove (`!f:real^1->real^N s. COUNTABLE(components s) /\ f has_bounded_variation_on s ==> negligible {x | x IN s /\ ~(f differentiable at x)}`, REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [UNIONS_COMPONENTS] THEN REWRITE_TAC[INTER_UNIONS] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE `s INTER {x | P x} = {x | x IN s /\ P x}`] THEN MATCH_MP_TAC LEBESGUE_DIFFERENTIATION_THEOREM THEN REWRITE_TAC[IS_INTERVAL_CONNECTED_1] THEN ASM_MESON_TAC[IN_COMPONENTS_CONNECTED; IN_COMPONENTS_SUBSET; HAS_BOUNDED_VARIATION_ON_SUBSET]);; let LEBESGUE_DIFFERENTIATION_THEOREM_INCREASING = prove (`!f s. is_interval s /\ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f x) <= drop(f y)) ==> negligible {x | x IN s /\ ~(f differentiable at x)}`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[LOCALLY_NEGLIGIBLE_ALT] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP IS_INTERVAL_LOCALLY_COMPACT_INTERVAL) THEN REWRITE_TAC[locally] THEN DISCH_THEN(MP_TAC o SPECL [`s:real^1->bool`; `x:real^1`]) THEN ASM_REWRITE_TAC[OPEN_IN_REFL] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[OPEN_IN_OPEN; MESON[] `(?u. (?t. open t /\ u = s INTER t) /\ P u) <=> ?t. open t /\ P(s INTER t)`] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^1->bool` THEN ASM_REWRITE_TAC[IN_INTER; LEFT_IMP_EXISTS_THM; IMP_CONJ; IN_ELIM_THM] THEN X_GEN_TAC `i:real^1->bool` THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`] THEN DISCH_THEN SUBST1_TAC THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x | x IN interval[a,b] /\ ~((f:real^1->real^1) differentiable at x)}` THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC LEBESGUE_DIFFERENTIATION_THEOREM_COMPACT THEN MATCH_MP_TAC INCREASING_BOUNDED_VARIATION THEN ASM SET_TAC[]);; let LEBESGUE_DIFFERENTIATION_THEOREM_DECREASING = prove (`!f s. is_interval s /\ (!x y. x IN s /\ y IN s /\ drop x <= drop y ==> drop(f y) <= drop(f x)) ==> negligible {x | x IN s /\ ~(f differentiable at x)}`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(--) o (f:real^1->real^1)`; `s:real^1->bool`] LEBESGUE_DIFFERENTIATION_THEOREM_INCREASING) THEN ASM_REWRITE_TAC[o_THM; DROP_NEG; REAL_LE_NEG2] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REPEAT(STRIP_TAC THEN ASM_REWRITE_TAC[]) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP DIFFERENTIABLE_NEG) THEN ASM_REWRITE_TAC[o_THM; VECTOR_NEG_NEG; ETA_AX]);; (* ------------------------------------------------------------------------- *) (* Forms of absolute continuity of the indefinite (absolute) integral. *) (* ------------------------------------------------------------------------- *) let ABSOLUTELY_CONTINUOUS_INTEGRAL = prove (`!f:real^M->real^N s e. f absolutely_integrable_on s /\ &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> norm(integral t f) < e`, ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\x. if x IN s then (f:real^M->real^N) x else vec 0`; `(:real^M)`; `e / &2`] ABSOLUTELY_INTEGRABLE_APPROXIMATE_CONTINUOUS) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV; REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `e / &2 / B` THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF] THEN X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC THEN TRANS_TAC REAL_LET_TRANS `drop(integral t (\x. lift(norm((if x IN s then f x else vec 0) - g x)) + lift(norm((g:real^M->real^N) x))))` THEN SUBGOAL_THEN `(g:real^M->real^N) absolutely_integrable_on t /\ (\x. if x IN s then (f:real^M->real^N) x else vec 0) absolutely_integrable_on t` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `(:real^M)` THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; SUBSET_UNIV]; ALL_TAC] THEN SUBGOAL_THEN `(f:real^M->real^N) absolutely_integrable_on t` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] ABSOLUTELY_INTEGRABLE_EQ)) THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [ABSOLUTELY_INTEGRABLE_NORM; ABSOLUTELY_INTEGRABLE_ADD; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_SUB] THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[LIFT_DROP; DROP_ADD] THEN COND_CASES_TAC THENL [CONV_TAC NORM_ARITH; ASM SET_TAC[]]; ALL_TAC] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [ABSOLUTELY_INTEGRABLE_NORM; INTEGRAL_ADD; DROP_ADD; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_SUB] THEN MATCH_MP_TAC(REAL_ARITH `x < e / &2 /\ y < e / &2 ==> x + y < e`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `norm(integral s (f:real^M->real^1)) < e / &2 ==> drop(integral t f) <= norm(integral s f) ==> drop(integral t f) < e / &2`)) THEN REWRITE_TAC[NORM_REAL; GSYM drop] THEN MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= abs y`) THEN MATCH_MP_TAC INTEGRAL_SUBSET_DROP_LE THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM; IN_UNIV; SUBSET_UNIV; LIFT_DROP; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ABSOLUTELY_INTEGRABLE_SUB] THEN REWRITE_TAC[NORM_POS_LE]; TRANS_TAC REAL_LET_TRANS `drop(integral t (\x:real^M. lift B))` THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_SIMP_TAC[LIFT_DROP; ABSOLUTELY_INTEGRABLE_NORM; INTEGRABLE_ON_CONST; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; ASM_SIMP_TAC[LIFT_EQ_CMUL; INTEGRAL_CMUL; INTEGRABLE_ON_CONST; INTEGRAL_MEASURE] THEN REWRITE_TAC[DROP_CMUL; DROP_VEC; REAL_MUL_RID] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ]]]);; let ABSOLUTELY_SETCONTINUOUS_INDEFINITE_INTEGRAL = prove (`!f:real^M->real^N s. f absolutely_integrable_on s /\ lebesgue_measurable s ==> (\k. integral k f) absolutely_setcontinuous_on s`, REPEAT STRIP_TAC THEN REWRITE_TAC[ABSOLUTELY_SETCONTINUOUS_ON_ALT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `e:real` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_INTEGRAL)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`d:(real^M->bool)->bool`; `t:real^M->bool`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] INTEGRAL_COMBINE_DIVISION_TOPDOWN)) THEN ANTS_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_INTEGRABLE_ON_LEBESGUE_MEASURABLE_SUBSET)) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_IMP_LEBESGUE_MEASURABLE THEN ASM_MESON_TAC[MEASURABLE_ELEMENTARY]; DISCH_THEN(SUBST1_TAC o SYM) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_ELEMENTARY]; ASM_MESON_TAC[MEASURE_ELEMENTARY]]]);; let ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_RIGHT = prove (`!f:real^1->real^N a b. f absolutely_integrable_on interval[a,b] ==> (\x. integral(interval[a,x]) f) absolutely_continuous_on interval[a,b]`, REPEAT STRIP_TAC THEN REWRITE_TAC[absolutely_continuous_on] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] ABSOLUTELY_SETCONTINUOUS_INDEFINITE_INTEGRAL)) THEN REWRITE_TAC[LEBESGUE_MEASURABLE_INTERVAL] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] ABSOLUTELY_SETCONTINUOUS_COMPARISON) THEN SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY] THEN MAP_EVERY X_GEN_TAC [`c:real^1`; `d:real^1`] THEN REWRITE_TAC[INTERVAL_NE_EMPTY_1; SUBSET_INTERVAL_1] THEN STRIP_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(NORM_ARITH `y + z:real^N = x ==> norm(x - y) <= norm z`) THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] INTEGRABLE_ON_SUBINTERVAL) o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN ASM_REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC);; let ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_LEFT = prove (`!f:real^1->real^N a b. f absolutely_integrable_on interval[a,b] ==> (\x. integral(interval[x,b]) f) absolutely_continuous_on interval[a,b]`, REPEAT STRIP_TAC THEN REWRITE_TAC[absolutely_continuous_on] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] ABSOLUTELY_SETCONTINUOUS_INDEFINITE_INTEGRAL)) THEN REWRITE_TAC[LEBESGUE_MEASURABLE_INTERVAL] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] ABSOLUTELY_SETCONTINUOUS_COMPARISON) THEN SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY] THEN MAP_EVERY X_GEN_TAC [`c:real^1`; `d:real^1`] THEN REWRITE_TAC[INTERVAL_NE_EMPTY_1; SUBSET_INTERVAL_1] THEN STRIP_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(NORM_ARITH `z + x:real^N = y ==> norm(x - y) <= norm z`) THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] INTEGRABLE_ON_SUBINTERVAL) o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN ASM_REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC);; let FUNDAMENTAL_THEOREM_OF_CALCULUS_ABSOLUTELY_CONTINUOUS = prove (`!f:real^1->real^N f' s a b. negligible s /\ drop a <= drop b /\ f absolutely_continuous_on interval[a,b] /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f'(x)) (at x within interval[a,b])) ==> (f' has_integral (f(b) - f(a))) (interval[a,b])`, REPEAT STRIP_TAC THEN MATCH_MP_TAC FUNDAMENTAL_THEOREM_OF_CALCULUS_BARTLE THEN EXISTS_TAC `s:real^1->bool` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [absolutely_continuous_on]) THEN REWRITE_TAC[absolutely_setcontinuous_on] THEN DISCH_THEN(MP_TAC o SPEC `e / &2 / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^1->bool`; `d:real`] MEASURABLE_OUTER_OPEN) THEN ASM_SIMP_TAC[MEASURE_EQ_0; NEGLIGIBLE_IMP_MEASURABLE; REAL_ADD_LID] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^1->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. if x IN s then u else ball(x:real^1,&1)` THEN CONJ_TAC THENL [REWRITE_TAC[gauge] THEN X_GEN_TAC `x:real^1` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_01] THEN ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `p:(real^1#(real^1->bool))->bool` THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC VSUM_NORM_TRIANGLE THEN FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o REWRITE_RULE[tagged_partial_division_of]) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE SND (p:(real^1#(real^1->bool))->bool)` o MATCH_MP(MESON[] `(!x y. P x y) ==> !x. P x (UNIONS x)`)) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [ASM_MESON_TAC[PARTIAL_DIVISION_OF_TAGGED_DIVISION]; REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_PAIR_THM] THEN ASM_MESON_TAC[tagged_partial_division_of; SUBSET]; TRANS_TAC REAL_LET_TRANS `measure(u:real^1->bool)` THEN ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LE_TRANS `measure(UNIONS (IMAGE SND (p:(real^1#(real^1->bool))->bool)))` THEN CONJ_TAC THENL [MATCH_MP_TAC(REAL_ARITH `x = y ==> y <= x`) THEN MATCH_MP_TAC MEASURE_ELEMENTARY THEN ASM_MESON_TAC[PARTIAL_DIVISION_OF_TAGGED_DIVISION]; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_ELEMENTARY; PARTIAL_DIVISION_OF_TAGGED_DIVISION]; REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_PAIR_THM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM] THEN REWRITE_TAC[AND_FORALL_THM; IMP_IMP] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN SET_TAC[]]]]; W(MP_TAC o PART_MATCH (lhand o rand) (ISPEC `SND` SUM_IMAGE_GEN) o lhand o rand o snd) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(REAL_ARITH `y <= &2 * x ==> x < e / &2 ==> y <= e`) THEN REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_LE THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `k:real^1->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `?y z:real^1. k = interval[y,z] /\ x IN interval[y,z]` MP_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [tagged_partial_division_of]) THEN ASM_MESON_TAC[]; DISCH_THEN(REPEAT_TCL CHOOSE_THEN (CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC))] THEN ASM_CASES_TAC `interval[y:real^1,z] = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `sum {x | x IN (p:(real^1#(real^1->bool))->bool) /\ SND x = interval[y,z]} (\a. norm(f(z) - f(y):real^N))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_SIMP_TAC[FINITE_RESTRICT; FORALL_PAIR_THM; IN_ELIM_THM] THEN ASM_SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY; REAL_LE_REFL]; ASM_SIMP_TAC[SUM_CONST; FINITE_RESTRICT] THEN ASM_SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY]] THEN ASM_CASES_TAC `content(interval[y:real^1,z]) = &0` THENL [SUBGOAL_THEN `z:real^1 = y` (fun th -> REWRITE_TAC[th; VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL]) THEN ASM_MESON_TAC[DROP_EQ; CONTENT_EQ_0_1; REAL_LE_ANTISYM; INTERVAL_NE_EMPTY_1]; MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE]] THEN REWRITE_TAC[REAL_OF_NUM_LE] THEN MP_TAC(ISPECL [`p:(real^1#(real^1->bool))->bool`; `interval[a:real^1,b]`; `x:real^1`] TAGGED_PARTIAL_DIVISION_COMMON_POINT_BOUND) THEN ASM_REWRITE_TAC[DIMINDEX_1; EXP_1] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LE_TRANS) THEN MATCH_MP_TAC CARD_SUBSET THEN REWRITE_TAC[SUBSET; FORALL_PAIR_THM; IN_ELIM_THM; PAIR_EQ] THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN SET_TAC[]]);; let ABSOLUTELY_INTEGRABLE_BOUNDED_VARIATION_EQ = prove (`!f:real^1->real^N a b. f absolutely_integrable_on interval[a,b] <=> f integrable_on interval[a,b] /\ (\t. integral(interval[a,t]) f) has_bounded_variation_on interval[a,b]`, REPEAT GEN_TAC THEN REWRITE_TAC[has_bounded_variation_on; ABSOLUTELY_INTEGRABLE_BOUNDED_SETVARIATION_EQ] THEN ASM_CASES_TAC `(f:real^1->real^N) integrable_on interval[a,b]` THEN ASM_REWRITE_TAC[] THEN EQ_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_SETVARIATION_ON_EQ) THEN MAP_EVERY X_GEN_TAC [`c:real^1`; `d:real^1`] THEN STRIP_TAC THEN REWRITE_TAC[] THENL [CONV_TAC SYM_CONV; ALL_TAC] THEN ASM_SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY] THEN MATCH_MP_TAC(VECTOR_ARITH `b + c:real^N = a ==> a - b = c`) THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN REPEAT CONJ_TAC THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] INTEGRABLE_ON_SUBINTERVAL))) THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1; SUBSET_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC);; let ABSOLUTELY_INTEGRABLE_ABSOLUTELY_CONTINUOUS_DERIVATIVE = prove (`!f:real^1->real^N f' s a b. f absolutely_continuous_on interval[a,b] /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f'(x)) (at x within interval[a,b])) ==> f' absolutely_integrable_on interval[a,b]`, REPEAT STRIP_TAC THEN DISJ_CASES_TAC(REAL_ARITH `drop b <= drop a \/ drop a <= drop b`) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_ON_NULL; CONTENT_EQ_0_1] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_BOUNDED_VARIATION_EQ] THEN CONJ_TAC THENL [REWRITE_TAC[integrable_on] THEN EXISTS_TAC `(f:real^1->real^N) b - f a` THEN MATCH_MP_TAC FUNDAMENTAL_THEOREM_OF_CALCULUS_ABSOLUTELY_CONTINUOUS THEN EXISTS_TAC `s:real^1->bool` THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_ON_IMP_HAS_BOUNDED_VARIATION_ON THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_ON_EQ THEN EXISTS_TAC `\x. (f:real^1->real^N) x - f a` THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_SUB; ABSOLUTELY_CONTINUOUS_ON_CONST] THEN X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC FUNDAMENTAL_THEOREM_OF_CALCULUS_ABSOLUTELY_CONTINUOUS THEN EXISTS_TAC `s:real^1->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `interval[a:real^1,b]` THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; ASM_REAL_ARITH_TAC] THEN ASM_REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]]);; let ABSOLUTE_INTEGRAL_ABSOLUTELY_CONTINUOUS_DERIVATIVE_EQ = prove (`!f:real^1->real^N f' a b. f' absolutely_integrable_on interval[a,b] /\ (!x. x IN interval[a,b] ==> (f' has_integral (f x - f a)) (interval[a,x])) <=> f absolutely_continuous_on interval[a,b] /\ ?s. negligible s /\ !x. x IN interval [a,b] DIFF s ==> (f has_vector_derivative f' x) (at x within interval[a,b])`, REPEAT GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_ON_EQ THEN EXISTS_TAC `\x. (f:real^1->real^N) a + integral(interval[a,x]) f'` THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_RIGHT; ABSOLUTELY_CONTINUOUS_ON_ADD; ABSOLUTELY_CONTINUOUS_ON_CONST] THEN X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a + b:real^N = c <=> b = c - a`] THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_SIMP_TAC[]; FIRST_ASSUM(MP_TAC o MATCH_MP HAS_VECTOR_DERIVATIVE_INDEFINITE_INTEGRAL o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `s:real^1->bool` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[IN_DIFF] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^1` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `((\x:real^1. (f:real^1->real^N) a) has_vector_derivative (vec 0)) (at x within interval[a,b])` MP_TAC THENL [REWRITE_TAC[HAS_VECTOR_DERIVATIVE_CONST]; ALL_TAC] THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_VECTOR_DERIVATIVE_ADD) THEN REWRITE_TAC[VECTOR_ADD_LID] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HAS_VECTOR_DERIVATIVE_TRANSFORM_WITHIN)) THEN EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[REAL_LT_01] THEN X_GEN_TAC `y:real^1` THEN STRIP_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a + b:real^N = c <=> b = c - a`] THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_SIMP_TAC[]; MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ABSOLUTELY_CONTINUOUS_DERIVATIVE THEN ASM_MESON_TAC[]; MATCH_MP_TAC FUNDAMENTAL_THEOREM_OF_CALCULUS_ABSOLUTELY_CONTINUOUS THEN EXISTS_TAC `s:real^1->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET THEN EXISTS_TAC `interval[a:real^1,b]` THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; ASM_REAL_ARITH_TAC] THEN ASM_REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]]);; let ABSOLUTELY_INTEGRABLE_ABSOLUTELY_CONTINUOUS_DERIVATIVE_EQ = prove (`!f':real^1->real^N a b. f' absolutely_integrable_on interval[a,b] <=> ?f s. f absolutely_continuous_on interval[a,b] /\ negligible s /\ !x. x IN interval [a,b] DIFF s ==> (f has_vector_derivative f' x) (at x within interval[a,b])`, REPEAT GEN_TAC THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN REWRITE_TAC[GSYM ABSOLUTE_INTEGRAL_ABSOLUTELY_CONTINUOUS_DERIVATIVE_EQ] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN REWRITE_TAC[TAUT `(p <=> p /\ q) <=> p ==> q`] THEN DISCH_THEN(ASSUME_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE) THEN EXISTS_TAC `\t. integral(interval[a,t]) (f':real^1->real^N)` THEN REWRITE_TAC[INTEGRAL_REFL; VECTOR_SUB_RZERO] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] INTEGRABLE_ON_SUBINTERVAL)) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Characterizing absolutely continuous functions as Lebesgue integrals. *) (* ------------------------------------------------------------------------- *) let ABSOLUTELY_INTEGRABLE_INCREASING_DERIVATIVE = prove (`!f:real^1->real^1 a b. (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f x) <= drop(f y)) ==> ?s f'. negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x)) /\ f' absolutely_integrable_on interval[a,b] /\ (!x. x IN interval[a,b] ==> drop(integral(interval[a,x]) f') <= drop(f x) - drop(f a))`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `interval[a:real^1,b]`] LEBESGUE_DIFFERENTIATION_THEOREM_ALT) THEN ASM_SIMP_TAC[IS_INTERVAL_INTERVAL; INCREASING_BOUNDED_VARIATION] THEN DISCH_THEN(X_CHOOSE_THEN `s:real^1->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{a:real^1,b} UNION s` THEN ASM_SIMP_TAC[NEGLIGIBLE_UNION_EQ; NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY] THEN RULE_ASSUM_TAC (REWRITE_RULE[VECTOR_DIFFERENTIABLE; RIGHT_IMP_EXISTS_THM]) THEN FIRST_X_ASSUM(X_CHOOSE_TAC`f':real^1->real^1` o GEN_REWRITE_RULE I [SKOLEM_THM]) THEN EXISTS_TAC `f':real^1->real^1` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `interval[a:real^1,b] = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; ABSOLUTELY_INTEGRABLE_ON_EMPTY] THEN SUBGOAL_THEN `!c. c IN interval[a,b] ==> (f':real^1->real^1) absolutely_integrable_on interval[a,c] /\ drop(integral(interval[a,c]) f') <= drop(f c) - drop(f a)` (fun th -> ASM_MESON_TAC[ENDS_IN_INTERVAL; th]) THEN X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN SUBGOAL_THEN `~(interval[a:real^1,c] = {})` MP_TAC THENL [ASM_MESON_TAC[INTERVAL_NE_EMPTY_1; IN_INTERVAL_1]; ALL_TAC] THEN SUBGOAL_THEN `interval[a:real^1,c] SUBSET interval[a,b]` ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET_INTERVAL_1; IN_INTERVAL_1; REAL_LE_REFL]; ALL_TAC] THEN UNDISCH_TAC `negligible(s:real^1->bool)` THEN SUBGOAL_THEN `!x. x IN interval[a,c] DIFF s ==> ((f:real^1->real^1) has_vector_derivative f' x) (at x)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x y. x IN interval[a,c] /\ y IN interval[a,c] /\ drop x <= drop y ==> drop(f x) <= drop(f y)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`c:real^1`,`b:real^1`) THEN GEN_TAC THEN REPEAT DISCH_TAC THEN ABBREV_TAC `g = \x. if drop x < drop a then f(a) else if drop b < drop x then f(b) else (f:real^1->real^1) x` THEN SUBGOAL_THEN `(f':real^1->real^1) absolutely_integrable_on interval[a,b] /\ drop(integral(interval[a,b]) f') <= drop(g b) - drop(g a)` MP_TAC THENL [ALL_TAC; EXPAND_TAC "g" THEN REWRITE_TAC[REAL_LT_REFL] THEN ASM_REWRITE_TAC[GSYM INTERVAL_EQ_EMPTY_1]] THEN ABBREV_TAC `t = s UNION {a:real^1,b}` THEN SUBGOAL_THEN `negligible(t:real^1->bool)` MP_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN ASM_REWRITE_TAC[NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY]; ALL_TAC] THEN SUBGOAL_THEN `!x. x IN interval[a,b] DIFF t ==> ((g:real^1->real^1) has_vector_derivative f' x) (at x)` MP_TAC THENL [X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_DIFF] THEN EXPAND_TAC "t" THEN REWRITE_TAC[IN_UNION; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_TRANSFORM_WITHIN_OPEN THEN MAP_EVERY EXISTS_TAC [`f:real^1->real^1`; `interval(a:real^1,b)`] THEN REWRITE_TAC[OPEN_INTERVAL; CONJ_ASSOC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[OPEN_CLOSED_INTERVAL_1; IN_DIFF; IN_INSERT] THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; DE_MORGAN_THM; IN_INTERVAL_1] THEN REPEAT STRIP_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[GSYM REAL_NOT_LE] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x. drop b <= drop x ==> (g:real^1->real^1) x = g b` MP_TAC THENL [REPEAT STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN EXPAND_TAC "g" THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `!x y. drop x <= drop y ==> drop(g x) <= drop(g y)` MP_TAC THENL [REPEAT STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN EXPAND_TAC "g" THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL]) THEN FIRST_ASSUM(MP_TAC o SPECL [`a:real^1`; `b:real^1`]) THEN REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN UNDISCH_TAC `~(interval[a:real^1,b] = {})` THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`t:real^1->bool`,`s:real^1->bool`) THEN SPEC_TAC(`g:real^1->real^1`,`f:real^1->real^1`) THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN MP_TAC(ISPECL [`\n x. &n % ((f:real^1->real^1)(x + lift(inv(&n))) - f(x))`; `f':real^1->real^1`; `interval[a:real^1,b]`; `s:real^1->bool`; `drop(f(b:real^1)) - drop(f a)`] FATOU_STRONG) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[IN_DIFF] THEN REWRITE_TAC[has_derivative_at; has_vector_derivative] THEN DISCH_THEN(MP_TAC o SPEC `\n. x + lift(inv(&n))` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] (ISPEC `sequentially` LIM_COMPOSE_AT))) o CONJUNCT2) THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; o_DEF; VECTOR_ADD_SUB] THEN REWRITE_TAC[NORM_LIFT; REAL_ABS_INV; REAL_ABS_NUM; REAL_INV_INV] THEN REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; REAL_INV_0; VECTOR_MUL_LZERO] THEN ANTS_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST; SEQ_HARMONIC]; ALL_TAC] THEN GEN_REWRITE_TAC RAND_CONV [LIM_NULL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM) THEN MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[VECTOR_ARITH `n % (f' - (f + d)) - (n % (f' - f) - k):real^N = k - n % d`] THEN REWRITE_TAC[LIFT_DROP; VECTOR_MUL_ASSOC; VECTOR_SUB_EQ] THEN ASM_SIMP_TAC[REAL_MUL_RINV; REAL_OF_NUM_EQ; LE_1; VECTOR_MUL_LID]; MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[] THEN MATCH_MP_TAC INTEGRABLE_CMUL THEN MATCH_MP_TAC INTEGRABLE_SUB THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[INTEGRABLE_INCREASING_1]] THEN SUBGOAL_THEN `(f:real^1->real^1) integrable_on interval[a + lift(inv(&n)),b + lift(inv(&n))]` MP_TAC THENL [MATCH_MP_TAC INTEGRABLE_INCREASING_1 THEN ASM_SIMP_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPECL [`&1`; `lift(inv(&n))`] o MATCH_MP(REWRITE_RULE[IMP_CONJ] INTEGRABLE_AFFINITY)) THEN REWRITE_TAC[IMAGE_AFFINITY_INTERVAL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[real_div; REAL_MUL_LID; VECTOR_MUL_LID] THEN REWRITE_TAC[VECTOR_ARITH `(a + x) + --x:real^1 = a`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTERVAL_EQ_EMPTY_1]) THEN ASM_REWRITE_TAC[DROP_ADD; REAL_ARITH `x + i < y + i <=> ~(y <= x)`] THEN ASM_MESON_TAC[INTERVAL_NE_EMPTY_1]; MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[DROP_CMUL; DROP_SUB] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_POS; REAL_SUB_LE] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[DROP_ADD; REAL_LE_ADDR; LIFT_DROP; REAL_LE_INV_EQ] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB] THEN MP_TAC(ISPECL [`f:real^1->real^1`; `integral (interval[a + lift(inv(&n)),b + lift(inv(&n))]) f:real^1`; `interval[a + lift(inv(&n)),b + lift(inv(&n))]`; `&1`; `lift(inv(&n))`] HAS_INTEGRAL_AFFINITY) THEN REWRITE_TAC[DIMINDEX_1] THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL; IMAGE_AFFINITY_INTERVAL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[VECTOR_MUL_LID; INTERVAL_EQ_EMPTY_1] THEN REWRITE_TAC[DROP_ADD; LIFT_DROP] THEN COND_CASES_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[VECTOR_ARITH `(a + x) + --x:real^N = a`]] THEN ANTS_TAC THENL [ASM_MESON_TAC[INTEGRABLE_INCREASING_1]; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPEC `&n` o MATCH_MP HAS_INTEGRAL_CMUL) THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB] THEN W(MP_TAC o PART_MATCH (lhand o rand) INTEGRAL_SUB o rand o lhand o snd) THEN SUBGOAL_THEN `!u v. (f:real^1->real^1) integrable_on interval[u,v]` ASSUME_TAC THENL [ASM_MESON_TAC[INTEGRABLE_INCREASING_1]; ALL_TAC] THEN ASM_SIMP_TAC[INTEGRABLE_CMUL; INTEGRAL_CMUL] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; DROP_CMUL] THEN MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b + lift(&1 / &n)`; `a + lift(&1 / &n)`] INTEGRAL_COMBINE) THEN ASM_REWRITE_TAC[VECTOR_ARITH `a + b:real^N = b <=> b = b - a`] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[DROP_ADD; REAL_LE_ADDR; REAL_LE_RADD] THEN SIMP_TAC[real_div; REAL_MUL_LID; REAL_LE_INV_EQ; LIFT_DROP; REAL_POS] THEN ASM_MESON_TAC[INTERVAL_NE_EMPTY_1]; DISCH_THEN(SUBST1_TAC o MATCH_MP (VECTOR_ARITH `a + b:real^N = c ==> b = c - a`))] THEN MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b + lift(&1 / &n)`; `b:real^1`] INTEGRAL_COMBINE) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[DROP_ADD; REAL_LE_ADDR; REAL_LE_RADD] THEN SIMP_TAC[real_div; REAL_MUL_LID; REAL_LE_INV_EQ; LIFT_DROP; REAL_POS] THEN ASM_MESON_TAC[INTERVAL_NE_EMPTY_1]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[VECTOR_ARITH `(i + b) - a - i:real^N = b - a`] THEN REWRITE_TAC[DROP_SUB; REAL_SUB_LDISTRIB] THEN MATCH_MP_TAC(REAL_ARITH `a <= b /\ d <= c ==> a - c <= b - d`) THEN CONJ_TAC THENL [TRANS_TAC REAL_LE_TRANS `&n * drop(integral (interval[b,b + lift (&1 / &n)]) (\x. (f:real^1->real^1) b))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_POS] THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[INTEGRABLE_CONST; IN_INTERVAL_1] THEN ASM_MESON_TAC[REAL_LE_REFL]; REWRITE_TAC[INTEGRAL_CONST] THEN SIMP_TAC[CONTENT_1; DROP_ADD; LIFT_DROP; REAL_LE_ADDR; REAL_LE_DIV; REAL_POS] THEN REWRITE_TAC[DROP_CMUL] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_OF_NUM_LE]) THEN CONV_TAC REAL_FIELD]; TRANS_TAC REAL_LE_TRANS `&n * drop(integral (interval[a,a + lift (&1 / &n)]) (\x. (f:real^1->real^1) a))` THEN CONJ_TAC THENL [REWRITE_TAC[INTEGRAL_CONST] THEN SIMP_TAC[CONTENT_1; DROP_ADD; LIFT_DROP; REAL_LE_ADDR; REAL_LE_DIV; REAL_POS] THEN REWRITE_TAC[DROP_CMUL] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_OF_NUM_LE]) THEN CONV_TAC REAL_FIELD; MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_POS] THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_SIMP_TAC[INTEGRABLE_CONST; IN_INTERVAL_1]]]);; let ABSOLUTELY_INTEGRABLE_INCREASING_DERIVATIVE_ALT = prove (`!f:real^1->real^1 f' a b s. (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f x) <= drop(f y)) /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x)) ==> f' absolutely_integrable_on interval[a,b] /\ (!x. x IN interval[a,b] ==> drop(integral(interval[a,x]) f') <= drop(f x) - drop(f a))`, REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_INCREASING_DERIVATIVE) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^1->bool`; `g:real^1->real^1`] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN interval[a,b] DIFF (s UNION t) ==> (f':real^1->real^1) x = g x` ASSUME_TAC THENL [X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN MATCH_MP_TAC VECTOR_DERIVATIVE_UNIQUE_AT THEN MAP_EVERY EXISTS_TAC [`f:real^1->real^1`; `x:real^1`] THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] ABSOLUTELY_INTEGRABLE_SPIKE) THEN EXISTS_TAC `g:real^1->real^1` THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `s UNION t:real^1->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ]; X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `drop(integral(interval[a,c]) (g:real^1->real^1))` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE THEN EXISTS_TAC `s UNION t:real^1->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN SUBGOAL_THEN `interval[a:real^1,c] SUBSET interval[a,b]` MP_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1; REAL_LE_REFL]; ASM SET_TAC[]]]);; let ABSOLUTELY_INTEGRABLE_BOUNDED_VARIATION_DERIVATIVE = prove (`!f:real^1->real^N a b. f has_bounded_variation_on interval[a,b] ==> ?f' s. negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x)) /\ f' absolutely_integrable_on interval[a,b]`, let lemma = prove (`!f:real^1->real^1 a b. f has_bounded_variation_on interval[a,b] ==> ?f' s. negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x)) /\ f' absolutely_integrable_on interval[a,b]`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_BOUNDED_VARIATION_DARBOUX]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:real^1->real^1`; `h:real^1->real^1`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`h:real^1->real^1`; `a:real^1`; `b:real^1`] ABSOLUTELY_INTEGRABLE_INCREASING_DERIVATIVE) THEN MP_TAC(ISPECL [`g:real^1->real^1`; `a:real^1`; `b:real^1`] ABSOLUTELY_INTEGRABLE_INCREASING_DERIVATIVE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`s:real^1->bool`; `g':real^1->real^1`] THEN STRIP_TAC THEN MAP_EVERY X_GEN_TAC [`t:real^1->bool`; `h':real^1->real^1`] THEN STRIP_TAC THEN EXISTS_TAC `\x. (g':real^1->real^1) x - h' x` THEN EXISTS_TAC `s UNION t:real^1->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_SUB] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM FUN_EQ_THM]) THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_SUB THEN ASM SET_TAC[]) in REPEAT STRIP_TAC THEN MP_TAC(MATCH_MP MONO_FORALL (GEN `i:num` (MATCH_MP MONO_IMP (CONJ (TAUT `1 <= i /\ i <= dimindex(:N) ==> 1 <= i /\ i <= dimindex(:N)`) (SPECL [`\x. lift((f:real^1->real^N)x$i)`; `a:real^1`; `b:real^1`] lemma))))) THEN ASM_REWRITE_TAC[GSYM HAS_BOUNDED_VARIATION_ON_COMPONENTWISE] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`g:num->real^1->real^1`; `s:num->real^1->bool`] THEN DISCH_TAC THEN EXISTS_TAC `(\x. lambda i. drop((g:num->real^1->real^1) i x)) :real^1->real^N` THEN EXISTS_TAC `UNIONS (IMAGE (s:num->real^1->bool) (1..dimindex(:N)))` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN ASM_SIMP_TAC[IN_NUMSEG; FINITE_IMAGE; FORALL_IN_IMAGE; FINITE_NUMSEG]; X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_DIFF; UNIONS_IMAGE; IN_NUMSEG; IN_ELIM_THM] THEN REWRITE_TAC[MESON[] `~(?x. P x /\ Q x) <=> (!x. P x ==> ~Q x)`] THEN STRIP_TAC THEN REWRITE_TAC[has_vector_derivative] THEN ONCE_REWRITE_TAC[HAS_DERIVATIVE_COMPONENTWISE_AT] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ASM_SIMP_TAC[IN_DIFF; has_vector_derivative] THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; LAMBDA_BETA] THEN REWRITE_TAC[LIFT_CMUL; LIFT_DROP]; ONCE_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_COMPONENTWISE] THEN ASM_SIMP_TAC[LAMBDA_BETA; LIFT_DROP; ETA_AX]]);; let ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_EQ = prove (`!f:real^1->real^N a b. f absolutely_continuous_on interval[a,b] <=> ?f'. f' absolutely_integrable_on interval[a,b] /\ !x. x IN interval[a,b] ==> (f' has_integral (f x - f a)) (interval[a,x])`, REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_ON_IMP_HAS_BOUNDED_VARIATION_ON)) THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_BOUNDED_VARIATION_DERIVATIVE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f':real^1->real^N` THEN DISCH_THEN(X_CHOOSE_THEN `s:real^1->bool` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN MATCH_MP_TAC FUNDAMENTAL_THEOREM_OF_CALCULUS_ABSOLUTELY_CONTINUOUS THEN EXISTS_TAC `s:real^1->bool` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_INTERVAL_1; REAL_LE_REFL] THEN ASM_MESON_TAC[IN_INTERVAL_1]; X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_AT_WITHIN THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]; FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_RIGHT) THEN MP_TAC(ISPECL [`interval[a:real^1,b]`; `(f:real^1->real^N) a`] ABSOLUTELY_CONTINUOUS_ON_CONST) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_CONTINUOUS_ON_ADD) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_ON_EQ) THEN REWRITE_TAC[VECTOR_ARITH `a + i:real^N = x <=> i = x - a`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_SIMP_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Variation in terms of integral in a more general setting. *) (* ------------------------------------------------------------------------- *) let ABSOLUTE_INTEGRAL_NORM_DERIVATIVE_LE_VARIATION = prove (`!f:real^1->real^N a b. f has_bounded_variation_on interval[a,b] ==> ?s f'. negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x)) /\ f' absolutely_integrable_on interval[a,b] /\ !c. c IN interval[a,b] ==> drop(integral (interval[a,c]) (\x. lift(norm(f' x)))) <= vector_variation (interval[a,c]) f`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_BOUNDED_VARIATION_DERIVATIVE) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f':real^1->real^N`; `s:real^1->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`\x. lift(vector_variation (interval[a,x]) (f:real^1->real^N))`; `a:real^1`; `b:real^1`] ABSOLUTELY_INTEGRABLE_INCREASING_DERIVATIVE) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC VECTOR_VARIATION_MONOTONE THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_ON_SUBSET)); ALL_TAC] THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[LIFT_DROP; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`t:real^1->bool`; `v':real^1->real^1`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`a INSERT b INSERT s UNION t:real^1->bool`; `f':real^1->real^N`] THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; NEGLIGIBLE_INSERT] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN SUBGOAL_THEN `interval[a:real^1,c] SUBSET interval[a,b]` ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (ASSUME `c IN interval[a:real^1,b]`)) THEN SIMP_TAC[VECTOR_VARIATION_ON_NULL; CONTENT_EQ_0_1; REAL_LE_REFL; BOUNDED_INTERVAL; REAL_SUB_RZERO] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC INTEGRAL_DROP_LE_AE THEN EXISTS_TAC `a INSERT b INSERT s UNION t:real^1->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; NEGLIGIBLE_INSERT] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE THEN ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL]; MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL]; ALL_TAC] THEN X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_DIFF; IN_INSERT; IN_UNION; LIFT_DROP; DE_MORGAN_THM] THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `a <= abs b /\ &0 <= b ==> a <= b`) THEN SUBGOAL_THEN `eventually (\y:real^1. y IN interval[a,b]) (at x)` ASSUME_TAC THENL [MP_TAC(ISPEC `interval(a:real^1,b)` OPEN_CONTAINS_BALL) THEN REWRITE_TAC[OPEN_INTERVAL; EVENTUALLY_AT] THEN DISCH_THEN(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[OPEN_CLOSED_INTERVAL_1; IN_DIFF; IN_INSERT] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `d:real` THEN REWRITE_TAC[SUBSET; IN_BALL; IN_DIFF] THEN MESON_TAC[DIST_SYM]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM NORM_1] THEN MATCH_MP_TAC NORM_VECTOR_DERIVATIVES_LE_AT THEN MAP_EVERY EXISTS_TAC [`f:real^1->real^N`; `\x. lift (vector_variation (interval[a,x]) (f:real^1->real^N))`; `x:real^1`] THEN REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_MONO)) THEN REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT] THEN SUBGOAL_THEN `!u v. u IN interval[a,b] /\ v IN interval[a,b] ==> norm((f:real^1->real^N) u - f v) <= abs(vector_variation (interval[a,u]) f - vector_variation (interval[a,v]) f)` (fun th -> MP_TAC th THEN ASM SET_TAC[]) THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [REWRITE_TAC[NORM_SUB; REAL_ABS_SUB; CONJ_SYM]; ALL_TAC] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN REPEAT STRIP_TAC THEN TRANS_TAC REAL_LE_TRANS `vector_variation (interval[u,v]) (f:real^1->real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC VECTOR_VARIATION_GE_NORM_FUNCTION THEN ONCE_REWRITE_TAC[SEGMENT_SYM] THEN ASM_REWRITE_TAC[SEGMENT_1; SUBSET_REFL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_ON_SUBSET)) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC(REAL_ARITH `a + b = c ==> b <= abs(a - c)`) THEN MATCH_MP_TAC VECTOR_VARIATION_COMBINE THEN REPEAT CONJ_TAC THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_ON_SUBSET))) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]; REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`)) THEN ASM_REWRITE_TAC[IN_DIFF; HAS_VECTOR_DERIVATIVE_AT_1D] THEN REPEAT(ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] LIM_DROP_LBOUND)) THEN REWRITE_TAC[TRIVIAL_LIMIT_AT; DROP_CMUL; DROP_SUB; LIFT_DROP] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_MONO)) THEN X_GEN_TAC `y:real^1` THEN REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `x IN interval[a:real^1,b]` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISJ_CASES_TAC(REAL_ARITH `drop x <= drop y \/ drop y <= drop x`) THENL [ALL_TAC; ONCE_REWRITE_TAC[GSYM REAL_NEG_SUB] THEN SIMP_TAC[REAL_INV_NEG; REAL_MUL_LNEG; REAL_MUL_RNEG; REAL_NEG_NEG]] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ; REAL_SUB_LE] THEN (MATCH_MP_TAC VECTOR_VARIATION_MONOTONE THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_ON_SUBSET)); ALL_TAC]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC]);; let ABSOLUTE_INTEGRAL_NORM_DERIVATIVE_LE_VARIATION_ALT = prove (`!f:real^1->real^N f' a b s. f has_bounded_variation_on interval[a,b] /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x within interval[a,b])) ==> f' absolutely_integrable_on interval[a,b] /\ !c. c IN interval[a,b] ==> drop(integral (interval[a,c]) (\x. lift(norm(f' x)))) <= vector_variation (interval[a,c]) f`, REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTE_INTEGRAL_NORM_DERIVATIVE_LE_VARIATION) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`t:real^1->bool`; `g:real^1->real^N`] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. x IN interval[a,b] DIFF (a INSERT b INSERT (s UNION t)) ==> (f':real^1->real^N) x = g x` ASSUME_TAC THENL [X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN MATCH_MP_TAC VECTOR_DERIVATIVE_UNIQUE_AT THEN MAP_EVERY EXISTS_TAC [`f:real^1->real^N`; `x:real^1`] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN SUBGOAL_THEN `((f:real^1->real^N) has_vector_derivative f' x) (at x within interval[a,b])` MP_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP] THEN SIMP_TAC[HAS_VECTOR_DERIVATIVE_WITHIN_1D; HAS_VECTOR_DERIVATIVE_AT_1D] THEN MATCH_MP_TAC LIM_WITHIN_INTERIOR THEN REWRITE_TAC[INTERIOR_INTERVAL] THEN REWRITE_TAC[OPEN_CLOSED_INTERVAL_1] THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] ABSOLUTELY_INTEGRABLE_SPIKE) THEN EXISTS_TAC `g:real^1->real^N` THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `a INSERT b INSERT (s UNION t):real^1->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; NEGLIGIBLE_INSERT]; X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `drop(integral(interval[a,c]) (\x. lift(norm((g:real^1->real^N) x))))` THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE THEN EXISTS_TAC `a INSERT b INSERT (s UNION t):real^1->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; NEGLIGIBLE_INSERT] THEN SUBGOAL_THEN `interval[a:real^1,c] SUBSET interval[a,b]` MP_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1; REAL_LE_REFL]; ASM SET_TAC[]]]);; let VECTOR_VARIATION_INTEGRAL_NORM_DERIVATIVE_GEN = prove (`!f:real^1->real^N f' a b s. f absolutely_continuous_on interval[a,b] /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x within interval[a,b])) ==> f' absolutely_integrable_on interval[a,b] /\ vector_variation (interval[a,b]) f = drop(integral (interval[a,b]) (\x. lift(norm(f' x))))`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_MESON_TAC[ABSOLUTELY_INTEGRABLE_ABSOLUTELY_CONTINUOUS_DERIVATIVE]; DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_SET_VARIATION)] THEN REWRITE_TAC[vector_variation] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC SET_VARIATION_EQ THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN SIMP_TAC[INTERVAL_NE_EMPTY_1; INTERVAL_LOWERBOUND_1; INTERVAL_UPPERBOUND_1] THEN STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC FUNDAMENTAL_THEOREM_OF_CALCULUS_ABSOLUTELY_CONTINUOUS THEN EXISTS_TAC `s:real^1->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_MESON_TAC[ABSOLUTELY_CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN ASM_MESON_TAC[HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET; SUBSET; IN_DIFF]);; let ABSOLUTELY_CONTINUOUS_VECTOR_VARIATION = prove (`!f:real^1->real^N a b. f has_bounded_variation_on interval[a,b] /\ (\x. lift(vector_variation (interval [a,x]) f)) absolutely_continuous_on interval[a,b] <=> f absolutely_continuous_on interval[a,b]`, REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_COMPARISON)) THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN STRIP_TAC THEN REWRITE_TAC[dist] THEN TRANS_TAC REAL_LE_TRANS `vector_variation(interval[x,y]) (f:real^1->real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC VECTOR_VARIATION_GE_NORM_FUNCTION THEN ONCE_REWRITE_TAC[SEGMENT_SYM] THEN ASM_SIMP_TAC[SEGMENT_1; REAL_LT_IMP_LE; SUBSET_REFL]; REWRITE_TAC[NORM_LIFT; GSYM LIFT_SUB] THEN MATCH_MP_TAC(REAL_ARITH `x + y = z ==> y <= abs(x - z)`) THEN MATCH_MP_TAC VECTOR_VARIATION_COMBINE THEN REPEAT CONJ_TAC] THEN TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_VARIATION_ON_SUBSET))) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_HAS_BOUNDED_VARIATION_ON; BOUNDED_INTERVAL]; DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_BOUNDED_VARIATION_DERIVATIVE)] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f':real^1->real^N`; `s:real^1->bool`] THEN STRIP_TAC THEN ASM_CASES_TAC `interval[a:real^1,b] = {}` THEN ASM_REWRITE_TAC[ABSOLUTELY_CONTINUOUS_ON_EMPTY] THEN REWRITE_TAC[ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_EQ] THEN EXISTS_TAC `\x. lift(norm((f':real^1->real^N) x))` THEN SIMP_TAC[VECTOR_VARIATION_ON_NULL; CONTENT_EQ_0_1; REAL_LE_REFL; BOUNDED_INTERVAL; REAL_SUB_RZERO; LIFT_NUM; VECTOR_SUB_RZERO] THEN MATCH_MP_TAC(MESON[] `b IN interval[a,b] /\ (!c. c IN interval[a,b] ==> P(interval[a,c]) /\ Q c) ==> P(interval[a,b]) /\ !c. c IN interval[a,b] ==> Q c`) THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL; HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; TAUT `(p ==> q) ==> (p /\ q /\ r <=> p /\ r)`] THEN X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN W(MP_TAC o PART_MATCH (rand o rand) VECTOR_VARIATION_INTEGRAL_NORM_DERIVATIVE_GEN o rand o snd) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM]] THEN SUBGOAL_THEN `interval[a:real^1,c] SUBSET interval[a,b]` ASSUME_TAC THENL [REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[HAS_VECTOR_DERIVATIVE_AT_WITHIN; SUBSET; IN_DIFF; ABSOLUTELY_CONTINUOUS_ON_SUBSET]]]);; (* ------------------------------------------------------------------------- *) (* Converses to a couple of theorems above: equality holds only if the *) (* function concerned is absolutely continuous. *) (* ------------------------------------------------------------------------- *) let INCREASING_FTC_AE_IMP_ABSOLUTELY_CONTINUOUS = prove (`!f f' a b s. (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f x) <= drop(f y)) /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x)) /\ integral (interval[a,b]) f' = f(b) - f(a) ==> f absolutely_continuous_on interval[a,b]`, REPEAT STRIP_TAC THEN REWRITE_TAC[ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_EQ] THEN EXISTS_TAC `f':real^1->real^1` THEN REWRITE_TAC[RIGHT_AND_FORALL_THM] THEN X_GEN_TAC `c:real^1` THEN MP_TAC(ISPECL [`f:real^1->real^1`; `f':real^1->real^1`] ABSOLUTELY_INTEGRABLE_INCREASING_DERIVATIVE_ALT) THEN DISCH_THEN(fun th -> MP_TAC(ISPECL [`a:real^1`; `c:real^1`; `s:real^1->bool`] th) THEN MP_TAC(ISPECL [`c:real^1`; `b:real^1`; `s:real^1->bool`] th) THEN MP_TAC(ISPECL [`a:real^1`; `b:real^1`; `s:real^1->bool`] th)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `c IN interval[a:real^1,b]` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `interval[a:real^1,c] SUBSET interval[a,b] /\ interval[c,b] SUBSET interval[a,b]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1; REAL_LE_REFL]; ALL_TAC] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `b:real^1`))] THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; REAL_LE_REFL]; DISCH_TAC] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `c:real^1`))] THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; REAL_LE_REFL]; DISCH_TAC] THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM DROP_EQ]) THEN REWRITE_TAC[GSYM DROP_EQ; DROP_SUB] THEN MATCH_MP_TAC(REAL_ARITH `!j. i <= c - a /\ j <= b - c /\ i + j = k ==> k = b - a ==> i = c - a`) THEN EXISTS_TAC `drop(integral(interval[c:real^1,b]) f')` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM DROP_ADD; DROP_EQ] THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN ASM_REWRITE_TAC[GSYM IN_INTERVAL_1]);; let VECTOR_VARIATION_INTEGRAL_NORM_DERIVATIVE_REV = prove (`!f:real^1->real^N f' a b s. f has_bounded_variation_on interval[a,b] /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x within interval[a,b])) /\ vector_variation (interval[a,b]) f = drop(integral (interval[a,b]) (\x. lift(norm(f' x)))) ==> f absolutely_continuous_on interval[a,b]`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_CONTINUOUS_VECTOR_VARIATION] THEN ASM_REWRITE_TAC[ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_EQ] THEN EXISTS_TAC `\x. lift(norm((f':real^1->real^N) x))` THEN SIMP_TAC[VECTOR_VARIATION_ON_NULL; CONTENT_EQ_0_1; REAL_LE_REFL; BOUNDED_INTERVAL; REAL_SUB_RZERO; LIFT_NUM; VECTOR_SUB_RZERO] THEN REWRITE_TAC[RIGHT_AND_FORALL_THM] THEN X_GEN_TAC `c:real^1` THEN MP_TAC(ISPECL [`f:real^1->real^N`; `f':real^1->real^N`] ABSOLUTE_INTEGRAL_NORM_DERIVATIVE_LE_VARIATION_ALT) THEN DISCH_THEN(fun th -> MP_TAC(ISPECL [`a:real^1`; `c:real^1`; `s:real^1->bool`] th) THEN MP_TAC(ISPECL [`c:real^1`; `b:real^1`; `s:real^1->bool`] th) THEN MP_TAC(ISPECL [`a:real^1`; `b:real^1`; `s:real^1->bool`] th)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `c IN interval[a:real^1,b]` THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_NORM] THEN SUBGOAL_THEN `interval[a:real^1,c] SUBSET interval[a,b] /\ interval[c,b] SUBSET interval[a,b]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; SUBSET_INTERVAL_1; REAL_LE_REFL]; ALL_TAC] THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET; SUBSET; IN_DIFF; HAS_BOUNDED_VARIATION_ON_SUBSET]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `b:real^1`))] THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; REAL_LE_REFL]; DISCH_TAC] THEN ANTS_TAC THENL [ASM_MESON_TAC[HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET; SUBSET; IN_DIFF; HAS_BOUNDED_VARIATION_ON_SUBSET]; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `c:real^1`))] THEN ANTS_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; REAL_LE_REFL]; DISCH_TAC] THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN MP_TAC(ISPECL [`\x. lift(norm((f':real^1->real^N) x))`; `a:real^1`; `b:real^1`; `c:real^1`] INTEGRAL_COMBINE) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN MP_TAC(ISPECL [`f:real^1->real^N`; `a:real^1`; `b:real^1`; `c:real^1`] VECTOR_VARIATION_COMBINE) THEN ASM_REWRITE_TAC[GSYM IN_INTERVAL_1] THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; DROP_ADD] THEN ASM_REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Banach-Zarecki and related results. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE, LEBESGUE_MEASURABLE_ABSOLUTELY_CONTINUOUS_IMAGE, ABSOLUTELY_CONTINUOUS_IMP_BANACH_SPROPERTY = let lemma1 = prove (`!f:real^1->real^1 s e. f absolutely_continuous_on s /\ is_interval s /\ closed s /\ &0 < e ==> ?r. &0 < r /\ !t. t SUBSET s /\ measurable t /\ measure t < r ==> ?u. IMAGE f t SUBSET u /\ measurable u /\ measure u < e`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [absolutely_continuous_on]) THEN REWRITE_TAC[absolutely_setcontinuous_on] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `r / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN X_GEN_TAC `t:real^1->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL [`t:real^1->bool`; `r / &2`] MEASURABLE_OUTER_CLOSED_INTERVALS) THEN ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `ds:(real^1->bool)->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `!k. k IN ds ==> ~(k INTER s = {}) ==> ?u v. u IN k /\ v IN k /\ u IN s /\ v IN s /\ IMAGE (f:real^1->real^1) (k INTER s) SUBSET interval[f u,f v]` MP_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `(!k. k IN d ==> ~(k = {}) /\ ?a b. k = interval[a,b]) ==> (!a b. interval[a,b] IN d /\ ~(interval[a,b] = {}) ==> P(interval[a,b])) ==> !k. k IN d ==> P k`)) THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `?u v. IMAGE (f:real^1->real^1) (interval[a,b] INTER s) = interval[u,v]` MP_TAC THENL [REWRITE_TAC[GSYM CONNECTED_COMPACT_INTERVAL_1] THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE; MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE] THEN ASM_SIMP_TAC[COMPACT_INTERVAL; COMPACT_INTER_CLOSED; GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_INTER; IS_INTERVAL_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `s:real^1->bool` THEN REWRITE_TAC[INTER_SUBSET] THEN MATCH_MP_TAC UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_UNIFORMLY_CONTINUOUS]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^1`; `v:real^1`] THEN DISCH_TAC THEN SUBGOAL_THEN `u IN IMAGE (f:real^1->real^1) (interval[a,b] INTER s) /\ v IN IMAGE (f:real^1->real^1) (interval[a,b] INTER s)` MP_TAC THENL [ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN ASM SET_TAC[]; REWRITE_TAC[IN_IMAGE; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM]] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN ASM SET_TAC[]]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [IMP_IMP; RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:(real^1->bool)->real^1`; `v:(real^1->bool)->real^1`] THEN DISCH_TAC THEN EXISTS_TAC `UNIONS (IMAGE (\k. interval[(f:real^1->real^1)(u k),f(v k)]) {k:real^1->bool | k IN ds /\ ~(k INTER s = {})})` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(MESON[REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`] `&0 < e /\ measurable s /\ measure s <= e / &2 ==> measurable s /\ measure s < e`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_RESTRICT] THEN REWRITE_TAC[FORALL_IN_IMAGE; MEASURABLE_INTERVAL] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `dd:(real^1->bool)->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `IMAGE (\k:real^1->bool. interval[(f:real^1->real^1) (u k),f(v k)]) dd = IMAGE (\k. interval[lift(min (drop(f(interval_lowerbound k))) (drop(f(interval_upperbound k)))), lift(max (drop(f(interval_lowerbound k))) (drop(f(interval_upperbound k))))]) (IMAGE (\k. interval[lift(min (drop(u k)) (drop(v k))), lift(max (drop(u k)) (drop(v k)))]) dd)` SUBST1_TAC THENL [REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> g x = f x) ==> IMAGE f s = IMAGE g s`) THEN SIMP_TAC[INTERVAL_LOWERBOUND_1; INTERVAL_UPPERBOUND_1; LIFT_DROP; REAL_ARITH `min a b <= max a b`] THEN X_GEN_TAC `k:real^1->bool` THEN DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `~(s = {}) /\ (~(s = {}) ==> t = s) ==> t = s`) THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[EQ_INTERVAL_1; INTERVAL_NE_EMPTY_1] THEN ASM_CASES_TAC `drop(u(k:real^1->bool)) <= drop(v k)` THEN ASM_SIMP_TAC[REAL_ARITH `a <= b ==> min a b = a /\ max a b = b`; REAL_ARITH `~(a <= b) ==> min a b = b /\ max a b = a`; LIFT_DROP] THEN SIMP_TAC[real_max; real_min; LIFT_DROP] THEN REAL_ARITH_TAC; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ASM_SIMP_TAC[MEASURABLE_INTERVAL; FINITE_IMAGE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`IMAGE (\k:real^1->bool. interval[lift(min (drop(u k)) (drop(v k))), lift(max (drop(u k)) (drop(v k)))]) dd`; `UNIONS(IMAGE (\k:real^1->bool. interval[lift(min (drop(u k)) (drop(v k))), lift(max (drop(u k)) (drop(v k)))]) dd)`]) THEN ANTS_TAC THENL [REWRITE_TAC[division_of]; MATCH_MP_TAC(REAL_ARITH `x <= y ==> y < e ==> x <= e`) THEN MATCH_MP_TAC SUM_LE THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN SIMP_TAC[INTERVAL_LOWERBOUND_1; INTERVAL_UPPERBOUND_1; LIFT_DROP; REAL_ARITH `min a b <= max a b`; MEASURE_INTERVAL_1] THEN REWRITE_TAC[NORM_1; DROP_SUB] THEN REAL_ARITH_TAC] THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN REPEAT CONJ_TAC THENL [GEN_TAC THEN DISCH_TAC THEN REPEAT CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[INTERVAL_NE_EMPTY_1; LIFT_DROP] THEN REAL_ARITH_TAC; MESON_TAC[]]; REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN X_GEN_TAC `k:real^1->bool` THEN DISCH_TAC THEN X_GEN_TAC `l:real^1->bool` THEN DISCH_TAC THEN ASM_CASES_TAC `l:real^1->bool = k` THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`k:real^1->bool`; `l:real^1->bool`]) THEN ASM_REWRITE_TAC[GSYM INTERIOR_INTER] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> t = {} ==> s = {}`) THEN MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC(SET_RULE `s SUBSET s' /\ t SUBSET t' ==> s INTER t SUBSET s' INTER t'`) THEN CONJ_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) INTERVAL_SUBSET_IS_INTERVAL o snd) THEN (ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_ELIM_THM]) THEN ASM_MESON_TAC[IS_INTERVAL_INTERVAL]; DISCH_THEN SUBST1_TAC THEN DISJ2_TAC]) THEN REWRITE_TAC[real_min; real_max] THEN REPEAT(COND_CASES_TAC THEN REWRITE_TAC[LIFT_DROP]) THEN ASM SET_TAC[]; REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERVAL_SUBSET_IS_INTERVAL] THEN DISJ2_TAC THEN REWRITE_TAC[real_min; real_max] THEN REPEAT(COND_CASES_TAC THEN REWRITE_TAC[LIFT_DROP]) THEN ASM SET_TAC[]; FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `m <= t + r / &2 ==> t < r / &2 /\ x <= m ==> x < r`)) THEN ASM_REWRITE_TAC[] THEN W(MP_TAC o PART_MATCH (lhs o rand) SUM_IMAGE_NONZERO o lhand o snd) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [MAP_EVERY X_GEN_TAC [`k:real^1->bool`; `l:real^1->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`k:real^1->bool`; `l:real^1->bool`]) THEN ASM_REWRITE_TAC[CONTENT_EQ_0_INTERIOR; GSYM INTERIOR_INTER] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> t = {} ==> s = {}`) THEN MATCH_MP_TAC SUBSET_INTERIOR THEN REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL [FIRST_ASSUM(SUBST1_TAC o SYM); ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) INTERVAL_SUBSET_IS_INTERVAL o snd) THEN (ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_ELIM_THM]) THEN ASM_MESON_TAC[IS_INTERVAL_INTERVAL]; DISCH_THEN SUBST1_TAC THEN DISJ2_TAC]) THEN REWRITE_TAC[real_min; real_max] THEN REPEAT(COND_CASES_TAC THEN REWRITE_TAC[LIFT_DROP]) THEN ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN TRANS_TAC REAL_LE_TRANS `sum (dd:(real^1->bool)->bool) content` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[o_DEF] THEN X_GEN_TAC `k:real^1->bool` THEN ASM_CASES_TAC `?a b:real^1. k = interval[a,b]` THENL [FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC); ASM SET_TAC[]] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC CONTENT_SUBSET THEN SIMP_TAC[INTERVAL_SUBSET_IS_INTERVAL; IS_INTERVAL_INTERVAL] THEN DISJ2_TAC THEN REWRITE_TAC[real_min; real_max] THEN REPEAT(COND_CASES_TAC THEN REWRITE_TAC[LIFT_DROP]) THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`dd:(real^1->bool)->bool`; `UNIONS dd:real^1->bool`] HAS_MEASURE_ELEMENTARY) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[division_of] THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(SUBST1_TAC o SYM o MATCH_MP MEASURE_UNIQUE) THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_ELIM_THM]) THEN ASM_MESON_TAC[MEASURABLE_INTERVAL]]) in let lemma2 = prove (`!f:real^1->real^1 s e. f absolutely_continuous_on s /\ is_interval s /\ &0 < e ==> ?r. &0 < r /\ !t. t SUBSET s /\ measurable t /\ measure t < r ==> ?u. IMAGE f t SUBSET u /\ measurable u /\ measure u < e`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] ABSOLUTELY_CONTINUOUS_EXTENDS_TO_CLOSURE)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `g:real^1->real^1` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`g:real^1->real^1`; `closure s:real^1->bool`; `e:real`] lemma1) THEN ASM_REWRITE_TAC[CLOSED_CLOSURE] THEN ANTS_TAC THENL [ASM_MESON_TAC[IS_INTERVAL_CONVEX_1; CONVEX_CLOSURE]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `t:real^1->bool` THEN MP_TAC(ISPEC `s:real^1->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]) in let lemma3 = prove (`!f:real^1->real^N s e. f absolutely_continuous_on s /\ is_interval s /\ &0 < e ==> ?r. &0 < r /\ !t. t SUBSET s /\ measurable t /\ measure t < r ==> ?u. IMAGE f t SUBSET u /\ measurable u /\ measure u < e`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `1 <= dimindex(:N)` MP_TAC THENL [REWRITE_TAC[DIMINDEX_GE_1]; REWRITE_TAC[ARITH_RULE `1 <= n <=> 2 <= n \/ n = 1`]] THEN STRIP_TAC THENL [EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN X_GEN_TAC `t:real^1->bool` THEN STRIP_TAC THEN EXISTS_TAC `IMAGE (f:real^1->real^N) s` THEN ASM_SIMP_TAC[IMAGE_SUBSET] THEN MP_TAC(ISPECL [`f:real^1->real^N`; `s:real^1->bool`] NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE_LOWDIM) THEN ASM_SIMP_TAC[NEGLIGIBLE_EQ_MEASURE_0]; ALL_TAC] THEN MP_TAC(ISPECL [`(:real^N)`; `(:real^1)`] ISOMETRIES_SUBSPACES) THEN ASM_SIMP_TAC[SUBSPACE_UNIV; DIM_UNIV; DIMINDEX_1] THEN REWRITE_TAC[IN_UNIV; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`h:real^N->real^1`; `k:real^1->real^N`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(h:real^N->real^1) o (f:real^1->real^N)`; `s:real^1->bool`; `e:real`] lemma2) THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_COMPOSE_LINEAR] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `t:real^1->bool` THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^1->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `IMAGE (k:real^1->real^N) u` THEN REPEAT CONJ_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_o]) THEN ASM SET_TAC[]; MATCH_MP_TAC MEASURABLE_LINEAR_IMAGE_GEN THEN ASM_REWRITE_TAC[DIMINDEX_1; LE_REFL]; ASM_SIMP_TAC[MEASURE_ISOMETRY; DIMINDEX_1]]) in let NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE = prove (`!f:real^1->real^N s t. f absolutely_continuous_on s /\ is_interval s /\ negligible t /\ t SUBSET s ==> negligible(IMAGE f t)`, REPEAT STRIP_TAC THEN REWRITE_TAC[NEGLIGIBLE_OUTER] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^1->real^N`; `s:real^1->bool`; `e:real`] lemma3) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[NEGLIGIBLE_EQ_MEASURE_0]) in let LEBESGUE_MEASURABLE_ABSOLUTELY_CONTINUOUS_IMAGE = prove (`!f:real^1->real^N s t. f absolutely_continuous_on s /\ is_interval s /\ lebesgue_measurable t /\ t SUBSET s ==> lebesgue_measurable(IMAGE f t)`, MESON_TAC[PRESERVES_LEBESGUE_MEASURABLE_IFF_PRESERVES_NEGLIGIBLE; NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE; ABSOLUTELY_CONTINUOUS_ON_IMP_UNIFORMLY_CONTINUOUS; UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS]) in let ABSOLUTELY_CONTINUOUS_IMP_BANACH_SPROPERTY = prove (`!f:real^1->real^N s e. f absolutely_continuous_on s /\ is_interval s /\ &0 < e ==> ?d. &0 < d /\ !t. t SUBSET s /\ measurable t /\ measure t < d ==> measurable(IMAGE f t) /\ measure(IMAGE f t) < e`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP lemma3) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `t:real^1->bool` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `u:real^N->bool` THEN ASM_MESON_TAC[LEBESGUE_MEASURABLE_ABSOLUTELY_CONTINUOUS_IMAGE; MEASURABLE_IMP_LEBESGUE_MEASURABLE]; DISCH_TAC THEN TRANS_TAC REAL_LET_TRANS `measure(u:real^N->bool)` THEN ASM_SIMP_TAC[MEASURE_SUBSET]]) in NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE, LEBESGUE_MEASURABLE_ABSOLUTELY_CONTINUOUS_IMAGE, ABSOLUTELY_CONTINUOUS_IMP_BANACH_SPROPERTY;; let MEASURABLE_ABSOLUTELY_CONTINUOUS_IMAGE = prove (`!f:real^1->real^N s t. f absolutely_continuous_on s /\ is_interval s /\ measurable t /\ t SUBSET s ==> measurable(IMAGE f t)`, REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN REWRITE_TAC[IMP_IMP] THEN MP_TAC(ISPECL [`f:real^1->real^N`; `s:real^1->bool`; `&1`] ABSOLUTELY_CONTINUOUS_IMP_BANACH_SPROPERTY) THEN ASM_REWRITE_TAC[REAL_LT_01; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN X_GEN_TAC `t:real^1->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `?n. measure(t:real^1->bool) < &2 pow n * d` ASSUME_TAC THENL [ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ] THEN MATCH_MP_TAC REAL_ARCH_POW THEN CONV_TAC REAL_RAT_REDUCE_CONV; FIRST_X_ASSUM(MP_TAC o check (is_conj o concl)) THEN FIRST_X_ASSUM(MP_TAC o check (is_exists o concl)) THEN SPEC_TAC(`t:real^1->bool`,`t:real^1->bool`) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM]] THEN MATCH_MP_TAC num_INDUCTION THEN ASM_SIMP_TAC[real_pow; REAL_MUL_LID] THEN FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl)) THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[IMP_IMP] THEN DISCH_TAC THEN X_GEN_TAC `t:real^1->bool` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HALF_MEASURES) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^1->bool`; `v:real^1->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `v:real^1->bool` th) THEN MP_TAC(SPEC `u:real^1->bool` th)) THEN ASM_REWRITE_TAC[REAL_ARITH `x / &2 < p * d <=> x < (&2 * p) * d`] THEN REPEAT(ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC]) THEN EXPAND_TAC "t" THEN REWRITE_TAC[IMAGE_UNION] THEN ASM_SIMP_TAC[MEASURABLE_UNION]);; let BANACH_ZARECKI = prove (`!f:real^1->real^1 a b. f absolutely_continuous_on interval[a,b] <=> f continuous_on interval[a,b] /\ f has_bounded_variation_on interval[a,b] /\ !t. t SUBSET interval[a,b] /\ negligible t ==> negligible(IMAGE f t)`, REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL [ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_HAS_BOUNDED_VARIATION_ON; ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS; IS_INTERVAL_INTERVAL; BOUNDED_INTERVAL] THEN ASM_MESON_TAC[NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE; IS_INTERVAL_INTERVAL]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_BOUNDED_VARIATION_DERIVATIVE) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`f':real^1->real^1`; `n:real^1->bool`] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_NORM) THEN DISCH_THEN(MP_TAC o MATCH_MP ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_RIGHT) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] ABSOLUTELY_CONTINUOUS_COMPARISON) THEN MAP_EVERY X_GEN_TAC [`s:real^1`; `t:real^1`] THEN STRIP_TAC THEN REWRITE_TAC[dist] THEN TRANS_TAC REAL_LE_TRANS `norm(integral(interval[s,t]) (\x. lift(norm((f':real^1->real^1) x))))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC(NORM_ARITH `a + c:real^N = b ==> norm c <= norm(a - b)`) THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC INTEGRABLE_SUBINTERVAL THEN MAP_EVERY EXISTS_TAC [`a:real^1`; `b:real^1`] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE] THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC] THEN TRANS_TAC REAL_LE_TRANS `measure(IMAGE (f:real^1->real^1) (interval[s,t]))` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM(CONJUNCT1 MEASURE_SEGMENT_1)] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_SEGMENT] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN REWRITE_TAC[COMPACT_INTERVAL]; REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN ASM_SIMP_TAC[FUN_IN_IMAGE; ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1; REAL_LT_IMP_LE] THEN REWRITE_TAC[GSYM CONNECTED_CONVEX_1] THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN REWRITE_TAC[CONNECTED_INTERVAL]] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `measure(IMAGE (f:real^1->real^1) (interval[s,t] DIFF n))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN FIRST_X_ASSUM(MP_TAC o SPEC `n INTER interval[a:real^1,b]`) THEN ANTS_TAC THENL [ASM_MESON_TAC[INTER_SUBSET; NEGLIGIBLE_SUBSET]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN SUBGOAL_THEN `interval[s:real^1,t] SUBSET interval[a,b]` MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC MEASURE_DIFFERENTIABLE_IMAGE THEN EXISTS_TAC `\x h. drop h % (f':real^1->real^1) x` THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; MEASURABLE_DIFF; MEASURABLE_INTERVAL; NEGLIGIBLE_IMP_MEASURABLE] THEN SIMP_TAC[DET_1; matrix; LAMBDA_BETA; DIMINDEX_1; ARITH; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; drop] THEN REWRITE_TAC[REAL_MUL_LID; GSYM drop; GSYM NORM_1] THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_DIFF; IN_INTERVAL_1]) THEN REWRITE_TAC[GSYM has_vector_derivative] THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_AT_WITHIN THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `interval[s:real^1,t]` THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SUBGOAL_THEN `interval[s:real^1,t] SUBSET interval[a,b]` MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_SUBINTERVAL THEN EXISTS_TAC `interval[a:real^1,b]` THEN ASM_REWRITE_TAC[SUBSET_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]; REWRITE_TAC[NORM_1] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> x <= abs y`) THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_SPIKE_SET THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]]);; let BANACH_ZARECKI_GEN = prove (`!f:real^1->real^1 s. is_interval s /\ bounded s ==> (f absolutely_continuous_on s <=> f continuous_on s /\ f has_bounded_variation_on s /\ !t. t SUBSET s /\ negligible t ==> negligible(IMAGE f t))`, REPEAT GEN_TAC THEN STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL [ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_HAS_BOUNDED_VARIATION_ON; ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS] THEN ASM_MESON_TAC[NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE]; ALL_TAC] THEN SUBGOAL_THEN `?a b:real^1. closure s = interval[a,b]` STRIP_ASSUME_TAC THENL [ASM_REWRITE_TAC[GSYM CONNECTED_COMPACT_INTERVAL_1; COMPACT_CLOSURE] THEN MATCH_MP_TAC CONNECTED_CLOSURE THEN ASM_REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1]; ALL_TAC] THEN SUBGOAL_THEN `?g:real^1->real^1. g continuous_on closure s /\ !x. x IN s ==> g x = f x` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `(f:real^1->real^1) uniformly_continuous_on s` MP_TAC THENL [ASM_SIMP_TAC[CONTINUOUS_BV_IMP_UNIFORMLY_CONTINUOUS]; DISCH_THEN(MP_TAC o MATCH_MP UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE) THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS]]; MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_ON_EQ THEN EXISTS_TAC `g:real^1->real^1` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[a:real^1,b]` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[CLOSURE_SUBSET]] THEN REWRITE_TAC[BANACH_ZARECKI] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN REPEAT CONJ_TAC THENL [ASM_REWRITE_TAC[]; MATCH_MP_TAC HAS_BOUNDED_VARIATION_ON_CLOSURE THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HAS_BOUNDED_VARIATION_ON_EQ THEN EXISTS_TAC `f:real^1->real^1` THEN ASM_MESON_TAC[]; X_GEN_TAC `t:real^1->bool` THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^1->real^1) (a INSERT b INSERT (t DIFF {a,b}))` THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[IMAGE_CLAUSES; NEGLIGIBLE_INSERT] THEN FIRST_X_ASSUM(MP_TAC o SPEC `t DIFF {a:real^1,b}`) THEN SUBGOAL_THEN `t DIFF {a:real^1,b} SUBSET s` ASSUME_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET; SUBSET_DIFF]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]] THEN TRANS_TAC SUBSET_TRANS `interior(closure s):real^1->bool` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[INTERIOR_INTERVAL] THEN REWRITE_TAC[OPEN_CLOSED_INTERVAL_1] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s DIFF u SUBSET t DIFF u`) THEN ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET_TRANS]; ASM_MESON_TAC[CONVEX_INTERIOR_CLOSURE; IS_INTERVAL_CONVEX_1; INTERIOR_SUBSET]]]]);; let ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV_GEN = prove (`!f:real^1->real^N s t. is_interval s /\ bounded s /\ f continuous_on s /\ f has_bounded_variation_on s /\ COUNTABLE t /\ (!x. x IN s DIFF t ==> f differentiable (at x within s)) ==> f absolutely_continuous_on s`, REPEAT GEN_TAC THEN ASM_CASES_TAC `COUNTABLE(t:real^1->bool)` THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT; HAS_BOUNDED_VARIATION_ON_COMPONENTWISE; DIFFERENTIABLE_COMPONENTWISE_WITHIN; ABSOLUTELY_CONTINUOUS_ON_COMPONENTWISE] THEN REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_FORALL_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN ABBREV_TAC `g = \x. lift((f:real^1->real^N) x$i)` THEN FIRST_X_ASSUM(K ALL_TAC o SYM) THEN STRIP_TAC THEN ASM_SIMP_TAC[BANACH_ZARECKI_GEN] THEN X_GEN_TAC `c:real^1->bool` THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (g:real^1->real^1) ((c DIFF t) UNION t)` THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[NEGLIGIBLE_UNION_EQ; IMAGE_UNION] THEN ASM_SIMP_TAC[NEGLIGIBLE_COUNTABLE; COUNTABLE_IMAGE] THEN MATCH_MP_TAC NEGLIGIBLE_DIFFERENTIABLE_IMAGE_NEGLIGIBLE THEN REWRITE_TAC[LE_REFL] THEN CONJ_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET; SUBSET_DIFF]; ALL_TAC] THEN REWRITE_TAC[differentiable_on; IN_DIFF] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ASM_MESON_TAC[DIFFERENTIABLE_WITHIN_SUBSET; SUBSET; IN_DIFF]);; let ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV = prove (`!f:real^1->real^N a b. f differentiable_on interval[a,b] /\ f has_bounded_variation_on interval[a,b] ==> f absolutely_continuous_on interval[a,b]`, REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV_GEN THEN EXISTS_TAC `{}:real^1->bool` THEN ASM_REWRITE_TAC[COUNTABLE_EMPTY; DIFF_EMPTY; GSYM differentiable_on] THEN ASM_SIMP_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_ON] THEN REWRITE_TAC[IS_INTERVAL_INTERVAL; BOUNDED_INTERVAL]);; let ABSOLUTELY_CONTINUOUS_ON_COMPOSE = prove (`!f:real^1->real^N g s t. is_interval s /\ bounded s /\ is_interval t /\ bounded t /\ f absolutely_continuous_on t /\ g absolutely_continuous_on s /\ IMAGE g s SUBSET t ==> ((f o g) absolutely_continuous_on s <=> (f o g) has_bounded_variation_on s)`, REPEAT GEN_TAC THEN DISCH_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_HAS_BOUNDED_VARIATION_ON] THEN ONCE_REWRITE_TAC[ABSOLUTELY_CONTINUOUS_ON_COMPONENTWISE; HAS_BOUNDED_VARIATION_ON_COMPONENTWISE] THEN POP_ASSUM MP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o funpow 4 RAND_CONV o LAND_CONV) [ABSOLUTELY_CONTINUOUS_ON_COMPONENTWISE] THEN REWRITE_TAC[RIGHT_AND_FORALL_THM; LEFT_AND_FORALL_THM] THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i /\ i <= dimindex(:N)` THEN ASM_REWRITE_TAC[o_DEF] THEN SUBGOAL_THEN `!h. (\x:real^1. lift(((f:real^1->real^N)(h x))$i)) = (\x. lift(f x$i)) o h` (fun th -> ONCE_REWRITE_TAC[th]) THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN SPEC_TAC(`\x. lift((f:real^1->real^N)x$i)`,`f:real^1->real^1`) THEN POP_ASSUM(K ALL_TAC) THEN REWRITE_TAC[GSYM I_DEF; I_O_ID] THEN GEN_TAC THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_SIMP_TAC[BANACH_ZARECKI_GEN] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_COMPOSE; CONTINUOUS_ON_SUBSET]; REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Indefinite integral of any gauge integrable function at least has the *) (* Luzin N property. Proof (with easy gap fixed up) from Bartle's "A Modern *) (* Theory of Integration" Th 14.20, apparently following Dongfu and Shipan: *) (* "Henstock integrals and Lusin's condition (N)" in Real Analysis Exchange. *) (* ------------------------------------------------------------------------- *) let NEGLIGIBLE_IMAGE_INDEFINITE_INTEGRAL = prove (`!f:real^1->real^1 s a b. f integrable_on interval[a,b] /\ negligible s /\ s SUBSET interval[a,b] ==> negligible (IMAGE (\c. integral(interval[a,c]) f) s)`, SUBGOAL_THEN `!f:real^1->real^1 s a b. f integrable_on interval[a,b] /\ negligible s /\ s SUBSET interval(a,b) ==> negligible (IMAGE (\c. integral(interval[a,c]) f) s)` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN REPEAT(GEN_REWRITE_TAC BINOP_CONV [SWAP_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_TAC THEN X_GEN_TAC `s:real^1->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `s DIFF {a:real^1,b}`) THEN ASM_SIMP_TAC[NEGLIGIBLE_DIFF; OPEN_CLOSED_INTERVAL_1] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP] THEN MATCH_MP_TAC NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (\c. integral(interval [a,c]) (f:real^1->real^1)) {a,b}` THEN REWRITE_TAC[IMAGE_CLAUSES; NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY] THEN SET_TAC[]] THEN SUBGOAL_THEN `!f:real^1->real^1 s a b. f integrable_on interval[a,b] /\ negligible s /\ s SUBSET interval(a,b) /\ (!x. x IN s ==> f x = vec 0) ==> negligible (IMAGE (\c. integral(interval[a,c]) f) s)` MP_TAC THENL [ALL_TAC; REPEAT(GEN_REWRITE_TAC BINOP_CONV [SWAP_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> GEN_TAC THEN STRIP_TAC THEN MP_TAC(SPEC `\x. if ~(x IN s) then (f:real^1->real^1) x else vec 0` th)) THEN ASM_SIMP_TAC[] THEN ANTS_TAC THENL [UNDISCH_TAC `(f:real^1->real^1) integrable_on interval[a,b]` THEN MATCH_MP_TAC INTEGRABLE_SPIKE; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> g x = f x) ==> IMAGE g s = IMAGE f s`) THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC INTEGRAL_SPIKE] THEN EXISTS_TAC `s:real^1->bool` THEN ASM SET_TAC[]] THEN SUBGOAL_THEN `!f:real^1->real^1 s a b. f integrable_on interval[a,b] /\ negligible s /\ s SUBSET interval(a,b) /\ (!x. x IN s ==> f x = vec 0) /\ (!x r. x IN s /\ &0 < r /\ cball(x,r) SUBSET interval[a,b] ==> ~ ?k. IMAGE (\c. integral(interval[a,c]) (f:real^1->real^1)) (cball(x,r)) SUBSET {k}) ==> negligible (IMAGE (\c. integral(interval[a,c]) f) s)` MP_TAC THENL [REPEAT STRIP_TAC; MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN REPEAT(GEN_REWRITE_TAC BINOP_CONV [SWAP_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_TAC THEN X_GEN_TAC `s:real^1->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{x | x IN s /\ !r. x IN s /\ &0 < r /\ cball(x,r) SUBSET interval[a,b] ==> ~ ?k. IMAGE (\c. integral(interval[a,c]) (f:real^1->real^1)) (cball(x,r)) SUBSET {k}}`) THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `s:real^1->bool` THEN ASM_REWRITE_TAC[SUBSET_RESTRICT]; MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC NEGLIGIBLE_SYMDIFF_EQ THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (\c. integral (interval [a,c]) (f:real^1->real^1)) {x | x IN s /\ ?r k. &0 < r /\ cball(x,r) SUBSET interval[a,b] /\ IMAGE (\c. integral (interval [a,c]) f) (cball (x,r)) SUBSET {k}}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE; ABBREV_TAC `triv x r <=> ?k. IMAGE (\c. integral (interval[a,c]) (f:real^1->real^1)) (cball (x,r)) SUBSET {k}` THEN ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN SET_TAC[]] THEN MP_TAC(SPEC `drop o (\c. integral (interval[a,c]) (f:real^1->real^1)) o lift` COUNTABLE_LOCAL_MAXIMA) THEN DISCH_THEN(MP_TAC o ISPEC `lift` o MATCH_MP COUNTABLE_IMAGE) THEN ONCE_REWRITE_TAC[SET_RULE `IMAGE f {g x | P x} = IMAGE (\x. f(g x)) {x | P x}`] THEN REWRITE_TAC[o_THM; LIFT_DROP] THEN REWRITE_TAC[SET_RULE `IMAGE (\x. f(lift x)) s = IMAGE f (IMAGE lift s)`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] COUNTABLE_SUBSET) THEN MATCH_MP_TAC IMAGE_SUBSET THEN GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[IN_ELIM_THM; IN_IMAGE] THEN REWRITE_TAC[MESON[LIFT_DROP] `x = lift y <=> drop x = y`] THEN REWRITE_TAC[UNWIND_THM1] THEN X_GEN_TAC `x:real^1` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP; GSYM DROP_SUB] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:real` THEN DISCH_THEN(X_CHOOSE_THEN `k:real^1` MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o CONJUNCT2)) THEN ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; GSYM NORM_1; IN_SING] THEN DISCH_THEN(fun th -> X_GEN_TAC `x':real^1` THEN DISCH_TAC THEN MP_TAC(SPEC `x:real^1` th) THEN MP_TAC(SPEC `x':real^1` th)) THEN ASM_SIMP_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[IN_CBALL; REAL_LT_IMP_LE; REAL_LE_REFL; NORM_ARITH `dist(x:real^N,x') = norm(x' - x)`]]] THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN ONCE_REWRITE_TAC[MESON[REAL_ARITH `(&0 < e <=> &0 < &4 * e) /\ &4 * e / &4 = e`] `((!e. &0 < e ==> P e) <=> (!e. &0 < e ==> P(&4 * e)))`] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_INTEGRAL_INTEGRAL]) THEN REWRITE_TAC[has_integral] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^1->real^1->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `!x r. x IN s /\ cball(x,r) SUBSET interval[a,b] ==> ?c t. IMAGE (\c. integral(interval[a,c]) (f:real^1->real^1)) (cball(x,r)) = cball(c,t)` MP_TAC THENL [REWRITE_TAC[CBALL_INTERVAL; GSYM EXISTS_LIFT; GSYM FORALL_LIFT] THEN REWRITE_TAC[MESON[VECTOR_ARITH `inv(&2) % (c + d) - inv(&2) % (d - c):real^N = c /\ inv(&2) % (c + d) + inv(&2) % (d - c):real^N = d`] `(?c t:real^1. P (c - t) (c + t)) <=> (?c d. P c d)`] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM IS_INTERVAL_COMPACT; IS_INTERVAL_CONNECTED_1] THEN CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE; MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE] THEN REWRITE_TAC[COMPACT_INTERVAL; CONNECTED_INTERVAL] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET)) THEN MATCH_MP_TAC INDEFINITE_INTEGRAL_CONTINUOUS_RIGHT THEN ASM_REWRITE_TAC[]; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`c:real^1->real->real^1`; `t:real^1->real->real`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`{(x,r) | x IN s /\ &0 < r /\ cball(x,r) SUBSET interval[a,b] /\ cball(x,r) SUBSET (g:real^1->real^1->bool)(x)}`; `\(x,r). (c:real^1->real->real^1) x r`; `\(x,r). (t:real^1->real->real) x r`; `IMAGE (\c. integral (interval [a,c]) (f:real^1->real^1)) s`] VITALI_COVERING_THEOREM_CBALLS) THEN REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC] THEN SUBGOAL_THEN `!x r. x IN s /\ &0 < r /\ cball(x,r) SUBSET interval[a,b] ==> &0 < (t:real^1->real->real) x r` ASSUME_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^1`; `r:real`] THEN STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^1`; `r:real`])) THEN ASM_REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_ARITH `~(&0 < r) <=> r < &0 \/ r = &0`] THEN STRIP_TAC THEN ASM_SIMP_TAC[CBALL_SING; CBALL_EMPTY] THEN SET_TAC[]; ALL_TAC] THEN ANTS_TAC THENL [ASM_SIMP_TAC[] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN X_GEN_TAC `r:real` THEN DISCH_TAC THEN EXISTS_TAC `x:real^1` THEN ASM_REWRITE_TAC[] THEN MP_TAC(fst(EQ_IMP_RULE(ISPEC `(g:real^1->real^1->bool) x INTER {y | y IN interval(a:real^1,b) /\ integral (interval[a,y]) f IN ball(integral (interval[a,x]) f:real^1,r / &2)}` OPEN_CONTAINS_CBALL))) THEN RULE_ASSUM_TAC(REWRITE_RULE[gauge]) THEN ANTS_TAC THENL [MATCH_MP_TAC OPEN_INTER THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONTINUOUS_OPEN_PREIMAGE THEN REWRITE_TAC[OPEN_INTERVAL; OPEN_BALL] THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[a:real^1,b]` THEN REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED] THEN MATCH_MP_TAC INDEFINITE_INTEGRAL_CONTINUOUS_RIGHT THEN ASM_REWRITE_TAC[]; DISCH_THEN(MP_TAC o SPEC `x:real^1`)] THEN ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM; CENTRE_IN_BALL; REAL_HALF] THEN ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN X_GEN_TAC `r':real` THEN REWRITE_TAC[SUBSET_INTER] THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM CONJ_ASSOC] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [MP_TAC(ISPECL [`a:real^1`; `b:real^1`] INTERVAL_OPEN_SUBSET_CLOSED) THEN ASM SET_TAC[]; DISCH_TAC] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^1`; `r':real`])) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `x:real^1` THEN ASM_SIMP_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE]; FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `c SUBSET {y | y IN s /\ f y IN t} ==> IMAGE f c SUBSET t`)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[BALL_1; SUBSET_INTERVAL_1] THEN REWRITE_TAC[DROP_ADD; DROP_SUB; LIFT_DROP] THEN ASM_REAL_ARITH_TAC]; DISCH_THEN(X_CHOOSE_THEN `k:real^1#real->bool` MP_TAC)] THEN REWRITE_TAC[pairwise; FORALL_PAIR_THM; SUBSET; IN_ELIM_PAIR_THM] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[GSYM SUBSET] THEN SUBGOAL_THEN `k:real^1#real->bool = {x,y | (x,y) IN k}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_ELIM_PAIR_THM; FORALL_PAIR_THM]; ONCE_REWRITE_TAC[SET_RULE `{f y | y IN {g x y | P x y}} = {f(g x y) | P x y}`]] THEN REWRITE_TAC[] THEN DISCH_TAC THEN EXISTS_TAC `(IMAGE (\x. integral (interval [a,x]) (f:real^1->real^1)) s DIFF UNIONS {cball (c x r,t x r) | x,r IN k}) UNION UNIONS {cball ((c:real^1->real->real^1) x r,t x r) | x,r IN k}` THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `measurable(UNIONS{cball((c:real^1->real->real^1) x r,t x r) | x,r IN k}) /\ measure(UNIONS {cball (c x r,t x r) | x,r IN k}) <= &4 * e` MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] MEASURABLE_NEGLIGIBLE_SYMDIFF); MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `IMAGE (\x. integral (interval [a,x]) (f:real^1->real^1)) s DIFF UNIONS {cball ((c:real^1->real->real^1) x r,t x r) | x,r IN k}` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN REWRITE_TAC[FORALL_IN_GSPEC; MEASURABLE_CBALL] THEN SUBGOAL_THEN `{cball ((c:real^1->real->real^1) x r,t x r) | x,r IN k} = IMAGE (\(x,r). cball (c x r,t x r)) k` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_ELIM_THM; EXISTS_PAIR_THM; IN_IMAGE] THEN MESON_TAC[]; ASM_SIMP_TAC[COUNTABLE_IMAGE]] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM SUBSET]) THEN X_GEN_TAC `l:real^1#real->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ASM_REWRITE_TAC[FORALL_PAIR_THM; MEASURABLE_CBALL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN TRANS_TAC REAL_LE_TRANS `sum l (\(x,r). (&2 * (t:real^1->real->real) x r) pow dimindex(:1))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_CBALL_BOUND THEN MATCH_MP_TAC REAL_LT_IMP_LE THEN ASM SET_TAC[]; REWRITE_TAC[REAL_POW_1; DIMINDEX_1]] THEN SUBGOAL_THEN `!x:real^1 r. (x,r) IN l ==> ?u v. x IN interval[u,v] /\ interval[u,v] SUBSET cball(x,r) /\ t x r <= norm(integral (interval[u,v]) (f:real^1->real^1))` MP_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^1`; `r:real`] THEN STRIP_TAC THEN SUBGOAL_THEN `(x:real^1,r:real) IN k` (ANTE_RES_THEN MP_TAC) THENL [ASM SET_TAC[]; STRIP_TAC] THEN FIRST_X_ASSUM(K ALL_TAC o SPECL [`x:real^1`; `r:real`; `x:real^1`]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^1`; `r:real`]) THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC]) THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `IMAGE f s = t ==> !y. y IN t ==> ?x. x IN s /\ f x = y`)) THEN DISCH_THEN(fun th -> MP_TAC(SPEC `(c:real^1->real->real^1) x r - lift(t x r)` th) THEN MP_TAC(SPEC `(c:real^1->real->real^1) x r + lift(t x r)` th)) THEN REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(x:real^N,x + r) = norm r`] THEN REWRITE_TAC[NORM_ARITH `dist(x:real^N,x - r) = norm r`] THEN ASM_SIMP_TAC[NORM_LIFT; REAL_ARITH `&0 < x ==> abs x <= x`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^1` ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `v:real^1` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[IMP_IMP; GSYM DROP_EQ; DROP_ADD; DROP_SUB; LIFT_DROP] THEN DISCH_THEN(MP_TAC o SPEC `drop(integral (interval[a,x]) (f:real^1->real^1))` o MATCH_MP (REAL_ARITH `u = x + r /\ v = x - r ==> !y. r <= abs(y - u) \/ r <= abs(y - v)`)) THEN MAP_EVERY UNDISCH_TAC [`dist(x:real^1,v) <= r`; `dist(x:real^1,u) <= r`] THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`v:real^1`; `u:real^1`] THEN ONCE_REWRITE_TAC[TAUT `p ==> q ==> r ==> s <=> r ==> p /\ q ==> s`] THEN MATCH_MP_TAC(MESON[] `(!u v. R u v ==> R v u) /\ (!u v. P u ==> R u v) ==> !u v. P u \/ P v ==> R u v`) THEN CONJ_TAC THENL [MESON_TAC[]; REPEAT STRIP_TAC] THEN DISJ_CASES_TAC(REAL_ARITH `drop u <= drop x \/ drop x <= drop u`) THENL [MAP_EVERY EXISTS_TAC [`u:real^1`; `x:real^1`]; MAP_EVERY EXISTS_TAC [`x:real^1`; `u:real^1`]] THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY_1] THEN (CONJ_TAC THENL [MATCH_MP_TAC(MESON[SUBSET_TRANS; INTERVAL_SUBSET_SEGMENT_1] `segment[a:real^1,b] SUBSET t ==> interval[a,b] SUBSET t`) THEN SIMP_TAC[CONVEX_CONTAINS_SEGMENT_IMP; CONVEX_CBALL] THEN ASM_REWRITE_TAC[IN_CBALL; DIST_REFL] THEN ASM_REAL_ARITH_TAC; ALL_TAC]) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS)) THEN REWRITE_TAC[NORM_1] THENL [MATCH_MP_TAC(REAL_ARITH `u + d = x ==> abs(x - u) <= abs d`); MATCH_MP_TAC(REAL_ARITH `x + d = u ==> abs(x - u) <= abs d`)] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_ADD; LIFT_DROP] THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN ASM_REWRITE_TAC[] THEN (SUBGOAL_THEN `x IN interval(a:real^1,b)` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTERVAL_1] THEN STRIP_TAC]) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN UNDISCH_TAC `cball(x:real^1,r) SUBSET interval [a,b]` THEN REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `u:real^1`) THEN ASM_REWRITE_TAC[IN_CBALL; IN_INTERVAL_1] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] INTEGRABLE_ON_SUBINTERVAL)) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`u:real^1->real->real^1`; `v:real^1->real->real^1`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`; `g:real^1->real^1->bool`; `e:real`] HENSTOCK_LEMMA_PART2) THEN ASM_REWRITE_TAC[DIMINDEX_1; REAL_MUL_LID] THEN DISCH_THEN(MP_TAC o SPEC `IMAGE (\(x:real^1,r:real). x,interval[u x r:real^1,v x r]) l`) THEN REWRITE_TAC[tagged_partial_division_of; GSYM CONJ_ASSOC] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN ANTS_TAC THENL [REWRITE_TAC[IN_IMAGE; EXISTS_PAIR_THM; PAIR_EQ] THEN ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!x r x' r'. (x,r) IN l /\ (x',r') IN l /\ ~(x = x' /\ r = r') ==> DISJOINT (interval[(u:real^1->real->real^1) x r,v x r]) (interval[u x' r',v x' r'])` ASSUME_TAC THENL [MAP_EVERY X_GEN_TAC [`x:real^1`; `r:real`; `x':real^1`; `r':real`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^1`; `r:real`; `x':real^1`; `r':real`]) THEN ASM_REWRITE_TAC[PAIR_EQ] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(SET_RULE `(?f. IMAGE f s SUBSET t /\ IMAGE f s' SUBSET t') ==> DISJOINT t t' ==> DISJOINT s s'`) THEN EXISTS_TAC `\c. integral (interval [a,c]) (f:real^1->real^1)` THEN CONJ_TAC THEN MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ IMAGE f t = v ==> IMAGE f s SUBSET v`) THENL [EXISTS_TAC `cball(x:real^1,r)`; EXISTS_TAC `cball(x':real^1,r')`] THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[IN_IMAGE; EXISTS_PAIR_THM; PAIR_EQ] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(SET_RULE `interior s SUBSET s /\ interior t SUBSET t /\ DISJOINT s t ==> interior s INTER interior t = {}`) THEN REWRITE_TAC[INTERIOR_SUBSET] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[]; ALL_TAC] THEN ANTS_TAC THENL [REWRITE_TAC[fine; IN_IMAGE; EXISTS_PAIR_THM; PAIR_EQ] THEN SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN ASM SET_TAC[]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_IMAGE o lhand o lhand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[FORALL_PAIR_THM; PAIR_EQ] THEN ASM SET_TAC[]; MATCH_MP_TAC(REAL_ARITH `x <= &2 * q ==> p = q ==> p <= &2 * e ==> x <= &4 * e`) THEN REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FORALL_PAIR_THM; o_THM] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `r:real`] THEN STRIP_TAC THEN SUBGOAL_THEN `(f:real^1->real^1) x = vec 0` SUBST1_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[VECTOR_MUL_RZERO]] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_POS] THEN ASM_SIMP_TAC[NORM_ARITH `norm(vec 0 - x:real^N) = norm x`]]);; (* ------------------------------------------------------------------------- *) (* More refined ways of deducing increasing/decreasing/constant status *) (* from the sign of a derivative that may not hold on a set of exceptions. *) (* ------------------------------------------------------------------------- *) let POSITIVE_AE_DERIVATIVE_IMP_NONDECREASING = prove (`!f f' a b s. f continuous_on interval[a,b] /\ interior(IMAGE f s) = {} /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x) /\ &0 < drop(f' x)) ==> !x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f x) <= drop(f y)`, let lemma1 = prove (`!f f' a b s. drop a <= drop b /\ f continuous_on interval[a,b] /\ interior(IMAGE f s) = {} /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x) /\ &0 < drop(f' x)) ==> drop(f a) <= drop(f b)`, REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN SUBGOAL_THEN `~(interval(f b,f a) SUBSET IMAGE (f:real^1->real^1) s)` MP_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP SUBSET_INTERIOR) THEN ASM_REWRITE_TAC[SUBSET_EMPTY; INTERVAL_NE_EMPTY_1; INTERIOR_INTERVAL]; REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_INTERVAL_1] THEN STRIP_TAC] THEN MP_TAC(ISPECL [`drop`; `{x | x IN interval[a,b] /\ (f:real^1->real^1) x = y}`] CONTINUOUS_ATTAINS_SUP) THEN REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID; IN_ELIM_THM] THEN ANTS_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[GSYM IN_SING] THEN MATCH_MP_TAC PROPER_MAP_FROM_COMPACT THEN EXISTS_TAC `(:real^1)` THEN ASM_REWRITE_TAC[SUBSET_UNIV; COMPACT_INTERVAL] THEN REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM CLOSED_IN; CLOSED_SING]; REWRITE_TAC[SET_RULE `~({x | x IN s /\ f x = y} = {}) <=> y IN IMAGE f s`] THEN SUBGOAL_THEN `connected (IMAGE (f:real^1->real^1) (interval[a,b]))` MP_TAC THENL [ASM_MESON_TAC[CONNECTED_CONTINUOUS_IMAGE; CONNECTED_INTERVAL]; REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_1]] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_IN_IMAGE] THEN EXISTS_TAC `b:real^1` THEN ASM_SIMP_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN EXISTS_TAC `a:real^1` THEN ASM_SIMP_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN ASM_REAL_ARITH_TAC]; DISCH_THEN(X_CHOOSE_THEN `c:real^1` MP_TAC) THEN ASM_CASES_TAC `(c:real^1) IN s` THENL [ASM SET_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `c IN interval[a:real^1,b]` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^1`) THEN ASM_REWRITE_TAC[IN_DIFF] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `c IN interval(a:real^1,b)` ASSUME_TAC THENL [SIMP_TAC[OPEN_CLOSED_INTERVAL_1; IN_INSERT; NOT_IN_EMPTY; IN_DIFF] THEN ASM_MESON_TAC[REAL_LT_REFL]; MATCH_MP_TAC(TAUT `F ==> p`)]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [has_vector_derivative]) THEN REWRITE_TAC[has_derivative_at; LIM_AT; DIST_0] THEN DISCH_THEN(MP_TAC o SPEC `drop(f'(c:real^1))` o CONJUNCT2) THEN ASM_REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN SIMP_TAC[dist; REAL_LT_LDIV_EQ] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SUBGOAL_THEN `~(ball(c:real^1,d) INTER interval(c,b) = {})` MP_TAC THENL [SIMP_TAC[BALL_1; DISJOINT_INTERVAL_1; DROP_SUB; DROP_ADD; LIFT_DROP] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; NOT_FORALL_THM; NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^1` THEN SIMP_TAC[IN_INTER; IN_BALL; ONCE_REWRITE_RULE[DIST_SYM] dist] THEN ASM_CASES_TAC `x:real^1 = c` THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ; IN_INTERVAL_1] THEN STRIP_TAC THEN GEN_REWRITE_TAC (funpow 3 RAND_CONV) [NORM_1] THEN ASM_SIMP_TAC[DROP_SUB; real_abs; REAL_SUB_LE; REAL_LT_IMP_LE] THEN REWRITE_TAC[NORM_1; DROP_ADD; DROP_SUB; DROP_CMUL] THEN MATCH_MP_TAC (REAL_ARITH `~(y < x) ==> ~(abs(x - (y + a * b)) < b * a)`) THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `x:real^1`; `b:real^1`; `drop y`; `1`] IVT_DECREASING_COMPONENT_ON_1) THEN ASM_SIMP_TAC[DIMINDEX_1; LE_REFL; GSYM drop; REAL_LT_IMP_LE; NOT_IMP] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_INTERVAL_1; REAL_LE_REFL] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_INTERVAL_1; DROP_EQ] THEN DISCH_THEN(X_CHOOSE_THEN `z:real^1` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:real^1`) THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC]) in REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_CASES_TAC `interval[a:real^1,b] = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY_1]) THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma1 THEN MAP_EVERY EXISTS_TAC [`f':real^1->real^1`; `s:real^1->bool`] THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; X_GEN_TAC `w:real^1` THEN REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; IN_DIFF] THEN ASM_REAL_ARITH_TAC]);; let POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LT_GEN = prove (`!f:real^1->real^1 f' a b s. f continuous_on interval[a,b] /\ interior s = {} /\ interior(IMAGE f s) = {} /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x) /\ &0 < drop(f' x)) ==> !x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x < drop y ==> drop(f x) < drop(f y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `f':real^1->real^1`; `a:real^1`; `b:real^1`; `s:real^1->bool`] POSITIVE_AE_DERIVATIVE_IMP_NONDECREASING) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_LT_LE] THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN SUBGOAL_THEN `!w. w IN interval[x,y] ==> (f:real^1->real^1) w = f x` ASSUME_TAC THENL [REWRITE_TAC[IN_INTERVAL_1; GSYM DROP_EQ; GSYM REAL_LE_ANTISYM] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(SUBST1_TAC o SYM); ALL_TAC] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC; SUBGOAL_THEN `~(interval(x:real^1,y) SUBSET s)` MP_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP SUBSET_INTERIOR) THEN ASM_REWRITE_TAC[INTERIOR_INTERVAL; SUBSET_EMPTY] THEN ASM_REWRITE_TAC[INTERVAL_NE_EMPTY_1]; REWRITE_TAC[SUBSET] THEN X_GEN_TAC `z:real^1` THEN REWRITE_TAC[IN_INTERVAL_1] THEN STRIP_TAC THEN ASM_CASES_TAC `(z:real^1) IN s` THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `(f has_vector_derivative f'(z:real^1)) (at z) /\ &0 < drop(f' z)` MP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_DIFF; IN_INTERVAL_1] THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC; DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `x = &0 ==> &0 < x ==> F`)] THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM] THEN MATCH_MP_TAC VECTOR_DERIVATIVE_UNIQUE_AT THEN MAP_EVERY EXISTS_TAC [`f:real^1->real^1`; `z:real^1`] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_TRANSFORM_WITHIN_OPEN THEN EXISTS_TAC `\w:real^1. (f:real^1->real^1) x` THEN EXISTS_TAC `interval(x:real^1,y)` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; OPEN_INTERVAL; HAS_VECTOR_DERIVATIVE_CONST] THEN ASM_MESON_TAC[IN_INTERVAL_1; REAL_LT_IMP_LE]]]);; let POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LT = prove (`!f:real^1->real^1 f' a b s. f absolutely_continuous_on interval[a,b] /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x) /\ &0 < drop(f' x)) ==> !x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x < drop y ==> drop(f x) < drop(f y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LT_GEN THEN MAP_EVERY EXISTS_TAC [`f':real^1->real^1`; `interval[a:real^1,b] INTER s`] THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS; IS_INTERVAL_INTERVAL] THEN ASM_REWRITE_TAC[SET_RULE `i DIFF (i INTER s) = i DIFF s`] THEN CONJ_TAC THEN MATCH_MP_TAC NEGLIGIBLE_EMPTY_INTERIOR THENL [ALL_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE)) THEN REWRITE_TAC[INTER_SUBSET; IS_INTERVAL_INTERVAL]] THEN ASM_MESON_TAC[NEGLIGIBLE_SUBSET; INTER_SUBSET]);; let POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LE_GEN = prove (`!f:real^1->real^1 f' a b s. f continuous_on interval[a,b] /\ negligible(IMAGE f s) /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x) /\ &0 <= drop(f' x)) ==> !x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f x) <= drop(f y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC POSITIVE_AE_DERIVATIVE_IMP_NONDECREASING THEN EXISTS_TAC `f':real^1->real^1` THEN EXISTS_TAC `s UNION {x | x IN interval[a,b] DIFF s /\ (f':real^1->real^1) x = vec 0}` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `x IN i DIFF (s UNION {x | x IN i DIFF s /\ P x}) <=> x IN i /\ ~P x /\ ~(x IN s)`] THEN ASM_SIMP_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`; IN_DIFF] THEN SIMP_TAC[GSYM DROP_EQ; DROP_VEC] THEN MATCH_MP_TAC NEGLIGIBLE_EMPTY_INTERIOR THEN ASM_REWRITE_TAC[IMAGE_UNION; NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC BABY_SARD THEN EXISTS_TAC `\x h. drop h % (f':real^1->real^1) x` THEN REWRITE_TAC[LE_REFL; GSYM has_vector_derivative] THEN ASM_SIMP_TAC[IN_ELIM_THM; IN_DIFF; HAS_VECTOR_DERIVATIVE_AT_WITHIN] THEN REWRITE_TAC[GSYM DET_EQ_0_RANK; DET_1] THEN SIMP_TAC[matrix; LAMBDA_BETA; LE_REFL; DIMINDEX_1] THEN SIMP_TAC[GSYM drop; DROP_CMUL; REAL_MUL_RZERO]);; let POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LE = prove (`!f:real^1->real^1 f' a b s. f absolutely_continuous_on interval[a,b] /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative f' x) (at x) /\ &0 <= drop(f' x)) ==> !x y. x IN interval[a,b] /\ y IN interval[a,b] /\ drop x <= drop y ==> drop(f x) <= drop(f y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LE_GEN THEN MAP_EVERY EXISTS_TAC [`f':real^1->real^1`; `interval[a:real^1,b] INTER s`] THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS; IS_INTERVAL_INTERVAL] THEN ASM_REWRITE_TAC[SET_RULE `i DIFF (i INTER s) = i DIFF s`] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE)) THEN REWRITE_TAC[INTER_SUBSET; IS_INTERVAL_INTERVAL] THEN ASM_MESON_TAC[NEGLIGIBLE_SUBSET; INTER_SUBSET]);; let ZERO_AE_DERIVATIVE_IMP_CONSTANT_GEN = prove (`!f:real^1->real^1 a b s. f continuous_on interval[a,b] /\ negligible(IMAGE f s) /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative (vec 0)) (at x)) ==> !x. x IN interval[a,b] ==> f x = f a`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real^1`; `(\x. vec 0):real^1->real^1`] POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LE_GEN) THEN MP_TAC(ISPECL [`(--) o (f:real^1->real^1)`; `(--) o (\x. vec 0):real^1->real^1`] POSITIVE_AE_DERIVATIVE_IMP_INCREASING_LE_GEN) THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `b:real^1`; `s:real^1->bool`]) THEN ASM_SIMP_TAC[CONTINUOUS_ON_COMPOSE; IMAGE_o; LINEAR_CONTINUOUS_ON; LINEAR_NEGATION; NEGLIGIBLE_LINEAR_IMAGE] THEN ASM_SIMP_TAC[DROP_VEC; REAL_LE_REFL; o_DEF; HAS_VECTOR_DERIVATIVE_NEG] THEN REWRITE_TAC[DROP_NEG; DROP_VEC; REAL_LE_REFL; REAL_NEG_0; REAL_LE_NEG2] THEN REWRITE_TAC[AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPECL [`a:real^1`; `x:real^1`]) THEN RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN REWRITE_TAC[GSYM DROP_EQ; IN_INTERVAL_1; GSYM REAL_LE_ANTISYM] THEN ASM_REAL_ARITH_TAC);; let ZERO_AE_DERIVATIVE_IMP_CONSTANT = prove (`!f:real^1->real^1 a b s. f absolutely_continuous_on interval[a,b] /\ negligible s /\ (!x. x IN interval[a,b] DIFF s ==> (f has_vector_derivative (vec 0)) (at x)) ==> !x. x IN interval[a,b] ==> f x = f a`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC ZERO_AE_DERIVATIVE_IMP_CONSTANT_GEN THEN EXISTS_TAC `interval[a:real^1,b] INTER s` THEN ASM_SIMP_TAC[ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS; IS_INTERVAL_INTERVAL] THEN ASM_REWRITE_TAC[SET_RULE `i DIFF (i INTER s) = i DIFF s`] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE)) THEN REWRITE_TAC[INTER_SUBSET; IS_INTERVAL_INTERVAL] THEN ASM_MESON_TAC[NEGLIGIBLE_SUBSET; INTER_SUBSET]);; (* ------------------------------------------------------------------------- *) (* Convergence in measure implies convergence AE of a subsequence. *) (* ------------------------------------------------------------------------- *) let CONVERGENCE_IN_MEASURE = prove (`!f:num->real^M->real^N g s. (!n. f n measurable_on s) /\ (!e. &0 < e ==> eventually (\n. ?t. {x | x IN s /\ dist(f n x,g x) >= e} SUBSET t /\ measurable t /\ measure t < e) sequentially) ==> ?r t. (!m n:num. m < n ==> r m < r n) /\ negligible t /\ t SUBSET s /\ !x. x IN s DIFF t ==> ((\n. f (r n) x) --> g x) sequentially`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?r. (!n. ?t. {x | x IN s /\ dist(f (r n) x,(g:real^M->real^N) x) >= inv(&2 pow n)} SUBSET t /\ measurable t /\ measure t < inv(&2 pow n)) /\ (!n. r n :num < r(SUC n))` MP_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `&1`); MAP_EVERY X_GEN_TAC [`n:num`; `p:num`] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `inv(&2 pow (SUC n))`)] THEN ASM_REWRITE_TAC[REAL_LT_01; REAL_LT_INV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THENL [MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN DISCH_THEN(MP_TAC o SPEC `m:num`) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[LE_REFL]; DISCH_THEN(X_CHOOSE_THEN `m:num` (MP_TAC o SPEC `m + p + 1:num`)) THEN DISCH_THEN(fun th -> EXISTS_TAC `m + p + 1:num` THEN MP_TAC th) THEN REWRITE_TAC[LE_ADD; ARITH_RULE `p < m + p + 1`]]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN X_GEN_TAC `t:num->real^M->bool` THEN STRIP_TAC] THEN EXISTS_TAC `s INTER INTERS {UNIONS {(t:num->real^M->bool) k | n <= k} | n IN (:num)}` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC TRANSITIVE_STEPWISE_LT THEN ASM_REWRITE_TAC[] THEN ARITH_TAC; MATCH_MP_TAC NEGLIGIBLE_INTER THEN DISJ2_TAC THEN SIMP_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e / &2`] REAL_ARCH_POW_INV) THEN ASM_REWRITE_TAC[REAL_POW_INV; REAL_HALF] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `UNIONS {(t:num->real^M->bool) k | N <= k}` THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `x IN s ==> INTERS s SUBSET x`) THEN SET_TAC[]; ALL_TAC] THEN REWRITE_TAC[LE_EXISTS; SET_RULE `{f n | ?d. n = N + d} = {f(N + n) | n IN (:num)}`] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `n:num` THEN TRANS_TAC REAL_LE_TRANS `sum(0..n) (\k. inv(&2 pow (N + k)))` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE_NUMSEG THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN REWRITE_TAC[REAL_POW_ADD; REAL_INV_MUL; SUM_LMUL; GSYM REAL_POW_INV] THEN REWRITE_TAC[SUM_GP; CONJUNCT1 LT] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[real_div; REAL_MUL_LID; REAL_INV_INV] THEN REWRITE_TAC[REAL_ARITH `x * y * &2 <= e <=> y * x <= e / &2`] THEN REWRITE_TAC[REAL_POW_INV] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `n < e / &2 ==> &0 <= x * n ==> (&1 - x) * n <= e / &2`)) THEN REWRITE_TAC[GSYM REAL_INV_MUL; REAL_LE_INV_EQ; GSYM REAL_POW_ADD] THEN SIMP_TAC[REAL_POW_LE; REAL_POS]; REWRITE_TAC[INTER_SUBSET]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[SET_RULE `s DIFF (s INTER t) = s DIFF t`] THEN REWRITE_TAC[IN_DIFF; INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[NOT_FORALL_THM; NOT_EXISTS_THM] THEN REWRITE_TAC[TAUT `~(a /\ b) <=> a ==> ~b`] THEN DISCH_THEN(X_CHOOSE_THEN `N:num` (LABEL_TAC "*")) THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN ASM_REWRITE_TAC[REAL_POW_INV] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_THEN `M:num` STRIP_ASSUME_TAC) THEN EXISTS_TAC `N + M:num` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `n:num`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [SUBSET]) THEN DISCH_THEN(MP_TAC o SPECL [`n:num`; `x:real^M`]) THEN ASM_REWRITE_TAC[IN_ELIM_THM; real_ge; REAL_NOT_LE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LT_TRANS) THEN TRANS_TAC REAL_LET_TRANS `inv(&2 pow M)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_POW_INV] THEN MATCH_MP_TAC REAL_POW_MONO_INV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_ARITH_TAC]);; let CONVERGENCE_IN_MEASURE_UNIQUE = prove (`!f:num->real^M->real^N g h s. (!n. f n measurable_on s) /\ (!e. &0 < e ==> eventually (\n. ?t. {x | x IN s /\ dist(f n x,g x) >= e} SUBSET t /\ measurable t /\ measure t < e) sequentially) /\ (!e. &0 < e ==> eventually (\n. ?t. {x | x IN s /\ dist(f n x,h x) >= e} SUBSET t /\ measurable t /\ measure t < e) sequentially) ==> negligible {x | x IN s /\ ~(g x = h x)}`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "g") (LABEL_TAC "h")) THEN MP_TAC(ISPECL [`f:num->real^M->real^N`; `g:real^M->real^N`; `s:real^M->bool`] CONVERGENCE_IN_MEASURE) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`r:num->num`; `t:real^M->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`(f:num->real^M->real^N) o (r:num->num)`; `h:real^M->real^N`; `s:real^M->bool`] CONVERGENCE_IN_MEASURE) THEN ASM_REWRITE_TAC[o_THM] THEN ANTS_TAC THENL [X_GEN_TAC `e:real` THEN DISCH_TAC THEN REMOVE_THEN "h" (MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `r:num->num` o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] EVENTUALLY_SUBSEQUENCE)) THEN ASM_REWRITE_TAC[o_DEF]; REWRITE_TAC[LEFT_IMP_EXISTS_THM; o_THM]] THEN MAP_EVERY X_GEN_TAC [`r':num->num`; `t':real^M->bool`] THEN STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `t UNION t':real^M->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; SET_RULE `{x | x IN s /\ ~P x} SUBSET t <=> !x. x IN s DIFF t ==> P x`] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[SET_RULE `x IN s DIFF (t UNION t') <=> x IN s DIFF t /\ x IN s DIFF t'`] THEN STRIP_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `((\n. (f:num->real^M->real^N) n x) o (r:num->num)) o (r':num->num)` THEN REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL [MATCH_MP_TAC LIM_SUBSEQUENCE; ALL_TAC] THEN ASM_SIMP_TAC[o_DEF]);; (* ------------------------------------------------------------------------- *) (* Fubini-type results for measure. *) (* ------------------------------------------------------------------------- *) let FUBINI_MEASURE = prove (`!s:real^(M,N)finite_sum->bool. measurable s ==> negligible {x | ~measurable {y | pastecart x y IN s}} /\ ((\x. lift(measure {y | pastecart x y IN s})) has_integral lift(measure s)) UNIV`, let MEASURE_PASTECART_INTERVAL = prove (`!a b:real^(M,N)finite_sum. (!x. measurable {y | pastecart x y IN interval[a,b]}) /\ ((\x. lift(measure {y | pastecart x y IN interval[a,b]})) has_integral lift(measure(interval[a,b]))) UNIV`, REWRITE_TAC[FORALL_PASTECART] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `c:real^N`; `b:real^M`; `d:real^N`] THEN REWRITE_TAC[GSYM PCROSS_INTERVAL; PASTECART_IN_PCROSS] THEN REWRITE_TAC[SET_RULE `{x | P /\ Q x} = if P then {x | Q x} else {}`] THEN REWRITE_TAC[COND_RAND; SET_RULE `{x | x IN s} = s`] THEN REWRITE_TAC[MEASURABLE_INTERVAL; MEASURABLE_EMPTY; COND_ID] THEN REWRITE_TAC[MEASURE_EMPTY; LIFT_NUM; HAS_INTEGRAL_RESTRICT_UNIV] THEN REWRITE_TAC[PCROSS_INTERVAL; MEASURE_INTERVAL; CONTENT_PASTECART] THEN REWRITE_TAC[LIFT_CMUL; HAS_INTEGRAL_CONST]) in let MEASURE_PASTECART_ELEMENTARY = prove (`!s:real^(M,N)finite_sum->bool. (?d. d division_of s) ==> (!x. measurable {y | pastecart x y IN s}) /\ ((\x. lift(measure {y | pastecart x y IN s})) has_integral lift(measure s)) UNIV`, let lemma = prove (`{x | f x IN UNIONS s} = UNIONS {{x | f x IN d} | d IN s}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in GEN_TAC THEN REWRITE_TAC[division_of; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:(real^(M,N)finite_sum->bool)->bool` THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[lemma] THEN CONJ_TAC THENL [X_GEN_TAC `s:real^M` THEN MATCH_MP_TAC MEASURABLE_UNIONS THEN ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `k:real^(M,N)finite_sum->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `?a b:real^(M,N)finite_sum. k = interval[a,b]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[MEASURE_PASTECART_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `((\x. vsum d (\k. lift(measure {y | pastecart x y IN k}))) has_integral vsum d (\k:real^(M,N)finite_sum->bool. lift(measure k))) UNIV` MP_TAC THENL [MATCH_MP_TAC HAS_INTEGRAL_VSUM THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `k:real^(M,N)finite_sum->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `?a b:real^(M,N)finite_sum. k = interval[a,b]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[MEASURE_PASTECART_INTERVAL]; ALL_TAC] THEN MATCH_MP_TAC(MESON[HAS_INTEGRAL_SPIKE] `!t. negligible t /\ a = b /\ (!x. x IN s DIFF t ==> g x = f x) ==> (f has_integral a) s ==> (g has_integral b) s`) THEN EXISTS_TAC `UNIONS { {x | (x:real^M)$i = fstcart(interval_lowerbound k:real^(M,N)finite_sum)$i} | i IN 1..dimindex(:M) /\ k IN d} UNION UNIONS { {x | x$i = fstcart(interval_upperbound k)$i} | i IN 1..dimindex(:M) /\ k IN d}` THEN CONJ_TAC THENL [REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN CONJ_TAC THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[CONJ_SYM] FINITE_PRODUCT_DEPENDENT; FINITE_NUMSEG] THEN SIMP_TAC[FORALL_IN_GSPEC; NEGLIGIBLE_STANDARD_HYPERPLANE; IN_NUMSEG]; REWRITE_TAC[IN_DIFF; IN_UNIV]] THEN REWRITE_TAC[REWRITE_RULE[o_DEF] (GSYM LIFT_SUM); FUN_EQ_THM; LIFT_EQ] THEN CONJ_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS; GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS_IMAGE] THEN ASM_REWRITE_TAC[GSYM HAS_MEASURE_MEASURE] THEN (CONJ_TAC THENL [ASM_MESON_TAC[MEASURE_PASTECART_INTERVAL; MEASURABLE_INTERVAL]; ALL_TAC]) THEN MAP_EVERY X_GEN_TAC [`k:real^(M,N)finite_sum->bool`; `l:real^(M,N)finite_sum->bool`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`k:real^(M,N)finite_sum->bool`; `l:real^(M,N)finite_sum->bool`]) THEN ASM_REWRITE_TAC[GSYM INTERIOR_INTER] THEN (SUBGOAL_THEN `?a b:real^(M,N)finite_sum c d:real^(M,N)finite_sum. k = interval[a,b] /\ l = interval[c,d]` MP_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN SIMP_TAC[LEFT_IMP_EXISTS_THM; NEGLIGIBLE_CONVEX_INTERIOR; CONVEX_INTER; CONVEX_INTERVAL] THEN REWRITE_TAC[FORALL_PASTECART; GSYM PCROSS_INTERVAL; PASTECART_IN_PCROSS] THEN ONCE_REWRITE_TAC[SET_RULE `{x | P /\ Q x} INTER {x | R /\ S x} = {x | P /\ R} INTER {x | Q x /\ S x}`] THEN REWRITE_TAC[INTER_PCROSS; INTERIOR_PCROSS; GSYM INTER] THEN REWRITE_TAC[SET_RULE `{x | P} = if P then UNIV else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[NEGLIGIBLE_EMPTY; INTER_EMPTY; INTER_UNIV] THEN SIMP_TAC[NEGLIGIBLE_CONVEX_INTERIOR; CONVEX_INTER; CONVEX_INTERVAL] THEN REWRITE_TAC[PCROSS_EQ_EMPTY; TAUT `(if p then q else T) <=> p ==> q`] THEN REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN SIMP_TAC[] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [IN_UNION]) THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; DE_MORGAN_THM; NOT_EXISTS_THM] THEN DISCH_THEN(CONJUNCTS_THEN(fun th -> MP_TAC(SPEC `l:real^(M,N)finite_sum->bool` th) THEN MP_TAC(SPEC `k:real^(M,N)finite_sum->bool` th))) THEN REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[PCROSS_INTERVAL]) THEN REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THEN ASM_REWRITE_TAC[TAUT `~a \/ b <=> a ==> b`] THEN ASM_SIMP_TAC[INTERVAL_LOWERBOUND_NONEMPTY; INTERVAL_UPPERBOUND_NONEMPTY; FSTCART_PASTECART] THEN REPLICATE_TAC 3 (GEN_REWRITE_TAC I [IMP_IMP]) THEN MATCH_MP_TAC(TAUT `(a ==> c ==> ~b) ==> a ==> b ==> c ==> d`) THEN REWRITE_TAC[IN_INTERVAL; INTERVAL_NE_EMPTY; AND_FORALL_THM; INTERIOR_INTERVAL; IMP_IMP; INTER_INTERVAL] THEN MATCH_MP_TAC MONO_FORALL THEN SIMP_TAC[LAMBDA_BETA] THEN GEN_TAC THEN ONCE_REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN REAL_ARITH_TAC) in let MEASURE_PASTECART_OPEN_MEASURABLE = prove (`!s:real^(M,N)finite_sum->bool. open s /\ measurable s ==> negligible {x | ~measurable {y | pastecart x y IN s}} /\ ((\x. lift(measure {y | pastecart x y IN s})) has_integral lift(measure s)) UNIV`, let lemur = prove (`UNIONS {{y | pastecart x y IN g n} | n IN (:num)} = {y | pastecart x y IN UNIONS {g n | n IN (:num)}}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in GEN_TAC THEN STRIP_TAC THEN FIRST_ASSUM(X_CHOOSE_THEN `g:num->real^(M,N)finite_sum->bool` STRIP_ASSUME_TAC o MATCH_MP OPEN_COUNTABLE_LIMIT_ELEMENTARY) THEN SUBGOAL_THEN `!n:num. g n SUBSET (s:real^(M,N)finite_sum->bool)` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`\n:num x:real^M. lift(measure {y:real^N | pastecart x y IN (g n)})`; `(:real^M)`] BEPPO_LEVI_MONOTONE_CONVERGENCE_INCREASING) THEN MP_TAC(GEN `n:num` (ISPEC `(g:num->real^(M,N)finite_sum->bool) n` MEASURE_PASTECART_ELEMENTARY)) THEN ASM_REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL; FORALL_AND_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; LIFT_DROP] THEN ANTS_TAC THENL [CONJ_TAC THENL [REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[MEASURE_PASTECART_ELEMENTARY]; ALL_TAC]) THEN ASM SET_TAC[]; REWRITE_TAC[bounded; FORALL_IN_GSPEC; NORM_LIFT] THEN EXISTS_TAC `measure(s:real^(M,N)finite_sum->bool)` THEN GEN_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x <= y ==> abs x <= y`) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_POS_LE; MATCH_MP_TAC MEASURE_SUBSET] THEN ASM_MESON_TAC[MEASURABLE_ELEMENTARY]]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`f:real^M->real^1`; `t:real^M->bool`] THEN STRIP_TAC THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN SUBGOAL_THEN `!x:real^M. ~(x IN t) ==> {y:real^N | pastecart x y IN s} has_measure drop(f x)` ASSUME_TAC THENL [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONVERGENT_IMP_BOUNDED) THEN REWRITE_TAC[BOUNDED_POS; FORALL_IN_IMAGE; IN_UNIV; NORM_LIFT] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`\n. {y | pastecart x y IN (g:num->real^(M,N)finite_sum->bool) n}`; `B:real`] HAS_MEASURE_NESTED_UNIONS) THEN ASM_SIMP_TAC[lemur; REAL_ARITH `abs x <= B ==> x <= B`] THEN ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN ASM_REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE; GSYM LIFT_EQ] THEN ASM_MESON_TAC[LIM_UNIQUE; TRIVIAL_LIMIT_SEQUENTIALLY; LIFT_DROP]; CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN REWRITE_TAC[measurable] THEN ASM SET_TAC[]; MATCH_MP_TAC HAS_INTEGRAL_SPIKE THEN MAP_EVERY EXISTS_TAC [`f:real^M->real^1`; `t:real^M->bool`] THEN ASM_REWRITE_TAC[NEGLIGIBLE; IN_DIFF; IN_UNIV] THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURE_UNIQUE]; ALL_TAC] THEN ASM_REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `\k. lift(measure ((g:num->real^(M,N)finite_sum->bool) k))` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN MP_TAC(ISPECL [`g:num->real^(M,N)finite_sum->bool`; `measure(s:real^(M,N)finite_sum->bool)`] HAS_MEASURE_NESTED_UNIONS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_MESON_TAC[MEASURABLE_ELEMENTARY; MEASURE_SUBSET]]]) in let MEASURE_PASTECART_COMPACT = prove (`!s:real^(M,N)finite_sum->bool. compact s ==> (!x. measurable {y | pastecart x y IN s}) /\ ((\x. lift(measure {y | pastecart x y IN s})) has_integral lift(measure s)) UNIV`, GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [GEN_TAC THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN REWRITE_TAC[BOUNDED_POS; FORALL_IN_GSPEC] THEN MESON_TAC[NORM_LE_PASTECART; REAL_LE_TRANS]; MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CONTINUOUS_PASTECART; CONTINUOUS_CONST; CONTINUOUS_AT_ID]]; DISCH_TAC] THEN SUBGOAL_THEN `?t:real^(M,N)finite_sum->bool. open t /\ measurable t /\ s SUBSET t` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[BOUNDED_SUBSET_BALL; COMPACT_IMP_BOUNDED; MEASURABLE_BALL; OPEN_BALL]; ALL_TAC] THEN MP_TAC(ISPEC `t:real^(M,N)finite_sum->bool` MEASURE_PASTECART_OPEN_MEASURABLE) THEN MP_TAC(ISPEC `t DIFF s:real^(M,N)finite_sum->bool` MEASURE_PASTECART_OPEN_MEASURABLE) THEN ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_COMPACT; OPEN_DIFF; COMPACT_IMP_CLOSED; MEASURE_DIFF_SUBSET; IMP_IMP] THEN DISCH_THEN(CONJUNCTS_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN REWRITE_TAC[LIFT_SUB; IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_INTEGRAL_SUB) THEN REWRITE_TAC[VECTOR_ARITH `t - (t - s):real^1 = s`] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HAS_INTEGRAL_SPIKE)) THEN EXISTS_TAC `{x | ~measurable {y | pastecart x y IN t DIFF s}} UNION {x:real^M | ~measurable {y:real^N | pastecart x y IN t}}` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; IN_DIFF; IN_UNIV] THEN X_GEN_TAC `x:real^M` THEN SIMP_TAC[IN_UNION; IN_ELIM_THM; DE_MORGAN_THM] THEN STRIP_TAC THEN REWRITE_TAC[LIFT_EQ; GSYM LIFT_SUB] THEN ONCE_REWRITE_TAC[REAL_ARITH `a:real = b - c <=> c = b - a`] THEN REWRITE_TAC[SET_RULE `{y | pastecart x y IN t /\ ~(pastecart x y IN s)} = {y | pastecart x y IN t} DIFF {y | pastecart x y IN s}`] THEN MATCH_MP_TAC MEASURE_DIFF_SUBSET THEN ASM SET_TAC[]) in GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN `?f. (!n. compact(f n) /\ f n SUBSET s /\ measurable(f n) /\ measure s < measure(f n) + inv(&n + &1)) /\ (!n. (f:num->real^(M,N)finite_sum->bool) n SUBSET f(SUC n))` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_INNER_COMPACT THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `t:real^(M,N)finite_sum->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^(M,N)finite_sum->bool`; `inv(&(SUC n) + &1)`] MEASURABLE_INNER_COMPACT) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^(M,N)finite_sum->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t UNION u:real^(M,N)finite_sum->bool` THEN ASM_SIMP_TAC[COMPACT_UNION; UNION_SUBSET; MEASURABLE_UNION] THEN REWRITE_TAC[SUBSET_UNION] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `s < a + e ==> a <= b ==> s < b + e`)) THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_UNION; SUBSET_UNION]; ALL_TAC] THEN SUBGOAL_THEN `?g. (!n. open(g n) /\ s SUBSET g n /\ measurable(g n) /\ measure(g n) < measure s + inv(&n + &1)) /\ (!n. (g:num->real^(M,N)finite_sum->bool) (SUC n) SUBSET g n)` STRIP_ASSUME_TAC THENL [MATCH_MP_TAC DEPENDENT_CHOICE THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_OUTER_OPEN THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`n:num`; `t:real^(M,N)finite_sum->bool`] THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^(M,N)finite_sum->bool`; `inv(&(SUC n) + &1)`] MEASURABLE_OUTER_OPEN) THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^(M,N)finite_sum->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `t INTER u:real^(M,N)finite_sum->bool` THEN ASM_SIMP_TAC[OPEN_INTER; SUBSET_INTER; MEASURABLE_INTER] THEN REWRITE_TAC[INTER_SUBSET] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `a < s + e ==> b <= a ==> b < s + e`)) THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_INTER; INTER_SUBSET]; ALL_TAC] THEN MP_TAC(ISPECL [`\n:num x:real^M. lift(measure {y:real^N | pastecart x y IN (g n)}) - lift(measure {y:real^N | pastecart x y IN (f n)})`; `(:real^M)`] BEPPO_LEVI_MONOTONE_CONVERGENCE_DECREASING_AE) THEN MP_TAC(GEN `n:num` (ISPEC `(f:num->real^(M,N)finite_sum->bool) n` MEASURE_PASTECART_COMPACT)) THEN MP_TAC(GEN `n:num` (ISPEC `(g:num->real^(M,N)finite_sum->bool) n` MEASURE_PASTECART_OPEN_MEASURABLE)) THEN ASM_REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL; FORALL_AND_THM] THEN STRIP_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; DROP_SUB; LIFT_DROP] THEN ASM_SIMP_TAC[INTEGRABLE_SUB; INTEGRAL_SUB] THEN ANTS_TAC THENL [CONJ_TAC THENL [X_GEN_TAC `n:num` THEN EXISTS_TAC `{x:real^M | ~measurable {y:real^N | pastecart x y IN g n}} UNION {x:real^M | ~measurable {y | pastecart x y IN g (SUC n)}}` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; IN_UNION; DE_MORGAN_THM] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `f <= f' /\ g' <= g ==> g' - f' <= g - f`) THEN CONJ_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN EXISTS_TAC `measure((g:num->real^(M,N)finite_sum->bool) 0) - measure((f:num->real^(M,N)finite_sum->bool) 0)` THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT] THEN MATCH_MP_TAC(REAL_ARITH `!s. f' <= s /\ s <= g' /\ f <= f' /\ g' <= g ==> abs(g' - f') <= g - f`) THEN EXISTS_TAC `measure(s:real^(M,N)finite_sum->bool)` THEN REPEAT CONJ_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN MP_TAC(ARITH_RULE `0 <= n`) THEN SPEC_TAC(`n:num`,`n:num`) THEN SPEC_TAC(`0`,`m:num`) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN SET_TAC[]]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN MAP_EVERY X_GEN_TAC [`h:real^M->real^1`; `k:real^M->bool`] THEN STRIP_TAC THEN SUBGOAL_THEN `?t. negligible t /\ (!n x. ~(x IN t) ==> measurable {y:real^N | pastecart x y IN g n}) /\ (!x. ~(x IN t) ==> ((\k. lift(measure {y | pastecart x y IN g k}) - lift(measure {y:real^N | pastecart x y IN f k})) --> vec 0) sequentially) /\ (!x. ~(x IN t) ==> (h:real^M->real^1) x = vec 0)` MP_TAC THENL [MP_TAC(ISPECL [`\x. if x IN UNIONS{ {x | ~measurable {y:real^N | pastecart x y IN g n}} | n IN (:num)} UNION k then vec 0 else (h:real^M->real^1) x`; `(:real^M)`] HAS_INTEGRAL_NEGLIGIBLE_EQ) THEN REWRITE_TAC[IN_UNIV; DIMINDEX_1; FORALL_1] THEN ANTS_TAC THENL [X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN COND_CASES_TAC THEN REWRITE_TAC[VEC_COMPONENT; REAL_LE_REFL] THEN FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [DE_MORGAN_THM]) THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_COMPONENT_LBOUND) THEN EXISTS_TAC `\k:num. lift(measure {y | pastecart x y IN (g:num->real^(M,N)finite_sum->bool) k}) - lift(measure {y | pastecart x y IN (f:num->real^(M,N)finite_sum->bool) k})` THEN REWRITE_TAC[DIMINDEX_1; TRIVIAL_LIMIT_SEQUENTIALLY; LE_REFL] THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[GSYM drop; DROP_SUB; LIFT_DROP] THEN REWRITE_TAC[REAL_SUB_LE] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[UNIONS_GSPEC]) THEN ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN ANTS_TAC THENL [MATCH_MP_TAC HAS_INTEGRAL_SPIKE THEN EXISTS_TAC `h:real^M->real^1` THEN EXISTS_TAC `UNIONS{ {x | ~measurable {y | pastecart x y IN (g:num->real^(M,N)finite_sum->bool) n}} | n IN (:num)} UNION k` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; IN_DIFF; IN_UNION; IN_UNIV] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IN_UNIV] NEGLIGIBLE_COUNTABLE_UNIONS) THEN ASM_REWRITE_TAC[]; MESON_TAC[]; ASM_REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `\k. lift(measure((g:num->real^(M,N)finite_sum->bool) k)) - lift(measure((f:num->real^(M,N)finite_sum->bool) k))` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN REWRITE_TAC[LIM_SEQUENTIALLY; GSYM LIFT_SUB; DIST_0; NORM_LIFT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e / &2` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `!s d. f <= s /\ s <= g /\ s < f + d /\ g < s + d /\ d <= e / &2 ==> abs(g - f) < e`) THEN EXISTS_TAC `measure(s:real^(M,N)finite_sum->bool)` THEN EXISTS_TAC `inv(&n + &1)` THEN ASM_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURE_SUBSET]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `inv(&N)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC]; DISCH_TAC THEN EXISTS_TAC `{x | ~((if x IN UNIONS {{x | ~measurable {y | pastecart x y IN g n}} | n | T} UNION k then vec 0 else (h:real^M->real^1) x) = vec 0)} UNION UNIONS {{x | ~measurable {y | pastecart x y IN (g:num->real^(M,N)finite_sum->bool) n}} | n | T} UNION k` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN ASM_SIMP_TAC[IN_UNION; DE_MORGAN_THM] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[IN_UNIV] NEGLIGIBLE_COUNTABLE_UNIONS) THEN ASM_REWRITE_TAC[]; CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]]]; FIRST_X_ASSUM(K ALL_TAC o SPEC `x:real^M`) THEN STRIP_TAC] THEN SUBGOAL_THEN `!x:real^M. ~(x IN t) ==> measurable {y:real^N | pastecart x y IN s}` ASSUME_TAC THENL [REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[MEASURABLE_INNER_OUTER] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `x:real^M` th) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [LIM_SEQUENTIALLY]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_SIMP_TAC[DIST_0] THEN DISCH_THEN(X_CHOOSE_THEN `N:num` (MP_TAC o SPEC `N:num`)) THEN REWRITE_TAC[LE_REFL; GSYM LIFT_SUB; NORM_LIFT] THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`{y | pastecart x y IN (f:num->real^(M,N)finite_sum->bool) N}`; `{y | pastecart x y IN (g:num->real^(M,N)finite_sum->bool) N}`] THEN ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[REAL_ABS_SUB] THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `t:real^M->bool` THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`\n:num x:real^M. lift(measure {y:real^N | pastecart x y IN (g n)})`; `\x:real^M. lift(measure {y:real^N | pastecart x y IN s})`; `(:real^M)`; `t:real^M->bool`] MONOTONE_CONVERGENCE_DECREASING_AE) THEN ASM_REWRITE_TAC[LIFT_DROP; IN_UNIV; IN_DIFF] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[IN_DIFF] THEN ASM SET_TAC[]; X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `x:real^M` th) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [LIM_SEQUENTIALLY]) THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_SIMP_TAC[DIST_0] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[DIST_LIFT; GSYM dist] THEN MATCH_MP_TAC(REAL_ARITH `f <= s /\ s <= g ==> abs(g - f) < e ==> abs(g - s) < e`) THEN CONJ_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[IN_DIFF] THEN ASM SET_TAC[]; REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN EXISTS_TAC `measure((g:num->real^(M,N)finite_sum->bool) 0)` THEN ASM_SIMP_TAC[NORM_LIFT; real_abs; MEASURE_POS_LE] THEN X_GEN_TAC `m:num` THEN MP_TAC(ARITH_RULE `0 <= m`) THEN SPEC_TAC(`m:num`,`m:num`) THEN SPEC_TAC(`0`,`n:num`) THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN REPEAT(CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC]) THEN GEN_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[]]; STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN EXISTS_TAC `\k. lift(measure((g:num->real^(M,N)finite_sum->bool) k))` THEN ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN REWRITE_TAC[LIM_SEQUENTIALLY; DIST_LIFT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC `e:real` REAL_ARCH_INV) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH `!d. g < s + d /\ s <= g /\ d < e ==> abs(g - s) < e`) THEN EXISTS_TAC `inv(&n + &1)` THEN ASM_SIMP_TAC[MEASURE_SUBSET] THEN TRANS_TAC REAL_LET_TRANS `inv(&N)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC]);; let FUBINI_MEASURE_ALT = prove (`!s:real^(M,N)finite_sum->bool. measurable s ==> negligible {y | ~measurable {x | pastecart x y IN s}} /\ ((\y. lift(measure {x | pastecart x y IN s})) has_integral lift(measure s)) UNIV`, GEN_TAC THEN DISCH_TAC THEN MP_TAC(ISPEC `IMAGE (\z. pastecart (sndcart z) (fstcart z)) (s:real^(M,N)finite_sum->bool)` FUBINI_MEASURE) THEN MP_TAC(ISPEC `\z:real^(M,N)finite_sum. pastecart (sndcart z) (fstcart z)` HAS_MEASURE_ISOMETRY) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL [REWRITE_TAC[DIMINDEX_FINITE_SUM; ADD_SYM] THEN SIMP_TAC[LINEAR_PASTECART; LINEAR_FSTCART; LINEAR_SNDCART] THEN SIMP_TAC[FORALL_PASTECART; NORM_EQ; GSYM NORM_POW_2; SQNORM_PASTECART] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; REAL_ADD_AC]; DISCH_TAC THEN ASM_REWRITE_TAC[measurable; measure] THEN ASM_REWRITE_TAC[GSYM measurable; GSYM measure] THEN REWRITE_TAC[IN_IMAGE; EXISTS_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_INJ] THEN REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; UNWIND_THM1]]);; let FUBINI_LEBESGUE_MEASURABLE = prove (`!s:real^(M,N)finite_sum->bool. lebesgue_measurable s ==> negligible {x | ~lebesgue_measurable {y | pastecart x y IN s}}`, let lemma = prove (`{x | ?n. P n x} = UNIONS {{x | P n x} | n IN (:num)}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_COUNTABLE_INTERVALS] THEN X_GEN_TAC `m:num` THEN REWRITE_TAC[LEBESGUE_MEASURABLE_MEASURABLE_ON_COUNTABLE_SUBINTERVALS] THEN REWRITE_TAC[INTER; IN_ELIM_THM; NOT_FORALL_THM; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[lemma] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `n:num` THEN MP_TAC(ISPEC `(s:real^(M,N)finite_sum->bool) INTER (interval[--vec m,vec m] PCROSS interval[--vec n,vec n])` FUBINI_MEASURE) THEN ANTS_TAC THENL [REWRITE_TAC[PCROSS_INTERVAL] THEN ASM_MESON_TAC[LEBESGUE_MEASURABLE_MEASURABLE_ON_SUBINTERVALS]; DISCH_THEN(MP_TAC o CONJUNCT1)] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INTER; PASTECART_IN_PCROSS] THEN ASM_CASES_TAC `(x:real^M) IN interval[--vec m,vec m]` THEN ASM_REWRITE_TAC[EMPTY_GSPEC; MEASURABLE_EMPTY]);; let FUBINI_LEBESGUE_MEASURABLE_ALT = prove (`!s:real^(M,N)finite_sum->bool. lebesgue_measurable s ==> negligible {y | ~lebesgue_measurable {x | pastecart x y IN s}}`, let lemma = prove (`{x | ?n. P n x} = UNIONS {{x | P n x} | n IN (:num)}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[NEGLIGIBLE_ON_COUNTABLE_INTERVALS] THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[LEBESGUE_MEASURABLE_MEASURABLE_ON_COUNTABLE_SUBINTERVALS] THEN REWRITE_TAC[INTER; IN_ELIM_THM; NOT_FORALL_THM; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[lemma] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `m:num` THEN MP_TAC(ISPEC `(s:real^(M,N)finite_sum->bool) INTER (interval[--vec m,vec m] PCROSS interval[--vec n,vec n])` FUBINI_MEASURE_ALT) THEN ANTS_TAC THENL [REWRITE_TAC[PCROSS_INTERVAL] THEN ASM_MESON_TAC[LEBESGUE_MEASURABLE_MEASURABLE_ON_SUBINTERVALS]; DISCH_THEN(MP_TAC o CONJUNCT1)] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_INTER; PASTECART_IN_PCROSS] THEN ASM_CASES_TAC `(y:real^N) IN interval[--vec n,vec n]` THEN ASM_REWRITE_TAC[EMPTY_GSPEC; MEASURABLE_EMPTY]);; let FUBINI_NEGLIGIBLE = prove (`!s. negligible s ==> negligible {x:real^M | ~negligible {y:real^N | pastecart x y IN s}}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_MEASURE o MATCH_MP NEGLIGIBLE_IMP_MEASURABLE) THEN ASM_SIMP_TAC[MEASURE_EQ_0; LIFT_NUM; IMP_CONJ] THEN DISCH_TAC THEN MP_TAC(ISPECL [`\x:real^M. lift (measure {y:real^N | pastecart x y IN s})`; `(:real^M)`; `{x:real^M | ~measurable {y:real^N | pastecart x y IN s}}`] HAS_INTEGRAL_NEGLIGIBLE_EQ_AE) THEN ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; IN_DIFF; IN_ELIM_THM] THEN SIMP_TAC[IMP_IMP; FORALL_1; DIMINDEX_1; GSYM drop; LIFT_DROP; IN_UNIV] THEN ASM_SIMP_TAC[MEASURE_POS_LE; IMP_CONJ] THEN DISCH_THEN(K ALL_TAC) THEN UNDISCH_TAC `negligible {x:real^M | ~measurable {y:real^N | pastecart x y IN s}}` THEN REWRITE_TAC[IMP_IMP; GSYM NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNION; GSYM DROP_EQ] THEN REWRITE_TAC[LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[HAS_MEASURE_MEASURE; GSYM HAS_MEASURE_0] THEN SET_TAC[]);; let FUBINI_NEGLIGIBLE_ALT = prove (`!s. negligible s ==> negligible {y:real^N | ~negligible {x:real^M | pastecart x y IN s}}`, let lemma = prove (`!s:real^(M,N)finite_sum->bool. negligible s ==> negligible (IMAGE (\z. pastecart (sndcart z) (fstcart z)) s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_LINEAR_IMAGE_GEN THEN ASM_REWRITE_TAC[DIMINDEX_FINITE_SUM; ADD_SYM; LE_REFL] THEN REWRITE_TAC[linear; FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART; FSTCART_ADD; SNDCART_ADD; FSTCART_CMUL; SNDCART_CMUL; GSYM PASTECART_ADD; GSYM PASTECART_CMUL]) in GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_NEGLIGIBLE) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[IN_IMAGE; EXISTS_PASTECART; PASTECART_INJ; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM] THEN REWRITE_TAC[UNWIND_THM1; UNWIND_THM2]);; let NEGLIGIBLE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. negligible(s PCROSS t) <=> negligible s \/ negligible t`, REPEAT(STRIP_TAC ORELSE EQ_TAC) THENL [FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_NEGLIGIBLE) THEN REWRITE_TAC[PASTECART_IN_PCROSS] THEN REWRITE_TAC[SET_RULE `{y | P /\ Q y} = if P then {y | Q y} else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[NEGLIGIBLE_EMPTY] THEN ASM_CASES_TAC `negligible(t:real^N->bool)` THEN ASM_REWRITE_TAC[SET_RULE `~(if P then F else T) = P`; SET_RULE `{x | x IN s} = s`]; ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN REWRITE_TAC[FORALL_PASTECART; GSYM PCROSS_INTERVAL; INTER_PCROSS] THEN MAP_EVERY X_GEN_TAC [`aa:real^M`; `a:real^N`; `bb:real^M`; `b:real^N`] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `(s:real^M->bool) PCROSS interval[a:real^N,b]` THEN REWRITE_TAC[SUBSET_PCROSS; INTER_SUBSET] THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`s:real^M->bool`; `e / (content(interval[a:real^N,b]) + &1)`] MEASURABLE_OUTER_CLOSED_INTERVALS) THEN ASM_SIMP_TAC[NEGLIGIBLE_IMP_MEASURABLE; REAL_LT_DIV; CONTENT_POS_LE; MEASURE_EQ_0; REAL_ADD_LID; REAL_ARITH `&0 <= x ==> &0 < x + &1`] THEN DISCH_THEN(X_CHOOSE_THEN `d:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `UNIONS { (k:real^M->bool) PCROSS interval[a:real^N,b] | k IN d}` THEN ASM_REWRITE_TAC[GSYM PCROSS_UNIONS; SUBSET_PCROSS; SUBSET_REFL] THEN REWRITE_TAC[PCROSS_UNIONS] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL; PCROSS_INTERVAL]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `D:(real^M->bool)->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL; PCROSS_INTERVAL; SUBSET]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)] THEN TRANS_TAC REAL_LE_TRANS `sum D (\k:real^M->bool. measure k * content(interval[a:real^N,b]))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `?u v:real^M. k = interval[u,v]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET]; ASM_REWRITE_TAC[]] THEN ASM_REWRITE_TAC[PCROSS_INTERVAL; MEASURE_INTERVAL; CONTENT_PASTECART]; REWRITE_TAC[SUM_RMUL]] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x * (y + &1) <= e ==> x * y <= e`) THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_POS_LE THEN ASM_MESON_TAC[MEASURE_POS_LE; SUBSET; MEASURABLE_INTERVAL]; SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_ARITH `&0 <= x ==> &0 < x + &1`; CONTENT_POS_LE]] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN TRANS_TAC REAL_LE_TRANS `measure(UNIONS D:real^M->bool)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[SUBSET_UNIONS] THEN ASM_MESON_TAC[MEASURABLE_UNIONS; MEASURABLE_INTERVAL; SUBSET]] THEN TRANS_TAC EQ_TRANS `sum (D:(real^M->bool)->bool) content` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[MEASURE_INTERVAL; SUBSET]; CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_ELEMENTARY THEN REWRITE_TAC[division_of] THEN ASM SET_TAC[]]; ONCE_REWRITE_TAC[NEGLIGIBLE_ON_INTERVALS] THEN REWRITE_TAC[FORALL_PASTECART; GSYM PCROSS_INTERVAL; INTER_PCROSS] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `aa:real^N`; `b:real^M`; `bb:real^N`] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `interval[a:real^M,b] PCROSS (t:real^N->bool)` THEN REWRITE_TAC[SUBSET_PCROSS; INTER_SUBSET] THEN REWRITE_TAC[NEGLIGIBLE_OUTER_LE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPECL [`t:real^N->bool`; `e / (content(interval[a:real^M,b]) + &1)`] MEASURABLE_OUTER_CLOSED_INTERVALS) THEN ASM_SIMP_TAC[NEGLIGIBLE_IMP_MEASURABLE; REAL_LT_DIV; CONTENT_POS_LE; MEASURE_EQ_0; REAL_ADD_LID; REAL_ARITH `&0 <= x ==> &0 < x + &1`] THEN DISCH_THEN(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN EXISTS_TAC `UNIONS { interval[a:real^M,b] PCROSS (k:real^N->bool) | k IN d}` THEN ASM_REWRITE_TAC[GSYM PCROSS_UNIONS; SUBSET_PCROSS; SUBSET_REFL] THEN REWRITE_TAC[PCROSS_UNIONS] THEN MATCH_MP_TAC MEASURE_COUNTABLE_UNIONS_LE_STRONG_GEN THEN ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL; PCROSS_INTERVAL]; ALL_TAC] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN X_GEN_TAC `D:(real^N->bool)->bool` THEN STRIP_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) MEASURE_UNIONS_LE_IMAGE o lhand o snd) THEN ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE] THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURABLE_INTERVAL; PCROSS_INTERVAL; SUBSET]; MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)] THEN TRANS_TAC REAL_LE_TRANS `sum D (\k:real^N->bool. content(interval[a:real^M,b]) * measure k)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN SUBGOAL_THEN `?u v:real^N. k = interval[u,v]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[SUBSET]; ASM_REWRITE_TAC[]] THEN ASM_REWRITE_TAC[PCROSS_INTERVAL; MEASURE_INTERVAL; CONTENT_PASTECART]; REWRITE_TAC[SUM_LMUL]] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x * (y + &1) <= e ==> y * x <= e`) THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_POS_LE THEN ASM_MESON_TAC[MEASURE_POS_LE; SUBSET; MEASURABLE_INTERVAL]; SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_ARITH `&0 <= x ==> &0 < x + &1`; CONTENT_POS_LE]] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS)) THEN TRANS_TAC REAL_LE_TRANS `measure(UNIONS D:real^N->bool)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_EQ_IMP_LE; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[SUBSET_UNIONS] THEN ASM_MESON_TAC[MEASURABLE_UNIONS; MEASURABLE_INTERVAL; SUBSET]] THEN TRANS_TAC EQ_TRANS `sum (D:(real^N->bool)->bool) content` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[MEASURE_INTERVAL; SUBSET]; CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_ELEMENTARY THEN REWRITE_TAC[division_of] THEN ASM SET_TAC[]]]);; let FUBINI_TONELLI_MEASURE = prove (`!s:real^(M,N)finite_sum->bool. lebesgue_measurable s ==> (measurable s <=> negligible {x | ~measurable {y | pastecart x y IN s}} /\ (\x. lift(measure {y | pastecart x y IN s})) integrable_on UNIV)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ASM_MESON_TAC[FUBINI_MEASURE; integrable_on]; STRIP_TAC] THEN MP_TAC(ISPECL [`\n. s INTER ball(vec 0:real^(M,N)finite_sum,&n)`; `drop(integral (:real^M) (\x. lift (measure {y:real^N | pastecart x y IN s})))`] MEASURABLE_NESTED_UNIONS) THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_BALL; GSYM REAL_OF_NUM_SUC; SUBSET_BALL; REAL_ARITH `x <= x + &1`; SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`] THEN ANTS_TAC THENL [X_GEN_TAC `n:num` THEN MP_TAC(SPEC `s INTER ball(vec 0:real^(M,N)finite_sum,&n)` FUBINI_MEASURE) THEN ASM_SIMP_TAC[MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE; MEASURABLE_BALL; HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC INTEGRAL_DROP_LE_AE THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `{x:real^M | ~measurable {y:real^N | pastecart x y IN s}} UNION {x:real^M | ~measurable {y:real^N | pastecart x y IN s INTER ball (vec 0,&n)}}` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; IN_DIFF; IN_UNIV; DE_MORGAN_THM; IN_UNION; IN_ELIM_THM; LIFT_DROP] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[UNIONS_GSPEC; IN_INTER; IN_BALL_0; IN_UNIV] THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN MESON_TAC[REAL_ARCH_LT]]);; let FUBINI_TONELLI_MEASURE_ALT = prove (`!s:real^(M,N)finite_sum->bool. lebesgue_measurable s ==> (measurable s <=> negligible {y | ~measurable {x | pastecart x y IN s}} /\ (\y. lift(measure {x | pastecart x y IN s})) integrable_on UNIV)`, GEN_TAC THEN DISCH_TAC THEN MP_TAC(ISPEC `IMAGE (\z. pastecart (sndcart z) (fstcart z)) (s:real^(M,N)finite_sum->bool)` FUBINI_TONELLI_MEASURE) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN; LINEAR_PASTECART; LINEAR_FSTCART; LINEAR_SNDCART; DIMINDEX_FINITE_SUM; ARITH_RULE `m + n:num <= n + m`] THEN MP_TAC(ISPEC `\z:real^(M,N)finite_sum. pastecart (sndcart z) (fstcart z)` HAS_MEASURE_ISOMETRY) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL [REWRITE_TAC[DIMINDEX_FINITE_SUM; ADD_SYM] THEN SIMP_TAC[LINEAR_PASTECART; LINEAR_FSTCART; LINEAR_SNDCART] THEN SIMP_TAC[FORALL_PASTECART; NORM_EQ; GSYM NORM_POW_2; SQNORM_PASTECART] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; REAL_ADD_AC]; DISCH_TAC THEN ASM_REWRITE_TAC[measurable; measure] THEN ASM_REWRITE_TAC[GSYM measurable; GSYM measure] THEN REWRITE_TAC[IN_IMAGE; EXISTS_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_INJ] THEN REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; UNWIND_THM1]]);; let FUBINI_TONELLI_NEGLIGIBLE = prove (`!s:real^(M,N)finite_sum->bool. lebesgue_measurable s ==> (negligible s <=> negligible {x | ~negligible {y | pastecart x y IN s}})`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[FUBINI_NEGLIGIBLE] THEN DISCH_TAC THEN REWRITE_TAC[NEGLIGIBLE_EQ_MEASURE_0] THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[FUBINI_TONELLI_MEASURE] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; CONTRAPOS_THM; NEGLIGIBLE_IMP_MEASURABLE]; MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE)]; DISCH_TAC THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP FUBINI_MEASURE) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_UNIQUE)) THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE] THEN EXISTS_TAC `(\x. vec 0):real^M->real^1` THEN EXISTS_TAC `{x:real^M | ~negligible {y:real^N | pastecart x y IN s}}` THEN ASM_REWRITE_TAC[INTEGRABLE_0; IN_DIFF; IN_UNIV; IN_ELIM_THM] THEN SIMP_TAC[MEASURE_EQ_0; GSYM DROP_EQ; DROP_VEC; LIFT_DROP; HAS_INTEGRAL_0]);; let FUBINI_TONELLI_NEGLIGIBLE_ALT = prove (`!s:real^(M,N)finite_sum->bool. lebesgue_measurable s ==> (negligible s <=> negligible {y | ~negligible {x | pastecart x y IN s}})`, GEN_TAC THEN DISCH_TAC THEN MP_TAC(ISPEC `IMAGE (\z. pastecart (sndcart z) (fstcart z)) (s:real^(M,N)finite_sum->bool)` FUBINI_TONELLI_NEGLIGIBLE) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN; LINEAR_PASTECART; LINEAR_FSTCART; LINEAR_SNDCART; DIMINDEX_FINITE_SUM; ARITH_RULE `m + n:num <= n + m`] THEN MP_TAC(ISPEC `\z:real^(M,N)finite_sum. pastecart (sndcart z) (fstcart z)` HAS_MEASURE_ISOMETRY) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL [REWRITE_TAC[DIMINDEX_FINITE_SUM; ADD_SYM] THEN SIMP_TAC[LINEAR_PASTECART; LINEAR_FSTCART; LINEAR_SNDCART] THEN SIMP_TAC[FORALL_PASTECART; NORM_EQ; GSYM NORM_POW_2; SQNORM_PASTECART] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; REAL_ADD_AC]; DISCH_TAC THEN ASM_REWRITE_TAC[HAS_MEASURE_0] THEN ASM_REWRITE_TAC[GSYM HAS_MEASURE_0] THEN REWRITE_TAC[IN_IMAGE; EXISTS_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_INJ] THEN REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; UNWIND_THM1]]);; let LEBESGUE_MEASURABLE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. lebesgue_measurable(s PCROSS t) <=> negligible s \/ negligible t \/ (lebesgue_measurable s /\ lebesgue_measurable t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `negligible(s:real^M->bool)` THENL [ASM_MESON_TAC[NEGLIGIBLE_PCROSS; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]; ASM_REWRITE_TAC[]] THEN ASM_CASES_TAC `negligible(t:real^N->bool)` THENL [ASM_MESON_TAC[NEGLIGIBLE_PCROSS; NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE]; ASM_REWRITE_TAC[]] THEN REWRITE_TAC[lebesgue_measurable; measurable_on; IN_UNIV] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [MAP_EVERY X_GEN_TAC [`k:real^(M,N)finite_sum->bool`; `g:num->real^(M,N)finite_sum->real^1`] THEN STRIP_TAC THEN FIRST_ASSUM(fun th -> ASSUME_TAC(MATCH_MP FUBINI_NEGLIGIBLE th) THEN ASSUME_TAC(MATCH_MP FUBINI_NEGLIGIBLE_ALT th)) THEN SUBGOAL_THEN `~(s SUBSET {x:real^M | ~negligible {y:real^N | pastecart x y IN k}})` MP_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET]; ALL_TAC] THEN SUBGOAL_THEN `~(t SUBSET {y:real^N | ~negligible {x:real^M | pastecart x y IN k}})` MP_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_SUBSET]; ALL_TAC] THEN REWRITE_TAC[SUBSET; NOT_FORALL_THM; NOT_IMP; IN_ELIM_THM] THEN DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `x:real^M` STRIP_ASSUME_TAC) THEN EXISTS_TAC `{x:real^M | pastecart x (y:real^N) IN k}` THEN EXISTS_TAC `\n x. (g:num->real^(M,N)finite_sum->real^1) n (pastecart x y)` THEN EXISTS_TAC `{y:real^N | pastecart (x:real^M) y IN k}` THEN EXISTS_TAC `\n y. (g:num->real^(M,N)finite_sum->real^1) n (pastecart x y)` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THEN (CONJ_TAC THENL [GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC]) THENL [X_GEN_TAC `u:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (u:real^M) (y:real^N)`); X_GEN_TAC `v:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `pastecart (x:real^M) (v:real^N)`)] THEN ASM_REWRITE_TAC[indicator; PASTECART_IN_PCROSS]; MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `f:num->real^M->real^1`; `v:real^N->bool`; `g:num->real^N->real^1`] THEN STRIP_TAC THEN EXISTS_TAC `u PCROSS (:real^N) UNION (:real^M) PCROSS v` THEN EXISTS_TAC `\n:num z:real^(M,N)finite_sum. lift(drop(f n (fstcart z)) * drop(g n (sndcart z)))` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ; NEGLIGIBLE_PCROSS] THEN CONJ_TAC THENL [GEN_TAC THEN REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP] THEN CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; REWRITE_TAC[FORALL_PASTECART; IN_UNION; PASTECART_IN_PCROSS] THEN REWRITE_TAC[IN_UNIV; DE_MORGAN_THM; LIFT_CMUL; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN STRIP_TAC THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN SUBGOAL_THEN `indicator (s PCROSS t) (pastecart x y) = drop(indicator s (x:real^M)) % indicator t (y:real^N)` SUBST1_TAC THENL [REWRITE_TAC[indicator; PASTECART_IN_PCROSS] THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(y:real^N) IN t`] THEN ASM_REWRITE_TAC[GSYM DROP_EQ; DROP_CMUL; DROP_VEC] THEN CONV_TAC REAL_RAT_REDUCE_CONV; MATCH_MP_TAC LIM_MUL THEN REWRITE_TAC[o_DEF; LIFT_DROP] THEN ASM_SIMP_TAC[]]]]);; let MEASURABLE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. measurable(s PCROSS t) <=> negligible s \/ negligible t \/ (measurable s /\ measurable t)`, REPEAT GEN_TAC THEN ASM_CASES_TAC `negligible(s:real^M->bool)` THENL [ASM_MESON_TAC[NEGLIGIBLE_PCROSS; NEGLIGIBLE_IMP_MEASURABLE]; ASM_REWRITE_TAC[]] THEN ASM_CASES_TAC `negligible(t:real^N->bool)` THENL [ASM_MESON_TAC[NEGLIGIBLE_PCROSS; NEGLIGIBLE_IMP_MEASURABLE]; ASM_REWRITE_TAC[]] THEN ASM_CASES_TAC `lebesgue_measurable((s:real^M->bool) PCROSS (t:real^N->bool))` THENL [ASM_SIMP_TAC[FUBINI_TONELLI_MEASURE; PASTECART_IN_PCROSS]; ASM_MESON_TAC[LEBESGUE_MEASURABLE_PCROSS; MEASURABLE_IMP_LEBESGUE_MEASURABLE]] THEN REWRITE_TAC[SET_RULE `{x | P /\ x IN s} = if P then s else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURABLE_EMPTY; MEASURE_EMPTY] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[LIFT_NUM; INTEGRABLE_RESTRICT_UNIV; INTEGRABLE_ON_CONST] THEN REWRITE_TAC[SET_RULE `{x | if x IN s then P else F} = if P then s else {}`] THEN ASM_CASES_TAC `measurable(s:real^M->bool)` THEN ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `measurable(t:real^N->bool)` THEN ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY] THEN REWRITE_TAC[GSYM DROP_EQ; DROP_VEC; LIFT_DROP] THEN ASM_MESON_TAC[NEGLIGIBLE_EQ_MEASURE_0]);; let HAS_MEASURE_PCROSS = prove (`!s:real^M->bool t:real^N->bool a b. s has_measure a /\ t has_measure b ==> (s PCROSS t) has_measure (a * b)`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `(s:real^M->bool) PCROSS (t:real^N->bool)` FUBINI_MEASURE) THEN REWRITE_TAC[MEASURABLE_PCROSS; PASTECART_IN_PCROSS] THEN ANTS_TAC THENL [ASM_MESON_TAC[measurable]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `{y | P /\ y IN s} = if P then s else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURABLE_EMPTY; MEASURE_EMPTY] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[LIFT_NUM; INTEGRABLE_RESTRICT_UNIV; INTEGRABLE_ON_CONST] THEN REWRITE_TAC[SET_RULE `{x | if x IN s then P else F} = if P then s else {}`] THEN REWRITE_TAC[HAS_INTEGRAL_RESTRICT_UNIV] THEN STRIP_TAC THEN REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_PCROSS] THEN CONJ_TAC THENL [ASM_MESON_TAC[measurable]; ALL_TAC] THEN REWRITE_TAC[GSYM LIFT_EQ] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_UNIQUE)) THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]) THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN ASM_REWRITE_TAC[LIFT_EQ_CMUL] THEN MATCH_MP_TAC HAS_INTEGRAL_CMUL THEN REWRITE_TAC[GSYM LIFT_EQ_CMUL] THEN ONCE_REWRITE_TAC[GSYM HAS_INTEGRAL_RESTRICT_UNIV] THEN ASM_REWRITE_TAC[GSYM HAS_MEASURE; HAS_MEASURE_MEASURABLE_MEASURE]);; let MEASURE_PCROSS = prove (`!s:real^M->bool t:real^N->bool. measurable s /\ measurable t ==> measure(s PCROSS t) = measure s * measure t`, MESON_TAC[HAS_MEASURE_MEASURABLE_MEASURE; HAS_MEASURE_PCROSS]);; (* ------------------------------------------------------------------------- *) (* Relate the measurability of a function and of its ordinate set. *) (* ------------------------------------------------------------------------- *) let LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LE = prove (`!f:real^M->real^N k. f measurable_on (:real^M) ==> lebesgue_measurable {pastecart x (y:real^N) | y$k <= (f x)$k}`, let lemma = prove (`!x y. x <= y <=> !q. rational q /\ y < q ==> x < q`, REPEAT GEN_TAC THEN EQ_TAC THENL [MESON_TAC[REAL_LET_TRANS]; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LE; NOT_FORALL_THM; NOT_IMP; REAL_NOT_LT] THEN MESON_TAC[RATIONAL_BETWEEN; REAL_LT_IMP_LE]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `{pastecart (x:real^M) (y:real^N) | y$k <= (f x:real^N)$k} = INTERS {{pastecart x y | (f x)$k < q ==> y$k < q} | q IN rational}` SUBST1_TAC THENL [REWRITE_TAC[INTERS_GSPEC; EXTENSION; FORALL_PASTECART] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM] THEN ONCE_REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM] THEN MESON_TAC[lemma; IN]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_INTERS THEN SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; COUNTABLE_RATIONAL] THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN ONCE_REWRITE_TAC[SET_RULE `{f x y | P x y ==> Q x y} = {f x y | Q x y} UNION {f x y | ~(P x y)}`] THEN X_GEN_TAC `q:real` THEN REWRITE_TAC[IN] THEN DISCH_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN REWRITE_TAC[REAL_NOT_LT; GSYM PCROSS; LEBESGUE_MEASURABLE_PCROSS; SET_RULE `{f x y |x,y| P x} = {f x y | x IN {x | P x} /\ y IN UNIV}`; SET_RULE `{f x y |x,y| Q y} = {f x y | x IN UNIV /\ y IN {x | Q x}}`] THEN CONJ_TAC THEN REPEAT DISJ2_TAC THEN REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV] THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_OPEN THEN REWRITE_TAC[drop; OPEN_HALFSPACE_COMPONENT_LT]; ONCE_REWRITE_TAC[SET_RULE `{x | q <= (f x)$k} = {x | f x IN {y | q <= y$k}}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_PREIMAGE_CLOSED THEN ASM_REWRITE_TAC[drop; GSYM real_ge; CLOSED_HALFSPACE_COMPONENT_GE]]);; let LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LT = prove (`!f:real^M->real^N k. f measurable_on (:real^M) ==> lebesgue_measurable {pastecart x (y:real^N) | y$k < (f x)$k}`, REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `f < y <=> ~(--f <= --y)`] THEN MP_TAC(ISPECL [`(--) o (f:real^M->real^N)`; `k:num`] LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LE) THEN ANTS_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_COMPOSE_CONTINUOUS THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN SIMP_TAC[CONTINUOUS_ON_NEG; CONTINUOUS_ON_ID]; ALL_TAC] THEN MP_TAC(ISPEC `\z:real^(M,N)finite_sum. pastecart (fstcart z) (--sndcart z)` LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; PASTECART_INJ; VECTOR_EQ_NEG2; GSYM PASTECART_EQ] THEN ANTS_TAC THENL [REWRITE_TAC[linear; PASTECART_EQ; FSTCART_PASTECART; SNDCART_PASTECART; FSTCART_ADD; FSTCART_CMUL; SNDCART_ADD; SNDCART_CMUL] THEN VECTOR_ARITH_TAC; DISCH_THEN(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th])] THEN GEN_REWRITE_TAC LAND_CONV [GSYM LEBESGUE_MEASURABLE_COMPL] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[SET_RULE `UNIV DIFF s = t <=> s = UNIV DIFF t`] THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_ELIM_PASTECART_THM; o_DEF; FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[VECTOR_NEG_COMPONENT; REAL_NEG_NEG] THEN MESON_TAC[FSTCART_PASTECART; SNDCART_PASTECART; VECTOR_NEG_NEG]);; let LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LE_EQ, LEBESGUE_MEASURABLE_FUNCTION_ORTHANT_SET_LE_EQ = (CONJ_PAIR o prove) (`(!f:real^M->real^N. f measurable_on (:real^M) <=> !k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {pastecart x (y:real^N) | y$k <= (f x)$k}) /\ (!f:real^M->real^N. f measurable_on (:real^M) <=> lebesgue_measurable {pastecart x (y:real^N) | !k. 1 <= k /\ k <= dimindex(:N) ==> y$k <= (f x)$k})`, REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ (q ==> r) /\ (r ==> p) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THEN DISCH_TAC THENL [ASM_SIMP_TAC[LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LE]; SUBGOAL_THEN `{ pastecart x y | !k. 1 <= k /\ k <= dimindex(:N) ==> (y:real^N)$k <= (f:real^M->real^N) x$k } = INTERS {{ pastecart x y | (y:real^N)$k <= (f:real^M->real^N) x$k} | k IN 1..dimindex(:N)}` SUBST1_TAC THENL [REWRITE_TAC[INTERS_GSPEC; EXTENSION; IN_ELIM_THM; IN_NUMSEG] THEN REWRITE_TAC[FORALL_PASTECART; PASTECART_INJ] THEN MESON_TAC[]; MATCH_MP_TAC LEBESGUE_MEASURABLE_INTERS THEN SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; IN_NUMSEG]]; MP_TAC(ISPECL [`f:real^M->real^N`; `{y | lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex (:N) ==> (y:real^N)$k <= (f:real^M->real^N) x$k}}`] MEASURABLE_ON_PREIMAGE_ORTHANT_GE_DENSE) THEN ASM_REWRITE_TAC[IN_ELIM_THM; real_ge] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_LEBESGUE_MEASURABLE_ALT) THEN REWRITE_TAC[SET_RULE `{x | ~P x} = UNIV DIFF {x | P x}`] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM] THEN REWRITE_TAC[SET_RULE `s = UNIV <=> UNIV DIFF s = {}`] THEN REWRITE_TAC[GSYM INTERIOR_COMPLEMENT; NEGLIGIBLE_EMPTY_INTERIOR]]);; let LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LT_EQ, LEBESGUE_MEASURABLE_FUNCTION_ORTHANT_SET_LT_EQ = (CONJ_PAIR o prove) (`(!f:real^M->real^N. f measurable_on (:real^M) <=> !k. 1 <= k /\ k <= dimindex(:N) ==> lebesgue_measurable {pastecart x (y:real^N) | y$k < (f x)$k}) /\ (!f:real^M->real^N. f measurable_on (:real^M) <=> lebesgue_measurable {pastecart x (y:real^N) | !k. 1 <= k /\ k <= dimindex(:N) ==> y$k < (f x)$k})`, REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ (q ==> r) /\ (r ==> p) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THEN DISCH_TAC THENL [ASM_SIMP_TAC[LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LT]; SUBGOAL_THEN `{ pastecart x y | !k. 1 <= k /\ k <= dimindex(:N) ==> (y:real^N)$k < (f:real^M->real^N) x$k } = INTERS {{ pastecart x y | (y:real^N)$k < (f:real^M->real^N) x$k} | k IN 1..dimindex(:N)}` SUBST1_TAC THENL [REWRITE_TAC[INTERS_GSPEC; EXTENSION; IN_ELIM_THM; IN_NUMSEG] THEN REWRITE_TAC[FORALL_PASTECART; PASTECART_INJ] THEN MESON_TAC[]; MATCH_MP_TAC LEBESGUE_MEASURABLE_INTERS THEN SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN ASM_SIMP_TAC[FORALL_IN_IMAGE; IN_NUMSEG]]; MP_TAC(ISPECL [`f:real^M->real^N`; `{y | lebesgue_measurable {x | !k. 1 <= k /\ k <= dimindex (:N) ==> (y:real^N)$k < (f:real^M->real^N) x$k}}`] MEASURABLE_ON_PREIMAGE_ORTHANT_GT_DENSE) THEN ASM_REWRITE_TAC[IN_ELIM_THM; real_gt] THEN DISCH_THEN MATCH_MP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_LEBESGUE_MEASURABLE_ALT) THEN REWRITE_TAC[SET_RULE `{x | ~P x} = UNIV DIFF {x | P x}`] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM] THEN REWRITE_TAC[SET_RULE `s = UNIV <=> UNIV DIFF s = {}`] THEN REWRITE_TAC[GSYM INTERIOR_COMPLEMENT; NEGLIGIBLE_EMPTY_INTERIOR]]);; let NEGLIGIBLE_MEASURABLE_FUNCTION_GRAPH = prove (`!f:real^M->real^N. f measurable_on (:real^M) ==> negligible {pastecart x y | f x = y}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_DISJOINT_TRANSLATES THEN EXISTS_TAC `{pastecart (vec 0:real^M) x | x IN (:real^N)}` THEN EXISTS_TAC `vec 0:real^(M,N)finite_sum` THEN REPEAT CONJ_TAC THENL [SUBGOAL_THEN `{pastecart x y | (f:real^M->real^N) x = y} = INTERS {{pastecart x y | y$i <= (f x)$i} DIFF {pastecart x y | y$i < (f x)$i} | i IN 1..dimindex(:N)}` SUBST1_TAC THENL [REWRITE_TAC[CART_EQ; INTERS_GSPEC; EXTENSION; FORALL_PASTECART] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM; IN_NUMSEG] THEN ONCE_REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[IN_ELIM_PASTECART_THM; IN_DIFF; REAL_NOT_LT] THEN REWRITE_TAC[REAL_LE_ANTISYM] THEN MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_INTERS THEN SIMP_TAC[FINITE_IMAGE; SIMPLE_IMAGE; FINITE_NUMSEG] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_DIFF THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LE; LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LT]; MATCH_MP_TAC CONNECTED_IMP_PERFECT THEN REWRITE_TAC[GSYM PCROSS; SET_RULE `{f a x | x IN s} = {f w x | w IN {a} /\ x IN s}`] THEN REWRITE_TAC[GSYM PASTECART_VEC; PASTECART_IN_PCROSS] THEN REWRITE_TAC[CONNECTED_SING; CONNECTED_PCROSS_EQ; CONNECTED_UNIV] THEN REWRITE_TAC[IN_SING; IN_UNIV] THEN MATCH_MP_TAC(SET_RULE `!a b. a IN s /\ b IN s /\ ~(a = b) ==> ~(?a. s = {a})`) THEN EXISTS_TAC `pastecart (vec 0:real^M) (vec 0:real^N)` THEN EXISTS_TAC `pastecart (vec 0:real^M) (vec 1:real^N)` THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_SING; IN_UNIV] THEN REWRITE_TAC[PASTECART_INJ; VEC_EQ; ARITH_EQ]; REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV; PASTECART_INJ] THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV; PASTECART_INJ; FORALL_IN_IMAGE; SET_RULE `DISJOINT s t <=> !x. x IN s ==> !y. y IN t ==> ~(x = y)`] THEN REWRITE_TAC[PASTECART_ADD; VECTOR_ADD_LID; PASTECART_INJ] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x':real^M`; `y':real^N`] THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(SUBST1_TAC o SYM)) THEN ASM_CASES_TAC `x':real^M = x` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `~(a:real^N = b)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN CONV_TAC VECTOR_ARITH]);; (* ------------------------------------------------------------------------- *) (* Hence relate integrals and "area under curve" for functions into R^+. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_IFF_LEBESGUE_MEASURABLE_UNDER_CURVE = prove (`!f:real^N->real^1. (!x. &0 <= drop(f x)) ==> (f measurable_on (:real^N) <=> lebesgue_measurable { pastecart x y | y IN interval[vec 0,f x]})`, REPEAT STRIP_TAC THEN REWRITE_TAC[LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LE_EQ] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; IN_INTERVAL_1; GSYM drop; DROP_VEC] THEN EQ_TAC THEN DISCH_TAC THENL [SUBGOAL_THEN `{pastecart x y | &0 <= drop y /\ drop y <= drop (f x)} = (:real^N) PCROSS {y | &0 <= drop y} INTER {pastecart (x:real^N) y | drop y <= drop (f x)}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; FORALL_PASTECART; PASTECART_IN_PCROSS; IN_INTER; IN_ELIM_PASTECART_THM] THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM]; MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_PCROSS; LEBESGUE_MEASURABLE_UNIV] THEN SIMP_TAC[LEBESGUE_MEASURABLE_CLOSED; GSYM real_ge; drop; CLOSED_HALFSPACE_COMPONENT_GE]]; SUBGOAL_THEN `{pastecart (x:real^N) y | drop y <= drop (f x)} = {pastecart x y | &0 <= drop y /\ drop y <= drop (f x)} UNION (:real^N) PCROSS {y | drop y < &0}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNION; IN_ELIM_PASTECART_THM] THEN REWRITE_TAC[IN_UNIV; IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_NOT_LE; REAL_LT_IMP_LE; REAL_LE_TRANS]; MATCH_MP_TAC LEBESGUE_MEASURABLE_UNION THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_PCROSS; LEBESGUE_MEASURABLE_UNIV] THEN SIMP_TAC[LEBESGUE_MEASURABLE_OPEN; drop; OPEN_HALFSPACE_COMPONENT_LT]]]);; let INTEGRABLE_IFF_MEASURABLE_UNDER_CURVE = prove (`!f:real^N->real^1. (!x. &0 <= drop(f x)) ==> (f integrable_on (:real^N) <=> measurable { pastecart x y | y IN interval[vec 0,f x]})`, REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) FUBINI_TONELLI_MEASURE o snd) THEN REWRITE_TAC[IN_ELIM_PASTECART_THM; SET_RULE `{x | x IN s} = s`] THEN ASM_SIMP_TAC[MEASURE_INTERVAL_1; DROP_VEC; REAL_SUB_RZERO; LIFT_DROP] THEN REWRITE_TAC[MEASURABLE_INTERVAL; EMPTY_GSPEC; NEGLIGIBLE_EMPTY] THEN ASM_REWRITE_TAC[ETA_AX] THEN DISCH_THEN MATCH_MP_TAC THEN SUBGOAL_THEN `{pastecart x y | y IN interval [vec 0,f x]} = {pastecart x y | drop y <= drop(f x)} INTER (:real^N) PCROSS {x | &0 <= drop x}` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_INTER; IN_ELIM_PASTECART_THM; PASTECART_IN_PCROSS; IN_UNIV] THEN REWRITE_TAC[IN_INTERVAL_1; IN_ELIM_THM; DROP_VEC; CONJ_SYM]; MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN REWRITE_TAC[drop] THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_FUNCTION_ORDINATE_SET_LE; INTEGRABLE_IMP_MEASURABLE; LEBESGUE_MEASURABLE_PCROSS] THEN REPEAT DISJ2_TAC THEN REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_CLOSED THEN REWRITE_TAC[drop; GSYM real_ge; CLOSED_HALFSPACE_COMPONENT_GE]]; FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_MEASURE) THEN REWRITE_TAC[IN_ELIM_PASTECART_THM; SET_RULE `{x | x IN s} = s`] THEN ASM_SIMP_TAC[MEASURE_INTERVAL_1; DROP_VEC; REAL_SUB_RZERO; LIFT_DROP] THEN REWRITE_TAC[ETA_AX; GSYM LIFT_EQ] THEN MESON_TAC[integrable_on]]);; let HAS_INTEGRAL_MEASURE_UNDER_CURVE = prove (`!f:real^N->real^1 m. (!x. &0 <= drop(f x)) ==> ((f has_integral lift m) (:real^N) <=> { pastecart x y | y IN interval[vec 0,f x]} has_measure m)`, REPEAT STRIP_TAC THEN REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE; HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN MATCH_MP_TAC(TAUT `(p <=> p') /\ (p /\ p' ==> (q <=> q')) ==> (p /\ q <=> p' /\ q')`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[INTEGRABLE_IFF_MEASURABLE_UNDER_CURVE]; STRIP_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_MEASURE) THEN REWRITE_TAC[IN_ELIM_PASTECART_THM; SET_RULE `{x | x IN s} = s`] THEN ASM_REWRITE_TAC[MEASURE_INTERVAL_1; DROP_VEC; REAL_SUB_RZERO; LIFT_DROP] THEN REWRITE_TAC[ETA_AX; GSYM LIFT_EQ] THEN ASM_MESON_TAC[integrable_on; INTEGRAL_UNIQUE]);; (* ------------------------------------------------------------------------- *) (* Some miscellanous lemmas. *) (* ------------------------------------------------------------------------- *) let MEASURABLE_ON_COMPOSE_FSTCART = prove (`!f:real^M->real^P. f measurable_on (:real^M) ==> (\z:real^(M,N)finite_sum. f(fstcart z)) measurable_on (:real^(M,N)finite_sum)`, GEN_TAC THEN REWRITE_TAC[measurable_on; LEFT_IMP_EXISTS_THM; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `g:num->real^M->real^P`] THEN STRIP_TAC THEN EXISTS_TAC `(k:real^M->bool) PCROSS (:real^N)` THEN EXISTS_TAC `(\n z. g n (fstcart z)):num->real^(M,N)finite_sum->real^P` THEN ASM_REWRITE_TAC[NEGLIGIBLE_PCROSS; FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNIV; FSTCART_PASTECART; SNDCART_PASTECART] THEN GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_FSTCART; LINEAR_CONTINUOUS_ON] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]);; let MEASURABLE_ON_COMPOSE_SNDCART = prove (`!f:real^N->real^P. f measurable_on (:real^N) ==> (\z:real^(M,N)finite_sum. f(sndcart z)) measurable_on (:real^(M,N)finite_sum)`, GEN_TAC THEN REWRITE_TAC[measurable_on; LEFT_IMP_EXISTS_THM; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`k:real^N->bool`; `g:num->real^N->real^P`] THEN STRIP_TAC THEN EXISTS_TAC `(:real^M) PCROSS (k:real^N->bool)` THEN EXISTS_TAC `(\n z. g n (sndcart z)):num->real^(M,N)finite_sum->real^P` THEN ASM_REWRITE_TAC[NEGLIGIBLE_PCROSS; FORALL_PASTECART; PASTECART_IN_PCROSS; IN_UNIV; SNDCART_PASTECART; SNDCART_PASTECART] THEN GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_CONTINUOUS_ON] THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]);; let MEASURABLE_ON_COMPOSE_SUB = prove (`!f:real^M->real^N. f measurable_on (:real^M) ==> (\z. f(fstcart z - sndcart z)) measurable_on (:real^(M,M)finite_sum)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(\z. f(fstcart z - sndcart z)):real^(M,M)finite_sum->real^N = (\z. f(fstcart z)) o (\z. pastecart (fstcart z - sndcart z) (sndcart z))` SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART]; W(MP_TAC o PART_MATCH (lhs o rand) MEASURABLE_ON_LINEAR_IMAGE_EQ_GEN o snd)] THEN REWRITE_TAC[FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART] THEN ANTS_TAC THENL [REWRITE_TAC[PASTECART_INJ] THEN CONJ_TAC THENL [MATCH_MP_TAC LINEAR_PASTECART; CONV_TAC VECTOR_ARITH] THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_FSTCART; LINEAR_COMPOSE_SUB]; DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `(:real^(M,M)finite_sum)` THEN ASM_SIMP_TAC[MEASURABLE_ON_COMPOSE_FSTCART; SUBSET_UNIV] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN THEN REWRITE_TAC[LE_REFL; LEBESGUE_MEASURABLE_UNIV] THEN MATCH_MP_TAC LINEAR_PASTECART THEN SIMP_TAC[LINEAR_SNDCART; LINEAR_FSTCART; LINEAR_COMPOSE_SUB]]);; (* ------------------------------------------------------------------------- *) (* Fubini for absolute integrability. *) (* ------------------------------------------------------------------------- *) let FUBINI_ABSOLUTELY_INTEGRABLE = prove (`!f:real^(M,N)finite_sum->real^P. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> negligible {x | ~((\y. f(pastecart x y)) absolutely_integrable_on (:real^N))} /\ ((\x. integral (:real^N) (\y. f(pastecart x y))) has_integral integral (:real^(M,N)finite_sum) f) (:real^M)`, let lemma = prove (`{x | ~(!i. i IN k ==> P i x)} = UNIONS {{x | ~P i x} | i IN k}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in let assoclemma = prove (`!P:real^(M,N)finite_sum->real^P->bool. {pastecart x y | P x y} has_measure m ==> {pastecart x (pastecart y z) | P (pastecart x y) z} has_measure m`, GEN_TAC THEN MP_TAC(ISPECL [`\z. pastecart (fstcart(fstcart z):real^M) (pastecart (sndcart(fstcart z):real^N) (sndcart z:real^P))`; `{pastecart (x:real^(M,N)finite_sum) (y:real^P) | P x y}`; `m:real`] HAS_MEASURE_ISOMETRY) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; ADD_AC] THEN ANTS_TAC THENL [CONJ_TAC THENL [REPEAT(MATCH_MP_TAC LINEAR_PASTECART THEN CONJ_TAC) THEN REWRITE_TAC[GSYM o_DEF] THEN REPEAT(MATCH_MP_TAC LINEAR_COMPOSE THEN CONJ_TAC) THEN REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]; SIMP_TAC[FORALL_PASTECART; NORM_EQ; GSYM NORM_POW_2; SQNORM_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART; REAL_ADD_AC]]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART; IN_ELIM_THM; EXISTS_PASTECART; PASTECART_INJ] THEN MESON_TAC[]]) in let FUBINI_LEMMA = prove (`!f:real^(M,N)finite_sum->real^1. f integrable_on (:real^(M,N)finite_sum) /\ (!x. &0 <= drop(f x)) ==> negligible {x | ~((f o pastecart x) integrable_on (:real^N))} /\ ((\x. integral (:real^N) (f o pastecart x)) has_integral integral (:real^(M,N)finite_sum) f) (:real^M)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPEC `f:real^(M,N)finite_sum->real^1` INTEGRABLE_IFF_MEASURABLE_UNDER_CURVE) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `measurable { pastecart x (pastecart y z) | z IN interval[vec 0,(f:real^(M,N)finite_sum->real^1) (pastecart x y)] }` ASSUME_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [measurable]) THEN REWRITE_TAC[measurable] THEN MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[assoclemma]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_MEASURE) THEN REWRITE_TAC[IN_ELIM_THM; PASTECART_INJ] THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM1] THEN REWRITE_TAC[SET_RULE `{x | ?y z. P y z /\ x = pastecart y z} = {pastecart y z | P y z}`] THEN MP_TAC(GEN `x:real^M` (ISPEC `(f:real^(M,N)finite_sum->real^1) o pastecart x` INTEGRABLE_IFF_MEASURABLE_UNDER_CURVE)) THEN ASM_REWRITE_TAC[o_DEF] THEN DISCH_THEN(ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[] `y = z /\ ((f has_integral y) s ==> (g has_integral y) s) ==> (f has_integral y) s ==> (g has_integral z) s`) THEN CONJ_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_SIMP_TAC[HAS_INTEGRAL_MEASURE_UNDER_CURVE] THEN ASM_REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_UNIQUE THEN MATCH_MP_TAC assoclemma THEN ASM_REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE]; MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HAS_INTEGRAL_SPIKE)) THEN EXISTS_TAC `{x | ~((\y. (f:real^(M,N)finite_sum->real^1) (pastecart x y)) integrable_on (:real^N))}` THEN ASM_REWRITE_TAC[IN_UNIV; IN_DIFF; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN ASM_SIMP_TAC[HAS_INTEGRAL_MEASURE_UNDER_CURVE] THEN ASM_SIMP_TAC[GSYM HAS_MEASURE_MEASURE]]) in let FUBINI_1 = prove (`!f:real^(M,N)finite_sum->real^1. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> negligible {x | ~((f o pastecart x) absolutely_integrable_on (:real^N))} /\ ((\x. integral (:real^N) (f o pastecart x)) has_integral integral (:real^(M,N)finite_sum) f) (:real^M)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MAP_EVERY ABBREV_TAC [`g = \x:real^(M,N)finite_sum. lift (max (&0) (drop(f x)))`; `h = \x:real^(M,N)finite_sum. --(lift (min (&0) (drop(f x))))`] THEN SUBGOAL_THEN `!x:real^(M,N)finite_sum. &0 <= drop(g x) /\ &0 <= drop(h x)` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["g"; "h"] THEN REWRITE_TAC[DROP_NEG; LIFT_DROP] THEN REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `(g:real^(M,N)finite_sum->real^1) absolutely_integrable_on UNIV /\ (h:real^(M,N)finite_sum->real^1) absolutely_integrable_on UNIV` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["g"; "h"] THEN REWRITE_TAC[] THEN CONJ_TAC THEN TRY(MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NEG) THENL [MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_MAX_1; MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_MIN_1] THEN ASM_REWRITE_TAC[LIFT_DROP; ETA_AX; LIFT_NUM] THEN REWRITE_TAC[ABSOLUTELY_INTEGRABLE_0]; ALL_TAC] THEN SUBGOAL_THEN `(f:real^(M,N)finite_sum->real^1) = \x. g x - h x` SUBST1_TAC THENL [MAP_EVERY EXPAND_TAC ["g"; "h"] THEN REWRITE_TAC[FUN_EQ_THM; GSYM DROP_EQ; LIFT_DROP; DROP_SUB; DROP_NEG] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(ISPEC `h:real^(M,N)finite_sum->real^1` FUBINI_LEMMA) THEN MP_TAC(ISPEC `g:real^(M,N)finite_sum->real^1` FUBINI_LEMMA) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN ONCE_REWRITE_TAC[TAUT `p /\ q ==> r /\ s ==> t <=> p /\ r ==> q /\ s ==> t`] THEN REWRITE_TAC[GSYM NEGLIGIBLE_UNION_EQ; o_DEF] THEN DISCH_TAC THEN DISCH_THEN(ASSUME_TAC o MATCH_MP HAS_INTEGRAL_SUB) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNION; GSYM DE_MORGAN_THM] THEN REWRITE_TAC[CONTRAPOS_THM; o_DEF] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN ASM_REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop; IN_UNIV]; ASM_SIMP_TAC[INTEGRAL_SUB; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] HAS_INTEGRAL_SPIKE))) THEN FIRST_ASSUM(fun th -> EXISTS_TAC(rand(concl th)) THEN CONJ_TAC THENL [ACCEPT_TAC th; ALL_TAC]) THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_UNION; IN_ELIM_THM] THEN SIMP_TAC[DE_MORGAN_THM; INTEGRAL_SUB]]) in REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_COMPONENTWISE] THEN REWRITE_TAC[GSYM IN_NUMSEG; lemma] THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG] THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG]; DISCH_TAC THEN ONCE_REWRITE_TAC[HAS_INTEGRAL_COMPONENTWISE]] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ABSOLUTELY_INTEGRABLE_COMPONENTWISE]) THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_1) THEN SIMP_TAC[o_DEF] THEN DISCH_THEN(MP_TAC o CONJUNCT2) THEN ASM_SIMP_TAC[LIFT_INTEGRAL_COMPONENT; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC] HAS_INTEGRAL_SPIKE)) THEN FIRST_ASSUM(fun th -> EXISTS_TAC(rand(concl th)) THEN CONJ_TAC THENL [ACCEPT_TAC th; ALL_TAC]) THEN REWRITE_TAC[IN_UNIV; IN_DIFF; IN_ELIM_THM] THEN ASM_SIMP_TAC[LIFT_INTEGRAL_COMPONENT; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]);; let FUBINI_ABSOLUTELY_INTEGRABLE_ALT = prove (`!f:real^(M,N)finite_sum->real^P. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> negligible {y | ~((\x. f(pastecart x y)) absolutely_integrable_on (:real^M))} /\ ((\y. integral (:real^M) (\x. f(pastecart x y))) has_integral integral (:real^(M,N)finite_sum) f) (:real^N)`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM ABSOLUTELY_INTEGRABLE_PASTECART_SYM_UNIV]) THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN REWRITE_TAC[INTEGRAL_PASTECART_SYM_UNIV]);; let FUBINI_INTEGRAL = prove (`!f:real^(M,N)finite_sum->real^P. f absolutely_integrable_on UNIV ==> integral UNIV f = integral UNIV (\x. integral UNIV (\y. f(pastecart x y)))`, REPEAT STRIP_TAC THEN FIRST_ASSUM (MP_TAC o CONJUNCT2 o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN DISCH_THEN(SUBST1_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN REFL_TAC);; let FUBINI_INTEGRAL_ALT = prove (`!f:real^(M,N)finite_sum->real^P. f absolutely_integrable_on UNIV ==> integral UNIV f = integral UNIV (\y. integral UNIV (\x. f(pastecart x y)))`, REPEAT STRIP_TAC THEN FIRST_ASSUM (MP_TAC o CONJUNCT2 o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE_ALT) THEN DISCH_THEN(SUBST1_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN REFL_TAC);; let FUBINI_HAS_ABSOLUTE_INTEGRAL = prove (`!f:real^(M,N)finite_sum->real^P. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> negligible {x | ~((\y. f(pastecart x y)) absolutely_integrable_on (:real^N))} /\ (\x. integral (:real^N) (\y. f(pastecart x y))) absolutely_integrable_on (:real^M) /\ integral (:real^M) (\x. integral (:real^N) (\y. f(pastecart x y))) = integral (:real^(M,N)finite_sum) f`, GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_NORM) THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN ASM_REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_UNIV; INTEGRABLE_IMP_MEASURABLE] THEN MAP_EVERY ABBREV_TAC [`n1 = {x | ~((\y. (f:real^(M,N)finite_sum->real^P)(pastecart x y)) absolutely_integrable_on (:real^N))} `; `n2 = {x | ~((\y. lift(norm((f:real^(M,N)finite_sum->real^P) (pastecart x y)))) absolutely_integrable_on (:real^N))}`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] ABSOLUTELY_INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `(:real^M) DIFF (n1 UNION n2)` THEN REWRITE_TAC[SET_RULE `(s DIFF UNIV) UNION UNIV DIFF s = UNIV DIFF s`] THEN REWRITE_TAC[COMPL_COMPL] THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\x. integral (:real^N) (\y. lift(norm((f:real^(M,N)finite_sum->real^P) (pastecart x y))))` THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_UNION; DE_MORGAN_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_IMP_MEASURABLE THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `n1 UNION n2:real^M->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN SET_TAC[]; MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `(:real^M)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `n1 UNION n2:real^M->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN SET_TAC[]; X_GEN_TAC `x:real^M` THEN MAP_EVERY EXPAND_TAC ["n1"; "n2"] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[LIFT_DROP; REAL_LE_REFL] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]]);; let FUBINI_HAS_ABSOLUTE_INTEGRAL_ALT = prove (`!f:real^(M,N)finite_sum->real^P. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> negligible {y | ~((\x. f(pastecart x y)) absolutely_integrable_on (:real^M))} /\ (\y. integral (:real^M) (\x. f(pastecart x y))) absolutely_integrable_on (:real^N) /\ integral (:real^N) (\y. integral (:real^M) (\x. f(pastecart x y))) = integral (:real^(M,N)finite_sum) f`, GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE_ALT) THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_NORM) THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE_ALT) THEN ASM_REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN STRIP_TAC THEN ASM_SIMP_TAC[IN_UNIV; INTEGRABLE_IMP_MEASURABLE] THEN MAP_EVERY ABBREV_TAC [`n1 = {y | ~((\x. (f:real^(M,N)finite_sum->real^P)(pastecart x y)) absolutely_integrable_on (:real^M))} `; `n2 = {y | ~((\x. lift(norm((f:real^(M,N)finite_sum->real^P) (pastecart x y)))) absolutely_integrable_on (:real^M))}`] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] ABSOLUTELY_INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `(:real^N) DIFF (n1 UNION n2)` THEN REWRITE_TAC[SET_RULE `(s DIFF UNIV) UNION UNIV DIFF s = UNIV DIFF s`] THEN REWRITE_TAC[COMPL_COMPL] THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\y. integral (:real^M) (\x. lift(norm((f:real^(M,N)finite_sum->real^P) (pastecart x y))))` THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_UNION; DE_MORGAN_THM] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_IMP_MEASURABLE THEN MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `n1 UNION n2:real^N->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN SET_TAC[]; MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN EXISTS_TAC `(:real^N)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `n1 UNION n2:real^N->bool` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN SET_TAC[]; X_GEN_TAC `x:real^N` THEN MAP_EVERY EXPAND_TAC ["n1"; "n2"] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REWRITE_TAC[LIFT_DROP; REAL_LE_REFL] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]]);; let FUBINI_INTEGRAL_SWAP = prove (`!f. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> integral (:real^M) (\x. integral (:real^N) (\y. f (pastecart x y))):real^P = integral (:real^N) (\y. integral (:real^M) (\x. f (pastecart x y)))`, MESON_TAC[FUBINI_INTEGRAL; FUBINI_INTEGRAL_ALT]);; let FUBINI_HAS_INTEGRAL_SWAP = prove (`!f. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> ((\x. integral (:real^N) (\y. f (pastecart x y))) has_integral integral (:real^N) (\y. integral (:real^M) (\x. f (pastecart x y))):real^P) (:real^M)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE_ALT) THEN MESON_TAC[INTEGRAL_UNIQUE]);; let FUBINI_HAS_INTEGRAL_SWAP_ALT = prove (`!f. f absolutely_integrable_on (:real^(M,N)finite_sum) ==> ((\y. integral (:real^M) (\x. f (pastecart x y))) has_integral integral (:real^M) (\x. integral (:real^N) (\y. f (pastecart x y))):real^P) (:real^N)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE_ALT) THEN MESON_TAC[INTEGRAL_UNIQUE]);; let FUBINI_INTEGRAL_INTERVAL = prove (`!f:real^(M,N)finite_sum->real^P a b c d. f absolutely_integrable_on interval[pastecart a c,pastecart b d] ==> integral (interval[pastecart a c,pastecart b d]) f = integral (interval[a,b]) (\x. integral (interval[c,d]) (\y. f(pastecart x y)))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_INTEGRAL) THEN REWRITE_TAC[INTEGRAL_RESTRICT_UNIV] THEN DISCH_THEN SUBST1_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM INTEGRAL_RESTRICT_UNIV] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN GEN_TAC THEN REWRITE_TAC[PASTECART_IN_PCROSS; GSYM PCROSS_INTERVAL] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[INTEGRAL_0] THEN REWRITE_TAC[INTEGRAL_RESTRICT_UNIV]);; let FUBINI_INTEGRAL_INTERVAL_ALT = prove (`!f:real^(M,N)finite_sum->real^P a b c d. f absolutely_integrable_on interval[pastecart a c,pastecart b d] ==> integral (interval[pastecart a c,pastecart b d]) f = integral (interval[c,d]) (\y. integral (interval[a,b]) (\x. f(pastecart x y)))`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_INTEGRAL_ALT) THEN REWRITE_TAC[INTEGRAL_RESTRICT_UNIV] THEN DISCH_THEN SUBST1_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM INTEGRAL_RESTRICT_UNIV] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN GEN_TAC THEN REWRITE_TAC[PASTECART_IN_PCROSS; GSYM PCROSS_INTERVAL] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[INTEGRAL_0] THEN REWRITE_TAC[INTEGRAL_RESTRICT_UNIV]);; let INTEGRAL_PASTECART_CONTINUOUS = prove (`!f:real^(M,N)finite_sum->real^P a b c d. f continuous_on interval[pastecart a c,pastecart b d] ==> integral (interval[pastecart a c,pastecart b d]) f = integral (interval[a,b]) (\x. integral (interval[c,d]) (\y. f(pastecart x y)))`, SIMP_TAC[FUBINI_INTEGRAL_INTERVAL; ABSOLUTELY_INTEGRABLE_CONTINUOUS]);; let INTEGRAL_SWAP_CONTINUOUS = prove (`!f:real^M->real^N->real^P a b c d. (\z. f (fstcart z) (sndcart z)) continuous_on interval[pastecart a c,pastecart b d] ==> integral (interval[a,b]) (\x. integral (interval[c,d]) (f x)) = integral (interval[c,d]) (\y. integral (interval[a,b]) (\x. f x y))`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_CONTINUOUS) THEN FIRST_X_ASSUM(fun th -> MP_TAC(MATCH_MP FUBINI_INTEGRAL_INTERVAL_ALT th) THEN MP_TAC(MATCH_MP FUBINI_INTEGRAL_INTERVAL th)) THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[ETA_AX]);; let FUBINI_TONELLI = prove (`!f:real^(M,N)finite_sum->real^P. f measurable_on (:real^(M,N)finite_sum) ==> (f absolutely_integrable_on (:real^(M,N)finite_sum) <=> negligible {x | ~((\y. f(pastecart x y)) absolutely_integrable_on (:real^N))} /\ (\x. integral (:real^N) (\y. lift(norm(f(pastecart x y))))) integrable_on (:real^M))`, REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL [FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN FIRST_ASSUM(MP_TAC o MATCH_MP ABSOLUTELY_INTEGRABLE_NORM) THEN DISCH_THEN(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_ASSUM(ACCEPT_TAC o MATCH_MP HAS_INTEGRAL_INTEGRABLE); ALL_TAC] THEN ASM_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_MEASURABLE] THEN ABBREV_TAC `g = \n x. if x IN interval[--vec n,vec n] then lift(min (norm ((f:real^(M,N)finite_sum->real^P) x)) (&n)) else vec 0` THEN SUBGOAL_THEN `!n. (g:num->real^(M,N)finite_sum->real^1) n absolutely_integrable_on UNIV` ASSUME_TAC THENL [X_GEN_TAC `n:num` THEN EXPAND_TAC "g" THEN REWRITE_TAC[] THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN REWRITE_TAC[IN_UNIV; DIMINDEX_1; FORALL_1] THEN REWRITE_TAC[COND_RAND; COND_RATOR; GSYM drop; LIFT_DROP; DROP_VEC] THEN CONJ_TAC THENL [CONV_TAC NORM_ARITH; ALL_TAC] THEN MATCH_MP_TAC INTEGRABLE_CASES THEN REWRITE_TAC[INTEGRABLE_0; IN_UNIV; SET_RULE `{x | x IN s} = s`] THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN EXISTS_TAC `\x:real^(M,N)finite_sum. lift(&n)` THEN REWRITE_TAC[INTEGRABLE_CONST; NORM_LIFT; LIFT_DROP] THEN SIMP_TAC[NORM_POS_LE; REAL_ARITH `&0 <= x ==> abs(min x (&n)) <= &n`] THEN MP_TAC(ISPECL [`\x. lift(norm((f:real^(M,N)finite_sum->real^P) x))`; `\x:real^(M,N)finite_sum. lift(&n)`; `interval[--vec n:real^(M,N)finite_sum,vec n]`] MEASURABLE_ON_MIN) THEN ANTS_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `(:real^(M,N)finite_sum)` THEN REWRITE_TAC[SUBSET_UNIV; LEBESGUE_MEASURABLE_INTERVAL] THEN ASM_SIMP_TAC[MEASURABLE_ON_NORM; MEASURABLE_ON_CONST]; MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[FUN_EQ_THM; CART_EQ; LAMBDA_BETA] THEN REWRITE_TAC[DIMINDEX_1; LIFT_DROP; FORALL_1; GSYM drop]]; ALL_TAC] THEN MP_TAC(ISPECL [`g:num->real^(M,N)finite_sum->real^1`; `\x. lift(norm((f:real^(M,N)finite_sum->real^P) x))`; `(:real^(M,N)finite_sum)`] MONOTONE_CONVERGENCE_INCREASING) THEN ANTS_TAC THENL [ALL_TAC; SIMP_TAC[]] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; IN_UNIV] THEN REPEAT CONJ_TAC THENL [REPEAT GEN_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[LIFT_DROP]) THEN REWRITE_TAC[REAL_LE_REFL; DROP_VEC; GSYM REAL_OF_NUM_SUC] THEN TRY(CONV_TAC NORM_ARITH) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (TAUT `~p ==> p ==> q`)) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN REWRITE_TAC[SUBSET_INTERVAL; VEC_COMPONENT; VECTOR_NEG_COMPONENT] THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN REPEAT STRIP_TAC THEN REAL_ARITH_TAC; X_GEN_TAC `z:real^(M,N)finite_sum` THEN MATCH_MP_TAC LIM_EVENTUALLY THEN MP_TAC(ISPEC `&1 + max (norm z) (norm((f:real^(M,N)finite_sum->real^P) z))` REAL_ARCH_SIMPLE) THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE] THEN DISCH_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[] THEN COND_CASES_TAC THENL [AP_TERM_TAC THEN REWRITE_TAC[REAL_ARITH `min a b = a <=> a <= b`] THEN ASM_REAL_ARITH_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (TAUT `~p ==> p ==> q`)) THEN REWRITE_TAC[IN_INTERVAL; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN REWRITE_TAC[GSYM REAL_ABS_BOUNDS] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `abs(x$i) <= norm(x:real^N) /\ norm x <= a ==> abs(x$i) <= a`) THEN REWRITE_TAC[COMPONENT_LE_NORM] THEN ASM_REAL_ARITH_TAC]; MP_TAC(GEN `n:num` (ISPEC `(g:num->real^(M,N)finite_sum->real^1) n` FUBINI_ABSOLUTELY_INTEGRABLE)) THEN ASM_REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP INTEGRAL_UNIQUE (SPEC `n:num` th))]) THEN REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN EXISTS_TAC `drop(integral (:real^M) (\x. lift(norm(integral (:real^N) (\y. lift(norm( (f:real^(M,N)finite_sum->real^P) (pastecart x y))))))))` THEN X_GEN_TAC `n:num` THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL_AE THEN EXISTS_TAC `{x | ~((\y. (f:real^(M,N)finite_sum->real^P)(pastecart x y)) absolutely_integrable_on (:real^N))} UNION {x | ~((\y. (g:num->real^(M,N)finite_sum->real^1) n (pastecart x y)) absolutely_integrable_on (:real^N))}` THEN ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[integrable_on]; MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE_AE THEN EXISTS_TAC `{x | ~((\y. (f:real^(M,N)finite_sum->real^P)(pastecart x y)) absolutely_integrable_on (:real^N))}` THEN ASM_REWRITE_TAC[IN_DIFF; IN_UNIV] THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP; DIMINDEX_1; FORALL_1; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[absolutely_integrable_on; GSYM drop] THEN STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_DROP_POS THEN ASM_REWRITE_TAC[LIFT_DROP; NORM_POS_LE]; X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_UNION; IN_ELIM_THM; DE_MORGAN_THM] THEN STRIP_TAC THEN REWRITE_TAC[LIFT_DROP] THEN MATCH_MP_TAC(REAL_ARITH `drop a <= norm a /\ x <= drop a==> x <= norm a`) THEN CONJ_TAC THENL [REWRITE_TAC[drop; NORM_REAL] THEN REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN RULE_ASSUM_TAC(REWRITE_RULE[absolutely_integrable_on]) THEN ASM_REWRITE_TAC[LIFT_DROP; REAL_LE_REFL; IN_UNIV] THEN X_GEN_TAC `y:real^N` THEN EXPAND_TAC "g" THEN COND_CASES_TAC THEN REWRITE_TAC[NORM_0; NORM_POS_LE] THEN REWRITE_TAC[NORM_LIFT] THEN CONV_TAC NORM_ARITH]]);; let FUBINI_TONELLI_ALT = prove (`!f:real^(M,N)finite_sum->real^P. f measurable_on (:real^(M,N)finite_sum) ==> (f absolutely_integrable_on (:real^(M,N)finite_sum) <=> negligible {y | ~((\x. f(pastecart x y)) absolutely_integrable_on (:real^M))} /\ (\y. integral (:real^M) (\x. lift(norm(f(pastecart x y))))) integrable_on (:real^N))`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `(f:real^(M,N)finite_sum->real^P) o (\z. pastecart (sndcart z) (fstcart z))` FUBINI_TONELLI) THEN REWRITE_TAC[] THEN ANTS_TAC THENL [W(MP_TAC o PART_MATCH (lhand o rand) MEASURABLE_ON_LINEAR_IMAGE_EQ_GEN o snd) THEN ASM_REWRITE_TAC[DIMINDEX_FINITE_SUM; ADD_SYM] THEN ANTS_TAC THENL [SIMP_TAC[linear; FORALL_PASTECART; FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_INJ; FSTCART_ADD; SNDCART_ADD; FSTCART_CMUL; SNDCART_CMUL] THEN REWRITE_TAC[GSYM PASTECART_ADD; GSYM PASTECART_CMUL]; DISCH_THEN SUBST1_TAC THEN POP_ASSUM MP_TAC THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_UNIV] THEN REWRITE_TAC[EXISTS_PASTECART; FORALL_PASTECART] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN MESON_TAC[]]; REWRITE_TAC[ABSOLUTELY_INTEGRABLE_PASTECART_SYM_UNIV; o_DEF; FSTCART_PASTECART; SNDCART_PASTECART]]);; let HAS_DOUBLE_INTEGRAL_PCROSS = prove (`!bop:real^P->real^Q->real^R f:real^M->real^P g:real^N->real^Q s t. bilinear bop /\ f absolutely_integrable_on s /\ g absolutely_integrable_on t ==> ((\z. bop (f(fstcart z)) (g(sndcart z))) has_integral bop (integral s f) (integral t g)) (s PCROSS t)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ONCE_REWRITE_TAC[GSYM HAS_INTEGRAL_RESTRICT_UNIV] THEN ONCE_REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV] THEN ONCE_REWRITE_TAC[GSYM INTEGRAL_RESTRICT_UNIV] THEN ABBREV_TAC `f':real^M->real^P = \x. if x IN s then f x else vec 0` THEN ABBREV_TAC `g':real^N->real^Q = \x. if x IN t then g x else vec 0` THEN SUBGOAL_THEN `(\x:real^(M,N)finite_sum. if x IN s PCROSS t then (bop:real^P->real^Q->real^R) (f (fstcart x)) (g (sndcart x)) else vec 0) = (\x. bop (f'(fstcart x)) (g'(sndcart x)))` SUBST1_TAC THENL [MAP_EVERY EXPAND_TAC ["f'"; "g'"] THEN REWRITE_TAC[FUN_EQ_THM; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP BILINEAR_LZERO) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP BILINEAR_RZERO) THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN MAP_EVERY ASM_CASES_TAC [`(x:real^M) IN s`; `(y:real^N) IN t`] THEN ASM_REWRITE_TAC[]; UNDISCH_TAC `bilinear(bop:real^P->real^Q->real^R)` THEN POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`f':real^M->real^P`,`f:real^M->real^P`) THEN SPEC_TAC(`g':real^N->real^Q`,`g:real^N->real^Q`) THEN REPEAT STRIP_TAC] THEN ABBREV_TAC `h = \z. (bop:real^P->real^Q->real^R) ((f:real^M->real^P) (fstcart z)) ((g:real^N->real^Q) (sndcart z))` THEN SUBGOAL_THEN `(h:real^(M,N)finite_sum->real^R) measurable_on UNIV` ASSUME_TAC THENL [EXPAND_TAC "h" THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_BILINEAR)) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_COMPOSE_FSTCART; MATCH_MP_TAC MEASURABLE_ON_COMPOSE_SNDCART] THEN RULE_ASSUM_TAC(REWRITE_RULE[ABSOLUTELY_INTEGRABLE_MEASURABLE]) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(h:real^(M,N)finite_sum->real^R) absolutely_integrable_on UNIV` ASSUME_TAC THENL [MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\z. lift(B * norm((f:real^M->real^P) (fstcart z)) * norm((g:real^N->real^Q) (sndcart z)))` THEN REWRITE_TAC[IN_UNIV; LIFT_DROP; LIFT_CMUL; DROP_CMUL] THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_CMUL; EXPAND_TAC "h" THEN REWRITE_TAC[DROP_CMUL; LIFT_DROP] THEN ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN W(MP_TAC o PART_MATCH (lhand o rand) FUBINI_TONELLI_ALT o snd) THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ANTS_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_MUL THEN CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_ON_NORM THENL [MATCH_MP_TAC MEASURABLE_ON_COMPOSE_FSTCART; MATCH_MP_TAC MEASURABLE_ON_COMPOSE_SNDCART] THEN RULE_ASSUM_TAC(REWRITE_RULE[ABSOLUTELY_INTEGRABLE_MEASURABLE]) THEN ASM_REWRITE_TAC[]; DISCH_THEN SUBST1_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s = {} ==> negligible s`) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN X_GEN_TAC `y:real^N` THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_CMUL THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC INTEGRABLE_EQ THEN REWRITE_TAC[NORM_MUL; LIFT_CMUL; REAL_ABS_NORM; NORM_LIFT; IN_UNIV] THEN EXISTS_TAC `\y. drop(integral (:real^M) (\x. lift(norm((f:real^M->real^P) x)))) % lift(norm((g:real^N->real^Q) y))` THEN CONJ_TAC THENL [X_GEN_TAC `y:real^N` THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN REWRITE_TAC[LIFT_DROP] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_CMUL THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN MATCH_MP_TAC INTEGRABLE_CMUL THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN ASM_SIMP_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN FIRST_ASSUM(MP_TAC o MATCH_MP FUBINI_ABSOLUTELY_INTEGRABLE) THEN EXPAND_TAC "h" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN MP_TAC(GEN `x:real^P` (ISPECL [`g:real^N->real^Q`; `(:real^N)`; `(bop:real^P->real^Q->real^R) x`] INTEGRAL_LINEAR)) THEN RULE_ASSUM_TAC(REWRITE_RULE[bilinear; ETA_AX]) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; o_DEF] THEN DISCH_THEN(K ALL_TAC) THEN MP_TAC(GEN `y:real^Q` (ISPECL [`f:real^M->real^P`; `(:real^M)`; `\x. (bop:real^P->real^Q->real^R) x y`] INTEGRAL_LINEAR)) THEN RULE_ASSUM_TAC(REWRITE_RULE[bilinear; ETA_AX]) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; o_DEF]);; (* ------------------------------------------------------------------------- *) (* Some versions of Fubini where we stay in a fixed space R^n. *) (* ------------------------------------------------------------------------- *) let FUBINI_NEGLIGIBLE_REPLACEMENTS = prove (`!k s:real^N->bool. lebesgue_measurable s ==> (negligible s <=> negligible { lift a | ~negligible { x:real^N | (lambda i. if i = k then a else x$i) IN s}})`, let lemma0 = prove (`!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> linear((\x. lambda k. x$swap(i,j)k):real^N->real^N)`, SIMP_TAC[linear; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA]) in let lemma1 = prove (`!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> ((\x. lambda k. x$swap(i,j)k):real^N->real^N) o ((\x. lambda k. x$swap(i,j)k):real^N->real^N) = I`, SIMP_TAC[CART_EQ; FUN_EQ_THM; o_THM; I_THM; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[swap] THEN REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA])) in let lemma2 = prove (`!i j s. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> (negligible (IMAGE ((\x. lambda k. x$swap(i,j)k):real^N->real^N) s) <=> negligible s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_LINEAR_IMAGE_EQ THEN ASM_SIMP_TAC[lemma0] THEN MATCH_MP_TAC(MESON[] `(!x. n(n x) = x) ==> !x y. n x = n y ==> x = y`) THEN ASM_SIMP_TAC[REWRITE_RULE[o_DEF; FUN_EQ_THM; I_DEF] lemma1]) in let lemma3 = prove (`!s. negligible s <=> negligible(s PCROSS (:real^1))`, REWRITE_TAC[NEGLIGIBLE_PCROSS; NOT_NEGLIGIBLE_UNIV]) in let lemma4 = prove (`!s:real^(N,1)finite_sum->bool. lebesgue_measurable s ==> (negligible s <=> negligible {lift a | ~negligible { x:real^(N,1)finite_sum | (lambda i. if i = dimindex(:N) + 1 then a else x$i) IN s}})`, REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP FUBINI_TONELLI_NEGLIGIBLE_ALT) THEN AP_TERM_TAC THEN GEN_REWRITE_TAC RAND_CONV [SIMPLE_IMAGE_GEN] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL [MESON_TAC[LIFT_DROP]; X_GEN_TAC `y:real`] THEN REWRITE_TAC[IN_ELIM_THM] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [lemma3] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `z:real^1`] THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; pastecart; DIMINDEX_FINITE_SUM; DIMINDEX_1; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_CASES_TAC `i = dimindex(:N) + 1` THEN ASM_REWRITE_TAC[ARITH_RULE `~(n + 1 <= n) /\ (n + 1) - n = 1`] THEN REWRITE_TAC[GSYM drop; LIFT_DROP] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_ARITH_TAC) in let lemma5 = prove (`!k s:real^(N,1)finite_sum->bool. lebesgue_measurable s /\ 1 <= k /\ k <= dimindex(:N) ==> (negligible s <=> negligible { lift a | ~negligible { x:real^(N,1)finite_sum | (lambda i. if i = k then a else x$i) IN s}})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`k:num`; `dimindex(:N) + 1`] (INST_TYPE [`:(N,1)finite_sum`,`:N`] lemma2)) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL [REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; DISCH_THEN(fun th -> ONCE_REWRITE_TAC[GSYM th])] THEN W(MP_TAC o PART_MATCH (lhand o rand) lemma4 o lhand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN THEN ASM_REWRITE_TAC[LE_REFL] THEN MATCH_MP_TAC lemma0 THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `a:real` THEN AP_TERM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `(!x. f(f x) = x) ==> (!y. P y ==> ?x. f x = y)`) THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM] lemma1) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN MATCH_MP_TAC(SET_RULE `(!x. f(f x) = x) /\ (f a IN s <=> Q) ==> (a IN IMAGE f s <=> Q)`) THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM] lemma1) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; AP_THM_TAC THEN AP_TERM_TAC] THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = k` THEN ASM_REWRITE_TAC[swap] THEN (W(MP_TAC o PART_MATCH (lhs o rand) LAMBDA_BETA o lhand o snd) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN SUBST1_TAC THEN SIMP_TAC[]]) THEN COND_CASES_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhs o rand) LAMBDA_BETA o lhand o snd) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN ASM_ARITH_TAC) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `1 <= k /\ k <= dimindex(:N)` THENL [GEN_REWRITE_TAC LAND_CONV [lemma3] THEN MP_TAC(ISPECL [`k:num`; `(s:real^N->bool) PCROSS (:real^1)`] lemma5) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_PCROSS; LEBESGUE_MEASURABLE_UNIV] THEN DISCH_THEN SUBST1_TAC THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `a:real` THEN AP_TERM_TAC THEN GEN_REWRITE_TAC RAND_CONV [lemma3] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; FORALL_PASTECART] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM PASTECART_FST_SND] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV; IN_ELIM_THM] THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[fstcart; CART_EQ; LAMBDA_BETA; DIMINDEX_FINITE_SUM; DIMINDEX_1; pastecart; ARITH_RULE `i <= n ==> i <= n + 1`]; SUBGOAL_THEN `!a x. (lambda i. if i = k then a else x$i):real^N = x` (fun th -> REWRITE_TAC[th]) THENL [SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[SET_RULE `{lift a | p} = if p then IMAGE lift UNIV else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[IMAGE_LIFT_UNIV; NOT_NEGLIGIBLE_UNIV; NEGLIGIBLE_EMPTY] THEN ASM_CASES_TAC `negligible(s:real^N->bool)` THEN ASM_REWRITE_TAC[SET_RULE `{x | x IN s} = s`]]);; let FUBINI_NEGLIGIBLE_REPLACEMENTS_ALT = prove (`!k s:real^N->bool. lebesgue_measurable s ==> (negligible s <=> negligible { x:real^N | ~negligible { lift a | (lambda i. if i = k then a else x$i) IN s}})`, let lemma0 = prove (`!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> linear((\x. lambda k. x$swap(i,j)k):real^N->real^N)`, SIMP_TAC[linear; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; LAMBDA_BETA]) in let lemma1 = prove (`!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> ((\x. lambda k. x$swap(i,j)k):real^N->real^N) o ((\x. lambda k. x$swap(i,j)k):real^N->real^N) = I`, SIMP_TAC[CART_EQ; FUN_EQ_THM; o_THM; I_THM; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[swap] THEN REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA])) in let lemma2 = prove (`!i j s. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> (negligible (IMAGE ((\x. lambda k. x$swap(i,j)k):real^N->real^N) s) <=> negligible s)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_LINEAR_IMAGE_EQ THEN ASM_SIMP_TAC[lemma0] THEN MATCH_MP_TAC(MESON[] `(!x. n(n x) = x) ==> !x y. n x = n y ==> x = y`) THEN ASM_SIMP_TAC[REWRITE_RULE[o_DEF; FUN_EQ_THM; I_DEF] lemma1]) in let lemma3 = prove (`!s. negligible s <=> negligible(s PCROSS (:real^1))`, REWRITE_TAC[NEGLIGIBLE_PCROSS; NOT_NEGLIGIBLE_UNIV]) in let lemma4 = prove (`!s:real^(N,1)finite_sum->bool. lebesgue_measurable s ==> (negligible s <=> negligible { x:real^(N,1)finite_sum | ~negligible { lift a | (lambda i. if i = dimindex(:N) + 1 then a else x$i) IN s}})`, REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP FUBINI_TONELLI_NEGLIGIBLE) THEN GEN_REWRITE_TAC LAND_CONV [lemma3] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `z:real^1`] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC RAND_CONV [SIMPLE_IMAGE_GEN] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL [MESON_TAC[LIFT_DROP]; X_GEN_TAC `y:real`] THEN REWRITE_TAC[IN_ELIM_THM] THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; pastecart; DIMINDEX_FINITE_SUM; DIMINDEX_1; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_CASES_TAC `i = dimindex(:N) + 1` THEN ASM_REWRITE_TAC[ARITH_RULE `~(n + 1 <= n) /\ (n + 1) - n = 1`] THEN REWRITE_TAC[GSYM drop; LIFT_DROP] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_ARITH_TAC) in let lemma5 = prove (`!k s:real^(N,1)finite_sum->bool. lebesgue_measurable s /\ 1 <= k /\ k <= dimindex(:N) ==> (negligible s <=> negligible { x:real^(N,1)finite_sum | ~negligible { lift a | (lambda i. if i = k then a else x$i) IN s}})`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`k:num`; `dimindex(:N) + 1`] (INST_TYPE [`:(N,1)finite_sum`,`:N`] lemma2)) THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN ANTS_TAC THENL [REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; DISCH_THEN(fun th -> ONCE_REWRITE_TAC[GSYM th])] THEN W(MP_TAC o PART_MATCH (lhand o rand) lemma4 o lhand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN THEN ASM_REWRITE_TAC[LE_REFL] THEN MATCH_MP_TAC lemma0 THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `(!x. f(f x) = x) ==> (!y. P y ==> ?x. f x = y)`) THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM] lemma1) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^(N,1)finite_sum` THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `a:real` THEN MATCH_MP_TAC(SET_RULE `(!x. f(f x) = x) /\ (f a IN s <=> Q) ==> (a IN IMAGE f s <=> Q)`) THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MATCH_MP_TAC(REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM] lemma1) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ASM_ARITH_TAC; AP_THM_TAC THEN AP_TERM_TAC] THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = k` THEN ASM_REWRITE_TAC[swap] THEN (W(MP_TAC o PART_MATCH (lhs o rand) LAMBDA_BETA o lhand o snd) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN SUBST1_TAC THEN SIMP_TAC[]]) THEN COND_CASES_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhs o rand) LAMBDA_BETA o lhand o snd) THEN REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN AP_TERM_TAC THEN ASM_ARITH_TAC) in REPEAT STRIP_TAC THEN ASM_CASES_TAC `1 <= k /\ k <= dimindex(:N)` THENL [ONCE_REWRITE_TAC[lemma3] THEN MP_TAC(ISPECL [`k:num`; `(s:real^N->bool) PCROSS (:real^1)`] lemma5) THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_PCROSS; LEBESGUE_MEASURABLE_UNIV] THEN DISCH_THEN SUBST1_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN REPEAT GEN_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM PASTECART_FST_SND] THEN REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV] THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[fstcart; CART_EQ; LAMBDA_BETA; DIMINDEX_FINITE_SUM; DIMINDEX_1; pastecart; ARITH_RULE `i <= n ==> i <= n + 1`]; SUBGOAL_THEN `!a x. (lambda i. if i = k then a else x$i):real^N = x` (fun th -> REWRITE_TAC[th]) THENL [SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[SET_RULE `{lift a | p} = if p then IMAGE lift UNIV else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[IMAGE_LIFT_UNIV; NOT_NEGLIGIBLE_UNIV; NEGLIGIBLE_EMPTY] THEN AP_TERM_TAC THEN SET_TAC[]]);; let FUBINI_NEGLIGIBLE_OFFSET = prove (`!s v:real^N. lebesgue_measurable s ==> (negligible s <=> negligible { x | ~negligible {t | (x + drop t % v) IN s}})`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `v:real^N = vec 0` THENL [ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN REWRITE_TAC[SET_RULE `{a | P} = if P then UNIV else {}`] THEN ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[NOT_NEGLIGIBLE_UNIV; NEGLIGIBLE_EMPTY] THEN AP_TERM_TAC THEN SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPECL [`norm(v:real^N) % basis 1:real^N`; `v:real^N`] ORTHOGONAL_TRANSFORMATION_EXISTS) THEN SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN REWRITE_TAC[REAL_ABS_NORM; REAL_MUL_RID; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_INVERSE_o) THEN REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^N->real^N` THEN STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN MP_TAC(ISPECL [`1`; `IMAGE (g:real^N->real^N) s`] FUBINI_NEGLIGIBLE_REPLACEMENTS_ALT) THEN ASM_SIMP_TAC[LEBESGUE_MEASURABLE_LINEAR_IMAGE_GEN; LE_REFL] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL [ASM_MESON_TAC[NEGLIGIBLE_LINEAR_IMAGE_EQ; ORTHOGONAL_TRANSFORMATION_INJECTIVE]; ALL_TAC] THEN SUBGOAL_THEN `!x. {t | (x + drop t % v:real^N) IN s} = {t | ((g:real^N->real^N) x + (drop t * norm(v)) % basis 1) IN IMAGE g s}` (fun th -> ONCE_REWRITE_TAC[th]) THENL [X_GEN_TAC `x:real^N` THEN MATCH_MP_TAC(SET_RULE `(!t. f(b t) = a t) /\ (!x. f(g x) = x) /\ (!y. g(f y) = y) ==> {t | a t IN s} = {t | b t IN IMAGE g s}`) THEN ASM_SIMP_TAC[LINEAR_ADD] THEN ASM_MESON_TAC[LINEAR_CMUL; VECTOR_MUL_ASSOC]; REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM DROP_CMUL)]] THEN SUBGOAL_THEN `!P. {t:real^1 | P(norm(v:real^N) % t)} = IMAGE (\x. inv(norm v) % x) {t | P(t)}` (fun th -> REWRITE_TAC[th]) THENL [GEN_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [X_GEN_TAC `y:real^1` THEN DISCH_TAC THEN EXISTS_TAC `norm(v:real^N) % y:real^1`; ALL_TAC] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_MUL_RINV; NORM_EQ_0; VECTOR_MUL_LID]; ALL_TAC] THEN ASM_SIMP_TAC[NEGLIGIBLE_LINEAR_IMAGE_EQ; VECTOR_MUL_LCANCEL; NORM_EQ_0; REAL_INV_EQ_0; LINEAR_SCALING] THEN SUBGOAL_THEN `!P. {x | P(g x)} = IMAGE (f:real^N->real^N) {x | P x}` MP_TAC THENL [ASM SET_TAC[]; SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC)] THEN W(MP_TAC o PART_MATCH (lhand o rand) NEGLIGIBLE_LINEAR_IMAGE_EQ o rand o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_INJECTIVE]; DISCH_THEN SUBST1_TAC] THEN SUBGOAL_THEN `!s x. {t | (x + drop t % basis 1) IN s} = IMAGE (\y. --(x$1 % basis 1) + y) {t | ((lambda i. if i = 1 then drop t else x$i):real^N) IN s}` (fun th -> ONCE_REWRITE_TAC[th]) THENL [REPEAT GEN_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[EXISTS_REFL; VECTOR_ARITH `--a + x:real^N = y <=> x = y + a`] THEN GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM; DROP_ADD; DROP_NEG] THEN AP_THM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; LAMBDA_BETA; DROP_CMUL; DROP_BASIS; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN ASM_CASES_TAC `k = 1` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; REWRITE_TAC[NEGLIGIBLE_TRANSLATION_EQ]] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN AP_TERM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC LAND_CONV [SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[IN_ELIM_THM; LIFT_DROP] THEN MESON_TAC[LIFT_DROP]);; (* ------------------------------------------------------------------------- *) (* Some basic results about convolution. *) (* ------------------------------------------------------------------------- *) let HAS_INTEGRAL_CONVOLUTION_SYM = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P i x. ((\y. bop (f(x - y)) (g y)) has_integral i) UNIV <=> ((\y. bop (f y) (g(x - y))) has_integral i) UNIV`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM REFLECT_UNIV] THEN REWRITE_TAC[GSYM HAS_INTEGRAL_REFLECT_GEN] THEN MP_TAC(ISPEC `--x:real^M` TRANSLATION_UNIV) THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SYM th]) THEN REWRITE_TAC[GSYM HAS_INTEGRAL_TRANSLATION] THEN REWRITE_TAC[VECTOR_ARITH `x - --(--x + y):real^M = y`] THEN REWRITE_TAC[VECTOR_ARITH `--(--x + y):real^N = x - y`]);; let INTEGRABLE_CONVOLUTION_SYM = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P x. (\y. bop (f(x - y)) (g y)) integrable_on UNIV <=> (\y. bop (f y) (g(x - y))) integrable_on UNIV`, REWRITE_TAC[integrable_on; HAS_INTEGRAL_CONVOLUTION_SYM]);; let INTEGRAL_CONVOLUTION_SYM = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P x. integral UNIV (\y. bop (f(x - y)) (g y)) = integral UNIV (\y. bop (f y) (g(x - y)))`, REWRITE_TAC[integral; HAS_INTEGRAL_CONVOLUTION_SYM]);; let ABSOLUTELY_INTEGRABLE_CONVOLUTION_SYM = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P x. (\y. bop (f(x - y)) (g y)) absolutely_integrable_on UNIV <=> (\y. bop (f y) (g(x - y))) absolutely_integrable_on UNIV`, REPEAT GEN_TAC THEN REWRITE_TAC[absolutely_integrable_on] THEN BINOP_TAC THEN REWRITE_TAC[INTEGRABLE_CONVOLUTION_SYM] THEN MP_TAC(ISPECL [`\x y. lift(norm((bop:real^N->real^P->real^Q) x y))`; `f:real^M->real^N`; `g:real^M->real^P`; `x:real^M`] INTEGRABLE_CONVOLUTION_SYM) THEN SIMP_TAC[]);; let MEASURABLE_ON_CONVOLUTION = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P x. bilinear bop /\ f measurable_on (:real^M) /\ g measurable_on (:real^M) ==> (\y. bop (f(x - y)) (g y)) measurable_on (:real^M)`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_BILINEAR)) THEN ASM_REWRITE_TAC[VECTOR_SUB; MEASURABLE_ON_REFLECT; REFLECT_UNIV] THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`; `x:real^M`] MEASURABLE_ON_TRANSLATION) THEN ASM_REWRITE_TAC[TRANSLATION_UNIV]);; let ABSOLUTELY_INTEGRABLE_CONVOLUTION_AE, HAS_DOUBLE_INTEGRAL_CONVOLUTION = (CONJ_PAIR o prove) (`(!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P. bilinear bop /\ f absolutely_integrable_on (:real^M) /\ g absolutely_integrable_on (:real^M) ==> ?t. negligible t /\ !x. ~(x IN t) ==> (\y. bop (f(x - y)) (g y)) absolutely_integrable_on (:real^M)) /\ (!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P. bilinear bop /\ f absolutely_integrable_on (:real^M) /\ g absolutely_integrable_on (:real^M) ==> ((\x. integral (:real^M) (\y. bop (f(x - y)) (g y))) has_integral bop (integral (:real^M) f) (integral (:real^M) g)) (:real^M))`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN REWRITE_TAC[TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN STRIP_TAC THEN ABBREV_TAC `h = \z. (bop:real^N->real^P->real^Q) ((f:real^M->real^N) (fstcart z - sndcart z)) (g(sndcart z))` THEN SUBGOAL_THEN `(h:real^(M,M)finite_sum->real^Q) measurable_on UNIV` ASSUME_TAC THENL [EXPAND_TAC "h" THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ] MEASURABLE_ON_BILINEAR)) THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_COMPOSE_SUB; MATCH_MP_TAC MEASURABLE_ON_COMPOSE_SNDCART] THEN RULE_ASSUM_TAC(REWRITE_RULE[ABSOLUTELY_INTEGRABLE_MEASURABLE]) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(h:real^(M,M)finite_sum->real^Q) absolutely_integrable_on UNIV` ASSUME_TAC THENL [MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\z. lift(B * norm((f:real^M->real^N) (fstcart z - sndcart z)) * norm((g:real^M->real^P) (sndcart z)))` THEN REWRITE_TAC[IN_UNIV; LIFT_DROP; LIFT_CMUL; DROP_CMUL] THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_CMUL; EXPAND_TAC "h" THEN REWRITE_TAC[DROP_CMUL; LIFT_DROP] THEN ASM_REWRITE_TAC[]] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN W(MP_TAC o PART_MATCH (lhand o rand) FUBINI_TONELLI_ALT o snd) THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ANTS_TAC THENL [MATCH_MP_TAC MEASURABLE_ON_MUL THEN CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_ON_NORM THENL [MATCH_MP_TAC MEASURABLE_ON_COMPOSE_SUB; MATCH_MP_TAC MEASURABLE_ON_COMPOSE_SNDCART] THEN RULE_ASSUM_TAC(REWRITE_RULE[ABSOLUTELY_INTEGRABLE_MEASURABLE]) THEN ASM_REWRITE_TAC[]; DISCH_THEN SUBST1_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s = {} ==> negligible s`) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_CMUL THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN SUBST1_TAC(GSYM(ISPEC `y:real^M` TRANSLATION_UNIV)) THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN ASM_REWRITE_TAC[VECTOR_ADD_SUB; ETA_AX]; ALL_TAC] THEN MATCH_MP_TAC INTEGRABLE_EQ THEN REWRITE_TAC[NORM_MUL; LIFT_CMUL; REAL_ABS_NORM; NORM_LIFT; IN_UNIV] THEN EXISTS_TAC `\y. drop(integral (:real^M) (\x. lift(norm((f:real^M->real^N)(x - y))))) % lift(norm((g:real^M->real^P) y))` THEN CONJ_TAC THENL [X_GEN_TAC `y:real^M` THEN REWRITE_TAC[GSYM LIFT_CMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN REWRITE_TAC[LIFT_DROP] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_CMUL THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN SUBST1_TAC(GSYM(ISPEC `y:real^M` TRANSLATION_UNIV)) THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN ASM_REWRITE_TAC[VECTOR_ADD_SUB; ETA_AX]; ALL_TAC] THEN MATCH_MP_TAC INTEGRABLE_EQ THEN EXISTS_TAC `\y. drop(integral (:real^M) (\x. lift(norm((f:real^M->real^N) x)))) % lift(norm((g:real^M->real^P) y))` THEN REWRITE_TAC[IN_UNIV] THEN CONJ_TAC THENL [X_GEN_TAC `y:real^M` THEN AP_THM_TAC THEN AP_TERM_TAC THEN MP_TAC(ISPEC `y:real^M` TRANSLATION_UNIV) THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[GSYM INTEGRAL_TRANSLATION] THEN ASM_REWRITE_TAC[VECTOR_ADD_SUB; ETA_AX]; MATCH_MP_TAC INTEGRABLE_CMUL THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_NORM THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN MP_TAC(ISPEC `h:real^(M,M)finite_sum->real^Q` FUBINI_ABSOLUTELY_INTEGRABLE) THEN EXPAND_TAC "h" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[] `P s ==> (negligible s ==> ?t. negligible t /\ P t)`) THEN SET_TAC[]; ALL_TAC] THEN MP_TAC(ISPEC `h:real^(M,M)finite_sum->real^Q` FUBINI_ABSOLUTELY_INTEGRABLE_ALT) THEN EXPAND_TAC "h" THEN REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(ASSUME_TAC o CONJUNCT2) THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_UNIQUE)) THEN MATCH_MP_TAC HAS_INTEGRAL_EQ THEN EXISTS_TAC `\y:real^M. (bop:real^N->real^P->real^Q) (integral (:real^M) (\x. f(x - y))) (g y)` THEN REWRITE_TAC[IN_UNIV] THEN CONJ_TAC THENL [X_GEN_TAC `y:real^M` THEN CONV_TAC SYM_CONV THEN MP_TAC(ISPECL [`\x. (f:real^M->real^N)(x - y)`; `(:real^M)`; `\x. (bop:real^N->real^P->real^Q) x (g(y:real^M))`] INTEGRAL_LINEAR) THEN RULE_ASSUM_TAC(REWRITE_RULE[bilinear]) THEN ASM_REWRITE_TAC[o_DEF] THEN DISCH_THEN MATCH_MP_TAC THEN SUBST1_TAC(GSYM(ISPEC `y:real^M` TRANSLATION_UNIV)) THEN REWRITE_TAC[GSYM INTEGRABLE_TRANSLATION] THEN REWRITE_TAC[VECTOR_ADD_SUB; ETA_AX] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]; ALL_TAC] THEN MATCH_MP_TAC HAS_INTEGRAL_EQ THEN EXISTS_TAC `\y:real^M. (bop:real^N->real^P->real^Q) (integral (:real^M) f) (g y)` THEN REWRITE_TAC[IN_UNIV] THEN CONJ_TAC THENL [X_GEN_TAC `y:real^M` THEN AP_THM_TAC THEN AP_TERM_TAC THEN MP_TAC(ISPEC `y:real^M` TRANSLATION_UNIV) THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN REWRITE_TAC[GSYM INTEGRAL_TRANSLATION] THEN ASM_REWRITE_TAC[VECTOR_ADD_SUB; ETA_AX]; MP_TAC(GEN `y:real^P` (ISPECL [`g:real^M->real^P`; `y:real^P`; `(:real^M)`; `\z. (bop:real^N->real^P->real^Q) (integral (:real^M) f) z`] HAS_INTEGRAL_LINEAR)) THEN RULE_ASSUM_TAC(REWRITE_RULE[bilinear]) THEN ASM_REWRITE_TAC[o_DEF] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE]]);; let DOUBLE_INTEGRABLE_CONVOLUTION = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P. bilinear bop /\ f absolutely_integrable_on (:real^M) /\ g absolutely_integrable_on (:real^M) ==> (\x. integral (:real^M) (\y. bop (f(x - y)) (g y))) integrable_on (:real^M)`, REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP HAS_DOUBLE_INTEGRAL_CONVOLUTION) THEN REWRITE_TAC[integrable_on] THEN MESON_TAC[]);; let DOUBLE_INTEGRAL_CONVOLUTION = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P. bilinear bop /\ f absolutely_integrable_on (:real^M) /\ g absolutely_integrable_on (:real^M) ==> integral (:real^M) (\x. integral (:real^M) (\y. bop (f(x - y)) (g y))) = bop (integral (:real^M) f) (integral (:real^M) g)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC HAS_DOUBLE_INTEGRAL_CONVOLUTION THEN ASM_REWRITE_TAC[]);; let ABSOLUTELY_INTEGRABLE_CONVOLUTION_L2 = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P x. bilinear bop /\ f measurable_on (:real^M) /\ g measurable_on (:real^M) /\ (\x. lift(norm(f x) pow 2)) absolutely_integrable_on (:real^M) /\ (\x. lift(norm(g x) pow 2)) absolutely_integrable_on (:real^M) ==> (\y. bop (f(x - y)) (g y)) absolutely_integrable_on (:real^M)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN ASM_SIMP_TAC[MEASURABLE_ON_CONVOLUTION; IN_UNIV] THEN FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\y. B / &2 % lift(norm((f:real^M->real^N) (x - y)) pow 2 + norm((g:real^M->real^P) y) pow 2)` THEN CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_CMUL THEN REWRITE_TAC[LIFT_ADD] THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ADD THEN ASM_REWRITE_TAC[] THEN SUBST1_TAC(GSYM(ISPEC `x:real^M` TRANSLATION_UNIV)) THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REFLECT_UNIV] THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_REFLECT_GEN] THEN ASM_REWRITE_TAC[VECTOR_ARITH `x - (x + --y):real^N = y`]; X_GEN_TAC `y:real^M` THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH lhand th o lhand o snd)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[DROP_CMUL; LIFT_DROP; REAL_ARITH `B * x * y <= B / &2 * (x pow 2 + y pow 2) <=> &0 <= B * (x - y) pow 2`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LE_POW_2; REAL_LT_IMP_LE]]);; let ABSOLUTELY_INTEGRABLE_CONVOLUTION_L1_LINF = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P x. bilinear bop /\ f absolutely_integrable_on (:real^M) /\ g measurable_on (:real^M) /\ bounded(IMAGE g (:real^M)) ==> (\y. bop (f(x - y)) (g y)) absolutely_integrable_on (:real^M)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE; IN_UNIV] THEN X_GEN_TAC `C:real` THEN STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN EXISTS_TAC `\y. B % C % lift(norm((f:real^M->real^N) (x - y)))` THEN ASM_SIMP_TAC[MEASURABLE_ON_CONVOLUTION; INTEGRABLE_IMP_MEASURABLE; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE] THEN CONJ_TAC THENL [REPEAT(MATCH_MP_TAC INTEGRABLE_CMUL) THEN SUBST1_TAC(SYM(ISPEC `x:real^M` TRANSLATION_UNIV)) THEN REWRITE_TAC[GSYM INTEGRABLE_TRANSLATION] THEN ONCE_REWRITE_TAC[GSYM REFLECT_UNIV] THEN REWRITE_TAC[GSYM INTEGRABLE_REFLECT_GEN; VECTOR_ARITH `x - (x + --y):real^N = y`] THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE]; X_GEN_TAC `y:real^M` THEN REWRITE_TAC[IN_UNIV; DROP_CMUL; LIFT_DROP] THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH lhand th o lhand o snd)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REWRITE_TAC[NORM_POS_LE]]);; let ABSOLUTELY_INTEGRABLE_CONVOLUTION_LINF_L1 = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P x. bilinear bop /\ f measurable_on (:real^M) /\ bounded(IMAGE f (:real^M)) /\ g absolutely_integrable_on (:real^M) ==> (\y. bop (f(x - y)) (g y)) absolutely_integrable_on (:real^M)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[ABSOLUTELY_INTEGRABLE_CONVOLUTION_SYM] THEN MATCH_MP_TAC(REWRITE_RULE[BILINEAR_SWAP] (ISPEC `\x y. (bop:real^N->real^P->real^Q) y x` ABSOLUTELY_INTEGRABLE_CONVOLUTION_L1_LINF)) THEN ASM_REWRITE_TAC[]);; let CONTINUOUS_ON_CONVOLUTION_L1_LINF = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P. bilinear bop /\ f absolutely_integrable_on (:real^M) /\ g measurable_on (:real^M) /\ bounded(IMAGE g (:real^M)) ==> (\x. integral (:real^M) (\y. bop (f(x - y)) (g y))) continuous_on (:real^M)`, REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE; IN_UNIV] THEN X_GEN_TAC `C:real` THEN STRIP_TAC THEN ONCE_REWRITE_TAC[LIM_NULL] THEN MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\z. drop(integral (:real^M) (\y. B % C % lift(norm(f(z - y) - f(x - y):real^N))))` THEN REWRITE_TAC[WITHIN_UNIV; EVENTUALLY_AT] THEN CONJ_TAC THENL [EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; GSYM DIST_NZ] THEN X_GEN_TAC `z:real^M` THEN STRIP_TAC THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_CONVOLUTION_L1_LINF; ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; GSYM INTEGRAL_SUB] THEN MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC INTEGRABLE_SUB THEN CONJ_TAC THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_CONVOLUTION_L1_LINF THEN ASM_REWRITE_TAC[]; REPEAT(MATCH_MP_TAC INTEGRABLE_CMUL) THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THENL [SUBST1_TAC(SYM(ISPEC `z:real^M` TRANSLATION_UNIV)); SUBST1_TAC(SYM(ISPEC `x:real^M` TRANSLATION_UNIV))] THEN ONCE_REWRITE_TAC[GSYM REFLECT_UNIV] THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_REFLECT_GEN; GSYM ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN ASM_REWRITE_TAC[VECTOR_ARITH `x - (x + --y):real^N = y`; ETA_AX]; X_GEN_TAC `y:real^M` THEN REWRITE_TAC[IN_UNIV; DROP_CMUL; LIFT_DROP] THEN ASM_SIMP_TAC[GSYM BILINEAR_LSUB] THEN FIRST_X_ASSUM(fun t -> W(MP_TAC o PART_MATCH lhand t o lhand o snd)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REWRITE_TAC[NORM_POS_LE]]; SUBGOAL_THEN `!z w. integral (:real^M) (\y. B % C % lift(norm((f:real^M->real^N)(z - y) - f (w - y)))) = B % C % integral (:real^M) (\y. lift(norm((f(z - y) - f(w - y)))))` (fun th -> REWRITE_TAC[th]) THENL [REPEAT GEN_TAC THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN MATCH_MP_TAC INTEGRAL_CMUL THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_LIFT_NORM_INTEGRABLE THEN MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_SUB THEN CONJ_TAC THENL [SUBST1_TAC(SYM(ISPEC `z:real^M` TRANSLATION_UNIV)); SUBST1_TAC(SYM(ISPEC `w:real^M` TRANSLATION_UNIV))] THEN ONCE_REWRITE_TAC[GSYM REFLECT_UNIV] THEN REWRITE_TAC[GSYM ABSOLUTELY_INTEGRABLE_REFLECT_GEN; GSYM ABSOLUTELY_INTEGRABLE_TRANSLATION] THEN ASM_REWRITE_TAC[VECTOR_ARITH `x - (x + --y):real^N = y`; ETA_AX]; REWRITE_TAC[LIFT_CMUL; LIFT_DROP] THEN REPEAT(MATCH_MP_TAC LIM_NULL_CMUL) THEN ONCE_REWRITE_TAC[LIM_AT_ZERO] THEN REWRITE_TAC[] THEN SUBST1_TAC(SYM(ISPEC `x:real^M` TRANSLATION_UNIV)) THEN REWRITE_TAC[GSYM INTEGRAL_TRANSLATION; VECTOR_ARITH `(x + z) - (x + y):real^N = z + --y`] THEN REWRITE_TAC[VECTOR_ARITH `x - (x + y):real^N = --y`] THEN ONCE_REWRITE_TAC[GSYM REFLECT_UNIV] THEN REWRITE_TAC[GSYM INTEGRAL_REFLECT_GEN; VECTOR_NEG_NEG] THEN MATCH_MP_TAC CONTINUOUS_ON_ABSOLUTELY_INTEGRABLE_TRANSLATION_NORM THEN ASM_REWRITE_TAC[]]]);; let CONTINUOUS_ON_CONVOLUTION_LINF_L1 = prove (`!bop:real^N->real^P->real^Q f:real^M->real^N g:real^M->real^P. bilinear bop /\ f measurable_on (:real^M) /\ bounded(IMAGE f (:real^M)) /\ g absolutely_integrable_on (:real^M) ==> (\x. integral (:real^M) (\y. bop (f(x - y)) (g y))) continuous_on (:real^M)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[INTEGRAL_CONVOLUTION_SYM] THEN MATCH_MP_TAC(REWRITE_RULE[BILINEAR_SWAP] (ISPEC `\x y. (bop:real^N->real^P->real^Q) y x` CONTINUOUS_ON_CONVOLUTION_L1_LINF)) THEN ASM_REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* Some Steinhaus variants with two different sets of positive measure. *) (* ------------------------------------------------------------------------- *) let STEINHAUS_SUMS = prove (`!s t:real^N->bool. lebesgue_measurable s /\ ~negligible s /\ lebesgue_measurable t /\ ~negligible t ==> ~(interior {x + y | x IN s /\ y IN t} = {})`, SUBGOAL_THEN `!s t:real^N->bool. measurable s /\ &0 < measure s /\ measurable t /\ &0 < measure t ==> ~(interior {x + y | x IN s /\ y IN t} = {})` ASSUME_TAC THENL [REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`\x y. lift(drop x * drop y)`; `indicator(s:real^N->bool)`; `indicator(t:real^N->bool)`] HAS_DOUBLE_INTEGRAL_CONVOLUTION) THEN ASM_SIMP_TAC[ABSOLUTELY_INTEGRABLE_ON_INDICATOR; INTER_UNIV; INTEGRAL_INDICATOR; LIFT_CMUL; LIFT_DROP; BILINEAR_DROP_MUL] THEN REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `drop`) THEN REWRITE_TAC[DROP_CMUL; LIFT_DROP] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[] `x = y ==> &0 < y ==> &0 < x`)) THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN DISCH_THEN(MP_TAC o MATCH_MP REAL_LT_IMP_NZ) THEN REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] INTEGRAL_EQ_0)) THEN REWRITE_TAC[NOT_FORALL_THM; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`\x y:real^1. drop x % y`; `indicator(s:real^N->bool)`; `indicator(t:real^N->bool)`] CONTINUOUS_ON_CONVOLUTION_L1_LINF) THEN ASM_REWRITE_TAC[BILINEAR_DROP_MUL; ABSOLUTELY_INTEGRABLE_ON_INDICATOR; INTER_UNIV; GSYM lebesgue_measurable] THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE] THEN REWRITE_TAC[bounded; FORALL_IN_IMAGE; IN_UNIV; NORM_1] THEN ANTS_TAC THENL [MESON_TAC[DROP_INDICATOR_ABS_LE_1]; ALL_TAC] THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; WITHIN_UNIV] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN REWRITE_TAC[IN_UNIV] THEN REWRITE_TAC[continuous_at] THEN DISCH_THEN(MP_TAC o SPEC `norm(integral (:real^N) (\y. drop(indicator s (x - y)) % indicator t y))`) THEN ASM_REWRITE_TAC[NORM_POS_LT] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTERIOR] THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `d:real`] THEN ASM_REWRITE_TAC[SUBSET; IN_BALL; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ANTS_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH `dist(a:real^N,b) < norm b ==> ~(a = vec 0)`)) THEN DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] INTEGRAL_EQ_0)) THEN REWRITE_TAC[NOT_FORALL_THM; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `z:real^N` THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN REWRITE_TAC[DE_MORGAN_THM; GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN REWRITE_TAC[indicator] THEN ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN REWRITE_TAC[VEC_EQ; ARITH_EQ] THEN REWRITE_TAC[TAUT `(if p then F else T) <=> ~p`] THEN STRIP_TAC THEN MAP_EVERY EXISTS_TAC [`y - z:real^N`; `z:real^N`] THEN ASM_REWRITE_TAC[] THEN CONV_TAC VECTOR_ARITH; REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPEC `t:real^N->bool` NEGLIGIBLE_ON_INTERVALS) THEN MP_TAC(ISPEC `s:real^N->bool` NEGLIGIBLE_ON_INTERVALS) THEN ASM_REWRITE_TAC[NOT_FORALL_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`c:real^N`; `d:real^N`] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`s INTER interval[a:real^N,b]`; `t INTER interval[c:real^N,d]`]) THEN ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_INTERVAL; MEASURABLE_LEBESGUE_MEASURABLE_INTER_MEASURABLE] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> ~(s = {}) ==> ~(t = {})`) THEN MATCH_MP_TAC SUBSET_INTERIOR THEN SET_TAC[]]);; let STEINHAUS_DIFFS = prove (`!s t:real^N->bool. lebesgue_measurable s /\ ~negligible s /\ lebesgue_measurable t /\ ~negligible t ==> ~(interior {x - y | x IN s /\ y IN t} = {})`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `IMAGE (--) (t:real^N->bool)`] STEINHAUS_SUMS) THEN ASM_SIMP_TAC[LINEAR_NEGATION; VECTOR_EQ_NEG2; NEGLIGIBLE_LINEAR_IMAGE_EQ; LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ] THEN REWRITE_TAC[SET_RULE `{x + y:real^N | x IN s /\ y IN IMAGE f t} = {x + f y | x IN s /\ y IN t}`] THEN REWRITE_TAC[GSYM VECTOR_SUB]);; (* ------------------------------------------------------------------------- *) (* More refined Ostrowski-style theorems about midpoint convexity. *) (* ------------------------------------------------------------------------- *) let MIDPOINT_CONVEX_IMP_CONTINUOUS_OSTROWSKI = prove (`!f:real^N->real s t B. convex s /\ open s /\ (!x y. x IN s /\ y IN s ==> f(midpoint (x,y)) <= (f x + f y) / &2) /\ t SUBSET s /\ lebesgue_measurable t /\ ~negligible t /\ (!x. x IN t ==> f x <= B) ==> (lift o f) continuous_on s`, let lemma = prove (`!f:real^N->real s u a b. a IN s /\ convex s /\ open s /\ open u /\ u SUBSET s /\ ~(u = {}) /\ (!x y. x IN s /\ y IN s ==> f(midpoint (x,y)) <= (f x + f y) / &2) /\ (!x. x IN u ==> f x <= b) ==> ?v c. a IN v /\ open v /\ v SUBSET s /\ !x. x IN v ==> f x <= c`, REPEAT GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN GEOM_ORIGIN_TAC `z:real^N` THEN REPEAT GEN_TAC THEN ABBREV_TAC `g = \x. (f:real^N->real) (z + x)` THEN RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM]) THEN ASM_REWRITE_TAC[] THEN POP_ASSUM(K ALL_TAC) THEN SPEC_TAC(`a:real^N`,`y:real^N`) THEN SPEC_TAC(`g:real^N->real`,`f:real^N->real`) THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `y:real^N = vec 0` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `?z:real^N p n. z IN s /\ p < 2 EXP n /\ &p / &2 pow n % z = y` STRIP_ASSUME_TAC THENL [MP_TAC(ISPECL [`{r | r IN interval(vec 0,vec 1) /\ inv(drop r) % (y:real^N) IN s}`; `{inv (&2 pow n) % x:real^1 | n,x | !i. 1 <= i /\ i <= dimindex (:1) ==> integer (x$i)}`] OPEN_INTER_CLOSURE_EQ_EMPTY) THEN REWRITE_TAC[CLOSURE_DYADIC_RATIONALS; INTER_UNIV] THEN MATCH_MP_TAC(TAUT `p /\ ~q /\ (~r ==> s) ==> (p ==> (q <=> r)) ==> s`) THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_OPEN_PREIMAGE THEN ASM_REWRITE_TAC[OPEN_INTERVAL] THEN MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN SIMP_TAC[IN_INTERVAL_1; REAL_LT_IMP_NZ; LIFT_DROP; DROP_VEC] THEN REWRITE_TAC[CONTINUOUS_ON_ID]; REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN UNDISCH_TAC `open(s:real^N->bool)` THEN REWRITE_TAC[OPEN_CONTAINS_CBALL; SUBSET; IN_CBALL] THEN DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(MP_TAC o SPEC `lift(inv(&1 + e / norm(y:real^N)))`) THEN REWRITE_TAC[IN_INTERVAL_1; DROP_VEC; LIFT_DROP; REAL_INV_INV] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[REAL_LT_INV_EQ] THEN MATCH_MP_TAC REAL_LT_ADD THEN ASM_SIMP_TAC[NORM_POS_LT; REAL_LT_01; REAL_LT_DIV]; MATCH_MP_TAC REAL_INV_LT_1 THEN ASM_SIMP_TAC[REAL_LT_ADDR; NORM_POS_LT; REAL_LT_01; REAL_LT_DIV]; FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID] THEN REWRITE_TAC[NORM_ARITH `dist(y:real^N,y + x) = norm x`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC]; DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `~(r INTER s = {}) ==> ?x. x IN s /\ x IN r`)) THEN REWRITE_TAC[EXISTS_IN_GSPEC; DIMINDEX_1; FORALL_1] THEN REWRITE_TAC[GSYM drop; EXISTS_LIFT; LIFT_DROP; IN_ELIM_THM; DROP_CMUL; IN_INTERVAL_1; DROP_VEC; REAL_INV_MUL; REAL_INV_INV] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`n:num`; `x:real`] THEN ASM_CASES_TAC `&0 < x` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; INTEGER_POS] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `p:num` SUBST_ALL_TAC) THEN MAP_EVERY EXISTS_TAC [`(&2 pow n * inv(&p)) % y:real^N`; `p:num`; `n:num`] THEN ASM_REWRITE_TAC[GSYM REAL_OF_NUM_LT; GSYM REAL_OF_NUM_POW] THEN ASM_REWRITE_TAC[VECTOR_MUL_RCANCEL; VECTOR_ARITH `a % b % x:real^N = x <=> (a * b) % x = &1 % x`] THEN MP_TAC(SPEC `n:num` REAL_LT_POW2) THEN UNDISCH_TAC `&0 < &p` THEN SPEC_TAC(`&2 pow n`,`k:real`) THEN CONV_TAC REAL_FIELD]; MAP_EVERY EXISTS_TAC [`IMAGE (\x:real^N. y + x) (IMAGE (\x. (&1 - &p / &2 pow n) % x) u)`; `&(2 EXP n - p) / &2 pow n * b + &p / &2 pow n * f(z:real^N)`] THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[IN_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `vec 0:real^N` THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID]; ALL_TAC] THEN CONJ_TAC THENL [REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN ONCE_REWRITE_TAC[VECTOR_ADD_SYM] THEN MATCH_MP_TAC OPEN_AFFINITY THEN ASM_REWRITE_TAC[REAL_ARITH `&1 - x = &0 <=> x = &1`] THEN MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN MATCH_MP_TAC(REAL_FIELD `&0 <= x /\ x < y ==> ~(x / y = &1)`) THEN ASM_REWRITE_TAC[REAL_POS; REAL_OF_NUM_POW; REAL_OF_NUM_LT]; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; AND_FORALL_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN u` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`] MIDPOINT_CONVEX_DYADIC_RATIONALS) THEN ASM_SIMP_TAC[MIDPOINT_IN_CONVEX] THEN DISCH_THEN(MP_TAC o SPECL [`n:num`; `2 EXP n - p`; `p:num`; `x:real^N`; `z:real^N`]) THEN ASM_SIMP_TAC[ARITH_RULE `p:num < n ==> (n - p) + p = n`] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `&(2 EXP n - p) / &2 pow n % x + y:real^N = y + (&1 - &p / &2 pow n) % x` (fun th -> REWRITE_TAC[th]) THENL [ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; LT_IMP_LE; REAL_ARITH `(a - b) / c:real = a / c - b / c`] THEN SIMP_TAC[REAL_OF_NUM_POW; REAL_DIV_REFL; REAL_OF_NUM_EQ; EXP_EQ_0; ARITH_EQ] THEN CONV_TAC VECTOR_ARITH; MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[REAL_LE_RADD] THEN MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_DIV THEN REWRITE_TAC[REAL_OF_NUM_POW; REAL_POS]]]) in REPEAT STRIP_TAC THEN SUBGOAL_THEN `?u. u SUBSET s /\ open u /\ ~(u = {}) /\ !x. x IN u ==> (f:real^N->real)(x) <= B` STRIP_ASSUME_TAC THENL [EXISTS_TAC `interior {midpoint(x,y):real^N | x IN t /\ y IN t}` THEN REWRITE_TAC[OPEN_INTERIOR] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(MESON[SUBSET_TRANS; INTERIOR_SUBSET] `s SUBSET t ==> interior s SUBSET t`) THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN ASM_MESON_TAC[MIDPOINT_IN_CONVEX; SUBSET]; REWRITE_TAC[midpoint; SET_RULE `{inv(&2) % (x + y):real^N | P x y} = IMAGE (\x. inv(&2) % x) {x + y | P x y}`] THEN SIMP_TAC[INTERIOR_INJECTIVE_LINEAR_IMAGE; LINEAR_SCALING; VECTOR_ARITH `inv(&2) % x:real^N = inv(&2) % y <=> x = y`] THEN REWRITE_TAC[IMAGE_EQ_EMPTY] THEN MATCH_MP_TAC STEINHAUS_SUMS THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC(SET_RULE `interior s SUBSET s /\ (!x. x IN s ==> P x) ==> (!x:real^N. x IN interior s ==> P x)`) THEN REWRITE_TAC[INTERIOR_SUBSET; FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC(REAL_ARITH `x <= b /\ y <= b ==> (x + y) / &2 <= b`) THEN ASM SET_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`; `u:real^N->bool`; `a:real^N`; `B:real`] lemma) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`v:real^N->bool`; `C:real`] THEN STRIP_TAC THEN REWRITE_TAC[continuous_at; o_THM; DIST_LIFT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN REWRITE_TAC[TAUT `~(p /\ q) <=> p ==> ~q`] THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [NOT_FORALL_THM] THEN REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; REAL_NOT_LT] THEN DISCH_THEN(LABEL_TAC "A") THEN MP_TAC(ISPEC `s:real^N->bool` OPEN_CONTAINS_BALL) THEN ASM_REWRITE_TAC[SUBSET; IN_BALL] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `!d. &0 < d ==> ?x. x IN s /\ dist(a,x) < d /\ dist(a,x) < r /\ (f:real^N->real) a + e <= f x` (LABEL_TAC "B") THENL [X_GEN_TAC `d:real` THEN DISCH_TAC THEN REMOVE_THEN "A" (MP_TAC o SPEC `min d r:real`) THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH `e <= abs(x - a) ==> a + e <= x \/ x + e <= a`)) THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN EXISTS_TAC `&2 % a - x:real^N` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[NORM_ARITH `dist(a:real^N,&2 % a - x) = dist(x,a)`] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `&2 % a - x:real^N`]) THEN REWRITE_TAC[midpoint; VECTOR_ARITH `inv(&2) % (x + (&2 % a - x)):real^N = a`] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[]; ASM_REAL_ARITH_TAC] THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[NORM_ARITH `dist(a:real^N,&2 % a - x) = dist(x,a)`] THEN ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN SUBGOAL_THEN `!n d. &0 < d ==> ?x. x IN s /\ dist(a,x) < d /\ dist(a,x) < r /\ &2 pow n * e <= (f:real^N->real) x - f a` ASSUME_TAC THENL [MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[real_pow] THEN CONJ_TAC THENL [ASM_MESON_TAC[REAL_ARITH `&1 * e <= x - a <=> a + e <= x`]; ALL_TAC] THEN X_GEN_TAC `n:num` THEN DISCH_THEN(fun th -> X_GEN_TAC `d:real` THEN DISCH_TAC THEN MP_TAC(SPEC `min (d / &2) (r / &2)` th)) THEN ASM_REWRITE_TAC[REAL_HALF; REAL_LT_MIN] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `&2 % x - a:real^N` THEN CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[NORM_ARITH `dist(a:real^N,&2 % x - a) = &2 * dist(a,x)`; REAL_ARITH `&2 * x < r <=> x < r / &2`] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `n * e <= x ==> &2 * x <= y ==> (&2 * n) * e <= y`)) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `&2 % x - a:real^N`]) THEN REWRITE_TAC[midpoint; VECTOR_ARITH `inv(&2) % (a + (&2 % x - a)):real^N = x`] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[]; ASM_REAL_ARITH_TAC] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN UNDISCH_TAC `dist(a:real^N,x) < r / &2` THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN MP_TAC(SPEC `(C - (f:real^N->real) a) / e` REAL_ARCH_POW2) THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ] THEN DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN UNDISCH_TAC `open(v:real^N->bool)` THEN REWRITE_TAC[OPEN_CONTAINS_BALL; SUBSET; IN_BALL] THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `d:real`]) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(f:real^N->real) x <= C` MP_TAC THENL [ASM_SIMP_TAC[]; ASM_REAL_ARITH_TAC]);; let MIDPOINT_CONVEX_IMP_CONVEX_OSTROWSKI = prove (`!f:real^N->real s t B. convex s /\ open s /\ t SUBSET s /\ lebesgue_measurable t /\ ~negligible t /\ (!x y. x IN s /\ y IN s ==> f(midpoint (x,y)) <= (f x + f y) / &2) /\ (!x. x IN t ==> f x <= B) ==> f convex_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MIDPOINT_CONVEX THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MIDPOINT_CONVEX_IMP_CONTINUOUS_OSTROWSKI THEN ASM_MESON_TAC[]);; let MEASURABLE_MIDPOINT_CONVEX_IMP_CONTINUOUS = prove (`!f:real^N->real s. (lift o f) measurable_on s /\ open s /\ (!x y. x IN s /\ y IN s ==> f(midpoint (x,y)) <= (f x + f y) / &2) ==> (lift o f) continuous_on s`, REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `\n. {x | x IN ball(a,r) /\ (f:real^N->real) x <= &n}` NEGLIGIBLE_COUNTABLE_UNIONS) THEN GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN ANTS_TAC THENL [SUBGOAL_THEN `~negligible(ball(a:real^N,r))` MP_TAC THENL [ASM_SIMP_TAC[OPEN_NOT_NEGLIGIBLE; OPEN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE]; REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP] THEN AP_TERM_TAC THEN REWRITE_TAC[UNIONS_GSPEC; EXTENSION; IN_UNIV; IN_ELIM_THM] THEN MESON_TAC[REAL_ARCH_SIMPLE]; REWRITE_TAC[NOT_FORALL_THM] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC)] THEN MP_TAC(ISPECL [`f:real^N->real`; `ball(a:real^N,r)`; `{x:real^N | x IN ball (a,r) /\ f x <= &n}`; `&n:real`] MIDPOINT_CONVEX_IMP_CONTINUOUS_OSTROWSKI) THEN SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_BALL] THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[CENTRE_IN_BALL]] THEN ASM_REWRITE_TAC[SUBSET_RESTRICT; CONVEX_BALL] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MP_TAC(ISPECL [`lift o (f:real^N->real)`; `ball(a:real^N,r)`] MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_LE) THEN REWRITE_TAC[LEBESGUE_MEASURABLE_BALL; DIMINDEX_1; FORALL_1] THEN REWRITE_TAC[GSYM drop; o_THM; LIFT_DROP] THEN MATCH_MP_TAC(TAUT `(q ==> r) /\ p ==> (p <=> q) ==> r`) THEN SIMP_TAC[] THEN MATCH_MP_TAC MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[LEBESGUE_MEASURABLE_BALL]);; let MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_GEN = prove (`!f:real^N->real s. (lift o f) measurable_on s /\ convex s /\ (!x y. x IN s /\ y IN s ==> f(midpoint (x,y)) <= (f x + f y) / &2) ==> !x y u v. x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ u + v = &1 /\ (segment[x,y] SUBSET frontier s ==> x = y) ==> f (u % x + v % y) <= u * f x + v * f y`, REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `interior s:real^N->bool`] MEASURABLE_MIDPOINT_CONVEX_IMP_CONTINUOUS) THEN REWRITE_TAC[OPEN_INTERIOR] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET]] THEN MATCH_MP_TAC MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[INTERIOR_SUBSET] THEN SIMP_TAC[LEBESGUE_MEASURABLE_OPEN; OPEN_INTERIOR]; DISCH_TAC] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `v:real`; `u:real`] THEN ASM_CASES_TAC `v = &1 - u` THENL [FIRST_X_ASSUM SUBST_ALL_TAC; ASM_REAL_ARITH_TAC] THEN REWRITE_TAC[REAL_ARITH `&1 - u + u = &1`] THEN ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[VECTOR_ARITH `(&1 - u) % a + u % a:real^N = a`; REAL_ARITH `x <= (&1 - u) * x + u * x`] THEN STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC [`u = &0`; `u = &1`] THEN ASM_REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LZERO; VECTOR_MUL_LID; VECTOR_ADD_RID; REAL_MUL_LZERO; REAL_ADD_RID; REAL_LE_REFL; REAL_MUL_LID; REAL_SUB_REFL; REAL_ADD_LID; VECTOR_ADD_LID] THEN SUBGOAL_THEN `!v. v IN closure (interval(midpoint(vec 0,lift u),midpoint(lift u,vec 1)) INTER {inv (&2 pow n) % x:real^1 | n,x | !i. 1 <= i /\ i <= dimindex (:1) ==> integer (x$i)}) ==> lift(f((&1 - drop v) % a + drop v % b:real^N) - ((&1 - drop v) * f a + drop v * f b)) IN {x | x$1 <= &0}` MP_TAC THENL [MATCH_MP_TAC FORALL_IN_CLOSURE THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE] THEN SIMP_TAC[CLOSURE_DYADIC_RATIONALS_IN_OPEN_SET; OPEN_INTERVAL] THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `interval[midpoint(vec 0,lift u),midpoint(lift u,vec 1)]` THEN SIMP_TAC[CLOSURE_INTERVAL] THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN REWRITE_TAC[LIFT_SUB; LIFT_ADD; LIFT_CMUL; LIFT_DROP] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN SIMP_TAC[CONTINUOUS_ON_MUL; CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; o_DEF; LIFT_DROP; CONTINUOUS_ON_ID; LIFT_SUB; CONTINUOUS_ON_SUB] THEN REPLICATE_TAC 2 (ONCE_REWRITE_TAC[GSYM o_DEF]) THEN REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_MUL; CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; o_DEF; LIFT_DROP; CONTINUOUS_ON_ID; LIFT_SUB; CONTINUOUS_ON_SUB] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; FORALL_LIFT] THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; midpoint; DROP_CMUL; DROP_ADD; DROP_VEC] THEN X_GEN_TAC `w:real` THEN STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN interior s` THENL [REWRITE_TAC[VECTOR_ARITH `(&1 - w) % a + w % b:real^N = b - (&1 - w) % (b - a)`] THEN MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_SIMP_TAC[CLOSURE_INC] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `(b:real^N) IN interior s` THENL [REWRITE_TAC[VECTOR_ARITH `(&1 - w) % a + w % b:real^N = a - w % (a - b)`] THEN MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN ASM_SIMP_TAC[CLOSURE_INC] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `~(interior s:real^N->bool = {})` ASSUME_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `~(t SUBSET frontier s) ==> t SUBSET s /\ s DIFF frontier s = interior s ==> ~(interior s = {})`)) THEN ASM_SIMP_TAC[SET_DIFF_FRONTIER; SEGMENT_SUBSET_CONVEX]; ALL_TAC] THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`; `(&1 - w) % a + w % b:real^N`] SEGMENT_SUBSET_RELATIVE_FRONTIER_CONVEX) THEN ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR] THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; frontier; IN_DIFF] THEN ASM_SIMP_TAC[CLOSURE_INC] THEN MATCH_MP_TAC(SET_RULE `x IN t /\ t SUBSET s ==> ~(x IN t /\ x IN s /\ ~r) ==> r`) THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `w:real` THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[CONVEX_CONTAINS_OPEN_SEGMENT; SUBSET_TRANS; CLOSURE_SUBSET]]; REWRITE_TAC[SET_RULE `(!x. x IN s INTER t ==> P x) <=> (!x. x IN t ==> x IN s ==> P x)`] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[DIMINDEX_1; FORALL_1; FORALL_LIFT; GSYM drop; LIFT_DROP] THEN MAP_EVERY X_GEN_TAC [`n:num`; `p:real`] THEN DISCH_TAC THEN REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; midpoint; DROP_ADD; LIFT_DROP; DROP_VEC; IN_ELIM_THM] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN REWRITE_TAC[REAL_ARITH `x - y <= &0 <=> x <= y`] THEN STRIP_TAC THEN SUBGOAL_THEN `&0 < p / &2 pow n /\ p / &2 pow n < &1` MP_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_LT_POW2] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID] THEN STRIP_TAC THEN UNDISCH_TAC `integer p` THEN ASM_SIMP_TAC[INTEGER_POS; REAL_LT_IMP_LE] THEN DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST_ALL_TAC) THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`] MIDPOINT_CONVEX_DYADIC_RATIONALS) THEN ASM_SIMP_TAC[CONVEX_ON_IMP_MIDPOINT_CONVEX; MIDPOINT_IN_CONVEX] THEN DISCH_THEN(MP_TAC o SPECL [`n:num`; `2 EXP n - m`; `m:num`; `a:real^N`; `b:real^N`]) THEN SUBGOAL_THEN `m <= 2 EXP n` ASSUME_TAC THENL [ASM_SIMP_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_POW; REAL_LT_IMP_LE]; ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_POW]] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN REWRITE_TAC[REAL_ARITH `(a - b) / c:real = a / c - b / c`] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_POW_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ]]; SIMP_TAC[CLOSURE_DYADIC_RATIONALS_IN_OPEN_SET; OPEN_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `lift u`) THEN REWRITE_TAC[IN_ELIM_THM; LIFT_DROP; REAL_ARITH `a - b <= &0 <=> a <= b`; GSYM drop; LIFT_DROP] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[CLOSURE_INTERVAL] THEN REWRITE_TAC[INTERVAL_EQ_EMPTY_1; midpoint; DROP_CMUL; DROP_ADD; DROP_VEC; LIFT_DROP] THEN COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_CMUL; DROP_ADD; DROP_VEC] THEN ASM_REAL_ARITH_TAC]);; let MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_OPEN = prove (`!f:real^N->real s. (lift o f) measurable_on s /\ convex s /\ open s /\ (!x y. x IN s /\ y IN s ==> f(midpoint (x,y)) <= (f x + f y) / &2) ==> f convex_on s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`] MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_GEN) THEN ASM_REWRITE_TAC[convex_on] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[frontier] THEN MATCH_MP_TAC(SET_RULE `a IN t /\ a IN segment[a,b] ==> segment[a,b] SUBSET s DIFF t ==> p`) THEN ASM_SIMP_TAC[ENDS_IN_SEGMENT; INTERIOR_OPEN]);; let MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_1D = prove (`!f:real^1->real s. (lift o f) measurable_on s /\ convex s /\ (!x y. x IN s /\ y IN s ==> f(midpoint (x,y)) <= (f x + f y) / &2) ==> f convex_on s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^1->real`; `s:real^1->bool`] MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_GEN) THEN ASM_REWRITE_TAC[convex_on] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET)) THEN ASM_SIMP_TAC[CARD_FRONTIER_INTERVAL_1; IS_INTERVAL_CONVEX_1] THEN ASM_REWRITE_TAC[FINITE_SEGMENT]);; let MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_CBALL = prove (`!f:real^N->real a r. (lift o f) measurable_on cball(a,r) /\ (!x y. x IN cball(a,r) /\ y IN cball(a,r) ==> f(midpoint (x,y)) <= (f x + f y) / &2) ==> f convex_on cball(a,r)`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `cball(a:real^N,r)`] MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_GEN) THEN ASM_REWRITE_TAC[convex_on; CONVEX_CBALL] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[FRONTIER_CBALL] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `segment[x,y] SUBSET s ==> x IN segment[x,y] /\ y IN segment[x,y] /\ midpoint(x,y) IN segment[x,y] ==> x IN s /\ y IN s /\ midpoint(x,y) IN s`)) THEN REWRITE_TAC[MIDPOINT_IN_SEGMENT; ENDS_IN_SEGMENT] THEN POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_SPHERE_0] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(MESON[DIFFERENT_NORM_3_COLLINEAR_POINTS] `(~(x = y) ==> m IN segment(x,y)) ==> norm x = r /\ norm y = r /\ norm m = r ==> x = y`) THEN REWRITE_TAC[MIDPOINT_IN_SEGMENT]);; let OSTROWSKI_THEOREM = prove (`!f:real^M->real^N B s. (!x y. f(x + y) = f(x) + f(y)) /\ (!x. x IN s ==> norm(f x) <= B) /\ measurable s /\ &0 < measure s ==> linear f`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[LINEAR_COMPONENTWISE] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[LINEAR_CONVEX_ON_1] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`vec 0:real^M`; `vec 0:real^M`]) THEN SIMP_TAC[VECTOR_ADD_LID; VECTOR_ARITH `a:real^N = a + a <=> a = vec 0`; VEC_COMPONENT; LIFT_NUM]; ALL_TAC] THEN SUBGOAL_THEN `!x. (f:real^M->real^N)(inv(&2) % x) = inv(&2) % f x` ASSUME_TAC THENL [GEN_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`inv(&2) % x:real^M`; `inv(&2) % x:real^M`]) THEN REWRITE_TAC[VECTOR_ARITH `inv(&2) % x + inv(&2) % x:real^N = x`] THEN CONV_TAC VECTOR_ARITH; CONJ_TAC THEN MATCH_MP_TAC MIDPOINT_CONVEX_IMP_CONVEX_OSTROWSKI THEN MAP_EVERY EXISTS_TAC [`s:real^M->bool`; `B:real`] THEN REWRITE_TAC[o_THM; LIFT_DROP; CONVEX_UNIV; OPEN_UNIV; SUBSET_UNIV] THEN ASM_SIMP_TAC[MEASURABLE_IMP_LEBESGUE_MEASURABLE; GSYM MEASURABLE_MEASURE_POS_LT] THEN ASM_REWRITE_TAC[IN_UNIV; midpoint; VECTOR_ADD_LDISTRIB] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN (CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC]) THEN ASM_MESON_TAC[REAL_LE_TRANS; COMPONENT_LE_NORM; REAL_ARITH `abs x <= b <=> x <= b /\ --x <= b`]]);; let MEASURABLE_ADDITIVE_IMP_LINEAR = prove (`!f:real^M->real^N. f measurable_on (:real^M) /\ (!x y. f(x + y) = f(x) + f(y)) ==> linear f`, GEN_TAC THEN ONCE_REWRITE_TAC[MEASURABLE_ON_COMPONENTWISE; CART_EQ; LINEAR_COMPONENTWISE] THEN ONCE_REWRITE_TAC[MESON[] `(!x y z. P x y z) <=> (!z x y. P x y z)`] THEN REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN ASM_CASES_TAC `1 <= i /\ i <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN ABBREV_TAC `g = \x. lift((f:real^M->real^N) x$i)` THEN RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM]) THEN ASM_REWRITE_TAC[GSYM LIFT_EQ; VECTOR_ADD_COMPONENT; LIFT_ADD] THEN POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[LINEAR_CONVEX_ON_1] THEN STRIP_TAC THEN SUBGOAL_THEN `!x. (g:real^M->real^1)(inv(&2) % x) = inv(&2) % g x` ASSUME_TAC THENL [GEN_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`inv(&2) % x:real^M`; `inv(&2) % x:real^M`]) THEN REWRITE_TAC[VECTOR_ARITH `inv(&2) % x + inv(&2) % x:real^N = x`] THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`vec 0:real^M`; `vec 0:real^M`]) THEN REWRITE_TAC[VECTOR_ADD_LID] THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_OPEN THEN ASM_REWRITE_TAC[CONVEX_UNIV; OPEN_UNIV; o_DEF; LIFT_DROP; ETA_AX] THEN ASM_REWRITE_TAC[midpoint; VECTOR_ADD_LDISTRIB] THEN REWRITE_TAC[DROP_ADD; DROP_CMUL; IN_UNIV; LIFT_NEG; LIFT_DROP] THEN ASM_REWRITE_TAC[MEASURABLE_ON_NEG_EQ] THEN REAL_ARITH_TAC);; let CONTINUOUS_ADDITIVE_IMP_LINEAR = prove (`!f:real^M->real^N. f continuous_on (:real^M) /\ (!x y. f(x + y) = f(x) + f(y)) ==> linear f`, ASM_MESON_TAC[CONTINUOUS_IMP_MEASURABLE_ON; MEASURABLE_ADDITIVE_IMP_LINEAR]);; (* ------------------------------------------------------------------------- *) (* Rademacher's theorem (the "simple proof" from Nekvinda and Zajicek). *) (* ------------------------------------------------------------------------- *) let RADEMACHER_UNIV = prove (`!f:real^M->real^N. (?B. !x y. norm(f x - f y) <= B * norm(x - y)) ==> negligible {x | ~(f differentiable at x)}`, let lemma = prove (`{x | ~(!i. i IN s ==> P i x)} = UNIONS (IMAGE (\i. {x | ~(P i x)}) s)`, REWRITE_TAC[UNIONS_IMAGE] THEN SET_TAC[]) and lemma' = prove (`{x | !i. P i ==> R i x} = INTERS {{x | R i x} | P i}`, REWRITE_TAC[INTERS_GSPEC] THEN SET_TAC[]) and lemur = prove (`!a b c. c < a + b ==> ?e p r s. rational r /\ rational s /\ rational p /\ rational e /\ &0 < e /\ a >= r + e /\ b >= s + e /\ c <= p - e /\ p < r + s`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`&0`; `((a + b) - c) / &10`] RATIONAL_APPROXIMATION_ABOVE) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c + e:real`; `e:real`] RATIONAL_APPROXIMATION_ABOVE) THEN MP_TAC(ISPECL [`b - e:real`; `e:real`] RATIONAL_APPROXIMATION_BELOW) THEN MP_TAC(ISPECL [`a - e:real`; `e:real`] RATIONAL_APPROXIMATION_BELOW) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `s:real` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `p:real` STRIP_ASSUME_TAC) THEN MAP_EVERY EXISTS_TAC [`e:real`; `p:real`; `r:real`; `s:real`] THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC) in SUBGOAL_THEN `!f:real^M->real^1. (?B. !x y. norm(f x - f y) <= B * norm(x - y)) ==> negligible {x | ~(f differentiable at x)}` ASSUME_TAC THENL [ALL_TAC; X_GEN_TAC `f:real^M->real^N` THEN DISCH_THEN(X_CHOOSE_TAC `B:real`) THEN ONCE_REWRITE_TAC[DIFFERENTIABLE_COMPONENTWISE_AT] THEN REWRITE_TAC[GSYM IN_NUMSEG; lemma] THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE; IN_NUMSEG] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `B:real` THEN ASM_REWRITE_TAC[GSYM LIFT_SUB; GSYM VECTOR_SUB_COMPONENT; NORM_LIFT] THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS]] THEN REPEAT GEN_TAC THEN REWRITE_TAC[LIPSCHITZ_POS] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `(:real^M)` SEPARABLE) THEN REWRITE_TAC[SUBSET_UNIV; SET_RULE `UNIV SUBSET s <=> s = UNIV`] THEN DISCH_THEN(X_CHOOSE_THEN `qq:real^M->bool` STRIP_ASSUME_TAC) THEN ABBREV_TAC `dd = UNIONS { {x | ~(?l. ((\t. inv(drop t) % ((f:real^M->real^1)(x + drop t % v) - f x)) --> l) (at (vec 0)))} | v IN qq}` THEN SUBGOAL_THEN `!x v. ~(x IN dd) ==> ?l. ((\t. inv(drop t) % ((f:real^M->real^1)(x + drop t % v) - f x)) --> l) (at (vec 0))` MP_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[CONVERGENT_EQ_CAUCHY_AT] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `v:real^M` o MATCH_MP (SET_RULE `s = UNIV ==> !x. x IN s`)) THEN REWRITE_TAC[CLOSURE_APPROACHABLE] THEN DISCH_THEN(MP_TAC o SPEC `e / &4 / B`) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &4`] THEN DISCH_THEN(X_CHOOSE_THEN `u:real^M` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `~((x:real^M) IN dd)` THEN EXPAND_TAC "dd" THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `u:real^M`) THEN ASM_REWRITE_TAC[CONVERGENT_EQ_CAUCHY_AT; NOT_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `s:real^1` THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `t:real^1` THEN REWRITE_TAC[GSYM DIST_NZ] THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `norm(a:real^M - a') <= e / &4 /\ norm(b - b') <= e / &4 ==> dist(a,b) < e / &2 ==> dist(a',b') < e`) THEN REWRITE_TAC[VECTOR_ARITH `c % (y - x) - c % (z - x):real^M = c % (y - z)`] THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; GSYM NORM_1] THEN ASM_SIMP_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div); REAL_LE_LDIV_EQ; NORM_POS_LT] THEN CONJ_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH lhand th o lhand o snd)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[VECTOR_ARITH `(x + s % u) - (x + s % v):real^N = s % (u - v)`] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[NORM_MUL; GSYM NORM_1; GSYM REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; GSYM dist; REAL_LT_IMP_LE]; GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `f':real^M->real^M->real^1` THEN DISCH_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `dd UNION {x | ~(x IN dd) /\ ~(linear((f':real^M->real^M->real^1) x))}` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; IN_UNION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[DE_MORGAN_THM; differentiable] THEN STRIP_TAC THEN EXISTS_TAC `(f':real^M->real^M->real^1) x` THEN MATCH_MP_TAC GATEAUX_DERIVATIVE_LIPSCHITZ THEN EXISTS_TAC `(:real^M)` THEN ASM_SIMP_TAC[IN_UNIV; OPEN_UNIV] THEN ASM_MESON_TAC[]] THEN SUBGOAL_THEN `!v. lebesgue_measurable {x | ~(?l. ((\t. inv(drop t) % ((f:real^M->real^1)(x + drop t % v) - f x)) --> l) (at (vec 0)))}` ASSUME_TAC THENL [X_GEN_TAC `v:real^M` THEN ONCE_REWRITE_TAC[GSYM LEBESGUE_MEASURABLE_COMPL] THEN REWRITE_TAC[SET_RULE `UNIV DIFF {x | ~P x} = {x | x IN UNIV /\ P x}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_POINTS_OF_CONVERGENCE THEN REWRITE_TAC[LEBESGUE_MEASURABLE_UNIV; CONTINUOUS_ON_CONST] THEN X_GEN_TAC `t:real^1` THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST]; ALL_TAC] THEN MATCH_MP_TAC LIPSCHITZ_IMP_CONTINUOUS_ON THEN EXISTS_TAC `B:real` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN CONJ_TAC THENL [EXPAND_TAC "dd" THEN REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN X_GEN_TAC `v:real^M` THEN DISCH_TAC THEN MP_TAC(GEN `s:real^M->bool` (ISPECL [`s:real^M->bool`; `v:real^M`] FUBINI_NEGLIGIBLE_OFFSET)) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o snd)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s:real^N->bool = {} ==> negligible s`) THEN REWRITE_TAC[SET_RULE `{x | ~P x} = {} <=> !x. P x`] THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[IN_ELIM_THM] THEN GEN_REWRITE_TAC I [NEGLIGIBLE_ON_INTERVALS] THEN MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`] THEN REWRITE_TAC[SET_RULE `{x | P x} INTER s = {x | x IN s /\ P x}`] THEN MP_TAC(ISPECL [`\h. (f:real^M->real^1) (y + drop h % v)`; `interval[a:real^1,b]`] LEBESGUE_DIFFERENTIATION_THEOREM) THEN REWRITE_TAC[IS_INTERVAL_INTERVAL] THEN ANTS_TAC THENL [MATCH_MP_TAC LIPSCHITZ_IMP_HAS_BOUNDED_VARIATION THEN EXISTS_TAC `B * norm(v:real^M)` THEN REWRITE_TAC[BOUNDED_INTERVAL] THEN MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN DISCH_TAC THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH lhand th o lhand o snd)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN REWRITE_TAC[NORM_MUL; GSYM DROP_SUB; GSYM NORM_1; VECTOR_ARITH `(x + a % v) - (x + b % v):real^M = (a - b) % v`] THEN REWRITE_TAC[REAL_LE_REFL; REAL_MUL_AC]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `h:real^1` THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN REWRITE_TAC[IN_ELIM_THM; IN_UNIV; VECTOR_DIFFERENTIABLE] THEN REWRITE_TAC[HAS_VECTOR_DERIVATIVE_AT_1D] THEN REPLICATE_TAC 3 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `l:real^1` THEN REWRITE_TAC[] THEN GEN_REWRITE_TAC RAND_CONV [LIM_AT_ZERO] THEN REWRITE_TAC[VECTOR_ADD_SUB; DROP_ADD; VECTOR_ADD_RDISTRIB] THEN REWRITE_TAC[GSYM VECTOR_ADD_ASSOC]]; ALL_TAC] THEN SUBGOAL_THEN `!c x v. ~(x IN dd) ==> (f':real^M->real^M->real^1) x (c % v) = c % f' x v` ASSUME_TAC THENL [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(ISPECL [`x:real^M`; `c % v:real^M`] th) THEN MP_TAC(ISPECL [`x:real^M`; `v:real^M`] th)) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> p /\ r ==> q ==> s`] LIM_UNIQUE) THEN REWRITE_TAC[TRIVIAL_LIMIT_AT] THEN ASM_CASES_TAC `c = &0` THENL [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO] THEN REWRITE_TAC[VECTOR_ADD_RID; VECTOR_SUB_REFL; VECTOR_MUL_RZERO] THEN REWRITE_TAC[LIM_CONST]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o SPEC `c:real` o MATCH_MP LIM_CMUL) THEN DISCH_THEN(MP_TAC o ISPECL [`at(vec 0:real^1)`; `\x:real^1. c % x`] o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT] (REWRITE_RULE[CONJ_ASSOC] LIM_COMPOSE_AT))) THEN ASM_REWRITE_TAC[o_DEF; EVENTUALLY_AT; VECTOR_MUL_EQ_0] THEN SIMP_TAC[GSYM DIST_NZ] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LT_01]] THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^1 = c % vec 0`) THEN MATCH_MP_TAC LIM_CMUL THEN REWRITE_TAC[VECTOR_MUL_RZERO; LIM_AT_ID]; REWRITE_TAC[DROP_CMUL; REAL_INV_MUL; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_ASSOC; REAL_MUL_RINV; REAL_MUL_LID] THEN REWRITE_TAC[REAL_ARITH `drop x * y = y * drop x`]]; ALL_TAC] THEN SUBGOAL_THEN `!v w r s p e. p < r + s /\ &0 < e ==> negligible {x | ~(x IN dd) /\ drop((f':real^M->real^M->real^1) x v) >= r + e /\ drop(f' x w) >= s + e /\ drop(f' x (v + w)) <= p - e}` (LABEL_TAC "*") THENL [ALL_TAC; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS { UNIONS {UNIONS {{x | ~(x IN dd) /\ drop((f':real^M->real^M->real^1) x v) >= r + e /\ drop(f' x w) >= s + e /\ drop(f' x (v + w)) <= p - e} |r,s| r IN rational /\ s IN {s | s IN rational /\ p < r + s}} |e,p| e IN {e | e IN rational /\ &0 < e} /\ p IN rational} | v IN qq UNION IMAGE (--) qq /\ w IN qq UNION IMAGE (--) qq}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[COUNTABLE_PRODUCT_DEPENDENT; COUNTABLE_UNION; COUNTABLE_IMAGE; FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN REPLICATE_TAC 2 (MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS_GEN THEN ASM_SIMP_TAC[COUNTABLE_PRODUCT_DEPENDENT; COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_RATIONAL; FORALL_IN_GSPEC] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [IN_ELIM_THM] THEN STRIP_TAC) THEN ASM_SIMP_TAC[]; ALL_TAC] THEN REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_SIMP_TAC[linear; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN DISCH_TAC THEN SUBGOAL_THEN `(f':real^M->real^M->real^1) x continuous_on (:real^M)` MP_TAC THENL [MATCH_MP_TAC LIPSCHITZ_IMP_CONTINUOUS_ON THEN EXISTS_TAC `B:real` THEN MAP_EVERY X_GEN_TAC [`w:real^M`; `z:real^M`] THEN DISCH_TAC THEN MATCH_MP_TAC(ISPEC `at (vec 0:real^1)` LIM_NORM_UBOUND) THEN EXISTS_TAC `\t. inv(drop t) % ((f:real^M->real^1)(x + drop t % w) - f x) - inv(drop t) % (f(x + drop t % z) - f x)` THEN ASM_SIMP_TAC[LIM_SUB; TRIVIAL_LIMIT_AT; EVENTUALLY_AT] THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN X_GEN_TAC `t:real^1` THEN REWRITE_TAC[GSYM DIST_NZ] THEN STRIP_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; VECTOR_ARITH `a % (y - x) - a % (z - x):real^M = a % (y - z)`] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; GSYM REAL_ABS_NZ; GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM] THEN ONCE_REWRITE_TAC[REAL_ARITH `(a * b) * c:real = a * c * b`] THEN REWRITE_TAC[GSYM NORM_MUL] THEN FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH lhand th o lhand o snd)) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN AP_TERM_TAC THEN CONV_TAC VECTOR_ARITH; ALL_TAC] THEN REWRITE_TAC[continuous_on; IN_UNIV] THEN GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `norm((f':real^M->real^M->real^1) x (u + v) - (f' x u + f' x v)) / &3`) THEN REWRITE_TAC[REAL_ARITH `&0 < x / &3 <=> &0 < x`] THEN ASM_SIMP_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN DISCH_THEN(fun th -> MP_TAC(SPEC `u + v:real^M` th) THEN MP_TAC(SPEC `v:real^M` th) THEN MP_TAC(SPEC `u:real^M` th)) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `du:real` THEN STRIP_TAC THEN X_GEN_TAC `dv:real` THEN STRIP_TAC THEN X_GEN_TAC `duv:real` THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `s = UNIV ==> !x. x IN s`)) THEN REWRITE_TAC[CLOSURE_APPROACHABLE] THEN DISCH_THEN(fun th -> MP_TAC(ISPECL [`v:real^M`; `min dv (duv / &2)`] th) THEN MP_TAC(ISPECL [`u:real^M`; `min du (duv / &2)`] th)) THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_HALF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u':real^M` THEN STRIP_TAC THEN X_GEN_TAC `v':real^M` THEN STRIP_TAC THEN SUBGOAL_THEN `~((f':real^M->real^M->real^1) x (u' + v') = f' x u' + f' x v')` MP_TAC THENL [MATCH_MP_TAC(NORM_ARITH `!u v uv. norm(uv' - uv) < norm(uv - (u + v)) / &3 /\ norm(u' - u) < norm(uv - (u + v)) / &3 /\ norm(v' - v) < norm(uv - (u + v)) / &3 ==> ~(uv':real^M = u' + v')`) THEN MAP_EVERY EXISTS_TAC [`(f':real^M->real^M->real^1) x u`; `(f':real^M->real^M->real^1) x v`; `(f':real^M->real^M->real^1) x (u + v)`] THEN REPEAT CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM dist] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY UNDISCH_TAC [`dist(u':real^M,u) < duv / &2`; `dist(v':real^M,v) < duv / &2`] THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM DROP_EQ] THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `~(x = y) <=> x < y \/ y < x`] THEN REWRITE_TAC[DROP_ADD; REAL_ARITH `y + z < x <=> --x < --y + --z`] THEN STRIP_TAC THENL [MAP_EVERY EXISTS_TAC [`u':real^M`; `v':real^M`]; MAP_EVERY EXISTS_TAC [`--u':real^M`; `--v':real^M`]] THEN (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN REWRITE_TAC[VECTOR_ARITH `--a + --b:real^N = --(a + b)`] THEN REWRITE_TAC[VECTOR_ARITH `--x:real^M = --(&1) % x`] THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[VECTOR_MUL_LNEG; VECTOR_MUL_LID; DROP_NEG] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; IN; GSYM CONJ_ASSOC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP lemur) THEN MESON_TAC[]] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `v + w:real^M = vec 0` THENL [MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s = {} ==> negligible s`) THEN ASM_REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN X_GEN_TAC `y:real^M` THEN ASM_CASES_TAC `(y:real^M) IN dd` THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP (VECTOR_ARITH `v + w:real^N = vec 0 ==> w = --(&1) % v`)) THEN SUBST1_TAC(VECTOR_ARITH `vec 0:real^M = &0 % v`) THEN ASM_SIMP_TAC[DROP_CMUL] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x | ~(x IN dd) /\ eventually (\t. (drop(f(x + drop t % v)) - drop(f(x:real^M))) / drop t >= r /\ (drop(f(x + drop t % w)) - drop(f x)) / drop t >= s /\ (drop(f(x + drop t % (v + w))) - drop(f x)) / drop t <= p) (at (vec 0))}` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN dd` THEN ASM_REWRITE_TAC[EVENTUALLY_AND] THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THEN REWRITE_TAC[real_gt] THEN DISCH_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `v:real^M`]); FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `w:real^M`]); FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `v + w:real^M`])] THEN ASM_REWRITE_TAC[tendsto] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN REWRITE_TAC[DIST_1; DROP_CMUL; DROP_SUB] THEN ASM_REAL_ARITH_TAC] THEN SUBGOAL_THEN `!e. &0 < e ==> negligible {x | ~(x IN dd) /\ !t. ~(t = vec 0) /\ norm(t) < e ==> (drop(f(x + drop t % v)) - drop(f x)) / drop t >= r /\ (drop(f(x + drop t % w)) - drop(f x)) / drop t >= s /\ (drop(f(x + drop t % (v + w:real^M))) - drop(f x)) / drop t <= p}` ASSUME_TAC THENL [ALL_TAC; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS { {x | ~(x IN dd) /\ !t. ~(t = vec 0) /\ norm(t) < inv(&n + &1) ==> (drop(f(x + drop t % v)) - drop(f x)) / drop t >= r /\ (drop(f(x + drop t % w)) - drop(f x)) / drop t >= s /\ (drop(f(x + drop t % (v + w:real^M))) - drop(f x)) / drop t <= p} | n IN (:num)}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN GEN_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC; REWRITE_TAC[SUBSET; IN_UNIV; UNIONS_GSPEC; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN dd` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[EVENTUALLY_AT] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN REWRITE_TAC[DIST_0; NORM_POS_LT] THEN MESON_TAC[REAL_LT_TRANS]]] THEN UNDISCH_THEN `&0 < e` (K ALL_TAC) THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(GEN `s:real^M->bool` (ISPECL [`s:real^M->bool`; `v + w:real^M`] FUBINI_NEGLIGIBLE_OFFSET)) THEN DISCH_THEN(fun th -> W(MP_TAC o PART_MATCH (lhand o rand) th o snd)) THEN ANTS_TAC THENL [REWRITE_TAC[SET_RULE `{x | ~(x IN s) /\ Q x} = (UNIV DIFF s) INTER {x | Q x}`] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_INTER THEN CONJ_TAC THENL [REWRITE_TAC[LEBESGUE_MEASURABLE_COMPL] THEN EXPAND_TAC "dd" THEN REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_COUNTABLE_UNIONS THEN ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE]; ALL_TAC] THEN REWRITE_TAC[lemma'] THEN MATCH_MP_TAC LEBESGUE_MEASURABLE_CLOSED THEN MATCH_MP_TAC CLOSED_INTERS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN REWRITE_TAC[real_ge] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN REPEAT(MATCH_MP_TAC CLOSED_INTER THEN CONJ_TAC) THEN ONCE_REWRITE_TAC[MESON[LIFT_DROP] `&0 <= x <=> &0 <= drop(lift x)`] THEN ONCE_REWRITE_TAC[SET_RULE `{x | &0 <= drop(f x)} = {x | x IN UNIV /\ f x IN {y | &0 <= drop y}}`] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN REWRITE_TAC[CLOSED_UNIV; drop; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_COMPONENT_GE] THEN REWRITE_TAC[GSYM drop; LIFT_SUB; LIFT_DROP; LIFT_CMUL; ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN (CONJ_TAC THENL [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST]; ALL_TAC]) THEN MATCH_MP_TAC LIPSCHITZ_IMP_CONTINUOUS_ON THEN ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC] THEN MATCH_MP_TAC(MESON[NEGLIGIBLE_EMPTY] `s = {} ==> negligible s`) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN X_GEN_TAC `z:real^M` THEN MATCH_MP_TAC(MESON[NEGLIGIBLE_SUBSET] `{x | P x /\ Q x} SUBSET {x | Q x} /\ negligible {x | Q x} ==> negligible {x | P x /\ Q x}`) THEN CONJ_TAC THENL [SIMP_TAC[IN_ELIM_THM; SUBSET]; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_COUNTABLE THEN MATCH_MP_TAC DISCRETE_IMP_COUNTABLE THEN MATCH_MP_TAC(MESON[] `(?e. &0 < e /\ !x y. x IN s /\ y IN s /\ ~(x = y) ==> e <= norm(y - x)) ==> !x:real^M. x IN s ==> ?e. &0 < e /\ !y. y IN s /\ ~(y = x) ==> e <= norm(y - x)`) THEN EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[FORALL_LIFT] THEN MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [REWRITE_TAC[NORM_SUB; CONJ_ACI; EQ_SYM_EQ]; ALL_TAC] THEN REWRITE_TAC[FORALL_DROP; LIFT_DROP; IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`r:real^1`; `s:real^1`] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN ABBREV_TAC `x = z + drop r % (v + w):real^M` THEN ABBREV_TAC `y = z + drop s % (v + w):real^M` THEN REWRITE_TAC[GSYM REAL_NOT_LT; NORM_1; DROP_SUB] THEN DISCH_TAC THEN ABBREV_TAC `t:real^1 = s - r` THEN SUBGOAL_THEN `~(t = vec 0) /\ &0 < drop t /\ norm(t) < e` STRIP_ASSUME_TAC THENL [EXPAND_TAC "t" THEN REWRITE_TAC[VECTOR_SUB_EQ; DROP_SUB; NORM_1; REAL_SUB_LT] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_LT_LE] THEN ASM_REWRITE_TAC[DROP_EQ]; ALL_TAC] THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 (MP_TAC o SPEC `t:real^1`) (MP_TAC o SPEC `--t:real^1`)) THEN SUBGOAL_THEN `x + drop t % (v + w):real^M = y /\ y + drop(--t) % w = x + drop t % v` STRIP_ASSUME_TAC THENL [MAP_EVERY EXPAND_TAC ["x"; "y"; "t"] THEN REWRITE_TAC[DROP_SUB; DROP_NEG] THEN CONV_TAC VECTOR_ARITH; ASM_REWRITE_TAC[VECTOR_NEG_EQ_0; NORM_NEG]] THEN REWRITE_TAC[DROP_NEG; real_div; REAL_INV_NEG; REAL_MUL_RNEG] THEN ASM_REAL_ARITH_TAC);; let RADEMACHER = prove (`!f:real^M->real^N s. (?B. !x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> negligible {x | x IN s /\ ~(f differentiable (at x within s))}`, REPEAT GEN_TAC THEN REWRITE_TAC[LIPSCHITZ_ON_POS] THEN DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `B:real`] KIRSZBRAUN) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `g:real^M->real^N` THEN STRIP_TAC THEN MP_TAC(ISPEC `g:real^M->real^N` RADEMACHER_UNIV) THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM; IMP_CONJ; CONTRAPOS_THM] THEN X_GEN_TAC `x:real^M` THEN REPEAT DISCH_TAC THEN MATCH_MP_TAC DIFFERENTIABLE_TRANSFORM_WITHIN THEN MAP_EVERY EXISTS_TAC [`g:real^M->real^N`; `&1`] THEN ASM_SIMP_TAC[DIFFERENTIABLE_AT_WITHIN; REAL_LT_01]);; let RADEMACHER_OPEN = prove (`!f:real^M->real^N s. open s /\ (?B. !x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> negligible {x | x IN s /\ ~(f differentiable (at x))}`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o MATCH_MP RADEMACHER) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN MATCH_MP_TAC(SET_RULE `(!x. P x ==> (Q x <=> R x)) ==> {x | P x /\ ~R x} SUBSET {x | P x /\ ~Q x}`) THEN ASM_SIMP_TAC[DIFFERENTIABLE_WITHIN_OPEN]);; let RADEMACHER_GEN = prove (`!f:real^M->real^N s. negligible(frontier s) /\ (?B. !x y. x IN s /\ y IN s ==> norm(f x - f y) <= B * norm(x - y)) ==> negligible {x | x IN s /\ ~(f differentiable (at x))}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `frontier s UNION {x | x IN interior s /\ ~((f:real^M->real^N) differentiable (at x))}` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN MATCH_MP_TAC RADEMACHER_OPEN THEN ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET; OPEN_INTERIOR]; REWRITE_TAC[GSYM SET_DIFF_FRONTIER] THEN SET_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Stepanov's theorem (Maly's proof, slightly generalized by localization). *) (* ------------------------------------------------------------------------- *) let STEPANOV_GEN = prove (`!f:real^M->real^N s. lebesgue_measurable s ==> negligible {x | x IN s /\ (?B. eventually (\y. norm(f x - f y) / norm(x - y) <= B) (at x within s)) /\ ~(f differentiable at x within s)}`, let lemma = prove (`{x | x IN s /\ P x /\ ?i. i IN k /\ R i x} = UNIONS {{x | x IN s /\ P x /\ R i x} | i IN k}`, REWRITE_TAC[UNIONS_GSPEC] THEN SET_TAC[]) in SUBGOAL_THEN `!f:real^M->real^1 s. lebesgue_measurable s ==> negligible {x | x IN s /\ (?B. eventually (\y. norm(f x - f y) / norm(x - y) <= B) (at x within s)) /\ ~(f differentiable at x within s)}` MP_TAC THENL [ALL_TAC; ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `s:real^M->bool` THEN DISCH_TAC THEN X_GEN_TAC `f:real^M->real^N` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[DIFFERENTIABLE_COMPONENTWISE_WITHIN] THEN REWRITE_TAC[NOT_FORALL_THM; GSYM IN_NUMSEG; NOT_IMP; lemma] THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FINITE_NUMSEG; FORALL_IN_IMAGE] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\x. lift((f:real^M->real^N) x$i)`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THEN REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `B:real` THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN X_GEN_TAC `y:real^M` THEN ASM_CASES_TAC `y:real^M = x` THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT; GSYM VECTOR_SUB_COMPONENT] THEN REWRITE_TAC[COMPONENT_LE_NORM]] THEN REPEAT STRIP_TAC THEN ABBREV_TAC `t = {x | x IN s /\ ?B. eventually (\y. norm(f x - f y:real^1) / norm(x - y) <= B) (at (x:real^M) within s)}` THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP (SET_RULE `{x | P x /\ Q x} = t ==> {x | P x /\ Q x /\ R x} = {x | x IN t /\ R x}`) th]) THEN ABBREV_TAC `us = { ball(x,e) | &0 < e /\ rational e /\ (!i. 1 <= i /\ i <= dimindex(:M) ==> rational(x$i)) /\ bounded (IMAGE (f:real^M->real^1) (ball(x,e) INTER s))}` THEN SUBGOAL_THEN `!x. x IN t ==> ?B. eventually (\y. norm(f x - f y:real^1) / norm(x - y) <= B) (at (x:real^M) within s)` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM] THEN REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `B:real^M->real` THEN DISCH_TAC THEN SUBGOAL_THEN `!x:real^M. x IN t ==> INFINITE {c | c IN us /\ x IN c /\ !y. y IN c INTER s ==> norm(f x - f y:real^1) <= B x * norm(x - y)}` ASSUME_TAC THENL [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[EVENTUALLY_WITHIN] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `x IN closure {x:real^M | !i. 1 <= i /\ i <= dimindex (:M) ==> rational (x$i)}` MP_TAC THENL [ASM_REWRITE_TAC[CLOSURE_RATIONAL_COORDINATES; IN_UNIV]; REWRITE_TAC[CLOSURE_APPROACHABLE; IN_ELIM_THM]] THEN DISCH_THEN(MP_TAC o SPEC `d / &3`) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN(X_CHOOSE_THEN `q:real^M` STRIP_ASSUME_TAC)] THEN MATCH_MP_TAC INFINITE_SUPERSET THEN EXISTS_TAC `{ball(q:real^M,e) | rational e /\ d / &3 < e /\ e < &2 * d / &3}` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC INFINITE_IMAGE THEN CONJ_TAC THENL [MATCH_MP_TAC INFINITE_RATIONAL_IN_RANGE; REWRITE_TAC[IN_ELIM_THM; EQ_BALLS]] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN EXPAND_TAC "us" THEN REWRITE_TAC[IN_ELIM_THM; IN_BALL] THEN REPEAT CONJ_TAC THENL [MAP_EVERY EXISTS_TAC [`q:real^M`; `e:real`] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[bounded; FORALL_IN_IMAGE] THEN EXISTS_TAC `norm((f:real^M->real^1) x) + (abs(B x) + &1) * d` THEN X_GEN_TAC `y:real^M` THEN REWRITE_TAC[IN_INTER; IN_BALL] THEN STRIP_TAC THEN ASM_CASES_TAC `y:real^M = x` THEN ASM_SIMP_TAC[REAL_LE_ADDR; REAL_LT_MUL; REAL_LT_IMP_LE; REAL_ARITH `&0 < abs B + &1`] THEN MATCH_MP_TAC(NORM_ARITH `!a. norm(x - y:real^M) <= a /\ a <= b ==> norm(x) <= norm(y) + b`) THEN EXISTS_TAC `B(x) * norm(y - x:real^M)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[GSYM DIST_NZ]; TRANS_TAC REAL_LE_TRANS `(abs(B x) + &1) * norm(y - x:real^M)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN REAL_ARITH_TAC; MATCH_MP_TAC REAL_LE_LMUL]]; ASM_REAL_ARITH_TAC; X_GEN_TAC `y:real^M` THEN REWRITE_TAC[IN_BALL; IN_INTER] THEN STRIP_TAC THEN ASM_CASES_TAC `y:real^M = x` THEN ASM_SIMP_TAC[VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL; GSYM REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[GSYM DIST_NZ]] THEN MAP_EVERY UNDISCH_TAC [`dist(q:real^M,x) < d / &3`; `dist(q:real^M,y) < e`; `e < &2 * d / &3`] THEN CONV_TAC NORM_ARITH]; ALL_TAC] THEN ASM_CASES_TAC `t:real^M->bool = {}` THEN ASM_REWRITE_TAC[EMPTY_GSPEC; NOT_IN_EMPTY; NEGLIGIBLE_EMPTY] THEN MP_TAC(ISPEC `us:(real^M->bool)->bool` COUNTABLE_AS_IMAGE) THEN ANTS_TAC THENL [CONJ_TAC THENL [EXPAND_TAC "us" THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `{ball(a,e) | (a:real^M) IN {y | !i. 1 <= i /\ i <= dimindex(:M) ==> rational(y$i)} /\ e IN rational}` THEN CONJ_TAC THENL [MATCH_MP_TAC COUNTABLE_PRODUCT_DEPENDENT THEN REWRITE_TAC[COUNTABLE_RATIONAL_COORDINATES; COUNTABLE_RATIONAL]; SET_TAC[]]; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_TAC `a:real^M`) THEN FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; INFINITE; EMPTY_GSPEC; FINITE_EMPTY]]; DISCH_THEN(X_CHOOSE_THEN `b:num->real^M->bool` (ASSUME_TAC o SYM))] THEN ABBREV_TAC `u = \i x. lift(inf {drop((g:real^M->real^1)(x)) |g| (!x. x IN b i INTER s ==> drop(f x) <= drop(g x)) /\ (!x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(g x - g y) <= &i * norm(x - y))})` THEN ABBREV_TAC `v = \i x. lift(sup {drop((g:real^M->real^1)(x)) |g| (!x. x IN b i INTER s ==> drop(g x) <= drop(f x)) /\ (!x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(g x - g y) <= &i * norm(x - y))})` THEN SUBGOAL_THEN `!i x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(u i x - u i y:real^1) <= &i * norm(x - y:real^M)` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "u" THEN REWRITE_TAC[] THEN MATCH_MP_TAC LIPSCHITZ_ON_INF THEN SIMP_TAC[REAL_POS] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN EXISTS_TAC `drop(f(x:real^M))` THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `!i x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(v i x - v i y:real^1) <= &i * norm(x - y:real^M)` ASSUME_TAC THENL [GEN_TAC THEN EXPAND_TAC "v" THEN REWRITE_TAC[] THEN MATCH_MP_TAC LIPSCHITZ_ON_SUP THEN SIMP_TAC[REAL_POS] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN EXISTS_TAC `drop(f(x:real^M))` THEN ASM_MESON_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `(!i:num x:real^M. x IN b i INTER s ==> drop(f x) <= drop(u i x)) /\ (!i z g. z IN b i INTER s /\ (!x. x IN b i INTER s ==> drop (f x) <= drop (g x)) /\ (!x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(g x - g y) <= &i * norm (x - y)) ==> drop(u i z) <= drop(g z))` STRIP_ASSUME_TAC THENL [REWRITE_TAC[AND_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`i:num`; `x:real^M`] THEN ASM_CASES_TAC `(x:real^M) IN b(i:num) INTER s` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `{drop((g:real^M->real^1)(x)) |g| (!x. x IN b i INTER s ==> drop(f x) <= drop(g x)) /\ (!x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(g x - g y) <= &i * norm(x - y))}` INF) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP (MESON[] `(\i x. f i x) = v ==> !i x. f i x = v i x`))) THEN GEN_REWRITE_TAC I [IMP_IMP] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM DROP_EQ] THEN REWRITE_TAC[LIFT_DROP] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN EXISTS_TAC `drop((f:real^M->real^1) x)` THEN ASM_SIMP_TAC[FORALL_IN_GSPEC] THEN MATCH_MP_TAC(SET_RULE `(?x. P x) ==> ~({f x | P x} = {})`) THEN SUBGOAL_THEN `bounded(IMAGE (f:real^M->real^1) (b(i:num) INTER s))` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[BOUNDED_POS]] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE; NORM_1] THEN X_GEN_TAC `C:real` THEN STRIP_TAC THEN EXISTS_TAC `\x:real^M. lift C` THEN REWRITE_TAC[VECTOR_SUB_REFL; DROP_VEC; REAL_ABS_NUM; LIFT_DROP] THEN SIMP_TAC[REAL_LE_MUL; REAL_POS; NORM_POS_LE] THEN ASM_MESON_TAC[REAL_ARITH `abs x <= C ==> x <= C`]; REWRITE_TAC[FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `(!i:num x:real^M. x IN b i INTER s ==> drop(v i x) <= drop(f x)) /\ (!i z g. z IN b i INTER s /\ (!x. x IN b i INTER s ==> drop (g x) <= drop (f x)) /\ (!x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(g x - g y) <= &i * norm (x - y)) ==> drop(g z) <= drop(v i z))` STRIP_ASSUME_TAC THENL [REWRITE_TAC[AND_FORALL_THM] THEN MAP_EVERY X_GEN_TAC [`i:num`; `x:real^M`] THEN ASM_CASES_TAC `(x:real^M) IN b(i:num) INTER s` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPEC `{drop((g:real^M->real^1)(x)) |g| (!x. x IN b i INTER s ==> drop(g x) <= drop(f x)) /\ (!x y. x IN b i INTER s /\ y IN b i INTER s ==> norm(g x - g y) <= &i * norm(x - y))}` SUP) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP (MESON[] `(\i x. f i x) = v ==> !i x. f i x = v i x`))) THEN GEN_REWRITE_TAC I [IMP_IMP] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM DROP_EQ] THEN REWRITE_TAC[LIFT_DROP] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN EXISTS_TAC `drop((f:real^M->real^1) x)` THEN ASM_SIMP_TAC[FORALL_IN_GSPEC] THEN MATCH_MP_TAC(SET_RULE `(?x. P x) ==> ~({f x | P x} = {})`) THEN SUBGOAL_THEN `bounded(IMAGE (f:real^M->real^1) (b(i:num) INTER s))` MP_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[BOUNDED_POS]] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_IN_IMAGE; NORM_1] THEN X_GEN_TAC `C:real` THEN STRIP_TAC THEN EXISTS_TAC `\x:real^M. lift(--C)` THEN REWRITE_TAC[VECTOR_SUB_REFL; DROP_VEC; REAL_ABS_NUM; LIFT_DROP] THEN SIMP_TAC[REAL_LE_MUL; REAL_POS; NORM_POS_LE] THEN ASM_MESON_TAC[REAL_ARITH `abs x <= C ==> --C <= x`]; REWRITE_TAC[FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `!a. a IN t ==> ?i. a IN b i /\ (u:num->real^M->real^1) i a = v i a` ASSUME_TAC THENL [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN SUBGOAL_THEN `(x:real^M) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN X_CHOOSE_TAC `M:num` (SPEC `(B:real^M->real) x` REAL_ARCH_SIMPLE) THEN SUBGOAL_THEN `INFINITE({i | x IN b i /\ !y. y IN b i INTER s ==> norm((f:real^M->real^1) x - f y) <= B x * norm(x - y)} DIFF (0..M))` MP_TAC THENL [MATCH_MP_TAC INFINITE_DIFF_FINITE THEN REWRITE_TAC[FINITE_NUMSEG] THEN REWRITE_TAC[INFINITE] THEN DISCH_THEN(MP_TAC o ISPEC `b:num->real^M->bool` o MATCH_MP FINITE_IMAGE) THEN REWRITE_TAC[GSYM INFINITE] THEN MATCH_MP_TAC INFINITE_SUPERSET THEN EXISTS_TAC `{c | c IN us /\ (x:real^M) IN c /\ !y. y IN c INTER s ==> norm(f x - f y:real^1) <= B x * norm(x - y)}` THEN ASM_SIMP_TAC[] THEN SUBST1_TAC(SYM(ASSUME `IMAGE (b:num->real^M->bool) (:num) = us`)) THEN SET_TAC[]; DISCH_THEN(MP_TAC o MATCH_MP (MESON[FINITE_EMPTY; INFINITE] `INFINITE s ==> ~(s = {})`))] THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_DIFF; IN_ELIM_THM; IN_NUMSEG] THEN REWRITE_TAC[LE_0; NOT_LE] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM DROP_EQ; GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[IN_INTER; REAL_LE_TRANS]] THEN TRANS_TAC REAL_LE_TRANS `drop(f(x:real^M))` THEN CONJ_TAC THENL [SUBGOAL_THEN `f x = (\y:real^M. f x + &i % lift(norm(y - x))) x` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; LIFT_NUM; VECTOR_MUL_RZERO; VECTOR_ADD_RID]; ALL_TAC]; SUBGOAL_THEN `f x = (\y:real^M. f x - &i % lift(norm(y - x))) x` SUBST1_TAC THENL [REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; LIFT_NUM; VECTOR_MUL_RZERO; VECTOR_SUB_RZERO]; ALL_TAC]] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTER] THEN REWRITE_TAC[VECTOR_ARITH `(x - a) - (x - b):real^M = --(a - b)`; VECTOR_ARITH `(x + a) - (x + b):real^M = a - b`] THEN REWRITE_TAC[NORM_NEG; GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN REWRITE_TAC[REAL_ABS_NUM; GSYM LIFT_SUB; NORM_LIFT] THEN (CONJ_TAC THENL [ALL_TAC; REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_POS] THEN CONV_TAC NORM_ARITH]) THEN REWRITE_TAC[DROP_ADD; DROP_SUB] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THENL [MATCH_MP_TAC(REAL_ARITH `abs(y - x) <= e ==> y <= x + e`); MATCH_MP_TAC(REAL_ARITH `abs(y - x) <= e ==> x - e <= y`)] THEN REWRITE_TAC[GSYM DROP_SUB; GSYM NORM_1; DROP_CMUL; LIFT_DROP] THEN ASM_CASES_TAC `y:real^M = x` THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL; REAL_MUL_RZERO; REAL_LE_REFL; NORM_0] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN TRANS_TAC REAL_LE_TRANS `&M` THEN ASM_SIMP_TAC[REAL_OF_NUM_LE; LT_IMP_LE] THEN TRANS_TAC REAL_LE_TRANS `(B:real^M->real) x` THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTER]; ALL_TAC] THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS {{x | x IN b i INTER s /\ ~((u i:real^M->real^1) differentiable (at x within s))} UNION {x | x IN b i INTER s /\ ~((v i:real^M->real^1) differentiable (at x within s))} | i IN (:num)} UNION {x | x IN s /\ ~(((\e. lift(measure(s INTER ball(x,drop e)) / measure(ball(x,drop e)))) --> vec 1) (at (vec 0) within {t | &0 < drop t}))}` THEN CONJ_TAC THENL [REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[NEGLIGIBLE_UNION_EQ] THEN CONJ_TAC THEN MATCH_MP_TAC(MESON[] `negligible {x | x IN b INTER s /\ P x (b INTER s)} /\ {x | x IN b INTER s /\ P x (b INTER s)} = {x | x IN b INTER s /\ P x s} ==> negligible {x | x IN b INTER s /\ P x s}`) THEN (CONJ_TAC THENL [MATCH_MP_TAC RADEMACHER THEN ASM_MESON_TAC[]; ALL_TAC]) THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN MATCH_MP_TAC(TAUT `(p ==> (q <=> r)) ==> (p /\ ~q <=> p /\ ~r)`) THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN REWRITE_TAC[differentiable; has_derivative_within] THEN AP_TERM_TAC THEN ABS_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC LIM_WITHIN_INTERIOR_INTER THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> interior s = s ==> x IN interior s`)) THEN MATCH_MP_TAC INTERIOR_OPEN THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE b UNIV = u ==> (!x. x IN u ==> open x) ==> open(b i)`)) THEN EXPAND_TAC "us" THEN REWRITE_TAC[FORALL_IN_GSPEC; OPEN_BALL]; FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_DENSITY_THEOREM_LIFT_BALL) THEN DISCH_THEN(X_CHOOSE_THEN `k:real^M->bool` (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*"))) THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[]]; ALL_TAC] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `a:real^M` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN SUBGOAL_THEN `(a:real^M) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN ASM_REWRITE_TAC[IN_UNION; DE_MORGAN_THM; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN SUBGOAL_THEN `?i. a IN b i /\ (u:num->real^M->real^1) i a = v i a` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `u = v ==> !f. v <= f /\ f <= u ==> u = f /\ v = f`) o GEN_REWRITE_RULE I [GSYM DROP_EQ]) THEN DISCH_THEN(MP_TAC o SPEC `drop(f(a:real^M))`) THEN ASM_SIMP_TAC[IN_INTER; DROP_EQ] THEN STRIP_TAC THEN REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV; NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; DE_MORGAN_THM] THEN REWRITE_TAC[differentiable; IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `u':real^M->real^1` THEN DISCH_TAC THEN X_GEN_TAC `v':real^M->real^1` THEN DISCH_TAC THEN EXISTS_TAC `u':real^M->real^1` THEN SUBGOAL_THEN `v':real^M->real^1 = u'` SUBST_ALL_TAC THENL [ALL_TAC; MAP_EVERY UNDISCH_TAC [`((u:num->real^M->real^1) i has_derivative u') (at a within s)`; `((v:num->real^M->real^1) i has_derivative u') (at a within s)`] THEN ASM_REWRITE_TAC[has_derivative_within] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[IMP_IMP] THEN REWRITE_TAC[tendsto; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[GSYM EVENTUALLY_AND] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MP) THEN REWRITE_TAC[EVENTUALLY_WITHIN] THEN SUBGOAL_THEN `a IN interior((b:num->real^M->bool) i)` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> interior s = s ==> x IN interior s`)) THEN MATCH_MP_TAC INTERIOR_OPEN THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE b UNIV = u ==> (!x. x IN u ==> open x) ==> open(b i)`)) THEN EXPAND_TAC "us" THEN REWRITE_TAC[FORALL_IN_GSPEC; OPEN_BALL]; REWRITE_TAC[IN_INTERIOR]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[GSYM DIST_NZ] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN REWRITE_TAC[DIST_0; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN REWRITE_TAC[NORM_1; DROP_SUB] THEN MATCH_MP_TAC(REAL_ARITH `v <= f /\ f <= u ==> abs(v - x) < e /\ abs(u - x) < e ==> abs(f - x) < e`) THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTER] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] IN_BALL]] THEN REWRITE_TAC[FUN_EQ_THM] THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN GEN_REWRITE_TAC I [GSYM FUN_EQ_THM] THEN MATCH_MP_TAC DIFFERENTIAL_ZERO_MAXMIN_DENSITY THEN MAP_EVERY EXISTS_TAC [`\x. (v:num->real^M->real^1) i x - u i x`; `s:real^M->bool`; `a:real^M`] THEN ASM_SIMP_TAC[HAS_DERIVATIVE_SUB; ETA_AX] THEN DISJ2_TAC THEN REWRITE_TAC[DROP_SUB; REAL_ARITH `v - u <= a - a <=> v <= u`] THEN REWRITE_TAC[EVENTUALLY_WITHIN] THEN SUBGOAL_THEN `a IN interior((b:num->real^M->bool) i)` MP_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `x IN s ==> interior s = s ==> x IN interior s`)) THEN MATCH_MP_TAC INTERIOR_OPEN THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `IMAGE b UNIV = u ==> (!x. x IN u ==> open x) ==> open(b i)`)) THEN EXPAND_TAC "us" THEN REWRITE_TAC[FORALL_IN_GSPEC; OPEN_BALL]; REWRITE_TAC[IN_INTERIOR]] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN TRANS_TAC REAL_LE_TRANS `drop((f:real^M->real^1) x)` THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_INTER] THEN FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] IN_BALL]);; let STEPANOV = prove (`!f:real^M->real^N s. open s ==> negligible {x | x IN s /\ (?B. eventually (\y. norm(f x - f y) / norm(x - y) <= B) (at x)) /\ ~(f differentiable at x)}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LEBESGUE_MEASURABLE_OPEN) THEN DISCH_THEN(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP STEPANOV_GEN) THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN ASM_SIMP_TAC[DIFFERENTIABLE_WITHIN_OPEN; EVENTUALLY_WITHIN_OPEN]);; let STEPANOV_UNIV = prove (`!f:real^M->real^N. negligible {x | (?B. eventually (\y. norm(f x - f y) / norm(x - y) <= B) (at x)) /\ ~(f differentiable at x)}`, GEN_TAC THEN MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`] STEPANOV) THEN ASM_REWRITE_TAC[OPEN_UNIV; IN_UNIV]);;