Datasets:

hoskinson-center
/

proof-pile

	(* ========================================================================= *)
	(* L_p spaces for functions R^m->R^n based on an arbitrary set. *)
	(* ========================================================================= *)

	needs "Multivariate/realanalysis.ml";;

	(* ------------------------------------------------------------------------- *)
	(* The space L_p of measurable functions f with \|f\|^p integrable (on s). *)
	(* ------------------------------------------------------------------------- *)

	let lspace = new_definition
	`lspace s p =
	{f:real^M->real^N \| f measurable_on s /\
	(\x. lift(norm(f x) rpow p)) integrable_on s}`;;

	let LSPACE_ZERO = prove
	(`!s. lspace s (&0) =
	if measurable s then {f:real^M->real^N \| f measurable_on s} else {}`,
	REWRITE_TAC[lspace; RPOW_POW; real_pow; NORM_0; LIFT_NUM] THEN
	GEN_TAC THEN REWRITE_TAC[INTEGRABLE_ON_CONST; VEC_EQ; ARITH_EQ] THEN
	ASM_CASES_TAC `measurable(s:real^M->bool)` THEN
	ASM_REWRITE_TAC[] THEN SET_TAC[]);;

	let LSPACE_CONST = prove
	(`!s p c. measurable s ==> (\x. c) IN lspace s p`,
	SIMP_TAC[lspace; IN_ELIM_THM; INTEGRABLE_ON_CONST;
	INTEGRABLE_IMP_MEASURABLE]);;

	let LSPACE_0 = prove
	(`!s p. ~(p = &0) ==> (\x. vec 0) IN lspace s p`,
	SIMP_TAC[lspace; IN_ELIM_THM; NORM_0; RPOW_ZERO; LIFT_NUM] THEN
	SIMP_TAC[INTEGRABLE_IMP_MEASURABLE; INTEGRABLE_0]);;

	let LSPACE_CMUL = prove
	(`!s p c f:real^M->real^N.
	f IN lspace s p ==> (\x. c % f x) IN lspace s p`,
	REPEAT GEN_TAC THEN REWRITE_TAC[lspace; IN_ELIM_THM] THEN
	SIMP_TAC[NORM_MUL; RPOW_MUL; NORM_POS_LE; LIFT_CMUL] THEN
	SIMP_TAC[MEASURABLE_ON_CMUL; INTEGRABLE_CMUL]);;

	let LSPACE_NEG = prove
	(`!s p f:real^M->real^N. f IN lspace s p ==> (\x. --(f x)) IN lspace s p`,
	REWRITE_TAC[VECTOR_ARITH `--x:real^N = --(&1) % x`; LSPACE_CMUL]);;

	let LSPACE_ADD = prove
	(`!s p f g:real^M->real^N.
	&0 <= p /\ f IN lspace s p /\ g IN lspace s p
	==> (\x. f(x) + g(x)) IN lspace s p`,
	REPEAT GEN_TAC THEN REWRITE_TAC[REAL_LE_LT] THEN ASM_CASES_TAC `p = &0` THEN
	ASM_REWRITE_TAC[] THENL
	[REWRITE_TAC[LSPACE_ZERO] THEN
	ASM_CASES_TAC `measurable(s:real^M->bool)` THEN
	ASM_REWRITE_TAC[NOT_IN_EMPTY; IN_ELIM_THM; MEASURABLE_ON_ADD];
	ALL_TAC] THEN
	REWRITE_TAC[lspace; IN_ELIM_THM] THEN
	REPEAT STRIP_TAC THEN ASM_SIMP_TAC[MEASURABLE_ON_ADD] THEN
	MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
	EXISTS_TAC `\x. lift(&2 rpow p * (norm((f:real^M->real^N) x) rpow p +
	norm((g:real^M->real^N) x) rpow p))` THEN
	REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
	[SUBGOAL_THEN
	`(\x:real^M. lift(norm(f x + g x:real^N) rpow p)) =
	(lift o (\y. y rpow p) o drop) o (\x. lift(norm(f x + g x)))`
	SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM; LIFT_DROP]; ALL_TAC] THEN
	MATCH_MP_TAC MEASURABLE_ON_COMPOSE_CONTINUOUS_0 THEN REPEAT CONJ_TAC THENL
	[MATCH_MP_TAC MEASURABLE_ON_NORM THEN
	MATCH_MP_TAC MEASURABLE_ON_ADD THEN ASM_REWRITE_TAC[];
	ONCE_REWRITE_TAC[GSYM IMAGE_LIFT_UNIV] THEN
	REWRITE_TAC[GSYM REAL_CONTINUOUS_ON] THEN
	MATCH_MP_TAC REAL_CONTINUOUS_ON_RPOW THEN ASM_REAL_ARITH_TAC;
	ASM_SIMP_TAC[o_THM; DROP_VEC; RPOW_ZERO; REAL_LT_IMP_NZ] THEN
	REWRITE_TAC[LIFT_NUM]];
	REWRITE_TAC[LIFT_CMUL; LIFT_ADD] THEN MATCH_MP_TAC INTEGRABLE_CMUL THEN
	MATCH_MP_TAC INTEGRABLE_ADD THEN ASM_REWRITE_TAC[];
	X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
	REWRITE_TAC[NORM_LIFT; REAL_ABS_NORM; LIFT_DROP] THEN
	MATCH_MP_TAC(REAL_ARITH
	`(&0 <= norm(f + g:real^N) rpow p /\ &0 <= norm f /\ &0 <= norm g /\
	norm(f + g) rpow p <= (norm f + norm g) rpow p) /\
	(&0 <= norm f /\ &0 <= norm g ==> (norm f + norm g) rpow p <= e)
	==> abs(norm(f + g) rpow p) <= e`) THEN
	CONJ_TAC THENL
	[ASM_SIMP_TAC[NORM_POS_LE; RPOW_POS_LE; RPOW_LE2; NORM_TRIANGLE; RPOW_LE2;
	REAL_LT_IMP_LE];
	SPEC_TAC(`norm((g:real^M->real^N) x)`,`z:real`) THEN
	SPEC_TAC(`norm((f:real^M->real^N) x)`,`w:real`) THEN
	MATCH_MP_TAC REAL_WLOG_LE THEN
	CONJ_TAC THENL [MESON_TAC[REAL_ADD_SYM]; ALL_TAC] THEN
	REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
	EXISTS_TAC `(&2 * z) rpow p` THEN CONJ_TAC THENL
	[MATCH_MP_TAC RPOW_LE2 THEN ASM_REAL_ARITH_TAC;
	ASM_SIMP_TAC[RPOW_MUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN
	ASM_SIMP_TAC[REAL_LE_ADDL; RPOW_POS_LE; REAL_POS]]]]);;

	let LSPACE_SUB = prove
	(`!s p f g:real^M->real^N.
	&0 <= p /\ f IN lspace s p /\ g IN lspace s p
	==> (\x. f(x) - g(x)) IN lspace s p`,
	SIMP_TAC[VECTOR_SUB; LSPACE_ADD; LSPACE_NEG]);;

	let LSPACE_IMP_INTEGRABLE = prove
	(`!s p f. f IN lspace s p ==> (\x. lift(norm(f x) rpow p)) integrable_on s`,
	SIMP_TAC[lspace; IN_ELIM_THM]);;

	let LSPACE_NORM = prove
	(`!s p f. f IN lspace s p ==> (\x. lift(norm(f x))) IN lspace s p`,
	REWRITE_TAC[lspace; IN_ELIM_THM] THEN
	SIMP_TAC[NORM_LIFT; REAL_ABS_NORM; MEASURABLE_ON_NORM]);;

	let LSPACE_VSUM = prove
	(`!s p f:A->real^M->real^N t.
	&0 < p /\ FINITE t /\ (!i. i IN t ==> (f i) IN lspace s p)
	==> (\x. vsum t (\i. f i x)) IN lspace s p`,
	REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
	REPEAT GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
	MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
	ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; LSPACE_0; REAL_LT_IMP_NZ] THEN
	ASM_SIMP_TAC[LSPACE_ADD; REAL_LT_IMP_LE; ETA_AX; IN_INSERT]);;

	let LSPACE_MAX = prove
	(`!s p k f:real^M->real^N g:real^M->real^N.
	f IN lspace s p /\ g IN lspace s p /\ &0 < p
	==> ((\x. lambda i. max (f x$i) (g x$i)):real^M->real^N) IN lspace s p`,
	REWRITE_TAC[lspace; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
	ASM_SIMP_TAC[MEASURABLE_ON_MAX] THEN
	MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
	EXISTS_TAC
	`\x. lift(&(dimindex(:N)) rpow p *
	max (norm((f:real^M->real^N) x) rpow p)
	(norm((g:real^M->real^N) x) rpow p))` THEN
	ASM_SIMP_TAC[MEASURABLE_ON_LIFT_RPOW; MEASURABLE_ON_NORM;
	MEASURABLE_ON_MAX] THEN
	CONJ_TAC THENL
	[REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC INTEGRABLE_CMUL THEN
	MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN
	MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_MAX_1 THEN
	CONJ_TAC THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN
	ASM_REWRITE_TAC[] THEN
	REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
	REWRITE_TAC[IMP_IMP; DIMINDEX_1; FORALL_1; GSYM drop; LIFT_DROP] THEN
	SIMP_TAC[RPOW_POS_LE; NORM_POS_LE];
	X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
	ASM_SIMP_TAC[REAL_MAX_RPOW; NORM_POS_LE; REAL_LT_IMP_LE] THEN
	REWRITE_TAC[GSYM RPOW_MUL; NORM_LIFT; REAL_ABS_RPOW; REAL_ABS_NORM] THEN
	REWRITE_TAC[LIFT_DROP] THEN MATCH_MP_TAC RPOW_LE2 THEN
	ASM_SIMP_TAC[REAL_LT_IMP_LE; NORM_POS_LE] 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
	MATCH_MP_TAC SUM_BOUND THEN
	SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; LAMBDA_BETA] THEN
	REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH
	`abs(x) <= y /\ abs(x') <= y' ==> abs(max x x') <= max y y'`) THEN
	ASM_SIMP_TAC[COMPONENT_LE_NORM]]);;

	let LSPACE_MIN = prove
	(`!s p k f:real^M->real^N g:real^M->real^N.
	f IN lspace s p /\ g IN lspace s p /\ &0 < p
	==> ((\x. lambda i. min (f x$i) (g x$i)):real^M->real^N) IN lspace s p`,
	REWRITE_TAC[lspace; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
	ASM_SIMP_TAC[MEASURABLE_ON_MIN] THEN
	MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
	EXISTS_TAC
	`\x. lift(&(dimindex(:N)) rpow p *
	max (norm((f:real^M->real^N) x) rpow p)
	(norm((g:real^M->real^N) x) rpow p))` THEN
	ASM_SIMP_TAC[MEASURABLE_ON_LIFT_RPOW; MEASURABLE_ON_NORM;
	MEASURABLE_ON_MIN] THEN
	CONJ_TAC THENL
	[REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC INTEGRABLE_CMUL THEN
	MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN
	MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_MAX_1 THEN
	CONJ_TAC THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN
	ASM_REWRITE_TAC[] THEN
	REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
	REWRITE_TAC[IMP_IMP; DIMINDEX_1; FORALL_1; GSYM drop; LIFT_DROP] THEN
	SIMP_TAC[RPOW_POS_LE; NORM_POS_LE];
	X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
	ASM_SIMP_TAC[REAL_MAX_RPOW; NORM_POS_LE; REAL_LT_IMP_LE] THEN
	REWRITE_TAC[GSYM RPOW_MUL; NORM_LIFT; REAL_ABS_RPOW; REAL_ABS_NORM] THEN
	REWRITE_TAC[LIFT_DROP] THEN MATCH_MP_TAC RPOW_LE2 THEN
	ASM_SIMP_TAC[REAL_LT_IMP_LE; NORM_POS_LE] 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
	MATCH_MP_TAC SUM_BOUND THEN
	SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; LAMBDA_BETA] THEN
	REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH
	`abs(x) <= y /\ abs(x') <= y' ==> abs(min x x') <= max y y'`) THEN
	ASM_SIMP_TAC[COMPONENT_LE_NORM]]);;

	let LSPACE_BOUNDED_MEASURABLE = prove
	(`!s p f:real^M->real^N g:real^M->real^P.
	&0 < p /\ f measurable_on s /\ g IN lspace s p /\
	(!x. x IN s ==> norm(f x) <= norm(g x))
	==> f IN lspace s p`,
	REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[lspace; IN_ELIM_THM] THEN
	MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
	EXISTS_TAC `\x. lift(norm((g:real^M->real^P) x) rpow p)` THEN
	ASM_SIMP_TAC[LSPACE_IMP_INTEGRABLE] THEN
	ASM_SIMP_TAC[MEASURABLE_ON_LIFT_RPOW; MEASURABLE_ON_NORM] THEN
	REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_LIFT; LIFT_DROP] THEN
	REWRITE_TAC[REAL_ABS_RPOW; REAL_ABS_NORM] THEN
	ASM_SIMP_TAC[RPOW_LE2; REAL_LT_IMP_LE; NORM_POS_LE]);;

	let LSPACE_BOUNDED_MEASURABLE_SIMPLE = prove
	(`!s p f:real^M->real^N.
	&0 < p /\ f measurable_on s /\ measurable s /\ bounded(IMAGE f s)
	==> f IN lspace s p`,
	REPEAT STRIP_TAC THEN
	MATCH_MP_TAC(INST_TYPE [`:1`,`:P`] LSPACE_BOUNDED_MEASURABLE) THEN
	MATCH_MP_TAC(MESON[] `(?x. P(\a. lift x)) ==> (?x. P x)`) THEN
	ASM_SIMP_TAC[LSPACE_CONST; NORM_LIFT] THEN
	FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN
	MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
	SIMP_TAC[real_abs; REAL_LT_IMP_LE]);;

	let LSPACE_INTEGRABLE_PRODUCT = prove
	(`!s p q f:real^M->real^N g:real^M->real^N.
	&0 < p /\ &0 < q /\ inv(p) + inv(q) = &1 /\
	f IN lspace s p /\ g IN lspace s q
	==> (\x. lift(norm(f x) * norm(g x))) integrable_on s`,
	REWRITE_TAC[lspace; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
	MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
	EXISTS_TAC `\x. lift(norm((f:real^M->real^N) x) rpow p / p) +
	lift(norm((g:real^M->real^N) x) rpow q / q)` THEN
	REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
	[REWRITE_TAC[LIFT_CMUL] THEN
	GEN_REWRITE_TAC (LAND_CONV o ABS_CONV o LAND_CONV)
	[GSYM LIFT_DROP] THEN
	MATCH_MP_TAC MEASURABLE_ON_DROP_MUL THEN
	CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_ON_NORM THEN ASM_REWRITE_TAC[];
	MATCH_MP_TAC INTEGRABLE_ADD THEN
	REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN
	REWRITE_TAC[LIFT_CMUL] THEN CONJ_TAC THEN MATCH_MP_TAC INTEGRABLE_CMUL THEN
	ASM_REWRITE_TAC[];
	REWRITE_TAC[NORM_LIFT; REAL_ABS_MUL; REAL_ABS_NORM; LIFT_DROP;
	DROP_ADD] THEN
	REPEAT STRIP_TAC THEN MATCH_MP_TAC YOUNG_INEQUALITY THEN
	ASM_REWRITE_TAC[NORM_POS_LE]]);;

	let LSPACE_1 = prove
	(`!f:real^M->real^N s. f IN lspace s (&1) <=> f absolutely_integrable_on s`,
	REWRITE_TAC[ABSOLUTELY_INTEGRABLE_MEASURABLE; lspace; IN_ELIM_THM] THEN
	REWRITE_TAC[RPOW_POW; REAL_POW_1]);;

	let LSPACE_MONO = prove
	(`!f:real^M->real^N s p q.
	f IN lspace s q /\ measurable s /\ &0 < p /\ p <= q
	==> f IN lspace s p`,
	REWRITE_TAC[lspace; IN_ELIM_THM] THEN
	REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
	MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
	EXISTS_TAC `\x. lift(max (&1) (norm((f:real^M->real^N) x) rpow q))` THEN
	ASM_SIMP_TAC[MEASURABLE_ON_LIFT_RPOW; MEASURABLE_ON_NORM] THEN CONJ_TAC THENL
	[MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN
	MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_MAX_1 THEN
	CONJ_TAC THEN MATCH_MP_TAC NONNEGATIVE_ABSOLUTELY_INTEGRABLE THEN
	ASM_SIMP_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; INTEGRABLE_ON_CONST] THEN
	REWRITE_TAC[IMP_IMP; DIMINDEX_1; FORALL_1; GSYM drop; LIFT_DROP] THEN
	SIMP_TAC[RPOW_POS_LE; NORM_POS_LE; REAL_POS];
	X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
	REWRITE_TAC[NORM_LIFT; LIFT_DROP; REAL_ABS_RPOW; REAL_ABS_NORM] THEN
	DISJ_CASES_TAC(ISPECL [`&1`; `norm((f:real^M->real^N) x)`] REAL_LE_TOTAL)
	THENL
	[MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= max z y`) THEN
	MATCH_MP_TAC RPOW_MONO_LE THEN ASM_REWRITE_TAC[];
	MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= max y z`) THEN
	MATCH_MP_TAC RPOW_1_LE THEN REWRITE_TAC[NORM_POS_LE] THEN
	ASM_REAL_ARITH_TAC]]);;

	let LSPACE_INCLUSION = prove
	(`!s p q. measurable s /\ &0 < p /\ p <= q
	==> (lspace s q :(real^M->real^N)->bool) SUBSET (lspace s p)`,
	REWRITE_TAC[SUBSET] THEN REPEAT STRIP_TAC THEN
	MATCH_MP_TAC LSPACE_MONO THEN EXISTS_TAC `q:real` THEN
	ASM_REWRITE_TAC[]);;

	(* ------------------------------------------------------------------------- *)
	(* The corresponding seminorm; Hoelder and Minkowski inequalities. *)
	(* ------------------------------------------------------------------------- *)

	let lnorm = new_definition
	`lnorm s p f = drop(integral s (\x. lift(norm(f x) rpow p))) rpow (inv p)`;;

	let LNORM_0 = prove
	(`!s p. ~(p = &0) ==> lnorm s p (\x. vec 0) = &0`,
	REPEAT STRIP_TAC THEN
	ASM_REWRITE_TAC[lnorm; NORM_0; RPOW_ZERO] THEN
	ASM_REWRITE_TAC[LIFT_NUM; INTEGRAL_0; DROP_VEC; RPOW_ZERO; REAL_INV_EQ_0]);;

	let LNORM_CONST = prove
	(`!s p c:real^N.
	measurable s /\ &0 < p
	==> lnorm s p (\x:real^M. c) = measure s rpow (inv p) * norm c`,
	SIMP_TAC[lnorm; INTEGRAL_CONST_GEN; DROP_CMUL; LIFT_DROP] THEN
	SIMP_TAC[RPOW_RPOW; NORM_POS_LE; RPOW_MUL] THEN
	SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ; RPOW_POW; REAL_POW_1]);;

	let LNORM_MONO = prove
	(`!f:real^M->real^N g:real^M->real^P s t p.
	&0 <= p /\ f IN lspace s p /\ g IN lspace s p /\
	negligible t /\ (!x. x IN s DIFF t ==> norm(f x) <= norm(g x))
	==> lnorm s p f <= lnorm s p g`,
	REWRITE_TAC[lspace; lnorm; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
	MATCH_MP_TAC RPOW_LE2 THEN ASM_REWRITE_TAC[REAL_LE_INV_EQ] THEN
	ASM_SIMP_TAC[INTEGRAL_DROP_POS; LIFT_DROP; RPOW_POS_LE; NORM_POS_LE] THEN
	MATCH_MP_TAC INTEGRAL_DROP_LE_AE THEN
	EXISTS_TAC `t:real^M->bool` THEN ASM_REWRITE_TAC[LIFT_DROP] THEN
	ASM_SIMP_TAC[RPOW_LE2; NORM_POS_LE]);;

	let LNORM_NEG = prove
	(`!s p f:real^M->real^N. lnorm s p (\x. --(f x)) = lnorm s p f`,
	REWRITE_TAC[lnorm; NORM_NEG]);;

	let LNORM_MUL = prove
	(`!s p f c. f IN lspace s p /\ ~(p = &0)
	==> lnorm s p (\x. c % f x) = abs(c) * lnorm s p f`,
	REPEAT STRIP_TAC THEN REWRITE_TAC[lnorm; NORM_MUL; RPOW_MUL; LIFT_CMUL] THEN
	ASM_SIMP_TAC[INTEGRAL_CMUL; LSPACE_IMP_INTEGRABLE] THEN
	REWRITE_TAC[DROP_CMUL; RPOW_MUL] THEN
	AP_THM_TAC THEN AP_TERM_TAC THEN
	ASM_SIMP_TAC[RPOW_RPOW; REAL_ABS_POS; REAL_MUL_RINV] THEN
	REWRITE_TAC[RPOW_POW; REAL_POW_1]);;

	let LNORM_EQ_0 = prove
	(`!s p f. ~(p = &0) /\ f IN lspace s p
	==> (lnorm s p f = &0 <=>
	negligible {x \| x IN s /\ ~(f x = vec 0)})`,
	REWRITE_TAC[lspace; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
	ASM_SIMP_TAC[lnorm; RPOW_EQ_0; REAL_INV_EQ_0] THEN
	REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN
	ASM_SIMP_TAC[INTEGRAL_EQ_HAS_INTEGRAL] THEN
	SIMP_TAC[HAS_INTEGRAL_NEGLIGIBLE_EQ; lift; LAMBDA_BETA; NORM_POS_LE;
	RPOW_POS_LE] THEN
	AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
	SIMP_TAC[IN_ELIM_THM; CART_EQ; LAMBDA_BETA] THEN
	REWRITE_TAC[FORALL_1; DIMINDEX_1; VEC_COMPONENT] THEN
	ASM_REWRITE_TAC[RPOW_EQ_0; NORM_EQ_0; CART_EQ; VEC_COMPONENT]);;

	let LNORM_POS_LE = prove
	(`!s p f. f IN lspace s p ==> &0 <= lnorm s p f`,
	SIMP_TAC[lspace; IN_ELIM_THM; lnorm] THEN REPEAT STRIP_TAC THEN
	MATCH_MP_TAC RPOW_POS_LE THEN MATCH_MP_TAC INTEGRAL_DROP_POS THEN
	ASM_SIMP_TAC[LIFT_DROP; NORM_POS_LE; RPOW_POS_LE]);;

	let LNORM_NORM = prove
	(`!s p f. lnorm s p (\x. lift(norm(f x))) = lnorm s p f`,
	REWRITE_TAC[lnorm; NORM_LIFT; REAL_ABS_NORM]);;

	let LNORM_RPOW = prove
	(`!s p f:real^M->real^N.
	f IN lspace s p /\ ~(p = &0)
	==> (lnorm s p f) rpow p =
	drop(integral s (\x. lift(norm(f x) rpow p)))`,
	REPEAT STRIP_TAC THEN REWRITE_TAC[lnorm] THEN
	ASM_SIMP_TAC[INTEGRAL_DROP_POS; LIFT_DROP; NORM_POS_LE; RPOW_RPOW;
	LSPACE_IMP_INTEGRABLE; RPOW_POS_LE] THEN
	ASM_SIMP_TAC[REAL_MUL_LINV; RPOW_POW; REAL_POW_1]);;

	let INTEGRAL_LNORM_RPOW = prove
	(`!s p f:real^M->real^N.
	f IN lspace s p /\ ~(p = &0)
	==> integral s (\x. lift(norm(f x) rpow p)) =
	lift((lnorm s p f) rpow p)`,
	SIMP_TAC[GSYM DROP_EQ; LIFT_DROP; LNORM_RPOW]);;

	let HOELDER_INEQUALITY = prove
	(`!s p q f:real^M->real^N g:real^M->real^N.
	&0 < p /\ &0 < q /\ inv(p) + inv(q) = &1 /\
	f IN lspace s p /\ g IN lspace s q
	==> drop(integral s (\x. lift(norm(f x) * norm(g x))))
	<= lnorm s p f * lnorm s q g`,
	MP_TAC LSPACE_INTEGRABLE_PRODUCT 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 DISCH_TAC THEN
	SUBGOAL_THEN `&0 <= lnorm s p (f:real^M->real^N) /\
	&0 <= lnorm s q (g:real^M->real^N)`
	MP_TAC THENL [ASM_SIMP_TAC[LNORM_POS_LE]; REWRITE_TAC[IMP_CONJ]] THEN
	REPEAT
	(GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
	DISCH_THEN(DISJ_CASES_THEN2 MP_TAC ASSUME_TAC) THENL
	[ASM_SIMP_TAC[LNORM_EQ_0; REAL_LT_IMP_NZ] THEN REPEAT DISCH_TAC THEN
	MATCH_MP_TAC(REAL_ARITH `&0 <= y /\ x = &0 ==> x <= y`) THEN
	ASM_SIMP_TAC[REAL_LE_MUL; LNORM_POS_LE; GSYM LIFT_EQ; LIFT_DROP] THEN
	ASM_SIMP_TAC[INTEGRAL_EQ_HAS_INTEGRAL; LIFT_NUM] THEN
	SIMP_TAC[HAS_INTEGRAL_NEGLIGIBLE_EQ; lift; LAMBDA_BETA; NORM_POS_LE;
	REAL_LE_MUL] THEN
	FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
	NEGLIGIBLE_SUBSET)) THEN
	SIMP_TAC[CART_EQ; SUBSET; IN_ELIM_THM; LAMBDA_BETA] THEN
	REWRITE_TAC[DIMINDEX_1; FORALL_1; VEC_COMPONENT] THEN
	REWRITE_TAC[REAL_ENTIRE; CART_EQ; NORM_EQ_0; VEC_COMPONENT] THEN
	MESON_TAC[];
	ALL_TAC]) THEN
	GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN
	ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_LT_MUL] THEN
	REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN
	REWRITE_TAC[GSYM DROP_CMUL] THEN ASM_SIMP_TAC[GSYM INTEGRAL_CMUL] THEN
	REWRITE_TAC[REAL_INV_MUL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
	EXISTS_TAC `drop(integral s
	(\x. lift(norm(inv(lnorm s p f) % (f:real^M->real^N) x) rpow p / p +
	norm(inv(lnorm s q g) % (g:real^M->real^N) x) rpow q / q)))` THEN
	CONJ_TAC THENL
	[MATCH_MP_TAC INTEGRAL_DROP_LE THEN
	ASM_SIMP_TAC[LIFT_DROP; INTEGRABLE_CMUL] THEN CONJ_TAC THENL
	[REWRITE_TAC[LIFT_ADD] THEN MATCH_MP_TAC INTEGRABLE_ADD THEN
	REWRITE_TAC[NORM_MUL; RPOW_MUL] THEN
	REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN
	ASM_SIMP_TAC[LSPACE_IMP_INTEGRABLE; INTEGRABLE_CMUL; LIFT_CMUL];
	REWRITE_TAC[DROP_CMUL; LIFT_DROP; NORM_MUL; REAL_ABS_INV] THEN
	ASM_SIMP_TAC[real_abs; LNORM_POS_LE; REAL_LT_IMP_NZ] THEN
	ONCE_REWRITE_TAC[REAL_ARITH
	`(a * b) * (c * d:real) = (a * c) * (b * d)`] THEN
	REPEAT STRIP_TAC THEN MATCH_MP_TAC YOUNG_INEQUALITY THEN
	ASM_SIMP_TAC[REAL_LE_MUL; NORM_POS_LE; LNORM_POS_LE; REAL_LE_INV_EQ]];
	REWRITE_TAC[LIFT_ADD; NORM_MUL; LIFT_CMUL; RPOW_MUL] THEN
	REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN
	REWRITE_TAC[LIFT_CMUL; VECTOR_MUL_ASSOC] THEN
	ASM_SIMP_TAC[INTEGRAL_ADD; INTEGRABLE_CMUL; INTEGRAL_CMUL;
	LSPACE_IMP_INTEGRABLE; REAL_ABS_INV] THEN
	ASM_SIMP_TAC[REAL_ARITH `&0 < x ==> abs x = x`; RPOW_INV] THEN
	ASM_SIMP_TAC[INTEGRAL_LNORM_RPOW; REAL_LT_IMP_NZ] THEN
	REWRITE_TAC[DROP_ADD; DROP_CMUL; LIFT_DROP] THEN
	ASM_SIMP_TAC[GSYM REAL_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ;
	RPOW_POS_LT] THEN
	ASM_REWRITE_TAC[REAL_MUL_RID; REAL_LE_REFL]]);;

	let HOELDER_INEQUALITY_FULL = prove
	(`!s p q f:real^M->real^N g:real^M->real^N.
	&0 < p /\ &0 < q /\ inv(p) + inv(q) = &1 /\
	f IN lspace s p /\ g IN lspace s q
	==> (\x. lift(norm(f x) * norm(g x))) integrable_on s /\
	drop(integral s (\x. lift(norm(f x) * norm(g x))))
	<= lnorm s p f * lnorm s q g`,
	REPEAT GEN_TAC THEN DISCH_TAC THEN
	FIRST_ASSUM(ASSUME_TAC o MATCH_MP LSPACE_INTEGRABLE_PRODUCT) THEN
	ASM_SIMP_TAC[HOELDER_INEQUALITY]);;

	let LNORM_TRIANGLE = prove
	(`!s p f:real^M->real^N g:real^M->real^N.
	f IN lspace s p /\ g IN lspace s p /\ &1 <= p
	==> lnorm s p (\x. f x + g x) <= lnorm s p f + lnorm s p g`,
	REPEAT STRIP_TAC THEN ASM_CASES_TAC `p = &1` THENL
	[FIRST_X_ASSUM SUBST_ALL_TAC THEN
	ASM_SIMP_TAC[lnorm;
	MESON[RPOW_POW; REAL_POW_1; REAL_INV_1] `x rpow (inv(&1)) = x`;
	GSYM DROP_ADD; GSYM INTEGRAL_ADD; LSPACE_IMP_INTEGRABLE] THEN
	MATCH_MP_TAC INTEGRAL_DROP_LE_MEASURABLE THEN
	ASM_SIMP_TAC[LSPACE_IMP_INTEGRABLE; INTEGRABLE_ADD] THEN
	REWRITE_TAC[RPOW_POW; REAL_POW_1; LIFT_DROP; DROP_ADD] THEN
	REWRITE_TAC[NORM_POS_LE; NORM_TRIANGLE] THEN
	MATCH_MP_TAC MEASURABLE_ON_NORM THEN MATCH_MP_TAC MEASURABLE_ON_ADD THEN
	RULE_ASSUM_TAC(REWRITE_RULE[lspace; IN_ELIM_THM]) THEN
	ASM_REWRITE_TAC[];
	ALL_TAC] THEN
	SUBGOAL_THEN `&1 < p` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
	SUBGOAL_THEN `&0 <= lnorm s p (\x. (f:real^M->real^N) x + g x)` MP_TAC THENL
	[ASM_SIMP_TAC[LNORM_POS_LE; LSPACE_ADD; REAL_ARITH `&1 <= p ==> &0 <= p`];
	GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
	STRIP_TAC THEN ASM_SIMP_TAC[LNORM_POS_LE; REAL_LE_ADD]] THEN
	MATCH_MP_TAC REAL_LE_LCANCEL_IMP THEN
	EXISTS_TAC `lnorm s p (\x. (f:real^M->real^N) x + g x) rpow (p - &1)` THEN
	ASM_SIMP_TAC[RPOW_POS_LT] THEN
	GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM REAL_POW_1] THEN
	ASM_SIMP_TAC[GSYM RPOW_POW; GSYM RPOW_ADD] THEN
	ASM_SIMP_TAC[LSPACE_ADD; LNORM_RPOW; REAL_ARITH `p - &1 + &1 = p`;
	REAL_ARITH `&1 <= p ==> &0 <= p /\ ~(p = &0)`] THEN
	CONV_TAC(LAND_CONV(SUBS_CONV[REAL_ARITH `p = &1 + (p - &1)`])) THEN
	ASM_SIMP_TAC[RPOW_ADD_ALT; NORM_POS_LE; REAL_ARITH
	`&1 <= p ==> &1 + p - &1 = &0 ==> p - &1 = &0`] THEN
	REWRITE_TAC[RPOW_POW; REAL_POW_1] THEN
	MP_TAC(ISPECL
	[`s:real^M->bool`; `p:real`; `p / (p - &1)`;
	`\x. lift(norm((g:real^M->real^N) x))`;
	`\x. lift(norm((f:real^M->real^N)(x) + g(x)) rpow (p - &1))`]
	HOELDER_INEQUALITY_FULL) THEN
	MP_TAC(ISPECL
	[`s:real^M->bool`; `p:real`; `p / (p - &1)`;
	`\x. lift(norm((f:real^M->real^N) x))`;
	`\x. lift(norm((f:real^M->real^N)(x) + g(x)) rpow (p - &1))`]
	HOELDER_INEQUALITY_FULL) THEN
	ASM_SIMP_TAC[LSPACE_NORM; REAL_LT_DIV; REAL_SUB_LT;
	REAL_ARITH `&1 < p ==> &0 < p`;
	REAL_FIELD `&1 < p ==> inv(p) + inv(p / (p - &1)) = &1`] THEN
	MATCH_MP_TAC(TAUT
	`p /\ (q ==> r ==> s) ==> (p ==> q) ==> (p ==> r) ==> s`) THEN
	CONJ_TAC THENL
	[SIMP_TAC[lspace; IN_ELIM_THM; NORM_LIFT; REAL_ABS_NORM; REAL_ABS_RPOW;
	RPOW_RPOW; NORM_POS_LE] THEN
	ASM_SIMP_TAC[REAL_FIELD `&1 < p ==> (p - &1) * p / (p - &1) = p`] THEN
	ASM_SIMP_TAC[LSPACE_IMP_INTEGRABLE; LSPACE_ADD;
	REAL_ARITH `&1 < p ==> &0 <= p`] THEN
	MATCH_MP_TAC MEASURABLE_ON_LIFT_RPOW THEN
	CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
	SUBGOAL_THEN `((\x. f x + g x):real^M->real^N) IN lspace s p` MP_TAC THENL
	[ASM_SIMP_TAC[LSPACE_ADD; REAL_ARITH `&1 < p ==> &0 <= p`];
	SIMP_TAC[lspace; IN_ELIM_THM; MEASURABLE_ON_NORM]];
	ALL_TAC] THEN
	REWRITE_TAC[NORM_LIFT; REAL_ABS_NORM; LNORM_NORM; REAL_ABS_RPOW] THEN
	MATCH_MP_TAC(TAUT
	`(p1 /\ p2 ==> b1 /\ b2 ==> c) ==> p1 /\ b1 ==> p2 /\ b2 ==> c`) THEN
	STRIP_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP REAL_LE_ADD2) THEN
	ASM_SIMP_TAC[GSYM DROP_ADD; GSYM INTEGRAL_ADD] THEN
	SUBGOAL_THEN
	`lnorm s (p / (p - &1)) (\x. lift(norm (f x + g x) rpow (p - &1))) =
	lnorm s p (\x. (f:real^M->real^N) x + g x) rpow (p - &1)`
	SUBST1_TAC THENL
	[REWRITE_TAC[lnorm] THEN
	ASM_SIMP_TAC[RPOW_RPOW; INTEGRAL_DROP_POS; LIFT_DROP; NORM_POS_LE;
	NORM_LIFT; REAL_ABS_NORM; REAL_ABS_RPOW] THEN
	ASM_SIMP_TAC[REAL_FIELD `&1 < p ==> (p - &1) * p / (p - &1) = p`] THEN
	REWRITE_TAC[REAL_INV_DIV] THEN REWRITE_TAC[real_div] THEN
	ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN MATCH_MP_TAC(GSYM RPOW_RPOW) THEN
	MATCH_MP_TAC INTEGRAL_DROP_POS THEN
	ASM_SIMP_TAC[LIFT_DROP; RPOW_POS_LE; NORM_POS_LE; LSPACE_IMP_INTEGRABLE;
	LSPACE_ADD; REAL_ARITH `&1 < p ==> &0 <= p`];
	ALL_TAC] THEN
	MATCH_MP_TAC(REAL_ARITH
	`i2 <= i1 ==> i1 <= f * y + g * y ==> i2 <= y * (f + g)`) THEN
	MATCH_MP_TAC INTEGRAL_DROP_LE_MEASURABLE THEN
	ASM_SIMP_TAC[INTEGRABLE_ADD] THEN CONJ_TAC THENL
	[MATCH_MP_TAC MEASURABLE_ON_LIFT_MUL THEN CONJ_TAC THENL
	[ALL_TAC;
	MATCH_MP_TAC MEASURABLE_ON_LIFT_RPOW THEN
	CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC]] THEN
	(SUBGOAL_THEN `((\x. f x + g x):real^M->real^N) IN lspace s p` MP_TAC THENL
	[ASM_SIMP_TAC[LSPACE_ADD; REAL_ARITH `&1 < p ==> &0 <= p`];
	SIMP_TAC[lspace; IN_ELIM_THM; MEASURABLE_ON_NORM]]);
	REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; LIFT_DROP; DROP_ADD] THEN
	SIMP_TAC[NORM_TRIANGLE; REAL_LE_RMUL; NORM_POS_LE; RPOW_POS_LE;
	REAL_LE_MUL]]);;

	let VSUM_LNORM = prove
	(`!s p f:A->real^M->real^N t.
	&1 <= p /\ FINITE t /\ (!i. i IN t ==> (f i) IN lspace s p)
	==> lnorm s p (\x. vsum t (\i. f i x)) <= sum t (\i. lnorm s p (f i))`,
	REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
	REPEAT GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
	MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
	ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; LNORM_0; REAL_LE_REFL;
	REAL_ARITH `&1 <= p ==> ~(p = &0)`] THEN
	MAP_EVERY X_GEN_TAC [`i:A`; `u:A->bool`] THEN
	REWRITE_TAC[IN_INSERT] THEN
	DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
	ASM_SIMP_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
	MATCH_MP_TAC(REAL_ARITH `a <= x + y ==> y <= z ==> a <= x + z`) THEN
	W(MP_TAC o PART_MATCH (lhand o rand) LNORM_TRIANGLE o lhand o snd) THEN
	ASM_SIMP_TAC[ETA_AX; LSPACE_VSUM; REAL_ARITH `&1 <= p ==> &0 < p`]);;

	(* ------------------------------------------------------------------------- *)
	(* Completeness (Riesz-Fischer). *)
	(* ------------------------------------------------------------------------- *)

	let LSPACE_SUMMABLE_UNIV = prove
	(`!f:num->real^M->real^N p s.
	&1 <= p /\
	(!i. f i IN lspace s p) /\
	real_summable (:num) (\i. lnorm s p (f i))
	==> ?g. g IN lspace s p /\
	!e. &0 < e ==> eventually
	(\n. lnorm s p (\x. vsum (0..n) (\i. f i x) -
	g(x)) < e)
	sequentially`,
	REPEAT STRIP_TAC THEN
	FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_SUMS_INFSUM]) THEN
	ABBREV_TAC `M = real_infsum (:num) (\i. lnorm s p (f i:real^M->real^N))` THEN
	DISCH_TAC THEN
	ABBREV_TAC
	`g = \n x:real^M. vsum(0..n) (\i. lift(norm(f i x:real^N)))` THEN
	SUBGOAL_THEN `!n:num. lnorm s p (g n:real^M->real^1) <= M` ASSUME_TAC THENL
	[GEN_TAC THEN EXPAND_TAC "g" THEN
	W(MP_TAC o PART_MATCH (lhand o rand) VSUM_LNORM o lhand o snd) THEN
	ASM_SIMP_TAC[FINITE_NUMSEG; LSPACE_NORM; ETA_AX] THEN
	MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
	REWRITE_TAC[LNORM_NORM] THEN EXPAND_TAC "M" THEN
	GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SET_RULE `s = UNIV INTER s`] THEN
	REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC REAL_PARTIAL_SUMS_LE_INFSUM THEN
	ASM_SIMP_TAC[LNORM_POS_LE];
	ALL_TAC] THEN
	SUBGOAL_THEN `!n:num. (g n:real^M->real^1) IN lspace s p` ASSUME_TAC THENL
	[GEN_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[] THEN
	MATCH_MP_TAC LSPACE_VSUM THEN
	CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[FINITE_NUMSEG]] THEN
	ASM_SIMP_TAC[LSPACE_NORM; ETA_AX];
	ALL_TAC] THEN
	SUBGOAL_THEN `!n:num x:real^M. &0 <= drop(g n x)` ASSUME_TAC THENL
	[REPEAT GEN_TAC THEN EXPAND_TAC "g" THEN
	SIMP_TAC[DROP_VSUM; FINITE_NUMSEG; LIFT_DROP] THEN
	MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN
	REWRITE_TAC[o_DEF; LIFT_DROP; NORM_POS_LE];
	ALL_TAC] THEN
	MP_TAC(ISPECL [`\i:num x:real^M. lift(drop(g i x) rpow p)`; `s:real^M->bool`]
	BEPPO_LEVI_MONOTONE_CONVERGENCE_INCREASING) THEN
	REWRITE_TAC[LIFT_DROP] THEN ANTS_TAC THENL
	[MATCH_MP_TAC(TAUT `b /\ a /\ c ==> a /\ b /\ c`) THEN CONJ_TAC THENL
	[REPEAT STRIP_TAC THEN EXPAND_TAC "g" THEN
	SIMP_TAC[DROP_VSUM; FINITE_NUMSEG] THEN
	MATCH_MP_TAC RPOW_LE2 THEN REPEAT CONJ_TAC THENL
	[MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN
	REWRITE_TAC[o_DEF; LIFT_DROP; NORM_POS_LE];
	SIMP_TAC[SUM_CLAUSES_NUMSEG; LE_0; REAL_LE_ADDR] THEN
	REWRITE_TAC[o_DEF; LIFT_DROP; NORM_POS_LE];
	ASM_REAL_ARITH_TAC];
	ALL_TAC] THEN
	SUBGOAL_THEN
	`!k x. drop((g:num->real^M->real^1) k x) = norm(g k x)`
	(fun th -> REWRITE_TAC[th])
	THENL
	[REPEAT GEN_TAC THEN REWRITE_TAC[NORM_REAL; GSYM drop] THEN
	ASM_REWRITE_TAC[real_abs];
	ALL_TAC] THEN
	ASM_SIMP_TAC[LSPACE_IMP_INTEGRABLE; ETA_AX] THEN
	REWRITE_TAC[bounded] THEN EXISTS_TAC `M rpow p` THEN
	REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `n:num` THEN
	DISCH_THEN(K ALL_TAC) THEN
	ASM_SIMP_TAC[INTEGRAL_LNORM_RPOW; ETA_AX;
	REAL_ARITH `&1 <= p ==> ~(p = &0)`] THEN
	REWRITE_TAC[NORM_LIFT; REAL_ABS_RPOW] THEN
	MATCH_MP_TAC RPOW_LE2 THEN
	ASM_SIMP_TAC[REAL_ARITH `&1 <= p ==> &0 <= p`] THEN
	MATCH_MP_TAC(REAL_ARITH
	`&0 <= x /\ x <= a ==> &0 <= abs x /\ abs x <= a`) THEN
	ASM_SIMP_TAC[LNORM_POS_LE];
	ALL_TAC] THEN
	REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
	MAP_EVERY X_GEN_TAC [`hp:real^M->real^1`; `k:real^M->bool`] THEN
	STRIP_TAC THEN
	ABBREV_TAC `h:real^M->real^1 = \x. lift(drop(hp x) rpow (inv p))` THEN
	SUBGOAL_THEN
	`!x. x IN s DIFF k ==> ((\i. g i x) --> ((h:real^M->real^1) x)) sequentially`
	ASSUME_TAC THENL
	[X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
	MP_TAC(ISPECL
	[`lift o (\x. x rpow (inv p)) o drop`;
	`sequentially`; `\i. lift(drop((g:num->real^M->real^1) i x) rpow p)`;
	`(hp:real^M->real^1) x`]
	LIM_CONTINUOUS_FUNCTION) THEN
	ASM_SIMP_TAC[] THEN ANTS_TAC THENL
	[GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM LIFT_DROP] THEN
	REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS_ATREAL] THEN
	MATCH_MP_TAC REAL_CONTINUOUS_AT_RPOW THEN
	REWRITE_TAC[REAL_LE_INV_EQ] THEN ASM_REAL_ARITH_TAC;
	ALL_TAC] THEN
	EXPAND_TAC "h" THEN REWRITE_TAC[o_DEF; LIFT_DROP] THEN
	ASM_SIMP_TAC[RPOW_RPOW; REAL_MUL_RINV;
	REAL_ARITH `&1 <= p ==> ~(p = &0)`] THEN
	REWRITE_TAC[RPOW_POW; REAL_POW_1; LIFT_DROP; ETA_AX];
	ALL_TAC] THEN
	SUBGOAL_THEN
	`!x. x IN s DIFF k ==> summable (:num) (\i. (f:num->real^M->real^N) i x)`
	MP_TAC THENL
	[REPEAT STRIP_TAC THEN MATCH_MP_TAC SERIES_LIFT_ABSCONV_IMP_CONV THEN
	REWRITE_TAC[summable] THEN EXISTS_TAC `(h:real^M->real^1) x` THEN
	REWRITE_TAC[sums; INTER_UNIV] THEN
	RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM]) THEN
	ASM_SIMP_TAC[];
	ALL_TAC] THEN
	REWRITE_TAC[summable] THEN
	GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
	REWRITE_TAC[SKOLEM_THM] THEN
	MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^M->real^N` THEN
	DISCH_TAC THEN
	SUBGOAL_THEN
	`!n x. x IN s DIFF k
	==> norm(vsum (0..n) (\i. (f:num->real^M->real^N) i x)) <= drop(h x)`
	ASSUME_TAC THENL
	[REPEAT STRIP_TAC THEN
	MATCH_MP_TAC VSUM_NORM_TRIANGLE THEN
	REWRITE_TAC[FINITE_NUMSEG] THEN
	GEN_REWRITE_TAC LAND_CONV [GSYM LIFT_DROP] THEN
	SIMP_TAC[LIFT_SUM; FINITE_NUMSEG] THEN
	MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_LBOUND) THEN
	EXISTS_TAC `\n. vsum (0..n)
	(\i. lift(norm((f:num->real^M->real^N) i x)))` THEN
	REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL
	[RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM]) THEN ASM_SIMP_TAC[IN_DIFF];
	REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `n:num` THEN
	X_GEN_TAC `m:num` THEN DISCH_TAC THEN
	SIMP_TAC[DROP_VSUM; FINITE_NUMSEG; o_DEF; LIFT_DROP] THEN
	MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
	REWRITE_TAC[SUBSET; IN_NUMSEG; NORM_POS_LE; FINITE_NUMSEG] THEN
	UNDISCH_TAC `n:num <= m` THEN ARITH_TAC];
	ALL_TAC] THEN
	SUBGOAL_THEN
	`!x. x IN s DIFF k ==> norm((l:real^M->real^N) x) <= drop(h x)`
	ASSUME_TAC THENL
	[REPEAT STRIP_TAC THEN
	MATCH_MP_TAC(ISPEC `sequentially` LIM_NORM_UBOUND) THEN
	EXISTS_TAC `\n. vsum ((:num) INTER (0..n))
	(\i. (f:num->real^M->real^N) i x)` THEN
	ASM_SIMP_TAC[IN_DIFF; GSYM sums; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
	MATCH_MP_TAC ALWAYS_EVENTUALLY THEN ASM_SIMP_TAC[INTER_UNIV];
	ALL_TAC] THEN
	MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
	[REWRITE_TAC[lspace; IN_ELIM_THM] THEN
	MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
	[MATCH_MP_TAC MEASURABLE_ON_LIMIT THEN
	EXISTS_TAC `\n x. vsum (0..n) (\i. (f:num->real^M->real^N) i x)` THEN
	EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN
	ONCE_REWRITE_TAC[SET_RULE `0..n = UNIV INTER (0..n)`] THEN
	ASM_REWRITE_TAC[GSYM sums] THEN GEN_TAC THEN
	REWRITE_TAC[INTER_UNIV] THEN MATCH_MP_TAC MEASURABLE_ON_VSUM THEN
	RULE_ASSUM_TAC(REWRITE_RULE[lspace; IN_ELIM_THM]) THEN
	ASM_REWRITE_TAC[FINITE_NUMSEG];
	DISCH_TAC] THEN
	MATCH_MP_TAC MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
	EXISTS_TAC
	`\x. if x IN k then lift(norm(l x:real^N) rpow p)
	else (hp:real^M->real^1) x` THEN
	ASM_SIMP_TAC[MEASURABLE_ON_LIFT_RPOW; MEASURABLE_ON_NORM; ETA_AX;
	REAL_ARITH `&1 <= p ==> &0 < p`] THEN
	CONJ_TAC THENL
	[UNDISCH_TAC `(hp:real^M->real^1) integrable_on s` THEN
	MATCH_MP_TAC INTEGRABLE_SPIKE THEN
	EXISTS_TAC `k:real^M->bool` THEN ASM_SIMP_TAC[IN_DIFF];
	REWRITE_TAC[NORM_LIFT; REAL_ABS_RPOW; REAL_ABS_NORM] THEN
	GEN_TAC THEN DISCH_TAC THEN COND_CASES_TAC THEN
	REWRITE_TAC[LIFT_DROP; REAL_LE_REFL] THEN
	MATCH_MP_TAC REAL_LE_TRANS THEN
	EXISTS_TAC `drop(h(x:real^M)) rpow p` THEN CONJ_TAC THENL
	[MATCH_MP_TAC RPOW_LE2 THEN ASM_SIMP_TAC[NORM_POS_LE; IN_DIFF] THEN
	ASM_REAL_ARITH_TAC;
	EXPAND_TAC "h" THEN REWRITE_TAC[LIFT_DROP] THEN
	MATCH_MP_TAC(REAL_ARITH `x = y pow 1 ==> x <= y`) THEN
	MATCH_MP_TAC EQ_TRANS THEN
	EXISTS_TAC `drop(hp(x:real^M)) rpow (inv p * p)` THEN CONJ_TAC THENL
	[MATCH_MP_TAC RPOW_RPOW THEN
	MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_LBOUND) THEN
	EXISTS_TAC `\k. lift(drop((g:num->real^M->real^1) k x) rpow p)` THEN
	ASM_SIMP_TAC[IN_DIFF; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
	ASM_SIMP_TAC[LIFT_DROP; RPOW_POS_LE; EVENTUALLY_TRUE];
	ASM_SIMP_TAC[REAL_MUL_LINV; REAL_ARITH `&1 <= p ==> ~(p = &0)`] THEN
	REWRITE_TAC[RPOW_POW]]]];
	DISCH_TAC] THEN
	SUBGOAL_THEN `!x:real^M. x IN s DIFF k ==> &0 <= drop(h x)` ASSUME_TAC THENL
	[ASM_MESON_TAC[REAL_LE_TRANS; NORM_POS_LE]; ALL_TAC] THEN
	SUBGOAL_THEN `!x:real^M. x IN s DIFF k ==> &0 <= drop(hp x)` ASSUME_TAC THENL
	[REPEAT STRIP_TAC THEN
	MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_LBOUND) THEN
	EXISTS_TAC `\k. lift(drop((g:num->real^M->real^1) k x) rpow p)` THEN
	ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIFT_DROP; RPOW_POS_LE] THEN
	REWRITE_TAC[EVENTUALLY_TRUE];
	ALL_TAC] THEN
	MP_TAC(ISPECL
	[`\n x. lift(norm(vsum (0..n) (\i. (f:num->real^M->real^N) i x) - l x)
	rpow p)`;
	`(\x. vec 0):real^M->real^1`;
	`\x:real^M. &2 rpow p % lift(drop(h x) rpow p)`;
	`s DIFF k:real^M->bool`]
	DOMINATED_CONVERGENCE) THEN
	REWRITE_TAC[lnorm; INTEGRAL_0; REAL_INTEGRAL_0; INTEGRABLE_0] THEN
	ANTS_TAC THENL
	[REPEAT CONJ_TAC THENL
	[X_GEN_TAC `n:num` THEN
	MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN
	EXISTS_TAC `s:real^M->bool` THEN CONJ_TAC THENL
	[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
	NEGLIGIBLE_SUBSET)) THEN SET_TAC[];
	MATCH_MP_TAC LSPACE_IMP_INTEGRABLE THEN
	MATCH_MP_TAC LSPACE_SUB THEN ASM_REWRITE_TAC[ETA_AX] THEN
	CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
	MATCH_MP_TAC LSPACE_VSUM THEN
	ASM_REWRITE_TAC[FINITE_NUMSEG] THEN ASM_REAL_ARITH_TAC];
	MATCH_MP_TAC INTEGRABLE_CMUL THEN EXPAND_TAC "h" THEN
	REWRITE_TAC[LIFT_DROP] THEN
	MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE) THEN
	EXISTS_TAC `hp:real^M->real^1` THEN
	EXISTS_TAC `{}:real^M->bool` THEN
	ASM_SIMP_TAC[DIFF_EMPTY; NEGLIGIBLE_EMPTY; RPOW_RPOW] THEN
	ASM_SIMP_TAC[REAL_MUL_LINV; REAL_ARITH `&1 <= p ==> ~(p = &0)`] THEN
	REWRITE_TAC[LIFT_DROP; RPOW_POW; REAL_POW_1] THEN
	UNDISCH_TAC `(hp:real^M->real^1) integrable_on s` 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[];
	REWRITE_TAC[DROP_CMUL; GSYM RPOW_MUL; LIFT_DROP] THEN
	REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_REAL; GSYM drop] THEN
	REWRITE_TAC[REAL_ABS_NORM; LIFT_DROP; REAL_ABS_RPOW] THEN
	MATCH_MP_TAC RPOW_LE2 THEN REWRITE_TAC[NORM_POS_LE] THEN
	CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
	MATCH_MP_TAC(NORM_ARITH
	`norm(x:real^N) <= a /\ norm(y) <= a ==> norm(x - y) <= &2 * a`) THEN
	ASM_SIMP_TAC[];
	X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN
	MATCH_MP_TAC LIM_NULL_RPOW THEN
	CONJ_TAC THENL [REWRITE_TAC[o_DEF]; ASM_REAL_ARITH_TAC] THEN
	REWRITE_TAC[GSYM LIM_NULL_NORM] THEN REWRITE_TAC[GSYM LIM_NULL] THEN
	RULE_ASSUM_TAC(REWRITE_RULE[sums; INTER_UNIV]) THEN
	ASM_SIMP_TAC[]];
	GEN_REWRITE_TAC (LAND_CONV o RATOR_CONV o LAND_CONV o ABS_CONV)
	[GSYM LIFT_DROP] THEN
	DISCH_THEN(MP_TAC o MATCH_MP
	(REWRITE_RULE[IMP_CONJ; o_DEF] LIM_NULL_RPOW)) THEN
	DISCH_THEN(MP_TAC o SPEC `inv p:real`) THEN
	ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN
	ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
	REWRITE_TAC[tendsto; DIST_0; NORM_REAL; GSYM drop; LIFT_DROP] THEN
	MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
	MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
	MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
	SUBGOAL_THEN
	`!f:real^M->real^1. integral (s DIFF k) f = integral s f`
	MP_TAC THENL [ALL_TAC; SIMP_TAC[REAL_ARITH `abs(x) < e ==> x < e`]] THEN
	GEN_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 LSPACE_SUMMABLE = prove
	(`!f:num->real^M->real^N p s t.
	&1 <= p /\
	(!i. i IN t ==> f i IN lspace s p) /\
	real_summable t (\i. lnorm s p (f i))
	==> ?g. g IN lspace s p /\
	((\n. lnorm s p (\x. vsum (t INTER (0..n)) (\i. f i x) - g x))
	---> &0) sequentially`,
	REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUMMABLE_RESTRICT] THEN
	REWRITE_TAC[] THEN STRIP_TAC THEN
	MP_TAC(ISPECL
	[`(\n:num x. if n IN t then f n x else vec 0):num->real^M->real^N`;
	`p:real`; `s:real^M->bool`] LSPACE_SUMMABLE_UNIV) THEN
	ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
	[CONJ_TAC THENL
	[X_GEN_TAC `i:num` THEN ASM_CASES_TAC `(i:num) IN t` THEN
	ASM_SIMP_TAC[LSPACE_0; ETA_AX; REAL_ARITH `&1 <= p ==> ~(p = &0)`];
	FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [real_summable]) THEN
	REWRITE_TAC[real_summable] THEN MATCH_MP_TAC MONO_EXISTS THEN
	GEN_TAC 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 COND_CASES_TAC THEN
	ASM_SIMP_TAC[ETA_AX; LNORM_0; REAL_ARITH `&1 <= p ==> ~(p = &0)`]];
	MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^N` THEN
	ASM_CASES_TAC `(g:real^M->real^N) IN lspace s p` THEN
	ASM_REWRITE_TAC[tendsto_real] THEN
	MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
	MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
	MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
	X_GEN_TAC `n:num` THEN REWRITE_TAC[REAL_SUB_RZERO] THEN
	MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x = y ==> x < e ==> abs y < e`) THEN
	CONJ_TAC THENL
	[MATCH_MP_TAC LNORM_POS_LE THEN MATCH_MP_TAC LSPACE_SUB THEN
	ASM_SIMP_TAC[REAL_ARITH `&1 <= p ==> &0 <= p`] THEN
	MATCH_MP_TAC LSPACE_VSUM THEN
	ASM_SIMP_TAC[FINITE_NUMSEG; REAL_ARITH `&1 <= p ==> &0 < p`] THEN
	X_GEN_TAC `i:num` THEN ASM_CASES_TAC `(i:num) IN t` THEN
	ASM_SIMP_TAC[ETA_AX; LSPACE_0; REAL_ARITH `&1 <= p ==> ~(p = &0)`];
	AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN
	X_GEN_TAC `x:real^M` THEN REWRITE_TAC[GSYM VSUM_RESTRICT_SET] THEN
	REWRITE_TAC[SET_RULE `s INTER t = {x \| x IN t /\ x IN s}`]]]);;

	let RIESZ_FISCHER = prove
	(`!f:num->real^M->real^N p s.
	&1 <= p /\ (!n. (f n) IN lspace s p) /\
	(!e. &0 < e
	==> ?N. !m n. m >= N /\ n >= N
	==> lnorm s p (\x. f m x - f n x) < e)
	==> ?g. g IN lspace s p /\
	!e. &0 < e
	==> ?N. !n. n >= N
	==> lnorm s p (\x. f n x - g x) < e`,
	REPEAT STRIP_TAC THEN
	SUBGOAL_THEN
	`?k:num->num.
	(!n. k n < k (SUC n)) /\
	(!n. lnorm s p ((\x. f (k(SUC n)) x - f (k n) x):real^M->real^N)
	< inv(&2 pow n))`
	STRIP_ASSUME_TAC THENL
	[FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPEC `inv(&2 pow n)`) THEN
	REWRITE_TAC[REAL_LT_INV_EQ; REAL_LT_POW2; SKOLEM_THM] THEN
	DISCH_THEN(X_CHOOSE_TAC `N:num->num`) THEN
	MP_TAC(prove_recursive_functions_exist num_RECURSION
	`k 0 = N 0 /\
	!n. k(SUC n) = MAX (k n + 1) (MAX (N n) (N(SUC n)))`) THEN
	MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
	ASM_REWRITE_TAC[ARITH_RULE `n < MAX (n + 1) m`] THEN
	FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
	CONJ_TAC THENL [ARITH_TAC; SPEC_TAC(`n:num`,`n:num`)] THEN
	INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN ARITH_TAC;
	ALL_TAC] THEN
	MP_TAC(ISPECL
	[`\n x. f (k(SUC n)) x - (f:num->real^M->real^N) (k n) x`;
	`p:real`; `s:real^M->bool`] LSPACE_SUMMABLE_UNIV) THEN
	ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
	[ASM_SIMP_TAC[LSPACE_SUB; ETA_AX; REAL_ARITH `&1 <= p ==> &0 <= p`] THEN
	MATCH_MP_TAC REAL_SUMMABLE_COMPARISON THEN
	EXISTS_TAC `\n. inv(&2) pow n` THEN CONJ_TAC THENL
	[MATCH_MP_TAC REAL_SUMMABLE_GP THEN CONV_TAC REAL_RAT_REDUCE_CONV;
	EXISTS_TAC `0` THEN X_GEN_TAC `n:num` THEN DISCH_THEN(K ALL_TAC) THEN
	REWRITE_TAC[GSYM REAL_INV_POW] THEN
	MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x < y ==> abs x <= y`) THEN
	ASM_SIMP_TAC[LNORM_POS_LE; LSPACE_SUB; ETA_AX;
	REAL_ARITH `&1 <= p ==> &0 <= p`]];
	DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` MP_TAC) THEN
	DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*")) THEN
	EXISTS_TAC `\x. (g:real^M->real^N) x + f (k 0:num) x` THEN
	ASM_SIMP_TAC[LSPACE_ADD; ETA_AX; REAL_ARITH `&1 <= p ==> &0 <= p`] THEN
	X_GEN_TAC `e:real` THEN DISCH_TAC THEN
	REMOVE_THEN "*" (MP_TAC o SPEC `e / &2`) THEN
	ASM_REWRITE_TAC[REAL_HALF; EVENTUALLY_SEQUENTIALLY] THEN
	REWRITE_TAC[ADD1; VSUM_DIFFS_ALT; LE_0] THEN
	DISCH_THEN(X_CHOOSE_THEN `N1:num` (LABEL_TAC "+")) THEN
	FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
	ASM_REWRITE_TAC[REAL_HALF; GE] THEN
	DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN
	EXISTS_TAC `MAX N1 N2` THEN X_GEN_TAC `n:num` THEN
	REWRITE_TAC[ARITH_RULE `MAX N1 N2 <= n <=> N1 <= n /\ N2 <= n`] THEN
	STRIP_TAC THEN REMOVE_THEN "+" (MP_TAC o SPEC `n:num`) THEN
	FIRST_X_ASSUM(MP_TAC o SPECL [`k(n + 1):num`; `n:num`]) THEN
	ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
	[MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `n + 1` THEN
	CONJ_TAC THENL [ASM_ARITH_TAC; SPEC_TAC(`n + 1`,`m:num`)] THEN
	INDUCT_TAC THEN REWRITE_TAC[LE_0] THEN
	MATCH_MP_TAC(ARITH_RULE
	`m <= k m /\ k m < k(SUC m) ==> SUC m <= k(SUC m)`) THEN
	ASM_REWRITE_TAC[];
	REPEAT DISCH_TAC THEN
	ONCE_REWRITE_TAC[VECTOR_ARITH
	`f n x - (g x + f (k 0) x):real^N =
	(f (k (n + 1)) x - f (k 0) x - g x) +
	--(f (k (n + 1)) x - f n x)`] THEN
	W(MP_TAC o PART_MATCH (lhand o rand) LNORM_TRIANGLE o lhand o snd) THEN
	ASM_SIMP_TAC[LSPACE_SUB; LSPACE_NEG; ETA_AX;
	REAL_ARITH `&1 <= p ==> &0 <= p`] THEN
	MATCH_MP_TAC(REAL_ARITH
	`x < e / &2 /\ y < e / &2 ==> z <= x + y ==> z < e`) THEN
	ASM_SIMP_TAC[LNORM_NEG; LSPACE_SUB; ETA_AX;
	REAL_ARITH `&1 <= p ==> &0 <= p`]]]);;

	(* ------------------------------------------------------------------------- *)
	(* A sort of dominated convergence theorem for L_p spaces. *)
	(* ------------------------------------------------------------------------- *)

	let LSPACE_DOMINATED_CONVERGENCE = prove
	(`!f:num->real^M->real^N g h:real^M->real^N s p k.
	&0 < p /\
	(!n. (f n) IN lspace s p) /\ h IN lspace s p /\
	(!n x. x IN s ==> norm(f n x) <= norm(h x)) /\
	negligible k /\
	(!x. x IN s DIFF k ==> ((\n. f n x) --> g(x)) sequentially)
	==> g IN lspace s p /\
	((\n. lnorm s p (\x. f n x - g x)) ---> &0) sequentially`,
	REPEAT GEN_TAC THEN STRIP_TAC THEN
	MP_TAC(ISPECL
	[`\n x. lift(norm((f:num->real^M->real^N) n x) rpow p)`;
	`\x. lift(norm((g:real^M->real^N) x) rpow p)`;
	`\x. lift(norm((h:real^M->real^N) x) rpow p)`;
	`s DIFF k:real^M->bool`] DOMINATED_CONVERGENCE) THEN
	REWRITE_TAC[] THEN ANTS_TAC THENL
	[REPEAT CONJ_TAC THENL
	[X_GEN_TAC `k:num` THEN
	FIRST_ASSUM(MP_TAC o MATCH_MP LSPACE_IMP_INTEGRABLE o SPEC `k:num`) 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[];
	FIRST_ASSUM(MP_TAC o MATCH_MP LSPACE_IMP_INTEGRABLE) 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[];
	MAP_EVERY X_GEN_TAC [`k:num`; `x:real^M`] THEN
	REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN
	REWRITE_TAC[NORM_LIFT; REAL_ABS_RPOW; REAL_ABS_NORM; LIFT_DROP] THEN
	MATCH_MP_TAC RPOW_LE2 THEN ASM_SIMP_TAC[NORM_POS_LE; REAL_LT_IMP_LE];
	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_THEN(MP_TAC o ISPEC
	`(lift o (\x. x rpow p) o drop) o (lift o (norm:real^N->real))` o
	MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN
	ASM_SIMP_TAC[o_THM; DROP_VEC; RPOW_ZERO; REAL_LT_IMP_NZ; LIFT_NUM] THEN
	REWRITE_TAC[o_THM; LIFT_DROP] THEN DISCH_THEN MATCH_MP_TAC THEN
	MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN
	REWRITE_TAC[CONTINUOUS_AT_LIFT_NORM] THEN
	GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM LIFT_DROP] THEN
	REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS_ATREAL] THEN
	MATCH_MP_TAC REAL_CONTINUOUS_AT_RPOW THEN
	REWRITE_TAC[REAL_LE_INV_EQ] THEN ASM_REAL_ARITH_TAC];
	STRIP_TAC] THEN
	MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
	[REWRITE_TAC[lspace; IN_ELIM_THM] THEN CONJ_TAC THENL
	[MATCH_MP_TAC MEASURABLE_ON_LIMIT THEN
	EXISTS_TAC `f:num->real^M->real^N` THEN
	EXISTS_TAC `k:real^M->bool` THEN ASM_REWRITE_TAC[] THEN
	RULE_ASSUM_TAC(REWRITE_RULE[lspace; IN_ELIM_THM]) THEN ASM_REWRITE_TAC[];
	FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE
	[TAUT `a ==> b ==> c <=> b ==> a ==> c`] INTEGRABLE_SPIKE_SET)) THEN
	FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
	NEGLIGIBLE_SUBSET)) THEN SET_TAC[]];
	DISCH_TAC] THEN
	SUBGOAL_THEN
	`!x. x IN s DIFF k
	==> norm((g:real^M->real^N) x) <= norm((h:real^M->real^N) x)`
	ASSUME_TAC THENL
	[X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN
	MATCH_MP_TAC(ISPEC `sequentially` LIM_NORM_UBOUND) THEN
	EXISTS_TAC `\n. (f:num->real^M->real^N) n x` THEN
	ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
	MATCH_MP_TAC ALWAYS_EVENTUALLY THEN
	RULE_ASSUM_TAC(REWRITE_RULE[IN_DIFF]) THEN ASM_SIMP_TAC[];
	ALL_TAC] THEN
	MP_TAC(ISPECL
	[`\n x. lift(norm((f:num->real^M->real^N) n x - g x) rpow p)`;
	`(\x. vec 0):real^M->real^1`;
	`\x. lift(norm(&2 % (h:real^M->real^N) x) rpow p)`;
	`s DIFF k:real^M->bool`] DOMINATED_CONVERGENCE) THEN
	REWRITE_TAC[] THEN ANTS_TAC THENL
	[REPEAT CONJ_TAC THENL
	[X_GEN_TAC `k:num` THEN
	SUBGOAL_THEN `(\x. (f:num->real^M->real^N) k x - g x) IN lspace s p`
	MP_TAC THENL
	[ASM_SIMP_TAC[LSPACE_SUB; REAL_LT_IMP_LE; ETA_AX]; ALL_TAC] THEN
	DISCH_THEN(MP_TAC o MATCH_MP LSPACE_IMP_INTEGRABLE) THEN
	REWRITE_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[];
	REWRITE_TAC[NORM_MUL; RPOW_MUL; LIFT_CMUL] THEN
	MATCH_MP_TAC INTEGRABLE_CMUL THEN
	UNDISCH_TAC `(h:real^M->real^N) IN lspace s p` THEN
	DISCH_THEN(MP_TAC o MATCH_MP LSPACE_IMP_INTEGRABLE) 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[];
	MAP_EVERY X_GEN_TAC [`k:num`; `x:real^M`] THEN
	REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN
	REWRITE_TAC[NORM_LIFT; REAL_ABS_RPOW; REAL_ABS_NORM; LIFT_DROP] THEN
	MATCH_MP_TAC RPOW_LE2 THEN ASM_SIMP_TAC[NORM_POS_LE; REAL_LT_IMP_LE] THEN
	MATCH_MP_TAC(NORM_ARITH
	`norm(x:real^N) <= norm(z) /\ norm(y) <= norm z
	==> norm(x - y) <= norm(&2 % z:real^N)`) THEN
	ASM_SIMP_TAC[IN_DIFF];
	X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
	UNDISCH_TAC
	`!x. x IN s DIFF k
	==> ((\n. (f:num->real^M->real^N) n x) --> g x) sequentially` THEN
	DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN
	GEN_REWRITE_TAC LAND_CONV [LIM_NULL] THEN
	DISCH_THEN(MP_TAC o ISPEC
	`(lift o (\x. x rpow p) o drop) o (lift o (norm:real^N->real))` o
	MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN
	ASM_SIMP_TAC[o_THM; DROP_VEC; RPOW_ZERO; REAL_LT_IMP_NZ; LIFT_NUM] THEN
	ASM_SIMP_TAC[NORM_0; RPOW_ZERO; REAL_LT_IMP_NZ; LIFT_DROP; LIFT_NUM] THEN
	DISCH_THEN MATCH_MP_TAC THEN
	MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN
	REWRITE_TAC[CONTINUOUS_AT_LIFT_NORM] THEN
	GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM LIFT_DROP] THEN
	REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS_ATREAL] THEN
	MATCH_MP_TAC REAL_CONTINUOUS_AT_RPOW THEN
	REWRITE_TAC[REAL_LE_INV_EQ] THEN ASM_REAL_ARITH_TAC];
	DISCH_THEN(MP_TAC o CONJUNCT2)] THEN
	REWRITE_TAC[INTEGRAL_0; TENDSTO_REAL; lnorm; o_DEF; LIFT_DROP; LIFT_NUM] THEN
	DISCH_THEN(MP_TAC o ISPEC `lift o (\x. x rpow inv p) o drop` o
	MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] LIM_CONTINUOUS_FUNCTION)) THEN
	ASM_SIMP_TAC[o_THM; DROP_VEC; RPOW_ZERO; REAL_LT_IMP_NZ; LIFT_NUM] THEN
	ASM_SIMP_TAC[REAL_INV_EQ_0; REAL_LT_IMP_NZ; LIFT_NUM] THEN ANTS_TAC THENL
	[GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM LIFT_DROP] THEN
	REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS_ATREAL] THEN
	MATCH_MP_TAC REAL_CONTINUOUS_AT_RPOW THEN
	REWRITE_TAC[REAL_LE_INV_EQ] THEN ASM_REAL_ARITH_TAC;
	ALL_TAC] THEN
	MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM) THEN
	MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN
	X_GEN_TAC `k:num` THEN REWRITE_TAC[VECTOR_SUB_EQ] THEN
	AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC 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[]);;

	(* ------------------------------------------------------------------------- *)
	(* Approximation of functions in L_p by bounded ones and continuous ones, *)
	(* and (for bounded domain sets) by purely polynomial ones. *)
	(* ------------------------------------------------------------------------- *)

	let LSPACE_APPROXIMATE_BOUNDED = prove
	(`!f:real^M->real^N s p e.
	&0 < p /\ measurable s /\ f IN lspace s p /\ &0 < e
	==> ?g. g IN lspace s p /\
	bounded (IMAGE g s) /\
	lnorm s p (\x. f x - g x) < e`,
	REPEAT STRIP_TAC THEN
	MP_TAC(ISPECL
	[`(\n x. (lambda i. max (--(&n)) (min (&n) ((f:real^M->real^N)(x)$i))))
	:num->real^M->real^N`;
	`f:real^M->real^N`;
	`f:real^M->real^N`;
	`s:real^M->bool`; `p:real`; `{}:real^M->bool`]
	LSPACE_DOMINATED_CONVERGENCE) THEN
	ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY] THEN
	MATCH_MP_TAC(TAUT
	`b /\ c /\ a /\ (a /\ d ==> e)
	==> (a /\ b /\ c ==> d) ==> e`) THEN
	REPEAT CONJ_TAC THENL
	[REPEAT STRIP_TAC THEN MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN
	SIMP_TAC[LAMBDA_BETA] THEN REAL_ARITH_TAC;
	X_GEN_TAC `x:real^M` THEN REWRITE_TAC[DIFF_EMPTY] THEN DISCH_TAC THEN
	MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
	MP_TAC(ISPEC
	`sup(IMAGE (\i. abs((f:real^M->real^N) x$i)) (1..dimindex(:N)))`
	REAL_ARCH_SIMPLE) THEN
	MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
	SIMP_TAC[REAL_SUP_LE_FINITE; FINITE_NUMSEG; NUMSEG_EMPTY;
	NOT_LT; DIMINDEX_GE_1; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
	SIMP_TAC[FORALL_IN_IMAGE; IN_NUMSEG; CART_EQ; LAMBDA_BETA] THEN
	DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
	X_GEN_TAC `i:num` THEN STRIP_TAC THEN
	MATCH_MP_TAC(REAL_ARITH
	`abs(x) <= n ==> max (--n) (min n x) = x`) THEN
	ASM_MESON_TAC[REAL_OF_NUM_LE; REAL_LE_TRANS];
	X_GEN_TAC `n:num` THEN
	MP_TAC(ISPECL
	[`s:real^M->bool`; `p:real`; `vec n:real^N`] LSPACE_CONST) THEN
	ASM_REWRITE_TAC[] THEN
	UNDISCH_TAC `(f:real^M->real^N) IN lspace s p` THEN
	REWRITE_TAC[IMP_IMP] THEN
	DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE
	[TAUT `a /\ b /\ c ==> d <=> a /\ b ==> c ==> d`] LSPACE_MIN)) THEN
	ASM_REWRITE_TAC[] THEN
	MP_TAC(ISPECL
	[`s:real^M->bool`; `p:real`; `--vec n:real^N`] LSPACE_CONST) THEN
	ASM_REWRITE_TAC[IMP_IMP] THEN
	DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE
	[TAUT `a /\ b /\ c ==> d <=> a /\ b ==> c ==> d`] LSPACE_MAX)) THEN
	ASM_REWRITE_TAC[] THEN
	MATCH_MP_TAC(MESON[] `x = y ==> x IN s ==> y IN s`) THEN
	SIMP_TAC[FUN_EQ_THM; CART_EQ; LAMBDA_BETA; VEC_COMPONENT;
	VECTOR_NEG_COMPONENT] THEN REAL_ARITH_TAC;
	DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
	REWRITE_TAC[REALLIM_SEQUENTIALLY] 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; REAL_SUB_RZERO] 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 CONJ_TAC THENL
	[REWRITE_TAC[bounded; FORALL_IN_IMAGE] THEN
	EXISTS_TAC `&(dimindex(:N)) * &n` THEN
	X_GEN_TAC `x:real^M` 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
	GEN_REWRITE_TAC
	(RAND_CONV o LAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN
	MATCH_MP_TAC SUM_BOUND THEN
	SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; LAMBDA_BETA] THEN REAL_ARITH_TAC;
	MATCH_MP_TAC(REAL_ARITH `abs(x) < e ==> x < e`) THEN
	ONCE_REWRITE_TAC[GSYM LNORM_NEG] THEN
	ASM_REWRITE_TAC[VECTOR_NEG_SUB]]]);;

	let LSPACE_APPROXIMATE_CONTINUOUS = prove
	(`!f:real^M->real^N s p e.
	&1 <= p /\ measurable s /\ f IN lspace s p /\ &0 < e
	==> ?g. g continuous_on (:real^M) /\
	g IN lspace s p /\
	lnorm s p (\x. f x - g x) < e`,
	REPEAT STRIP_TAC THEN
	FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REAL_ARITH `&1 <= p ==> &0 < p`)) THEN
	MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`; `p:real`; `e / &2`]
	LSPACE_APPROXIMATE_BOUNDED) THEN
	ASM_REWRITE_TAC[REAL_HALF] THEN
	DISCH_THEN(X_CHOOSE_THEN `h: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] 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. x IN s ==> 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
	[RULE_ASSUM_TAC(REWRITE_RULE[lspace; IN_ELIM_THM]) THEN ASM_REWRITE_TAC[];
	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) IN lspace s p` ASSUME_TAC THENL
	[X_GEN_TAC `n:num` THEN
	MATCH_MP_TAC(INST_TYPE [`:N`,`:P`] LSPACE_BOUNDED_MEASURABLE) THEN
	EXISTS_TAC `(\x. B % vec 1):real^M->real^N` THEN
	ASM_SIMP_TAC[LSPACE_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
	[`g:num->real^M->real^N`; `h:real^M->real^N`;
	`(\x. B % vec 1):real^M->real^N`;
	`s:real^M->bool`; `p:real`; `k:real^M->bool`]
	LSPACE_DOMINATED_CONVERGENCE) THEN
	ASM_SIMP_TAC[LSPACE_CONST] THEN
	REWRITE_TAC[REALLIM_SEQUENTIALLY; REAL_SUB_RZERO] 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] THEN DISCH_TAC THEN
	EXISTS_TAC `(g:num->real^M->real^N) n` THEN
	ASM_REWRITE_TAC[] THEN
	SUBGOAL_THEN
	`(\x. f x - (g:num->real^M->real^N) n x) =
	(\x. (f x - h x) + --(g n x - h x))`
	SUBST1_TAC THENL [SIMP_TAC[FUN_EQ_THM] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN
	W(MP_TAC o PART_MATCH (lhand o rand) LNORM_TRIANGLE o lhand o snd) THEN
	ASM_SIMP_TAC[LSPACE_SUB; ETA_AX; REAL_LT_IMP_LE; LSPACE_NEG] THEN
	MATCH_MP_TAC(REAL_ARITH
	`y < e / &2 /\ z < e / &2 ==> x <= y + z ==> x < e`) THEN
	ASM_SIMP_TAC[LNORM_NEG; REAL_ARITH `abs x < e ==> x < e`]);;

	let LSPACE_APPROXIMATE_VECTOR_POLYNOMIAL_FUNCTION = prove
	(`!f:real^M->real^N s p e.
	&1 <= p /\ bounded s /\ measurable s /\ f IN lspace s p /\ &0 < e
	==> ?g. vector_polynomial_function g /\
	g IN lspace s p /\
	lnorm s p (\x. f x - g x) < e`,
	REPEAT STRIP_TAC THEN
	MP_TAC(ISPECL
	[`f:real^M->real^N`; `s:real^M->bool`; `p:real`; `e / &2`]
	LSPACE_APPROXIMATE_CONTINUOUS) THEN
	ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_HALF] THEN
	X_GEN_TAC `g:real^M->real^N` THEN STRIP_TAC THEN
	MP_TAC(ISPECL [`g:real^M->real^N`; `closure s:real^M->bool`;
	`e / &2 / (measure(s:real^M->bool) rpow (inv p) + &1)`]
	STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION) THEN
	ASM_REWRITE_TAC[REAL_HALF; COMPACT_CLOSURE] THEN ANTS_TAC THENL
	[CONJ_TAC THENL
	[ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV];
	MATCH_MP_TAC REAL_LT_DIV THEN ASM_REWRITE_TAC[REAL_HALF] THEN
	MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> &0 < x + &1`) THEN
	ASM_SIMP_TAC[RPOW_POS_LE; MEASURE_POS_LE]];
	MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^M->real^N` THEN
	STRIP_TAC THEN ASM_REWRITE_TAC[]] THEN
	MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
	[MATCH_MP_TAC LSPACE_BOUNDED_MEASURABLE_SIMPLE THEN
	CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ASM_REWRITE_TAC[]] THEN
	ASM_SIMP_TAC[CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET;
	MEASURABLE_IMP_LEBESGUE_MEASURABLE;
	CONTINUOUS_ON_VECTOR_POLYNOMIAL_FUNCTION] THEN
	MATCH_MP_TAC BOUNDED_SUBSET THEN
	EXISTS_TAC `IMAGE (h:real^M->real^N) (closure s)` THEN
	SIMP_TAC[IMAGE_SUBSET; CLOSURE_SUBSET] THEN
	MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN
	MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
	ASM_SIMP_TAC[CONTINUOUS_ON_VECTOR_POLYNOMIAL_FUNCTION; COMPACT_CLOSURE];
	DISCH_TAC] THEN
	TRANS_TAC REAL_LET_TRANS
	`lnorm s p (\x. (f:real^M->real^N) x - g x) +
	lnorm s p (\x. g x - h x)` THEN
	CONJ_TAC THENL
	[W(MP_TAC o PART_MATCH (rand o rand) LNORM_TRIANGLE o rand o snd) THEN
	ASM_SIMP_TAC[LSPACE_SUB; REAL_ARITH `&1 <= p ==> &0 <= p`] THEN
	REWRITE_TAC[VECTOR_ARITH `(f - g) + (g - h):real^N = f - h`];
	FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
	`x < e / &2 ==> y <= e / &2 ==> x + y < e`))] THEN
	TRANS_TAC REAL_LE_TRANS
	`lnorm (s:real^M->bool) p
	(\x. lift(e / &2 / (measure s rpow inv p + &1)))` THEN
	CONJ_TAC THENL
	[MATCH_MP_TAC LNORM_MONO THEN EXISTS_TAC `{}:real^M->bool` THEN
	REWRITE_TAC[NEGLIGIBLE_EMPTY; DIFF_EMPTY] THEN
	CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
	ASM_SIMP_TAC[LSPACE_SUB; LSPACE_CONST; REAL_ARITH `&1 <= p ==> &0 <= p`;
	NORM_LIFT; REAL_ARITH `x < y ==> x <= abs y`;
	REWRITE_RULE[SUBSET] CLOSURE_SUBSET];
	ASM_SIMP_TAC[LNORM_CONST; REAL_ARITH `&1 <= p ==> &0 < p`] THEN
	REWRITE_TAC[NORM_LIFT; REAL_ABS_DIV; REAL_ABS_NUM] THEN
	ASM_SIMP_TAC[REAL_ARITH
	`&0 < e ==> x * abs e / &2 / y = (x * e / &2) / y`] THEN
	ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4)
	[MEASURE_POS_LE; RPOW_POS_LE; REAL_LE_LDIV_EQ;
	REAL_ARITH `abs x = if &0 < x then x else --x`;
	REAL_ARITH `&0 <= x ==> &0 < x + &1`] THEN
	REWRITE_TAC[REAL_ARITH `m * e / &2 <= e / &2 * n <=> e * m <= e * n`] THEN
	ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN REAL_ARITH_TAC]);;