(* ========================================================================= *) (* Determinant and trace of a square matrix. *) (* *) (* (c) Copyright, John Harrison 1998-2008 *) (* ========================================================================= *) needs "Multivariate/vectors.ml";; needs "Library/permutations.ml";; needs "Library/floor.ml";; needs "Library/products.ml";; prioritize_real();; (* ------------------------------------------------------------------------- *) (* Trace of a matrix (this is relatively easy). *) (* ------------------------------------------------------------------------- *) let trace = new_definition `(trace:real^N^N->real) A = sum(1..dimindex(:N)) (\i. A$i$i)`;; let TRACE_0 = prove (`trace(mat 0) = &0`, SIMP_TAC[trace; mat; LAMBDA_BETA; SUM_0]);; let TRACE_I = prove (`trace(mat 1 :real^N^N) = &(dimindex(:N))`, SIMP_TAC[trace; mat; LAMBDA_BETA; SUM_CONST_NUMSEG; REAL_MUL_RID] THEN AP_TERM_TAC THEN ARITH_TAC);; let TRACE_ADD = prove (`!A B:real^N^N. trace(A + B) = trace(A) + trace(B)`, SIMP_TAC[trace; matrix_add; SUM_ADD_NUMSEG; LAMBDA_BETA]);; let TRACE_SUB = prove (`!A B:real^N^N. trace(A - B) = trace(A) - trace(B)`, SIMP_TAC[trace; matrix_sub; SUM_SUB_NUMSEG; LAMBDA_BETA]);; let TRACE_CMUL = prove (`!c A:real^N^N. trace(c %% A) = c * trace A`, REWRITE_TAC[trace; MATRIX_CMUL_COMPONENT; SUM_LMUL]);; let TRACE_NEG = prove (`!A:real^N^N. trace(--A) = --(trace A)`, REWRITE_TAC[trace; MATRIX_NEG_COMPONENT; SUM_NEG]);; let TRACE_MUL_SYM = prove (`!A B:real^N^M. trace(A ** B) = trace(B ** A)`, REPEAT GEN_TAC THEN SIMP_TAC[trace; matrix_mul; LAMBDA_BETA] THEN GEN_REWRITE_TAC RAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[REAL_MUL_SYM]);; let TRACE_TRANSP = prove (`!A:real^N^N. trace(transp A) = trace A`, SIMP_TAC[trace; transp; LAMBDA_BETA]);; let TRACE_SIMILAR = prove (`!A:real^N^N U:real^N^N. invertible U ==> trace(matrix_inv U ** A ** U) = trace A`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[TRACE_MUL_SYM] THEN ASM_SIMP_TAC[GSYM MATRIX_MUL_ASSOC; MATRIX_INV; MATRIX_MUL_RID]);; let TRACE_MUL_CYCLIC = prove (`!A:real^P^M B C:real^M^N. trace(A ** B ** C) = trace(B ** C ** A)`, REPEAT GEN_TAC THEN REWRITE_TAC[MATRIX_MUL_ASSOC] THEN GEN_REWRITE_TAC RAND_CONV [TRACE_MUL_SYM] THEN REWRITE_TAC[MATRIX_MUL_ASSOC]);; (* ------------------------------------------------------------------------- *) (* Definition of determinant. *) (* ------------------------------------------------------------------------- *) let det = new_definition `det(A:real^N^N) = sum { p | p permutes 1..dimindex(:N) } (\p. sign(p) * product (1..dimindex(:N)) (\i. A$i$(p i)))`;; (* ------------------------------------------------------------------------- *) (* A few general lemmas we need below. *) (* ------------------------------------------------------------------------- *) let IN_DIMINDEX_SWAP = prove (`!m n j. 1 <= m /\ m <= dimindex(:N) /\ 1 <= n /\ n <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> 1 <= swap(m,n) j /\ swap(m,n) j <= dimindex(:N)`, REWRITE_TAC[swap] THEN ARITH_TAC);; let LAMBDA_BETA_PERM = prove (`!p i. p permutes 1..dimindex(:N) /\ 1 <= i /\ i <= dimindex(:N) ==> ((lambda) g :A^N) $ p(i) = g(p i)`, ASM_MESON_TAC[LAMBDA_BETA; PERMUTES_IN_IMAGE; IN_NUMSEG]);; let PRODUCT_PERMUTE = prove (`!f p s. p permutes s ==> product s f = product s (f o p)`, REWRITE_TAC[product] THEN MATCH_MP_TAC ITERATE_PERMUTE THEN REWRITE_TAC[MONOIDAL_REAL_MUL]);; let PRODUCT_PERMUTE_NUMSEG = prove (`!f p m n. p permutes m..n ==> product(m..n) f = product(m..n) (f o p)`, MESON_TAC[PRODUCT_PERMUTE; FINITE_NUMSEG]);; let REAL_MUL_SUM = prove (`!s t f g. FINITE s /\ FINITE t ==> sum s f * sum t g = sum s (\i. sum t (\j. f(i) * g(j)))`, SIMP_TAC[SUM_LMUL] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[SUM_LMUL]);; let REAL_MUL_SUM_NUMSEG = prove (`!m n p q. sum(m..n) f * sum(p..q) g = sum(m..n) (\i. sum(p..q) (\j. f(i) * g(j)))`, SIMP_TAC[REAL_MUL_SUM; FINITE_NUMSEG]);; (* ------------------------------------------------------------------------- *) (* Basic determinant properties. *) (* ------------------------------------------------------------------------- *) let DET_CMUL = prove (`!A:real^N^N c. det(c %% A) = c pow dimindex(:N) * det A`, REPEAT GEN_TAC THEN SIMP_TAC[det; MATRIX_CMUL_COMPONENT; PRODUCT_MUL; FINITE_NUMSEG] THEN SIMP_TAC[PRODUCT_CONST_NUMSEG_1; GSYM SUM_LMUL] THEN REWRITE_TAC[REAL_MUL_AC]);; let DET_NEG = prove (`!A:real^N^N. det(--A) = --(&1) pow dimindex(:N) * det A`, REWRITE_TAC[MATRIX_NEG_MINUS1; DET_CMUL]);; let DET_TRANSP = prove (`!A:real^N^N. det(transp A) = det A`, GEN_TAC THEN REWRITE_TAC[det] THEN GEN_REWRITE_TAC LAND_CONV [SUM_PERMUTATIONS_INVERSE] THEN MATCH_MP_TAC SUM_EQ THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN BINOP_TAC THENL [ASM_MESON_TAC[SIGN_INVERSE; PERMUTATION_PERMUTES; FINITE_NUMSEG]; ALL_TAC] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM(MATCH_MP PERMUTES_IMAGE th)]) THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `product(1..dimindex(:N)) ((\i. (transp A:real^N^N)$i$inverse p(i)) o p)` THEN CONJ_TAC THENL [MATCH_MP_TAC PRODUCT_IMAGE THEN ASM_MESON_TAC[FINITE_NUMSEG; PERMUTES_INJECTIVE; PERMUTES_INVERSE]; MATCH_MP_TAC PRODUCT_EQ THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN SIMP_TAC[transp; LAMBDA_BETA; o_THM] THEN FIRST_ASSUM(MP_TAC o MATCH_MP PERMUTES_INVERSES_o) THEN SIMP_TAC[FUN_EQ_THM; I_THM; o_THM] THEN STRIP_TAC THEN ASM_SIMP_TAC[PERMUTES_IN_NUMSEG; LAMBDA_BETA_PERM; LAMBDA_BETA]]);; let DET_LOWERTRIANGULAR = prove (`!A:real^N^N. (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ i < j ==> A$i$j = &0) ==> det(A) = product(1..dimindex(:N)) (\i. A$i$i)`, REPEAT STRIP_TAC THEN REWRITE_TAC[det] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum {I} (\p. sign p * product(1..dimindex(:N)) (\i. (A:real^N^N)$i$p(i)))` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUM_SING; SIGN_I; REAL_MUL_LID; I_THM]] THEN MATCH_MP_TAC SUM_SUPERSET THEN SIMP_TAC[IN_SING; FINITE_RULES; SUBSET; IN_ELIM_THM; PERMUTES_I] THEN X_GEN_TAC `p:num->num` THEN STRIP_TAC THEN ASM_REWRITE_TAC[PRODUCT_EQ_0_NUMSEG; REAL_ENTIRE; SIGN_NZ] THEN MP_TAC(SPECL [`p:num->num`; `1..dimindex(:N)`] PERMUTES_NUMSET_LE) THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG; NOT_LT]);; let DET_UPPERTRIANGULAR = prove (`!A:real^N^N. (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ j < i ==> A$i$j = &0) ==> det(A) = product(1..dimindex(:N)) (\i. A$i$i)`, REPEAT STRIP_TAC THEN REWRITE_TAC[det] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum {I} (\p. sign p * product(1..dimindex(:N)) (\i. (A:real^N^N)$i$p(i)))` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[SUM_SING; SIGN_I; REAL_MUL_LID; I_THM]] THEN MATCH_MP_TAC SUM_SUPERSET THEN SIMP_TAC[IN_SING; FINITE_RULES; SUBSET; IN_ELIM_THM; PERMUTES_I] THEN X_GEN_TAC `p:num->num` THEN STRIP_TAC THEN ASM_REWRITE_TAC[PRODUCT_EQ_0_NUMSEG; REAL_ENTIRE; SIGN_NZ] THEN MP_TAC(SPECL [`p:num->num`; `1..dimindex(:N)`] PERMUTES_NUMSET_GE) THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG; NOT_LT]);; let DET_I = prove (`det(mat 1 :real^N^N) = &1`, MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `product(1..dimindex(:N)) (\i. (mat 1:real^N^N)$i$i)` THEN CONJ_TAC THENL [MATCH_MP_TAC DET_LOWERTRIANGULAR; MATCH_MP_TAC PRODUCT_EQ_1_NUMSEG] THEN SIMP_TAC[mat; LAMBDA_BETA] THEN MESON_TAC[LT_REFL]);; let DET_0 = prove (`det(mat 0 :real^N^N) = &0`, MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `product(1..dimindex(:N)) (\i. (mat 0:real^N^N)$i$i)` THEN CONJ_TAC THENL [MATCH_MP_TAC DET_LOWERTRIANGULAR; REWRITE_TAC[PRODUCT_EQ_0_NUMSEG] THEN EXISTS_TAC `1`] THEN SIMP_TAC[mat; LAMBDA_BETA; COND_ID; DIMINDEX_GE_1; LE_REFL]);; let DET_PERMUTE_ROWS = prove (`!A:real^N^N p. p permutes 1..dimindex(:N) ==> det(lambda i. A$p(i)) = sign(p) * det(A)`, REWRITE_TAC[det] THEN SIMP_TAC[LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN SIMP_TAC[GSYM SUM_LMUL; FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [MATCH_MP SUM_PERMUTATIONS_COMPOSE_R th]) THEN MATCH_MP_TAC SUM_EQ THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN X_GEN_TAC `q:num->num` THEN REWRITE_TAC[IN_ELIM_THM; REAL_MUL_ASSOC] THEN DISCH_TAC THEN BINOP_TAC THENL [ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_MESON_TAC[SIGN_COMPOSE; PERMUTATION_PERMUTES; FINITE_NUMSEG]; ALL_TAC] THEN MP_TAC(MATCH_MP PERMUTES_INVERSE (ASSUME `p permutes 1..dimindex(:N)`)) THEN DISCH_THEN(fun th -> GEN_REWRITE_TAC LAND_CONV [MATCH_MP PRODUCT_PERMUTE_NUMSEG th]) THEN MATCH_MP_TAC PRODUCT_EQ THEN REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[PERMUTES_INVERSES]);; let DET_PERMUTE_COLUMNS = prove (`!A:real^N^N p. p permutes 1..dimindex(:N) ==> det((lambda i j. A$i$p(j)):real^N^N) = sign(p) * det(A)`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (funpow 2 RAND_CONV) [GSYM DET_TRANSP] THEN FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC [GSYM(MATCH_MP DET_PERMUTE_ROWS th)]) THEN GEN_REWRITE_TAC RAND_CONV [GSYM DET_TRANSP] THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; transp; LAMBDA_BETA; LAMBDA_BETA_PERM]);; let DET_IDENTICAL_ROWS = prove (`!A:real^N^N i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) /\ row i A = row j A ==> det A = &0`, REPEAT STRIP_TAC THEN MP_TAC(SPECL [`A:real^N^N`; `swap(i:num,j:num)`] DET_PERMUTE_ROWS) THEN ASM_SIMP_TAC[PERMUTES_SWAP; IN_NUMSEG; SIGN_SWAP] THEN MATCH_MP_TAC(REAL_ARITH `a = b ==> b = -- &1 * a ==> a = &0`) THEN AP_TERM_TAC THEN FIRST_X_ASSUM(MP_TAC o SYM) THEN SIMP_TAC[row; CART_EQ; LAMBDA_BETA] THEN REWRITE_TAC[swap] THEN ASM_MESON_TAC[]);; let DET_IDENTICAL_COLUMNS = prove (`!A:real^N^N i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) /\ column i A = column j A ==> det A = &0`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM DET_TRANSP] THEN MATCH_MP_TAC DET_IDENTICAL_ROWS THEN ASM_MESON_TAC[ROW_TRANSP]);; let DET_ZERO_ROW = prove (`!A:real^N^N i. 1 <= i /\ i <= dimindex(:N) /\ row i A = vec 0 ==> det A = &0`, SIMP_TAC[det; row; CART_EQ; LAMBDA_BETA; VEC_COMPONENT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ_0 THEN REWRITE_TAC[IN_ELIM_THM; REAL_ENTIRE; SIGN_NZ] THEN REPEAT STRIP_TAC THEN SIMP_TAC[PRODUCT_EQ_0; FINITE_NUMSEG; IN_NUMSEG] THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]);; let DET_ZERO_COLUMN = prove (`!A:real^N^N i. 1 <= i /\ i <= dimindex(:N) /\ column i A = vec 0 ==> det A = &0`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM DET_TRANSP] THEN MATCH_MP_TAC DET_ZERO_ROW THEN ASM_MESON_TAC[ROW_TRANSP]);; let DET_ROW_ADD = prove (`!a b c k. 1 <= k /\ k <= dimindex(:N) ==> det((lambda i. if i = k then a + b else c i):real^N^N) = det((lambda i. if i = k then a else c i):real^N^N) + det((lambda i. if i = k then b else c i):real^N^N)`, SIMP_TAC[det; LAMBDA_BETA; GSYM SUM_ADD; FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN REWRITE_TAC[GSYM REAL_ADD_LDISTRIB] THEN AP_TERM_TAC THEN SUBGOAL_THEN `1..dimindex(:N) = k INSERT ((1..dimindex(:N)) DELETE k)` SUBST1_TAC THENL [ASM_MESON_TAC[INSERT_DELETE; IN_NUMSEG]; ALL_TAC] THEN SIMP_TAC[PRODUCT_CLAUSES; FINITE_DELETE; FINITE_NUMSEG; IN_DELETE] THEN MATCH_MP_TAC(REAL_RING `c = a + b /\ y = x:real /\ z = x ==> c * x = a * y + b * z`) THEN REWRITE_TAC[VECTOR_ADD_COMPONENT] THEN CONJ_TAC THEN MATCH_MP_TAC PRODUCT_EQ THEN SIMP_TAC[IN_DELETE; FINITE_DELETE; FINITE_NUMSEG]);; let DET_ROW_MUL = prove (`!a b c k. 1 <= k /\ k <= dimindex(:N) ==> det((lambda i. if i = k then c % a else b i):real^N^N) = c * det((lambda i. if i = k then a else b i):real^N^N)`, SIMP_TAC[det; LAMBDA_BETA; GSYM SUM_LMUL; FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `1..dimindex(:N) = k INSERT ((1..dimindex(:N)) DELETE k)` SUBST1_TAC THENL [ASM_MESON_TAC[INSERT_DELETE; IN_NUMSEG]; ALL_TAC] THEN SIMP_TAC[PRODUCT_CLAUSES; FINITE_DELETE; FINITE_NUMSEG; IN_DELETE] THEN MATCH_MP_TAC(REAL_RING `cp = c * p /\ p1 = p2:real ==> s * cp * p1 = c * s * p * p2`) THEN REWRITE_TAC[VECTOR_MUL_COMPONENT] THEN MATCH_MP_TAC PRODUCT_EQ THEN SIMP_TAC[IN_DELETE; FINITE_DELETE; FINITE_NUMSEG]);; let DET_ROW_OPERATION = prove (`!A:real^N^N i. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> det(lambda k. if k = i then row i A + c % row j A else row k A) = det A`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[DET_ROW_ADD; DET_ROW_MUL] THEN MATCH_MP_TAC(REAL_RING `a = b /\ d = &0 ==> a + c * d = b`) THEN CONJ_TAC THENL [AP_TERM_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA; CART_EQ] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[row; LAMBDA_BETA; CART_EQ]; MATCH_MP_TAC DET_IDENTICAL_ROWS THEN MAP_EVERY EXISTS_TAC [`i:num`; `j:num`] THEN ASM_SIMP_TAC[row; LAMBDA_BETA; CART_EQ]]);; let DET_ROW_SPAN = prove (`!A:real^N^N i x. 1 <= i /\ i <= dimindex(:N) /\ x IN span {row j A | 1 <= j /\ j <= dimindex(:N) /\ ~(j = i)} ==> det(lambda k. if k = i then row i A + x else row k A) = det A`, GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN MATCH_MP_TAC SPAN_INDUCT_ALT THEN CONJ_TAC THENL [AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; VECTOR_ADD_RID] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[row; LAMBDA_BETA]; ALL_TAC] THEN REPEAT GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `j:num`) (SUBST_ALL_TAC o SYM)) THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a + c % x + y:real^N = (a + y) + c % x`] THEN ABBREV_TAC `z = row i (A:real^N^N) + y` THEN ASM_SIMP_TAC[DET_ROW_MUL; DET_ROW_ADD] THEN MATCH_MP_TAC(REAL_RING `d = &0 ==> a + c * d = a`) THEN MATCH_MP_TAC DET_IDENTICAL_ROWS THEN MAP_EVERY EXISTS_TAC [`i:num`; `j:num`] THEN ASM_SIMP_TAC[row; LAMBDA_BETA; CART_EQ]);; (* ------------------------------------------------------------------------- *) (* May as well do this, though it's a bit unsatisfactory since it ignores *) (* exact duplicates by considering the rows/columns as a set. *) (* ------------------------------------------------------------------------- *) let DET_DEPENDENT_ROWS = prove (`!A:real^N^N. dependent(rows A) ==> det A = &0`, GEN_TAC THEN REWRITE_TAC[dependent; rows; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN GEN_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM_CASES_TAC `?i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) /\ row i (A:real^N^N) = row j A` THENL [ASM_MESON_TAC[DET_IDENTICAL_ROWS]; ALL_TAC] THEN MP_TAC(SPECL [`A:real^N^N`; `i:num`; `--(row i (A:real^N^N))`] DET_ROW_SPAN) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SPAN_NEG THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN]) THEN MATCH_MP_TAC(TAUT `a = b ==> a ==> b`) THEN REWRITE_TAC[IN] THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_DELETE; IN_ELIM_THM] THEN ASM_MESON_TAC[]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC DET_ZERO_ROW THEN EXISTS_TAC `i:num` THEN ASM_SIMP_TAC[row; LAMBDA_BETA; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_NEG_COMPONENT; VEC_COMPONENT] THEN REAL_ARITH_TAC]);; let DET_DEPENDENT_COLUMNS = prove (`!A:real^N^N. dependent(columns A) ==> det A = &0`, MESON_TAC[DET_DEPENDENT_ROWS; ROWS_TRANSP; DET_TRANSP]);; (* ------------------------------------------------------------------------- *) (* Multilinearity and the multiplication formula. *) (* ------------------------------------------------------------------------- *) let DET_LINEAR_ROW_VSUM = prove (`!a c s k. FINITE s /\ 1 <= k /\ k <= dimindex(:N) ==> det((lambda i. if i = k then vsum s a else c i):real^N^N) = sum s (\j. det((lambda i. if i = k then a(j) else c i):real^N^N))`, GEN_TAC THEN GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[VSUM_CLAUSES; SUM_CLAUSES; DET_ROW_ADD] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC DET_ZERO_ROW THEN EXISTS_TAC `k:num` THEN ASM_SIMP_TAC[row; LAMBDA_BETA; CART_EQ; VEC_COMPONENT]);; let BOUNDED_FUNCTIONS_BIJECTIONS_1 = prove (`!p. p IN {(y,g) | y IN s /\ g IN {f | (!i. 1 <= i /\ i <= k ==> f i IN s) /\ (!i. ~(1 <= i /\ i <= k) ==> f i = i)}} ==> (\(y,g) i. if i = SUC k then y else g(i)) p IN {f | (!i. 1 <= i /\ i <= SUC k ==> f i IN s) /\ (!i. ~(1 <= i /\ i <= SUC k) ==> f i = i)} /\ (\h. h(SUC k),(\i. if i = SUC k then i else h(i))) ((\(y,g) i. if i = SUC k then y else g(i)) p) = p`, REWRITE_TAC[FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN CONV_TAC(REDEPTH_CONV GEN_BETA_CONV) THEN REWRITE_TAC[IN_ELIM_THM] THEN MAP_EVERY X_GEN_TAC [`y:num`; `h:num->num`] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[LE]; ASM_MESON_TAC[LE; ARITH_RULE `~(1 <= i /\ i <= SUC k) ==> ~(i = SUC k)`]; REWRITE_TAC[PAIR_EQ; FUN_EQ_THM] THEN ASM_MESON_TAC[ARITH_RULE `~(SUC k <= k)`]]);; let BOUNDED_FUNCTIONS_BIJECTIONS_2 = prove (`!h. h IN {f | (!i. 1 <= i /\ i <= SUC k ==> f i IN s) /\ (!i. ~(1 <= i /\ i <= SUC k) ==> f i = i)} ==> (\h. h(SUC k),(\i. if i = SUC k then i else h(i))) h IN {(y,g) | y IN s /\ g IN {f | (!i. 1 <= i /\ i <= k ==> f i IN s) /\ (!i. ~(1 <= i /\ i <= k) ==> f i = i)}} /\ (\(y,g) i. if i = SUC k then y else g(i)) ((\h. h(SUC k),(\i. if i = SUC k then i else h(i))) h) = h`, REWRITE_TAC[IN_ELIM_PAIR_THM] THEN CONV_TAC(REDEPTH_CONV GEN_BETA_CONV) THEN REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `h:num->num` THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC; ASM_MESON_TAC[ARITH_RULE `i <= k ==> i <= SUC k /\ ~(i = SUC k)`]; ASM_MESON_TAC[ARITH_RULE `i <= SUC k /\ ~(i = SUC k) ==> i <= k`]; REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[LE_REFL]]);; let FINITE_BOUNDED_FUNCTIONS = prove (`!s k. FINITE s ==> FINITE {f | (!i. 1 <= i /\ i <= k ==> f(i) IN s) /\ (!i. ~(1 <= i /\ i <= k) ==> f(i) = i)}`, REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN GEN_TAC THEN DISCH_TAC THEN INDUCT_TAC THENL [REWRITE_TAC[ARITH_RULE `~(1 <= i /\ i <= 0)`] THEN SIMP_TAC[GSYM FUN_EQ_THM; SET_RULE `{x | x = y} = {y}`; FINITE_RULES]; ALL_TAC] THEN UNDISCH_TAC `FINITE(s:num->bool)` THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[TAUT `a ==> b ==> c <=> b /\ a ==> c`] THEN DISCH_THEN(MP_TAC o MATCH_MP FINITE_PRODUCT) THEN DISCH_THEN(MP_TAC o ISPEC `\(y:num,g) i. if i = SUC k then y else g(i)` o MATCH_MP FINITE_IMAGE) THEN MATCH_MP_TAC(TAUT `a = b ==> a ==> b`) THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE] THEN X_GEN_TAC `h:num->num` THEN EQ_TAC THENL [STRIP_TAC THEN ASM_SIMP_TAC[BOUNDED_FUNCTIONS_BIJECTIONS_1]; ALL_TAC] THEN DISCH_TAC THEN EXISTS_TAC `(\h. h(SUC k),(\i. if i = SUC k then i else h(i))) h` THEN PURE_ONCE_REWRITE_TAC[CONJ_SYM] THEN CONV_TAC (RAND_CONV SYM_CONV) THEN MATCH_MP_TAC BOUNDED_FUNCTIONS_BIJECTIONS_2 THEN ASM_REWRITE_TAC[]);; let DET_LINEAR_ROWS_VSUM_LEMMA = prove (`!s k a c. FINITE s /\ k <= dimindex(:N) ==> det((lambda i. if i <= k then vsum s (a i) else c i):real^N^N) = sum {f | (!i. 1 <= i /\ i <= k ==> f(i) IN s) /\ !i. ~(1 <= i /\ i <= k) ==> f(i) = i} (\f. det((lambda i. if i <= k then a i (f i) else c i) :real^N^N))`, let lemma = prove (`(lambda i. if i <= 0 then x(i) else y(i)) = (lambda i. y i)`, SIMP_TAC[CART_EQ; ARITH; LAMBDA_BETA; ARITH_RULE `1 <= k ==> ~(k <= 0)`]) in ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN GEN_TAC THEN DISCH_TAC THEN INDUCT_TAC THENL [REWRITE_TAC[lemma; LE_0] THEN GEN_TAC THEN REWRITE_TAC[ARITH_RULE `~(1 <= i /\ i <= 0)`] THEN REWRITE_TAC[GSYM FUN_EQ_THM; SET_RULE `{x | x = y} = {y}`] THEN REWRITE_TAC[SUM_SING]; ALL_TAC] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN ASM_SIMP_TAC[ARITH_RULE `SUC k <= n ==> k <= n`] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [LE] THEN REWRITE_TAC[TAUT `(if a \/ b then c else d) = (if a then c else if b then c else d)`] THEN ASM_SIMP_TAC[DET_LINEAR_ROW_VSUM; ARITH_RULE `1 <= SUC k`] THEN ONCE_REWRITE_TAC[TAUT `(if a then b else if c then d else e) = (if c then (if a then b else d) else (if a then b else e))`] THEN ASM_SIMP_TAC[ARITH_RULE `i <= k ==> ~(i = SUC k)`] THEN ASM_SIMP_TAC[SUM_SUM_PRODUCT; FINITE_BOUNDED_FUNCTIONS] THEN MATCH_MP_TAC SUM_EQ_GENERAL_INVERSES THEN EXISTS_TAC `\(y:num,g) i. if i = SUC k then y else g(i)` THEN EXISTS_TAC `\h. h(SUC k),(\i. if i = SUC k then i else h(i))` THEN CONJ_TAC THENL [ACCEPT_TAC BOUNDED_FUNCTIONS_BIJECTIONS_2; ALL_TAC] THEN X_GEN_TAC `p:num#(num->num)` THEN DISCH_THEN(STRIP_ASSUME_TAC o MATCH_MP BOUNDED_FUNCTIONS_BIJECTIONS_1) THEN ASM_REWRITE_TAC[] THEN SPEC_TAC(`p:num#(num->num)`,`q:num#(num->num)`) THEN REWRITE_TAC[FORALL_PAIR_THM] THEN CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN MAP_EVERY X_GEN_TAC [`y:num`; `g:num->num`] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_MESON_TAC[LE; ARITH_RULE `~(SUC k <= k)`]);; let DET_LINEAR_ROWS_VSUM = prove (`!s a. FINITE s ==> det((lambda i. vsum s (a i)):real^N^N) = sum {f | (!i. 1 <= i /\ i <= dimindex(:N) ==> f(i) IN s) /\ !i. ~(1 <= i /\ i <= dimindex(:N)) ==> f(i) = i} (\f. det((lambda i. a i (f i)):real^N^N))`, let lemma = prove (`(lambda i. if i <= dimindex(:N) then x(i) else y(i)):real^N^N = (lambda i. x(i))`, SIMP_TAC[CART_EQ; LAMBDA_BETA]) in REPEAT STRIP_TAC THEN MP_TAC(SPECL [`s:num->bool`; `dimindex(:N)`] DET_LINEAR_ROWS_VSUM_LEMMA) THEN ASM_REWRITE_TAC[LE_REFL; lemma] THEN SIMP_TAC[]);; let MATRIX_MUL_VSUM_ALT = prove (`!A:real^N^N B:real^N^N. A ** B = lambda i. vsum (1..dimindex(:N)) (\k. A$i$k % B$k)`, SIMP_TAC[matrix_mul; CART_EQ; LAMBDA_BETA; VECTOR_MUL_COMPONENT; VSUM_COMPONENT]);; let DET_ROWS_MUL = prove (`!a c. det((lambda i. c(i) % a(i)):real^N^N) = product(1..dimindex(:N)) (\i. c(i)) * det((lambda i. a(i)):real^N^N)`, REPEAT GEN_TAC THEN SIMP_TAC[det; LAMBDA_BETA] THEN SIMP_TAC[GSYM SUM_LMUL; FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN MATCH_MP_TAC SUM_EQ THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_RING `b = c * d ==> s * b = c * s * d`) THEN SIMP_TAC[GSYM PRODUCT_MUL_NUMSEG] THEN MATCH_MP_TAC PRODUCT_EQ_NUMSEG THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG; VECTOR_MUL_COMPONENT]);; let DET_MUL = prove (`!A B:real^N^N. det(A ** B) = det(A) * det(B)`, REPEAT GEN_TAC THEN REWRITE_TAC[MATRIX_MUL_VSUM_ALT] THEN SIMP_TAC[DET_LINEAR_ROWS_VSUM; FINITE_NUMSEG] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum {p | p permutes 1..dimindex(:N)} (\f. det (lambda i. (A:real^N^N)$i$f i % (B:real^N^N)$f i))` THEN CONJ_TAC THENL [REWRITE_TAC[DET_ROWS_MUL] THEN MATCH_MP_TAC SUM_SUPERSET THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN CONJ_TAC THENL [MESON_TAC[permutes; IN_NUMSEG]; ALL_TAC] THEN X_GEN_TAC `f:num->num` THEN REWRITE_TAC[permutes; IN_NUMSEG] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN REWRITE_TAC[REAL_ENTIRE] THEN DISJ2_TAC THEN MATCH_MP_TAC DET_IDENTICAL_ROWS THEN MP_TAC(ISPECL [`1..dimindex(:N)`; `f:num->num`] SURJECTIVE_IFF_INJECTIVE) THEN ASM_REWRITE_TAC[SUBSET; IN_NUMSEG; FINITE_NUMSEG; FORALL_IN_IMAGE] THEN MATCH_MP_TAC(TAUT `(~b ==> c) /\ (b ==> ~a) ==> (a <=> b) ==> c`) THEN CONJ_TAC THENL [REWRITE_TAC[NOT_FORALL_THM] THEN REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; row; NOT_IMP]; ALL_TAC] THEN DISCH_TAC THEN SUBGOAL_THEN `!x y. (f:num->num)(x) = f(y) ==> x = y` ASSUME_TAC THENL [REPEAT GEN_TAC THEN ASM_CASES_TAC `1 <= x /\ x <= dimindex(:N)` THEN ASM_CASES_TAC `1 <= y /\ y <= dimindex(:N)` THEN ASM_MESON_TAC[]; ALL_TAC] THEN ASM_MESON_TAC[]; ALL_TAC] THEN SIMP_TAC[det; REAL_MUL_SUM; FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN MATCH_MP_TAC SUM_EQ THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [MATCH_MP SUM_PERMUTATIONS_COMPOSE_R (MATCH_MP PERMUTES_INVERSE th)]) THEN MATCH_MP_TAC SUM_EQ THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN X_GEN_TAC `q:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN REWRITE_TAC[o_THM] THEN ONCE_REWRITE_TAC[AC REAL_MUL_AC `(p * x) * (q * y) = (p * q) * (x * y)`] THEN BINOP_TAC THENL [SUBGOAL_THEN `sign(q o inverse p) = sign(p:num->num) * sign(q:num->num)` (fun t -> SIMP_TAC[REAL_MUL_ASSOC; SIGN_IDEMPOTENT; REAL_MUL_LID; t]) THEN ASM_MESON_TAC[SIGN_COMPOSE; PERMUTES_INVERSE; PERMUTATION_PERMUTES; FINITE_NUMSEG; SIGN_INVERSE; REAL_MUL_SYM]; ALL_TAC] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [MATCH_MP PRODUCT_PERMUTE_NUMSEG (ASSUME `p permutes 1..dimindex(:N)`)] THEN SIMP_TAC[GSYM PRODUCT_MUL; FINITE_NUMSEG] THEN MATCH_MP_TAC PRODUCT_EQ_NUMSEG THEN ASM_SIMP_TAC[LAMBDA_BETA; LAMBDA_BETA_PERM; o_THM] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `(A:real^N^N)$i$p(i) * (B:real^N^N)$p(i)$q(i)` THEN CONJ_TAC THENL [ASM_MESON_TAC[VECTOR_MUL_COMPONENT; PERMUTES_IN_IMAGE; IN_NUMSEG]; ASM_MESON_TAC[PERMUTES_INVERSES]]);; let DET_LINEAR_ROWS = prove (`!f:real^N->real^N A:real^N^N. linear f ==> det(lambda i. f(A$i)) = det(matrix f) * det A`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM DET_TRANSP] THEN REWRITE_TAC[GSYM DET_MUL] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN GEN_REWRITE_TAC LAND_CONV [GSYM DET_TRANSP] THEN REWRITE_TAC[matrix_mul; matrix_vector_mul; transp] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA]);; (* ------------------------------------------------------------------------- *) (* Relation to invertibility. *) (* ------------------------------------------------------------------------- *) let INVERTIBLE_DET_NZ = prove (`!A:real^N^N. invertible(A) <=> ~(det A = &0)`, GEN_TAC THEN EQ_TAC THENL [REWRITE_TAC[INVERTIBLE_RIGHT_INVERSE; LEFT_IMP_EXISTS_THM] THEN GEN_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `det:real^N^N->real`) THEN REWRITE_TAC[DET_MUL; DET_I] THEN CONV_TAC REAL_RING; ALL_TAC] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[INVERTIBLE_RIGHT_INVERSE] THEN REWRITE_TAC[MATRIX_RIGHT_INVERTIBLE_INDEPENDENT_ROWS] THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`c:num->real`; `i:num`] THEN STRIP_TAC THEN MP_TAC(SPECL [`A:real^N^N`; `i:num`; `--(row i (A:real^N^N))`] DET_ROW_SPAN) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `--(row i (A:real^N^N)) = vsum ((1..dimindex(:N)) DELETE i) (\j. inv(c i) % c j % row j A)` SUBST1_TAC THENL [ASM_SIMP_TAC[VSUM_DELETE_CASES; FINITE_NUMSEG; IN_NUMSEG; VSUM_LMUL] THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG; FINITE_DELETE; IN_DELETE] THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN REPEAT(MATCH_MP_TAC SPAN_MUL) THEN MATCH_MP_TAC(CONJUNCT1 SPAN_CLAUSES) THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[]; ALL_TAC] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC DET_ZERO_ROW THEN EXISTS_TAC `i:num` THEN ASM_SIMP_TAC[row; CART_EQ; LAMBDA_BETA; VEC_COMPONENT; VECTOR_ARITH `x + --x:real^N = vec 0`]);; let DET_EQ_0 = prove (`!A:real^N^N. det(A) = &0 <=> ~invertible(A)`, REWRITE_TAC[INVERTIBLE_DET_NZ]);; let DET_MATRIX_INV = prove (`!A:real^N^N. det(matrix_inv A) = inv(det A)`, GEN_TAC THEN ASM_CASES_TAC `invertible(A:real^N^N)` THENL [MATCH_MP_TAC(REAL_FIELD `a * b = &1 ==> a = inv b`) THEN ASM_SIMP_TAC[GSYM DET_MUL; MATRIX_INV; DET_I]; ASM_MESON_TAC[DET_EQ_0; INVERTIBLE_MATRIX_INV; REAL_INV_0]]);; let MATRIX_MUL_LINV = prove (`!A:real^N^N. ~(det A = &0) ==> matrix_inv A ** A = mat 1`, SIMP_TAC[MATRIX_INV; DET_EQ_0]);; let MATRIX_MUL_RINV = prove (`!A:real^N^N. ~(det A = &0) ==> A ** matrix_inv A = mat 1`, SIMP_TAC[MATRIX_INV; DET_EQ_0]);; let DET_MATRIX_EQ_0 = prove (`!f:real^N->real^N. linear f ==> (det(matrix f) = &0 <=> ~(?g. linear g /\ f o g = I /\ g o f = I))`, SIMP_TAC[DET_EQ_0; MATRIX_INVERTIBLE]);; let DET_MATRIX_EQ_0_LEFT = prove (`!f:real^N->real^N. linear f ==> (det(matrix f) = &0 <=> ~(?g. linear g /\ g o f = I))`, SIMP_TAC[DET_MATRIX_EQ_0] THEN MESON_TAC[LINEAR_INVERSE_LEFT]);; let DET_MATRIX_EQ_0_RIGHT = prove (`!f:real^N->real^N. linear f ==> (det(matrix f) = &0 <=> ~(?g. linear g /\ f o g = I))`, SIMP_TAC[DET_MATRIX_EQ_0] THEN MESON_TAC[LINEAR_INVERSE_LEFT]);; let DET_EQ_0_RANK = prove (`!A:real^N^N. det A = &0 <=> rank A < dimindex(:N)`, REWRITE_TAC[DET_EQ_0; INVERTIBLE_LEFT_INVERSE; GSYM FULL_RANK_INJECTIVE; MATRIX_LEFT_INVERTIBLE_INJECTIVE] THEN GEN_TAC THEN MP_TAC(ISPEC `A:real^N^N` RANK_BOUND) THEN ARITH_TAC);; let RANK_EQ_FULL_DET = prove (`!A:real^N^N. rank A = dimindex(:N) <=> ~(det A = &0)`, GEN_TAC THEN MP_TAC(ISPEC `A:real^N^N` RANK_BOUND) THEN SIMP_TAC[DET_EQ_0_RANK; NOT_LT; GSYM LE_ANTISYM; ARITH_RULE `MIN n n = n`]);; let INVERTIBLE_COVARIANCE_RANK = prove (`!A:real^N^M. invertible(transp A ** A) <=> rank A = dimindex(:N)`, REWRITE_TAC[INVERTIBLE_DET_NZ; GSYM RANK_EQ_FULL_DET; RANK_GRAM]);; let HOMOGENEOUS_LINEAR_EQUATIONS_DET = prove (`!A:real^N^N. (?x. ~(x = vec 0) /\ A ** x = vec 0) <=> det A = &0`, GEN_TAC THEN REWRITE_TAC[MATRIX_NONFULL_LINEAR_EQUATIONS_EQ; DET_EQ_0_RANK] THEN MATCH_MP_TAC(ARITH_RULE `r <= MIN N N ==> (~(r = N) <=> r < N)`) THEN REWRITE_TAC[RANK_BOUND]);; let INVERTIBLE_MATRIX_MUL = prove (`!A:real^N^N B:real^N^N. invertible(A ** B) <=> invertible A /\ invertible B`, REWRITE_TAC[INVERTIBLE_DET_NZ; DET_MUL; DE_MORGAN_THM; REAL_ENTIRE]);; let MATRIX_INV_MUL = prove (`!A:real^N^N B:real^N^N. invertible A /\ invertible B ==> matrix_inv(A ** B) = matrix_inv B ** matrix_inv A`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MATRIX_INV_UNIQUE THEN ONCE_REWRITE_TAC[MATRIX_MUL_ASSOC] THEN GEN_REWRITE_TAC (BINOP_CONV o LAND_CONV o LAND_CONV) [GSYM MATRIX_MUL_ASSOC] THEN ASM_SIMP_TAC[MATRIX_MUL_LINV; DET_EQ_0; MATRIX_MUL_RID; MATRIX_MUL_RINV]);; let DET_SIMILAR = prove (`!S:real^N^N A. invertible S ==> det(matrix_inv S ** A ** S) = det A`, REWRITE_TAC[INVERTIBLE_DET_NZ; DET_MUL; DET_MATRIX_INV] THEN CONV_TAC REAL_FIELD);; let INVERTIBLE_NEARBY_ONORM = prove (`!A B:real^N^N. invertible A /\ onorm(\x. (B - A) ** x) < inv(onorm(\x. matrix_inv A ** x)) ==> invertible B`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM ONORM_NEG] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_LNEG; MATRIX_NEG_SUB] THEN DISCH_TAC THEN ABBREV_TAC `S = matrix_inv(A:real^N^N) ** (A - B)` THEN SUBGOAL_THEN `B = (A:real^N^N) ** (mat 1 - S:real^N^N)` SUBST1_TAC THENL [EXPAND_TAC "S" THEN REWRITE_TAC[MATRIX_SUB_LDISTRIB; MATRIX_MUL_ASSOC] THEN ASM_SIMP_TAC[MATRIX_INV; MATRIX_MUL_RID; MATRIX_MUL_LID] THEN REWRITE_TAC[MATRIX_SUB; MATRIX_NEG_ADD] THEN REWRITE_TAC[MATRIX_ADD_RNEG; MATRIX_ADD_ASSOC; MATRIX_ADD_LID] THEN REWRITE_TAC[MATRIX_NEG_NEG]; ASM_REWRITE_TAC[INVERTIBLE_MATRIX_MUL]] THEN REWRITE_TAC[INVERTIBLE_LEFT_INVERSE; MATRIX_LEFT_INVERTIBLE_KER] THEN X_GEN_TAC `x:real^N` THEN REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_RDISTRIB; VECTOR_SUB_EQ] THEN CONV_TAC(LAND_CONV SYM_CONV) THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LID] THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN MP_TAC(ISPECL [`\x:real^N. matrix_inv(A:real^N^N) ** x`; `\x:real^N. (A - B:real^N^N) ** x`] ONORM_COMPOSE) THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_LINEAR; o_DEF; MATRIX_VECTOR_MUL_ASSOC] THEN REWRITE_TAC[REAL_NOT_LE] THEN TRANS_TAC REAL_LTE_TRANS `&1` THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN W(MP_TAC o PART_MATCH (rand o rand) REAL_LT_RDIV_EQ o snd) THEN ASM_REWRITE_TAC[real_div; REAL_MUL_LID] THEN DISCH_THEN MATCH_MP_TAC THEN SIMP_TAC[ONORM_POS_LT; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[GSYM MATRIX_EQ_0; MATRIX_INV_EQ_0] THEN ASM_MESON_TAC[INVERTIBLE_MAT]; MP_TAC(ISPEC `\x:real^N. (S:real^N^N) ** x` ONORM) THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N` o CONJUNCT1) THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM REAL_MUL_LID] THEN ASM_SIMP_TAC[REAL_LE_RMUL_EQ; NORM_POS_LT]]);; let INVERTIBLE_NEARBY = prove (`!A:real^N^N. invertible A ==> ?e. &0 < e /\ !B. onorm(\x. (B - A) ** x) < e ==> invertible B`, REPEAT STRIP_TAC THEN EXISTS_TAC `inv(onorm(\x. matrix_inv(A:real^N^N) ** x))` THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[INVERTIBLE_NEARBY_ONORM]] THEN SIMP_TAC[REAL_LT_INV_EQ; ONORM_POS_LT; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[GSYM MATRIX_EQ_0; MATRIX_INV_EQ_0] THEN ASM_MESON_TAC[INVERTIBLE_MAT]);; (* ------------------------------------------------------------------------- *) (* Cramer's rule. *) (* ------------------------------------------------------------------------- *) let CRAMER_LEMMA_TRANSP = prove (`!A:real^N^N x:real^N. 1 <= k /\ k <= dimindex(:N) ==> det((lambda i. if i = k then vsum(1..dimindex(:N)) (\i. x$i % row i A) else row i A):real^N^N) = x$k * det A`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `1..dimindex(:N) = k INSERT ((1..dimindex(:N)) DELETE k)` SUBST1_TAC THENL [ASM_MESON_TAC[INSERT_DELETE; IN_NUMSEG]; ALL_TAC] THEN SIMP_TAC[VSUM_CLAUSES; FINITE_NUMSEG; FINITE_DELETE; IN_DELETE] THEN REWRITE_TAC[VECTOR_ARITH `(x:real^N)$k % row k (A:real^N^N) + s = (x$k - &1) % row k A + row k A + s`] THEN W(MP_TAC o PART_MATCH (lhs o rand) DET_ROW_ADD o lhand o snd) THEN ASM_SIMP_TAC[DET_ROW_MUL] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC(REAL_RING `d = d' /\ e = d' ==> (c - &1) * d + e = c * d'`) THEN CONJ_TAC THENL [AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA; row]; MATCH_MP_TAC DET_ROW_SPAN THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG; FINITE_DELETE; IN_DELETE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC(CONJUNCT1 SPAN_CLAUSES) THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[]]);; let CRAMER_LEMMA = prove (`!A:real^N^N x:real^N. 1 <= k /\ k <= dimindex(:N) ==> det((lambda i j. if j = k then (A**x)$i else A$i$j):real^N^N) = x$k * det(A)`, REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[MATRIX_MUL_VSUM] THEN FIRST_ASSUM(MP_TAC o SYM o SPECL [`transp(A:real^N^N)`; `x:real^N`] o MATCH_MP CRAMER_LEMMA_TRANSP) THEN REWRITE_TAC[DET_TRANSP] THEN DISCH_THEN SUBST1_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM DET_TRANSP] THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; transp; LAMBDA_BETA; MATRIX_MUL_VSUM; row; column; COND_COMPONENT; VECTOR_MUL_COMPONENT; VSUM_COMPONENT]);; let CRAMER = prove (`!A:real^N^N x b. ~(det(A) = &0) ==> (A ** x = b <=> x = lambda k. det((lambda i j. if j = k then b$i else A$i$j):real^N^N) / det(A))`, GEN_TAC THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(MESON[] `(?x. p(x)) /\ (!x. p(x) ==> x = a) ==> !x. p(x) <=> x = a`) THEN CONJ_TAC THENL [MP_TAC(SPEC `A:real^N^N` INVERTIBLE_DET_NZ) THEN ASM_MESON_TAC[invertible; MATRIX_VECTOR_MUL_ASSOC; MATRIX_VECTOR_MUL_LID]; GEN_TAC THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[CART_EQ; CRAMER_LEMMA; LAMBDA_BETA; REAL_FIELD `~(z = &0) ==> (x = y / z <=> x * z = y)`]]);; (* ------------------------------------------------------------------------- *) (* Variants of Cramer's rule for matrix-matrix multiplication. *) (* ------------------------------------------------------------------------- *) let CRAMER_MATRIX_LEFT = prove (`!A:real^N^N X:real^N^N B:real^N^N. ~(det A = &0) ==> (X ** A = B <=> X = lambda k l. det((lambda i j. if j = l then B$k$i else A$j$i):real^N^N) / det A)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[CART_EQ] THEN ASM_SIMP_TAC[MATRIX_MUL_COMPONENT; CRAMER; DET_TRANSP] THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPLICATE_TAC 2 (AP_TERM_TAC THEN ABS_TAC THEN AP_TERM_TAC) THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; transp]);; let CRAMER_MATRIX_RIGHT = prove (`!A:real^N^N X:real^N^N B:real^N^N. ~(det A = &0) ==> (A ** X = B <=> X = lambda k l. det((lambda i j. if j = k then B$i$l else A$i$j):real^N^N) / det A)`, REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM TRANSP_EQ] THEN REWRITE_TAC[MATRIX_TRANSP_MUL] THEN ASM_SIMP_TAC[CRAMER_MATRIX_LEFT; DET_TRANSP] THEN GEN_REWRITE_TAC LAND_CONV [GSYM TRANSP_EQ] THEN REWRITE_TAC[TRANSP_TRANSP] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; transp] THEN REPEAT(GEN_TAC THEN STRIP_TAC) THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; transp]);; let CRAMER_MATRIX_RIGHT_INVERSE = prove (`!A:real^N^N A':real^N^N. A ** A' = mat 1 <=> ~(det A = &0) /\ A' = lambda k l. det((lambda i j. if j = k then if i = l then &1 else &0 else A$i$j):real^N^N) / det A`, REPEAT GEN_TAC THEN ASM_CASES_TAC `det(A:real^N^N) = &0` THENL [ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o AP_TERM `det:real^N^N->real`) THEN ASM_REWRITE_TAC[DET_MUL; DET_I] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[CRAMER_MATRIX_RIGHT] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT(GEN_TAC THEN STRIP_TAC) THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA; mat]]);; let CRAMER_MATRIX_LEFT_INVERSE = prove (`!A:real^N^N A':real^N^N. A' ** A = mat 1 <=> ~(det A = &0) /\ A' = lambda k l. det((lambda i j. if j = l then if i = k then &1 else &0 else A$j$i):real^N^N) / det A`, REPEAT GEN_TAC THEN ASM_CASES_TAC `det(A:real^N^N) = &0` THENL [ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o AP_TERM `det:real^N^N->real`) THEN ASM_REWRITE_TAC[DET_MUL; DET_I] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[CRAMER_MATRIX_LEFT] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT(GEN_TAC THEN STRIP_TAC) THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA; mat] THEN MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* Cofactors and their relationship to inverse matrices. *) (* ------------------------------------------------------------------------- *) let cofactor = new_definition `(cofactor:real^N^N->real^N^N) A = lambda i j. det((lambda k l. if k = i /\ l = j then &1 else if k = i \/ l = j then &0 else A$k$l):real^N^N)`;; let COFACTOR_TRANSP = prove (`!A:real^N^N. cofactor(transp A) = transp(cofactor A)`, SIMP_TAC[cofactor; CART_EQ; LAMBDA_BETA; transp] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM DET_TRANSP] THEN AP_TERM_TAC THEN SIMP_TAC[cofactor; CART_EQ; LAMBDA_BETA; transp] THEN MESON_TAC[]);; let COFACTOR_COLUMN = prove (`!A:real^N^N. cofactor A = lambda i j. det((lambda k l. if l = j then if k = i then &1 else &0 else A$k$l):real^N^N)`, GEN_TAC THEN CONV_TAC SYM_CONV THEN SIMP_TAC[cofactor; CART_EQ; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN REWRITE_TAC[det] THEN MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN GEN_TAC THEN DISCH_TAC THEN AP_TERM_TAC THEN ASM_CASES_TAC `(p:num->num) i = j` THENL [MATCH_MP_TAC PRODUCT_EQ THEN X_GEN_TAC `k:num` THEN SIMP_TAC[IN_NUMSEG; LAMBDA_BETA] THEN STRIP_TAC THEN SUBGOAL_THEN `(p:num->num) k IN 1..dimindex(:N)` MP_TAC THENL [ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]; SIMP_TAC[LAMBDA_BETA; IN_NUMSEG] THEN STRIP_TAC] THEN ASM_CASES_TAC `(p:num->num) k = j` THEN ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; MATCH_MP_TAC(REAL_ARITH `s = &0 /\ t = &0 ==> s = t`) THEN ASM_SIMP_TAC[PRODUCT_EQ_0; FINITE_NUMSEG] THEN CONJ_TAC THEN EXISTS_TAC `inverse (p:num->num) j` THEN ASM_SIMP_TAC[IN_NUMSEG; LAMBDA_BETA] THEN (SUBGOAL_THEN `inverse(p:num->num) j IN 1..dimindex(:N)` MP_TAC THENL [ASM_MESON_TAC[PERMUTES_IN_IMAGE; PERMUTES_INVERSE; IN_NUMSEG]; SIMP_TAC[LAMBDA_BETA; IN_NUMSEG] THEN STRIP_TAC] THEN SUBGOAL_THEN `(p:num->num)(inverse p j) = j` SUBST1_TAC THENL [ASM_MESON_TAC[PERMUTES_INVERSES; IN_NUMSEG]; ASM_SIMP_TAC[LAMBDA_BETA] THEN ASM_MESON_TAC[PERMUTES_INVERSE_EQ]])]);; let COFACTOR_ROW = prove (`!A:real^N^N. cofactor A = lambda i j. det((lambda k l. if k = i then if l = j then &1 else &0 else A$k$l):real^N^N)`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM TRANSP_EQ] THEN REWRITE_TAC[GSYM COFACTOR_TRANSP] THEN SIMP_TAC[COFACTOR_COLUMN; CART_EQ; LAMBDA_BETA; transp] THEN REPEAT STRIP_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM DET_TRANSP] THEN AP_TERM_TAC THEN SIMP_TAC[cofactor; CART_EQ; LAMBDA_BETA; transp]);; let MATRIX_RIGHT_INVERSE_COFACTOR = prove (`!A:real^N^N A':real^N^N. A ** A' = mat 1 <=> ~(det A = &0) /\ A' = inv(det A) %% transp(cofactor A)`, REPEAT GEN_TAC THEN REWRITE_TAC[CRAMER_MATRIX_RIGHT_INVERSE] THEN ASM_CASES_TAC `det(A:real^N^N) = &0` THEN ASM_REWRITE_TAC[] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; MATRIX_CMUL_COMPONENT] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN X_GEN_TAC `l:num` THEN STRIP_TAC THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div] THEN AP_TERM_TAC THEN ASM_SIMP_TAC[transp; COFACTOR_COLUMN; LAMBDA_BETA] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA]);; let MATRIX_LEFT_INVERSE_COFACTOR = prove (`!A:real^N^N A':real^N^N. A' ** A = mat 1 <=> ~(det A = &0) /\ A' = inv(det A) %% transp(cofactor A)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[MATRIX_LEFT_RIGHT_INVERSE] THEN REWRITE_TAC[MATRIX_RIGHT_INVERSE_COFACTOR]);; let MATRIX_INV_COFACTOR = prove (`!A. ~(det A = &0) ==> matrix_inv A = inv(det A) %% transp(cofactor A)`, GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP MATRIX_MUL_LINV) THEN SIMP_TAC[MATRIX_LEFT_INVERSE_COFACTOR]);; let COFACTOR_MATRIX_INV = prove (`!A:real^N^N. ~(det A = &0) ==> cofactor A = det(A) %% transp(matrix_inv A)`, SIMP_TAC[MATRIX_INV_COFACTOR; TRANSP_MATRIX_CMUL; TRANSP_TRANSP] THEN SIMP_TAC[MATRIX_CMUL_ASSOC; REAL_MUL_RINV; MATRIX_CMUL_LID]);; let COFACTOR_I = prove (`cofactor(mat 1:real^N^N) = mat 1`, SIMP_TAC[COFACTOR_MATRIX_INV; DET_I; REAL_OF_NUM_EQ; ARITH_EQ] THEN REWRITE_TAC[MATRIX_INV_I; MATRIX_CMUL_LID; TRANSP_MAT]);; let DET_COFACTOR_EXPANSION = prove (`!A:real^N^N i. 1 <= i /\ i <= dimindex(:N) ==> det A = sum (1..dimindex(:N)) (\j. A$i$j * (cofactor A)$i$j)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[COFACTOR_COLUMN; LAMBDA_BETA; det] THEN REWRITE_TAC[GSYM SUM_LMUL] THEN W(MP_TAC o PART_MATCH (lhand o rand) SUM_SWAP o rand o snd) THEN ANTS_TAC THENL [SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG]; ALL_TAC] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN GEN_TAC THEN DISCH_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `a * s * p:real = s * a * p`] THEN REWRITE_TAC[SUM_LMUL] THEN AP_TERM_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum (1..dimindex (:N)) (\j. (A:real^N^N)$i$j * product (inverse p j INSERT ((1..dimindex(:N)) DELETE (inverse p j))) (\k. if k = inverse p j then if k = i then &1 else &0 else A$k$(p k)))` THEN CONJ_TAC THENL [SIMP_TAC[PRODUCT_CLAUSES; FINITE_DELETE; FINITE_PERMUTATIONS; FINITE_NUMSEG; IN_DELETE] THEN SUBGOAL_THEN `!j. inverse (p:num->num) j = i <=> j = p i` (fun th -> REWRITE_TAC[th]) THENL [ASM_MESON_TAC[PERMUTES_INVERSES; IN_NUMSEG]; ALL_TAC] THEN REWRITE_TAC[REAL_ARITH `x * (if p then &1 else &0) * y = if p then x * y else &0`] THEN SIMP_TAC[SUM_DELTA] THEN COND_CASES_TAC THENL [ALL_TAC; ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]] THEN SUBGOAL_THEN `1..dimindex(:N) = i INSERT ((1..dimindex(:N)) DELETE i)` (fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [th]) THENL [ASM_SIMP_TAC[IN_NUMSEG; SET_RULE `s = x INSERT (s DELETE x) <=> x IN s`]; SIMP_TAC[PRODUCT_CLAUSES; FINITE_DELETE; FINITE_NUMSEG; IN_DELETE] THEN AP_TERM_TAC THEN MATCH_MP_TAC(MESON[PRODUCT_EQ] `s = t /\ (!x. x IN t ==> f x = g x) ==> product s f = product t g`) THEN SIMP_TAC[IN_DELETE] THEN ASM_MESON_TAC[PERMUTES_INVERSES; IN_NUMSEG]]; MATCH_MP_TAC SUM_EQ_NUMSEG THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN MATCH_MP_TAC(MESON[PRODUCT_EQ] `s = t /\ (!x. x IN t ==> f x = g x) ==> product s f = product t g`) THEN CONJ_TAC THENL [REWRITE_TAC[SET_RULE `x INSERT (s DELETE x) = s <=> x IN s`] THEN ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG; PERMUTES_INVERSE]; X_GEN_TAC `k:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN SUBGOAL_THEN `(p:num->num) k IN 1..dimindex(:N)` MP_TAC THENL [ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_NUMSEG]; ALL_TAC] THEN SIMP_TAC[LAMBDA_BETA; IN_NUMSEG] THEN ASM_MESON_TAC[PERMUTES_INVERSES; IN_NUMSEG]]]);; let MATRIX_MUL_RIGHT_COFACTOR = prove (`!A:real^N^N. A ** transp(cofactor A) = det(A) %% mat 1`, GEN_TAC THEN SIMP_TAC[CART_EQ; MATRIX_CMUL_COMPONENT; mat; matrix_mul; LAMBDA_BETA; transp] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `i':num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[GSYM DET_COFACTOR_EXPANSION; REAL_MUL_RID] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `det((lambda k l. if k = i' then (A:real^N^N)$i$l else A$k$l):real^N^N)` THEN CONJ_TAC THENL [MP_TAC(GEN `A:real^N^N` (ISPECL [`A:real^N^N`; `i':num`] DET_COFACTOR_EXPANSION)) THEN ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `j:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN AP_TERM_TAC THEN ASM_SIMP_TAC[cofactor; LAMBDA_BETA] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN ASM_MESON_TAC[]; REWRITE_TAC[REAL_MUL_RZERO] THEN MATCH_MP_TAC DET_IDENTICAL_ROWS THEN MAP_EVERY EXISTS_TAC [`i:num`;` i':num`] THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA; row]]);; let MATRIX_MUL_LEFT_COFACTOR = prove (`!A:real^N^N. transp(cofactor A) ** A = det(A) %% mat 1`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM TRANSP_EQ] THEN REWRITE_TAC[MATRIX_TRANSP_MUL] THEN ONCE_REWRITE_TAC[GSYM COFACTOR_TRANSP] THEN REWRITE_TAC[MATRIX_MUL_RIGHT_COFACTOR; TRANSP_MATRIX_CMUL] THEN REWRITE_TAC[DET_TRANSP; TRANSP_MAT]);; let COFACTOR_CMUL = prove (`!A:real^N^N c. cofactor(c %% A) = c pow (dimindex(:N) - 1) %% cofactor A`, REPEAT GEN_TAC THEN SIMP_TAC[CART_EQ; cofactor; LAMBDA_BETA; MATRIX_CMUL_COMPONENT] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN REWRITE_TAC[det; GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `p:num->num` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `a * b * c:real = b * a * c`] THEN AP_TERM_TAC THEN SUBGOAL_THEN `1..dimindex (:N) = i INSERT ((1..dimindex (:N)) DELETE i)` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_INSERT; IN_NUMSEG; IN_DELETE] THEN ASM_ARITH_TAC; SIMP_TAC[PRODUCT_CLAUSES; FINITE_DELETE; FINITE_NUMSEG; IN_DELETE]] THEN SUBGOAL_THEN `1 <= (p:num->num) i /\ p i <= dimindex(:N)` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP PERMUTES_IMAGE) THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_NUMSEG] THEN ASM SET_TAC[]; ASM_SIMP_TAC[LAMBDA_BETA]] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO] THEN SUBGOAL_THEN `dimindex(:N) - 1 = CARD((1..dimindex(:N)) DELETE i)` SUBST1_TAC THENL [ASM_SIMP_TAC[CARD_DELETE; FINITE_NUMSEG; IN_NUMSEG; CARD_NUMSEG_1]; ASM_SIMP_TAC[REAL_MUL_LID; GSYM PRODUCT_CONST; FINITE_NUMSEG; FINITE_DELETE; GSYM PRODUCT_MUL]] THEN MATCH_MP_TAC PRODUCT_EQ THEN X_GEN_TAC `k:num` THEN REWRITE_TAC[IN_DELETE; IN_NUMSEG] THEN STRIP_TAC THEN SUBGOAL_THEN `1 <= (p:num->num) k /\ p k <= dimindex(:N)` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o MATCH_MP PERMUTES_IMAGE) THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_NUMSEG] THEN ASM SET_TAC[]; ASM_SIMP_TAC[LAMBDA_BETA] THEN REAL_ARITH_TAC]);; let COFACTOR_0 = prove (`cofactor(mat 0:real^N^N) = if dimindex(:N) = 1 then mat 1 else mat 0`, MP_TAC(ISPECL [`mat 1:real^N^N`; `&0`] COFACTOR_CMUL) THEN REWRITE_TAC[MATRIX_CMUL_LZERO; COFACTOR_I; REAL_POW_ZERO] THEN DISCH_THEN SUBST1_TAC THEN SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> (n - 1 = 0 <=> n = 1)`] THEN COND_CASES_TAC THEN REWRITE_TAC[MATRIX_CMUL_LZERO; MATRIX_CMUL_LID]);; (* ------------------------------------------------------------------------- *) (* Explicit formulas for low dimensions. *) (* ------------------------------------------------------------------------- *) let PRODUCT_1 = prove (`product(1..1) f = f(1)`, REWRITE_TAC[PRODUCT_SING_NUMSEG]);; let PRODUCT_2 = prove (`!t. product(1..2) t = t(1) * t(2)`, REWRITE_TAC[num_CONV `2`; PRODUCT_CLAUSES_NUMSEG] THEN REWRITE_TAC[PRODUCT_SING_NUMSEG; ARITH; REAL_MUL_ASSOC]);; let PRODUCT_3 = prove (`!t. product(1..3) t = t(1) * t(2) * t(3)`, REWRITE_TAC[num_CONV `3`; num_CONV `2`; PRODUCT_CLAUSES_NUMSEG] THEN REWRITE_TAC[PRODUCT_SING_NUMSEG; ARITH; REAL_MUL_ASSOC]);; let PRODUCT_4 = prove (`!t. product(1..4) t = t(1) * t(2) * t(3) * t(4)`, REWRITE_TAC[num_CONV `4`; num_CONV `3`; num_CONV `2`; PRODUCT_CLAUSES_NUMSEG] THEN REWRITE_TAC[PRODUCT_SING_NUMSEG; ARITH; REAL_MUL_ASSOC]);; let DET_1_GEN = prove (`!A:real^N^N. dimindex(:N) = 1 ==> det A = A$1$1`, REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[det; PERMUTES_SING; NUMSEG_SING] THEN REWRITE_TAC[SUM_SING; SET_RULE `{x | x = a} = {a}`; PRODUCT_SING] THEN REWRITE_TAC[SIGN_I; I_THM] THEN REAL_ARITH_TAC);; let DET_1 = prove (`!A:real^1^1. det A = A$1$1`, SIMP_TAC[DET_1_GEN; DIMINDEX_1]);; let DET_2 = prove (`!A:real^2^2. det A = A$1$1 * A$2$2 - A$1$2 * A$2$1`, GEN_TAC THEN REWRITE_TAC[det; DIMINDEX_2] THEN CONV_TAC(LAND_CONV(RATOR_CONV(ONCE_DEPTH_CONV NUMSEG_CONV))) THEN SIMP_TAC[SUM_OVER_PERMUTATIONS_INSERT; FINITE_INSERT; FINITE_EMPTY; ARITH_EQ; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[PERMUTES_EMPTY; SUM_SING; SET_RULE `{x | x = a} = {a}`] THEN REWRITE_TAC[SWAP_REFL; I_O_ID] THEN REWRITE_TAC[GSYM(NUMSEG_CONV `1..2`); SUM_2] THEN SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY; ARITH_EQ; IN_INSERT; NOT_IN_EMPTY] THEN SIMP_TAC[SIGN_COMPOSE; PERMUTATION_SWAP] THEN REWRITE_TAC[SIGN_SWAP; ARITH] THEN REWRITE_TAC[PRODUCT_2] THEN REWRITE_TAC[o_THM; swap; ARITH] THEN REAL_ARITH_TAC);; let DET_3 = prove (`!A:real^3^3. det(A) = A$1$1 * A$2$2 * A$3$3 + A$1$2 * A$2$3 * A$3$1 + A$1$3 * A$2$1 * A$3$2 - A$1$1 * A$2$3 * A$3$2 - A$1$2 * A$2$1 * A$3$3 - A$1$3 * A$2$2 * A$3$1`, GEN_TAC THEN REWRITE_TAC[det; DIMINDEX_3] THEN CONV_TAC(LAND_CONV(RATOR_CONV(ONCE_DEPTH_CONV NUMSEG_CONV))) THEN SIMP_TAC[SUM_OVER_PERMUTATIONS_INSERT; FINITE_INSERT; FINITE_EMPTY; ARITH_EQ; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[PERMUTES_EMPTY; SUM_SING; SET_RULE `{x | x = a} = {a}`] THEN REWRITE_TAC[SWAP_REFL; I_O_ID] THEN REWRITE_TAC[GSYM(NUMSEG_CONV `1..3`); SUM_3] THEN SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY; ARITH_EQ; IN_INSERT; NOT_IN_EMPTY] THEN SIMP_TAC[SIGN_COMPOSE; PERMUTATION_SWAP] THEN REWRITE_TAC[SIGN_SWAP; ARITH] THEN REWRITE_TAC[PRODUCT_3] THEN REWRITE_TAC[o_THM; swap; ARITH] THEN REAL_ARITH_TAC);; let DET_4 = prove (`!A:real^4^4. det(A) = A$1$1 * A$2$2 * A$3$3 * A$4$4 + A$1$1 * A$2$3 * A$3$4 * A$4$2 + A$1$1 * A$2$4 * A$3$2 * A$4$3 + A$1$2 * A$2$1 * A$3$4 * A$4$3 + A$1$2 * A$2$3 * A$3$1 * A$4$4 + A$1$2 * A$2$4 * A$3$3 * A$4$1 + A$1$3 * A$2$1 * A$3$2 * A$4$4 + A$1$3 * A$2$2 * A$3$4 * A$4$1 + A$1$3 * A$2$4 * A$3$1 * A$4$2 + A$1$4 * A$2$1 * A$3$3 * A$4$2 + A$1$4 * A$2$2 * A$3$1 * A$4$3 + A$1$4 * A$2$3 * A$3$2 * A$4$1 - A$1$1 * A$2$2 * A$3$4 * A$4$3 - A$1$1 * A$2$3 * A$3$2 * A$4$4 - A$1$1 * A$2$4 * A$3$3 * A$4$2 - A$1$2 * A$2$1 * A$3$3 * A$4$4 - A$1$2 * A$2$3 * A$3$4 * A$4$1 - A$1$2 * A$2$4 * A$3$1 * A$4$3 - A$1$3 * A$2$1 * A$3$4 * A$4$2 - A$1$3 * A$2$2 * A$3$1 * A$4$4 - A$1$3 * A$2$4 * A$3$2 * A$4$1 - A$1$4 * A$2$1 * A$3$2 * A$4$3 - A$1$4 * A$2$2 * A$3$3 * A$4$1 - A$1$4 * A$2$3 * A$3$1 * A$4$2`, let lemma = prove (`(sum {3,4} f = f 3 + f 4) /\ (sum {2,3,4} f = f 2 + f 3 + f 4)`, SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN REWRITE_TAC[ARITH_EQ; IN_INSERT; NOT_IN_EMPTY] THEN REAL_ARITH_TAC) in GEN_TAC THEN REWRITE_TAC[det; DIMINDEX_4] THEN CONV_TAC(LAND_CONV(RATOR_CONV(ONCE_DEPTH_CONV NUMSEG_CONV))) THEN SIMP_TAC[SUM_OVER_PERMUTATIONS_INSERT; FINITE_INSERT; FINITE_EMPTY; ARITH_EQ; IN_INSERT; NOT_IN_EMPTY] THEN REWRITE_TAC[PERMUTES_EMPTY; SUM_SING; SET_RULE `{x | x = a} = {a}`] THEN REWRITE_TAC[SWAP_REFL; I_O_ID] THEN REWRITE_TAC[GSYM(NUMSEG_CONV `1..4`); SUM_4; lemma] THEN SIMP_TAC[SIGN_COMPOSE; PERMUTATION_SWAP; PERMUTATION_COMPOSE] THEN REWRITE_TAC[SIGN_SWAP; ARITH] THEN REWRITE_TAC[PRODUCT_4] THEN REWRITE_TAC[o_THM; swap; ARITH] THEN REAL_ARITH_TAC);; let COFACTOR_1_GEN = prove (`!A:real^N^N. dimindex(:N) = 1 ==> cofactor A = mat 1`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CART_EQ; mat; cofactor; LAMBDA_BETA; DET_1_GEN; ARITH] THEN REWRITE_TAC[LE_ANTISYM] THEN MESON_TAC[]);; let COFACTOR_1 = prove (`!A:real^1^1. cofactor A = mat 1`, SIMP_TAC[COFACTOR_1_GEN; DIMINDEX_1]);; (* ------------------------------------------------------------------------- *) (* Disjoint or subset-related halfspaces and hyperplanes are parallel. *) (* ------------------------------------------------------------------------- *) let DISJOINT_HYPERPLANES_IMP_COLLINEAR = prove (`!a b:real^N c d. DISJOINT {x | a dot x = c} {x | b dot x = d} ==> collinear {vec 0, a, b}`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `DISJOINT {x:real^N | a dot x = c} {x | b dot x = d} ==> !u v. a dot (u % a + v % b) = c /\ b dot (u % a + v % b) = d ==> F`)) THEN REWRITE_TAC[DOT_RADD; DOT_RMUL] THEN GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN MP_TAC(ISPECL [`vector[vector[(a:real^N) dot a; a dot b]; vector[a dot b; b dot b]]:real^2^2`; `vector[c;d]:real^2`] MATRIX_FULL_LINEAR_EQUATIONS) THEN REWRITE_TAC[RANK_EQ_FULL_DET] THEN SIMP_TAC[CART_EQ; DIMINDEX_2; MATRIX_VECTOR_MUL_COMPONENT; ARITH; VECTOR_2; FORALL_2; DOT_2; EXISTS_VECTOR_2; DET_2] THEN MATCH_MP_TAC MONO_IMP THEN CONJ_TAC THENL [REWRITE_TAC[CONTRAPOS_THM]; MESON_TAC[DOT_SYM; REAL_MUL_SYM]] THEN REWRITE_TAC[REAL_ARITH `a - b * b = &0 <=> b pow 2 = a`] THEN REWRITE_TAC[DOT_CAUCHY_SCHWARZ_EQUAL]);; let DISJOINT_HALFSPACES_IMP_COLLINEAR = prove (`(!a b:real^N c d. DISJOINT {x | a dot x < c} {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x < c} {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x < c} {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x < c} {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x < c} {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x <= c} {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x <= c} {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x <= c} {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x <= c} {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x <= c} {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x = c} {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x = c} {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x = c} {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x = c} {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x = c} {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x >= c} {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x >= c} {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x >= c} {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x >= c} {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x >= c} {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x > c} {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x > c} {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x > c} {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x > c} {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. DISJOINT {x | a dot x > c} {x | b dot x > d} ==> collinear {vec 0, a, b})`, let lemma = prove (`(!a b:real^N. collinear {vec 0,--a,b} <=> collinear{vec 0,a,b}) /\ (!a b:real^N. collinear {vec 0,a,--b} <=> collinear{vec 0,a,b})`, REWRITE_TAC[COLLINEAR_LEMMA_ALT; VECTOR_NEG_EQ_0] THEN REWRITE_TAC[VECTOR_ARITH `b:real^N = c % --a <=> b = --c % a`; VECTOR_ARITH `--b:real^N = c % a <=> b = --c % a`] THEN REWRITE_TAC[MESON[REAL_NEG_NEG] `(?x:real. P(--x)) <=> ?x. P x`]) in REWRITE_TAC[REAL_ARITH `x >= d <=> --x <= --d`; REAL_ARITH `x > d <=> --x < --d`] THEN REWRITE_TAC[GSYM DOT_LNEG] THEN REPEAT STRIP_TAC THEN REPLICATE_TAC 2 (TRY(FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `DISJOINT {x | a dot x <= b} t ==> (!x y. x < y ==> x <= y) ==> DISJOINT {x | a dot x < b} t`)) THEN REWRITE_TAC[REAL_LT_IMP_LE] THEN DISCH_TAC) THEN RULE_ASSUM_TAC(ONCE_REWRITE_RULE[DISJOINT_SYM])) THEN REPLICATE_TAC 2 (TRY(FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE `DISJOINT {x | a dot x < b} t ==> b - &1 < b ==> DISJOINT {x | a dot x = b - &1} t`)) THEN REWRITE_TAC[ARITH_RULE `c - &1 < c`] THEN DISCH_TAC) THEN RULE_ASSUM_TAC(ONCE_REWRITE_RULE[DISJOINT_SYM])) THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP DISJOINT_HYPERPLANES_IMP_COLLINEAR) THEN REWRITE_TAC[lemma]);; let SUBSET_HALFSPACES_IMP_COLLINEAR = prove (`(!a b:real^N c d. {x | a dot x < c} SUBSET {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x < c} SUBSET {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x < c} SUBSET {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x < c} SUBSET {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x < c} SUBSET {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x <= c} SUBSET {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x <= c} SUBSET {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x <= c} SUBSET {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x <= c} SUBSET {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x <= c} SUBSET {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x = c} SUBSET {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x = c} SUBSET {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x = c} SUBSET {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x = c} SUBSET {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x = c} SUBSET {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x >= c} SUBSET {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x >= c} SUBSET {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x >= c} SUBSET {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x >= c} SUBSET {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x >= c} SUBSET {x | b dot x > d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x > c} SUBSET {x | b dot x < d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x > c} SUBSET {x | b dot x <= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x > c} SUBSET {x | b dot x = d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x > c} SUBSET {x | b dot x >= d} ==> collinear {vec 0, a, b}) /\ (!a b:real^N c d. {x | a dot x > c} SUBSET {x | b dot x > d} ==> collinear {vec 0, a, b})`, REWRITE_TAC[SET_RULE `s SUBSET {x | P x} <=> DISJOINT s {x | ~P x}`] THEN REWRITE_TAC[REAL_ARITH `(~(x < a) <=> x >= a) /\ (~(x <= a) <=> x > a) /\ (~(x = a) <=> x > a \/ x < a) /\ (~(x > a) <=> x <= a) /\ (~(x >= a) <=> x < a)`] THEN REWRITE_TAC[SET_RULE `DISJOINT s {x | P x \/ Q x} <=> DISJOINT s {x | P x} /\ DISJOINT s {x | Q x}`] THEN REPEAT CONJ_TAC THEN REPEAT GEN_TAC THEN TRY(DISCH_THEN(MP_TAC o CONJUNCT1)) THEN REWRITE_TAC[DISJOINT_HALFSPACES_IMP_COLLINEAR]);; let SUBSET_HYPERPLANES = prove (`!a b a' b'. {x | a dot x = b} SUBSET {x | a' dot x = b'} <=> {x | a dot x = b} = {} \/ {x | a' dot x = b'} = (:real^N) \/ {x | a dot x = b} = {x | a' dot x = b'}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `{x:real^N | a dot x = b} = {}` THEN ASM_REWRITE_TAC[EMPTY_SUBSET] THEN ASM_CASES_TAC `{x | a' dot x = b'} = (:real^N)` THEN ASM_REWRITE_TAC[SUBSET_UNIV] THEN RULE_ASSUM_TAC(REWRITE_RULE [HYPERPLANE_EQ_EMPTY; HYPERPLANE_EQ_UNIV]) THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN ASM_CASES_TAC `{x:real^N | a dot x = b} SUBSET {x | a' dot x = b'}` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`a:real^N`; `a':real^N`; `b:real`; `b':real`] (el 12 (CONJUNCTS SUBSET_HALFSPACES_IMP_COLLINEAR))) THEN ASM_REWRITE_TAC[COLLINEAR_LEMMA_ALT] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_SIMP_TAC[DOT_LZERO] THENL [SET_TAC[]; STRIP_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `c:real` SUBST_ALL_TAC) THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN ASM_CASES_TAC `c % a:real^N = vec 0` THEN ASM_SIMP_TAC[DOT_LZERO] THENL [SET_TAC[]; POP_ASSUM MP_TAC] THEN SIMP_TAC[VECTOR_MUL_EQ_0; DE_MORGAN_THM; DOT_LMUL; REAL_FIELD `~(c = &0) ==> (c * a = b <=> a = b / c)`] THEN STRIP_TAC THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN DISCH_THEN(MP_TAC o SPEC `(b / (a dot a)) % a:real^N`) THEN ASM_SIMP_TAC[DOT_RMUL; REAL_DIV_RMUL; DOT_EQ_0]);; (* ------------------------------------------------------------------------- *) (* Existence of the characteristic polynomial. *) (* ------------------------------------------------------------------------- *) let EIGENVALUES_CHARACTERISTIC_ALT = prove (`!A:real^N^N c. (?v. ~(v = vec 0) /\ A ** v = c % v) <=> det(A - c %% mat 1) = &0`, REWRITE_TAC[GSYM HOMOGENEOUS_LINEAR_EQUATIONS_DET] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN REWRITE_TAC[MATRIX_VECTOR_LMUL; VECTOR_SUB_EQ; MATRIX_VECTOR_MUL_LID]);; let EIGENVALUES_CHARACTERISTIC = prove (`!A:real^N^N c. (?v. ~(v = vec 0) /\ A ** v = c % v) <=> det(c %% mat 1 - A) = &0`, ONCE_REWRITE_TAC[GSYM MATRIX_NEG_SUB] THEN ASM_REWRITE_TAC[EIGENVALUES_CHARACTERISTIC_ALT; DET_NEG] THEN REWRITE_TAC[REAL_ENTIRE; REAL_POW_EQ_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let INVERTIBLE_EIGENVALUES = prove (`!A:real^N^N. invertible(A) <=> !c v. A ** v = c % v /\ ~(v = vec 0) ==> ~(c = &0)`, GEN_TAC THEN REWRITE_TAC[LEFT_FORALL_IMP_THM] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[EIGENVALUES_CHARACTERISTIC_ALT; INVERTIBLE_DET_NZ] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[FORALL_UNWIND_THM2; MATRIX_CMUL_LZERO; MATRIX_SUB_RZERO]);; let CHARACTERISTIC_POLYNOMIAL = prove (`!A:real^N^N. ?a. a(dimindex(:N)) = &1 /\ !x. det(x %% mat 1 - A) = sum (0..dimindex(:N)) (\i. a i * x pow i)`, GEN_TAC THEN REWRITE_TAC[det] THEN SUBGOAL_THEN `!p n. IMAGE p (1..dimindex(:N)) SUBSET 1..dimindex(:N) /\ n <= dimindex(:N) ==> ?a. a n = (if !i. 1 <= i /\ i <= n ==> p i = i then &1 else &0) /\ !x. product (1..n) (\i. (x %% mat 1 - A:real^N^N)$i$p i) = sum (0..n) (\i. a i * x pow i)` MP_TAC THENL [GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN INDUCT_TAC THEN REWRITE_TAC[PRODUCT_CLAUSES_NUMSEG] THEN REWRITE_TAC[LE_0; ARITH_EQ; ARITH_RULE `1 <= SUC n`] THENL [EXISTS_TAC `\i. if i = 0 then &1 else &0` THEN SIMP_TAC[real_pow; REAL_MUL_LID; ARITH_RULE `1 <= i ==> ~(i <= 0)`; SUM_CLAUSES_NUMSEG]; DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN ASM_SIMP_TAC[ARITH_RULE `SUC n <= N ==> n <= N`] THEN DISCH_THEN(X_CHOOSE_THEN `a:num->real` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[MATRIX_SUB_COMPONENT; MATRIX_CMUL_COMPONENT] THEN ASSUME_TAC(ARITH_RULE `1 <= SUC n`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN DISCH_THEN(MP_TAC o SPEC `SUC n`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_SIMP_TAC[MAT_COMPONENT] THEN ASM_CASES_TAC `p(SUC n) = SUC n` THEN ASM_REWRITE_TAC[] THENL [ALL_TAC; EXISTS_TAC `\i. if i <= n then --((A:real^N^N)$(SUC n)$(p(SUC n))) * a i else &0` THEN SIMP_TAC[SUM_CLAUSES_NUMSEG; LE_0; ARITH_RULE `~(SUC n <= n)`] THEN CONJ_TAC THENL [COND_CASES_TAC THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `SUC n`) THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_RID; GSYM SUM_RMUL] THEN GEN_TAC THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN REWRITE_TAC[] THEN REAL_ARITH_TAC]] THEN REWRITE_TAC[REAL_SUB_LDISTRIB; REAL_MUL_RID] THEN REWRITE_TAC[GSYM SUM_RMUL] THEN EXISTS_TAC `\i. (if i = 0 then &0 else a(i - 1)) - (if i = SUC n then &0 else (A:real^N^N)$(SUC n)$(SUC n) * a i)` THEN ASM_REWRITE_TAC[NOT_SUC; LE; SUC_SUB1; REAL_SUB_RZERO] THEN CONJ_TAC THENL [ASM_MESON_TAC[LE_REFL]; ALL_TAC] THEN REWRITE_TAC[REAL_SUB_RDISTRIB; SUM_SUB_NUMSEG] THEN GEN_TAC THEN BINOP_TAC THENL [SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE `0 <= SUC n`] THEN REWRITE_TAC[ADD1; SUM_OFFSET; ARITH_RULE `~(i + 1 = 0)`; ADD_SUB] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_POW_ADD; REAL_POW_1; REAL_ADD_LID]; SIMP_TAC[SUM_CLAUSES_NUMSEG; LE_0; REAL_MUL_LZERO; REAL_ADD_RID] THEN SIMP_TAC[ARITH_RULE `i <= n ==> ~(i = SUC n)`]] THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN REWRITE_TAC[REAL_ADD_LID; REAL_MUL_AC]]; GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `dimindex(:N)`) THEN REWRITE_TAC[LE_REFL] 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 `a:(num->num)->num->real` THEN DISCH_TAC] THEN EXISTS_TAC `\i:num. sum {p | p permutes 1..dimindex(:N)} (\p. sign p * a p i)` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MP_TAC(ISPECL [`\p:num->num. sign p * a p (dimindex(:N))`; `{p | p permutes 1..dimindex(:N)}`; `I:num->num`] SUM_DELETE) THEN SIMP_TAC[IN_ELIM_THM; PERMUTES_I; FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN MATCH_MP_TAC(REAL_ARITH `k = &1 /\ s' = &0 ==> s' = s - k ==> s = &1`) THEN CONJ_TAC THENL [FIRST_X_ASSUM(MP_TAC o SPEC `I:num->num`) THEN SIMP_TAC[IMAGE_I; SUBSET_REFL; SIGN_I; I_THM; REAL_MUL_LID]; MATCH_MP_TAC SUM_EQ_0 THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM; IN_DELETE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `p:num->num`) THEN ANTS_TAC THENL [ASM_MESON_TAC[PERMUTES_IMAGE; SUBSET_REFL]; ALL_TAC] THEN COND_CASES_TAC THEN SIMP_TAC[REAL_MUL_RZERO] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [permutes]) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [FUN_EQ_THM]) THEN REWRITE_TAC[IN_NUMSEG; I_THM] THEN ASM_MESON_TAC[]]; X_GEN_TAC `x:real` THEN REWRITE_TAC[GSYM SUM_RMUL] THEN W(MP_TAC o PART_MATCH (lhs o rand) SUM_SWAP o rand o snd) THEN SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC; SUM_LMUL] THEN AP_TERM_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `p:num->num`) THEN ANTS_TAC THENL [ASM_MESON_TAC[PERMUTES_IMAGE; SUBSET_REFL]; SIMP_TAC[]]]);; let FINITE_EIGENVALUES = prove (`!A:real^N^N. FINITE {c | ?v. ~(v = vec 0) /\ A ** v = c % v}`, GEN_TAC THEN REWRITE_TAC[EIGENVALUES_CHARACTERISTIC] THEN MP_TAC(ISPEC `A:real^N^N` CHARACTERISTIC_POLYNOMIAL) THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_POLYFUN_FINITE_ROOTS] THEN EXISTS_TAC `dimindex(:N)` THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_0; LE_REFL] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Grassmann-Plucker relations for n = 2, n = 3 and n = 4. *) (* I have a proof of the general n case but the proof is a bit long and the *) (* result doesn't seem generally useful enough to go in the main theories. *) (* ------------------------------------------------------------------------- *) let GRASSMANN_PLUCKER_2 = prove (`!x1 x2 y1 y2:real^2. det(vector[x1;x2]) * det(vector[y1;y2]) = det(vector[y1;x2]) * det(vector[x1;y2]) + det(vector[y2;x2]) * det(vector[y1;x1])`, REWRITE_TAC[DET_2; VECTOR_2] THEN REAL_ARITH_TAC);; let GRASSMANN_PLUCKER_3 = prove (`!x1 x2 x3 y1 y2 y3:real^3. det(vector[x1;x2;x3]) * det(vector[y1;y2;y3]) = det(vector[y1;x2;x3]) * det(vector[x1;y2;y3]) + det(vector[y2;x2;x3]) * det(vector[y1;x1;y3]) + det(vector[y3;x2;x3]) * det(vector[y1;y2;x1])`, REWRITE_TAC[DET_3; VECTOR_3] THEN REAL_ARITH_TAC);; let GRASSMANN_PLUCKER_4 = prove (`!x1 x2 x3 x4:real^4 y1 y2 y3 y4:real^4. det(vector[x1;x2;x3;x4]) * det(vector[y1;y2;y3;y4]) = det(vector[y1;x2;x3;x4]) * det(vector[x1;y2;y3;y4]) + det(vector[y2;x2;x3;x4]) * det(vector[y1;x1;y3;y4]) + det(vector[y3;x2;x3;x4]) * det(vector[y1;y2;x1;y4]) + det(vector[y4;x2;x3;x4]) * det(vector[y1;y2;y3;x1])`, REWRITE_TAC[DET_4; VECTOR_4] THEN REAL_ARITH_TAC);; (* ------------------------------------------------------------------------- *) (* Determinants of integer matrices. *) (* ------------------------------------------------------------------------- *) let INTEGER_PRODUCT = prove (`!f s. (!x. x IN s ==> integer(f x)) ==> integer(product s f)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC PRODUCT_CLOSED THEN ASM_REWRITE_TAC[INTEGER_CLOSED]);; let INTEGER_SIGN = prove (`!p. integer(sign p)`, SIMP_TAC[sign; COND_RAND; INTEGER_CLOSED; COND_ID]);; let INTEGER_DET = prove (`!M:real^N^N. (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> integer(M$i$j)) ==> integer(det M)`, REPEAT STRIP_TAC THEN REWRITE_TAC[det] THEN MATCH_MP_TAC INTEGER_SUM THEN X_GEN_TAC `p:num->num` THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN MATCH_MP_TAC INTEGER_MUL THEN REWRITE_TAC[INTEGER_SIGN] THEN MATCH_MP_TAC INTEGER_PRODUCT THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[IN_NUMSEG; permutes]);; (* ------------------------------------------------------------------------- *) (* Diagonal matrices (for arbitrary rectangular matrix, not just square). *) (* ------------------------------------------------------------------------- *) let diagonal_matrix = new_definition `diagonal_matrix(A:real^N^M) <=> !i j. 1 <= i /\ i <= dimindex(:M) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> A$i$j = &0`;; let DIAGONAL_MATRIX = prove (`!A:real^N^N. diagonal_matrix A <=> A = (lambda i j. if i = j then A$i$j else &0)`, SIMP_TAC[CART_EQ; LAMBDA_BETA; diagonal_matrix] THEN MESON_TAC[]);; let DIAGONAL_MATRIX_MAT = prove (`!m. diagonal_matrix(mat m:real^N^N)`, SIMP_TAC[mat; diagonal_matrix; LAMBDA_BETA]);; let TRANSP_DIAGONAL_MATRIX = prove (`!A:real^N^N. diagonal_matrix A ==> transp A = A`, GEN_TAC THEN REWRITE_TAC[diagonal_matrix; CART_EQ; TRANSP_COMPONENT] THEN STRIP_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = j` THEN ASM_SIMP_TAC[]);; let DIAGONAL_IMP_SYMMETRIC_MATRIX = prove (`!A:real^N^N. diagonal_matrix A ==> symmetric_matrix A`, REWRITE_TAC[symmetric_matrix; TRANSP_DIAGONAL_MATRIX]);; let DIAGONAL_MATRIX_ADD = prove (`!A B:real^N^M. diagonal_matrix A /\ diagonal_matrix B ==> diagonal_matrix(A + B)`, SIMP_TAC[diagonal_matrix; MATRIX_ADD_COMPONENT; REAL_ADD_LID; REAL_ADD_RID]);; let DIAGONAL_MATRIX_CMUL = prove (`!A:real^N^M c. diagonal_matrix A ==> diagonal_matrix(c %% A)`, SIMP_TAC[diagonal_matrix; MATRIX_CMUL_COMPONENT; REAL_MUL_RZERO]);; let MATRIX_MUL_DIAGONAL = prove (`!A:real^N^N B:real^N^N. diagonal_matrix A /\ diagonal_matrix B ==> A ** B = lambda i j. A$i$j * B$i$j`, REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I [DIAGONAL_MATRIX])) THEN SIMP_TAC[CART_EQ; matrix_mul; LAMBDA_BETA] THEN ONCE_REWRITE_TAC[MESON[REAL_MUL_LZERO; REAL_MUL_RZERO] `(if p then a else &0) * (if q then b else &0) = if q then (if p then a * b else &0) else &0`] THEN SIMP_TAC[SUM_DELTA; IN_NUMSEG; COND_ID; SUM_0]);; let DIAGONAL_MATRIX_MUL_COMPONENT = prove (`!A:real^N^N B:real^N^N i j. diagonal_matrix A /\ diagonal_matrix B /\ 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> (A ** B)$i$j = A$i$j * B$i$j`, ASM_SIMP_TAC[MATRIX_MUL_DIAGONAL; LAMBDA_BETA]);; let DIAGONAL_MATRIX_MUL = prove (`!A:real^N^N B:real^N^N. diagonal_matrix A /\ diagonal_matrix B ==> diagonal_matrix(A ** B)`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [diagonal_matrix] THEN SIMP_TAC[DIAGONAL_MATRIX_MUL_COMPONENT] THEN SIMP_TAC[diagonal_matrix; REAL_MUL_LZERO]);; let DIAGONAL_MATRIX_MUL_EQ = prove (`!A:real^M^N B:real^N^M. diagonal_matrix (A ** B) <=> pairwise (\i j. orthogonal (row i A) (column j B)) (1..dimindex(:N))`, REWRITE_TAC[diagonal_matrix; matrix_mul; pairwise] THEN SIMP_TAC[LAMBDA_BETA; IN_NUMSEG; orthogonal; dot; row; column] THEN REWRITE_TAC[GSYM CONJ_ASSOC]);; let DIAGONAL_MATRIX_INV_EXPLICIT = prove (`!A:real^N^N. diagonal_matrix A ==> matrix_inv A = lambda i j. inv(A$i$j)`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MATRIX_INV_UNIQUE_STRONG THEN REWRITE_TAC[symmetric_matrix] THEN SUBGOAL_THEN `diagonal_matrix((lambda i j. inv((A:real^N^N)$i$j)):real^N^N)` ASSUME_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[diagonal_matrix]) THEN ASM_SIMP_TAC[diagonal_matrix; LAMBDA_BETA; REAL_INV_0]; ASM_SIMP_TAC[DIAGONAL_MATRIX_MUL_COMPONENT; CART_EQ; LAMBDA_BETA; TRANSP_COMPONENT; DIAGONAL_MATRIX_MUL]] THEN MP_TAC(ISPEC `A:real^N^N` DIAGONAL_MATRIX) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN SIMP_TAC[LAMBDA_BETA] THEN REPEAT CONJ_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_INV_0; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN REWRITE_TAC[REAL_INV_EQ_0; REAL_RING `a * b * a = a <=> b * a = &1 \/ a = &0`] THEN CONV_TAC REAL_FIELD);; let DIAGONAL_MATRIX_INV_COMPONENT = prove (`!A:real^N^N i j. diagonal_matrix A /\ 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> (matrix_inv A)$i$j = inv(A$i$j)`, ASM_SIMP_TAC[DIAGONAL_MATRIX_INV_EXPLICIT; LAMBDA_BETA]);; let DIAGONAL_MATRIX_INV = prove (`!A:real^N^N. diagonal_matrix(matrix_inv A) <=> diagonal_matrix A`, SUBGOAL_THEN `!A:real^N^N. diagonal_matrix A ==> diagonal_matrix(matrix_inv A)` MP_TAC THENL [REPEAT STRIP_TAC; MESON_TAC[MATRIX_INV_INV]] THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP DIAGONAL_MATRIX_INV_EXPLICIT) THEN POP_ASSUM MP_TAC THEN SIMP_TAC[diagonal_matrix; LAMBDA_BETA] THEN REWRITE_TAC[REAL_INV_0]);; let DET_DIAGONAL = prove (`!A:real^N^N. diagonal_matrix A ==> det(A) = product(1..dimindex(:N)) (\i. A$i$i)`, REWRITE_TAC[diagonal_matrix] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC DET_LOWERTRIANGULAR THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[LT_REFL]);; let INVERTIBLE_DIAGONAL_MATRIX = prove (`!D:real^N^N. diagonal_matrix D ==> (invertible D <=> !i. 1 <= i /\ i <= dimindex(:N) ==> ~(D$i$i = &0))`, SIMP_TAC[INVERTIBLE_DET_NZ; DET_DIAGONAL] THEN SIMP_TAC[PRODUCT_EQ_0; FINITE_NUMSEG; IN_NUMSEG] THEN MESON_TAC[]);; let COMMUTING_WITH_DIAGONAL_MATRIX = prove (`!A D:real^N^N. diagonal_matrix D ==> (A ** D = D ** A <=> !i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> A$i$j = &0 \/ D$i$i = D$j$j)`, REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o REWRITE_RULE[DIAGONAL_MATRIX]) THEN SIMP_TAC[CART_EQ; matrix_mul; LAMBDA_BETA] THEN REWRITE_TAC[MESON[REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_MUL_SYM] `(if a = b then x else &0) * y = (if b = a then x * y else &0) /\ y * (if a = b then x else &0) = (if a = b then x * y else &0)`] THEN SIMP_TAC[SUM_DELTA; IN_NUMSEG; REAL_EQ_MUL_RCANCEL] THEN MESON_TAC[]);; let RANK_DIAGONAL_MATRIX = prove (`!A:real^N^N. diagonal_matrix A ==> rank A = CARD {i | i IN 1..dimindex(:N) /\ ~(A$i$i = &0)}`, REPEAT STRIP_TAC THEN REWRITE_TAC[RANK_DIM_IM; GSYM SPAN_STDBASIS] THEN SIMP_TAC[GSYM SPAN_LINEAR_IMAGE; MATRIX_VECTOR_MUL_LINEAR; DIM_SPAN] THEN REWRITE_TAC[GSYM IN_NUMSEG; SIMPLE_IMAGE; GSYM IMAGE_o; o_DEF] THEN TRANS_TAC EQ_TRANS `dim {(A:real^N^N)$i$i % basis i:real^N | i IN 1..dimindex(:N)}` THEN CONJ_TAC THENL [AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. x IN s ==> f x = g x) ==> IMAGE f s = {g x | x IN s}`) THEN FIRST_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I [DIAGONAL_MATRIX]) THEN SIMP_TAC[matrix_vector_mul; LAMBDA_BETA; IN_NUMSEG; CART_EQ] THEN ONCE_REWRITE_TAC[MESON[REAL_MUL_LZERO] `(if i = j then a else &0) * b = if j = i then a * b else &0`] THEN SIMP_TAC[SUM_DELTA; IN_NUMSEG; BASIS_COMPONENT; VECTOR_MUL_COMPONENT] THEN MESON_TAC[REAL_MUL_RZERO]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `dim {(A:real^N^N)$i$i % basis i:real^N |i| i IN 1..dimindex(:N) /\ ~(A$i$i = &0)}` THEN CONJ_TAC THENL [MATCH_MP_TAC(MESON[DIM_INSERT_0] `(vec 0:real^N) INSERT s = (vec 0:real^N) INSERT t ==> dim s = dim t`) THEN MATCH_MP_TAC(SET_RULE `t SUBSET s /\ (!x. x IN s ==> ~(x IN t) ==> x = a) ==> a INSERT s = a INSERT t`) THEN CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[FORALL_IN_GSPEC]] THEN SIMP_TAC[VECTOR_MUL_EQ_0; IN_ELIM_THM; BASIS_NONZERO; IN_NUMSEG] THEN SET_TAC[]; ALL_TAC] THEN TRANS_TAC EQ_TRANS `dim{basis i:real^N | i IN 1..dimindex(:N) /\ ~((A:real^N^N)$i$i = &0)}` THEN CONJ_TAC THENL [MATCH_MP_TAC SPAN_EQ_DIM THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN CONJ_TAC THEN MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN REWRITE_TAC[SUBSPACE_SPAN] THEN REWRITE_TAC[FORALL_IN_GSPEC; SUBSET; IN_NUMSEG] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THENL [ALL_TAC; SUBGOAL_THEN `basis i:real^N = inv((A:real^N^N)$i$i) % A$i$i % basis i` (fun th -> GEN_REWRITE_TAC LAND_CONV [th]) THENL [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID]; ALL_TAC]] THEN MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhs o rand) DIM_EQ_CARD o lhand o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC(MATCH_MP (REWRITE_RULE[IMP_CONJ] INDEPENDENT_MONO) INDEPENDENT_STDBASIS) THEN REWRITE_TAC[IN_NUMSEG] THEN SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SIMPLE_IMAGE_GEN] THEN MATCH_MP_TAC CARD_IMAGE_INJ THEN SIMP_TAC[FINITE_RESTRICT; FINITE_NUMSEG; IN_ELIM_THM; IN_NUMSEG] THEN REWRITE_TAC[IMP_CONJ] THEN SIMP_TAC[BASIS_INJ_EQ]);; let ONORM_DIAGONAL_MATRIX = prove (`!A:real^N^N. diagonal_matrix A ==> onorm(\x. A ** x) = sup {abs(A$i$i) | 1 <= i /\ i <= dimindex(:N)}`, REPEAT STRIP_TAC THEN REWRITE_TAC[onorm] THEN MATCH_MP_TAC SUP_EQ THEN X_GEN_TAC `b:real` THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN EQ_TAC THEN DISCH_TAC THENL [X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `basis i:real^N`) THEN ASM_SIMP_TAC[NORM_BASIS; MATRIX_VECTOR_MUL_BASIS] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[COMPONENT_LE_NORM; REAL_LE_TRANS] `norm(x) <= b ==> !i. abs(x$i) <= b`)) THEN DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_SIMP_TAC[column; LAMBDA_BETA]; X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN TRANS_TAC REAL_LE_TRANS `norm(b % x:real^N)` THEN CONJ_TAC THENL [MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN FIRST_X_ASSUM(SUBST_ALL_TAC o GEN_REWRITE_RULE I [DIAGONAL_MATRIX]) THEN FIRST_X_ASSUM(K ALL_TAC o SYM) THEN POP_ASSUM MP_TAC THEN SIMP_TAC[LAMBDA_BETA; MATRIX_VECTOR_MUL_COMPONENT; dot] THEN REWRITE_TAC[COND_RAND; COND_RATOR; REAL_MUL_LZERO] THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN SIMP_TAC[SUM_DELTA; IN_NUMSEG] THEN REWRITE_TAC[REAL_ABS_MUL; VECTOR_MUL_COMPONENT] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN MATCH_MP_TAC(REAL_ARITH `x <= b ==> x <= abs b`) THEN ASM_SIMP_TAC[]; ASM_REWRITE_TAC[NORM_MUL] THEN FIRST_X_ASSUM(MP_TAC o SPEC `1`) THEN ASM_REWRITE_TAC[DIMINDEX_GE_1; LE_REFL] THEN REAL_ARITH_TAC]]);; (* ------------------------------------------------------------------------- *) (* Positive semidefinite matrices. *) (* ------------------------------------------------------------------------- *) let positive_semidefinite = new_definition `positive_semidefinite(A:real^N^N) <=> symmetric_matrix A /\ !x. &0 <= x dot (A ** x)`;; let POSITIVE_SEMIDEFINITE_IMP_SYMMETRIC_MATRIX = prove (`!A:real^N^N. positive_semidefinite A ==> symmetric_matrix A`, SIMP_TAC[positive_semidefinite]);; let POSITIVE_SEMIDEFINITE_IMP_SYMMETRIC = prove (`!A:real^N^N. positive_semidefinite A ==> transp A = A`, REWRITE_TAC[GSYM symmetric_matrix; POSITIVE_SEMIDEFINITE_IMP_SYMMETRIC_MATRIX]);; let POSITIVE_SEMIDEFINITE_ADD = prove (`!A B:real^N^N. positive_semidefinite A /\ positive_semidefinite B ==> positive_semidefinite(A + B)`, SIMP_TAC[positive_semidefinite; SYMMETRIC_MATRIX_ADD] THEN SIMP_TAC[MATRIX_VECTOR_MUL_ADD_RDISTRIB; DOT_RADD; REAL_LE_ADD]);; let POSITIVE_SEMIDEFINITE_CMUL = prove (`!c A:real^N^N. positive_semidefinite A /\ &0 <= c ==> positive_semidefinite(c %% A)`, SIMP_TAC[positive_semidefinite; SYMMETRIC_MATRIX_CMUL] THEN SIMP_TAC[MATRIX_VECTOR_LMUL; DOT_RMUL; REAL_LE_MUL]);; let POSITIVE_SEMIDEFINITE_TRANSP = prove (`!A:real^N^N. positive_semidefinite(transp A) <=> positive_semidefinite A`, REWRITE_TAC[positive_semidefinite; symmetric_matrix] THEN MESON_TAC[TRANSP_TRANSP]);; let POSITIVE_SEMIDEFINITE_COVARIANCE = prove (`!A:real^N^M. positive_semidefinite(transp A ** A)`, REWRITE_TAC[positive_semidefinite; symmetric_matrix; MATRIX_TRANSP_MUL; TRANSP_TRANSP] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_ASSOC] THEN ONCE_REWRITE_TAC[GSYM DOT_LMUL_MATRIX] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_TRANSP; DOT_POS_LE]);; let POSITIVE_SEMIDEFINITE_SIMILAR = prove (`!A B:real^N^M. positive_semidefinite A ==> positive_semidefinite(transp B ** A ** B)`, REWRITE_TAC[positive_semidefinite; symmetric_matrix] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[MATRIX_TRANSP_MUL; TRANSP_TRANSP; GSYM MATRIX_MUL_ASSOC] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_ASSOC] THEN REWRITE_TAC[GSYM DOT_LMUL_MATRIX; GSYM MATRIX_VECTOR_MUL_TRANSP] THEN ASM_REWRITE_TAC[DOT_LMUL_MATRIX]);; let POSITIVE_SEMIDEFINITE_SIMILAR_EQ = prove (`!A B:real^N^N. invertible B ==> (positive_semidefinite (transp B ** A ** B) <=> positive_semidefinite A)`, REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[POSITIVE_SEMIDEFINITE_SIMILAR] THEN DISCH_THEN(MP_TAC o ISPEC `matrix_inv B:real^N^N` o MATCH_MP POSITIVE_SEMIDEFINITE_SIMILAR) THEN ASM_SIMP_TAC[GSYM MATRIX_MUL_ASSOC; MATRIX_INV; MATRIX_MUL_RID] THEN REWRITE_TAC[MATRIX_MUL_ASSOC; GSYM MATRIX_TRANSP_MUL] THEN ASM_SIMP_TAC[MATRIX_INV; TRANSP_MAT; MATRIX_MUL_LID]);; let POSITIVE_SEMIDEFINITE_DIAGONAL_MATRIX = prove (`!D:real^N^N. diagonal_matrix D /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= D$i$i) ==> positive_semidefinite D`, SIMP_TAC[positive_semidefinite; DIAGONAL_IMP_SYMMETRIC_MATRIX] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I [DIAGONAL_MATRIX]) THEN SIMP_TAC[matrix_vector_mul; LAMBDA_BETA; dot] THEN SIMP_TAC[COND_RATOR; COND_RAND; REAL_MUL_LZERO] THEN CONV_TAC(RAND_CONV(ONCE_DEPTH_CONV SYM_CONV)) THEN SIMP_TAC[SUM_DELTA] THEN MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `x * d * x:real = d * x * x`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[REAL_LE_SQUARE]);; let POSITIVE_SEMIDEFINITE_DIAGONAL_MATRIX_EQ = prove (`!D:real^N^N. diagonal_matrix D ==> (positive_semidefinite D <=> !i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= D$i$i)`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[POSITIVE_SEMIDEFINITE_DIAGONAL_MATRIX] THEN REWRITE_TAC[positive_semidefinite] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `basis i:real^N`) THEN ASM_SIMP_TAC[DOT_BASIS; MATRIX_VECTOR_MUL_BASIS; column; LAMBDA_BETA]);; let DIAGONAL_POSITIVE_SEMIDEFINITE = prove (`!A:real^N^N i. positive_semidefinite A /\ 1 <= i /\ i <= dimindex(:N) ==> &0 <= A$i$i`, REWRITE_TAC[positive_semidefinite] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `basis i:real^N`) THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_BASIS; column; DOT_BASIS; LAMBDA_BETA]);; let TRACE_POSITIVE_SEMIDEFINITE = prove (`!A:real^N^N. positive_semidefinite A ==> &0 <= trace A`, SIMP_TAC[trace; SUM_POS_LE_NUMSEG; DIAGONAL_POSITIVE_SEMIDEFINITE]);; let TRACE_LE_MUL_SQUARES = prove (`!A B:real^N^N. symmetric_matrix A /\ symmetric_matrix B ==> trace((A ** B) ** (A ** B)) <= trace((A ** A) ** (B ** B))`, REWRITE_TAC[symmetric_matrix] THEN REPEAT STRIP_TAC THEN MP_TAC (ISPEC `A ** B - B ** A:real^N^N` POSITIVE_SEMIDEFINITE_COVARIANCE) THEN DISCH_THEN(MP_TAC o MATCH_MP TRACE_POSITIVE_SEMIDEFINITE) THEN REWRITE_TAC[MATRIX_TRANSP_MUL; TRANSP_MATRIX_SUB; MATRIX_SUB_LDISTRIB] THEN ASM_REWRITE_TAC[MATRIX_SUB_RDISTRIB; TRACE_SUB] THEN MATCH_MP_TAC(REAL_ARITH `a = y /\ d = y /\ b = x /\ c = x ==> &0 <= a - b - (c - d) ==> x <= y`) THEN REWRITE_TAC[GSYM MATRIX_MUL_ASSOC] THEN REPEAT CONJ_TAC THEN REPEAT(GEN_REWRITE_TAC LAND_CONV [TRACE_MUL_SYM] THEN REWRITE_TAC[GSYM MATRIX_MUL_ASSOC]));; let POSITIVE_SEMIDEFINITE_ZERO_FORM = prove (`!A:real^N^N. positive_semidefinite A /\ x dot (A ** x) = &0 ==> A ** x = vec 0`, let lemma = prove (`(!t. &0 <= a + b * t) ==> b = &0`, ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN DISCH_THEN(MP_TAC o SPEC `--(a + &1) / b`) THEN ASM_SIMP_TAC[REAL_DIV_LMUL] THEN REAL_ARITH_TAC) in REWRITE_TAC[positive_semidefinite; symmetric_matrix] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN `t:real` o SPEC `(A:real^N^N) ** x + t % x`) THEN REWRITE_TAC[MATRIX_VECTOR_MUL_ADD_LDISTRIB; DOT_RADD] THEN REWRITE_TAC[DOT_LADD; MATRIX_VECTOR_MUL_RMUL; DOT_LMUL] THEN REWRITE_TAC[DOT_RMUL] THEN SUBGOAL_THEN `x dot (A ** A ** x) = ((A:real^N^N) ** x) dot (A ** x)` SUBST1_TAC THENL [ASM_REWRITE_TAC[GSYM DOT_LMUL_MATRIX; VECTOR_MATRIX_MUL_TRANSP]; ASM_REWRITE_TAC[REAL_ARITH `(a + t * b) + t * b + t * t * &0 = a + (&2 * b) * t`]] THEN DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN REWRITE_TAC[REAL_ENTIRE; DOT_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ]);; let POSITIVE_SEMIDEFINITE_ZERO_FORM_EQ = prove (`!A:real^N^N. positive_semidefinite A ==> (x dot (A ** x) = &0 <=> A ** x = vec 0)`, REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN ASM_SIMP_TAC[DOT_RZERO; POSITIVE_SEMIDEFINITE_ZERO_FORM]);; let POSITIVE_SEMIDEFINITE_1_GEN = prove (`!A:real^N^N. dimindex(:N) = 1 ==> (positive_semidefinite A <=> &0 <= A$1$1)`, REPEAT STRIP_TAC THEN SIMP_TAC[positive_semidefinite; symmetric_matrix; transp; CART_EQ; dot] THEN ASM_SIMP_TAC[LAMBDA_BETA; ARITH; MATRIX_VECTOR_MUL_COMPONENT] THEN ASM_REWRITE_TAC[FORALL_1; SUM_1; dot] THEN REWRITE_TAC[REAL_ARITH `x * a * x:real = a * x pow 2`] THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LE_MUL; REAL_LE_POW_2]] THEN DISCH_THEN(MP_TAC o SPEC `basis 1:real^N`) THEN SIMP_TAC[BASIS_COMPONENT; ARITH; DIMINDEX_GE_1; LE_REFL] THEN REAL_ARITH_TAC);; let POSITIVE_SEMIDEFINITE_1 = prove (`!A:real^1^1. positive_semidefinite A <=> &0 <= A$1$1`, GEN_TAC THEN MATCH_MP_TAC POSITIVE_SEMIDEFINITE_1_GEN THEN REWRITE_TAC[DIMINDEX_1]);; let POSITIVE_SEMIDEFINITE_SUBMATRIX_2 = prove (`!A:real^N^N i j. positive_semidefinite A /\ 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> positive_semidefinite (vector[vector[A$i$i;A$i$j]; vector[A$j$i;A$j$j]]:real^2^2)`, REWRITE_TAC[positive_semidefinite; symmetric_matrix] THEN REPEAT STRIP_TAC THENL [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN SIMP_TAC[CART_EQ; transp; LAMBDA_BETA; DIMINDEX_2; VECTOR_2; ARITH; FORALL_2] THEN ASM_MESON_TAC[]; SIMP_TAC[DOT_2; VECTOR_2; matrix_vector_mul; DIMINDEX_2; LAMBDA_BETA; ARITH; SUM_2]] THEN ASM_CASES_TAC `j:num = i` THENL [ASM_REWRITE_TAC[REAL_ARITH `x * (a * x + a * y) + y * (a * x + a * y):real = a * (x + y) pow 2`] THEN MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_LE_POW_2] THEN MATCH_MP_TAC DIAGONAL_POSITIVE_SEMIDEFINITE THEN ASM_REWRITE_TAC[positive_semidefinite; symmetric_matrix]; FIRST_X_ASSUM(MP_TAC o SPEC `(lambda m. if m = i then (x:real^2)$1 else if m = j then (x:real^2)$2 else &0):real^N`) THEN SIMP_TAC[matrix_vector_mul; LAMBDA_BETA] THEN REPLICATE_TAC 2 (REPLICATE_TAC 2 (ONCE_REWRITE_TAC[COND_RAND]) THEN SIMP_TAC[SUM_CASES; FINITE_NUMSEG; SUM_DELTA; REAL_MUL_RZERO] THEN ASM_SIMP_TAC[SET_RULE `P a ==> {x | P x /\ x = a} = {a}`; IN_NUMSEG; IN_ELIM_THM; SUM_SING] THEN SIMP_TAC[dot; LAMBDA_BETA] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM])]);; (* ------------------------------------------------------------------------- *) (* The Frobenius norm and associated inner product, which turn out to be the *) (* usual Euclidean versions modulo flattening. *) (* ------------------------------------------------------------------------- *) let DOT_VECTORIZE = prove (`!A B:real^N^M. vectorize A dot vectorize B = trace(transp A ** B)`, REPEAT GEN_TAC THEN SIMP_TAC[dot; trace; matrix_mul; transp; LAMBDA_BETA] THEN SIMP_TAC[SUM_SUM_PRODUCT; FINITE_NUMSEG] THEN SIMP_TAC[VECTORIZE_COMPONENT; DIMINDEX_FINITE_PROD] THEN MATCH_MP_TAC SUM_EQ_GENERAL_INVERSES THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN EXISTS_TAC `\k. (1 + (k - 1) MOD dimindex(:N)),(1 + (k - 1) DIV dimindex(:N))` THEN EXISTS_TAC `\(i,j). (j - 1) * dimindex(:N) + i` THEN REWRITE_TAC[IN_ELIM_PAIR_THM; PAIR_EQ; IN_NUMSEG] THEN CONJ_TAC THENL [MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN CONJ_TAC THENL [CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN TRANS_TAC LE_TRANS `(j - 1) * dimindex(:N) + dimindex(:N)` THEN ASM_REWRITE_TAC[LE_ADD_LCANCEL] THEN REWRITE_TAC[ARITH_RULE `x * n + n = (x + 1) * n`] THEN ASM_SIMP_TAC[SUB_ADD; LE_MULT_RCANCEL]; CONJ_TAC THEN MATCH_MP_TAC(ARITH_RULE `1 <= i /\ j = i - 1 ==> 1 + j = i`) THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC MOD_UNIQ THEN EXISTS_TAC `j - 1` THEN ASM_ARITH_TAC; MATCH_MP_TAC DIV_UNIQ THEN EXISTS_TAC `i - 1` THEN ASM_ARITH_TAC]]; X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[LE_ADD] THEN SIMP_TAC[DIVISION; DIMINDEX_GE_1; LE_1; ADD_SUB2; RDIV_LT_EQ; ARITH_RULE `1 <= n ==> (1 + m <= n <=> m < n)`] THEN CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN MATCH_MP_TAC(ARITH_RULE `1 <= x /\ x - 1 = q * n + r /\ r < n ==> q * n + 1 + r = x`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC DIVISION THEN SIMP_TAC[DIMINDEX_GE_1; LE_1]]);; let NORM_VECTORIZE_TRANSP = prove (`!A:real^N^M. norm(vectorize(transp A)) = norm(vectorize A)`, REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_EQ; DOT_VECTORIZE; TRANSP_TRANSP] THEN MATCH_ACCEPT_TAC TRACE_MUL_SYM);; let COMPATIBLE_NORM_VECTORIZE = prove (`!A:real^N^M x. norm(A ** x) <= norm(vectorize A) * norm x`, REPEAT GEN_TAC THEN SIMP_TAC[NORM_LE_SQUARE; REAL_LE_MUL; NORM_POS_LE] THEN REWRITE_TAC[dot] THEN SIMP_TAC[MATRIX_MUL_DOT; LAMBDA_BETA] THEN TRANS_TAC REAL_LE_TRANS `sum (1..dimindex(:M)) (\i. norm((A:real^N^M)$i) pow 2 * norm(x:real^N) pow 2)` THEN CONJ_TAC THENL [MATCH_MP_TAC SUM_LE_NUMSEG THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM REAL_POW_MUL; GSYM REAL_POW_2] THEN REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_ABS_MUL; REAL_ABS_NORM] THEN REWRITE_TAC[NORM_CAUCHY_SCHWARZ_ABS]; REWRITE_TAC[SUM_RMUL; REAL_POW_MUL] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_LE_POW_2; NORM_POW_2; DOT_VECTORIZE] THEN ONCE_REWRITE_TAC[TRACE_MUL_SYM] THEN REWRITE_TAC[trace] THEN MATCH_MP_TAC SUM_LE_NUMSEG THEN SIMP_TAC[transp; matrix_mul; LAMBDA_BETA; dot; REAL_LE_REFL]]);; let ONORM_LE_NORM_VECTORIZE = prove (`!A:real^M^N. onorm(\x. A ** x) <= norm(vectorize A)`, GEN_TAC THEN MATCH_MP_TAC (CONJUNCT2(MATCH_MP ONORM (SPEC_ALL MATRIX_VECTOR_MUL_LINEAR))) THEN REWRITE_TAC[COMPATIBLE_NORM_VECTORIZE]);; let NORM_VECTORIZE_POW_2 = prove (`!A:real^N^M. norm(vectorize A) pow 2 = sum(1..dimindex(:M)) (\i. norm(A$i) pow 2)`, GEN_TAC THEN REWRITE_TAC[NORM_POW_2; DOT_VECTORIZE] THEN SIMP_TAC[trace; transp; matrix_mul; dot; LAMBDA_BETA] THEN GEN_REWRITE_TAC LAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[]);; let NORM_VECTORIZE_MUL_LE = prove (`!A:real^N^P B:real^M^N. norm(vectorize(A ** B)) <= norm(vectorize A) * norm(vectorize B)`, REPEAT GEN_TAC THEN SIMP_TAC[NORM_LE_SQUARE; REAL_LE_MUL; NORM_POS_LE] THEN REWRITE_TAC[GSYM NORM_POW_2; NORM_VECTORIZE_POW_2] THEN SIMP_TAC[MATRIX_MUL_COMPONENT; REAL_POW_MUL] THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [NORM_VECTORIZE_POW_2] THEN REWRITE_TAC[GSYM SUM_RMUL] THEN MATCH_MP_TAC SUM_LE_NUMSEG THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM REAL_POW_MUL] THEN REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_ABS_MUL; REAL_ABS_NORM] THEN MESON_TAC[COMPATIBLE_NORM_VECTORIZE; NORM_VECTORIZE_TRANSP; REAL_MUL_SYM]);; let NORM_VECTORIZE_HADAMARD_LE = prove (`!A:real^N^M B:real^N^M. norm(vectorize((lambda i j. A$i$j * B$i$j):real^N^M)) <= norm(vectorize A) * norm(vectorize B)`, REPEAT GEN_TAC THEN SIMP_TAC[NORM_LE_SQUARE; REAL_LE_MUL; NORM_POS_LE] THEN REWRITE_TAC[DOT_VECTORIZE; REAL_POW_MUL; NORM_POW_2] THEN SIMP_TAC[transp; matrix_mul; trace; LAMBDA_BETA] THEN SIMP_TAC[SUM_SUM_PRODUCT; FINITE_NUMSEG] THEN W(MP_TAC o PART_MATCH (rand o rand) SUM_MUL_BOUND o rand o snd) THEN SIMP_TAC[FINITE_PRODUCT_DEPENDENT; FINITE_NUMSEG; FORALL_IN_GSPEC] THEN REWRITE_TAC[REAL_LE_SQUARE] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[REAL_MUL_AC]);; let TRACE_COVARIANCE_POS_LE = prove (`!A:real^M^N. &0 <= trace(transp A ** A)`, SIMP_TAC[POSITIVE_SEMIDEFINITE_COVARIANCE; TRACE_POSITIVE_SEMIDEFINITE]);; let TRACE_COVARIANCE_EQ_0 = prove (`!A:real^M^N. trace(transp A ** A) = &0 <=> A = mat 0`, REWRITE_TAC[GSYM DOT_VECTORIZE; DOT_EQ_0; VECTORIZE_EQ_0]);; let TRACE_COVARIANCE_POS_LT = prove (`!A:real^M^N. &0 < trace(transp A ** A) <=> ~(A = mat 0)`, MESON_TAC[REAL_LT_LE; TRACE_COVARIANCE_POS_LE; TRACE_COVARIANCE_EQ_0]);; let TRACE_COVARIANCE_CAUCHY_SCHWARZ = prove (`!A B:real^M^N. trace(transp A ** B) <= sqrt(trace(transp A ** A)) * sqrt(trace(transp B ** B))`, REWRITE_TAC[GSYM DOT_VECTORIZE; GSYM vector_norm; NORM_CAUCHY_SCHWARZ]);; let TRACE_COVARIANCE_CAUCHY_SCHWARZ_ABS = prove (`!A B:real^M^N. abs(trace(transp A ** B)) <= sqrt(trace(transp A ** A)) * sqrt(trace(transp B ** B))`, REWRITE_TAC[GSYM DOT_VECTORIZE; GSYM vector_norm; NORM_CAUCHY_SCHWARZ_ABS]);; let TRACE_COVARIANCE_CAUCHY_SCHWARZ_SQUARE = prove (`!A B:real^M^N. trace(transp A ** B) pow 2 <= trace(transp A ** A) * trace(transp B ** B)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_POW2_ABS] THEN MATCH_MP_TAC REAL_RSQRT_LE THEN SIMP_TAC[REAL_ABS_POS; REAL_LE_MUL; TRACE_COVARIANCE_POS_LE] THEN REWRITE_TAC[TRACE_COVARIANCE_CAUCHY_SCHWARZ_ABS; SQRT_MUL]);; (* ------------------------------------------------------------------------- *) (* Positive definite matrices. *) (* ------------------------------------------------------------------------- *) let positive_definite = new_definition `positive_definite(A:real^N^N) <=> symmetric_matrix A /\ !x. ~(x = vec 0) ==> &0 < x dot (A ** x)`;; let POSITIVE_DEFINITE_IMP_SYMMETRIC_MATRIX = prove (`!A:real^N^N. positive_definite A ==> symmetric_matrix A`, SIMP_TAC[positive_definite]);; let POSITIVE_DEFINITE_IMP_SYMMETRIC = prove (`!A:real^N^N. positive_definite A ==> transp A = A`, REWRITE_TAC[GSYM symmetric_matrix; POSITIVE_DEFINITE_IMP_SYMMETRIC_MATRIX]);; let POSITIVE_DEFINITE_POSITIVE_SEMIDEFINITE = prove (`!A:real^N^N. positive_definite A <=> positive_semidefinite A /\ invertible A`, GEN_TAC THEN REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`; positive_definite; FORALL_AND_THM; TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN SIMP_TAC[MESON[] `P a ==> ((!x:real^N. ~(x = a) ==> P x) <=> (!x. P x))`; DOT_LZERO; REAL_LE_REFL] THEN REWRITE_TAC[CONJ_ASSOC; GSYM positive_semidefinite] THEN ASM_CASES_TAC `positive_semidefinite(A:real^N^N)` THEN ASM_SIMP_TAC[POSITIVE_SEMIDEFINITE_ZERO_FORM_EQ] THEN REWRITE_TAC[GSYM HOMOGENEOUS_LINEAR_EQUATIONS_DET; INVERTIBLE_DET_NZ] THEN MESON_TAC[]);; let POSITIVE_DEFINITE_SIMILAR_EQ = prove (`!A B:real^N^N. positive_definite(transp B ** A ** B) <=> invertible B /\ positive_definite A`, REPEAT GEN_TAC THEN REWRITE_TAC[POSITIVE_DEFINITE_POSITIVE_SEMIDEFINITE] THEN REWRITE_TAC[INVERTIBLE_MATRIX_MUL; INVERTIBLE_TRANSP] THEN MESON_TAC[POSITIVE_SEMIDEFINITE_SIMILAR_EQ]);; let POSITIVE_DEFINITE_1_GEN = prove (`!A:real^N^N. dimindex(:N) = 1 ==> (positive_definite A <=> &0 < A$1$1)`, REPEAT STRIP_TAC THEN REWRITE_TAC[positive_definite; symmetric_matrix; transp; CART_EQ; dot] THEN ASM_SIMP_TAC[LAMBDA_BETA; ARITH; MATRIX_VECTOR_MUL_COMPONENT] THEN ASM_REWRITE_TAC[FORALL_1; SUM_1; dot; VEC_COMPONENT] THEN REWRITE_TAC[REAL_ARITH `x * a * x:real = a * x pow 2`] THEN REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LT_MUL; REAL_LT_POW_2]] THEN DISCH_THEN(MP_TAC o SPEC `basis 1:real^N`) THEN SIMP_TAC[BASIS_COMPONENT; ARITH; DIMINDEX_GE_1; LE_REFL] THEN REAL_ARITH_TAC);; let POSITIVE_DEFINITE_1 = prove (`!A:real^1^1. positive_definite A <=> &0 < A$1$1`, GEN_TAC THEN MATCH_MP_TAC POSITIVE_DEFINITE_1_GEN THEN REWRITE_TAC[DIMINDEX_1]);; let POSITIVE_DEFINITE_IMP_INVERTIBLE = prove (`!A:real^N^N. positive_definite A ==> invertible A`, SIMP_TAC[POSITIVE_DEFINITE_POSITIVE_SEMIDEFINITE]);; let POSITIVE_DEFINITE_IMP_POSITIVE_SEMIDEFINITE = prove (`!A:real^N^N. positive_definite A ==> positive_semidefinite A`, SIMP_TAC[POSITIVE_DEFINITE_POSITIVE_SEMIDEFINITE]);; let POSITIVE_SEMIDEFINITE_POSITIVE_DEFINITE_ADD = prove (`!A B:real^N^N. positive_semidefinite A /\ positive_definite B ==> positive_definite(A + B)`, SIMP_TAC[positive_definite; positive_semidefinite; SYMMETRIC_MATRIX_ADD] THEN SIMP_TAC[MATRIX_VECTOR_MUL_ADD_RDISTRIB; DOT_RADD; REAL_LET_ADD]);; let POSITIVE_DEFINITE_POSITIVE_SEMIDEFINITE_ADD = prove (`!A B:real^N^N. positive_definite A /\ positive_semidefinite B ==> positive_definite(A + B)`, SIMP_TAC[positive_definite; positive_semidefinite; SYMMETRIC_MATRIX_ADD] THEN SIMP_TAC[MATRIX_VECTOR_MUL_ADD_RDISTRIB; DOT_RADD; REAL_LTE_ADD]);; let POSITIVE_DEFINITE_ADD = prove (`!A B:real^N^N. positive_definite A /\ positive_definite B ==> positive_definite(A + B)`, SIMP_TAC[positive_definite; SYMMETRIC_MATRIX_ADD] THEN SIMP_TAC[MATRIX_VECTOR_MUL_ADD_RDISTRIB; DOT_RADD; REAL_LT_ADD]);; let POSITIVE_DEFINITE_CMUL = prove (`!c A:real^N^N. positive_definite A /\ &0 < c ==> positive_definite(c %% A)`, SIMP_TAC[positive_definite; SYMMETRIC_MATRIX_CMUL] THEN SIMP_TAC[MATRIX_VECTOR_LMUL; DOT_RMUL; REAL_LT_MUL]);; let NEARBY_POSITIVE_DEFINITE_MATRIX_GEN = prove (`!A:real^N^N B x. positive_semidefinite A /\ positive_definite B /\ &0 < x ==> positive_definite(A + x %% B)`, SIMP_TAC[POSITIVE_SEMIDEFINITE_POSITIVE_DEFINITE_ADD; POSITIVE_DEFINITE_CMUL]);; let POSITIVE_DEFINITE_TRANSP = prove (`!A:real^N^N. positive_definite(transp A) <=> positive_definite A`, REWRITE_TAC[positive_definite; symmetric_matrix] THEN MESON_TAC[TRANSP_TRANSP]);; let POSITIVE_DEFINITE_COVARIANCE = prove (`!A:real^N^N. positive_definite(transp A ** A) <=> invertible A`, REWRITE_TAC[POSITIVE_DEFINITE_POSITIVE_SEMIDEFINITE; POSITIVE_SEMIDEFINITE_COVARIANCE] THEN REWRITE_TAC[INVERTIBLE_MATRIX_MUL; INVERTIBLE_TRANSP]);; let POSITIVE_DEFINITE_SIMILAR = prove (`!A B:real^N^N. positive_definite A /\ invertible B ==> positive_definite(transp B ** A ** B)`, SIMP_TAC[POSITIVE_DEFINITE_POSITIVE_SEMIDEFINITE; POSITIVE_SEMIDEFINITE_SIMILAR; INVERTIBLE_MATRIX_MUL; INVERTIBLE_TRANSP]);; let POSITIVE_DEFINITE_DIAGONAL_MATRIX = prove (`!D:real^N^N. diagonal_matrix D /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 < D$i$i) ==> positive_definite D`, SIMP_TAC[positive_definite; DIAGONAL_IMP_SYMMETRIC_MATRIX] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I [DIAGONAL_MATRIX]) THEN SIMP_TAC[matrix_vector_mul; LAMBDA_BETA; dot] THEN SIMP_TAC[COND_RATOR; COND_RAND; REAL_MUL_LZERO] THEN CONV_TAC(RAND_CONV(ONCE_DEPTH_CONV SYM_CONV)) THEN SIMP_TAC[SUM_DELTA] THEN MATCH_MP_TAC SUM_POS_LT THEN REWRITE_TAC[REAL_ARITH `x * d * x:real = d * x * x`] THEN ASM_SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; REAL_LE_MUL; REAL_LE_SQUARE; REAL_LT_IMP_LE] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; VEC_COMPONENT] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_SIMP_TAC[GSYM REAL_POW_2; REAL_LT_MUL; REAL_LT_POW_2]);; let POSITIVE_DEFINITE_DIAGONAL_MATRIX_EQ = prove (`!D:real^N^N. diagonal_matrix D ==> (positive_definite D <=> !i. 1 <= i /\ i <= dimindex(:N) ==> &0 < D$i$i)`, REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[POSITIVE_DEFINITE_DIAGONAL_MATRIX] THEN REWRITE_TAC[positive_definite] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `basis i:real^N`) THEN ASM_SIMP_TAC[DOT_BASIS; MATRIX_VECTOR_MUL_BASIS; column; LAMBDA_BETA; BASIS_NONZERO]);; let DIAGONAL_POSITIVE_DEFINITE = prove (`!A:real^N^N i. positive_definite A /\ 1 <= i /\ i <= dimindex(:N) ==> &0 < A$i$i`, REWRITE_TAC[positive_definite] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `basis i:real^N`) THEN ASM_SIMP_TAC[MATRIX_VECTOR_MUL_BASIS; column; DOT_BASIS; LAMBDA_BETA; BASIS_NONZERO]);; let TRACE_POSITIVE_DEFINITE = prove (`!A:real^N^N. positive_definite A ==> &0 < trace A`, SIMP_TAC[trace; SUM_POS_LT_ALL; DIAGONAL_POSITIVE_DEFINITE; IN_NUMSEG; FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1]);; let POSITIVE_DEFINITE_MAT = prove (`!m. positive_definite(mat m:real^N^N) <=> 0 < m`, SIMP_TAC[POSITIVE_DEFINITE_DIAGONAL_MATRIX_EQ; DIAGONAL_MATRIX_MAT] THEN SIMP_TAC[mat; LAMBDA_BETA; REAL_OF_NUM_LT] THEN MESON_TAC[LE_REFL; DIMINDEX_GE_1]);; let POSITIVE_DEFINITE_ID = prove (`positive_definite(mat 1:real^N^N)`, REWRITE_TAC[POSITIVE_DEFINITE_MAT; ARITH]);; let POSITIVE_SEMIDEFINITE_MAT = prove (`!m. positive_semidefinite(mat m:real^N^N)`, SIMP_TAC[POSITIVE_SEMIDEFINITE_DIAGONAL_MATRIX_EQ; DIAGONAL_MATRIX_MAT] THEN SIMP_TAC[mat; LAMBDA_BETA; REAL_POS] THEN MESON_TAC[LE_REFL; DIMINDEX_GE_1]);; let NEARBY_POSITIVE_DEFINITE_MATRIX = prove (`!A:real^N^N x. positive_semidefinite A /\ &0 < x ==> positive_definite(A + x %% mat 1)`, SIMP_TAC[NEARBY_POSITIVE_DEFINITE_MATRIX_GEN; POSITIVE_DEFINITE_ID]);; let POSITIVE_SEMIDEFINITE_ANTISYM = prove (`!A:real^N^N. positive_semidefinite A /\ positive_semidefinite(--A) <=> A = mat 0`, GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[POSITIVE_SEMIDEFINITE_MAT; MATRIX_NEG_0] THEN ASM_SIMP_TAC[MATRIX_EQ_0; GSYM POSITIVE_SEMIDEFINITE_ZERO_FORM_EQ] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[positive_semidefinite] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LNEG; DOT_RNEG; IMP_IMP] THEN DISCH_THEN(CONJUNCTS_THEN (MP_TAC o CONJUNCT2)) THEN REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);; let LOEWNER_ORDER_ANTISYM = prove (`!(A:real^N^N) B. positive_semidefinite(A - B) /\ positive_semidefinite(B - A) <=> A = B`, REPEAT GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM MATRIX_SUB_EQ] THEN GEN_REWRITE_TAC RAND_CONV [GSYM POSITIVE_SEMIDEFINITE_ANTISYM] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN CONV_TAC MATRIX_ARITH);; (* ------------------------------------------------------------------------- *) (* Hadamard's inequality. *) (* ------------------------------------------------------------------------- *) let HADAMARD_INEQUALITY_ROW = prove (`!A:real^N^N. abs(det A) <= product(1..dimindex(:N)) (\i. norm(row i A))`, GEN_TAC THEN ABBREV_TAC `a = \i. (A:real^N^N)$i` THEN (MP_TAC o DISCH_ALL o instantiate_casewise_recursion) `?b. !j. b j :real^N = a j - vsum(1..j-1) (\i. (a j dot b i) / (b i dot b i) % b i)` THEN ANTS_TAC THENL [EXISTS_TAC `(<):num->num->bool` THEN REWRITE_TAC[WF_num] THEN MATCH_MP_TAC ADMISSIBLE_IMP_SUPERADMISSIBLE THEN REWRITE_TAC[admissible] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[IN_NUMSEG; ARITH_RULE `1 <= x /\ x <= y - 1 ==> x < y`]; DISCH_THEN(STRIP_ASSUME_TAC o GSYM)] THEN ABBREV_TAC `B:real^N^N = lambda i. b i` THEN TRANS_TAC REAL_LE_TRANS `abs(det(B:real^N^N))` THEN CONJ_TAC THENL [SUBGOAL_THEN `!n. det((lambda i. if i <= n then b i else a i):real^N^N) = det(A:real^N^N)` (MP_TAC o SPEC `dimindex(:N)`) THENL [MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL [AP_TERM_TAC THEN EXPAND_TAC "a" THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN SIMP_TAC[ARITH_RULE `1 <= n ==> ~(n <= 0)`]; X_GEN_TAC `n:num` THEN DISCH_THEN(SUBST1_TAC o SYM)] THEN ASM_CASES_TAC `dimindex(:N) <= n` THENL [AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_ARITH_TAC; FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (ARITH_RULE `~(n <= k) ==> SUC k <= n`))] THEN MP_TAC(ISPECL [`(lambda i. if i <= n then b i else a i):real^N^N`; `SUC n`; `--vsum (1..SUC n - 1) (\i. (a (SUC n) dot b i) / (b i dot b i) % b i):real^N`] DET_ROW_SPAN) THEN ASM_REWRITE_TAC[row; LAMBDA_ETA; ARITH_RULE `1 <= SUC n`] THEN ANTS_TAC THENL [MATCH_MP_TAC SPAN_NEG THEN MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `i:num` THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN SIMP_TAC[LAMBDA_BETA] THEN ASM_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [CART_EQ] THEN X_GEN_TAC `k:num` THEN SIMP_TAC[LAMBDA_BETA] THEN STRIP_TAC THEN ASM_CASES_TAC `SUC n = k` THEN ASM_SIMP_TAC[LE_REFL; LAMBDA_BETA; GSYM VECTOR_SUB; ARITH_RULE `SUC n = k ==> ~(k <= n)`] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_ARITH_TAC]; DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [CART_EQ] THEN EXPAND_TAC "B" THEN SIMP_TAC[LAMBDA_BETA]]; ALL_TAC] THEN SUBGOAL_THEN `!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> orthogonal (b i:real^N) (b j)` ASSUME_TAC THENL [ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[ORTHOGONAL_SYM]; ALL_TAC] THEN GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN REWRITE_TAC[IMP_IMP] THEN REWRITE_TAC[ARITH_RULE `j < n /\ 1 <= n /\ n <= N /\ 1 <= j /\ j <= N /\ ~(n = j) <=> (1 <= n /\ n <= N) /\ (1 <= j /\ j <= N /\ j < n)`] THEN MATCH_MP_TAC num_WF THEN CONV_TAC NUM_REDUCE_CONV THEN X_GEN_TAC `n:num` THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_FORALL_THM] THEN REWRITE_TAC[IMP_IMP] THEN DISCH_TAC THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM o SPEC `n:num`) THEN REWRITE_TAC[orthogonal; DOT_LSUB; REAL_SUB_0] THEN SIMP_TAC[DOT_LSUM; FINITE_NUMSEG; DOT_LMUL] THEN TRANS_TAC EQ_TRANS `sum(1..n-1) (\j. if j = m then (a n:real^N) dot (b m) else &0)` THEN CONJ_TAC THENL [REWRITE_TAC[SUM_DELTA; IN_NUMSEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC; MATCH_MP_TAC SUM_EQ_NUMSEG THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_CASES_TAC `(b:num->real^N) m = vec 0` THEN ASM_REWRITE_TAC[DOT_RZERO; REAL_MUL_RZERO] THEN ASM_SIMP_TAC[DOT_EQ_0; REAL_DIV_RMUL]; CONV_TAC SYM_CONV THEN REWRITE_TAC[REAL_ENTIRE] THEN DISJ2_TAC THEN REWRITE_TAC[GSYM orthogonal] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE `~(m:num = n) ==> n < m \/ m < n`)) THENL [ALL_TAC; ONCE_REWRITE_TAC[ORTHOGONAL_SYM]] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC]]; ALL_TAC] THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N) ==> norm(b i:real^N) <= norm(a i:real^N)` ASSUME_TAC THENL [X_GEN_TAC `i:num` THEN STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `i:num`) THEN REWRITE_TAC[NORM_LE; VECTOR_ARITH `(x - y:real^N) dot (x - y) = (x dot x + y dot y) - &2 * x dot y`] THEN REWRITE_TAC[REAL_ARITH `(a + b) - x <= a <=> b <= x`] THEN SIMP_TAC[DOT_RSUM; FINITE_NUMSEG; DOT_RMUL; GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_LE_NUMSEG THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x = y ==> y <= &2 * x`) THEN CONJ_TAC THENL [ONCE_REWRITE_TAC[REAL_ARITH `x / y * x:real = (x * x) / y`] THEN MATCH_MP_TAC REAL_LE_DIV THEN REWRITE_TAC[REAL_LE_SQUARE; DOT_POS_LE]; AP_TERM_TAC] THEN TRANS_TAC EQ_TRANS `sum(1..i-1) (\k. if k = j then (a i:real^N) dot (b j) else &0)` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[SUM_DELTA; IN_NUMSEG]; ALL_TAC] THEN SIMP_TAC[DOT_LSUM; FINITE_NUMSEG] THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN REWRITE_TAC[DOT_LMUL] THEN ASM_CASES_TAC `(b:num->real^N) j = vec 0` THEN ASM_REWRITE_TAC[DOT_RZERO; REAL_MUL_RZERO; COND_ID] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[DOT_EQ_0; REAL_DIV_RMUL] THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[REAL_ENTIRE] THEN DISJ2_TAC THEN REWRITE_TAC[GSYM orthogonal] THEN FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE `~(m:num = n) ==> n < m \/ m < n`)) THENL [ALL_TAC; ONCE_REWRITE_TAC[ORTHOGONAL_SYM]] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `product(1..dimindex(:N)) (\i. norm(b i:real^N))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC PRODUCT_LE_NUMSEG THEN REWRITE_TAC[NORM_POS_LE; row; LAMBDA_ETA] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN TRANS_TAC REAL_LE_TRANS `norm((a:num->real^N) i)` THEN ASM_SIMP_TAC[] THEN EXPAND_TAC "a" THEN REWRITE_TAC[REAL_LE_REFL]] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= y /\ abs x <= abs y ==> abs x <= y`) THEN SIMP_TAC[PRODUCT_POS_LE_NUMSEG; NORM_POS_LE; REAL_LE_SQUARE_ABS] THEN REWRITE_TAC[REAL_POW_2; GSYM PRODUCT_MUL_NUMSEG] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM DET_TRANSP] THEN REWRITE_TAC[GSYM DET_MUL] THEN W(MP_TAC o PART_MATCH (lhand o rand) DET_DIAGONAL o lhand o snd) THEN SIMP_TAC[DIAGONAL_MATRIX_MUL_EQ; pairwise; GSYM ROW_TRANSP; IN_NUMSEG] THEN EXPAND_TAC "B" THEN SIMP_TAC[TRANSP_TRANSP; row; LAMBDA_ETA; LAMBDA_BETA] THEN ASM_REWRITE_TAC[GSYM CONJ_ASSOC] THEN DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC PRODUCT_EQ_NUMSEG THEN EXPAND_TAC "B" THEN REWRITE_TAC[transp; GSYM REAL_POW_2] THEN SIMP_TAC[matrix_mul; NORM_POW_2; dot; LAMBDA_BETA; dot]);; let HADAMARD_INEQUALITY_COLUMN = prove (`!A:real^N^N. abs(det A) <= product(1..dimindex(:N)) (\i. norm(column i A))`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM DET_TRANSP] THEN SIMP_TAC[GSYM ROW_TRANSP; HADAMARD_INEQUALITY_ROW]);; (* ------------------------------------------------------------------------- *) (* Orthogonality of a transformation and matrix. *) (* ------------------------------------------------------------------------- *) let orthogonal_transformation = new_definition `orthogonal_transformation(f:real^N->real^N) <=> linear f /\ !v w. f(v) dot f(w) = v dot w`;; let ORTHOGONAL_TRANSFORMATION = prove (`!f. orthogonal_transformation f <=> linear f /\ !v. norm(f v) = norm(v)`, GEN_TAC THEN REWRITE_TAC[orthogonal_transformation] THEN EQ_TAC THENL [MESON_TAC[vector_norm]; SIMP_TAC[DOT_NORM] THEN MESON_TAC[LINEAR_ADD]]);; let ORTHOGONAL_ORTHOGONAL_TRANSFORMATION = prove (`!f x y:real^N. orthogonal_transformation f ==> (orthogonal (f x) (f y) <=> orthogonal x y)`, SIMP_TAC[orthogonal; orthogonal_transformation]);; let ORTHOGONAL_TRANSFORMATION_COMPOSE = prove (`!f g. orthogonal_transformation f /\ orthogonal_transformation g ==> orthogonal_transformation(f o g)`, SIMP_TAC[orthogonal_transformation; LINEAR_COMPOSE; o_THM]);; let ORTHOGONAL_TRANSFORMATION_NEG = prove (`!f:real^N->real^N. orthogonal_transformation(\x. --(f x)) <=> orthogonal_transformation f`, REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; LINEAR_COMPOSE_NEG_EQ; NORM_NEG]);; let ORTHOGONAL_TRANSFORMATION_LINEAR = prove (`!f:real^N->real^N. orthogonal_transformation f ==> linear f`, SIMP_TAC[orthogonal_transformation]);; let ORTHOGONAL_TRANSFORMATION_INJECTIVE = prove (`!f:real^N->real^N. orthogonal_transformation f ==> !x y. f x = f y ==> x = y`, SIMP_TAC[LINEAR_INJECTIVE_0; ORTHOGONAL_TRANSFORMATION; GSYM NORM_EQ_0]);; let ORTHOGONAL_TRANSFORMATION_SURJECTIVE = prove (`!f:real^N->real^N. orthogonal_transformation f ==> !y. ?x. f x = y`, MESON_TAC[LINEAR_INJECTIVE_IMP_SURJECTIVE; ORTHOGONAL_TRANSFORMATION_INJECTIVE; orthogonal_transformation]);; let orthogonal_matrix = new_definition `orthogonal_matrix(Q:real^N^N) <=> transp(Q) ** Q = mat 1 /\ Q ** transp(Q) = mat 1`;; let ORTHOGONAL_MATRIX = prove (`orthogonal_matrix(Q:real^N^N) <=> transp(Q) ** Q = mat 1`, MESON_TAC[MATRIX_LEFT_RIGHT_INVERSE; orthogonal_matrix]);; let ORTHOGONAL_MATRIX_ALT = prove (`!A:real^N^N. orthogonal_matrix A <=> A ** transp A = mat 1`, MESON_TAC[MATRIX_LEFT_RIGHT_INVERSE; orthogonal_matrix]);; let ORTHOGONAL_MATRIX_TRANSP = prove (`!A:real^N^N. orthogonal_matrix(transp A) <=> orthogonal_matrix A`, REWRITE_TAC[orthogonal_matrix; TRANSP_TRANSP; CONJ_ACI]);; let ORTHOGONAL_MATRIX_TRANSP_LMUL = prove (`!P:real^N^N. orthogonal_matrix P ==> transp P ** P = mat 1`, REWRITE_TAC[ORTHOGONAL_MATRIX]);; let ORTHOGONAL_MATRIX_TRANSP_RMUL = prove (`!P:real^N^N. orthogonal_matrix P ==> P ** transp P = mat 1`, REWRITE_TAC[ORTHOGONAL_MATRIX_ALT]);; let NORM_VECTORIZE_ORTHOGONAL_MATRIX_RMUL = prove (`!A:real^N^N P:real^N^N. orthogonal_matrix P ==> norm(vectorize(A ** P)) = norm(vectorize A)`, REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_EQ; DOT_VECTORIZE; MATRIX_TRANSP_MUL] THEN GEN_REWRITE_TAC LAND_CONV [TRACE_MUL_SYM] THEN ONCE_REWRITE_TAC[MATRIX_MUL_ASSOC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV) [GSYM MATRIX_MUL_ASSOC] THEN ASM_SIMP_TAC[ORTHOGONAL_MATRIX_TRANSP_RMUL; MATRIX_MUL_RID] THEN MATCH_ACCEPT_TAC TRACE_MUL_SYM);; let NORM_VECTORIZE_ORTHOGONAL_MATRIX_LMUL = prove (`!A:real^N^N P:real^N^N. orthogonal_matrix P ==> norm(vectorize(P ** A)) = norm(vectorize A)`, REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM NORM_VECTORIZE_TRANSP] THEN REWRITE_TAC[MATRIX_TRANSP_MUL] THEN MATCH_MP_TAC NORM_VECTORIZE_ORTHOGONAL_MATRIX_RMUL THEN ASM_REWRITE_TAC[ORTHOGONAL_MATRIX_TRANSP]);; let ORTHOGONAL_MATRIX_ID = prove (`orthogonal_matrix(mat 1)`, REWRITE_TAC[orthogonal_matrix; TRANSP_MAT; MATRIX_MUL_LID]);; let ORTHOGONAL_MATRIX_MUL = prove (`!A B. orthogonal_matrix A /\ orthogonal_matrix B ==> orthogonal_matrix(A ** B)`, REWRITE_TAC[orthogonal_matrix; MATRIX_TRANSP_MUL] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM MATRIX_MUL_ASSOC] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [MATRIX_MUL_ASSOC] THEN ASM_REWRITE_TAC[MATRIX_MUL_LID; MATRIX_MUL_RID]);; let ORTHOGONAL_TRANSFORMATION_MATRIX = prove (`!f:real^N->real^N. orthogonal_transformation f <=> linear f /\ orthogonal_matrix(matrix f)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [REWRITE_TAC[orthogonal_transformation; ORTHOGONAL_MATRIX] THEN STRIP_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`basis i:real^N`; `basis j:real^N`]) THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN REWRITE_TAC[DOT_MATRIX_VECTOR_MUL] THEN ABBREV_TAC `A = transp (matrix f) ** matrix(f:real^N->real^N)` THEN ASM_SIMP_TAC[matrix_mul; columnvector; rowvector; basis; LAMBDA_BETA; SUM_DELTA; DIMINDEX_1; LE_REFL; dot; IN_NUMSEG; mat; MESON[REAL_MUL_LID; REAL_MUL_LZERO; REAL_MUL_RID; REAL_MUL_RZERO] `(if b then &1 else &0) * x = (if b then x else &0) /\ x * (if b then &1 else &0) = (if b then x else &0)`]; REWRITE_TAC[orthogonal_matrix; ORTHOGONAL_TRANSFORMATION; NORM_EQ] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN ASM_REWRITE_TAC[DOT_MATRIX_VECTOR_MUL] THEN SIMP_TAC[DOT_MATRIX_PRODUCT; MATRIX_MUL_LID]]);; let ORTHOGONAL_MATRIX_TRANSFORMATION = prove (`!A:real^N^N. orthogonal_matrix A <=> orthogonal_transformation(\x. A ** x)`, REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[MATRIX_OF_MATRIX_VECTOR_MUL]);; let ORTHOGONAL_MATRIX_MATRIX = prove (`!f:real^N->real^N. orthogonal_transformation f ==> orthogonal_matrix(matrix f)`, SIMP_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX]);; let ORTHOGONAL_MATRIX_NORM_EQ = prove (`!A. orthogonal_matrix A <=> !x. norm(A ** x) = norm x`, REWRITE_TAC[ORTHOGONAL_MATRIX_TRANSFORMATION; MATRIX_VECTOR_MUL_LINEAR; ORTHOGONAL_TRANSFORMATION]);; let ORTHOGONAL_MATRIX_NORM = prove (`!A x:real^N. orthogonal_matrix A ==> norm(A ** x) = norm x`, SIMP_TAC[ORTHOGONAL_MATRIX_TRANSFORMATION; ORTHOGONAL_TRANSFORMATION]);; let DET_ORTHOGONAL_MATRIX = prove (`!Q. orthogonal_matrix Q ==> det(Q) = &1 \/ det(Q) = -- &1`, GEN_TAC THEN REWRITE_TAC[REAL_RING `x = &1 \/ x = -- &1 <=> x * x = &1`] THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV) [GSYM DET_TRANSP] THEN SIMP_TAC[GSYM DET_MUL; orthogonal_matrix; DET_I]);; let ORTHOGONAL_MATRIX_IMP_INVERTIBLE = prove (`!A:real^N^N. orthogonal_matrix A ==> invertible A`, GEN_TAC THEN REWRITE_TAC[INVERTIBLE_DET_NZ] THEN DISCH_THEN(MP_TAC o MATCH_MP DET_ORTHOGONAL_MATRIX) THEN REAL_ARITH_TAC);; let MATRIX_MUL_LTRANSP_DOT_COLUMN = prove (`!A:real^N^M. transp A ** A = (lambda i j. (column i A) dot (column j A))`, SIMP_TAC[matrix_mul; CART_EQ; LAMBDA_BETA; transp; dot; column]);; let MATRIX_MUL_RTRANSP_DOT_ROW = prove (`!A:real^N^M. A ** transp A = (lambda i j. (row i A) dot (row j A))`, SIMP_TAC[matrix_mul; CART_EQ; LAMBDA_BETA; transp; dot; row]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS = prove (`!A:real^N^N. orthogonal_matrix A <=> (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(column i A) = &1) /\ (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> orthogonal (column i A) (column j A))`, REWRITE_TAC[ORTHOGONAL_MATRIX] THEN SIMP_TAC[MATRIX_MUL_LTRANSP_DOT_COLUMN; CART_EQ; mat; LAMBDA_BETA] THEN REWRITE_TAC[orthogonal; NORM_EQ_1] THEN MESON_TAC[]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS = prove (`!A:real^N^N. orthogonal_matrix A <=> (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(row i A) = &1) /\ (!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) /\ ~(i = j) ==> orthogonal (row i A) (row j A))`, ONCE_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSP] THEN SIMP_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS; COLUMN_TRANSP]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_INDEXED = prove (`!A:real^N^N. orthogonal_matrix A <=> (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(row i A) = &1) /\ pairwise (\i j. orthogonal (row i A) (row j A)) (1..dimindex(:N))`, REPEAT GEN_TAC THEN REWRITE_TAC[ORTHOGONAL_MATRIX_ALT] THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; pairwise; MAT_COMPONENT] THEN SIMP_TAC[MATRIX_MUL_RTRANSP_DOT_ROW; IN_NUMSEG; LAMBDA_BETA] THEN REWRITE_TAC[NORM_EQ_SQUARE; REAL_POS] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[orthogonal] THEN MESON_TAC[]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_PAIRWISE = prove (`!A:real^N^N. orthogonal_matrix A <=> CARD(rows A) = dimindex(:N) /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(row i A) = &1) /\ pairwise orthogonal (rows A)`, REWRITE_TAC[rows; ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_INDEXED] THEN GEN_TAC THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[PAIRWISE_IMAGE; GSYM numseg] THEN MATCH_MP_TAC(TAUT `(p ==> (q <=> r /\ s)) ==> (p /\ q <=> r /\ p /\ s)`) THEN DISCH_TAC THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN SIMP_TAC[CARD_IMAGE_EQ_INJ; FINITE_NUMSEG] THEN REWRITE_TAC[pairwise; IN_NUMSEG] THEN ASM_MESON_TAC[ORTHOGONAL_REFL; NORM_ARITH `~(norm(vec 0:real^N) = &1)`]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_SPAN = prove (`!A:real^N^N. orthogonal_matrix A <=> span(rows A) = (:real^N) /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(row i A) = &1) /\ pairwise orthogonal (rows A)`, GEN_TAC THEN REWRITE_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_PAIRWISE] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC(SET_RULE `UNIV SUBSET s ==> s = UNIV`) THEN MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN ASM_REWRITE_TAC[DIM_UNIV; SUBSET_UNIV; LE_REFL]; CONV_TAC SYM_CONV THEN REWRITE_TAC[GSYM DIM_UNIV] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[DIM_SPAN] THEN MATCH_MP_TAC DIM_EQ_CARD] THEN MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN ASM_REWRITE_TAC[rows; IN_ELIM_THM] THEN ASM_MESON_TAC[NORM_ARITH `~(norm(vec 0:real^N) = &1)`]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS_INDEXED = prove (`!A:real^N^N. orthogonal_matrix A <=> (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(column i A) = &1) /\ pairwise (\i j. orthogonal (column i A) (column j A)) (1..dimindex(:N))`, ONCE_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSP] THEN REWRITE_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_INDEXED] THEN SIMP_TAC[ROW_TRANSP; ROWS_TRANSP; pairwise; IN_NUMSEG]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS_PAIRWISE = prove (`!A:real^N^N. orthogonal_matrix A <=> CARD(columns A) = dimindex(:N) /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(column i A) = &1) /\ pairwise orthogonal (columns A)`, ONCE_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSP] THEN REWRITE_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_PAIRWISE] THEN SIMP_TAC[ROW_TRANSP; ROWS_TRANSP]);; let ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS_SPAN = prove (`!A:real^N^N. orthogonal_matrix A <=> span(columns A) = (:real^N) /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(column i A) = &1) /\ pairwise orthogonal (columns A)`, ONCE_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSP] THEN REWRITE_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_SPAN] THEN SIMP_TAC[ROW_TRANSP; ROWS_TRANSP]);; let ORTHOGONAL_MATRIX_2 = prove (`!A:real^2^2. orthogonal_matrix A <=> A$1$1 pow 2 + A$2$1 pow 2 = &1 /\ A$1$2 pow 2 + A$2$2 pow 2 = &1 /\ A$1$1 * A$1$2 + A$2$1 * A$2$2 = &0`, SIMP_TAC[orthogonal_matrix; CART_EQ; matrix_mul; LAMBDA_BETA; TRANSP_COMPONENT; MAT_COMPONENT] THEN REWRITE_TAC[DIMINDEX_2; FORALL_2; SUM_2] THEN CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING);; let ORTHOGONAL_MATRIX_2_ALT = prove (`!A:real^2^2. orthogonal_matrix A <=> A$1$1 pow 2 + A$2$1 pow 2 = &1 /\ (A$1$1 = A$2$2 /\ A$1$2 = --(A$2$1) \/ A$1$1 = --(A$2$2) /\ A$1$2 = A$2$1)`, REWRITE_TAC[ORTHOGONAL_MATRIX_2] THEN CONV_TAC REAL_RING);; let ORTHOGONAL_MATRIX_INV = prove (`!A:real^N^N. orthogonal_matrix A ==> matrix_inv A = transp A`, MESON_TAC[orthogonal_matrix; MATRIX_INV_UNIQUE]);; let ORTHOGONAL_MATRIX_INV_EQ = prove (`!A:real^N^N. orthogonal_matrix(matrix_inv A) <=> orthogonal_matrix A`, MATCH_MP_TAC(MESON[] `(!x. f(f x) = x) /\ (!x. P x ==> P(f x)) ==> (!x. P(f x) <=> P x)`) THEN REWRITE_TAC[MATRIX_INV_INV] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP ORTHOGONAL_MATRIX_INV) THEN ASM_REWRITE_TAC[ORTHOGONAL_MATRIX_TRANSP]);; let ORTHOGONAL_TRANSFORMATION_ORTHOGONAL_EIGENVECTORS = prove (`!f:real^N->real^N v w a b. orthogonal_transformation f /\ f v = a % v /\ f w = b % w /\ ~(a = b) ==> orthogonal v w`, REWRITE_TAC[orthogonal_transformation] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPECL [`v:real^N`; `v:real^N`] th) THEN MP_TAC(SPECL [`v:real^N`; `w:real^N`] th) THEN MP_TAC(SPECL [`w:real^N`; `w:real^N`] th)) THEN ASM_REWRITE_TAC[DOT_LMUL; DOT_RMUL; orthogonal] THEN REWRITE_TAC[REAL_MUL_ASSOC; REAL_RING `x * y = y <=> x = &1 \/ y = &0`] THEN REWRITE_TAC[DOT_EQ_0] THEN ASM_CASES_TAC `v:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_LZERO] THEN ASM_CASES_TAC `w:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_RZERO] THEN ASM_CASES_TAC `(v:real^N) dot w = &0` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `~(a:real = b)` THEN CONV_TAC REAL_RING);; let ORTHOGONAL_MATRIX_ORTHOGONAL_EIGENVECTORS = prove (`!A:real^N^N v w a b. orthogonal_matrix A /\ A ** v = a % v /\ A ** w = b % w /\ ~(a = b) ==> orthogonal v w`, REWRITE_TAC[ORTHOGONAL_MATRIX_TRANSFORMATION; ORTHOGONAL_TRANSFORMATION_ORTHOGONAL_EIGENVECTORS]);; let ORTHOGONAL_TRANSFORMATION_ID = prove (`orthogonal_transformation(\x. x)`, REWRITE_TAC[orthogonal_transformation; LINEAR_ID]);; let ORTHOGONAL_TRANSFORMATION_I = prove (`orthogonal_transformation I`, REWRITE_TAC[I_DEF; ORTHOGONAL_TRANSFORMATION_ID]);; let ORTHOGONAL_TRANSFORMATION_1_GEN = prove (`!f:real^N->real^N. dimindex(:N) = 1 ==> (orthogonal_transformation f <=> f = I \/ f = (--))`, REPEAT STRIP_TAC THEN REWRITE_TAC[I_DEF] THEN GEN_REWRITE_TAC (funpow 3 RAND_CONV) [GSYM ETA_AX] THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ID; ORTHOGONAL_TRANSFORMATION_NEG] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ORTHOGONAL_TRANSFORMATION]) THEN ASM_SIMP_TAC[LINEAR_1_GEN] THEN DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[NORM_MUL] THEN DISCH_THEN(MP_TAC o SPEC `basis 1:real^N`) THEN SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL; DIMINDEX_1] THEN REWRITE_TAC[REAL_ARITH `abs x * &1 = &1 <=> x = &1 \/ x = -- &1`] THEN MATCH_MP_TAC MONO_OR THEN SIMP_TAC[FUN_EQ_THM] THEN REPEAT STRIP_TAC THEN CONV_TAC VECTOR_ARITH);; let ORTHOGONAL_MATRIX_1 = prove (`!m:real^N^N. dimindex(:N) = 1 ==> (orthogonal_matrix m <=> m = mat 1 \/ m = --mat 1)`, REWRITE_TAC[ORTHOGONAL_MATRIX_TRANSFORMATION] THEN SIMP_TAC[ORTHOGONAL_TRANSFORMATION_1_GEN] THEN REWRITE_TAC[MATRIX_EQ; FUN_EQ_THM] THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LID; MATRIX_VECTOR_MUL_LNEG] THEN REWRITE_TAC[I_THM]);; let MATRIX_INV_ORTHOGONAL_LMUL = prove (`!U A:real^M^N. orthogonal_matrix U ==> matrix_inv(U ** A) = matrix_inv A ** matrix_inv U`, REPEAT STRIP_TAC THEN MATCH_MP_TAC MATRIX_INV_UNIQUE_STRONG THEN REWRITE_TAC[symmetric_matrix] THEN REWRITE_TAC[MATRIX_TRANSP_MUL; GSYM MATRIX_MUL_ASSOC] THEN ASM_SIMP_TAC[ORTHOGONAL_MATRIX_INV; TRANSP_TRANSP] THEN REWRITE_TAC[MESON[MATRIX_MUL_ASSOC] `(A:real^M^N) ** transp U ** U ** (B:real^P^M) = A ** (transp U ** U) ** B`] THEN RULE_ASSUM_TAC(REWRITE_RULE[orthogonal_matrix]) THEN ASM_REWRITE_TAC[MATRIX_MUL_LID] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM orthogonal_matrix]) THEN ASM_SIMP_TAC[MATRIX_MUL_LCANCEL; ORTHOGONAL_MATRIX_IMP_INVERTIBLE] THEN REWRITE_TAC[MATRIX_MUL_ASSOC] THEN ASM_SIMP_TAC[MATRIX_MUL_RCANCEL; ORTHOGONAL_MATRIX_IMP_INVERTIBLE; ORTHOGONAL_MATRIX_TRANSP] THEN REWRITE_TAC[GSYM MATRIX_TRANSP_MUL; GSYM MATRIX_MUL_ASSOC] THEN REWRITE_TAC[REWRITE_RULE[symmetric_matrix] SYMMETRIC_MATRIX_INV_LMUL; REWRITE_RULE[symmetric_matrix] SYMMETRIC_MATRIX_INV_RMUL; MATRIX_INV_MUL_INNER; MATRIX_INV_MUL_OUTER]);; let MATRIX_INV_ORTHOGONAL_RMUL = prove (`!U A:real^M^N. orthogonal_matrix U ==> matrix_inv(A ** U) = matrix_inv U ** matrix_inv A`, ONCE_REWRITE_TAC[GSYM TRANSP_EQ; GSYM ORTHOGONAL_MATRIX_TRANSP] THEN SIMP_TAC[TRANSP_MATRIX_INV; MATRIX_TRANSP_MUL; MATRIX_INV_ORTHOGONAL_LMUL]);; let ORTHOGONAL_TRANSFORMATION_EQ_ADJOINT_LEFT = prove (`!f:real^N->real^N. orthogonal_transformation f <=> linear f /\ adjoint f o f = I`, GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; I_THM; o_THM] THEN EQ_TAC THENL [REWRITE_TAC[orthogonal_transformation] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP ADJOINT_WORKS th]) THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[VECTOR_EQ_LDOT]; STRIP_TAC THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION] THEN REWRITE_TAC[NORM_EQ] THEN FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP ADJOINT_WORKS th]) THEN ASM_REWRITE_TAC[]]);; let ORTHOGONAL_TRANSFORMATION_EQ_ADJOINT_RIGHT = prove (`!f:real^N->real^N. orthogonal_transformation f <=> linear f /\ f o adjoint f = I`, GEN_TAC THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_EQ_ADJOINT_LEFT] THEN MESON_TAC[ADJOINT_LINEAR; LINEAR_INVERSE_LEFT]);; let ORTHOGONAL_TRANSFORMATION_EQ_ADJOINT = prove (`!f:real^N->real^N. orthogonal_transformation f <=> linear f /\ adjoint f o f = I /\ f o adjoint f = I`, MESON_TAC[ORTHOGONAL_TRANSFORMATION_EQ_ADJOINT_LEFT; ORTHOGONAL_TRANSFORMATION_EQ_ADJOINT_RIGHT]);; let ORTHOGONAL_TRANSFORMATION_ADJOINT = prove (`!f:real^N->real^N. orthogonal_transformation f ==> orthogonal_transformation(adjoint f)`, REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_EQ_ADJOINT_LEFT] THEN SIMP_TAC[ADJOINT_ADJOINT; ADJOINT_LINEAR] THEN MESON_TAC[ADJOINT_LINEAR; LINEAR_INVERSE_LEFT]);; let ORTHOGONAL_TRANSFORMATION_ADJOINT_EQ = (`!f:real^N->real^N. linear f ==> (orthogonal_transformation(adjoint f) <=> orthogonal_transformation f)`, MESON_TAC[ORTHOGONAL_TRANSFORMATION_ADJOINT; ADJOINT_LINEAR; ADJOINT_ADJOINT]);; let ONORM_ORTHOGONAL_TRANSFORMATION = prove (`!f:real^N->real^N. orthogonal_transformation f ==> onorm f = &1`, SIMP_TAC[ORTHOGONAL_TRANSFORMATION; onorm] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SUP_UNIQUE THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `c:real` THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN DISCH_THEN(MP_TAC o SPEC `basis 1:real^N`) THEN SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL]);; let ONORM_ORTHOGONAL_MATRIX = prove (`!A:real^N^N. orthogonal_matrix A ==> onorm(\x. A ** x) = &1`, REWRITE_TAC[ORTHOGONAL_MATRIX_TRANSFORMATION] THEN REWRITE_TAC[ONORM_ORTHOGONAL_TRANSFORMATION]);; (* ------------------------------------------------------------------------- *) (* Linearity of scaling, and hence isometry, that preserves origin. *) (* ------------------------------------------------------------------------- *) let SCALING_LINEAR = prove (`!f:real^M->real^N c. (f(vec 0) = vec 0) /\ (!x y. dist(f x,f y) = c * dist(x,y)) ==> linear(f)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `!v w. ((f:real^M->real^N) v) dot (f w) = c pow 2 * (v dot w)` ASSUME_TAC THENL [FIRST_ASSUM(MP_TAC o GEN `v:real^M` o SPECL [`v:real^M`; `vec 0 :real^M`]) THEN REWRITE_TAC[dist] THEN ASM_REWRITE_TAC[VECTOR_SUB_RZERO] THEN DISCH_TAC THEN ASM_REWRITE_TAC[DOT_NORM_SUB; GSYM dist] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[linear; VECTOR_EQ] THEN ASM_REWRITE_TAC[DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL] THEN REAL_ARITH_TAC);; let ISOMETRY_LINEAR = prove (`!f:real^M->real^N. (f(vec 0) = vec 0) /\ (!x y. dist(f x,f y) = dist(x,y)) ==> linear(f)`, MESON_TAC[SCALING_LINEAR; REAL_MUL_LID]);; let ISOMETRY_IMP_AFFINITY = prove (`!f:real^M->real^N. (!x y. dist(f x,f y) = dist(x,y)) ==> ?h. linear h /\ !x. f(x) = f(vec 0) + h(x)`, REPEAT STRIP_TAC THEN EXISTS_TAC `\x. (f:real^M->real^N) x - f(vec 0)` THEN REWRITE_TAC[VECTOR_ARITH `a + (x - a):real^N = x`] THEN MATCH_MP_TAC ISOMETRY_LINEAR THEN REWRITE_TAC[VECTOR_SUB_REFL] THEN ASM_REWRITE_TAC[NORM_ARITH `dist(x - a:real^N,y - a) = dist(x,y)`]);; (* ------------------------------------------------------------------------- *) (* An orthogonality-preserving linear map is a similarity. *) (* ------------------------------------------------------------------------- *) let ORTHOGONALITY_PRESERVING_IMP_SCALING = prove (`!f:real^M->real^N. linear f /\ (!x y. orthogonal x y ==> orthogonal (f x) (f y)) ==> ?c. &0 <= c /\ !x. norm(f x) = c * norm(x)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c. &0 <= c /\ !i. 1 <= i /\ i <= dimindex(:M) ==> norm((f:real^M->real^N)(basis i)) = c` MP_TAC THENL [MATCH_MP_TAC(MESON[] `(!x. A(f x)) /\ (?x. P x) /\ (!i j. P i /\ P j ==> f i = f j) ==> ?c. A c /\ !x. P x ==> f x = c`) THEN REWRITE_TAC[NORM_POS_LE] THEN CONJ_TAC THENL [EXISTS_TAC `1` THEN REWRITE_TAC[LE_REFL; DIMINDEX_GE_1]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = j` THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o ISPECL [`basis i + basis j:real^M`; `basis i - basis j:real^M`]) THEN ASM_SIMP_TAC[orthogonal; LINEAR_ADD; LINEAR_SUB; VECTOR_ARITH `(x + y:real^M) dot (x - y) = x dot x - y dot y`] THEN ASM_SIMP_TAC[GSYM NORM_POW_2; REAL_SUB_0; NORM_BASIS] THEN REWRITE_TAC[NORM_POW_2; GSYM NORM_EQ]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real` THEN STRIP_TAC THEN ASM_SIMP_TAC[NORM_EQ_SQUARE; NORM_POS_LE; REAL_LE_MUL] THEN X_GEN_TAC `x:real^M` THEN REWRITE_TAC[GSYM NORM_POW_2] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV o RAND_CONV) [GSYM BASIS_EXPANSION] THEN ASM_SIMP_TAC[LINEAR_VSUM; FINITE_NUMSEG; o_DEF; LINEAR_CMUL] THEN W(MP_TAC o PART_MATCH (lhand o rand) NORM_VSUM_PYTHAGOREAN o lhand o snd) THEN REWRITE_TAC[pairwise; IN_NUMSEG; ORTHOGONAL_MUL; FINITE_NUMSEG] THEN ASM_SIMP_TAC[ORTHOGONAL_BASIS_BASIS] THEN DISCH_THEN SUBST1_TAC THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[NORM_MUL; REAL_POW_MUL; SUM_RMUL; REAL_POW2_ABS] THEN REWRITE_TAC[REAL_POW_2; GSYM dot; GSYM NORM_POW_2]]);; let ORTHOGONALITY_PRESERVING_EQ_SIMILARITY_ALT, ORTHOGONALITY_PRESERVING_EQ_SIMILARITY = (CONJ_PAIR o prove) (`(!f:real^N->real^N. linear f /\ (!x y. orthogonal x y ==> orthogonal (f x) (f y)) <=> ?c g. &0 <= c /\ orthogonal_transformation g /\ f = \z. c % g z) /\ (!f:real^N->real^N. linear f /\ (!x y. orthogonal x y ==> orthogonal (f x) (f y)) <=> ?c g. orthogonal_transformation g /\ f = \z. c % g z)`, REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(TAUT `(q ==> r) /\ (r ==> p) /\ (p ==> q) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THENL [ASM_MESON_TAC[]; STRIP_TAC THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR; LINEAR_COMPOSE_CMUL] THEN ASM_SIMP_TAC[ORTHOGONAL_MUL; ORTHOGONAL_ORTHOGONAL_TRANSFORMATION]; DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONALITY_PRESERVING_IMP_SCALING) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real` THEN ASM_CASES_TAC `c = &0` THENL [ASM_SIMP_TAC[REAL_MUL_LZERO; FUN_EQ_THM; NORM_EQ_0] THEN DISCH_TAC THEN EXISTS_TAC `\x:real^N. x` THEN REWRITE_TAC[VECTOR_MUL_LZERO; ORTHOGONAL_TRANSFORMATION_ID]; STRIP_TAC THEN EXISTS_TAC `\x. inv(c) % (f:real^N->real^N) x` THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; FUN_EQ_THM] THEN ASM_SIMP_TAC[LINEAR_COMPOSE_CMUL; NORM_MUL; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RINV; VECTOR_MUL_LID; REAL_ABS_INV] THEN ASM_REWRITE_TAC[real_abs; REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_MUL_LID]]]);; (* ------------------------------------------------------------------------- *) (* Hence another formulation of orthogonal transformation. *) (* ------------------------------------------------------------------------- *) let ORTHOGONAL_TRANSFORMATION_ISOMETRY = prove (`!f:real^N->real^N. orthogonal_transformation f <=> (f(vec 0) = vec 0) /\ (!x y. dist(f x,f y) = dist(x,y))`, GEN_TAC THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION] THEN EQ_TAC THENL [MESON_TAC[LINEAR_0; LINEAR_SUB; dist]; STRIP_TAC] THEN ASM_SIMP_TAC[ISOMETRY_LINEAR] THEN X_GEN_TAC `x:real^N` THEN FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `vec 0:real^N`]) THEN ASM_REWRITE_TAC[dist; VECTOR_SUB_RZERO]);; (* ------------------------------------------------------------------------- *) (* Can extend an isometry from unit sphere. *) (* ------------------------------------------------------------------------- *) let ISOMETRY_SPHERE_EXTEND = prove (`!f:real^N->real^N. (!x. norm(x) = &1 ==> norm(f x) = &1) /\ (!x y. norm(x) = &1 /\ norm(y) = &1 ==> dist(f x,f y) = dist(x,y)) ==> ?g. orthogonal_transformation g /\ (!x. norm(x) = &1 ==> g(x) = f(x))`, let lemma = prove (`!x:real^N y:real^N x':real^N y':real^N x0 y0 x0' y0'. x = norm(x) % x0 /\ y = norm(y) % y0 /\ x' = norm(x) % x0' /\ y' = norm(y) % y0' /\ norm(x0) = &1 /\ norm(x0') = &1 /\ norm(y0) = &1 /\ norm(y0') = &1 /\ norm(x0' - y0') = norm(x0 - y0) ==> norm(x' - y') = norm(x - y)`, REPEAT GEN_TAC THEN MAP_EVERY ABBREV_TAC [`a = norm(x:real^N)`; `b = norm(y:real^N)`] THEN REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[NORM_EQ; NORM_EQ_1] THEN REWRITE_TAC[DOT_LSUB; DOT_RSUB; DOT_LMUL; DOT_RMUL] THEN REWRITE_TAC[DOT_SYM] THEN CONV_TAC REAL_RING) in REPEAT STRIP_TAC THEN EXISTS_TAC `\x. if x = vec 0 then vec 0 else norm(x) % (f:real^N->real^N)(inv(norm x) % x)` THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ISOMETRY] THEN SIMP_TAC[VECTOR_MUL_LID; REAL_INV_1] THEN CONJ_TAC THENL [ALL_TAC; MESON_TAC[NORM_0; REAL_ARITH `~(&1 = &0)`]] THEN REPEAT GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN REWRITE_TAC[dist; VECTOR_SUB_LZERO; VECTOR_SUB_RZERO; NORM_NEG; NORM_MUL; REAL_ABS_NORM] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_EQ_RDIV_EQ; NORM_POS_LT] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_LT_IMP_NZ; NORM_EQ_0] THEN TRY(FIRST_X_ASSUM MATCH_MP_TAC) THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0] THEN MATCH_MP_TAC lemma THEN MAP_EVERY EXISTS_TAC [`inv(norm x) % x:real^N`; `inv(norm y) % y:real^N`; `(f:real^N->real^N) (inv (norm x) % x)`; `(f:real^N->real^N) (inv (norm y) % y)`] THEN REWRITE_TAC[NORM_MUL; VECTOR_MUL_ASSOC; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_MUL_RINV; NORM_EQ_0] THEN ASM_REWRITE_TAC[GSYM dist; VECTOR_MUL_LID] THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[NORM_MUL; VECTOR_MUL_ASSOC; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_MUL_LINV; REAL_MUL_RINV; NORM_EQ_0]);; let ORTHOGONAL_TRANSFORMATION_INVERSE_o = prove (`!f:real^N->real^N. orthogonal_transformation f ==> ?g. orthogonal_transformation g /\ g o f = I /\ f o g = I`, REPEAT STRIP_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_INJECTIVE) THEN MP_TAC(ISPEC `f:real^N->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `g:real^N->real^N`] LINEAR_INVERSE_LEFT) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION] THEN X_GEN_TAC `v:real^N` THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `norm((f:real^N->real^N)((g:real^N->real^N) v))` THEN CONJ_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION]; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM]) THEN ASM_REWRITE_TAC[]);; let ORTHOGONAL_TRANSFORMATION_INVERSE = prove (`!f:real^N->real^N. orthogonal_transformation f ==> ?g. orthogonal_transformation g /\ (!x. g(f x) = x) /\ (!y. f(g y) = y)`, GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_INVERSE_o) THEN REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM]);; let ONORM_COMPOSE_ORTHOGONAL_TRANSFORMATION_LEFT = prove (`!f g. orthogonal_transformation f ==> onorm(f o g) = onorm g`, SIMP_TAC[ORTHOGONAL_TRANSFORMATION; onorm; o_DEF]);; let ONORM_COMPOSE_ORTHOGONAL_TRANSFORMATION_RIGHT = prove (`!f g. orthogonal_transformation g ==> onorm(f o g) = onorm f`, REPEAT STRIP_TAC THEN REWRITE_TAC[onorm; o_DEF] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_INVERSE_o) THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]);; (* ------------------------------------------------------------------------- *) (* Reading operator norms off eigenvalue bases or diagonalizations. *) (* ------------------------------------------------------------------------- *) let SQNORM_LE_MAX_EIGENVECTOR_SPAN = prove (`!(f:real^N->real^N) b c x l. linear f /\ pairwise orthogonal b /\ (!x. x IN b ==> f x = c x % x /\ c x pow 2 <= l) /\ x IN span b ==> norm(f x) pow 2 <= l * norm x pow 2`, REPEAT GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP PAIRWISE_ORTHOGONAL_IMP_FINITE) THEN ASM_SIMP_TAC[SPAN_FINITE; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `v:real^N->real` THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[LINEAR_VSUM; o_DEF; LINEAR_CMUL] THEN W(MP_TAC o PART_MATCH (lhand o rand) NORM_VSUM_PYTHAGOREAN o lhand o snd) THEN W(MP_TAC o PART_MATCH(lhand o rand) NORM_VSUM_PYTHAGOREAN o rand o rand o rand o snd) THEN ASM_REWRITE_TAC[] THEN REPEAT(ANTS_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN REWRITE_TAC[pairwise; ORTHOGONAL_MUL] THEN ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC]) THEN REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] VECTOR_MUL_ASSOC] THEN REWRITE_TAC[GSYM VECTOR_MUL_ASSOC] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [NORM_MUL] THEN REWRITE_TAC[REAL_POW_MUL] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_SIMP_TAC[REAL_POW2_ABS; REAL_LE_POW_2]);; let NORM_LE_MAX_EIGENVECTOR_SPAN = prove (`!(f:real^N->real^N) b c x l. linear f /\ pairwise orthogonal b /\ (!x. x IN b ==> f x = c x % x /\ abs(c x) <= l) /\ x IN span b ==> norm(f x) <= l * norm x`, REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N->bool = {}` THENL [ASM_REWRITE_TAC[SPAN_EMPTY; IN_SING] THEN MESON_TAC[LINEAR_0; NORM_0; REAL_MUL_RZERO; REAL_LE_REFL]; STRIP_TAC] THEN GEN_REWRITE_TAC I [NORM_LE_SQUARE] THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN ASM_MESON_TAC[REAL_ABS_POS; REAL_LE_TRANS; MEMBER_NOT_EMPTY]; REWRITE_TAC[REAL_POW_MUL; GSYM NORM_POW_2]] THEN MATCH_MP_TAC SQNORM_LE_MAX_EIGENVECTOR_SPAN THEN MAP_EVERY EXISTS_TAC [`b:real^N->bool`; `c:real^N->real`] THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_ABS_ABS] THEN ASM_REAL_ARITH_TAC);; let ONORM_EQ_MAX_EIGENVECTOR = prove (`!(f:real^N->real^N) b c. linear f /\ pairwise orthogonal b /\ span b = (:real^N) /\ ~(vec 0 IN b) /\ (!x. x IN b ==> f x = c x % x) ==> onorm f = sup {abs(c x) | x IN b}`, REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N->bool = {}` THENL [ASM_REWRITE_TAC[SPAN_EMPTY] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s = UNIV ==> (?x. ~(x IN s)) ==> P`)) THEN EXISTS_TAC `vec 1:real^N` THEN REWRITE_TAC[VEC_EQ; IN_SING; ARITH_EQ]; STRIP_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM]] THEN CONJ_TAC THENL [ASM_SIMP_TAC[ONORM_LE_EQ] THEN GEN_TAC THEN MATCH_MP_TAC NORM_LE_MAX_EIGENVECTOR_SPAN THEN MAP_EVERY EXISTS_TAC [`b:real^N->bool`; `c:real^N->real`] THEN ASM_REWRITE_TAC[IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_SIMP_TAC[REAL_LE_SUP_FINITE; SIMPLE_IMAGE; FINITE_IMAGE; IMAGE_EQ_EMPTY; PAIRWISE_ORTHOGONAL_IMP_FINITE] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[REAL_LE_REFL]; MATCH_MP_TAC REAL_SUP_LE THEN ASM_SIMP_TAC[SIMPLE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_RCANCEL_IMP THEN EXISTS_TAC `norm(x:real^N)` THEN REWRITE_TAC[NORM_POS_LT] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN TRANS_TAC REAL_LE_TRANS `norm((f:real^N->real^N) x)` THEN ASM_SIMP_TAC[ONORM; NORM_MUL; REAL_LE_REFL]]);; let ONORM_ORTHOGONAL_MATRIX_MUL_LEFT = prove (`!(A:real^N^N) (P:real^N^N). orthogonal_matrix P ==> onorm (\x. (P ** A) ** x) = onorm(\x. A ** x)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(\x. ((P:real^N^N) ** (A:real^N^N)) ** x) = (\x. P ** x) o (\x. A ** x)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; MATRIX_VECTOR_MUL_ASSOC]; ALL_TAC] THEN MATCH_MP_TAC ONORM_COMPOSE_ORTHOGONAL_TRANSFORMATION_LEFT THEN ASM_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSFORMATION]);; let ONORM_ORTHOGONAL_MATRIX_MUL_RIGHT = prove (`!(A:real^N^N) (P:real^N^N). orthogonal_matrix P ==> onorm (\x. (A ** P) ** x) = onorm(\x. A ** x)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `(\x. ((A:real^N^N) ** (P:real^N^N)) ** x) = (\x. A ** x) o (\x. P ** x)` SUBST1_TAC THENL [REWRITE_TAC[o_DEF; MATRIX_VECTOR_MUL_ASSOC]; ALL_TAC] THEN MATCH_MP_TAC ONORM_COMPOSE_ORTHOGONAL_TRANSFORMATION_RIGHT THEN ASM_REWRITE_TAC[GSYM ORTHOGONAL_MATRIX_TRANSFORMATION]);; let ONORM_DIAGONALIZED_MATRIX = prove (`!(A:real^N^N) D P. orthogonal_matrix P /\ diagonal_matrix D /\ transp P ** D ** P = A ==> onorm(\x. A ** x) = sup {abs(D$i$i) | 1 <= i /\ i <= dimindex (:N)}`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[ONORM_ORTHOGONAL_MATRIX_MUL_LEFT; ORTHOGONAL_MATRIX_TRANSP; ONORM_ORTHOGONAL_MATRIX_MUL_RIGHT] THEN ASM_SIMP_TAC[ONORM_DIAGONAL_MATRIX]);; let ONORM_DIAGONALIZED_COVARIANCE_MATRIX = prove (`!(A:real^N^N) D P. orthogonal_matrix P /\ diagonal_matrix D /\ transp P ** D ** P = transp A ** A ==> onorm(\x. A ** x) = sqrt(sup {abs(D$i$i) | 1 <= i /\ i <= dimindex (:N)})`, REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SQRT_UNIQUE THEN SIMP_TAC[ONORM_POS_LE; MATRIX_VECTOR_MUL_LINEAR] THEN REWRITE_TAC[GSYM ONORM_COVARIANCE] THEN MATCH_MP_TAC ONORM_DIAGONALIZED_MATRIX THEN ASM_MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* We can find an orthogonal matrix taking any unit vector to any other. *) (* ------------------------------------------------------------------------- *) let ORTHOGONAL_MATRIX_EXISTS_BASIS = prove (`!a:real^N. norm(a) = &1 ==> ?A. orthogonal_matrix A /\ A**(basis 1) = a`, REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP VECTOR_IN_ORTHONORMAL_BASIS) THEN REWRITE_TAC[HAS_SIZE] THEN DISCH_THEN(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] FINITE_INDEX_NUMSEG_SPECIAL) THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN REWRITE_TAC[TAUT `a /\ b ==> c <=> c \/ ~a \/ ~b`] THEN DISCH_THEN(X_CHOOSE_THEN `f:num->real^N` (CONJUNCTS_THEN2 ASSUME_TAC (CONJUNCTS_THEN2 (ASSUME_TAC o SYM) ASSUME_TAC))) THEN EXISTS_TAC `(lambda i j. ((f j):real^N)$i):real^N^N` THEN SIMP_TAC[CART_EQ; LAMBDA_BETA; matrix_vector_mul; BASIS_COMPONENT; IN_NUMSEG] THEN ONCE_REWRITE_TAC[COND_RAND] THEN SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA] THEN ASM_REWRITE_TAC[IN_NUMSEG; REAL_MUL_RID; LE_REFL; DIMINDEX_GE_1] THEN REWRITE_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS] THEN SIMP_TAC[column; LAMBDA_BETA] THEN CONJ_TAC THENL [X_GEN_TAC `i:num` THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `norm((f:num->real^N) i)` THEN CONJ_TAC THENL [AP_TERM_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA]; ASM_MESON_TAC[IN_IMAGE; IN_NUMSEG]]; MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN SUBGOAL_THEN `orthogonal ((f:num->real^N) i) (f j)` MP_TAC THENL [ASM_MESON_TAC[pairwise; IN_IMAGE; IN_NUMSEG]; ALL_TAC] THEN MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA]]);; let ORTHOGONAL_TRANSFORMATION_EXISTS_1 = prove (`!a b:real^N. norm(a) = &1 /\ norm(b) = &1 ==> ?f. orthogonal_transformation f /\ f a = b`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `b:real^N` ORTHOGONAL_MATRIX_EXISTS_BASIS) THEN MP_TAC(ISPEC `a:real^N` ORTHOGONAL_MATRIX_EXISTS_BASIS) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `A:real^N^N` (STRIP_ASSUME_TAC o GSYM)) THEN DISCH_THEN(X_CHOOSE_THEN `B:real^N^N` (STRIP_ASSUME_TAC o GSYM)) THEN EXISTS_TAC `\x:real^N. ((B:real^N^N) ** transp(A:real^N^N)) ** x` THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX; MATRIX_VECTOR_MUL_LINEAR; MATRIX_OF_MATRIX_VECTOR_MUL] THEN ASM_SIMP_TAC[ORTHOGONAL_MATRIX_MUL; ORTHOGONAL_MATRIX_TRANSP] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_ASSOC] THEN AP_TERM_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[ORTHOGONAL_MATRIX]) THEN ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_ASSOC; MATRIX_VECTOR_MUL_LID]);; let ORTHOGONAL_TRANSFORMATION_EXISTS = prove (`!a b:real^N. norm(a) = norm(b) ==> ?f. orthogonal_transformation f /\ f a = b`, REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N = vec 0` THEN ASM_SIMP_TAC[NORM_0; NORM_EQ_0] THENL [MESON_TAC[ORTHOGONAL_TRANSFORMATION_ID]; ALL_TAC] THEN ASM_CASES_TAC `a:real^N = vec 0` THENL [ASM_MESON_TAC[NORM_0; NORM_EQ_0]; ALL_TAC] THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(norm a) % a:real^N`; `inv(norm b) % b:real^N`] ORTHOGONAL_TRANSFORMATION_EXISTS_1) THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[NORM_EQ_0; REAL_MUL_LINV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP LINEAR_CMUL o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN ASM_REWRITE_TAC[VECTOR_ARITH `a % x:real^N = a % y <=> a % (x - y) = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0; VECTOR_SUB_EQ]);; (* ------------------------------------------------------------------------- *) (* Or indeed, taking any subspace to another of suitable dimension. *) (* ------------------------------------------------------------------------- *) let ORTHOGONAL_TRANSFORMATION_INTO_SUBSPACE = prove (`!s t:real^N->bool. subspace s /\ subspace t /\ dim s <= dim t ==> ?f. orthogonal_transformation f /\ IMAGE f s SUBSET t`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `t:real^N->bool` ORTHONORMAL_BASIS_SUBSPACE) THEN MP_TAC(ISPEC `s:real^N->bool` ORTHONORMAL_BASIS_SUBSPACE) THEN ASM_REWRITE_TAC[HAS_SIZE] THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`c:real^N->bool`; `(:real^N)`] ORTHONORMAL_EXTENSION) THEN MP_TAC(ISPECL [`b:real^N->bool`; `(:real^N)`] ORTHONORMAL_EXTENSION) THEN ASM_REWRITE_TAC[UNION_UNIV; SPAN_UNIV; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b':real^N->bool` THEN STRIP_TAC THEN X_GEN_TAC `c':real^N->bool` THEN STRIP_TAC THEN SUBGOAL_THEN `independent(b UNION b':real^N->bool) /\ independent(c UNION c':real^N->bool)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN ASM_REWRITE_TAC[IN_UNION] THEN ASM_MESON_TAC[NORM_ARITH `~(norm(vec 0:real^N) = &1)`]; ALL_TAC] THEN SUBGOAL_THEN `FINITE(b UNION b':real^N->bool) /\ FINITE(c UNION c':real^N->bool)` MP_TAC THENL [ASM_SIMP_TAC[PAIRWISE_ORTHOGONAL_IMP_FINITE]; REWRITE_TAC[FINITE_UNION] THEN STRIP_TAC] THEN SUBGOAL_THEN `?f:real^N->real^N. (!x y. x IN b UNION b' /\ y IN b UNION b' ==> (f x = f y <=> x = y)) /\ IMAGE f b SUBSET c /\ IMAGE f (b UNION b') SUBSET c UNION c'` (X_CHOOSE_THEN `fb:real^N->real^N` STRIP_ASSUME_TAC) THENL [MP_TAC(ISPECL [`b:real^N->bool`; `c:real^N->bool`] CARD_LE_INJ) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; INJECTIVE_ON_ALT] THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN MP_TAC(ISPECL [`b':real^N->bool`; `(c UNION c') DIFF IMAGE (f:real^N->real^N) b`] CARD_LE_INJ) THEN ANTS_TAC THENL [ASM_SIMP_TAC[FINITE_UNION; FINITE_DIFF] THEN W(MP_TAC o PART_MATCH (lhs o rand) CARD_DIFF o rand o snd) THEN ASM_REWRITE_TAC[FINITE_UNION] THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN MATCH_MP_TAC(ARITH_RULE `a + b:num = c ==> a <= c - b`) THEN W(MP_TAC o PART_MATCH (lhs o rand) CARD_IMAGE_INJ o rand o lhs o snd) THEN ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC] THEN W(MP_TAC o PART_MATCH (rhs o rand) CARD_UNION o lhs o snd) THEN ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [UNION_COMM] THEN MATCH_MP_TAC(MESON[LE_ANTISYM] `(FINITE s /\ CARD s <= CARD t) /\ (FINITE t /\ CARD t <= CARD s) ==> CARD s = CARD t`) THEN CONJ_TAC THEN MATCH_MP_TAC INDEPENDENT_SPAN_BOUND THEN ASM_REWRITE_TAC[FINITE_UNION; SUBSET_UNIV]; DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `\x. if x IN b then (f:real^N->real^N) x else g x` THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM SET_TAC[]]; ALL_TAC] THEN MP_TAC(ISPECL [`fb:real^N->real^N`; `b UNION b':real^N->bool`] LINEAR_INDEPENDENT_EXTEND) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION]; REWRITE_TAC[SYM(ASSUME `span b:real^N->bool = s`)] THEN ASM_SIMP_TAC[GSYM SPAN_LINEAR_IMAGE] THEN REWRITE_TAC[SYM(ASSUME `span c:real^N->bool = t`)] THEN MATCH_MP_TAC SPAN_MONO THEN ASM SET_TAC[]] THEN SUBGOAL_THEN `!v. v IN UNIV ==> norm((f:real^N->real^N) v) = norm v` (fun th -> ASM_MESON_TAC[th; IN_UNIV]) THEN UNDISCH_THEN `span (b UNION b') = (:real^N)` (SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[SPAN_FINITE; FINITE_UNION; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`z:real^N`; `u:real^N->real`] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[LINEAR_VSUM; FINITE_UNION] THEN REWRITE_TAC[o_DEF; NORM_EQ_SQUARE; NORM_POS_LE; GSYM NORM_POW_2] THEN ASM_SIMP_TAC[LINEAR_CMUL] THEN W(MP_TAC o PART_MATCH (lhand o rand) NORM_VSUM_PYTHAGOREAN o rand o snd) THEN W(MP_TAC o PART_MATCH (lhand o rand) NORM_VSUM_PYTHAGOREAN o lhand o rand o snd) THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM_SIMP_TAC[pairwise; ORTHOGONAL_CLAUSES; FINITE_UNION] THEN ANTS_TAC THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[ORTHOGONAL_MUL] THEN REPEAT DISJ2_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[]; REPEAT(DISCH_THEN SUBST1_TAC) THEN ASM_SIMP_TAC[NORM_MUL] THEN MATCH_MP_TAC SUM_EQ THEN ASM SET_TAC[]]);; let ORTHOGONAL_TRANSFORMATION_ONTO_SUBSPACE = prove (`!s t:real^N->bool. subspace s /\ subspace t /\ dim s = dim t ==> ?f. orthogonal_transformation f /\ IMAGE f s = t`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] ORTHOGONAL_TRANSFORMATION_INTO_SUBSPACE) THEN ASM_REWRITE_TAC[LE_REFL] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `span(IMAGE (f:real^N->real^N) s) = span t` MP_TAC THENL [MATCH_MP_TAC DIM_EQ_SPAN THEN ASM_REWRITE_TAC[] THEN W(MP_TAC o PART_MATCH (lhs o rand) DIM_INJECTIVE_LINEAR_IMAGE o rand o snd) THEN ASM_MESON_TAC[LE_REFL; orthogonal_transformation; ORTHOGONAL_TRANSFORMATION_INJECTIVE]; ASM_SIMP_TAC[SPAN_LINEAR_IMAGE; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN ASM_SIMP_TAC[SPAN_OF_SUBSPACE]]);; (* ------------------------------------------------------------------------- *) (* Rotation, reflection, rotoinversion. *) (* ------------------------------------------------------------------------- *) let rotation_matrix = new_definition `rotation_matrix Q <=> orthogonal_matrix Q /\ det(Q) = &1`;; let rotoinversion_matrix = new_definition `rotoinversion_matrix Q <=> orthogonal_matrix Q /\ det(Q) = -- &1`;; let ORTHOGONAL_ROTATION_OR_ROTOINVERSION = prove (`!Q. orthogonal_matrix Q <=> rotation_matrix Q \/ rotoinversion_matrix Q`, MESON_TAC[rotation_matrix; rotoinversion_matrix; DET_ORTHOGONAL_MATRIX]);; let ROTATION_MATRIX_1 = prove (`!m:real^N^N. dimindex(:N) = 1 ==> (rotation_matrix m <=> m = mat 1)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[ORTHOGONAL_MATRIX_1; rotation_matrix] THEN ASM_CASES_TAC `m:real^N^N = mat 1` THEN ASM_REWRITE_TAC[DET_I] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[DET_NEG; REAL_POW_ONE; DET_I] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let ROTOINVERSION_MATRIX_1 = prove (`!m:real^N^N. dimindex(:N) = 1 ==> (rotoinversion_matrix m <=> m = --mat 1)`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[ORTHOGONAL_MATRIX_1; rotoinversion_matrix] THEN ASM_CASES_TAC `m:real^N^N = --mat 1` THEN ASM_REWRITE_TAC[DET_NEG; DET_I; REAL_POW_ONE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[DET_I] THEN CONV_TAC REAL_RAT_REDUCE_CONV);; let ROTATION_MATRIX_2 = prove (`!A:real^2^2. rotation_matrix A <=> A$1$1 pow 2 + A$2$1 pow 2 = &1 /\ A$1$1 = A$2$2 /\ A$1$2 = --(A$2$1)`, REWRITE_TAC[rotation_matrix; ORTHOGONAL_MATRIX_2; DET_2] THEN CONV_TAC REAL_RING);; (* ------------------------------------------------------------------------- *) (* Slightly stronger results giving rotation, but only in >= 2 dimensions. *) (* ------------------------------------------------------------------------- *) let ROTATION_MATRIX_EXISTS_BASIS = prove (`!a:real^N. 2 <= dimindex(:N) /\ norm(a) = &1 ==> ?A. rotation_matrix A /\ A**(basis 1) = a`, REPEAT STRIP_TAC THEN FIRST_X_ASSUM(X_CHOOSE_THEN `A:real^N^N` STRIP_ASSUME_TAC o MATCH_MP ORTHOGONAL_MATRIX_EXISTS_BASIS) THEN FIRST_ASSUM(DISJ_CASES_TAC o GEN_REWRITE_RULE I [ORTHOGONAL_ROTATION_OR_ROTOINVERSION]) THENL [ASM_MESON_TAC[]; ALL_TAC] THEN EXISTS_TAC `transp(lambda i. if i = dimindex(:N) then -- &1 % transp A$i else (transp A:real^N^N)$i):real^N^N` THEN REWRITE_TAC[rotation_matrix; DET_TRANSP] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[ORTHOGONAL_MATRIX_TRANSP]; SIMP_TAC[DET_ROW_MUL; DIMINDEX_GE_1; LE_REFL] THEN MATCH_MP_TAC(REAL_ARITH `x = -- &1 ==> -- &1 * x = &1`) THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [rotoinversion_matrix]) THEN DISCH_THEN(SUBST1_TAC o SYM o CONJUNCT2) THEN GEN_REWRITE_TAC RAND_CONV [GSYM DET_TRANSP] THEN AP_TERM_TAC THEN SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN MESON_TAC[]; FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN SIMP_TAC[matrix_vector_mul; LAMBDA_BETA; CART_EQ; transp; BASIS_COMPONENT] THEN ONCE_REWRITE_TAC[REAL_ARITH `x * (if p then &1 else &0) = if p then x else &0`] THEN ASM_SIMP_TAC[ARITH_RULE `2 <= n ==> ~(1 = n)`; LAMBDA_BETA]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM ORTHOGONAL_MATRIX_TRANSP]) THEN SPEC_TAC(`transp(A:real^N^N)`,`B:real^N^N`) THEN GEN_TAC THEN SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N) ==> row i ((lambda i. if i = dimindex(:N) then -- &1 % B$i else (B:real^N^N)$i):real^N^N) = if i = dimindex(:N) then --(row i B) else row i B` ASSUME_TAC THENL [SIMP_TAC[row; LAMBDA_BETA; LAMBDA_ETA; VECTOR_MUL_LID; VECTOR_MUL_LNEG]; ASM_SIMP_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS] THEN ASM_MESON_TAC[ORTHOGONAL_LNEG; ORTHOGONAL_RNEG; NORM_NEG]]);; let ROTATION_EXISTS_1 = prove (`!a b:real^N. 2 <= dimindex(:N) /\ norm(a) = &1 /\ norm(b) = &1 ==> ?f. orthogonal_transformation f /\ det(matrix f) = &1 /\ f a = b`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `b:real^N` ROTATION_MATRIX_EXISTS_BASIS) THEN MP_TAC(ISPEC `a:real^N` ROTATION_MATRIX_EXISTS_BASIS) THEN ASM_REWRITE_TAC[rotation_matrix] THEN DISCH_THEN(X_CHOOSE_THEN `A:real^N^N` (CONJUNCTS_THEN2 STRIP_ASSUME_TAC (ASSUME_TAC o SYM))) THEN DISCH_THEN(X_CHOOSE_THEN `B:real^N^N` (CONJUNCTS_THEN2 STRIP_ASSUME_TAC (ASSUME_TAC o SYM))) THEN EXISTS_TAC `\x:real^N. ((B:real^N^N) ** transp(A:real^N^N)) ** x` THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX; MATRIX_VECTOR_MUL_LINEAR; MATRIX_OF_MATRIX_VECTOR_MUL; DET_MUL; DET_TRANSP] THEN ASM_SIMP_TAC[ORTHOGONAL_MATRIX_MUL; ORTHOGONAL_MATRIX_TRANSP] THEN REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_ASSOC; REAL_MUL_LID] THEN AP_TERM_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[ORTHOGONAL_MATRIX]) THEN ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_ASSOC; MATRIX_VECTOR_MUL_LID]);; let ROTATION_EXISTS = prove (`!a b:real^N. 2 <= dimindex(:N) /\ norm(a) = norm(b) ==> ?f. orthogonal_transformation f /\ det(matrix f) = &1 /\ f a = b`, REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N = vec 0` THEN ASM_SIMP_TAC[NORM_0; NORM_EQ_0] THENL [MESON_TAC[ORTHOGONAL_TRANSFORMATION_ID; MATRIX_ID; DET_I]; ALL_TAC] THEN ASM_CASES_TAC `a:real^N = vec 0` THENL [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_ID; MATRIX_ID; DET_I; NORM_0; NORM_EQ_0]; ALL_TAC] THEN DISCH_TAC THEN MP_TAC(ISPECL [`inv(norm a) % a:real^N`; `inv(norm b) % b:real^N`] ROTATION_EXISTS_1) THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[NORM_EQ_0; REAL_MUL_LINV] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP LINEAR_CMUL o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN ASM_REWRITE_TAC[VECTOR_ARITH `a % x:real^N = a % y <=> a % (x - y) = vec 0`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0; VECTOR_SUB_EQ]);; let ROTATION_RIGHTWARD_LINE = prove (`!a:real^N k. 1 <= k /\ k <= dimindex(:N) ==> ?b f. orthogonal_transformation f /\ (2 <= dimindex(:N) ==> det(matrix f) = &1) /\ f(b % basis k) = a /\ &0 <= b`, REPEAT STRIP_TAC THEN EXISTS_TAC `norm(a:real^N)` THEN ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT; LE_REFL; DIMINDEX_GE_1; REAL_MUL_RID; NORM_POS_LE; LT_IMP_LE; LTE_ANTISYM] THEN REWRITE_TAC[ARITH_RULE `2 <= n <=> 1 <= n /\ ~(n = 1)`; DIMINDEX_GE_1] THEN ASM_CASES_TAC `dimindex(:N) = 1` THEN ASM_REWRITE_TAC[] THENL [MATCH_MP_TAC ORTHOGONAL_TRANSFORMATION_EXISTS; MATCH_MP_TAC ROTATION_EXISTS] THEN ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; LE_REFL; DIMINDEX_GE_1] THEN REWRITE_TAC[REAL_ABS_NORM; REAL_MUL_RID] THEN MATCH_MP_TAC(ARITH_RULE `~(n = 1) /\ 1 <= n ==> 2 <= n`) THEN ASM_REWRITE_TAC[DIMINDEX_GE_1]);; (* ------------------------------------------------------------------------- *) (* In 3 dimensions, a rotation is indeed about an "axis". *) (* ------------------------------------------------------------------------- *) let EULER_ROTATION_THEOREM = prove (`!A:real^3^3. rotation_matrix A ==> ?v:real^3. ~(v = vec 0) /\ A ** v = v`, REPEAT STRIP_TAC THEN MP_TAC(ISPEC `A - mat 1:real^3^3` HOMOGENEOUS_LINEAR_EQUATIONS_DET) THEN REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_RDISTRIB; VECTOR_SUB_EQ; MATRIX_VECTOR_MUL_LID] THEN DISCH_THEN SUBST1_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[rotation_matrix; orthogonal_matrix; DET_3] THEN SIMP_TAC[CART_EQ; FORALL_3; MAT_COMPONENT; DIMINDEX_3; LAMBDA_BETA; ARITH; MATRIX_SUB_COMPONENT; MAT_COMPONENT; SUM_3; matrix_mul; transp; matrix_vector_mul] THEN CONV_TAC REAL_RING);; let EULER_ROTOINVERSION_THEOREM = prove (`!A:real^3^3. rotoinversion_matrix A ==> ?v:real^3. ~(v = vec 0) /\ A ** v = --v`, REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_ARITH `a:real^N = --v <=> a + v = vec 0`] THEN MP_TAC(ISPEC `A + mat 1:real^3^3` HOMOGENEOUS_LINEAR_EQUATIONS_DET) THEN REWRITE_TAC[MATRIX_VECTOR_MUL_ADD_RDISTRIB; MATRIX_VECTOR_MUL_LID] THEN DISCH_THEN SUBST1_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[rotoinversion_matrix; orthogonal_matrix; DET_3] THEN SIMP_TAC[CART_EQ; FORALL_3; MAT_COMPONENT; DIMINDEX_3; LAMBDA_BETA; ARITH; MATRIX_ADD_COMPONENT; MAT_COMPONENT; SUM_3; matrix_mul; transp; matrix_vector_mul] THEN CONV_TAC REAL_RING);; (* ------------------------------------------------------------------------- *) (* We can always rotate so that a hyperplane is "horizontal". *) (* ------------------------------------------------------------------------- *) let ROTATION_LOWDIM_HORIZONTAL = prove (`!s:real^N->bool. dim s < dimindex(:N) ==> ?f. orthogonal_transformation f /\ det(matrix f) = &1 /\ (IMAGE f s) SUBSET {z | z$(dimindex(:N)) = &0}`, GEN_TAC THEN ASM_CASES_TAC `dim(s:real^N->bool) = 0` THENL [RULE_ASSUM_TAC(REWRITE_RULE[DIM_EQ_0]) THEN DISCH_TAC THEN EXISTS_TAC `\x:real^N. x` THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ID; MATRIX_ID; DET_I] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE `s SUBSET {a} ==> a IN t ==> IMAGE (\x. x) s SUBSET t`)) THEN SIMP_TAC[IN_ELIM_THM; VEC_COMPONENT; LE_REFL; DIMINDEX_GE_1]; DISCH_TAC] THEN SUBGOAL_THEN `2 <= dimindex(:N)` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN MP_TAC(ISPECL [`a:real^N`; `norm(a:real^N) % basis(dimindex(:N)):real^N`] ROTATION_EXISTS) THEN ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; LE_REFL; DIMINDEX_GE_1] THEN REWRITE_TAC[REAL_ABS_NORM; REAL_MUL_RID] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN SUBGOAL_THEN `(f:real^N->real^N) x dot (f a) = &0` MP_TAC THENL [RULE_ASSUM_TAC(REWRITE_RULE[orthogonal_transformation]) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[DOT_SYM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN ASM_SIMP_TAC[SPAN_SUPERSET; IN_ELIM_THM]; ASM_SIMP_TAC[DOT_BASIS; LE_REFL; DIMINDEX_GE_1; DOT_RMUL] THEN ASM_REWRITE_TAC[REAL_ENTIRE; NORM_EQ_0]]);; let ORTHOGONAL_TRANSFORMATION_LOWDIM_HORIZONTAL = prove (`!s:real^N->bool. dim s < dimindex(:N) ==> ?f. orthogonal_transformation f /\ (IMAGE f s) SUBSET {z | z$(dimindex(:N)) = &0}`, GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP ROTATION_LOWDIM_HORIZONTAL) THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[]);; let ORTHOGONAL_TRANSFORMATION_BETWEEN_ORTHOGONAL_SETS = prove (`!v:num->real^N w k. pairwise (\i j. orthogonal (v i) (v j)) k /\ pairwise (\i j. orthogonal (w i) (w j)) k /\ (!i. i IN k ==> norm(v i) = norm(w i)) ==> ?f. orthogonal_transformation f /\ (!i. i IN k ==> f(v i) = w i)`, let lemma1 = prove (`!v:num->real^N n. pairwise (\i j. orthogonal (v i) (v j)) (1..n) /\ (!i. 1 <= i /\ i <= n ==> norm(v i) = &1) ==> ?f. orthogonal_transformation f /\ (!i. 1 <= i /\ i <= n ==> f(basis i) = v i)`, REWRITE_TAC[pairwise; IN_NUMSEG; GSYM CONJ_ASSOC] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `pairwise orthogonal (IMAGE (v:num->real^N) (1..n))` ASSUME_TAC THENL [REWRITE_TAC[PAIRWISE_IMAGE] THEN ASM_SIMP_TAC[pairwise; IN_NUMSEG]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_ORTHOGONAL_INDEPENDENT)) THEN REWRITE_TAC[SET_RULE `~(a IN IMAGE f s) <=> !x. x IN s ==> ~(f x = a)`] THEN ANTS_TAC THENL [REWRITE_TAC[IN_NUMSEG] THEN ASM_MESON_TAC[NORM_0; REAL_ARITH `~(&1 = &0)`]; DISCH_THEN(MP_TAC o CONJUNCT2 o MATCH_MP INDEPENDENT_BOUND)] THEN SUBGOAL_THEN `!i j. 1 <= i /\ i <= n /\ 1 <= j /\ j <= n /\ ~(i = j) ==> ~(v i:real^N = v j)` ASSUME_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_REFL; NORM_0; REAL_ARITH `~(&1 = &0)`]; ALL_TAC] THEN SUBGOAL_THEN `CARD(IMAGE (v:num->real^N) (1..n)) = n` ASSUME_TAC THENL [W(MP_TAC o PART_MATCH (lhs o rand) CARD_IMAGE_INJ o lhs o snd) THEN ASM_REWRITE_TAC[CARD_NUMSEG_1; IN_NUMSEG; FINITE_NUMSEG] THEN ASM_MESON_TAC[]; ASM_REWRITE_TAC[] THEN DISCH_TAC] THEN SUBGOAL_THEN `?w:num->real^N. pairwise (\i j. orthogonal (w i) (w j)) (1..dimindex(:N)) /\ (!i. 1 <= i /\ i <= dimindex(:N) ==> norm(w i) = &1) /\ (!i. 1 <= i /\ i <= n ==> w i = v i)` STRIP_ASSUME_TAC THENL [ALL_TAC; EXISTS_TAC `(\x. vsum(1..dimindex(:N)) (\i. x$i % w i)):real^N->real^N` THEN SIMP_TAC[BASIS_COMPONENT; IN_NUMSEG; COND_RATOR; COND_RAND] THEN REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_LZERO; VSUM_DELTA] THEN ASM_SIMP_TAC[IN_NUMSEG] THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[LE_TRANS]] THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX] THEN CONJ_TAC THENL [MATCH_MP_TAC LINEAR_COMPOSE_VSUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN REWRITE_TAC[linear; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[matrix; column; ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS] THEN SIMP_TAC[LAMBDA_BETA; LAMBDA_ETA; BASIS_COMPONENT; IN_NUMSEG] THEN SIMP_TAC[COND_RATOR; COND_RAND; VECTOR_MUL_LZERO; VSUM_DELTA] THEN SIMP_TAC[IN_NUMSEG; orthogonal; dot; LAMBDA_BETA; NORM_EQ_SQUARE] THEN REWRITE_TAC[VECTOR_MUL_LID; GSYM dot; GSYM NORM_EQ_SQUARE] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise; IN_NUMSEG; orthogonal]) THEN ASM_SIMP_TAC[]] THEN FIRST_ASSUM(MP_TAC o SPEC `(:real^N)` o MATCH_MP (REWRITE_RULE[IMP_CONJ] ORTHONORMAL_EXTENSION)) THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG; UNION_UNIV; SPAN_UNIV] THEN DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`n+1..dimindex(:N)`; `t:real^N->bool`] CARD_EQ_BIJECTION) THEN ANTS_TAC THENL [REWRITE_TAC[FINITE_NUMSEG] THEN MP_TAC(ISPECL [`(:real^N)`; `IMAGE v (1..n) UNION t:real^N->bool`] BASIS_CARD_EQ_DIM) THEN ASM_REWRITE_TAC[SUBSET_UNIV] THEN ANTS_TAC THENL [MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN ASM_REWRITE_TAC[IN_UNION; DE_MORGAN_THM; IN_NUMSEG] THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG; SET_RULE `~(x IN s) <=> !y. y IN s ==> ~(y = x)`] THEN ASM_MESON_TAC[NORM_0; REAL_ARITH `~(&1 = &0)`]; ALL_TAC] THEN ASM_SIMP_TAC[FINITE_UNION; IMP_CONJ; FINITE_IMAGE; CARD_UNION; SET_RULE `t INTER s = {} <=> DISJOINT s t`] THEN DISCH_TAC THEN DISCH_TAC THEN REWRITE_TAC[CARD_NUMSEG; DIM_UNIV] THEN ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[CONJ_ASSOC; SET_RULE `(!x. x IN s ==> f x IN t) /\ (!y. y IN t ==> ?x. x IN s /\ f x = y) <=> t = IMAGE f s`] THEN REWRITE_TAC[GSYM CONJ_ASSOC; LEFT_IMP_EXISTS_THM; IN_NUMSEG] THEN X_GEN_TAC `w:num->real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC) THEN REWRITE_TAC[ARITH_RULE `n + 1 <= x <=> n < x`; CONJ_ASSOC] THEN ONCE_REWRITE_TAC[TAUT `p /\ q ==> r <=> p /\ ~r ==> ~q`] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN STRIP_TAC THEN REWRITE_TAC[TAUT `p /\ ~r ==> ~q <=> p /\ q ==> r`] THEN EXISTS_TAC `\i. if i <= n then (v:num->real^N) i else w i` THEN SIMP_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[FORALL_IN_IMAGE; IN_NUMSEG]) THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[ARITH_RULE `~(i <= n) ==> n + 1 <= i`]] THEN REWRITE_TAC[pairwise] THEN MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[ORTHOGONAL_SYM]; ALL_TAC] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN DISCH_TAC THEN ASM_CASES_TAC `j:num <= n` THEN ASM_REWRITE_TAC[IN_NUMSEG] THENL [COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN ASM_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `i:num <= n` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN UNDISCH_TAC `pairwise orthogonal (IMAGE (v:num->real^N) (1..n) UNION IMAGE w (n+1..dimindex (:N)))` THEN REWRITE_TAC[pairwise] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `(w:num->real^N) j`) THENL [DISCH_THEN(MP_TAC o SPEC `(v:num->real^N) i`); DISCH_THEN(MP_TAC o SPEC `(w:num->real^N) i`)] THEN ASM_REWRITE_TAC[IN_UNION; IN_IMAGE; IN_NUMSEG] THEN DISCH_THEN MATCH_MP_TAC THENL [CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL [ASM_MESON_TAC[ARITH_RULE `~(x <= n) ==> n + 1 <= x`]; ALL_TAC]; ASM_MESON_TAC[ARITH_RULE `~(x <= n) ==> n + 1 <= x /\ n < x`]] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DISJOINT]) THEN REWRITE_TAC[SET_RULE `IMAGE w t INTER IMAGE v s = {} <=> !i j. i IN s /\ j IN t ==> ~(v i = w j)`] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN ASM_ARITH_TAC) in let lemma2 = prove (`!v:num->real^N w k. pairwise (\i j. orthogonal (v i) (v j)) k /\ pairwise (\i j. orthogonal (w i) (w j)) k /\ (!i. i IN k ==> norm(v i) = norm(w i)) /\ (!i. i IN k ==> ~(v i = vec 0) /\ ~(w i = vec 0)) ==> ?f. orthogonal_transformation f /\ (!i. i IN k ==> f(v i) = w i)`, REPEAT STRIP_TAC THEN SUBGOAL_THEN `FINITE(k:num->bool)` MP_TAC THENL [SUBGOAL_THEN `pairwise orthogonal (IMAGE (v:num->real^N) k)` ASSUME_TAC THENL [REWRITE_TAC[PAIRWISE_IMAGE] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM_SIMP_TAC[pairwise]; ALL_TAC] THEN FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] PAIRWISE_ORTHOGONAL_INDEPENDENT)) THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP INDEPENDENT_IMP_FINITE) THEN MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC FINITE_IMAGE_INJ_EQ THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN ASM_MESON_TAC[ORTHOGONAL_REFL]; ALL_TAC] THEN DISCH_THEN(MP_TAC o GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> p /\ q /\ ~s ==> ~r`] THEN DISCH_THEN(X_CHOOSE_THEN `n:num->num` MP_TAC) THEN REWRITE_TAC[IN_NUMSEG] THEN GEN_REWRITE_TAC I [IMP_CONJ] THEN DISCH_THEN(fun th -> DISCH_THEN SUBST_ALL_TAC THEN ASSUME_TAC th) THEN RULE_ASSUM_TAC(REWRITE_RULE [PAIRWISE_IMAGE; FORALL_IN_IMAGE; IN_NUMSEG]) THEN MP_TAC(ISPECL [`\i. inv(norm(w(n i))) % (w:num->real^N) ((n:num->num) i)`; `CARD(k:num->bool)`] lemma1) THEN MP_TAC(ISPECL [`\i. inv(norm(v(n i))) % (v:num->real^N) ((n:num->num) i)`; `CARD(k:num->bool)`] lemma1) THEN ASM_SIMP_TAC[NORM_MUL; REAL_MUL_LINV; NORM_EQ_0; REAL_ABS_INV; REAL_ABS_NORM; pairwise; orthogonal; IN_NUMSEG] THEN RULE_ASSUM_TAC(REWRITE_RULE[pairwise; orthogonal; IN_NUMSEG]) THEN ASM_SIMP_TAC[DOT_LMUL; DOT_RMUL; REAL_ENTIRE; FORALL_IN_IMAGE] THEN DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC) THEN MP_TAC(ISPEC `f:real^N->real^N` ORTHOGONAL_TRANSFORMATION_INVERSE) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `f':real^N->real^N` STRIP_ASSUME_TAC) THEN EXISTS_TAC `(g:real^N->real^N) o (f':real^N->real^N)` THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE; IN_NUMSEG] THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN REWRITE_TAC[o_THM] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `(g:real^N->real^N) (norm((w:num->real^N)(n(i:num))) % basis i)` THEN CONJ_TAC THENL [AP_TERM_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[] `(!x. f'(f x) = x) ==> f x = y ==> f' y = x`)); ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE[orthogonal_transformation]) THEN ASM_SIMP_TAC[LINEAR_CMUL; VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_MUL_RINV; NORM_EQ_0; VECTOR_MUL_LID]) in REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`v:num->real^N`; `w:num->real^N`; `{i | i IN k /\ ~((v:num->real^N) i = vec 0)}`] lemma2) THEN ASM_SIMP_TAC[IN_ELIM_THM; CONJ_ASSOC] THEN ANTS_TAC THENL [CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[NORM_EQ_0]] THEN CONJ_TAC THEN MATCH_MP_TAC PAIRWISE_MONO THEN EXISTS_TAC `k:num->bool` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[orthogonal_transformation] THEN GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN ASM_CASES_TAC `(v:num->real^N) i = vec 0` THEN ASM_SIMP_TAC[] THEN ASM_MESON_TAC[LINEAR_0; NORM_EQ_0]]);; (* ------------------------------------------------------------------------- *) (* Reflection of a vector about 0 along a line. *) (* ------------------------------------------------------------------------- *) let reflect_along = new_definition `reflect_along v (x:real^N) = x - (&2 * (x dot v) / (v dot v)) % v`;; let REFLECT_ALONG_ADD = prove (`!v x y:real^N. reflect_along v (x + y) = reflect_along v x + reflect_along v y`, REPEAT GEN_TAC THEN REWRITE_TAC[reflect_along; VECTOR_ARITH `x - a % v + y - b % v:real^N = (x + y) - (a + b) % v`] THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[DOT_LADD] THEN REAL_ARITH_TAC);; let REFLECT_ALONG_MUL = prove (`!v a x:real^N. reflect_along v (a % x) = a % reflect_along v x`, REWRITE_TAC[reflect_along; DOT_LMUL; REAL_ARITH `&2 * (a * x) / y = a * &2 * x / y`] THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC]);; let LINEAR_REFLECT_ALONG = prove (`!v:real^N. linear(reflect_along v)`, REWRITE_TAC[linear; REFLECT_ALONG_ADD; REFLECT_ALONG_MUL]);; let REFLECT_ALONG_0 = prove (`!v:real^N. reflect_along v (vec 0) = vec 0`, REWRITE_TAC[MATCH_MP LINEAR_0 (SPEC_ALL LINEAR_REFLECT_ALONG)]);; let REFLECT_ALONG_NEG = prove (`!v x:real^N. reflect_along v (--x) = --(reflect_along v x)`, MESON_TAC[LINEAR_REFLECT_ALONG; LINEAR_NEG]);; let REFLECT_ALONG_REFL = prove (`!v:real^N. reflect_along v v = --v`, GEN_TAC THEN ASM_CASES_TAC `v:real^N = vec 0` THEN ASM_REWRITE_TAC[VECTOR_NEG_0; REFLECT_ALONG_0] THEN REWRITE_TAC[reflect_along] THEN ASM_SIMP_TAC[REAL_DIV_REFL; DOT_EQ_0] THEN VECTOR_ARITH_TAC);; let REFLECT_ALONG_INVOLUTION = prove (`!v x:real^N. reflect_along v (reflect_along v x) = x`, REWRITE_TAC[reflect_along; DOT_LSUB; VECTOR_MUL_EQ_0; VECTOR_ARITH `x - a % v - b % v:real^N = x <=> (a + b) % v = vec 0`] THEN REWRITE_TAC[DOT_LMUL; GSYM DOT_EQ_0] THEN CONV_TAC REAL_FIELD);; let REFLECT_ALONG_GALOIS = prove (`!v p q:real^N. reflect_along v p = q <=> p = reflect_along v q`, MESON_TAC[REFLECT_ALONG_INVOLUTION]);; let REFLECT_ALONG_EQ_0 = prove (`!v x:real^N. reflect_along v x = vec 0 <=> x = vec 0`, MESON_TAC[REFLECT_ALONG_0; REFLECT_ALONG_INVOLUTION]);; let ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG = prove (`!v:real^N. orthogonal_transformation(reflect_along v)`, GEN_TAC THEN ASM_CASES_TAC `v:real^N = vec 0` THENL [GEN_REWRITE_TAC RAND_CONV [GSYM ETA_AX] THEN ASM_REWRITE_TAC[reflect_along; VECTOR_MUL_RZERO; VECTOR_SUB_RZERO; ORTHOGONAL_TRANSFORMATION_ID]; REWRITE_TAC[ORTHOGONAL_TRANSFORMATION] THEN REWRITE_TAC[LINEAR_REFLECT_ALONG; NORM_EQ] THEN REWRITE_TAC[reflect_along; VECTOR_ARITH `(a - b:real^N) dot (a - b) = (a dot a + b dot b) - &2 * a dot b`] THEN REWRITE_TAC[DOT_LMUL; DOT_RMUL] THEN X_GEN_TAC `w:real^N` THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM DOT_EQ_0]) THEN CONV_TAC REAL_FIELD]);; let REFLECT_ALONG_EQ_SELF = prove (`!v x:real^N. reflect_along v x = x <=> orthogonal v x`, REPEAT GEN_TAC THEN REWRITE_TAC[reflect_along; orthogonal] THEN REWRITE_TAC[VECTOR_ARITH `x - a:real^N = x <=> a = vec 0`] THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN ASM_CASES_TAC `v:real^N = vec 0` THEN ASM_SIMP_TAC[DOT_LZERO; DOT_SYM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM DOT_EQ_0]) THEN CONV_TAC REAL_FIELD);; let REFLECT_ALONG_ZERO = prove (`reflect_along (vec 0:real^N) = I`, REWRITE_TAC[FUN_EQ_THM; I_THM; REFLECT_ALONG_EQ_SELF; ORTHOGONAL_0]);; let REFLECT_ALONG_LINEAR_IMAGE = prove (`!f:real^M->real^N v x. linear f /\ (!x. norm(f x) = norm x) ==> reflect_along (f v) (f x) = f(reflect_along v x)`, REWRITE_TAC[reflect_along] THEN SIMP_TAC[PRESERVES_NORM_PRESERVES_DOT; LINEAR_SUB; LINEAR_CMUL]);; add_linear_invariants [REFLECT_ALONG_LINEAR_IMAGE];; let REFLECT_ALONG_SCALE = prove (`!c v x:real^N. ~(c = &0) ==> reflect_along (c % v) x = reflect_along v x`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `v:real^N = vec 0` THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO; REFLECT_ALONG_ZERO] THEN REWRITE_TAC[reflect_along; VECTOR_MUL_ASSOC] THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[DOT_RMUL] THEN REWRITE_TAC[DOT_LMUL] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM DOT_EQ_0]) THEN POP_ASSUM MP_TAC THEN CONV_TAC REAL_FIELD);; let REFLECT_ALONG_NEGATION = prove (`!v:real^N. reflect_along (--v) = reflect_along v`, REWRITE_TAC[FUN_EQ_THM; VECTOR_NEG_MINUS1] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC REFLECT_ALONG_SCALE THEN REAL_ARITH_TAC);; let REFLECT_ALONG_1D = prove (`!v x:real^N. dimindex(:N) = 1 ==> reflect_along v x = if v = vec 0 then x else --x`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[reflect_along; dot; SUM_1; CART_EQ; FORALL_1] THEN REWRITE_TAC[VEC_COMPONENT; COND_RATOR; COND_RAND] THEN SIMP_TAC[VECTOR_NEG_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_SUB_COMPONENT; REAL_MUL_RZERO] THEN CONV_TAC REAL_FIELD);; let REFLECT_ALONG_BASIS = prove (`!x:real^N k. 1 <= k /\ k <= dimindex(:N) ==> reflect_along (basis k) x = x - (&2 * x$k) % basis k`, SIMP_TAC[reflect_along; DOT_BASIS; BASIS_COMPONENT; REAL_DIV_1]);; let MATRIX_REFLECT_ALONG_BASIS = prove (`!k. 1 <= k /\ k <= dimindex(:N) ==> matrix(reflect_along (basis k)):real^N^N = lambda i j. if i = k /\ j = k then --(&1) else if i = j then &1 else &0`, SIMP_TAC[CART_EQ; LAMBDA_BETA; matrix; REFLECT_ALONG_BASIS; VECTOR_SUB_COMPONENT; BASIS_COMPONENT; VECTOR_MUL_COMPONENT] THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_CASES_TAC `i:num = j` THEN ASM_REWRITE_TAC[] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN ASM_REAL_ARITH_TAC);; let ROTOINVERSION_MATRIX_REFLECT_ALONG = prove (`!v:real^N. ~(v = vec 0) ==> rotoinversion_matrix(matrix(reflect_along v))`, REPEAT STRIP_TAC THEN REWRITE_TAC[rotoinversion_matrix] THEN CONJ_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG]; ALL_TAC] THEN ABBREV_TAC `w:real^N = inv(norm v) % v` THEN SUBGOAL_THEN `reflect_along (v:real^N) = reflect_along w` SUBST1_TAC THENL [EXPAND_TAC "w" THEN REWRITE_TAC[FUN_EQ_THM] THEN ASM_SIMP_TAC[REFLECT_ALONG_SCALE; REAL_INV_EQ_0; NORM_EQ_0]; SUBGOAL_THEN `norm(w:real^N) = &1` MP_TAC THENL [EXPAND_TAC "w" THEN SIMP_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN MATCH_MP_TAC REAL_MUL_LINV THEN ASM_REWRITE_TAC[NORM_EQ_0]; POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`w:real^N`,`v:real^N`)]] THEN X_GEN_TAC `v:real^N` THEN ASM_CASES_TAC `v:real^N = vec 0` THEN ASM_REWRITE_TAC[NORM_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_TAC THEN MP_TAC(ISPECL [`v:real^N`; `basis 1:real^N`] ORTHOGONAL_TRANSFORMATION_EXISTS) THEN ASM_SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `matrix(reflect_along v) = transp(matrix(f:real^N->real^N)) ** matrix(reflect_along (f v)) ** matrix f` SUBST1_TAC THENL [UNDISCH_THEN `(f:real^N->real^N) v = basis 1` (K ALL_TAC) THEN REWRITE_TAC[MATRIX_EQ; GSYM MATRIX_VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[MATRIX_WORKS; LINEAR_REFLECT_ALONG; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN X_GEN_TAC `x:real^N` THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `(transp(matrix(f:real^N->real^N)) ** matrix f) ** (reflect_along v x:real^N)` THEN CONJ_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_MATRIX; MATRIX_VECTOR_MUL_LID; ORTHOGONAL_TRANSFORMATION_MATRIX]; REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[MATRIX_WORKS; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC REFLECT_ALONG_LINEAR_IMAGE THEN ASM_REWRITE_TAC[GSYM ORTHOGONAL_TRANSFORMATION]]; ASM_REWRITE_TAC[DET_MUL; DET_TRANSP] THEN MATCH_MP_TAC(REAL_RING `(x = &1 \/ x = -- &1) /\ y = a ==> x * y * x = a`) THEN CONJ_TAC THENL [ASM_MESON_TAC[DET_ORTHOGONAL_MATRIX; ORTHOGONAL_TRANSFORMATION_MATRIX]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhs o rand) DET_UPPERTRIANGULAR o lhand o snd) THEN SIMP_TAC[MATRIX_REFLECT_ALONG_BASIS; DIMINDEX_GE_1; LE_REFL] THEN SIMP_TAC[LAMBDA_BETA; ARITH_RULE `j < i ==> ~(i = j) /\ ~(i = 1 /\ j = 1)`] THEN DISCH_THEN(K ALL_TAC) THEN SIMP_TAC[PRODUCT_CLAUSES_LEFT; DIMINDEX_GE_1] THEN MATCH_MP_TAC(REAL_RING `x = &1 ==> a * x = a`) THEN MATCH_MP_TAC PRODUCT_EQ_1 THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC]);; let DET_MATRIX_REFLECT_ALONG = prove (`!v:real^N. det(matrix(reflect_along v)) = if v = vec 0 then &1 else --(&1)`, GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REFLECT_ALONG_ZERO] THEN REWRITE_TAC[MATRIX_I; DET_I] THEN FIRST_ASSUM(MP_TAC o MATCH_MP ROTOINVERSION_MATRIX_REFLECT_ALONG) THEN SIMP_TAC[rotoinversion_matrix]);; let REFLECT_ALONG_BASIS_COMPONENT = prove (`!x:real^N i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> reflect_along (basis i) x$j = if j = i then --(x$j) else x$j`, SIMP_TAC[REFLECT_ALONG_BASIS; VECTOR_SUB_COMPONENT] THEN SIMP_TAC[VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);; let REFLECT_BASIS_ALONG_BASIS = prove (`!i j. 1 <= i /\ i <= dimindex(:N) /\ 1 <= j /\ j <= dimindex(:N) ==> reflect_along (basis i:real^N) (basis j) = if i = j then --(basis j) else basis j`, REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[CART_EQ; REFLECT_ALONG_BASIS_COMPONENT; BASIS_COMPONENT; VECTOR_NEG_COMPONENT] THEN ASM_MESON_TAC[REAL_NEG_0]);; let NORM_REFLECT_ALONG = prove (`!v x:real^N. norm(reflect_along v x) = norm x`, MESON_TAC[ORTHOGONAL_TRANSFORMATION; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG]);; let REFLECT_ALONG_EQ = prove (`!v x y:real^N. reflect_along v x = reflect_along v y <=> x = y`, MESON_TAC[ORTHOGONAL_TRANSFORMATION_INJECTIVE; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG]);; let REFLECT_ALONG_SURJECTIVE = prove (`!v y:real^N. ?x. reflect_along v x = y`, MESON_TAC[REFLECT_ALONG_INVOLUTION]);; let REFLECT_ALONG_SWITCH = prove (`!a b:real^N. norm a = norm b /\ ~(a = b) ==> reflect_along (b - a) a = b /\ reflect_along (b - a) b = a`, REPEAT GEN_TAC THEN STRIP_TAC THEN SIMP_TAC[reflect_along; DOT_RSUB] THEN REWRITE_TAC[real_div; VECTOR_ARITH `(a - c % (b - a):real^N = b <=> (&1 + c) % (b - a) = vec 0) /\ (b - c % (b - a):real^N = a <=> (&1 - c) % (b - a) = vec 0)`] THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN MATCH_MP_TAC(REAL_FIELD `~(d = &0) /\ x + y = &0 /\ y - x = d ==> &1 + &2 * x * inv d = &0 /\ &1 - &2 * y * inv d = &0`) THEN ASM_REWRITE_TAC[GSYM DOT_RSUB; DOT_EQ_0; VECTOR_SUB_EQ] THEN ASM_REWRITE_TAC[DOT_RSUB; GSYM NORM_POW_2; DOT_LSUB] THEN REWRITE_TAC[DOT_SYM] THEN REAL_ARITH_TAC);; let ROTOINVERSION_EXISTS_GEN = prove (`!s a b:real^N. subspace s /\ a IN s /\ b IN s /\ ~(a = b) /\ norm a = norm b ==> ?f. orthogonal_transformation f /\ IMAGE f s = s /\ (!x. orthogonal a x /\ orthogonal b x ==> f x = x) /\ det (matrix f) = -- &1 /\ f a = b /\ f b = a`, REPEAT STRIP_TAC THEN EXISTS_TAC `reflect_along (b - a:real^N)` THEN REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG] THEN ASM_REWRITE_TAC[DET_MATRIX_REFLECT_ALONG; VECTOR_SUB_EQ] THEN ASM_SIMP_TAC[REFLECT_ALONG_SWITCH] THEN CONJ_TAC THENL [MATCH_MP_TAC(SET_RULE `(!x. f(f x) = x) /\ (!x. x IN s ==> f x IN s) ==> IMAGE f s = s`) THEN REWRITE_TAC[REFLECT_ALONG_INVOLUTION] THEN REWRITE_TAC[reflect_along] THEN ASM_SIMP_TAC[SUBSPACE_SUB; SUBSPACE_MUL]; REWRITE_TAC[ONCE_REWRITE_RULE[DOT_SYM] orthogonal] THEN SIMP_TAC[reflect_along; DOT_RSUB] THEN REWRITE_TAC[real_div; REAL_SUB_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN REWRITE_TAC[VECTOR_ARITH `x - &0 % y:real^N = x`]]);; let ORTHOGONAL_TRANSFORMATION_EXISTS_GEN = prove (`!s a b:real^N. subspace s /\ a IN s /\ b IN s /\ norm a = norm b ==> ?f. orthogonal_transformation f /\ IMAGE f s = s /\ (!x. orthogonal a x /\ orthogonal b x ==> f x = x) /\ f a = b /\ f b = a`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL [EXISTS_TAC `\x:real^N. x` THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ID; IMAGE_ID]; MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`] ROTOINVERSION_EXISTS_GEN) THEN ASM_REWRITE_TAC[] THEN MESON_TAC[]]);; (* ------------------------------------------------------------------------- *) (* All orthogonal transformations are a composition of reflections. *) (* ------------------------------------------------------------------------- *) let ORTHOGONAL_TRANSFORMATION_GENERATED_BY_REFLECTIONS = prove (`!f:real^N->real^N n. orthogonal_transformation f /\ dimindex(:N) <= dim {x | f x = x} + n ==> ?l. LENGTH l <= n /\ ALL (\v. ~(v = vec 0)) l /\ f = ITLIST (\v h. reflect_along v o h) l I`, ONCE_REWRITE_TAC[GSYM SWAP_FORALL_THM] THEN INDUCT_TAC THENL [REWRITE_TAC[CONJUNCT1 LE; LENGTH_EQ_NIL; ADD_CLAUSES; UNWIND_THM2] THEN SIMP_TAC[DIM_SUBSET_UNIV; ARITH_RULE `a:num <= b ==> (b <= a <=> a = b)`; ITLIST; DIM_EQ_FULL; orthogonal_transformation] THEN SIMP_TAC[SPAN_OF_SUBSPACE; SUBSPACE_LINEAR_FIXED_POINTS; IMP_CONJ] THEN REWRITE_TAC[EXTENSION; IN_UNIV; IN_ELIM_THM] THEN SIMP_TAC[FUN_EQ_THM; I_THM; ALL]; REPEAT STRIP_TAC THEN ASM_CASES_TAC `!x:real^N. f x = x` THENL [EXISTS_TAC `[]:(real^N) list` THEN ASM_REWRITE_TAC[ITLIST; FUN_EQ_THM; I_THM; ALL; LENGTH; LE_0]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM])] THEN DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN ABBREV_TAC `v:real^N = inv(&2) % (f a - a)` THEN FIRST_X_ASSUM (MP_TAC o SPEC `reflect_along v o (f:real^N->real^N)`) THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG; ORTHOGONAL_TRANSFORMATION_COMPOSE] THEN ANTS_TAC THENL [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE `a <= d + SUC n ==> d < d' ==> a <= d' + n`)) THEN MATCH_MP_TAC DIM_PSUBSET THEN REWRITE_TAC[PSUBSET_ALT] THEN SUBGOAL_THEN `!y:real^N. dist(y,f a) = dist(y,a) ==> reflect_along v y = y` ASSUME_TAC THENL [REWRITE_TAC[dist; NORM_EQ_SQUARE; NORM_POS_LE; NORM_POW_2] THEN REWRITE_TAC[VECTOR_ARITH `(y - b:real^N) dot (y - b) = (y dot y + b dot b) - &2 * y dot b`] THEN REWRITE_TAC[REAL_ARITH `(y + aa) - &2 * a = (y + bb) - &2 * b <=> a - b = inv(&2) * (aa - bb)`] THEN RULE_ASSUM_TAC(REWRITE_RULE[orthogonal_transformation]) THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_MUL_RZERO] THEN EXPAND_TAC "v" THEN REWRITE_TAC[GSYM DOT_RSUB; reflect_along] THEN SIMP_TAC[DOT_RMUL; real_div; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_RZERO]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC SPAN_MONO THEN SIMP_TAC[SUBSET; IN_ELIM_THM; o_THM] THEN ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_ISOMETRY]; ALL_TAC] THEN EXISTS_TAC `a:real^N` THEN ASM_SIMP_TAC[SUBSPACE_LINEAR_FIXED_POINTS; SPAN_OF_SUBSPACE; ORTHOGONAL_TRANSFORMATION_LINEAR; IN_ELIM_THM] THEN MATCH_MP_TAC SPAN_SUPERSET THEN REWRITE_TAC[IN_ELIM_THM; o_THM] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `reflect_along (v:real^N) (midpoint(f a,a) + v)` THEN CONJ_TAC THENL [AP_TERM_TAC; REWRITE_TAC[REFLECT_ALONG_ADD] THEN ASM_SIMP_TAC[DIST_MIDPOINT; REFLECT_ALONG_REFL]] THEN EXPAND_TAC "v" THEN REWRITE_TAC[midpoint] THEN VECTOR_ARITH_TAC; DISCH_THEN(X_CHOOSE_THEN `l:(real^N)list` STRIP_ASSUME_TAC) THEN EXISTS_TAC `CONS (v:real^N) l` THEN ASM_REWRITE_TAC[ALL; LENGTH; LE_SUC; VECTOR_SUB_EQ; ITLIST] THEN EXPAND_TAC "v" THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ] THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `(o)(reflect_along (v:real^N)):(real^N->real^N)->(real^N->real^N)`) THEN REWRITE_TAC[FUN_EQ_THM; o_THM; REFLECT_ALONG_INVOLUTION]]]);; let ORTHOGONAL_TRANSFORMATION_REFLECT_INDUCT = prove (`!P:(real^N->real^N)->bool. P I /\ (!f a. orthogonal_transformation f /\ ~(a = vec 0) /\ P f ==> P(reflect_along a o f)) ==> !f. orthogonal_transformation f ==> P f`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`f:real^N->real^N`; `dimindex(:N)`] ORTHOGONAL_TRANSFORMATION_GENERATED_BY_REFLECTIONS) THEN ASM_REWRITE_TAC[ONCE_REWRITE_RULE[ADD_SYM] LE_ADD] THEN DISCH_THEN(X_CHOOSE_THEN `l:(real^N)list` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `orthogonal_transformation(f:real^N->real^N)` THEN MATCH_MP_TAC(TAUT `p /\ q ==> p ==> q`) THEN FIRST_X_ASSUM SUBST1_TAC THEN UNDISCH_TAC `ALL (\v:real^N. ~(v = vec 0)) l` THEN UNDISCH_THEN `LENGTH(l:(real^N)list) <= dimindex(:N)` (K ALL_TAC) THEN SPEC_TAC(`l:(real^N)list`,`l:(real^N)list`) THEN MATCH_MP_TAC list_INDUCT THEN REWRITE_TAC[ALL; ITLIST] THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_I] THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE; ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG]);; (* ------------------------------------------------------------------------- *) (* Extract scaling, translation and linear invariance theorems. *) (* For the linear case, chain through some basic consequences automatically, *) (* e.g. norm-preserving and linear implies injective. *) (* ------------------------------------------------------------------------- *) let SCALING_THEOREMS v = let th1 = UNDISCH(snd(EQ_IMP_RULE(ISPEC v NORM_POS_LT))) in let t = rand(concl th1) in end_itlist CONJ (map (C MP th1 o SPEC t) (!scaling_theorems));; let TRANSLATION_INVARIANTS x = end_itlist CONJ (mapfilter (ISPEC x) (!invariant_under_translation));; let USABLE_CONCLUSION f ths th = let ith = PURE_REWRITE_RULE[RIGHT_FORALL_IMP_THM] (ISPEC f th) in let bod = concl ith in let cjs = conjuncts(fst(dest_imp bod)) in let ths = map (fun t -> find(fun th -> aconv (concl th) t) ths) cjs in GEN_ALL(MP ith (end_itlist CONJ ths));; let LINEAR_INVARIANTS = let sths = (CONJUNCTS o prove) (`(!f:real^M->real^N. linear f /\ (!x. norm(f x) = norm x) ==> (!x y. f x = f y ==> x = y)) /\ (!f:real^N->real^N. linear f /\ (!x. norm(f x) = norm x) ==> (!y. ?x. f x = y)) /\ (!f:real^N->real^N. linear f /\ (!x y. f x = f y ==> x = y) ==> (!y. ?x. f x = y)) /\ (!f:real^N->real^N. linear f /\ (!y. ?x. f x = y) ==> (!x y. f x = f y ==> x = y))`, CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN SIMP_TAC[GSYM LINEAR_SUB; GSYM NORM_EQ_0]; MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE; ORTHOGONAL_TRANSFORMATION_INJECTIVE; ORTHOGONAL_TRANSFORMATION; LINEAR_SURJECTIVE_IFF_INJECTIVE]]) in fun f ths -> let ths' = ths @ mapfilter (USABLE_CONCLUSION f ths) sths in end_itlist CONJ (mapfilter (USABLE_CONCLUSION f ths') (!invariant_under_linear));; (* ------------------------------------------------------------------------- *) (* Tactic to pick WLOG a particular point as the origin. The conversion form *) (* assumes it's the outermost universal variable; the tactic is more general *) (* and allows any free or outer universally quantified variable. The list *) (* "avoid" is the points not to translate. There is also a tactic to help in *) (* proving new translation theorems, which uses similar machinery. *) (* ------------------------------------------------------------------------- *) let GEOM_ORIGIN_CONV,GEOM_TRANSLATE_CONV = let pth = prove (`!a:real^N. a = a + vec 0 /\ {} = IMAGE (\x. a + x) {} /\ {} = IMAGE (IMAGE (\x. a + x)) {} /\ (:real^N) = IMAGE (\x. a + x) (:real^N) /\ (:real^N->bool) = IMAGE (IMAGE (\x. a + x)) (:real^N->bool) /\ [] = MAP (\x. a + x) []`, REWRITE_TAC[IMAGE_CLAUSES; VECTOR_ADD_RID; MAP] THEN REWRITE_TAC[SET_RULE `UNIV = IMAGE f UNIV <=> !y. ?x. f x = y`] THEN REWRITE_TAC[SURJECTIVE_IMAGE] THEN REWRITE_TAC[VECTOR_ARITH `a + y:real^N = x <=> y = x - a`; EXISTS_REFL]) and qth = prove (`!a:real^N. ((!P. (!x. P x) <=> (!x. P (a + x))) /\ (!P. (?x. P x) <=> (?x. P (a + x))) /\ (!Q. (!s. Q s) <=> (!s. Q(IMAGE (\x. a + x) s))) /\ (!Q. (?s. Q s) <=> (?s. Q(IMAGE (\x. a + x) s))) /\ (!Q. (!s. Q s) <=> (!s. Q(IMAGE (IMAGE (\x. a + x)) s))) /\ (!Q. (?s. Q s) <=> (?s. Q(IMAGE (IMAGE (\x. a + x)) s))) /\ (!P. (!g:real^1->real^N. P g) <=> (!g. P ((\x. a + x) o g))) /\ (!P. (?g:real^1->real^N. P g) <=> (?g. P ((\x. a + x) o g))) /\ (!P. (!g:num->real^N. P g) <=> (!g. P ((\x. a + x) o g))) /\ (!P. (?g:num->real^N. P g) <=> (?g. P ((\x. a + x) o g))) /\ (!Q. (!l. Q l) <=> (!l. Q(MAP (\x. a + x) l))) /\ (!Q. (?l. Q l) <=> (?l. Q(MAP (\x. a + x) l)))) /\ ((!P. {x | P x} = IMAGE (\x. a + x) {x | P(a + x)}) /\ (!Q. {s | Q s} = IMAGE (IMAGE (\x. a + x)) {s | Q(IMAGE (\x. a + x) s)}) /\ (!R. {l | R l} = IMAGE (MAP (\x. a + x)) {l | R(MAP (\x. a + x) l)}))`, GEN_TAC THEN MATCH_MP_TAC QUANTIFY_SURJECTION_HIGHER_THM THEN X_GEN_TAC `y:real^N` THEN EXISTS_TAC `y - a:real^N` THEN VECTOR_ARITH_TAC) in let GEOM_ORIGIN_CONV avoid tm = let x,tm0 = dest_forall tm in let th0 = ISPEC x pth in let x' = genvar(type_of x) in let ith = ISPEC x' qth in let th1 = PARTIAL_EXPAND_QUANTS_CONV avoid (ASSUME(concl ith)) tm0 in let th2 = CONV_RULE(RAND_CONV(SUBS_CONV(CONJUNCTS th0))) th1 in let th3 = INST[x,x'] (PROVE_HYP ith th2) in let ths = TRANSLATION_INVARIANTS x in let thr = REFL x in let th4 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV) [BETA_THM;ADD_ASSUM(concl thr) ths] th3 in let th5 = MK_FORALL x (PROVE_HYP thr th4) in GEN_REWRITE_RULE (RAND_CONV o TRY_CONV) [FORALL_SIMP] th5 and GEOM_TRANSLATE_CONV avoid a tm = let cth = CONJUNCT2(ISPEC a pth) and vth = ISPEC a qth in let th1 = PARTIAL_EXPAND_QUANTS_CONV avoid (ASSUME(concl vth)) tm in let th2 = CONV_RULE(RAND_CONV(SUBS_CONV(CONJUNCTS cth))) th1 in let th3 = PROVE_HYP vth th2 in let ths = TRANSLATION_INVARIANTS a in let thr = REFL a in let th4 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV) [BETA_THM;ADD_ASSUM(concl thr) ths] th3 in PROVE_HYP thr th4 in GEOM_ORIGIN_CONV,GEOM_TRANSLATE_CONV;; let GEN_GEOM_ORIGIN_TAC x avoid (asl,w as gl) = let avs,bod = strip_forall w and avs' = subtract (frees w) (freesl(map (concl o snd) asl)) in (MAP_EVERY X_GEN_TAC avs THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) (rev(subtract (avs@avs') [x])) THEN SPEC_TAC(x,x) THEN CONV_TAC(GEOM_ORIGIN_CONV avoid)) gl;; let GEOM_ORIGIN_TAC x = GEN_GEOM_ORIGIN_TAC x [];; let GEOM_TRANSLATE_TAC avoid (asl,w) = let a,bod = dest_forall w in let n = length(fst(strip_forall bod)) in (X_GEN_TAC a THEN CONV_TAC(funpow n BINDER_CONV (LAND_CONV(GEOM_TRANSLATE_CONV avoid a))) THEN REWRITE_TAC[]) (asl,w);; (* ------------------------------------------------------------------------- *) (* Rename existential variables in conclusion to fresh genvars. *) (* ------------------------------------------------------------------------- *) let EXISTS_GENVAR_RULE = let rec rule vs th = match vs with [] -> th | v::ovs -> let x,bod = dest_exists(concl th) in let th1 = rule ovs (ASSUME bod) in let th2 = SIMPLE_CHOOSE x (SIMPLE_EXISTS x th1) in PROVE_HYP th (CONV_RULE (GEN_ALPHA_CONV v) th2) in fun th -> rule (map (genvar o type_of) (fst(strip_exists(concl th)))) th;; (* ------------------------------------------------------------------------- *) (* Rotate so that WLOG some point is a +ve multiple of basis vector k. *) (* For general N, it's better to use k = 1 so the side-condition can be *) (* discharged. For dimensions 1, 2 and 3 anything will work automatically. *) (* Could generalize by asking the user to prove theorem 1 <= k <= N. *) (* ------------------------------------------------------------------------- *) let GEOM_BASIS_MULTIPLE_RULE = let pth = prove (`!f. orthogonal_transformation (f:real^N->real^N) ==> (vec 0 = f(vec 0) /\ {} = IMAGE f {} /\ {} = IMAGE (IMAGE f) {} /\ (:real^N) = IMAGE f (:real^N) /\ (:real^N->bool) = IMAGE (IMAGE f) (:real^N->bool) /\ [] = MAP f []) /\ ((!P. (!x. P x) <=> (!x. P (f x))) /\ (!P. (?x. P x) <=> (?x. P (f x))) /\ (!Q. (!s. Q s) <=> (!s. Q (IMAGE f s))) /\ (!Q. (?s. Q s) <=> (?s. Q (IMAGE f s))) /\ (!Q. (!s. Q s) <=> (!s. Q (IMAGE (IMAGE f) s))) /\ (!Q. (?s. Q s) <=> (?s. Q (IMAGE (IMAGE f) s))) /\ (!P. (!g:real^1->real^N. P g) <=> (!g. P (f o g))) /\ (!P. (?g:real^1->real^N. P g) <=> (?g. P (f o g))) /\ (!P. (!g:num->real^N. P g) <=> (!g. P (f o g))) /\ (!P. (?g:num->real^N. P g) <=> (?g. P (f o g))) /\ (!Q. (!l. Q l) <=> (!l. Q(MAP f l))) /\ (!Q. (?l. Q l) <=> (?l. Q(MAP f l)))) /\ ((!P. {x | P x} = IMAGE f {x | P(f x)}) /\ (!Q. {s | Q s} = IMAGE (IMAGE f) {s | Q(IMAGE f s)}) /\ (!R. {l | R l} = IMAGE (MAP f) {l | R(MAP f l)}))`, REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_SURJECTIVE) THEN CONJ_TAC THENL [REWRITE_TAC[IMAGE_CLAUSES; MAP] THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN CONJ_TAC THENL [ASM_MESON_TAC[LINEAR_0]; ALL_TAC] THEN REWRITE_TAC[SET_RULE `UNIV = IMAGE f UNIV <=> !y. ?x. f x = y`] THEN ASM_REWRITE_TAC[SURJECTIVE_IMAGE]; MATCH_MP_TAC QUANTIFY_SURJECTION_HIGHER_THM THEN ASM_REWRITE_TAC[]]) and oth = prove (`!f:real^N->real^N. orthogonal_transformation f /\ (2 <= dimindex(:N) ==> det(matrix f) = &1) ==> linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y) /\ (!x. norm(f x) = norm x) /\ (2 <= dimindex(:N) ==> det(matrix f) = &1)`, GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL [ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR]; ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_INJECTIVE]; ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE]; ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION]]) and arithconv = REWRITE_CONV[DIMINDEX_1; DIMINDEX_2; DIMINDEX_3; ARITH_RULE `1 <= 1`; DIMINDEX_GE_1] THENC NUM_REDUCE_CONV in fun k tm -> let x,bod = dest_forall tm in let th0 = ISPECL [x; mk_small_numeral k] ROTATION_RIGHTWARD_LINE in let th1 = EXISTS_GENVAR_RULE (MP th0 (EQT_ELIM(arithconv(lhand(concl th0))))) in let [a;f],tm1 = strip_exists(concl th1) in let th_orth,th2 = CONJ_PAIR(ASSUME tm1) in let th_det,th2a = CONJ_PAIR th2 in let th_works,th_zero = CONJ_PAIR th2a in let thc,thq = CONJ_PAIR(PROVE_HYP th2 (UNDISCH(ISPEC f pth))) in let th3 = CONV_RULE(RAND_CONV(SUBS_CONV(GSYM th_works::CONJUNCTS thc))) (EXPAND_QUANTS_CONV(ASSUME(concl thq)) bod) in let th4 = PROVE_HYP thq th3 in let thps = CONJUNCTS(MATCH_MP oth (CONJ th_orth th_det)) in let th5 = LINEAR_INVARIANTS f thps in let th6 = PROVE_HYP th_orth (GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV) [BETA_THM; th5] th4) in let ntm = mk_forall(a,mk_imp(concl th_zero,rand(concl th6))) in let th7 = MP(SPEC a (ASSUME ntm)) th_zero in let th8 = DISCH ntm (EQ_MP (SYM th6) th7) in if intersect (frees(concl th8)) [a;f] = [] then let th9 = PROVE_HYP th1 (itlist SIMPLE_CHOOSE [a;f] th8) in let th10 = DISCH ntm (GEN x (UNDISCH th9)) in let a' = variant (frees(concl th10)) (mk_var(fst(dest_var x),snd(dest_var a))) in CONV_RULE(LAND_CONV (GEN_ALPHA_CONV a')) th10 else let mtm = list_mk_forall([a;f],mk_imp(hd(hyp th8),rand(concl th6))) in let th9 = EQ_MP (SYM th6) (UNDISCH(SPECL [a;f] (ASSUME mtm))) in let th10 = itlist SIMPLE_CHOOSE [a;f] (DISCH mtm th9) in let th11 = GEN x (PROVE_HYP th1 th10) in MATCH_MP MONO_FORALL th11;; let GEOM_BASIS_MULTIPLE_TAC k l (asl,w as gl) = let avs,bod = strip_forall w and avs' = subtract (frees w) (freesl(map (concl o snd) asl)) in (MAP_EVERY X_GEN_TAC avs THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) (rev(subtract (avs@avs') [l])) THEN SPEC_TAC(l,l) THEN W(MATCH_MP_TAC o GEOM_BASIS_MULTIPLE_RULE k o snd)) gl;; (* ------------------------------------------------------------------------- *) (* Create invariance theorems automatically, in simple cases. If there are *) (* any nested quantifiers, this will need surjectivity. It's often possible *) (* to prove a stronger theorem by more delicate manual reasoning, so this *) (* isn't used nearly as often as GEOM_TRANSLATE_CONV / GEOM_TRANSLATE_TAC. *) (* As a small step, some ad-hoc rewrites analogous to FORALL_IN_IMAGE are *) (* tried if the first step doesn't finish the goal, but it's very ad hoc. *) (* ------------------------------------------------------------------------- *) let GEOM_TRANSFORM_TAC = let cth0 = prove (`!f:real^M->real^N. linear f ==> vec 0 = f(vec 0) /\ {} = IMAGE f {} /\ {} = IMAGE (IMAGE f) {}`, REWRITE_TAC[IMAGE_CLAUSES] THEN MESON_TAC[LINEAR_0]) and cth1 = prove (`!f:real^M->real^N. (!y. ?x. f x = y) ==> (:real^N) = IMAGE f (:real^M) /\ (:real^N->bool) = IMAGE (IMAGE f) (:real^M->bool)`, REWRITE_TAC[SET_RULE `UNIV = IMAGE f UNIV <=> !y. ?x. f x = y`] THEN REWRITE_TAC[SURJECTIVE_IMAGE]) and sths = (CONJUNCTS o prove) (`(!f:real^M->real^N. linear f /\ (!x. norm(f x) = norm x) ==> (!x y. f x = f y ==> x = y)) /\ (!f:real^N->real^N. linear f /\ (!x. norm(f x) = norm x) ==> (!y. ?x. f x = y)) /\ (!f:real^N->real^N. linear f /\ (!x y. f x = f y ==> x = y) ==> (!y. ?x. f x = y)) /\ (!f:real^N->real^N. linear f /\ (!y. ?x. f x = y) ==> (!x y. f x = f y ==> x = y))`, CONJ_TAC THENL [ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN SIMP_TAC[GSYM LINEAR_SUB; GSYM NORM_EQ_0]; MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE; ORTHOGONAL_TRANSFORMATION_INJECTIVE; ORTHOGONAL_TRANSFORMATION; LINEAR_SURJECTIVE_IFF_INJECTIVE]]) and aths = (CONJUNCTS o prove) (`(!f s P. (!y. y IN IMAGE f s ==> P y) <=> (!x. x IN s ==> P(f x))) /\ (!f s P. (!u. u IN IMAGE (IMAGE f) s ==> P u) <=> (!t. t IN s ==> P(IMAGE f t))) /\ (!f s P. (?y. y IN IMAGE f s /\ P y) <=> (?x. x IN s /\ P(f x))) /\ (!f s P. (?u. u IN IMAGE (IMAGE f) s /\ P u) <=> (?t. t IN s /\ P(IMAGE f t)))`, SET_TAC[]) in fun avoid (asl,w as gl) -> let f,wff = dest_forall w in let vs,bod = strip_forall wff in let ant,cons = dest_imp bod in let hths = CONJUNCTS(ASSUME ant) in let fths = hths @ mapfilter (USABLE_CONCLUSION f hths) sths in let cths = mapfilter (USABLE_CONCLUSION f fths) [cth0; cth1] and vconv = try let vth = USABLE_CONCLUSION f fths QUANTIFY_SURJECTION_HIGHER_THM in PROVE_HYP vth o PARTIAL_EXPAND_QUANTS_CONV avoid (ASSUME(concl vth)) with Failure _ -> ALL_CONV and bths = LINEAR_INVARIANTS f fths in (MAP_EVERY X_GEN_TAC (f::vs) THEN DISCH_TAC THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) cths THEN CONV_TAC(LAND_CONV vconv) THEN GEN_REWRITE_TAC (LAND_CONV o REDEPTH_CONV) [bths] THEN REWRITE_TAC[] THEN REWRITE_TAC(mapfilter (ADD_ASSUM ant o ISPEC f) aths) THEN GEN_REWRITE_TAC (LAND_CONV o REDEPTH_CONV) [bths] THEN REWRITE_TAC[]) gl;; (* ------------------------------------------------------------------------- *) (* Scale so that a chosen vector has size 1. Generates a conjunction of *) (* two formulas, one for the zero case (which one hopes is trivial) and *) (* one just like the original goal but with a norm(...) = 1 assumption. *) (* ------------------------------------------------------------------------- *) let GEOM_NORMALIZE_RULE = let pth = prove (`!a:real^N. ~(a = vec 0) ==> vec 0 = norm(a) % vec 0 /\ a = norm(a) % inv(norm a) % a /\ {} = IMAGE (\x. norm(a) % x) {} /\ {} = IMAGE (IMAGE (\x. norm(a) % x)) {} /\ (:real^N) = IMAGE (\x. norm(a) % x) (:real^N) /\ (:real^N->bool) = IMAGE (IMAGE (\x. norm(a) % x)) (:real^N->bool) /\ [] = MAP (\x. norm(a) % x) []`, REWRITE_TAC[IMAGE_CLAUSES; VECTOR_MUL_ASSOC; VECTOR_MUL_RZERO; MAP] THEN SIMP_TAC[NORM_EQ_0; REAL_MUL_RINV; VECTOR_MUL_LID] THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[SET_RULE `UNIV = IMAGE f UNIV <=> !y. ?x. f x = y`] THEN ASM_REWRITE_TAC[SURJECTIVE_IMAGE] THEN X_GEN_TAC `y:real^N` THEN EXISTS_TAC `inv(norm(a:real^N)) % y:real^N` THEN ASM_SIMP_TAC[VECTOR_MUL_ASSOC; NORM_EQ_0; REAL_MUL_RINV; VECTOR_MUL_LID]) and qth = prove (`!a:real^N. ~(a = vec 0) ==> ((!P. (!r:real. P r) <=> (!r. P(norm a * r))) /\ (!P. (?r:real. P r) <=> (?r. P(norm a * r))) /\ (!P. (!x:real^N. P x) <=> (!x. P (norm(a) % x))) /\ (!P. (?x:real^N. P x) <=> (?x. P (norm(a) % x))) /\ (!Q. (!s:real^N->bool. Q s) <=> (!s. Q(IMAGE (\x. norm(a) % x) s))) /\ (!Q. (?s:real^N->bool. Q s) <=> (?s. Q(IMAGE (\x. norm(a) % x) s))) /\ (!Q. (!s:(real^N->bool)->bool. Q s) <=> (!s. Q(IMAGE (IMAGE (\x. norm(a) % x)) s))) /\ (!Q. (?s:(real^N->bool)->bool. Q s) <=> (?s. Q(IMAGE (IMAGE (\x. norm(a) % x)) s))) /\ (!P. (!g:real^1->real^N. P g) <=> (!g. P ((\x. norm(a) % x) o g))) /\ (!P. (?g:real^1->real^N. P g) <=> (?g. P ((\x. norm(a) % x) o g))) /\ (!P. (!g:num->real^N. P g) <=> (!g. P ((\x. norm(a) % x) o g))) /\ (!P. (?g:num->real^N. P g) <=> (?g. P ((\x. norm(a) % x) o g))) /\ (!Q. (!l. Q l) <=> (!l. Q(MAP (\x:real^N. norm(a) % x) l))) /\ (!Q. (?l. Q l) <=> (?l. Q(MAP (\x:real^N. norm(a) % x) l)))) /\ ((!P. {x:real^N | P x} = IMAGE (\x. norm(a) % x) {x | P(norm(a) % x)}) /\ (!Q. {s:real^N->bool | Q s} = IMAGE (IMAGE (\x. norm(a) % x)) {s | Q(IMAGE (\x. norm(a) % x) s)}) /\ (!R. {l:(real^N)list | R l} = IMAGE (MAP (\x:real^N. norm(a) % x)) {l | R(MAP (\x:real^N. norm(a) % x) l)}))`, GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `(a /\ b) /\ c /\ d ==> (a /\ b /\ c) /\ d`) THEN CONJ_TAC THENL [ASM_MESON_TAC[NORM_EQ_0; REAL_FIELD `~(x = &0) ==> x * inv x * a = a`]; MP_TAC(ISPEC `\x:real^N. norm(a:real^N) % x` (INST_TYPE [`:real^1`,`:C`] QUANTIFY_SURJECTION_HIGHER_THM)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[SURJECTIVE_SCALING; NORM_EQ_0]]) and lth = prove (`(!b:real^N. ~(b = vec 0) ==> (P(b) <=> Q(inv(norm b) % b))) ==> P(vec 0) /\ (!b. norm(b) = &1 ==> Q b) ==> (!b. P b)`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `b:real^N = vec 0` THEN ASM_SIMP_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0]) in fun avoid tm -> let x,tm0 = dest_forall tm in let cth = UNDISCH(ISPEC x pth) and vth = UNDISCH(ISPEC x qth) in let th1 = ONCE_REWRITE_CONV[cth] tm0 in let th2 = CONV_RULE (RAND_CONV (PARTIAL_EXPAND_QUANTS_CONV avoid vth)) th1 in let th3 = SCALING_THEOREMS x in let th3' = (end_itlist CONJ (map (fun th -> let avs,_ = strip_forall(concl th) in let gvs = map (genvar o type_of) avs in GENL gvs (SPECL gvs th)) (CONJUNCTS th3))) in let th4 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV) [BETA_THM; th3'] th2 in MATCH_MP lth (GEN x (DISCH_ALL th4));; let GEN_GEOM_NORMALIZE_TAC x avoid (asl,w as gl) = let avs,bod = strip_forall w and avs' = subtract (frees w) (freesl(map (concl o snd) asl)) in (MAP_EVERY X_GEN_TAC avs THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) (rev(subtract (avs@avs') [x])) THEN SPEC_TAC(x,x) THEN W(MATCH_MP_TAC o GEOM_NORMALIZE_RULE avoid o snd)) gl;; let GEOM_NORMALIZE_TAC x = GEN_GEOM_NORMALIZE_TAC x [];; (* ------------------------------------------------------------------------- *) (* Add invariance theorems for collinearity. *) (* ------------------------------------------------------------------------- *) let COLLINEAR_TRANSLATION_EQ = prove (`!a s. collinear (IMAGE (\x. a + x) s) <=> collinear s`, REWRITE_TAC[collinear] THEN GEOM_TRANSLATE_TAC["u"]);; add_translation_invariants [COLLINEAR_TRANSLATION_EQ];; let COLLINEAR_TRANSLATION = prove (`!s a. collinear s ==> collinear (IMAGE (\x. a + x) s)`, REWRITE_TAC[COLLINEAR_TRANSLATION_EQ]);; let COLLINEAR_LINEAR_IMAGE = prove (`!f s. collinear s /\ linear f ==> collinear(IMAGE f s)`, REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[collinear; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN ASM_MESON_TAC[LINEAR_SUB; LINEAR_CMUL]);; let COLLINEAR_LINEAR_IMAGE_EQ = prove (`!f s. linear f /\ (!x y. f x = f y ==> x = y) ==> (collinear (IMAGE f s) <=> collinear s)`, MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COLLINEAR_LINEAR_IMAGE));; add_linear_invariants [COLLINEAR_LINEAR_IMAGE_EQ];; (* ------------------------------------------------------------------------- *) (* Take a theorem "th" with outer universal quantifiers involving real^N *) (* and a theorem "dth" asserting |- dimindex(:M) <= dimindex(:N) and *) (* return a theorem replacing type :N by :M in th. Neither N or M need be a *) (* type variable. *) (* ------------------------------------------------------------------------- *) let GEOM_DROP_DIMENSION_RULE = let oth = prove (`!f:real^M->real^N. linear f /\ (!x. norm(f x) = norm x) ==> linear f /\ (!x y. f x = f y ==> x = y) /\ (!x. norm(f x) = norm x)`, MESON_TAC[PRESERVES_NORM_INJECTIVE]) and cth = prove (`linear(f:real^M->real^N) ==> vec 0 = f(vec 0) /\ {} = IMAGE f {} /\ {} = IMAGE (IMAGE f) {} /\ [] = MAP f []`, REWRITE_TAC[IMAGE_CLAUSES; MAP; GSYM LINEAR_0]) in fun dth th -> let ath = GEN_ALL th and eth = MATCH_MP ISOMETRY_UNIV_UNIV dth and avoid = variables(concl th) in let f,bod = dest_exists(concl eth) in let fimage = list_mk_icomb "IMAGE" [f] and fmap = list_mk_icomb "MAP" [f] and fcompose = list_mk_icomb "o" [f] in let fimage2 = list_mk_icomb "IMAGE" [fimage] in let lin,iso = CONJ_PAIR(ASSUME bod) in let olduniv = rand(rand(concl dth)) and newuniv = rand(lhand(concl dth)) in let oldty = fst(dest_fun_ty(type_of olduniv)) and newty = fst(dest_fun_ty(type_of newuniv)) in let newvar v = let n,t = dest_var v in variant avoid (mk_var(n,tysubst[newty,oldty] t)) in let newterm v = try let v' = newvar v in tryfind (fun f -> mk_comb(f,v')) [f;fimage;fmap;fcompose;fimage2] with Failure _ -> v in let specrule th = let v = fst(dest_forall(concl th)) in SPEC (newterm v) th in let sth = SUBS(CONJUNCTS(MATCH_MP cth lin)) ath in let fth = SUBS[SYM(MATCH_MP LINEAR_0 lin)] (repeat specrule sth) in let thps = CONJUNCTS(MATCH_MP oth (ASSUME bod)) in let th5 = LINEAR_INVARIANTS f thps in let th6 = GEN_REWRITE_RULE REDEPTH_CONV [th5] fth in let th7 = PROVE_HYP eth (SIMPLE_CHOOSE f th6) in GENL (map newvar (fst(strip_forall(concl ath)))) th7;; (* ------------------------------------------------------------------------- *) (* Transfer theorems automatically between same-dimension spaces. *) (* Given dth = A |- dimindex(:M) = dimindex(:N) *) (* and a theorem th involving variables of type real^N *) (* returns a corresponding theorem mapped to type real^M with assumptions A. *) (* ------------------------------------------------------------------------- *) let GEOM_EQUAL_DIMENSION_RULE = let bth = prove (`dimindex(:M) = dimindex(:N) ==> ?f:real^M->real^N. (linear f /\ (!y. ?x. f x = y)) /\ (!x. norm(f x) = norm x)`, REWRITE_TAC[SET_RULE `(!y. ?x. f x = y) <=> IMAGE f UNIV = UNIV`] THEN DISCH_TAC THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN MATCH_MP_TAC ISOMETRY_UNIV_SUBSPACE THEN REWRITE_TAC[SUBSPACE_UNIV; DIM_UNIV] THEN FIRST_ASSUM ACCEPT_TAC) and pth = prove (`!f:real^M->real^N. linear f /\ (!y. ?x. f x = y) ==> (vec 0 = f(vec 0) /\ {} = IMAGE f {} /\ {} = IMAGE (IMAGE f) {} /\ (:real^N) = IMAGE f (:real^M) /\ (:real^N->bool) = IMAGE (IMAGE f) (:real^M->bool) /\ [] = MAP f []) /\ ((!P. (!x. P x) <=> (!x. P (f x))) /\ (!P. (?x. P x) <=> (?x. P (f x))) /\ (!Q. (!s. Q s) <=> (!s. Q (IMAGE f s))) /\ (!Q. (?s. Q s) <=> (?s. Q (IMAGE f s))) /\ (!Q. (!s. Q s) <=> (!s. Q (IMAGE (IMAGE f) s))) /\ (!Q. (?s. Q s) <=> (?s. Q (IMAGE (IMAGE f) s))) /\ (!P. (!g:real^1->real^N. P g) <=> (!g. P (f o g))) /\ (!P. (?g:real^1->real^N. P g) <=> (?g. P (f o g))) /\ (!P. (!g:num->real^N. P g) <=> (!g. P (f o g))) /\ (!P. (?g:num->real^N. P g) <=> (?g. P (f o g))) /\ (!Q. (!l. Q l) <=> (!l. Q(MAP f l))) /\ (!Q. (?l. Q l) <=> (?l. Q(MAP f l)))) /\ ((!P. {x | P x} = IMAGE f {x | P(f x)}) /\ (!Q. {s | Q s} = IMAGE (IMAGE f) {s | Q(IMAGE f s)}) /\ (!R. {l | R l} = IMAGE (MAP f) {l | R(MAP f l)}))`, GEN_TAC THEN SIMP_TAC[SET_RULE `UNIV = IMAGE f UNIV <=> (!y. ?x. f x = y)`; SURJECTIVE_IMAGE] THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[QUANTIFY_SURJECTION_HIGHER_THM] THEN REWRITE_TAC[IMAGE_CLAUSES; MAP] THEN MESON_TAC[LINEAR_0]) in fun dth th -> let eth = EXISTS_GENVAR_RULE (MATCH_MP bth dth) in let f,bod = dest_exists(concl eth) in let lsth,neth = CONJ_PAIR(ASSUME bod) in let cth,qth = CONJ_PAIR(MATCH_MP pth lsth) in let th1 = CONV_RULE (EXPAND_QUANTS_CONV qth THENC SUBS_CONV(CONJUNCTS cth)) th in let ith = LINEAR_INVARIANTS f (neth::CONJUNCTS lsth) in let th2 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV) [BETA_THM;ith] th1 in let th3 = GEN f (DISCH bod th2) in MP (CONV_RULE (REWR_CONV LEFT_FORALL_IMP_THM) th3) eth;;