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(* ========================================================================= *)
(* The Cayley-Hamilton theorem (for real matrices). *)
(* ========================================================================= *)
needs "Multivariate/complexes.ml";;
needs "Multivariate/msum.ml";;
(* ------------------------------------------------------------------------- *)
(* Powers of a square matrix (mpow). *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("mpow",(24,"left"));;
let mpow = define
`(!A:real^N^N. A mpow 0 = (mat 1 :real^N^N)) /\
(!A:real^N^N n. A mpow (SUC n) = A ** A mpow n)`;;
let MPOW_ADD = prove
(`!A:real^N^N m n. A mpow (m + n) = A mpow m ** A mpow n`,
GEN_TAC THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[ADD_CLAUSES; mpow; MATRIX_MUL_LID] THEN
REWRITE_TAC[MATRIX_MUL_ASSOC]);;
let MPOW_1 = prove
(`!A:real^N^N. A mpow 1 = A`,
REWRITE_TAC[num_CONV `1`; mpow] THEN
REWRITE_TAC[SYM(num_CONV `1`); MATRIX_MUL_RID]);;
let MPOW_SUC = prove
(`!A:real^N^N n. A mpow (SUC n) = A mpow n ** A`,
REWRITE_TAC[ADD1; MPOW_ADD; MPOW_1]);;
(* ------------------------------------------------------------------------- *)
(* The main lemma underlying the proof. *)
(* ------------------------------------------------------------------------- *)
let MATRIC_POLYFUN_EQ_0 = prove
(`!n A:num->real^N^M.
(!x. msum(0..n) (\i. x pow i %% A i) = mat 0) <=>
(!i. i IN 0..n ==> A i = mat 0)`,
SIMP_TAC[CART_EQ; MSUM_COMPONENT; MAT_COMPONENT; LAMBDA_BETA;
FINITE_NUMSEG; COND_ID;
ONCE_REWRITE_RULE[REAL_MUL_SYM] MATRIX_CMUL_COMPONENT] THEN
REWRITE_TAC[MESON[]
`(!x i. P i ==> !j. Q j ==> R x i j) <=>
(!i. P i ==> !j. Q j ==> !x. R x i j)`] THEN
REWRITE_TAC[REAL_POLYFUN_EQ_0] THEN MESON_TAC[]);;
let MATRIC_POLY_LEMMA = prove
(`!(A:real^N^N) B (C:real^N^N) n.
(!x. msum (0..n) (\i. (x pow i) %% B i) ** (A - x %% mat 1) = C)
==> C = mat 0`,
SIMP_TAC[GSYM MSUM_MATRIX_RMUL; FINITE_NUMSEG; MATRIX_SUB_LDISTRIB] THEN
REWRITE_TAC[MATRIX_MUL_RMUL] THEN ONCE_REWRITE_TAC[MATRIX_MUL_LMUL] THEN
ONCE_REWRITE_TAC[MATRIX_CMUL_ASSOC] THEN
REWRITE_TAC[GSYM(CONJUNCT2 real_pow)] THEN
SIMP_TAC[MSUM_SUB; FINITE_NUMSEG] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!x. msum(0..SUC n)
(\i. x pow i %% (((if i = 0 then (--C:real^N^N) else mat 0) +
(if i <= n then B i ** (A:real^N^N) else mat 0)) -
(if i = 0 then mat 0 else B(i - 1) ** mat 1))) =
mat 0`
MP_TAC THENL
[SIMP_TAC[MATRIX_CMUL_SUB_LDISTRIB; MSUM_SUB; FINITE_NUMSEG;
MATRIX_CMUL_ADD_LDISTRIB; MSUM_ADD] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MATRIX_CMUL_RZERO] THEN
ONCE_REWRITE_TAC[MESON[]
`(if p then mat 0 else x) = (if ~p then x else mat 0)`] THEN
REWRITE_TAC[GSYM MSUM_RESTRICT_SET; IN_NUMSEG] THEN
REWRITE_TAC[ARITH_RULE `(0 <= i /\ i <= SUC n) /\ i = 0 <=> i = 0`;
ARITH_RULE `(0 <= i /\ i <= SUC n) /\ i <= n <=> 0 <= i /\ i <= n`;
ARITH_RULE `(0 <= i /\ i <= SUC n) /\ ~(i = 0) <=>
1 <= i /\ i <= SUC n`] THEN
REWRITE_TAC[SING_GSPEC; GSYM numseg; MSUM_SING; real_pow] THEN
REWRITE_TAC[MATRIX_CMUL_LID] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC ONCE_DEPTH_CONV [GSYM th]) THEN
REWRITE_TAC[MATRIX_NEG_SUB] THEN REWRITE_TAC[MATRIX_SUB; AC MATRIX_ADD_AC
`(((A:real^N^N) + --B) + B) + C = (--B + B) + A + C`] THEN
REWRITE_TAC[MATRIX_ADD_LNEG; MATRIX_ADD_LID] THEN
REWRITE_TAC[num_CONV `1`] THEN REWRITE_TAC[ADD1; MSUM_OFFSET] THEN
REWRITE_TAC[ADD_CLAUSES; ADD_SUB; MATRIX_ADD_RNEG];
REWRITE_TAC[MATRIC_POLYFUN_EQ_0; IN_NUMSEG; LE_0] THEN DISCH_TAC THEN
SUBGOAL_THEN `!i:num. B(n - i) = (mat 0:real^N^N)` MP_TAC THENL
[MATCH_MP_TAC num_INDUCTION THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `SUC n`) THEN
REWRITE_TAC[LE_REFL; SUB_0; NOT_SUC; ARITH_RULE `~(SUC n <= n)`] THEN
REWRITE_TAC[MATRIX_ADD_LID; SUC_SUB1; MATRIX_MUL_RID] THEN
REWRITE_TAC[MATRIX_SUB_LZERO; MATRIX_NEG_EQ_0];
X_GEN_TAC `m:num` THEN DISCH_TAC THEN
DISJ_CASES_TAC(ARITH_RULE `n - SUC m = n - m \/ m < n`) THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `n - m:num`) THEN
ASM_SIMP_TAC[ARITH_RULE `m < n ==> ~(n - m = 0)`;
ARITH_RULE `n - m <= SUC n /\ n - m <= n`] THEN
REWRITE_TAC[MATRIX_MUL_LZERO; MATRIX_ADD_LID; MATRIX_SUB_LZERO] THEN
REWRITE_TAC[MATRIX_MUL_RID; MATRIX_NEG_EQ_0] THEN
ASM_SIMP_TAC[ARITH_RULE `n - m - 1 = n - SUC m`]];
DISCH_THEN(MP_TAC o SPEC `n:num`) THEN REWRITE_TAC[SUB_REFL] THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `0`) THEN
ASM_REWRITE_TAC[LE_0; MATRIX_MUL_LZERO; MATRIX_ADD_RID] THEN
REWRITE_TAC[MATRIX_SUB_RZERO; MATRIX_NEG_EQ_0]]]);;
(* ------------------------------------------------------------------------- *)
(* Show that cofactor and determinant are n-1 and n degree polynomials. *)
(* ------------------------------------------------------------------------- *)
let POLYFUN_N_CONST = prove
(`!c n. ?b. !x. c = sum(0..n) (\i. b i * x pow i)`,
REPEAT STRIP_TAC THEN
EXISTS_TAC `\i. if i = 0 then c else &0` THEN
REWRITE_TAC[COND_RAND; COND_RATOR; REAL_MUL_LZERO] THEN
REWRITE_TAC[GSYM SUM_RESTRICT_SET; IN_NUMSEG] THEN
REWRITE_TAC[ARITH_RULE `(0 <= i /\ i <= n) /\ i = 0 <=> i = 0`] THEN
REWRITE_TAC[SING_GSPEC; SUM_SING; real_pow; REAL_MUL_RID]);;
let POLYFUN_N_ADD = prove
(`!f g. (?b. !x. f(x) = sum(0..n) (\i. b i * x pow i)) /\
(?c. !x. g(x) = sum(0..n) (\i. c i * x pow i))
==> ?d. !x. f(x) + g(x) = sum(0..n) (\i. d i * x pow i)`,
REPEAT STRIP_TAC THEN
EXISTS_TAC `\i. (b:num->real) i + c i` THEN
ASM_REWRITE_TAC[SUM_ADD_NUMSEG; REAL_ADD_RDISTRIB]);;
let POLYFUN_N_CMUL = prove
(`!f c. (?b. !x. f(x) = sum(0..n) (\i. b i * x pow i))
==> ?b. !x. c * f(x) = sum(0..n) (\i. b i * x pow i)`,
REPEAT STRIP_TAC THEN
EXISTS_TAC `\i. c * (b:num->real) i` THEN
ASM_REWRITE_TAC[SUM_LMUL; GSYM REAL_MUL_ASSOC]);;
let POLYFUN_N_SUM = prove
(`!f s. FINITE s /\
(!a. a IN s ==> ?b. !x. f x a = sum(0..n) (\i. b i * x pow i))
==> ?b. !x. sum s (f x) = sum(0..n) (\i. b i * x pow i)`,
GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[SUM_CLAUSES; FORALL_IN_INSERT; NOT_IN_EMPTY; POLYFUN_N_CONST] THEN
REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN
MATCH_MP_TAC POLYFUN_N_ADD THEN ASM_SIMP_TAC[]);;
let POLYFUN_N_PRODUCT = prove
(`!f s n. FINITE s /\
(!a:A. a IN s ==> ?c d. !x. f x a = c + d * x) /\ CARD(s) <= n
==> ?b. !x. product s (f x) = sum(0..n) (\i. b i * x pow i)`,
GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[PRODUCT_CLAUSES; POLYFUN_N_CONST; FORALL_IN_INSERT] THEN
REPEAT GEN_TAC THEN DISCH_THEN(fun th ->
DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_SIMP_TAC[CARD_CLAUSES] THEN
INDUCT_TAC THENL [ARITH_TAC; REWRITE_TAC[LE_SUC]] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `b:num->real`) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `c:real` (X_CHOOSE_TAC `d:real`)) THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `\i. (if i <= n then c * b i else &0) +
(if ~(i = 0) then d * b(i - 1) else &0)` THEN
X_GEN_TAC `x:real` THEN
REWRITE_TAC[REAL_ADD_RDISTRIB; SUM_ADD_NUMSEG] THEN
REWRITE_TAC[COND_RAND; COND_RATOR; GSYM SUM_LMUL; REAL_MUL_LZERO] THEN
REWRITE_TAC[GSYM SUM_RESTRICT_SET; IN_NUMSEG] THEN
REWRITE_TAC[ARITH_RULE
`((0 <= i /\ i <= SUC n) /\ i <= n <=> 0 <= i /\ i <= n) /\
((0 <= i /\ i <= SUC n) /\ ~(i = 0) <=> 1 <= i /\ i <= SUC n)`] THEN
REWRITE_TAC[GSYM numseg] THEN
REWRITE_TAC[MESON[num_CONV `1`; ADD1] `1..SUC n = 0+1..n+1`] THEN
REWRITE_TAC[SUM_OFFSET; ADD_SUB; REAL_POW_ADD] THEN
BINOP_TAC THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN REAL_ARITH_TAC);;
let COFACTOR_ENTRY_AS_POLYFUN = prove
(`!A:real^N^N x i j.
1 <= i /\ i <= dimindex(:N) /\
1 <= j /\ j <= dimindex(:N)
==> ?c. !x. cofactor(A - x %% mat 1)$i$j =
sum(0..dimindex(:N)-1) (\i. c(i) * x pow i)`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[cofactor; LAMBDA_BETA; det] THEN
MATCH_MP_TAC POLYFUN_N_SUM THEN
SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG; FORALL_IN_GSPEC] THEN
X_GEN_TAC `p:num->num` THEN DISCH_TAC THEN
MATCH_MP_TAC POLYFUN_N_CMUL THEN
SUBGOAL_THEN `1..dimindex(:N) = i INSERT ((1..dimindex(:N)) DELETE i)`
SUBST1_TAC THENL
[REWRITE_TAC[EXTENSION; IN_INSERT; IN_DELETE; IN_NUMSEG] THEN ASM_ARITH_TAC;
SIMP_TAC[PRODUCT_CLAUSES; FINITE_DELETE; FINITE_NUMSEG]] THEN
ASM_REWRITE_TAC[IN_DELETE; IN_NUMSEG] THEN
MATCH_MP_TAC POLYFUN_N_CMUL THEN
MATCH_MP_TAC POLYFUN_N_PRODUCT THEN
SIMP_TAC[CARD_DELETE; FINITE_DELETE; FINITE_NUMSEG] THEN
ASM_REWRITE_TAC[IN_NUMSEG; IN_DELETE; CARD_NUMSEG_1; LE_REFL] THEN
X_GEN_TAC `k:num` 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
ASM_SIMP_TAC[IN_NUMSEG; LAMBDA_BETA] THEN STRIP_TAC THEN
ASM_CASES_TAC `(p:num->num) k = j` THEN ASM_REWRITE_TAC[] THENL
[REPEAT(EXISTS_TAC `&0`) THEN REAL_ARITH_TAC; ALL_TAC] THEN
ASM_SIMP_TAC[MATRIX_SUB_COMPONENT; MATRIX_CMUL_COMPONENT; MAT_COMPONENT] THEN
REWRITE_TAC[REAL_ARITH `a - x * d:real = a + (--d) * x`] THEN MESON_TAC[]);;
let DETERMINANT_AS_POLYFUN = prove
(`!A:real^N^N.
?c. !x. det(A - x %% mat 1) =
sum(0..dimindex(:N)) (\i. c(i) * x pow i)`,
GEN_TAC THEN REWRITE_TAC[det] THEN
MATCH_MP_TAC POLYFUN_N_SUM THEN
SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG; FORALL_IN_GSPEC] THEN
X_GEN_TAC `p:num->num` THEN DISCH_TAC THEN
MATCH_MP_TAC POLYFUN_N_CMUL THEN MATCH_MP_TAC POLYFUN_N_PRODUCT THEN
SIMP_TAC[FINITE_NUMSEG; CARD_NUMSEG_1; LE_REFL; IN_NUMSEG] THEN
X_GEN_TAC `k:num` 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
ASM_SIMP_TAC[IN_NUMSEG; LAMBDA_BETA] THEN STRIP_TAC THEN
ASM_SIMP_TAC[MATRIX_SUB_COMPONENT; MATRIX_CMUL_COMPONENT; MAT_COMPONENT] THEN
REWRITE_TAC[REAL_ARITH `a - x * d:real = a + (--d) * x`] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Hence define characteristic polynomial coefficients. *)
(* ------------------------------------------------------------------------- *)
let char_poly = new_specification ["char_poly"]
(REWRITE_RULE[SKOLEM_THM] DETERMINANT_AS_POLYFUN);;
(* ------------------------------------------------------------------------- *)
(* Now the Cayley-Hamilton proof. *)
(* ------------------------------------------------------------------------- *)
let COFACTOR_AS_MATRIC_POLYNOMIAL = prove
(`!A:real^N^N. ?C.
!x. cofactor(A - x %% mat 1) =
msum(0..dimindex(:N)-1) (\i. x pow i %% C i)`,
GEN_TAC THEN SIMP_TAC[CART_EQ; MSUM_COMPONENT; FINITE_NUMSEG] THEN
MP_TAC(ISPEC `A:real^N^N` COFACTOR_ENTRY_AS_POLYFUN) THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[IMP_IMP] THEN REWRITE_TAC[LAMBDA_SKOLEM] THEN
DISCH_THEN(X_CHOOSE_THEN `c:(num->real)^N^N` ASSUME_TAC) THEN
EXISTS_TAC `(\i. lambda j k. ((c:(num->real)^N^N)$j$k) i):num->real^N^N` THEN
MAP_EVERY X_GEN_TAC [`x:real`; `i:num`] THEN STRIP_TAC THEN
X_GEN_TAC `j:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN
MATCH_MP_TAC SUM_EQ_NUMSEG THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[MATRIX_CMUL_COMPONENT; LAMBDA_BETA] THEN REAL_ARITH_TAC);;
let MATRIC_POWER_DIFFERENCE = prove
(`!A:real^N^N x n.
A mpow (SUC n) - x pow (SUC n) %% mat 1 =
msum (0..n) (\i. x pow i %% A mpow (n - i)) ** (A - x %% mat 1)`,
GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THENL
[REWRITE_TAC[MSUM_CLAUSES_NUMSEG; real_pow; SUB_0; mpow] THEN
REWRITE_TAC[MATRIX_MUL_RID; MATRIX_CMUL_LID; MATRIX_MUL_LID] THEN
REWRITE_TAC[REAL_MUL_RID];
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
`(A mpow SUC n - x pow SUC n %% mat 1) ** A +
(x pow (SUC n) %% mat 1 :real^N^N) ** (A - x %% mat 1:real^N^N)` THEN
CONJ_TAC THENL
[GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [MPOW_SUC] THEN
REWRITE_TAC[MATRIX_SUB_RDISTRIB; MATRIX_SUB_LDISTRIB] THEN
REWRITE_TAC[MATRIX_SUB; MATRIX_MUL_LMUL; MATRIX_MUL_LID] THEN
REWRITE_TAC[GSYM MATRIX_ADD_ASSOC] THEN AP_TERM_TAC THEN
REWRITE_TAC[MATRIX_ADD_ASSOC; MATRIX_ADD_LNEG; MATRIX_ADD_LID] THEN
REWRITE_TAC[real_pow; MATRIX_CMUL_ASSOC] THEN REWRITE_TAC[REAL_MUL_AC];
ASM_REWRITE_TAC[MSUM_CLAUSES_NUMSEG; LE_0] THEN
REWRITE_TAC[SUB_REFL; mpow; MATRIX_ADD_RDISTRIB] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
SIMP_TAC[GSYM MSUM_MATRIX_RMUL; FINITE_NUMSEG] THEN
MATCH_MP_TAC MSUM_EQ THEN REWRITE_TAC[FINITE_NUMSEG] THEN
X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN
ASM_SIMP_TAC[MATRIX_MUL_LMUL] THEN AP_TERM_TAC THEN
ASM_SIMP_TAC[ARITH_RULE `i <= n ==> SUC n - i = SUC(n - i)`] THEN
REWRITE_TAC[MPOW_SUC; GSYM MATRIX_MUL_ASSOC] THEN AP_TERM_TAC THEN
REWRITE_TAC[MATRIX_SUB_LDISTRIB; MATRIX_SUB_RDISTRIB] THEN
REWRITE_TAC[MATRIX_MUL_RMUL; MATRIX_MUL_LMUL] THEN
REWRITE_TAC[MATRIX_MUL_LID; MATRIX_MUL_RID]]]);;
let MATRIC_CHARPOLY_DIFFERENCE = prove
(`!A:real^N^N. ?B.
!x. msum(0..dimindex(:N)) (\i. char_poly A i %% A mpow i) -
sum(0..dimindex(:N)) (\i. char_poly A i * x pow i) %% mat 1 =
msum(0..(dimindex(:N)-1)) (\i. x pow i %% B i) ** (A - x %% mat 1)`,
GEN_TAC THEN SPEC_TAC(`dimindex(:N)`,`n:num`) THEN
SPEC_TAC(`char_poly(A:real^N^N)`,`c:num->real`) THEN
GEN_TAC THEN INDUCT_TAC THEN
SIMP_TAC[MSUM_CLAUSES_NUMSEG; SUM_CLAUSES_NUMSEG; LE_0] THENL
[EXISTS_TAC `(\i. mat 0):num->real^N^N` THEN
CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[MSUM_CLAUSES_NUMSEG] THEN
REWRITE_TAC[real_pow; MATRIX_MUL_LMUL; MATRIX_MUL_LZERO; mpow;
REAL_MUL_RID; MATRIX_CMUL_RZERO; MATRIX_SUB_REFL];
FIRST_X_ASSUM(X_CHOOSE_TAC `B:num->real^N^N`) THEN
REWRITE_TAC[MATRIX_SUB; MATRIX_NEG_ADD; MATRIX_CMUL_ADD_RDISTRIB] THEN
ONCE_REWRITE_TAC[AC MATRIX_ADD_AC
`(A + B) + (C + D):real^N^N = (A + C) + (B + D)`] THEN
ASM_REWRITE_TAC[GSYM MATRIX_SUB] THEN
REWRITE_TAC[GSYM MATRIX_CMUL_ASSOC; GSYM MATRIX_CMUL_SUB_LDISTRIB] THEN
REWRITE_TAC[MATRIC_POWER_DIFFERENCE; SUC_SUB1] THEN
EXISTS_TAC `(\i. (if i <= n - 1 then B i else mat 0) +
c(SUC n) %% A mpow (n - i)):num->real^N^N` THEN
X_GEN_TAC `x:real` THEN REWRITE_TAC[MATRIX_CMUL_ADD_LDISTRIB] THEN
SIMP_TAC[MSUM_ADD; FINITE_NUMSEG; MATRIX_ADD_RDISTRIB] THEN
REWRITE_TAC[GSYM MATRIX_MUL_LMUL] THEN
BINOP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THENL
[REWRITE_TAC[COND_RAND; COND_RATOR; MATRIX_CMUL_RZERO] THEN
REWRITE_TAC[GSYM MSUM_RESTRICT_SET; IN_NUMSEG] THEN
REWRITE_TAC[numseg; ARITH_RULE
`(0 <= i /\ i <= n) /\ i <= n - 1 <=> 0 <= i /\ i <= n - 1`];
SIMP_TAC[GSYM MSUM_LMUL; FINITE_NUMSEG; MATRIX_CMUL_ASSOC] THEN
REWRITE_TAC[REAL_MUL_SYM]]]);;
let CAYLEY_HAMILTON = prove
(`!A:real^N^N. msum(0..dimindex(:N)) (\i. char_poly A i %% A mpow i) = mat 0`,
GEN_TAC THEN MATCH_MP_TAC MATRIC_POLY_LEMMA THEN MATCH_MP_TAC(MESON[]
`!g. (!x. g x = k) /\ (?a b c. !x. f a b c x = g x)
==> ?a b c. !x. f a b c x = k`) THEN
EXISTS_TAC
`\x. (msum(0..dimindex(:N)) (\i. char_poly A i %% (A:real^N^N) mpow i) -
sum(0..dimindex(:N)) (\i. char_poly A i * x pow i) %% mat 1) +
sum(0..dimindex(:N)) (\i. char_poly A i * x pow i) %% mat 1` THEN
REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[MATRIX_SUB; GSYM MATRIX_ADD_ASSOC; MATRIX_ADD_LNEG] THEN
REWRITE_TAC[MATRIX_ADD_RID];
X_CHOOSE_THEN `B:num->real^N^N` (fun th -> ONCE_REWRITE_TAC[th])
(ISPEC `A:real^N^N` MATRIC_CHARPOLY_DIFFERENCE) THEN
REWRITE_TAC[GSYM char_poly; GSYM MATRIX_MUL_LEFT_COFACTOR] THEN
REWRITE_TAC[GSYM MATRIX_ADD_RDISTRIB] THEN
REWRITE_TAC[GSYM COFACTOR_TRANSP; TRANSP_MATRIX_SUB] THEN
REWRITE_TAC[TRANSP_MATRIX_CMUL; TRANSP_MAT] THEN
X_CHOOSE_THEN `C:num->real^N^N` (fun th -> ONCE_REWRITE_TAC[th])
(ISPEC `transp A:real^N^N` COFACTOR_AS_MATRIC_POLYNOMIAL) THEN
MAP_EVERY EXISTS_TAC
[`A:real^N^N`; `(\i. B i + C i):num->real^N^N`; `dimindex(:N)-1`] THEN
SIMP_TAC[GSYM MSUM_ADD; FINITE_NUMSEG; MATRIX_CMUL_ADD_LDISTRIB]]);;
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