statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
comm_mxM f g g' : comm_mx f g -> comm_mx f g' -> comm_mx f (g *m g'). | Proof. by rewrite /comm_mx mulmxA => ->; rewrite -!mulmxA => ->. Qed. | Lemma | comm_mxM | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"comm_mx",
"mulmxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_sum I (s : seq I) (P : pred I) (F : I -> 'M[R]_n) (f : 'M[R]_n) :
(forall i : I, P i -> comm_mx f (F i)) -> comm_mx f (\sum_(i <- s | P i) F i). | Proof. by move=> comm_mxfF; elim/big_ind: _ => // g h; apply: comm_mxD. Qed. | Lemma | comm_mx_sum | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_ind",
"comm_mx",
"comm_mxD",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mxP f g : reflect (comm_mx f g) (comm_mxb f g). | Proof. exact: eqP. Qed. | Lemma | comm_mxP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"comm_mx",
"comm_mxb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_comm_mx fs | := (all2rel comm_mxb fs). | Notation | all_comm_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"all2rel",
"comm_mxb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_comm_mxP fs :
reflect {in fs &, forall f g, f *m g = g *m f} (all_comm_mx fs). | Proof. by apply: (iffP allrelP) => fsP ? ? ? ?; apply/eqP/fsP. Qed. | Lemma | all_comm_mxP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
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"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"all_comm_mx",
"allrelP",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_comm_mx1 f : all_comm_mx [:: f]. | Proof. by rewrite /comm_mxb all2rel1. Qed. | Lemma | all_comm_mx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"all2rel1",
"all_comm_mx",
"comm_mxb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_comm_mx2P f g : reflect (f *m g = g *m f) (all_comm_mx [:: f; g]). | Proof. by rewrite /comm_mxb /= all2rel2 ?eqxx //; exact: eqP. Qed. | Lemma | all_comm_mx2P | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"all2rel2",
"all_comm_mx",
"comm_mxb",
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_comm_mx_cons f fs :
all_comm_mx (f :: fs) = all (comm_mxb f) fs && all_comm_mx fs. | Proof. by rewrite /comm_mxb /= all2rel_cons //= eqxx. Qed. | Lemma | all_comm_mx_cons | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"all",
"all2rel_cons",
"all_comm_mx",
"comm_mxb",
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mxE : comm_mx = @GRing.comm _. | Proof. by []. Qed. | Lemma | comm_mxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"comm",
"comm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_comm_mx | := (allrel comm_mxb). | Notation | all_comm_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"allrel",
"comm_mxb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_mul A B : (A *m B)^T = B^T *m A^T. | Proof.
rewrite trmx_mul_rev; apply/matrixP=> k i; rewrite !mxE.
by apply: eq_bigr => j _; rewrite mulrC.
Qed. | Lemma | trmx_mul | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"matrixP",
"mulrC",
"mxE",
"trmx_mul_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemxAr a A B : a *: (A *m B) = A *m (a *: B). | Proof. by apply: trmx_inj; rewrite trmx_mul !linearZ /= trmx_mul scalemxAl. Qed. | Lemma | scalemxAr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"linearZ",
"scalemxAl",
"trmx_inj",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_is_scalable A : scalable (@mulmx _ m n p A). | Proof. by move=> a B; rewrite scalemxAr. Qed. | Fact | mulmx_is_scalable | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulmx",
"scalable",
"scalemxAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mulmx A : 'M[R]_(n * p, m * p) | := lin_mx (mulmx A). | Definition | lin_mulmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lin_mx",
"mulmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mulmx_is_semilinear : semilinear lin_mulmx. | Proof.
by split=> [a A|A B]; apply/row_matrixP=> i; rewrite (linearZ, linearD) /=;
rewrite !rowE !mul_rV_lin /= -(linearZ, linearD) /= (scalemxAl, mulmxDl).
Qed. | Fact | lin_mulmx_is_semilinear | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"bigop",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"lin_mulmx",
"linearD",
"linearZ",
"mul_rV_lin",
"mulmxDl",
"rowE",
"row_matrixP",
"scalemxAl",
"semilinear",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mul_row u : 'M[R]_(m * n, n) | := lin1_mx (mulmx u \o vec_mx). | Definition | lin_mul_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lin1_mx",
"mulmx",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mul_row_is_semilinear : semilinear lin_mul_row. | Proof.
by split=> [a u|u v]; apply/row_matrixP=> i; rewrite (linearZ, linearD) /=;
rewrite !rowE !mul_rV_lin1 /= (mulmxDl, scalemxAl).
Qed. | Fact | lin_mul_row_is_semilinear | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"lin_mul_row",
"linearD",
"linearZ",
"mul_rV_lin1",
"mulmxDl",
"rowE",
"row_matrixP",
"scalemxAl",
"semilinear",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_vec_lin_row A u : mxvec A *m lin_mul_row u = u *m A. | Proof. by rewrite mul_rV_lin1 /= mxvecK. Qed. | Lemma | mul_vec_lin_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"order",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lin_mul_row",
"mul_rV_lin1",
"mxvec",
"mxvecK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mxC n (d e : 'rV[R]_n) :
diag_mx d *m diag_mx e = diag_mx e *m diag_mx d. | Proof.
by rewrite !mulmx_diag; congr (diag_mx _); apply/rowP=> i; rewrite !mxE mulrC.
Qed. | Lemma | diag_mxC | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"diag_mx",
"mulmx_diag",
"mulrC",
"mxE",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_comm n (d e : 'rV[R]_n) : comm_mx (diag_mx d) (diag_mx e). | Proof. exact: diag_mxC. Qed. | Lemma | diag_mx_comm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"comm_mx",
"diag_mx",
"diag_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mxC m n a (A : 'M[R]_(m, n)) : A *m a%:M = a%:M *m A. | Proof.
rewrite -!diag_const_mx mul_mx_diag mul_diag_mx.
by apply/matrixP => i j; rewrite !mxE mulrC.
Qed. | Lemma | scalar_mxC | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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... | [
"apply",
"diag_const_mx",
"matrixP",
"mul_diag_mx",
"mul_mx_diag",
"mulrC",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_scalar n a (A : 'M[R]_n) : comm_mx A a%:M. | Proof. exact: scalar_mxC. Qed. | Lemma | comm_mx_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"comm_mx",
"scalar_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_scalar_mx n a (A : 'M[R]_n) : comm_mx a%:M A. | Proof. exact/comm_mx_sym/comm_mx_scalar. Qed. | Lemma | comm_scalar_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"comm_mx",
"comm_mx_scalar",
"comm_mx_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_mulC m n (A : 'M[R]_(m, n)) B : \tr (A *m B) = \tr (B *m A). | Proof.
have expand_trM C D: \tr (C *m D) = \sum_i \sum_j C i j * D j i.
by apply: eq_bigr => i _; rewrite mxE.
rewrite !{}expand_trM exchange_big /=.
by do 2!apply: eq_bigr => ? _; apply: mulrC.
Qed. | Lemma | mxtrace_mulC | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"exchange_big",
"mulrC",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec_dotmul m n (A : 'M[R]_(m, n)) u v :
mxvec (u^T *m v) *m (mxvec A)^T = u *m A *m v^T. | Proof.
transitivity (\sum_i \sum_j (u 0 i * A i j *: row j v^T)).
apply/rowP=> i; rewrite {i}ord1 mxE (reindex _ (curry_mxvec_bij _ _)) /=.
rewrite pair_bigA summxE; apply: eq_bigr => [[i j]] /= _.
by rewrite !mxE !mxvecE mxE big_ord1 mxE mulrAC.
rewrite mulmx_sum_row exchange_big; apply: eq_bigr => j _ /=.
by re... | Lemma | mxvec_dotmul | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_ord1",
"curry_mxvec_bij",
"eq_bigr",
"exchange_big",
"mulmx_sum_row",
"mulrAC",
"mxE",
"mxvec",
"mxvecE",
"ord1",
"pair_bigA",
"reindex",
"row",
"rowP",
"scaler_suml",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mx_scalar m n a (A : 'M[R]_(m, n)) : A *m a%:M = a *: A. | Proof. by rewrite scalar_mxC mul_scalar_mx. Qed. | Lemma | mul_mx_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mul_scalar_mx",
"scalar_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_is_linear n : linear (@diag_mx R n). | Proof. exact: linearP. Qed. | Fact | diag_mx_is_linear | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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... | [
"diag_mx",
"linear",
"linearP"
] | Diagonal matrices | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mulmxN m n p (A : 'M[R]_(m, n)) (B : 'M_(n, p)) : A *m (- B) = - (A *m B). | Proof. exact: raddfN. Qed. | Lemma | mulmxN | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"finalg",
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... | [
"raddfN"
] | Scalar matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mulNmx m n p (A : 'M[R]_(m, n)) (B : 'M_(n, p)) : - A *m B = - (A *m B). | Proof. exact: (raddfN (mulmxr _)). Qed. | Lemma | mulNmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"zmodp",
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... | [
"mulmxr",
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxBl m n p (A1 A2 : 'M[R]_(m, n)) (B : 'M_(n, p)) :
(A1 - A2) *m B = A1 *m B - A2 *m B. | Proof. exact: (raddfB (mulmxr _)). Qed. | Lemma | mulmxBl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"finalg",
"zmodp",
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... | [
"mulmxr",
"raddfB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxBr m n p (A : 'M[R]_(m, n)) (B1 B2 : 'M_(n, p)) :
A *m (B1 - B2) = A *m B1 - A *m B2. | Proof. exact: raddfB. Qed. | Lemma | mulmxBr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"raddfB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
copid_mx {n} r : 'M[R]_n | := 1%:M - pid_mx r. | Definition | copid_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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... | [
"pid_mx"
] | Partial identity matrix (used in rank decomposition). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mul_copid_mx_pid m n r :
r <= m -> copid_mx r *m pid_mx r = 0 :> 'M_(m, n). | Proof. by move=> le_r_m; rewrite mulmxBl mul1mx pid_mx_id ?subrr. Qed. | Lemma | mul_copid_mx_pid | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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... | [
"copid_mx",
"mul1mx",
"mulmxBl",
"pid_mx",
"pid_mx_id",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_pid_mx_copid m n r :
r <= n -> pid_mx r *m copid_mx r = 0 :> 'M_(m, n). | Proof. by move=> le_r_n; rewrite mulmxBr mulmx1 pid_mx_id ?subrr. Qed. | Lemma | mul_pid_mx_copid | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"finalg",
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"GRing.Theory",
... | [
"copid_mx",
"mulmx1",
"mulmxBr",
"pid_mx",
"pid_mx_id",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
copid_mx_id n r : r <= n -> copid_mx r *m copid_mx r = copid_mx r :> 'M_n. | Proof.
by move=> le_r_n; rewrite mulmxBl mul1mx mul_pid_mx_copid // oppr0 addr0.
Qed. | Lemma | copid_mx_id | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"GRing.Theory",
... | [
"addr0",
"copid_mx",
"mul1mx",
"mul_pid_mx_copid",
"mulmxBl",
"oppr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxr_is_linear m n p B : linear (@mulmxr R m n p B). | Proof. exact: linearP. Qed. | Fact | mulmxr_is_linear | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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... | [
"linear",
"linearP",
"mulmxr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mulmxr_is_linear m n p : linear (@lin_mulmxr R m n p). | Proof. exact: linearP. Qed. | Fact | lin_mulmxr_is_linear | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finalg",
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"GRing.Theory",
... | [
"lin_mulmxr",
"linear",
"linearP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_is_scalar n : scalar (@mxtrace R n). | Proof. exact: scalarP. Qed. | Fact | mxtrace_is_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
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"finalg",
"zmodp",
"GRing.Theory",
... | [
"mxtrace",
"scalar",
"scalarP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
determinant n (A : 'M_n) : R | :=
\sum_(s : 'S_n) (-1) ^+ s * \prod_i A i (s i). | Definition | determinant | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
"ssrnat",
"seq",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | The determinant, in one line with the Leibniz Formula | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cofactor n A (i j : 'I_n) : R | :=
(-1) ^+ (i + j) * determinant (row' i (col' j A)). | Definition | cofactor | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
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... | [
"col'",
"determinant",
"row'"
] | The cofactor of a matrix on the indexes i and j | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
adjugate_key : unit. | Proof. by []. Qed. | Fact | adjugate_key | algebra | algebra/matrix.v | [
"HB",
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"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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... | [
"unit"
] | The adjugate matrix : defined as the transpose of the matrix of cofactors | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
adjugate n (A : 'M_n) | := \matrix[adjugate_key]_(i, j) cofactor A j i. | Definition | adjugate | algebra | algebra/matrix.v | [
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"mathcomp",
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... | [
"adjugate_key",
"cofactor",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'\det' A" | := (determinant A) : ring_scope. | Notation | '\det' A | algebra | algebra/matrix.v | [
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"mathcomp",
"ssreflect",
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... | [
"determinant"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'\adj' A" | := (adjugate A) : ring_scope. | Notation | '\adj' A | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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... | [
"adjugate"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_map_mx n' (A : 'M_n') : \det A^f = f (\det A). | Proof.
rewrite rmorph_sum //; apply: eq_bigr => s _.
rewrite rmorphM /= rmorph_sign rmorph_prod; congr (_ * _).
by apply: eq_bigr => i _; rewrite mxE.
Qed. | Lemma | det_map_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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... | [
"apply",
"eq_bigr",
"mxE",
"n'",
"rmorphM",
"rmorph_prod",
"rmorph_sign",
"rmorph_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cofactor_map_mx (A : 'M_n) i j : cofactor A^f i j = f (cofactor A i j). | Proof. by rewrite rmorphM /= rmorph_sign -det_map_mx map_row' map_col'. Qed. | Lemma | cofactor_map_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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... | [
"cofactor",
"det_map_mx",
"map_col'",
"map_row'",
"rmorphM",
"rmorph_sign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_adj (A : 'M_n) : (\adj A)^f = \adj A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE cofactor_map_mx. Qed. | Lemma | map_mx_adj | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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... | [
"apply",
"cofactor_map_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_copid_mx n r : (copid_mx r)^f = copid_mx r :> 'M_n. | Proof. by rewrite map_mxB map_mx1 map_pid_mx. Qed. | Lemma | map_copid_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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... | [
"copid_mx",
"map_mx1",
"map_mxB",
"map_pid_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mxN f g : comm_mx f g -> comm_mx f (- g). | Proof. by rewrite /comm_mx mulmxN mulNmx => ->. Qed. | Lemma | comm_mxN | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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... | [
"comm_mx",
"mulNmx",
"mulmxN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mxN1 f : comm_mx f (- 1%:M). | Proof. exact/comm_mxN/comm_mx1. Qed. | Lemma | comm_mxN1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"comm_mx",
"comm_mx1",
"comm_mxN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mxB f g g' : comm_mx f g -> comm_mx f g' -> comm_mx f (g - g'). | Proof. by move=> fg fg'; apply/comm_mxD => //; apply/comm_mxN. Qed. | Lemma | comm_mxB | algebra | algebra/matrix.v | [
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"comm_mxN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mulmx_is_linear m n p : linear (@lin_mulmx R m n p). | Proof. exact: linearP. Qed. | Fact | lin_mulmx_is_linear | algebra | algebra/matrix.v | [
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"mathcomp",
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mul_row_is_linear m n : linear (@lin_mul_row R m n). | Proof. exact: linearP. Qed. | Fact | lin_mul_row_is_linear | algebra | algebra/matrix.v | [
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"mathcomp",
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"linear",
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
determinant_multilinear n (A B C : 'M[R]_n) i0 b c :
row i0 A = b *: row i0 B + c *: row i0 C ->
row' i0 B = row' i0 A ->
row' i0 C = row' i0 A ->
\det A = b * \det B + c * \det C. | Proof.
rewrite -[_ + _](row_id 0); move/row_eq=> ABC.
move/row'_eq=> BA; move/row'_eq=> CA.
rewrite !big_distrr -big_split; apply: eq_bigr => s _ /=.
rewrite -!(mulrCA (_ ^+s)) -mulrDr; congr (_ * _).
rewrite !(bigD1 i0 (_ : predT i0)) //= {}ABC !mxE mulrDl !mulrA.
by congr (_ * _ + _ * _); apply: eq_bigr => i i0i; rew... | Lemma | determinant_multilinear | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"bigD1",
"big_distrr",
"big_split",
"eq_bigr",
"i0",
"mulrA",
"mulrCA",
"mulrDl",
"mulrDr",
"mxE",
"row",
"row'",
"row'_eq",
"row_eq",
"row_id"
] | The theory of determinants | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
determinant_alternate n (A : 'M[R]_n) i1 i2 :
i1 != i2 -> A i1 =1 A i2 -> \det A = 0. | Proof.
move=> neq_i12 eqA12; pose t := tperm i1 i2.
have oddMt s: (t * s)%g = ~~ s :> bool by rewrite odd_permM odd_tperm neq_i12.
rewrite [\det A](bigID (@odd_perm _)) /=.
apply: canLR (subrK _) _; rewrite add0r -sumrN.
rewrite (reindex_inj (mulgI t)); apply: eq_big => //= s.
rewrite oddMt => /negPf->; rewrite mulN1r ... | Lemma | determinant_alternate | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"add0r",
"apply",
"bigID",
"eq_big",
"eq_bigr",
"mul1r",
"mulN1r",
"mulgI",
"odd_perm",
"odd_permM",
"odd_tperm",
"permM",
"reindex_inj",
"reindex_perm",
"subrK",
"sumrN",
"tperm",
"tpermK",
"tpermP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_tr n (A : 'M[R]_n) : \det A^T = \det A. | Proof.
rewrite [\det A^T](reindex_inj invg_inj) /=.
apply: eq_bigr => s _ /=; rewrite !odd_permV (reindex_perm s) /=.
by congr (_ * _); apply: eq_bigr => i _; rewrite mxE permK.
Qed. | Lemma | det_tr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"invg_inj",
"mxE",
"odd_permV",
"permK",
"reindex_inj",
"reindex_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_perm n (s : 'S_n) : \det (perm_mx s) = (-1) ^+ s :> R. | Proof.
rewrite [\det _](bigD1 s) //= big1 => [i _|]; first by rewrite /= !mxE eqxx.
rewrite mulr1 big1 ?addr0 => //= t Dst.
case: (pickP (fun i => s i != t i)) => [i ist | Est].
by rewrite (bigD1 i) // mulrCA /= !mxE (negPf ist) mul0r.
by case/eqP: Dst; apply/permP => i; move/eqP: (Est i).
Qed. | Lemma | det_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addr0",
"apply",
"big1",
"bigD1",
"eqxx",
"mul0r",
"mulr1",
"mulrCA",
"mxE",
"permP",
"perm_mx",
"pickP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det1 n : \det (1%:M : 'M[R]_n) = 1. | Proof. by rewrite -perm_mx1 det_perm odd_perm1. Qed. | Lemma | det1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_perm",
"odd_perm1",
"perm_mx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_mx00 (A : 'M[R]_0) : \det A = 1. | Proof. by rewrite flatmx0 -(flatmx0 1%:M) det1. Qed. | Lemma | det_mx00 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det1",
"flatmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
detZ n a (A : 'M[R]_n) : \det (a *: A) = a ^+ n * \det A. | Proof.
rewrite big_distrr /=; apply: eq_bigr => s _; rewrite mulrCA; congr (_ * _).
rewrite -[n in a ^+ n]card_ord -prodr_const -big_split /=.
by apply: eq_bigr=> i _; rewrite mxE.
Qed. | Lemma | detZ | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_distrr",
"big_split",
"card_ord",
"eq_bigr",
"mulrCA",
"mxE",
"prodr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det0 n' : \det (0 : 'M[R]_n'.+1) = 0. | Proof. by rewrite -(scale0r 0) detZ exprS !mul0r. Qed. | Lemma | det0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"detZ",
"exprS",
"mul0r",
"n'",
"scale0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_scalar n a : \det (a%:M : 'M[R]_n) = a ^+ n. | Proof. by rewrite -{1}(mulr1 a) -scale_scalar_mx detZ det1 mulr1. Qed. | Lemma | det_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det1",
"detZ",
"mulr1",
"scale_scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_scalar1 a : \det (a%:M : 'M[R]_1) = a. | Proof. exact: det_scalar. Qed. | Lemma | det_scalar1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_scalar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_mx11 (M : 'M[R]_1) : \det M = M 0 0. | Proof. by rewrite {1}[M]mx11_scalar det_scalar. Qed. | Lemma | det_mx11 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_scalar",
"mx11_scalar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_mulmx n (A B : 'M[R]_n) : \det (A *m B) = \det A * \det B. | Proof.
rewrite big_distrl /=.
pose F := ('I_n ^ n)%type; pose AB s i j := A i j * B j (s i).
transitivity (\sum_(f : F) \sum_(s : 'S_n) (-1) ^+ s * \prod_i AB s i (f i)).
rewrite exchange_big; apply: eq_bigr => /= s _; rewrite -big_distrr /=.
congr (_ * _); rewrite -(bigA_distr_bigA (AB s)) /=.
by apply: eq_bigr ... | Lemma | det_mulmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"add0r",
"addrC",
"apply",
"big1",
"bigA_distr_bigA",
"bigID",
"big_distrl",
"big_distrr",
"big_split",
"determinant_alternate",
"eq_big",
"eq_bigr",
"exchange_big",
"injectivePn",
"injectiveb",
"insubd",
"insubdK",
"mulgI",
"mulrA",
"mulrACA",
"mulrC",
"mulrCA",
"mxE",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
detM n' (A B : 'M[R]_n'.+1) : \det (A * B) = \det A * \det B. | Proof. exact: det_mulmx. Qed. | Lemma | detM | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_mulmx",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expand_cofactor n (A : 'M[R]_n) i j :
cofactor A i j =
\sum_(s : 'S_n | s i == j) (-1) ^+ s * \prod_(k | i != k) A k (s k). | Proof.
case: n A i j => [|n] A i0 j0; first by case: i0.
rewrite (reindex (lift_perm i0 j0)).
pose ulsf i (s : 'S_n.+1) k := odflt k (unlift (s i) (s (lift i k))).
have ulsfK i (s : 'S_n.+1) k: lift (s i) (ulsf i s k) = s (lift i k).
rewrite /ulsf; have:= neq_lift i k.
by rewrite -(can_eq (permK s)) => /unl... | Lemma | expand_cofactor | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big1",
"big_distrr",
"can_eq",
"cofactor",
"eq_big",
"eqxx",
"i0",
"last",
"lift",
"liftK",
"lift_perm",
"lift_perm_id",
"lift_perm_lift",
"mulrA",
"mxE",
"neq_lift",
"oddD",
"odd_lift_perm",
"permE",
"permK",
"permP",
"pickP",
"reindex",
"signr_addb",
"si... | Laplace expansion lemma | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
expand_det_row n (A : 'M[R]_n) i0 :
\det A = \sum_j A i0 j * cofactor A i0 j. | Proof.
rewrite /(\det A) (partition_big (fun s : 'S_n => s i0) predT) //=.
apply: eq_bigr => j0 _; rewrite expand_cofactor big_distrr /=.
apply: eq_bigr => s /eqP Dsi0.
rewrite mulrCA (bigID (pred1 i0)) /= big_pred1_eq Dsi0; congr (_ * (_ * _)).
by apply: eq_bigl => i; rewrite eq_sym.
Qed. | Lemma | expand_det_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"bigID",
"big_distrr",
"big_pred1_eq",
"cofactor",
"eq_bigl",
"eq_bigr",
"eq_sym",
"expand_cofactor",
"i0",
"mulrCA",
"partition_big",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cofactor_tr n (A : 'M[R]_n) i j : cofactor A^T i j = cofactor A j i. | Proof.
rewrite /cofactor addnC; congr (_ * _).
rewrite -tr_row' -tr_col' det_tr; congr (\det _).
by apply/matrixP=> ? ?; rewrite !mxE.
Qed. | Lemma | cofactor_tr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addnC",
"apply",
"cofactor",
"det_tr",
"matrixP",
"mxE",
"tr_col'",
"tr_row'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cofactorZ n a (A : 'M[R]_n) i j :
cofactor (a *: A) i j = a ^+ n.-1 * cofactor A i j. | Proof. by rewrite {1}/cofactor !linearZ detZ mulrCA mulrA. Qed. | Lemma | cofactorZ | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"cofactor",
"detZ",
"linearZ",
"mulrA",
"mulrCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expand_det_col n (A : 'M[R]_n) j0 :
\det A = \sum_i (A i j0 * cofactor A i j0). | Proof.
rewrite -det_tr (expand_det_row _ j0).
by under eq_bigr do rewrite cofactor_tr mxE.
Qed. | Lemma | expand_det_col | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"cofactor",
"cofactor_tr",
"det_tr",
"eq_bigr",
"expand_det_row",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_adj n (A : 'M[R]_n) : (\adj A)^T = \adj A^T. | Proof. by apply/matrixP=> i j; rewrite !mxE cofactor_tr. Qed. | Lemma | trmx_adj | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"cofactor_tr",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adjZ n a (A : 'M[R]_n) : \adj (a *: A) = a^+n.-1 *: \adj A. | Proof. by apply/matrixP=> i j; rewrite !mxE cofactorZ. Qed. | Lemma | adjZ | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"cofactorZ",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mx_adj n (A : 'M[R]_n) : A *m \adj A = (\det A)%:M. | Proof.
apply/matrixP=> i1 i2 /[!mxE]; have [->|Di] := eqVneq.
rewrite (expand_det_row _ i2) //=.
by apply: eq_bigr => j _; congr (_ * _); rewrite mxE.
pose B := \matrix_(i, j) (if i == i2 then A i1 j else A i j).
have EBi12: B i1 =1 B i2 by move=> j; rewrite /= !mxE eqxx (negPf Di).
rewrite -[_ *+ _](determinant_al... | Lemma | mul_mx_adj | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"determinant_alternate",
"eqVneq",
"eq_bigr",
"eq_sym",
"eqxx",
"expand_det_row",
"matrixP",
"mxE",
"neq_lift"
] | Cramer Rule : adjugate on the left | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mul_adj_mx n (A : 'M[R]_n) : \adj A *m A = (\det A)%:M. | Proof.
by apply: trmx_inj; rewrite trmx_mul trmx_adj mul_mx_adj det_tr tr_scalar_mx.
Qed. | Lemma | mul_adj_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"det_tr",
"mul_mx_adj",
"tr_scalar_mx",
"trmx_adj",
"trmx_inj",
"trmx_mul"
] | Cramer rule : adjugate on the right | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
adj1 n : \adj (1%:M) = 1%:M :> 'M[R]_n. | Proof. by rewrite -{2}(det1 n) -mul_adj_mx mulmx1. Qed. | Lemma | adj1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det1",
"mul_adj_mx",
"mulmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx1C n (A B : 'M[R]_n) : A *m B = 1%:M -> B *m A = 1%:M. | Proof.
move=> AB1; pose A' := \det B *: \adj A.
suffices kA: A' *m A = 1%:M by rewrite -[B]mul1mx -kA -(mulmxA A') AB1 mulmx1.
by rewrite -scalemxAl mul_adj_mx scale_scalar_mx mulrC -det_mulmx AB1 det1.
Qed. | Lemma | mulmx1C | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"A'",
"det1",
"det_mulmx",
"mul1mx",
"mul_adj_mx",
"mulmx1",
"mulmxA",
"mulrC",
"scale_scalar_mx",
"scalemxAl"
] | Left inverses are right inverses. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
det_ublock n1 n2 Aul (Aur : 'M[R]_(n1, n2)) Adr :
\det (block_mx Aul Aur 0 Adr) = \det Aul * \det Adr. | Proof.
elim: n1 => [|n1 IHn1] in Aul Aur *.
have ->: Aul = 1%:M by apply/matrixP=> i [].
rewrite det1 mul1r; congr (\det _); apply/matrixP=> i j.
by do 2![rewrite !mxE; case: splitP => [[]|k] //=; move/val_inj=> <- {k}].
rewrite (expand_det_col _ (lshift n2 0)) big_split_ord /=.
rewrite addrC big1 1?simp => [i _|... | Lemma | det_ublock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addrC",
"apply",
"big1",
"big_distrl",
"big_split_ord",
"block_mx",
"block_mxEdl",
"block_mxEul",
"col'Kl",
"col'_col_mx",
"det1",
"eq_bigr",
"expand_det_col",
"lshift",
"matrixP",
"mul1r",
"mulrA",
"mxE",
"raddf0",
"row'Ku",
"row'_row_mx",
"simp",
"splitP",
"val_inj"
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_lblock n1 n2 Aul (Adl : 'M[R]_(n2, n1)) Adr :
\det (block_mx Aul 0 Adl Adr) = \det Aul * \det Adr. | Proof. by rewrite -det_tr tr_block_mx trmx0 det_ublock !det_tr. Qed. | Lemma | det_lblock | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"det_tr",
"det_ublock",
"tr_block_mx",
"trmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_trig n (A : 'M[R]_n) : is_trig_mx A -> \det A = \prod_(i < n) A i i. | Proof.
elim/trigsqmx_ind => [|k x c B Bt IHB]; first by rewrite ?big_ord0 ?det_mx00.
rewrite det_lblock big_ord_recl det_mx11 IHB//; congr (_ * _).
by rewrite -[ord0](lshift0 _ 0) block_mxEul.
by apply: eq_bigr => i; rewrite -!rshift1 block_mxEdr.
Qed. | Lemma | det_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_ord0",
"big_ord_recl",
"block_mxEdr",
"block_mxEul",
"det_lblock",
"det_mx00",
"det_mx11",
"eq_bigr",
"is_trig_mx",
"lshift0",
"ord0",
"rshift1",
"trigsqmx_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_diag n (d : 'rV[R]_n) : \det (diag_mx d) = \prod_i d 0 i. | Proof. by rewrite det_trig//; apply: eq_bigr => i; rewrite !mxE eqxx. Qed. | Lemma | det_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"det_trig",
"diag_mx",
"eq_bigr",
"eqxx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx1_min (R : comNzRingType) m n (A : 'M[R]_(m, n)) B :
A *m B = 1%:M -> m <= n. | Proof.
move=> AB1; rewrite leqNgt; apply/negP=> /subnKC; rewrite addSnnS.
move: (_ - _)%N => m' def_m; move: AB1; rewrite -{m}def_m in A B *.
rewrite -(vsubmxK A) -(hsubmxK B) mul_col_row scalar_mx_block.
case/eq_block_mx=> /mulmx1C BlAu1 AuBr0 _ => /eqP/idPn[].
by rewrite -[_ B]mul1mx -BlAu1 -mulmxA AuBr0 !mulmx0 eq_s... | Lemma | mulmx1_min | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addSnnS",
"apply",
"eq_block_mx",
"eq_sym",
"hsubmxK",
"leqNgt",
"mul1mx",
"mul_col_row",
"mulmx0",
"mulmx1C",
"mulmxA",
"oner_neq0",
"scalar_mx_block",
"subnKC",
"vsubmxK"
] | Only tall matrices have inverses. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
unitmx : pred 'M[R]_n | := fun A => \det A \is a GRing.unit. | Definition | unitmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmx A | := if A \in unitmx then (\det A)^-1 *: \adj A else A. | Definition | invmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitmxE A : (A \in unitmx) = (\det A \is a GRing.unit). | Proof. by []. Qed. | Lemma | unitmxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitmx1 : 1%:M \in unitmx. | Proof. by rewrite unitmxE det1 unitr1. Qed. | Lemma | unitmx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det1",
"unitmx",
"unitmxE",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitmx_perm s : perm_mx s \in unitmx. | Proof. by rewrite unitmxE det_perm unitrX ?unitrN ?unitr1. Qed. | Lemma | unitmx_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_perm",
"perm_mx",
"unitmx",
"unitmxE",
"unitr1",
"unitrN",
"unitrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitmx_tr A : (A^T \in unitmx) = (A \in unitmx). | Proof. by rewrite unitmxE det_tr. Qed. | Lemma | unitmx_tr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_tr",
"unitmx",
"unitmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitmxZ a A : a \is a GRing.unit -> (a *: A \in unitmx) = (A \in unitmx). | Proof. by move=> Ua; rewrite !unitmxE detZ unitrM unitrX. Qed. | Lemma | unitmxZ | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"detZ",
"unit",
"unitmx",
"unitmxE",
"unitrM",
"unitrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmx1 : invmx 1%:M = 1%:M. | Proof. by rewrite /invmx det1 invr1 scale1r adj1 if_same. Qed. | Lemma | invmx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"adj1",
"det1",
"invmx",
"invr1",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmxZ a A : a *: A \in unitmx -> invmx (a *: A) = a^-1 *: invmx A. | Proof.
rewrite /invmx !unitmxE detZ unitrM => /andP[Ua U_A].
rewrite Ua U_A adjZ !scalerA invrM {U_A}//=.
case: (posnP n) A => [-> | n_gt0] A; first by rewrite flatmx0 [_ *: _]flatmx0.
rewrite unitrX_pos // in Ua; rewrite -[_ * _](mulrK Ua) mulrC -!mulrA.
by rewrite -exprSr prednK // !mulrA divrK ?unitrX.
Qed. | Lemma | invmxZ | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"adjZ",
"detZ",
"divrK",
"exprSr",
"flatmx0",
"invmx",
"invrM",
"mulrA",
"mulrC",
"mulrK",
"n_gt0",
"posnP",
"prednK",
"scalerA",
"unitmx",
"unitmxE",
"unitrM",
"unitrX",
"unitrX_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmx_scalar a : invmx a%:M = a^-1%:M. | Proof.
case Ua: (a%:M \in unitmx).
by rewrite -scalemx1 in Ua *; rewrite invmxZ // invmx1 scalemx1.
rewrite /invmx Ua; have [->|n_gt0] := posnP n; first by rewrite ![_%:M]flatmx0.
by rewrite unitmxE det_scalar unitrX_pos // in Ua; rewrite invr_out ?Ua.
Qed. | Lemma | invmx_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_scalar",
"flatmx0",
"invmx",
"invmx1",
"invmxZ",
"invr_out",
"n_gt0",
"posnP",
"scalemx1",
"unitmx",
"unitmxE",
"unitrX_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVmx : {in unitmx, left_inverse 1%:M invmx mulmx}. | Proof.
by move=> A nsA; rewrite /invmx nsA -scalemxAl mul_adj_mx scale_scalar_mx mulVr.
Qed. | Lemma | mulVmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"invmx",
"mulVr",
"mul_adj_mx",
"mulmx",
"scale_scalar_mx",
"scalemxAl",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxV : {in unitmx, right_inverse 1%:M invmx mulmx}. | Proof.
by move=> A nsA; rewrite /invmx nsA -scalemxAr mul_mx_adj scale_scalar_mx mulVr.
Qed. | Lemma | mulmxV | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"invmx",
"mulVr",
"mul_mx_adj",
"mulmx",
"scale_scalar_mx",
"scalemxAr",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulKmx m : {in unitmx, @left_loop _ 'M_(n, m) invmx mulmx}. | Proof. by move=> A uA /= B; rewrite mulmxA mulVmx ?mul1mx. Qed. | Lemma | mulKmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"invmx",
"mul1mx",
"mulVmx",
"mulmx",
"mulmxA",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulKVmx m : {in unitmx, @rev_left_loop _ 'M_(n, m) invmx mulmx}. | Proof. by move=> A uA /= B; rewrite mulmxA mulmxV ?mul1mx. Qed. | Lemma | mulKVmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"invmx",
"mul1mx",
"mulmx",
"mulmxA",
"mulmxV",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxK m : {in unitmx, @right_loop 'M_(m, n) _ invmx mulmx}. | Proof. by move=> A uA /= B; rewrite -mulmxA mulmxV ?mulmx1. Qed. | Lemma | mulmxK | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"invmx",
"mulmx",
"mulmx1",
"mulmxA",
"mulmxV",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxKV m : {in unitmx, @rev_right_loop 'M_(m, n) _ invmx mulmx}. | Proof. by move=> A uA /= B; rewrite -mulmxA mulVmx ?mulmx1. Qed. | Lemma | mulmxKV | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"invmx",
"mulVmx",
"mulmx",
"mulmx1",
"mulmxA",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_inv A : \det (invmx A) = (\det A)^-1. | Proof.
case uA: (A \in unitmx); last by rewrite /invmx uA invr_out ?negbT.
by apply: (mulrI uA); rewrite -det_mulmx mulmxV ?divrr ?det1.
Qed. | Lemma | det_inv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"det1",
"det_mulmx",
"divrr",
"invmx",
"invr_out",
"last",
"mulmxV",
"mulrI",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitmx_inv A : (invmx A \in unitmx) = (A \in unitmx). | Proof. by rewrite !unitmxE det_inv unitrV. Qed. | Lemma | unitmx_inv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"det_inv",
"invmx",
"unitmx",
"unitmxE",
"unitrV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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