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 |
|---|---|---|---|---|---|---|---|---|---|---|
tr_tperm_mx n i1 i2 : (tperm_mx i1 i2)^T = tperm_mx i1 i2 :> 'M_n. | Proof. by rewrite tr_perm_mx tpermV. Qed. | Lemma | tr_tperm_mx | 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",
... | [
"tpermV",
"tperm_mx",
"tr_perm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_mx1 n : perm_mx 1 = 1%:M :> 'M_n. | Proof. exact: row_perm1. Qed. | Lemma | perm_mx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"perm_mx",
"row_perm1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_mxM n (s t : 'S_n) : perm_mx (s * t) = perm_mx s *m perm_mx t. | Proof. by rewrite -row_permE -row_permM. Qed. | Lemma | perm_mxM | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"perm_mx",
"row_permE",
"row_permM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_perm_mx n (A : 'M_n) | := [exists s, A == perm_mx s]. | Definition | is_perm_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"perm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_perm_mxP n (A : 'M_n) :
reflect (exists s, A = perm_mx s) (is_perm_mx A). | Proof. by apply: (iffP existsP) => [] [s /eqP]; exists s. Qed. | Lemma | is_perm_mxP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"countalg",
"finalg",
"zmodp",
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... | [
"apply",
"existsP",
"is_perm_mx",
"perm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_mx_is_perm n (s : 'S_n) : is_perm_mx (perm_mx s). | Proof. by apply/is_perm_mxP; exists s. Qed. | Lemma | perm_mx_is_perm | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_perm_mx",
"is_perm_mxP",
"perm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_perm_mx1 n : is_perm_mx (1%:M : 'M_n). | Proof. by rewrite -perm_mx1 perm_mx_is_perm. Qed. | Lemma | is_perm_mx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"is_perm_mx",
"perm_mx1",
"perm_mx_is_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_perm_mxMl n (A B : 'M_n) :
is_perm_mx A -> is_perm_mx (A *m B) = is_perm_mx B. | Proof.
case/is_perm_mxP=> s ->.
apply/is_perm_mxP/is_perm_mxP=> [[t def_t] | [t ->]]; last first.
by exists (s * t)%g; rewrite perm_mxM.
exists (s^-1 * t)%g.
by rewrite perm_mxM -def_t -!row_permE -row_permM mulVg row_perm1.
Qed. | Lemma | is_perm_mxMl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_perm_mx",
"is_perm_mxP",
"last",
"mulVg",
"perm_mxM",
"row_perm1",
"row_permE",
"row_permM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_perm_mx_tr n (A : 'M_n) : is_perm_mx A^T = is_perm_mx A. | Proof.
apply/is_perm_mxP/is_perm_mxP=> [[t def_t] | [t ->]]; exists t^-1%g.
by rewrite -tr_perm_mx -def_t trmxK.
by rewrite tr_perm_mx.
Qed. | Lemma | is_perm_mx_tr | 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",
"is_perm_mx",
"is_perm_mxP",
"tr_perm_mx",
"trmxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_perm_mxMr n (A B : 'M_n) :
is_perm_mx B -> is_perm_mx (A *m B) = is_perm_mx A. | Proof.
case/is_perm_mxP=> s ->.
rewrite -[s]invgK -col_permE -is_perm_mx_tr tr_col_perm row_permE.
by rewrite is_perm_mxMl (perm_mx_is_perm, is_perm_mx_tr).
Qed. | Lemma | is_perm_mxMr | 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",
... | [
"col_permE",
"invgK",
"is_perm_mx",
"is_perm_mxMl",
"is_perm_mxP",
"is_perm_mx_tr",
"perm_mx_is_perm",
"row_permE",
"tr_col_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_key : unit. | Proof. by []. Qed. | Fact | pid_mx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | Partial identity matrix (used in rank decomposition). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pid_mx {m n} r : 'M[R]_(m, n) | :=
\matrix[pid_mx_key]_(i, j) ((i == j :> nat) && (i < r))%:R. | Definition | pid_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix",
"nat",
"pid_mx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_0 m n : pid_mx 0 = 0 :> 'M_(m, n). | Proof. by apply/matrixP=> i j; rewrite !mxE andbF. Qed. | Lemma | pid_mx_0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"countalg",
"finalg",
"zmodp",
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... | [
"apply",
"matrixP",
"mxE",
"pid_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_1 r : pid_mx r = 1%:M :> 'M_r. | Proof. by apply/matrixP=> i j; rewrite !mxE ltn_ord andbT. Qed. | Lemma | pid_mx_1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"ltn_ord",
"matrixP",
"mxE",
"pid_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_row n r : pid_mx r = row_mx 1%:M 0 :> 'M_(r, r + n). | Proof.
apply/matrixP=> i j; rewrite !mxE ltn_ord andbT.
by case: split_ordP => j' ->; rewrite !mxE// (val_eqE (lshift n i)) eq_shift.
Qed. | Lemma | pid_mx_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
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"finalg",
"zmodp",
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... | [
"apply",
"eq_shift",
"lshift",
"ltn_ord",
"matrixP",
"mxE",
"pid_mx",
"row_mx",
"split_ordP",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_col m r : pid_mx r = col_mx 1%:M 0 :> 'M_(r + m, r). | Proof.
apply/matrixP=> i j; rewrite !mxE andbC.
by case: split_ordP => ? ->; rewrite !mxE//.
Qed. | Lemma | pid_mx_col | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"zmodp",
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... | [
"apply",
"col_mx",
"matrixP",
"mxE",
"pid_mx",
"split_ordP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_block m n r : pid_mx r = block_mx 1%:M 0 0 0 :> 'M_(r + m, r + n). | Proof.
apply/matrixP=> i j; rewrite !mxE row_mx0 andbC.
do ![case: split_ordP => ? -> /[!mxE]//].
by rewrite (val_eqE (lshift n _)) eq_shift.
Qed. | Lemma | pid_mx_block | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
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... | [
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"block_mx",
"eq_shift",
"lshift",
"matrixP",
"mxE",
"pid_mx",
"row_mx0",
"split_ordP",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_pid_mx m n r : (pid_mx r)^T = pid_mx r :> 'M_(n, m). | Proof. by apply/matrixP=> i j /[!mxE]; case: eqVneq => // ->. Qed. | Lemma | tr_pid_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eqVneq",
"matrixP",
"mxE",
"pid_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_minv m n r : pid_mx (minn m r) = pid_mx r :> 'M_(m, n). | Proof. by apply/matrixP=> i j; rewrite !mxE leq_min ltn_ord. Qed. | Lemma | pid_mx_minv | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"leq_min",
"ltn_ord",
"matrixP",
"minn",
"mxE",
"pid_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_minh m n r : pid_mx (minn n r) = pid_mx r :> 'M_(m, n). | Proof. by apply: trmx_inj; rewrite !tr_pid_mx pid_mx_minv. Qed. | Lemma | pid_mx_minh | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"minn",
"pid_mx",
"pid_mx_minv",
"tr_pid_mx",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_pid_mx m n p q r :
(pid_mx q : 'M_(m, n)) *m (pid_mx r : 'M_(n, p)) = pid_mx (minn n (minn q r)). | Proof.
apply/matrixP=> i k; rewrite !mxE !leq_min.
have [le_n_i | lt_i_n] := leqP n i.
rewrite andbF big1 // => j _.
by rewrite -pid_mx_minh !mxE leq_min ltnNge le_n_i andbF mul0r.
rewrite (bigD1 (Ordinal lt_i_n)) //= big1 ?addr0 => [j|].
by rewrite -val_eqE /= !mxE eq_sym -natrM => /negPf->.
by rewrite !mxE eqxx... | Lemma | mul_pid_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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... | [
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"mulnb",
"mxE",
"natrM",
"pid_mx",
"pid_mx_minh",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mx_id m n p r :
r <= n -> (pid_mx r : 'M_(m, n)) *m (pid_mx r : 'M_(n, p)) = pid_mx r. | Proof. by move=> le_r_n; rewrite mul_pid_mx minnn (minn_idPr _). Qed. | Lemma | pid_mx_id | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"minn_idPr",
"minnn",
"mul_pid_mx",
"pid_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mxErow m n (le_mn : m <= n) :
pid_mx m = rowsub (widen_ord le_mn) 1%:M. | Proof. by apply/matrixP=> i j; rewrite !mxE -!val_eqE/= ltn_ord andbT. Qed. | Lemma | pid_mxErow | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
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"finalg",
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... | [
"apply",
"ltn_ord",
"matrixP",
"mxE",
"pid_mx",
"rowsub",
"val_eqE",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pid_mxEcol m n (le_mn : m <= n) :
pid_mx n = colsub (widen_ord le_mn) 1%:M. | Proof. by apply/matrixP=> i j; rewrite !mxE -!val_eqE/= ltn_ord andbT. Qed. | Lemma | pid_mxEcol | algebra | algebra/matrix.v | [
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"structures",
"mathcomp",
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"eqtype",
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... | [
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"val_eqE",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mx_row m n p1 p2 (A : 'M_(m, n)) (Bl : 'M_(n, p1)) (Br : 'M_(n, p2)) :
A *m row_mx Bl Br = row_mx (A *m Bl) (A *m Br). | Proof.
apply/matrixP=> i k; rewrite !mxE.
by case defk: (split k) => /[!mxE]; under eq_bigr do rewrite mxE defk.
Qed. | Lemma | mul_mx_row | 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",
"eq_bigr",
"matrixP",
"mxE",
"row_mx",
"split"
] | Block products; we cover all 1 x 2, 2 x 1, and 2 x 2 block products. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mul_col_mx m1 m2 n p (Au : 'M_(m1, n)) (Ad : 'M_(m2, n)) (B : 'M_(n, p)) :
col_mx Au Ad *m B = col_mx (Au *m B) (Ad *m B). | Proof.
apply/matrixP=> i k; rewrite !mxE.
by case defi: (split i) => /[!mxE]; under eq_bigr do rewrite mxE defi.
Qed. | Lemma | mul_col_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Ad",
"apply",
"col_mx",
"eq_bigr",
"matrixP",
"mxE",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_row_col m n1 n2 p (Al : 'M_(m, n1)) (Ar : 'M_(m, n2))
(Bu : 'M_(n1, p)) (Bd : 'M_(n2, p)) :
row_mx Al Ar *m col_mx Bu Bd = Al *m Bu + Ar *m Bd. | Proof.
apply/matrixP=> i k; rewrite !mxE big_split_ord /=.
congr (_ + _); apply: eq_bigr => j _; first by rewrite row_mxEl col_mxEu.
by rewrite row_mxEr col_mxEd.
Qed. | Lemma | mul_row_col | 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",
"big_split_ord",
"col_mx",
"col_mxEd",
"col_mxEu",
"eq_bigr",
"matrixP",
"mxE",
"row_mx",
"row_mxEl",
"row_mxEr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_col_row m1 m2 n p1 p2 (Au : 'M_(m1, n)) (Ad : 'M_(m2, n))
(Bl : 'M_(n, p1)) (Br : 'M_(n, p2)) :
col_mx Au Ad *m row_mx Bl Br
= block_mx (Au *m Bl) (Au *m Br) (Ad *m Bl) (Ad *m Br). | Proof. by rewrite mul_col_mx !mul_mx_row. Qed. | Lemma | mul_col_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Ad",
"block_mx",
"col_mx",
"mul_col_mx",
"mul_mx_row",
"row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_row_block m n1 n2 p1 p2 (Al : 'M_(m, n1)) (Ar : 'M_(m, n2))
(Bul : 'M_(n1, p1)) (Bur : 'M_(n1, p2))
(Bdl : 'M_(n2, p1)) (Bdr : 'M_(n2, p2)) :
row_mx Al Ar *m block_mx Bul Bur Bdl Bdr
= row_mx (Al *m Bul + Ar *m Bdl) (Al *m Bur + Ar *m Bdr). | Proof. by rewrite block_mxEh mul_mx_row !mul_row_col. Qed. | Lemma | mul_row_block | 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",
... | [
"block_mx",
"block_mxEh",
"mul_mx_row",
"mul_row_col",
"row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_block_col m1 m2 n1 n2 p (Aul : 'M_(m1, n1)) (Aur : 'M_(m1, n2))
(Adl : 'M_(m2, n1)) (Adr : 'M_(m2, n2))
(Bu : 'M_(n1, p)) (Bd : 'M_(n2, p)) :
block_mx Aul Aur Adl Adr *m col_mx Bu Bd
= col_mx (Aul *m Bu + Aur *m Bd) (Adl *m Bu + Adr *m Bd). | Proof. by rewrite mul_col_mx !mul_row_col. Qed. | Lemma | mul_block_col | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"col_mx",
"mul_col_mx",
"mul_row_col"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_block m1 m2 n1 n2 p1 p2 (Aul : 'M_(m1, n1)) (Aur : 'M_(m1, n2))
(Adl : 'M_(m2, n1)) (Adr : 'M_(m2, n2))
(Bul : 'M_(n1, p1)) (Bur : 'M_(n1, p2))
(Bdl : 'M_(n2, p1)) (Bdr : 'M_(n2, p2)) :
block_mx Aul Aur A... | Proof. by rewrite mul_col_mx !mul_row_block. Qed. | Lemma | mulmx_block | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"mul_col_mx",
"mul_row_block"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_lsub m n p k (A : 'M_(m, n)) (B : 'M_(n, p + k)) :
A *m lsubmx B = lsubmx (A *m B). | Proof. by rewrite !lsubmxEsub mulmx_colsub. Qed. | Lemma | mulmx_lsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lsubmx",
"lsubmxEsub",
"mulmx_colsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_rsub m n p k (A : 'M_(m, n)) (B : 'M_(n, p + k)) :
A *m rsubmx B = rsubmx (A *m B). | Proof. by rewrite !rsubmxEsub mulmx_colsub. Qed. | Lemma | mulmx_rsub | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulmx_colsub",
"rsubmx",
"rsubmxEsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_usub_mx m k n p (A : 'M_(m + k, n)) (B : 'M_(n, p)) :
usubmx A *m B = usubmx (A *m B). | Proof. by rewrite !usubmxEsub mul_rowsub_mx. Qed. | Lemma | mul_usub_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mul_rowsub_mx",
"usubmx",
"usubmxEsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_dsub_mx m k n p (A : 'M_(m + k, n)) (B : 'M_(n, p)) :
dsubmx A *m B = dsubmx (A *m B). | Proof. by rewrite !dsubmxEsub mul_rowsub_mx. Qed. | Lemma | mul_dsub_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"dsubmx",
"dsubmxEsub",
"mul_rowsub_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace1 : \tr (1%:M : 'M[R]_n) = n%:R. | Proof. exact: mxtrace_scalar. Qed. | Lemma | mxtrace1 | 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",
... | [
"mxtrace_scalar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtraceZ a (A : 'M_n) : \tr (a *: A) = a * \tr A. | Proof. by rewrite mulr_sumr; apply: eq_bigr=> i _; rewrite mxE. Qed. | Lemma | mxtraceZ | 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",
"eq_bigr",
"mulr_sumr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
swizzle_mx_is_scalable m n p q f g k :
scalable (@swizzle_mx R m n p q f g k). | Proof. by move=> a A; apply/matrixP=> i j; rewrite !mxE. Qed. | Fact | swizzle_mx_is_scalable | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"scalable",
"swizzle_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SwizzleLin op | := (GRing.Linear.copy op (swizzle_mx _ _ _)). | Notation | SwizzleLin | 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",
... | [
"Linear",
"copy",
"swizzle_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec_is_scalable m n | := can2_scalable (@vec_mxK R m n) mxvecK. | Definition | mxvec_is_scalable | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"can2_scalable",
"mxvecK",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_sum_delta n (u : 'rV_n) : u = \sum_(j < n) u 0 j *: delta_mx 0 j. | Proof. by rewrite [u in LHS]matrix_sum_delta big_ord1. Qed. | Lemma | row_sum_delta | 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",
... | [
"big_ord1",
"delta_mx",
"matrix_sum_delta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_row_mx m n1 n2 a (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
a *: row_mx A1 A2 = row_mx (a *: A1) (a *: A2). | Proof. by split_mxE. Qed. | Lemma | scale_row_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",
... | [
"row_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_col_mx m1 m2 n a (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
a *: col_mx A1 A2 = col_mx (a *: A1) (a *: A2). | Proof. by split_mxE. Qed. | Lemma | scale_col_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_block_mx m1 m2 n1 n2 a (Aul : 'M_(m1, n1)) (Aur : 'M_(m1, n2))
(Adl : 'M_(m2, n1)) (Adr : 'M_(m2, n2)) :
a *: block_mx Aul Aur Adl Adr
= block_mx (a *: Aul) (a *: Aur) (a *: Adl) (a *: Adr). | Proof. by rewrite scale_col_mx !scale_row_mx. Qed. | Lemma | scale_block_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"finset",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"scale_col_mx",
"scale_row_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_is_scalable n : scalable (@diag_mx R n). | Proof. by move=> a A; apply/matrixP=> i j; rewrite !mxE mulrnAr. Qed. | Fact | diag_mx_is_scalable | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"diag_mx",
"matrixP",
"mulrnAr",
"mxE",
"scalable"
] | Diagonal matrices | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
diag_mx_sum_delta n (d : 'rV_n) :
diag_mx d = \sum_i d 0 i *: delta_mx i i. | Proof.
apply/matrixP=> i j; rewrite summxE (bigD1_ord i) //= !mxE eqxx /=.
by rewrite eq_sym mulr_natr big1 ?addr0 // => i'; rewrite !mxE eq_liftF mulr0.
Qed. | Lemma | diag_mx_sum_delta | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addr0",
"apply",
"big1",
"bigD1_ord",
"delta_mx",
"diag_mx",
"eq_liftF",
"eq_sym",
"eqxx",
"matrixP",
"mulr0",
"mulr_natr",
"mxE",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_diag_mx n (d : 'rV_n) i : row i (diag_mx d) = d 0 i *: delta_mx 0 i. | Proof. by apply/rowP => j; rewrite !mxE eqxx eq_sym mulr_natr. Qed. | Lemma | row_diag_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",
... | [
"apply",
"delta_mx",
"diag_mx",
"eq_sym",
"eqxx",
"mulr_natr",
"mxE",
"row",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_scalar_mx n a1 a2 : a1 *: a2%:M = (a1 * a2)%:M :> 'M_n. | Proof. by apply/matrixP=> i j; rewrite !mxE mulrnAr. Qed. | Lemma | scale_scalar_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",
... | [
"a1",
"a2",
"apply",
"matrixP",
"mulrnAr",
"mxE"
] | Scalar matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
scalemx1 n a : a *: 1%:M = a%:M :> 'M_n. | Proof. by rewrite scale_scalar_mx mulr1. Qed. | Lemma | scalemx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulr1",
"scale_scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_sum_delta n a : a%:M = \sum_i a *: delta_mx i i :> 'M_n. | Proof.
by rewrite -diag_const_mx diag_mx_sum_delta; under eq_bigr do rewrite mxE.
Qed. | Lemma | scalar_mx_sum_delta | 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",
... | [
"delta_mx",
"diag_const_mx",
"diag_mx_sum_delta",
"eq_bigr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx1_sum_delta n : 1%:M = \sum_i delta_mx i i :> 'M[R]_n. | Proof. by rewrite [1%:M]scalar_mx_sum_delta -scaler_sumr scale1r. Qed. | Lemma | mx1_sum_delta | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"delta_mx",
"scalar_mx_sum_delta",
"scale1r",
"scaler_sumr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_sum_row m n (u : 'rV_m) (A : 'M_(m, n)) :
u *m A = \sum_i u 0 i *: row i A. | Proof. by apply/rowP => j /[!(mxE, summxE)]; apply: eq_bigr => i _ /[!mxE]. Qed. | Lemma | mulmx_sum_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_bigr",
"mxE",
"row",
"rowP",
"summxE"
] | Right scaling associativity requires a commutative ring | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mul_scalar_mx m n a (A : 'M_(m, n)) : a%:M *m A = a *: A. | Proof.
by rewrite -diag_const_mx mul_diag_mx; apply/matrixP=> i j; rewrite !mxE.
Qed. | Lemma | mul_scalar_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"diag_const_mx",
"matrixP",
"mul_diag_mx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxE : mulmx = *%R. | Proof. by []. Qed. | Lemma | mulmxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
idmxE : 1%:M = 1 :> 'M_n. | Proof. by []. Qed. | Lemma | idmxE | 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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_monoid_morphism : monoid_morphism (@scalar_mx R n). | Proof. by split=> //; apply: scalar_mxM. Qed. | Fact | scalar_mx_is_monoid_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"monoid_morphism",
"scalar_mx",
"scalar_mxM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_multiplicative | := scalar_mx_is_monoid_morphism. | Definition | scalar_mx_is_multiplicative | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"scalar_mx_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin1_mx_key : unit. | Proof. by []. Qed. | Fact | lin1_mx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin1_mx (f : 'rV[R]_m -> 'rV[R]_n) | :=
\matrix[lin1_mx_key]_(i, j) f (delta_mx 0 i) 0 j. | Definition | lin1_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"delta_mx",
"lin1_mx_key",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_rV_lin1 u : u *m lin1_mx f = f u. | Proof.
rewrite [u in RHS]matrix_sum_delta big_ord1 linear_sum; apply/rowP=> i.
by rewrite mxE summxE; apply: eq_bigr => j _; rewrite linearZ !mxE.
Qed. | Lemma | mul_rV_lin1 | 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",
"big_ord1",
"eq_bigr",
"lin1_mx",
"linearZ",
"linear_sum",
"matrix_sum_delta",
"mxE",
"rowP",
"summxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mx (f : 'M[R]_(m1, n1) -> 'M[R]_(m2, n2)) | :=
lin1_mx (mxvec \o f \o vec_mx). | Definition | lin_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"fingroup",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lin1_mx",
"mxvec",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_rV_lin u : u *m lin_mx f = mxvec (f (vec_mx u)). | Proof. exact: mul_rV_lin1. Qed. | Lemma | mul_rV_lin | 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",
... | [
"lin_mx",
"mul_rV_lin1",
"mxvec",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_vec_lin A : mxvec A *m lin_mx f = mxvec (f A). | Proof. by rewrite mul_rV_lin mxvecK. Qed. | Lemma | mul_vec_lin | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lin_mx",
"mul_rV_lin",
"mxvec",
"mxvecK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rV_lin u : vec_mx (u *m lin_mx f) = f (vec_mx u). | Proof. by rewrite mul_rV_lin mxvecK. Qed. | Lemma | mx_rV_lin | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lin_mx",
"mul_rV_lin",
"mxvecK",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_vec_lin A : vec_mx (mxvec A *m lin_mx f) = f A. | Proof. by rewrite mul_rV_lin !mxvecK. Qed. | Lemma | mx_vec_lin | 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",
... | [
"lin_mx",
"mul_rV_lin",
"mxvec",
"mxvecK",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxr B A | := mulmx A B. | Definition | mulmxr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxr_is_semilinear B : semilinear (mulmxr B). | Proof. by split=> [a A|A1 A2]; rewrite /= (mulmxDl, scalemxAl). Qed. | Fact | mulmxr_is_semilinear | 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",
... | [
"mulmxDl",
"mulmxr",
"scalemxAl",
"semilinear",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mulmxr B | := lin_mx (mulmxr B). | Definition | lin_mulmxr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"lin_mx",
"mulmxr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_mulmxr_is_semilinear : semilinear lin_mulmxr. | Proof.
split=> [a A|A B]; apply/row_matrixP; case/mxvec_indexP=> i j;
rewrite (linearZ, linearD) /= !rowE !mul_rV_lin /= vec_mx_delta;
rewrite -(linearZ, linearD) 1?mulmxDr //=.
congr mxvec; apply/row_matrixP=> k.
rewrite linearZ /= !row_mul rowE mul_delta_mx_cond.
by case: (k == i); [rewrite -!rowE linearZ | rewri... | Fact | lin_mulmxr_is_semilinear | 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",
"lin_mulmxr",
"linearD",
"linearZ",
"mul0mx",
"mul_delta_mx_cond",
"mul_rV_lin",
"mulmxDr",
"mxvec",
"mxvec_indexP",
"raddf0",
"rowE",
"row_matrixP",
"row_mul",
"semilinear",
"split",
"vec_mx_delta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lift0_mx A : 'M_(1 + n) | := block_mx 1 0 0 A. | Definition | lift0_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
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"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lift0_mx_perm s : lift0_mx (perm_mx s) = perm_mx (lift0_perm s). | Proof.
apply/matrixP=> /= i j; rewrite !mxE split1 /=; case: unliftP => [i'|] -> /=.
rewrite lift0_perm_lift !mxE split1 /=.
by case: unliftP => [j'|] ->; rewrite ?(inj_eq (lift_inj _)) /= !mxE.
rewrite lift0_perm0 !mxE split1 /=.
by case: unliftP => [j'|] ->; rewrite /= mxE.
Qed. | Lemma | lift0_mx_perm | 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",
"inj_eq",
"lift0_mx",
"lift0_perm",
"lift0_perm0",
"lift0_perm_lift",
"lift_inj",
"matrixP",
"mxE",
"perm_mx",
"split1",
"unliftP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lift0_mx_is_perm s : is_perm_mx (lift0_mx (perm_mx s)). | Proof. by rewrite lift0_mx_perm perm_mx_is_perm. Qed. | Lemma | lift0_mx_is_perm | 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",
... | [
"is_perm_mx",
"lift0_mx",
"lift0_mx_perm",
"perm_mx",
"perm_mx_is_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp_block_diag_mx m n (A: 'M_m.+1) (B : 'M_n.+1) k :
(block_mx A 0 0 B) ^+ k = block_mx (A ^+ k) 0 0 (B ^+ k). | Proof.
elim: k=> [|k IHk]; first by rewrite !expr0 -scalar_mx_block.
rewrite !exprS IHk [LHS](mulmx_block A _ _ _ (A ^+ k)).
by rewrite !mulmx0 !mul0mx !add0r !addr0.
Qed. | Lemma | exp_block_diag_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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... | [
"add0r",
"addr0",
"block_mx",
"expr0",
"exprS",
"mul0mx",
"mulmx0",
"mulmx_block",
"scalar_mx_block"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"a %:M" | := (scalar_mx a) : ring_scope. | Notation | a %:M | algebra | algebra/matrix.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_mul_rev (R : pzSemiRingType) m n p
(A : 'M[R]_(m, n)) (B : 'M[R]_(n, p)) :
(A *m B)^T = (B : 'M[R^c]_(n, p))^T *m (A : 'M[R^c]_(m, n))^T. | Proof. by apply/matrixP=> k i /[!mxE]; apply: eq_bigr => j _ /[!mxE]. Qed. | Lemma | trmx_mul_rev | 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",
"matrixP",
"mxE"
] | Non-commutative transpose requires multiplication in the converse ring. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
n | := n'.+1. | Notation | n | 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",
... | [
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_nonzero1 : 1%:M != 0 :> 'M[R]_n. | Proof. by apply/eqP=> /matrixP/(_ 0 0)/eqP; rewrite !mxE oner_eq0. Qed. | Lemma | matrix_nonzero1 | 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",
"matrixP",
"mxE",
"oner_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxZ a A : (a *: A)^f = f a *: A^f. | Proof. by apply/matrixP=> i j; rewrite !mxE rmorphM. Qed. | Lemma | map_mxZ | 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",
"matrixP",
"mxE",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxM A B : (A *m B)^f = A^f *m B^f :> 'M_(m, p). | Proof.
apply/matrixP=> i k; rewrite !mxE rmorph_sum //.
by apply: eq_bigr => j; rewrite !mxE rmorphM.
Qed. | Lemma | map_mxM | 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",
"matrixP",
"mxE",
"rmorphM",
"rmorph_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_delta_mx i j : (delta_mx i j)^f = delta_mx i j :> 'M_(m, n). | Proof. by apply/matrixP=> i' j'; rewrite !mxE rmorph_nat. Qed. | Lemma | map_delta_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",
"delta_mx",
"matrixP",
"mxE",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_diag_mx d : (diag_mx d)^f = diag_mx d^f :> 'M_n. | Proof. by apply/matrixP=> i j; rewrite !mxE rmorphMn. Qed. | Lemma | map_diag_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",
"diag_mx",
"matrixP",
"mxE",
"rmorphMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_scalar_mx a : a%:M^f = (f a)%:M :> 'M_n. | Proof. by apply/matrixP=> i j; rewrite !mxE rmorphMn. Qed. | Lemma | map_scalar_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",
"matrixP",
"mxE",
"rmorphMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx1 : 1%:M^f = 1%:M :> 'M_n. | Proof. by rewrite map_scalar_mx rmorph1. Qed. | Lemma | map_mx1 | 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",
... | [
"map_scalar_mx",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_perm_mx (s : 'S_n) : (perm_mx s)^f = perm_mx s. | Proof. by apply/matrixP=> i j; rewrite !mxE rmorph_nat. Qed. | Lemma | map_perm_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",
"matrixP",
"mxE",
"perm_mx",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_tperm_mx (i1 i2 : 'I_n) : (tperm_mx i1 i2)^f = tperm_mx i1 i2. | Proof. exact: map_perm_mx. Qed. | Lemma | map_tperm_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",
... | [
"map_perm_mx",
"tperm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_pid_mx r : (pid_mx r)^f = pid_mx r :> 'M_(m, n). | Proof. by apply/matrixP=> i j; rewrite !mxE rmorph_nat. Qed. | Lemma | map_pid_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",
"matrixP",
"mxE",
"pid_mx",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trace_map_mx (A : 'M_n) : \tr A^f = f (\tr A). | Proof. by rewrite rmorph_sum; apply: eq_bigr => i _; rewrite mxE. Qed. | Lemma | trace_map_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",
"eq_bigr",
"mxE",
"rmorph_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_lin1_mx m n (g : 'rV_m -> 'rV_n) gf :
(forall v, (g v)^f = gf v^f) -> (lin1_mx g)^f = lin1_mx gf. | Proof.
by move=> def_gf; apply/matrixP => i j; rewrite !mxE -map_delta_mx -def_gf mxE.
Qed. | Lemma | map_lin1_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",
"lin1_mx",
"map_delta_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_lin_mx m1 n1 m2 n2 (g : 'M_(m1, n1) -> 'M_(m2, n2)) gf :
(forall A, (g A)^f = gf A^f) -> (lin_mx g)^f = lin_mx gf. | Proof.
move=> def_gf; apply: map_lin1_mx => A /=.
by rewrite map_mxvec def_gf map_vec_mx.
Qed. | Lemma | map_lin_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",
"lin_mx",
"map_lin1_mx",
"map_mxvec",
"map_vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_is_monoid_morphism n : monoid_morphism (map_mx f : 'M_n -> 'M_n). | Proof. by split; [apply: map_mx1 | apply: map_mxM]. Qed. | Fact | map_mx_is_monoid_morphism | 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",
"map_mx",
"map_mx1",
"map_mxM",
"monoid_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_is_multiplicative | := map_mx_is_monoid_morphism. | Definition | map_mx_is_multiplicative | 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",
... | [
"map_mx_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx f g : Prop | := f *m g = g *m f. | Definition | 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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mxb f g : bool | := f *m g == g *m f. | Definition | comm_mxb | 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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_sym f g : comm_mx f g -> comm_mx g f. | Proof. by rewrite /comm_mx. Qed. | Lemma | comm_mx_sym | 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",
... | [
"comm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_refl f : comm_mx f f. | Proof. by []. Qed. | Lemma | comm_mx_refl | 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",
... | [
"comm_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx0 f : comm_mx f 0. | Proof. by rewrite /comm_mx mulmx0 mul0mx. Qed. | Lemma | comm_mx0 | 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",
... | [
"comm_mx",
"mul0mx",
"mulmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm0mx f : comm_mx 0 f. | Proof. by rewrite /comm_mx mulmx0 mul0mx. Qed. | Lemma | comm0mx | 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",
... | [
"comm_mx",
"mul0mx",
"mulmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx1 f : comm_mx f 1%:M. | Proof. by rewrite /comm_mx mulmx1 mul1mx. Qed. | Lemma | comm_mx1 | 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",
... | [
"comm_mx",
"mul1mx",
"mulmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm1mx f : comm_mx 1%:M f. | Proof. by rewrite /comm_mx mulmx1 mul1mx. Qed. | Lemma | comm1mx | 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",
... | [
"comm_mx",
"mul1mx",
"mulmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mxD f g g' : comm_mx f g -> comm_mx f g' -> comm_mx f (g + g'). | Proof. by rewrite /comm_mx mulmxDl mulmxDr => -> ->. Qed. | Lemma | comm_mxD | 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",
... | [
"comm_mx",
"mulmxDl",
"mulmxDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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