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 |
|---|---|---|---|---|---|---|---|---|---|---|
submx0 m n (A : 'M_(m, n)) : (A <= (0 : 'M_n))%MS = (A == 0). | Proof. by apply/idP/eqP=> [|->]; [apply: submx0null | apply: sub0mx]. Qed. | Lemma | submx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"sub0mx",
"submx0null"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt0mx m n (A : 'M_(m, n)) : ((0 : 'M_n) < A)%MS = (A != 0). | Proof. by rewrite /ltmx sub0mx submx0. Qed. | Lemma | lt0mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx",
"sub0mx",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmx0 m n (A : 'M[F]_(m, n)) : (A < (0 : 'M_n))%MS = false. | Proof. by rewrite /ltmx sub0mx andbF. Qed. | Lemma | ltmx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx0P m n (A : 'M_(m, n)) : reflect (A = 0) (A == (0 : 'M_n))%MS. | Proof. by rewrite submx0 sub0mx andbT; apply: eqP. Qed. | Lemma | eqmx0P | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"sub0mx",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_eq0 m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :=: B)%MS -> (A == 0) = (B == 0). | Proof. by move=> eqAB; rewrite -!submx0 eqAB. Qed. | Lemma | eqmx_eq0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addmx_sub m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m1, n)) (C : 'M_(m2, n)) :
(A <= C)%MS -> (B <= C)%MS -> ((A + B)%R <= C)%MS. | Proof.
by case/submxP=> A' ->; case/submxP=> B' ->; rewrite -mulmxDl submxMl.
Qed. | Lemma | addmx_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"A'",
"mulmxDl",
"submxMl",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowsub_sub m1 m2 n (f : 'I_m2 -> 'I_m1) (A : 'M_(m1, n)) :
(rowsub f A <= A)%MS. | Proof. by rewrite rowsubE mulmx_sub. Qed. | Lemma | rowsub_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mulmx_sub",
"rowsub",
"rowsubE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
summx_sub m1 m2 n (B : 'M_(m2, n))
I (r : seq I) (P : pred I) (A_ : I -> 'M_(m1, n)) :
(forall i, P i -> A_ i <= B)%MS -> ((\sum_(i <- r | P i) A_ i)%R <= B)%MS. | Proof.
by move=> leAB; elim/big_ind: _ => // [|C D]; [apply/sub0mx | apply/addmx_sub].
Qed. | Lemma | summx_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addmx_sub",
"apply",
"big_ind",
"seq",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalemx_sub m1 m2 n a (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A <= B)%MS -> (a *: A <= B)%MS. | Proof. by case/submxP=> A' ->; rewrite scalemxAl submxMl. Qed. | Lemma | scalemx_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"A'",
"scalemxAl",
"submxMl",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_sub m n i (A : 'M_(m, n)) : (row i A <= A)%MS. | Proof. exact: rowsub_sub. Qed. | Lemma | row_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"row",
"rowsub_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_row_sub m n v (A : 'M_(m, n)) i : row i A = v -> (v <= A)%MS. | Proof. by move <-; rewrite row_sub. Qed. | Lemma | eq_row_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"row",
"row_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz_row_sub m n (A : 'M_(m, n)) : (nz_row A <= A)%MS. | Proof. by rewrite /nz_row; case: pickP => [i|] _; rewrite ?row_sub ?sub0mx. Qed. | Lemma | nz_row_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"nz_row",
"pickP",
"row_sub",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_subP m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (forall i, row i A <= B)%MS (A <= B)%MS. | Proof.
apply: (iffP idP) => [sAB i|sAB].
by apply: submx_trans sAB; apply: row_sub.
rewrite submxE; apply/eqP/row_matrixP=> i; apply/eqP.
by rewrite row_mul row0 -submxE.
Qed. | Lemma | row_subP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"row",
"row0",
"row_matrixP",
"row_mul",
"row_sub",
"submxE",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rV_subP m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (forall v : 'rV_n, v <= A -> v <= B)%MS (A <= B)%MS. | Proof.
apply: (iffP idP) => [sAB v Av | sAB]; first exact: submx_trans sAB.
by apply/row_subP=> i; rewrite sAB ?row_sub.
Qed. | Lemma | rV_subP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"row_sub",
"row_subP",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_subPn m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (exists i, ~~ (row i A <= B)%MS) (~~ (A <= B)%MS). | Proof. by rewrite (sameP row_subP forallP); apply: forallPn. Qed. | Lemma | row_subPn | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"forallP",
"forallPn",
"row",
"row_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_rVP n (u v : 'rV[F]_n) : reflect (exists a, u = a *: v) (u <= v)%MS. | Proof.
apply: (iffP submxP) => [[w ->] | [a ->]].
by exists (w 0 0); rewrite -mul_scalar_mx -mx11_scalar.
by exists a%:M; rewrite mul_scalar_mx.
Qed. | Lemma | sub_rVP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"mul_scalar_mx",
"mx11_scalar",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_rV n (v : 'rV[F]_n) : \rank v = (v != 0). | Proof.
case: eqP => [-> | nz_v]; first by rewrite mxrank0.
by apply/eqP; rewrite eqn_leq rank_leq_row lt0n mxrank_eq0; apply/eqP.
Qed. | Lemma | rank_rV | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqn_leq",
"lt0n",
"mxrank0",
"mxrank_eq0",
"rank",
"rank_leq_row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowV0Pn m n (A : 'M_(m, n)) :
reflect (exists2 v : 'rV_n, v <= A & v != 0)%MS (A != 0). | Proof.
rewrite -submx0; apply: (iffP idP) => [| [v svA]]; last first.
by rewrite -submx0; apply: contra (submx_trans _).
by case/row_subPn=> i; rewrite submx0; exists (row i A); rewrite ?row_sub.
Qed. | Lemma | rowV0Pn | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"last",
"row",
"row_sub",
"row_subPn",
"submx0",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowV0P m n (A : 'M_(m, n)) :
reflect (forall v : 'rV_n, v <= A -> v = 0)%MS (A == 0). | Proof.
rewrite -[A == 0]negbK; case: rowV0Pn => IH.
by right; case: IH => v svA nzv IH; case/eqP: nzv; apply: IH.
by left=> v svA; apply/eqP/idPn=> nzv; case: IH; exists v.
Qed. | Lemma | rowV0P | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"rowV0Pn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_full m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
row_full B -> (A <= B)%MS. | Proof.
by rewrite submxE /cokermx => /eqnP->; rewrite /copid_mx pid_mx_1 subrr !mulmx0.
Qed. | Lemma | submx_full | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cokermx",
"copid_mx",
"eqnP",
"mulmx0",
"pid_mx_1",
"row_full",
"submxE",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_fullP m n (A : 'M_(m, n)) :
reflect (exists B, B *m A = 1%:M) (row_full A). | Proof.
apply: (iffP idP) => [Afull | [B kA]].
by exists (1%:M *m pinvmx A); apply: mulmxKpV (submx_full _ Afull).
by rewrite [_ A]eqn_leq rank_leq_col (mulmx1_min_rank B 1%:M) ?mulmx1.
Qed. | Lemma | row_fullP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqn_leq",
"mulmx1",
"mulmx1_min_rank",
"mulmxKpV",
"pinvmx",
"rank_leq_col",
"row_full",
"submx_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_full_inj m n p A : row_full A -> injective (@mulmx F m n p A). | Proof.
case/row_fullP=> A' A'K; apply: can_inj (mulmx A') _ => B.
by rewrite mulmxA A'K mul1mx.
Qed. | Lemma | row_full_inj | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"A'",
"apply",
"mul1mx",
"mulmx",
"mulmxA",
"row_full",
"row_fullP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_freeP m n (A : 'M_(m, n)) :
reflect (exists B, A *m B = 1%:M) (row_free A). | Proof.
rewrite /row_free -mxrank_tr.
apply: (iffP row_fullP) => [] [B kA];
by exists B^T; rewrite -trmx1 -kA trmx_mul ?trmxK.
Qed. | Lemma | row_freeP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"mxrank_tr",
"row_free",
"row_fullP",
"trmx1",
"trmxK",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_free_inj m n p A : row_free A -> injective ((@mulmx F m n p)^~ A). | Proof.
case/row_freeP=> A' AK; apply: can_inj (mulmx^~ A') _ => B.
by rewrite -mulmxA AK mulmx1.
Qed. | Lemma | row_free_inj | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"A'",
"apply",
"mulmx",
"mulmx1",
"mulmxA",
"row_free",
"row_freeP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_free_injr m n p A : row_free A -> injective (mulmxr A) | :=
@row_free_inj m n p A. | Definition | row_free_injr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mulmxr",
"row_free",
"row_free_inj"
] | but which is canonically additive | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
row_free_unit n (A : 'M_n) : row_free A = (A \in unitmx). | Proof.
apply/row_fullP/idP=> [[A'] | uA]; first by case/mulmx1_unit.
by exists (invmx A); rewrite mulVmx.
Qed. | Lemma | row_free_unit | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"A'",
"apply",
"invmx",
"mulVmx",
"mulmx1_unit",
"row_free",
"row_fullP",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_full_unit n (A : 'M_n) : row_full A = (A \in unitmx). | Proof. exact: row_free_unit. Qed. | Lemma | row_full_unit | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"row_free_unit",
"row_full",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_unit n (A : 'M_n) : A \in unitmx -> \rank A = n. | Proof. by rewrite -row_full_unit => /eqnP. Qed. | Lemma | mxrank_unit | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqnP",
"rank",
"row_full_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank1 n : \rank (1%:M : 'M_n) = n. | Proof. exact: mxrank_unit. Qed. | Lemma | mxrank1 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxrank_unit",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_delta m n i j : \rank (delta_mx i j : 'M_(m, n)) = 1. | Proof.
apply/eqP; rewrite eqn_leq lt0n mxrank_eq0.
rewrite -{1}(mul_delta_mx (0 : 'I_1)) mulmx_max_rank.
by apply/eqP; move/matrixP; move/(_ i j); move/eqP; rewrite !mxE !eqxx oner_eq0.
Qed. | Lemma | mxrank_delta | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"delta_mx",
"eqn_leq",
"eqxx",
"lt0n",
"matrixP",
"mul_delta_mx",
"mulmx_max_rank",
"mxE",
"mxrank_eq0",
"oner_eq0",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrankS m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A <= B)%MS -> \rank A <= \rank B. | Proof. by case/submxP=> D ->; rewrite mxrankM_maxr. Qed. | Lemma | mxrankS | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxrankM_maxr",
"rank",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx1 m n (A : 'M_(m, n)) : (A <= 1%:M)%MS. | Proof. by rewrite submx_full // row_full_unit unitmx1. Qed. | Lemma | submx1 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"row_full_unit",
"submx_full",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub1mx m n (A : 'M_(m, n)) : (1%:M <= A)%MS = row_full A. | Proof.
apply/idP/idP; last exact: submx_full.
by move/mxrankS; rewrite mxrank1 col_leq_rank.
Qed. | Lemma | sub1mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"col_leq_rank",
"last",
"mxrank1",
"mxrankS",
"row_full",
"submx_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmx1 m n (A : 'M_(m, n)) : (A < 1%:M)%MS = ~~ row_full A. | Proof. by rewrite /ltmx sub1mx submx1. Qed. | Lemma | ltmx1 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx",
"row_full",
"sub1mx",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt1mx m n (A : 'M_(m, n)) : (1%:M < A)%MS = false. | Proof. by rewrite /ltmx submx1 andbF. Qed. | Lemma | lt1mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pinvmxE n (A : 'M[F]_n) : A \in unitmx -> pinvmx A = invmx A. | Proof.
move=> A_unit; apply: (@row_free_inj _ _ _ A); rewrite ?row_free_unit//.
by rewrite -[pinvmx _]mul1mx mulmxKpV ?sub1mx ?row_full_unit// mulVmx.
Qed. | Lemma | pinvmxE | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"invmx",
"mul1mx",
"mulVmx",
"mulmxKpV",
"pinvmx",
"row_free_inj",
"row_free_unit",
"row_full_unit",
"sub1mx",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVpmx m n (A : 'M[F]_(m, n)) : row_full A -> pinvmx A *m A = 1%:M. | Proof. by move=> fA; rewrite -[pinvmx _]mul1mx mulmxKpV// sub1mx. Qed. | Lemma | mulVpmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"fA",
"mul1mx",
"mulmxKpV",
"pinvmx",
"row_full",
"sub1mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pinvmx_free m n (A : 'M[F]_(m, n)) : row_full A -> row_free (pinvmx A). | Proof. by move=> /mulVpmx pAA1; apply/row_freeP; exists A. Qed. | Lemma | pinvmx_free | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"mulVpmx",
"pinvmx",
"row_free",
"row_freeP",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pinvmx_full m n (A : 'M[F]_(m, n)) : row_free A -> row_full (pinvmx A). | Proof. by move=> /mulmxVp ApA1; apply/row_fullP; exists A. Qed. | Lemma | pinvmx_full | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"mulmxVp",
"pinvmx",
"row_free",
"row_full",
"row_fullP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmxP m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (A :=: B)%MS (A == B)%MS. | Proof.
apply: (iffP andP) => [[sAB sBA] | eqAB]; last by rewrite !eqAB.
split=> [|m3 C]; first by apply/eqP; rewrite eqn_leq !mxrankS.
split; first by apply/idP/idP; apply: submx_trans.
by apply/idP/idP=> sC; apply: submx_trans sC _.
Qed. | Lemma | eqmxP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqn_leq",
"last",
"mxrankS",
"split",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rV_eqP m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (forall u : 'rV_n, (u <= A) = (u <= B))%MS (A == B)%MS. | Proof.
apply: (iffP idP) => [eqAB u | eqAB]; first by rewrite (eqmxP eqAB).
by apply/andP; split; apply/rV_subP=> u; rewrite eqAB.
Qed. | Lemma | rV_eqP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxP",
"rV_subP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxP (m n : nat) (A B : 'M[F]_(m, n)) :
reflect (forall u : 'rV_m, u *m A = u *m B) (A == B). | Proof.
apply: (iffP eqP) => [-> //|eqAB].
apply: (@row_full_inj _ _ _ 1%:M); first by rewrite row_full_unit unitmx1.
by apply/row_matrixP => i; rewrite !row_mul eqAB.
Qed. | Lemma | mulmxP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"nat",
"row_full_inj",
"row_full_unit",
"row_matrixP",
"row_mul",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_refl m1 n (A : 'M_(m1, n)) : (A :=: A)%MS. | Proof. by []. Qed. | Lemma | eqmx_refl | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_sym m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :=: B)%MS -> (B :=: A)%MS. | Proof. by move=> eqAB; split=> [|m3 C]; rewrite !eqAB. Qed. | Lemma | eqmx_sym | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_trans m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A :=: B)%MS -> (B :=: C)%MS -> (A :=: C)%MS. | Proof. by move=> eqAB eqBC; split=> [|m4 D]; rewrite !eqAB !eqBC. Qed. | Lemma | eqmx_trans | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_rank m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A == B)%MS -> \rank A = \rank B. | Proof. by move/eqmxP->. Qed. | Lemma | eqmx_rank | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqmxP",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_eqmx m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :=: B)%MS ->
forall C : 'M_(m3, n), (((A < C) = (B < C))%MS * ((C < A) = (C < B))%MS)%type. | Proof. by move=> eqAB C; rewrite /ltmx !eqAB. Qed. | Lemma | lt_eqmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmxMr m1 m2 n p (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(n, p)) :
(A :=: B)%MS -> (A *m C :=: B *m C)%MS. | Proof. by move=> eqAB; apply/eqmxP; rewrite !submxMr ?eqAB. Qed. | Lemma | eqmxMr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxP",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmxMfull m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
row_full A -> (A *m B :=: B)%MS. | Proof.
case/row_fullP=> A' A'A; apply/eqmxP; rewrite submxMl /=.
by apply/submxP; exists A'; rewrite mulmxA A'A mul1mx.
Qed. | Lemma | eqmxMfull | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"A'",
"apply",
"eqmxP",
"mul1mx",
"mulmxA",
"row_full",
"row_fullP",
"submxMl",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx0 m n : ((0 : 'M[F]_(m, n)) :=: (0 : 'M_n))%MS. | Proof. by apply/eqmxP; rewrite !sub0mx. Qed. | Lemma | eqmx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxP",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_scale m n a (A : 'M_(m, n)) : a != 0 -> (a *: A :=: A)%MS. | Proof.
move=> nz_a; apply/eqmxP; rewrite scalemx_sub //.
by rewrite -{1}[A]scale1r -(mulVf nz_a) -scalerA scalemx_sub.
Qed. | Lemma | eqmx_scale | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxP",
"mulVf",
"scale1r",
"scalemx_sub",
"scalerA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_opp m n (A : 'M_(m, n)) : (- A :=: A)%MS. | Proof.
by rewrite -scaleN1r; apply: eqmx_scale => //; rewrite oppr_eq0 oner_eq0.
Qed. | Lemma | eqmx_opp | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmx_scale",
"oner_eq0",
"oppr_eq0",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxMfree m1 m2 n p (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(n, p)) :
row_free C -> (A *m C <= B *m C)%MS = (A <= B)%MS. | Proof.
case/row_freeP=> C' C_C'_1; apply/idP/idP=> sAB; last exact: submxMr.
by rewrite -[A]mulmx1 -[B]mulmx1 -C_C'_1 !mulmxA submxMr.
Qed. | Lemma | submxMfree | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"last",
"mulmx1",
"mulmxA",
"row_free",
"row_freeP",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmxMfree m1 m2 n p (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(n, p)) :
row_free C -> (A *m C :=: B *m C)%MS -> (A :=: B)%MS. | Proof.
by move=> Cfree eqAB; apply/eqmxP; move/eqmxP: eqAB; rewrite !submxMfree.
Qed. | Lemma | eqmxMfree | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxP",
"row_free",
"submxMfree"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrankMfree m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
row_free B -> \rank (A *m B) = \rank A. | Proof.
by move=> Bfree; rewrite -mxrank_tr trmx_mul eqmxMfull /row_full mxrank_tr.
Qed. | Lemma | mxrankMfree | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqmxMfull",
"mxrank_tr",
"rank",
"row_free",
"row_full",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_row_base m n (A : 'M_(m, n)) : (row_base A :=: A)%MS. | Proof.
apply/eqmxP/andP; split; apply/submxP.
exists (pid_mx (\rank A) *m invmx (col_ebase A)).
by rewrite -{8}[A]mulmx_ebase !mulmxA mulmxKV // pid_mx_id.
exists (col_ebase A *m pid_mx (\rank A)).
by rewrite mulmxA -(mulmxA _ _ (pid_mx _)) pid_mx_id // mulmx_ebase.
Qed. | Lemma | eq_row_base | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"col_ebase",
"eqmxP",
"invmx",
"mulmxA",
"mulmxKV",
"mulmx_ebase",
"pid_mx",
"pid_mx_id",
"rank",
"row_base",
"split",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_base0 (m n : nat) : row_base (0 : 'M[F]_(m, n)) = 0. | Proof. by apply/eqmx0P; rewrite !eq_row_base !sub0mx. Qed. | Lemma | row_base0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eq_row_base",
"eqmx0P",
"nat",
"row_base",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qidmx_eq1 n (A : 'M_n) : qidmx A = (A == 1%:M). | Proof. by rewrite /qidmx eqxx pid_mx_1. Qed. | Let | qidmx_eq1 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqxx",
"pid_mx_1",
"qidmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_witnessP m n (A : 'M_(m, n)) :
equivmx A (row_full A) (genmx_witness A). | Proof.
rewrite /equivmx qidmx_eq1 /genmx_witness.
case fullA: (row_full A); first by rewrite eqxx sub1mx submx1 fullA.
set B := _ *m _; have defB : (B == A)%MS.
apply/andP; split; apply/submxP.
exists (pid_mx (\rank A) *m invmx (col_ebase A)).
by rewrite -{3}[A]mulmx_ebase !mulmxA mulmxKV // pid_mx_id.
exis... | Let | genmx_witnessP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"col_ebase",
"equivmx",
"eqxx",
"genmx_witness",
"invmx",
"mulmxA",
"mulmxKV",
"mulmx_ebase",
"negb_add",
"pid_mx",
"pid_mx_id",
"qidmx_eq1",
"rank",
"row_full",
"split",
"sub1mx",
"submx1",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmxE m n (A : 'M_(m, n)) : (<<A>> :=: A)%MS. | Proof.
by rewrite unlock; apply/eqmxP; case/andP: (chooseP (genmx_witnessP A)).
Qed. | Lemma | genmxE | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"chooseP",
"eqmxP",
"genmx_witnessP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_genmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :=: B -> <<A>> = <<B>>)%MS. | Proof.
move=> eqAB; rewrite unlock.
have{} eqAB: equivmx A (row_full A) =1 equivmx B (row_full B).
by move=> C; rewrite /row_full /equivmx !eqAB.
rewrite (eq_choose eqAB) (choose_id _ (genmx_witnessP B)) //.
by rewrite -eqAB genmx_witnessP.
Qed. | Lemma | eq_genmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"choose_id",
"eq_choose",
"equivmx",
"genmx_witnessP",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmxP m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (<<A>> = <<B>>)%MS (A == B)%MS. | Proof.
apply: (iffP idP) => eqAB; first exact: eq_genmx (eqmxP _).
by rewrite -!(genmxE A) eqAB !genmxE andbb.
Qed. | Lemma | genmxP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eq_genmx",
"eqmxP",
"genmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx0 m n : <<0 : 'M_(m, n)>>%MS = 0. | Proof. by apply/eqP; rewrite -submx0 genmxE sub0mx. Qed. | Lemma | genmx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"genmxE",
"sub0mx",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx1 n : <<1%:M : 'M_n>>%MS = 1%:M. | Proof.
rewrite unlock; case/andP: (chooseP (@genmx_witnessP n n 1%:M)) => _ /eqP.
by rewrite qidmx_eq1 row_full_unit unitmx1 => /eqP.
Qed. | Lemma | genmx1 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"chooseP",
"genmx_witnessP",
"qidmx_eq1",
"row_full_unit",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_id m n (A : 'M_(m, n)) : (<<<<A>>>> = <<A>>)%MS. | Proof. exact/eq_genmx/genmxE. Qed. | Lemma | genmx_id | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eq_genmx",
"genmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_base_free m n (A : 'M_(m, n)) : row_free (row_base A). | Proof. by apply/eqnP; rewrite eq_row_base. Qed. | Lemma | row_base_free | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eq_row_base",
"eqnP",
"row_base",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_gen m n (A : 'M_(m, n)) : \rank <<A>>%MS = \rank A. | Proof. by rewrite genmxE. Qed. | Lemma | mxrank_gen | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"genmxE",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_base_full m n (A : 'M_(m, n)) : row_full (col_base A). | Proof.
apply/row_fullP; exists (pid_mx (\rank A) *m invmx (col_ebase A)).
by rewrite !mulmxA mulmxKV // pid_mx_id // pid_mx_1.
Qed. | Lemma | col_base_full | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"col_base",
"col_ebase",
"invmx",
"mulmxA",
"mulmxKV",
"pid_mx",
"pid_mx_1",
"pid_mx_id",
"rank",
"row_full",
"row_fullP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_leqif_sup m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A <= B)%MS -> \rank A <= \rank B ?= iff (B <= A)%MS. | Proof.
move=> sAB; split; first by rewrite mxrankS.
apply/idP/idP=> [| sBA]; last by rewrite eqn_leq !mxrankS.
case/submxP: sAB => D ->; set r := \rank B; rewrite -(mulmx_base B) mulmxA.
rewrite mxrankMfree // => /row_fullP[E kE].
by rewrite -[rB in _ *m rB]mul1mx -kE -(mulmxA E) (mulmxA _ E) submxMl.
Qed. | Lemma | mxrank_leqif_sup | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqn_leq",
"last",
"mul1mx",
"mulmxA",
"mulmx_base",
"mxrankMfree",
"mxrankS",
"rank",
"row_fullP",
"split",
"submxMl",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_leqif_eq m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A <= B)%MS -> \rank A <= \rank B ?= iff (A == B)%MS. | Proof. by move=> sAB; rewrite sAB; apply: mxrank_leqif_sup. Qed. | Lemma | mxrank_leqif_eq | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"mxrank_leqif_sup",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmxErank m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A < B)%MS = (A <= B)%MS && (\rank A < \rank B). | Proof.
by apply: andb_id2l => sAB; rewrite (ltn_leqif (mxrank_leqif_sup sAB)).
Qed. | Lemma | ltmxErank | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"ltn_leqif",
"mxrank_leqif_sup",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_ltmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A < B)%MS -> \rank A < \rank B. | Proof. by rewrite ltmxErank => /andP[]. Qed. | Lemma | rank_ltmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmxErank",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_cast m1 m2 n (A : 'M_(m1, n)) e :
((castmx e A : 'M_(m2, n)) :=: A)%MS. | Proof. by case: e A; case: m2 / => A e; rewrite castmx_id. Qed. | Lemma | eqmx_cast | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"castmx",
"castmx_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_full_castmx m1 m2 n (A : 'M_(m1, n)) e :
row_full (castmx e A : 'M_(m2, n)) = row_full A. | Proof. exact/eq_row_full/eqmx_cast. Qed. | Lemma | row_full_castmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"castmx",
"eq_row_full",
"eqmx_cast",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_free_castmx m1 m2 n (A : 'M_(m1, n)) e :
row_free (castmx e A : 'M_(m2, n)) = row_free A. | Proof. by rewrite /row_free eqmx_cast; congr (_ == _); rewrite e.1. Qed. | Lemma | row_free_castmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"castmx",
"eqmx_cast",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_conform m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(conform_mx A B :=: A \/ conform_mx A B :=: B)%MS. | Proof.
case: (eqVneq m2 m1) => [-> | neqm12] in B *.
by right; rewrite conform_mx_id.
by left; rewrite nonconform_mx ?neqm12.
Qed. | Lemma | eqmx_conform | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"conform_mx",
"conform_mx_id",
"eqVneq",
"nonconform_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_sum_nop m n (A : 'M_(m, n)) : (addsmx_nop A :=: A)%MS. | Proof.
case: (eqmx_conform <<A>>%MS A) => // eq_id_gen.
exact: eqmx_trans (genmxE A).
Qed. | Let | eqmx_sum_nop | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_nop",
"eqmx_conform",
"eqmx_trans",
"genmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowsub_comp_sub (m n p q : nat) f (g : 'I_n -> 'I_p) (A : 'M_(m, q)) :
(rowsub (f \o g) A <= rowsub f A)%MS. | Proof. by rewrite rowsub_comp rowsubE mulmx_sub. Qed. | Lemma | rowsub_comp_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mulmx_sub",
"nat",
"rowsub",
"rowsubE",
"rowsub_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_rowsub (m n p q : nat) (h : 'I_n -> 'I_p) f g (A : 'M_(m, q)) :
f =1 g \o h -> (rowsub f A <= rowsub g A)%MS. | Proof. by move=> /eq_rowsub->; rewrite rowsub_comp_sub. Qed. | Lemma | submx_rowsub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eq_rowsub",
"nat",
"rowsub",
"rowsub_comp_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_rowsub_comp_perm (m1 m2 n : nat) (s : 'S_m2) f (A : 'M_(m1, n)) :
(rowsub (f \o s) A :=: rowsub f A)%MS. | Proof.
rewrite rowsub_comp rowsubE; apply: eqmxMfull.
by rewrite -perm_mxEsub row_full_unit unitmx_perm.
Qed. | Lemma | eqmx_rowsub_comp_perm | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eqmxMfull",
"nat",
"perm_mxEsub",
"row_full_unit",
"rowsub",
"rowsubE",
"rowsub_comp",
"unitmx_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_rowsub_comp (m n p q : nat) f (g : 'I_n -> 'I_p) (A : 'M_(m, q)) :
p <= n -> injective g -> (rowsub (f \o g) A :=: rowsub f A)%MS. | Proof.
move=> leq_pn g_inj; have eq_np : n == p by rewrite eqn_leq leq_pn (inj_leq g).
rewrite (eqP eq_np) in g g_inj *.
rewrite (eq_rowsub (f \o (perm g_inj))); first by move=> i; rewrite /= permE.
exact: eqmx_rowsub_comp_perm.
Qed. | Lemma | eqmx_rowsub_comp | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eq_rowsub",
"eqmx_rowsub_comp_perm",
"eqn_leq",
"inj_leq",
"nat",
"permE",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_rowsub (m n p q : nat) (h : 'I_n -> 'I_p) f g (A : 'M_(m, q)) :
injective h -> p <= n -> f =1 g \o h -> (rowsub f A :=: rowsub g A)%MS. | Proof. by move=> leq_pn h_inj /eq_rowsub->; apply: eqmx_rowsub_comp. Qed. | Lemma | eqmx_rowsub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eq_rowsub",
"eqmx_rowsub_comp",
"nat",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mx_sub m3 (C : 'M_(m3, n)) :
(col_mx A B <= C)%MS = (A <= C)%MS && (B <= C)%MS. | Proof.
rewrite !submxE mul_col_mx -col_mx0.
by apply/eqP/andP; [case/eq_col_mx=> -> -> | case; do 2!move/eqP->].
Qed. | Lemma | col_mx_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"col_mx",
"col_mx0",
"eq_col_mx",
"mul_col_mx",
"submxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmxE : (A + B :=: col_mx A B)%MS. | Proof.
have:= submx_refl (col_mx A B); rewrite col_mx_sub; case/andP=> sAS sBS.
rewrite unlock; do 2?case: eqP => [AB0 | _]; last exact: genmxE.
by apply/eqmxP; rewrite !eqmx_sum_nop sBS col_mx_sub AB0 sub0mx /=.
by apply/eqmxP; rewrite !eqmx_sum_nop sAS col_mx_sub AB0 sub0mx andbT /=.
Qed. | Lemma | addsmxE | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"col_mx",
"col_mx_sub",
"eqmxP",
"eqmx_sum_nop",
"genmxE",
"last",
"sub0mx",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_sub m3 (C : 'M_(m3, n)) :
(A + B <= C)%MS = (A <= C)%MS && (B <= C)%MS. | Proof. by rewrite addsmxE col_mx_sub. Qed. | Lemma | addsmx_sub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxE",
"col_mx_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmxSl : (A <= A + B)%MS. | Proof. by have:= submx_refl (A + B)%MS; rewrite addsmx_sub; case/andP. Qed. | Lemma | addsmxSl | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_sub",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmxSr : (B <= A + B)%MS. | Proof. by have:= submx_refl (A + B)%MS; rewrite addsmx_sub; case/andP. Qed. | Lemma | addsmxSr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_sub",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_idPr : reflect (A + B :=: B)%MS (A <= B)%MS. | Proof.
have:= @eqmxP _ _ _ (A + B)%MS B.
by rewrite addsmxSr addsmx_sub submx_refl !andbT.
Qed. | Lemma | addsmx_idPr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxSr",
"addsmx_sub",
"eqmxP",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_idPl : reflect (A + B :=: A)%MS (B <= A)%MS. | Proof.
have:= @eqmxP _ _ _ (A + B)%MS A.
by rewrite addsmxSl addsmx_sub submx_refl !andbT.
Qed. | Lemma | addsmx_idPl | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxSl",
"addsmx_sub",
"eqmxP",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adds0mx m1 m2 n (B : 'M_(m2, n)) : ((0 : 'M_(m1, n)) + B :=: B)%MS. | Proof. by apply/eqmxP; rewrite addsmx_sub sub0mx addsmxSr /= andbT. Qed. | Lemma | adds0mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxSr",
"addsmx_sub",
"apply",
"eqmxP",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx0 m1 m2 n (A : 'M_(m1, n)) : (A + (0 : 'M_(m2, n)) :=: A)%MS. | Proof. by apply/eqmxP; rewrite addsmx_sub sub0mx addsmxSl /= !andbT. Qed. | Lemma | addsmx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxSl",
"addsmx_sub",
"apply",
"eqmxP",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_nop_eq0 m n (A : 'M_(m, n)) : (addsmx_nop A == 0) = (A == 0). | Proof. by rewrite -!submx0 eqmx_sum_nop. Qed. | Let | addsmx_nop_eq0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_nop",
"eqmx_sum_nop",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_nop0 m n : addsmx_nop (0 : 'M_(m, n)) = 0. | Proof. by apply/eqP; rewrite addsmx_nop_eq0. Qed. | Let | addsmx_nop0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_nop",
"addsmx_nop_eq0",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_nop_id n (A : 'M_n) : addsmx_nop A = A. | Proof. exact: conform_mx_id. Qed. | Let | addsmx_nop_id | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_nop",
"conform_mx_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmxC m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) : (A + B = B + A)%MS. | Proof.
have: (A + B == B + A)%MS.
by apply/andP; rewrite !addsmx_sub andbC -addsmx_sub andbC -addsmx_sub.
move/genmxP; rewrite [@addsmx]unlock -!submx0 !submx0.
by do 2!case: eqP => [// -> | _]; rewrite ?genmx_id ?addsmx_nop0.
Qed. | Lemma | addsmxC | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_nop0",
"addsmx_sub",
"apply",
"genmxP",
"genmx_id",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adds0mx_id m1 n (B : 'M_n) : ((0 : 'M_(m1, n)) + B)%MS = B. | Proof. by rewrite unlock eqxx addsmx_nop_id. Qed. | Lemma | adds0mx_id | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_nop_id",
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx0_id m2 n (A : 'M_n) : (A + (0 : 'M_(m2, n)))%MS = A. | Proof. by rewrite addsmxC adds0mx_id. Qed. | Lemma | addsmx0_id | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"adds0mx_id",
"addsmxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmxA m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A + (B + C) = A + B + C)%MS. | Proof.
have: (A + (B + C) :=: A + B + C)%MS.
by apply/eqmxP/andP; rewrite !addsmx_sub -andbA andbA -!addsmx_sub.
rewrite {1 3}[in @addsmx _ m1]unlock [in @addsmx _ n]unlock !addsmx_nop_id -!submx0.
rewrite !addsmx_sub ![@addsmx]unlock -!submx0; move/eq_genmx.
by do 3!case: (_ <= 0)%MS; rewrite //= !genmx_id.
Qed. | Lemma | addsmxA | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmx_nop_id",
"addsmx_sub",
"apply",
"eq_genmx",
"eqmxP",
"genmx_id",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmxMr m1 m2 n p (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(n, p)) :
((A + B)%MS *m C :=: A *m C + B *m C)%MS. | Proof. by apply/eqmxP; rewrite !addsmxE -!mul_col_mx !submxMr ?addsmxE. Qed. | Lemma | addsmxMr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxE",
"apply",
"eqmxP",
"mul_col_mx",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmxS m1 m2 m3 m4 n (A : 'M_(m1, n)) (B : 'M_(m2, n))
(C : 'M_(m3, n)) (D : 'M_(m4, n)) :
(A <= C -> B <= D -> A + B <= C + D)%MS. | Proof.
move=> sAC sBD.
by rewrite addsmx_sub {1}addsmxC !(submx_trans _ (addsmxSr _ _)).
Qed. | Lemma | addsmxS | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxC",
"addsmxSr",
"addsmx_sub",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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