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 |
|---|---|---|---|---|---|---|---|---|---|---|
addmx_sub_adds m m1 m2 n (A : 'M_(m, n)) (B : 'M_(m, n))
(C : 'M_(m1, n)) (D : 'M_(m2, n)) :
(A <= C -> B <= D -> (A + B)%R <= C + D)%MS. | Proof.
move=> sAC; move/(addsmxS sAC); apply: submx_trans.
by rewrite addmx_sub ?addsmxSl ?addsmxSr.
Qed. | Lemma | addmx_sub_adds | 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",
"addsmxS",
"addsmxSl",
"addsmxSr",
"apply",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_addKl n m1 m2 (A : 'M_(m1, n)) (B C : 'M_(m2, n)) :
(B <= A)%MS -> (A + (B + C)%R :=: A + C)%MS. | Proof.
move=> sBA; apply/eqmxP; rewrite !addsmx_sub !addsmxSl.
by rewrite -{3}[C](addKr B) !addmx_sub_adds ?eqmx_opp.
Qed. | Lemma | addsmx_addKl | 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"... | [
"addKr",
"addmx_sub_adds",
"addsmxSl",
"addsmx_sub",
"apply",
"eqmxP",
"eqmx_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_addKr n m1 m2 (A B : 'M_(m1, n)) (C : 'M_(m2, n)) :
(B <= C)%MS -> ((A + B)%R + C :=: A + C)%MS. | Proof. by rewrite -!(addsmxC C) addrC; apply: addsmx_addKl. Qed. | Lemma | addsmx_addKr | 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"... | [
"addrC",
"addsmxC",
"addsmx_addKl",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adds_eqmx 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. by move=> eqAC eqBD; apply/eqmxP; rewrite !addsmxS ?eqAC ?eqBD. Qed. | Lemma | adds_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"... | [
"addsmxS",
"apply",
"eqmxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_adds m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(<<(A + B)%MS>> = <<A>> + <<B>>)%MS. | Proof.
rewrite -(eq_genmx (adds_eqmx (genmxE A) (genmxE B))).
by rewrite [@addsmx]unlock !addsmx_nop_id !(fun_if (@genmx _ _ _)) !genmx_id.
Qed. | Lemma | genmx_adds | 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"... | [
"adds_eqmx",
"addsmx_nop_id",
"eq_genmx",
"genmxE",
"genmx_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_addsmxP m1 m2 m3 n
(A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
reflect (exists u, A = u.1 *m B + u.2 *m C) (A <= B + C)%MS. | Proof.
apply: (iffP idP) => [|[u ->]]; last by rewrite addmx_sub_adds ?submxMl.
rewrite addsmxE; case/submxP=> u ->; exists (lsubmx u, rsubmx u).
by rewrite -mul_row_col hsubmxK.
Qed. | Lemma | sub_addsmxP | 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_adds",
"addsmxE",
"apply",
"hsubmxK",
"last",
"lsubmx",
"mul_row_col",
"rsubmx",
"submxMl",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_sums P n (B_ : I -> 'M_n) :
<<(\sum_(i | P i) B_ i)%MS>>%MS = (\sum_(i | P i) <<B_ i>>)%MS. | Proof. exact: (big_morph _ (@genmx_adds n n n) (@genmx0 n n)). Qed. | Lemma | genmx_sums | 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"... | [
"big_morph",
"genmx0",
"genmx_adds"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumsmx_sup i0 P m n (A : 'M_(m, n)) (B_ : I -> 'M_n) :
P i0 -> (A <= B_ i0)%MS -> (A <= \sum_(i | P i) B_ i)%MS. | Proof.
by move=> Pi0 sAB; apply: submx_trans sAB _; rewrite (bigD1 i0) // addsmxSl.
Qed. | Lemma | sumsmx_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"... | [
"Pi0",
"addsmxSl",
"apply",
"bigD1",
"i0",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumsmx_subP P m n (A_ : I -> 'M_n) (B : 'M_(m, n)) :
reflect (forall i, P i -> A_ i <= B)%MS (\sum_(i | P i) A_ i <= B)%MS. | Proof.
apply: (iffP idP) => [sAB i Pi | sAB].
by apply: submx_trans sAB; apply: sumsmx_sup Pi _.
by elim/big_rec: _ => [|i Ai Pi sAiB]; rewrite ?sub0mx // addsmx_sub sAB.
Qed. | Lemma | sumsmx_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"... | [
"addsmx_sub",
"apply",
"big_rec",
"sub0mx",
"submx_trans",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
summx_sub_sums P m n (A : I -> 'M[F]_(m, n)) B :
(forall i, P i -> A i <= B i)%MS ->
((\sum_(i | P i) A i)%R <= \sum_(i | P i) B i)%MS. | Proof.
by move=> sAB; apply: summx_sub => i Pi; rewrite (sumsmx_sup i) ?sAB.
Qed. | Lemma | summx_sub_sums | 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",
"summx_sub",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumsmxS P n (A B : I -> 'M[F]_n) :
(forall i, P i -> A i <= B i)%MS ->
(\sum_(i | P i) A i <= \sum_(i | P i) B i)%MS. | Proof.
by move=> sAB; apply/sumsmx_subP=> i Pi; rewrite (sumsmx_sup i) ?sAB.
Qed. | Lemma | sumsmxS | 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",
"sumsmx_subP",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_sums P n (A B : I -> 'M[F]_n) :
(forall i, P i -> A i :=: B i)%MS ->
(\sum_(i | P i) A i :=: \sum_(i | P i) B i)%MS. | Proof. by move=> eqAB; apply/eqmxP; rewrite !sumsmxS // => i; move/eqAB->. Qed. | Lemma | eqmx_sums | 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",
"sumsmxS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_sums_genmxP P m n p (A : 'M_(m, p)) (B_ : I -> 'M_(n, p)) :
reflect (exists u_ : I -> 'M_(m, n), A = \sum_(i | P i) u_ i *m B_ i)
(A <= \sum_(i | P i) <<B_ i>>)%MS. | Proof.
apply: (iffP idP) => [| [u_ ->]]; last first.
by apply: summx_sub_sums => i _; rewrite genmxE; apply: submxMl.
have [b] := ubnP #|P|; elim: b => // b IHb in P A *.
case: (pickP P) => [i Pi | P0 _]; last first.
rewrite big_pred0 //; move/submx0null->.
by exists (fun _ => 0); rewrite big_pred0.
rewrite (card... | Lemma | sub_sums_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"... | [
"P0",
"adds_eqmx",
"apply",
"bigD1",
"big_pred0",
"cardD1x",
"eq_bigr",
"eqmx_refl",
"eqxx",
"genmxE",
"last",
"pickP",
"sub_addsmxP",
"submx0null",
"submxMl",
"summx_sub_sums",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_sumsmxP P m n (A : 'M_(m, n)) (B_ : I -> 'M_n) :
reflect (exists u_, A = \sum_(i | P i) u_ i *m B_ i)
(A <= \sum_(i | P i) B_ i)%MS. | Proof.
by rewrite -(eqmx_sums (fun _ _ => genmxE _)); apply/sub_sums_genmxP.
Qed. | Lemma | sub_sumsmxP | 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_sums",
"genmxE",
"sub_sums_genmxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumsmxMr_gen P m n A (B : 'M[F]_(m, n)) :
((\sum_(i | P i) A i)%MS *m B :=: \sum_(i | P i) <<A i *m B>>)%MS. | Proof.
apply/eqmxP/andP; split; last first.
by apply/sumsmx_subP=> i Pi; rewrite genmxE submxMr ?(sumsmx_sup i).
have [u ->] := sub_sumsmxP _ _ _ (submx_refl (\sum_(i | P i) A i)%MS).
by rewrite mulmx_suml summx_sub_sums // => i _; rewrite genmxE -mulmxA submxMl.
Qed. | Lemma | sumsmxMr_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"... | [
"apply",
"eqmxP",
"genmxE",
"last",
"mulmxA",
"mulmx_suml",
"split",
"sub_sumsmxP",
"submxMl",
"submxMr",
"submx_refl",
"summx_sub_sums",
"sumsmx_subP",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumsmxMr P n (A_ : I -> 'M[F]_n) (B : 'M_n) :
((\sum_(i | P i) A_ i)%MS *m B :=: \sum_(i | P i) (A_ i *m B))%MS. | Proof.
by apply: eqmx_trans (sumsmxMr_gen _ _ _) (eqmx_sums _) => i _; apply: genmxE.
Qed. | Lemma | sumsmxMr | 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_sums",
"eqmx_trans",
"genmxE",
"sumsmxMr_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_pid_mx m n r : r <= m -> r <= n -> \rank (pid_mx r : 'M_(m, n)) = r. | Proof.
do 2!move/subnKC <-; rewrite pid_mx_block block_mxEv row_mx0 -addsmxE addsmx0.
by rewrite -mxrank_tr tr_row_mx trmx0 trmx1 -addsmxE addsmx0 mxrank1.
Qed. | Lemma | rank_pid_mx | 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"... | [
"addsmx0",
"addsmxE",
"block_mxEv",
"mxrank1",
"mxrank_tr",
"pid_mx",
"pid_mx_block",
"rank",
"row_mx0",
"subnKC",
"tr_row_mx",
"trmx0",
"trmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_copid_mx n r : r <= n -> \rank (copid_mx r : 'M_n) = (n - r)%N. | Proof.
move/subnKC <-; rewrite /copid_mx pid_mx_block scalar_mx_block.
rewrite opp_block_mx !oppr0 add_block_mx !addr0 subrr block_mxEv row_mx0.
rewrite -addsmxE adds0mx -mxrank_tr tr_row_mx trmx0 trmx1.
by rewrite -addsmxE adds0mx mxrank1 addKn.
Qed. | Lemma | rank_copid_mx | 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"... | [
"addKn",
"add_block_mx",
"addr0",
"adds0mx",
"addsmxE",
"block_mxEv",
"copid_mx",
"mxrank1",
"mxrank_tr",
"opp_block_mx",
"oppr0",
"pid_mx_block",
"rank",
"row_mx0",
"scalar_mx_block",
"subnKC",
"subrr",
"tr_row_mx",
"trmx0",
"trmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_compl m n (A : 'M_(m, n)) : \rank A^C = (n - \rank A)%N. | Proof. by rewrite mxrankMfree ?row_free_unit ?rank_copid_mx. Qed. | Lemma | mxrank_compl | 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"... | [
"mxrankMfree",
"rank",
"rank_copid_mx",
"row_free_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_ker m n (A : 'M_(m, n)) : \rank (kermx A) = (m - \rank A)%N. | Proof. by rewrite mxrankMfree ?row_free_unit ?unitmx_inv ?rank_copid_mx. Qed. | Lemma | mxrank_ker | 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"... | [
"kermx",
"mxrankMfree",
"rank",
"rank_copid_mx",
"row_free_unit",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermx_eq0 n m (A : 'M_(m, n)) : (kermx A == 0) = row_free A. | Proof. by rewrite -mxrank_eq0 mxrank_ker subn_eq0 row_leq_rank. Qed. | Lemma | kermx_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"... | [
"kermx",
"mxrank_eq0",
"mxrank_ker",
"row_free",
"row_leq_rank",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_coker m n (A : 'M_(m, n)) : \rank (cokermx A) = (n - \rank A)%N. | Proof. by rewrite eqmxMfull ?row_full_unit ?unitmx_inv ?rank_copid_mx. Qed. | Lemma | mxrank_coker | 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",
"eqmxMfull",
"rank",
"rank_copid_mx",
"row_full_unit",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cokermx_eq0 n m (A : 'M_(m, n)) : (cokermx A == 0) = row_full A. | Proof. by rewrite -mxrank_eq0 mxrank_coker subn_eq0 col_leq_rank. Qed. | Lemma | cokermx_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"... | [
"cokermx",
"col_leq_rank",
"mxrank_coker",
"mxrank_eq0",
"row_full",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_ker m n (A : 'M_(m, n)) : kermx A *m A = 0. | Proof.
by rewrite -{2}[A]mulmx_ebase !mulmxA mulmxKV // mul_copid_mx_pid ?mul0mx.
Qed. | Lemma | mulmx_ker | 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"... | [
"kermx",
"mul0mx",
"mul_copid_mx_pid",
"mulmxA",
"mulmxKV",
"mulmx_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxKV_ker m n p (A : 'M_(n, p)) (B : 'M_(m, n)) :
B *m A = 0 -> B *m col_ebase A *m kermx A = B. | Proof.
rewrite mulmxA mulmxBr mulmx1 mulmxBl mulmxK //.
rewrite -{1}[A]mulmx_ebase !mulmxA => /(canRL (mulmxK (row_ebase_unit A))).
rewrite mul0mx // => BA0; apply: (canLR (addrK _)).
by rewrite -(pid_mx_id _ _ n (rank_leq_col A)) mulmxA BA0 !mul0mx addr0.
Qed. | Lemma | mulmxKV_ker | 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"... | [
"addr0",
"addrK",
"apply",
"col_ebase",
"kermx",
"mul0mx",
"mulmx1",
"mulmxA",
"mulmxBl",
"mulmxBr",
"mulmxK",
"mulmx_ebase",
"pid_mx_id",
"rank_leq_col",
"row_ebase_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_kermxP p m n (A : 'M_(m, n)) (B : 'M_(p, m)) :
reflect (B *m A = 0) (B <= kermx A)%MS. | Proof.
apply: (iffP submxP) => [[D ->]|]; first by rewrite -mulmxA mulmx_ker mulmx0.
by move/mulmxKV_ker; exists (B *m col_ebase A).
Qed. | Lemma | sub_kermxP | 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",
"kermx",
"mulmx0",
"mulmxA",
"mulmxKV_ker",
"mulmx_ker",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_kermx p m n (A : 'M_(m, n)) (B : 'M_(p, m)) :
(B <= kermx A)%MS = (B *m A == 0). | Proof. exact/sub_kermxP/eqP. Qed. | Lemma | sub_kermx | 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"... | [
"kermx",
"sub_kermxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermx0 m n : (kermx (0 : 'M_(m, n)) :=: 1%:M)%MS. | Proof. by apply/eqmxP; rewrite submx1/= sub_kermx mulmx0. Qed. | Lemma | kermx0 | 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",
"kermx",
"mulmx0",
"sub_kermx",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_free_eq0 m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
row_free B -> (A *m B == 0) = (A == 0). | Proof. by rewrite -sub_kermx -kermx_eq0 => /eqP->; rewrite submx0. Qed. | Lemma | mulmx_free_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"... | [
"kermx_eq0",
"row_free",
"sub_kermx",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_row_free m n (A : 'M_(m, n)) :
(forall v : 'rV_m, v *m A = 0 -> v = 0) -> row_free A. | Proof.
move=> Ainj; rewrite -kermx_eq0; apply/eqP/row_matrixP => i.
by rewrite row0; apply/Ainj; rewrite -row_mul mulmx_ker row0.
Qed. | Lemma | inj_row_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",
"kermx_eq0",
"mulmx_ker",
"row0",
"row_free",
"row_matrixP",
"row_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_freePn m n (M : 'M[F]_(m, n)) :
reflect (exists i, (row i M <= row' i M)%MS) (~~ row_free M). | Proof.
rewrite -kermx_eq0; apply: (iffP (rowV0Pn _)) => [|[i0 /submxP[D rM]]].
move=> [v /sub_kermxP vM_eq0 /rV0Pn[i0 vi0_neq0]]; exists i0.
have := vM_eq0; rewrite mulmx_sum_row (bigD1_ord i0)//=.
move=> /(canRL (addrK _))/(canRL (scalerK _))->//.
rewrite sub0r scalerN -scaleNr scalemx_sub// summx_sub// => l _... | Lemma | row_freePn | 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"... | [
"addrC",
"addrK",
"apply",
"big1",
"bigD1_ord",
"i0",
"kermx_eq0",
"last",
"liftK",
"mulmx_sum_row",
"mxE",
"oner_eq0",
"oppr_eq0",
"rV0Pn",
"row",
"row'",
"rowV0Pn",
"row_free",
"row_rowsub",
"row_sub",
"scaleN1r",
"scaleNr",
"scalemx_sub",
"scalerK",
"scalerN",
"s... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
negb_row_free m n (M : 'M[F]_(m, n)) :
~~ row_free M = [exists i, (row i M <= row' i M)%MS]. | Proof. exact/row_freePn/existsP. Qed. | Lemma | negb_row_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"... | [
"existsP",
"row",
"row'",
"row_free",
"row_freePn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx0_rank_max m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
A *m B = 0 -> \rank A + \rank B <= n. | Proof.
move=> AB0; rewrite -{3}(subnK (rank_leq_row B)) leq_add2r.
by rewrite -mxrank_ker mxrankS // sub_kermx AB0.
Qed. | Lemma | mulmx0_rank_max | 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"... | [
"leq_add2r",
"mxrankS",
"mxrank_ker",
"rank",
"rank_leq_row",
"sub_kermx",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_Frobenius m n p q (A : 'M_(m, n)) B (C : 'M_(p, q)) :
\rank (A *m B) + \rank (B *m C) <= \rank B + \rank (A *m B *m C). | Proof.
rewrite -{2}(mulmx_base (A *m B)) -mulmxA (eqmxMfull _ (col_base_full _)).
set C2 := row_base _ *m C.
rewrite -{1}(subnK (rank_leq_row C2)) -(mxrank_ker C2) addnAC leq_add2r.
rewrite addnC -{1}(mulmx_base B) -mulmxA eqmxMfull //.
set C1 := _ *m C; rewrite -{2}(subnKC (rank_leq_row C1)) leq_add2l -mxrank_ker.
rew... | Lemma | mxrank_Frobenius | 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"... | [
"addnAC",
"addnC",
"col_base_full",
"eq_row_base",
"eqmxMfull",
"leq_add2l",
"leq_add2r",
"mulmxA",
"mulmx_base",
"mulmx_ker",
"mxrankMfree",
"mxrankS",
"mxrank_ker",
"rank",
"rank_leq_row",
"row_base",
"row_base_free",
"sub_kermx",
"submxMl",
"submxP",
"subnK",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_mul_min m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
\rank A + \rank B - n <= \rank (A *m B). | Proof.
by have:= mxrank_Frobenius A 1%:M B; rewrite mulmx1 mul1mx mxrank1 leq_subLR.
Qed. | Lemma | mxrank_mul_min | 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"... | [
"leq_subLR",
"mul1mx",
"mulmx1",
"mxrank1",
"mxrank_Frobenius",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_compl_full m n (A : 'M_(m, n)) : row_full (A + A^C)%MS. | Proof.
rewrite /row_full addsmxE; apply/row_fullP.
exists (row_mx (pinvmx A) (cokermx A)); rewrite mul_row_col.
rewrite -{2}[A]mulmx_ebase -!mulmxA mulKmx // -mulmxDr !mulmxA.
by rewrite pid_mx_id ?copid_mx_id // -mulmxDl addrC subrK mul1mx mulVmx.
Qed. | Lemma | addsmx_compl_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"... | [
"addrC",
"addsmxE",
"apply",
"cokermx",
"copid_mx_id",
"mul1mx",
"mulKmx",
"mulVmx",
"mul_row_col",
"mulmxA",
"mulmxDl",
"mulmxDr",
"mulmx_ebase",
"pid_mx_id",
"pinvmx",
"row_full",
"row_fullP",
"row_mx",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_capmx_gen m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A <= capmx_gen B C)%MS = (A <= B)%MS && (A <= C)%MS. | Proof.
apply/idP/andP=> [sAI | [/submxP[B' ->{A}] /submxP[C' eqBC']]].
rewrite !(submx_trans sAI) ?submxMl // /capmx_gen.
have:= mulmx_ker (col_mx B C); set K := kermx _.
rewrite -{1}[K]hsubmxK mul_row_col; move/(canRL (addrK _))->.
by rewrite add0r -mulNmx submxMl.
have: (row_mx B' (- C') <= kermx (col_mx B... | Lemma | sub_capmx_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"... | [
"add0r",
"addrK",
"apply",
"capmx_gen",
"col_mx",
"eq_row_mx",
"hsubmxK",
"kermx",
"mulNmx",
"mul_mx_row",
"mul_row_col",
"mulmxA",
"mulmx_ker",
"row_mx",
"sub_kermx",
"submxMl",
"submxP",
"submx_trans",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_witnessP m n (A : 'M_(m, n)) : equivmx A (qidmx A) (capmx_witness A). | Proof.
rewrite /equivmx qidmx_eq1 /qidmx /capmx_witness.
rewrite -sub1mx; case s1A: (1%:M <= A)%MS => /=; last first.
rewrite !genmxE submx_refl /= -negb_add; apply: contra {s1A}(negbT s1A).
have [<- | _] := eqP; first by rewrite genmxE.
by case: eqP A => //= -> A /eqP ->; rewrite pid_mx_1.
case: (m =P n) => [-> ... | Let | capmx_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",
"capmx_witness",
"conform_mx_id",
"equivmx",
"eqxx",
"genmxE",
"last",
"negb_add",
"nonconform_mx",
"pid_mx_1",
"qidmx",
"qidmx_eq1",
"sub1mx",
"submx1",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_normP m n (A : 'M_(m, n)) : equivmx_spec A (qidmx A) (capmx_norm A). | Proof. by case/andP: (chooseP (capmx_witnessP A)) => /eqmxP defN /eqP. Qed. | Let | capmx_normP | 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"... | [
"capmx_norm",
"capmx_witnessP",
"chooseP",
"eqmxP",
"equivmx_spec",
"qidmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_norm_eq m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
qidmx A = qidmx B -> (A == B)%MS -> capmx_norm A = capmx_norm B. | Proof.
move=> eqABid /eqmxP eqAB.
have{eqABid} eqAB: equivmx A (qidmx A) =1 equivmx B (qidmx B).
by move=> C; rewrite /equivmx eqABid !eqAB.
rewrite {1}/capmx_norm (eq_choose eqAB).
by apply: choose_id; first rewrite -eqAB; apply: capmx_witnessP.
Qed. | Let | capmx_norm_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",
"capmx_norm",
"capmx_witnessP",
"choose_id",
"eq_choose",
"eqmxP",
"equivmx",
"qidmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_nopP m n (A : 'M_(m, n)) : equivmx_spec A (qidmx A) (capmx_nop A). | Proof.
rewrite /capmx_nop; case: (eqVneq m n) => [-> | ne_mn] in A *.
by rewrite conform_mx_id.
by rewrite nonconform_mx ?ne_mn //; apply: capmx_normP.
Qed. | Let | capmx_nopP | 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",
"capmx_nop",
"capmx_normP",
"conform_mx_id",
"eqVneq",
"equivmx_spec",
"nonconform_mx",
"qidmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_qidmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
qidmx B -> (A <= B)%MS. | Proof.
rewrite /qidmx => idB; apply: {A}submx_trans (submx1 A) _.
by case: eqP B idB => [-> _ /eqP-> | _ B]; rewrite (=^~ sub1mx, pid_mx_1).
Qed. | Let | sub_qidmx | 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",
"pid_mx_1",
"qidmx",
"sub1mx",
"submx1",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qidmx_cap m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
qidmx (A :&: B)%MS = qidmx A && qidmx B. | Proof.
rewrite unlock -sub1mx.
case idA: (qidmx A); case idB: (qidmx B); try by rewrite capmx_nopP.
case s1B: (_ <= B)%MS; first by rewrite capmx_normP.
apply/idP=> /(sub_qidmx 1%:M).
by rewrite capmx_normP sub_capmx_gen s1B andbF.
Qed. | Let | qidmx_cap | 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",
"capmx_nopP",
"capmx_normP",
"qidmx",
"sub1mx",
"sub_capmx_gen",
"sub_qidmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_eq_norm m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
qidmx A = qidmx B -> (A :&: B)%MS = capmx_norm (A :&: B)%MS. | Proof.
move=> eqABid; rewrite unlock -sub1mx {}eqABid.
have norm_id m (C : 'M_(m, n)) (N := capmx_norm C) : capmx_norm N = N.
by apply: capmx_norm_eq; rewrite ?capmx_normP ?andbb.
case idB: (qidmx B); last by case: ifP; rewrite norm_id.
rewrite /capmx_nop; case: (eqVneq m2 n) => [-> | neqm2n] in B idB *.
have idN :... | Let | capmx_eq_norm | 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",
"capmx_nop",
"capmx_norm",
"capmx_normP",
"capmx_norm_eq",
"conform_mx_id",
"eqVneq",
"last",
"nonconform_mx",
"qidmx",
"qidmx_eq1",
"sub1mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxE m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :&: B :=: capmx_gen A B)%MS. | Proof.
rewrite unlock -sub1mx; apply/eqmxP.
have:= submx_refl (capmx_gen A B); rewrite !sub_capmx_gen => /andP[sIA sIB].
case idA: (qidmx A); first by rewrite !capmx_nopP submx_refl sub_qidmx.
case idB: (qidmx B); first by rewrite !capmx_nopP submx_refl sub_qidmx.
case s1B: (1%:M <= B)%MS; rewrite !capmx_normP ?sub_cap... | Lemma | capmxE | 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",
"capmx_gen",
"capmx_nopP",
"capmx_normP",
"eqmxP",
"qidmx",
"sub1mx",
"sub_capmx_gen",
"sub_qidmx",
"submx1",
"submx_refl",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxSl m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) : (A :&: B <= A)%MS. | Proof. by rewrite capmxE submxMl. Qed. | Lemma | capmxSl | 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"... | [
"capmxE",
"submxMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_capmx m m1 m2 n (A : 'M_(m, n)) (B : 'M_(m1, n)) (C : 'M_(m2, n)) :
(A <= B :&: C)%MS = (A <= B)%MS && (A <= C)%MS. | Proof. by rewrite capmxE sub_capmx_gen. Qed. | Lemma | sub_capmx | 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"... | [
"capmxE",
"sub_capmx_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxC m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) : (A :&: B = B :&: A)%MS. | Proof.
have [eqAB|] := eqVneq (qidmx A) (qidmx B).
rewrite (capmx_eq_norm eqAB) (capmx_eq_norm (esym eqAB)).
apply: capmx_norm_eq; first by rewrite !qidmx_cap andbC.
by apply/andP; split; rewrite !sub_capmx andbC -sub_capmx.
by rewrite negb_eqb !unlock => /addbP <-; case: (qidmx A).
Qed. | Lemma | capmxC | 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",
"capmx_eq_norm",
"capmx_norm_eq",
"eqVneq",
"negb_eqb",
"qidmx",
"qidmx_cap",
"split",
"sub_capmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxSr m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) : (A :&: B <= B)%MS. | Proof. by rewrite capmxC capmxSl. Qed. | Lemma | capmxSr | 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"... | [
"capmxC",
"capmxSl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_idPr n m1 m2 (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (A :&: B :=: B)%MS (B <= A)%MS. | Proof.
have:= @eqmxP _ _ _ (A :&: B)%MS B.
by rewrite capmxSr sub_capmx submx_refl !andbT.
Qed. | Lemma | capmx_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"... | [
"capmxSr",
"eqmxP",
"sub_capmx",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_idPl n m1 m2 (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (A :&: B :=: A)%MS (A <= B)%MS. | Proof. by rewrite capmxC; apply: capmx_idPr. Qed. | Lemma | capmx_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"... | [
"apply",
"capmxC",
"capmx_idPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxS 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.
by move=> sAC sBD; rewrite sub_capmx {1}capmxC !(submx_trans (capmxSr _ _)).
Qed. | Lemma | capmxS | 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"... | [
"capmxC",
"capmxSr",
"sub_capmx",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cap_eqmx 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. by move=> eqAC eqBD; apply/eqmxP; rewrite !capmxS ?eqAC ?eqBD. Qed. | Lemma | cap_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"... | [
"apply",
"capmxS",
"eqmxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxMr m1 m2 n p (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(n, p)) :
((A :&: B) *m C <= A *m C :&: B *m C)%MS. | Proof. by rewrite sub_capmx !submxMr ?capmxSl ?capmxSr. Qed. | Lemma | capmxMr | 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"... | [
"capmxSl",
"capmxSr",
"sub_capmx",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cap0mx m1 m2 n (A : 'M_(m2, n)) : ((0 : 'M_(m1, n)) :&: A)%MS = 0. | Proof. exact: submx0null (capmxSl _ _). Qed. | Lemma | cap0mx | 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"... | [
"capmxSl",
"submx0null"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx0 m1 m2 n (A : 'M_(m1, n)) : (A :&: (0 : 'M_(m2, n)))%MS = 0. | Proof. exact: submx0null (capmxSr _ _). Qed. | Lemma | capmx0 | 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"... | [
"capmxSr",
"submx0null"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxT m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
row_full B -> (A :&: B :=: A)%MS. | Proof.
rewrite -sub1mx => s1B; apply/eqmxP.
by rewrite capmxSl sub_capmx submx_refl (submx_trans (submx1 A)).
Qed. | Lemma | capmxT | 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",
"capmxSl",
"eqmxP",
"row_full",
"sub1mx",
"sub_capmx",
"submx1",
"submx_refl",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capTmx m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
row_full A -> (A :&: B :=: B)%MS. | Proof. by move=> Afull; apply/eqmxP; rewrite capmxC !capmxT ?andbb. Qed. | Lemma | capTmx | 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",
"capmxC",
"capmxT",
"eqmxP",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_nop_id n (A : 'M_n) : capmx_nop A = A. | Proof. by rewrite /capmx_nop conform_mx_id. Qed. | Let | capmx_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"... | [
"capmx_nop",
"conform_mx_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cap1mx n (A : 'M_n) : (1%:M :&: A = A)%MS. | Proof. by rewrite unlock qidmx_eq1 eqxx capmx_nop_id. Qed. | Lemma | cap1mx | 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"... | [
"capmx_nop_id",
"eqxx",
"qidmx_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx1 n (A : 'M_n) : (A :&: 1%:M = A)%MS. | Proof. by rewrite capmxC cap1mx. Qed. | Lemma | capmx1 | 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"... | [
"cap1mx",
"capmxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_cap m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
<<A :&: B>>%MS = (<<A>> :&: <<B>>)%MS. | Proof.
rewrite -(eq_genmx (cap_eqmx (genmxE A) (genmxE B))).
case idAB: (qidmx <<A>> || qidmx <<B>>)%MS.
rewrite [@capmx]unlock !capmx_nop_id !(fun_if (@genmx _ _ _)) !genmx_id.
by case: (qidmx _) idAB => //= ->.
case idA: (qidmx _) idAB => //= idB; rewrite {2}capmx_eq_norm ?idA //.
set C := (_ :&: _)%MS; have eq_i... | Lemma | genmx_cap | 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",
"cap_eqmx",
"capmx_eq_norm",
"capmx_nop_id",
"capmx_norm",
"capmx_witnessP",
"choose_id",
"eq_genmx",
"genmx1",
"genmxE",
"genmxP",
"genmx_id",
"genmx_witnessP",
"qidmx",
"qidmx_cap",
"qidmx_eq1",
"row_full",
"sub1mx",
"sub_capmx",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmxA m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A :&: (B :&: C) = A :&: B :&: C)%MS. | Proof.
rewrite (capmxC A B) capmxC; wlog idA: m1 m3 A C / qidmx A.
move=> IH; case idA: (qidmx A); first exact: IH.
case idC: (qidmx C); first by rewrite -IH.
rewrite (@capmx_eq_norm n m3) ?qidmx_cap ?idA ?idC ?andbF //.
rewrite capmx_eq_norm ?qidmx_cap ?idA ?idC ?andbF //.
apply: capmx_norm_eq; first by rewr... | Lemma | capmxA | 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",
"capmxC",
"capmxS",
"capmx_eq_norm",
"capmx_nopP",
"capmx_nop_id",
"capmx_norm_eq",
"eqVneq",
"qidmx",
"qidmx_cap",
"split",
"sub_capmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcapmx_inf i0 P m n (A_ : I -> 'M_n) (B : 'M_(m, n)) :
P i0 -> (A_ i0 <= B -> \bigcap_(i | P i) A_ i <= B)%MS. | Proof. by move=> Pi0; apply: submx_trans; rewrite (bigD1 i0) // capmxSl. Qed. | Lemma | bigcapmx_inf | 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"... | [
"Pi0",
"apply",
"bigD1",
"capmxSl",
"i0",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_bigcapmxP P m n (A : 'M_(m, n)) (B_ : I -> 'M_n) :
reflect (forall i, P i -> A <= B_ i)%MS (A <= \bigcap_(i | P i) B_ i)%MS. | Proof.
apply: (iffP idP) => [sAB i Pi | sAB].
by apply: (submx_trans sAB); rewrite (bigcapmx_inf Pi).
by elim/big_rec: _ => [|i Pi C sAC]; rewrite ?submx1 // sub_capmx sAB.
Qed. | Lemma | sub_bigcapmxP | 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",
"big_rec",
"bigcapmx_inf",
"sub_capmx",
"submx1",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_bigcap P n (A_ : I -> 'M_n) :
(<<\bigcap_(i | P i) A_ i>> = \bigcap_(i | P i) <<A_ i>>)%MS. | Proof. exact: (big_morph _ (@genmx_cap n n n) (@genmx1 n)). Qed. | Lemma | genmx_bigcap | 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"... | [
"big_morph",
"genmx1",
"genmx_cap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_modl m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A <= C -> A + (B :&: C) :=: (A + B) :&: C)%MS. | Proof.
move=> sAC; set D := ((A + B) :&: C)%MS; apply/eqmxP.
rewrite sub_capmx addsmxS ?capmxSl // addsmx_sub sAC capmxSr /=.
have: (D <= B + A)%MS by rewrite addsmxC capmxSl.
case/sub_addsmxP=> u defD; rewrite defD addrC addmx_sub_adds ?submxMl //.
rewrite sub_capmx submxMl -[_ *m B](addrK (u.2 *m A)) -defD.
by rewrit... | Lemma | matrix_modl | 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",
"addmx_sub_adds",
"addrC",
"addrK",
"addsmxC",
"addsmxS",
"addsmx_sub",
"apply",
"capmxSl",
"capmxSr",
"eqmxP",
"eqmx_opp",
"mulmx_sub",
"sub_addsmxP",
"sub_capmx",
"submxMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_modr m1 m2 m3 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(C <= A -> (A :&: B) + C :=: A :&: (B + C))%MS. | Proof. by rewrite !(capmxC A) -!(addsmxC C); apply: matrix_modl. Qed. | Lemma | matrix_modr | 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",
"apply",
"capmxC",
"matrix_modl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_compl m n (A : 'M_(m, n)) : (A :&: A^C)%MS = 0. | Proof.
set D := (A :&: A^C)%MS; have: (D <= D)%MS by [].
rewrite sub_capmx andbC => /andP[/submxP[B defB]].
rewrite submxE => /eqP; rewrite defB -!mulmxA mulKVmx ?copid_mx_id //.
by rewrite mulmxA => ->; rewrite mul0mx.
Qed. | Lemma | capmx_compl | 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"... | [
"copid_mx_id",
"mul0mx",
"mulKVmx",
"mulmxA",
"sub_capmx",
"submxE",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_mul_ker m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
(\rank (A *m B) + \rank (A :&: kermx B))%N = \rank A. | Proof.
apply/eqP; set K := kermx B; set C := (A :&: K)%MS.
rewrite -(eqmxMr B (eq_row_base A)); set K' := _ *m B.
rewrite -{2}(subnKC (rank_leq_row K')) -mxrank_ker eqn_add2l.
rewrite -(mxrankMfree _ (row_base_free A)) mxrank_leqif_sup; last first.
by rewrite sub_capmx -(eq_row_base A) submxMl sub_kermx -mulmxA mulmx... | Lemma | mxrank_mul_ker | 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",
"capmxSl",
"capmxSr",
"eq_row_base",
"eqmxMr",
"eqn_add2l",
"kermx",
"last",
"mulmxA",
"mulmx_ker",
"mxrankMfree",
"mxrank_ker",
"mxrank_leqif_sup",
"rank",
"rank_leq_row",
"row_base",
"row_base_free",
"sub_capmx",
"sub_kermx",
"submxMl",
"submxMr",
"submxP",
"su... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_injP m n p (A : 'M_(m, n)) (f : 'M_(n, p)) :
reflect (\rank (A *m f) = \rank A) ((A :&: kermx f)%MS == 0). | Proof.
rewrite -mxrank_eq0 -(eqn_add2l (\rank (A *m f))).
by rewrite mxrank_mul_ker addn0 eq_sym; apply: eqP.
Qed. | Lemma | mxrank_injP | 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"... | [
"addn0",
"apply",
"eq_sym",
"eqn_add2l",
"kermx",
"mxrank_eq0",
"mxrank_mul_ker",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_disjoint_sum m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :&: B)%MS = 0 -> \rank (A + B)%MS = (\rank A + \rank B)%N. | Proof.
move=> AB0; pose Ar := row_base A; pose Br := row_base B.
have [Afree Bfree]: row_free Ar /\ row_free Br by rewrite !row_base_free.
have: (Ar :&: Br <= A :&: B)%MS by rewrite capmxS ?eq_row_base.
rewrite {}AB0 submx0 -mxrank_eq0 capmxE mxrankMfree //.
set Cr := col_mx Ar Br; set Crl := lsubmx _; rewrite mxrank_e... | Lemma | mxrank_disjoint_sum | 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"... | [
"add0r",
"adds_eqmx",
"addsmxE",
"apply",
"capmxE",
"capmxS",
"col_mx",
"eq_row_base",
"eqn_leq",
"hsubmxK",
"kermx",
"lsubmx",
"mul0mx",
"mul_row_col",
"mulmx_ker",
"mxrank0",
"mxrankMfree",
"mxrank_eq0",
"mxrank_ker",
"rank",
"rank_leq_row",
"row_base",
"row_base_free",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diffmxE m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :\: B :=: A :&: (capmx_gen A B)^C)%MS. | Proof. by rewrite unlock; apply/eqmxP; rewrite !genmxE !capmxE andbb. Qed. | Lemma | diffmxE | 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",
"capmxE",
"capmx_gen",
"eqmxP",
"genmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_diff m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(<<A :\: B>> = A :\: B)%MS. | Proof. by rewrite [@diffmx]unlock genmx_id. Qed. | Lemma | genmx_diff | 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"... | [
"genmx_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diffmxSl m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) : (A :\: B <= A)%MS. | Proof. by rewrite diffmxE capmxSl. Qed. | Lemma | diffmxSl | 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"... | [
"capmxSl",
"diffmxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_diff m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
((A :\: B) :&: B)%MS = 0. | Proof.
apply/eqP; pose C := capmx_gen A B; rewrite -submx0 -(capmx_compl C).
by rewrite sub_capmx -capmxE sub_capmx andbAC -sub_capmx -diffmxE -sub_capmx.
Qed. | Lemma | capmx_diff | 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",
"capmxE",
"capmx_compl",
"capmx_gen",
"diffmxE",
"sub_capmx",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_diff_cap_eq m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :\: B + A :&: B :=: A)%MS. | Proof.
apply/eqmxP; rewrite addsmx_sub capmxSl diffmxSl /=.
set C := (A :\: B)%MS; set D := capmx_gen A B.
suffices sACD: (A <= C + D)%MS.
by rewrite (submx_trans sACD) ?addsmxS ?capmxE.
have:= addsmx_compl_full D; rewrite /row_full addsmxE.
case/row_fullP=> U /(congr1 (mulmx A)); rewrite mulmx1.
rewrite -[U]hsubmxK ... | Lemma | addsmx_diff_cap_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"... | [
"addmx_sub",
"addrC",
"addrK",
"addsmxE",
"addsmxS",
"addsmx_compl_full",
"addsmx_sub",
"apply",
"capmxE",
"capmxSl",
"capmx_gen",
"diffmxE",
"diffmxSl",
"eqmxP",
"hsubmxK",
"mulNmx",
"mul_row_col",
"mulmx",
"mulmx1",
"mulmxA",
"mulmxDr",
"mulmx_sub",
"row_full",
"row_f... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_cap_compl m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(\rank (A :&: B) + \rank (A :\: B))%N = \rank A. | Proof.
rewrite addnC -mxrank_disjoint_sum ?addsmx_diff_cap_eq //.
by rewrite (capmxC A) capmxA capmx_diff cap0mx.
Qed. | Lemma | mxrank_cap_compl | 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"... | [
"addnC",
"addsmx_diff_cap_eq",
"cap0mx",
"capmxA",
"capmxC",
"capmx_diff",
"mxrank_disjoint_sum",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_sum_cap m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(\rank (A + B) + \rank (A :&: B) = \rank A + \rank B)%N. | Proof.
set C := (A :&: B)%MS; set D := (A :\: B)%MS.
have rDB: \rank (A + B)%MS = \rank (D + B)%MS.
apply/eqP; rewrite mxrank_leqif_sup; last by rewrite addsmxS ?diffmxSl.
by rewrite addsmx_sub addsmxSr -(addsmx_diff_cap_eq A B) addsmxS ?capmxSr.
rewrite {1}rDB mxrank_disjoint_sum ?capmx_diff //.
by rewrite addnC a... | Lemma | mxrank_sum_cap | 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"... | [
"addnA",
"addnC",
"addsmxS",
"addsmxSr",
"addsmx_diff_cap_eq",
"addsmx_sub",
"apply",
"capmxSr",
"capmx_diff",
"diffmxSl",
"last",
"mxrank_cap_compl",
"mxrank_disjoint_sum",
"mxrank_leqif_sup",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_adds_leqif m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
\rank (A + B) <= \rank A + \rank B ?= iff (A :&: B <= (0 : 'M_n))%MS. | Proof.
rewrite -mxrank_sum_cap; split; first exact: leq_addr.
by rewrite addnC (@eqn_add2r _ 0) eq_sym mxrank_eq0 -submx0.
Qed. | Lemma | mxrank_adds_leqif | 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"... | [
"addnC",
"eq_sym",
"eqn_add2r",
"leq_addr",
"mxrank_eq0",
"mxrank_sum_cap",
"rank",
"split",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_col_mx0 m n p (A : 'M_(m, n)) :
\rank (col_mx A (0 : 'M_(p, n))) = \rank A. | Proof. by rewrite -addsmxE addsmx0. Qed. | Lemma | rank_col_mx0 | 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"... | [
"addsmx0",
"addsmxE",
"col_mx",
"rank"
] | rank of block matrices with 0s inside | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rank_col_0mx m n p (A : 'M_(m, n)) :
\rank (col_mx (0 : 'M_(p, n)) A) = \rank A. | Proof. by rewrite -addsmxE adds0mx. Qed. | Lemma | rank_col_0mx | 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",
"addsmxE",
"col_mx",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_row_mx0 m n p (A : 'M_(m, n)) :
\rank (row_mx A (0 : 'M_(m, p))) = \rank A. | Proof. by rewrite -mxrank_tr -[RHS]mxrank_tr tr_row_mx trmx0 rank_col_mx0. Qed. | Lemma | rank_row_mx0 | 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"... | [
"mxrank_tr",
"rank",
"rank_col_mx0",
"row_mx",
"tr_row_mx",
"trmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_row_0mx m n p (A : 'M_(m, n)) :
\rank (row_mx (0 : 'M_(m, p)) A) = \rank A. | Proof. by rewrite -mxrank_tr -[RHS]mxrank_tr tr_row_mx trmx0 rank_col_0mx. Qed. | Lemma | rank_row_0mx | 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"... | [
"mxrank_tr",
"rank",
"rank_col_0mx",
"row_mx",
"tr_row_mx",
"trmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_diag_block_mx m n p q
(A : 'M_(m, n)) (B : 'M_(p, q)) :
\rank (block_mx A 0 0 B) = (\rank A + \rank B)%N. | Proof.
rewrite block_mxEv -addsmxE mxrank_disjoint_sum ?rank_row_mx0 ?rank_row_0mx//.
apply/eqP/rowV0P => v; rewrite sub_capmx => /andP[/submxP[x ->]].
rewrite mul_mx_row mulmx0 => /submxP[y]; rewrite mul_mx_row mulmx0.
by move=> /eq_row_mx[-> _]; rewrite row_mx0.
Qed. | Lemma | rank_diag_block_mx | 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",
"block_mx",
"block_mxEv",
"eq_row_mx",
"mul_mx_row",
"mulmx0",
"mxrank_disjoint_sum",
"rank",
"rank_row_0mx",
"rank_row_mx0",
"rowV0P",
"row_mx0",
"sub_capmx",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_mx_sub m n U V (W : 'M_(m, n)) : (W *m proj_mx U V <= U)%MS. | Proof. by rewrite !mulmx_sub // -addsmxE addsmx0. Qed. | Lemma | proj_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"... | [
"addsmx0",
"addsmxE",
"mulmx_sub",
"proj_mx"
] | Subspace projection matrix | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
proj_mx_compl_sub m n U V (W : 'M_(m, n)) :
(W <= U + V -> W - W *m proj_mx U V <= V)%MS. | Proof.
rewrite addsmxE => sWUV; rewrite mulmxA -{1}(mulmxKpV sWUV) -mulmxBr.
by rewrite mulmx_sub // opp_col_mx add_col_mx subrr subr0 -addsmxE adds0mx.
Qed. | Lemma | proj_mx_compl_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"... | [
"add_col_mx",
"adds0mx",
"addsmxE",
"mulmxA",
"mulmxBr",
"mulmxKpV",
"mulmx_sub",
"opp_col_mx",
"proj_mx",
"subr0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_mx_id m n U V (W : 'M_(m, n)) :
(U :&: V = 0)%MS -> (W <= U)%MS -> W *m proj_mx U V = W. | Proof.
move=> dxUV sWU; apply/eqP; rewrite -subr_eq0 -submx0 -dxUV.
rewrite sub_capmx addmx_sub ?eqmx_opp ?proj_mx_sub //= -eqmx_opp opprB.
by rewrite proj_mx_compl_sub // (submx_trans sWU) ?addsmxSl.
Qed. | Lemma | proj_mx_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"... | [
"addmx_sub",
"addsmxSl",
"apply",
"eqmx_opp",
"opprB",
"proj_mx",
"proj_mx_compl_sub",
"proj_mx_sub",
"sub_capmx",
"submx0",
"submx_trans",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_mx_0 m n U V (W : 'M_(m, n)) :
(U :&: V = 0)%MS -> (W <= V)%MS -> W *m proj_mx U V = 0. | Proof.
move=> dxUV sWV; apply/eqP; rewrite -submx0 -dxUV.
rewrite sub_capmx proj_mx_sub /= -[_ *m _](subrK W) addmx_sub // -eqmx_opp.
by rewrite opprB proj_mx_compl_sub // (submx_trans sWV) ?addsmxSr.
Qed. | Lemma | proj_mx_0 | 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",
"addsmxSr",
"apply",
"eqmx_opp",
"opprB",
"proj_mx",
"proj_mx_compl_sub",
"proj_mx_sub",
"sub_capmx",
"submx0",
"submx_trans",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_proj_mx m n U V (W : 'M_(m, n)) :
(U :&: V = 0)%MS -> (W <= U + V)%MS ->
W *m proj_mx U V + W *m proj_mx V U = W. | Proof.
move=> dxUV sWUV; apply/eqP; rewrite -subr_eq0 -submx0 -dxUV.
rewrite -addrA sub_capmx {2}addrCA -!(opprB W).
by rewrite !{1}addmx_sub ?proj_mx_sub ?eqmx_opp ?proj_mx_compl_sub // addsmxC.
Qed. | Lemma | add_proj_mx | 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",
"addrA",
"addrCA",
"addsmxC",
"apply",
"eqmx_opp",
"opprB",
"proj_mx",
"proj_mx_compl_sub",
"proj_mx_sub",
"sub_capmx",
"submx0",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_mx_proj n (U V : 'M_n) :
let P := proj_mx U V in (U :&: V = 0)%MS -> P *m P = P. | Proof.
by move=> P dxUV; rewrite -[P in P *m _]mul1mx proj_mx_id ?proj_mx_sub ?mul1mx.
Qed. | Lemma | proj_mx_proj | 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"... | [
"mul1mx",
"proj_mx",
"proj_mx_id",
"proj_mx_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
complete_unitmx m n (U : 'M_(m, n)) (f : 'M_n) :
\rank (U *m f) = \rank U -> {g : 'M_n | g \in unitmx & U *m f = U *m g}. | Proof.
move=> injfU; pose V := <<U>>%MS; pose W := V *m f.
pose g := proj_mx V (V^C)%MS *m f + cokermx V *m row_ebase W.
have defW: V *m g = W.
rewrite mulmxDr mulmxA proj_mx_id ?genmxE ?capmx_compl //.
by rewrite mulmxA mulmx_coker mul0mx addr0.
exists g; last first.
have /submxP[u ->]: (U <= V)%MS by rewrite ge... | Lemma | complete_unitmx | 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"... | [
"add0r",
"addr0",
"addrC",
"apply",
"capmx_compl",
"cokermx",
"col_ebase",
"col_ebase_unit",
"copid_mx_id",
"eqmxMr",
"genmxE",
"invmx",
"last",
"mul0mx",
"mul1mx",
"mulKmx",
"mulVmx",
"mulmxA",
"mulmxDl",
"mulmxDr",
"mulmxK",
"mulmx_coker",
"mulmx_ebase",
"proj_mx",
... | Completing a partially injective matrix to get a unit matrix. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eqmxMunitP m n (U V : 'M_(m, n)) :
reflect (exists2 P, P \in unitmx & U = P *m V) (U == V)%MS. | Proof.
apply: (iffP eqmxP) => [eqUV | [P Punit ->]]; last first.
by apply/eqmxMfull; rewrite row_full_unit.
have [D defU]: exists D, U = D *m V by apply/submxP; rewrite eqUV.
have{eqUV} [Pt Pt_unit defUt]: {Pt | Pt \in unitmx & V^T *m D^T = V^T *m Pt}.
by apply/complete_unitmx; rewrite -trmx_mul -defU !mxrank_tr eq... | Lemma | eqmxMunitP | 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",
"complete_unitmx",
"defU",
"eqmxMfull",
"eqmxP",
"last",
"mxrank_tr",
"row_full_unit",
"submxP",
"trmxK",
"trmx_inj",
"trmx_mul",
"unitmx",
"unitmx_tr"
] | iff they differ only by a change of basis. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_rank_unitmx m1 m2 n (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
\rank U = \rank V -> {f : 'M_n | f \in unitmx & V :=: U *m f}%MS. | Proof.
move=> eqrUV; pose f := invmx (row_ebase <<U>>%MS) *m row_ebase <<V>>%MS.
have defUf: (<<U>> *m f :=: <<V>>)%MS.
rewrite -[<<U>>%MS]mulmx_ebase mulmxA mulmxK ?row_ebase_unit // -mulmxA.
rewrite genmxE eqrUV -genmxE -{3}[<<V>>%MS]mulmx_ebase -mulmxA.
move: (pid_mx _ *m _) => W; apply/eqmxP.
by rewrite !eq... | Lemma | eq_rank_unitmx | 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_unit",
"complete_unitmx",
"eqmxMfull",
"eqmxMr",
"eqmxP",
"genmxE",
"invmx",
"mulmxA",
"mulmxK",
"mulmx_ebase",
"pid_mx",
"rank",
"row_ebase",
"row_ebase_unit",
"row_full_unit",
"unitmx"
] | Mapping between two subspaces with the same dimension. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
maxrankfun : 'I_m ^ \rank A | :=
[arg max_(f > finfun (widen_ord (rank_leq_row A))) \rank (rowsub f A)]. | Definition | maxrankfun | 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"... | [
"rank",
"rank_leq_row",
"rowsub",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxf | := maxrankfun. | Notation | mxf | 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"... | [
"maxrankfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxrowsub_free : row_free (rowsub mxf A). | Proof.
rewrite /mxf; case: arg_maxnP => //= f _ fM; apply/negP => /negP rfA.
have [i NriA] : exists i, ~~ (row i A <= rowsub f A)%MS.
by apply/row_subPn; apply: contraNN rfA => /mxrankS; rewrite row_leq_rank.
have [j rjfA] : exists j, (row (f j) A <= rowsub (f \o lift j) A)%MS.
case/row_freePn: rfA => j.
by rewri... | Lemma | maxrowsub_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",
"arg_maxnP",
"eqVneq",
"eq_row_sub",
"eq_sym",
"eqxx",
"fM",
"ffunE",
"last",
"lift",
"ltmxE",
"ltmxErank",
"ltnNge",
"mxf",
"mxrankS",
"mxsub_comp",
"neq_lift",
"rank",
"row",
"row'Esub",
"row_free",
"row_freePn",
"row_leq_rank",
"row_rowsub",
"row_subP",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_maxrowsub : (rowsub mxf A :=: A)%MS. | Proof.
apply/eqmxP; rewrite -(eq_leqif (mxrank_leqif_eq _))//; last first.
exact: maxrowsub_free.
apply/row_subP => i; apply/submxP; exists (delta_mx 0 (mxf i)).
by rewrite -rowE; apply/rowP => j; rewrite !mxE.
Qed. | Lemma | eq_maxrowsub | 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",
"delta_mx",
"eq_leqif",
"eqmxP",
"last",
"maxrowsub_free",
"mxE",
"mxf",
"mxrank_leqif_eq",
"rowE",
"rowP",
"row_subP",
"rowsub",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxrankfun_inj : injective mxf. | Proof.
move=> i j eqAij; have /row_free_inj := maxrowsub_free.
move=> /(_ 1) /(_ (delta_mx 0 i) (delta_mx 0 j)).
rewrite -!rowE !row_rowsub eqAij => /(_ erefl) /matrixP /(_ 0 i) /eqP.
by rewrite !mxE !eqxx/=; case: (i =P j); rewrite // oner_eq0.
Qed. | Lemma | maxrankfun_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"... | [
"delta_mx",
"eqxx",
"matrixP",
"maxrowsub_free",
"mxE",
"mxf",
"oner_eq0",
"rowE",
"row_free_inj",
"row_rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxrowsub_full : row_full (rowsub mxf A). | Proof. by rewrite /row_full eq_maxrowsub. Qed. | Lemma | maxrowsub_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"... | [
"eq_maxrowsub",
"mxf",
"row_full",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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