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 |
|---|---|---|---|---|---|---|---|---|---|---|
map2_ulsubmx : map2_mx (ulsubmx B) (ulsubmx B') = ulsubmx (map2_mx B B'). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_ulsubmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"map2_mx",
"matrixP",
"mxE",
"ulsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_ursubmx : map2_mx (ursubmx B) (ursubmx B') = ursubmx (map2_mx B B'). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_ursubmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"map2_mx",
"matrixP",
"mxE",
"ursubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_dlsubmx : map2_mx (dlsubmx B) (dlsubmx B') = dlsubmx (map2_mx B B'). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_dlsubmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"dlsubmx",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_drsubmx : map2_mx (drsubmx B) (drsubmx B') = drsubmx (map2_mx B B'). | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | map2_drsubmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"drsubmx",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_map2_mx (f g : R -> S -> T) (M : 'M[R]_(m, n)) (M' : 'M[S]_(m, n)) :
(forall i j, f (M i j) (M' i j) = g (M i j) (M' i j)) ->
map2_mx f M M' = map2_mx g M M'. | Proof. by move=> fg; apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | eq_in_map2_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_map2_mx (f g : R -> S -> T) : f =2 g ->
@map2_mx _ _ _ f m n =2 @map2_mx _ _ _ g m n. | Proof. by move=> eq_fg M M'; apply/eq_in_map2_mx. Qed. | Lemma | eq_map2_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_in_map2_mx",
"map2_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx_left_in (f : R -> R -> R) (M : 'M_(m, n)) (M' : 'M_(m, n)) :
(forall i j, f (M i j) (M' i j) = M i j) -> map2_mx f M M' = M. | Proof. by move=> fM; apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | map2_mx_left_in | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"fM",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx_left (f : R -> R -> R) : f =2 (fun x _ => x) ->
forall (M : 'M_(m, n)) (M' : 'M_(m, n)), map2_mx f M M' = M. | Proof. by move=> fl M M'; rewrite map2_mx_left_in// =>i j; rewrite fl. Qed. | Lemma | map2_mx_left | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"map2_mx",
"map2_mx_left_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx_right_in (f : R -> R -> R) (M : 'M_(m, n)) (M' : 'M_(m, n)) :
(forall i j, f (M i j) (M' i j) = M' i j) -> map2_mx f M M' = M'. | Proof. by move=> fM; apply/matrixP => i j; rewrite !mxE. Qed. | Lemma | map2_mx_right_in | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"fM",
"map2_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx_right (f : R -> R -> R) : f =2 (fun _ x => x) ->
forall (M : 'M_(m, n)) (M' : 'M_(m, n)), map2_mx f M M' = M'. | Proof. by move=> fr M M'; rewrite map2_mx_right_in// =>i j; rewrite fr. Qed. | Lemma | map2_mx_right | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"map2_mx",
"map2_mx_right_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mxA {opm : Monoid.law idm} : associative (@map2_mx _ _ _ opm m n). | Proof. by move=> A B C; apply/matrixP=> i j; rewrite !mxE Monoid.mulmA. Qed. | Lemma | map2_mxA | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"idm",
"law",
"map2_mx",
"matrixP",
"mulmA",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_1mx {opm : Monoid.law idm} :
left_id (const_mx idm) (@map2_mx _ _ _ opm m n). | Proof. by move=> A; apply/matrixP=> i j; rewrite !mxE Monoid.mul1m. Qed. | Lemma | map2_1mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"idm",
"law",
"map2_mx",
"matrixP",
"mul1m",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx1 {opm : Monoid.law idm} :
right_id (const_mx idm) (@map2_mx _ _ _ opm m n). | Proof. by move=> A; apply/matrixP=> i j; rewrite !mxE Monoid.mulm1. Qed. | Lemma | map2_mx1 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"idm",
"law",
"map2_mx",
"matrixP",
"mulm1",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mxC {opm : Monoid.com_law idm} :
commutative (@map2_mx _ _ _ opm m n). | Proof. by move=> A B; apply/matrixP=> i j; rewrite !mxE Monoid.mulmC. Qed. | Lemma | map2_mxC | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"com_law",
"idm",
"map2_mx",
"matrixP",
"mulmC",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_0mx {opm : Monoid.mul_law idm} :
left_zero (const_mx idm) (@map2_mx _ _ _ opm m n). | Proof. by move=> A; apply/matrixP=> i j; rewrite !mxE Monoid.mul0m. Qed. | Lemma | map2_0mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"idm",
"map2_mx",
"matrixP",
"mul0m",
"mul_law",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mx0 {opm : Monoid.mul_law idm} :
right_zero (const_mx idm) (@map2_mx _ _ _ opm m n). | Proof. by move=> A; apply/matrixP=> i j; rewrite !mxE Monoid.mulm0. Qed. | Lemma | map2_mx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"idm",
"map2_mx",
"matrixP",
"mul_law",
"mulm0",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mxDl {mul : T -> T -> T} {add : Monoid.add_law idm mul} :
left_distributive (@map2_mx _ _ _ mul m n) (@map2_mx _ _ _ add m n). | Proof. by move=> A B C; apply/matrixP=> i j; rewrite !mxE Monoid.mulmDl. Qed. | Lemma | map2_mxDl | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"add",
"add_law",
"apply",
"idm",
"map2_mx",
"matrixP",
"mul",
"mulmDl",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map2_mxDr {mul : T -> T -> T} {add : Monoid.add_law idm mul} :
right_distributive (@map2_mx _ _ _ mul m n) (@map2_mx _ _ _ add m n). | Proof. by move=> A B C; apply/matrixP=> i j; rewrite !mxE Monoid.mulmDr. Qed. | Lemma | map2_mxDr | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"add",
"add_law",
"apply",
"idm",
"map2_mx",
"matrixP",
"mul",
"mulmDr",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addmx_key : unit. | Proof. by []. Qed. | Fact | addmx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addmx | := @map2_mx V V V +%R m n. | Definition | addmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"map2_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addmxA : associative addmx | := map2_mxA. | Definition | addmxA | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addmx",
"map2_mxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addmxC : commutative addmx | := map2_mxC. | Definition | addmxC | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addmx",
"map2_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add0mx : left_id (const_mx 0) addmx | := map2_1mx. | Definition | add0mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addmx",
"const_mx",
"map2_1mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxnE A d i j : (A *+ d) i j = A i j *+ d. | Proof. by elim: d => [|d IHd]; rewrite ?mulrS mxE ?IHd. Qed. | Lemma | mulmxnE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mulrS",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
summxE I r (P : pred I) (E : I -> 'M_(m, n)) i j :
(\sum_(k <- r | P k) E k) i j = \sum_(k <- r | P k) E k i j. | Proof. by apply: (big_morph (fun A => A i j)) => [A B|]; rewrite mxE. Qed. | Lemma | summxE | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"big_morph",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_mx_is_nmod_morphism : nmod_morphism const_mx. | Proof. by split=> [|a b]; apply/matrixP => // i j; rewrite !mxE. Qed. | Fact | const_mx_is_nmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"matrixP",
"mxE",
"nmod_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_mx_is_semi_additive | := const_mx_is_nmod_morphism. | Definition | const_mx_is_semi_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"const_mx_is_nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
swizzle_mx k (A : 'M[V]_(m, n)) | :=
\matrix[k]_(i, j) A (f i j) (g i j). | Definition | swizzle_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
swizzle_mx_is_nmod_morphism k : nmod_morphism (swizzle_mx k). | Proof. by split=> [|A B]; apply/matrixP => i j; rewrite !mxE. Qed. | Fact | swizzle_mx_is_nmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE",
"nmod_morphism",
"split",
"swizzle_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
swizzle_mx_is_semi_additive | := swizzle_mx_is_nmod_morphism. | Definition | swizzle_mx_is_semi_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"swizzle_mx_is_nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SwizzleAdd op | := (GRing.Additive.copy op (swizzle_mx _ _ _)). | Notation | SwizzleAdd | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"copy",
"swizzle_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
flatmx0 n : all_equal_to (0 : 'M_(0, n)). | Proof. by move=> A; apply/matrixP=> [] []. Qed. | Lemma | flatmx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
thinmx0 n : all_equal_to (0 : 'M_(n, 0)). | Proof. by move=> A; apply/matrixP=> i []. Qed. | Lemma | thinmx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx0 m n : (0 : 'M_(m, n))^T = 0. | Proof. exact: trmx_const. Qed. | Lemma | trmx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"trmx_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row0 m n i0 : row i0 (0 : 'M_(m, n)) = 0. | Proof. exact: row_const. Qed. | Lemma | row0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"i0",
"row",
"row_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col0 m n j0 : col j0 (0 : 'M_(m, n)) = 0. | Proof. exact: col_const. Qed. | Lemma | col0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col",
"col_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvec_eq0 m n (A : 'M_(m, n)) : (mxvec A == 0) = (A == 0). | Proof. by rewrite (can2_eq mxvecK vec_mxK) raddf0. Qed. | Lemma | mxvec_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"can2_eq",
"mxvec",
"mxvecK",
"raddf0",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vec_mx_eq0 m n (v : 'rV_(m * n)) : (vec_mx v == 0) = (v == 0). | Proof. by rewrite (can2_eq vec_mxK mxvecK) raddf0. Qed. | Lemma | vec_mx_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"can2_eq",
"mxvecK",
"raddf0",
"vec_mx",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mx0 m n1 n2 : row_mx 0 0 = 0 :> 'M_(m, n1 + n2). | Proof. exact: row_mx_const. Qed. | Lemma | row_mx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"row_mx",
"row_mx_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mx0 m1 m2 n : col_mx 0 0 = 0 :> 'M_(m1 + m2, n). | Proof. exact: col_mx_const. Qed. | Lemma | col_mx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mx",
"col_mx_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mx0 m1 m2 n1 n2 : block_mx 0 0 0 0 = 0 :> 'M_(m1 + m2, n1 + n2). | Proof. exact: block_mx_const. Qed. | Lemma | block_mx0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"block_mx_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_row_mx m n1 n2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) B1 B2 :
row_mx A1 A2 + row_mx B1 B2 = row_mx (A1 + B1) (A2 + B2). | Proof. by split_mxE. Qed. | Lemma | add_row_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"row_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_col_mx m1 m2 n (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) B1 B2 :
col_mx A1 A2 + col_mx B1 B2 = col_mx (A1 + B1) (A2 + B2). | Proof. by split_mxE. Qed. | Lemma | add_col_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mx",
"split_mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_block_mx m1 m2 n1 n2 (Aul : 'M_(m1, n1)) Aur Adl (Adr : 'M_(m2, n2))
Bul Bur Bdl Bdr :
let A := block_mx Aul Aur Adl Adr in let B := block_mx Bul Bur Bdl Bdr in
A + B = block_mx (Aul + Bul) (Aur + Bur) (Adl + Bdl) (Adr + Bdr). | Proof. by rewrite /= add_col_mx !add_row_mx. Qed. | Lemma | add_block_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"add_col_mx",
"add_row_mx",
"block_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_mx_eq0 (m n1 n2 : nat) (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)):
(row_mx A1 A2 == 0) = (A1 == 0) && (A2 == 0). | Proof.
apply/eqP/andP; last by case=> /eqP-> /eqP->; rewrite row_mx0.
by rewrite -row_mx0 => /eq_row_mx [-> ->].
Qed. | Lemma | row_mx_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_row_mx",
"last",
"nat",
"row_mx",
"row_mx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_mx_eq0 (m1 m2 n : nat) (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)):
(col_mx A1 A2 == 0) = (A1 == 0) && (A2 == 0). | Proof. by rewrite - !trmx0 tr_col_mx row_mx_eq0. Qed. | Lemma | col_mx_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"col_mx",
"inj_eq",
"nat",
"row_mx_eq0",
"tr_col_mx",
"trmx0",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
block_mx_eq0 m1 m2 n1 n2 (Aul : 'M_(m1, n1)) Aur Adl (Adr : 'M_(m2, n2)) :
(block_mx Aul Aur Adl Adr == 0) =
[&& Aul == 0, Aur == 0, Adl == 0 & Adr == 0]. | Proof. by rewrite col_mx_eq0 !row_mx_eq0 !andbA. Qed. | Lemma | block_mx_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"col_mx_eq0",
"row_mx_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_eq0 m n (A : 'M_(m, n)) : (A^T == 0) = (A == 0). | Proof. by rewrite -trmx0 (inj_eq trmx_inj). Qed. | Lemma | trmx_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"inj_eq",
"trmx0",
"trmx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix_eq0 m n (A : 'M_(m, n)) :
(A == 0) = [forall i, forall j, A i j == 0]. | Proof.
apply/eqP/'forall_'forall_eqP => [-> i j|A_eq0]; first by rewrite !mxE.
by apply/matrixP => i j; rewrite A_eq0 !mxE.
Qed. | Lemma | matrix_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix0Pn m n (A : 'M_(m, n)) : reflect (exists i j, A i j != 0) (A != 0). | Proof.
by rewrite matrix_eq0; apply/(iffP forallPn) => -[i /forallPn]; exists i.
Qed. | Lemma | matrix0Pn | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"forallPn",
"matrix_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rV0Pn n (v : 'rV_n) : reflect (exists i, v 0 i != 0) (v != 0). | Proof.
apply: (iffP (matrix0Pn _)) => [[i [j]]|[j]]; last by exists 0, j.
by rewrite ord1; exists j.
Qed. | Lemma | rV0Pn | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"last",
"matrix0Pn",
"ord1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cV0Pn n (v : 'cV_n) : reflect (exists i, v i 0 != 0) (v != 0). | Proof.
apply: (iffP (matrix0Pn _)) => [[i] [j]|[i]]; last by exists i, 0.
by rewrite ord1; exists i.
Qed. | Lemma | cV0Pn | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"last",
"matrix0Pn",
"ord1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz_row m n (A : 'M_(m, n)) | :=
oapp (fun i => row i A) 0 [pick i | row i A != 0]. | Definition | nz_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"pick",
"row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz_row_eq0 m n (A : 'M_(m, n)) : (nz_row A == 0) = (A == 0). | Proof.
rewrite /nz_row; symmetry; case: pickP => [i /= nzAi | Ai0].
by rewrite (negPf nzAi); apply: contraTF nzAi => /eqP->; rewrite row0 eqxx.
by rewrite eqxx; apply/eqP/row_matrixP=> i; move/eqP: (Ai0 i) ->; rewrite row0.
Qed. | Lemma | nz_row_eq0 | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eqxx",
"nz_row",
"pickP",
"row0",
"row_matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_mx m n (A : 'M[V]_(m, n)) | :=
[forall i : 'I__, forall j : 'I__, (i != j :> nat) ==> (A i j == 0)]. | Definition | is_diag_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_mxP m n (A : 'M[V]_(m, n)) :
reflect (forall i j : 'I__, i != j :> nat -> A i j = 0) (is_diag_mx A). | Proof. by apply: (iffP 'forall_'forall_implyP) => /(_ _ _ _)/eqP. Qed. | Lemma | is_diag_mxP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_diag_mx",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx0_is_diag m n : is_diag_mx (0 : 'M[V]_(m, n)). | Proof. by apply/is_diag_mxP => i j _; rewrite mxE. Qed. | Lemma | mx0_is_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_diag_mx",
"is_diag_mxP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx11_is_diag (M : 'M_1) : is_diag_mx M. | Proof. by apply/is_diag_mxP => i j; rewrite !ord1 eqxx. Qed. | Lemma | mx11_is_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eqxx",
"is_diag_mx",
"is_diag_mxP",
"ord1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_trig_mx m n (A : 'M[V]_(m, n)) | :=
[forall i : 'I__, forall j : 'I__, (i < j)%N ==> (A i j == 0)]. | Definition | is_trig_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_trig_mxP m n (A : 'M[V]_(m, n)) :
reflect (forall i j : 'I__, (i < j)%N -> A i j = 0) (is_trig_mx A). | Proof. by apply: (iffP 'forall_'forall_implyP) => /(_ _ _ _)/eqP. Qed. | Lemma | is_trig_mxP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_trig_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_mx_is_trig m n (A : 'M[V]_(m, n)) : is_diag_mx A -> is_trig_mx A. | Proof.
by move=> /is_diag_mxP A_eq0; apply/is_trig_mxP=> i j lt_ij;
rewrite A_eq0// ltn_eqF.
Qed. | Lemma | is_diag_mx_is_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_diag_mx",
"is_diag_mxP",
"is_trig_mx",
"is_trig_mxP",
"ltn_eqF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx0_is_trig m n : is_trig_mx (0 : 'M[V]_(m, n)). | Proof. by apply/is_trig_mxP => i j _; rewrite mxE. Qed. | Lemma | mx0_is_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_trig_mx",
"is_trig_mxP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx11_is_trig (M : 'M_1) : is_trig_mx M. | Proof. by apply/is_trig_mxP => i j; rewrite !ord1 ltnn. Qed. | Lemma | mx11_is_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_trig_mx",
"is_trig_mxP",
"ltnn",
"ord1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_mxEtrig m n (A : 'M[V]_(m, n)) :
is_diag_mx A = is_trig_mx A && is_trig_mx A^T. | Proof.
apply/is_diag_mxP/andP => [Adiag|[/is_trig_mxP Atrig /is_trig_mxP ATtrig]].
by split; apply/is_trig_mxP => i j lt_ij; rewrite ?mxE ?Adiag//;
[rewrite ltn_eqF|rewrite gtn_eqF].
by move=> i j; case: ltngtP => // [/Atrig|/ATtrig]; rewrite ?mxE.
Qed. | Lemma | is_diag_mxEtrig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"gtn_eqF",
"is_diag_mx",
"is_diag_mxP",
"is_trig_mx",
"is_trig_mxP",
"ltn_eqF",
"ltngtP",
"mxE",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_trmx m n (A : 'M[V]_(m, n)) : is_diag_mx A^T = is_diag_mx A. | Proof. by rewrite !is_diag_mxEtrig trmxK andbC. Qed. | Lemma | is_diag_trmx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"is_diag_mx",
"is_diag_mxEtrig",
"trmxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ursubmx_trig m1 m2 n1 n2 (A : 'M[V]_(m1 + m2, n1 + n2)) :
m1 <= n1 -> is_trig_mx A -> ursubmx A = 0. | Proof.
move=> leq_m1_n1 /is_trig_mxP Atrig; apply/matrixP => i j.
by rewrite !mxE Atrig//= ltn_addr// (@leq_trans m1).
Qed. | Lemma | ursubmx_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_trig_mx",
"is_trig_mxP",
"leq_trans",
"ltn_addr",
"matrixP",
"mxE",
"ursubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dlsubmx_diag m1 m2 n1 n2 (A : 'M[V]_(m1 + m2, n1 + n2)) :
n1 <= m1 -> is_diag_mx A -> dlsubmx A = 0. | Proof.
move=> leq_m2_n2 /is_diag_mxP Adiag; apply/matrixP => i j.
by rewrite !mxE Adiag// gtn_eqF//= ltn_addr// (@leq_trans n1).
Qed. | Lemma | dlsubmx_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"dlsubmx",
"gtn_eqF",
"is_diag_mx",
"is_diag_mxP",
"leq_trans",
"ltn_addr",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ulsubmx_trig m1 m2 n1 n2 (A : 'M[V]_(m1 + m2, n1 + n2)) :
is_trig_mx A -> is_trig_mx (ulsubmx A). | Proof.
move=> /is_trig_mxP Atrig; apply/is_trig_mxP => i j lt_ij.
by rewrite !mxE Atrig.
Qed. | Lemma | ulsubmx_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_trig_mx",
"is_trig_mxP",
"mxE",
"ulsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drsubmx_trig m1 m2 n1 n2 (A : 'M[V]_(m1 + m2, n1 + n2)) :
m1 <= n1 -> is_trig_mx A -> is_trig_mx (drsubmx A). | Proof.
move=> leq_m1_n1 /is_trig_mxP Atrig; apply/is_trig_mxP => i j lt_ij.
by rewrite !mxE Atrig//= -addnS leq_add.
Qed. | Lemma | drsubmx_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addnS",
"apply",
"drsubmx",
"is_trig_mx",
"is_trig_mxP",
"leq_add",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ulsubmx_diag m1 m2 n1 n2 (A : 'M[V]_(m1 + m2, n1 + n2)) :
is_diag_mx A -> is_diag_mx (ulsubmx A). | Proof.
rewrite !is_diag_mxEtrig trmx_ulsub.
by move=> /andP[/ulsubmx_trig-> /ulsubmx_trig->].
Qed. | Lemma | ulsubmx_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"is_diag_mx",
"is_diag_mxEtrig",
"trmx_ulsub",
"ulsubmx",
"ulsubmx_trig"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drsubmx_diag m1 m2 n1 n2 (A : 'M[V]_(m1 + m2, n1 + n2)) :
m1 = n1 -> is_diag_mx A -> is_diag_mx (drsubmx A). | Proof.
move=> eq_m1_n1 /is_diag_mxP Adiag; apply/is_diag_mxP => i j neq_ij.
by rewrite !mxE Adiag//= eq_m1_n1 eqn_add2l.
Qed. | Lemma | drsubmx_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"drsubmx",
"eqn_add2l",
"is_diag_mx",
"is_diag_mxP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_trig_block_mx m1 m2 n1 n2 ul ur dl dr : m1 = n1 ->
@is_trig_mx (m1 + m2) (n1 + n2) (block_mx ul ur dl dr) =
[&& ur == 0, is_trig_mx ul & is_trig_mx dr]. | Proof.
move=> eq_m1_n1; rewrite {}eq_m1_n1 in ul ur dl dr *.
apply/is_trig_mxP/and3P => [Atrig|]; last first.
move=> [/eqP-> /is_trig_mxP ul_trig /is_trig_mxP dr_trig] i j; rewrite !mxE.
do 2![case: split_ordP => ? ->; rewrite ?mxE//=] => lt_ij; rewrite ?ul_trig//.
move: lt_ij; rewrite ltnNge -ltnS.
by rewr... | Lemma | is_trig_block_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"addnS",
"apply",
"block_mx",
"eq_shift",
"is_trig_mx",
"is_trig_mxP",
"last",
"leq_addr",
"leq_trans",
"lshift",
"ltnNge",
"ltnS",
"ltn_add2l",
"ltn_ord",
"matrixP",
"mxE",
"rshift",
"split",
"split_ordP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trigmx_ind (P : forall m n, 'M_(m, n) -> Type) :
(forall m, P m 0 0) ->
(forall n, P 0 n 0) ->
(forall m n x c A, is_trig_mx A ->
P m n A -> P (1 + m)%N (1 + n)%N (block_mx x 0 c A)) ->
forall m n A, is_trig_mx A -> P m n A. | Proof.
move=> P0l P0r PS m n A; elim: A => {m n} [m|n|m n xx r c] A PA;
do ?by rewrite (flatmx0, thinmx0); by [apply: P0l|apply: P0r].
by rewrite is_trig_block_mx => // /and3P[/eqP-> _ Atrig]; apply: PS (PA _).
Qed. | Lemma | trigmx_ind | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"block_mx",
"flatmx0",
"is_trig_block_mx",
"is_trig_mx",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trigsqmx_ind (P : forall n, 'M[V]_n -> Type) : (P 0 0) ->
(forall n x c A, is_trig_mx A -> P n A -> P (1 + n)%N (block_mx x 0 c A)) ->
forall n A, is_trig_mx A -> P n A. | Proof.
move=> P0 PS n A; elim/sqmx_ind: A => {n} [|n x r c] A PA.
by rewrite thinmx0; apply: P0.
by rewrite is_trig_block_mx => // /and3P[/eqP-> _ Atrig]; apply: PS (PA _).
Qed. | Lemma | trigsqmx_ind | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"P0",
"apply",
"block_mx",
"is_trig_block_mx",
"is_trig_mx",
"sqmx_ind",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_diag_block_mx m1 m2 n1 n2 ul ur dl dr : m1 = n1 ->
@is_diag_mx (m1 + m2) (n1 + n2) (block_mx ul ur dl dr) =
[&& ur == 0, dl == 0, is_diag_mx ul & is_diag_mx dr]. | Proof.
move=> eq_m1_n1.
rewrite !is_diag_mxEtrig tr_block_mx !is_trig_block_mx// trmx_eq0.
by rewrite andbACA -!andbA; congr [&& _, _, _ & _]; rewrite andbCA.
Qed. | Lemma | is_diag_block_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"is_diag_mx",
"is_diag_mxEtrig",
"is_trig_block_mx",
"tr_block_mx",
"trmx_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagmx_ind (P : forall m n, 'M_(m, n) -> Type) :
(forall m, P m 0 0) ->
(forall n, P 0 n 0) ->
(forall m n x c A, is_diag_mx A ->
P m n A -> P (1 + m)%N (1 + n)%N (block_mx x 0 c A)) ->
forall m n A, is_diag_mx A -> P m n A. | Proof.
move=> P0l P0r PS m n A Adiag; have Atrig := is_diag_mx_is_trig Adiag.
elim/trigmx_ind: Atrig Adiag => // {}m {}n r c {}A _ PA.
rewrite is_diag_block_mx => // /and4P[_ /eqP-> _ Adiag].
exact: PS (PA _).
Qed. | Lemma | diagmx_ind | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"block_mx",
"is_diag_block_mx",
"is_diag_mx",
"is_diag_mx_is_trig",
"trigmx_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagsqmx_ind (P : forall n, 'M[V]_n -> Type) :
(P 0 0) ->
(forall n x c A, is_diag_mx A -> P n A -> P (1 + n)%N (block_mx x 0 c A)) ->
forall n A, is_diag_mx A -> P n A. | Proof.
move=> P0 PS n A; elim/sqmx_ind: A => [|{}n x r c] A PA.
by rewrite thinmx0; apply: P0.
rewrite is_diag_block_mx => // /and4P[/eqP-> /eqP-> _ Adiag].
exact: PS (PA _).
Qed. | Lemma | diagsqmx_ind | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"P0",
"apply",
"block_mx",
"is_diag_block_mx",
"is_diag_mx",
"sqmx_ind",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_key : unit. | Proof. by []. Qed. | Fact | diag_mx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | Diagonal matrices | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
diag_mx n (d : 'rV[V]_n) | :=
\matrix[diag_mx_key]_(i, j) (d 0 i *+ (i == j)). | Definition | diag_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"diag_mx_key",
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_diag_mx n (d : 'rV_n) : (diag_mx d)^T = diag_mx d. | Proof. by apply/matrixP=> i j /[!mxE]; case: eqVneq => // ->. Qed. | Lemma | tr_diag_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"diag_mx",
"eqVneq",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_is_nmod_morphism n : nmod_morphism (@diag_mx n). | Proof.
by split=> [|A B]; apply/matrixP => i j; rewrite !mxE ?mul0rn// mulrnDl.
Qed. | Fact | diag_mx_is_nmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"diag_mx",
"matrixP",
"mul0rn",
"mulrnDl",
"mxE",
"nmod_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_is_semi_additive | := diag_mx_is_nmod_morphism. | Definition | diag_mx_is_semi_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"diag_mx_is_nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_row m n (l : 'rV_n) (r : 'rV_m) :
diag_mx (row_mx l r) = block_mx (diag_mx l) 0 0 (diag_mx r). | Proof.
apply/matrixP => i j.
by do ?[rewrite !mxE; case: split_ordP => ? ->]; rewrite mxE eq_shift.
Qed. | Lemma | diag_mx_row | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"block_mx",
"diag_mx",
"eq_shift",
"matrixP",
"mxE",
"row_mx",
"split_ordP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mxP n (A : 'M[V]_n) :
reflect (exists d : 'rV_n, A = diag_mx d) (is_diag_mx A). | Proof.
apply: (iffP (is_diag_mxP A)) => [Adiag|[d ->] i j neq_ij]; last first.
by rewrite !mxE -val_eqE (negPf neq_ij).
exists (\row_i A i i); apply/matrixP => i j; rewrite !mxE.
by case: (altP (i =P j)) => [->|/Adiag->].
Qed. | Lemma | diag_mxP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"diag_mx",
"is_diag_mx",
"is_diag_mxP",
"last",
"matrixP",
"mxE",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_is_diag n (r : 'rV[V]_n) : is_diag_mx (diag_mx r). | Proof. by apply/diag_mxP; exists r. Qed. | Lemma | diag_mx_is_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"diag_mx",
"diag_mxP",
"is_diag_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_mx_is_trig n (r : 'rV[V]_n) : is_trig_mx (diag_mx r). | Proof. exact/is_diag_mx_is_trig/diag_mx_is_diag. Qed. | Lemma | diag_mx_is_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"diag_mx",
"diag_mx_is_diag",
"is_diag_mx_is_trig",
"is_trig_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_key : unit. | Proof. by []. Qed. | Fact | scalar_mx_key | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx x : 'M[V]_n | :=
\matrix[scalar_mx_key]_(i , j) (x *+ (i == j)). | Definition | scalar_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix",
"scalar_mx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x %:M" | := (scalar_mx x) : ring_scope. | Notation | x %:M | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diag_const_mx a : diag_mx (const_mx a) = a%:M :> 'M_n. | Proof. by apply/matrixP=> i j; rewrite !mxE. Qed. | Lemma | diag_const_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"const_mx",
"diag_mx",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tr_scalar_mx a : (a%:M)^T = a%:M. | Proof. by apply/matrixP=> i j; rewrite !mxE eq_sym. Qed. | Lemma | tr_scalar_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eq_sym",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_nmod_morphism : nmod_morphism scalar_mx. | Proof. by split=> [|a b]; rewrite -!diag_const_mx ?raddf0// !raddfD. Qed. | Fact | scalar_mx_is_nmod_morphism | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"diag_const_mx",
"nmod_morphism",
"raddf0",
"raddfD",
"scalar_mx",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_semi_additive | := scalar_mx_is_nmod_morphism. | Definition | scalar_mx_is_semi_additive | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"scalar_mx_is_nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_scalar_mx (A : 'M[V]_n) | :=
if insub 0 is Some i then A == (A i i)%:M else true. | Definition | is_scalar_mx | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"insub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_scalar_mxP A : reflect (exists a, A = a%:M) (is_scalar_mx A). | Proof.
rewrite /is_scalar_mx; case: insubP => [i _ _ | ].
by apply: (iffP eqP) => [|[a ->]]; [exists (A i i) | rewrite mxE eqxx].
rewrite -eqn0Ngt => /eqP n0; left; exists 0.
by rewrite raddf0; rewrite n0 in A *; rewrite [A]flatmx0.
Qed. | Lemma | is_scalar_mxP | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"eqn0Ngt",
"eqxx",
"flatmx0",
"insubP",
"is_scalar_mx",
"mxE",
"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_scalar a : is_scalar_mx a%:M. | Proof. by apply/is_scalar_mxP; exists a. Qed. | Lemma | scalar_mx_is_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_scalar_mx",
"is_scalar_mxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx0_is_scalar : is_scalar_mx 0. | Proof. by apply/is_scalar_mxP; exists 0; rewrite raddf0. Qed. | Lemma | mx0_is_scalar | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_scalar_mx",
"is_scalar_mxP",
"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_diag a : is_diag_mx a%:M. | Proof. by rewrite -diag_const_mx diag_mx_is_diag. Qed. | Lemma | scalar_mx_is_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"diag_const_mx",
"diag_mx_is_diag",
"is_diag_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_scalar_mx_is_diag A : is_scalar_mx A -> is_diag_mx A. | Proof. by move=> /is_scalar_mxP[a ->]; apply: scalar_mx_is_diag. Qed. | Lemma | is_scalar_mx_is_diag | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"apply",
"is_diag_mx",
"is_scalar_mx",
"is_scalar_mxP",
"scalar_mx_is_diag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_is_trig a : is_trig_mx a%:M. | Proof. by rewrite is_diag_mx_is_trig// scalar_mx_is_diag. Qed. | Lemma | scalar_mx_is_trig | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"is_diag_mx_is_trig",
"is_trig_mx",
"scalar_mx_is_diag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.