fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
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.
HB.instance Definition _ {opm : Monoid.com_law idm} :=
SemiGroup.isCommutativeLaw.Build 'M_(m, n) (@map2_mx _ _ _ opm _ _) map2_mxC.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map2_mxC
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map2_0mx
| |
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.
HB.instance Definition _ {opm : Monoid.mul_law idm} :=
Monoid.isMulLaw.Build 'M_(m, n) (const_mx idm) (@map2_mx _ _ _ opm _ _)
map2_0mx map2_mx0.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map2_mx0
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map2_mxDl
| |
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.
HB.instance Definition _ {mul : T -> T -> T} {add : Monoid.add_law idm mul} :=
Monoid.isAddLaw.Build 'M_(m, n)
(@map2_mx _ _ _ mul _ _) (@map2_mx _ _ _ add _ _)
map2_mxDl map2_mxDr.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map2_mxDr
| |
addmx:= @map2_mx V V V +%R m n.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
addmx
| |
addmxA: associative addmx := map2_mxA.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
addmxA
| |
addmxC: commutative addmx := map2_mxC.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
addmxC
| |
add0mx: left_id (const_mx 0) addmx := map2_1mx.
HB.instance Definition _ := GRing.isNmodule.Build 'M[V]_(m, n)
addmxA addmxC add0mx.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
add0mx
| |
mulmxnEA d i j : (A *+ d) i j = A i j *+ d.
Proof. by elim: d => [|d IHd]; rewrite ?mulrS mxE ?IHd. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mulmxnE
| |
summxEI 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.
Fact const_mx_is_nmod_morphism : nmod_morphism const_mx.
Proof. by split=> [|a b]; apply/matrixP => // i j; rewrite !mxE. Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `const_mx_is_nmod_morphism` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
summxE
| |
const_mx_is_semi_additive:= const_mx_is_nmod_morphism.
HB.instance Definition _ := GRing.isNmodMorphism.Build V 'M[V]_(m, n) const_mx
const_mx_is_nmod_morphism.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
const_mx_is_semi_additive
| |
swizzle_mxk (A : 'M[V]_(m, n)) :=
\matrix[k]_(i, j) A (f i j) (g i j).
Fact swizzle_mx_is_nmod_morphism k : nmod_morphism (swizzle_mx k).
Proof. by split=> [|A B]; apply/matrixP => i j; rewrite !mxE. Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `swizzle_mx_is_nmod_morphism` instead")]
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
swizzle_mx
| |
swizzle_mx_is_semi_additive:= swizzle_mx_is_nmod_morphism.
HB.instance Definition _ k := GRing.isNmodMorphism.Build 'M_(m, n) 'M_(p, q)
(swizzle_mx k) (swizzle_mx_is_nmod_morphism k).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
swizzle_mx_is_semi_additive
| |
Definition_ m n := SwizzleAdd (@trmx V m n).
HB.instance Definition _ m n i := SwizzleAdd (@row V m n i).
HB.instance Definition _ m n j := SwizzleAdd (@col V m n j).
HB.instance Definition _ m n i := SwizzleAdd (@row' V m n i).
HB.instance Definition _ m n j := SwizzleAdd (@col' V m n j).
HB.instance Definition _ m n m' n' f g := SwizzleAdd (@mxsub V m n m' n' f g).
HB.instance Definition _ m n s := SwizzleAdd (@row_perm V m n s).
HB.instance Definition _ m n s := SwizzleAdd (@col_perm V m n s).
HB.instance Definition _ m n i1 i2 := SwizzleAdd (@xrow V m n i1 i2).
HB.instance Definition _ m n j1 j2 := SwizzleAdd (@xcol V m n j1 j2).
HB.instance Definition _ m n1 n2 := SwizzleAdd (@lsubmx V m n1 n2).
HB.instance Definition _ m n1 n2 := SwizzleAdd (@rsubmx V m n1 n2).
HB.instance Definition _ m1 m2 n := SwizzleAdd (@usubmx V m1 m2 n).
HB.instance Definition _ m1 m2 n := SwizzleAdd (@dsubmx V m1 m2 n).
HB.instance Definition _ m n := SwizzleAdd (@vec_mx V m n).
HB.instance Definition _ m n := GRing.isNmodMorphism.Build 'M_(m, n) 'rV_(m * n)
mxvec (can2_nmod_morphism (@vec_mxK V m n) mxvecK).
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
Definition
| |
flatmx0n : all_equal_to (0 : 'M_(0, n)).
Proof. by move=> A; apply/matrixP=> [] []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
flatmx0
| |
thinmx0n : all_equal_to (0 : 'M_(n, 0)).
Proof. by move=> A; apply/matrixP=> i []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
thinmx0
| |
trmx0m n : (0 : 'M_(m, n))^T = 0.
Proof. exact: trmx_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
trmx0
| |
row0m n i0 : row i0 (0 : 'M_(m, n)) = 0.
Proof. exact: row_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
row0
| |
col0m n j0 : col j0 (0 : 'M_(m, n)) = 0.
Proof. exact: col_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
col0
| |
mxvec_eq0m n (A : 'M_(m, n)) : (mxvec A == 0) = (A == 0).
Proof. by rewrite (can2_eq mxvecK vec_mxK) raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxvec_eq0
| |
vec_mx_eq0m n (v : 'rV_(m * n)) : (vec_mx v == 0) = (v == 0).
Proof. by rewrite (can2_eq vec_mxK mxvecK) raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
vec_mx_eq0
| |
row_mx0m n1 n2 : row_mx 0 0 = 0 :> 'M_(m, n1 + n2).
Proof. exact: row_mx_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
row_mx0
| |
col_mx0m1 m2 n : col_mx 0 0 = 0 :> 'M_(m1 + m2, n).
Proof. exact: col_mx_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
col_mx0
| |
block_mx0m1 m2 n1 n2 : block_mx 0 0 0 0 = 0 :> 'M_(m1 + m2, n1 + n2).
Proof. exact: block_mx_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
block_mx0
| |
split_mxE:= apply/matrixP=> i j; do ![rewrite mxE | case: split => ?].
|
Ltac
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
split_mxE
| |
add_row_mxm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
add_row_mx
| |
add_col_mxm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
add_col_mx
| |
add_block_mxm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
add_block_mx
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
row_mx_eq0
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
col_mx_eq0
| |
block_mx_eq0m1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
block_mx_eq0
| |
trmx_eq0m n (A : 'M_(m, n)) : (A^T == 0) = (A == 0).
Proof. by rewrite -trmx0 (inj_eq trmx_inj). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
trmx_eq0
| |
matrix_eq0m 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
matrix_eq0
| |
matrix0Pnm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
matrix0Pn
| |
rV0Pnn (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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
rV0Pn
| |
cV0Pnn (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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
cV0Pn
| |
nz_rowm n (A : 'M_(m, n)) :=
oapp (fun i => row i A) 0 [pick i | row i A != 0].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
nz_row
| |
nz_row_eq0m 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
nz_row_eq0
| |
is_diag_mxm n (A : 'M[V]_(m, n)) :=
[forall i : 'I__, forall j : 'I__, (i != j :> nat) ==> (A i j == 0)].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_diag_mx
| |
is_diag_mxPm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_diag_mxP
| |
mx0_is_diagm n : is_diag_mx (0 : 'M[V]_(m, n)).
Proof. by apply/is_diag_mxP => i j _; rewrite mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mx0_is_diag
| |
mx11_is_diag(M : 'M_1) : is_diag_mx M.
Proof. by apply/is_diag_mxP => i j; rewrite !ord1 eqxx. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mx11_is_diag
| |
is_trig_mxm n (A : 'M[V]_(m, n)) :=
[forall i : 'I__, forall j : 'I__, (i < j)%N ==> (A i j == 0)].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_trig_mx
| |
is_trig_mxPm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_trig_mxP
| |
is_diag_mx_is_trigm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_diag_mx_is_trig
| |
mx0_is_trigm n : is_trig_mx (0 : 'M[V]_(m, n)).
Proof. by apply/is_trig_mxP => i j _; rewrite mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mx0_is_trig
| |
mx11_is_trig(M : 'M_1) : is_trig_mx M.
Proof. by apply/is_trig_mxP => i j; rewrite !ord1 ltnn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mx11_is_trig
| |
is_diag_mxEtrigm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_diag_mxEtrig
| |
is_diag_trmxm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_diag_trmx
| |
ursubmx_trigm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
ursubmx_trig
| |
dlsubmx_diagm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
dlsubmx_diag
| |
ulsubmx_trigm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
ulsubmx_trig
| |
drsubmx_trigm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
drsubmx_trig
| |
ulsubmx_diagm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
ulsubmx_diag
| |
drsubmx_diagm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
drsubmx_diag
| |
is_trig_block_mxm1 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 rewrite (leq_trans (ltn_ord _))// -addnS leq_addr.
by rewrite dr_trig//; move: lt_ij; rewrite ltn_add2l.
split.
- apply/eqP/matrixP => i j; have := Atrig (lshift _ i) (rshift _ j).
rewrite !mxE; case: split_ordP => k /eqP; rewrite eq_shift// ?mxE.
case: split_ordP => l /eqP; rewrite eq_shift// ?mxE => /eqP-> /eqP<- <- //.
by rewrite /= (leq_trans (ltn_ord _)) ?leq_addr.
- apply/is_trig_mxP => i j lt_ij; have := Atrig (lshift _ i) (lshift _ j).
rewrite !mxE; case: split_ordP => k /eqP; rewrite eq_shift// ?mxE.
by case: split_ordP => l /eqP; rewrite eq_shift// ?mxE => /eqP<- /eqP<- ->.
- apply/is_trig_mxP => i j lt_ij; have := Atrig (rshift _ i) (rshift _ j).
rewrite !mxE; case: split_ordP => k /eqP; rewrite eq_shift// ?mxE.
case: split_ordP => l /eqP; rewrite eq_shift// ?mxE => /eqP<- /eqP<- -> //.
by rewrite /= ltn_add2l.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_trig_block_mx
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
trigmx_ind
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
trigsqmx_ind
| |
is_diag_block_mxm1 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_diag_block_mx
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diagmx_ind
| |
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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diagsqmx_ind
| |
diag_mxn (d : 'rV[V]_n) :=
\matrix[diag_mx_key]_(i, j) (d 0 i *+ (i == j)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diag_mx
| |
tr_diag_mxn (d : 'rV_n) : (diag_mx d)^T = diag_mx d.
Proof. by apply/matrixP=> i j /[!mxE]; case: eqVneq => // ->. Qed.
Fact 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.
#[deprecated(since="mathcomp 2.5.0",
note="use `diag_mx_is_nmod_morphism` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
tr_diag_mx
| |
diag_mx_is_semi_additive:= diag_mx_is_nmod_morphism.
HB.instance Definition _ n := GRing.isNmodMorphism.Build 'rV_n 'M_n (@diag_mx n)
(@diag_mx_is_nmod_morphism n).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diag_mx_is_semi_additive
| |
diag_mx_rowm 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diag_mx_row
| |
diag_mxPn (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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diag_mxP
| |
diag_mx_is_diagn (r : 'rV[V]_n) : is_diag_mx (diag_mx r).
Proof. by apply/diag_mxP; exists r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diag_mx_is_diag
| |
diag_mx_is_trign (r : 'rV[V]_n) : is_trig_mx (diag_mx r).
Proof. exact/is_diag_mx_is_trig/diag_mx_is_diag. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diag_mx_is_trig
| |
scalar_mxx : 'M[V]_n :=
\matrix[scalar_mx_key]_(i , j) (x *+ (i == j)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
scalar_mx
| |
diag_const_mxa : diag_mx (const_mx a) = a%:M :> 'M_n.
Proof. by apply/matrixP=> i j; rewrite !mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
diag_const_mx
| |
tr_scalar_mxa : (a%:M)^T = a%:M.
Proof. by apply/matrixP=> i j; rewrite !mxE eq_sym. Qed.
Fact scalar_mx_is_nmod_morphism : nmod_morphism scalar_mx.
Proof. by split=> [|a b]; rewrite -!diag_const_mx ?raddf0// !raddfD. Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `scalar_mx_is_nmod_morphism` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
tr_scalar_mx
| |
scalar_mx_is_semi_additive:= scalar_mx_is_nmod_morphism.
HB.instance Definition _ := GRing.isNmodMorphism.Build V 'M_n scalar_mx
scalar_mx_is_nmod_morphism.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
scalar_mx_is_semi_additive
| |
is_scalar_mx(A : 'M[V]_n) :=
if insub 0 is Some i then A == (A i i)%:M else true.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_scalar_mx
| |
is_scalar_mxPA : 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
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_scalar_mxP
| |
scalar_mx_is_scalara : is_scalar_mx a%:M.
Proof. by apply/is_scalar_mxP; exists a. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
scalar_mx_is_scalar
| |
mx0_is_scalar: is_scalar_mx 0.
Proof. by apply/is_scalar_mxP; exists 0; rewrite raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mx0_is_scalar
| |
scalar_mx_is_diaga : is_diag_mx a%:M.
Proof. by rewrite -diag_const_mx diag_mx_is_diag. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
scalar_mx_is_diag
| |
is_scalar_mx_is_diagA : is_scalar_mx A -> is_diag_mx A.
Proof. by move=> /is_scalar_mxP[a ->]; apply: scalar_mx_is_diag. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_scalar_mx_is_diag
| |
scalar_mx_is_triga : is_trig_mx a%:M.
Proof. by rewrite is_diag_mx_is_trig// scalar_mx_is_diag. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
scalar_mx_is_trig
| |
is_scalar_mx_is_trigA : is_scalar_mx A -> is_trig_mx A.
Proof. by move=> /is_scalar_mx_is_diag /is_diag_mx_is_trig. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
is_scalar_mx_is_trig
| |
mx11_scalar(A : 'M_1) : A = (A 0 0)%:M.
Proof. by apply/rowP=> j; rewrite ord1 mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mx11_scalar
| |
scalar_mx_blockn1 n2 a : a%:M = block_mx a%:M 0 0 a%:M :> 'M_(n1 + n2).
Proof.
apply/matrixP=> i j; rewrite !mxE.
by do 2![case: split_ordP => ? ->; rewrite !mxE]; rewrite ?eq_shift.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
scalar_mx_block
| |
mxtrace(A : 'M[V]_n) := \sum_i A i i.
Local Notation "'\tr' A" := (mxtrace A) : ring_scope.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtrace
| |
mxtrace_trA : \tr A^T = \tr A.
Proof. by apply: eq_bigr=> i _; rewrite mxE. Qed.
Fact mxtrace_is_nmod_morphism : nmod_morphism mxtrace.
Proof.
split=> [|A B]; first by apply: big1 => i; rewrite mxE.
by rewrite -big_split /=; apply: eq_bigr => i _; rewrite mxE.
Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `mxtrace_is_nmod_morphism` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtrace_tr
| |
mxtrace_is_semi_additive:= mxtrace_is_nmod_morphism.
HB.instance Definition _ := GRing.isNmodMorphism.Build 'M_n V mxtrace
mxtrace_is_nmod_morphism.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtrace_is_semi_additive
| |
mxtrace0: \tr 0 = 0. Proof. exact: raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtrace0
| |
mxtraceDA B : \tr (A + B) = \tr A + \tr B. Proof. exact: raddfD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtraceD
| |
mxtrace_diagD : \tr (diag_mx D) = \sum_j D 0 j.
Proof. by apply: eq_bigr => j _; rewrite mxE eqxx. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtrace_diag
| |
mxtrace_scalara : \tr a%:M = a *+ n.
Proof.
rewrite -diag_const_mx mxtrace_diag; under eq_bigr do rewrite mxE.
by rewrite sumr_const card_ord.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtrace_scalar
| |
trace_mx11(A : 'M_1) : \tr A = A 0 0.
Proof. by rewrite [A in LHS]mx11_scalar mxtrace_scalar. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
trace_mx11
| |
mxtrace_blockn1 n2 (Aul : 'M_n1) Aur Adl (Adr : 'M_n2) :
\tr (block_mx Aul Aur Adl Adr) = \tr Aul + \tr Adr.
Proof.
rewrite /(\tr _) big_split_ord /=.
by congr (_ + _); under eq_bigr do rewrite (block_mxEul, block_mxEdr).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
mxtrace_block
| |
map_mx0: 0^f = 0 :> 'M_(m, n).
Proof. by rewrite map_const_mx raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map_mx0
| |
map_mxDA B : (A + B)^f = A^f + B^f.
Proof. by apply/matrixP=> i j; rewrite !mxE raddfD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map_mxD
| |
map_mx_sum:= big_morph _ map_mxD map_mx0.
HB.instance Definition _ :=
GRing.isNmodMorphism.Build 'M[aR]_(m, n) 'M[rR]_(m, n) (map_mx f)
(map_mx0, map_mxD).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
map_mx_sum
| |
oppmx:= @map_mx V V -%R m n.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
oppmx
| |
addNmx: left_inverse (const_mx 0) oppmx (@addmx V m n).
Proof. by move=> A; apply/matrixP=> i j; rewrite !mxE addNr. Qed.
HB.instance Definition _ := GRing.Nmodule_isZmodule.Build 'M[V]_(m, n) addNmx.
#[deprecated(since="mathcomp 2.5.0", note="use `raddfB` instead")]
Fact const_mx_is_zmod_morphism : zmod_morphism const_mx.
Proof. exact: raddfB. Qed.
#[deprecated(since="mathcomp 2.5.0", note="use `raddfB` instead"),
warning="-deprecated"]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
addNmx
| |
const_mx_is_additive:= const_mx_is_zmod_morphism.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
const_mx_is_additive
| |
swizzle_mx_is_additive:= swizzle_mx_is_zmod_morphism.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
swizzle_mx_is_additive
| |
split_mxE:= apply/matrixP=> i j; do ![rewrite mxE | case: split => ?].
|
Ltac
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
split_mxE
|
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