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 |
|---|---|---|---|---|---|---|
block_mxKdr: drsubmx A = Adr.
Proof. by rewrite /drsubmx col_mxKd row_mxKr. 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_mxKdr
| |
block_mxEv: A = col_mx (row_mx Aul Aur) (row_mx Adl Adr).
Proof. by []. 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_mxEv
| |
trmx_ulsub: (ulsubmx A)^T = ulsubmx A^T.
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
|
trmx_ulsub
| |
trmx_ursub: (ursubmx A)^T = dlsubmx A^T.
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
|
trmx_ursub
| |
trmx_dlsub: (dlsubmx A)^T = ursubmx A^T.
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
|
trmx_dlsub
| |
trmx_drsub: (drsubmx A)^T = drsubmx A^T.
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
|
trmx_drsub
| |
tr_block_mx:
(block_mx Aul Aur Adl Adr)^T = block_mx Aul^T Adl^T Aur^T Adr^T.
Proof.
rewrite -[_^T]submxK -trmx_ulsub -trmx_ursub -trmx_dlsub -trmx_drsub.
by rewrite block_mxKul block_mxKur block_mxKdl block_mxKdr.
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
|
tr_block_mx
| |
block_mxEh:
block_mx Aul Aur Adl Adr = row_mx (col_mx Aul Adl) (col_mx Aur Adr).
Proof. by apply: trmx_inj; rewrite tr_block_mx tr_row_mx 2!tr_col_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
|
block_mxEh
| |
block_mxAm1 m2 m3 n1 n2 n3
(A11 : 'M_(m1, n1)) (A12 : 'M_(m1, n2)) (A13 : 'M_(m1, n3))
(A21 : 'M_(m2, n1)) (A22 : 'M_(m2, n2)) (A23 : 'M_(m2, n3))
(A31 : 'M_(m3, n1)) (A32 : 'M_(m3, n2)) (A33 : 'M_(m3, n3)) :
let cast := (esym (addnA m1 m2 m3), esym (addnA n1 n2 n3)) in
let row1 := row_mx A12 A13 in let col1 := col_mx A21 A31 in
let row3 := row_mx A31 A32 in let col3 := col_mx A13 A23 in
block_mx A11 row1 col1 (block_mx A22 A23 A32 A33)
= castmx cast (block_mx (block_mx A11 A12 A21 A22) col3 row3 A33).
Proof.
rewrite /= block_mxEh !col_mxA -cast_row_mx -block_mxEv -block_mxEh.
rewrite block_mxEv block_mxEh !row_mxA -cast_col_mx -block_mxEh -block_mxEv.
by rewrite castmx_comp etrans_id.
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_mxA
| |
block_mxAx:= block_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
|
block_mxAx
| |
row_indm (P : forall n, 'M[R]_(m, n) -> Type) :
(forall A, P 0 A) ->
(forall n c A, P n A -> P (1 + n)%N (row_mx c A)) ->
forall n A, P n A.
Proof.
move=> P0 PS; elim=> [//|n IHn] A.
by rewrite -[n.+1]/(1 + n)%N in A *; rewrite -[A]hsubmxK; apply: PS.
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_ind
| |
col_indn (P : forall m, 'M[R]_(m, n) -> Type) :
(forall A, P 0 A) ->
(forall m r A, P m A -> P (1 + m)%N (col_mx r A)) ->
forall m A, P m A.
Proof.
move=> P0 PS; elim=> [//|m IHm] A.
by rewrite -[m.+1]/(1 + m)%N in A *; rewrite -[A]vsubmxK; apply: PS.
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_ind
| |
mx_ind(P : forall m n, 'M[R]_(m, n) -> Type) :
(forall m A, P m 0 A) ->
(forall n A, P 0 n A) ->
(forall m n x r c A, P m n A -> P (1 + m)%N (1 + n)%N (block_mx x r c A)) ->
forall m n A, P m n A.
Proof.
move=> P0l P0r PS; elim=> [|m IHm] [|n] A; do ?by [apply: P0l|apply: P0r].
by rewrite -[A](@submxK 1 _ 1); apply: PS.
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
|
mx_ind
| |
matrix_rect:= mx_ind.
|
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
|
matrix_rect
| |
matrix_rec:= mx_ind.
|
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
|
matrix_rec
| |
matrix_ind:= mx_ind.
|
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
|
matrix_ind
| |
sqmx_ind(P : forall n, 'M[R]_n -> Type) :
(forall A, P 0 A) ->
(forall n x r c A, P n A -> P (1 + n)%N (block_mx x r c A)) ->
forall n A, P n A.
Proof.
by move=> P0 PS; elim=> [//|n IHn] A; rewrite -[A](@submxK 1 _ 1); apply: PS.
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
|
sqmx_ind
| |
ringmx_ind(P : forall n, 'M[R]_n.+1 -> Type) :
(forall x, P 0 x) ->
(forall n x (r : 'rV_n.+1) (c : 'cV_n.+1) A,
P n A -> P (1 + n)%N (block_mx x r c A)) ->
forall n A, P n A.
Proof.
by move=> P0 PS; elim=> [//|n IHn] A; rewrite -[A](@submxK 1 _ 1); apply: PS.
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
|
ringmx_ind
| |
mxsub_ind(weight : forall m n, 'M[R]_(m, n) -> nat)
(sub : forall m n m' n', ('I_m' -> 'I_m) -> ('I_n' -> 'I_n) -> Prop)
(P : forall m n, 'M[R]_(m, n) -> Type) :
(forall m n (A : 'M[R]_(m, n)),
(forall m' n' f g, weight m' n' (mxsub f g A) < weight m n A ->
sub m n m' n' f g ->
P m' n' (mxsub f g A)) -> P m n A) ->
forall m n A, P m n A.
Proof.
move=> Psub m n A; have [k] := ubnP (weight m n A).
elim: k => [//|k IHk] in m n A *.
rewrite ltnS => lt_A_k; apply: Psub => m' n' f g lt_A'_A ?.
by apply: IHk; apply: leq_trans lt_A_k.
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
|
mxsub_ind
| |
mxvec_cast: #|{:'I_m * 'I_n}| = (m * n)%N.
Proof. by rewrite card_prod !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
|
mxvec_cast
| |
mxvec_index(i : 'I_m) (j : 'I_n) :=
cast_ord mxvec_cast (enum_rank (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
|
mxvec_index
| |
is_mxvec_index: 'I_(m * n) -> Type :=
isMxvecIndex i j : is_mxvec_index (mxvec_index i j).
|
Variant
|
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_mxvec_index
| |
mxvec_indexPk : is_mxvec_index k.
Proof.
rewrite -[k](cast_ordK (esym mxvec_cast)) esymK.
by rewrite -[_ k]enum_valK; case: (enum_val _).
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_indexP
| |
pair_of_mxvec_indexk (i_k : is_mxvec_index k) :=
let: isMxvecIndex i j := i_k in (i, j).
|
Coercion
|
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
|
pair_of_mxvec_index
| |
mxvec(A : 'M[R]_(m, n)) :=
castmx (erefl _, mxvec_cast) (\row_k A (enum_val k).1 (enum_val k).2).
Fact vec_mx_key : unit. Proof. by []. Qed.
|
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
|
mxvec
| |
vec_mx(u : 'rV[R]_(m * n)) :=
\matrix[vec_mx_key]_(i, j) u 0 (mxvec_index 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
|
vec_mx
| |
mxvecEA i j : mxvec A 0 (mxvec_index i j) = A i j.
Proof. by rewrite castmxE mxE cast_ordK enum_rankK. 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
|
mxvecE
| |
mxvecK: cancel mxvec vec_mx.
Proof. by move=> A; apply/matrixP=> i j; rewrite mxE mxvecE. 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
|
mxvecK
| |
vec_mxK: cancel vec_mx mxvec.
Proof.
by move=> u; apply/rowP=> k; case/mxvec_indexP: k => i j; rewrite mxvecE 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
|
vec_mxK
| |
curry_mxvec_bij: {on 'I_(m * n), bijective (uncurry mxvec_index)}.
Proof.
exists (enum_val \o cast_ord (esym mxvec_cast)) => [[i j] _ | k _] /=.
by rewrite cast_ordK enum_rankK.
by case/mxvec_indexP: k => i j /=; rewrite cast_ordK enum_rankK.
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
|
curry_mxvec_bij
| |
colsubg := (mxsub id g).
|
Notation
|
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
|
colsub
| |
rowsubf := (mxsub f id).
Arguments eq_mxsub [R m n m' n' f] f' [g] g' _.
Arguments eq_rowsub [R m n m' f] f' _.
Arguments eq_colsub [R m n n' g] g' _.
|
Notation
|
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
|
rowsub
| |
map_mxm n (A : 'M_(m, n)) := \matrix[map_mx_key]_(i, j) f (A 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
|
map_mx
| |
map_trmx: A^f^T = A^T^f.
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
|
map_trmx
| |
map_const_mxa : (const_mx a)^f = const_mx (f 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
|
map_const_mx
| |
map_rowi : (row i A)^f = row i A^f.
Proof. by apply/rowP=> 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
|
map_row
| |
map_colj : (col j A)^f = col j A^f.
Proof. by apply/colP=> i; 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
|
map_col
| |
map_row'i0 : (row' i0 A)^f = row' i0 A^f.
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
|
map_row'
| |
map_col'j0 : (col' j0 A)^f = col' j0 A^f.
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
|
map_col'
| |
map_mxsubm' n' g h : (@mxsub _ _ _ m' n' g h A)^f = mxsub g h A^f.
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
|
map_mxsub
| |
map_row_perms : (row_perm s A)^f = row_perm s A^f.
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
|
map_row_perm
| |
map_col_perms : (col_perm s A)^f = col_perm s A^f.
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
|
map_col_perm
| |
map_xrowi1 i2 : (xrow i1 i2 A)^f = xrow i1 i2 A^f.
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
|
map_xrow
| |
map_xcolj1 j2 : (xcol j1 j2 A)^f = xcol j1 j2 A^f.
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
|
map_xcol
| |
map_castmxm' n' c : (castmx c A)^f = castmx c A^f :> 'M_(m', n').
Proof. by apply/matrixP=> i j; rewrite !(castmxE, 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
|
map_castmx
| |
map_conform_mxm' n' (B : 'M_(m', n')) :
(conform_mx B A)^f = conform_mx B^f A^f.
Proof.
move: B; have [[<- <-] B|] := eqVneq (m, n) (m', n').
by rewrite !conform_mx_id.
by rewrite negb_and => neq_mn B; rewrite !nonconform_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
|
map_conform_mx
| |
map_mxvec: (mxvec A)^f = mxvec A^f.
Proof. by apply/rowP=> i; rewrite !(castmxE, 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
|
map_mxvec
| |
map_vec_mx(v : 'rV_(m * n)) : (vec_mx v)^f = vec_mx v^f.
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
|
map_vec_mx
| |
map_row_mx: (row_mx Aul Aur)^f = row_mx Aul^f Aur^f.
Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. 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_row_mx
| |
map_col_mx: (col_mx Aul Adl)^f = col_mx Aul^f Adl^f.
Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. 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_col_mx
| |
map_block_mx:
(block_mx Aul Aur Adl Adr)^f = block_mx Aul^f Aur^f Adl^f Adr^f.
Proof. by apply/matrixP=> i j; do 3![rewrite !mxE //; case: split => ?]. 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_block_mx
| |
map_lsubmx: (lsubmx Bh)^f = lsubmx Bh^f.
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
|
map_lsubmx
| |
map_rsubmx: (rsubmx Bh)^f = rsubmx Bh^f.
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
|
map_rsubmx
| |
map_usubmx: (usubmx Bv)^f = usubmx Bv^f.
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
|
map_usubmx
| |
map_dsubmx: (dsubmx Bv)^f = dsubmx Bv^f.
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
|
map_dsubmx
| |
map_ulsubmx: (ulsubmx B)^f = ulsubmx B^f.
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
|
map_ulsubmx
| |
map_ursubmx: (ursubmx B)^f = ursubmx B^f.
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
|
map_ursubmx
| |
map_dlsubmx: (dlsubmx B)^f = dlsubmx B^f.
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
|
map_dlsubmx
| |
map_drsubmx: (drsubmx B)^f = drsubmx B^f.
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
|
map_drsubmx
| |
map_mx_comp(f : R -> S) (g : S -> T)
(M : 'M_(m, n)) : M ^ (g \o f) = (M ^ f) ^ g.
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
|
map_mx_comp
| |
eq_in_map_mx(g f : R -> S) (M : 'M_(m, n)) :
(forall i j, f (M i j) = g (M i j)) -> M ^ f = M ^ g.
Proof. by move=> fg; 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
|
eq_in_map_mx
| |
eq_map_mx(g f : R -> S) : f =1 g ->
forall (M : 'M_(m, n)), M ^ f = M ^ g.
Proof. by move=> eq_fg M; apply/eq_in_map_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
|
eq_map_mx
| |
map_mx_id_in(f : R -> R) (M : 'M_(m, n)) :
(forall i j, f (M i j) = M i j) -> M ^ f = M.
Proof. by move=> fM; 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
|
map_mx_id_in
| |
map_mx_id(f : R -> R) : f =1 id -> forall M : 'M_(m, n), M ^ f = M.
Proof. by move=> fid M; rewrite map_mx_id_in. 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_mx_id
| |
map2_mxm n (A : 'M_(m, n)) (B : 'M_(m, n)) :=
\matrix[map2_mx_key]_(i, j) f (A i j) (B 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
|
map2_mx
| |
map2_trmx: (map2_mx A B)^T = map2_mx A^T B^T.
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
|
map2_trmx
| |
map2_const_mxa b :
map2_mx (const_mx a) (const_mx b) = const_mx (f a b) :> '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
|
map2_const_mx
| |
map2_rowi : map2_mx (row i A) (row i B) = row i (map2_mx A B).
Proof. by apply/rowP=> 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
|
map2_row
| |
map2_colj : map2_mx (col j A) (col j B) = col j (map2_mx A B).
Proof. by apply/colP=> i; 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
|
map2_col
| |
map2_row'i0 : map2_mx (row' i0 A) (row' i0 B) = row' i0 (map2_mx A B).
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
|
map2_row'
| |
map2_col'j0 : map2_mx (col' j0 A) (col' j0 B) = col' j0 (map2_mx A B).
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
|
map2_col'
| |
map2_mxsubm' n' g h :
map2_mx (@mxsub _ _ _ m' n' g h A) (@mxsub _ _ _ m' n' g h B) =
mxsub g h (map2_mx A B).
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
|
map2_mxsub
| |
map2_row_perms :
map2_mx (row_perm s A) (row_perm s B) = row_perm s (map2_mx A B).
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
|
map2_row_perm
| |
map2_col_perms :
map2_mx (col_perm s A) (col_perm s B) = col_perm s (map2_mx A B).
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
|
map2_col_perm
| |
map2_xrowi1 i2 :
map2_mx (xrow i1 i2 A) (xrow i1 i2 B) = xrow i1 i2 (map2_mx A B).
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
|
map2_xrow
| |
map2_xcolj1 j2 :
map2_mx (xcol j1 j2 A) (xcol j1 j2 B) = xcol j1 j2 (map2_mx A B).
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
|
map2_xcol
| |
map2_castmxm' n' c :
map2_mx (castmx c A) (castmx c B) = castmx c (map2_mx A B) :> 'M_(m', n').
Proof. by apply/matrixP=> i j; rewrite !(castmxE, 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
|
map2_castmx
| |
map2_conform_mxm' n' (A' : 'M_(m', n')) (B' : 'M_(m', n')) :
map2_mx (conform_mx A' A) (conform_mx B' B) =
conform_mx (map2_mx A' B') (map2_mx A B).
Proof.
move: A' B'; have [[<- <-] A' B'|] := eqVneq (m, n) (m', n').
by rewrite !conform_mx_id.
by rewrite negb_and => neq_mn A' B'; rewrite !nonconform_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
|
map2_conform_mx
| |
map2_mxvec: map2_mx (mxvec A) (mxvec B) = mxvec (map2_mx A B).
Proof. by apply/rowP=> i; rewrite !(castmxE, 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
|
map2_mxvec
| |
map2_vec_mx(v : 'rV_(m * n)) (w : 'rV_(m * n)) :
map2_mx (vec_mx v) (vec_mx w) = vec_mx (map2_mx v w).
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
|
map2_vec_mx
| |
map2_row_mx:
map2_mx (row_mx Aul Aur) (row_mx A'ul A'ur) =
row_mx (map2_mx Aul A'ul) (map2_mx Aur A'ur).
Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. 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_row_mx
| |
map2_col_mx:
map2_mx (col_mx Aul Adl) (col_mx A'ul A'dl) =
col_mx (map2_mx Aul A'ul) (map2_mx Adl A'dl).
Proof. by apply/matrixP=> i j; do 2![rewrite !mxE //; case: split => ?]. 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_col_mx
| |
map2_block_mx:
map2_mx (block_mx Aul Aur Adl Adr) (block_mx A'ul A'ur A'dl A'dr) =
block_mx
(map2_mx Aul A'ul) (map2_mx Aur A'ur) (map2_mx Adl A'dl) (map2_mx Adr A'dr).
Proof. by apply/matrixP=> i j; do 3![rewrite !mxE //; case: split => ?]. 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_block_mx
| |
map2_lsubmx: map2_mx (lsubmx Bh) (lsubmx B'h) = lsubmx (map2_mx Bh B'h).
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
|
map2_lsubmx
| |
map2_rsubmx: map2_mx (rsubmx Bh) (rsubmx B'h) = rsubmx (map2_mx Bh B'h).
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
|
map2_rsubmx
| |
map2_usubmx: map2_mx (usubmx Bv) (usubmx B'v) = usubmx (map2_mx Bv B'v).
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
|
map2_usubmx
| |
map2_dsubmx: map2_mx (dsubmx Bv) (dsubmx B'v) = dsubmx (map2_mx Bv B'v).
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
|
map2_dsubmx
| |
map2_ulsubmx: map2_mx (ulsubmx B) (ulsubmx B') = ulsubmx (map2_mx B B').
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
|
map2_ulsubmx
| |
map2_ursubmx: map2_mx (ursubmx B) (ursubmx B') = ursubmx (map2_mx B B').
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
|
map2_ursubmx
| |
map2_dlsubmx: map2_mx (dlsubmx B) (dlsubmx B') = dlsubmx (map2_mx B B').
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
|
map2_dlsubmx
| |
map2_drsubmx: map2_mx (drsubmx B) (drsubmx B') = drsubmx (map2_mx B B').
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
|
map2_drsubmx
| |
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
|
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
|
eq_in_map2_mx
| |
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
|
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
|
eq_map2_mx
| |
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
|
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_mx_left_in
| |
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
|
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_mx_left
| |
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
|
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_mx_right_in
| |
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
|
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_mx_right
| |
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
|
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_mxA
| |
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
|
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_1mx
| |
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.
HB.instance Definition _ {opm : Monoid.law idm} :=
Monoid.isLaw.Build 'M_(m, n) (const_mx idm) (@map2_mx _ _ _ opm _ _)
map2_mxA map2_1mx map2_mx1.
|
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_mx1
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.