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 |
|---|---|---|---|---|---|---|
col_idn j (V : 'cV_n) : col j V = V.
Proof. by apply/colP=> i; rewrite mxE [j]ord1. 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_id
| |
row_eqm1 m2 n i1 i2 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row i1 A1 = row i2 A2 -> A1 i1 =1 A2 i2.
Proof. by move/rowP=> eqA12 j; have /[!mxE] := eqA12 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
|
row_eq
| |
col_eqm n1 n2 j1 j2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col j1 A1 = col j2 A2 -> A1^~ j1 =1 A2^~ j2.
Proof. by move/colP=> eqA12 i; have /[!mxE] := eqA12 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
|
col_eq
| |
row'_eqm n i0 (A B : 'M_(m, n)) :
row' i0 A = row' i0 B -> {in predC1 i0, A =2 B}.
Proof.
move=> /matrixP eqAB' i /[!inE]/[1!eq_sym]/unlift_some[i' -> _] j.
by have /[!mxE] := eqAB' i' 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
|
row'_eq
| |
col'_eqm n j0 (A B : 'M_(m, n)) :
col' j0 A = col' j0 B -> forall i, {in predC1 j0, A i =1 B i}.
Proof.
move=> /matrixP eqAB' i j /[!inE]/[1!eq_sym]/unlift_some[j' -> _].
by have /[!mxE] := eqAB' i 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
|
col'_eq
| |
tr_rowm n i0 (A : 'M_(m, n)) : (row i0 A)^T = col i0 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
|
tr_row
| |
tr_row'm n i0 (A : 'M_(m, n)) : (row' i0 A)^T = col' i0 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
|
tr_row'
| |
tr_colm n j0 (A : 'M_(m, n)) : (col j0 A)^T = row j0 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
|
tr_col
| |
tr_col'm n j0 (A : 'M_(m, n)) : (col' j0 A)^T = row' j0 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
|
tr_col'
| |
mxsub_compm1 m2 m3 n1 n2 n3
(f : 'I_m2 -> 'I_m1) (f' : 'I_m3 -> 'I_m2)
(g : 'I_n2 -> 'I_n1) (g' : 'I_n3 -> 'I_n2) (A : 'M_(m1, n1)) :
mxsub (f \o f') (g \o g') A = mxsub f' g' (mxsub f g A).
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
|
mxsub_comp
| |
rowsub_compm1 m2 m3 n
(f : 'I_m2 -> 'I_m1) (f' : 'I_m3 -> 'I_m2) (A : 'M_(m1, n)) :
rowsub (f \o f') A = rowsub f' (rowsub f A).
Proof. exact: mxsub_comp. 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
|
rowsub_comp
| |
colsub_compm n n2 n3
(g : 'I_n2 -> 'I_n) (g' : 'I_n3 -> 'I_n2) (A : 'M_(m, n)) :
colsub (g \o g') A = colsub g' (colsub g A).
Proof. exact: mxsub_comp. 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
|
colsub_comp
| |
mxsubrcm1 m2 n n2 f g (A : 'M_(m1, n)) :
mxsub f g A = rowsub f (colsub g A) :> 'M_(m2, n2).
Proof. exact: mxsub_comp. 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
|
mxsubrc
| |
mxsubcrm1 m2 n n2 f g (A : 'M_(m1, n)) :
mxsub f g A = colsub g (rowsub f A) :> 'M_(m2, n2).
Proof. exact: mxsub_comp. 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
|
mxsubcr
| |
rowsub_castm1 m2 n (eq_m : m1 = m2) (A : 'M_(m2, n)) :
rowsub (cast_ord eq_m) A = castmx (esym eq_m, erefl) A.
Proof. by case: _ / eq_m in A *; apply: (mxsub_eq_id (cast_ord_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
|
rowsub_cast
| |
colsub_castm n1 n2 (eq_n : n1 = n2) (A : 'M_(m, n2)) :
colsub (cast_ord eq_n) A = castmx (erefl, esym eq_n) A.
Proof. by case: _ / eq_n in A *; apply: (mxsub_eq_id _ (cast_ord_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
|
colsub_cast
| |
mxsub_castm1 m2 n1 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) A :
mxsub (cast_ord eq_m) (cast_ord eq_n) A = castmx (esym eq_m, esym eq_n) A.
Proof. by rewrite mxsubrc rowsub_cast colsub_cast 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
|
mxsub_cast
| |
castmxEsubm1 m2 n1 n2 (eq_mn : (m1 = m2) * (n1 = n2)) A :
castmx eq_mn A = mxsub (cast_ord (esym eq_mn.1)) (cast_ord (esym eq_mn.2)) A.
Proof. by rewrite mxsub_cast !esymK; case: eq_mn. 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
|
castmxEsub
| |
trmx_mxsubm1 m2 n1 n2 f g (A : 'M_(m1, n1)) :
(mxsub f g A)^T = mxsub g f A^T :> 'M_(n2, m2).
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_mxsub
| |
row_mxsubm1 m2 n1 n2
(f : 'I_m2 -> 'I_m1) (g : 'I_n2 -> 'I_n1) (A : 'M_(m1, n1)) i :
row i (mxsub f g A) = row (f i) (colsub g A).
Proof. by rewrite !rowEsub -!mxsub_comp. 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_mxsub
| |
col_mxsubm1 m2 n1 n2
(f : 'I_m2 -> 'I_m1) (g : 'I_n2 -> 'I_n1) (A : 'M_(m1, n1)) i :
col i (mxsub f g A) = col (g i) (rowsub f A).
Proof. by rewrite !colEsub -!mxsub_comp. 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_mxsub
| |
row_rowsubm1 m2 n (f : 'I_m2 -> 'I_m1) (A : 'M_(m1, n)) i :
row i (rowsub f A) = row (f i) A.
Proof. by rewrite row_mxsub mxsub_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
|
row_rowsub
| |
col_colsubm n1 n2 (g : 'I_n2 -> 'I_n1) (A : 'M_(m, n1)) i :
col i (colsub g A) = col (g i) A.
Proof. by rewrite col_mxsub mxsub_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
|
col_colsub
| |
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
| |
row_mx(A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) : 'M[R]_(m, n1 + n2) :=
\matrix[row_mx_key]_(i, j)
match split j with inl j1 => A1 i j1 | inr j2 => A2 i j2 end.
Fact col_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
|
row_mx
| |
col_mx(A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) : 'M[R]_(m1 + m2, n) :=
\matrix[col_mx_key]_(i, j)
match split i with inl i1 => A1 i1 j | inr i2 => A2 i2 j end.
|
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
|
col_mx
| |
lsubmx(A : 'M[R]_(m, n1 + n2)) :=
\matrix[lsubmx_key]_(i, j) A i (lshift n2 j).
Fact rsubmx_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
|
lsubmx
| |
rsubmx(A : 'M[R]_(m, n1 + n2)) :=
\matrix[rsubmx_key]_(i, j) A i (rshift n1 j).
Fact usubmx_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
|
rsubmx
| |
usubmx(A : 'M[R]_(m1 + m2, n)) :=
\matrix[usubmx_key]_(i, j) A (lshift m2 i) j.
Fact dsubmx_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
|
usubmx
| |
dsubmx(A : 'M[R]_(m1 + m2, n)) :=
\matrix[dsubmx_key]_(i, j) A (rshift m1 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
|
dsubmx
| |
row_mxElA1 A2 i j : row_mx A1 A2 i (lshift n2 j) = A1 i j.
Proof. by rewrite mxE (unsplitK (inl _ _)). 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_mxEl
| |
row_mxKlA1 A2 : lsubmx (row_mx A1 A2) = A1.
Proof. by apply/matrixP=> i j; rewrite mxE row_mxEl. 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_mxKl
| |
row_mxErA1 A2 i j : row_mx A1 A2 i (rshift n1 j) = A2 i j.
Proof. by rewrite mxE (unsplitK (inr _ _)). 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_mxEr
| |
row_mxKrA1 A2 : rsubmx (row_mx A1 A2) = A2.
Proof. by apply/matrixP=> i j; rewrite mxE row_mxEr. 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_mxKr
| |
hsubmxKA : row_mx (lsubmx A) (rsubmx A) = A.
Proof. by apply/matrixP=> i j /[!mxE]; case: split_ordP => k -> /[!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
|
hsubmxK
| |
col_mxEuA1 A2 i j : col_mx A1 A2 (lshift m2 i) j = A1 i j.
Proof. by rewrite mxE (unsplitK (inl _ _)). 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_mxEu
| |
col_mxKuA1 A2 : usubmx (col_mx A1 A2) = A1.
Proof. by apply/matrixP=> i j; rewrite mxE col_mxEu. 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_mxKu
| |
col_mxEdA1 A2 i j : col_mx A1 A2 (rshift m1 i) j = A2 i j.
Proof. by rewrite mxE (unsplitK (inr _ _)). 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_mxEd
| |
col_mxKdA1 A2 : dsubmx (col_mx A1 A2) = A2.
Proof. by apply/matrixP=> i j; rewrite mxE col_mxEd. 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_mxKd
| |
lsubmxEsub: lsubmx = colsub (lshift _).
Proof. by rewrite /lsubmx /mxsub !unlock. 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
|
lsubmxEsub
| |
rsubmxEsub: rsubmx = colsub (@rshift _ _).
Proof. by rewrite /rsubmx /mxsub !unlock. 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
|
rsubmxEsub
| |
usubmxEsub: usubmx = rowsub (lshift _).
Proof. by rewrite /usubmx /mxsub !unlock. 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
|
usubmxEsub
| |
dsubmxEsub: dsubmx = rowsub (@rshift _ _).
Proof. by rewrite /dsubmx /mxsub !unlock. 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
|
dsubmxEsub
| |
eq_row_mxA1 A2 B1 B2 : row_mx A1 A2 = row_mx B1 B2 -> A1 = B1 /\ A2 = B2.
Proof.
move=> eqAB; move: (congr1 lsubmx eqAB) (congr1 rsubmx eqAB).
by rewrite !(row_mxKl, 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
|
eq_row_mx
| |
eq_col_mxA1 A2 B1 B2 : col_mx A1 A2 = col_mx B1 B2 -> A1 = B1 /\ A2 = B2.
Proof.
move=> eqAB; move: (congr1 usubmx eqAB) (congr1 dsubmx eqAB).
by rewrite !(col_mxKu, col_mxKd).
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_col_mx
| |
lsubmx_const(r : R) : lsubmx (const_mx r : 'M_(m, n1 + n2)) = const_mx r.
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
|
lsubmx_const
| |
rsubmx_const(r : R) : rsubmx (const_mx r : 'M_(m, n1 + n2)) = const_mx r.
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
|
rsubmx_const
| |
row_mx_consta : row_mx (const_mx a) (const_mx a) = const_mx a.
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
|
row_mx_const
| |
col_mx_consta : col_mx (const_mx a) (const_mx a) = const_mx a.
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
|
col_mx_const
| |
row_usubmxA i : row i (usubmx A) = row (lshift m2 i) A.
Proof. by apply/rowP=> j; rewrite !mxE; congr (A _ _); apply/val_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
|
row_usubmx
| |
row_dsubmxA i : row i (dsubmx A) = row (rshift m1 i) A.
Proof. by apply/rowP=> j; rewrite !mxE; congr (A _ _); apply/val_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
|
row_dsubmx
| |
col_lsubmxA i : col i (lsubmx A) = col (lshift n2 i) A.
Proof. by apply/colP=> j; rewrite !mxE; congr (A _ _); apply/val_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
|
col_lsubmx
| |
col_rsubmxA i : col i (rsubmx A) = col (rshift n1 i) A.
Proof. by apply/colP=> j; rewrite !mxE; congr (A _ _); apply/val_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
|
col_rsubmx
| |
row_thin_mxm n (A : 'M_(m,0)) (B : 'M_(m,n)) : row_mx A B = B.
Proof.
apply/matrixP=> i j; rewrite mxE; case: splitP=> [|k H]; first by case.
by congr fun_of_matrix; exact: val_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
|
row_thin_mx
| |
col_flat_mxm n (A : 'M_(0,n)) (B : 'M_(m,n)) : col_mx A B = B.
Proof.
apply/matrixP=> i j; rewrite mxE; case: splitP => [|k H]; first by case.
by congr fun_of_matrix; exact: val_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
|
col_flat_mx
| |
trmx_lsubm n1 n2 (A : 'M_(m, n1 + n2)) : (lsubmx A)^T = usubmx A^T.
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
|
trmx_lsub
| |
trmx_rsubm n1 n2 (A : 'M_(m, n1 + n2)) : (rsubmx A)^T = dsubmx A^T.
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
|
trmx_rsub
| |
tr_row_mxm n1 n2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
(row_mx A1 A2)^T = col_mx A1^T A2^T.
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
|
tr_row_mx
| |
tr_col_mxm1 m2 n (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
(col_mx A1 A2)^T = row_mx A1^T A2^T.
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
|
tr_col_mx
| |
trmx_usubm1 m2 n (A : 'M_(m1 + m2, n)) : (usubmx A)^T = lsubmx A^T.
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
|
trmx_usub
| |
trmx_dsubm1 m2 n (A : 'M_(m1 + m2, n)) : (dsubmx A)^T = rsubmx A^T.
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
|
trmx_dsub
| |
vsubmxKm1 m2 n (A : 'M_(m1 + m2, n)) : col_mx (usubmx A) (dsubmx A) = A.
Proof. by apply: trmx_inj; rewrite tr_col_mx trmx_usub trmx_dsub hsubmxK. 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
|
vsubmxK
| |
cast_row_mxm m' n1 n2 (eq_m : m = m') A1 A2 :
castmx (eq_m, erefl _) (row_mx A1 A2)
= row_mx (castmx (eq_m, erefl n1) A1) (castmx (eq_m, erefl n2) A2).
Proof. by case: m' / eq_m. 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
|
cast_row_mx
| |
cast_col_mxm1 m2 n n' (eq_n : n = n') A1 A2 :
castmx (erefl _, eq_n) (col_mx A1 A2)
= col_mx (castmx (erefl m1, eq_n) A1) (castmx (erefl m2, eq_n) A2).
Proof. by case: n' / eq_n. 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
|
cast_col_mx
| |
row_mxAm n1 n2 n3 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) (A3 : 'M_(m, n3)) :
let cast := (erefl m, esym (addnA n1 n2 n3)) in
row_mx A1 (row_mx A2 A3) = castmx cast (row_mx (row_mx A1 A2) A3).
Proof.
apply: (canRL (castmxKV _ _)); apply/matrixP=> i j.
rewrite castmxE !mxE cast_ord_id; case: splitP => j1 /= def_j.
have: (j < n1 + n2) && (j < n1) by rewrite def_j lshift_subproof /=.
by move: def_j; do 2![case: splitP => // ? ->; rewrite ?mxE] => /ord_inj->.
case: splitP def_j => j2 ->{j} def_j /[!mxE].
have: ~~ (j2 < n1) by rewrite -leqNgt def_j leq_addr.
have: j1 < n2 by rewrite -(ltn_add2l n1) -def_j.
by move: def_j; do 2![case: splitP => // ? ->] => /addnI/val_inj->.
have: ~~ (j1 < n2) by rewrite -leqNgt -(leq_add2l n1) -def_j leq_addr.
by case: splitP def_j => // ? ->; rewrite addnA => /addnI/val_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
|
row_mxA
| |
row_mxAx:= row_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
|
row_mxAx
| |
col_mxAm1 m2 m3 n (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) (A3 : 'M_(m3, n)) :
let cast := (esym (addnA m1 m2 m3), erefl n) in
col_mx A1 (col_mx A2 A3) = castmx cast (col_mx (col_mx A1 A2) A3).
Proof. by apply: trmx_inj; rewrite trmx_cast !tr_col_mx -row_mxA. 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_mxA
| |
col_mxAx:= col_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
|
col_mxAx
| |
row_row_mxm n1 n2 i0 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
row i0 (row_mx A1 A2) = row_mx (row i0 A1) (row i0 A2).
Proof.
by apply/matrixP=> i j /[!mxE]; case: (split j) => j' /[1!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
|
row_row_mx
| |
col_col_mxm1 m2 n j0 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
col j0 (col_mx A1 A2) = col_mx (col j0 A1) (col j0 A2).
Proof. by apply: trmx_inj; rewrite !(tr_col, tr_col_mx, row_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
|
col_col_mx
| |
row'_row_mxm n1 n2 i0 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
row' i0 (row_mx A1 A2) = row_mx (row' i0 A1) (row' i0 A2).
Proof.
by apply/matrixP=> i j /[!mxE]; case: (split j) => j' /[1!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
|
row'_row_mx
| |
col'_col_mxm1 m2 n j0 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
col' j0 (col_mx A1 A2) = col_mx (col' j0 A1) (col' j0 A2).
Proof. by apply: trmx_inj; rewrite !(tr_col', tr_col_mx, row'_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
|
col'_col_mx
| |
colKlm n1 n2 j1 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col (lshift n2 j1) (row_mx A1 A2) = col j1 A1.
Proof. by apply/matrixP=> i j; rewrite !(row_mxEl, 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
|
colKl
| |
colKrm n1 n2 j2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col (rshift n1 j2) (row_mx A1 A2) = col j2 A2.
Proof. by apply/matrixP=> i j; rewrite !(row_mxEr, 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
|
colKr
| |
rowKum1 m2 n i1 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row (lshift m2 i1) (col_mx A1 A2) = row i1 A1.
Proof. by apply/matrixP=> i j; rewrite !(col_mxEu, 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
|
rowKu
| |
rowKdm1 m2 n i2 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row (rshift m1 i2) (col_mx A1 A2) = row i2 A2.
Proof. by apply/matrixP=> i j; rewrite !(col_mxEd, 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
|
rowKd
| |
col'Klm n1 n2 j1 (A1 : 'M_(m, n1.+1)) (A2 : 'M_(m, n2)) :
col' (lshift n2 j1) (row_mx A1 A2) = row_mx (col' j1 A1) A2.
Proof.
apply/matrixP=> i /= j; symmetry; rewrite 2!mxE; case: split_ordP => j' ->.
by rewrite mxE -(row_mxEl _ A2); congr (row_mx _ _ _); apply: ord_inj.
rewrite -(row_mxEr A1); congr (row_mx _ _ _); apply: ord_inj => /=.
by rewrite /bump -ltnS -addSn ltn_addr.
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'Kl
| |
row'Kum1 m2 n i1 (A1 : 'M_(m1.+1, n)) (A2 : 'M_(m2, n)) :
row' (lshift m2 i1) (@col_mx m1.+1 m2 n A1 A2) = col_mx (row' i1 A1) A2.
Proof.
by apply: trmx_inj; rewrite tr_col_mx !(@tr_row' _.+1) (@tr_col_mx _.+1) col'Kl.
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'Ku
| |
mx'_castm n : 'I_n -> (m + n.-1)%N = (m + n).-1.
Proof. by case=> j /ltn_predK <-; rewrite addnS. 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'_cast
| |
col'Krm n1 n2 j2 (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) :
col' (rshift n1 j2) (@row_mx m n1 n2 A1 A2)
= castmx (erefl m, mx'_cast n1 j2) (row_mx A1 (col' j2 A2)).
Proof.
apply/matrixP=> i j; symmetry; rewrite castmxE mxE cast_ord_id.
case: splitP => j' /= def_j.
rewrite mxE -(row_mxEl _ A2); congr (row_mx _ _ _); apply: ord_inj.
by rewrite /= def_j /bump leqNgt ltn_addr.
rewrite 2!mxE -(row_mxEr A1); congr (row_mx _ _ _ _); apply: ord_inj.
by rewrite /= def_j /bump leq_add2l addnCA.
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'Kr
| |
row'Kdm1 m2 n i2 (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) :
row' (rshift m1 i2) (col_mx A1 A2)
= castmx (mx'_cast m1 i2, erefl n) (col_mx A1 (row' i2 A2)).
Proof. by apply: trmx_inj; rewrite trmx_cast !(tr_row', tr_col_mx) col'Kr. 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'Kd
| |
block_mxAul Aur Adl Adr : 'M_(m1 + m2, n1 + n2) :=
col_mx (row_mx Aul Aur) (row_mx Adl Adr).
|
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_mx
| |
eq_block_mxAul Aur Adl Adr Bul Bur Bdl Bdr :
block_mx Aul Aur Adl Adr = block_mx Bul Bur Bdl Bdr ->
[/\ Aul = Bul, Aur = Bur, Adl = Bdl & Adr = Bdr].
Proof. by case/eq_col_mx; do 2!case/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
|
eq_block_mx
| |
block_mx_consta :
block_mx (const_mx a) (const_mx a) (const_mx a) (const_mx a) = const_mx a.
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
|
block_mx_const
| |
ulsubmx:= lsubmx (usubmx A).
|
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
|
ulsubmx
| |
ursubmx:= rsubmx (usubmx A).
|
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
|
ursubmx
| |
dlsubmx:= lsubmx (dsubmx A).
|
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
|
dlsubmx
| |
drsubmx:= rsubmx (dsubmx A).
|
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
|
drsubmx
| |
submxK: block_mx ulsubmx ursubmx dlsubmx drsubmx = A.
Proof. by rewrite /block_mx !hsubmxK vsubmxK. 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
|
submxK
| |
ulsubmxEsub: ulsubmx = mxsub (lshift _) (lshift _) A.
Proof. by rewrite /ulsubmx lsubmxEsub usubmxEsub -mxsub_comp. 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
|
ulsubmxEsub
| |
dlsubmxEsub: dlsubmx = mxsub (@rshift _ _) (lshift _) A.
Proof. by rewrite /dlsubmx lsubmxEsub dsubmxEsub -mxsub_comp. 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
|
dlsubmxEsub
| |
ursubmxEsub: ursubmx = mxsub (lshift _) (@rshift _ _) A.
Proof. by rewrite /ursubmx rsubmxEsub usubmxEsub -mxsub_comp. 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
|
ursubmxEsub
| |
drsubmxEsub: drsubmx = mxsub (@rshift _ _) (@rshift _ _) A.
Proof. by rewrite /drsubmx rsubmxEsub dsubmxEsub -mxsub_comp. 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
|
drsubmxEsub
| |
block_mxEuli j : A (lshift m2 i) (lshift n2 j) = Aul i j.
Proof. by rewrite col_mxEu row_mxEl. 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_mxEul
| |
block_mxKul: ulsubmx A = Aul.
Proof. by rewrite /ulsubmx col_mxKu row_mxKl. 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_mxKul
| |
block_mxEuri j : A (lshift m2 i) (rshift n1 j) = Aur i j.
Proof. by rewrite col_mxEu row_mxEr. 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_mxEur
| |
block_mxKur: ursubmx A = Aur.
Proof. by rewrite /ursubmx col_mxKu 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_mxKur
| |
block_mxEdli j : A (rshift m1 i) (lshift n2 j) = Adl i j.
Proof. by rewrite col_mxEd row_mxEl. 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_mxEdl
| |
block_mxKdl: dlsubmx A = Adl.
Proof. by rewrite /dlsubmx col_mxKd row_mxKl. 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_mxKdl
| |
block_mxEdri j : A (rshift m1 i) (rshift n1 j) = Adr i j.
Proof. by rewrite col_mxEd row_mxEr. 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_mxEdr
|
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