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 |
|---|---|---|---|---|---|---|
num_gt_mina x y :
Num.min x%:num y%:num < a = (x%:num < a) || (y%:num < a).
Proof. by rewrite -comparable_gt_min// real_comparable. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
num_gt_min
| |
num_abs_lea x : 0 <= a -> (`|a|%:nng <= x) = (a <= x%:num).
Proof. by move=> a0; rewrite -num_le//= ger0_norm. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
num_abs_le
| |
num_abs_lta x : 0 <= a -> (`|a|%:nng < x) = (a < x%:num).
Proof. by move=> a0; rewrite -num_lt/= ger0_norm. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
num_abs_lt
| |
itvnum_subdef: num_spec (Itv.Real (Interval l u)) x.
Proof. by apply/and3P. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
itvnum_subdef
| |
ItvNum: num_def R (Itv.Real (Interval l u)) := Itv.mk itvnum_subdef.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
ItvNum
| |
itvreal_subdef: num_spec (Itv.Real (Interval l u)) x.
Proof. by apply/and3P; split; first exact: num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
itvreal_subdef
| |
ItvReal: num_def R (Itv.Real (Interval l u)) :=
Itv.mk itvreal_subdef.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
ItvReal
| |
itv01_subdef: num_spec (Itv.Real `[0%Z, 1%Z]) x.
Proof. by apply/and3P; split; rewrite ?bnd_simp// ger0_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
itv01_subdef
| |
Itv01: num_def R (Itv.Real `[0%Z, 1%Z]) := Itv.mk itv01_subdef.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
Itv01
| |
posnum_subdef: num_spec (Itv.Real `]0, +oo[) x.
Proof. by apply/and3P; rewrite /= gtr0_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
posnum_subdef
| |
PosNum: {posnum R} := Itv.mk posnum_subdef.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
PosNum
| |
nngnum_subdef: num_spec (Itv.Real `[0, +oo[) x.
Proof. by apply/and3P; rewrite /= ger0_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
nngnum_subdef
| |
NngNum: {nonneg R} := Itv.mk nngnum_subdef.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
NngNum
| |
posnum_spec(R : numDomainType) (x : R) :
R -> bool -> bool -> bool -> Type :=
| IsPosnum (p : {posnum R}) : posnum_spec x (p%:num) false true true.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
posnum_spec
| |
posnumP(R : numDomainType) (x : R) : 0 < x ->
posnum_spec x x (x == 0) (0 <= x) (0 < x).
Proof.
move=> x_gt0; case: real_ltgt0P (x_gt0) => []; rewrite ?gtr0_real // => _ _.
by rewrite -[x]/(PosNum x_gt0)%:num; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
posnumP
| |
nonneg_spec(R : numDomainType) (x : R) : R -> bool -> Type :=
| IsNonneg (p : {nonneg R}) : nonneg_spec x (p%:num) true.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
nonneg_spec
| |
nonnegP(R : numDomainType) (x : R) : 0 <= x -> nonneg_spec x x (0 <= x).
Proof. by move=> xge0; rewrite xge0 -[x]/(NngNum xge0)%:num; constructor. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
nonnegP
| |
s_of_pq(p q : {i01 R}) : {i01 R} :=
(1 - ((1 - p%:num)%:i01%:num * (1 - q%:num)%:i01%:num))%:i01.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
s_of_pq
| |
s_of_p0(p : {i01 R}) : s_of_pq p 0%:i01 = p.
Proof. by apply/val_inj; rewrite /= subr0 mulr1 subKr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
s_of_p0
| |
matrix: predArgType := Matrix of {ffun 'I_m * 'I_n -> R}.
|
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
|
matrix
| |
mx_valA := let: Matrix g := A in g.
HB.instance Definition _ := [isNew for mx_val].
|
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
|
mx_val
| |
fun_of_matrixA (i : 'I_m) (j : 'I_n) := mx_val 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
|
fun_of_matrix
| |
fun_of_matrix: matrix >-> Funclass.
|
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
|
fun_of_matrix
| |
matrix_of_funR (m n : nat) (k : unit) (F : 'I_m -> 'I_n -> R) :=
@Matrix R m n [ffun ij => F ij.1 ij.2].
|
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_of_fun
| |
matrix_unlockable:= Unlockable matrix_of_fun.unlock.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From 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_unlockable
| |
mxEk F : matrix_of_fun k F =2 F.
Proof. by move=> i j; rewrite unlock /fun_of_matrix /= ffunE. 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
|
mxE
| |
matrixP(A B : matrix R m n) : A =2 B <-> A = B.
Proof.
rewrite /fun_of_matrix; split=> [/= eqAB | -> //].
by apply/val_inj/ffunP=> [[i j]]; apply: eqAB.
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
|
matrixP
| |
eq_mxk F1 F2 : (F1 =2 F2) -> matrix_of_fun k F1 = matrix_of_fun k F2.
Proof. by move=> eq_F; apply/matrixP => i j; rewrite !mxE eq_F. 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_mx
| |
Definition_ (R : eqType) m n := [Equality of 'M[R]_(m, n) by <:].
HB.instance Definition _ (R : choiceType) m n := [Choice of 'M[R]_(m, n) by <:].
HB.instance Definition _ (R : countType) m n := [Countable of 'M[R]_(m, n) by <:].
HB.instance Definition _ (R : finType) m n := [Finite of 'M[R]_(m, n) by <:].
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
Definition
| |
card_mx(F : finType) m n : (#|{: 'M[F]_(m, n)}| = #|F| ^ (m * n))%N.
Proof. by rewrite card_sub card_ffun 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
|
card_mx
| |
const_mxm n a : 'M[R]_(m, n) := \matrix[const_mx_key]_(i, j) a.
Arguments const_mx {m n}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
const_mx
| |
castmxm' n' (eq_mn : (m = m') * (n = n')) A : 'M_(m', n') :=
let: erefl in _ = m' := eq_mn.1 return 'M_(m', n') in
let: erefl in _ = n' := eq_mn.2 return 'M_(m, n') in 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
|
castmx
| |
conform_mxm' n' B A :=
match m =P m', n =P n' with
| ReflectT eq_m, ReflectT eq_n => castmx (eq_m, eq_n) A
| _, _ => B
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
|
conform_mx
| |
trmxA := \matrix[trmx_key]_(i, j) A j i.
|
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
|
trmx
| |
row_perm(s : 'S_m) A := \matrix[row_perm_key]_(i, j) A (s i) j.
Fact col_perm_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_perm
| |
col_perm(s : 'S_n) A := \matrix[col_perm_key]_(i, j) A i (s 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
|
col_perm
| |
xrowi1 i2 := row_perm (tperm i1 i2).
|
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
|
xrow
| |
xcolj1 j2 := col_perm (tperm j1 j2).
|
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
|
xcol
| |
rowi0 A := \row_j A i0 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
|
row
| |
colj0 A := \col_i A i j0.
|
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
| |
row'i0 A := \matrix_(i, j) A (lift i0 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
|
row'
| |
col'j0 A := \matrix_(i, j) A i (lift j0 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
|
col'
| |
mxsubm' n' f g A := \matrix_(i < m', j < n') A (f i) (g j).
Local Notation colsub g := (mxsub id g).
Local Notation rowsub f := (mxsub f id).
|
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
|
mxsub
| |
castmx_constm' n' (eq_mn : (m = m') * (n = n')) a :
castmx eq_mn (const_mx a) = const_mx a.
Proof. by case: eq_mn; case: m' /; case: 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
|
castmx_const
| |
trmx_consta : trmx (const_mx a) = const_mx 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
|
trmx_const
| |
row_perm_consts a : row_perm s (const_mx a) = const_mx 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
|
row_perm_const
| |
col_perm_consts a : col_perm s (const_mx a) = const_mx 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
|
col_perm_const
| |
xrow_consti1 i2 a : xrow i1 i2 (const_mx a) = const_mx a.
Proof. exact: row_perm_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
xrow_const
| |
xcol_constj1 j2 a : xcol j1 j2 (const_mx a) = const_mx a.
Proof. exact: col_perm_const. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
xcol_const
| |
rowP(u v : 'rV[R]_n) : u 0 =1 v 0 <-> u = v.
Proof. by split=> [eq_uv | -> //]; apply/matrixP=> i; rewrite 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
|
rowP
| |
rowKu_ i0 : row i0 (\matrix_i u_ i) = u_ i0.
Proof. by apply/rowP=> 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
|
rowK
| |
row_matrixPA B : (forall i, row i A = row i B) <-> A = B.
Proof.
split=> [eqAB | -> //]; apply/matrixP=> i j.
by move/rowP/(_ j): (eqAB 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
|
row_matrixP
| |
colP(u v : 'cV[R]_m) : u^~ 0 =1 v^~ 0 <-> u = v.
Proof. by split=> [eq_uv | -> //]; apply/matrixP=> i j; rewrite 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
|
colP
| |
row_consti0 a : row i0 (const_mx a) = const_mx a.
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
|
row_const
| |
col_constj0 a : col j0 (const_mx a) = const_mx a.
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
|
col_const
| |
row'_consti0 a : row' i0 (const_mx a) = const_mx 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
|
row'_const
| |
col'_constj0 a : col' j0 (const_mx a) = const_mx 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
|
col'_const
| |
col_perm1A : col_perm 1 A = A.
Proof. by apply/matrixP=> i j; rewrite mxE perm1. 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_perm1
| |
row_perm1A : row_perm 1 A = A.
Proof. by apply/matrixP=> i j; rewrite mxE perm1. 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_perm1
| |
col_permMs t A : col_perm (s * t) A = col_perm s (col_perm t A).
Proof. by apply/matrixP=> i j; rewrite !mxE permM. 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_permM
| |
row_permMs t A : row_perm (s * t) A = row_perm s (row_perm t A).
Proof. by apply/matrixP=> i j; rewrite !mxE permM. 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_permM
| |
col_row_permCs t A :
col_perm s (row_perm t A) = row_perm t (col_perm s 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
|
col_row_permC
| |
rowEsubi : row i = rowsub (fun=> i). 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
|
rowEsub
| |
colEsubj : col j = colsub (fun=> j). 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
|
colEsub
| |
row'Esubi : row' i = rowsub (lift i). 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
|
row'Esub
| |
col'Esubj : col' j = colsub (lift j). 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
|
col'Esub
| |
row_permEsubs : row_perm s = rowsub s.
Proof. by rewrite /row_perm /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
|
row_permEsub
| |
col_permEsubs : col_perm s = colsub s.
Proof. by rewrite /col_perm /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
|
col_permEsub
| |
xrowEsubi1 i2 : xrow i1 i2 = rowsub (tperm i1 i2).
Proof. exact: row_permEsub. 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
|
xrowEsub
| |
xcolEsubj1 j2 : xcol j1 j2 = colsub (tperm j1 j2).
Proof. exact: col_permEsub. 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
|
xcolEsub
| |
mxsub_id: mxsub id id =1 id.
Proof. by move=> A; 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_id
| |
eq_mxsubm' n' f f' g g' : f =1 f' -> g =1 g' ->
@mxsub m' n' f g =1 mxsub f' g'.
Proof. by move=> eq_f eq_g A; apply/matrixP => i j; rewrite !mxE eq_f eq_g. 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_mxsub
| |
eq_rowsubm' (f f' : 'I_m' -> 'I_m) : f =1 f' -> rowsub f =1 rowsub f'.
Proof. by move=> /eq_mxsub; apply. 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_rowsub
| |
eq_colsubn' (g g' : 'I_n' -> 'I_n) : g =1 g' -> colsub g =1 colsub g'.
Proof. by move=> /eq_mxsub; apply. 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_colsub
| |
mxsub_eq_idf g : f =1 id -> g =1 id -> mxsub f g =1 id.
Proof. by move=> fid gid A; rewrite (eq_mxsub fid gid) 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
|
mxsub_eq_id
| |
mxsub_eq_colsubn' f g : f =1 id -> @mxsub _ n' f g =1 colsub g.
Proof. by move=> f_id; apply: eq_mxsub. 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_eq_colsub
| |
mxsub_eq_rowsubm' f g : g =1 id -> @mxsub m' _ f g =1 rowsub f.
Proof. exact: eq_mxsub. 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_eq_rowsub
| |
mxsub_ffunlm' n' f g : @mxsub m' n' (finfun f) g =1 mxsub f g.
Proof. by apply: eq_mxsub => // i; rewrite ffunE. 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_ffunl
| |
mxsub_ffunrm' n' f g : @mxsub m' n' f (finfun g) =1 mxsub f g.
Proof. by apply: eq_mxsub => // i; rewrite ffunE. 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_ffunr
| |
mxsub_ffunm' n' f g : @mxsub m' n' (finfun f) (finfun g) =1 mxsub f g.
Proof. by move=> A; rewrite mxsub_ffunl mxsub_ffunr. 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_ffun
| |
mxsub_constm' n' f g a : @mxsub m' n' f g (const_mx a) = const_mx 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_const
| |
castmx_idm n erefl_mn (A : 'M_(m, n)) : castmx erefl_mn A = A.
Proof. by case: erefl_mn => e_m e_n; rewrite [e_m]eq_axiomK [e_n]eq_axiomK. 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
|
castmx_id
| |
castmx_compm1 n1 m2 n2 m3 n3 (eq_m1 : m1 = m2) (eq_n1 : n1 = n2)
(eq_m2 : m2 = m3) (eq_n2 : n2 = n3) A :
castmx (eq_m2, eq_n2) (castmx (eq_m1, eq_n1) A)
= castmx (etrans eq_m1 eq_m2, etrans eq_n1 eq_n2) A.
Proof.
by case: m2 / eq_m1 eq_m2; case: m3 /; case: n2 / eq_n1 eq_n2; case: n3 /.
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
|
castmx_comp
| |
castmxKm1 n1 m2 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) :
cancel (castmx (eq_m, eq_n)) (castmx (esym eq_m, esym eq_n)).
Proof. by case: m2 / eq_m; case: n2 / 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
|
castmxK
| |
castmxKVm1 n1 m2 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) :
cancel (castmx (esym eq_m, esym eq_n)) (castmx (eq_m, eq_n)).
Proof. by case: m2 / eq_m; case: n2 / 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
|
castmxKV
| |
castmx_symm1 n1 m2 n2 (eq_m : m1 = m2) (eq_n : n1 = n2) A1 A2 :
A1 = castmx (eq_m, eq_n) A2 -> A2 = castmx (esym eq_m, esym eq_n) A1.
Proof. by move/(canLR (castmxK _ _)). 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
|
castmx_sym
| |
eq_castmxm1 n1 m2 n2 (eq_mn eq_mn' : (m1 = m2) * (n1 = n2)) :
castmx eq_mn =1 castmx eq_mn'.
Proof.
case: eq_mn eq_mn' => [em en] [em' en'] A.
by apply: (canRL (castmxKV _ _)); rewrite castmx_comp castmx_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
|
eq_castmx
| |
castmxEm1 n1 m2 n2 (eq_mn : (m1 = m2) * (n1 = n2)) A i j :
castmx eq_mn A i j =
A (cast_ord (esym eq_mn.1) i) (cast_ord (esym eq_mn.2) j).
Proof.
by do [case: eq_mn; case: m2 /; case: n2 /] in A i j *; rewrite !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
|
castmxE
| |
conform_mx_idm n (B A : 'M_(m, n)) : conform_mx B A = A.
Proof. by rewrite /conform_mx; do 2!case: eqP => // *; rewrite castmx_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
|
conform_mx_id
| |
nonconform_mxm m' n n' (B : 'M_(m', n')) (A : 'M_(m, n)) :
(m != m') || (n != n') -> conform_mx B A = B.
Proof. by rewrite /conform_mx; do 2!case: eqP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
nonconform_mx
| |
conform_castmxm1 n1 m2 n2 m3 n3
(e_mn : (m2 = m3) * (n2 = n3)) (B : 'M_(m1, n1)) A :
conform_mx B (castmx e_mn A) = conform_mx B A.
Proof. by do [case: e_mn; case: m3 /; case: n3 /] in A *. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun finset fingroup perm order div",
"From mathcomp Require Import prime binomial ssralg countalg finalg zmodp bigop"
] |
algebra/matrix.v
|
conform_castmx
| |
trmxKm n : cancel (@trmx m n) (@trmx n m).
Proof. by move=> A; 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
|
trmxK
| |
trmx_injm n : injective (@trmx m n).
Proof. exact: can_inj (@trmxK m 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
|
trmx_inj
| |
trmx_castm1 n1 m2 n2 (eq_mn : (m1 = m2) * (n1 = n2)) A :
(castmx eq_mn A)^T = castmx (eq_mn.2, eq_mn.1) A^T.
Proof.
by case: eq_mn => eq_m eq_n; apply/matrixP=> i j; rewrite !(mxE, castmxE).
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_cast
| |
trmx_conformm' n' m n (B : 'M_(m', n')) (A : 'M_(m, n)) :
(conform_mx B A)^T = conform_mx B^T A^T.
Proof.
rewrite /conform_mx; do !case: eqP; rewrite ?mxE// => en em.
by rewrite trmx_cast.
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_conform
| |
tr_row_permm n s (A : 'M_(m, n)) : (row_perm s A)^T = col_perm s 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_perm
| |
tr_col_permm n s (A : 'M_(m, n)) : (col_perm s A)^T = row_perm s 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_perm
| |
tr_xrowm n i1 i2 (A : 'M_(m, n)) : (xrow i1 i2 A)^T = xcol i1 i2 A^T.
Proof. exact: tr_row_perm. 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_xrow
| |
tr_xcolm n j1 j2 (A : 'M_(m, n)) : (xcol j1 j2 A)^T = xrow j1 j2 A^T.
Proof. exact: tr_col_perm. 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_xcol
| |
row_idn i (V : 'rV_n) : row i V = V.
Proof. by apply/rowP=> j; rewrite mxE [i]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
|
row_id
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.