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 |
|---|---|---|---|---|---|---|
mul1q: left_id one mul.
Proof. by move=> x; rewrite -[x]reprK !piE mul1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] |
algebra/ring_quotient.v
|
mul1q
| |
mulq_addl: left_distributive mul +%R.
Proof.
move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK.
by apply/eqP; rewrite piE /= mulrDl equiv_refl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] |
algebra/ring_quotient.v
|
mulq_addl
| |
nonzero1q: one != 0.
Proof. by rewrite piE equivE subr0 idealr1. Qed.
#[export]
HB.instance Definition _ := GRing.Zmodule_isComNzRing.Build (quot idealI)
mulqA mulqC mul1q mulq_addl nonzero1q.
#[export]
HB.instance Definition _ := @isNzRingQuotient.Build
R (equiv idealI) 0 -%R +%R 1%R *%R (quot idealI) (lock _) pi_mul.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] |
algebra/ring_quotient.v
|
nonzero1q
| |
rquot_IdomainAxiom(x y : {quot I}): x * y = 0 -> (x == 0) || (y == 0).
Proof.
by move=> /eqP; rewrite -[x]reprK -[y]reprK !piE !equivE !subr0 prime_idealrM.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] |
algebra/ring_quotient.v
|
rquot_IdomainAxiom
| |
eq_map_mx_id(R : nzRingType) m n (M : 'M[R]_(m, n)) (f : R -> R) :
f =1 id -> M ^ f = M.
Proof. by move=> /eq_map_mx->; rewrite map_mx_id. Qed.
HB.mixin Record isInvolutive (R : nzRingType) (f : R -> R) :=
{ involutive_subproof : involutive f }.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
eq_map_mx_id
| |
DefinitionInvolutiveRMorphism (R : nzRingType) :=
{ f of @GRing.RMorphism R R f & @isInvolutive R f }.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Definition
| |
Definition_ := isInvolutive.Build _ _ idfunK.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Definition
| |
rmorphK(f : involutive_rmorphism R) : involutive f.
Proof. by move: f => [? [? ? []]]. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
rmorphK
| |
Definition_ :=
isInvolutive.Build _ conjC conjCfun_involutive.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Definition
| |
map_mxCK{C : numClosedFieldType} m n (A : 'M[C]_(m, n)) :
(A ^ conjC) ^ conjC = A.
Proof. by apply/matrixP=> i j; rewrite !mxE conjCK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_mxCK
| |
RecordisBilinear (R : nzRingType) (U U' : lmodType R) (V : zmodType)
(s : R -> V -> V) (s' : R -> V -> V) (f : U -> U' -> V) := {
zmod_morphisml_subproof : forall u', zmod_morphism (f ^~ u') ;
zmod_morphismr_subproof : forall u, zmod_morphism (f u) ;
linearl_subproof : forall u', scalable_for s (f ^~ u') ;
linearr_subproof : forall u, scalable_for s' (f u)
}.
#[short(type="bilinear")]
HB.structure Definition Bilinear (R : nzRingType) (U U' : lmodType R)
(V : zmodType) (s : R -> V -> V) (s' : R -> V -> V) :=
{f of isBilinear R U U' V s s' f}.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Record
| |
bilinear_for(R : nzRingType) (U U' : lmodType R) (V : zmodType)
(s : GRing.Scale.law R V) (s' : GRing.Scale.law R V) (f : U -> U' -> V) :=
((forall u', GRing.linear_for (s : R -> V -> V) (f ^~ u'))
* (forall u, GRing.linear_for s' (f u)))%type.
HB.factory Record bilinear_isBilinear (R : nzRingType) (U U' : lmodType R)
(V : zmodType) (s : GRing.Scale.law R V) (s' : GRing.Scale.law R V)
(f : U -> U' -> V) := {
bilinear_subproof : bilinear_for s s' f
}.
HB.builders Context R U U' V s s' f of bilinear_isBilinear R U U' V s s' f.
HB.instance Definition _ := isBilinear.Build R U U' V s s' f
(fun u' => zmod_morphism_linear (bilinear_subproof.1 u'))
(fun u => zmod_morphism_linear (bilinear_subproof.2 u))
(fun u' => scalable_linear (bilinear_subproof.1 u'))
(fun u => scalable_linear (bilinear_subproof.2 u)).
HB.end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
bilinear_for
| |
mapUUV:= (@Bilinear.type R U U' V s s').
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
mapUUV
| |
map_class:= mapUUV.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_class
| |
map_at_left(a : R) := mapUUV.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_at_left
| |
map_at_right(b : R) := mapUUV.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_at_right
| |
map_at_both(a b : R) := mapUUV.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_at_both
| |
map_for_lefta s_a :=
MapForLeft {map_for_left_map : mapUUV; _ : s a = s_a }.
|
Structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_for_left
| |
map_for_rightb s'_b :=
MapForRight {map_for_right_map : mapUUV; _ : s' b = s'_b }.
|
Structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_for_right
| |
map_for_botha b s_a s'_b :=
MapForBoth {map_for_both_map : mapUUV; _ : s a = s_a ; _ : s' b = s'_b }.
|
Structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
map_for_both
| |
unify_map_at_lefta (f : map_at_left a) :=
MapForLeft f (erefl (s a)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
unify_map_at_left
| |
unify_map_at_rightb (f : map_at_right b) :=
MapForRight f (erefl (s' b)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
unify_map_at_right
| |
unify_map_at_botha b (f : map_at_both a b) :=
MapForBoth f (erefl (s a)) (erefl (s' b)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
unify_map_at_both
| |
wrapped:= Wrap {unwrap : mapUUV}.
|
Structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
wrapped
| |
wrap(f : map_class) := Wrap f.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
wrap
| |
Definition_ (R : nzRingType) (U U' : lmodType R) (V : zmodType)
(s : R -> V -> V) (s' : R -> V -> V)
(f : {bilinear U -> U' -> V | s & s'}) (u : U)
:= @GRing.isZmodMorphism.Build U' V (f u) (@zmod_morphismr_subproof _ _ _ _ _ _ f u).
#[non_forgetful_inheritance]
HB.instance Definition _ (R : nzRingType) (U U' : lmodType R) (V : zmodType)
(s : R -> V -> V) (s' : R -> V -> V) (f : @bilinear R U U' V s s') (u : U)
:= @GRing.isScalable.Build _ _ _ _ (f u) (@linearr_subproof _ _ _ _ _ _ f u).
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Definition
| |
applyr_headt (f : U -> U' -> V) u v := let: tt := t in f v u.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
applyr_head
| |
applyr:= (applyr_head tt).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
applyr
| |
Bilinear.map_for_left_map : Bilinear.map_for_left >-> Bilinear.type.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.map_for_right_map : Bilinear.map_for_right >-> Bilinear.type.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.map_for_both_map : Bilinear.map_for_both >-> Bilinear.type.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.unify_map_at_left : Bilinear.map_at_left >-> Bilinear.map_for_left.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.unify_map_at_right : Bilinear.map_at_right >-> Bilinear.map_for_right.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.unify_map_at_both : Bilinear.map_at_both >-> Bilinear.map_for_both.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.unify_map_at_left.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.unify_map_at_right.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.unify_map_at_both.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.unwrap : Bilinear.wrapped >-> Bilinear.type.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.wrap : Bilinear.map_class >-> Bilinear.wrapped.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
Bilinear.wrap.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Bilinear
| |
linear0r: f z 0 = 0. Proof. by rewrite raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linear0r
| |
linearNr: {morph f z : x / - x}. Proof. exact: raddfN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearNr
| |
linearDr: {morph f z : x y / x + y}. Proof. exact: raddfD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearDr
| |
linearBr: {morph f z : x y / x - y}. Proof. exact: raddfB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearBr
| |
linearMnrn : {morph f z : x / x *+ n}. Proof. exact: raddfMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearMnr
| |
linearMNnrn : {morph f z : x / x *- n}. Proof. exact: raddfMNn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearMNnr
| |
linear_sumrI r (P : pred I) E :
f z (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f z (E i).
Proof. exact: raddf_sum. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linear_sumr
| |
linearZr_LR: scalable_for s' (f z). Proof. exact: linearZ_LR. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearZr_LR
| |
linearPra : {morph f z : u v / a *: u + v >-> s' a u + v}.
Proof. exact: linearP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearPr
| |
applyrEx : applyr f x =1 f^~ x. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
applyrE
| |
Definition_ :=
GRing.isZmodMorphism.Build _ _ (applyr f z) (@zmod_morphisml_subproof _ _ _ _ _ _ f z).
HB.instance Definition _ :=
GRing.isScalable.Build _ _ _ _ (applyr f z) (@linearl_subproof _ _ _ _ _ _ f z).
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Definition
| |
linear0l: f 0 z = 0. Proof. by rewrite -applyrE raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linear0l
| |
linearNl: {morph f^~ z : x / - x}.
Proof. by move=> ?; rewrite -applyrE raddfN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearNl
| |
linearDl: {morph f^~ z : x y / x + y}.
Proof. by move=> ? ?; rewrite -applyrE raddfD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearDl
| |
linearBl: {morph f^~ z : x y / x - y}.
Proof. by move=> ? ?; rewrite -applyrE raddfB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearBl
| |
linearMnln : {morph f^~ z : x / x *+ n}.
Proof. by move=> ?; rewrite -applyrE raddfMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearMnl
| |
linearMNnln : {morph f^~ z : x / x *- n}.
Proof. by move=> ?; rewrite -applyrE raddfMNn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearMNnl
| |
linear_sumlzI r (P : pred I) E :
f (\sum_(i <- r | P i) E i) z = \sum_(i <- r | P i) f (E i) z.
Proof. by rewrite -applyrE raddf_sum. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linear_sumlz
| |
linearZl_LR: scalable_for s (f ^~ z).
Proof. by move=> ? ?; rewrite -applyrE linearZ_LR. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearZl_LR
| |
linearPla : {morph f^~ z : u v / a *: u + v >-> s a u + v}.
Proof. by move=> ? ?; rewrite -applyrE linearP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearPl
| |
linearZlz (c : S) (a : R) (h_c := h c)
(f : Bilinear.map_for_left U U' s s' a h_c) u :
f (a *: u) z = h_c (Bilinear.wrap f u z).
Proof. by rewrite linearZl_LR; case: f => f /= ->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearZl
| |
linearZrz c' b (h'_c' := h' c')
(f : Bilinear.map_for_right U U' s s' b h'_c') u :
f z (b *: u) = h'_c' (Bilinear.wrap f z u).
Proof. by rewrite linearZr_LR; case: f => f /= ->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearZr
| |
linearZlrc c' a b (h_c := h c) (h'_c' := h' c')
(f : Bilinear.map_for_both U U' s s' a b h_c h'_c') u v :
f (a *: u) (b *: v) = h_c (h'_c' (Bilinear.wrap f u v)).
Proof. by rewrite linearZl_LR linearZ_LR; case: f => f /= -> ->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearZlr
| |
linearZrlc c' a b (h_c := h c) (h'_c' := h' c')
(f : Bilinear.map_for_both U U' s s' a b h_c h'_c') u v :
f (a *: u) (b *: v) = h'_c' (h_c (Bilinear.wrap f u v)).
Proof. by rewrite linearZ_LR/= linearZl_LR; case: f => f /= -> ->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearZrl
| |
rev_mulmx(R : nzRingType) m n p := [revop mulmxr of @mulmx R m n p].
*)
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
rev_mulmx
| |
mulmx_is_bilinear(R : comNzRingType) m n p : bilinear_for
(GRing.Scale.Law.clone _ _ *:%R _) (GRing.Scale.Law.clone _ _ *:%R _)
(@mulmx R m n p).
Proof.
split=> [u'|u] a x y /=.
- by rewrite mulmxDl scalemxAl.
- by rewrite mulmxDr scalemxAr.
Qed.
HB.instance Definition _ (R : comNzRingType) m n p := bilinear_isBilinear.Build R
[the lmodType R of 'M[R]_(m, n)] [the lmodType R of 'M[R]_(n, p)]
[the zmodType of 'M[R]_(m, p)] _ _ (@mulmx R m n p)
(mulmx_is_bilinear R m n p).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
mulmx_is_bilinear
| |
formu v := (u *m M *m (v ^t theta)) 0 0.
Local Notation "''[' u , v ]" := (form u%R v%R) : ring_scope.
Local Notation "''[' u ]" := '[u, u] : ring_scope.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
form
| |
form0lu : '[0, u] = 0. Proof. by rewrite /form !mul0mx mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
form0l
| |
form0ru : '[u, 0] = 0.
Proof. by rewrite /form trmx0 map_mx0 mulmx0 mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
form0r
| |
formDlu v w : '[u + v, w] = '[u, w] + '[v, w].
Proof. by rewrite /form !mulmxDl mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
formDl
| |
formDru v w : '[u, v + w] = '[u, v] + '[u, w].
Proof. by rewrite /form linearD !map_mxD !mulmxDr mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
formDr
| |
formZra u v : '[u, a *: v] = theta a * '[u, v].
Proof. by rewrite /form !(linearZ, map_mxZ) /= mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
formZr
| |
formZla u v : '[a *: u, v] = a * '[u, v].
Proof.
by do !rewrite /form -[_ *: _ *m _]/(mulmxr _ _) linearZ /=; rewrite mxE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
formZl
| |
formNlu v : '[- u, v] = - '[u, v].
Proof. by rewrite -scaleN1r formZl mulN1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
formNl
| |
formNru v : '[u, - v] = - '[u, v].
Proof. by rewrite -scaleN1r formZr rmorphN1 mulN1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
formNr
| |
formeei j : '['e_i, 'e_j] = M i j.
Proof.
rewrite /form -rowE -map_trmx map_delta_mx -[M in LHS]trmxK.
by rewrite -tr_col -trmx_mul -rowE !mxE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
formee
| |
form0_eq0: M = 0 -> forall u v, '[u, v] = 0.
Proof. by rewrite/form=> -> u v; rewrite mulmx0 mul0mx mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
form0_eq0
| |
RecordisHermitianSesquilinear (R : nzRingType) (U : lmodType R)
(eps : bool) (theta : R -> R) (f : U -> U -> R) := {
hermitian_subproof : forall x y : U, f x y = (-1) ^+ eps * theta (f y x)
}.
HB.structure Definition Hermitian (R : nzRingType) (U : lmodType R)
(eps : bool) (theta : R -> R) :=
{f of @Bilinear R U U _ ( *%R ) (theta \; *%R) f &
@isHermitianSesquilinear R U eps theta f}.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Record
| |
Definition_ (R : nzRingType) (U : lmodType R)
(eps : bool) (theta : R -> R) (f : {hermitian U for eps & theta}) (u : U) :=
@GRing.isZmodMorphism.Build _ _ (f u) (@zmod_morphismr_subproof _ _ _ _ _ _ f u).
#[non_forgetful_inheritance]
HB.instance Definition _ (R : nzRingType) (U : lmodType R)
(eps : bool) (theta : R -> R) (f : {hermitian U for eps & theta}) (u : U) :=
@GRing.isScalable.Build _ _ _ _ (f u) (@linearr_subproof _ _ _ _ _ _ f u).
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Definition
| |
axiom(f : U -> U -> R) :=
forall x y : U, f x y = (-1) ^+ eps * theta (f y x).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
axiom
| |
class_of(f : U -> U -> R) : Prop := Class {
base : Bilinear.class_of ( *%R) (theta \; *%R) f;
mixin : axiom f
}.*)
|
Record
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
class_of
| |
linearr(u : U) := Linear (base class u).
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearr
| |
additivel(u' : U) := @GRing.Additive.Pack _ _ (Phant (U -> R))
(applyr cF u') (Bilinear.basel (base class) u').
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
additivel
| |
linearl(u' : U) := @GRing.Linear.Pack _ _ _ _ (Phant (U -> R))
(applyr cF u') (Bilinear.basel (base class) u').
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearl
| |
bilinear:= @Bilinear.Pack _ _ _ _ _ _ (Phant (U -> U -> R)) cF (base class).*)
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
bilinear
| |
base: class_of >-> bilmorphism_for.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
base
| |
apply: map >-> Funclass.
|
Coercion
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
apply
| |
hermitian_for:= Hermitian.axiom.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
hermitian_for
| |
HermitianfM := (pack (Phant _) fM idfun).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
Hermitian
| |
additiver.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
additiver
| |
linearr.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearr
| |
additivel.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
additivel
| |
linearl.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
linearl
| |
bilinear.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
bilinear
| |
hermapplyr:= (@applyr_head _ _ _ _ tt).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
hermapplyr
| |
orthomx{R : fieldType} (theta : R -> R) n m M (B : 'M_(m, n)) : 'M_n :=
kermx (M *m (B ^t theta)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
orthomx
| |
sesqui:=
[qualify M : 'M_n | M == ((-1) ^+ eps_theta.1) *: M ^t eps_theta.2].
Fact sesqui_key : pred_key sesqui. Proof. by []. Qed.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
sesqui
| |
sesqui_keyed:= KeyedQualifier sesqui_key.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
sesqui_keyed
| |
sesquiE: (M \is (eps, theta).-sesqui) = (M == (-1) ^+ eps *: M ^t theta).
Proof. by rewrite qualifE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
sesquiE
| |
sesquiP: reflect (M = (-1) ^+ eps *: M ^t theta)
(M \is (eps, theta).-sesqui).
Proof. by rewrite sesquiE; exact/eqP. Qed.
Hypotheses (thetaK : involutive theta) (M_sesqui : M \is (eps, theta).-sesqui).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple bigop ssralg finset fingroup",
"From mathcomp Require Import zmodp poly order ssrnum matrix mxalgebra vector"
] |
algebra/sesquilinear.v
|
sesquiP
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.