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 |
|---|---|---|---|---|---|---|
trmx_sesqui: M^T = (-1) ^+ eps *: M ^ theta.
Proof.
rewrite [in LHS](sesquiP _) // -mul_scalar_mx trmx_mul.
by rewrite tr_scalar_mx mul_mx_scalar map_trmx trmxK.
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
|
trmx_sesqui
| |
maptrmx_sesqui: M^t theta = (-1) ^+ eps *: M.
Proof.
by rewrite trmx_sesqui map_mxZ rmorph_sign -map_mx_comp eq_map_mx_id.
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
|
maptrmx_sesqui
| |
formCu v : '[u, v] = (-1) ^+ eps * theta '[v, u].
Proof.
rewrite /form [M in LHS](sesquiP _) // -mulmxA !mxE rmorph_sum mulr_sumr.
apply: eq_bigr => /= i _; rewrite !(mxE, mulr_sumr, mulr_suml, rmorph_sum).
apply: eq_bigr => /= j _; rewrite !mxE !rmorphM mulrCA -!mulrA.
by congr (_ * _); rewrite mulrA mulrC /= thetaK.
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
|
formC
| |
form_eq0Cu v : ('[u, v] == 0) = ('[v, u] == 0).
Proof. by rewrite formC mulf_eq0 signr_eq0 /= fmorph_eq0. 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
|
form_eq0C
| |
orthom (B : 'M_(m, n)) := orthomx theta M B.
Local Notation "B ^_|_" := (ortho B) : ring_scope.
Local Notation "A '_|_ B" := (A%MS <= B^_|_)%MS : 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
|
ortho
| |
normalEu v : (u '_|_ v) = ('[u, v] == 0).
Proof.
by rewrite (sameP sub_kermxP eqP) mulmxA [_ *m _^t _]mx11_scalar fmorph_eq0.
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
|
normalE
| |
form_eq0P{u v} : reflect ('[u, v] = 0) (u '_|_ v).
Proof. by rewrite normalE; apply/eqP. 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
|
form_eq0P
| |
normalPp q (A : 'M_(p, n)) (B :'M_(q, n)) :
reflect (forall (u v : 'rV_n), (u <= A)%MS -> (v <= B)%MS -> u '_|_ v)
(A '_|_ B).
Proof.
apply: (iffP idP) => AnB.
move=> u v uA vB; rewrite (submx_trans uA) // (submx_trans AnB) //.
apply/sub_kermxP; have /submxP [w ->] := vB.
rewrite trmx_mul map_mxM !mulmxA -[kermx _ *m _ *m _]mulmxA.
by rewrite [kermx _ *m _](sub_kermxP _) // mul0mx.
apply/rV_subP => u /AnB /(_ _) /sub_kermxP uMv; apply/sub_kermxP.
suff: forall m (v : 'rV[R]_m),
(forall i, v *m 'e_i ^t theta = 0 :> 'M_1) -> v = 0.
apply => i; rewrite !mulmxA -!mulmxA -map_mxM -trmx_mul uMv //.
by apply/submxP; exists 'e_i.
move=> /= m v Hv; apply: (can_inj (@trmxK _ _ _)).
rewrite trmx0; apply/row_matrixP=> i; rewrite row0 rowE.
apply: (can_inj (@trmxK _ _ _)); rewrite trmx0 trmx_mul trmxK.
by rewrite -(map_delta_mx theta) map_trmx Hv.
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
|
normalP
| |
normalCp q (A : 'M_(p, n)) (B : 'M_(q, n)) : (A '_|_ B) = (B '_|_ A).
Proof.
gen have nC : p q A B / A '_|_ B -> B '_|_ A; last by apply/idP/idP; apply/nC.
move=> AnB; apply/normalP => u v ? ?; rewrite normalE.
rewrite formC mulf_eq0 ?fmorph_eq0 ?signr_eq0 /=.
by rewrite -normalE (normalP _ _ AnB).
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
|
normalC
| |
normal_ortho_mxp (A : 'M_(p, n)) : ((A^_|_) '_|_ A).
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
|
normal_ortho_mx
| |
normal_mx_orthop (A : 'M_(p, n)) : (A '_|_ (A^_|_)).
Proof. by rewrite normalC. 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
|
normal_mx_ortho
| |
rank_normalu : (\rank (u ^_|_) >= n.-1)%N.
Proof.
rewrite mxrank_ker -subn1 leq_sub2l //.
by rewrite (leq_trans (mxrankM_maxr _ _)) // rank_leq_col.
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
|
rank_normal
| |
rad:= 1%:M^_|_.
|
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
|
rad
| |
rad_ker: rad = kermx M.
Proof. by rewrite /rad /ortho /orthomx trmx1 map_mx1 mulmx1. 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
|
rad_ker
| |
formDdu v : u '_|_ v -> '[u + v] = '[u] + '[v].
Proof.
move=> uNv; rewrite formDl !formDr ['[v, u]]formC.
by rewrite ['[u, v]](form_eq0P _) // rmorph0 mulr0 addr0 add0r.
Qed.
|
Theorem
|
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
|
formDd
| |
formZa u : '[a *: u]= (a * theta a) * '[u].
Proof. by rewrite formZl formZr mulrA. 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
|
formZ
| |
formNu : '[- u] = '[u].
Proof. by rewrite formNr formNl opprK. 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
|
formN
| |
form_signm u : '[(-1) ^+ m *: u] = '[u].
Proof. by rewrite -signr_odd scaler_sign; case: odd; rewrite ?formN. 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
|
form_sign
| |
formDu v : let d := '[u, v] in
'[u + v] = '[u] + '[v] + (d + (-1) ^+ eps * theta d).
Proof. by rewrite formDl !formDr ['[v, _]]formC [_ + '[v]]addrC addrACA. 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
|
formD
| |
formBu v : let d := '[u, v] in
'[u - v] = '[u] + '[v] - (d + (-1) ^+ eps * theta d).
Proof. by rewrite formD formN !formNr rmorphN mulrN -opprD. 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
|
formB
| |
formBdu v : u '_|_ v -> '[u - v] = '[u] + '[v].
Proof.
by move=> uTv; rewrite formDd ?formN // normalE formNr oppr_eq0 -normalE.
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
|
formBd
| |
symmetric_form:= (false, idfun).-sesqui.
|
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
|
symmetric_form
| |
skew:= (true, idfun).-sesqui.
|
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
|
skew
| |
hermitian:= (false, conjC).-sesqui.
HB.mixin Record isDotProduct (R : numDomainType) (U : lmodType R)
(op : U -> U -> R) := { neq0_dnorm_gt0 : forall u, u != 0 -> 0 < op u u }.
HB.structure Definition Dot (R : numDomainType) (U : lmodType R)
(theta : R -> R) :=
{op of isDotProduct R U op & @Hermitian R U false theta op}.
|
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
| |
Definition_ (R : numDomainType) (U : lmodType R)
(theta : R -> R) (f : {dot U for theta}) (u : U) :=
@GRing.isZmodMorphism.Build _ _ (f u) (@zmod_morphismr_subproof _ _ _ _ _ _ f u).
#[non_forgetful_inheritance]
HB.instance Definition _ (R : numDomainType) (U : lmodType R)
(theta : R -> R) (f : {dot U for 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
| |
base: class_of >-> Hermitian.class_of.
|
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
| |
DotfM := (pack 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
|
Dot
| |
is_dot:= Dot.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
|
is_dot
| |
is_skew(R : nzRingType) (eps : bool) (theta : R -> R)
(U : lmodType R) (form : {hermitian U for eps & theta}) :=
(eps = true) /\ (theta =1 id).
|
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
|
is_skew
| |
is_sym(R : nzRingType) (eps : bool) (theta : R -> R)
(U : lmodType R) (form : {hermitian U for eps & theta}) :=
(eps = false) /\ (theta =1 id).
|
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
|
is_sym
| |
is_hermsym(R : nzRingType) (eps : bool) (theta : R -> R)
(U : lmodType R) (form : {hermitian U for eps & theta}) :=
(eps = false).
|
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
|
is_hermsym
| |
hermCu v : '[u, v] = (-1) ^+ eps * theta '[v, u].
Proof. by move: form => [? [[? ? ? ?] []]] /=. 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
|
hermC
| |
hnormNu : '[- u] = '[u].
Proof. by rewrite linearNl linearNr opprK. 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
|
hnormN
| |
hnorm_signn u : '[(-1) ^+ n *: u] = '[u].
Proof. by rewrite -signr_odd scaler_sign; case: (odd n); rewrite ?hnormN. 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
|
hnorm_sign
| |
hnormDu v :
let d := '[u, v] in '[u + v] = '[u] + '[v] + (d + (-1) ^+ eps * theta d).
Proof. by rewrite /= addrAC -hermC linearDl 2!linearDr !addrA. 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
|
hnormD
| |
hnormBu v :
let d := '[u, v] in '[u - v] = '[u] + '[v] - (d + (-1) ^+ eps * theta d).
Proof.
by rewrite /= hnormD hnormN linearNr addrA rmorphN mulrN opprD addrA.
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
|
hnormB
| |
hnormDdu v : '[u, v] = 0 -> '[u + v] = '[u] + '[v].
Proof. by move=> ouv; rewrite hnormD ouv rmorph0 mulr0 !addr0. 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
|
hnormDd
| |
hnormBdu v : '[u, v] = 0 -> '[u - v] = '[u] + '[v].
Proof.
by move=> ouv; rewrite hnormDd ?hnormN// linearNr [X in - X]ouv oppr0.
Qed.
Local Notation "u '_|_ v" := ('[u, v] == 0) : ring_scope.
|
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
|
hnormBd
| |
ortho_rec(s1 s2 : seq U) :=
all [pred u | all [pred v | u '_|_ v] s2] s1.
|
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
|
ortho_rec
| |
pair_ortho_rec(s : seq U) :=
if s is v :: s' then ortho_rec [:: v] s' && pair_ortho_rec s' else true.
|
Fixpoint
|
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
|
pair_ortho_rec
| |
pairwise_orthogonals := (0 \notin s) && pair_ortho_rec s.
|
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
|
pairwise_orthogonal
| |
orthogonals1 s2 := (@ortho_rec s1 s2).
Arguments orthogonal : simpl never.
|
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
|
orthogonal
| |
orthogonal_consu us vs :
orthogonal (u :: us) vs = orthogonal [:: u] vs && orthogonal us vs.
Proof. by rewrite /orthogonal /= andbT. 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
|
orthogonal_cons
| |
orthonormals := all [pred v | '[v] == 1] s && pair_ortho_rec s.
|
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
|
orthonormal
| |
orthonormal_not0S : orthonormal S -> 0 \notin S.
Proof.
by case/andP=> /allP S1 _; rewrite (contra (S1 _)) //= linear0r eq_sym oner_eq0.
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
|
orthonormal_not0
| |
orthonormalES :
orthonormal S = all [pred phi | '[phi] == 1] S && pairwise_orthogonal S.
Proof. by rewrite -(andb_idl (@orthonormal_not0 S)) andbCA. 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
|
orthonormalE
| |
orthonormal_orthogonalS : orthonormal S -> pairwise_orthogonal S.
Proof. by rewrite orthonormalE => /andP[_]. 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
|
orthonormal_orthogonal
| |
isometrytau := forall u v, form1 (tau u) (tau v) = form2 u%R v%R.
|
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
|
isometry
| |
isometry_from_tomD tau mR :=
prop_in2 mD (inPhantom (isometry tau)) /\
prop_in1 mD (inPhantom (forall u, in_mem (tau u) mR)).
Local Notation "{ 'in' D , 'isometry' tau , 'to' R }" :=
(isometry_from_to (mem D) tau (mem R))
(format "{ 'in' D , 'isometry' tau , 'to' R }")
: type_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
|
isometry_from_to
| |
herm_eq0Cu v : ('[u, v] == 0) = ('[v, u] == 0).
Proof. by rewrite hermC mulf_eq0 signr_eq0 /= fmorph_eq0. 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
|
herm_eq0C
| |
orthovV := (\bigcap_(i < \dim V) lker (alpha (vbasis V)`_i))%VS.
Local Notation "U '_|_ V" := (U <= orthov V)%VS : vspace_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
|
orthov
| |
mem_orthovPnV u : reflect (exists2 v, v \in V & '[u, v] != 0) (u \notin orthov V).
Proof.
apply: (iffP idP) => [u_orthovV|[v /coord_vbasis-> uvNorthov]]; last first.
apply/subv_bigcapP => uP.
rewrite linear_sumr big1 ?eqxx//= in uvNorthov.
move=> i _; have := uP i isT.
by rewrite -memvE memv_ker lfunE/= linearZr/= => /eqP/= ->; rewrite mulr0.
suff /existsP [i ui_neq0] : [exists i : 'I_(\dim V), '[u, (vbasis V)`_i] != 0].
by exists (vbasis V)`_i => //; rewrite vbasis_mem ?mem_nth ?size_tuple.
apply: contraNT u_orthovV; rewrite negb_exists => /forallP ui_eq0.
apply/subv_bigcapP => i _.
by rewrite -memvE memv_ker lfunE /= -[_ == _]negbK.
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
|
mem_orthovPn
| |
mem_orthovPV u : reflect {in V, forall v, '[u, v] = 0} (u \in orthov V).
Proof.
apply: (iffP idP) => [/mem_orthovPn orthovNu v vV|/(_ _ _)/eqP orthov_u].
by apply/eqP/negP=> /negP Northov_uv; apply: orthovNu; exists v.
by apply/mem_orthovPn => -[v /orthov_u->].
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
|
mem_orthovP
| |
orthov1Eu : orthov <[u]> = lker (alpha u).
Proof.
apply/eqP; rewrite eqEsubv; apply/andP.
split; apply/subvP=> v; rewrite memv_ker lfunE /=.
by move=> /mem_orthovP-> //; rewrite ?memv_line.
move=> vu_eq0; apply/mem_orthovP => w /vlineP[k->].
by apply/eqP; rewrite linearZ mulf_eq0 vu_eq0 orbT.
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
|
orthov1E
| |
orthovPU V : reflect {in U & V, forall u v, '[u, v] = 0} (U '_|_ V)%VS.
Proof.
apply: (iffP subvP); last by move=> H ??; apply/mem_orthovP=> ??; apply: H.
by move=> /(_ _ _)/mem_orthovP; move=> H ????; apply: H.
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
|
orthovP
| |
orthov_symU V : (U '_|_ V)%VS = (V '_|_ U)%VS.
Proof. by apply/orthovP/orthovP => eq0 ????; apply/eqP; rewrite herm_eq0C eq0. 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
|
orthov_sym
| |
mem_orthov1v u : (u \in orthov <[v]>) = ('[u, v] == 0).
Proof. by rewrite orthov1E memv_ker lfunE. 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
|
mem_orthov1
| |
orthov11u v : (<[u]> '_|_ <[v]>)%VS = ('[u, v] == 0).
Proof. exact: mem_orthov1. 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
|
orthov11
| |
mem_orthov1_symv u : (u \in orthov <[v]>) = (v \in orthov <[u]>).
Proof. exact: orthov_sym. 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
|
mem_orthov1_sym
| |
orthov0: orthov 0 = fullv.
Proof.
apply/eqP; rewrite eqEsubv subvf.
apply/subvP => x _; rewrite mem_orthov1.
by rewrite linear0r.
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
|
orthov0
| |
mem_orthov_symV u : (u \in orthov V) = (V <= orthov <[u]>)%VS.
Proof. exact: orthov_sym. 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
|
mem_orthov_sym
| |
leq_dim_orthov1u V : ((\dim V).-1 <= \dim (V :&: orthov <[u]>))%N.
Proof.
rewrite -(limg_ker_dim (alpha u) V) -orthov1E.
have := dimvS (subvf (alpha u @: V)); rewrite dimvf addnC.
by case: (\dim _) => [|[]] // _; rewrite leq_pred.
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
|
leq_dim_orthov1
| |
dim_img_form_eq1u V : u \notin orthov V -> \dim (alpha u @: V)%VS = 1%N.
Proof.
move=> /mem_orthovPn [v vV Northov_uv]; apply/eqP; rewrite eqn_leq /=.
rewrite -[1%N as X in (_ <= X)%N](dimvf [the vectType F of F^o]) dimvS ?subvf//=.
have := @dimvS _ _ <['[v, u] : F^o]> (alpha u @: V).
rewrite -memvE dim_vline herm_eq0C Northov_uv; apply.
by apply/memv_imgP; exists v; rewrite ?memvf// !lfunE /=.
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
|
dim_img_form_eq1
| |
eq_dim_orthov1u V : u \notin orthov V -> (\dim V).-1 = \dim (V :&: orthov <[u]>).
Proof.
rewrite -(limg_ker_dim (alpha u) V) => /dim_img_form_eq1->.
by rewrite -orthov1E addn1.
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
|
eq_dim_orthov1
| |
dim_img_form_eq0u V : u \in orthov V -> \dim (alpha u @: V)%VS = 0%N.
Proof. by move=> uV; apply/eqP; rewrite dimv_eq0 -lkerE -orthov1E orthov_sym. 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
|
dim_img_form_eq0
| |
neq_dim_orthov1u V : (\dim V > 0)%N ->
u \in orthov V -> ((\dim V).-1 < \dim (V :&: orthov <[u]>))%N.
Proof.
move=> V_gt0; rewrite -(limg_ker_dim (alpha u) V) -orthov1E => u_in.
rewrite dim_img_form_eq0 // addn0 (capv_idPl _) 1?orthov_sym //.
by case: (\dim _) V_gt0.
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
|
neq_dim_orthov1
| |
leqif_dim_orthov1u V : (\dim V > 0)%N ->
((\dim V).-1 <= \dim (V :&: orthov <[u]>) ?= iff (u \notin orthov V))%N.
Proof.
move=> Vr_gt0; apply/leqifP.
by case: (boolP (u \in _)) => /= [/neq_dim_orthov1->|/eq_dim_orthov1->].
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
|
leqif_dim_orthov1
| |
leqif_dim_orthov1_fullu : (n > 0)%N ->
((\dim {:vT}).-1 <= \dim (orthov <[u]>) ?= iff (u \notin orthov fullv))%N.
Proof.
by move=> n_gt0; have := @leqif_dim_orthov1 u fullv; rewrite capfv; apply.
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
|
leqif_dim_orthov1_full
| |
orthogonal1Pu v : reflect ('[u, v] = 0) (orthogonal form [:: u] [:: v]).
Proof. by rewrite /orthogonal /= !andbT; apply: eqP. 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
|
orthogonal1P
| |
orthogonalPus vs :
reflect {in us & vs, forall u v, '[u, v] = 0} (orthogonal form us vs).
Proof.
apply: (iffP allP) => ousvs u => [v /ousvs/allP opus /opus/eqP // | /ousvs opus].
by apply/allP=> v /= /opus->.
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
|
orthogonalP
| |
orthogonal_opprS R : orthogonal form S (map -%R R) = orthogonal form S R.
Proof.
wlog suffices IH: S R / orthogonal form S R -> orthogonal form S (map -%R R).
by apply/idP/idP=> /IH; rewrite ?mapK //; apply: opprK.
move/orthogonalP=> oSR; apply/orthogonalP=> xi1 _ Sxi1 /mapP[xi2 Rxi2 ->].
by rewrite linearNr /= oSR ?oppr0.
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
|
orthogonal_oppr
| |
orthogonalEus vs : (orthogonal form us vs) = (<<us>> '_|_ <<vs>>)%VS.
Proof.
apply/orthogonalP/orthovP => uvsP u v; last first.
by move=> uus vvs; rewrite uvsP // memv_span.
rewrite -[us]in_tupleE -[vs]in_tupleE => /coord_span-> /coord_span->.
rewrite linear_sumr big1 //= => i _.
rewrite linear_sumlz big1 //= => j _.
by rewrite linearZlr/= uvsP ?mulr0// mem_nth.
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
|
orthogonalE
| |
orthovEU V : (U '_|_ V)%VS = orthogonal form (vbasis U) (vbasis V).
Proof. by rewrite orthogonalE !(span_basis (vbasisP _)). 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
|
orthovE
| |
radv:= (orthov fullv).
|
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
|
radv
| |
orthoDvU V W : (U + V '_|_ W)%VS = (U '_|_ W)%VS && (V '_|_ W)%VS.
Proof. by rewrite subv_add. 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
|
orthoDv
| |
orthovDU V W : (U '_|_ V + W)%VS = (U '_|_ V)%VS && (U '_|_ W)%VS.
Proof. by rewrite ![(U '_|_ _)%VS]orthov_sym orthoDv. 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
|
orthovD
| |
nondegenerate:= radv == 0%VS.
|
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
|
nondegenerate
| |
is_psymplectic:= [/\ nondegenerate, is_skew form &
2 \in [pchar F] -> forall u, '[u, u] = 0].
|
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
|
is_psymplectic
| |
is_porthogonal:= [/\ nondegenerate, is_sym form &
2 \in [pchar F] -> forall u, '[u, u] = 0].
|
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
|
is_porthogonal
| |
is_unitary:= nondegenerate /\ (is_hermsym form).
|
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
|
is_unitary
| |
is_symplectic:= is_psymplectic (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use is_porthogonal instead.")]
|
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
|
is_symplectic
| |
is_orthogonal:= is_porthogonal (only parsing).
Arguments orthogonalP {F eps theta vT form us vs}.
Arguments orthovP {F eps theta vT form U V}.
Arguments mem_orthovPn {F eps theta vT form V u}.
Arguments mem_orthovP {F eps theta vT form V u}.
|
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
|
is_orthogonal
| |
dnorm_geiff0u : 0 <= '[u] ?= iff (u == 0).
Proof.
by apply/leifP; have [->|uN0] := altP eqP; rewrite ?linear0r ?neq0_dnorm_gt0.
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
|
dnorm_geiff0
| |
dnorm_ge0u : 0 <= '[u]. Proof. by rewrite dnorm_geiff0. 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
|
dnorm_ge0
| |
dnorm_eq0u : ('[u] == 0) = (u == 0).
Proof. by rewrite -dnorm_geiff0 eq_sym. 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
|
dnorm_eq0
| |
dnorm_gt0u : (0 < '[u]) = (u != 0).
Proof. by rewrite lt_def dnorm_eq0 dnorm_ge0 andbT. 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
|
dnorm_gt0
| |
sqrt_dnorm_ge0u : 0 <= sqrtC '[u].
Proof. by rewrite sqrtC_ge0 dnorm_ge0. 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
|
sqrt_dnorm_ge0
| |
sqrt_dnorm_eq0u : (sqrtC '[u] == 0) = (u == 0).
Proof. by rewrite sqrtC_eq0 dnorm_eq0. 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
|
sqrt_dnorm_eq0
| |
sqrt_dnorm_gt0u : (sqrtC '[u] > 0) = (u != 0).
Proof. by rewrite sqrtC_gt0 dnorm_gt0. 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
|
sqrt_dnorm_gt0
| |
dnormZa u : '[a *: u]= `|a| ^+ 2 * '[u].
Proof. by rewrite linearZl_LR linearZr_LR/= mulrA normCK. 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
|
dnormZ
| |
dnormDu v : let d := '[u, v] in '[u + v] = '[u] + '[v] + (d + d^*).
Proof. by rewrite hnormD mul1r. 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
|
dnormD
| |
dnormBu v : let d := '[u, v] in '[u - v] = '[u] + '[v] - (d + d^*).
Proof. by rewrite hnormB mul1r. 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
|
dnormB
| |
pairwise_orthogonalPS :
reflect (uniq (0 :: S)
/\ {in S &, forall phi psi, phi != psi -> '[phi, psi] = 0})
(pairwise_orthogonal form S).
Proof.
rewrite /pairwise_orthogonal /=; case notS0: (~~ _); last by right; case.
elim: S notS0 => [|phi S IH] /=; first by left.
rewrite inE eq_sym andbT => /norP[nz_phi {}/IH IH].
have [opS | not_opS] := allP; last first.
right=> [[/andP[notSp _] opS]]; case: not_opS => psi Spsi /=.
by rewrite opS ?mem_head 1?mem_behead // (memPnC notSp).
rewrite (contra (opS _)) /= ?dnorm_eq0 //.
apply: (iffP IH) => [] [uniqS oSS]; last first.
by split=> //; apply: sub_in2 oSS => psi Spsi; apply: mem_behead.
split=> // psi xi; rewrite !inE => /predU1P[-> // | Spsi].
by case/predU1P=> [-> | /opS] /eqP.
case/predU1P=> [-> _ | Sxi /oSS-> //].
apply/eqP; rewrite hermC.
by move: (opS psi Spsi) => /= /eqP ->; rewrite rmorph0 mulr0.
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
|
pairwise_orthogonalP
| |
pairwise_orthogonal_catR S :
pairwise_orthogonal form (R ++ S) =
[&& pairwise_orthogonal form R, pairwise_orthogonal form S & orthogonal form R S].
Proof.
rewrite /pairwise_orthogonal mem_cat negb_or -!andbA; do !bool_congr.
elim: R => [|phi R /= ->]; rewrite ?andbT// all_cat -!andbA /=.
by do !bool_congr.
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
|
pairwise_orthogonal_cat
| |
orthonormal_catR S :
orthonormal form (R ++ S) =
[&& orthonormal form R, orthonormal form S & orthogonal form R S].
Proof.
rewrite !orthonormalE pairwise_orthogonal_cat all_cat -!andbA.
by do !bool_congr.
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
|
orthonormal_cat
| |
orthonormalPS :
reflect (uniq S /\ {in S &, forall phi psi, '[phi, psi] = (phi == psi)%:R})
(orthonormal form S).
Proof.
rewrite orthonormalE; have [/= normS | not_normS] := allP; last first.
by right=> [[_ o1S]]; case: not_normS => phi Sphi; rewrite /= o1S ?eqxx.
apply: (iffP (pairwise_orthogonalP S)) => [] [uniqS oSS].
split=> // [|phi psi]; first by case/andP: uniqS.
by have [-> _ /normS/eqP | /oSS] := altP eqP.
split=> // [|phi psi Sphi Spsi /negbTE]; last by rewrite oSS // => ->.
by rewrite /= (contra (normS _)) // linear0r eq_sym oner_eq0.
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
|
orthonormalP
| |
sub_orthonormalS1 S2 :
{subset S1 <= S2} -> uniq S1 -> orthonormal form S2 -> orthonormal form S1.
Proof.
move=> sS12 uniqS1 /orthonormalP[_ oS1].
by apply/orthonormalP; split; last apply: sub_in2 sS12 _ _.
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
|
sub_orthonormal
| |
orthonormal2Pphi psi :
reflect [/\ '[phi, psi] = 0, '[phi] = 1 & '[psi] = 1]
(orthonormal form [:: phi; psi]).
Proof.
rewrite /orthonormal /= !andbT andbC.
by apply: (iffP and3P) => [] []; do 3!move/eqP->.
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
|
orthonormal2P
| |
sub_pairwise_orthogonalS1 S2 :
{subset S1 <= S2} -> uniq S1 ->
pairwise_orthogonal form S2 -> pairwise_orthogonal form S1.
Proof.
move=> sS12 uniqS1 /pairwise_orthogonalP[/andP[notS2_0 _] oS2].
apply/pairwise_orthogonalP; rewrite /= (contra (sS12 0)) //.
by split=> //; apply: sub_in2 oS2.
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
|
sub_pairwise_orthogonal
|
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