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Coq-MathComp

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form0_eq0 : M = 0 -> forall u v, '[u, v] = 0.
Proof. by rewrite/form=> -> u v; rewrite mulmx0 mul0mx mxE. Qed.
Lemma
form0_eq0
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "form", "mul0mx", "mulmx0", "mxE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'hermitian' U 'for' eps & theta }"
:= (@Hermitian.type _ U eps theta) (format "{ 'hermitian' U 'for' eps & theta }") : ring_scope.
Notation
{ 'hermitian' U 'for' eps & theta }
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "theta", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthomx {R : fieldType} (theta : R -> R) n m M (B : 'M_(m, n)) : 'M_n
:= kermx (M *m (B ^t theta)).
Definition
orthomx
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "kermx", "theta" ]
Module Exports. Notation "{ 'hermitian' U 'for' eps & theta }" := (map eps theta (Phant U)) (format "{ 'hermitian' U 'for' eps & theta }") : ring_scope. Coercion base : class_of >-> bilmorphism_for. Coercion apply : map >-> Funclass. Notation "[ 'hermitian' 'of' f 'as' g ]" := (@clone _ _ _ _ _ _ f g _ idfun idf...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sesqui
:= [qualify M : 'M_n | M == ((-1) ^+ eps_theta.1) *: M ^t eps_theta.2].
Definition
sesqui
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sesqui_key : pred_key sesqui.
Proof. by []. Qed.
Fact
sesqui_key
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sesqui_keyed
:= KeyedQualifier sesqui_key.
Canonical
sesqui_keyed
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui_key" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"eps_theta .-sesqui"
:= (sesqui eps_theta).
Notation
eps_theta .-sesqui
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''[' u , v ]"
:= (form theta M u%R v%R) : ring_scope.
Notation
''[' u , v ]
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "form", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sesquiE : (M \is (eps, theta).-sesqui) = (M == (-1) ^+ eps *: M ^t theta).
Proof. by rewrite qualifE. Qed.
Lemma
sesquiE
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sesquiP : reflect (M = (-1) ^+ eps *: M ^t theta) (M \is (eps, theta).-sesqui).
Proof. by rewrite sesquiE; exact/eqP. Qed.
Lemma
sesquiP
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui", "sesquiE", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
(thetaK : involutive theta) (M_sesqui : M \is (eps, theta).-sesqui).
Hypotheses
thetaK
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
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
trmx_sesqui
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "map_trmx", "mul_mx_scalar", "mul_scalar_mx", "sesquiP", "theta", "tr_scalar_mx", "trmxK", "trmx_mul" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
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
maptrmx_sesqui
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "eq_map_mx_id", "map_mxZ", "map_mx_comp", "rmorph_sign", "theta", "trmx_sesqui" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
formC u 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
formC
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "eq_bigr", "form", "mulmxA", "mulrA", "mulrC", "mulrCA", "mulr_suml", "mulr_sumr", "mxE", "rmorphM", "rmorph_sum", "sesquiP", "theta", "thetaK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
form_eq0C u v : ('[u, v] == 0) = ('[v, u] == 0).
Proof. by rewrite formC mulf_eq0 signr_eq0 /= fmorph_eq0. Qed.
Lemma
form_eq0C
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "fmorph_eq0", "formC", "mulf_eq0", "signr_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ortho m (B : 'M_(m, n))
:= orthomx theta M B.
Definition
ortho
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "orthomx", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"B ^_|_"
:= (ortho B) : ring_scope.
Notation
B ^_|_
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "ortho" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A '_|_ B"
:= (A%MS <= B^_|_)%MS : ring_scope.
Notation
A '_|_ B
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalE u v : (u '_|_ v) = ('[u, v] == 0).
Proof. by rewrite (sameP sub_kermxP eqP) mulmxA [_ *m _^t _]mx11_scalar fmorph_eq0. Qed.
Lemma
normalE
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "fmorph_eq0", "mulmxA", "mx11_scalar", "sub_kermxP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
form_eq0P {u v} : reflect ('[u, v] = 0) (u '_|_ v).
Proof. by rewrite normalE; apply/eqP. Qed.
Lemma
form_eq0P
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "normalE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalP p 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/...
Lemma
normalP
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "kermx", "map_delta_mx", "map_mxM", "map_trmx", "mul0mx", "mulmxA", "rV_subP", "row0", "rowE", "row_matrixP", "sub_kermxP", "submxP", "submx_trans", "theta", "trmx0", "trmxK", "trmx_mul", "vB" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalC p 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
normalC
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "fmorph_eq0", "formC", "gen", "last", "mulf_eq0", "normalE", "normalP", "signr_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_ortho_mx p (A : 'M_(p, n)) : ((A^_|_) '_|_ A).
Proof. by []. Qed.
Lemma
normal_ortho_mx
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_mx_ortho p (A : 'M_(p, n)) : (A '_|_ (A^_|_)).
Proof. by rewrite normalC. Qed.
Lemma
normal_mx_ortho
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "normalC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_normal u : (\rank (u ^_|_) >= n.-1)%N.
Proof. rewrite mxrank_ker -subn1 leq_sub2l //. by rewrite (leq_trans (mxrankM_maxr _ _)) // rank_leq_col. Qed.
Lemma
rank_normal
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "leq_sub2l", "leq_trans", "mxrankM_maxr", "mxrank_ker", "rank", "rank_leq_col", "subn1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rad
:= 1%:M^_|_.
Definition
rad
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rad_ker : rad = kermx M.
Proof. by rewrite /rad /ortho /orthomx trmx1 map_mx1 mulmx1. Qed.
Lemma
rad_ker
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "kermx", "map_mx1", "mulmx1", "ortho", "orthomx", "rad", "trmx1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
formDd u 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
formDd
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "add0r", "addr0", "formC", "formDl", "formDr", "form_eq0P", "mulr0", "rmorph0" ]
Pythagoras
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
formZ a u : '[a *: u]= (a * theta a) * '[u].
Proof. by rewrite formZl formZr mulrA. Qed.
Lemma
formZ
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "formZl", "formZr", "mulrA", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
formN u : '[- u] = '[u].
Proof. by rewrite formNr formNl opprK. Qed.
Lemma
formN
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "formNl", "formNr", "opprK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
form_sign m u : '[(-1) ^+ m *: u] = '[u].
Proof. by rewrite -signr_odd scaler_sign; case: odd; rewrite ?formN. Qed.
Lemma
form_sign
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "formN", "odd", "scaler_sign", "signr_odd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
formD u 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
formD
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "addrACA", "addrC", "formC", "formDl", "formDr", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
formB u 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
formB
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "formD", "formN", "formNr", "mulrN", "opprD", "rmorphN", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
formBd u v : u '_|_ v -> '[u - v] = '[u] + '[v].
Proof. by move=> uTv; rewrite formDd ?formN // normalE formNr oppr_eq0 -normalE. Qed.
Lemma
formBd
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "formDd", "formN", "formNr", "normalE", "oppr_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"eps_theta .-sesqui"
:= (sesqui _ eps_theta) : ring_scope.
Notation
eps_theta .-sesqui
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
symmetric_form
:= (false, idfun).-sesqui.
Notation
symmetric_form
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
skew
:= (true, idfun).-sesqui.
Notation
skew
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "sesqui" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hermitian
:= (false, conjC).-sesqui.
Notation
hermitian
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "conjC", "sesqui" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'dot' U 'for' theta }"
:= (@Dot.type _ U theta) (format "{ 'dot' U 'for' theta }") : ring_scope.
Notation
{ 'dot' U 'for' theta }
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "theta", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'symmetric' U }"
:= ({hermitian U for false & idfun}) (format "{ 'symmetric' U }") : ring_scope.
Notation
{ 'symmetric' U }
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "hermitian" ]
Notation "{ 'dot' U 'for' theta }" := (map theta (Phant U)) (format "{ 'dot' U 'for' theta }") : ring_scope. Coercion base : class_of >-> Hermitian.class_of. Coercion apply : map >-> Funclass. Notation "[ 'dot' 'of' f 'as' g ]" := (@clone _ _ _ _ _ f g _ idfun idfun) (format "[ 'dot' 'of' f 'as' g ]") : form...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'skew_symmetric' U }"
:= ({hermitian U for true & idfun}) (format "{ 'skew_symmetric' U }") : ring_scope.
Notation
{ 'skew_symmetric' U }
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "hermitian" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'hermitian_sym' U 'for' theta }"
:= ({hermitian U for false & theta}) (format "{ 'hermitian_sym' U 'for' theta }") : ring_scope.
Notation
{ 'hermitian_sym' U 'for' theta }
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "hermitian", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_skew (R : nzRingType) (eps : bool) (theta : R -> R) (U : lmodType R) (form : {hermitian U for eps & theta})
:= (eps = true) /\ (theta =1 id).
Definition
is_skew
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "form", "hermitian", "id", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_sym (R : nzRingType) (eps : bool) (theta : R -> R) (U : lmodType R) (form : {hermitian U for eps & theta})
:= (eps = false) /\ (theta =1 id).
Definition
is_sym
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "form", "hermitian", "id", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_hermsym (R : nzRingType) (eps : bool) (theta : R -> R) (U : lmodType R) (form : {hermitian U for eps & theta})
:= (eps = false).
Definition
is_hermsym
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "form", "hermitian", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''[' u ]"
:= '[u, u]%R : ring_scope.
Notation
''[' u ]
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hermC u v : '[u, v] = (-1) ^+ eps * theta '[v, u].
Proof. by move: form => [? [[? ? ? ?] []]] /=. Qed.
Lemma
hermC
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "form", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hnormN u : '[- u] = '[u].
Proof. by rewrite linearNl linearNr opprK. Qed.
Lemma
hnormN
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "linearNl", "linearNr", "opprK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hnorm_sign n u : '[(-1) ^+ n *: u] = '[u].
Proof. by rewrite -signr_odd scaler_sign; case: (odd n); rewrite ?hnormN. Qed.
Lemma
hnorm_sign
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "hnormN", "odd", "scaler_sign", "signr_odd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hnormD u 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
hnormD
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "addrA", "addrAC", "hermC", "linearDl", "linearDr", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hnormB u 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
hnormB
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "addrA", "hnormD", "hnormN", "linearNr", "mulrN", "opprD", "rmorphN", "theta" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hnormDd u v : '[u, v] = 0 -> '[u + v] = '[u] + '[v].
Proof. by move=> ouv; rewrite hnormD ouv rmorph0 mulr0 !addr0. Qed.
Lemma
hnormDd
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "addr0", "hnormD", "mulr0", "rmorph0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
hnormBd u v : '[u, v] = 0 -> '[u - v] = '[u] + '[v].
Proof. by move=> ouv; rewrite hnormDd ?hnormN// linearNr [X in - X]ouv oppr0. Qed.
Lemma
hnormBd
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "hnormDd", "hnormN", "linearNr", "oppr0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"u '_|_ v"
:= ('[u, v] == 0) : ring_scope.
Notation
u '_|_ v
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ortho_rec (s1 s2 : seq U)
:= all [pred u | all [pred v | u '_|_ v] s2] s1.
Definition
ortho_rec
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "all", "s1", "s2", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_ortho_rec (s : seq U)
:= if s is v :: s' then ortho_rec [:: v] s' && pair_ortho_rec s' else true.
Fixpoint
pair_ortho_rec
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "ortho_rec", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairwise_orthogonal s
:= (0 \notin s) && pair_ortho_rec s.
Definition
pairwise_orthogonal
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "pair_ortho_rec" ]
We exclude 0 from pairwise orthogonal sets.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthogonal s1 s2
:= (@ortho_rec s1 s2).
Definition
orthogonal
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "ortho_rec", "s1", "s2" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthogonal_cons u us vs : orthogonal (u :: us) vs = orthogonal [:: u] vs && orthogonal us vs.
Proof. by rewrite /orthogonal /= andbT. Qed.
Lemma
orthogonal_cons
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "orthogonal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthonormal s
:= all [pred v | '[v] == 1] s && pair_ortho_rec s.
Definition
orthonormal
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "all", "pair_ortho_rec" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthonormal_not0 S : orthonormal S -> 0 \notin S.
Proof. by case/andP=> /allP S1 _; rewrite (contra (S1 _)) //= linear0r eq_sym oner_eq0. Qed.
Lemma
orthonormal_not0
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "S1", "allP", "eq_sym", "linear0r", "oner_eq0", "orthonormal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthonormalE S : orthonormal S = all [pred phi | '[phi] == 1] S && pairwise_orthogonal S.
Proof. by rewrite -(andb_idl (@orthonormal_not0 S)) andbCA. Qed.
Lemma
orthonormalE
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "all", "orthonormal", "orthonormal_not0", "pairwise_orthogonal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthonormal_orthogonal S : orthonormal S -> pairwise_orthogonal S.
Proof. by rewrite orthonormalE => /andP[_]. Qed.
Lemma
orthonormal_orthogonal
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "orthonormal", "orthonormalE", "pairwise_orthogonal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''[' u , v ]_1"
:= (form1 u%R v%R) : ring_scope.
Notation
''[' u , v ]_1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''[' u , v ]_2"
:= (form2 u%R v%R) : ring_scope.
Notation
''[' u , v ]_2
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''[' u ]_1"
:= (form1 u%R u%R) : ring_scope.
Notation
''[' u ]_1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''[' u ]_2"
:= (form2 u%R u%R): ring_scope.
Notation
''[' u ]_2
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isometry tau
:= forall u v, form1 (tau u) (tau v) = form2 u%R v%R.
Definition
isometry
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isometry_from_to mD tau mR
:= prop_in2 mD (inPhantom (isometry tau)) /\ prop_in1 mD (inPhantom (forall u, in_mem (tau u) mR)).
Definition
isometry_from_to
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "isometry" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'in' D , 'isometry' tau , 'to' R }"
:= (isometry_from_to (mem D) tau (mem R)) (format "{ 'in' D , 'isometry' tau , 'to' R }") : type_scope.
Notation
{ 'in' D , 'isometry' tau , 'to' R }
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "isometry_from_to" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
herm_eq0C u v : ('[u, v] == 0) = ('[v, u] == 0).
Proof. by rewrite hermC mulf_eq0 signr_eq0 /= fmorph_eq0. Qed.
Lemma
herm_eq0C
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "fmorph_eq0", "hermC", "mulf_eq0", "signr_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
n
:= \dim {:vT}.
Let
n
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "dim", "vT" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
alpha v
:= (linfun (applyr form v : vT -> F^o)).
Let
alpha
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "applyr", "form", "linfun", "vT" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthov V
:= (\bigcap_(i < \dim V) lker (alpha (vbasis V)`_i))%VS.
Definition
orthov
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "alpha", "dim", "lker", "vbasis" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"U '_|_ V"
:= (U <= orthov V)%VS : vspace_scope.
Notation
U '_|_ V
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "orthov" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_orthovPn V 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...
Lemma
mem_orthovPn
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "big1", "coord_vbasis", "dim", "eqxx", "existsP", "forallP", "last", "lfunE", "linearZr", "linear_sumr", "mem_nth", "memvE", "memv_ker", "mulr0", "negb_exists", "orthov", "size_tuple", "subv_bigcapP", "vbasis", "vbasis_mem" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_orthovP V 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
mem_orthovP
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "mem_orthovPn", "orthov" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthov1E u : 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
orthov1E
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "alpha", "apply", "eqEsubv", "lfunE", "linearZ", "lker", "mem_orthovP", "memv_ker", "memv_line", "mulf_eq0", "orthov", "split", "subvP", "vlineP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthovP U 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
orthovP
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "last", "mem_orthovP", "subvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthov_sym U V : (U '_|_ V)%VS = (V '_|_ U)%VS.
Proof. by apply/orthovP/orthovP => eq0 ????; apply/eqP; rewrite herm_eq0C eq0. Qed.
Lemma
orthov_sym
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "herm_eq0C", "orthovP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_orthov1 v u : (u \in orthov <[v]>) = ('[u, v] == 0).
Proof. by rewrite orthov1E memv_ker lfunE. Qed.
Lemma
mem_orthov1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "lfunE", "memv_ker", "orthov", "orthov1E" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthov11 u v : (<[u]> '_|_ <[v]>)%VS = ('[u, v] == 0).
Proof. exact: mem_orthov1. Qed.
Lemma
orthov11
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "mem_orthov1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_orthov1_sym v u : (u \in orthov <[v]>) = (v \in orthov <[u]>).
Proof. exact: orthov_sym. Qed.
Lemma
mem_orthov1_sym
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "orthov", "orthov_sym" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthov0 : orthov 0 = fullv.
Proof. apply/eqP; rewrite eqEsubv subvf. apply/subvP => x _; rewrite mem_orthov1. by rewrite linear0r. Qed.
Lemma
orthov0
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "eqEsubv", "fullv", "linear0r", "mem_orthov1", "orthov", "subvP", "subvf" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_orthov_sym V u : (u \in orthov V) = (V <= orthov <[u]>)%VS.
Proof. exact: orthov_sym. Qed.
Lemma
mem_orthov_sym
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "orthov", "orthov_sym" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_dim_orthov1 u 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
leq_dim_orthov1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "addnC", "alpha", "dim", "dimvS", "dimvf", "leq_pred", "limg_ker_dim", "orthov", "orthov1E", "subvf" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_img_form_eq1 u 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...
Lemma
dim_img_form_eq1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "alpha", "apply", "dim", "dim_vline", "dimvS", "dimvf", "eqn_leq", "herm_eq0C", "lfunE", "mem_orthovPn", "memvE", "memv_imgP", "memvf", "orthov", "subvf" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_dim_orthov1 u 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
eq_dim_orthov1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "addn1", "alpha", "dim", "dim_img_form_eq1", "limg_ker_dim", "orthov", "orthov1E" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_img_form_eq0 u 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
dim_img_form_eq0
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "alpha", "apply", "dim", "dimv_eq0", "lkerE", "orthov", "orthov1E", "orthov_sym" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
neq_dim_orthov1 u 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
neq_dim_orthov1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "addn0", "alpha", "capv_idPl", "dim", "dim_img_form_eq0", "limg_ker_dim", "orthov", "orthov1E", "orthov_sym" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
leqif_dim_orthov1 u 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
leqif_dim_orthov1
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "dim", "eq_dim_orthov1", "leqifP", "neq_dim_orthov1", "orthov" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
leqif_dim_orthov1_full u : (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
leqif_dim_orthov1_full
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "capfv", "dim", "fullv", "leqif_dim_orthov1", "n_gt0", "orthov", "vT" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthogonal1P u v : reflect ('[u, v] = 0) (orthogonal form [:: u] [:: v]).
Proof. by rewrite /orthogonal /= !andbT; apply: eqP. Qed.
Lemma
orthogonal1P
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "form", "orthogonal" ]
Link between orthov and orthovgonality of sequences
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthogonalP us 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
orthogonalP
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "allP", "apply", "form", "orthogonal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthogonal_oppr S 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
orthogonal_oppr
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "form", "linearNr", "map", "mapK", "mapP", "oppr0", "opprK", "orthogonal", "orthogonalP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthogonalE us 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
orthogonalE
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "apply", "big1", "coord_span", "form", "in_tupleE", "last", "linearZlr", "linear_sumlz", "linear_sumr", "mem_nth", "memv_span", "mulr0", "orthogonal", "orthogonalP", "orthovP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthovE U V : (U '_|_ V)%VS = orthogonal form (vbasis U) (vbasis V).
Proof. by rewrite orthogonalE !(span_basis (vbasisP _)). Qed.
Lemma
orthovE
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "form", "orthogonal", "orthogonalE", "span_basis", "vbasis", "vbasisP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
radv
:= (orthov fullv).
Notation
radv
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "fullv", "orthov" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthoDv U V W : (U + V '_|_ W)%VS = (U '_|_ W)%VS && (V '_|_ W)%VS.
Proof. by rewrite subv_add. Qed.
Lemma
orthoDv
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "subv_add" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
orthovD U V W : (U '_|_ V + W)%VS = (U '_|_ V)%VS && (U '_|_ W)%VS.
Proof. by rewrite ![(U '_|_ _)%VS]orthov_sym orthoDv. Qed.
Lemma
orthovD
algebra
algebra/sesquilinear.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "bigop", "order", "nmodule", "rings_modules_and_algebras", "divalg", "orderedzmod", "numdomain", "numfield", "matrix", "mxalgebra", "vector", "GRin...
[ "orthoDv", "orthov_sym" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d