statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
cfunD1E phi : (phi \in 'CF(G, G^#)) = (phi 1%g == 0). | Proof. by rewrite cfun_onD1 cfun_onG. Qed. | Lemma | cfunD1E | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onD1",
"cfun_onG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfunGid : 'CF(G, G) = 'CF(G)%VS. | Proof. by apply/vspaceP=> phi; rewrite cfun_onG memvf. Qed. | Lemma | cfunGid | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfun_onG",
"memvf",
"vspaceP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_onS A B phi : B \subset A -> phi \in 'CF(G, B) -> phi \in 'CF(G, A). | Proof. by rewrite !cfun_onE => sBA /subset_trans->. Qed. | Lemma | cfun_onS | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onE",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_complement A :
A <| G -> ('CF(G, A) + 'CF(G, G :\: A)%SET = 'CF(G))%VS. | Proof.
case/andP=> sAG nAG; rewrite -cfunGid [rhs in _ = rhs]cfun_on_sum.
rewrite (bigID (fun B => B \subset A)) /=.
congr (_ + _)%VS; rewrite cfun_on_sum; apply: eq_bigl => /= xG.
rewrite andbAC; apply/esym/andb_idr=> /andP[/imsetP[x Gx ->] _].
by rewrite class_subG.
rewrite -andbA; apply: andb_id2l => /imsetP[x G... | Lemma | cfun_complement | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"bigID",
"cfunGid",
"cfun_on_sum",
"class_subG",
"class_sub_norm",
"eq_bigl",
"imsetP",
"inE",
"normG",
"normsD",
"rhs",
"sAG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjCE phi x : ( phi^* )%CF x = (phi x)^*. | Proof. by rewrite cfunE. Qed. | Lemma | cfConjCE | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjCK : involutive (fun phi => phi^* )%CF. | Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE /= conjCK. Qed. | Lemma | cfConjCK | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfunE",
"cfunP",
"conjCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_cfun1 : ( 1^* )%CF = 1 :> 'CF(G). | Proof. exact: rmorph1. Qed. | Lemma | cfConjC_cfun1 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_is_group phi : group_set (cfker phi). | Proof.
apply/group_setP; split=> [|x y]; rewrite !inE ?group1.
by apply/forallP=> y; rewrite mul1g.
case/andP=> Gx /forallP-Kx /andP[Gy /forallP-Ky]; rewrite groupM //.
by apply/forallP=> z; rewrite -mulgA (eqP (Kx _)) Ky.
Qed. | Fact | cfker_is_group | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"forallP",
"group1",
"groupM",
"group_set",
"group_setP",
"inE",
"mul1g",
"mulgA",
"split"
] | Class function kernel and faithful class functions | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfker_group phi | := Group (cfker_is_group phi). | Canonical | cfker_group | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker_is_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_sub phi : cfker phi \subset G. | Proof. by rewrite /cfker setIdE subsetIl. Qed. | Lemma | cfker_sub | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker",
"setIdE",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_norm phi : G \subset 'N(cfker phi). | Proof.
apply/subsetP=> z Gz; have phiJz := cfunJ phi _ (groupVr Gz).
rewrite inE; apply/subsetP=> _ /imsetP[x /setIdP[Gx /forallP-Kx] ->].
rewrite inE groupJ //; apply/forallP=> y.
by rewrite -(phiJz y) -phiJz conjMg conjgK Kx.
Qed. | Lemma | cfker_norm | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfunJ",
"conjMg",
"conjgK",
"forallP",
"groupJ",
"groupVr",
"imsetP",
"inE",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_normal phi : cfker phi <| G. | Proof. by rewrite /normal cfker_sub cfker_norm. Qed. | Lemma | cfker_normal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker",
"cfker_norm",
"cfker_sub",
"normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfkerMl phi x y : x \in cfker phi -> phi (x * y)%g = phi y. | Proof. by case/setIdP=> _ /eqfunP->. Qed. | Lemma | cfkerMl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker",
"eqfunP",
"setIdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfkerMr phi x y : x \in cfker phi -> phi (y * x)%g = phi y. | Proof.
by move=> Kx; rewrite conjgC cfkerMl ?cfunJ ?(subsetP (cfker_sub phi)).
Qed. | Lemma | cfkerMr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker",
"cfkerMl",
"cfker_sub",
"cfunJ",
"conjgC",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker1 phi x : x \in cfker phi -> phi x = phi 1%g. | Proof. by move=> Kx; rewrite -[x]mulg1 cfkerMl. Qed. | Lemma | cfker1 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker",
"cfkerMl",
"mulg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_cfun0 : @cfker _ G 0 = G. | Proof.
apply/setP=> x; rewrite !inE andb_idr // => Gx; apply/forallP=> y.
by rewrite !cfunE.
Qed. | Lemma | cfker_cfun0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfunE",
"forallP",
"inE",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_add phi psi : cfker phi :&: cfker psi \subset cfker (phi + psi). | Proof.
apply/subsetP=> x /setIP[Kphi_x Kpsi_x]; have [Gx _] := setIdP Kphi_x.
by rewrite inE Gx; apply/forallP=> y; rewrite !cfunE !cfkerMl.
Qed. | Lemma | cfker_add | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfkerMl",
"cfunE",
"forallP",
"inE",
"setIP",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_sum I r (P : pred I) (Phi : I -> 'CF(G)) :
G :&: \bigcap_(i <- r | P i) cfker (Phi i)
\subset cfker (\sum_(i <- r | P i) Phi i). | Proof.
elim/big_rec2: _ => [|i K psi Pi sK_psi]; first by rewrite setIT cfker_cfun0.
by rewrite setICA; apply: subset_trans (setIS _ sK_psi) (cfker_add _ _).
Qed. | Lemma | cfker_sum | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"big_rec2",
"cfker",
"cfker_add",
"cfker_cfun0",
"setICA",
"setIS",
"setIT",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_scale a phi : cfker phi \subset cfker (a *: phi). | Proof.
apply/subsetP=> x Kphi_x; have [Gx _] := setIdP Kphi_x.
by rewrite inE Gx; apply/forallP=> y; rewrite !cfunE cfkerMl.
Qed. | Lemma | cfker_scale | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfkerMl",
"cfunE",
"forallP",
"inE",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_scale_nz a phi : a != 0 -> cfker (a *: phi) = cfker phi. | Proof.
move=> nz_a; apply/eqP.
by rewrite eqEsubset -{2}(scalerK nz_a phi) !cfker_scale.
Qed. | Lemma | cfker_scale_nz | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfker_scale",
"eqEsubset",
"scalerK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_opp phi : cfker (- phi) = cfker phi. | Proof. by rewrite -scaleN1r cfker_scale_nz // oppr_eq0 oner_eq0. Qed. | Lemma | cfker_opp | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker",
"cfker_scale_nz",
"oner_eq0",
"oppr_eq0",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_cfun1 : @cfker _ G 1 = G. | Proof.
apply/setP=> x; rewrite !inE andb_idr // => Gx; apply/forallP=> y.
by rewrite !cfun1E groupMl.
Qed. | Lemma | cfker_cfun1 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfun1E",
"forallP",
"groupMl",
"inE",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_mul phi psi : cfker phi :&: cfker psi \subset cfker (phi * psi). | Proof.
apply/subsetP=> x /setIP[Kphi_x Kpsi_x]; have [Gx _] := setIdP Kphi_x.
by rewrite inE Gx; apply/forallP=> y; rewrite !cfunE !cfkerMl.
Qed. | Lemma | cfker_mul | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfkerMl",
"cfunE",
"forallP",
"inE",
"setIP",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_prod I r (P : pred I) (Phi : I -> 'CF(G)) :
G :&: \bigcap_(i <- r | P i) cfker (Phi i)
\subset cfker (\prod_(i <- r | P i) Phi i). | Proof.
elim/big_rec2: _ => [|i K psi Pi sK_psi]; first by rewrite setIT cfker_cfun1.
by rewrite setICA; apply: subset_trans (setIS _ sK_psi) (cfker_mul _ _).
Qed. | Lemma | cfker_prod | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"big_rec2",
"cfker",
"cfker_cfun1",
"cfker_mul",
"setICA",
"setIS",
"setIT",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfaithfulE phi : cfaithful phi = (cfker phi \subset [1]). | Proof. by []. Qed. | Lemma | cfaithfulE | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfaithful",
"cfker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotE phi psi :
'[phi, psi] = #|G|%:R^-1 * \sum_(x in G) phi x * (psi x)^*. | Proof. by []. Qed. | Lemma | cfdotE | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotElr A B phi psi :
phi \in 'CF(G, A) -> psi \in 'CF(G, B) ->
'[phi, psi] = #|G|%:R^-1 * \sum_(x in A :&: B) phi x * (psi x)^*. | Proof.
move=> Aphi Bpsi; rewrite (big_setID G)/= cfdotE (big_setID (A :&: B))/= setIC.
congr (_ * (_ + _)); rewrite !big1 // => x /setDP[_]; last first.
by move/cfun0->; rewrite mul0r.
rewrite inE; case/nandP=> notABx; first by rewrite (cfun_on0 Aphi) ?mul0r.
by rewrite (cfun_on0 Bpsi) // rmorph0 mulr0.
Qed. | Lemma | cfdotElr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"big1",
"big_setID",
"cfdotE",
"cfun0",
"cfun_on0",
"inE",
"last",
"mul0r",
"mulr0",
"rmorph0",
"setDP",
"setIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotEl A phi psi :
phi \in 'CF(G, A) ->
'[phi, psi] = #|G|%:R^-1 * \sum_(x in A) phi x * (psi x)^*. | Proof. by move=> Aphi; rewrite (cfdotElr Aphi (cfun_onT psi)) setIT. Qed. | Lemma | cfdotEl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotElr",
"cfun_onT",
"setIT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotEr A phi psi :
psi \in 'CF(G, A) ->
'[phi, psi] = #|G|%:R^-1 * \sum_(x in A) phi x * (psi x)^*. | Proof. by move=> Apsi; rewrite (cfdotElr (cfun_onT phi) Apsi) setTI. Qed. | Lemma | cfdotEr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotElr",
"cfun_onT",
"setTI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_complement A phi psi :
phi \in 'CF(G, A) -> psi \in 'CF(G, G :\: A) -> '[phi, psi] = 0. | Proof.
move=> Aphi A'psi; rewrite (cfdotElr Aphi A'psi).
by rewrite setDE setICA setICr setI0 big_set0 mulr0.
Qed. | Lemma | cfdot_complement | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"big_set0",
"cfdotElr",
"mulr0",
"setDE",
"setI0",
"setICA",
"setICr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnormE A phi :
phi \in 'CF(G, A) -> '[phi] = #|G|%:R^-1 * (\sum_(x in A) `|phi x| ^+ 2). | Proof. by move/cfdotEl->; rewrite (eq_bigr _ (fun _ _ => normCK _)). Qed. | Lemma | cfnormE | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotEl",
"eq_bigr",
"normCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_cfdotl A phi1 phi2 psi :
psi \in 'CF(G, A) -> {in A, phi1 =1 phi2} -> '[phi1, psi] = '[phi2, psi]. | Proof.
move/cfdotEr=> eq_dot eq_phi; rewrite !eq_dot; congr (_ * _).
by apply: eq_bigr => x Ax; rewrite eq_phi.
Qed. | Lemma | eq_cfdotl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdotEr",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_cfuni A B :
A <| G -> B <| G -> '['1_A, '1_B]_G = #|A :&: B|%:R / #|G|%:R. | Proof.
move=> nsAG nsBG; rewrite (cfdotElr (cfuni_on G A) (cfuni_on G B)) mulrC.
congr (_ / _); rewrite -sumr_const; apply: eq_bigr => x /setIP[Ax Bx].
by rewrite !cfuniE // Ax Bx mul1r rmorph1.
Qed. | Lemma | cfdot_cfuni | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdotElr",
"cfuniE",
"cfuni_on",
"eq_bigr",
"mul1r",
"mulrC",
"rmorph1",
"setIP",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm1 : '[1]_G = 1. | Proof. by rewrite cfdot_cfuni ?genGid // setIid divff ?neq0CG. Qed. | Lemma | cfnorm1 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdot_cfuni",
"divff",
"genGid",
"neq0CG",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotrE psi phi : cfdotr psi phi = '[phi, psi]. | Proof. by []. Qed. | Lemma | cfdotrE | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotr_is_linear xi : linear (cfdotr xi : 'CF(G) -> algC^o). | Proof.
move=> a phi psi; rewrite scalerAr -mulrDr; congr (_ * _).
rewrite linear_sum -big_split; apply: eq_bigr => x _ /=.
by rewrite !cfunE mulrDl -mulrA.
Qed. | Lemma | cfdotr_is_linear | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"apply",
"big_split",
"cfdotr",
"cfunE",
"eq_bigr",
"linear",
"linear_sum",
"mulrA",
"mulrDl",
"mulrDr",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot0l xi : '[0, xi] = 0. | Proof. by rewrite -cfdotrE linear0. Qed. | Lemma | cfdot0l | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotrE",
"linear0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotNl xi phi : '[- phi, xi] = - '[phi, xi]. | Proof. by rewrite -!cfdotrE linearN. Qed. | Lemma | cfdotNl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotrE",
"linearN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotDl xi phi psi : '[phi + psi, xi] = '[phi, xi] + '[psi, xi]. | Proof. by rewrite -!cfdotrE linearD. Qed. | Lemma | cfdotDl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotrE",
"linearD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotBl xi phi psi : '[phi - psi, xi] = '[phi, xi] - '[psi, xi]. | Proof. by rewrite -!cfdotrE linearB. Qed. | Lemma | cfdotBl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotrE",
"linearB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotMnl xi phi n : '[phi *+ n, xi] = '[phi, xi] *+ n. | Proof. by rewrite -!cfdotrE linearMn. Qed. | Lemma | cfdotMnl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotrE",
"linearMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_suml xi I r (P : pred I) (phi : I -> 'CF(G)) :
'[\sum_(i <- r | P i) phi i, xi] = \sum_(i <- r | P i) '[phi i, xi]. | Proof. by rewrite -!cfdotrE linear_sum. Qed. | Lemma | cfdot_suml | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotrE",
"linear_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotZl xi a phi : '[a *: phi, xi] = a * '[phi, xi]. | Proof. by rewrite -!cfdotrE linearZ. Qed. | Lemma | cfdotZl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotrE",
"linearZ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotC phi psi : '[phi, psi] = ('[psi, phi])^*. | Proof.
rewrite /cfdot rmorphM /= fmorphV rmorph_nat rmorph_sum; congr (_ * _).
by apply: eq_bigr=> x _; rewrite rmorphM /= conjCK mulrC.
Qed. | Lemma | cfdotC | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdot",
"conjCK",
"eq_bigr",
"fmorphV",
"mulrC",
"rmorphM",
"rmorph_nat",
"rmorph_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_cfdotr A phi psi1 psi2 :
phi \in 'CF(G, A) -> {in A, psi1 =1 psi2} -> '[phi, psi1] = '[phi, psi2]. | Proof. by move=> Aphi /eq_cfdotl eq_dot; rewrite cfdotC eq_dot // -cfdotC. Qed. | Lemma | eq_cfdotr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotC",
"eq_cfdotl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotBr xi phi psi : '[xi, phi - psi] = '[xi, phi] - '[xi, psi]. | Proof. by rewrite !(cfdotC xi) -rmorphB cfdotBl. Qed. | Lemma | cfdotBr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotBl",
"cfdotC",
"rmorphB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot0r xi : '[xi, 0] = 0. | Proof. exact: raddf0. Qed. | Lemma | cfdot0r | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"raddf0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotNr xi phi : '[xi, - phi] = - '[xi, phi]. | Proof. exact: raddfN. Qed. | Lemma | cfdotNr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotDr xi phi psi : '[xi, phi + psi] = '[xi, phi] + '[xi, psi]. | Proof. exact: raddfD. Qed. | Lemma | cfdotDr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"raddfD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotMnr xi phi n : '[xi, phi *+ n] = '[xi, phi] *+ n. | Proof. exact: raddfMn. Qed. | Lemma | cfdotMnr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"raddfMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_sumr xi I r (P : pred I) (phi : I -> 'CF(G)) :
'[xi, \sum_(i <- r | P i) phi i] = \sum_(i <- r | P i) '[xi, phi i]. | Proof. exact: raddf_sum. Qed. | Lemma | cfdot_sumr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"raddf_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotZr a xi phi : '[xi, a *: phi] = a^* * '[xi, phi]. | Proof. by rewrite !(cfdotC xi) cfdotZl rmorphM. Qed. | Lemma | cfdotZr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotC",
"cfdotZl",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_cfAut (u : {rmorphism algC -> algC}) phi psi :
{in image psi G, {morph u : x / x^*}} ->
'[cfAut u phi, cfAut u psi] = u '[phi, psi]. | Proof.
move=> uC; rewrite rmorphM /= fmorphV rmorph_nat rmorph_sum; congr (_ * _).
by apply: eq_bigr => x Gx; rewrite !cfunE rmorphM /= uC ?map_f ?mem_enum.
Qed. | Lemma | cfdot_cfAut | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"apply",
"cfAut",
"cfunE",
"eq_bigr",
"fmorphV",
"image",
"map_f",
"mem_enum",
"rmorphM",
"rmorph_nat",
"rmorph_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_conjC phi psi : '[phi^*, psi^*] = '[phi, psi]^*. | Proof. by rewrite cfdot_cfAut. Qed. | Lemma | cfdot_conjC | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdot_cfAut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_conjCl phi psi : '[phi^*, psi] = '[phi, psi^*]^*. | Proof. by rewrite -cfdot_conjC cfConjCK. Qed. | Lemma | cfdot_conjCl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfConjCK",
"cfdot_conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_conjCr phi psi : '[phi, psi^*] = '[phi^*, psi]^*. | Proof. by rewrite -cfdot_conjC cfConjCK. Qed. | Lemma | cfdot_conjCr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfConjCK",
"cfdot_conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_ge0 phi : 0 <= '[phi]. | Proof.
by rewrite mulr_ge0 ?invr_ge0 ?ler0n ?sumr_ge0 // => x _; apply: mul_conjC_ge0.
Qed. | Lemma | cfnorm_ge0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"invr_ge0",
"ler0n",
"mul_conjC_ge0",
"mulr_ge0",
"sumr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_eq0 phi : ('[phi] == 0) = (phi == 0). | Proof.
apply/idP/eqP=> [|->]; last by rewrite cfdot0r.
rewrite mulf_eq0 invr_eq0 (negbTE (neq0CG G)) /= => /eqP/psumr_eq0P phi0.
apply/cfun_inP=> x Gx; apply/eqP; rewrite cfunE -mul_conjC_eq0.
by rewrite phi0 // => y _; apply: mul_conjC_ge0.
Qed. | Lemma | cfnorm_eq0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdot0r",
"cfunE",
"cfun_inP",
"invr_eq0",
"last",
"mul_conjC_eq0",
"mul_conjC_ge0",
"mulf_eq0",
"neq0CG",
"psumr_eq0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_gt0 phi : ('[phi] > 0) = (phi != 0). | Proof. by rewrite lt_def cfnorm_ge0 cfnorm_eq0 andbT. Qed. | Lemma | cfnorm_gt0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfnorm_eq0",
"cfnorm_ge0",
"lt_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_cfnorm_ge0 phi : 0 <= sqrtC '[phi]. | Proof. by rewrite sqrtC_ge0 cfnorm_ge0. Qed. | Lemma | sqrt_cfnorm_ge0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfnorm_ge0",
"sqrtC",
"sqrtC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_cfnorm_eq0 phi : (sqrtC '[phi] == 0) = (phi == 0). | Proof. by rewrite sqrtC_eq0 cfnorm_eq0. Qed. | Lemma | sqrt_cfnorm_eq0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfnorm_eq0",
"sqrtC",
"sqrtC_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_cfnorm_gt0 phi : (sqrtC '[phi] > 0) = (phi != 0). | Proof. by rewrite sqrtC_gt0 cfnorm_gt0. Qed. | Lemma | sqrt_cfnorm_gt0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfnorm_gt0",
"sqrtC",
"sqrtC_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnormZ a phi : '[a *: phi]= `|a| ^+ 2 * '[phi]_G. | Proof. by rewrite cfdotZl cfdotZr mulrA normCK. Qed. | Lemma | cfnormZ | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotZl",
"cfdotZr",
"mulrA",
"normCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnormN phi : '[- phi] = '[phi]. | Proof. by rewrite cfdotNl raddfN opprK. Qed. | Lemma | cfnormN | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotNl",
"opprK",
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_sign n phi : '[(-1) ^+ n *: phi] = '[phi]. | Proof. by rewrite -signr_odd scaler_sign; case: (odd n); rewrite ?cfnormN. Qed. | Lemma | cfnorm_sign | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfnormN",
"odd",
"scaler_sign",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnormD phi psi :
let d := '[phi, psi] in '[phi + psi] = '[phi] + '[psi] + ( d + d^* ). | Proof. by rewrite /= addrAC -cfdotC cfdotDl !cfdotDr !addrA. Qed. | Lemma | cfnormD | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"addrA",
"addrAC",
"cfdotC",
"cfdotDl",
"cfdotDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnormB phi psi :
let d := '[phi, psi] in '[phi - psi] = '[phi] + '[psi] - ( d + d^* ). | Proof. by rewrite /= cfnormD cfnormN cfdotNr rmorphN -opprD. Qed. | Lemma | cfnormB | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotNr",
"cfnormD",
"cfnormN",
"opprD",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnormDd phi psi : '[phi, psi] = 0 -> '[phi + psi] = '[phi] + '[psi]. | Proof. by move=> ophipsi; rewrite cfnormD ophipsi rmorph0 !addr0. Qed. | Lemma | cfnormDd | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"addr0",
"cfnormD",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnormBd phi psi : '[phi, psi] = 0 -> '[phi - psi] = '[phi] + '[psi]. | Proof.
by move=> ophipsi; rewrite cfnormDd ?cfnormN // cfdotNr ophipsi oppr0.
Qed. | Lemma | cfnormBd | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotNr",
"cfnormDd",
"cfnormN",
"oppr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_conjC phi : '[phi^*] = '[phi]. | Proof. by rewrite cfdot_conjC geC0_conj // cfnorm_ge0. Qed. | Lemma | cfnorm_conjC | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdot_conjC",
"cfnorm_ge0",
"geC0_conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfCauchySchwarz phi psi :
`|'[phi, psi]| ^+ 2 <= '[phi] * '[psi] ?= iff ~~ free (phi :: psi). | Proof.
rewrite free_cons span_seq1 seq1_free -negb_or negbK orbC.
have [-> | nz_psi] /= := eqVneq psi 0.
by apply/leifP; rewrite !cfdot0r normCK mul0r mulr0.
without loss ophi: phi / '[phi, psi] = 0.
move=> IHo; pose a := '[phi, psi] / '[psi]; pose phi1 := phi - a *: psi.
have ophi: '[phi1, psi] = 0.
by rewri... | Lemma | cfCauchySchwarz | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"add0r",
"apply",
"cfdot0r",
"cfdotBl",
"cfdotDl",
"cfdotZl",
"cfdotZr",
"cfnormDd",
"cfnormZ",
"cfnorm_eq0",
"cfnorm_ge0",
"divfK",
"eqVneq",
"eq_sym",
"exprMn",
"free",
"free_cons",
"ger0_norm",
"last",
"leifBLR",
"leifP",
"memv_line",
"mul0r",
"mulf_eq0",
"mulr0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfCauchySchwarz_sqrt phi psi :
`|'[phi, psi]| <= sqrtC '[phi] * sqrtC '[psi] ?= iff ~~ free (phi :: psi). | Proof.
rewrite -(sqrCK (normr_ge0 _)) -sqrtCM ?qualifE/= ?cfnorm_ge0 //.
rewrite (mono_in_leif (@ler_sqrtC _)) 1?rpredM ?qualifE/= ?cfnorm_ge0 //;
[ exact: O.. | exact: cfCauchySchwarz ].
Qed. | Lemma | cfCauchySchwarz_sqrt | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfCauchySchwarz",
"cfnorm_ge0",
"free",
"ler_sqrtC",
"mono_in_leif",
"normr_ge0",
"rpredM",
"sqrCK",
"sqrtC",
"sqrtCM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cf_triangle_leif phi psi :
sqrtC '[phi + psi] <= sqrtC '[phi] + sqrtC '[psi]
?= iff ~~ free (phi :: psi) && (0 <= coord [tuple psi] 0 phi). | Proof.
rewrite -(mono_in_leif ler_sqr) ?rpredD ?qualifE/= ?sqrtC_ge0 ?cfnorm_ge0 //;
[ exact: O.. |].
rewrite andbC sqrrD !sqrtCK addrAC cfnormD (mono_leif (lerD2l _)).
rewrite -mulr_natr -[_ + _](divfK (negbT (eqC_nat 2 0))) -/('Re _).
rewrite (mono_leif (ler_pM2r _)) ?ltr0n //.
have:= leif_trans (leif_Re_Creal '[ph... | Lemma | cf_triangle_leif | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Re",
"ReE",
"addrAC",
"apply",
"cfCauchySchwarz_sqrt",
"cfdot0r",
"cfdotZl",
"cfnormD",
"cfnorm_ge0",
"cfnorm_gt0",
"coord",
"coord0",
"coord_free",
"divfK",
"eqC_nat",
"eqVneq",
"eqxx",
"free",
"free_cons",
"leif_Re_Creal",
"leif_trans",
"lerD2l",
"ler_pM2r",
"ler_sqr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_cons phi R S :
orthogonal (phi :: R) S = orthogonal phi S && orthogonal R S. | Proof. by rewrite /orthogonal /= andbT. Qed. | Lemma | orthogonal_cons | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoP phi psi : reflect ('[phi, psi] = 0) (orthogonal phi psi). | Proof. by rewrite /orthogonal /= !andbT; apply: eqP. Qed. | Lemma | orthoP | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonalP S R :
reflect {in S & R, forall phi psi, '[phi, psi] = 0} (orthogonal S R). | Proof.
apply: (iffP allP) => oSR phi => [psi /oSR/allP opS /opS/eqP // | /oSR opS].
by apply/allP=> psi /= /opS->.
Qed. | Lemma | orthogonalP | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"allP",
"apply",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoPl phi S :
reflect {in S, forall psi, '[phi, psi] = 0} (orthogonal phi S). | Proof.
by rewrite [orthogonal _ S]andbT /=; apply: (iffP allP) => ophiS ? /ophiS/eqP.
Qed. | Lemma | orthoPl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"allP",
"apply",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_sym : symmetric (@orthogonal _ G). | Proof.
apply: symmetric_from_pre => R S /orthogonalP oRS.
by apply/orthogonalP=> phi psi Rpsi Sphi; rewrite cfdotC oRS ?rmorph0.
Qed. | Lemma | orthogonal_sym | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdotC",
"orthogonal",
"orthogonalP",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthoPr S psi :
reflect {in S, forall phi, '[phi, psi] = 0} (orthogonal S psi). | Proof.
rewrite orthogonal_sym.
by apply: (iffP orthoPl) => oSpsi phi Sphi; rewrite cfdotC oSpsi ?conjC0.
Qed. | Lemma | orthoPr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdotC",
"conjC0",
"orthoPl",
"orthogonal",
"orthogonal_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_orthogonal R1 R2 S1 S2 :
R1 =i R2 -> S1 =i S2 -> orthogonal R1 S1 = orthogonal R2 S2. | Proof.
move=> eqR eqS; rewrite [orthogonal _ _](eq_all_r eqR).
by apply: eq_all => psi /=; apply: eq_all_r.
Qed. | Lemma | eq_orthogonal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"R1",
"R2",
"S1",
"S2",
"apply",
"eq_all",
"eq_all_r",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_catl R1 R2 S :
orthogonal (R1 ++ R2) S = orthogonal R1 S && orthogonal R2 S. | Proof. exact: all_cat. Qed. | Lemma | orthogonal_catl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"R1",
"R2",
"all_cat",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_catr R S1 S2 :
orthogonal R (S1 ++ S2) = orthogonal R S1 && orthogonal R S2. | Proof. by rewrite !(orthogonal_sym R) orthogonal_catl. Qed. | Lemma | orthogonal_catr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"orthogonal",
"orthogonal_catl",
"orthogonal_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
span_orthogonal S1 S2 phi1 phi2 :
orthogonal S1 S2 -> phi1 \in <<S1>>%VS -> phi2 \in <<S2>>%VS ->
'[phi1, phi2] = 0. | Proof.
move/orthogonalP=> oS12; do 2!move/(@coord_span _ _ _ (in_tuple _))->.
rewrite cfdot_suml big1 // => i _; rewrite cfdot_sumr big1 // => j _.
by rewrite cfdotZl cfdotZr oS12 ?mem_nth ?mulr0.
Qed. | Lemma | span_orthogonal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"big1",
"cfdotZl",
"cfdotZr",
"cfdot_suml",
"cfdot_sumr",
"coord_span",
"in_tuple",
"mem_nth",
"mulr0",
"orthogonal",
"orthogonalP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_split S beta :
{X : 'CF(G) & X \in <<S>>%VS &
{Y | [/\ beta = X + Y, '[X, Y] = 0 & orthogonal Y S]}}. | Proof.
suffices [X S_X [Y -> oYS]]:
{X : _ & X \in <<S>>%VS & {Y | beta = X + Y & orthogonal Y S}}.
- exists X => //; exists Y.
by rewrite cfdotC (span_orthogonal oYS) ?memv_span1 ?conjC0.
elim: S beta => [|phi S IHS] beta.
by exists 0; last exists beta; rewrite ?mem0v ?add0r.
have [[U S_U [V -> oVS]] [X S_X [Y -... | Lemma | orthogonal_split | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"add0r",
"addKr",
"addrA",
"addrC",
"apply",
"cfdot0r",
"cfdotBl",
"cfdotC",
"cfdotDr",
"cfdotZl",
"cfnorm_eq0",
"conjC0",
"divfK",
"eqVneq",
"inE",
"last",
"mem0v",
"mem_head",
"memvB",
"memvZ",
"memv_add",
"memv_line",
"memv_span",
"memv_span1",
"mul0r",
"mulr0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_orthogonal M (nu : 'CF(G) -> 'CF(M)) S R (A : {pred 'CF(G)}) :
{in A &, isometry nu} -> {subset S <= A} -> {subset R <= A} ->
orthogonal (map nu S) (map nu R) = orthogonal S R. | Proof.
move=> Inu sSA sRA; rewrite [orthogonal _ _]all_map.
apply: eq_in_all => phi Sphi; rewrite /= all_map.
by apply: eq_in_all => psi Rpsi; rewrite /= Inu ?(sSA phi) ?(sRA psi).
Qed. | Lemma | map_orthogonal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Inu",
"all_map",
"apply",
"eq_in_all",
"isometry",
"map",
"orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_oppr S R : orthogonal S (map -%R R) = orthogonal S R. | Proof.
wlog suffices IH: S R / orthogonal S R -> orthogonal 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 cfdotNr oSR ?oppr0.
Qed. | Lemma | orthogonal_oppr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdotNr",
"map",
"mapK",
"mapP",
"oppr0",
"opprK",
"orthogonal",
"orthogonalP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_oppl S R : orthogonal (map -%R S) R = orthogonal S R. | Proof. by rewrite -!(orthogonal_sym R) orthogonal_oppr. Qed. | Lemma | orthogonal_oppl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"map",
"orthogonal",
"orthogonal_oppr",
"orthogonal_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairwise_orthogonalP S :
reflect (uniq (0 :: S)
/\ {in S &, forall phi psi, phi != psi -> '[phi, psi] = 0})
(pairwise_orthogonal 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_b... | Lemma | pairwise_orthogonalP | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"allP",
"apply",
"cfdotC",
"cfnorm_eq0",
"conjC_eq0",
"eq_sym",
"inE",
"last",
"memPnC",
"mem_behead",
"mem_head",
"notS0",
"oSS",
"pairwise_orthogonal",
"predU1P",
"split",
"uniq",
"uniqS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairwise_orthogonal_cat R S :
pairwise_orthogonal (R ++ S) =
[&& pairwise_orthogonal R, pairwise_orthogonal S & orthogonal R S]. | Proof.
rewrite /pairwise_orthogonal mem_cat negb_or -!andbA; do !bool_congr.
elim: R => [|phi R /= ->]; rewrite ?andbT // orthogonal_cons all_cat -!andbA /=.
by do !bool_congr.
Qed. | Lemma | pairwise_orthogonal_cat | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"all_cat",
"mem_cat",
"orthogonal",
"orthogonal_cons",
"pairwise_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_pairwise_orthogonal R S :
perm_eq R S -> pairwise_orthogonal R = pairwise_orthogonal S. | Proof.
apply: catCA_perm_subst R S => R S S'.
rewrite !pairwise_orthogonal_cat !orthogonal_catr (orthogonal_sym R S) -!andbA.
by do !bool_congr.
Qed. | Lemma | eq_pairwise_orthogonal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"catCA_perm_subst",
"orthogonal_catr",
"orthogonal_sym",
"pairwise_orthogonal",
"pairwise_orthogonal_cat",
"perm_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_pairwise_orthogonal S1 S2 :
{subset S1 <= S2} -> uniq S1 ->
pairwise_orthogonal S2 -> pairwise_orthogonal 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 | sub_pairwise_orthogonal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"apply",
"pairwise_orthogonal",
"pairwise_orthogonalP",
"split",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_free S : pairwise_orthogonal S -> free S. | Proof.
case/pairwise_orthogonalP=> [/=/andP[notS0 uniqS] oSS].
rewrite -(in_tupleE S); apply/freeP => a aS0 i.
have S_i: S`_i \in S by apply: mem_nth.
have /eqP: '[S`_i, 0]_G = 0 := cfdot0r _.
rewrite -{2}aS0 raddf_sum /= (bigD1 i) //= big1 => [j neq_ji|].
by rewrite cfdotZr oSS ?mulr0 ?mem_nth // eq_sym nth_uniq.
re... | Lemma | orthogonal_free | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"addr0",
"apply",
"big1",
"bigD1",
"cfdot0r",
"cfdotZr",
"cfnorm_eq0",
"conjC_eq0",
"eq_sym",
"free",
"freeP",
"in_tupleE",
"mem_nth",
"mulf_eq0",
"mulr0",
"notS0",
"nth_uniq",
"oSS",
"pairwise_orthogonal",
"pairwise_orthogonalP",
"pred2P",
"raddf_sum",
"uniqS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_pairwise_orthogonal S p :
pairwise_orthogonal S -> pairwise_orthogonal (filter p S). | Proof.
move=> orthoS; apply: sub_pairwise_orthogonal (orthoS).
exact: mem_subseq (filter_subseq p S).
exact/filter_uniq/free_uniq/orthogonal_free.
Qed. | Lemma | filter_pairwise_orthogonal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"filter",
"filter_subseq",
"filter_uniq",
"free_uniq",
"mem_subseq",
"orthogonal_free",
"pairwise_orthogonal",
"sub_pairwise_orthogonal"
] | 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 _)) //= cfdot0r eq_sym oner_eq0.
Qed. | Lemma | orthonormal_not0 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"allP",
"cfdot0r",
"eq_sym",
"oner_eq0",
"orthonormal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormal_cat R S :
orthonormal (R ++ S) = [&& orthonormal R, orthonormal S & orthogonal R S]. | Proof.
rewrite !orthonormalE pairwise_orthogonal_cat all_cat -!andbA.
by do !bool_congr.
Qed. | Lemma | orthonormal_cat | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"all_cat",
"orthogonal",
"orthonormal",
"orthonormalE",
"pairwise_orthogonal_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_orthonormal R S : perm_eq R S -> orthonormal R = orthonormal S. | Proof.
move=> eqRS; rewrite !orthonormalE (eq_all_r (perm_mem eqRS)).
by rewrite (eq_pairwise_orthogonal eqRS).
Qed. | Lemma | eq_orthonormal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"eq_all_r",
"eq_pairwise_orthogonal",
"orthonormal",
"orthonormalE",
"perm_eq",
"perm_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormal_free S : orthonormal S -> free S. | Proof. by move/orthonormal_orthogonal/orthogonal_free. Qed. | Lemma | orthonormal_free | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"free",
"orthogonal_free",
"orthonormal",
"orthonormal_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormalP S :
reflect (uniq S /\ {in S &, forall phi psi, '[phi, psi]_G = (phi == psi)%:R})
(orthonormal 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] := eqVneq.
split=> // ... | Lemma | orthonormalP | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"allP",
"apply",
"cfdot0r",
"eqVneq",
"eq_sym",
"eqxx",
"last",
"oSS",
"oner_eq0",
"orthonormal",
"orthonormalE",
"pairwise_orthogonalP",
"split",
"uniq",
"uniqS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_orthonormal S1 S2 :
{subset S1 <= S2} -> uniq S1 -> orthonormal S2 -> orthonormal S1. | Proof.
move=> sS12 uniqS1 /orthonormalP[_ oS1].
by apply/orthonormalP; split; last apply: sub_in2 sS12 _ _.
Qed. | Lemma | sub_orthonormal | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"apply",
"last",
"orthonormal",
"orthonormalP",
"split",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormal2P phi psi :
reflect [/\ '[phi, psi] = 0, '[phi] = 1 & '[psi] = 1]
(orthonormal [:: phi; psi]). | Proof.
rewrite /orthonormal /= !andbT andbC.
by apply: (iffP and3P) => [] []; do 3!move/eqP->.
Qed. | Lemma | orthonormal2P | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"orthonormal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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