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 |
|---|---|---|---|---|---|---|---|---|---|---|
cfunM_onI A B phi psi :
phi \in 'CF(G, A) -> psi \in 'CF(G, B) -> phi * psi \in 'CF(G, A :&: B). | Proof.
rewrite !cfun_onE => Aphi Bpsi; apply/subsetP=> x; rewrite !inE cfunE mulf_eq0.
by case/norP=> /(subsetP Aphi)-> /(subsetP Bpsi).
Qed. | Lemma | cfunM_onI | 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",
"cfun_onE",
"inE",
"mulf_eq0",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfunM_on A phi psi :
phi \in 'CF(G, A) -> psi \in 'CF(G, A) -> phi * psi \in 'CF(G, A). | Proof. by move=> Aphi Bpsi; rewrite -[A]setIid cfunM_onI. Qed. | Lemma | cfunM_on | 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",
"... | [
"cfunM_onI",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprodKey : unit. | Proof. by []. Qed. | Fact | cfSdprodKey | 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",
"... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod | :=
locked_with cfSdprodKey
(cfMorph \o cfIsom (tagged (sdprod_isom defG)) : 'CF(H) -> 'CF(G)). | Definition | cfSdprod | 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",
"... | [
"cfIsom",
"cfMorph",
"cfSdprodKey",
"defG",
"sdprod_isom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_unlockable | := [unlockable of cfSdprod]. | Canonical | cfSdprod_unlockable | 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",
"... | [
"cfSdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_is_zmod_morphism : zmod_morphism cfSdprod. | Proof. rewrite unlock; exact: raddfB. Qed. | Lemma | cfSdprod_is_zmod_morphism | 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",
"... | [
"cfSdprod",
"raddfB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_is_additive | := cfSdprod_is_zmod_morphism. | Definition | cfSdprod_is_additive | 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",
"... | [
"cfSdprod_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_is_monoid_morphism : monoid_morphism cfSdprod. | Proof. rewrite unlock; exact: (rmorph1 _, rmorphM _). Qed. | Lemma | cfSdprod_is_monoid_morphism | 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",
"... | [
"cfSdprod",
"monoid_morphism",
"rmorph1",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_is_multiplicative | :=
(fun g => (g.2,g.1)) cfSdprod_is_monoid_morphism. | Definition | cfSdprod_is_multiplicative | 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",
"... | [
"cfSdprod_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_is_scalable : scalable cfSdprod. | Proof. rewrite unlock; exact: linearZ_LR. Qed. | Lemma | cfSdprod_is_scalable | 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",
"... | [
"cfSdprod",
"linearZ_LR",
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod1 phi : cfSdprod phi 1%g = phi 1%g. | Proof. by rewrite unlock /= cfMorph1 cfIsom1. Qed. | Lemma | cfSdprod1 | 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",
"... | [
"cfIsom1",
"cfMorph1",
"cfSdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nsKG : K <| G. | Proof. by have [] := sdprod_context defG. Qed. | Let | nsKG | 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",
"... | [
"defG",
"sdprod_context"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sHG : H \subset G. | Proof. by have [] := sdprod_context defG. Qed. | Let | sHG | 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",
"... | [
"defG",
"sdprod_context"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sKG : K \subset G. | Proof. by have [] := andP nsKG. Qed. | Let | sKG | 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",
"... | [
"nsKG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_sdprod phi : K \subset cfker (cfSdprod phi). | Proof. by rewrite unlock_with cfker_mod. Qed. | Lemma | cfker_sdprod | 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",
"... | [
"cfSdprod",
"cfker",
"cfker_mod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprodEr phi : {in H, cfSdprod phi =1 phi}. | Proof. by move=> y Hy; rewrite unlock cfModE ?cfIsomE ?(subsetP sHG). Qed. | Lemma | cfSdprodEr | 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",
"... | [
"cfIsomE",
"cfModE",
"cfSdprod",
"sHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprodE phi : {in K & H, forall x y, cfSdprod phi (x * y)%g = phi y}. | Proof.
by move=> x y Kx Hy; rewrite /= cfkerMl ?(subsetP (cfker_sdprod _)) ?cfSdprodEr.
Qed. | Lemma | cfSdprodE | 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",
"... | [
"cfSdprod",
"cfSdprodEr",
"cfkerMl",
"cfker_sdprod",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprodK : cancel cfSdprod 'Res[H]. | Proof. by move=> phi; apply/cfun_inP=> x Hx; rewrite cfResE ?cfSdprodEr. Qed. | Lemma | cfSdprodK | 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",
"cfResE",
"cfSdprod",
"cfSdprodEr",
"cfun_inP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_inj : injective cfSdprod. | Proof. exact: can_inj cfSdprodK. Qed. | Lemma | cfSdprod_inj | 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",
"... | [
"cfSdprod",
"cfSdprodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_eq1 phi : (cfSdprod phi == 1) = (phi == 1). | Proof. exact: rmorph_eq1 cfSdprod_inj. Qed. | Lemma | cfSdprod_eq1 | 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",
"... | [
"cfSdprod",
"cfSdprod_inj",
"rmorph_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_sdprodK phi : K \subset cfker phi -> cfSdprod ('Res[H] phi) = phi. | Proof.
move=> kerK; apply/cfun_inP=> _ /(mem_sdprod defG)[x [y [Kx Hy -> _]]].
by rewrite cfSdprodE // cfResE // cfkerMl ?(subsetP kerK).
Qed. | Lemma | cfRes_sdprodK | 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",
"cfResE",
"cfSdprod",
"cfSdprodE",
"cfker",
"cfkerMl",
"cfun_inP",
"defG",
"mem_sdprod",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_cfker phi : K ><| cfker phi = cfker (cfSdprod phi). | Proof.
have [skerH [_ _ nKH tiKH]] := (cfker_sub phi, sdprodP defG).
rewrite unlock cfker_morph ?normal_norm // cfker_isom restrmEsub //=.
rewrite -(sdprod_modl defG) ?sub_cosetpre //=; congr (_ ><| _).
by rewrite quotientK ?(subset_trans skerH) // -group_modr //= setIC tiKH mul1g.
Qed. | Lemma | sdprod_cfker | 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",
"... | [
"cfSdprod",
"cfker",
"cfker_isom",
"cfker_morph",
"cfker_sub",
"defG",
"group_modr",
"mul1g",
"nKH",
"normal_norm",
"quotientK",
"restrmEsub",
"sdprodP",
"sdprod_modl",
"setIC",
"sub_cosetpre",
"subset_trans",
"tiKH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_sdprod phi : #[cfSdprod phi]%CF = #[phi]%CF. | Proof. exact: cforder_inj_rmorph cfSdprod_inj. Qed. | Lemma | cforder_sdprod | 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",
"... | [
"cfSdprod",
"cfSdprod_inj",
"cforder_inj_rmorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex_dprod R idx (op : Monoid.com_law idx) (F : gT -> R) :
\big[op/idx]_(g in G) F g =
\big[op/idx]_(k in K) \big[op/idx]_(h in H) F (k * h)%g. | Proof.
have /mulgmP/misomP[fM /isomP[injf im_f]] := KxH.
rewrite pair_big_dep -im_f morphimEdom big_imset; first exact/injmP.
by apply: eq_big => [][x y]; rewrite ?inE.
Qed. | Lemma | reindex_dprod | 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",
"... | [
"KxH",
"apply",
"big_imset",
"com_law",
"eq_big",
"fM",
"gT",
"inE",
"injf",
"injmP",
"isomP",
"misomP",
"morphimEdom",
"mulgmP",
"pair_big_dep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodr | := cfSdprod (dprodWsd KxH). | Definition | cfDprodr | 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",
"... | [
"KxH",
"cfSdprod",
"dprodWsd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodl | := cfSdprod (dprodWsdC KxH). | Definition | cfDprodl | 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",
"... | [
"KxH",
"cfSdprod",
"dprodWsdC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod phi psi | := cfDprodl phi * cfDprodr psi. | Definition | cfDprod | 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",
"... | [
"cfDprodl",
"cfDprodr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodl1 phi : cfDprodl phi 1%g = phi 1%g. | Proof. exact: cfSdprod1. Qed. | Lemma | cfDprodl1 | 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",
"... | [
"cfDprodl",
"cfSdprod1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodr1 psi : cfDprodr psi 1%g = psi 1%g. | Proof. exact: cfSdprod1. Qed. | Lemma | cfDprodr1 | 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",
"... | [
"cfDprodr",
"cfSdprod1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod1 phi psi : cfDprod phi psi 1%g = phi 1%g * psi 1%g. | Proof. by rewrite cfunE /= !cfSdprod1. Qed. | Lemma | cfDprod1 | 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",
"... | [
"cfDprod",
"cfSdprod1",
"cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodl_eq1 phi : (cfDprodl phi == 1) = (phi == 1). | Proof. exact: cfSdprod_eq1. Qed. | Lemma | cfDprodl_eq1 | 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",
"... | [
"cfDprodl",
"cfSdprod_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodr_eq1 psi : (cfDprodr psi == 1) = (psi == 1). | Proof. exact: cfSdprod_eq1. Qed. | Lemma | cfDprodr_eq1 | 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",
"... | [
"cfDprodr",
"cfSdprod_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_cfun1r phi : cfDprod phi 1 = cfDprodl phi. | Proof. by rewrite /cfDprod rmorph1 mulr1. Qed. | Lemma | cfDprod_cfun1r | 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",
"... | [
"cfDprod",
"cfDprodl",
"mulr1",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_cfun1l psi : cfDprod 1 psi = cfDprodr psi. | Proof. by rewrite /cfDprod rmorph1 mul1r. Qed. | Lemma | cfDprod_cfun1l | 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",
"... | [
"cfDprod",
"cfDprodr",
"mul1r",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_cfun1 : cfDprod 1 1 = 1. | Proof. by rewrite cfDprod_cfun1l rmorph1. Qed. | Lemma | cfDprod_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",
"... | [
"cfDprod",
"cfDprod_cfun1l",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_split phi psi : cfDprod phi psi = cfDprod phi 1 * cfDprod 1 psi. | Proof. by rewrite cfDprod_cfun1l cfDprod_cfun1r. Qed. | Lemma | cfDprod_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",
"... | [
"cfDprod",
"cfDprod_cfun1l",
"cfDprod_cfun1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nsKG : K <| G. | Proof. by have [] := dprod_normal2 KxH. Qed. | Let | nsKG | 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",
"... | [
"KxH",
"dprod_normal2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nsHG : H <| G. | Proof. by have [] := dprod_normal2 KxH. Qed. | Let | nsHG | 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",
"... | [
"KxH",
"dprod_normal2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cKH : H \subset 'C(K). | Proof. by have [] := dprodP KxH. Qed. | Let | cKH | 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",
"... | [
"KxH",
"dprodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sKG | := normal_sub nsKG. | Let | sKG | 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",
"... | [
"normal_sub",
"nsKG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sHG | := normal_sub nsHG. | Let | sHG | 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",
"... | [
"normal_sub",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodlK : cancel cfDprodl 'Res[K]. | Proof. exact: cfSdprodK. Qed. | Lemma | cfDprodlK | 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",
"... | [
"cfDprodl",
"cfSdprodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodrK : cancel cfDprodr 'Res[H]. | Proof. exact: cfSdprodK. Qed. | Lemma | cfDprodrK | 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",
"... | [
"cfDprodr",
"cfSdprodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_dprodl phi : cfker phi \x H = cfker (cfDprodl phi). | Proof.
by rewrite dprodC -sdprod_cfker dprodEsd // centsC (centsS (cfker_sub _)).
Qed. | Lemma | cfker_dprodl | 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",
"... | [
"centsC",
"centsS",
"cfDprodl",
"cfker",
"cfker_sub",
"dprodC",
"dprodEsd",
"sdprod_cfker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_dprodr psi : K \x cfker psi = cfker (cfDprodr psi). | Proof. by rewrite -sdprod_cfker dprodEsd // (subset_trans (cfker_sub _)). Qed. | Lemma | cfker_dprodr | 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",
"... | [
"cfDprodr",
"cfker",
"cfker_sub",
"dprodEsd",
"sdprod_cfker",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodEl phi : {in K & H, forall k h, cfDprodl phi (k * h)%g = phi k}. | Proof. by move=> k h Kk Hh /=; rewrite -(centsP cKH) // cfSdprodE. Qed. | Lemma | cfDprodEl | 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",
"... | [
"Hh",
"cKH",
"centsP",
"cfDprodl",
"cfSdprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodEr psi : {in K & H, forall k h, cfDprodr psi (k * h)%g = psi h}. | Proof. exact: cfSdprodE. Qed. | Lemma | cfDprodEr | 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",
"... | [
"cfDprodr",
"cfSdprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodE phi psi :
{in K & H, forall h k, cfDprod phi psi (h * k)%g = phi h * psi k}. | Proof. by move=> k h Kk Hh /=; rewrite cfunE cfDprodEl ?cfDprodEr. Qed. | Lemma | cfDprodE | 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",
"... | [
"Hh",
"cfDprod",
"cfDprodEl",
"cfDprodEr",
"cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_Resl phi psi : 'Res[K] (cfDprod phi psi) = psi 1%g *: phi. | Proof.
by apply/cfun_inP=> x Kx; rewrite cfunE cfResE // -{1}[x]mulg1 mulrC cfDprodE.
Qed. | Lemma | cfDprod_Resl | 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",
"cfDprod",
"cfDprodE",
"cfResE",
"cfunE",
"cfun_inP",
"mulg1",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_Resr phi psi : 'Res[H] (cfDprod phi psi) = phi 1%g *: psi. | Proof.
by apply/cfun_inP=> y Hy; rewrite cfunE cfResE // -{1}[y]mul1g cfDprodE.
Qed. | Lemma | cfDprod_Resr | 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",
"cfDprod",
"cfDprodE",
"cfResE",
"cfunE",
"cfun_inP",
"mul1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodKl (psi : 'CF(H)) : psi 1%g = 1 -> cancel (cfDprod^~ psi) 'Res. | Proof. by move=> psi1 phi; rewrite cfDprod_Resl psi1 scale1r. Qed. | Lemma | cfDprodKl | 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",
"... | [
"cfDprod",
"cfDprod_Resl",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodKr (phi : 'CF(K)) : phi 1%g = 1 -> cancel (cfDprod phi) 'Res. | Proof. by move=> phi1 psi; rewrite cfDprod_Resr phi1 scale1r. Qed. | Lemma | cfDprodKr | 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",
"... | [
"cfDprod",
"cfDprod_Resr",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_dprod phi psi :
cfker phi <*> cfker psi \subset cfker (cfDprod phi psi). | Proof.
rewrite -genM_join gen_subG; apply/subsetP=> _ /mulsgP[x y kKx kHy ->] /=.
have [[Kx _] [Hy _]] := (setIdP kKx, setIdP kHy).
have Gxy: (x * y)%g \in G by rewrite -(dprodW KxH) mem_mulg.
rewrite inE Gxy; apply/forallP=> g.
have [Gg | G'g] := boolP (g \in G); last by rewrite !cfun0 1?groupMl.
have{g Gg} [k [h [Kk ... | Lemma | cfker_dprod | 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",
"... | [
"Gg",
"Hh",
"KxH",
"apply",
"cKH",
"centsP",
"cfDprod",
"cfDprodE",
"cfker",
"cfkerMl",
"cfun0",
"dprodW",
"forallP",
"genM_join",
"gen_subG",
"groupM",
"groupMl",
"inE",
"last",
"mem_dprod",
"mem_mulg",
"mulgA",
"mulsgP",
"setIdP",
"subsetP"
] | or else phi != 0, psi != 0 and coprime #|K : cfker phi| #|H : cfker phi|. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfdot_dprod phi1 phi2 psi1 psi2 :
'[cfDprod phi1 psi1, cfDprod phi2 psi2] = '[phi1, phi2] * '[psi1, psi2]. | Proof.
rewrite !cfdotE mulrACA -invfM -natrM (dprod_card KxH); congr (_ * _).
rewrite big_distrl reindex_dprod /=; apply: eq_bigr => k Kk; rewrite big_distrr.
by apply: eq_bigr => h Hh /=; rewrite mulrACA -rmorphM !cfDprodE.
Qed. | Lemma | cfdot_dprod | 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",
"... | [
"Hh",
"KxH",
"apply",
"big_distrl",
"big_distrr",
"cfDprod",
"cfDprodE",
"cfdotE",
"dprod_card",
"eq_bigr",
"invfM",
"mulrACA",
"natrM",
"reindex_dprod",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodl_iso : isometry cfDprodl. | Proof.
by move=> phi1 phi2; rewrite -!cfDprod_cfun1r cfdot_dprod cfnorm1 mulr1.
Qed. | Lemma | cfDprodl_iso | 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",
"... | [
"cfDprod_cfun1r",
"cfDprodl",
"cfdot_dprod",
"cfnorm1",
"isometry",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodr_iso : isometry cfDprodr. | Proof.
by move=> psi1 psi2; rewrite -!cfDprod_cfun1l cfdot_dprod cfnorm1 mul1r.
Qed. | Lemma | cfDprodr_iso | 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",
"... | [
"cfDprod_cfun1l",
"cfDprodr",
"cfdot_dprod",
"cfnorm1",
"isometry",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_dprodl phi : #[cfDprodl phi]%CF = #[phi]%CF. | Proof. exact: cforder_sdprod. Qed. | Lemma | cforder_dprodl | 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",
"... | [
"cfDprodl",
"cforder_sdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_dprodr psi : #[cfDprodr psi]%CF = #[psi]%CF. | Proof. exact: cforder_sdprod. Qed. | Lemma | cforder_dprodr | 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",
"... | [
"cfDprodr",
"cforder_sdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodC (gT : finGroupType) (G K H : {group gT})
(KxH : K \x H = G) (HxK : H \x K = G) chi psi :
cfDprod KxH chi psi = cfDprod HxK psi chi. | Proof.
rewrite /cfDprod mulrC.
by congr (_ * _); congr (cfSdprod _ _); apply: eq_irrelevance.
Qed. | Lemma | cfDprodC | 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",
"... | [
"KxH",
"apply",
"cfDprod",
"cfSdprod",
"chi",
"eq_irrelevance",
"gT",
"group",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_subproof i :
gval (if P i then A i else 1%G) \x <<\bigcup_(j | P j && (j != i)) A j>> = G. | Proof.
have:= defG; rewrite fun_if big_mkcond (bigD1 i) // -big_mkcondl /= => defGi.
by have [[_ Gi' _ defGi']] := dprodP defGi; rewrite (bigdprodWY defGi') -defGi'.
Qed. | Fact | cfBigdprodi_subproof | 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",
"... | [
"bigD1",
"big_mkcond",
"big_mkcondl",
"bigdprodWY",
"defG",
"dprodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi i | := cfDprodl (cfBigdprodi_subproof i) \o 'Res[_, A i]. | Definition | cfBigdprodi | 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",
"... | [
"cfBigdprodi_subproof",
"cfDprodl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi1 i (phi : 'CF(A i)) : cfBigdprodi phi 1%g = phi 1%g. | Proof. by rewrite cfDprodl1 cfRes1. Qed. | Lemma | cfBigdprodi1 | 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",
"... | [
"cfBigdprodi",
"cfDprodl1",
"cfRes1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_eq1 i (phi : 'CF(A i)) :
P i -> (cfBigdprodi phi == 1) = (phi == 1). | Proof. by move=> Pi; rewrite cfSdprod_eq1 Pi cfRes_id. Qed. | Lemma | cfBigdprodi_eq1 | 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",
"... | [
"cfBigdprodi",
"cfRes_id",
"cfSdprod_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodiK i : P i -> cancel (@cfBigdprodi i) 'Res[A i]. | Proof.
move=> Pi phi; have:= cfDprodlK (cfBigdprodi_subproof i) ('Res phi).
by rewrite -[cfDprodl _ _]/(cfBigdprodi phi) Pi cfRes_id.
Qed. | Lemma | cfBigdprodiK | 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",
"... | [
"cfBigdprodi",
"cfBigdprodi_subproof",
"cfDprodl",
"cfDprodlK",
"cfRes_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_inj i : P i -> injective (@cfBigdprodi i). | Proof. by move/cfBigdprodiK; apply: can_inj. Qed. | Lemma | cfBigdprodi_inj | 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",
"cfBigdprodi",
"cfBigdprodiK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodEi i (phi : 'CF(A i)) x :
P i -> (forall j, P j -> x j \in A j) ->
cfBigdprodi phi (\prod_(j | P j) x j)%g = phi (x i). | Proof.
have [r big_r [Ur mem_r] _] := big_enumP P => Pi AxP.
have:= bigdprodWcp defG; rewrite -!big_r => defGr.
have{AxP} [r_i Axr]: i \in r /\ {in r, forall j, x j \in A j}.
by split=> [|j]; rewrite mem_r // => /AxP.
rewrite (perm_bigcprod defGr Axr (perm_to_rem r_i)) big_cons.
rewrite cfDprodEl ?Pi ?cfRes_id ?Axr /... | Lemma | cfBigdprodEi | 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_cons",
"big_enumP",
"big_seq",
"bigcupP",
"bigdprodWcp",
"cfBigdprodi",
"cfDprodEl",
"cfRes_id",
"defG",
"group_prod",
"mem_gen",
"mem_rem_uniq",
"perm_bigcprod",
"perm_to_rem",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_iso i : P i -> isometry (@cfBigdprodi i). | Proof. by move=> Pi phi psi; rewrite cfDprodl_iso Pi !cfRes_id. Qed. | Lemma | cfBigdprodi_iso | 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",
"... | [
"cfBigdprodi",
"cfDprodl_iso",
"cfRes_id",
"isometry"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprod (phi : forall i, 'CF(A i)) | :=
\prod_(i | P i) cfBigdprodi (phi i). | Definition | cfBigdprod | 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",
"... | [
"cfBigdprodi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodE phi x :
(forall i, P i -> x i \in A i) ->
cfBigdprod phi (\prod_(i | P i) x i)%g = \prod_(i | P i) phi i (x i). | Proof.
move=> Ax; rewrite prod_cfunE; first by rewrite -(bigdprodW defG) mem_prodg.
by apply: eq_bigr => i Pi; rewrite cfBigdprodEi.
Qed. | Lemma | cfBigdprodE | 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",
"bigdprodW",
"cfBigdprod",
"cfBigdprodEi",
"defG",
"eq_bigr",
"mem_prodg",
"prod_cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprod1 phi : cfBigdprod phi 1%g = \prod_(i | P i) phi i 1%g. | Proof. by rewrite prod_cfunE //; apply/eq_bigr=> i _; apply: cfBigdprodi1. Qed. | Lemma | cfBigdprod1 | 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",
"cfBigdprod",
"cfBigdprodi1",
"eq_bigr",
"prod_cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodK phi (Phi := cfBigdprod phi) i (a := phi i 1%g / Phi 1%g) :
Phi 1%g != 0 -> P i -> a != 0 /\ a *: 'Res[A i] Phi = phi i. | Proof.
move=> nzPhi Pi; split.
rewrite mulf_neq0 ?invr_eq0 // (contraNneq _ nzPhi) // => phi_i0.
by rewrite cfBigdprod1 (bigD1 i) //= phi_i0 mul0r.
apply/cfun_inP=> x Aix; rewrite cfunE cfResE ?sAG // mulrAC.
have {1}->: x = (\prod_(j | P j) (if j == i then x else 1))%g.
rewrite -big_mkcondr (big_pred1 i) ?eqxx /... | Lemma | cfBigdprodK | 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",
"bigD1",
"big_mkcondr",
"big_pred1",
"cfBigdprod",
"cfBigdprod1",
"cfBigdprodE",
"cfResE",
"cfunE",
"cfun_inP",
"contraNneq",
"eq_bigr",
"eqxx",
"invr_eq0",
"mul0r",
"mulfK",
"mulf_neq0",
"mulrA",
"mulrAC",
"mulrCA",
"sAG",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_bigdprod phi psi :
'[cfBigdprod phi, cfBigdprod psi] = \prod_(i | P i) '[phi i, psi i]. | Proof.
apply: canLR (mulKf (neq0CG G)) _; rewrite -(bigdprod_card defG).
rewrite (big_morph _ (@natrM _) (erefl _)) -big_split /=.
rewrite (eq_bigr _ (fun i _ => mulVKf (neq0CG _) _)) (big_distr_big_dep 1%g) /=.
set F := pfamily _ _ _; pose h (f : {ffun I -> gT}) := (\prod_(i | P i) f i)%g.
pose is_hK x f := forall f1,... | Lemma | cfdot_bigdprod | 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",
"... | [
"Dx",
"apply",
"big_distr_big_dep",
"big_morph",
"big_split",
"bigdprod_card",
"cfBigdprod",
"cfBigdprodE",
"defG",
"eq_big",
"eq_bigr",
"eqxx",
"f1",
"fK",
"familyP",
"ffunE",
"ffunP",
"fin_all_exists",
"gT",
"group_prod",
"last",
"mem_bigdprod",
"mulKf",
"mulVKf",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_iso aT rT (G D : {group aT}) (f : {morphism D >-> rT}) :
G \subset D -> isometry (cfMorph : 'CF(f @* G) -> 'CF(G)). | Proof.
move=> sGD phi psi; rewrite !cfdotE card_morphim (setIidPr sGD).
rewrite -(LagrangeI G ('ker f)) /= mulnC natrM invfM -mulrA.
congr (_ * _); apply: (canLR (mulKf (neq0CG _))).
rewrite mulr_sumr (partition_big_imset f) /= -morphimEsub //.
apply: eq_bigr => _ /morphimP[x Dx Gx ->].
rewrite -(card_rcoset _ x) mulr_... | Lemma | cfMorph_iso | 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",
"... | [
"Dx",
"LagrangeI",
"aT",
"apply",
"card_morphim",
"card_rcoset",
"cfMorph",
"cfMorphE",
"cfdotE",
"eq_big",
"eq_bigr",
"group",
"groupMr",
"groupV",
"inE",
"invfM",
"isometry",
"ker",
"last",
"mem_rcoset",
"morphimEsub",
"morphimP",
"morphism",
"mulKf",
"mulnC",
"mu... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_iso rT G (R : {group rT}) (f : {morphism G >-> rT}) :
forall isoG : isom G R f, isometry (cfIsom isoG). | Proof.
move=> isoG phi psi; rewrite unlock cfMorph_iso //; set G1 := _ @* R.
by rewrite -(isom_im (isom_sym isoG)) -/G1 in phi psi *; rewrite !cfRes_id.
Qed. | Lemma | cfIsom_iso | 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",
"... | [
"G1",
"cfIsom",
"cfMorph_iso",
"cfRes_id",
"group",
"isoG",
"isom",
"isom_im",
"isom_sym",
"isometry",
"morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_iso H G : H <| G -> isometry (@cfMod _ G H). | Proof. by case/andP=> _; apply: cfMorph_iso. Qed. | Lemma | cfMod_iso | 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",
"cfMod",
"cfMorph_iso",
"isometry"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_iso H G :
H <| G -> {in [pred phi | H \subset cfker phi] &, isometry (@cfQuo _ G H)}. | Proof.
by move=> nsHG phi psi sHkphi sHkpsi; rewrite -(cfMod_iso nsHG) !cfQuoK.
Qed. | Lemma | cfQuo_iso | 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",
"... | [
"cfMod_iso",
"cfQuo",
"cfQuoK",
"cfker",
"isometry",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_quo H G phi :
H <| G -> H \subset cfker phi -> '[phi / H] = '[phi]_G. | Proof. by move=> nsHG sHker; apply: cfQuo_iso. Qed. | Lemma | cfnorm_quo | 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",
"cfQuo_iso",
"cfker",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_iso K H G (defG : K ><| H = G) : isometry (cfSdprod defG). | Proof.
move=> phi psi; have [/andP[_ nKG] _ _ _ _] := sdprod_context defG.
by rewrite [cfSdprod _]locked_withE cfMorph_iso ?cfIsom_iso.
Qed. | Lemma | cfSdprod_iso | 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",
"... | [
"cfIsom_iso",
"cfMorph_iso",
"cfSdprod",
"defG",
"isometry",
"nKG",
"sdprod_context"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_cfInd (phi : 'CF(A)) | :=
[ffun x => if H \subset G then #|A|%:R^-1 * (\sum_(y in G) phi (x ^ y))
else #|G|%:R * '[phi, 1] *+ (x == 1%g)]. | Definition | ffun_cfInd | 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",
"... | [] | so that Frobenius reciprocity holds even in this degenerate case. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfInd_subproof phi : is_class_fun G (ffun_cfInd phi). | Proof.
apply: intro_class_fun => [x y Gx Gy | x H'x]; last first.
case: subsetP => [sHG | _]; last by rewrite (negPf (group1_contra H'x)).
rewrite big1 ?mulr0 // => y Gy; rewrite cfun0gen ?(contra _ H'x) //= => /sHG.
by rewrite memJ_norm ?(subsetP (normG _)).
rewrite conjg_eq1 (reindex_inj (mulgI y^-1)%g); congr ... | Fact | cfInd_subproof | 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",
"big1",
"cfun0gen",
"conjgM",
"conjg_eq1",
"eq_big",
"ffun_cfInd",
"group1_contra",
"groupMl",
"groupV",
"intro_class_fun",
"is_class_fun",
"last",
"memJ_norm",
"mulKVg",
"mulgI",
"mulr0",
"normG",
"reindex_inj",
"sHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd phi | := Cfun 1 (cfInd_subproof phi). | Definition | cfInd | 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",
"cfInd_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd_is_linear : linear cfInd. | Proof.
move=> c phi psi; apply/cfunP=> x; rewrite !cfunElock; case: ifP => _.
rewrite mulrCA -mulrDr [c * _]mulr_sumr -big_split /=.
by congr (_ * _); apply: eq_bigr => y _; rewrite !cfunE.
rewrite mulrnAr -mulrnDl !(mulrCA c) -!mulrDr [c * _]mulr_sumr -big_split /=.
by congr (_ * (_ * _) *+ _); apply: eq_bigr => y... | Lemma | cfInd_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",
"... | [
"apply",
"big_split",
"cfInd",
"cfunE",
"cfunElock",
"cfunP",
"eq_bigr",
"linear",
"mulrA",
"mulrCA",
"mulrDl",
"mulrDr",
"mulr_sumr",
"mulrnAr",
"mulrnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Ind[' B , A ]" | := (@cfInd B A) : ring_scope. | Notation | ''Ind[' B , A ] | 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",
"... | [
"cfInd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Ind[' B ]" | := 'Ind[B, _] : ring_scope. | Notation | ''Ind[' B ] | 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 | |
cfIndE (G H : {group gT}) phi x :
H \subset G -> 'Ind[G, H] phi x = #|H|%:R^-1 * (\sum_(y in G) phi (x ^ y)). | Proof. by rewrite cfunElock !genGid => ->. Qed. | Lemma | cfIndE | 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",
"... | [
"cfunElock",
"gT",
"genGid",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIndEout phi :
~~ (H \subset G) -> 'Ind[G] phi = (#|G|%:R * '[phi, 1]) *: '1_1%G. | Proof.
move/negPf=> not_sHG; apply/cfunP=> x; rewrite cfunE cfuniE ?normal1 // inE.
by rewrite mulr_natr cfunElock !genGid not_sHG.
Qed. | Lemma | cfIndEout | 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",
"cfunElock",
"cfunP",
"cfuniE",
"genGid",
"inE",
"mulr_natr",
"normal1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIndEsdprod (phi : 'CF(K)) x :
K ><| H = G -> 'Ind[G] phi x = \sum_(w in H) phi (x ^ w)%g. | Proof.
move=> defG; have [/andP[sKG _] _ mulKH nKH _] := sdprod_context defG.
rewrite cfIndE //; apply: canLR (mulKf (neq0CG _)) _; rewrite -mulKH mulr_sumr.
rewrite (set_partition_big _ (rcosets_partition_mul H K)) ?big_imset /=.
have [{}nKH /isomP[injf _]] := sdprod_isom defG.
apply: can_in_inj (fun Ky => invm in... | Lemma | cfIndEsdprod | 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_imset",
"cfIndE",
"cfunJ",
"conjgM",
"coset",
"coset_reprK",
"defG",
"eq_bigr",
"injf",
"invm",
"invmE",
"isomP",
"lcosetE",
"mulKf",
"mulgI",
"mulr_natl",
"mulr_sumr",
"nKH",
"neq0CG",
"norm_rlcoset",
"rcosetE",
"rcosets_partition_mul",
"repr",
"sKG",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd_on A phi :
H \subset G -> phi \in 'CF(H, A) -> 'Ind[G] phi \in 'CF(G, class_support A G). | Proof.
move=> sHG Af; apply/cfun_onP=> g AG'g; rewrite cfIndE ?big1 ?mulr0 // => h Gh.
apply: (cfun_on0 Af); apply: contra AG'g => Agh.
by rewrite -[g](conjgK h) memJ_class_support // groupV.
Qed. | Lemma | cfInd_on | 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",
"big1",
"cfIndE",
"cfun_on0",
"cfun_onP",
"class_support",
"conjgK",
"groupV",
"memJ_class_support",
"mulr0",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd_id phi : 'Ind[H] phi = phi. | Proof.
apply/cfun_inP=> x Hx; rewrite cfIndE // (eq_bigr _ (cfunJ phi x)) sumr_const.
by rewrite -[phi x *+ _]mulr_natl mulKf ?neq0CG.
Qed. | Lemma | cfInd_id | 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",
"cfIndE",
"cfunJ",
"cfun_inP",
"eq_bigr",
"mulKf",
"mulr_natl",
"neq0CG",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd_normal phi : H <| G -> 'Ind[G] phi \in 'CF(G, H). | Proof.
case/andP=> sHG nHG; apply: (cfun_onS (class_support_sub_norm (subxx _) nHG)).
by rewrite cfInd_on ?cfun_onG.
Qed. | Lemma | cfInd_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",
"... | [
"apply",
"cfInd_on",
"cfun_onG",
"cfun_onS",
"class_support_sub_norm",
"nHG",
"sHG",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd1 phi : H \subset G -> 'Ind[G] phi 1%g = #|G : H|%:R * phi 1%g. | Proof.
move=> sHG; rewrite cfIndE // natf_indexg // -mulrA mulrCA; congr (_ * _).
by rewrite mulr_natl -sumr_const; apply: eq_bigr => x; rewrite conj1g.
Qed. | Lemma | cfInd1 | 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",
"cfIndE",
"conj1g",
"eq_bigr",
"mulrA",
"mulrCA",
"mulr_natl",
"natf_indexg",
"sHG",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd_cfun1 : H <| G -> 'Ind[G, H] 1 = #|G : H|%:R *: '1_H. | Proof.
move=> nsHG; have [sHG nHG] := andP nsHG; rewrite natf_indexg // mulrC.
apply/cfunP=> x; rewrite cfIndE ?cfunE ?cfuniE // -mulrA; congr (_ * _).
rewrite mulr_natl -sumr_const; apply: eq_bigr => y Gy.
by rewrite cfun1E -{1}(normsP nHG y Gy) memJ_conjg.
Qed. | Lemma | cfInd_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",
"cfIndE",
"cfun1E",
"cfunE",
"cfunP",
"cfuniE",
"eq_bigr",
"memJ_conjg",
"mulrA",
"mulrC",
"mulr_natl",
"nHG",
"natf_indexg",
"normsP",
"nsHG",
"sHG",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_Ind_cfun1 : H <| G -> '['Ind[G, H] 1] = #|G : H|%:R. | Proof.
move=> nsHG; rewrite cfInd_cfun1 // cfnormZ normr_nat cfdot_cfuni // setIid.
by rewrite expr2 {2}natf_indexg ?normal_sub // !mulrA divfK ?mulfK ?neq0CG.
Qed. | Lemma | cfnorm_Ind_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",
"... | [
"cfInd_cfun1",
"cfdot_cfuni",
"cfnormZ",
"divfK",
"expr2",
"mulfK",
"mulrA",
"natf_indexg",
"neq0CG",
"normal_sub",
"normr_nat",
"nsHG",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIndInd phi :
K \subset G -> H \subset K -> 'Ind[G] ('Ind[K] phi) = 'Ind[G] phi. | Proof.
move=> sKG sHK; apply/cfun_inP=> x Gx; rewrite !cfIndE ?(subset_trans sHK) //.
apply: canLR (mulKf (neq0CG K)) _; rewrite mulr_sumr mulr_natl.
transitivity (\sum_(y in G) \sum_(z in K) #|H|%:R^-1 * phi ((x ^ y) ^ z)).
by apply: eq_bigr => y Gy; rewrite cfIndE // -mulr_sumr.
symmetry; rewrite exchange_big /= -s... | Lemma | cfIndInd | 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",
"cfIndE",
"cfun_inP",
"conjgM",
"eq_big",
"eq_bigr",
"exchange_big",
"groupMr",
"mulIg",
"mulKf",
"mulr_natl",
"mulr_sumr",
"neq0CG",
"reindex_inj",
"sHK",
"sKG",
"subsetP",
"subset_trans",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Frobenius_reciprocity phi psi : '[phi, 'Res[H] psi] = '['Ind[G] phi, psi]. | Proof.
have [sHG | not_sHG] := boolP (H \subset G); last first.
rewrite cfResEout // cfIndEout // cfdotZr cfdotZl mulrAC; congr (_ * _).
rewrite (cfdotEl _ (cfuni_on _ _)) mulVKf ?neq0CG // big_set1.
by rewrite cfuniE ?normal1 ?set11 ?mul1r.
transitivity (#|H|%:R^-1 * \sum_(x in G) phi x * (psi x)^* ).
rewrite ... | Lemma | Frobenius_reciprocity | 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",
"addrC",
"apply",
"astabsJ",
"big1",
"big_set1",
"big_setID",
"cfIndE",
"cfIndEout",
"cfResE",
"cfResEout",
"cfdotEl",
"cfdotZl",
"cfdotZr",
"cfun0",
"cfunJ",
"cfuniE",
"cfuni_on",
"eq_bigr",
"exchange_big",
"last",
"mul0r",
"mul1r",
"mulKf",
"mulVKf",
"mul... | This is Isaacs, Lemma (5.2). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfdot_Res_r | := Frobenius_reciprocity. | Definition | cfdot_Res_r | 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",
"... | [
"Frobenius_reciprocity"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_Res_l psi phi : '['Res[H] psi, phi] = '[psi, 'Ind[G] phi]. | Proof. by rewrite cfdotC cfdot_Res_r -cfdotC. Qed. | Lemma | cfdot_Res_l | 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",
"cfdot_Res_r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIndM phi psi: H \subset G ->
'Ind[G] (phi * ('Res[H] psi)) = 'Ind[G] phi * psi. | Proof.
move=> HsG; apply/cfun_inP=> x Gx; rewrite !cfIndE // !cfunE !cfIndE // -mulrA.
congr (_ * _); rewrite mulr_suml; apply: eq_bigr=> i iG; rewrite !cfunE.
case: (boolP (x ^ i \in H)) => xJi; last by rewrite cfun0gen ?mul0r ?genGid.
by rewrite !cfResE //; congr (_ * _); rewrite cfunJgen ?genGid.
Qed. | Lemma | cfIndM | 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",
"cfIndE",
"cfResE",
"cfun0gen",
"cfunE",
"cfunJgen",
"cfun_inP",
"eq_bigr",
"genGid",
"iG",
"last",
"mul0r",
"mulrA",
"mulr_suml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Ind[' G , H ]" | := (@cfInd _ G H) (only parsing) : ring_scope. | Notation | ''Ind[' G , H ] | 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",
"... | [
"cfInd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Ind[' G ]" | := 'Ind[G, _] : ring_scope. | Notation | ''Ind[' G ] | 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 |
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