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 |
|---|---|---|---|---|---|---|---|---|---|---|
cfun_mul phi psi | := Cfun 0 (cfun_mul_subproof phi psi). | Definition | cfun_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",
"... | [
"Cfun",
"cfun_mul_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_unit | := [pred phi : classfun | [forall x in G, phi x != 0]]. | Definition | cfun_unit | 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",
"... | [
"classfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_inv phi | :=
if phi \in cfun_unit then cfun_comp (invr0 _) phi else phi. | Definition | cfun_inv | 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_comp",
"cfun_unit",
"invr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_scale a | := cfun_comp (mulr0 a). | Definition | cfun_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",
"... | [
"cfun_comp",
"mulr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_addA : associative cfun_add. | Proof. by move=> phi psi xi; apply/cfunP=> x; rewrite !cfunE addrA. Qed. | Fact | cfun_addA | 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",
"apply",
"cfunE",
"cfunP",
"cfun_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_addC : commutative cfun_add. | Proof. by move=> phi psi; apply/cfunP=> x; rewrite !cfunE addrC. Qed. | Fact | cfun_addC | 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",
"... | [
"addrC",
"apply",
"cfunE",
"cfunP",
"cfun_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_add0 : left_id cfun_zero cfun_add. | Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE add0r. Qed. | Fact | cfun_add0 | 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",
"cfunE",
"cfunP",
"cfun_add",
"cfun_zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_addN : left_inverse cfun_zero cfun_opp cfun_add. | Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE addNr. Qed. | Fact | cfun_addN | 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",
"... | [
"addNr",
"apply",
"cfunE",
"cfunP",
"cfun_add",
"cfun_opp",
"cfun_zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_cfunE phi n x : (phi *+ n) x = phi x *+ n. | Proof. by elim: n => [|n IHn]; rewrite ?mulrS !cfunE ?IHn. Qed. | Lemma | muln_cfunE | 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",
"mulrS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_cfunE I r (P : pred I) (phi : I -> classfun) x :
(\sum_(i <- r | P i) phi i) x = \sum_(i <- r | P i) (phi i) x. | Proof. by elim/big_rec2: _ => [|i _ psi _ <-]; rewrite cfunE. Qed. | Lemma | sum_cfunE | 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_rec2",
"cfunE",
"classfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_mulA : associative cfun_mul. | Proof. by move=> phi psi xi; apply/cfunP=> x; rewrite !cfunE mulrA. Qed. | Fact | cfun_mulA | 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",
"cfun_mul",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_mulC : commutative cfun_mul. | Proof. by move=> phi psi; apply/cfunP=> x; rewrite !cfunE mulrC. Qed. | Fact | cfun_mulC | 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",
"cfun_mul",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_mul1 : left_id '1_G cfun_mul. | Proof.
by move=> phi; apply: cfun_in_genP => x Gx; rewrite !cfunE cfun1Egen Gx mul1r.
Qed. | Fact | cfun_mul1 | 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",
"cfun1Egen",
"cfunE",
"cfun_in_genP",
"cfun_mul",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_mulD : left_distributive cfun_mul cfun_add. | Proof. by move=> phi psi xi; apply/cfunP=> x; rewrite !cfunE mulrDl. Qed. | Fact | cfun_mulD | 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",
"cfun_add",
"cfun_mul",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_nz1 : '1_G != 0. | Proof.
by apply/eqP=> /cfunP/(_ 1%g)/eqP; rewrite cfun1Egen cfunE group1 oner_eq0.
Qed. | Fact | cfun_nz1 | 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",
"cfun1Egen",
"cfunE",
"cfunP",
"group1",
"oner_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_nzRingType : nzRingType | := classfun. | Definition | cfun_nzRingType | 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",
"... | [
"classfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_ringType | := (cfun_nzRingType) (only parsing). | Notation | cfun_ringType | 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_nzRingType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expS_cfunE phi n x : (phi ^+ n.+1) x = phi x ^+ n.+1. | Proof. by elim: n => //= n IHn; rewrite !cfunE IHn. Qed. | Lemma | expS_cfunE | 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 | |
cfun_mulV : {in cfun_unit, left_inverse 1 cfun_inv *%R}. | Proof.
move=> phi Uphi; rewrite /cfun_inv Uphi; apply/cfun_in_genP=> x Gx.
by rewrite !cfunE cfun1Egen Gx mulVf ?(forall_inP Uphi).
Qed. | Fact | cfun_mulV | 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",
"cfun1Egen",
"cfunE",
"cfun_in_genP",
"cfun_inv",
"cfun_unit",
"forall_inP",
"mulVf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_unitP phi psi : psi * phi = 1 -> phi \in cfun_unit. | Proof.
move/cfunP=> phiK; apply/forall_inP=> x Gx; rewrite -unitfE; apply/unitrP.
by exists (psi x); have:= phiK x; rewrite !cfunE cfun1Egen Gx mulrC.
Qed. | Fact | cfun_unitP | 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",
"cfun1Egen",
"cfunE",
"cfunP",
"cfun_unit",
"forall_inP",
"mulrC",
"unitfE",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_inv0id : {in [predC cfun_unit], cfun_inv =1 id}. | Proof. by rewrite /cfun_inv => phi /negbTE/= ->. Qed. | Fact | cfun_inv0id | 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_inv",
"cfun_unit",
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_scaleA a b phi :
cfun_scale a (cfun_scale b phi) = cfun_scale (a * b) phi. | Proof. by apply/cfunP=> x; rewrite !cfunE mulrA. Qed. | Fact | cfun_scaleA | 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",
"cfun_scale",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_scale1 : left_id 1 cfun_scale. | Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE mul1r. Qed. | Fact | cfun_scale1 | 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",
"cfun_scale",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_scaleDr : right_distributive cfun_scale +%R. | Proof. by move=> a phi psi; apply/cfunP=> x; rewrite !cfunE mulrDr. Qed. | Fact | cfun_scaleDr | 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",
"cfun_scale",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_scaleDl phi : {morph cfun_scale^~ phi : a b / a + b}. | Proof. by move=> a b; apply/cfunP=> x; rewrite !cfunE mulrDl. Qed. | Fact | cfun_scaleDl | 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",
"cfun_scale",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_scaleAl a phi psi : a *: (phi * psi) = (a *: phi) * psi. | Proof. by apply/cfunP=> x; rewrite !cfunE mulrA. Qed. | Fact | cfun_scaleAl | 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",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut | := cfun_comp (rmorph0 u). | Definition | 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",
"... | [
"cfun_comp",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_cfun1i A : cfAut '1_A = '1_A. | Proof. by apply/cfunP=> x; rewrite !cfunElock rmorph_nat. Qed. | Lemma | cfAut_cfun1i | 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",
"cfAut",
"cfunElock",
"cfunP",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutZ a phi : cfAut (a *: phi) = u a *: cfAut phi. | Proof. by apply/cfunP=> x; rewrite !cfunE rmorphM. Qed. | Lemma | cfAutZ | 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",
"cfAut",
"cfunE",
"cfunP",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_is_zmod_morphism : zmod_morphism cfAut. | Proof.
by move=> phi psi; apply/cfunP=> x; rewrite ?cfAut_cfun1i // !cfunE /= rmorphB.
Qed. | Lemma | cfAut_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",
"... | [
"apply",
"cfAut",
"cfAut_cfun1i",
"cfunE",
"cfunP",
"rmorphB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_is_additive | := cfAut_is_zmod_morphism. | Definition | cfAut_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",
"... | [
"cfAut_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_is_monoid_morphism : monoid_morphism cfAut. | Proof.
by split=> [|phi psi]; apply/cfunP=> x; rewrite ?cfAut_cfun1i // !cfunE rmorphM.
Qed. | Lemma | cfAut_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",
"... | [
"apply",
"cfAut",
"cfAut_cfun1i",
"cfunE",
"cfunP",
"monoid_morphism",
"rmorphM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_is_multiplicative | :=
(fun g => (g.2,g.1)) cfAut_is_monoid_morphism. | Definition | cfAut_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",
"... | [
"cfAut_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_cfun1 : cfAut 1 = 1. | Proof. exact: rmorph1. Qed. | Lemma | cfAut_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",
"... | [
"cfAut",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_scalable : scalable_for (u \; *:%R) cfAut. | Proof. by move=> a phi; apply/cfunP=> x; rewrite !cfunE rmorphM. Qed. | Lemma | cfAut_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",
"... | [
"apply",
"cfAut",
"cfunE",
"cfunP",
"rmorphM",
"scalable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_closed (S : seq classfun) | :=
{in S, forall phi, cfAut phi \in S}. | Definition | cfAut_closed | 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",
"... | [
"cfAut",
"classfun",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfReal phi | := cfAut conjC phi == phi. | Definition | cfReal | 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",
"... | [
"cfAut",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_subset (S1 S2 : seq classfun) | :=
[/\ uniq S1, {subset S1 <= S2} & cfAut_closed conjC S1]. | Definition | cfConjC_subset | 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",
"cfAut_closed",
"classfun",
"conjC",
"seq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_vect_iso : Vector.axiom #|classes G| classfun. | Proof.
exists (fun phi => \row_i phi (repr (enum_val i))) => [a phi psi|].
by apply/rowP=> i; rewrite !(mxE, cfunE).
set n := #|_|; pose eK x : 'I_n := enum_rank_in (classes1 _) (x ^: G).
have rV2vP v : is_class_fun G [ffun x => v (eK x) *+ (x \in G)].
apply: intro_class_fun => [x y Gx Gy | x /negbTE/=-> //].
by ... | Fact | cfun_vect_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",
"... | [
"Cfun",
"apply",
"axiom",
"cfunE",
"cfunJgen",
"cfun_in_genP",
"classGidl",
"classes",
"classes1",
"classfun",
"enum_rankK_in",
"enum_rank_in",
"enum_val",
"enum_valK_in",
"enum_valP",
"groupJ",
"groupJr",
"imsetP",
"intro_class_fun",
"is_class_fun",
"mem_classes",
"mxE",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_vectType : vectType _ | := classfun. | Definition | cfun_vectType | 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",
"... | [
"classfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_base A : #|classes B ::&: A|.-tuple classfun | :=
[tuple of [seq '1_xB | xB in classes B ::&: A]]. | Definition | cfun_base | 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",
"... | [
"classes",
"classfun",
"seq",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
classfun_on A | := <<cfun_base A>>%VS. | Definition | classfun_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",
"... | [
"cfun_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot phi psi | := #|B|%:R^-1 * \sum_(x in B) phi x * (psi x)^*. | Definition | cfdot | 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 | |
cfdotr psi phi | := cfdot phi psi. | Definition | 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",
"... | [
"cfdot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm phi | := cfdot phi phi. | Definition | cfnorm | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq_of_cfun phi | := [:: phi]. | Coercion | seq_of_cfun | 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 | |
cforder phi | := \big[lcmn/1]_(x in <<B>>) #[phi x]%C. | Definition | cforder | 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",
"... | [
"lcmn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''CF' ( G )" | := (classfun G) : type_scope. | Notation | ''CF' ( 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",
"... | [
"classfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''CF' ( G )" | := (@fullv _ (cfun_vectType G)) : vspace_scope. | Notation | ''CF' ( 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",
"... | [
"cfun_vectType",
"fullv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''1_' A" | := (cfun_indicator _ A) : ring_scope. | Notation | ''1_' 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",
"... | [
"cfun_indicator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''CF' ( G , A )" | := (classfun_on G A) : ring_scope. | Notation | ''CF' ( G , 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",
"... | [
"classfun_on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"1" | := (@GRing.one (cfun_nzRingType _)) (only parsing) : cfun_scope. | Notation | 1 | 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_nzRingType",
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi ^*" | := (cfAut conjC phi) : cfun_scope. | Notation | phi ^* | 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",
"... | [
"cfAut",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_closed | := (cfAut_closed conjC). | Notation | cfConjC_closed | 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",
"... | [
"cfAut_closed",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqcfP | := (@eqP (cfun_eqType _) _ _) (only parsing). | Notation | eqcfP | 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_eqType"
] | Workaround for overeager projection reduction. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"#[ phi ]" | := (cforder phi) : cfun_scope. | Notation | #[ phi ] | 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",
"... | [
"cforder"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' u , v ]_ G" | := (@cfdot _ G u v) (only parsing) : ring_scope. | Notation | ''[' u , v ]_ 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",
"... | [
"cfdot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' u , v ]" | := (cfdot u v) : ring_scope. | Notation | ''[' u , v ] | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' u ]_ G" | := '[u, u]_G (only parsing) : ring_scope. | Notation | ''[' u ]_ 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 | |
cfker phi | := [set x in D | [forall y, phi (x * y)%g == phi y]]. | Definition | 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",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfaithful phi | := cfker phi \subset [1]. | Definition | cfaithful | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ortho_rec S1 S2 | :=
all [pred phi | all [pred psi | '[phi, psi] == 0] S2] S1. | Definition | ortho_rec | 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",
"all"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal | := ortho_rec. | Definition | 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",
"... | [
"ortho_rec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_ortho_rec S | :=
if S is psi :: S' then ortho_rec psi S' && pair_ortho_rec S' else true. | Fixpoint | pair_ortho_rec | 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",
"... | [
"ortho_rec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairwise_orthogonal S | := (0 \notin S) && pair_ortho_rec S. | Definition | 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",
"... | [
"pair_ortho_rec"
] | We exclude 0 from pairwise orthogonal sets. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
orthonormal S | := all [pred psi | '[psi] == 1] S && pair_ortho_rec S. | Definition | 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",
"... | [
"all",
"pair_ortho_rec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry tau | := forall phi psi, '[tau phi, tau psi] = '[phi, psi]. | Definition | isometry | 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 | |
isometry_from_to mCFD tau mCFR | :=
prop_in2 mCFD (inPhantom (isometry tau))
/\ prop_in1 mCFD (inPhantom (forall phi, in_mem (tau phi) mCFR)). | Definition | isometry_from_to | 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",
"... | [
"isometry"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'in' CFD , 'isometry' tau , 'to' CFR }" | :=
(isometry_from_to (mem CFD) tau (mem CFR))
(format "{ 'in' CFD , 'isometry' tau , 'to' CFR }")
: type_scope. | Notation | { 'in' CFD , 'isometry' tau , 'to' CFR } | 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",
"... | [
"isometry_from_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''1_' A" | := (cfun_indicator G A). | Notation | ''1_' 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",
"... | [
"cfun_indicator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun0 phi x : x \notin G -> phi x = 0. | Proof. by rewrite -{1}(genGid G) => /(cfun0gen phi). Qed. | Lemma | 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",
"... | [
"cfun0gen",
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
support_cfun phi : support phi \subset G. | Proof. by apply/subsetP=> g; apply: contraR => /cfun0->. Qed. | Lemma | support_cfun | 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",
"cfun0",
"subsetP",
"support"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfunJ phi x y : y \in G -> phi (x ^ y) = phi x. | Proof. by rewrite -{1}(genGid G) => /(cfunJgen phi)->. Qed. | Lemma | cfunJ | 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",
"... | [
"cfunJgen",
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_repr phi x : phi (repr (x ^: G)) = phi x. | Proof. by have [y Gy ->] := repr_class G x; apply: cfunJ. Qed. | Lemma | cfun_repr | 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",
"cfunJ",
"repr",
"repr_class"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_inP phi psi : {in G, phi =1 psi} -> phi = psi. | Proof. by rewrite -{1}genGid => /cfun_in_genP. Qed. | Lemma | cfun_inP | 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_in_genP",
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfuniE A x : A <| G -> '1_A x = (x \in A)%:R. | Proof.
case/andP=> sAG nAG; rewrite cfunElock genGid.
by rewrite class_sub_norm // andb_idl // => /(subsetP sAG).
Qed. | Lemma | cfuniE | 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",
"class_sub_norm",
"genGid",
"sAG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
support_cfuni A : A <| G -> support '1_A =i A. | Proof. by move=> nsAG x; rewrite !inE cfuniE // pnatr_eq0 -lt0n lt0b. Qed. | Lemma | support_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",
"... | [
"cfuniE",
"inE",
"lt0b",
"lt0n",
"pnatr_eq0",
"support"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mul_cfuni A phi : A <| G -> {in A, phi * '1_A =1 phi}. | Proof. by move=> nsAG x Ax; rewrite cfunE cfuniE // Ax mulr1. Qed. | Lemma | eq_mul_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",
"... | [
"cfunE",
"cfuniE",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_cfuni A : A <| G -> {in A, '1_A =1 (1 : 'CF(G))}. | Proof. by rewrite -['1_A]mul1r; apply: eq_mul_cfuni. Qed. | Lemma | eq_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",
"eq_mul_cfuni",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfuniG : '1_G = 1. | Proof. by rewrite -[G in '1_G]genGid. Qed. | Lemma | cfuniG | 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",
"... | [
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun1E g : (1 : 'CF(G)) g = (g \in G)%:R. | Proof. by rewrite -cfuniG cfuniE. Qed. | Lemma | cfun1E | 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",
"... | [
"cfuniE",
"cfuniG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun11 : (1 : 'CF(G)) 1%g = 1. | Proof. by rewrite cfun1E group1. Qed. | Lemma | cfun11 | 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",
"... | [
"cfun1E",
"group1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_cfunE I r (P : pred I) (phi : I -> 'CF(G)) x :
x \in G -> (\prod_(i <- r | P i) phi i) x = \prod_(i <- r | P i) (phi i) x. | Proof.
by move=> Gx; elim/big_rec2: _ => [|i _ psi _ <-]; rewrite ?cfunE ?cfun1E ?Gx.
Qed. | Lemma | prod_cfunE | 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_rec2",
"cfun1E",
"cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp_cfunE phi n x : x \in G -> (phi ^+ n) x = phi x ^+ n. | Proof. by rewrite -[n]card_ord -!prodr_const; apply: prod_cfunE. Qed. | Lemma | exp_cfunE | 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",
"card_ord",
"prod_cfunE",
"prodr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_cfuni A B : '1_A * '1_B = '1_(A :&: B) :> 'CF(G). | Proof.
apply/cfunP=> g; rewrite !cfunElock -natrM mulnb subsetI.
by rewrite andbCA !andbA andbb.
Qed. | Lemma | mul_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",
"cfunElock",
"cfunP",
"mulnb",
"natrM",
"subsetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_classE x y : '1_(x ^: G) y = ((x \in G) && (y \in x ^: G))%:R. | Proof.
rewrite cfunElock genGid class_sub_norm ?class_norm //; congr (_ : bool)%:R.
by apply: andb_id2r => /imsetP[z Gz ->]; rewrite groupJr.
Qed. | Lemma | cfun_classE | 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",
"cfunElock",
"class_norm",
"class_sub_norm",
"genGid",
"groupJr",
"imsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_on_sum A :
'CF(G, A) = (\sum_(xG in classes G | xG \subset A) <['1_xG]>)%VS. | Proof.
by rewrite ['CF(G, A)]span_def big_image; apply: eq_bigl => xG; rewrite !inE.
Qed. | Lemma | cfun_on_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_image",
"classes",
"eq_bigl",
"inE",
"span_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_onP A phi :
reflect (forall x, x \notin A -> phi x = 0) (phi \in 'CF(G, A)). | Proof.
apply: (iffP idP) => [/coord_span-> x notAx | Aphi].
set b := cfun_base G A; rewrite sum_cfunE big1 // => i _; rewrite cfunE.
have /mapP[xG]: b`_i \in b by rewrite -tnth_nth mem_tnth.
rewrite mem_enum => /setIdP[/imsetP[y Gy ->] Ay] ->.
by rewrite cfun_classE Gy (contraNF (subsetP Ay x)) ?mulr0.
suffices... | Lemma | cfun_onP | 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",
"cfunE",
"cfunJ",
"cfun_base",
"cfun_classE",
"cfun_inP",
"cfun_on_sum",
"cfun_repr",
"class_eqP",
"class_refl",
"classes",
"contraNeq",
"coord_span",
"eqxx",
"imsetP",
"lt0b",
"lt0n",
"mapP",
"mem_classes",
"mem_enum",
"mem_tnth",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_on0 A phi x : phi \in 'CF(G, A) -> x \notin A -> phi x = 0. | Proof. by move/cfun_onP; apply. Qed. | Lemma | cfun_on0 | 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_onP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_by_classes (R : nzRingType) (F : gT -> R) :
{in G &, forall g h, F (g ^ h) = F g} ->
\sum_(g in G) F g = \sum_(xG in classes G) #|xG|%:R * F (repr xG). | Proof.
move=> FJ; rewrite {1}(partition_big _ _ ((@mem_classes gT)^~ G)) /=.
apply: eq_bigr => _ /imsetP[x Gx ->]; have [y Gy ->] := repr_class G x.
rewrite mulr_natl -sumr_const FJ {y Gy}//; apply/esym/eq_big=> y /=.
apply/idP/andP=> [xGy | [Gy /eqP<-]]; last exact: class_refl.
by rewrite (class_eqP xGy) (subsetP... | Lemma | sum_by_classes | 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",
"class_eqP",
"class_refl",
"class_subG",
"classes",
"eq_big",
"eq_bigr",
"gT",
"imsetP",
"last",
"mem_classes",
"mulr_natl",
"partition_big",
"repr",
"repr_class",
"subsetP",
"subxx",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_base_free A : free (cfun_base G A). | Proof.
have b_i (i : 'I_#|classes G ::&: A|) : (cfun_base G A)`_i = '1_(enum_val i).
by rewrite /enum_val -!tnth_nth tnth_map.
apply/freeP => s S0 i; move/cfunP/(_ (repr (enum_val i))): S0.
rewrite sum_cfunE (bigD1 i) //= big1 ?addr0 => [j|]; last first.
rewrite b_i !cfunE; have /setIdP[/imsetP[x Gx ->] _] := enum_... | Lemma | cfun_base_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",
"... | [
"S0",
"addr0",
"apply",
"big1",
"bigD1",
"cfunE",
"cfunP",
"cfun_base",
"cfun_classE",
"cfun_repr",
"class_eqP",
"class_refl",
"classes",
"contraNeq",
"enum_val",
"enum_valP",
"enum_val_inj",
"free",
"freeP",
"imsetP",
"inj_eq",
"last",
"lt0b",
"lt0n",
"mulf_eq0",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_cfun : \dim 'CF(G) = #|classes G|. | Proof. by rewrite dimvf /dim /= genGid. Qed. | Lemma | dim_cfun | 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",
"... | [
"classes",
"dim",
"dimvf",
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_cfun_on A : \dim 'CF(G, A) = #|classes G ::&: A|. | Proof. by rewrite (eqnP (cfun_base_free A)) size_tuple. Qed. | Lemma | dim_cfun_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",
"... | [
"cfun_base_free",
"classes",
"dim",
"eqnP",
"size_tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_cfun_on_abelian A : abelian G -> A \subset G -> \dim 'CF(G, A) = #|A|. | Proof.
move/abelian_classP=> cGG sAG; rewrite -(card_imset _ set1_inj) dim_cfun_on.
apply/eq_card=> xG; rewrite !inE.
apply/andP/imsetP=> [[/imsetP[x Gx ->] Ax] | [x Ax ->]] {xG}.
by rewrite cGG ?sub1set // in Ax *; exists x.
by rewrite -{1}(cGG x) ?mem_classes ?(subsetP sAG) ?sub1set.
Qed. | Lemma | dim_cfun_on_abelian | 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",
"... | [
"abelian",
"abelian_classP",
"apply",
"cGG",
"card_imset",
"dim",
"dim_cfun_on",
"eq_card",
"imsetP",
"inE",
"mem_classes",
"sAG",
"set1_inj",
"sub1set",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfuni_on A : '1_A \in 'CF(G, A). | Proof.
apply/cfun_onP=> x notAx; rewrite cfunElock genGid.
by case: andP => // [[_ s_xG_A]]; rewrite (subsetP s_xG_A) ?class_refl in notAx.
Qed. | Lemma | cfuni_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",
"cfunElock",
"cfun_onP",
"class_refl",
"genGid",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_cfuni_on A phi : phi * '1_A \in 'CF(G, A). | Proof.
by apply/cfun_onP=> x /(cfun_onP (cfuni_on A)) Ax0; rewrite cfunE Ax0 mulr0.
Qed. | Lemma | mul_cfuni_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",
"cfunE",
"cfun_onP",
"cfuni_on",
"mulr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_onE phi A : (phi \in 'CF(G, A)) = (support phi \subset A). | Proof. exact: (sameP cfun_onP supportP). Qed. | Lemma | cfun_onE | 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_onP",
"support",
"supportP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_onT phi : phi \in 'CF(G, [set: gT]). | Proof. by rewrite cfun_onE. Qed. | Lemma | cfun_onT | 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",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_onD1 phi A :
(phi \in 'CF(G, A^#)) = (phi \in 'CF(G, A)) && (phi 1%g == 0). | Proof.
by rewrite !cfun_onE -!(eq_subset (in_set (support _))) subsetD1 !inE negbK.
Qed. | Lemma | cfun_onD1 | 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",
"eq_subset",
"inE",
"in_set",
"subsetD1",
"support"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_onG phi : phi \in 'CF(G, G). | Proof. by rewrite cfun_onE support_cfun. Qed. | Lemma | cfun_onG | 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",
"support_cfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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