fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
factm_morphM: {in q @* G &, {morph ff : x y / x * y}}.
Proof.
move=> _ _ /morphimP[x Hx Gx ->] /morphimP[y Hy Gy ->].
by rewrite -morphM ?factmE ?groupM // morphM.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
factm_morphM
| |
factm_morphism:= Morphism factm_morphM.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
factm_morphism
| |
morphim_factm(A : {set aT}) : ff @* (q @* A) = f @* A.
Proof.
rewrite -morphim_comp /= {1}/morphim /= morphimGK //; last first.
by rewrite (subset_trans sKqKf) ?subsetIl.
apply/setP=> y; apply/morphimP/morphimP;
by case=> x Gx Ax ->{y}; exists x; rewrite //= factmE.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_factm
| |
morphpre_factm(C : {set rT}) : ff @*^-1 C = q @* (f @*^-1 C).
Proof.
apply/setP=> y /[!inE]/=; apply/andP/morphimP=> [[]|[x Hx]]; last first.
by case/morphpreP=> Gx Cfx ->; rewrite factmE ?imset_f ?inE ?Hx.
case/morphimP=> x Hx Gx ->; rewrite factmE //.
by exists x; rewrite // !inE Gx.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre_factm
| |
ker_factm: 'ker ff = q @* 'ker f.
Proof. exact: morphpre_factm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_factm
| |
injm_factm: 'injm f -> 'injm ff.
Proof. by rewrite ker_factm => /trivgP->; rewrite morphim1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_factm
| |
injm_factmP: reflect ('ker f = 'ker q) ('injm ff).
Proof.
rewrite ker_factm -morphimIdom sub_morphim_pre ?subsetIl //.
rewrite setIA (setIidPr sGH) (sameP setIidPr eqP) (setIidPl _) // eq_sym.
exact: eqP.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_factmP
| |
ker_factm_loc(K : {group aT}) : 'ker_(q @* K) ff = q @* 'ker_K f.
Proof. by rewrite ker_factm -morphimIG. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_factm_loc
| |
invm_subker: 'ker f \subset 'ker (idm G).
Proof. by rewrite ker_idm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
invm_subker
| |
invm:= factm invm_subker (subxx _).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
invm
| |
invm_morphism:= Eval hnf in [morphism of invm].
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
invm_morphism
| |
invmE: {in G, cancel f invm}.
Proof. exact: factmE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
invmE
| |
invmK: {in f @* G, cancel invm f}.
Proof. by move=> fx; case/morphimP=> x _ Gx ->; rewrite invmE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
invmK
| |
morphpre_invmA : invm @*^-1 A = f @* A.
Proof. by rewrite morphpre_factm morphpre_idm morphimIdom. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre_invm
| |
morphim_invmA : A \subset G -> invm @* (f @* A) = A.
Proof. by move=> sAG; rewrite morphim_factm morphim_idm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_invm
| |
morphim_invmEC : invm @* C = f @*^-1 C.
Proof.
rewrite -morphpreIdom -(morphim_invm (subsetIl _ _)).
by rewrite morphimIdom -morphpreIim morphpreK (subsetIl, morphimIdom).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_invmE
| |
injm_properA B :
A \subset G -> B \subset G -> (f @* A \proper f @* B) = (A \proper B).
Proof.
move=> dA dB; rewrite -morphpre_invm -(morphpre_invm B).
by rewrite morphpre_proper ?morphim_invm.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_proper
| |
injm_invm: 'injm invm.
Proof. by move/can_in_inj/injmP: invmK. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_invm
| |
ker_invm: 'ker invm = 1.
Proof. by move/trivgP: injm_invm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_invm
| |
im_invm: invm @* (f @* G) = G.
Proof. exact: morphim_invm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
im_invm
| |
ifactm:=
tag (domP [morphism of g \o invm injf] (morphpre_invm injf G)).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ifactm
| |
ifactmE: {in D, forall x, ifactm (f x) = g x}.
Proof.
rewrite /ifactm => x Dx; case: domP => f' /= [def_f' _ _ _].
by rewrite {f'}def_f' //= invmE.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ifactmE
| |
morphim_ifactm(A : {set gT}) :
A \subset D -> ifactm @* (f @* A) = g @* A.
Proof.
rewrite /ifactm => sAD; case: domP => _ /= [_ _ _ ->].
by rewrite morphim_comp morphim_invm.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_ifactm
| |
im_ifactm: G \subset D -> ifactm @* (f @* G) = g @* G.
Proof. exact: morphim_ifactm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
im_ifactm
| |
morphpre_ifactmC : ifactm @*^-1 C = f @* (g @*^-1 C).
Proof.
rewrite /ifactm; case: domP => _ /= [_ _ -> _].
by rewrite morphpre_comp morphpre_invm.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre_ifactm
| |
ker_ifactm: 'ker ifactm = f @* 'ker g.
Proof. exact: morphpre_ifactm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_ifactm
| |
injm_ifactm: 'injm g -> 'injm ifactm.
Proof. by rewrite ker_ifactm => /trivgP->; rewrite morphim1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_ifactm
| |
morphic(f : aT -> rT) :=
[forall u in [predX A & A], f (u.1 * u.2) == f u.1 * f u.2].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphic
| |
isomf := f @: A^# == B^#.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom
| |
misomf := morphic f && isom f.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
misom
| |
isog:= [exists f : {ffun aT -> rT}, misom f].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog
| |
morphicP: reflect {in A &, {morph f : x y / x * y}} (morphic f).
Proof.
apply: (iffP forallP) => [fM x y Ax Ay | fM [x y] /=].
by apply/eqP; have:= fM (x, y); rewrite inE /= Ax Ay.
by apply/implyP=> /andP[Ax Ay]; rewrite fM.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphicP
| |
morphmof morphic f := f : aT -> FinGroup.sort rT.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphm
| |
morphmEfM : morphm fM = f. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphmE
| |
morphm_morphismfM := @Morphism _ _ A (morphm fM) (morphicP fM).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphm_morphism
| |
misomPf : reflect {fM : morphic f & isom (morphm fM)} (misom f).
Proof. by apply: (iffP andP) => [] [fM fiso] //; exists fM. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
misomP
| |
misom_isogf : misom f -> isog.
Proof.
case/andP=> fM iso_f; apply/existsP; exists (finfun f).
apply/andP; split; last by rewrite /misom /isom !(eq_imset _ (ffunE f)).
by apply/forallP=> u; rewrite !ffunE; apply: forallP fM u.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
misom_isog
| |
isom_isog(D : {group aT}) (f : {morphism D >-> rT}) :
A \subset D -> isom f -> isog.
Proof.
move=> sAD isof; apply: (@misom_isog f); rewrite /misom isof andbT.
by apply/morphicP; apply: (sub_in2 (subsetP sAD) (morphM f)).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_isog
| |
isog_isom: isog -> {f : {morphism A >-> rT} | isom f}.
Proof.
by case/existsP/sigW=> f /misomP[fM isom_f]; exists (morphm_morphism fM).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_isom
| |
isomP(f : {morphism G >-> rT}) :
reflect ('injm f /\ f @* G = H) (isom G H f).
Proof.
apply: (iffP eqP) => [eqfGH | [injf <-]]; last first.
by rewrite -injmD1 // morphimEsub ?subsetDl.
split.
apply/subsetP=> x /morphpreP[Gx fx1]; have: f x \notin H^# by rewrite inE fx1.
by apply: contraR => ntx; rewrite -eqfGH imset_f // inE ntx.
rewrite morphimEdom -{2}(setD1K (group1 G)) imsetU eqfGH.
by rewrite imset_set1 morph1 setD1K.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isomP
| |
isogP:
reflect (exists2 f : {morphism G >-> rT}, 'injm f & f @* G = H) (G \isog H).
Proof.
apply: (iffP idP) => [/isog_isom[f /isomP[]] | [f injf fG]]; first by exists f.
by apply: (isom_isog f) => //; apply/isomP.
Qed.
Variable f : {morphism G >-> rT}.
Hypothesis isoGH : isom G H f.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isogP
| |
isom_inj: 'injm f. Proof. by have /isomP[] := isoGH. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_inj
| |
isom_im: f @* G = H. Proof. by have /isomP[] := isoGH. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_im
| |
isom_card: #|G| = #|H|.
Proof. by rewrite -isom_im card_injm ?isom_inj. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_card
| |
isom_sub_im: H \subset f @* G. Proof. by rewrite isom_im. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_sub_im
| |
isom_inv:= restrm isom_sub_im (invm isom_inj).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_inv
| |
morphim_isom(H : {group aT}) (K : {group rT}) :
H \subset G -> isom H K f -> f @* H = K.
Proof. by case/(restrmP f)=> g [gf _ _ <- //]; rewrite -gf; case/isomP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_isom
| |
sub_isom(A : {set aT}) (C : {set rT}) :
A \subset G -> f @* A = C -> 'injm f -> isom A C f.
Proof.
move=> sAG; case: (restrmP f sAG) => g [_ _ _ img] <-{C} injf.
rewrite /isom -morphimEsub ?morphimDG ?morphim1 //.
by rewrite subDset setUC subsetU ?sAG.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sub_isom
| |
sub_isog(A : {set aT}) : A \subset G -> 'injm f -> isog A (f @* A).
Proof. by move=> sAG injf; apply: (isom_isog f sAG); apply: sub_isom. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sub_isog
| |
restr_isom_to(A : {set aT}) (C R : {group rT}) (sAG : A \subset G) :
f @* A = C -> isom G R f -> isom A C (restrm sAG f).
Proof. by move=> defC /isomP[inj_f _]; apply: sub_isom. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
restr_isom_to
| |
restr_isom(A : {group aT}) (R : {group rT}) (sAG : A \subset G) :
isom G R f -> isom A (f @* A) (restrm sAG f).
Proof. exact: restr_isom_to. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
restr_isom
| |
idm_isom: isom G G (idm G).
Proof. exact: sub_isom (im_idm G) (injm_idm G). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
idm_isom
| |
isog_refl: G \isog G. Proof. exact: isom_isog idm_isom. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_refl
| |
card_isog: G \isog H -> #|G| = #|H|.
Proof. by case/isogP=> f injf <-; apply: isom_card (f) _; apply/isomP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
card_isog
| |
isog_abelian: G \isog H -> abelian G = abelian H.
Proof. by case/isogP=> f injf <-; rewrite injm_abelian. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_abelian
| |
trivial_isog: G :=: 1 -> H :=: 1 -> G \isog H.
Proof.
move=> -> ->; apply/isogP.
exists [morphism of @trivm gT hT 1]; rewrite /= ?morphim1 //.
by rewrite ker_trivm; apply: subxx.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
trivial_isog
| |
isog_eq1: G \isog H -> (G :==: 1) = (H :==: 1).
Proof. by move=> isoGH; rewrite !trivg_card1 card_isog. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_eq1
| |
isom_sym(f : {morphism G >-> hT}) (isoGH : isom G H f) :
isom H G (isom_inv isoGH).
Proof.
rewrite sub_isom 1?injm_restrm ?injm_invm // im_restrm.
by rewrite -(isom_im isoGH) im_invm.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_sym
| |
isog_symr: G \isog H -> H \isog G.
Proof. by case/isog_isom=> f /isom_sym/isom_isog->. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_symr
| |
isog_trans: G \isog H -> H \isog K -> G \isog K.
Proof.
case/isogP=> f injf <-; case/isogP=> g injg <-.
have defG: f @*^-1 (f @* G) = G by rewrite morphimGK ?subsetIl.
rewrite -morphim_comp -{1 8}defG.
by apply/isogP; exists [morphism of g \o f]; rewrite ?injm_comp.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_trans
| |
nclasses_isog: G \isog H -> #|classes G| = #|classes H|.
Proof. by case/isogP=> f injf <-; rewrite nclasses_injm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
nclasses_isog
| |
isog_sym: (G \isog H) = (H \isog G).
Proof. by apply/idP/idP; apply: isog_symr. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_sym
| |
isog_transl: G \isog H -> (G \isog K) = (H \isog K).
Proof.
by move=> iso; apply/idP/idP; apply: isog_trans; rewrite // -isog_sym.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_transl
| |
isog_transr: G \isog H -> (K \isog G) = (K \isog H).
Proof.
by move=> iso; apply/idP/idP; move/isog_trans; apply; rewrite // -isog_sym.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_transr
| |
homgrT aT (C : {set rT}) (D : {set aT}) :=
[exists (f : {ffun aT -> rT} | morphic D f), f @: D == C].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
homg
| |
homgPrT aT (C : {set rT}) (D : {set aT}) :
reflect (exists f : {morphism D >-> rT}, f @* D = C) (homg C D).
Proof.
apply: (iffP exists_eq_inP) => [[f fM <-] | [f <-]].
by exists (morphm_morphism fM); rewrite /morphim /= setIid.
exists (finfun f); first by apply/morphicP=> x y Dx Dy; rewrite !ffunE morphM.
by rewrite /morphim setIid; apply: eq_imset => x; rewrite ffunE.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
homgP
| |
morphim_homgaT rT (A D : {set aT}) (f : {morphism D >-> rT}) :
A \subset D -> homg (f @* A) A.
Proof.
move=> sAD; apply/homgP; exists (restrm_morphism sAD f).
by rewrite morphim_restrm setIid.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_homg
| |
leq_homgrT aT (C : {set rT}) (G : {group aT}) :
homg C G -> #|C| <= #|G|.
Proof. by case/homgP=> f <-; apply: leq_morphim. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
leq_homg
| |
homg_reflaT (A : {set aT}) : homg A A.
Proof. by apply/homgP; exists (idm_morphism A); rewrite im_idm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
homg_refl
| |
homg_transaT (B : {set aT}) rT (C : {set rT}) gT (G : {group gT}) :
homg C B -> homg B G -> homg C G.
Proof.
move=> homCB homBG; case/homgP: homBG homCB => fG <- /homgP[fK <-].
by rewrite -morphim_comp morphim_homg // -sub_morphim_pre.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
homg_trans
| |
isogEcardrT aT (G : {group rT}) (H : {group aT}) :
(G \isog H) = (homg G H) && (#|H| <= #|G|).
Proof.
rewrite isog_sym; apply/isogP/andP=> [[f injf <-] | []].
by rewrite leq_eqVlt eq_sym card_im_injm injf morphim_homg.
case/homgP=> f <-; rewrite leq_eqVlt eq_sym card_im_injm.
by rewrite ltnNge leq_morphim orbF; exists f.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isogEcard
| |
isog_homrT aT (G : {group rT}) (H : {group aT}) : G \isog H -> homg G H.
Proof. by rewrite isogEcard; case/andP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_hom
| |
isogEhomrT aT (G : {group rT}) (H : {group aT}) :
(G \isog H) = homg G H && homg H G.
Proof.
apply/idP/andP=> [isoGH | [homGH homHG]].
by rewrite !isog_hom // isog_sym.
by rewrite isogEcard homGH leq_homg.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isogEhom
| |
eq_homglgT aT rT (G : {group gT}) (H : {group aT}) (K : {group rT}) :
G \isog H -> homg G K = homg H K.
Proof.
by rewrite isogEhom => /andP[homGH homHG]; apply/idP/idP; apply: homg_trans.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
eq_homgl
| |
eq_homgrgT rT aT (G : {group gT}) (H : {group rT}) (K : {group aT}) :
G \isog H -> homg K G = homg K H.
Proof.
rewrite isogEhom => /andP[homGH homHG].
by apply/idP/idP=> homK; apply: homg_trans homK _.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
eq_homgr
| |
sgval_morphism:= Morphism (@sgvalM _ G).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sgval_morphism
| |
subg_morphism:= Morphism (@subgM _ G).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
subg_morphism
| |
injm_sgval: 'injm sgval.
Proof. exact/injmP/(in2W subg_inj). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_sgval
| |
injm_subg: 'injm (subg G).
Proof. exact/injmP/(can_in_inj subgK). Qed.
Hint Resolve injm_sgval injm_subg : core.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_subg
| |
ker_sgval: 'ker sgval = 1. Proof. exact/trivgP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_sgval
| |
ker_subg: 'ker (subg G) = 1. Proof. exact/trivgP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_subg
| |
im_subg: subg G @* G = [subg G].
Proof.
apply/eqP; rewrite -subTset morphimEdom.
by apply/subsetP=> u _; rewrite -(sgvalK u) imset_f ?subgP.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
im_subg
| |
sgval_subA : sgval @* A \subset G.
Proof. by apply/subsetP=> x; case/imsetP=> u _ ->; apply: subgP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sgval_sub
| |
sgvalmKA : subg G @* (sgval @* A) = A.
Proof.
apply/eqP; rewrite eqEcard !card_injm ?subsetT ?sgval_sub // leqnn andbT.
rewrite -morphim_comp; apply/subsetP=> _ /morphimP[v _ Av ->] /=.
by rewrite sgvalK.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sgvalmK
| |
subgmK(A : {set gT}) : A \subset G -> sgval @* (subg G @* A) = A.
Proof.
move=> sAG; apply/eqP; rewrite eqEcard !card_injm ?subsetT //.
rewrite leqnn andbT -morphim_comp morphimE /= morphpreT.
by apply/subsetP=> _ /morphimP[v Gv Av ->] /=; rewrite subgK.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
subgmK
| |
im_sgval: sgval @* [subg G] = G.
Proof. by rewrite -{2}im_subg subgmK. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
im_sgval
| |
isom_subg: isom G [subg G] (subg G).
Proof. by apply/isomP; rewrite im_subg. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_subg
| |
isom_sgval: isom [subg G] G sgval.
Proof. by apply/isomP; rewrite im_sgval. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isom_sgval
| |
isog_subg: isog G [subg G].
Proof. exact: isom_isog isom_subg. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
isog_subg
| |
perm_type: predArgType :=
Perm (pval : {ffun T -> T}) & injectiveb pval.
|
Inductive
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_type
| |
pvalp := let: Perm f _ := p in f.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
pval
| |
perm_of:= perm_type.
Identity Coercion type_of_perm : perm_of >-> perm_type.
HB.instance Definition _ := [isSub for pval].
HB.instance Definition _ := [Finite of perm_type by <:].
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_of
| |
perm_proof(f : T -> T) : injective f -> injectiveb (finfun f).
Proof.
by move=> f_inj; apply/injectiveP; apply: eq_inj f_inj _ => x; rewrite ffunE.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_proof
| |
perm_unlock:= Unlockable perm.unlock.
HB.lock Definition fun_of_perm T (u : perm_type T) : T -> T := val u.
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_unlock
| |
fun_of_perm_unlock:= Unlockable fun_of_perm.unlock.
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
fun_of_perm_unlock
| |
fun_of_perm: perm_type >-> Funclass.
|
Coercion
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
fun_of_perm
| |
permPs t : s =1 t <-> s = t.
Proof. by split=> [| -> //]; rewrite unlock => eq_sv; apply/val_inj/ffunP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permP
| |
pvalEs : pval s = s :> (T -> T).
Proof. by rewrite [@fun_of_perm]unlock. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
pvalE
| |
permEf f_inj : @perm T f f_inj =1 f.
Proof. by move=> x; rewrite -pvalE [@perm]unlock ffunE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permE
| |
perm_inj{s} : injective s.
Proof. by rewrite -!pvalE; apply: (injectiveP _ (valP s)). Qed.
Hint Resolve perm_inj : core.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_inj
|
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