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
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| docstring
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|---|---|---|---|---|---|---|
subact_is_action: is_action subact_dom subact.
Proof.
split=> [a u v eq_uv | u a b Na Nb]; apply: val_inj.
move/(congr1 val): eq_uv; rewrite !val_subact.
by case: (a \in _); first move/act_inj.
have Da := astabs_dom Na; have Db := astabs_dom Nb.
by rewrite !val_subact Na Nb groupM ?actMin.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
subact_is_action
| |
subaction:= Action subact_is_action.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
subaction
| |
astab_subactS : 'C(S | subaction) = subact_dom :&: 'C(val @: S | to).
Proof.
apply/setP=> a; rewrite inE in_setI; apply: andb_id2l => sDa.
have [Da _] := setIP sDa; rewrite !inE Da.
apply/subsetP/subsetP=> [cSa _ /imsetP[x Sx ->] | cSa x Sx] /[!inE].
by have:= cSa x Sx; rewrite inE -val_eqE val_subact sDa.
by have:= cSa _ (imset_f val Sx); rewrite inE -val_eqE val_subact sDa.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab_subact
| |
astabs_subactS : 'N(S | subaction) = subact_dom :&: 'N(val @: S | to).
Proof.
apply/setP=> a; rewrite inE in_setI; apply: andb_id2l => sDa.
have [Da _] := setIP sDa; rewrite !inE Da.
apply/subsetP/subsetP=> [nSa _ /imsetP[x Sx ->] | nSa x Sx] /[!inE].
by have /[1!inE]/(imset_f val) := nSa x Sx; rewrite val_subact sDa.
have /[1!inE]/imsetP[y Sy def_y] := nSa _ (imset_f val Sx).
by rewrite ((_ a =P y) _) // -val_eqE val_subact sDa def_y.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_subact
| |
afix_subactA :
A \subset subact_dom -> 'Fix_subaction(A) = val @^-1: 'Fix_to(A).
Proof.
move/subsetP=> sAD; apply/setP=> u.
rewrite !inE !(sameP setIidPl eqP); congr (_ == A).
apply/setP=> a /[!inE]; apply: andb_id2l => Aa.
by rewrite -val_eqE val_subact sAD.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afix_subact
| |
qact_dom:= 'N(rcosets H 'N(H) | to^*).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
qact_dom
| |
qact_dom_group:= [group of qact_dom].
Local Notation subdom := (subact_dom (coset_range H) to^*).
Fact qact_subdomE : subdom = qact_dom.
Proof. by congr 'N(_|_); apply/setP=> Hx; rewrite !inE genGid. Qed.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
qact_dom_group
| |
qact_proof: qact_dom \subset subdom.
Proof. by rewrite qact_subdomE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
qact_proof
| |
qact: coset_of H -> aT -> coset_of H := act (to^*^? \ qact_proof).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
qact
| |
quotient_action:= [action of qact].
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
quotient_action
| |
acts_qact_dom: [acts qact_dom, on 'N(H) | to].
Proof.
apply/subsetP=> a nNa; rewrite !inE (astabs_dom nNa); apply/subsetP=> x Nx.
have: H :* x \in rcosets H 'N(H) by rewrite -rcosetE imset_f.
rewrite inE -(astabs_act _ nNa) => /rcosetsP[y Ny defHy].
have: to x a \in H :* y by rewrite -defHy (imset_f (to^~a)) ?rcoset_refl.
by apply: subsetP; rewrite mul_subG ?sub1set ?normG.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_qact_dom
| |
qactEcondx a :
x \in 'N(H) ->
quotient_action (coset H x) a
= coset H (if a \in qact_dom then to x a else x).
Proof.
move=> Nx; apply: val_inj; rewrite val_subact //= qact_subdomE.
have: H :* x \in rcosets H 'N(H) by rewrite -rcosetE imset_f.
case nNa: (a \in _); rewrite // -(astabs_act _ nNa).
rewrite !val_coset ?(acts_act acts_qact_dom nNa) //=.
case/rcosetsP=> y Ny defHy; rewrite defHy; apply: rcoset_eqP.
by rewrite rcoset_sym -defHy (imset_f (_^~_)) ?rcoset_refl.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
qactEcond
| |
qactEx a :
x \in 'N(H) -> a \in qact_dom ->
quotient_action (coset H x) a = coset H (to x a).
Proof. by move=> Nx nNa; rewrite qactEcond ?nNa. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
qactE
| |
acts_quotient(A : {set aT}) (B : {set rT}) :
A \subset 'N_qact_dom(B | to) -> [acts A, on B / H | quotient_action].
Proof.
move=> nBA; apply: subset_trans {A}nBA _; apply/subsetP=> a /setIP[dHa nBa].
rewrite inE dHa inE; apply/subsetP=> _ /morphimP[x nHx Bx ->].
rewrite inE /= qactE //.
by rewrite mem_morphim ?(acts_act acts_qact_dom) ?(astabs_act _ nBa).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_quotient
| |
astabs_quotient(G : {group rT}) :
H <| G -> 'N(G / H | quotient_action) = 'N_qact_dom(G | to).
Proof.
move=> nsHG; have [_ nHG] := andP nsHG.
apply/eqP; rewrite eqEsubset acts_quotient // andbT.
apply/subsetP=> a nGa; have dHa := astabs_dom nGa; have [Da _]:= setIdP dHa.
rewrite inE dHa 2!inE Da; apply/subsetP=> x Gx; have nHx := subsetP nHG x Gx.
rewrite -(quotientGK nsHG) 2!inE (acts_act acts_qact_dom) ?nHx //= inE.
by rewrite -qactE // (astabs_act _ nGa) mem_morphim.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_quotient
| |
modactx (Ha : coset_of H) :=
if x \in range then to x (repr (D :&: Ha)) else x.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
modact
| |
modactEcondx a :
a \in dom -> modact x (coset H a) = (if x \in range then to x a else x).
Proof.
case/setIP=> Da Na; case: ifP => Cx; rewrite /modact Cx //.
rewrite val_coset // -group_modr ?sub1set //.
case: (repr _) / (repr_rcosetP (D :&: H) a) => a' Ha'.
by rewrite actMin ?(afixP Cx _ Ha') //; case/setIP: Ha'.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
modactEcond
| |
modactEx a :
a \in D -> a \in 'N(H) -> x \in range -> modact x (coset H a) = to x a.
Proof. by move=> Da Na Rx; rewrite modactEcond ?Rx // inE Da. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
modactE
| |
modact_is_action: is_action (D / H) modact.
Proof.
split=> [Ha x y | x Ha Hb]; last first.
case/morphimP=> a Na Da ->{Ha}; case/morphimP=> b Nb Db ->{Hb}.
rewrite -morphM //= !modactEcond // ?groupM ?(introT setIP _) //.
by case: ifP => Cx; rewrite ?(acts_dom, Cx, actMin, introT setIP _).
case: (set_0Vmem (D :&: Ha)) => [Da0 | [a /setIP[Da NHa]]].
by rewrite /modact Da0 repr_set0 !act1 !if_same.
have Na := subsetP (coset_norm _) _ NHa.
have NDa: a \in 'N_D(H) by rewrite inE Da.
rewrite -(coset_mem NHa) !modactEcond //.
do 2![case: ifP]=> Cy Cx // eqxy; first exact: act_inj eqxy.
by rewrite -eqxy acts_dom ?Cx in Cy.
by rewrite eqxy acts_dom ?Cy in Cx.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
modact_is_action
| |
mod_action:= Action modact_is_action.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
mod_action
| |
astabs_mod: 'N(S | mod_action) = 'N(S | to) / H.
Proof.
apply/setP=> Ha; apply/idP/morphimP=> [nSa | [a nHa nSa ->]].
case/morphimP: (astabs_dom nSa) => a nHa Da defHa.
exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE.
by have:= Sx; rewrite -(astabs_act x nSa) defHa /= modactE ?(subsetP fixSH).
have Da := astabs_dom nSa; rewrite !inE mem_quotient //; apply/subsetP=> x Sx.
by rewrite !inE /= modactE ?(astabs_act x nSa) ?(subsetP fixSH).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_mod
| |
astab_mod: 'C(S | mod_action) = 'C(S | to) / H.
Proof.
apply/setP=> Ha; apply/idP/morphimP=> [cSa | [a nHa cSa ->]].
case/morphimP: (astab_dom cSa) => a nHa Da defHa.
exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE.
by rewrite -{2}[x](astab_act cSa) // defHa /= modactE ?(subsetP fixSH).
have Da := astab_dom cSa; rewrite !inE mem_quotient //; apply/subsetP=> x Sx.
by rewrite !inE /= modactE ?(astab_act cSa) ?(subsetP fixSH).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab_mod
| |
afix_modG S :
H \subset 'C(S | to) -> G \subset 'N_D(H) ->
'Fix_(S | mod_action)(G / H) = 'Fix_(S | to)(G).
Proof.
move=> cSH /subsetIP[sGD nHG].
apply/eqP; rewrite eqEsubset !subsetI !subsetIl /= -!astabCin ?quotientS //.
have cfixH F: H \subset 'C(S :&: F | to).
by rewrite (subset_trans cSH) // astabS ?subsetIl.
rewrite andbC astab_mod ?quotientS //=; last by rewrite astabCin ?subsetIr.
by rewrite -(quotientSGK nHG) //= -astab_mod // astabCin ?quotientS ?subsetIr.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afix_mod
| |
modact_faithfulG S :
[faithful G / 'C_G(S | to), on S | mod_action 'C_G(S | to)].
Proof.
rewrite /faithful astab_mod ?subsetIr //=.
by rewrite -quotientIG ?subsetIr ?trivg_quotient.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
modact_faithful
| |
actperma := perm (act_inj to a).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actperm
| |
actpermM: {in D &, {morph actperm : a b / a * b}}.
Proof. by move=> a b Da Db; apply/permP=> x; rewrite permM !permE actMin. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actpermM
| |
actperm_morphism:= Morphism actpermM.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actperm_morphism
| |
actpermEa x : actperm a x = to x a.
Proof. by rewrite permE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actpermE
| |
actpermKx a : aperm x (actperm a) = to x a.
Proof. exact: actpermE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actpermK
| |
ker_actperm: 'ker actperm = 'C(setT | to).
Proof.
congr (_ :&: _); apply/setP=> a /[!inE]/=.
apply/eqP/subsetP=> [a1 x _ | a1]; first by rewrite inE -actpermE a1 perm1.
by apply/permP=> x; apply/eqP; have:= a1 x; rewrite !inE actpermE perm1 => ->.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
ker_actperm
| |
faithful_isom(A : {group aT}) S (nSA : actby_cond A S to) :
[faithful A, on S | to] -> isom A (actperm <[nSA]> @* A) (actperm <[nSA]>).
Proof.
by move=> ffulAS; apply/isomP; rewrite ker_actperm astab_actby setIT.
Qed.
Variables (A : {set aT}) (sAD : A \subset D).
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
faithful_isom
| |
ractpermE: actperm (to \ sAD) =1 actperm to.
Proof. by move=> a; apply/permP=> x; rewrite !permE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
ractpermE
| |
afix_ractB : 'Fix_(to \ sAD)(B) = 'Fix_to(B). Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afix_ract
| |
astab_ractS : 'C(S | to \ sAD) = 'C_A(S | to).
Proof. by rewrite setIA (setIidPl sAD). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab_ract
| |
astabs_ractS : 'N(S | to \ sAD) = 'N_A(S | to).
Proof. by rewrite setIA (setIidPl sAD). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_ract
| |
acts_ract(B : {set aT}) S :
[acts B, on S | to \ sAD] = (B \subset A) && [acts B, on S | to].
Proof. by rewrite astabs_ract subsetI. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_ract
| |
mactx a := phi a x.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
mact
| |
mact_is_action: is_action D mact.
Proof.
split=> [a x y | x a b Da Db]; first exact: perm_inj.
by rewrite /mact morphM //= permM.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
mact_is_action
| |
morph_action:= Action mact_is_action.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
morph_action
| |
mactEx a : morph_action x a = phi a x. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
mactE
| |
injm_faithful: 'injm phi -> [faithful D, on setT | morph_action].
Proof.
move/injmP=> phi_inj; apply/subsetP=> a /setIP[Da /astab_act a1].
apply/set1P/phi_inj => //; apply/permP=> x.
by rewrite morph1 perm1 -mactE a1 ?inE.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
injm_faithful
| |
perm_macta : actperm morph_action a = phi a.
Proof. by apply/permP=> x; rewrite permE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
perm_mact
| |
comp_actx e := to x (f e).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
comp_act
| |
comp_is_action: is_action (f @*^-1 D) comp_act.
Proof.
split=> [e | x e1 e2]; first exact: act_inj.
move=> /morphpreP[Be1 Dfe1] /morphpreP[Be2 Dfe2].
by rewrite /comp_act morphM ?actMin.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
comp_is_action
| |
comp_action:= Action comp_is_action.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
comp_action
| |
comp_actEx e : comp_action x e = to x (f e). Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
comp_actE
| |
afix_comp(A : {set gT}) :
A \subset B -> 'Fix_comp_action(A) = 'Fix_to(f @* A).
Proof.
move=> sAB; apply/setP=> x; rewrite !inE /morphim (setIidPr sAB).
apply/subsetP/subsetP; first by move=> + _ /imsetP[a + ->] => /[apply]/[!inE].
by move=> + a Aa => /(_ (f a)); rewrite !inE imset_f// => ->.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afix_comp
| |
astab_compS : 'C(S | comp_action) = f @*^-1 'C(S | to).
Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab_comp
| |
astabs_compS : 'N(S | comp_action) = f @*^-1 'N(S | to).
Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_comp
| |
aperm_is_action: is_action setT (@aperm rT).
Proof.
by apply: is_total_action => [x|x a b]; rewrite apermE (perm1, permM).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
aperm_is_action
| |
perm_action:= Action aperm_is_action.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
perm_action
| |
porbitEa : porbit a = orbit perm_action <[a]>%g.
Proof. by rewrite unlock. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
porbitE
| |
perm_act1Pa : reflect (forall x, aperm x a = x) (a == 1).
Proof.
apply: (iffP eqP) => [-> x | a1]; first exact: act1.
by apply/permP=> x; rewrite -apermE a1 perm1.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
perm_act1P
| |
perm_faithfulA : [faithful A, on setT | perm_action].
Proof.
apply/subsetP=> a /setIP[Da crTa].
by apply/set1P; apply/permP=> x; rewrite -apermE perm1 (astabP crTa) ?inE.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
perm_faithful
| |
actperm_idp : actperm perm_action p = p.
Proof. by apply/permP=> x; rewrite permE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actperm_id
| |
orbit_morphim_actperm(A : {set aT}) :
A \subset D -> orbit 'P (actperm to @* A) =1 orbit to A.
Proof.
move=> sAD x; rewrite morphimEsub // /orbit -imset_comp.
by apply: eq_imset => a //=; rewrite actpermK.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_morphim_actperm
| |
porbit_actperm(a : aT) :
a \in D -> porbit (actperm to a) =1 orbit to <[a]>.
Proof.
move=> Da x.
by rewrite porbitE -orbit_morphim_actperm ?cycle_subG ?morphim_cycle.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
porbit_actperm
| |
restr_perm:= actperm (<[subxx 'N(S | 'P)]>).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
restr_perm
| |
restr_perm_morphism:= [morphism of restr_perm].
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
restr_perm_morphism
| |
restr_perm_onp : perm_on S (restr_perm p).
Proof.
apply/subsetP=> x; apply: contraR => notSx.
by rewrite permE /= /actby (negPf notSx).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
restr_perm_on
| |
triv_restr_permp : p \notin 'N(S | 'P) -> restr_perm p = 1.
Proof.
move=> not_nSp; apply/permP=> x.
by rewrite !permE /= /actby (negPf not_nSp) andbF.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
triv_restr_perm
| |
restr_permE: {in 'N(S | 'P) & S, forall p, restr_perm p =1 p}.
Proof. by move=> y x nSp Sx; rewrite /= actpermE actbyE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
restr_permE
| |
ker_restr_perm: 'ker restr_perm = 'C(S | 'P).
Proof. by rewrite ker_actperm astab_actby setIT (setIidPr (astab_sub _ _)). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
ker_restr_perm
| |
im_restr_permp : restr_perm p @: S = S.
Proof. exact: im_perm_on (restr_perm_on p). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
im_restr_perm
| |
restr_perm_commutes : commute (restr_perm s) s.
Proof.
have [sC|/triv_restr_perm->] := boolP (s \in 'N(S | 'P)); last first.
exact: (commute_sym (commute1 _)).
apply/permP => x; have /= xsS := astabsP sC x; rewrite !permM.
have [xS|xNS] := boolP (x \in S); first by rewrite ?(restr_permE) ?xsS.
by rewrite !(out_perm (restr_perm_on _)) ?xsS.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
restr_perm_commute
| |
SymE: Sym S = 'C(~: S | 'P).
Proof.
apply/setP => s; rewrite inE; apply/idP/astabP => [sS x|/= S_id].
by rewrite inE /= apermE => /out_perm->.
by apply/subsetP => x; move=> /(contra_neqN (S_id _)); rewrite inE negbK.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
SymE
| |
Aut_inA (B : {set gT}) := 'N_A(B | 'P) / 'C_A(B | 'P).
Variables G H : {group gT}.
Hypothesis sHG: H \subset G.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
Aut_in
| |
Aut_restr_perma : a \in Aut G -> restr_perm H a \in Aut H.
Proof.
move=> AutGa.
case nHa: (a \in 'N(H | 'P)); last by rewrite triv_restr_perm ?nHa ?group1.
rewrite inE restr_perm_on; apply/morphicP=> x y Hx Hy /=.
by rewrite !restr_permE ?groupM // -(autmE AutGa) morphM ?(subsetP sHG).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
Aut_restr_perm
| |
restr_perm_Aut: restr_perm H @* Aut G \subset Aut H.
Proof.
by apply/subsetP=> a'; case/morphimP=> a _ AutGa ->{a'}; apply: Aut_restr_perm.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
restr_perm_Aut
| |
Aut_in_isog: Aut_in (Aut G) H \isog restr_perm H @* Aut G.
Proof.
rewrite /Aut_in -ker_restr_perm kerE -morphpreIdom -morphimIdom -kerE /=.
by rewrite setIA (setIC _ (Aut G)) first_isog_loc ?subsetIr.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
Aut_in_isog
| |
Aut_sub_fullP:
reflect (forall h : {morphism H >-> gT}, 'injm h -> h @* H = H ->
exists g : {morphism G >-> gT},
[/\ 'injm g, g @* G = G & {in H, g =1 h}])
(Aut_in (Aut G) H \isog Aut H).
Proof.
rewrite (isog_transl _ Aut_in_isog) /=; set rG := _ @* _.
apply: (iffP idP) => [iso_rG h injh hH| AutHinG].
have: aut injh hH \in rG; last case/morphimP=> g nHg AutGg def_g.
suffices ->: rG = Aut H by apply: Aut_aut.
by apply/eqP; rewrite eqEcard restr_perm_Aut /= (card_isog iso_rG).
exists (autm_morphism AutGg); rewrite injm_autm im_autm; split=> // x Hx.
by rewrite -(autE injh hH Hx) def_g actpermE actbyE.
suffices ->: rG = Aut H by apply: isog_refl.
apply/eqP; rewrite eqEsubset restr_perm_Aut /=.
apply/subsetP=> h AutHh; have hH := im_autm AutHh.
have [g [injg gG eq_gh]] := AutHinG _ (injm_autm AutHh) hH.
have [Ng AutGg]: aut injg gG \in 'N(H | 'P) /\ aut injg gG \in Aut G.
rewrite Aut_aut !inE; split=> //; apply/subsetP=> x Hx.
by rewrite inE /= /aperm autE ?(subsetP sHG) // -hH eq_gh ?mem_morphim.
apply/morphimP; exists (aut injg gG) => //; apply: (eq_Aut AutHh) => [|x Hx].
by rewrite (subsetP restr_perm_Aut) // mem_morphim.
by rewrite restr_permE //= /aperm autE ?eq_gh ?(subsetP sHG).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
Aut_sub_fullP
| |
astabs_Aut_isoma :
a \in Aut G -> (fGisom a \in 'N(f @* H | 'P)) = (a \in 'N(H | 'P)).
Proof.
move=> AutGa; rewrite !inE sub_morphim_pre // subsetI sHD /= /aperm.
rewrite !(sameP setIidPl eqP) !eqEsubset !subsetIl; apply: eq_subset_r => x.
rewrite !inE; apply: andb_id2l => Hx; have Gx: x \in G := subsetP sHG x Hx.
have Dax: a x \in D by rewrite (subsetP sGD) // Aut_closed.
by rewrite Aut_isomE // -!sub1set -morphim_set1 // injmSK ?sub1set.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_Aut_isom
| |
isom_restr_perma : a \in Aut G -> fHisom (inH a) = infH (fGisom a).
Proof.
move=> AutGa; case nHa: (a \in 'N(H | 'P)); last first.
by rewrite !triv_restr_perm ?astabs_Aut_isom ?nHa ?morph1.
apply: (eq_Aut (Aut_Aut_isom injf sHD _)) => [|fx Hfx /=].
by rewrite (Aut_restr_perm (morphimS f sHG)) ?Aut_Aut_isom.
have [x Dx Hx def_fx] := morphimP Hfx; have Gx := subsetP sHG x Hx.
rewrite {1}def_fx Aut_isomE ?(Aut_restr_perm sHG) //.
by rewrite !restr_permE ?astabs_Aut_isom // def_fx Aut_isomE.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
isom_restr_perm
| |
restr_perm_isom: isom (inH @* Aut G) (infH @* Aut (f @* G)) fHisom.
Proof.
apply: sub_isom; rewrite ?restr_perm_Aut ?injm_Aut_isom //=.
rewrite -(im_Aut_isom injf sGD) -!morphim_comp.
apply: eq_in_morphim; last exact: isom_restr_perm.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
restr_perm_isom
| |
injm_Aut_sub: Aut_in (Aut (f @* G)) (f @* H) \isog Aut_in (Aut G) H.
Proof.
do 2!rewrite isog_sym (isog_transl _ (Aut_in_isog _ _)).
by rewrite isog_sym (isom_isog _ _ restr_perm_isom) // restr_perm_Aut.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
injm_Aut_sub
| |
injm_Aut_full:
(Aut_in (Aut (f @* G)) (f @* H) \isog Aut (f @* H))
= (Aut_in (Aut G) H \isog Aut H).
Proof.
by rewrite (isog_transl _ injm_Aut_sub) (isog_transr _ (injm_Aut injf sHD)).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
injm_Aut_full
| |
is_groupAction(to : actT) :=
{in D, forall a, actperm to a \in Aut R}.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
is_groupAction
| |
groupAction:= GroupAction {gact :> actT; _ : is_groupAction gact}.
|
Structure
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
groupAction
| |
clone_groupActionto :=
let: GroupAction _ toA := to return {type of GroupAction for to} -> _ in
fun k => k toA : groupAction.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
clone_groupAction
| |
gact_rangeof groupAction D R := R.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gact_range
| |
gacentto A := 'Fix_(R | to)(D :&: A).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gacent
| |
acts_on_groupA S to := [acts A, on S | to] /\ S \subset R.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_on_group
| |
actby_cond_groupA S to : acts_on_group A S to -> actby_cond A S to :=
@proj1 _ _.
|
Coercion
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actby_cond_group
| |
acts_irreduciblyA S to :=
[min S of G | G :!=: 1 & [acts A, on G | to]].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_irreducibly
| |
actperm_Aut: is_groupAction R to. Proof. by case: to. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actperm_Aut
| |
im_actperm_Aut: actperm to @* D \subset Aut R.
Proof. by apply/subsetP=> _ /morphimP[a _ Da ->]; apply: actperm_Aut. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
im_actperm_Aut
| |
gact_outx a : a \in D -> x \notin R -> to x a = x.
Proof. by move=> Da Rx; rewrite -actpermE (out_Aut _ Rx) ?actperm_Aut. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gact_out
| |
gactM: {in D, forall a, {in R &, {morph to^~ a : x y / x * y}}}.
Proof.
move=> a Da /= x y; rewrite -!(actpermE to); apply: morphicP x y.
by rewrite Aut_morphic ?actperm_Aut.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gactM
| |
actmMa : {in R &, {morph actm to a : x y / x * y}}.
Proof. by rewrite /actm; case: ifP => //; apply: gactM. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actmM
| |
act_morphisma := Morphism (actmM a).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
act_morphism
| |
morphim_actm:
{in D, forall a (S : {set rT}), S \subset R -> actm to a @* S = to^* S a}.
Proof. by move=> a Da /= S sSR; rewrite /morphim /= actmEfun ?(setIidPr _). Qed.
Variables (a : aT) (A B : {set aT}) (S : {set rT}).
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
morphim_actm
| |
gacentIdom: 'C_(|to)(D :&: A) = 'C_(|to)(A).
Proof. by rewrite /gacent setIA setIid. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gacentIdom
| |
gacentIim: 'C_(R | to)(A) = 'C_(|to)(A).
Proof. by rewrite setIA setIid. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gacentIim
| |
gacentS: A \subset B -> 'C_(|to)(B) \subset 'C_(|to)(A).
Proof. by move=> sAB; rewrite !(setIS, afixS). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gacentS
| |
gacentU: 'C_(|to)(A :|: B) = 'C_(|to)(A) :&: 'C_(|to)(B).
Proof. by rewrite -setIIr -afixU -setIUr. Qed.
Hypotheses (Da : a \in D) (sAD : A \subset D) (sSR : S \subset R).
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gacentU
| |
gacentE: 'C_(|to)(A) = 'Fix_(R | to)(A).
Proof. by rewrite -{2}(setIidPr sAD). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gacentE
| |
gacent1E: 'C_(|to)[a] = 'Fix_(R | to)[a].
Proof. by rewrite /gacent [D :&: _](setIidPr _) ?sub1set. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gacent1E
| |
subgacentE: 'C_(S | to)(A) = 'Fix_(S | to)(A).
Proof. by rewrite gacentE setIA (setIidPl sSR). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
subgacentE
| |
subgacent1E: 'C_(S | to)[a] = 'Fix_(S | to)[a].
Proof. by rewrite gacent1E setIA (setIidPl sSR). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
subgacent1E
| |
gact1: {in D, forall a, to 1 a = 1}.
Proof. by move=> a Da; rewrite /= -actmE ?morph1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
gact1
|
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