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 |
|---|---|---|---|---|---|---|
orbit_rcoset_inA a x y :
A \subset D -> a \in D ->
(to y a \in orbit to (A :* a) x) = (y \in orbit to A x).
Proof.
move=> sAD Da; rewrite -orbit_inv_in ?mul_subG ?sub1set // invMg.
by rewrite invg_set1 orbit_lcoset_in ?inv_subG ?groupV ?actKin ?orbit_inv_in.
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_rcoset_in
| |
orbit_conjsg_inA a x y :
A \subset D -> a \in D ->
(to y a \in orbit to (A :^ a) (to x a)) = (y \in orbit to A x).
Proof.
move=> sAD Da; rewrite conjsgE.
by rewrite orbit_lcoset_in ?groupV ?mul_subG ?sub1set ?actKin ?orbit_rcoset_in.
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_conjsg_in
| |
orbit1PG x : reflect (orbit to G x = [set x]) (x \in 'Fix_to(G)).
Proof.
apply: (iffP afixP) => [xfix | xfix a Ga].
apply/eqP; rewrite eq_sym eqEsubset sub1set -{1}[x]act1 imset_f //=.
by apply/subsetP=> y; case/imsetP=> a Ga ->; rewrite inE xfix.
by apply/set1P; rewrite -xfix 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
|
orbit1P
| |
card_orbit1G x : #|orbit to G x| = 1%N -> orbit to G x = [set x].
Proof.
move=> orb1; apply/eqP; rewrite eq_sym eqEcard {}orb1 cards1.
by rewrite sub1set orbit_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
|
card_orbit1
| |
orbit_partitionG S :
[acts G, on S | to] -> partition (orbit to G @: S) S.
Proof.
move=> actsGS; have sGD := acts_dom actsGS.
have eqiG: {in S & &, equivalence_rel [rel x y | y \in orbit to G x]}.
by move=> x y z * /=; rewrite orbit_refl; split=> // /orbit_in_eqP->.
congr (partition _ _): (equivalence_partitionP eqiG).
apply: eq_in_imset => x Sx; apply/setP=> y.
by rewrite inE /= andb_idl // => /acts_in_orbit->.
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_partition
| |
orbit_transversalA S := transversal (orbit to A @: S) S.
|
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
|
orbit_transversal
| |
orbit_transversalPG S (P := orbit to G @: S)
(X := orbit_transversal G S) :
[acts G, on S | to] ->
[/\ is_transversal X P S, X \subset S,
{in X &, forall x y, (y \in orbit to G x) = (x == y)}
& forall x, x \in S -> exists2 a, a \in G & to x a \in X].
Proof.
move/orbit_partition; rewrite -/P => partP.
have [/eqP defS tiP _] := and3P partP.
have trXP: is_transversal X P S := transversalP partP.
have sXS: X \subset S := transversal_sub trXP.
split=> // [x y Xx Xy /= | x Sx].
have Sx := subsetP sXS x Xx.
rewrite -(inj_in_eq (pblock_inj trXP)) // eq_pblock ?defS //.
by rewrite (def_pblock tiP (imset_f _ Sx)) ?orbit_refl.
have /imsetP[y Xy defxG]: orbit to G x \in pblock P @: X.
by rewrite (pblock_transversal trXP) ?imset_f.
suffices /orbitP[a Ga def_y]: y \in orbit to G x by exists a; rewrite ?def_y.
by rewrite defxG mem_pblock defS (subsetP sXS).
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_transversalP
| |
group_set_astabS : group_set 'C(S | to).
Proof.
apply/group_setP; split=> [|a b cSa cSb].
by rewrite !inE group1; apply/subsetP=> x _; rewrite inE act1.
rewrite !inE groupM ?(@astab_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astab_dom _ _ _ to S) ?(astab_act _ Sx).
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
|
group_set_astab
| |
astab_groupS := group (group_set_astab S).
|
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
|
astab_group
| |
afix_gen_inA : A \subset D -> 'Fix_to(<<A>>) = 'Fix_to(A).
Proof.
move=> sAD; apply/eqP; rewrite eqEsubset afixS ?sub_gen //=.
by rewrite -astabCin gen_subG ?astabCin.
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_gen_in
| |
afix_cycle_ina : a \in D -> 'Fix_to(<[a]>) = 'Fix_to[a].
Proof. by move=> Da; rewrite afix_gen_in ?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
|
afix_cycle_in
| |
afixYinA B :
A \subset D -> B \subset D -> 'Fix_to(A <*> B) = 'Fix_to(A) :&: 'Fix_to(B).
Proof. by move=> sAD sBD; rewrite afix_gen_in ?afixU // subUset 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
|
afixYin
| |
afixMinG H :
G \subset D -> H \subset D -> 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).
Proof.
by move=> sGD sHD; rewrite -afix_gen_in ?mul_subG // genM_join afixYin.
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
|
afixMin
| |
sub_astab1_inA x :
A \subset D -> (A \subset 'C[x | to]) = (x \in 'Fix_to(A)).
Proof. by move=> sAD; rewrite astabCin ?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
|
sub_astab1_in
| |
group_set_astabsS : group_set 'N(S | to).
Proof.
apply/group_setP; split=> [|a b cSa cSb].
by rewrite !inE group1; apply/subsetP=> x Sx; rewrite inE act1.
rewrite !inE groupM ?(@astabs_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astabs_dom _ _ _ to S) ?astabs_act.
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
|
group_set_astabs
| |
astabs_groupS := group (group_set_astabs S).
|
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
|
astabs_group
| |
astab_normS : 'N(S | to) \subset 'N('C(S | to)).
Proof.
apply/subsetP=> a nSa; rewrite inE sub_conjg; apply/subsetP=> b cSb.
have [Da Db] := (astabs_dom nSa, astab_dom cSb).
rewrite mem_conjgV !inE groupJ //; apply/subsetP=> x Sx.
rewrite inE !actMin ?groupM ?groupV //.
by rewrite (astab_act cSb) ?actKVin ?astabs_act ?groupV.
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_norm
| |
astab_normalS : 'C(S | to) <| 'N(S | to).
Proof. by rewrite /normal astab_sub astab_norm. 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_normal
| |
acts_sub_orbitG S x :
[acts G, on S | to] -> (orbit to G x \subset S) = (x \in S).
Proof.
move/acts_act=> GactS.
apply/subsetP/idP=> [| Sx y]; first by apply; apply: orbit_refl.
by case/orbitP=> a Ga <-{y}; rewrite GactS.
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_sub_orbit
| |
acts_orbitG x : G \subset D -> [acts G, on orbit to G x | to].
Proof.
move/subsetP=> sGD; apply/subsetP=> a Ga; rewrite !inE sGD //.
apply/subsetP=> _ /imsetP[b Gb ->].
by rewrite inE -actMin ?sGD // imset_f ?groupM.
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_orbit
| |
acts_subnorm_fixA : [acts 'N_D(A), on 'Fix_to(D :&: A) | to].
Proof.
apply/subsetP=> a nAa; have [Da _] := setIP nAa; rewrite !inE Da.
apply/subsetP=> x Cx /[1!inE]; apply/afixP=> b DAb.
have [Db _]:= setIP DAb; rewrite -actMin // conjgCV actMin ?groupJ ?groupV //.
by rewrite /= (afixP Cx) // memJ_norm // groupV (subsetP (normsGI _ _) _ nAa).
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_subnorm_fix
| |
atrans_orbitG x : [transitive G, on orbit to G x | to].
Proof. by apply: imset_f; apply: orbit_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
|
atrans_orbit
| |
amove_acta : a \in G -> amove to G x (to x a) = 'C_G[x | to] :* a.
Proof.
move=> Ga; apply/setP=> b; have Da := ssGD Ga.
rewrite mem_rcoset !(inE, sub1set) !groupMr ?groupV //.
by case Gb: (b \in G); rewrite //= actMin ?groupV ?ssGD ?(canF_eq (actKVin 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
|
amove_act
| |
amove_orbit: amove to G x @: orbit to G x = rcosets 'C_G[x | to] G.
Proof.
apply/setP => Ha; apply/imsetP/rcosetsP=> [[y] | [a Ga ->]].
by case/imsetP=> b Gb -> ->{Ha y}; exists b => //; rewrite amove_act.
by rewrite -amove_act //; exists (to x a); first apply: mem_orbit.
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
|
amove_orbit
| |
amoveK:
{in orbit to G x, cancel (amove to G x) (fun Ca => to x (repr Ca))}.
Proof.
move=> _ /orbitP[a Ga <-]; rewrite amove_act //= -[G :&: _]/(gval _).
case: repr_rcosetP => b; rewrite !(inE, sub1set)=> /and3P[Gb _ xbx].
by rewrite actMin ?ssGD ?(eqP xbx).
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
|
amoveK
| |
orbit_stabilizer:
orbit to G x = [set to x (repr Ca) | Ca in rcosets 'C_G[x | to] G].
Proof.
rewrite -amove_orbit -imset_comp /=; apply/setP=> z.
by apply/idP/imsetP=> [xGz | [y xGy ->]]; first exists z; rewrite /= ?amoveK.
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_stabilizer
| |
act_reprK:
{in rcosets 'C_G[x | to] G, cancel (to x \o repr) (amove to G x)}.
Proof.
move=> _ /rcosetsP[a Ga ->] /=; rewrite amove_act ?rcoset_repr //.
rewrite -[G :&: _]/(gval _); case: repr_rcosetP => b /setIP[Gb _].
exact: groupM.
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
|
act_reprK
| |
card_orbit_inG x : G \subset D -> #|orbit to G x| = #|G : 'C_G[x | to]|.
Proof.
move=> sGD; rewrite orbit_stabilizer 1?card_in_imset //.
exact: can_in_inj (act_reprK _).
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
|
card_orbit_in
| |
card_orbit_in_stabG x :
G \subset D -> (#|orbit to G x| * #|'C_G[x | to]|)%N = #|G|.
Proof. by move=> sGD; rewrite mulnC card_orbit_in ?Lagrange ?subsetIl. 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
|
card_orbit_in_stab
| |
acts_sum_card_orbitG S :
[acts G, on S | to] -> \sum_(T in orbit to G @: S) #|T| = #|S|.
Proof. by move/orbit_partition/card_partition. 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_sum_card_orbit
| |
astab_setact_inS a : a \in D -> 'C(to^* S a | to) = 'C(S | to) :^ a.
Proof.
move=> Da; apply/setP=> b; rewrite mem_conjg !inE -mem_conjg conjGid //.
apply: andb_id2l => Db; rewrite sub_imset_pre; apply: eq_subset_r => x.
by rewrite !inE !actMin ?groupM ?groupV // invgK (canF_eq (actKVin 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
|
astab_setact_in
| |
astab1_act_inx a : a \in D -> 'C[to x a | to] = 'C[x | to] :^ a.
Proof. by move=> Da; rewrite -astab_setact_in // /setact imset_set1. 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
|
astab1_act_in
| |
Frobenius_CauchyG S : [acts G, on S | to] ->
\sum_(a in G) #|'Fix_(S | to)[a]| = (#|orbit to G @: S| * #|G|)%N.
Proof.
move=> GactS; have sGD := acts_dom GactS.
transitivity (\sum_(a in G) \sum_(x in 'Fix_(S | to)[a]) 1%N).
by apply: eq_bigr => a _; rewrite -sum1_card.
rewrite (exchange_big_dep [in S]) /= => [|a x _]; last by case/setIP.
rewrite (set_partition_big _ (orbit_partition GactS)) -sum_nat_const /=.
apply: eq_bigr => _ /imsetP[x Sx ->].
rewrite -(card_orbit_in_stab x sGD) -sum_nat_const.
apply: eq_bigr => y; rewrite orbit_in_sym // => /imsetP[a Ga defx].
rewrite defx astab1_act_in ?(subsetP sGD) //.
rewrite -{2}(conjGid Ga) -conjIg cardJg -sum1_card setIA (setIidPl sGD).
by apply: eq_bigl => b; rewrite !(sub1set, inE) -(acts_act GactS Ga) -defx Sx.
Qed.
|
Theorem
|
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
|
Frobenius_Cauchy
| |
atrans_dvd_index_inG S :
G \subset D -> [transitive G, on S | to] -> #|S| %| #|G : 'C_G(S | to)|.
Proof.
move=> sGD /imsetP[x Sx {1}->]; rewrite card_orbit_in //.
by rewrite indexgS // setIS // astabS // 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
|
atrans_dvd_index_in
| |
atrans_dvd_inG S :
G \subset D -> [transitive G, on S | to] -> #|S| %| #|G|.
Proof.
move=> sGD transG; apply: dvdn_trans (atrans_dvd_index_in sGD transG) _.
exact: dvdn_indexg.
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
|
atrans_dvd_in
| |
atransPinG S :
G \subset D -> [transitive G, on S | to] ->
forall x, x \in S -> orbit to G x = S.
Proof. by move=> sGD /imsetP[y _ ->] x; apply/orbit_in_eqP. 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
|
atransPin
| |
atransP2inG S :
G \subset D -> [transitive G, on S | to] ->
{in S &, forall x y, exists2 a, a \in G & y = to x a}.
Proof. by move=> sGD transG x y /(atransPin sGD transG) <- /imsetP. 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
|
atransP2in
| |
atrans_acts_inG S :
G \subset D -> [transitive G, on S | to] -> [acts G, on S | to].
Proof.
move=> sGD transG; apply/subsetP=> a Ga; rewrite !inE (subsetP sGD) //.
by apply/subsetP=> x /(atransPin sGD transG) <-; 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
|
atrans_acts_in
| |
subgroup_transitivePinG H S x :
x \in S -> H \subset G -> G \subset D -> [transitive G, on S | to] ->
reflect ('C_G[x | to] * H = G) [transitive H, on S | to].
Proof.
move=> Sx sHG sGD trG; have sHD := subset_trans sHG sGD.
apply: (iffP idP) => [trH | defG].
rewrite group_modr //; apply/setIidPl/subsetP=> a Ga.
have Sxa: to x a \in S by rewrite (acts_act (atrans_acts_in sGD trG)).
have [b Hb xab]:= atransP2in sHD trH Sxa Sx.
have Da := subsetP sGD a Ga; have Db := subsetP sHD b Hb.
rewrite -(mulgK b a) mem_mulg ?groupV // !inE groupM //= sub1set inE.
by rewrite actMin -?xab.
apply/imsetP; exists x => //; apply/setP=> y; rewrite -(atransPin sGD trG Sx).
apply/imsetP/imsetP=> [] [a]; last by exists a; first apply: (subsetP sHG).
rewrite -defG => /imset2P[c b /setIP[_ cxc] Hb ->] ->.
exists b; rewrite ?actMin ?(astab_dom cxc) ?(subsetP sHD) //.
by rewrite (astab_act cxc) ?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
|
subgroup_transitivePin
| |
actMx a b : to x (a * b) = to (to x a) b.
Proof. by rewrite actMin ?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
|
actM
| |
actK: right_loop inv to.
Proof. by move=> a; apply: actKin; rewrite 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
|
actK
| |
actKV: rev_right_loop inv to.
Proof. by move=> a; apply: actKVin; rewrite 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
|
actKV
| |
actXx a n : to x (a ^+ n) = iter n (to^~ a) x.
Proof. by elim: n => [|n /= <-]; rewrite ?act1 // -actM expgSr. 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
|
actX
| |
actCJa b x : to (to x a) b = to (to x b) (a ^ b).
Proof. by rewrite !actM actK. 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
|
actCJ
| |
actCJVa b x : to (to x a) b = to (to x (b ^ a^-1)) a.
Proof. by rewrite (actCJ _ a) conjgKV. 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
|
actCJV
| |
orbit_symG x y : (x \in orbit to G y) = (y \in orbit to G x).
Proof. exact/orbit_in_sym/subsetT. 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_sym
| |
orbit_transG x y z :
x \in orbit to G y -> y \in orbit to G z -> x \in orbit to G z.
Proof. exact/orbit_in_trans/subsetT. 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_trans
| |
orbit_eqPG x y :
reflect (orbit to G x = orbit to G y) (x \in orbit to G y).
Proof. exact/orbit_in_eqP/subsetT. 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_eqP
| |
orbit_translG x y z :
y \in orbit to G x -> (y \in orbit to G z) = (x \in orbit to G z).
Proof. exact/orbit_in_transl/subsetT. 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_transl
| |
orbit_actG a x: a \in G -> orbit to G (to x a) = orbit to G x.
Proof. exact/orbit_act_in/subsetT. 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_act
| |
orbit_actrG a x y :
a \in G -> (to y a \in orbit to G x) = (y \in orbit to G x).
Proof. by move/mem_orbit/orbit_transl; apply. 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_actr
| |
orbit_eq_memG x y :
(orbit to G x == orbit to G y) = (x \in orbit to G y).
Proof. exact: sameP eqP (orbit_eqP G x 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
|
orbit_eq_mem
| |
orbit_invA x y : (y \in orbit to A^-1 x) = (x \in orbit to A y).
Proof. by rewrite orbit_inv_in ?subsetT. 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_inv
| |
orbit_lcosetA a x : orbit to (a *: A) x = orbit to A (to x a).
Proof. by rewrite orbit_lcoset_in ?subsetT ?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
|
orbit_lcoset
| |
orbit_rcosetA a x y :
(to y a \in orbit to (A :* a) x) = (y \in orbit to A x).
Proof. by rewrite orbit_rcoset_in ?subsetT ?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
|
orbit_rcoset
| |
orbit_conjsgA a x y :
(to y a \in orbit to (A :^ a) (to x a)) = (y \in orbit to A x).
Proof. by rewrite orbit_conjsg_in ?subsetT ?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
|
orbit_conjsg
| |
astabPS a : reflect (forall x, x \in S -> to x a = x) (a \in 'C(S | to)).
Proof.
apply: (iffP idP) => [cSa x|cSa]; first exact: astab_act.
by rewrite !inE; apply/subsetP=> x Sx; rewrite inE cSa.
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
|
astabP
| |
astab1Px a : reflect (to x a = x) (a \in 'C[x | to]).
Proof. by rewrite !inE sub1set inE; apply: eqP. 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
|
astab1P
| |
sub_astab1A x : (A \subset 'C[x | to]) = (x \in 'Fix_to(A)).
Proof. by rewrite sub_astab1_in ?subsetT. 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
|
sub_astab1
| |
astabCA S : (A \subset 'C(S | to)) = (S \subset 'Fix_to(A)).
Proof. by rewrite astabCin ?subsetT. 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
|
astabC
| |
afix_cyclea : 'Fix_to(<[a]>) = 'Fix_to[a].
Proof. by rewrite afix_cycle_in ?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
|
afix_cycle
| |
afix_genA : 'Fix_to(<<A>>) = 'Fix_to(A).
Proof. by rewrite afix_gen_in ?subsetT. 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_gen
| |
afixMG H : 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).
Proof. by rewrite afixMin ?subsetT. 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
|
afixM
| |
astabsPS a :
reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).
Proof.
apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act.
by rewrite !inE; apply/subsetP=> x; rewrite inE nSa.
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
|
astabsP
| |
card_orbitG x : #|orbit to G x| = #|G : 'C_G[x | to]|.
Proof. by rewrite card_orbit_in ?subsetT. 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
|
card_orbit
| |
dvdn_orbitG x : #|orbit to G x| %| #|G|.
Proof. by rewrite card_orbit dvdn_indexg. 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
|
dvdn_orbit
| |
card_orbit_stabG x : (#|orbit to G x| * #|'C_G[x | to]|)%N = #|G|.
Proof. by rewrite mulnC card_orbit Lagrange ?subsetIl. 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
|
card_orbit_stab
| |
actsPA S : reflect {acts A, on S | to} [acts A, on S | to].
Proof.
apply: (iffP idP) => [nSA x|nSA]; first exact: acts_act.
by apply/subsetP=> a Aa /[!inE]; apply/subsetP=> x; rewrite inE nSA.
Qed.
Arguments actsP {A S}.
|
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
|
actsP
| |
setact_orbitA x b : to^* (orbit to A x) b = orbit to (A :^ b) (to x b).
Proof.
apply/setP=> y; apply/idP/idP=> /imsetP[_ /imsetP[a Aa ->] ->{y}].
by rewrite actCJ mem_orbit ?memJ_conjg.
by rewrite -actCJ mem_setact ?mem_orbit.
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
|
setact_orbit
| |
astab_setactS a : 'C(to^* S a | to) = 'C(S | to) :^ a.
Proof.
apply/setP=> b; rewrite mem_conjg.
apply/astabP/astabP=> stab x => [Sx|].
by rewrite conjgE invgK !actM stab ?actK //; apply/imsetP; exists x.
by case/imsetP=> y Sy ->{x}; rewrite -actM conjgCV actM stab.
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_setact
| |
astab1_actx a : 'C[to x a | to] = 'C[x | to] :^ a.
Proof. by rewrite -astab_setact /setact imset_set1. 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
|
astab1_act
| |
atransPG S : [transitive G, on S | to] ->
forall x, x \in S -> orbit to G x = S.
Proof. by case/imsetP=> x _ -> y; apply/orbit_eqP. 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
|
atransP
| |
atransP2G S : [transitive G, on S | to] ->
{in S &, forall x y, exists2 a, a \in G & y = to x a}.
Proof. by move=> GtrS x y /(atransP GtrS) <- /imsetP. 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
|
atransP2
| |
atrans_actsG S : [transitive G, on S | to] -> [acts G, on S | to].
Proof.
move=> GtrS; apply/subsetP=> a Ga; rewrite !inE.
by apply/subsetP=> x /(atransP GtrS) <-; 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
|
atrans_acts
| |
atrans_supgroupG H S :
G \subset H -> [transitive G, on S | to] ->
[transitive H, on S | to] = [acts H, on S | to].
Proof.
move=> sGH trG; apply/idP/idP=> [|actH]; first exact: atrans_acts.
case/imsetP: trG => x Sx defS; apply/imsetP; exists x => //.
by apply/eqP; rewrite eqEsubset acts_sub_orbit ?Sx // defS imsetS.
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
|
atrans_supgroup
| |
atrans_acts_cardG S :
[transitive G, on S | to] =
[acts G, on S | to] && (#|orbit to G @: S| == 1%N).
Proof.
apply/idP/andP=> [GtrS | [nSG]].
split; first exact: atrans_acts.
rewrite ((_ @: S =P [set S]) _) ?cards1 // eqEsubset sub1set.
apply/andP; split=> //; apply/subsetP=> _ /imsetP[x Sx ->].
by rewrite inE (atransP GtrS).
rewrite eqn_leq andbC lt0n => /andP[/existsP[X /imsetP[x Sx X_Gx]]].
rewrite (cardD1 X) {X}X_Gx imset_f // ltnS leqn0 => /eqP GtrS.
apply/imsetP; exists x => //; apply/eqP.
rewrite eqEsubset acts_sub_orbit // Sx andbT.
apply/subsetP=> y Sy; have:= card0_eq GtrS (orbit to G y).
by rewrite !inE /= imset_f // andbT => /eqP <-; apply: orbit_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
|
atrans_acts_card
| |
atrans_dvdG S : [transitive G, on S | to] -> #|S| %| #|G|.
Proof. by case/imsetP=> x _ ->; apply: dvdn_orbit. 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
|
atrans_dvd
| |
acts_fix_normA B : A \subset 'N(B) -> [acts A, on 'Fix_to(B) | to].
Proof.
move=> nAB; have:= acts_subnorm_fix to B; rewrite !setTI.
exact: subset_trans.
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_fix_norm
| |
faithfulPA S :
reflect (forall a, a \in A -> {in S, to^~ a =1 id} -> a = 1)
[faithful A, on S | to].
Proof.
apply: (iffP subsetP) => [Cto1 a Aa Ca | Cto1 a].
by apply/set1P; rewrite Cto1 // inE Aa; apply/astabP.
by case/setIP=> Aa /astabP Ca; apply/set1P; apply: Cto1.
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
|
faithfulP
| |
astab_trans_gcoreG S u :
[transitive G, on S | to] -> u \in S -> 'C(S | to) = gcore 'C[u | to] G.
Proof.
move=> transG Su; apply/eqP; rewrite eqEsubset.
rewrite gcore_max ?astabS ?sub1set //=; last first.
exact: subset_trans (atrans_acts transG) (astab_norm _ _).
apply/subsetP=> x cSx; apply/astabP=> uy.
case/(atransP2 transG Su) => y Gy ->{uy}.
by apply/astab1P; rewrite astab1_act (bigcapP cSx).
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_trans_gcore
| |
subgroup_transitivePG H S x :
x \in S -> H \subset G -> [transitive G, on S | to] ->
reflect ('C_G[x | to] * H = G) [transitive H, on S | to].
Proof. by move=> Sx sHG; apply: subgroup_transitivePin (subsetT G). Qed.
|
Theorem
|
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
|
subgroup_transitiveP
| |
trans_subnorm_fixPx G H S :
let C := 'C_G[x | to] in let T := 'Fix_(S | to)(H) in
[transitive G, on S | to] -> x \in S -> H \subset C ->
reflect ((H :^: G) ::&: C = H :^: C) [transitive 'N_G(H), on T | to].
Proof.
move=> C T trGS Sx sHC; have actGS := acts_act (atrans_acts trGS).
have:= sHC; rewrite subsetI sub_astab1 => /andP[sHG cHx].
have Tx: x \in T by rewrite inE Sx.
apply: (iffP idP) => [trN | trC].
apply/setP=> Ha; apply/setIdP/imsetP=> [[]|[a Ca ->{Ha}]]; last first.
by rewrite conj_subG //; case/setIP: Ca => Ga _; rewrite imset_f.
case/imsetP=> a Ga ->{Ha}; rewrite subsetI !sub_conjg => /andP[_ sHCa].
have Txa: to x a^-1 \in T.
by rewrite inE -sub_astab1 astab1_act actGS ?Sx ?groupV.
have [b] := atransP2 trN Tx Txa; case/setIP=> Gb nHb cxba.
exists (b * a); last by rewrite conjsgM (normP nHb).
by rewrite inE groupM //; apply/astab1P; rewrite actM -cxba actKV.
apply/imsetP; exists x => //; apply/setP=> y; apply/idP/idP=> [Ty|].
have [Sy cHy]:= setIP Ty; have [a Ga defy] := atransP2 trGS Sx Sy.
have: H :^ a^-1 \in H :^: C.
rewrite -trC inE subsetI imset_f 1?conj_subG ?groupV // sub_conjgV.
by rewrite -astab1_act -defy sub_astab1.
case/imsetP=> b /setIP[Gb /astab1P cxb] defHb.
rewrite defy -{1}cxb -actM mem_orbit // inE groupM //.
by apply/normP; rewrite conjsgM -defHb conjsgKV.
case/imsetP=> a /setIP[Ga nHa] ->{y}.
by rewrite inE actGS // Sx (acts_act (acts_fix_norm _) nHa).
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
|
trans_subnorm_fixP
| |
ractof A \subset D := act to.
Variable sAD : A \subset D.
|
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
|
ract
| |
ract_is_action: is_action A (ract sAD).
Proof.
rewrite /ract; case: to => f [injf fM].
by split=> // x; apply: (sub_in2 (subsetP 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
|
ract_is_action
| |
raction:= Action ract_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
|
raction
| |
ractE: raction =1 to. 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
|
ractE
| |
actby_cond(A : {set aT}) R (to : action D rT) : Prop :=
[acts A, on R | 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
|
actby_cond
| |
actbyA R to of actby_cond A R to :=
fun x a => if (x \in R) && (a \in A) then to x a else x.
Variables (A : {group aT}) (R : {set rT}) (to : action D rT).
Hypothesis nRA : actby_cond A R 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
|
actby
| |
actby_is_action: is_action A (actby nRA).
Proof.
rewrite /actby; split=> [a x y | x a b Aa Ab /=]; last first.
rewrite Aa Ab groupM // !andbT actMin ?(subsetP (acts_dom nRA)) //.
by case Rx: (x \in R); rewrite ?(acts_act nRA) ?Rx.
case Aa: (a \in A); rewrite ?andbF ?andbT //.
case Rx: (x \in R); case Ry: (y \in R) => // eqxy; first exact: act_inj eqxy.
by rewrite -eqxy (acts_act nRA Aa) Rx in Ry.
by rewrite eqxy (acts_act nRA Aa) Ry in Rx.
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
|
actby_is_action
| |
action_by:= Action actby_is_action.
Local Notation "<[nRA]>" := action_by : action_scope.
|
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
|
action_by
| |
actbyEx a : x \in R -> a \in A -> <[nRA]>%act x a = to x a.
Proof. by rewrite /= /actby => -> ->. 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
|
actbyE
| |
afix_actbyB : 'Fix_<[nRA]>(B) = ~: R :|: 'Fix_to(A :&: B).
Proof.
apply/setP=> x; rewrite !inE /= /actby.
case: (x \in R); last by apply/subsetP=> a _ /[!inE].
apply/subsetP/subsetP=> [cBx a | cABx a Ba] /[!inE].
by case/andP=> Aa /cBx; rewrite inE Aa.
by case: ifP => //= Aa; have:= cABx a; rewrite !inE Aa => ->.
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_actby
| |
astab_actbyS : 'C(S | <[nRA]>) = 'C_A(R :&: S | to).
Proof.
apply/setP=> a; rewrite setIA (setIidPl (acts_dom nRA)) !inE.
case Aa: (a \in A) => //=; apply/subsetP/subsetP=> cRSa x => [|Sx].
by case/setIP=> Rx /cRSa; rewrite !inE actbyE.
by have:= cRSa x; rewrite !inE /= /actby Aa Sx; case: (x \in R) => //; apply.
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_actby
| |
astabs_actbyS : 'N(S | <[nRA]>) = 'N_A(R :&: S | to).
Proof.
apply/setP=> a; rewrite setIA (setIidPl (acts_dom nRA)) !inE.
case Aa: (a \in A) => //=; apply/subsetP/subsetP=> nRSa x => [|Sx].
by case/setIP=> Rx /nRSa; rewrite !inE actbyE ?(acts_act nRA) ?Rx.
have:= nRSa x; rewrite !inE /= /actby Aa Sx ?(acts_act nRA) //.
by case: (x \in R) => //; apply.
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_actby
| |
acts_actby(B : {set aT}) S :
[acts B, on S | <[nRA]>] = (B \subset A) && [acts B, on R :&: S | to].
Proof. by rewrite astabs_actby 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_actby
| |
subact_dom:= 'N([set x | sP x] | 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
|
subact_dom
| |
subact_dom_group:= [group of subact_dom].
Implicit Type Na : {a | a \in subact_dom}.
|
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
|
subact_dom_group
| |
sub_act_proofu Na : sP (to (val u) (val Na)).
Proof. by case: Na => a /= /(astabs_act (val u)); rewrite !inE valP. 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
|
sub_act_proof
| |
subactu a :=
if insub a is Some Na then Sub _ (sub_act_proof u Na) else u.
|
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
|
subact
| |
val_subactu a :
val (subact u a) = if a \in subact_dom then to (val u) a else val u.
Proof.
by rewrite /subact -if_neg; case: insubP => [Na|] -> //=; rewrite SubK => ->.
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
|
val_subact
|
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