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 |
|---|---|---|---|---|---|---|
sub_astabQA H Bbar :
(A \subset 'C(Bbar | 'Q)) = (A \subset 'N(H)) && (A / H \subset 'C(Bbar)).
Proof.
rewrite astabQ -morphpreIdom subsetI; apply: andb_id2l => nHA.
by rewrite -sub_quotient_pre.
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_astabQ
| |
sub_astabQRA B H :
A \subset 'N(H) -> B \subset 'N(H) ->
(A \subset 'C(B / H | 'Q)) = ([~: A, B] \subset H).
Proof.
move=> nHA nHB; rewrite sub_astabQ nHA /= (sameP commG1P eqP).
by rewrite eqEsubset sub1G andbT -quotientR // quotient_sub1 // comm_subG.
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_astabQR
| |
astabQRA H : A \subset 'N(H) ->
'C(A / H | 'Q) = [set x in 'N(H) | [~: [set x], A] \subset H].
Proof.
move=> nHA; apply/setP=> x; rewrite astabQ -morphpreIdom 2!inE -astabQ.
by case nHx: (x \in _); rewrite //= -sub1set sub_astabQR ?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
|
astabQR
| |
quotient_astabQH Abar : 'C(Abar | 'Q) / H = 'C(Abar).
Proof. by rewrite astabQ cosetpreK. 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
|
quotient_astabQ
| |
conj_astabQA H x :
x \in 'N(H) -> 'C(A / H | 'Q) :^ x = 'C(A :^ x / H | 'Q).
Proof.
move=> nHx; apply/setP=> y; rewrite !astabQ mem_conjg !in_setI -mem_conjg.
rewrite -normJ (normP nHx) quotientJ //; apply/andb_id2l => nHy.
by rewrite !inE centJ morphJ ?groupV ?morphV // -mem_conjg.
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
|
conj_astabQ
| |
index_cent1x : #|G : 'C_G[x]| = #|x ^: G|.
Proof. by rewrite -astab1J -card_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
|
index_cent1
| |
classes_partition: partition (classes G) G.
Proof. by apply: orbit_partition; apply/actsP=> x Gx y; apply: groupJr. 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
|
classes_partition
| |
sum_card_class: \sum_(C in classes G) #|C| = #|G|.
Proof. by apply: acts_sum_card_orbit; apply/actsP=> x Gx y; apply: groupJr. 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
|
sum_card_class
| |
class_formula: \sum_(C in classes G) #|G : 'C_G[repr C]| = #|G|.
Proof.
rewrite -sum_card_class; apply: eq_bigr => _ /imsetP[x Gx ->].
have: x \in x ^: G by rewrite -{1}(conjg1 x) imset_f.
by case/mem_repr/imsetP=> y Gy ->; rewrite index_cent1 classGidl.
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
|
class_formula
| |
abelian_classP: reflect {in G, forall x, x ^: G = [set x]} (abelian G).
Proof.
rewrite /abelian -astabJ astabC.
by apply: (iffP subsetP) => cGG x Gx; apply/orbit1P; apply: cGG.
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
|
abelian_classP
| |
card_classes_abelian: abelian G = (#|classes G| == #|G|).
Proof.
have cGgt0 C: C \in classes G -> 1 <= #|C| ?= iff (#|C| == 1)%N.
by case/imsetP=> x _ ->; rewrite eq_sym -index_cent1.
rewrite -sum_card_class -sum1_card (leqif_sum cGgt0).
apply/abelian_classP/forall_inP=> [cGG _ /imsetP[x Gx ->]| cGG x Gx].
by rewrite cGG ?cards1.
apply/esym/eqP; rewrite eqEcard sub1set cards1 class_refl leq_eqVlt cGG //.
exact: 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
|
card_classes_abelian
| |
gacentQ(gT : finGroupType) (H : {group gT}) (A : {set gT}) :
'C_(|'Q)(A) = 'C(A / H).
Proof.
apply/setP=> Hx; case: (cosetP Hx) => x Nx ->{Hx}.
rewrite -sub_cent1 -astab1J astabC sub1set -(quotientInorm H A).
have defD: qact_dom 'J H = 'N(H) by rewrite qact_domE ?subsetT ?astabsJ.
rewrite !(inE, mem_quotient) //= defD setIC.
apply/subsetP/subsetP=> [cAx _ /morphimP[a Na Aa ->] | cAx a Aa].
by move/cAx: Aa; rewrite !inE qactE ?defD ?morphJ.
have [_ Na] := setIP Aa; move/implyP: (cAx (coset H a)); rewrite mem_morphim //.
by rewrite !inE qactE ?defD ?morphJ.
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
|
gacentQ
| |
autact:= act ('P \ subsetT (Aut 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
|
autact
| |
aut_action:= [action of autact].
|
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
|
aut_action
| |
autactKa : actperm aut_action a = 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
|
autactK
| |
autact_is_groupAction: is_groupAction G aut_action.
Proof. by move=> a Aa /=; rewrite autactK. 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
|
autact_is_groupAction
| |
aut_groupAction:= GroupAction autact_is_groupAction.
|
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
|
aut_groupAction
| |
perm_prime_atrans: [transitive <[c]>, on setT | 'P].
Proof.
apply/imsetP; suff /existsP[x] : [exists x, ~~ (#|orbit 'P <[c]> x| < #[c])].
move=> oxT; suff /eqP orbit_x : orbit 'P <[c]> x == setT by exists x.
by rewrite eqEcard subsetT cardsT -cc leqNgt.
apply/forallP => olT; have o1 x : #|orbit 'P <[c]> x| == 1%N.
by case/primeP: cp => _ /(_ _ (dvdn_orbit 'P _ x))/orP[]//; rewrite ltn_eqF.
suff c1 : c = 1%g by rewrite c1 ?order1 in (cp).
apply/permP => x; rewrite perm1; apply/set1P.
by rewrite -(card_orbit1 (eqP (o1 _))) (mem_orbit 'P) ?cycle_id.
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_prime_atrans
| |
perm_prime_orbitx : orbit 'P <[c]> x = [set: T].
Proof. by apply: atransP => //; apply: perm_prime_atrans. 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_prime_orbit
| |
perm_prime_astabx : 'C_<[c]>[x | 'P]%g = 1%g.
Proof.
by apply/card1_trivg/eqP; rewrite -(@eqn_pmul2l #|orbit 'P <[c]> x|)
?card_orbit_stab ?perm_prime_orbit ?cardsT ?muln1 ?prime_gt0// -cc.
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_prime_astab
| |
AutA := [set a | perm_on A a & morphic A a].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut
| |
Aut_morphicA a : a \in Aut A -> morphic A a.
Proof. by case/setIdP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_morphic
| |
out_AutA a x : a \in Aut A -> x \notin A -> a x = x.
Proof. by case/setIdP=> Aa _; apply: out_perm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
out_Aut
| |
eq_AutA : {in Aut A &, forall a b, {in A, a =1 b} -> a = b}.
Proof.
move=> a g Aa Ag /= eqag; apply/permP=> x.
by have [/eqag // | /out_Aut out] := boolP (x \in A); rewrite !out.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
eq_Aut
| |
autmA a (AutAa : a \in Aut A) := morphm (Aut_morphic AutAa).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
autm
| |
autmEA a (AutAa : a \in Aut A) : autm AutAa = a.
Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
autmE
| |
autm_morphismA a aM := Eval hnf in [morphism of @autm A a aM].
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
autm_morphism
| |
Aut_group_set: group_set (Aut G).
Proof.
apply/group_setP; split=> [|a b].
by rewrite inE perm_on1; apply/morphicP=> ? *; rewrite !permE.
rewrite !inE => /andP[Ga aM] /andP[Gb bM]; rewrite perm_onM //=.
apply/morphicP=> x y Gx Gy; rewrite !permM (morphicP aM) //.
by rewrite (morphicP bM) ?perm_closed.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_group_set
| |
Aut_group:= group Aut_group_set.
Variable (a : {perm gT}) (AutGa : a \in Aut G).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_group
| |
f:= (autm AutGa).
|
Notation
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
f
| |
fE:= (autmE AutGa).
|
Notation
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
fE
| |
injm_autm: 'injm f.
Proof. by apply/injmP; apply: in2W; apply: perm_inj. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
injm_autm
| |
ker_autm: 'ker f = 1. Proof. by move/trivgP: injm_autm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
ker_autm
| |
im_autm: f @* G = G.
Proof.
apply/setP=> x; rewrite morphimEdom (can_imset_pre _ (permK a)) inE.
by have /[1!inE] /andP[/perm_closed <-] := AutGa; rewrite permKV.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
im_autm
| |
Aut_closedx : x \in G -> a x \in G.
Proof. by move=> Gx; rewrite -im_autm; apply: mem_morphim. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_closed
| |
Aut1: Aut 1 = 1.
Proof.
apply/trivgP/subsetP=> a /= AutGa; apply/set1P.
apply: eq_Aut (AutGa) (group1 _) _ => _ /set1P->.
by rewrite -(autmE AutGa) morph1 perm1.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut1
| |
perm_in_inj: injective (fun x => if x \in A then f x else x).
Proof.
move=> x y /=; wlog Ay: x y / y \in A.
by move=> IH eqfxy; case: ifP (eqfxy); [symmetry | case: ifP => //]; auto.
rewrite Ay; case: ifP => [Ax | nAx def_x]; first exact: injf.
by case/negP: nAx; rewrite def_x (subsetP sBf) ?imset_f.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
perm_in_inj
| |
perm_in:= perm perm_in_inj.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
perm_in
| |
perm_in_on: perm_on A perm_in.
Proof.
by apply/subsetP=> x; rewrite inE /= permE; case: ifP => // _; case/eqP.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
perm_in_on
| |
perm_inE: {in A, perm_in =1 f}.
Proof. by move=> x Ax; rewrite /= permE Ax. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
perm_inE
| |
morphim_fixPA : A \subset G -> reflect (f @* A = A) (f @* A \subset A).
Proof.
rewrite /morphim => sAG; have:= eqEcard (f @: A) A.
rewrite (setIidPr sAG) card_in_imset ?leqnn ?andbT => [<-|]; first exact: eqP.
by move/injmP: injf; apply: sub_in2; apply/subsetP.
Qed.
Hypothesis Gf : f @* G = G.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
morphim_fixP
| |
aut_closed: f @: G \subset G.
Proof. by rewrite -morphimEdom; apply/morphim_fixP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
aut_closed
| |
aut:= perm_in (injmP injf) aut_closed.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
aut
| |
autE: {in G, aut =1 f}.
Proof. exact: perm_inE. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
autE
| |
morphic_aut: morphic G aut.
Proof. by apply/morphicP=> x y Gx Gy /=; rewrite !autE ?groupM // morphM. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
morphic_aut
| |
Aut_aut: aut \in Aut G.
Proof. by rewrite inE morphic_aut perm_in_on. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_aut
| |
imset_autEA : A \subset G -> aut @: A = f @* A.
Proof.
move=> sAG; rewrite /morphim (setIidPr sAG).
by apply: eq_in_imset; apply: sub_in1 autE; apply/subsetP.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
imset_autE
| |
preim_autEA : A \subset G -> aut @^-1: A = f @*^-1 A.
Proof.
move=> sAG; apply/setP=> x; rewrite !inE permE /=.
by case Gx: (x \in G) => //; apply/negP=> Ax; rewrite (subsetP sAG) in Gx.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
preim_autE
| |
Aut_isom_subproofa :
{a' | a' \in Aut (f @* G) & a \in Aut G -> {in G, a' \o f =1 f \o a}}.
Proof.
set Aut_a := autm (subgP (subg [Aut G] a)).
have aDom: 'dom (f \o Aut_a \o invm injf) = f @* G.
rewrite /dom /= morphpre_invm -morphpreIim; congr (f @* _).
by rewrite [_ :&: D](setIidPl _) ?injmK ?injm_autm ?im_autm.
have [af [def_af ker_af _ im_af]] := domP _ aDom.
have inj_a': 'injm af by rewrite ker_af !injm_comp ?injm_autm ?injm_invm.
have im_a': af @* (f @* G) = f @* G.
by rewrite im_af !morphim_comp morphim_invm // im_autm.
pose a' := aut inj_a' im_a'; exists a' => [|AutGa x Gx]; first exact: Aut_aut.
have Dx := domG Gx; rewrite /= [a' _]autE ?mem_morphim //.
by rewrite def_af /= invmE // autmE subgK.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_isom_subproof
| |
Aut_isoma := s2val (Aut_isom_subproof a).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_isom
| |
Aut_Aut_isoma : Aut_isom a \in Aut (f @* G).
Proof. by rewrite /Aut_isom; case: (Aut_isom_subproof a). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_Aut_isom
| |
Aut_isomEa : a \in Aut G -> {in G, forall x, Aut_isom a (f x) = f (a x)}.
Proof. by rewrite /Aut_isom; case: (Aut_isom_subproof a). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_isomE
| |
Aut_isomM: {in Aut G &, {morph Aut_isom: x y / x * y}}.
Proof.
move=> a b AutGa AutGb.
apply: (eq_Aut (Aut_Aut_isom _)); rewrite ?groupM ?Aut_Aut_isom // => fx.
case/morphimP=> x Dx Gx ->{fx}.
by rewrite permM !Aut_isomE ?groupM /= ?permM ?Aut_closed.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_isomM
| |
Aut_isom_morphism:= Morphism Aut_isomM.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_isom_morphism
| |
injm_Aut_isom: 'injm Aut_isom.
Proof.
apply/injmP=> a b AutGa AutGb eq_ab'; apply: (eq_Aut AutGa AutGb) => x Gx.
by apply: (injmP injf); rewrite ?domG ?Aut_closed // -!Aut_isomE //= eq_ab'.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
injm_Aut_isom
| |
im_Aut_isom: Aut_isom injf sGD @* Aut G = Aut (f @* G).
Proof.
apply/eqP; rewrite eqEcard; apply/andP; split.
by apply/subsetP=> _ /morphimP[a _ AutGa ->]; apply: Aut_Aut_isom.
have inj_isom' := injm_Aut_isom (injm_invm injf) (morphimS _ sGD).
rewrite card_injm ?injm_Aut_isom // -(card_injm inj_isom') ?subset_leq_card //.
apply/subsetP=> a /morphimP[a' _ AutfGa' def_a].
by rewrite -(morphim_invm injf sGD) def_a Aut_Aut_isom.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
im_Aut_isom
| |
Aut_isomP: isom (Aut G) (Aut (f @* G)) (Aut_isom injf sGD).
Proof. by apply/isomP; split; [apply: injm_Aut_isom | apply: im_Aut_isom]. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_isomP
| |
injm_Aut: Aut (f @* G) \isog Aut G.
Proof. by rewrite isog_sym (isom_isog _ _ Aut_isomP). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
injm_Aut
| |
conjgmof {set gT} := fun x y : gT => y ^ x.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conjgm
| |
conjgmEA x y : conjgm A x y = y ^ x. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conjgmE
| |
conjgm_morphismA x :=
@Morphism _ _ A (conjgm A x) (in2W (fun y z => conjMg y z x)).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conjgm_morphism
| |
morphim_conjA x B : conjgm A x @* B = (A :&: B) :^ x.
Proof. by []. Qed.
Variable G : {group gT}.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
morphim_conj
| |
injm_conjx : 'injm (conjgm G x).
Proof. by apply/injmP; apply: in2W; apply: conjg_inj. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
injm_conj
| |
conj_isomx : isom G (G :^ x) (conjgm G x).
Proof. by apply/isomP; rewrite morphim_conj setIid injm_conj. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conj_isom
| |
conj_isogx : G \isog G :^ x.
Proof. exact: isom_isog (conj_isom x). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conj_isog
| |
norm_conjg_imx : x \in 'N(G) -> conjgm G x @* G = G.
Proof. by rewrite morphimEdom; apply: normP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
norm_conjg_im
| |
norm_conj_isomx : x \in 'N(G) -> isom G G (conjgm G x).
Proof. by move/norm_conjg_im/restr_isom_to/(_ (conj_isom x))->. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
norm_conj_isom
| |
conj_autx := aut (injm_conj _) (norm_conjg_im (subgP (subg _ x))).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conj_aut
| |
norm_conj_autE: {in 'N(G) & G, forall x y, conj_aut x y = y ^ x}.
Proof. by move=> x y nGx Gy; rewrite /= autE //= subgK. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
norm_conj_autE
| |
conj_autE: {in G &, forall x y, conj_aut x y = y ^ x}.
Proof. by apply: sub_in11 norm_conj_autE => //; apply: subsetP (normG G). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conj_autE
| |
conj_aut_morphM: {in 'N(G) &, {morph conj_aut : x y / x * y}}.
Proof.
move=> x y nGx nGy; apply/permP=> z /=; rewrite permM.
case Gz: (z \in G); last by rewrite !permE /= !Gz.
by rewrite !norm_conj_autE // (conjgM, memJ_norm, groupM).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conj_aut_morphM
| |
conj_aut_morphism:= Morphism conj_aut_morphM.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
conj_aut_morphism
| |
ker_conj_aut: 'ker conj_aut = 'C(G).
Proof.
apply/setP=> x /[1!inE]; case nGx: (x \in 'N(G)); last first.
by symmetry; apply/idP=> cGx; rewrite (subsetP (cent_sub G)) in nGx.
rewrite 2!inE /=; apply/eqP/centP=> [cx1 y Gy | cGx].
by rewrite /commute (conjgC y) -norm_conj_autE // cx1 perm1.
apply/permP=> y; case Gy: (y \in G); last by rewrite !permE Gy.
by rewrite perm1 norm_conj_autE // conjgE -cGx ?mulKg.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
ker_conj_aut
| |
Aut_conj_autA : conj_aut @* A \subset Aut G.
Proof. by apply/subsetP=> _ /imsetP[x _ ->]; apply: Aut_aut. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
Aut_conj_aut
| |
characteristicA B :=
(A \subset B) && [forall f in Aut B, f @: A \subset A].
Infix "\char" := characteristic.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
characteristic
| |
charPH G :
let fixH (f : {morphism G >-> gT}) := 'injm f -> f @* G = G -> f @* H = H in
reflect [/\ H \subset G & forall f, fixH f] (H \char G).
Proof.
do [apply: (iffP andP) => -[sHG chHG]; split] => // [f injf Gf|].
by apply/morphim_fixP; rewrite // -imset_autE ?(forall_inP chHG) ?Aut_aut.
apply/forall_inP=> f Af; rewrite -(autmE Af) -morphimEsub //.
by rewrite chHG ?injm_autm ?im_autm.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
charP
| |
char1G : 1 \char G.
Proof. by apply/charP; split=> [|f _ _]; rewrite (sub1G, morphim1). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char1
| |
char_reflG : G \char G.
Proof. exact/charP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_refl
| |
char_transH G K : K \char H -> H \char G -> K \char G.
Proof.
case/charP=> sKH chKH; case/charP=> sHG chHG.
apply/charP; split=> [|f injf Gf]; first exact: subset_trans sHG.
rewrite -{1}(setIidPr sKH) -(morphim_restrm sHG) chKH //.
by rewrite ker_restrm; move/trivgP: injf => ->; apply: subsetIr.
by rewrite morphim_restrm setIid chHG.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_trans
| |
char_normsH G : H \char G -> 'N(G) \subset 'N(H).
Proof.
case/charP=> sHG chHG; apply/normsP=> x /normP-Nx.
have:= chHG [morphism of conjgm G x] => /=.
by rewrite !morphimEsub //=; apply; rewrite // injm_conj.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_norms
| |
char_subA B : A \char B -> A \subset B.
Proof. by case/andP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_sub
| |
char_norm_transH G A : H \char G -> A \subset 'N(G) -> A \subset 'N(H).
Proof. by move/char_norms=> nHnG nGA; apply: subset_trans nHnG. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_norm_trans
| |
char_normal_transH G K : K \char H -> H <| G -> K <| G.
Proof.
move=> chKH /andP[sHG nHG].
by rewrite /normal (subset_trans (char_sub chKH)) // (char_norm_trans chKH).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_normal_trans
| |
char_normalH G : H \char G -> H <| G.
Proof. by move/char_normal_trans; apply; apply/andP; rewrite normG. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_normal
| |
char_normH G : H \char G -> G \subset 'N(H).
Proof. by case/char_normal/andP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_norm
| |
charIG H K : H \char G -> K \char G -> H :&: K \char G.
Proof.
case/charP=> sHG chHG; case/charP=> _ chKG.
apply/charP; split=> [|f injf Gf]; first by rewrite subIset // sHG.
by rewrite morphimGI ?(chHG, chKG) //; apply: subset_trans (sub1G H).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
charI
| |
charYG H K : H \char G -> K \char G -> H <*> K \char G.
Proof.
case/charP=> sHG chHG; case/charP=> sKG chKG.
apply/charP; split=> [|f injf Gf]; first by rewrite gen_subG subUset sHG.
by rewrite morphim_gen ?(morphimU, subUset, sHG, chHG, chKG).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
charY
| |
charMG H K : H \char G -> K \char G -> H * K \char G.
Proof.
move=> chHG chKG; rewrite -norm_joinEl ?charY //.
exact: subset_trans (char_sub chHG) (char_norm chKG).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
charM
| |
lone_subgroup_charG H :
H \subset G -> (forall K, K \subset G -> K \isog H -> K \subset H) ->
H \char G.
Proof.
move=> sHG Huniq; apply/charP; split=> // f injf Gf; apply/eqP.
have{} injf: {in H &, injective f}.
by move/injmP: injf; apply: sub_in2; apply/subsetP.
have fH: f @* H = f @: H by rewrite /morphim (setIidPr sHG).
rewrite eqEcard {2}fH card_in_imset ?{}Huniq //=.
by rewrite -{3}Gf morphimS.
rewrite isog_sym; apply/isogP.
exists [morphism of restrm sHG f] => //=; first exact/injmP.
by rewrite morphimEdom fH.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
lone_subgroup_char
| |
injm_char(G H : {group aT}) :
G \subset D -> H \char G -> f @* H \char f @* G.
Proof.
move=> sGD /charP[sHG charH].
apply/charP; split=> [|g injg gfG]; first exact: morphimS.
have /domP[h [_ ker_h _ im_h]]: 'dom (invm injf \o g \o f) = G.
by rewrite /dom /= -(morphpreIim g) (setIidPl _) ?injmK // gfG morphimS.
have hH: h @* H = H.
apply: charH; first by rewrite ker_h !injm_comp ?injm_invm.
by rewrite im_h !morphim_comp gfG morphim_invm.
rewrite /= -{2}hH im_h !morphim_comp morphim_invmE morphpreK //.
by rewrite (subset_trans _ (morphimS f sGD)) //= -{3}gfG !morphimS.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
injm_char
| |
char_injm(G H : {group aT}) :
G \subset D -> H \subset D -> (f @* H \char f @* G) = (H \char G).
Proof.
move=> sGD sHD; apply/idP/idP; last exact: injm_char.
by move/(injm_char (injm_invm injf)); rewrite !morphim_invm ?morphimS // => ->.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat fintype",
"From mathcomp Require Import finset fingroup perm morphism"
] |
fingroup/automorphism.v
|
char_injm
| |
BuildG := (isStarMonoid.Build G) (only parsing).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
Build
| |
isMulBaseGroupG := (isStarMonoid G) (only parsing).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
isMulBaseGroup
| |
BuildG := (StarMonoid_isGroup.Build G) (only parsing).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
Build
| |
BaseFinGroup_isGroupG := (StarMonoid_isGroup G) (only parsing).
#[arg_sort, short(type="finStarMonoidType")]
HB.structure Definition FinStarMonoid := { G of StarMonoid G & Finite G }.
#[deprecated(since="mathcomp 2.5.0",
note="Use Algebra.finStarMonoidType instead.")]
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
BaseFinGroup_isGroup
| |
baseFinGroupType:= finStarMonoidType (only parsing).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
baseFinGroupType
| |
sort:= (FinStarMonoid.sort) (only parsing).
#[deprecated(since="mathcomp 2.5.0",
note="Use Algebra.FinStarMonoid.arg_sort instead.")]
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sort
| |
arg_sort:= (FinStarMonoid.arg_sort) (only parsing).
#[deprecated(since="mathcomp 2.5.0",
note="Use Algebra.FinStarMonoid.on instead.")]
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
arg_sort
| |
onM := (FinStarMonoid.on M) (only parsing).
#[deprecated(since="mathcomp 2.5.0",
note="Use Algebra.FinStarMonoid.copy instead.")]
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
on
| |
copyM N := (FinStarMonoid.copy M N) (only parsing).
#[deprecated(since="mathcomp 2.5.0",
note="Use Algebra.FinStarMonoid.clone instead.")]
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
copy
|
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