phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
"''C' ( S \| to )"	:= (astab_group to S) : Group_scope.	Notation	''C' ( S \| to )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' A ( S \| to )"	:= (setI_group A 'C(S \| to)) : Group_scope.	Notation	''C_' A ( S \| to )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setI_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( A ) ( S \| to )"	:= (setI_group A 'C(S \| to)) (only parsing) : Group_scope.	Notation	''C_' ( A ) ( S \| to )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setI_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C' [ x \| to ]"	:= (astab_group to [set x%g]) : Group_scope.	Notation	''C' [ x \| to ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' A [ x \| to ]"	:= (setI_group A 'C[x \| to]) : Group_scope.	Notation	''C_' A [ x \| to ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setI_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( A ) [ x \| to ]"	:= (setI_group A 'C[x \| to]) (only parsing) : Group_scope.	Notation	''C_' ( A ) [ x \| to ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setI_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''N' ( S \| to )"	:= (astabs_group to S) : Group_scope.	Notation	''N' ( S \| to )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabs_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''N_' A ( S \| to )"	:= (setI_group A 'N(S \| to)) : Group_scope.	Notation	''N_' A ( S \| to )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setI_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actM x a b : to x (a * b) = to (to x a) b.	Proof. by rewrite actMin ?inE. Qed.	Lemma	actM	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actMin", "inE", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actK : right_loop inv to.	Proof. by move=> a; apply: actKin; rewrite inE. Qed.	Lemma	actK	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actKin", "apply", "inE", "inv", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actKV : rev_right_loop inv to.	Proof. by move=> a; apply: actKVin; rewrite inE. Qed.	Lemma	actKV	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actKVin", "apply", "inE", "inv", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actX x a n : to x (a ^+ n) = iter n (to^~ a) x.	Proof. by elim: n => [\|n /= <-]; rewrite ?act1 // -actM expgSr. Qed.	Lemma	actX	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act1", "actM", "expgSr", "iter", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actCJ a b x : to (to x a) b = to (to x b) (a ^ b).	Proof. by rewrite !actM actK. Qed.	Lemma	actCJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actK", "actM", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actCJV a b x : to (to x a) b = to (to x (b ^ a^-1)) a.	Proof. by rewrite (actCJ _ a) conjgKV. Qed.	Lemma	actCJV	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actCJ", "conjgKV", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_sym G x y : (x \in orbit to G y) = (y \in orbit to G x).	Proof. exact/orbit_in_sym/subsetT. Qed.	Lemma	orbit_sym	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "orbit_in_sym", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_trans G 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	orbit_trans	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "orbit_in_trans", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_eqP G 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	orbit_eqP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "orbit_in_eqP", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_transl G 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	orbit_transl	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "orbit_in_transl", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_act G a x: a \in G -> orbit to G (to x a) = orbit to G x.	Proof. exact/orbit_act_in/subsetT. Qed.	Lemma	orbit_act	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "orbit_act_in", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_actr G 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	orbit_actr	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "mem_orbit", "orbit", "orbit_transl", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_eq_mem G 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	orbit_eq_mem	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "orbit_eqP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_inv A x y : (y \in orbit to A^-1 x) = (x \in orbit to A y).	Proof. by rewrite orbit_inv_in ?subsetT. Qed.	Lemma	orbit_inv	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "orbit_inv_in", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_lcoset A a x : orbit to (a *: A) x = orbit to A (to x a).	Proof. by rewrite orbit_lcoset_in ?subsetT ?inE. Qed.	Lemma	orbit_lcoset	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "inE", "orbit", "orbit_lcoset_in", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_rcoset A 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	orbit_rcoset	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "inE", "orbit", "orbit_rcoset_in", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_conjsg A 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	orbit_conjsg	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "inE", "orbit", "orbit_conjsg_in", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabP S 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	astabP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astab_act", "inE", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1P x a : reflect (to x a = x) (a \in 'C[x \| to]).	Proof. by rewrite !inE sub1set inE; apply: eqP. Qed.	Lemma	astab1P	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "inE", "sub1set", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_astab1 A x : (A \subset 'C[x \| to]) = (x \in 'Fix_to(A)).	Proof. by rewrite sub_astab1_in ?subsetT. Qed.	Lemma	sub_astab1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sub_astab1_in", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabC A S : (A \subset 'C(S \| to)) = (S \subset 'Fix_to(A)).	Proof. by rewrite astabCin ?subsetT. Qed.	Lemma	astabC	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabCin", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_cycle a : 'Fix_to(<[a]>) = 'Fix_to[a].	Proof. by rewrite afix_cycle_in ?inE. Qed.	Lemma	afix_cycle	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix_cycle_in", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_gen A : 'Fix_to(<<A>>) = 'Fix_to(A).	Proof. by rewrite afix_gen_in ?subsetT. Qed.	Lemma	afix_gen	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix_gen_in", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixM G H : 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).	Proof. by rewrite afixMin ?subsetT. Qed.	Lemma	afixM	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixMin", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsP S 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	astabsP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabs_act", "inE", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_orbit G x : #\|orbit to G x\| = #\|G : 'C_G[x \| to]\|.	Proof. by rewrite card_orbit_in ?subsetT. Qed.	Lemma	card_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "card_orbit_in", "orbit", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_orbit G x : #\|orbit to G x\| %\| #\|G\|.	Proof. by rewrite card_orbit dvdn_indexg. Qed.	Lemma	dvdn_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "card_orbit", "dvdn_indexg", "orbit", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_orbit_stab G x : (#\|orbit to G x\| * #\|'C_G[x \| to]\|)%N = #\|G\|.	Proof. by rewrite mulnC card_orbit Lagrange ?subsetIl. Qed.	Lemma	card_orbit_stab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Lagrange", "card_orbit", "mulnC", "orbit", "subsetIl", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actsP A 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.	Lemma	actsP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_act", "apply", "inE", "on", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setact_orbit A 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	setact_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actCJ", "apply", "imsetP", "memJ_conjg", "mem_orbit", "mem_setact", "orbit", "setP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_setact S 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	astab_setact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actK", "actM", "apply", "astabP", "conjgCV", "conjgE", "imsetP", "invgK", "mem_conjg", "setP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1_act x a : 'C[to x a \| to] = 'C[x \| to] :^ a.	Proof. by rewrite -astab_setact /setact imset_set1. Qed.	Lemma	astab1_act	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab_setact", "imset_set1", "setact", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atransP G 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	atransP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "imsetP", "on", "orbit", "orbit_eqP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atransP2 G 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	atransP2	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "atransP", "imsetP", "on", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_acts G 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	atrans_acts	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "atransP", "imset_f", "inE", "on", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_supgroup G 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	atrans_supgroup	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_sub_orbit", "apply", "atrans_acts", "eqEsubset", "imsetP", "imsetS", "on", "sGH", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_acts_card G 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...	Lemma	atrans_acts_card	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_sub_orbit", "apply", "atransP", "atrans_acts", "card0_eq", "cardD1", "cards1", "eqEsubset", "eqn_leq", "existsP", "imsetP", "imset_f", "inE", "leqn0", "lt0n", "ltnS", "on", "orbit", "orbit_refl", "split", "sub1set", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_dvd G S : [transitive G, on S \| to] -> #\|S\| %\| #\|G\|.	Proof. by case/imsetP=> x _ ->; apply: dvdn_orbit. Qed.	Lemma	atrans_dvd	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "dvdn_orbit", "imsetP", "on", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_fix_norm A 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	acts_fix_norm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_subnorm_fix", "on", "setTI", "subset_trans", "to" ]	This is Aschbacher (5.2)	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
faithfulP A 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	faithfulP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabP", "faithful", "id", "inE", "on", "set1P", "setIP", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_trans_gcore G 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 //=. 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	astab_trans_gcore	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astab1P", "astab1_act", "astabP", "astabS", "astab_norm", "atransP2", "atrans_acts", "bigcapP", "eqEsubset", "gcore", "gcore_max", "on", "sub1set", "subsetP", "subset_trans", "to" ]	This is the first part of Aschbacher (5.7)	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgroup_transitiveP G 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	subgroup_transitiveP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "on", "sHG", "subgroup_transitivePin", "subsetT", "to" ]	This is Aschbacher (5.20)	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trans_subnorm_fixP x 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 =...	Lemma	trans_subnorm_fixP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actKV", "actM", "acts_act", "acts_fix_norm", "apply", "astab1P", "astab1_act", "atransP2", "atrans_acts", "conj_subG", "conjsgKV", "conjsgM", "groupM", "groupV", "imsetP", "imset_f", "inE", "last", "mem_orbit", "normP", "on", "sHG", "setIP", "setIdP", "setP", "sub_...	This is Aschbacher (5.21)	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ract & A \subset D	:= act to.	Definition	ract	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	ract_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "fM", "injf", "is_action", "ract", "sAD", "split", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
raction	:= Action ract_is_action.	Canonical	raction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "ract_is_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ractE : raction =1 to.	Proof. by []. Qed.	Lemma	ractE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "raction", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to \ sAD"	:= (raction to sAD) (at level 50) : action_scope.	Notation	to \ sAD	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "raction", "sAD", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actby_cond (A : {set aT}) R (to : action D rT) : Prop	:= [acts A, on R \| to].	Definition	actby_cond	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aT", "action", "on", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actby A R to & actby_cond A R to	:= fun x a => if (x \in R) && (a \in A) then to x a else x.	Definition	actby	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby_cond", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nRA : actby_cond A R to.		Hypothesis	nRA	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby_cond", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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. b...	Lemma	actby_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actMin", "act_inj", "actby", "acts_act", "acts_dom", "groupM", "is_action", "last", "nRA", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
action_by	:= Action actby_is_action.	Canonical	action_by	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby_is_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<[nRA]>"	:= action_by : action_scope.	Notation	<[nRA]>	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "action_by" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actbyE x a : x \in R -> a \in A -> <[nRA]>%act x a = to x a.	Proof. by rewrite /= /actby => -> ->. Qed.	Lemma	actbyE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act", "actby", "nRA", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_actby B : '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	afix_actby	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby", "apply", "inE", "last", "nRA", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_actby S : '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	astab_actby	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby", "actbyE", "acts_dom", "apply", "inE", "nRA", "setIA", "setIP", "setIidPl", "setP", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_actby S : '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	astabs_actby	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby", "actbyE", "acts_act", "acts_dom", "apply", "inE", "nRA", "setIA", "setIP", "setIidPl", "setP", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	acts_actby	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aT", "astabs_actby", "nRA", "on", "subsetI", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<[ nRA ] >"	:= (action_by nRA) : action_scope.	Notation	<[ nRA ] >	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "action_by", "nRA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subact_dom	:= 'N([set x \| sP x] \| to).	Definition	subact_dom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subact_dom_group	:= [group of subact_dom].	Canonical	subact_dom_group	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "group", "subact_dom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_act_proof u Na : sP (to (val u) (val Na)).	Proof. by case: Na => a /= /(astabs_act (val u)); rewrite !inE valP. Qed.	Lemma	sub_act_proof	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabs_act", "inE", "to", "val", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subact u a	:= if insub a is Some Na then Sub _ (sub_act_proof u Na) else u.	Definition	subact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Sub", "insub", "sub_act_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_subact u 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	val_subact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "SubK", "insubP", "subact", "subact_dom", "to", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	subact_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actMin", "act_inj", "apply", "astabs_dom", "groupM", "is_action", "split", "subact", "subact_dom", "val", "val_inj", "val_subact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subaction	:= Action subact_is_action.	Canonical	subaction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "subact_is_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_subact S : '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	astab_subact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apply", "imsetP", "imset_f", "inE", "in_setI", "setIP", "setP", "subact_dom", "subaction", "subsetP", "to", "val", "val_eqE", "val_subact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_subact S : '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 rewrit...	Lemma	astabs_subact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apply", "imsetP", "imset_f", "inE", "in_setI", "setIP", "setP", "subact_dom", "subaction", "subsetP", "to", "val", "val_eqE", "val_subact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_subact A : 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	afix_subact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "inE", "sAD", "setIidPl", "setP", "subact_dom", "subsetP", "val", "val_eqE", "val_subact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to ^?"	:= (subaction _ to) (format "to ^?") : action_scope.	Notation	to ^?	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "subaction", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact_dom	:= 'N(rcosets H 'N(H) \| to^*).	Definition	qact_dom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "rcosets", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact_dom_group	:= [group of qact_dom].	Canonical	qact_dom_group	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "group", "qact_dom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subdom	:= (subact_dom (coset_range H) to^*).	Notation	subdom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "coset_range", "subact_dom", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact_subdomE : subdom = qact_dom.	Proof. by congr 'N(_\|_); apply/setP=> Hx; rewrite !inE genGid. Qed.	Fact	qact_subdomE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "genGid", "inE", "qact_dom", "setP", "subdom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact_proof : qact_dom \subset subdom.	Proof. by rewrite qact_subdomE. Qed.	Lemma	qact_proof	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "qact_dom", "qact_subdomE", "subdom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact : coset_of H -> aT -> coset_of H	:= act (to^*^? \ qact_proof).	Definition	qact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aT", "act", "coset_of", "qact_proof", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_action	:= [action of qact].	Canonical	quotient_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "action", "qact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 ?sub1se...	Lemma	acts_qact_dom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabs_act", "astabs_dom", "imset_f", "inE", "mul_subG", "normG", "on", "qact_dom", "rcosetE", "rcoset_refl", "rcosets", "rcosetsP", "sub1set", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qactEcond x 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_...	Lemma	qactEcond	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_act", "acts_qact_dom", "apply", "astabs_act", "coset", "imset_f", "qact_dom", "qact_subdomE", "quotient_action", "rcosetE", "rcoset_eqP", "rcoset_refl", "rcoset_sym", "rcosets", "rcosetsP", "to", "val_coset", "val_inj", "val_subact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qactE x 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	qactE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "coset", "qactEcond", "qact_dom", "quotient_action", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	acts_quotient	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aT", "acts_act", "acts_qact_dom", "apply", "astabs_act", "inE", "mem_morphim", "morphimP", "nBA", "on", "qactE", "quotient_action", "setIP", "subsetP", "subset_trans", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 //...	Lemma	astabs_quotient	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "acts_act", "acts_qact_dom", "acts_quotient", "apply", "astabs_act", "astabs_dom", "eqEsubset", "group", "inE", "mem_morphim", "nHG", "nsHG", "qactE", "quotientGK", "quotient_action", "setIdP", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to / H"	:= (quotient_action to H) : action_scope.	Notation	to / H	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "quotient_action", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dom	:= 'N_D(H).	Notation	dom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
range	:= 'Fix_to(D :&: H).	Notation	range	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_dom : {acts dom, on range \| to}	:= acts_act (acts_subnorm_fix to H).	Let	acts_dom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_act", "acts_subnorm_fix", "dom", "on", "range", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modact x (Ha : coset_of H)	:= if x \in range then to x (repr (D :&: Ha)) else x.	Definition	modact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "coset_of", "range", "repr", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modactEcond x 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	modactEcond	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actMin", "afixP", "coset", "dom", "group_modr", "modact", "range", "repr", "repr_rcosetP", "setIP", "sub1set", "to", "val_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modactE x 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	modactE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "coset", "inE", "modact", "modactEcond", "range", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 rew...	Lemma	modact_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "act1", "actMin", "act_inj", "acts_dom", "coset_mem", "coset_norm", "groupM", "inE", "is_action", "last", "modact", "modactEcond", "morphM", "morphimP", "repr_set0", "setIP", "set_0Vmem", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mod_action	:= Action modact_is_action.	Canonical	mod_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "modact_is_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C' ( S | to )"

:= (astab_group to S) : Group_scope.

Notation

''C' ( S | to )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' A ( S | to )"

:= (setI_group A 'C(S | to)) : Group_scope.

Notation

''C_' A ( S | to )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setI_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( A ) ( S | to )"

:= (setI_group A 'C(S | to)) (only parsing) : Group_scope.

Notation

''C_' ( A ) ( S | to )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setI_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C' [ x | to ]"

:= (astab_group to [set x%g]) : Group_scope.

Notation

''C' [ x | to ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' A [ x | to ]"

:= (setI_group A 'C[x | to]) : Group_scope.

Notation

''C_' A [ x | to ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setI_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( A ) [ x | to ]"

:= (setI_group A 'C[x | to]) (only parsing) : Group_scope.

Notation

''C_' ( A ) [ x | to ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setI_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''N' ( S | to )"

:= (astabs_group to S) : Group_scope.

Notation

''N' ( S | to )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabs_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''N_' A ( S | to )"

:= (setI_group A 'N(S | to)) : Group_scope.

Notation

''N_' A ( S | to )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setI_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actM x a b : to x (a * b) = to (to x a) b.

Proof. by rewrite actMin ?inE. Qed.

Lemma

actM

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actMin", "inE", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actK : right_loop inv to.

Proof. by move=> a; apply: actKin; rewrite inE. Qed.

Lemma

actK

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actKin", "apply", "inE", "inv", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actKV : rev_right_loop inv to.

Proof. by move=> a; apply: actKVin; rewrite inE. Qed.

Lemma

actKV

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actKVin", "apply", "inE", "inv", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actX x a n : to x (a ^+ n) = iter n (to^~ a) x.

Proof. by elim: n => [|n /= <-]; rewrite ?act1 // -actM expgSr. Qed.

Lemma

actX

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act1", "actM", "expgSr", "iter", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actCJ a b x : to (to x a) b = to (to x b) (a ^ b).

Proof. by rewrite !actM actK. Qed.

Lemma

actCJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actK", "actM", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actCJV a b x : to (to x a) b = to (to x (b ^ a^-1)) a.

Proof. by rewrite (actCJ _ a) conjgKV. Qed.

Lemma

actCJV

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actCJ", "conjgKV", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_sym G x y : (x \in orbit to G y) = (y \in orbit to G x).

Proof. exact/orbit_in_sym/subsetT. Qed.

Lemma

orbit_sym

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "orbit_in_sym", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_trans G 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

orbit_trans

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "orbit_in_trans", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_eqP G 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

orbit_eqP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "orbit_in_eqP", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_transl G 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

orbit_transl

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "orbit_in_transl", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_act G a x: a \in G -> orbit to G (to x a) = orbit to G x.

Proof. exact/orbit_act_in/subsetT. Qed.

Lemma

orbit_act

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "orbit_act_in", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_actr G 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

orbit_actr

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "mem_orbit", "orbit", "orbit_transl", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_eq_mem G 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

orbit_eq_mem

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "orbit_eqP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_inv A x y : (y \in orbit to A^-1 x) = (x \in orbit to A y).

Proof. by rewrite orbit_inv_in ?subsetT. Qed.

Lemma

orbit_inv

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "orbit_inv_in", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_lcoset A a x : orbit to (a *: A) x = orbit to A (to x a).

Proof. by rewrite orbit_lcoset_in ?subsetT ?inE. Qed.

Lemma

orbit_lcoset

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "inE", "orbit", "orbit_lcoset_in", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_rcoset A 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

orbit_rcoset

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "inE", "orbit", "orbit_rcoset_in", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_conjsg A 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

orbit_conjsg

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "inE", "orbit", "orbit_conjsg_in", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabP S 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

astabP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astab_act", "inE", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1P x a : reflect (to x a = x) (a \in 'C[x | to]).

Proof. by rewrite !inE sub1set inE; apply: eqP. Qed.

Lemma

astab1P

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "inE", "sub1set", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_astab1 A x : (A \subset 'C[x | to]) = (x \in 'Fix_to(A)).

Proof. by rewrite sub_astab1_in ?subsetT. Qed.

Lemma

sub_astab1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sub_astab1_in", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabC A S : (A \subset 'C(S | to)) = (S \subset 'Fix_to(A)).

Proof. by rewrite astabCin ?subsetT. Qed.

Lemma

astabC

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabCin", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_cycle a : 'Fix_to(<[a]>) = 'Fix_to[a].

Proof. by rewrite afix_cycle_in ?inE. Qed.

Lemma

afix_cycle

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afix_cycle_in", "inE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_gen A : 'Fix_to(<<A>>) = 'Fix_to(A).

Proof. by rewrite afix_gen_in ?subsetT. Qed.

Lemma

afix_gen

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afix_gen_in", "subsetT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixM G H : 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).

Proof. by rewrite afixMin ?subsetT. Qed.

Lemma

afixM

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afixMin", "subsetT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsP S 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

astabsP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabs_act", "inE", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_orbit G x : #|orbit to G x| = #|G : 'C_G[x | to]|.

Proof. by rewrite card_orbit_in ?subsetT. Qed.

Lemma

card_orbit

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "card_orbit_in", "orbit", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_orbit G x : #|orbit to G x| %| #|G|.

Proof. by rewrite card_orbit dvdn_indexg. Qed.

Lemma

dvdn_orbit

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "card_orbit", "dvdn_indexg", "orbit", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_orbit_stab G x : (#|orbit to G x| * #|'C_G[x | to]|)%N = #|G|.

Proof. by rewrite mulnC card_orbit Lagrange ?subsetIl. Qed.

Lemma

card_orbit_stab

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Lagrange", "card_orbit", "mulnC", "orbit", "subsetIl", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actsP A 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.

Lemma

actsP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_act", "apply", "inE", "on", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setact_orbit A 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

setact_orbit

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actCJ", "apply", "imsetP", "memJ_conjg", "mem_orbit", "mem_setact", "orbit", "setP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_setact S 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

astab_setact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actK", "actM", "apply", "astabP", "conjgCV", "conjgE", "imsetP", "invgK", "mem_conjg", "setP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1_act x a : 'C[to x a | to] = 'C[x | to] :^ a.

Proof. by rewrite -astab_setact /setact imset_set1. Qed.

Lemma

astab1_act

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab_setact", "imset_set1", "setact", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atransP G 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

atransP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "imsetP", "on", "orbit", "orbit_eqP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atransP2 G 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

atransP2

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "atransP", "imsetP", "on", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_acts G 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

atrans_acts

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "atransP", "imset_f", "inE", "on", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_supgroup G 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

atrans_supgroup

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_sub_orbit", "apply", "atrans_acts", "eqEsubset", "imsetP", "imsetS", "on", "sGH", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_acts_card G 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...

Lemma

atrans_acts_card

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_sub_orbit", "apply", "atransP", "atrans_acts", "card0_eq", "cardD1", "cards1", "eqEsubset", "eqn_leq", "existsP", "imsetP", "imset_f", "inE", "leqn0", "lt0n", "ltnS", "on", "orbit", "orbit_refl", "split", "sub1set", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_dvd G S : [transitive G, on S | to] -> #|S| %| #|G|.

Proof. by case/imsetP=> x _ ->; apply: dvdn_orbit. Qed.

Lemma

atrans_dvd

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "dvdn_orbit", "imsetP", "on", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_fix_norm A 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

acts_fix_norm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_subnorm_fix", "on", "setTI", "subset_trans", "to" ]

This is Aschbacher (5.2)

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

faithfulP A 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

faithfulP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabP", "faithful", "id", "inE", "on", "set1P", "setIP", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_trans_gcore G 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 //=. 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

astab_trans_gcore

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astab1P", "astab1_act", "astabP", "astabS", "astab_norm", "atransP2", "atrans_acts", "bigcapP", "eqEsubset", "gcore", "gcore_max", "on", "sub1set", "subsetP", "subset_trans", "to" ]

This is the first part of Aschbacher (5.7)

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgroup_transitiveP G 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

subgroup_transitiveP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "on", "sHG", "subgroup_transitivePin", "subsetT", "to" ]

This is Aschbacher (5.20)

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trans_subnorm_fixP x 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 =...

Lemma

trans_subnorm_fixP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actKV", "actM", "acts_act", "acts_fix_norm", "apply", "astab1P", "astab1_act", "atransP2", "atrans_acts", "conj_subG", "conjsgKV", "conjsgM", "groupM", "groupV", "imsetP", "imset_f", "inE", "last", "mem_orbit", "normP", "on", "sHG", "setIP", "setIdP", "setP", "sub_...

This is Aschbacher (5.21)

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ract & A \subset D

:= act to.

Definition

ract

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

ract_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "fM", "injf", "is_action", "ract", "sAD", "split", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

raction

:= Action ract_is_action.

Canonical

raction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "ract_is_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ractE : raction =1 to.

Proof. by []. Qed.

Lemma

ractE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "raction", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to \ sAD"

:= (raction to sAD) (at level 50) : action_scope.

Notation

to \ sAD

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "raction", "sAD", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actby_cond (A : {set aT}) R (to : action D rT) : Prop

:= [acts A, on R | to].

Definition

actby_cond

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aT", "action", "on", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actby A R to & actby_cond A R to

:= fun x a => if (x \in R) && (a \in A) then to x a else x.

Definition

actby

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby_cond", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nRA : actby_cond A R to.

Hypothesis

nRA

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby_cond", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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. b...

Lemma

actby_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actMin", "act_inj", "actby", "acts_act", "acts_dom", "groupM", "is_action", "last", "nRA", "split", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

action_by

:= Action actby_is_action.

Canonical

action_by

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby_is_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<[nRA]>"

:= action_by : action_scope.

Notation

<[nRA]>

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "action_by" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actbyE x a : x \in R -> a \in A -> <[nRA]>%act x a = to x a.

Proof. by rewrite /= /actby => -> ->. Qed.

Lemma

actbyE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act", "actby", "nRA", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_actby B : '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

afix_actby

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby", "apply", "inE", "last", "nRA", "setP", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_actby S : '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

astab_actby

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby", "actbyE", "acts_dom", "apply", "inE", "nRA", "setIA", "setIP", "setIidPl", "setP", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_actby S : '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

astabs_actby

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby", "actbyE", "acts_act", "acts_dom", "apply", "inE", "nRA", "setIA", "setIP", "setIidPl", "setP", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

acts_actby

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aT", "astabs_actby", "nRA", "on", "subsetI", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<[ nRA ] >"

:= (action_by nRA) : action_scope.

Notation

<[ nRA ] >

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "action_by", "nRA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subact_dom

:= 'N([set x | sP x] | to).

Definition

subact_dom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subact_dom_group

:= [group of subact_dom].

Canonical

subact_dom_group

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "group", "subact_dom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_act_proof u Na : sP (to (val u) (val Na)).

Proof. by case: Na => a /= /(astabs_act (val u)); rewrite !inE valP. Qed.

Lemma

sub_act_proof

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabs_act", "inE", "to", "val", "valP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subact u a

:= if insub a is Some Na then Sub _ (sub_act_proof u Na) else u.

Definition

subact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Sub", "insub", "sub_act_proof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_subact u 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

val_subact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "SubK", "insubP", "subact", "subact_dom", "to", "val" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

subact_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actMin", "act_inj", "apply", "astabs_dom", "groupM", "is_action", "split", "subact", "subact_dom", "val", "val_inj", "val_subact" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subaction

:= Action subact_is_action.

Canonical

subaction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "subact_is_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_subact S : '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

astab_subact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apply", "imsetP", "imset_f", "inE", "in_setI", "setIP", "setP", "subact_dom", "subaction", "subsetP", "to", "val", "val_eqE", "val_subact" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_subact S : '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 rewrit...

Lemma

astabs_subact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apply", "imsetP", "imset_f", "inE", "in_setI", "setIP", "setP", "subact_dom", "subaction", "subsetP", "to", "val", "val_eqE", "val_subact" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_subact A : 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

afix_subact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "inE", "sAD", "setIidPl", "setP", "subact_dom", "subsetP", "val", "val_eqE", "val_subact" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to ^?"

:= (subaction _ to) (format "to ^?") : action_scope.

Notation

to ^?

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "subaction", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact_dom

:= 'N(rcosets H 'N(H) | to^*).

Definition

qact_dom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "rcosets", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact_dom_group

:= [group of qact_dom].

Canonical

qact_dom_group

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "group", "qact_dom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subdom

:= (subact_dom (coset_range H) to^*).

Notation

subdom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "coset_range", "subact_dom", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact_subdomE : subdom = qact_dom.

Proof. by congr 'N(_|_); apply/setP=> Hx; rewrite !inE genGid. Qed.

Fact

qact_subdomE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "genGid", "inE", "qact_dom", "setP", "subdom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact_proof : qact_dom \subset subdom.

Proof. by rewrite qact_subdomE. Qed.

Lemma

qact_proof

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "qact_dom", "qact_subdomE", "subdom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact : coset_of H -> aT -> coset_of H

:= act (to^*^? \ qact_proof).

Definition

qact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aT", "act", "coset_of", "qact_proof", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_action

:= [action of qact].

Canonical

quotient_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "action", "qact" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 ?sub1se...

Lemma

acts_qact_dom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabs_act", "astabs_dom", "imset_f", "inE", "mul_subG", "normG", "on", "qact_dom", "rcosetE", "rcoset_refl", "rcosets", "rcosetsP", "sub1set", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qactEcond x 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_...

Lemma

qactEcond

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_act", "acts_qact_dom", "apply", "astabs_act", "coset", "imset_f", "qact_dom", "qact_subdomE", "quotient_action", "rcosetE", "rcoset_eqP", "rcoset_refl", "rcoset_sym", "rcosets", "rcosetsP", "to", "val_coset", "val_inj", "val_subact" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qactE x 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

qactE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "coset", "qactEcond", "qact_dom", "quotient_action", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

acts_quotient

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aT", "acts_act", "acts_qact_dom", "apply", "astabs_act", "inE", "mem_morphim", "morphimP", "nBA", "on", "qactE", "quotient_action", "setIP", "subsetP", "subset_trans", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 //...

Lemma

astabs_quotient

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "acts_act", "acts_qact_dom", "acts_quotient", "apply", "astabs_act", "astabs_dom", "eqEsubset", "group", "inE", "mem_morphim", "nHG", "nsHG", "qactE", "quotientGK", "quotient_action", "setIdP", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to / H"

:= (quotient_action to H) : action_scope.

Notation

to / H

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "quotient_action", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dom

:= 'N_D(H).

Notation

dom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

range

:= 'Fix_to(D :&: H).

Notation

range

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_dom : {acts dom, on range | to}

:= acts_act (acts_subnorm_fix to H).

Let

acts_dom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_act", "acts_subnorm_fix", "dom", "on", "range", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modact x (Ha : coset_of H)

:= if x \in range then to x (repr (D :&: Ha)) else x.

Definition

modact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "coset_of", "range", "repr", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modactEcond x 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

modactEcond

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actMin", "afixP", "coset", "dom", "group_modr", "modact", "range", "repr", "repr_rcosetP", "setIP", "sub1set", "to", "val_coset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modactE x 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

modactE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "coset", "inE", "modact", "modactEcond", "range", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 rew...

Lemma

modact_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "act1", "actMin", "act_inj", "acts_dom", "coset_mem", "coset_norm", "groupM", "inE", "is_action", "last", "modact", "modactEcond", "morphM", "morphimP", "repr_set0", "setIP", "set_0Vmem", "split", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mod_action

:= Action modact_is_action.

Canonical

mod_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "modact_is_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d