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	source_url stringclasses 1 value	commit stringclasses 1 value
gen_subG A G : (<<A>> \subset G) = (A \subset G).	Proof. apply/idP/idP=> [\|sAG]; first exact: subset_trans (subset_gen A). by apply/subsetP=> x /generatedP; apply. Qed.	Lemma	gen_subG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "generatedP", "sAG", "subsetP", "subset_gen", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genGid G : <<G>> = G.	Proof. by apply/eqP; rewrite eqEsubset gen_subG subset_gen andbT. Qed.	Lemma	genGid	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "gen_subG", "subset_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genGidG G : <<G>>%G = G.	Proof. by apply: val_inj; apply: genGid. Qed.	Lemma	genGidG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "genGid", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gen_set_id A : group_set A -> <<A>> = A.	Proof. by move=> gA; apply: (genGid (group gA)). Qed.	Lemma	gen_set_id	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "genGid", "group", "group_set" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genS A B : A \subset B -> <<A>> \subset <<B>>.	Proof. by move=> sAB; rewrite gen_subG sub_gen. Qed.	Lemma	genS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gen_subG", "sub_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gen0 : <<set0>> = 1 :> {set gT}.	Proof. by apply/eqP; rewrite eqEsubset sub1G gen_subG sub0set. Qed.	Lemma	gen0	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "gT", "gen_subG", "set0", "sub0set", "sub1G" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gen_expgs A : {n \| <<A>> = (1 \|: A) ^+ n}.	Proof. set B := (1 \|: A); pose N := #\|gT\|. have BsubG n : B ^+ n \subset <<A>>. by elim: n => [\|n IHn]; rewrite ?expgS ?mul_subG ?subUset ?sub1G ?subset_gen. have B_1 n : 1 \in B ^+ n. by elim: n => [\|n IHn]; rewrite ?set11 // expgS mulUg mul1g inE IHn. case: (pickP (fun i : 'I_N => B ^+ i.+1 \subset B ^+ i)) => [n...	Lemma	gen_expgs	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cards1", "eqEsubset", "expgS", "expgSr", "gT", "genS", "gen_set_id", "group_set", "inE", "last", "leq_ltn_trans", "ltnNge", "ltnW", "max_card", "mul1g", "mulSg", "mulUg", "mul_subG", "mulg1", "mulgA", "mulg_subl", "nat", "pickP", "proper", "proper_card", ...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gen_prodgP A x : reflect (exists n, exists2 c, forall i : 'I_n, c i \in A & x = \prod_i c i) (x \in <<A>>).	Proof. apply: (iffP idP) => [\|[n [c Ac ->]]]; last first. by apply: group_prod => i _; rewrite mem_gen ?Ac. have [n ->] := gen_expgs A; rewrite /natexp Monoid.iteropE /=. rewrite -[n]card_ord -big_const => /prodsgP[/= c Ac def_x]. have{Ac def_x} ->: x = \prod_(i \| c i \in A) c i. rewrite big_mkcond {x}def_x; apply...	Lemma	gen_prodgP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "big_const", "big_enumP", "big_mkcond", "big_tuple", "card_ord", "eq_bigr", "gen_expgs", "group_prod", "in_tuple", "iteropE", "last", "mem_gen", "mem_tnth", "natexp", "prodsgP", "setU1P", "size", "tnth", "val" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genD A B : A \subset <<A :\: B>> -> <<A :\: B>> = <<A>>.	Proof. by move=> sAB; apply/eqP; rewrite eqEsubset genS (subsetDl, gen_subG). Qed.	Lemma	genD	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "genS", "gen_subG", "subsetDl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genV A : <<A^-1>> = <<A>>.	Proof. apply/eqP; rewrite eqEsubset !gen_subG -!(invSg _ <<_>>) invgK. by rewrite !invGid !subset_gen. Qed.	Lemma	genV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "gen_subG", "invGid", "invSg", "invgK", "subset_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genJ A z : <<A :^z>> = <<A>> :^ z.	Proof. by apply/eqP; rewrite eqEsubset sub_conjg !gen_subG conjSg -?sub_conjg !sub_gen. Qed.	Lemma	genJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjSg", "eqEsubset", "gen_subG", "sub_conjg", "sub_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjYg A B z : (A <> B) :^z = A :^ z <> B :^ z.	Proof. by rewrite -genJ conjUg. Qed.	Lemma	conjYg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjUg", "genJ" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genD1 A x : x \in <<A :\ x>> -> <<A :\ x>> = <<A>>.	Proof. move=> gA'x; apply/eqP; rewrite eqEsubset genS; first by rewrite subsetDl. rewrite gen_subG; apply/subsetP=> y Ay. by case: (y =P x) => [-> //\|]; move/eqP=> nyx; rewrite mem_gen // !inE nyx. Qed.	Lemma	genD1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "genS", "gen_subG", "inE", "mem_gen", "subsetDl", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genD1id A : <<A^#>> = <<A>>.	Proof. by rewrite genD1 ?group1. Qed.	Lemma	genD1id	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "genD1", "group1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joingT	:= (@joing gT) (only parsing).	Notation	joingT	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gT", "joing" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinGT	:= (@joinG gT) (only parsing).	Notation	joinGT	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gT", "joinG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joingE A B : A <*> B = <<A :\|: B>>.	Proof. by []. Qed.	Lemma	joingE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinGE G H : (G * H)%G = (G <*> H)%G.	Proof. by []. Qed.	Lemma	joinGE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joingC : commutative joingT.	Proof. by move=> A B; rewrite /joing setUC. Qed.	Lemma	joingC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joing", "joingT", "setUC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_idr A B : A <> <<B>> = A <> B.	Proof. apply/eqP; rewrite eqEsubset gen_subG subUset gen_subG /=. by rewrite -subUset subset_gen genS // setUS // subset_gen. Qed.	Lemma	joing_idr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "genS", "gen_subG", "setUS", "subUset", "subset_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_idl A B : <<A>> <> B = A <> B.	Proof. by rewrite -!(joingC B) joing_idr. Qed.	Lemma	joing_idl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joingC", "joing_idr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_subl A B : A \subset A <*> B.	Proof. by rewrite sub_gen ?subsetUl. Qed.	Lemma	joing_subl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "sub_gen", "subsetUl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_subr A B : B \subset A <*> B.	Proof. by rewrite sub_gen ?subsetUr. Qed.	Lemma	joing_subr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "sub_gen", "subsetUr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join_subG A B G : (A <*> B \subset G) = (A \subset G) && (B \subset G).	Proof. by rewrite gen_subG subUset. Qed.	Lemma	join_subG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gen_subG", "subUset" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_idPl G A : reflect (G <*> A = G) (A \subset G).	Proof. apply: (iffP idP) => [sHG \| <-]; last by rewrite joing_subr. by rewrite joingE (setUidPl sHG) genGid. Qed.	Lemma	joing_idPl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "genGid", "joingE", "joing_subr", "last", "sHG", "setUidPl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_idPr A G : reflect (A <*> G = G) (A \subset G).	Proof. by rewrite joingC; apply: joing_idPl. Qed.	Lemma	joing_idPr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "joingC", "joing_idPl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_subP A B G : reflect (A \subset G /\ B \subset G) (A <*> B \subset G).	Proof. by rewrite join_subG; apply: andP. Qed.	Lemma	joing_subP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "join_subG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_sub A B C : A <*> B = C -> A \subset C /\ B \subset C.	Proof. by move <-; apply/joing_subP. Qed.	Lemma	joing_sub	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "joing_subP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genDU A B C : A \subset C -> <<C :\: A>> = <<B>> -> <<A :\|: B>> = <<C>>.	Proof. move=> sAC; rewrite -joingE -joing_idr => <- {B}; rewrite joing_idr. by congr <<_>>; rewrite setDE setUIr setUCr setIT; apply/setUidPr. Qed.	Lemma	genDU	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "joingE", "joing_idr", "setDE", "setIT", "setUCr", "setUIr", "setUidPr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joingA : associative joingT.	Proof. by move=> A B C; rewrite joing_idl joing_idr /joing setUA. Qed.	Lemma	joingA	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joing", "joingT", "joing_idl", "joing_idr", "setUA" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing1G G : 1 <*> G = G.	Proof. by rewrite -gen0 joing_idl /joing set0U genGid. Qed.	Lemma	joing1G	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gen0", "genGid", "joing", "joing_idl", "set0U" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joingG1 G : G <*> 1 = G.	Proof. by rewrite joingC joing1G. Qed.	Lemma	joingG1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joing1G", "joingC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
genM_join G H : <<G * H>> = G <*> H.	Proof. apply/eqP; rewrite eqEsubset gen_subG /= -{1}[G <*> H]mulGid. rewrite genS; first by rewrite subUset mulG_subl mulG_subr. by rewrite mulgSS ?(sub_gen, subsetUl, subsetUr). Qed.	Lemma	genM_join	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "genS", "gen_subG", "mulG_subl", "mulG_subr", "mulGid", "mulgSS", "subUset", "sub_gen", "subsetUl", "subsetUr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulG_subG G H K : (G * H \subset K) = (G \subset K) && (H \subset K).	Proof. by rewrite -gen_subG genM_join join_subG. Qed.	Lemma	mulG_subG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "genM_join", "gen_subG", "join_subG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulGsubP K H G : reflect (K \subset G /\ H \subset G) (K * H \subset G).	Proof. by rewrite mulG_subG; apply: andP. Qed.	Lemma	mulGsubP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "mulG_subG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulG_sub K H A : K * H = A -> K \subset A /\ H \subset A.	Proof. by move <-; rewrite mulG_subl mulG_subr. Qed.	Lemma	mulG_sub	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mulG_subl", "mulG_subr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivMg G H : (G * H == 1) = (G :==: 1) && (H :==: 1).	Proof. by rewrite !eqEsubset -{2}[1]mulGid mulgSS ?sub1G // !andbT mulG_subG. Qed.	Lemma	trivMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "eqEsubset", "mulG_subG", "mulGid", "mulgSS", "sub1G" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_joingE G H : commute G H -> G <> H = G H.	Proof. by move/comm_group_setP=> gGH; rewrite -genM_join; apply: (genGid (group gGH)). Qed.	Lemma	comm_joingE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "comm_group_setP", "commute", "genGid", "genM_join", "group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinGC : commutative joinGT.	Proof. by move=> G H; apply: val_inj; apply: joingC. Qed.	Lemma	joinGC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "joinGT", "joingC", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinGA : associative joinGT.	Proof. by move=> G H K; apply: val_inj; apply: joingA. Qed.	Lemma	joinGA	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "joinGT", "joingA", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join1G : left_id 1%G joinGT.	Proof. by move=> G; apply: val_inj; apply: joing1G. Qed.	Lemma	join1G	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "joinGT", "joing1G", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinG1 : right_id 1%G joinGT.	Proof. by move=> G; apply: val_inj; apply: joingG1. Qed.	Lemma	joinG1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "joinGT", "joingG1", "val_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigprodGEgen I r (P : pred I) (F : I -> {set gT}) : (\prod_(i <- r \| P i) <<F i>>)%G :=: << \bigcup_(i <- r \| P i) F i >>.	Proof. elim/big_rec2: _ => /= [\|i A _ _ ->]; first by rewrite gen0. by rewrite joing_idl joing_idr. Qed.	Lemma	bigprodGEgen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "big_rec2", "gT", "gen0", "joing_idl", "joing_idr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigprodGE I r (P : pred I) (F : I -> {group gT}) : (\prod_(i <- r \| P i) F i)%G :=: << \bigcup_(i <- r \| P i) F i >>.	Proof. rewrite -bigprodGEgen /=; apply: congr_group. by apply: eq_bigr => i _; rewrite genGidG. Qed.	Lemma	bigprodGE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "bigprodGEgen", "congr_group", "eq_bigr", "gT", "genGidG", "group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_commg A B x y : x \in A -> y \in B -> [~ x, y] \in [~: A, B].	Proof. by move=> Ax By; rewrite mem_gen ?imset2_f. Qed.	Lemma	mem_commg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2_f", "mem_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commSg A B C : A \subset B -> [~: A, C] \subset [~: B, C].	Proof. by move=> sAC; rewrite genS ?imset2S. Qed.	Lemma	commSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "genS", "imset2S" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgS A B C : B \subset C -> [~: A, B] \subset [~: A, C].	Proof. by move=> sBC; rewrite genS ?imset2S. Qed.	Lemma	commgS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "genS", "imset2S" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgSS A B C D : A \subset B -> C \subset D -> [~: A, C] \subset [~: B, D].	Proof. by move=> sAB sCD; rewrite genS ?imset2S. Qed.	Lemma	commgSS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "genS", "imset2S" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
der1_subG G : [~: G, G] \subset G.	Proof. by rewrite gen_subG; apply/subsetP=> _ /imset2P[x y Gx Gy ->]; apply: groupR. Qed.	Lemma	der1_subG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "gen_subG", "groupR", "imset2P", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_subG A B G : A \subset G -> B \subset G -> [~: A, B] \subset G.	Proof. by move=> sAG sBG; apply: subset_trans (der1_subG G); apply: commgSS. Qed.	Lemma	comm_subG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "commgSS", "der1_subG", "sAG", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commGC A B : [~: A, B] = [~: B, A].	Proof. rewrite -[[~: A, B]]genV; congr <<_>>; apply/setP=> z; rewrite inE. by apply/imset2P/imset2P=> [] [x y Ax Ay]; last rewrite -{1}(invgK z); rewrite -invg_comm => /invg_inj->; exists y x. Qed.	Lemma	commGC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "genV", "imset2P", "inE", "invgK", "invg_comm", "invg_inj", "last", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsRg A B x : [~: A, B] :^ x = [~: A :^ x, B :^ x].	Proof. wlog suffices: A B x / [~: A, B] :^ x \subset [~: A :^ x, B :^ x]. move=> subJ; apply/eqP; rewrite eqEsubset subJ /= -sub_conjgV. by rewrite -{2}(conjsgK x A) -{2}(conjsgK x B). rewrite -genJ gen_subG; apply/subsetP=> _ /imsetP[_ /imset2P[y z Ay Bz ->] ->]. by rewrite conjRg mem_commg ?memJ_conjg. Qed.	Lemma	conjsRg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjRg", "conjsgK", "eqEsubset", "genJ", "gen_subG", "imset2P", "imsetP", "memJ_conjg", "mem_commg", "sub_conjgV", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle1 : <[1]> = [1 gT].	Proof. exact: genGid. Qed.	Lemma	cycle1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gT", "genGid" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
order1 : #[1 : gT] = 1%N.	Proof. by rewrite /order cycle1 cards1. Qed.	Lemma	order1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cards1", "cycle1", "gT", "order" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_id x : x \in <[x]>.	Proof. by rewrite mem_gen // set11. Qed.	Lemma	cycle_id	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mem_gen", "set11" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_cycle x i : x ^+ i \in <[x]>.	Proof. by rewrite groupX // cycle_id. Qed.	Lemma	mem_cycle	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle_id", "groupX" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_subG x G : (<[x]> \subset G) = (x \in G).	Proof. by rewrite gen_subG sub1set. Qed.	Lemma	cycle_subG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gen_subG", "sub1set" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_eq1 x : (<[x]> == 1) = (x == 1).	Proof. by rewrite eqEsubset sub1G andbT cycle_subG inE. Qed.	Lemma	cycle_eq1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle_subG", "eqEsubset", "inE", "sub1G" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orderE x : #[x] = #\|<[x]>\|.	Proof. by []. Qed.	Lemma	orderE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
order_eq1 x : (#[x] == 1%N) = (x == 1).	Proof. by rewrite -trivg_card1 cycle_eq1. Qed.	Lemma	order_eq1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle_eq1", "trivg_card1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
order_gt1 x : (#[x] > 1) = (x != 1).	Proof. by rewrite ltnNge -trivg_card_le1 cycle_eq1. Qed.	Lemma	order_gt1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle_eq1", "ltnNge", "trivg_card_le1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_traject x : <[x]> =i traject (mul x) 1 #[x].	Proof. set t := _ 1; apply: fsym; apply/subset_cardP; last first. by apply/subsetP=> _ /trajectP[i _ ->]; rewrite -iteropE mem_cycle. rewrite (card_uniqP _) ?size_traject //; case def_n: #[_] => // [n]. rewrite looping_uniq; apply: contraL (card_size (t n)) => /loopingP t_xi. rewrite -ltnNge size_traject -def_n ?subs...	Lemma	cycle_traject	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "card_size", "card_uniqP", "def_n", "eq_subset_r", "expgD", "genS", "gen_set_id", "group_setP", "inE", "in_set", "iteropE", "last", "loopingP", "looping_uniq", "ltnNge", "mem_cycle", "mul", "mulg1", "size_traject", "split", "sub1set", "subsetP", "subset_cardP",...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle2g x : #[x] = 2 -> <[x]> = [set 1; x].	Proof. by move=> ox; apply/setP=> y; rewrite cycle_traject ox !inE mulg1. Qed.	Lemma	cycle2g	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cycle_traject", "inE", "mulg1", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cyclePmin x y : y \in <[x]> -> {i \| i < #[x] & y = x ^+ i}.	Proof. rewrite cycle_traject; set tx := traject _ _ #[x] => tx_y; pose i := index y tx. have lt_i_x : i < #[x] by rewrite -index_mem size_traject in tx_y. by exists i; rewrite // [x ^+ i]iteropE /= -(nth_traject _ lt_i_x) nth_index. Qed.	Lemma	cyclePmin	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle_traject", "index", "index_mem", "iteropE", "nth_index", "nth_traject", "size_traject", "traject" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycleP x y : reflect (exists i, y = x ^+ i) (y \in <[x]>).	Proof. by apply: (iffP idP) => [/cyclePmin[i _]\|[i ->]]; [exists i \| apply: mem_cycle]. Qed.	Lemma	cycleP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cyclePmin", "mem_cycle" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expg_order x : x ^+ #[x] = 1.	Proof. have: uniq (traject (mul x) 1 #[x]). by apply/card_uniqP; rewrite size_traject -(eq_card (cycle_traject x)). case/cyclePmin: (mem_cycle x #[x]) => [] [//\|i] ltix. rewrite -(subnKC ltix) addSnnS /= expgD; move: (_ - _) => j x_j1. case/andP=> /trajectP[]; exists j; first exact: leq_addl. by apply: (mulgI (x ^+ i...	Lemma	expg_order	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "addSnnS", "apply", "card_uniqP", "cyclePmin", "cycle_traject", "eq_card", "expgD", "expgS", "iterS", "iterSr", "iteropE", "leq_addl", "mem_cycle", "mul", "mulg1", "mulgI", "size_traject", "subnKC", "traject", "trajectP", "uniq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expg_mod p k x : x ^+ p = 1 -> x ^+ (k %% p) = x ^+ k.	Proof. move=> xp. by rewrite {2}(divn_eq k p) expgD mulnC expgM xp expg1n mul1g. Qed.	Lemma	expg_mod	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "divn_eq", "expg1n", "expgD", "expgM", "mul1g", "mulnC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expg_mod_order x i : x ^+ (i %% #[x]) = x ^+ i.	Proof. by rewrite expg_mod // expg_order. Qed.	Lemma	expg_mod_order	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "expg_mod", "expg_order" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_expg x : x^-1 = x ^+ #[x].-1.	Proof. by apply/eqP; rewrite eq_invg_mul -expgS prednK ?expg_order. Qed.	Lemma	invg_expg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eq_invg_mul", "expgS", "expg_order", "prednK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg2id x : #[x] = 2 -> x^-1 = x.	Proof. by move=> ox; rewrite invg_expg ox. Qed.	Lemma	invg2id	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "invg_expg" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycleX x i : <[x ^+ i]> \subset <[x]>.	Proof. by rewrite cycle_subG; apply: mem_cycle. Qed.	Lemma	cycleX	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cycle_subG", "mem_cycle" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycleV x : <[x^-1]> = <[x]>.	Proof. by apply/eqP; rewrite eq_sym eqEsubset !cycle_subG groupV -groupV !cycle_id. Qed.	Lemma	cycleV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cycle_id", "cycle_subG", "eqEsubset", "eq_sym", "groupV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orderV x : #[x^-1] = #[x].	Proof. by rewrite /order cycleV. Qed.	Lemma	orderV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycleV", "order" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycleJ x y : <[x ^ y]> = <[x]> :^ y.	Proof. by rewrite -genJ conjg_set1. Qed.	Lemma	cycleJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjg_set1", "genJ" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orderJ x y : #[x ^ y] = #[x].	Proof. by rewrite /order cycleJ cardJg. Qed.	Lemma	orderJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardJg", "cycleJ", "order" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normP x A : reflect (A :^ x = A) (x \in 'N(A)).	Proof. suffices ->: (x \in 'N(A)) = (A :^ x == A) by apply: eqP. by rewrite eqEcard cardJg leqnn andbT inE. Qed.	Lemma	normP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cardJg", "eqEcard", "inE", "leqnn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_normaliser A : group_set 'N(A).	Proof. apply/group_setP; split=> [\|x y Nx Ny]; rewrite inE ?conjsg1 //. by rewrite conjsgM !(normP _). Qed.	Lemma	group_set_normaliser	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjsg1", "conjsgM", "group_set", "group_setP", "inE", "normP", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normaliser_group A	:= group (group_set_normaliser A).	Canonical	normaliser_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group", "group_set_normaliser" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsP A B : reflect {in A, normalised B} (A \subset 'N(B)).	Proof. apply: (iffP subsetP) => nBA x Ax; last by rewrite inE nBA //. by apply/normP; apply: nBA. Qed.	Lemma	normsP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "inE", "last", "nBA", "normP", "normalised", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memJ_norm x y A : x \in 'N(A) -> (y ^ x \in A) = (y \in A).	Proof. by move=> Nx; rewrite -{1}(normP Nx) memJ_conjg. Qed.	Lemma	memJ_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "memJ_conjg", "normP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms_cycle x y : (<[y]> \subset 'N(<[x]>)) = (x ^ y \in <[x]>).	Proof. by rewrite cycle_subG inE -cycleJ cycle_subG. Qed.	Lemma	norms_cycle	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycleJ", "cycle_subG", "inE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm1 : 'N(1) = setT :> {set gT}.	Proof. by apply/setP=> x; rewrite !inE conjs1g subxx. Qed.	Lemma	norm1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjs1g", "gT", "inE", "setP", "setT", "subxx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norms1 A : A \subset 'N(1).	Proof. by rewrite norm1 subsetT. Qed.	Lemma	norms1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "norm1", "subsetT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normCs A : 'N(~: A) = 'N(A).	Proof. by apply/setP=> x; rewrite -groupV !inE conjCg setCS sub_conjg. Qed.	Lemma	normCs	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjCg", "groupV", "inE", "setCS", "setP", "sub_conjg" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normG G : G \subset 'N(G).	Proof. by apply/normsP; apply: conjGid. Qed.	Lemma	normG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjGid", "normsP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normT : 'N([set: gT]) = [set: gT].	Proof. by apply/eqP; rewrite -subTset normG. Qed.	Lemma	normT	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "gT", "normG", "subTset" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normsG A G : A \subset G -> A \subset 'N(G).	Proof. by move=> sAG; apply: subset_trans (normG G). Qed.	Lemma	normsG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "normG", "sAG", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normC A B : A \subset 'N(B) -> commute A B.	Proof. move/subsetP=> nBA; apply/setP=> u. apply/mulsgP/mulsgP=> [[x y Ax By] \| [y x By Ax]] -> {u}. by exists (y ^ x^-1) x; rewrite -?conjgCV // memJ_norm // groupV nBA. by exists x (y ^ x); rewrite -?conjgC // memJ_norm // nBA. Qed.	Lemma	normC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "commute", "conjgC", "conjgCV", "groupV", "memJ_norm", "mulsgP", "nBA", "setP", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_joinEl G H : G \subset 'N(H) -> G <> H = G H.	Proof. by move/normC/comm_joingE. Qed.	Lemma	norm_joinEl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "comm_joingE", "normC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_joinEr G H : H \subset 'N(G) -> G <> H = G H.	Proof. by move/normC=> cHG; apply: comm_joingE. Qed.	Lemma	norm_joinEr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "comm_joingE", "normC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_rlcoset G x : x \in 'N(G) -> G :* x = x *: G.	Proof. by rewrite -sub1set => /normC. Qed.	Lemma	norm_rlcoset	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normC", "sub1set" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_mul G x y : x \in 'N(G) -> (G :* x) * (G :* y) = G :* (x * y).	Proof. move/norm_rlcoset=> GxxG. by rewrite mulgA -(mulgA _ _ G) -GxxG mulgA mulGid -mulgA mulg_set1. Qed.	Lemma	rcoset_mul	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mulGid", "mulgA", "mulg_set1", "norm_rlcoset" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normJ A x : 'N(A :^ x) = 'N(A) :^ x.	Proof. by apply/setP=> y; rewrite mem_conjg !inE -conjsgM conjgCV conjsgM conjSg. Qed.	Lemma	normJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "conjSg", "conjgCV", "conjsgM", "inE", "mem_conjg", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_conj_norm x A B : x \in 'N(A) -> (A \subset 'N(B :^ x)) = (A \subset 'N(B)).	Proof. by move=> Nx; rewrite normJ -sub_conjgV (normP _) ?groupV. Qed.	Lemma	norm_conj_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "groupV", "normJ", "normP", "sub_conjgV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_gen A : 'N(A) \subset 'N(<<A>>).	Proof. by apply/normsP=> x Nx; rewrite -genJ (normP Nx). Qed.	Lemma	norm_gen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "genJ", "normP", "normsP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_norm x G : G \subset 'N(x ^: G).	Proof. by apply/normsP=> y; apply: classGidr. Qed.	Lemma	class_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "classGidr", "normsP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_normal x G : x \in G -> x ^: G <\| G.	Proof. by move=> Gx; rewrite /normal class_norm class_subG. Qed.	Lemma	class_normal	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "class_norm", "class_subG", "normal" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_sub_norm G A x : G \subset 'N(A) -> (x ^: G \subset A) = (x \in A).	Proof. move=> nAG; apply/subsetP/idP=> [-> // \| Ax xy]; first exact: class_refl. by case/imsetP=> y Gy ->; rewrite memJ_norm ?(subsetP nAG). Qed.	Lemma	class_sub_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "class_refl", "imsetP", "memJ_norm", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_support_norm A G : G \subset 'N(class_support A G).	Proof. by apply/normsP; apply: class_supportGidr. Qed.	Lemma	class_support_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "class_support", "class_supportGidr", "normsP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_support_sub_norm A B G : A \subset G -> B \subset 'N(G) -> class_support A B \subset G.	Proof. move=> sAG nGB; rewrite class_supportEr. by apply/bigcupsP=> x Bx; rewrite -(normsP nGB x Bx) conjSg. Qed.	Lemma	class_support_sub_norm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "bigcupsP", "class_support", "class_supportEr", "conjSg", "normsP", "sAG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen_subG A G : (<<A>> \subset G) = (A \subset G).

Proof. apply/idP/idP=> [|sAG]; first exact: subset_trans (subset_gen A). by apply/subsetP=> x /generatedP; apply. Qed.

Lemma

gen_subG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "generatedP", "sAG", "subsetP", "subset_gen", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genGid G : <<G>> = G.

Proof. by apply/eqP; rewrite eqEsubset gen_subG subset_gen andbT. Qed.

Lemma

genGid

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "gen_subG", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genGidG G : <<G>>%G = G.

Proof. by apply: val_inj; apply: genGid. Qed.

Lemma

genGidG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "genGid", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen_set_id A : group_set A -> <<A>> = A.

Proof. by move=> gA; apply: (genGid (group gA)). Qed.

Lemma

gen_set_id

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "genGid", "group", "group_set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genS A B : A \subset B -> <<A>> \subset <<B>>.

Proof. by move=> sAB; rewrite gen_subG sub_gen. Qed.

Lemma

genS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gen_subG", "sub_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen0 : <<set0>> = 1 :> {set gT}.

Proof. by apply/eqP; rewrite eqEsubset sub1G gen_subG sub0set. Qed.

Lemma

gen0

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "gT", "gen_subG", "set0", "sub0set", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen_expgs A : {n | <<A>> = (1 |: A) ^+ n}.

Proof. set B := (1 |: A); pose N := #|gT|. have BsubG n : B ^+ n \subset <<A>>. by elim: n => [|n IHn]; rewrite ?expgS ?mul_subG ?subUset ?sub1G ?subset_gen. have B_1 n : 1 \in B ^+ n. by elim: n => [|n IHn]; rewrite ?set11 // expgS mulUg mul1g inE IHn. case: (pickP (fun i : 'I_N => B ^+ i.+1 \subset B ^+ i)) => [n...

Lemma

gen_expgs

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cards1", "eqEsubset", "expgS", "expgSr", "gT", "genS", "gen_set_id", "group_set", "inE", "last", "leq_ltn_trans", "ltnNge", "ltnW", "max_card", "mul1g", "mulSg", "mulUg", "mul_subG", "mulg1", "mulgA", "mulg_subl", "nat", "pickP", "proper", "proper_card", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen_prodgP A x : reflect (exists n, exists2 c, forall i : 'I_n, c i \in A & x = \prod_i c i) (x \in <<A>>).

Proof. apply: (iffP idP) => [|[n [c Ac ->]]]; last first. by apply: group_prod => i _; rewrite mem_gen ?Ac. have [n ->] := gen_expgs A; rewrite /natexp Monoid.iteropE /=. rewrite -[n]card_ord -big_const => /prodsgP[/= c Ac def_x]. have{Ac def_x} ->: x = \prod_(i | c i \in A) c i. rewrite big_mkcond {x}def_x; apply...

Lemma

gen_prodgP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "big_const", "big_enumP", "big_mkcond", "big_tuple", "card_ord", "eq_bigr", "gen_expgs", "group_prod", "in_tuple", "iteropE", "last", "mem_gen", "mem_tnth", "natexp", "prodsgP", "setU1P", "size", "tnth", "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genD A B : A \subset <<A :\: B>> -> <<A :\: B>> = <<A>>.

Proof. by move=> sAB; apply/eqP; rewrite eqEsubset genS (subsetDl, gen_subG). Qed.

Lemma

genD

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "genS", "gen_subG", "subsetDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genV A : <<A^-1>> = <<A>>.

Proof. apply/eqP; rewrite eqEsubset !gen_subG -!(invSg _ <<_>>) invgK. by rewrite !invGid !subset_gen. Qed.

Lemma

genV

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "gen_subG", "invGid", "invSg", "invgK", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genJ A z : <<A :^z>> = <<A>> :^ z.

Proof. by apply/eqP; rewrite eqEsubset sub_conjg !gen_subG conjSg -?sub_conjg !sub_gen. Qed.

Lemma

genJ

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjSg", "eqEsubset", "gen_subG", "sub_conjg", "sub_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjYg A B z : (A <*> B) :^z = A :^ z <*> B :^ z.

Proof. by rewrite -genJ conjUg. Qed.

Lemma

conjYg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "conjUg", "genJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genD1 A x : x \in <<A :\ x>> -> <<A :\ x>> = <<A>>.

Proof. move=> gA'x; apply/eqP; rewrite eqEsubset genS; first by rewrite subsetDl. rewrite gen_subG; apply/subsetP=> y Ay. by case: (y =P x) => [-> //|]; move/eqP=> nyx; rewrite mem_gen // !inE nyx. Qed.

Lemma

genD1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "genS", "gen_subG", "inE", "mem_gen", "subsetDl", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genD1id A : <<A^#>> = <<A>>.

Proof. by rewrite genD1 ?group1. Qed.

Lemma

genD1id

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "genD1", "group1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joingT

:= (@joing gT) (only parsing).

Notation

joingT

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gT", "joing" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinGT

:= (@joinG gT) (only parsing).

Notation

joinGT

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gT", "joinG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joingE A B : A <*> B = <<A :|: B>>.

Proof. by []. Qed.

Lemma

joingE

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinGE G H : (G * H)%G = (G <*> H)%G.

Proof. by []. Qed.

Lemma

joinGE

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joingC : commutative joingT.

Proof. by move=> A B; rewrite /joing setUC. Qed.

Lemma

joingC

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joing", "joingT", "setUC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_idr A B : A <*> <<B>> = A <*> B.

Proof. apply/eqP; rewrite eqEsubset gen_subG subUset gen_subG /=. by rewrite -subUset subset_gen genS // setUS // subset_gen. Qed.

Lemma

joing_idr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "genS", "gen_subG", "setUS", "subUset", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_idl A B : <<A>> <*> B = A <*> B.

Proof. by rewrite -!(joingC B) joing_idr. Qed.

Lemma

joing_idl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joingC", "joing_idr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_subl A B : A \subset A <*> B.

Proof. by rewrite sub_gen ?subsetUl. Qed.

Lemma

joing_subl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "sub_gen", "subsetUl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_subr A B : B \subset A <*> B.

Proof. by rewrite sub_gen ?subsetUr. Qed.

Lemma

joing_subr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "sub_gen", "subsetUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join_subG A B G : (A <*> B \subset G) = (A \subset G) && (B \subset G).

Proof. by rewrite gen_subG subUset. Qed.

Lemma

join_subG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gen_subG", "subUset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_idPl G A : reflect (G <*> A = G) (A \subset G).

Proof. apply: (iffP idP) => [sHG | <-]; last by rewrite joing_subr. by rewrite joingE (setUidPl sHG) genGid. Qed.

Lemma

joing_idPl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "genGid", "joingE", "joing_subr", "last", "sHG", "setUidPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_idPr A G : reflect (A <*> G = G) (A \subset G).

Proof. by rewrite joingC; apply: joing_idPl. Qed.

Lemma

joing_idPr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "joingC", "joing_idPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_subP A B G : reflect (A \subset G /\ B \subset G) (A <*> B \subset G).

Proof. by rewrite join_subG; apply: andP. Qed.

Lemma

joing_subP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "join_subG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_sub A B C : A <*> B = C -> A \subset C /\ B \subset C.

Proof. by move <-; apply/joing_subP. Qed.

Lemma

joing_sub

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "joing_subP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genDU A B C : A \subset C -> <<C :\: A>> = <<B>> -> <<A :|: B>> = <<C>>.

Proof. move=> sAC; rewrite -joingE -joing_idr => <- {B}; rewrite joing_idr. by congr <<_>>; rewrite setDE setUIr setUCr setIT; apply/setUidPr. Qed.

Lemma

genDU

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "joingE", "joing_idr", "setDE", "setIT", "setUCr", "setUIr", "setUidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joingA : associative joingT.

Proof. by move=> A B C; rewrite joing_idl joing_idr /joing setUA. Qed.

Lemma

joingA

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joing", "joingT", "joing_idl", "joing_idr", "setUA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing1G G : 1 <*> G = G.

Proof. by rewrite -gen0 joing_idl /joing set0U genGid. Qed.

Lemma

joing1G

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gen0", "genGid", "joing", "joing_idl", "set0U" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joingG1 G : G <*> 1 = G.

Proof. by rewrite joingC joing1G. Qed.

Lemma

joingG1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joing1G", "joingC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

genM_join G H : <<G * H>> = G <*> H.

Proof. apply/eqP; rewrite eqEsubset gen_subG /= -{1}[G <*> H]mulGid. rewrite genS; first by rewrite subUset mulG_subl mulG_subr. by rewrite mulgSS ?(sub_gen, subsetUl, subsetUr). Qed.

Lemma

genM_join

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "genS", "gen_subG", "mulG_subl", "mulG_subr", "mulGid", "mulgSS", "subUset", "sub_gen", "subsetUl", "subsetUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulG_subG G H K : (G * H \subset K) = (G \subset K) && (H \subset K).

Proof. by rewrite -gen_subG genM_join join_subG. Qed.

Lemma

mulG_subG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "genM_join", "gen_subG", "join_subG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulGsubP K H G : reflect (K \subset G /\ H \subset G) (K * H \subset G).

Proof. by rewrite mulG_subG; apply: andP. Qed.

Lemma

mulGsubP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "mulG_subG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulG_sub K H A : K * H = A -> K \subset A /\ H \subset A.

Proof. by move <-; rewrite mulG_subl mulG_subr. Qed.

Lemma

mulG_sub

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "mulG_subl", "mulG_subr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivMg G H : (G * H == 1) = (G :==: 1) && (H :==: 1).

Proof. by rewrite !eqEsubset -{2}[1]mulGid mulgSS ?sub1G // !andbT mulG_subG. Qed.

Lemma

trivMg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "eqEsubset", "mulG_subG", "mulGid", "mulgSS", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_joingE G H : commute G H -> G <*> H = G * H.

Proof. by move/comm_group_setP=> gGH; rewrite -genM_join; apply: (genGid (group gGH)). Qed.

Lemma

comm_joingE

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "comm_group_setP", "commute", "genGid", "genM_join", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinGC : commutative joinGT.

Proof. by move=> G H; apply: val_inj; apply: joingC. Qed.

Lemma

joinGC

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "joinGT", "joingC", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinGA : associative joinGT.

Proof. by move=> G H K; apply: val_inj; apply: joingA. Qed.

Lemma

joinGA

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "joinGT", "joingA", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join1G : left_id 1%G joinGT.

Proof. by move=> G; apply: val_inj; apply: joing1G. Qed.

Lemma

join1G

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "joinGT", "joing1G", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinG1 : right_id 1%G joinGT.

Proof. by move=> G; apply: val_inj; apply: joingG1. Qed.

Lemma

joinG1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "joinGT", "joingG1", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigprodGEgen I r (P : pred I) (F : I -> {set gT}) : (\prod_(i <- r | P i) <<F i>>)%G :=: << \bigcup_(i <- r | P i) F i >>.

Proof. elim/big_rec2: _ => /= [|i A _ _ ->]; first by rewrite gen0. by rewrite joing_idl joing_idr. Qed.

Lemma

bigprodGEgen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "big_rec2", "gT", "gen0", "joing_idl", "joing_idr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigprodGE I r (P : pred I) (F : I -> {group gT}) : (\prod_(i <- r | P i) F i)%G :=: << \bigcup_(i <- r | P i) F i >>.

Proof. rewrite -bigprodGEgen /=; apply: congr_group. by apply: eq_bigr => i _; rewrite genGidG. Qed.

Lemma

bigprodGE

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "bigprodGEgen", "congr_group", "eq_bigr", "gT", "genGidG", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_commg A B x y : x \in A -> y \in B -> [~ x, y] \in [~: A, B].

Proof. by move=> Ax By; rewrite mem_gen ?imset2_f. Qed.

Lemma

mem_commg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "imset2_f", "mem_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commSg A B C : A \subset B -> [~: A, C] \subset [~: B, C].

Proof. by move=> sAC; rewrite genS ?imset2S. Qed.

Lemma

commSg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "genS", "imset2S" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commgS A B C : B \subset C -> [~: A, B] \subset [~: A, C].

Proof. by move=> sBC; rewrite genS ?imset2S. Qed.

Lemma

commgS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "genS", "imset2S" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commgSS A B C D : A \subset B -> C \subset D -> [~: A, C] \subset [~: B, D].

Proof. by move=> sAB sCD; rewrite genS ?imset2S. Qed.

Lemma

commgSS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "genS", "imset2S" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

der1_subG G : [~: G, G] \subset G.

Proof. by rewrite gen_subG; apply/subsetP=> _ /imset2P[x y Gx Gy ->]; apply: groupR. Qed.

Lemma

der1_subG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "gen_subG", "groupR", "imset2P", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_subG A B G : A \subset G -> B \subset G -> [~: A, B] \subset G.

Proof. by move=> sAG sBG; apply: subset_trans (der1_subG G); apply: commgSS. Qed.

Lemma

comm_subG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "commgSS", "der1_subG", "sAG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commGC A B : [~: A, B] = [~: B, A].

Proof. rewrite -[[~: A, B]]genV; congr <<_>>; apply/setP=> z; rewrite inE. by apply/imset2P/imset2P=> [] [x y Ax Ay]; last rewrite -{1}(invgK z); rewrite -invg_comm => /invg_inj->; exists y x. Qed.

Lemma

commGC

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "genV", "imset2P", "inE", "invgK", "invg_comm", "invg_inj", "last", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsRg A B x : [~: A, B] :^ x = [~: A :^ x, B :^ x].

Proof. wlog suffices: A B x / [~: A, B] :^ x \subset [~: A :^ x, B :^ x]. move=> subJ; apply/eqP; rewrite eqEsubset subJ /= -sub_conjgV. by rewrite -{2}(conjsgK x A) -{2}(conjsgK x B). rewrite -genJ gen_subG; apply/subsetP=> _ /imsetP[_ /imset2P[y z Ay Bz ->] ->]. by rewrite conjRg mem_commg ?memJ_conjg. Qed.

Lemma

conjsRg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjRg", "conjsgK", "eqEsubset", "genJ", "gen_subG", "imset2P", "imsetP", "memJ_conjg", "mem_commg", "sub_conjgV", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle1 : <[1]> = [1 gT].

Proof. exact: genGid. Qed.

Lemma

cycle1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gT", "genGid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

order1 : #[1 : gT] = 1%N.

Proof. by rewrite /order cycle1 cards1. Qed.

Lemma

order1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cards1", "cycle1", "gT", "order" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_id x : x \in <[x]>.

Proof. by rewrite mem_gen // set11. Qed.

Lemma

cycle_id

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "mem_gen", "set11" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_cycle x i : x ^+ i \in <[x]>.

Proof. by rewrite groupX // cycle_id. Qed.

Lemma

mem_cycle

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycle_id", "groupX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_subG x G : (<[x]> \subset G) = (x \in G).

Proof. by rewrite gen_subG sub1set. Qed.

Lemma

cycle_subG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gen_subG", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_eq1 x : (<[x]> == 1) = (x == 1).

Proof. by rewrite eqEsubset sub1G andbT cycle_subG inE. Qed.

Lemma

cycle_eq1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycle_subG", "eqEsubset", "inE", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orderE x : #[x] = #|<[x]>|.

Proof. by []. Qed.

Lemma

orderE

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

order_eq1 x : (#[x] == 1%N) = (x == 1).

Proof. by rewrite -trivg_card1 cycle_eq1. Qed.

Lemma

order_eq1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycle_eq1", "trivg_card1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

order_gt1 x : (#[x] > 1) = (x != 1).

Proof. by rewrite ltnNge -trivg_card_le1 cycle_eq1. Qed.

Lemma

order_gt1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycle_eq1", "ltnNge", "trivg_card_le1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_traject x : <[x]> =i traject (mul x) 1 #[x].

Proof. set t := _ 1; apply: fsym; apply/subset_cardP; last first. by apply/subsetP=> _ /trajectP[i _ ->]; rewrite -iteropE mem_cycle. rewrite (card_uniqP _) ?size_traject //; case def_n: #[_] => // [n]. rewrite looping_uniq; apply: contraL (card_size (t n)) => /loopingP t_xi. rewrite -ltnNge size_traject -def_n ?subs...

Lemma

cycle_traject

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "card_size", "card_uniqP", "def_n", "eq_subset_r", "expgD", "genS", "gen_set_id", "group_setP", "inE", "in_set", "iteropE", "last", "loopingP", "looping_uniq", "ltnNge", "mem_cycle", "mul", "mulg1", "size_traject", "split", "sub1set", "subsetP", "subset_cardP",...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle2g x : #[x] = 2 -> <[x]> = [set 1; x].

Proof. by move=> ox; apply/setP=> y; rewrite cycle_traject ox !inE mulg1. Qed.

Lemma

cycle2g

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cycle_traject", "inE", "mulg1", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cyclePmin x y : y \in <[x]> -> {i | i < #[x] & y = x ^+ i}.

Proof. rewrite cycle_traject; set tx := traject _ _ #[x] => tx_y; pose i := index y tx. have lt_i_x : i < #[x] by rewrite -index_mem size_traject in tx_y. by exists i; rewrite // [x ^+ i]iteropE /= -(nth_traject _ lt_i_x) nth_index. Qed.

Lemma

cyclePmin

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycle_traject", "index", "index_mem", "iteropE", "nth_index", "nth_traject", "size_traject", "traject" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycleP x y : reflect (exists i, y = x ^+ i) (y \in <[x]>).

Proof. by apply: (iffP idP) => [/cyclePmin[i _]|[i ->]]; [exists i | apply: mem_cycle]. Qed.

Lemma

cycleP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cyclePmin", "mem_cycle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expg_order x : x ^+ #[x] = 1.

Proof. have: uniq (traject (mul x) 1 #[x]). by apply/card_uniqP; rewrite size_traject -(eq_card (cycle_traject x)). case/cyclePmin: (mem_cycle x #[x]) => [] [//|i] ltix. rewrite -(subnKC ltix) addSnnS /= expgD; move: (_ - _) => j x_j1. case/andP=> /trajectP[]; exists j; first exact: leq_addl. by apply: (mulgI (x ^+ i...

Lemma

expg_order

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "addSnnS", "apply", "card_uniqP", "cyclePmin", "cycle_traject", "eq_card", "expgD", "expgS", "iterS", "iterSr", "iteropE", "leq_addl", "mem_cycle", "mul", "mulg1", "mulgI", "size_traject", "subnKC", "traject", "trajectP", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expg_mod p k x : x ^+ p = 1 -> x ^+ (k %% p) = x ^+ k.

Proof. move=> xp. by rewrite {2}(divn_eq k p) expgD mulnC expgM xp expg1n mul1g. Qed.

Lemma

expg_mod

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "divn_eq", "expg1n", "expgD", "expgM", "mul1g", "mulnC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expg_mod_order x i : x ^+ (i %% #[x]) = x ^+ i.

Proof. by rewrite expg_mod // expg_order. Qed.

Lemma

expg_mod_order

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "expg_mod", "expg_order" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg_expg x : x^-1 = x ^+ #[x].-1.

Proof. by apply/eqP; rewrite eq_invg_mul -expgS prednK ?expg_order. Qed.

Lemma

invg_expg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eq_invg_mul", "expgS", "expg_order", "prednK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg2id x : #[x] = 2 -> x^-1 = x.

Proof. by move=> ox; rewrite invg_expg ox. Qed.

Lemma

invg2id

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "invg_expg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycleX x i : <[x ^+ i]> \subset <[x]>.

Proof. by rewrite cycle_subG; apply: mem_cycle. Qed.

Lemma

cycleX

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cycle_subG", "mem_cycle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycleV x : <[x^-1]> = <[x]>.

Proof. by apply/eqP; rewrite eq_sym eqEsubset !cycle_subG groupV -groupV !cycle_id. Qed.

Lemma

cycleV

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cycle_id", "cycle_subG", "eqEsubset", "eq_sym", "groupV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orderV x : #[x^-1] = #[x].

Proof. by rewrite /order cycleV. Qed.

Lemma

orderV

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycleV", "order" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycleJ x y : <[x ^ y]> = <[x]> :^ y.

Proof. by rewrite -genJ conjg_set1. Qed.

Lemma

cycleJ

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "conjg_set1", "genJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orderJ x y : #[x ^ y] = #[x].

Proof. by rewrite /order cycleJ cardJg. Qed.

Lemma

orderJ

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardJg", "cycleJ", "order" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normP x A : reflect (A :^ x = A) (x \in 'N(A)).

Proof. suffices ->: (x \in 'N(A)) = (A :^ x == A) by apply: eqP. by rewrite eqEcard cardJg leqnn andbT inE. Qed.

Lemma

normP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cardJg", "eqEcard", "inE", "leqnn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_normaliser A : group_set 'N(A).

Proof. apply/group_setP; split=> [|x y Nx Ny]; rewrite inE ?conjsg1 //. by rewrite conjsgM !(normP _). Qed.

Lemma

group_set_normaliser

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjsg1", "conjsgM", "group_set", "group_setP", "inE", "normP", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normaliser_group A

:= group (group_set_normaliser A).

Canonical

normaliser_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group", "group_set_normaliser" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsP A B : reflect {in A, normalised B} (A \subset 'N(B)).

Proof. apply: (iffP subsetP) => nBA x Ax; last by rewrite inE nBA //. by apply/normP; apply: nBA. Qed.

Lemma

normsP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "inE", "last", "nBA", "normP", "normalised", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memJ_norm x y A : x \in 'N(A) -> (y ^ x \in A) = (y \in A).

Proof. by move=> Nx; rewrite -{1}(normP Nx) memJ_conjg. Qed.

Lemma

memJ_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "memJ_conjg", "normP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms_cycle x y : (<[y]> \subset 'N(<[x]>)) = (x ^ y \in <[x]>).

Proof. by rewrite cycle_subG inE -cycleJ cycle_subG. Qed.

Lemma

norms_cycle

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycleJ", "cycle_subG", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm1 : 'N(1) = setT :> {set gT}.

Proof. by apply/setP=> x; rewrite !inE conjs1g subxx. Qed.

Lemma

norm1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjs1g", "gT", "inE", "setP", "setT", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norms1 A : A \subset 'N(1).

Proof. by rewrite norm1 subsetT. Qed.

Lemma

norms1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "norm1", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normCs A : 'N(~: A) = 'N(A).

Proof. by apply/setP=> x; rewrite -groupV !inE conjCg setCS sub_conjg. Qed.

Lemma

normCs

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjCg", "groupV", "inE", "setCS", "setP", "sub_conjg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normG G : G \subset 'N(G).

Proof. by apply/normsP; apply: conjGid. Qed.

Lemma

normG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjGid", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normT : 'N([set: gT]) = [set: gT].

Proof. by apply/eqP; rewrite -subTset normG. Qed.

Lemma

normT

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "gT", "normG", "subTset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normsG A G : A \subset G -> A \subset 'N(G).

Proof. by move=> sAG; apply: subset_trans (normG G). Qed.

Lemma

normsG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "normG", "sAG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normC A B : A \subset 'N(B) -> commute A B.

Proof. move/subsetP=> nBA; apply/setP=> u. apply/mulsgP/mulsgP=> [[x y Ax By] | [y x By Ax]] -> {u}. by exists (y ^ x^-1) x; rewrite -?conjgCV // memJ_norm // groupV nBA. by exists x (y ^ x); rewrite -?conjgC // memJ_norm // nBA. Qed.

Lemma

normC

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "commute", "conjgC", "conjgCV", "groupV", "memJ_norm", "mulsgP", "nBA", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_joinEl G H : G \subset 'N(H) -> G <*> H = G * H.

Proof. by move/normC/comm_joingE. Qed.

Lemma

norm_joinEl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "comm_joingE", "normC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_joinEr G H : H \subset 'N(G) -> G <*> H = G * H.

Proof. by move/normC=> cHG; apply: comm_joingE. Qed.

Lemma

norm_joinEr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "comm_joingE", "normC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_rlcoset G x : x \in 'N(G) -> G :* x = x *: G.

Proof. by rewrite -sub1set => /normC. Qed.

Lemma

norm_rlcoset

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "normC", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_mul G x y : x \in 'N(G) -> (G :* x) * (G :* y) = G :* (x * y).

Proof. move/norm_rlcoset=> GxxG. by rewrite mulgA -(mulgA _ _ G) -GxxG mulgA mulGid -mulgA mulg_set1. Qed.

Lemma

rcoset_mul

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "mulGid", "mulgA", "mulg_set1", "norm_rlcoset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normJ A x : 'N(A :^ x) = 'N(A) :^ x.

Proof. by apply/setP=> y; rewrite mem_conjg !inE -conjsgM conjgCV conjsgM conjSg. Qed.

Lemma

normJ

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "conjSg", "conjgCV", "conjsgM", "inE", "mem_conjg", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_conj_norm x A B : x \in 'N(A) -> (A \subset 'N(B :^ x)) = (A \subset 'N(B)).

Proof. by move=> Nx; rewrite normJ -sub_conjgV (normP _) ?groupV. Qed.

Lemma

norm_conj_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "groupV", "normJ", "normP", "sub_conjgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_gen A : 'N(A) \subset 'N(<<A>>).

Proof. by apply/normsP=> x Nx; rewrite -genJ (normP Nx). Qed.

Lemma

norm_gen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "genJ", "normP", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_norm x G : G \subset 'N(x ^: G).

Proof. by apply/normsP=> y; apply: classGidr. Qed.

Lemma

class_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "classGidr", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_normal x G : x \in G -> x ^: G <| G.

Proof. by move=> Gx; rewrite /normal class_norm class_subG. Qed.

Lemma

class_normal

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "class_norm", "class_subG", "normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_sub_norm G A x : G \subset 'N(A) -> (x ^: G \subset A) = (x \in A).

Proof. move=> nAG; apply/subsetP/idP=> [-> // | Ax xy]; first exact: class_refl. by case/imsetP=> y Gy ->; rewrite memJ_norm ?(subsetP nAG). Qed.

Lemma

class_sub_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "class_refl", "imsetP", "memJ_norm", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_support_norm A G : G \subset 'N(class_support A G).

Proof. by apply/normsP; apply: class_supportGidr. Qed.

Lemma

class_support_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "class_support", "class_supportGidr", "normsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_support_sub_norm A B G : A \subset G -> B \subset 'N(G) -> class_support A B \subset G.

Proof. move=> sAG nGB; rewrite class_supportEr. by apply/bigcupsP=> x Bx; rewrite -(normsP nGB x Bx) conjSg. Qed.

Lemma

class_support_sub_norm

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "bigcupsP", "class_support", "class_supportEr", "conjSg", "normsP", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d