phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
group1_contra x : x \notin G -> x != 1.	Proof. by apply: contraNneq => ->. Qed.	Lemma	group1_contra	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", "contraNneq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub1G : [1 gT] \subset G.	Proof. by rewrite sub1set. Qed.	Lemma	sub1G	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", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subG1 : (G \subset [1]) = (G :==: 1).	Proof. by rewrite eqEsubset sub1G andbT. Qed.	Lemma	subG1	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", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setI1g : 1 :&: G = 1.	Proof. exact: (setIidPl sub1G). Qed.	Lemma	setI1g	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" ]	[ "setIidPl", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setIg1 : G :&: 1 = 1.	Proof. exact: (setIidPr sub1G). Qed.	Lemma	setIg1	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" ]	[ "setIidPr", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subG1_contra H : G \subset H -> G :!=: 1 -> H :!=: 1.	Proof. by move=> sGH; rewrite -subG1; apply: contraNneq => <-. Qed.	Lemma	subG1_contra	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", "contraNneq", "sGH", "subG1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_group : repr G = 1.	Proof. by rewrite /repr group1. Qed.	Lemma	repr_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" ]	[ "group1", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardG_gt0 : 0 < #\|G\|.	Proof. by rewrite lt0n; apply/existsP; exists (1 : gT). Qed.	Lemma	cardG_gt0	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", "existsP", "gT", "lt0n" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexg_gt0 A : 0 < #\|G : A\|.	Proof. rewrite lt0n; apply/existsP; exists A. by rewrite -{2}[A]mulg1 -rcosetE; apply: imset_f. Qed.	Lemma	indexg_gt0	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", "existsP", "imset_f", "lt0n", "mulg1", "rcosetE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivgP : reflect (G :=: 1) (G \subset [1]).	Proof. by rewrite subG1; apply: eqP. Qed.	Lemma	trivgP	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", "subG1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivGP : reflect (G = 1%G) (G \subset [1]).	Proof. by rewrite subG1; apply: eqP. Qed.	Lemma	trivGP	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", "subG1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper1G : ([1] \proper G) = (G :!=: 1).	Proof. by rewrite properEneq sub1G andbT eq_sym. Qed.	Lemma	proper1G	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" ]	[ "eq_sym", "proper", "properEneq", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_one_group x : (x \in 1%G) = (x == 1).	Proof. by rewrite -[x \in _]/(x \in [set 1]) !inE. Qed.	Lemma	in_one_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" ]	[ "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inE	:= (in_one_group, inE).	Definition	inE	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" ]	[ "in_one_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivgPn : reflect (exists2 x, x \in G & x != 1) (G :!=: 1).	Proof. rewrite -subG1. by apply: (iffP subsetPn) => [] [x Gx x1]; exists x; rewrite ?inE in x1 *. Qed.	Lemma	trivgPn	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", "subG1", "subsetPn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_card_le1 : (G :==: 1) = (#\|G\| <= 1).	Proof. by rewrite eq_sym eqEcard cards1 sub1G. Qed.	Lemma	trivg_card_le1	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", "eqEcard", "eq_sym", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_card1 : (G :==: 1) = (#\|G\| == 1%N).	Proof. by rewrite trivg_card_le1 eqn_leq cardG_gt0 andbT. Qed.	Lemma	trivg_card1	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" ]	[ "cardG_gt0", "eqn_leq", "trivg_card_le1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardG_gt1 : (#\|G\| > 1) = (G :!=: 1).	Proof. by rewrite trivg_card_le1 ltnNge. Qed.	Lemma	cardG_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" ]	[ "ltnNge", "trivg_card_le1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_le1_trivg : #\|G\| <= 1 -> G :=: 1.	Proof. by rewrite -trivg_card_le1; move/eqP. Qed.	Lemma	card_le1_trivg	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" ]	[ "trivg_card_le1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card1_trivg : #\|G\| = 1%N -> G :=: 1.	Proof. by move=> G1; rewrite card_le1_trivg ?G1. Qed.	Lemma	card1_trivg	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" ]	[ "G1", "card_le1_trivg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulG_subl A : A \subset A * G.	Proof. exact: mulg_subl group1. Qed.	Lemma	mulG_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" ]	[ "group1", "mulg_subl" ]	Inclusion and product.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulG_subr A : A \subset (G * A).	Proof. exact: mulg_subr group1. Qed.	Lemma	mulG_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" ]	[ "group1", "mulg_subr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulGid : G * G = G.	Proof. by apply/eqP; rewrite eqEsubset mulG_subr andbT; case/andP: (valP G). Qed.	Lemma	mulGid	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", "mulG_subr", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulGS A B : (G * A \subset G * B) = (A \subset G * B).	Proof. apply/idP/idP; first exact: subset_trans (mulG_subr A). by move/(mulgS G); rewrite mulgA mulGid. Qed.	Lemma	mulGS	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_subr", "mulGid", "mulgA", "mulgS", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulSG A B : (A * G \subset B * G) = (A \subset B * G).	Proof. apply/idP/idP; first exact: subset_trans (mulG_subl A). by move/(mulSg G); rewrite -mulgA mulGid. Qed.	Lemma	mulSG	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_subl", "mulGid", "mulSg", "mulgA", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_subG A B : A \subset G -> B \subset G -> A * B \subset G.	Proof. by move=> sAG sBG; rewrite -mulGid mulgSS. Qed.	Lemma	mul_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" ]	[ "mulGid", "mulgSS", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_subG (I : Type) (r : seq I) (P : {pred I}) (F : I -> {set gT}) : (forall i, P i -> F i \subset G) -> \prod_(i <- r \| P i) F i \subset G.	Proof. move=> subFG; elim/big_rec: _ => [\|/= i A /subFG]; first by rewrite sub1set. exact: mul_subG. Qed.	Lemma	prod_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" ]	[ "big_rec", "gT", "mul_subG", "seq", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupM x y : x \in G -> y \in G -> x * y \in G.	Proof. by case/group_setP: (valP G) x y. Qed.	Lemma	groupM	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_setP", "valP" ]	Membership lemmas	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupX x n : x \in G -> x ^+ n \in G.	Proof. by move=> Gx; elim: n => [\|n IHn]; rewrite ?group1 // expgS groupM. Qed.	Lemma	groupX	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" ]	[ "expgS", "group1", "groupM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupVr x : x \in G -> x^-1 \in G.	Proof. move=> Gx; rewrite -(mul1g x^-1) -mem_rcoset ((G :* x =P G) _) //. by rewrite eqEcard card_rcoset leqnn mul_subG ?sub1set. Qed.	Lemma	groupVr	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" ]	[ "card_rcoset", "eqEcard", "leqnn", "mem_rcoset", "mul1g", "mul_subG", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupVl x : x^-1 \in G -> x \in G.	Proof. by move/groupVr; rewrite invgK. Qed.	Lemma	groupVl	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" ]	[ "groupVr", "invgK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupV x : (x^-1 \in G) = (x \in G).	Proof. by apply/idP/idP; [apply: groupVl \| apply: groupVr]. Qed.	Lemma	groupV	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", "groupVl", "groupVr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupMl x y : x \in G -> (x * y \in G) = (y \in G).	Proof. move=> Gx; apply/idP/idP=> [Gxy\|]; last exact: groupM. by rewrite -(mulKg x y) groupM ?groupVr. Qed.	Lemma	groupMl	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", "groupM", "groupVr", "last", "mulKg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupMr x y : x \in G -> (y * x \in G) = (y \in G).	Proof. by move=> Gx; rewrite -[_ \in G]groupV invMg groupMl groupV. Qed.	Lemma	groupMr	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" ]	[ "groupMl", "groupV", "invMg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_group	:= (group1, groupV, (groupMl, groupX)).	Definition	in_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" ]	[ "group1", "groupMl", "groupV", "groupX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupJ x y : x \in G -> y \in G -> x ^ y \in G.	Proof. by move=> Gx Gy; rewrite !in_group. Qed.	Lemma	groupJ	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" ]	[ "in_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupJr x y : y \in G -> (x ^ y \in G) = (x \in G).	Proof. by move=> Gy; rewrite groupMl (groupMr, groupV). Qed.	Lemma	groupJr	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" ]	[ "groupMl", "groupMr", "groupV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupR x y : x \in G -> y \in G -> [~ x, y] \in G.	Proof. by move=> Gx Gy; rewrite !in_group. Qed.	Lemma	groupR	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" ]	[ "in_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_prod I r (P : pred I) F : (forall i, P i -> F i \in G) -> \prod_(i <- r \| P i) F i \in G.	Proof. by move=> G_P; elim/big_ind: _ => //; apply: groupM. Qed.	Lemma	group_prod	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_ind", "groupM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invGid : G^-1 = G.	Proof. by apply/setP=> x; rewrite inE groupV. Qed.	Lemma	invGid	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", "groupV", "inE", "setP" ]	Inverse is an anti-morphism.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv_subG A : (A^-1 \subset G) = (A \subset G).	Proof. by rewrite -{1}invGid invSg. Qed.	Lemma	inv_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" ]	[ "invGid", "invSg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_lcoset x : (x : G)^-1 = G : x^-1.	Proof. by rewrite invMg invGid invg_set1. Qed.	Lemma	invg_lcoset	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" ]	[ "invGid", "invMg", "invg_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_rcoset x : (G :* x)^-1 = x^-1 *: G.	Proof. by rewrite invMg invGid invg_set1. Qed.	Lemma	invg_rcoset	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" ]	[ "invGid", "invMg", "invg_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memV_lcosetV x y : (y^-1 \in x^-1 : G) = (y \in G : x).	Proof. by rewrite -invg_rcoset memV_invg. Qed.	Lemma	memV_lcosetV	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_rcoset", "memV_invg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memV_rcosetV x y : (y^-1 \in G :* x^-1) = (y \in x *: G).	Proof. by rewrite -invg_lcoset memV_invg. Qed.	Lemma	memV_rcosetV	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_lcoset", "memV_invg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulSgGid A x : x \in A -> A \subset G -> A * G = G.	Proof. move=> Ax sAG; apply/eqP; rewrite eqEsubset -{2}mulGid mulSg //=. apply/subsetP=> y Gy; rewrite -(mulKVg x y) mem_mulg // groupMr // groupV. exact: (subsetP sAG). Qed.	Lemma	mulSgGid	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", "groupMr", "groupV", "mem_mulg", "mulGid", "mulKVg", "mulSg", "sAG", "subsetP" ]	Product idempotence	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulGSgid A x : x \in A -> A \subset G -> G * A = G.	Proof. rewrite -memV_invg -invSg invGid => Ax sAG. by apply: invg_inj; rewrite invMg invGid (mulSgGid Ax). Qed.	Lemma	mulGSgid	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", "invGid", "invMg", "invSg", "invg_inj", "memV_invg", "mulSgGid", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcoset_refl x : x \in x *: G.	Proof. by rewrite mem_lcoset mulVg group1. Qed.	Lemma	lcoset_refl	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" ]	[ "group1", "mem_lcoset", "mulVg" ]	Left cosets	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcoset_sym x y : (x \in y : G) = (y \in x : G).	Proof. by rewrite !mem_lcoset -groupV invMg invgK. Qed.	Lemma	lcoset_sym	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", "invMg", "invgK", "mem_lcoset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcoset_eqP {x y} : reflect (x : G = y : G) (x \in y *: G).	Proof. suffices <-: (x : G == y : G) = (x \in y *: G) by apply: eqP. by rewrite eqEsubset !mulSG !sub1set lcoset_sym andbb. Qed.	Lemma	lcoset_eqP	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", "lcoset_sym", "mulSG", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcoset_transl x y z : x \in y : G -> (x \in z : G) = (y \in z *: G).	Proof. by move=> Gyx; rewrite -2!(lcoset_sym z) (lcoset_eqP Gyx). Qed.	Lemma	lcoset_transl	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" ]	[ "lcoset_eqP", "lcoset_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcoset_trans x y z : x \in y : G -> y \in z : G -> x \in z *: G.	Proof. by move/lcoset_transl->. Qed.	Lemma	lcoset_trans	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" ]	[ "lcoset_transl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcoset_id x : x \in G -> x *: G = G.	Proof. by move=> Gx; rewrite (lcoset_eqP (_ : x \in 1 *: G)) mul1g. Qed.	Lemma	lcoset_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" ]	[ "lcoset_eqP", "mul1g" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_refl x : x \in G :* x.	Proof. by rewrite mem_rcoset mulgV group1. Qed.	Lemma	rcoset_refl	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" ]	[ "group1", "mem_rcoset", "mulgV" ]	Right cosets, with an elimination form for repr.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_sym x y : (x \in G :* y) = (y \in G :* x).	Proof. by rewrite -!memV_lcosetV lcoset_sym. Qed.	Lemma	rcoset_sym	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" ]	[ "lcoset_sym", "memV_lcosetV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_eqP {x y} : reflect (G :* x = G :* y) (x \in G :* y).	Proof. suffices <-: (G :* x == G :* y) = (x \in G :* y) by apply: eqP. by rewrite eqEsubset !mulGS !sub1set rcoset_sym andbb. Qed.	Lemma	rcoset_eqP	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", "mulGS", "rcoset_sym", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_transl x y z : x \in G :* y -> (x \in G :* z) = (y \in G :* z).	Proof. by move=> Gyx; rewrite -2!(rcoset_sym z) (rcoset_eqP Gyx). Qed.	Lemma	rcoset_transl	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" ]	[ "rcoset_eqP", "rcoset_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_trans x y z : x \in G :* y -> y \in G :* z -> x \in G :* z.	Proof. by move/rcoset_transl->. Qed.	Lemma	rcoset_trans	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" ]	[ "rcoset_transl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_id x : x \in G -> G :* x = G.	Proof. by move=> Gx; rewrite (rcoset_eqP (_ : x \in G :* 1)) mulg1. Qed.	Lemma	rcoset_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" ]	[ "mulg1", "rcoset_eqP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_repr_spec x : gT -> Type	:= RcosetReprSpec g : g \in G -> rcoset_repr_spec x (g * x).	Variant	rcoset_repr_spec	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" ]	Elimination form.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_repr_rcoset x : repr (G :* x) \in G :* x.	Proof. exact: mem_repr (rcoset_refl x). Qed.	Lemma	mem_repr_rcoset	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_repr", "rcoset_refl", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_rcosetP x : rcoset_repr_spec x (repr (G :* x)).	Proof. by rewrite -[repr _](mulgKV x); split; rewrite -mem_rcoset mem_repr_rcoset. Qed.	Lemma	repr_rcosetP	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_rcoset", "mem_repr_rcoset", "mulgKV", "rcoset_repr_spec", "repr", "split" ]	(weaker) primitive Coq algorithm for general (co)inductive type families.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_repr x : G :* (repr (G :* x)) = G :* x.	Proof. exact/rcoset_eqP/mem_repr_rcoset. Qed.	Lemma	rcoset_repr	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_repr_rcoset", "rcoset_eqP", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_rcosets A x : (G :* x \in rcosets G A) = (x \in G * A).	Proof. apply/rcosetsP/mulsgP=> [[a Aa /rcoset_eqP/rcosetP[g]] \| ]; first by exists g a. by case=> g a Gg Aa ->{x}; exists a; rewrite // rcosetM rcoset_id. Qed.	Lemma	mem_rcosets	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" ]	[ "Gg", "apply", "mulsgP", "rcosetM", "rcosetP", "rcoset_eqP", "rcoset_id", "rcosets", "rcosetsP" ]	Coset spaces.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_lcosets A x : (x : G \in lcosets G A) = (x \in A G).	Proof. rewrite -[LHS]memV_invg invg_lcoset invg_lcosets. by rewrite -[RHS]memV_invg invMg invGid mem_rcosets. Qed.	Lemma	mem_lcosets	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" ]	[ "invGid", "invMg", "invg_lcoset", "invg_lcosets", "lcosets", "memV_invg", "mem_rcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_setJ A x : group_set (A :^ x) = group_set A.	Proof. by rewrite /group_set mem_conjg conj1g -conjsMg conjSg. Qed.	Lemma	group_setJ	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" ]	[ "conj1g", "conjSg", "conjsMg", "group_set", "mem_conjg" ]	Conjugates.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_conjG x : group_set (G :^ x).	Proof. by rewrite group_setJ groupP. Qed.	Lemma	group_set_conjG	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" ]	[ "groupP", "group_set", "group_setJ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjG_group x	:= group (group_set_conjG x).	Canonical	conjG_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_conjG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjGid : {in G, normalised G}.	Proof. by move=> x Gx; apply/setP=> y; rewrite mem_conjg groupJr ?groupV. Qed.	Lemma	conjGid	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", "groupJr", "groupV", "mem_conjg", "normalised", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj_subG x A : x \in G -> A \subset G -> A :^ x \subset G.	Proof. by move=> Gx sAG; rewrite -(conjGid Gx) conjSg. Qed.	Lemma	conj_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" ]	[ "conjGid", "conjSg", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class1G : 1 ^: G = 1.	Proof. exact: class1g group1. Qed.	Lemma	class1G	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" ]	[ "class1g", "group1" ]	Classes	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classes1 : [1] \in classes G.	Proof. by rewrite -class1G mem_classes. Qed.	Lemma	classes1	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" ]	[ "class1G", "classes", "mem_classes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classGidl x y : y \in G -> (x ^ y) ^: G = x ^: G.	Proof. by move=> Gy; rewrite -class_lcoset lcoset_id. Qed.	Lemma	classGidl	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_lcoset", "lcoset_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classGidr x : {in G, normalised (x ^: G)}.	Proof. by move=> y Gy /=; rewrite -class_rcoset rcoset_id. Qed.	Lemma	classGidr	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_rcoset", "normalised", "rcoset_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_refl x : x \in x ^: G.	Proof. by apply/imsetP; exists 1; rewrite ?conjg1. Qed.	Lemma	class_refl	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", "conjg1", "imsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_eqP x y : reflect (x ^: G = y ^: G) (x \in y ^: G).	Proof. by apply: (iffP idP) => [/imsetP[z Gz ->] \| <-]; rewrite ?class_refl ?classGidl. Qed.	Lemma	class_eqP	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", "classGidl", "class_refl", "imsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_sym x y : (x \in y ^: G) = (y \in x ^: G).	Proof. by apply/idP/idP=> /class_eqP->. Qed.	Lemma	class_sym	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_eqP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_transl x y z : x \in y ^: G -> (x \in z ^: G) = (y \in z ^: G).	Proof. by rewrite -!(class_sym z) => /class_eqP->. Qed.	Lemma	class_transl	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_eqP", "class_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_trans x y z : x \in y ^: G -> y \in z ^: G -> x \in z ^: G.	Proof. by move/class_transl->. Qed.	Lemma	class_trans	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_transl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_class x : {y \| y \in G & repr (x ^: G) = x ^ y}.	Proof. set z := repr _; have: #\|[set y in G \| z == x ^ y]\| > 0. have: z \in x ^: G by apply: (mem_repr x). by case/imsetP=> y Gy ->; rewrite (cardD1 y) inE Gy eqxx. by move/card_mem_repr; move: (repr _) => y /setIdP[Gy /eqP]; exists y. Qed.	Lemma	repr_class	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", "cardD1", "card_mem_repr", "eqxx", "imsetP", "inE", "mem_repr", "repr", "setIdP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classG_eq1 x : (x ^: G == 1) = (x == 1).	Proof. apply/eqP/eqP=> [xG1 \| ->]; last exact: class1G. by have:= class_refl x; rewrite xG1 => /set1P. Qed.	Lemma	classG_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" ]	[ "apply", "class1G", "class_refl", "last", "set1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_subG x A : x \in G -> A \subset G -> x ^: A \subset G.	Proof. move=> Gx sAG; apply/subsetP=> _ /imsetP[y Ay ->]. by rewrite groupJ // (subsetP sAG). Qed.	Lemma	class_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", "groupJ", "imsetP", "sAG", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_classesP xG : reflect (repr xG \in G /\ xG = repr xG ^: G) (xG \in classes G).	Proof. apply: (iffP imsetP) => [[x Gx ->] \| []]; last by exists (repr xG). by have [y Gy ->] := repr_class x; rewrite classGidl ?groupJ. Qed.	Lemma	repr_classesP	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", "classGidl", "classes", "groupJ", "imsetP", "last", "repr", "repr_class" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_repr_classes xG : xG \in classes G -> repr xG \in xG.	Proof. by case/repr_classesP=> _ {2}->; apply: class_refl. Qed.	Lemma	mem_repr_classes	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", "classes", "repr", "repr_classesP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classes_gt0 : 0 < #\|classes G\|.	Proof. by rewrite (cardsD1 1) classes1. Qed.	Lemma	classes_gt0	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" ]	[ "cardsD1", "classes", "classes1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classes_gt1 : (#\|classes G\| > 1) = (G :!=: 1).	Proof. rewrite (cardsD1 1) classes1 ltnS lt0n cards_eq0. apply/set0Pn/trivgPn=> [[xG /setD1P[nt_xG]] \| [x Gx ntx]]. by case/imsetP=> x Gx def_xG; rewrite def_xG classG_eq1 in nt_xG; exists x. by exists (x ^: G); rewrite !inE classG_eq1 ntx; apply: imset_f. Qed.	Lemma	classes_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" ]	[ "apply", "cardsD1", "cards_eq0", "classG_eq1", "classes", "classes1", "imsetP", "imset_f", "inE", "lt0n", "ltnS", "set0Pn", "setD1P", "trivgPn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_class_support A x : x \in A -> x \in class_support A G.	Proof. by move=> Ax; rewrite -[x]conjg1 memJ_class_support. Qed.	Lemma	mem_class_support	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_support", "conjg1", "memJ_class_support" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_supportGidl A x : x \in G -> class_support (A :^ x) G = class_support A G.	Proof. by move=> Gx; rewrite -class_support_set1r -class_supportM lcoset_id. Qed.	Lemma	class_supportGidl	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_support", "class_supportM", "class_support_set1r", "lcoset_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_supportGidr A : {in G, normalised (class_support A G)}.	Proof. by move=> x Gx /=; rewrite -class_support_set1r -class_supportM rcoset_id. Qed.	Lemma	class_supportGidr	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_support", "class_supportM", "class_support_set1r", "normalised", "rcoset_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_support_subG A : A \subset G -> class_support A G \subset G.	Proof. by move=> sAG; rewrite class_supportEr; apply/bigcupsP=> x Gx; apply: conj_subG. Qed.	Lemma	class_support_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", "bigcupsP", "class_support", "class_supportEr", "conj_subG", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_class_support A : A \subset class_support A G.	Proof. by rewrite class_supportEr (bigcup_max 1) ?conjsg1. Qed.	Lemma	sub_class_support	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" ]	[ "bigcup_max", "class_support", "class_supportEr", "conjsg1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_support_id : class_support G G = G.	Proof. by apply/eqP; rewrite eqEsubset sub_class_support class_support_subG. Qed.	Lemma	class_support_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", "class_support", "class_support_subG", "eqEsubset", "sub_class_support" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_supportD1 A : (class_support A G)^# = cover (A^# :^: G).	Proof. rewrite cover_imset class_supportEr setDE big_distrl /=. by apply: eq_bigr => x _; rewrite -setDE conjD1g. Qed.	Lemma	class_supportD1	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_distrl", "class_support", "class_supportEr", "conjD1g", "cover", "cover_imset", "eq_bigr", "setDE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_of : predArgType	:= Subg x & x \in G.	Inductive	subg_of	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" ]	[]	the argument to a set.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sgval u	:= let: Subg x _ := u in x.	Definition	sgval	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
subg_of_Sub	:= Eval hnf in [isSub for sgval].	Definition	subg_of_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" ]	[ "sgval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgP u : sgval u \in G.	Proof. exact: valP. Qed.	Lemma	subgP	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" ]	[ "sgval", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_inj : injective sgval.	Proof. exact: val_inj. Qed.	Lemma	subg_inj	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" ]	[ "sgval", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
congr_subg u v : u = v -> sgval u = sgval v.	Proof. exact: congr1. Qed.	Lemma	congr_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" ]	[ "sgval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_one	:= Subg group1.	Definition	subg_one	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" ]	[ "group1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

group1_contra x : x \notin G -> x != 1.

Proof. by apply: contraNneq => ->. Qed.

Lemma

group1_contra

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", "contraNneq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub1G : [1 gT] \subset G.

Proof. by rewrite sub1set. Qed.

Lemma

sub1G

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", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subG1 : (G \subset [1]) = (G :==: 1).

Proof. by rewrite eqEsubset sub1G andbT. Qed.

Lemma

subG1

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", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setI1g : 1 :&: G = 1.

Proof. exact: (setIidPl sub1G). Qed.

Lemma

setI1g

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" ]

[ "setIidPl", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setIg1 : G :&: 1 = 1.

Proof. exact: (setIidPr sub1G). Qed.

Lemma

setIg1

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" ]

[ "setIidPr", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subG1_contra H : G \subset H -> G :!=: 1 -> H :!=: 1.

Proof. by move=> sGH; rewrite -subG1; apply: contraNneq => <-. Qed.

Lemma

subG1_contra

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", "contraNneq", "sGH", "subG1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_group : repr G = 1.

Proof. by rewrite /repr group1. Qed.

Lemma

repr_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" ]

[ "group1", "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardG_gt0 : 0 < #|G|.

Proof. by rewrite lt0n; apply/existsP; exists (1 : gT). Qed.

Lemma

cardG_gt0

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", "existsP", "gT", "lt0n" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexg_gt0 A : 0 < #|G : A|.

Proof. rewrite lt0n; apply/existsP; exists A. by rewrite -{2}[A]mulg1 -rcosetE; apply: imset_f. Qed.

Lemma

indexg_gt0

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", "existsP", "imset_f", "lt0n", "mulg1", "rcosetE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivgP : reflect (G :=: 1) (G \subset [1]).

Proof. by rewrite subG1; apply: eqP. Qed.

Lemma

trivgP

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", "subG1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivGP : reflect (G = 1%G) (G \subset [1]).

Proof. by rewrite subG1; apply: eqP. Qed.

Lemma

trivGP

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", "subG1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper1G : ([1] \proper G) = (G :!=: 1).

Proof. by rewrite properEneq sub1G andbT eq_sym. Qed.

Lemma

proper1G

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" ]

[ "eq_sym", "proper", "properEneq", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

in_one_group x : (x \in 1%G) = (x == 1).

Proof. by rewrite -[x \in _]/(x \in [set 1]) !inE. Qed.

Lemma

in_one_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" ]

[ "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inE

:= (in_one_group, inE).

Definition

inE

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" ]

[ "in_one_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivgPn : reflect (exists2 x, x \in G & x != 1) (G :!=: 1).

Proof. rewrite -subG1. by apply: (iffP subsetPn) => [] [x Gx x1]; exists x; rewrite ?inE in x1 *. Qed.

Lemma

trivgPn

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", "subG1", "subsetPn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_card_le1 : (G :==: 1) = (#|G| <= 1).

Proof. by rewrite eq_sym eqEcard cards1 sub1G. Qed.

Lemma

trivg_card_le1

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", "eqEcard", "eq_sym", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_card1 : (G :==: 1) = (#|G| == 1%N).

Proof. by rewrite trivg_card_le1 eqn_leq cardG_gt0 andbT. Qed.

Lemma

trivg_card1

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" ]

[ "cardG_gt0", "eqn_leq", "trivg_card_le1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardG_gt1 : (#|G| > 1) = (G :!=: 1).

Proof. by rewrite trivg_card_le1 ltnNge. Qed.

Lemma

cardG_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" ]

[ "ltnNge", "trivg_card_le1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_le1_trivg : #|G| <= 1 -> G :=: 1.

Proof. by rewrite -trivg_card_le1; move/eqP. Qed.

Lemma

card_le1_trivg

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" ]

[ "trivg_card_le1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card1_trivg : #|G| = 1%N -> G :=: 1.

Proof. by move=> G1; rewrite card_le1_trivg ?G1. Qed.

Lemma

card1_trivg

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" ]

[ "G1", "card_le1_trivg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulG_subl A : A \subset A * G.

Proof. exact: mulg_subl group1. Qed.

Lemma

mulG_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" ]

[ "group1", "mulg_subl" ]

Inclusion and product.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulG_subr A : A \subset (G * A).

Proof. exact: mulg_subr group1. Qed.

Lemma

mulG_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" ]

[ "group1", "mulg_subr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulGid : G * G = G.

Proof. by apply/eqP; rewrite eqEsubset mulG_subr andbT; case/andP: (valP G). Qed.

Lemma

mulGid

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", "mulG_subr", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulGS A B : (G * A \subset G * B) = (A \subset G * B).

Proof. apply/idP/idP; first exact: subset_trans (mulG_subr A). by move/(mulgS G); rewrite mulgA mulGid. Qed.

Lemma

mulGS

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_subr", "mulGid", "mulgA", "mulgS", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulSG A B : (A * G \subset B * G) = (A \subset B * G).

Proof. apply/idP/idP; first exact: subset_trans (mulG_subl A). by move/(mulSg G); rewrite -mulgA mulGid. Qed.

Lemma

mulSG

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_subl", "mulGid", "mulSg", "mulgA", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul_subG A B : A \subset G -> B \subset G -> A * B \subset G.

Proof. by move=> sAG sBG; rewrite -mulGid mulgSS. Qed.

Lemma

mul_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" ]

[ "mulGid", "mulgSS", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_subG (I : Type) (r : seq I) (P : {pred I}) (F : I -> {set gT}) : (forall i, P i -> F i \subset G) -> \prod_(i <- r | P i) F i \subset G.

Proof. move=> subFG; elim/big_rec: _ => [|/= i A /subFG]; first by rewrite sub1set. exact: mul_subG. Qed.

Lemma

prod_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" ]

[ "big_rec", "gT", "mul_subG", "seq", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupM x y : x \in G -> y \in G -> x * y \in G.

Proof. by case/group_setP: (valP G) x y. Qed.

Lemma

groupM

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_setP", "valP" ]

Membership lemmas

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupX x n : x \in G -> x ^+ n \in G.

Proof. by move=> Gx; elim: n => [|n IHn]; rewrite ?group1 // expgS groupM. Qed.

Lemma

groupX

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" ]

[ "expgS", "group1", "groupM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupVr x : x \in G -> x^-1 \in G.

Proof. move=> Gx; rewrite -(mul1g x^-1) -mem_rcoset ((G :* x =P G) _) //. by rewrite eqEcard card_rcoset leqnn mul_subG ?sub1set. Qed.

Lemma

groupVr

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" ]

[ "card_rcoset", "eqEcard", "leqnn", "mem_rcoset", "mul1g", "mul_subG", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupVl x : x^-1 \in G -> x \in G.

Proof. by move/groupVr; rewrite invgK. Qed.

Lemma

groupVl

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" ]

[ "groupVr", "invgK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupV x : (x^-1 \in G) = (x \in G).

Proof. by apply/idP/idP; [apply: groupVl | apply: groupVr]. Qed.

Lemma

groupV

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", "groupVl", "groupVr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupMl x y : x \in G -> (x * y \in G) = (y \in G).

Proof. move=> Gx; apply/idP/idP=> [Gxy|]; last exact: groupM. by rewrite -(mulKg x y) groupM ?groupVr. Qed.

Lemma

groupMl

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", "groupM", "groupVr", "last", "mulKg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupMr x y : x \in G -> (y * x \in G) = (y \in G).

Proof. by move=> Gx; rewrite -[_ \in G]groupV invMg groupMl groupV. Qed.

Lemma

groupMr

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" ]

[ "groupMl", "groupV", "invMg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

in_group

:= (group1, groupV, (groupMl, groupX)).

Definition

in_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" ]

[ "group1", "groupMl", "groupV", "groupX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupJ x y : x \in G -> y \in G -> x ^ y \in G.

Proof. by move=> Gx Gy; rewrite !in_group. Qed.

Lemma

groupJ

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" ]

[ "in_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupJr x y : y \in G -> (x ^ y \in G) = (x \in G).

Proof. by move=> Gy; rewrite groupMl (groupMr, groupV). Qed.

Lemma

groupJr

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" ]

[ "groupMl", "groupMr", "groupV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupR x y : x \in G -> y \in G -> [~ x, y] \in G.

Proof. by move=> Gx Gy; rewrite !in_group. Qed.

Lemma

groupR

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" ]

[ "in_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_prod I r (P : pred I) F : (forall i, P i -> F i \in G) -> \prod_(i <- r | P i) F i \in G.

Proof. by move=> G_P; elim/big_ind: _ => //; apply: groupM. Qed.

Lemma

group_prod

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_ind", "groupM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invGid : G^-1 = G.

Proof. by apply/setP=> x; rewrite inE groupV. Qed.

Lemma

invGid

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", "groupV", "inE", "setP" ]

Inverse is an anti-morphism.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inv_subG A : (A^-1 \subset G) = (A \subset G).

Proof. by rewrite -{1}invGid invSg. Qed.

Lemma

inv_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" ]

[ "invGid", "invSg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg_lcoset x : (x *: G)^-1 = G :* x^-1.

Proof. by rewrite invMg invGid invg_set1. Qed.

Lemma

invg_lcoset

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" ]

[ "invGid", "invMg", "invg_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg_rcoset x : (G :* x)^-1 = x^-1 *: G.

Proof. by rewrite invMg invGid invg_set1. Qed.

Lemma

invg_rcoset

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" ]

[ "invGid", "invMg", "invg_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memV_lcosetV x y : (y^-1 \in x^-1 *: G) = (y \in G :* x).

Proof. by rewrite -invg_rcoset memV_invg. Qed.

Lemma

memV_lcosetV

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_rcoset", "memV_invg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memV_rcosetV x y : (y^-1 \in G :* x^-1) = (y \in x *: G).

Proof. by rewrite -invg_lcoset memV_invg. Qed.

Lemma

memV_rcosetV

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_lcoset", "memV_invg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulSgGid A x : x \in A -> A \subset G -> A * G = G.

Proof. move=> Ax sAG; apply/eqP; rewrite eqEsubset -{2}mulGid mulSg //=. apply/subsetP=> y Gy; rewrite -(mulKVg x y) mem_mulg // groupMr // groupV. exact: (subsetP sAG). Qed.

Lemma

mulSgGid

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", "groupMr", "groupV", "mem_mulg", "mulGid", "mulKVg", "mulSg", "sAG", "subsetP" ]

Product idempotence

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulGSgid A x : x \in A -> A \subset G -> G * A = G.

Proof. rewrite -memV_invg -invSg invGid => Ax sAG. by apply: invg_inj; rewrite invMg invGid (mulSgGid Ax). Qed.

Lemma

mulGSgid

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", "invGid", "invMg", "invSg", "invg_inj", "memV_invg", "mulSgGid", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcoset_refl x : x \in x *: G.

Proof. by rewrite mem_lcoset mulVg group1. Qed.

Lemma

lcoset_refl

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" ]

[ "group1", "mem_lcoset", "mulVg" ]

Left cosets

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcoset_sym x y : (x \in y *: G) = (y \in x *: G).

Proof. by rewrite !mem_lcoset -groupV invMg invgK. Qed.

Lemma

lcoset_sym

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", "invMg", "invgK", "mem_lcoset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcoset_eqP {x y} : reflect (x *: G = y *: G) (x \in y *: G).

Proof. suffices <-: (x *: G == y *: G) = (x \in y *: G) by apply: eqP. by rewrite eqEsubset !mulSG !sub1set lcoset_sym andbb. Qed.

Lemma

lcoset_eqP

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", "lcoset_sym", "mulSG", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcoset_transl x y z : x \in y *: G -> (x \in z *: G) = (y \in z *: G).

Proof. by move=> Gyx; rewrite -2!(lcoset_sym z) (lcoset_eqP Gyx). Qed.

Lemma

lcoset_transl

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" ]

[ "lcoset_eqP", "lcoset_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcoset_trans x y z : x \in y *: G -> y \in z *: G -> x \in z *: G.

Proof. by move/lcoset_transl->. Qed.

Lemma

lcoset_trans

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" ]

[ "lcoset_transl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcoset_id x : x \in G -> x *: G = G.

Proof. by move=> Gx; rewrite (lcoset_eqP (_ : x \in 1 *: G)) mul1g. Qed.

Lemma

lcoset_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" ]

[ "lcoset_eqP", "mul1g" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_refl x : x \in G :* x.

Proof. by rewrite mem_rcoset mulgV group1. Qed.

Lemma

rcoset_refl

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" ]

[ "group1", "mem_rcoset", "mulgV" ]

Right cosets, with an elimination form for repr.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_sym x y : (x \in G :* y) = (y \in G :* x).

Proof. by rewrite -!memV_lcosetV lcoset_sym. Qed.

Lemma

rcoset_sym

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" ]

[ "lcoset_sym", "memV_lcosetV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_eqP {x y} : reflect (G :* x = G :* y) (x \in G :* y).

Proof. suffices <-: (G :* x == G :* y) = (x \in G :* y) by apply: eqP. by rewrite eqEsubset !mulGS !sub1set rcoset_sym andbb. Qed.

Lemma

rcoset_eqP

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", "mulGS", "rcoset_sym", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_transl x y z : x \in G :* y -> (x \in G :* z) = (y \in G :* z).

Proof. by move=> Gyx; rewrite -2!(rcoset_sym z) (rcoset_eqP Gyx). Qed.

Lemma

rcoset_transl

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" ]

[ "rcoset_eqP", "rcoset_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_trans x y z : x \in G :* y -> y \in G :* z -> x \in G :* z.

Proof. by move/rcoset_transl->. Qed.

Lemma

rcoset_trans

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" ]

[ "rcoset_transl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_id x : x \in G -> G :* x = G.

Proof. by move=> Gx; rewrite (rcoset_eqP (_ : x \in G :* 1)) mulg1. Qed.

Lemma

rcoset_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" ]

[ "mulg1", "rcoset_eqP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_repr_spec x : gT -> Type

:= RcosetReprSpec g : g \in G -> rcoset_repr_spec x (g * x).

Variant

rcoset_repr_spec

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" ]

Elimination form.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_repr_rcoset x : repr (G :* x) \in G :* x.

Proof. exact: mem_repr (rcoset_refl x). Qed.

Lemma

mem_repr_rcoset

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_repr", "rcoset_refl", "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_rcosetP x : rcoset_repr_spec x (repr (G :* x)).

Proof. by rewrite -[repr _](mulgKV x); split; rewrite -mem_rcoset mem_repr_rcoset. Qed.

Lemma

repr_rcosetP

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_rcoset", "mem_repr_rcoset", "mulgKV", "rcoset_repr_spec", "repr", "split" ]

(weaker) primitive Coq algorithm for general (co)inductive type families.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_repr x : G :* (repr (G :* x)) = G :* x.

Proof. exact/rcoset_eqP/mem_repr_rcoset. Qed.

Lemma

rcoset_repr

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_repr_rcoset", "rcoset_eqP", "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_rcosets A x : (G :* x \in rcosets G A) = (x \in G * A).

Proof. apply/rcosetsP/mulsgP=> [[a Aa /rcoset_eqP/rcosetP[g]] | ]; first by exists g a. by case=> g a Gg Aa ->{x}; exists a; rewrite // rcosetM rcoset_id. Qed.

Lemma

mem_rcosets

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" ]

[ "Gg", "apply", "mulsgP", "rcosetM", "rcosetP", "rcoset_eqP", "rcoset_id", "rcosets", "rcosetsP" ]

Coset spaces.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_lcosets A x : (x *: G \in lcosets G A) = (x \in A * G).

Proof. rewrite -[LHS]memV_invg invg_lcoset invg_lcosets. by rewrite -[RHS]memV_invg invMg invGid mem_rcosets. Qed.

Lemma

mem_lcosets

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" ]

[ "invGid", "invMg", "invg_lcoset", "invg_lcosets", "lcosets", "memV_invg", "mem_rcosets" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_setJ A x : group_set (A :^ x) = group_set A.

Proof. by rewrite /group_set mem_conjg conj1g -conjsMg conjSg. Qed.

Lemma

group_setJ

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" ]

[ "conj1g", "conjSg", "conjsMg", "group_set", "mem_conjg" ]

Conjugates.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_conjG x : group_set (G :^ x).

Proof. by rewrite group_setJ groupP. Qed.

Lemma

group_set_conjG

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" ]

[ "groupP", "group_set", "group_setJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjG_group x

:= group (group_set_conjG x).

Canonical

conjG_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_conjG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjGid : {in G, normalised G}.

Proof. by move=> x Gx; apply/setP=> y; rewrite mem_conjg groupJr ?groupV. Qed.

Lemma

conjGid

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", "groupJr", "groupV", "mem_conjg", "normalised", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_subG x A : x \in G -> A \subset G -> A :^ x \subset G.

Proof. by move=> Gx sAG; rewrite -(conjGid Gx) conjSg. Qed.

Lemma

conj_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" ]

[ "conjGid", "conjSg", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class1G : 1 ^: G = 1.

Proof. exact: class1g group1. Qed.

Lemma

class1G

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" ]

[ "class1g", "group1" ]

Classes

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classes1 : [1] \in classes G.

Proof. by rewrite -class1G mem_classes. Qed.

Lemma

classes1

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" ]

[ "class1G", "classes", "mem_classes" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classGidl x y : y \in G -> (x ^ y) ^: G = x ^: G.

Proof. by move=> Gy; rewrite -class_lcoset lcoset_id. Qed.

Lemma

classGidl

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_lcoset", "lcoset_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classGidr x : {in G, normalised (x ^: G)}.

Proof. by move=> y Gy /=; rewrite -class_rcoset rcoset_id. Qed.

Lemma

classGidr

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_rcoset", "normalised", "rcoset_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_refl x : x \in x ^: G.

Proof. by apply/imsetP; exists 1; rewrite ?conjg1. Qed.

Lemma

class_refl

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", "conjg1", "imsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_eqP x y : reflect (x ^: G = y ^: G) (x \in y ^: G).

Proof. by apply: (iffP idP) => [/imsetP[z Gz ->] | <-]; rewrite ?class_refl ?classGidl. Qed.

Lemma

class_eqP

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", "classGidl", "class_refl", "imsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_sym x y : (x \in y ^: G) = (y \in x ^: G).

Proof. by apply/idP/idP=> /class_eqP->. Qed.

Lemma

class_sym

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_eqP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_transl x y z : x \in y ^: G -> (x \in z ^: G) = (y \in z ^: G).

Proof. by rewrite -!(class_sym z) => /class_eqP->. Qed.

Lemma

class_transl

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_eqP", "class_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_trans x y z : x \in y ^: G -> y \in z ^: G -> x \in z ^: G.

Proof. by move/class_transl->. Qed.

Lemma

class_trans

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_transl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_class x : {y | y \in G & repr (x ^: G) = x ^ y}.

Proof. set z := repr _; have: #|[set y in G | z == x ^ y]| > 0. have: z \in x ^: G by apply: (mem_repr x). by case/imsetP=> y Gy ->; rewrite (cardD1 y) inE Gy eqxx. by move/card_mem_repr; move: (repr _) => y /setIdP[Gy /eqP]; exists y. Qed.

Lemma

repr_class

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", "cardD1", "card_mem_repr", "eqxx", "imsetP", "inE", "mem_repr", "repr", "setIdP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classG_eq1 x : (x ^: G == 1) = (x == 1).

Proof. apply/eqP/eqP=> [xG1 | ->]; last exact: class1G. by have:= class_refl x; rewrite xG1 => /set1P. Qed.

Lemma

classG_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" ]

[ "apply", "class1G", "class_refl", "last", "set1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_subG x A : x \in G -> A \subset G -> x ^: A \subset G.

Proof. move=> Gx sAG; apply/subsetP=> _ /imsetP[y Ay ->]. by rewrite groupJ // (subsetP sAG). Qed.

Lemma

class_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", "groupJ", "imsetP", "sAG", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_classesP xG : reflect (repr xG \in G /\ xG = repr xG ^: G) (xG \in classes G).

Proof. apply: (iffP imsetP) => [[x Gx ->] | []]; last by exists (repr xG). by have [y Gy ->] := repr_class x; rewrite classGidl ?groupJ. Qed.

Lemma

repr_classesP

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", "classGidl", "classes", "groupJ", "imsetP", "last", "repr", "repr_class" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_repr_classes xG : xG \in classes G -> repr xG \in xG.

Proof. by case/repr_classesP=> _ {2}->; apply: class_refl. Qed.

Lemma

mem_repr_classes

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", "classes", "repr", "repr_classesP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classes_gt0 : 0 < #|classes G|.

Proof. by rewrite (cardsD1 1) classes1. Qed.

Lemma

classes_gt0

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" ]

[ "cardsD1", "classes", "classes1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classes_gt1 : (#|classes G| > 1) = (G :!=: 1).

Proof. rewrite (cardsD1 1) classes1 ltnS lt0n cards_eq0. apply/set0Pn/trivgPn=> [[xG /setD1P[nt_xG]] | [x Gx ntx]]. by case/imsetP=> x Gx def_xG; rewrite def_xG classG_eq1 in nt_xG; exists x. by exists (x ^: G); rewrite !inE classG_eq1 ntx; apply: imset_f. Qed.

Lemma

classes_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" ]

[ "apply", "cardsD1", "cards_eq0", "classG_eq1", "classes", "classes1", "imsetP", "imset_f", "inE", "lt0n", "ltnS", "set0Pn", "setD1P", "trivgPn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_class_support A x : x \in A -> x \in class_support A G.

Proof. by move=> Ax; rewrite -[x]conjg1 memJ_class_support. Qed.

Lemma

mem_class_support

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_support", "conjg1", "memJ_class_support" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_supportGidl A x : x \in G -> class_support (A :^ x) G = class_support A G.

Proof. by move=> Gx; rewrite -class_support_set1r -class_supportM lcoset_id. Qed.

Lemma

class_supportGidl

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_support", "class_supportM", "class_support_set1r", "lcoset_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_supportGidr A : {in G, normalised (class_support A G)}.

Proof. by move=> x Gx /=; rewrite -class_support_set1r -class_supportM rcoset_id. Qed.

Lemma

class_supportGidr

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_support", "class_supportM", "class_support_set1r", "normalised", "rcoset_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_support_subG A : A \subset G -> class_support A G \subset G.

Proof. by move=> sAG; rewrite class_supportEr; apply/bigcupsP=> x Gx; apply: conj_subG. Qed.

Lemma

class_support_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", "bigcupsP", "class_support", "class_supportEr", "conj_subG", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_class_support A : A \subset class_support A G.

Proof. by rewrite class_supportEr (bigcup_max 1) ?conjsg1. Qed.

Lemma

sub_class_support

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" ]

[ "bigcup_max", "class_support", "class_supportEr", "conjsg1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_support_id : class_support G G = G.

Proof. by apply/eqP; rewrite eqEsubset sub_class_support class_support_subG. Qed.

Lemma

class_support_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", "class_support", "class_support_subG", "eqEsubset", "sub_class_support" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_supportD1 A : (class_support A G)^# = cover (A^# :^: G).

Proof. rewrite cover_imset class_supportEr setDE big_distrl /=. by apply: eq_bigr => x _; rewrite -setDE conjD1g. Qed.

Lemma

class_supportD1

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_distrl", "class_support", "class_supportEr", "conjD1g", "cover", "cover_imset", "eq_bigr", "setDE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_of : predArgType

:= Subg x & x \in G.

Inductive

subg_of

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" ]

[]

the argument to a set.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sgval u

:= let: Subg x _ := u in x.

Definition

sgval

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

subg_of_Sub

:= Eval hnf in [isSub for sgval].

Definition

subg_of_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" ]

[ "sgval" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgP u : sgval u \in G.

Proof. exact: valP. Qed.

Lemma

subgP

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" ]

[ "sgval", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_inj : injective sgval.

Proof. exact: val_inj. Qed.

Lemma

subg_inj

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" ]

[ "sgval", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

congr_subg u v : u = v -> sgval u = sgval v.

Proof. exact: congr1. Qed.

Lemma

congr_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" ]

[ "sgval" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_one

:= Subg group1.

Definition

subg_one

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" ]

[ "group1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d