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
lcoset_inj : right_injective (fun x A => x *: A).	Proof. by move=> x; apply: can_inj (lcosetK x). Qed.	Lemma	lcoset_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" ]	[ "apply", "lcosetK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosetS x A B : (x : A \subset x : B) = (A \subset B).	Proof. apply/idP/idP=> sAB; last exact: mulgS. by rewrite -(lcosetK x A) -(lcosetK x B) mulgS. Qed.	Lemma	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" ]	[ "apply", "last", "lcosetK", "mulgS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_lcoset x A B : (A \subset x : B) = (x^-1 : A \subset B).	Proof. by rewrite -(lcosetS x^-1) lcosetK. Qed.	Lemma	sub_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" ]	[ "lcosetK", "lcosetS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_lcosetV x A B : (A \subset x^-1 : B) = (x : A \subset B).	Proof. by rewrite sub_lcoset invgK. Qed.	Lemma	sub_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" ]	[ "invgK", "sub_lcoset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosetE A x : rcoset A x = A :* x.	Proof. by rewrite [_ * _]imset2_set1r. Qed.	Lemma	rcosetE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2_set1r", "rcoset" ]	Right cosets.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_rcoset A x : #\|A :* x\| = #\|A\|.	Proof. by rewrite -rcosetE (card_imset _ (mulIg _)). Qed.	Lemma	card_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" ]	[ "card_imset", "mulIg", "rcosetE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_rcoset A x y : (y \in A :* x) = (y * x^-1 \in A).	Proof. by rewrite -rcosetE [_ x](can_imset_pre A (mulgK _)) inE. Qed.	Lemma	mem_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" ]	[ "can_imset_pre", "inE", "mulgK", "rcosetE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosetP A x y : reflect (exists2 a, a \in A & y = a * x) (y \in A :* x).	Proof. by rewrite -rcosetE; apply: imsetP. Qed.	Lemma	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" ]	[ "apply", "imsetP", "rcosetE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosetsP A B C : reflect (exists2 x, x \in B & C = A :* x) (C \in rcosets A B).	Proof. by apply: (iffP imsetP) => [] [x Bx ->]; exists x; rewrite ?rcosetE. Qed.	Lemma	rcosetsP	finite_group	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", "imsetP", "rcosetE", "rcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosetM A x y : A :* (x * y) = A :* x :* y.	Proof. by rewrite -mulg_set1 mulgA. Qed.	Lemma	rcosetM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mulgA", "mulg_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset1 A : A :* 1 = A.	Proof. exact: mulg1. Qed.	Lemma	rcoset1	finite_group	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosetK : right_loop inv (fun A x => A :* x).	Proof. by move=> x A; rewrite -rcosetM mulgV mulg1. Qed.	Lemma	rcosetK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inv", "mulg1", "mulgV", "rcosetM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosetKV : rev_right_loop inv (fun A x => A :* x).	Proof. by move=> x A; rewrite -rcosetM mulVg mulg1. Qed.	Lemma	rcosetKV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inv", "mulVg", "mulg1", "rcosetM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_inj : left_injective (fun A x => A :* x).	Proof. by move=> x; apply: can_inj (rcosetK x). Qed.	Lemma	rcoset_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" ]	[ "apply", "rcosetK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosetS x A B : (A :* x \subset B :* x) = (A \subset B).	Proof. apply/idP/idP=> sAB; last exact: mulSg. by rewrite -(rcosetK x A) -(rcosetK x B) mulSg. Qed.	Lemma	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" ]	[ "apply", "last", "mulSg", "rcosetK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_rcoset x A B : (A \subset B :* x) = (A :* x ^-1 \subset B).	Proof. by rewrite -(rcosetS x^-1) rcosetK. Qed.	Lemma	sub_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" ]	[ "rcosetK", "rcosetS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_rcosetV x A B : (A \subset B :* x^-1) = (A :* x \subset B).	Proof. by rewrite sub_rcoset invgK. Qed.	Lemma	sub_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" ]	[ "invgK", "sub_rcoset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_lcosets A B : (lcosets A B)^-1 = rcosets A^-1 B^-1.	Proof. rewrite /A^-1/= -![_^-1](can_imset_pre _ invgK) -[RHS]imset_comp -imset_comp. by apply: eq_imset => x /=; rewrite lcosetE rcosetE invMg invg_set1. Qed.	Lemma	invg_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" ]	[ "apply", "can_imset_pre", "eq_imset", "imset_comp", "invMg", "invgK", "invg_set1", "lcosetE", "lcosets", "rcosetE", "rcosets" ]	Inverse maps lcosets to rcosets	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_preim A x : A :^ x = (conjg^~ x^-1) @^-1: A.	Proof. exact: can_imset_pre (conjgK _). Qed.	Lemma	conjg_preim	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "can_imset_pre", "conjg", "conjgK" ]	Conjugates.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_conjg A x y : (y \in A :^ x) = (y ^ x^-1 \in A).	Proof. by rewrite conjg_preim inE. Qed.	Lemma	mem_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" ]	[ "conjg_preim", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_conjgV A x y : (y \in A :^ x^-1) = (y ^ x \in A).	Proof. by rewrite mem_conjg invgK. Qed.	Lemma	mem_conjgV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "invgK", "mem_conjg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memJ_conjg A x y : (y ^ x \in A :^ x) = (y \in A).	Proof. by rewrite mem_conjg conjgK. Qed.	Lemma	memJ_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" ]	[ "conjgK", "mem_conjg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsgE A x : A :^ x = x^-1 : (A : x).	Proof. by apply/setP=> y; rewrite mem_lcoset mem_rcoset -mulgA mem_conjg. Qed.	Lemma	conjsgE	finite_group	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", "mem_conjg", "mem_lcoset", "mem_rcoset", "mulgA", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsg1 A : A :^ 1 = A.	Proof. by rewrite conjsgE invg1 mul1g mulg1. Qed.	Lemma	conjsg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjsgE", "invg1", "mul1g", "mulg1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsgM A x y : A :^ (x * y) = (A :^ x) :^ y.	Proof. by rewrite !conjsgE invMg -!mulg_set1 !mulgA. Qed.	Lemma	conjsgM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjsgE", "invMg", "mulgA", "mulg_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsgK : @right_loop _ gT inv conjugate.	Proof. by move=> x A; rewrite -conjsgM mulgV conjsg1. Qed.	Lemma	conjsgK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjsg1", "conjsgM", "conjugate", "gT", "inv", "mulgV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsgKV : @rev_right_loop _ gT inv conjugate.	Proof. by move=> x A; rewrite -conjsgM mulVg conjsg1. Qed.	Lemma	conjsgKV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjsg1", "conjsgM", "conjugate", "gT", "inv", "mulVg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsg_inj : @left_injective _ gT _ conjugate.	Proof. by move=> x; apply: can_inj (conjsgK x). Qed.	Lemma	conjsg_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" ]	[ "apply", "conjsgK", "conjugate", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardJg A x : #\|A :^ x\| = #\|A\|.	Proof. by rewrite (card_imset _ (conjg_inj x)). Qed.	Lemma	cardJg	finite_group	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_imset", "conjg_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjSg A B x : (A :^ x \subset B :^ x) = (A \subset B).	Proof. by rewrite !conjsgE lcosetS rcosetS. Qed.	Lemma	conjSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjsgE", "lcosetS", "rcosetS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
properJ A B x : (A :^ x \proper B :^ x) = (A \proper B).	Proof. by rewrite /proper !conjSg. Qed.	Lemma	properJ	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjSg", "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_conjg A B x : (A :^ x \subset B) = (A \subset B :^ x^-1).	Proof. by rewrite -(conjSg A _ x) conjsgKV. Qed.	Lemma	sub_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" ]	[ "conjSg", "conjsgKV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_conjgV A B x : (A :^ x^-1 \subset B) = (A \subset B :^ x).	Proof. by rewrite -(conjSg _ B x) conjsgKV. Qed.	Lemma	sub_conjgV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjSg", "conjsgKV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_set1 x y : [set x] :^ y = [set x ^ y].	Proof. by rewrite [_ :^ _]imset_set1. Qed.	Lemma	conjg_set1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjs1g x : 1 :^ x = 1.	Proof. by rewrite conjg_set1 conj1g. Qed.	Lemma	conjs1g	finite_group	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", "conjg_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsg_eq1 A x : (A :^ x == 1%g) = (A == 1%g).	Proof. by rewrite (canF_eq (conjsgK x)) conjs1g. Qed.	Lemma	conjsg_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" ]	[ "canF_eq", "conjs1g", "conjsgK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsMg A B x : (A * B) :^ x = A :^ x * B :^ x.	Proof. by rewrite !conjsgE !mulgA rcosetK. Qed.	Lemma	conjsMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjsgE", "mulgA", "rcosetK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjIg A B x : (A :&: B) :^ x = A :^ x :&: B :^ x.	Proof. by rewrite !conjg_preim preimsetI. Qed.	Lemma	conjIg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjg_preim", "preimsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj0g x : set0 :^ x = set0.	Proof. exact: imset0. Qed.	Lemma	conj0g	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset0", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjTg x : [set: gT] :^ x = [set: gT].	Proof. by rewrite conjg_preim preimsetT. Qed.	Lemma	conjTg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjg_preim", "gT", "preimsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigcapJ I r (P : pred I) (B : I -> {set gT}) x : \bigcap_(i <- r \| P i) (B i :^ x) = (\bigcap_(i <- r \| P i) B i) :^ x.	Proof. by rewrite (big_endo (conjugate^~ x)) => // [B1 B2\|]; rewrite (conjTg, conjIg). Qed.	Lemma	bigcapJ	finite_group	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_endo", "conjIg", "conjTg", "conjugate", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjUg A B x : (A :\|: B) :^ x = A :^ x :\|: B :^ x.	Proof. by rewrite !conjg_preim preimsetU. Qed.	Lemma	conjUg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjg_preim", "preimsetU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigcupJ I r (P : pred I) (B : I -> {set gT}) x : \bigcup_(i <- r \| P i) (B i :^ x) = (\bigcup_(i <- r \| P i) B i) :^ x.	Proof. rewrite (big_endo (conjugate^~ x)) => // [B1 B2\|]; first by rewrite conjUg. exact: imset0. Qed.	Lemma	bigcupJ	finite_group	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_endo", "conjUg", "conjugate", "gT", "imset0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjCg A x : (~: A) :^ x = ~: A :^ x.	Proof. by rewrite !conjg_preim preimsetC. Qed.	Lemma	conjCg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjg_preim", "preimsetC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjDg A B x : (A :\: B) :^ x = A :^ x :\: B :^ x.	Proof. by rewrite !setDE !(conjCg, conjIg). Qed.	Lemma	conjDg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjCg", "conjIg", "setDE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjD1g A x : A^# :^ x = (A :^ x)^#.	Proof. by rewrite conjDg conjs1g. Qed.	Lemma	conjD1g	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjDg", "conjs1g" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memJ_class x y A : y \in A -> x ^ y \in x ^: A.	Proof. exact: imset_f. Qed.	Lemma	memJ_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" ]	[ "imset_f" ]	Classes; not much for now.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classS x A B : A \subset B -> x ^: A \subset x ^: B.	Proof. exact: imsetS. Qed.	Lemma	classS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imsetS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_set1 x y : x ^: [set y] = [set x ^ y].	Proof. exact: imset_set1. Qed.	Lemma	class_set1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class1g x A : x \in A -> 1 ^: A = 1.	Proof. move=> Ax; apply/setP=> y. by apply/imsetP/set1P=> [[a Aa]\|] ->; last exists x; rewrite ?conj1g. 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" ]	[ "apply", "conj1g", "imsetP", "last", "set1P", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classVg x A : x^-1 ^: A = (x ^: A)^-1.	Proof. apply/setP=> xy; rewrite inE; apply/imsetP/imsetP=> [] [y Ay def_xy]. by rewrite def_xy conjVg invgK; exists y. by rewrite -[xy]invgK def_xy -conjVg; exists y. Qed.	Lemma	classVg	finite_group	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", "conjVg", "imsetP", "inE", "invgK", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_classes x A : x \in A -> x ^: A \in classes A.	Proof. exact: imset_f. Qed.	Lemma	mem_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" ]	[ "classes", "imset_f" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memJ_class_support A B x y : x \in A -> y \in B -> x ^ y \in class_support A B.	Proof. by move=> Ax By; apply: imset2_f. Qed.	Lemma	memJ_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" ]	[ "apply", "class_support", "imset2_f" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_supportM A B C : class_support A (B * C) = class_support (class_support A B) C.	Proof. apply/setP=> x; apply/imset2P/imset2P=> [[a y Aa] \| [y c]]. case/mulsgP=> b c Bb Cc -> ->{x y}. by exists (a ^ b) c; rewrite ?(imset2_f, conjgM). case/imset2P=> a b Aa Bb -> Cc ->{x y}. by exists a (b * c); rewrite ?(mem_mulg, conjgM). Qed.	Lemma	class_supportM	finite_group	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", "conjgM", "imset2P", "imset2_f", "mem_mulg", "mulsgP", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_support_set1l A x : class_support [set x] A = x ^: A.	Proof. exact: imset2_set1l. Qed.	Lemma	class_support_set1l	finite_group	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", "imset2_set1l" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_support_set1r A x : class_support A [set x] = A :^ x.	Proof. exact: imset2_set1r. Qed.	Lemma	class_support_set1r	finite_group	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", "imset2_set1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classM x A B : x ^: (A * B) = class_support (x ^: A) B.	Proof. by rewrite -!class_support_set1l class_supportM. Qed.	Lemma	classM	finite_group	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_set1l" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_lcoset x y A : x ^: (y *: A) = (x ^ y) ^: A.	Proof. by rewrite classM class_set1 class_support_set1l. Qed.	Lemma	class_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" ]	[ "classM", "class_set1", "class_support_set1l" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_rcoset x A y : x ^: (A :* y) = (x ^: A) :^ y.	Proof. by rewrite -class_support_set1r classM. Qed.	Lemma	class_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" ]	[ "classM", "class_support_set1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjugatesS A B C : B \subset C -> A :^: B \subset A :^: C.	Proof. exact: imsetS. Qed.	Lemma	conjugatesS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imsetS" ]	Conjugate set.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjugates_set1 A x : A :^: [set x] = [set A :^ x].	Proof. exact: imset_set1. Qed.	Lemma	conjugates_set1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjugates_conj A x B : (A :^ x) :^: B = A :^: (x *: B).	Proof. rewrite /conjugates [x *: B]imset2_set1l -imset_comp. by apply: eq_imset => y /=; rewrite conjsgM. Qed.	Lemma	conjugates_conj	finite_group	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", "conjsgM", "conjugates", "eq_imset", "imset2_set1l", "imset_comp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_supportEl A B : class_support A B = \bigcup_(x in A) x ^: B.	Proof. exact: curry_imset2l. Qed.	Lemma	class_supportEl	finite_group	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", "curry_imset2l" ]	Class support.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_supportEr A B : class_support A B = \bigcup_(x in B) A :^ x.	Proof. exact: curry_imset2r. Qed.	Lemma	class_supportEr	finite_group	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", "curry_imset2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set A	:= (1 \in A) && (A * A \subset A).	Definition	group_set	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]	Groups (at last!)	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_setP A : reflect (1 \in A /\ {in A & A, forall x y, x * y \in A}) (group_set A).	Proof. apply: (iffP andP) => [] [A1 AM]; split=> {A1}//. by move=> x y Ax Ay; apply: (subsetP AM); rewrite mem_mulg. by apply/subsetP=> _ /mulsgP[x y Ax Ay ->]; apply: AM. Qed.	Lemma	group_setP	finite_group	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", "group_set", "mem_mulg", "mulsgP", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_type : Type	:= Group { gval :> GroupSet.sort gT; _ : group_set gval }.	Structure	group_type	finite_group	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", "group_set", "sort" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_of : predArgType	:= group_type.	Definition	group_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" ]	[ "group_type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupT	:= group_of.	Notation	groupT	finite_group	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_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group (A : {set gT}) gA : groupT	:= @Group A gA.	Definition	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" ]	[ "gT", "groupT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone_group G	:= let: Group _ gP := G return {type of Group for G} -> groupT in fun k => k gP.	Definition	clone_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" ]	[ "groupT", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_inj : injective gval.	Proof. exact: val_inj. Qed.	Lemma	group_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" ]	[ "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupP (G : groupT) : group_set G.	Proof. by case: G. Qed.	Lemma	groupP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "groupT", "group_set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
congr_group (H K : groupT) : H = K -> H :=: K.	Proof. exact: congr1. Qed.	Lemma	congr_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" ]	[ "groupT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isgroupP A : reflect (exists G : groupT, A = G) (group_set A).	Proof. by apply: (iffP idP) => [gA \| [[B gB] -> //]]; exists (Group gA). Qed.	Lemma	isgroupP	finite_group	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", "groupT", "group_set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_one : group_set 1.	Proof. by rewrite /group_set set11 mulg1 subxx. Qed.	Lemma	group_set_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" ]	[ "group_set", "mulg1", "set11", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
one_group	:= group group_set_one.	Canonical	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" ]	[ "group", "group_set_one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1_group	:= @group [set 1] group_set_one.	Canonical	set1_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_one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_setT : group_set (setTfor gT).	Proof. by apply/group_setP; split=> [\|x y _ _]; rewrite inE. Qed.	Lemma	group_setT	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "gT", "group_set", "group_setP", "inE", "setTfor", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setT_group	:= group group_setT.	Canonical	setT_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_setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'group' gT }"	:= (group_of gT) (format "{ 'group' gT }") : type_scope.	Notation	{ 'group' gT }	finite_group	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", "group_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'group' 'of' G ]"	:= (clone_group (@group _ G)) (format "[ 'group' 'of' G ]") : form_scope.	Notation	[ 'group' 'of' G ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "clone_group", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"1"	:= (one_group _) : Group_scope.	Notation	1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "one_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 1 gT ]"	:= (1%G : {group gT}) : Group_scope.	Notation	[ 1 gT ]	finite_group	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", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'set' : gT ]"	:= (setT_group gT) : Group_scope.	Notation	[ 'set' : gT ]	finite_group	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", "setT_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
generated_unlockable	:= Unlockable generated.unlock.	Canonical	generated_unlockable	finite_group	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
gcore (gT : finGroupType) (A B : {set gT})	:= \bigcap_(x in B) A :^ x.	Definition	gcore	finite_group	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing (gT : finGroupType) (A B : {set gT})	:= generated (A :\|: B).	Definition	joing	finite_group	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commutator (gT : finGroupType) (A B : {set gT})	:= generated (commg_set A B).	Definition	commutator	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "commg_set", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle (gT : finGroupType) (x : gT)	:= generated [set x].	Definition	cycle	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
order (gT : finGroupType) (x : gT)	:= #\|cycle x\|.	Definition	order	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gsort gT	:= (FinStarMonoid.arg_sort gT%type) (only parsing).	Notation	gsort	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "arg_sort", "gT", "type" ]	Notation below.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< A >>"	:= (generated A) : group_scope.	Notation	<< A >>	finite_group	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
"<[ x ] >"	:= (cycle x) : group_scope.	Notation	<[ x ] >	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"#[ x ]"	:= (order x) : group_scope.	Notation	#[ x ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "order" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A <*> B"	:= (joing A B) : group_scope.	Notation	A <*> B	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joing" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ ~: A1 , A2 , .. , An ]"	:= (commutator .. (commutator A1 A2) .. An) : group_scope.	Notation	[ ~: A1 , A2 , .. , An ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "commutator" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sT	:= {set gT}.	Notation	sT	finite_group	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
valG : val G = G.	Proof. by []. Qed.	Lemma	valG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group1 : 1 \in G.	Proof. by case/group_setP: (valP G). Qed.	Lemma	group1	finite_group	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" ]	Non-triviality.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcoset_inj : right_injective (fun x A => x *: A).

Proof. by move=> x; apply: can_inj (lcosetK x). Qed.

Lemma

lcoset_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" ]

[ "apply", "lcosetK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcosetS x A B : (x *: A \subset x *: B) = (A \subset B).

Proof. apply/idP/idP=> sAB; last exact: mulgS. by rewrite -(lcosetK x A) -(lcosetK x B) mulgS. Qed.

Lemma

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" ]

[ "apply", "last", "lcosetK", "mulgS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_lcoset x A B : (A \subset x *: B) = (x^-1 *: A \subset B).

Proof. by rewrite -(lcosetS x^-1) lcosetK. Qed.

Lemma

sub_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" ]

[ "lcosetK", "lcosetS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_lcosetV x A B : (A \subset x^-1 *: B) = (x *: A \subset B).

Proof. by rewrite sub_lcoset invgK. Qed.

Lemma

sub_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" ]

[ "invgK", "sub_lcoset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosetE A x : rcoset A x = A :* x.

Proof. by rewrite [_ * _]imset2_set1r. Qed.

Lemma

rcosetE

finite_group

finite_group/fingroup.v

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

[ "imset2_set1r", "rcoset" ]

Right cosets.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_rcoset A x : #|A :* x| = #|A|.

Proof. by rewrite -rcosetE (card_imset _ (mulIg _)). Qed.

Lemma

card_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" ]

[ "card_imset", "mulIg", "rcosetE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_rcoset A x y : (y \in A :* x) = (y * x^-1 \in A).

Proof. by rewrite -rcosetE [_ x](can_imset_pre A (mulgK _)) inE. Qed.

Lemma

mem_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" ]

[ "can_imset_pre", "inE", "mulgK", "rcosetE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosetP A x y : reflect (exists2 a, a \in A & y = a * x) (y \in A :* x).

Proof. by rewrite -rcosetE; apply: imsetP. Qed.

Lemma

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" ]

[ "apply", "imsetP", "rcosetE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosetsP A B C : reflect (exists2 x, x \in B & C = A :* x) (C \in rcosets A B).

Proof. by apply: (iffP imsetP) => [] [x Bx ->]; exists x; rewrite ?rcosetE. Qed.

Lemma

rcosetsP

finite_group

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", "imsetP", "rcosetE", "rcosets" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosetM A x y : A :* (x * y) = A :* x :* y.

Proof. by rewrite -mulg_set1 mulgA. Qed.

Lemma

rcosetM

finite_group

finite_group/fingroup.v

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

[ "mulgA", "mulg_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset1 A : A :* 1 = A.

Proof. exact: mulg1. Qed.

Lemma

rcoset1

finite_group

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosetK : right_loop inv (fun A x => A :* x).

Proof. by move=> x A; rewrite -rcosetM mulgV mulg1. Qed.

Lemma

rcosetK

finite_group

finite_group/fingroup.v

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

[ "inv", "mulg1", "mulgV", "rcosetM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosetKV : rev_right_loop inv (fun A x => A :* x).

Proof. by move=> x A; rewrite -rcosetM mulVg mulg1. Qed.

Lemma

rcosetKV

finite_group

finite_group/fingroup.v

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

[ "inv", "mulVg", "mulg1", "rcosetM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_inj : left_injective (fun A x => A :* x).

Proof. by move=> x; apply: can_inj (rcosetK x). Qed.

Lemma

rcoset_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" ]

[ "apply", "rcosetK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosetS x A B : (A :* x \subset B :* x) = (A \subset B).

Proof. apply/idP/idP=> sAB; last exact: mulSg. by rewrite -(rcosetK x A) -(rcosetK x B) mulSg. Qed.

Lemma

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" ]

[ "apply", "last", "mulSg", "rcosetK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_rcoset x A B : (A \subset B :* x) = (A :* x ^-1 \subset B).

Proof. by rewrite -(rcosetS x^-1) rcosetK. Qed.

Lemma

sub_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" ]

[ "rcosetK", "rcosetS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_rcosetV x A B : (A \subset B :* x^-1) = (A :* x \subset B).

Proof. by rewrite sub_rcoset invgK. Qed.

Lemma

sub_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" ]

[ "invgK", "sub_rcoset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg_lcosets A B : (lcosets A B)^-1 = rcosets A^-1 B^-1.

Proof. rewrite /A^-1/= -![_^-1](can_imset_pre _ invgK) -[RHS]imset_comp -imset_comp. by apply: eq_imset => x /=; rewrite lcosetE rcosetE invMg invg_set1. Qed.

Lemma

invg_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" ]

[ "apply", "can_imset_pre", "eq_imset", "imset_comp", "invMg", "invgK", "invg_set1", "lcosetE", "lcosets", "rcosetE", "rcosets" ]

Inverse maps lcosets to rcosets

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_preim A x : A :^ x = (conjg^~ x^-1) @^-1: A.

Proof. exact: can_imset_pre (conjgK _). Qed.

Lemma

conjg_preim

finite_group

finite_group/fingroup.v

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

[ "can_imset_pre", "conjg", "conjgK" ]

Conjugates.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_conjg A x y : (y \in A :^ x) = (y ^ x^-1 \in A).

Proof. by rewrite conjg_preim inE. Qed.

Lemma

mem_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" ]

[ "conjg_preim", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_conjgV A x y : (y \in A :^ x^-1) = (y ^ x \in A).

Proof. by rewrite mem_conjg invgK. Qed.

Lemma

mem_conjgV

finite_group

finite_group/fingroup.v

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

[ "invgK", "mem_conjg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memJ_conjg A x y : (y ^ x \in A :^ x) = (y \in A).

Proof. by rewrite mem_conjg conjgK. Qed.

Lemma

memJ_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" ]

[ "conjgK", "mem_conjg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsgE A x : A :^ x = x^-1 *: (A :* x).

Proof. by apply/setP=> y; rewrite mem_lcoset mem_rcoset -mulgA mem_conjg. Qed.

Lemma

conjsgE

finite_group

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", "mem_conjg", "mem_lcoset", "mem_rcoset", "mulgA", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsg1 A : A :^ 1 = A.

Proof. by rewrite conjsgE invg1 mul1g mulg1. Qed.

Lemma

conjsg1

finite_group

finite_group/fingroup.v

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

[ "conjsgE", "invg1", "mul1g", "mulg1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsgM A x y : A :^ (x * y) = (A :^ x) :^ y.

Proof. by rewrite !conjsgE invMg -!mulg_set1 !mulgA. Qed.

Lemma

conjsgM

finite_group

finite_group/fingroup.v

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

[ "conjsgE", "invMg", "mulgA", "mulg_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsgK : @right_loop _ gT inv conjugate.

Proof. by move=> x A; rewrite -conjsgM mulgV conjsg1. Qed.

Lemma

conjsgK

finite_group

finite_group/fingroup.v

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

[ "conjsg1", "conjsgM", "conjugate", "gT", "inv", "mulgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsgKV : @rev_right_loop _ gT inv conjugate.

Proof. by move=> x A; rewrite -conjsgM mulVg conjsg1. Qed.

Lemma

conjsgKV

finite_group

finite_group/fingroup.v

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

[ "conjsg1", "conjsgM", "conjugate", "gT", "inv", "mulVg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsg_inj : @left_injective _ gT _ conjugate.

Proof. by move=> x; apply: can_inj (conjsgK x). Qed.

Lemma

conjsg_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" ]

[ "apply", "conjsgK", "conjugate", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardJg A x : #|A :^ x| = #|A|.

Proof. by rewrite (card_imset _ (conjg_inj x)). Qed.

Lemma

cardJg

finite_group

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_imset", "conjg_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjSg A B x : (A :^ x \subset B :^ x) = (A \subset B).

Proof. by rewrite !conjsgE lcosetS rcosetS. Qed.

Lemma

conjSg

finite_group

finite_group/fingroup.v

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

[ "conjsgE", "lcosetS", "rcosetS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

properJ A B x : (A :^ x \proper B :^ x) = (A \proper B).

Proof. by rewrite /proper !conjSg. Qed.

Lemma

properJ

finite_group

finite_group/fingroup.v

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

[ "conjSg", "proper" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_conjg A B x : (A :^ x \subset B) = (A \subset B :^ x^-1).

Proof. by rewrite -(conjSg A _ x) conjsgKV. Qed.

Lemma

sub_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" ]

[ "conjSg", "conjsgKV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_conjgV A B x : (A :^ x^-1 \subset B) = (A \subset B :^ x).

Proof. by rewrite -(conjSg _ B x) conjsgKV. Qed.

Lemma

sub_conjgV

finite_group

finite_group/fingroup.v

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

[ "conjSg", "conjsgKV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_set1 x y : [set x] :^ y = [set x ^ y].

Proof. by rewrite [_ :^ _]imset_set1. Qed.

Lemma

conjg_set1

finite_group

finite_group/fingroup.v

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

[ "imset_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjs1g x : 1 :^ x = 1.

Proof. by rewrite conjg_set1 conj1g. Qed.

Lemma

conjs1g

finite_group

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", "conjg_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsg_eq1 A x : (A :^ x == 1%g) = (A == 1%g).

Proof. by rewrite (canF_eq (conjsgK x)) conjs1g. Qed.

Lemma

conjsg_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" ]

[ "canF_eq", "conjs1g", "conjsgK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsMg A B x : (A * B) :^ x = A :^ x * B :^ x.

Proof. by rewrite !conjsgE !mulgA rcosetK. Qed.

Lemma

conjsMg

finite_group

finite_group/fingroup.v

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

[ "conjsgE", "mulgA", "rcosetK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjIg A B x : (A :&: B) :^ x = A :^ x :&: B :^ x.

Proof. by rewrite !conjg_preim preimsetI. Qed.

Lemma

conjIg

finite_group

finite_group/fingroup.v

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

[ "conjg_preim", "preimsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj0g x : set0 :^ x = set0.

Proof. exact: imset0. Qed.

Lemma

conj0g

finite_group

finite_group/fingroup.v

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

[ "imset0", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjTg x : [set: gT] :^ x = [set: gT].

Proof. by rewrite conjg_preim preimsetT. Qed.

Lemma

conjTg

finite_group

finite_group/fingroup.v

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

[ "conjg_preim", "gT", "preimsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigcapJ I r (P : pred I) (B : I -> {set gT}) x : \bigcap_(i <- r | P i) (B i :^ x) = (\bigcap_(i <- r | P i) B i) :^ x.

Proof. by rewrite (big_endo (conjugate^~ x)) => // [B1 B2|]; rewrite (conjTg, conjIg). Qed.

Lemma

bigcapJ

finite_group

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_endo", "conjIg", "conjTg", "conjugate", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjUg A B x : (A :|: B) :^ x = A :^ x :|: B :^ x.

Proof. by rewrite !conjg_preim preimsetU. Qed.

Lemma

conjUg

finite_group

finite_group/fingroup.v

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

[ "conjg_preim", "preimsetU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigcupJ I r (P : pred I) (B : I -> {set gT}) x : \bigcup_(i <- r | P i) (B i :^ x) = (\bigcup_(i <- r | P i) B i) :^ x.

Proof. rewrite (big_endo (conjugate^~ x)) => // [B1 B2|]; first by rewrite conjUg. exact: imset0. Qed.

Lemma

bigcupJ

finite_group

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_endo", "conjUg", "conjugate", "gT", "imset0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjCg A x : (~: A) :^ x = ~: A :^ x.

Proof. by rewrite !conjg_preim preimsetC. Qed.

Lemma

conjCg

finite_group

finite_group/fingroup.v

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

[ "conjg_preim", "preimsetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjDg A B x : (A :\: B) :^ x = A :^ x :\: B :^ x.

Proof. by rewrite !setDE !(conjCg, conjIg). Qed.

Lemma

conjDg

finite_group

finite_group/fingroup.v

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

[ "conjCg", "conjIg", "setDE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjD1g A x : A^# :^ x = (A :^ x)^#.

Proof. by rewrite conjDg conjs1g. Qed.

Lemma

conjD1g

finite_group

finite_group/fingroup.v

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

[ "conjDg", "conjs1g" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memJ_class x y A : y \in A -> x ^ y \in x ^: A.

Proof. exact: imset_f. Qed.

Lemma

memJ_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" ]

[ "imset_f" ]

Classes; not much for now.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classS x A B : A \subset B -> x ^: A \subset x ^: B.

Proof. exact: imsetS. Qed.

Lemma

classS

finite_group

finite_group/fingroup.v

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

[ "imsetS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_set1 x y : x ^: [set y] = [set x ^ y].

Proof. exact: imset_set1. Qed.

Lemma

class_set1

finite_group

finite_group/fingroup.v

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

[ "imset_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class1g x A : x \in A -> 1 ^: A = 1.

Proof. move=> Ax; apply/setP=> y. by apply/imsetP/set1P=> [[a Aa]|] ->; last exists x; rewrite ?conj1g. 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" ]

[ "apply", "conj1g", "imsetP", "last", "set1P", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classVg x A : x^-1 ^: A = (x ^: A)^-1.

Proof. apply/setP=> xy; rewrite inE; apply/imsetP/imsetP=> [] [y Ay def_xy]. by rewrite def_xy conjVg invgK; exists y. by rewrite -[xy]invgK def_xy -conjVg; exists y. Qed.

Lemma

classVg

finite_group

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", "conjVg", "imsetP", "inE", "invgK", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_classes x A : x \in A -> x ^: A \in classes A.

Proof. exact: imset_f. Qed.

Lemma

mem_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" ]

[ "classes", "imset_f" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memJ_class_support A B x y : x \in A -> y \in B -> x ^ y \in class_support A B.

Proof. by move=> Ax By; apply: imset2_f. Qed.

Lemma

memJ_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" ]

[ "apply", "class_support", "imset2_f" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_supportM A B C : class_support A (B * C) = class_support (class_support A B) C.

Proof. apply/setP=> x; apply/imset2P/imset2P=> [[a y Aa] | [y c]]. case/mulsgP=> b c Bb Cc -> ->{x y}. by exists (a ^ b) c; rewrite ?(imset2_f, conjgM). case/imset2P=> a b Aa Bb -> Cc ->{x y}. by exists a (b * c); rewrite ?(mem_mulg, conjgM). Qed.

Lemma

class_supportM

finite_group

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", "conjgM", "imset2P", "imset2_f", "mem_mulg", "mulsgP", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_support_set1l A x : class_support [set x] A = x ^: A.

Proof. exact: imset2_set1l. Qed.

Lemma

class_support_set1l

finite_group

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", "imset2_set1l" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_support_set1r A x : class_support A [set x] = A :^ x.

Proof. exact: imset2_set1r. Qed.

Lemma

class_support_set1r

finite_group

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", "imset2_set1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classM x A B : x ^: (A * B) = class_support (x ^: A) B.

Proof. by rewrite -!class_support_set1l class_supportM. Qed.

Lemma

classM

finite_group

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_set1l" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_lcoset x y A : x ^: (y *: A) = (x ^ y) ^: A.

Proof. by rewrite classM class_set1 class_support_set1l. Qed.

Lemma

class_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" ]

[ "classM", "class_set1", "class_support_set1l" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_rcoset x A y : x ^: (A :* y) = (x ^: A) :^ y.

Proof. by rewrite -class_support_set1r classM. Qed.

Lemma

class_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" ]

[ "classM", "class_support_set1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjugatesS A B C : B \subset C -> A :^: B \subset A :^: C.

Proof. exact: imsetS. Qed.

Lemma

conjugatesS

finite_group

finite_group/fingroup.v

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

[ "imsetS" ]

Conjugate set.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjugates_set1 A x : A :^: [set x] = [set A :^ x].

Proof. exact: imset_set1. Qed.

Lemma

conjugates_set1

finite_group

finite_group/fingroup.v

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

[ "imset_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjugates_conj A x B : (A :^ x) :^: B = A :^: (x *: B).

Proof. rewrite /conjugates [x *: B]imset2_set1l -imset_comp. by apply: eq_imset => y /=; rewrite conjsgM. Qed.

Lemma

conjugates_conj

finite_group

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", "conjsgM", "conjugates", "eq_imset", "imset2_set1l", "imset_comp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_supportEl A B : class_support A B = \bigcup_(x in A) x ^: B.

Proof. exact: curry_imset2l. Qed.

Lemma

class_supportEl

finite_group

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", "curry_imset2l" ]

Class support.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_supportEr A B : class_support A B = \bigcup_(x in B) A :^ x.

Proof. exact: curry_imset2r. Qed.

Lemma

class_supportEr

finite_group

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", "curry_imset2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set A

:= (1 \in A) && (A * A \subset A).

Definition

group_set

finite_group

finite_group/fingroup.v

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

[]

Groups (at last!)

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_setP A : reflect (1 \in A /\ {in A & A, forall x y, x * y \in A}) (group_set A).

Proof. apply: (iffP andP) => [] [A1 AM]; split=> {A1}//. by move=> x y Ax Ay; apply: (subsetP AM); rewrite mem_mulg. by apply/subsetP=> _ /mulsgP[x y Ax Ay ->]; apply: AM. Qed.

Lemma

group_setP

finite_group

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", "group_set", "mem_mulg", "mulsgP", "split", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_type : Type

:= Group { gval :> GroupSet.sort gT; _ : group_set gval }.

Structure

group_type

finite_group

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", "group_set", "sort" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_of : predArgType

:= group_type.

Definition

group_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" ]

[ "group_type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupT

:= group_of.

Notation

groupT

finite_group

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_of" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group (A : {set gT}) gA : groupT

:= @Group A gA.

Definition

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" ]

[ "gT", "groupT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone_group G

:= let: Group _ gP := G return {type of Group for G} -> groupT in fun k => k gP.

Definition

clone_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" ]

[ "groupT", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_inj : injective gval.

Proof. exact: val_inj. Qed.

Lemma

group_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" ]

[ "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupP (G : groupT) : group_set G.

Proof. by case: G. Qed.

Lemma

groupP

finite_group

finite_group/fingroup.v

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

[ "groupT", "group_set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

congr_group (H K : groupT) : H = K -> H :=: K.

Proof. exact: congr1. Qed.

Lemma

congr_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" ]

[ "groupT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isgroupP A : reflect (exists G : groupT, A = G) (group_set A).

Proof. by apply: (iffP idP) => [gA | [[B gB] -> //]]; exists (Group gA). Qed.

Lemma

isgroupP

finite_group

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", "groupT", "group_set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_one : group_set 1.

Proof. by rewrite /group_set set11 mulg1 subxx. Qed.

Lemma

group_set_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" ]

[ "group_set", "mulg1", "set11", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

one_group

:= group group_set_one.

Canonical

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" ]

[ "group", "group_set_one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1_group

:= @group [set 1] group_set_one.

Canonical

set1_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_one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_setT : group_set (setTfor gT).

Proof. by apply/group_setP; split=> [|x y _ _]; rewrite inE. Qed.

Lemma

group_setT

finite_group

finite_group/fingroup.v

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

[ "apply", "gT", "group_set", "group_setP", "inE", "setTfor", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setT_group

:= group group_setT.

Canonical

setT_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_setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'group' gT }"

:= (group_of gT) (format "{ 'group' gT }") : type_scope.

Notation

{ 'group' gT }

finite_group

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", "group_of" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'group' 'of' G ]"

:= (clone_group (@group _ G)) (format "[ 'group' 'of' G ]") : form_scope.

Notation

[ 'group' 'of' G ]

finite_group

finite_group/fingroup.v

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

[ "clone_group", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"1"

:= (one_group _) : Group_scope.

Notation

1

finite_group

finite_group/fingroup.v

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

[ "one_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 1 gT ]"

:= (1%G : {group gT}) : Group_scope.

Notation

[ 1 gT ]

finite_group

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", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'set' : gT ]"

:= (setT_group gT) : Group_scope.

Notation

[ 'set' : gT ]

finite_group

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", "setT_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

generated_unlockable

:= Unlockable generated.unlock.

Canonical

generated_unlockable

finite_group

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

gcore (gT : finGroupType) (A B : {set gT})

:= \bigcap_(x in B) A :^ x.

Definition

gcore

finite_group

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing (gT : finGroupType) (A B : {set gT})

:= generated (A :|: B).

Definition

joing

finite_group

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commutator (gT : finGroupType) (A B : {set gT})

:= generated (commg_set A B).

Definition

commutator

finite_group

finite_group/fingroup.v

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

[ "commg_set", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle (gT : finGroupType) (x : gT)

:= generated [set x].

Definition

cycle

finite_group

finite_group/fingroup.v

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

[ "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

order (gT : finGroupType) (x : gT)

:= #|cycle x|.

Definition

order

finite_group

finite_group/fingroup.v

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

[ "cycle", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gsort gT

:= (FinStarMonoid.arg_sort gT%type) (only parsing).

Notation

gsort

finite_group

finite_group/fingroup.v

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

[ "arg_sort", "gT", "type" ]

Notation below.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<< A >>"

:= (generated A) : group_scope.

Notation

<< A >>

finite_group

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

"<[ x ] >"

:= (cycle x) : group_scope.

Notation

<[ x ] >

finite_group

finite_group/fingroup.v

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

[ "cycle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"#[ x ]"

:= (order x) : group_scope.

Notation

#[ x ]

finite_group

finite_group/fingroup.v

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

[ "order" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A <*> B"

:= (joing A B) : group_scope.

Notation

A <*> B

finite_group

finite_group/fingroup.v

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

[ "joing" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ ~: A1 , A2 , .. , An ]"

:= (commutator .. (commutator A1 A2) .. An) : group_scope.

Notation

[ ~: A1 , A2 , .. , An ]

finite_group

finite_group/fingroup.v

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

[ "commutator" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sT

:= {set gT}.

Notation

sT

finite_group

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

valG : val G = G.

Proof. by []. Qed.

Lemma

valG

finite_group

finite_group/fingroup.v

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

[ "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group1 : 1 \in G.

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

Lemma

group1

finite_group

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" ]

Non-triviality.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d