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
conjg_eq1	:= conjg_eq1 (only parsing).	Notation	conjg_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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_prod	:= conjg_prod (only parsing).	Notation	conjg_prod	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgEl	:= commgEl (only parsing).	Notation	commgEl	finite_group	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
commgEr	:= commgEr (only parsing).	Notation	commgEr	finite_group	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
commgC	:= commgC (only parsing).	Notation	commgC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgCV	:= commgCV (only parsing).	Notation	commgCV	finite_group	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
conjRg	:= conjRg (only parsing).	Notation	conjRg	finite_group	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
invg_comm	:= invgR (only parsing).	Notation	invg_comm	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "invgR" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgP	:= commgP (only parsing).	Notation	commgP	finite_group	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
conjg_fixP	:= conjg_fixP (only parsing).	Notation	conjg_fixP	finite_group	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
commg1_sym	:= commg1_sym (only parsing).	Notation	commg1_sym	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commg1	:= commg1 (only parsing).	Notation	commg1	finite_group	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
comm1g	:= comm1g (only parsing).	Notation	comm1g	finite_group	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
commgg	:= commgg (only parsing).	Notation	commgg	finite_group	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
commgXg	:= commgXg (only parsing).	Notation	commgXg	finite_group	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
commgVg	:= commgVg (only parsing).	Notation	commgVg	finite_group	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
commgXVg	:= commgXVg (only parsing).	Notation	commgXVg	finite_group	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
repr A	:= if 1 \in A then 1 else odflt 1 [pick x in A].	Definition	repr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "pick" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_repr A x : x \in A -> repr A \in A.	Proof. by rewrite /repr; case: ifP => // _; case: pickP => // A0; rewrite [x \in A]A0. Qed.	Lemma	mem_repr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "pickP", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_mem_repr A : #\|A\| > 0 -> repr A \in A.	Proof. by rewrite lt0n => /existsP[x]; apply: mem_repr. Qed.	Lemma	card_mem_repr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "existsP", "lt0n", "mem_repr", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_set1 x : repr [set x] = x.	Proof. by apply/set1P/card_mem_repr; rewrite cards1. Qed.	Lemma	repr_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" ]	[ "apply", "card_mem_repr", "cards1", "repr", "set1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_set0 : repr set0 = 1.	Proof. by rewrite /repr; case: pickP => [x\|_] /[!inE]. Qed.	Lemma	repr_set0	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inE", "pickP", "repr", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_mulg A B	:= mul @2: (A, B).	Definition	set_mulg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mul" ]	Set-lifted group operations.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_invg A	:= inv @^-1: A.	Definition	set_invg	finite_group	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_mul1g : left_id [set 1] set_mulg.	Proof. move=> A; apply/setP=> y; apply/imset2P/idP=> [[_ x /set1P-> Ax ->] \| Ay]. by rewrite mul1g. by exists (1 : gT) y; rewrite ?(set11, mul1g). Qed.	Lemma	set_mul1g	finite_group	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", "imset2P", "mul1g", "set11", "set1P", "setP", "set_mulg" ]	The pre-group structure of group subsets.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_mulgA : associative set_mulg.	Proof. move=> A B C; apply/setP=> y. apply/imset2P/imset2P=> [[x1 z Ax1 /imset2P[x2 x3 Bx2 Cx3 ->] ->]\| [z x3]]. by exists (x1 * x2) x3; rewrite ?mulgA //; apply/imset2P; exists x1 x2. case/imset2P=> x1 x2 Ax1 Bx2 -> Cx3 ->. by exists x1 (x2 * x3); rewrite ?mulgA //; apply/imset2P; exists x2 x3. Qed.	Lemma	set_mulgA	finite_group	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", "imset2P", "mulgA", "setP", "set_mulg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_invgK : involutive set_invg.	Proof. by move=> A; apply/setP=> x; rewrite !inE invgK. Qed.	Lemma	set_invgK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "inE", "invgK", "setP", "set_invg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_invgM : {morph set_invg : A B / set_mulg A B >-> set_mulg B A}.	Proof. move=> A B; apply/setP=> z; rewrite inE. apply/imset2P/imset2P=> [[x y Ax By /(canRL invgK)->] \| [y x]]. by exists y^-1 x^-1; rewrite ?invMg // inE invgK. by rewrite !inE => By1 Ax1 ->; exists x^-1 y^-1; rewrite ?invMg. Qed.	Lemma	set_invgM	finite_group	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", "imset2P", "inE", "invMg", "invgK", "setP", "set_invg", "set_mulg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort (gT : finStarMonoidType)	:= {set gT}.	Definition	sort	finite_group	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
sort (gT : finStarMonoidType)	:= FinStarMonoid.arg_sort {set gT}.	Definition	sort	finite_group	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort (gT : finStarMonoidType)	:= Magma.sort {set gT}.	Definition	sort	finite_group	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
sort (gT : finStarMonoidType)	:= BaseGroup.sort {set gT}.	Definition	sort	finite_group	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
lcoset A x	:= mul x @: A.	Definition	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" ]	[ "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset A x	:= mul^~ x @: A.	Definition	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" ]	[ "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosets A B	:= lcoset A @: B.	Definition	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" ]	[ "lcoset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosets A B	:= rcoset A @: B.	Definition	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" ]	[ "rcoset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexg B A	:= #\|rcosets A B\|.	Definition	indexg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "rcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjugate A x	:= conjg^~ x @: A.	Definition	conjugate	finite_group	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjugates A B	:= conjugate A @: B.	Definition	conjugates	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjugate" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class x B	:= conjg x @: B.	Definition	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" ]	[ "conjg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classes A	:= class^~ A @: A.	Definition	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" ]	[ "class" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
class_support A B	:= conjg @2: (A, B).	Definition	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" ]	[ "conjg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commg_set A B	:= commg @2: (A, B).	Definition	commg_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" ]	[ "commg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normaliser A	:= [set x \| conjugate A x \subset A].	Definition	normaliser	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjugate" ]	keep all the Notation together.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser A	:= \bigcap_(x in A) normaliser [set x].	Definition	centraliser	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normaliser" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian A	:= A \subset centraliser A.	Definition	abelian	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centraliser" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal A B	:= (A \subset B) && (B \subset normaliser A).	Definition	normal	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "normaliser" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normalised A	:= forall x, conjugate A x = A.	Definition	normalised	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjugate" ]	the {in ...} form, as in abelian below.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centralises x A	:= forall y, y \in A -> commute x y.	Definition	centralises	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "commute" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centralised A	:= forall x, centralises x A.	Definition	centralised	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "centralises" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 1 gT ]"	:= (1 : {set 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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 1 ]"	:= [1 FinGroup.sort _] : 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" ]	[ "sort" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A ^#"	:= (A :\ 1) : 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 *: A"	:= ([set x%g] * A) : group_scope.	Notation	x *: 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
"A :* x"	:= (A * [set x%g]) : group_scope.	Notation	A :* 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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A :^ x"	:= (conjugate A x) : group_scope.	Notation	A :^ 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" ]	[ "conjugate" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x ^: B"	:= (class x B) : group_scope.	Notation	x ^: 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" ]	[ "class" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A :^: B"	:= (conjugates 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" ]	[ "conjugates" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"#\| B : A \|"	:= (indexg B A) : group_scope.	Notation	#\| B : 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" ]	[ "indexg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''N' ( A )"	:= (normaliser A) : group_scope.	Notation	''N' ( 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" ]	[ "normaliser" ]	No notation for the set commutator generator set commg_set.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''N_' G ( A )"	:= (G%g :&: 'N(A)) : group_scope.	Notation	''N_' G ( 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
"A <\| B"	:= (normal 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" ]	[ "normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C' ( A )"	:= (centraliser A) : group_scope.	Notation	''C' ( 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" ]	[ "centraliser" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' G ( A )"	:= (G%g :&: 'C(A)) : group_scope.	Notation	''C_' G ( 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
"''C_' ( G ) ( A )"	:= 'C_G(A) (only parsing) : group_scope.	Notation	''C_' ( G ) ( 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
"''C' [ x ]"	:= 'N([set x%g]) : group_scope.	Notation	''C' [ 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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' G [ x ]"	:= 'N_G([set x%g]) : group_scope.	Notation	''C_' G [ 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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( G ) [ x ]"	:= 'C_G[x] (only parsing) : group_scope.	Notation	''C_' ( G ) [ 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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulsgP A B x : reflect (imset2_spec mul (mem A) (fun _ => mem B) x) (x \in A * B).	Proof. exact: imset2P. Qed.	Lemma	mulsgP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2P", "imset2_spec", "mul" ]	only need to add the monotonicity rules.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_mulg A B x y : x \in A -> y \in B -> x * y \in A * B.	Proof. by move=> Ax By; apply/mulsgP; exists x y. Qed.	Lemma	mem_mulg	finite_group	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", "mulsgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodsgP (I : finType) (P : pred I) (A : I -> {set gT}) x : reflect (exists2 c, forall i, P i -> c i \in A i & x = \prod_(i \| P i) c i) (x \in \prod_(i \| P i) A i).	Proof. have [r big_r [Ur mem_r] _] := big_enumP P. pose inA c := all (fun i => c i \in A i); rewrite -big_r; set piAx := x \in _. suffices{big_r} IHr: reflect (exists2 c, inA c r & x = \prod_(i <- r) c i) piAx. apply: (iffP IHr) => -[c inAc ->]; do [exists c; last by rewrite big_r]. by move=> i Pi; rewrite (allP ...	Lemma	prodsgP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "all", "allP", "apply", "big_cons", "big_enumP", "eq_big_seq", "eqxx", "gT", "inA", "last", "memPn", "mem_mulg", "mulsgP", "set1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_prodg (I : finType) (P : pred I) (A : I -> {set gT}) c : (forall i, P i -> c i \in A i) -> \prod_(i \| P i) c i \in \prod_(i \| P i) A i.	Proof. by move=> Ac; apply/prodsgP; exists c. Qed.	Lemma	mem_prodg	finite_group	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", "prodsgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulSg A B C : A \subset B -> A * C \subset B * C.	Proof. exact: imset2Sl. Qed.	Lemma	mulSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2Sl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgS A B C : B \subset C -> A * B \subset A * C.	Proof. exact: imset2Sr. Qed.	Lemma	mulgS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2Sr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgSS A B C D : A \subset B -> C \subset D -> A * C \subset B * D.	Proof. exact: imset2S. Qed.	Lemma	mulgSS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2S" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg_subl A B : 1 \in B -> A \subset A * B.	Proof. by move=> B1; rewrite -{1}(mulg1 A) mulgS ?sub1set. Qed.	Lemma	mulg_subl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mulg1", "mulgS", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg_subr A B : 1 \in A -> B \subset A * B.	Proof. by move=> A1; rewrite -{1}(mul1g B) mulSg ?sub1set. Qed.	Lemma	mulg_subr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mul1g", "mulSg", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulUg A B C : (A :\|: B) * C = (A * C) :\|: (B * C).	Proof. exact: imset2Ul. Qed.	Lemma	mulUg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2Ul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgU A B C : A * (B :\|: C) = (A * B) :\|: (A * C).	Proof. exact: imset2Ur. Qed.	Lemma	mulgU	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "imset2Ur" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invUg A B : (A :\|: B)^-1 = A^-1 :\|: B^-1.	Proof. exact: preimsetU. Qed.	Lemma	invUg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "preimsetU" ]	Set (pointwise) inverse.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invIg A B : (A :&: B)^-1 = A^-1 :&: B^-1.	Proof. exact: preimsetI. Qed.	Lemma	invIg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "preimsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invDg A B : (A :\: B)^-1 = A^-1 :\: B^-1.	Proof. exact: preimsetD. Qed.	Lemma	invDg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "preimsetD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invCg A : (~: A)^-1 = ~: A^-1.	Proof. exact: preimsetC. Qed.	Lemma	invCg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "preimsetC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invSg A B : (A^-1 \subset B^-1) = (A \subset B).	Proof. by rewrite !(sameP setIidPl eqP) -invIg (inj_eq invg_inj). Qed.	Lemma	invSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inj_eq", "invIg", "invg_inj", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_invg x A : (x \in A^-1) = (x^-1 \in A).	Proof. by rewrite inE. Qed.	Lemma	mem_invg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memV_invg x A : (x^-1 \in A^-1) = (x \in A).	Proof. by rewrite inE invgK. Qed.	Lemma	memV_invg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inE", "invgK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_invg A : #\|A^-1\| = #\|A\|.	Proof. exact/card_preimset/invg_inj. Qed.	Lemma	card_invg	finite_group	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_preimset", "invg_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gE : 1 = [set 1] :> {set gT}.	Proof. by []. Qed.	Lemma	set1gE	finite_group	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" ]	Product with singletons.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gP x : reflect (x = 1) (x \in [1 gT]).	Proof. exact: set1P. Qed.	Lemma	set1gP	finite_group	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", "set1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg_set1 x y : [set x] :* y = [set x * y].	Proof. by rewrite [_ * _]imset2_set1l imset_set1. Qed.	Lemma	mulg_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" ]	[ "imset2_set1l", "imset_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_set1 x : [set x]^-1 = [set x^-1].	Proof. by apply/setP=> y; rewrite !inE inv_eq //; apply: invgK. Qed.	Lemma	invg_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" ]	[ "apply", "inE", "inv_eq", "invgK", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosetE A x : lcoset A x = x *: A.	Proof. by rewrite [_ * _]imset2_set1l. Qed.	Lemma	lcosetE	finite_group	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_set1l", "lcoset" ]	Left cosets.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_lcoset A x : #\|x *: A\| = #\|A\|.	Proof. by rewrite -lcosetE (card_imset _ (mulgI _)). Qed.	Lemma	card_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" ]	[ "card_imset", "lcosetE", "mulgI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_lcoset A x y : (y \in x : A) = (x^-1 y \in A).	Proof. by rewrite -lcosetE [_ x](can_imset_pre _ (mulKg _)) inE. Qed.	Lemma	mem_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" ]	[ "can_imset_pre", "inE", "lcosetE", "mulKg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosetP A x y : reflect (exists2 a, a \in A & y = x * a) (y \in x *: A).	Proof. by rewrite -lcosetE; apply: imsetP. Qed.	Lemma	lcosetP	finite_group	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", "lcosetE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosetsP A B C : reflect (exists2 x, x \in B & C = x *: A) (C \in lcosets A B).	Proof. by apply: (iffP imsetP) => [] [x Bx ->]; exists x; rewrite ?lcosetE. Qed.	Lemma	lcosetsP	finite_group	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", "lcosetE", "lcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosetM A x y : (x * y) : A = x : (y *: A).	Proof. by rewrite -mulg_set1 mulgA. Qed.	Lemma	lcosetM	finite_group	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
lcoset1 A : 1 *: A = A.	Proof. exact: mul1g. Qed.	Lemma	lcoset1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mul1g" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosetK : left_loop inv (fun x A => x *: A).	Proof. by move=> x A; rewrite -lcosetM mulVg mul1g. Qed.	Lemma	lcosetK	finite_group	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", "lcosetM", "mul1g", "mulVg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcosetKV : rev_left_loop inv (fun x A => x *: A).	Proof. by move=> x A; rewrite -lcosetM mulgV mul1g. Qed.	Lemma	lcosetKV	finite_group	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", "lcosetM", "mul1g", "mulgV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_eq1

:= conjg_eq1 (only parsing).

Notation

conjg_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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_prod

:= conjg_prod (only parsing).

Notation

conjg_prod

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commgEl

:= commgEl (only parsing).

Notation

commgEl

finite_group

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

commgEr

:= commgEr (only parsing).

Notation

commgEr

finite_group

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

commgC

:= commgC (only parsing).

Notation

commgC

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commgCV

:= commgCV (only parsing).

Notation

commgCV

finite_group

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

conjRg

:= conjRg (only parsing).

Notation

conjRg

finite_group

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

invg_comm

:= invgR (only parsing).

Notation

invg_comm

finite_group

finite_group/fingroup.v

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

[ "invgR" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commgP

:= commgP (only parsing).

Notation

commgP

finite_group

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

conjg_fixP

:= conjg_fixP (only parsing).

Notation

conjg_fixP

finite_group

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

commg1_sym

:= commg1_sym (only parsing).

Notation

commg1_sym

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commg1

:= commg1 (only parsing).

Notation

commg1

finite_group

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

comm1g

:= comm1g (only parsing).

Notation

comm1g

finite_group

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

commgg

:= commgg (only parsing).

Notation

commgg

finite_group

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

commgXg

:= commgXg (only parsing).

Notation

commgXg

finite_group

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

commgVg

:= commgVg (only parsing).

Notation

commgVg

finite_group

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

commgXVg

:= commgXVg (only parsing).

Notation

commgXVg

finite_group

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

repr A

:= if 1 \in A then 1 else odflt 1 [pick x in A].

Definition

repr

finite_group

finite_group/fingroup.v

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

[ "pick" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_repr A x : x \in A -> repr A \in A.

Proof. by rewrite /repr; case: ifP => // _; case: pickP => // A0; rewrite [x \in A]A0. Qed.

Lemma

mem_repr

finite_group

finite_group/fingroup.v

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

[ "pickP", "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_mem_repr A : #|A| > 0 -> repr A \in A.

Proof. by rewrite lt0n => /existsP[x]; apply: mem_repr. Qed.

Lemma

card_mem_repr

finite_group

finite_group/fingroup.v

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

[ "apply", "existsP", "lt0n", "mem_repr", "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_set1 x : repr [set x] = x.

Proof. by apply/set1P/card_mem_repr; rewrite cards1. Qed.

Lemma

repr_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" ]

[ "apply", "card_mem_repr", "cards1", "repr", "set1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_set0 : repr set0 = 1.

Proof. by rewrite /repr; case: pickP => [x|_] /[!inE]. Qed.

Lemma

repr_set0

finite_group

finite_group/fingroup.v

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

[ "inE", "pickP", "repr", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_mulg A B

:= mul @2: (A, B).

Definition

set_mulg

finite_group

finite_group/fingroup.v

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

[ "mul" ]

Set-lifted group operations.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_invg A

:= inv @^-1: A.

Definition

set_invg

finite_group

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_mul1g : left_id [set 1] set_mulg.

Proof. move=> A; apply/setP=> y; apply/imset2P/idP=> [[_ x /set1P-> Ax ->] | Ay]. by rewrite mul1g. by exists (1 : gT) y; rewrite ?(set11, mul1g). Qed.

Lemma

set_mul1g

finite_group

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", "imset2P", "mul1g", "set11", "set1P", "setP", "set_mulg" ]

The pre-group structure of group subsets.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_mulgA : associative set_mulg.

Proof. move=> A B C; apply/setP=> y. apply/imset2P/imset2P=> [[x1 z Ax1 /imset2P[x2 x3 Bx2 Cx3 ->] ->]| [z x3]]. by exists (x1 * x2) x3; rewrite ?mulgA //; apply/imset2P; exists x1 x2. case/imset2P=> x1 x2 Ax1 Bx2 -> Cx3 ->. by exists x1 (x2 * x3); rewrite ?mulgA //; apply/imset2P; exists x2 x3. Qed.

Lemma

set_mulgA

finite_group

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", "imset2P", "mulgA", "setP", "set_mulg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_invgK : involutive set_invg.

Proof. by move=> A; apply/setP=> x; rewrite !inE invgK. Qed.

Lemma

set_invgK

finite_group

finite_group/fingroup.v

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

[ "apply", "inE", "invgK", "setP", "set_invg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_invgM : {morph set_invg : A B / set_mulg A B >-> set_mulg B A}.

Proof. move=> A B; apply/setP=> z; rewrite inE. apply/imset2P/imset2P=> [[x y Ax By /(canRL invgK)->] | [y x]]. by exists y^-1 x^-1; rewrite ?invMg // inE invgK. by rewrite !inE => By1 Ax1 ->; exists x^-1 y^-1; rewrite ?invMg. Qed.

Lemma

set_invgM

finite_group

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", "imset2P", "inE", "invMg", "invgK", "setP", "set_invg", "set_mulg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort (gT : finStarMonoidType)

:= {set gT}.

Definition

sort

finite_group

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

sort (gT : finStarMonoidType)

:= FinStarMonoid.arg_sort {set gT}.

Definition

sort

finite_group

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort (gT : finStarMonoidType)

:= Magma.sort {set gT}.

Definition

sort

finite_group

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

sort (gT : finStarMonoidType)

:= BaseGroup.sort {set gT}.

Definition

sort

finite_group

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

lcoset A x

:= mul x @: A.

Definition

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" ]

[ "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset A x

:= mul^~ x @: A.

Definition

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" ]

[ "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcosets A B

:= lcoset A @: B.

Definition

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" ]

[ "lcoset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosets A B

:= rcoset A @: B.

Definition

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" ]

[ "rcoset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexg B A

:= #|rcosets A B|.

Definition

indexg

finite_group

finite_group/fingroup.v

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

[ "rcosets" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjugate A x

:= conjg^~ x @: A.

Definition

conjugate

finite_group

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjugates A B

:= conjugate A @: B.

Definition

conjugates

finite_group

finite_group/fingroup.v

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

[ "conjugate" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class x B

:= conjg x @: B.

Definition

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" ]

[ "conjg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classes A

:= class^~ A @: A.

Definition

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" ]

[ "class" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

class_support A B

:= conjg @2: (A, B).

Definition

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" ]

[ "conjg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commg_set A B

:= commg @2: (A, B).

Definition

commg_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" ]

[ "commg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normaliser A

:= [set x | conjugate A x \subset A].

Definition

normaliser

finite_group

finite_group/fingroup.v

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

[ "conjugate" ]

keep all the Notation together.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centraliser A

:= \bigcap_(x in A) normaliser [set x].

Definition

centraliser

finite_group

finite_group/fingroup.v

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

[ "normaliser" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian A

:= A \subset centraliser A.

Definition

abelian

finite_group

finite_group/fingroup.v

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

[ "centraliser" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal A B

:= (A \subset B) && (B \subset normaliser A).

Definition

normal

finite_group

finite_group/fingroup.v

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

[ "normaliser" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normalised A

:= forall x, conjugate A x = A.

Definition

normalised

finite_group

finite_group/fingroup.v

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

[ "conjugate" ]

the {in ...} form, as in abelian below.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centralises x A

:= forall y, y \in A -> commute x y.

Definition

centralises

finite_group

finite_group/fingroup.v

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

[ "commute" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centralised A

:= forall x, centralises x A.

Definition

centralised

finite_group

finite_group/fingroup.v

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

[ "centralises" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 1 gT ]"

:= (1 : {set 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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 1 ]"

:= [1 FinGroup.sort _] : 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" ]

[ "sort" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A ^#"

:= (A :\ 1) : 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 *: A"

:= ([set x%g] * A) : group_scope.

Notation

x *: 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

"A :* x"

:= (A * [set x%g]) : group_scope.

Notation

A :* 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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A :^ x"

:= (conjugate A x) : group_scope.

Notation

A :^ 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" ]

[ "conjugate" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x ^: B"

:= (class x B) : group_scope.

Notation

x ^: 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" ]

[ "class" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A :^: B"

:= (conjugates 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" ]

[ "conjugates" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"#| B : A |"

:= (indexg B A) : group_scope.

Notation

#| B : 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" ]

[ "indexg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''N' ( A )"

:= (normaliser A) : group_scope.

Notation

''N' ( 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" ]

[ "normaliser" ]

No notation for the set commutator generator set commg_set.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''N_' G ( A )"

:= (G%g :&: 'N(A)) : group_scope.

Notation

''N_' G ( 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

"A <| B"

:= (normal 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" ]

[ "normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C' ( A )"

:= (centraliser A) : group_scope.

Notation

''C' ( 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" ]

[ "centraliser" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' G ( A )"

:= (G%g :&: 'C(A)) : group_scope.

Notation

''C_' G ( 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

"''C_' ( G ) ( A )"

:= 'C_G(A) (only parsing) : group_scope.

Notation

''C_' ( G ) ( 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

"''C' [ x ]"

:= 'N([set x%g]) : group_scope.

Notation

''C' [ 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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' G [ x ]"

:= 'N_G([set x%g]) : group_scope.

Notation

''C_' G [ 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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( G ) [ x ]"

:= 'C_G[x] (only parsing) : group_scope.

Notation

''C_' ( G ) [ 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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulsgP A B x : reflect (imset2_spec mul (mem A) (fun _ => mem B) x) (x \in A * B).

Proof. exact: imset2P. Qed.

Lemma

mulsgP

finite_group

finite_group/fingroup.v

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

[ "imset2P", "imset2_spec", "mul" ]

only need to add the monotonicity rules.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_mulg A B x y : x \in A -> y \in B -> x * y \in A * B.

Proof. by move=> Ax By; apply/mulsgP; exists x y. Qed.

Lemma

mem_mulg

finite_group

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", "mulsgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodsgP (I : finType) (P : pred I) (A : I -> {set gT}) x : reflect (exists2 c, forall i, P i -> c i \in A i & x = \prod_(i | P i) c i) (x \in \prod_(i | P i) A i).

Proof. have [r big_r [Ur mem_r] _] := big_enumP P. pose inA c := all (fun i => c i \in A i); rewrite -big_r; set piAx := x \in _. suffices{big_r} IHr: reflect (exists2 c, inA c r & x = \prod_(i <- r) c i) piAx. apply: (iffP IHr) => -[c inAc ->]; do [exists c; last by rewrite big_r]. by move=> i Pi; rewrite (allP ...

Lemma

prodsgP

finite_group

finite_group/fingroup.v

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

[ "all", "allP", "apply", "big_cons", "big_enumP", "eq_big_seq", "eqxx", "gT", "inA", "last", "memPn", "mem_mulg", "mulsgP", "set1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_prodg (I : finType) (P : pred I) (A : I -> {set gT}) c : (forall i, P i -> c i \in A i) -> \prod_(i | P i) c i \in \prod_(i | P i) A i.

Proof. by move=> Ac; apply/prodsgP; exists c. Qed.

Lemma

mem_prodg

finite_group

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", "prodsgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulSg A B C : A \subset B -> A * C \subset B * C.

Proof. exact: imset2Sl. Qed.

Lemma

mulSg

finite_group

finite_group/fingroup.v

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

[ "imset2Sl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgS A B C : B \subset C -> A * B \subset A * C.

Proof. exact: imset2Sr. Qed.

Lemma

mulgS

finite_group

finite_group/fingroup.v

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

[ "imset2Sr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgSS A B C D : A \subset B -> C \subset D -> A * C \subset B * D.

Proof. exact: imset2S. Qed.

Lemma

mulgSS

finite_group

finite_group/fingroup.v

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

[ "imset2S" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulg_subl A B : 1 \in B -> A \subset A * B.

Proof. by move=> B1; rewrite -{1}(mulg1 A) mulgS ?sub1set. Qed.

Lemma

mulg_subl

finite_group

finite_group/fingroup.v

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

[ "mulg1", "mulgS", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulg_subr A B : 1 \in A -> B \subset A * B.

Proof. by move=> A1; rewrite -{1}(mul1g B) mulSg ?sub1set. Qed.

Lemma

mulg_subr

finite_group

finite_group/fingroup.v

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

[ "mul1g", "mulSg", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulUg A B C : (A :|: B) * C = (A * C) :|: (B * C).

Proof. exact: imset2Ul. Qed.

Lemma

mulUg

finite_group

finite_group/fingroup.v

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

[ "imset2Ul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgU A B C : A * (B :|: C) = (A * B) :|: (A * C).

Proof. exact: imset2Ur. Qed.

Lemma

mulgU

finite_group

finite_group/fingroup.v

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

[ "imset2Ur" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invUg A B : (A :|: B)^-1 = A^-1 :|: B^-1.

Proof. exact: preimsetU. Qed.

Lemma

invUg

finite_group

finite_group/fingroup.v

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

[ "preimsetU" ]

Set (pointwise) inverse.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invIg A B : (A :&: B)^-1 = A^-1 :&: B^-1.

Proof. exact: preimsetI. Qed.

Lemma

invIg

finite_group

finite_group/fingroup.v

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

[ "preimsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invDg A B : (A :\: B)^-1 = A^-1 :\: B^-1.

Proof. exact: preimsetD. Qed.

Lemma

invDg

finite_group

finite_group/fingroup.v

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

[ "preimsetD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invCg A : (~: A)^-1 = ~: A^-1.

Proof. exact: preimsetC. Qed.

Lemma

invCg

finite_group

finite_group/fingroup.v

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

[ "preimsetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite !(sameP setIidPl eqP) -invIg (inj_eq invg_inj). Qed.

Lemma

invSg

finite_group

finite_group/fingroup.v

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

[ "inj_eq", "invIg", "invg_inj", "setIidPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite inE. Qed.

Lemma

mem_invg

finite_group

finite_group/fingroup.v

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

[ "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite inE invgK. Qed.

Lemma

memV_invg

finite_group

finite_group/fingroup.v

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

[ "inE", "invgK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_invg A : #|A^-1| = #|A|.

Proof. exact/card_preimset/invg_inj. Qed.

Lemma

card_invg

finite_group

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_preimset", "invg_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gE : 1 = [set 1] :> {set gT}.

Proof. by []. Qed.

Lemma

set1gE

finite_group

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" ]

Product with singletons.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gP x : reflect (x = 1) (x \in [1 gT]).

Proof. exact: set1P. Qed.

Lemma

set1gP

finite_group

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", "set1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulg_set1 x y : [set x] :* y = [set x * y].

Proof. by rewrite [_ * _]imset2_set1l imset_set1. Qed.

Lemma

mulg_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" ]

[ "imset2_set1l", "imset_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg_set1 x : [set x]^-1 = [set x^-1].

Proof. by apply/setP=> y; rewrite !inE inv_eq //; apply: invgK. Qed.

Lemma

invg_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" ]

[ "apply", "inE", "inv_eq", "invgK", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcosetE A x : lcoset A x = x *: A.

Proof. by rewrite [_ * _]imset2_set1l. Qed.

Lemma

lcosetE

finite_group

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_set1l", "lcoset" ]

Left cosets.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite -lcosetE (card_imset _ (mulgI _)). Qed.

Lemma

card_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" ]

[ "card_imset", "lcosetE", "mulgI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite -lcosetE [_ x](can_imset_pre _ (mulKg _)) inE. Qed.

Lemma

mem_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" ]

[ "can_imset_pre", "inE", "lcosetE", "mulKg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Lemma

lcosetP

finite_group

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", "lcosetE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Lemma

lcosetsP

finite_group

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", "lcosetE", "lcosets" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite -mulg_set1 mulgA. Qed.

Lemma

lcosetM

finite_group

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

lcoset1 A : 1 *: A = A.

Proof. exact: mul1g. Qed.

Lemma

lcoset1

finite_group

finite_group/fingroup.v

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

[ "mul1g" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcosetK : left_loop inv (fun x A => x *: A).

Proof. by move=> x A; rewrite -lcosetM mulVg mul1g. Qed.

Lemma

lcosetK

finite_group

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", "lcosetM", "mul1g", "mulVg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lcosetKV : rev_left_loop inv (fun x A => x *: A).

Proof. by move=> x A; rewrite -lcosetM mulgV mul1g. Qed.

Lemma

lcosetKV

finite_group

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", "lcosetM", "mul1g", "mulgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d