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
"x ^ y"	:= (Conj x y) (in custom group_presentation at level 30, right associativity).	Notation	x ^ y	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x ^-1"	:= (Inv x) (in custom group_presentation at level 3).	Notation	x ^-1	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x ^- n"	:= (Inv (Exp x n)) (in custom group_presentation at level 29, n constr at level 28).	Notation	x ^- n	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ ~ x1 , x2 , .. , xn ]"	:= (Comm .. (Comm x1 x2) .. xn) (in custom group_presentation, x1, x2, xn at level 100).	Notation	[ ~ x1 , x2 , .. , xn ]	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x = y"	:= (Eq2 x y) (in custom group_presentation at level 70).	Notation	x = y	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x = y = z"	:= (Eq3 x y z) (in custom group_presentation at level 70, y at next level).	Notation	x = y = z	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Eq3", "next" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"r1 , r2 , .. , rn"	:= (And .. (And r1 r2) .. rn) (in custom group_presentation at level 200).	Notation	r1 , r2 , .. , rn	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "And", "r1", "r2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"( p )"	:= p (in custom group_presentation, p at level 200).	Notation	( p )	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"1"	:= Idx (in custom group_presentation).	Notation	1	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x"	:= x (in custom group_presentation at level 0, x ident).	Notation	x	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x : p"	:= (Generator (fun x => Cast p)) (in custom group_presentation, x ident, p custom group_presentation at level 200).	Notation	x : p	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Cast" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"H \homg 'Grp' p"	:= (hom H p) (at level 70, p at level 0, format "H \homg 'Grp' p") : group_scope.	Notation	H \homg 'Grp' p	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "hom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"H \isog 'Grp' p"	:= (iso H p) (at level 70, p at level 0, format "H \isog 'Grp' p") : group_scope.	Notation	H \isog 'Grp' p	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "iso" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"H \homg 'Grp' ( x : p )"	:= (hom H (fun x => Cast p)) (at level 70, x ident, p custom group_presentation at level 200, format "'[hv' H '/ ' \homg 'Grp' ( x : p ) ']'") : group_scope.	Notation	H \homg 'Grp' ( x : p )	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Cast", "hom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"H \isog 'Grp' ( x : p )"	:= (iso H (fun x => Cast p)) (at level 70, x ident, p custom group_presentation at level 200, format "'[hv' H '/ ' \isog 'Grp' ( x : p ) ']'") : group_scope.	Notation	H \isog 'Grp' ( x : p )	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Cast", "iso" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isoGrp_hom gT (G : {group gT}) p : G \isog Grp p -> G \homg Grp p.	Proof. by move <-; apply: homg_refl. Qed.	Lemma	isoGrp_hom	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Grp", "apply", "gT", "group", "homg", "homg_refl", "isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isoGrpP gT (G : {group gT}) p rT (H : {group rT}) : G \isog Grp p -> reflect (#\|H\| = #\|G\| /\ H \homg Grp p) (H \isog G).	Proof. move=> isoGp; apply: (iffP idP) => [isoGH \| [oH homHp]]. by rewrite (card_isog isoGH) -isoGp isog_hom. by rewrite isogEcard isoGp homHp /= oH. Qed.	Lemma	isoGrpP	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Grp", "apply", "card_isog", "gT", "group", "homg", "isoGH", "isog", "isogEcard", "isog_hom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homGrp_trans rT gT (H : {set rT}) (G : {group gT}) p : H \homg G -> G \homg Grp p -> H \homg Grp p.	Proof. case/homgP=> h <-{H}; rewrite /hom; move: {p}(p _) => p. have evalG e t: all [in G] e -> eval (map h e) t = h (eval e t). move=> Ge; apply: (@proj2 (eval e t \in G)); elim: t => /=. - move=> i; case: (leqP (size e) i) => [le_e_i \| lt_i_e]. by rewrite !nth_default ?size_map ?morph1. by rewrite (nth_...	Lemma	homGrp_trans	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Grp", "all", "allP", "and_rel", "apply", "cycle_subG", "env1", "eq_all_r", "eqxx", "eval", "existsP", "f1", "f2", "gT", "group", "groupJ", "groupM", "groupR", "groupV", "groupX", "hom", "homg", "homgP", "join_subG", "last", "leqP", "map", "map_rev", "mem_nth"...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_homGrp gT rT (G : {group gT}) (H : {group rT}) p : G \isog H -> (G \homg Grp p) = (H \homg Grp p).	Proof. by rewrite isogEhom => /andP[homGH homHG]; apply/idP/idP; apply: homGrp_trans. Qed.	Lemma	eq_homGrp	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Grp", "apply", "gT", "group", "homGrp_trans", "homg", "isog", "isogEhom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isoGrp_trans gT rT (G : {group gT}) (H : {group rT}) p : G \isog H -> H \isog Grp p -> G \isog Grp p.	Proof. by move=> isoGH isoHp kT K; rewrite -isoHp; apply: eq_homgr. Qed.	Lemma	isoGrp_trans	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Grp", "apply", "eq_homgr", "gT", "group", "isoGH", "isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
intro_isoGrp gT (G : {group gT}) p : G \homg Grp p -> (forall rT (H : {group rT}), H \homg Grp p -> H \homg G) -> G \isog Grp p.	Proof. move=> homGp freeG rT H. by apply/idP/idP=> [homHp\|]; [apply: homGrp_trans homGp \| apply: freeG]. Qed.	Lemma	intro_isoGrp	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Grp", "apply", "gT", "group", "homGrp_trans", "homg", "isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
H	:= <<A>>.	Notation	H	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_range	:= [pred B in rcosets H 'N(A)].	Definition	coset_range	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "rcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_of : Type	:= Coset { set_of_coset :> GroupSet.sort gT; _ : coset_range set_of_coset }.	Record	coset_of	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_range", "gT", "sort" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_one_proof : coset_range H.	Proof. by apply/rcosetsP; exists (1 : gT); rewrite (group1, mulg1). Qed.	Lemma	coset_one_proof	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset_range", "gT", "group1", "mulg1", "rcosetsP" ]	When A is a group, this is the largest possible quotient 'N(A) / A.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_one	:= Coset coset_one_proof.	Definition	coset_one	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_one_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nNH	:= subsetP (norm_gen A).	Let	nNH	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "norm_gen", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_range_mul (B C : coset_of) : coset_range (B * C).	Proof. case: B C => _ /= /rcosetsP[x Nx ->] [_ /= /rcosetsP[y Ny ->]]. by apply/rcosetsP; exists (x * y); rewrite !(groupM, rcoset_mul, nNH). Qed.	Lemma	coset_range_mul	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset_of", "coset_range", "groupM", "nNH", "rcoset_mul", "rcosetsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_mul B C	:= Coset (coset_range_mul B C).	Definition	coset_mul	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_range_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_range_inv (B : coset_of) : coset_range B^-1.	Proof. case: B => _ /= /rcosetsP[x Nx ->]; rewrite norm_rlcoset ?nNH // invg_lcoset. by apply/rcosetsP; exists x^-1; rewrite ?groupV. Qed.	Lemma	coset_range_inv	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset_of", "coset_range", "groupV", "invg_lcoset", "nNH", "norm_rlcoset", "rcosetsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_inv B	:= Coset (coset_range_inv B).	Definition	coset_inv	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_range_inv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_mulP : associative coset_mul.	Proof. by move=> B C D; apply: val_inj; rewrite /= mulgA. Qed.	Lemma	coset_mulP	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset_mul", "mulgA", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_oneP : left_id coset_one coset_mul.	Proof. case=> B coB; apply: val_inj => /=; case/rcosetsP: coB => x Hx ->{B}. by rewrite mulgA mulGid. Qed.	Lemma	coset_oneP	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset_mul", "coset_one", "mulGid", "mulgA", "rcosetsP", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_invP : left_inverse coset_one coset_inv coset_mul.	Proof. case=> B coB; apply: val_inj => /=; case/rcosetsP: coB => x Hx ->{B}. rewrite invg_rcoset -mulgA (mulgA H) mulGid. by rewrite norm_rlcoset ?nNH // -lcosetM mulVg mul1g. Qed.	Lemma	coset_invP	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset_inv", "coset_mul", "coset_one", "invg_rcoset", "lcosetM", "mul1g", "mulGid", "mulVg", "mulgA", "nNH", "norm_rlcoset", "rcosetsP", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset x : coset_of	:= insubd (1 : coset_of) (H :* x).	Definition	coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_of", "insubd" ]	Projection of the initial group type over the cosets groupType.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_coset_prim x : x \in 'N(A) -> coset x :=: H :* x.	Proof. by move=> Nx; rewrite val_insubd /= mem_rcosets -{1}(mul1g x) mem_mulg. Qed.	Lemma	val_coset_prim	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "mem_mulg", "mem_rcosets", "mul1g", "val_insubd" ]	the case where A is a group.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_morphM : {in 'N(A) &, {morph coset : x y / x * y}}.	Proof. move=> x y Nx Ny; apply: val_inj. by rewrite /= !val_coset_prim ?groupM //= rcoset_mul ?nNH. Qed.	Lemma	coset_morphM	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "groupM", "nNH", "rcoset_mul", "val_coset_prim", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_morphism	:= Morphism coset_morphM.	Canonical	coset_morphism	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_coset_prim : 'ker coset = 'N_H(A).	Proof. apply/setP=> z; rewrite !in_setI andbC 2!inE -val_eqE /=. case Nz: (z \in 'N(A)); rewrite ?andbF ?val_coset_prim // !andbT. by apply/eqP/idP=> [<-\| Az]; rewrite (rcoset_refl, rcoset_id). Qed.	Lemma	ker_coset_prim	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "inE", "in_setI", "ker", "rcoset_id", "rcoset_refl", "setP", "val_coset_prim", "val_eqE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_mem y xbar : y \in xbar -> coset y = xbar.	Proof. case: xbar => /= Hx NHx Hxy; apply: val_inj=> /=. case/rcosetsP: NHx (NHx) Hxy => x Nx -> NHx Hxy. by rewrite val_insubd /= (rcoset_eqP Hxy) NHx. Qed.	Lemma	coset_mem	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "rcoset_eqP", "rcosetsP", "val_inj", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_repr_coset xbar : repr xbar \in xbar.	Proof. by case: xbar => /= _ /rcosetsP[x _ ->]; apply: mem_repr_rcoset. Qed.	Lemma	mem_repr_coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "mem_repr_rcoset", "rcosetsP", "repr" ]	coset is an inverse to repr	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_coset1 : repr (1 : coset_of) = 1.	Proof. exact: repr_group. Qed.	Lemma	repr_coset1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_of", "repr", "repr_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_reprK : cancel (fun xbar => repr xbar) coset.	Proof. by move=> xbar; apply: coset_mem (mem_repr_coset xbar). Qed.	Lemma	coset_reprK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "coset_mem", "mem_repr_coset", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetP xbar : {x \| x \in 'N(A) & xbar = coset x}.	Proof. pose x := repr 'N_xbar(A). have [xbar_x Nx]: x \in xbar /\ x \in 'N(A). apply/setIP; rewrite {}/x; case: xbar => /= _ /rcosetsP[y Ny ->]. by apply: (mem_repr y); rewrite inE rcoset_refl. by exists x; last rewrite (coset_mem xbar_x). Qed.	Lemma	cosetP	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "coset_mem", "inE", "last", "mem_repr", "rcoset_refl", "rcosetsP", "repr", "setIP" ]	guarantee repr xbar \in 'N(A) when A is a group.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_id x : x \in A -> coset x = 1.	Proof. by move=> Ax; apply: coset_mem; apply: mem_gen. Qed.	Lemma	coset_id	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "coset_mem", "mem_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_coset : coset @* 'N(A) = setT.	Proof. by apply/setP=> xbar; case: (cosetP xbar) => x Nx ->; rewrite inE mem_morphim. Qed.	Lemma	im_coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "cosetP", "inE", "mem_morphim", "setP", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_im_coset (C : {set coset_of}) : C \subset coset @* 'N(A).	Proof. by rewrite im_coset subsetT. Qed.	Lemma	sub_im_coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "coset_of", "im_coset", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_proper C D : (coset @^-1 C \proper coset @^-1 D) = (C \proper D).	Proof. by rewrite morphpre_proper ?sub_im_coset. Qed.	Lemma	cosetpre_proper	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpre_proper", "proper", "sub_im_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient : {set coset_of}	:= coset @* Q.	Definition	quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "coset_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientE : quotient = coset @* Q.	Proof. by []. Qed.	Lemma	quotientE	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A / H"	:= (quotient A H) : group_scope.	Notation	A / H	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_group G A : {group coset_of A}	:= Eval hnf in [group of G / A].	Canonical	quotient_group	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_of", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_coset x : x \in 'N(H) -> coset H x :=: H :* x.	Proof. by move=> Nx; rewrite val_coset_prim // genGid. Qed.	Lemma	val_coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "genGid", "val_coset_prim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_default x : (x \in 'N(H)) = false -> coset H x = 1.	Proof. move=> Nx; apply: val_inj. by rewrite val_insubd /= mem_rcosets /= genGid mulSGid ?normG ?Nx. Qed.	Lemma	coset_default	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "genGid", "mem_rcosets", "mulSGid", "normG", "val_inj", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_norm xbar : xbar \subset 'N(H).	Proof. case: xbar => /= _ /rcosetsP[x Nx ->]. by rewrite genGid mul_subG ?sub1set ?normG. Qed.	Lemma	coset_norm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "genGid", "mul_subG", "normG", "rcosetsP", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_coset : 'ker (coset H) = H.	Proof. by rewrite ker_coset_prim genGid (setIidPl _) ?normG. Qed.	Lemma	ker_coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "genGid", "ker", "ker_coset_prim", "normG", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_idr x : x \in 'N(H) -> coset H x = 1 -> x \in H.	Proof. by move=> Nx Hx1; rewrite -ker_coset mem_morphpre //= Hx1 set11. Qed.	Lemma	coset_idr	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "ker_coset", "mem_morphpre", "set11" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_coset_norm xbar : repr xbar \in 'N(H).	Proof. exact: subsetP (coset_norm _) _ (mem_repr_coset _). Qed.	Lemma	repr_coset_norm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_norm", "mem_repr_coset", "repr", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
imset_coset G : coset H @: G = G / H.	Proof. apply/eqP; rewrite eqEsubset andbC imsetS ?subsetIr //=. apply/subsetP=> _ /imsetP[x Gx ->]. by case Nx: (x \in 'N(H)); rewrite ?(coset_default Nx) ?mem_morphim ?group1. Qed.	Lemma	imset_coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "coset_default", "eqEsubset", "group1", "imsetP", "imsetS", "mem_morphim", "subsetIr", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_quotient A : val @: (A / H) = rcosets H 'N_A(H).	Proof. apply/setP=> B; apply/imsetP/rcosetsP=> [[xbar Axbar]\|[x /setIP[Ax Nx]]] ->{B}. case/morphimP: Axbar => x Nx Ax ->{xbar}. by exists x; [rewrite inE Ax \| rewrite /= val_coset]. by exists (coset H x); [apply/morphimP; exists x \| rewrite /= val_coset]. Qed.	Lemma	val_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "imsetP", "inE", "morphimP", "rcosets", "rcosetsP", "setIP", "setP", "val", "val_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_quotient_subnorm A : #\|A / H\| = #\|'N_A(H) : H\|.	Proof. by rewrite -(card_imset _ val_inj) val_quotient. Qed.	Lemma	card_quotient_subnorm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "card_imset", "val_inj", "val_quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_quotient A : #\|A / H\| <= #\|A\|.	Proof. exact: leq_morphim. Qed.	Lemma	leq_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "leq_morphim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_quotient A : H :!=: 1 -> H \subset A -> #\|A / H\| < #\|A\|.	Proof. by move=> ntH sHA; rewrite ltn_morphim // ker_coset (setIidPr sHA) proper1G. Qed.	Lemma	ltn_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "ker_coset", "ltn_morphim", "proper1G", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_quotient A : A \subset 'N(H) -> #\|A / H\| = #\|A : H\|.	Proof. by move=> nHA; rewrite card_quotient_subnorm (setIidPl nHA). Qed.	Lemma	card_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "card_quotient_subnorm", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divg_normal G : H <\| G -> #\|G\| %/ #\|H\| = #\|G / H\|.	Proof. by case/andP=> sHG nHG; rewrite divgS ?card_quotient. Qed.	Lemma	divg_normal	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "card_quotient", "divgS", "nHG", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset1 : coset H 1 :=: H.	Proof. by rewrite morph1 /= genGid. Qed.	Lemma	coset1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "genGid", "morph1" ]	Variant of morph1; no specialization for other morph lemmas.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre1 : coset H @*^-1 1 = H.	Proof. by rewrite -kerE ker_coset. Qed.	Lemma	cosetpre1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "kerE", "ker_coset" ]	Variant of kerE.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_quotient : 'N(H) / H = setT.	Proof. exact: im_coset. Qed.	Lemma	im_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "im_coset", "setT" ]	morph[im\|pre]Iim are also covered by im_quotient.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientT : setT / H = setT.	Proof. by rewrite -im_quotient; apply: morphimT. Qed.	Lemma	quotientT	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "im_quotient", "morphimT", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientInorm A : 'N_A(H) / H = A / H.	Proof. by rewrite /quotient setIC morphimIdom. Qed.	Lemma	quotientInorm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimIdom", "quotient", "setIC" ]	Variant of morphimIdom.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_setIpre A D : (A :&: coset H @*^-1 D) / H = A / H :&: D.	Proof. exact: morphim_setIpre. Qed.	Lemma	quotient_setIpre	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphim_setIpre" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_quotient x G : x \in G -> coset H x \in G / H.	Proof. by move=> Gx; rewrite -imset_coset imset_f. Qed.	Lemma	mem_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "imset_coset", "imset_f" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientS A B : A \subset B -> A / H \subset B / H.	Proof. exact: morphimS. Qed.	Lemma	quotientS	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient0 : set0 / H = set0.	Proof. exact: morphim0. Qed.	Lemma	quotient0	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim0", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_set1 x : x \in 'N(H) -> [set x] / H = [set coset H x].	Proof. exact: morphim_set1. Qed.	Lemma	quotient_set1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphim_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient1 : 1 / H = 1.	Proof. exact: morphim1. Qed.	Lemma	quotient1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientV A : A^-1 / H = (A / H)^-1.	Proof. exact: morphimV. Qed.	Lemma	quotientV	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientMl A B : A \subset 'N(H) -> A * B / H = (A / H) * (B / H).	Proof. exact: morphimMl. Qed.	Lemma	quotientMl	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimMl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientMr A B : B \subset 'N(H) -> A * B / H = (A / H) * (B / H).	Proof. exact: morphimMr. Qed.	Lemma	quotientMr	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimMr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpreM C D : coset H @^-1 (C D) = coset H @^-1 C coset H @*^-1 D.	Proof. by rewrite morphpreMl ?sub_im_coset. Qed.	Lemma	cosetpreM	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpreMl", "sub_im_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientJ A x : x \in 'N(H) -> A :^ x / H = (A / H) :^ coset H x.	Proof. exact: morphimJ. Qed.	Lemma	quotientJ	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphimJ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientU A B : (A :\|: B) / H = A / H :\|: B / H.	Proof. exact: morphimU. Qed.	Lemma	quotientU	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientI A B : (A :&: B) / H \subset A / H :&: B / H.	Proof. exact: morphimI. Qed.	Lemma	quotientI	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientY A B : A \subset 'N(H) -> B \subset 'N(H) -> (A <> B) / H = (A / H) <> (B / H).	Proof. exact: morphimY. Qed.	Lemma	quotientY	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimY" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_homg A : A \subset 'N(H) -> homg (A / H) A.	Proof. exact: morphim_homg. Qed.	Lemma	quotient_homg	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "homg", "morphim_homg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_kerl x y : x \in H -> coset H (x * y) = coset H y.	Proof. move=> Hx; case Ny: (y \in 'N(H)); first by rewrite mkerl ?ker_coset. by rewrite !coset_default ?groupMl // (subsetP (normG H)). Qed.	Lemma	coset_kerl	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "coset_default", "groupMl", "ker_coset", "mkerl", "normG", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset_kerr x y : y \in H -> coset H (x * y) = coset H x.	Proof. move=> Hy; case Nx: (x \in 'N(H)); first by rewrite mkerr ?ker_coset. by rewrite !coset_default ?groupMr // (subsetP (normG H)). Qed.	Lemma	coset_kerr	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "coset_default", "groupMr", "ker_coset", "mkerr", "normG", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_kercosetP x y : x \in 'N(H) -> y \in 'N(H) -> reflect (coset H x = coset H y) (x \in H :* y).	Proof. by rewrite -{6}ker_coset; apply: rcoset_kerP. Qed.	Lemma	rcoset_kercosetP	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "ker_coset", "rcoset_kerP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
kercoset_rcoset x y : x \in 'N(H) -> y \in 'N(H) -> coset H x = coset H y -> exists2 z, z \in H & x = z * y.	Proof. by move=> Nx Ny eqfxy; rewrite -ker_coset; apply: ker_rcoset. Qed.	Lemma	kercoset_rcoset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "ker_coset", "ker_rcoset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientGI G A : H \subset G -> (G :&: A) / H = G / H :&: A / H.	Proof. by rewrite -{1}ker_coset; apply: morphimGI. Qed.	Lemma	quotientGI	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "ker_coset", "morphimGI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientIG A G : H \subset G -> (A :&: G) / H = A / H :&: G / H.	Proof. by rewrite -{1}ker_coset; apply: morphimIG. Qed.	Lemma	quotientIG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "ker_coset", "morphimIG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientD A B : A / H :\: B / H \subset (A :\: B) / H.	Proof. exact: morphimD. Qed.	Lemma	quotientD	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientD1 A : (A / H)^# \subset A^# / H.	Proof. exact: morphimD1. Qed.	Lemma	quotientD1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimD1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientDG A G : H \subset G -> (A :\: G) / H = A / H :\: G / H.	Proof. by rewrite -{1}ker_coset; apply: morphimDG. Qed.	Lemma	quotientDG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "ker_coset", "morphimDG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientK A : A \subset 'N(H) -> coset H @^-1 (A / H) = H A.	Proof. by rewrite -{8}ker_coset; apply: morphimK. Qed.	Lemma	quotientK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "ker_coset", "morphimK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientYK G : G \subset 'N(H) -> coset H @^-1 (G / H) = H <> G.	Proof. by move=> nHG; rewrite quotientK ?norm_joinEr. Qed.	Lemma	quotientYK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "nHG", "norm_joinEr", "quotientK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientGK G : H <\| G -> coset H @*^-1 (G / H) = G.	Proof. by case/andP; rewrite -{1}ker_coset; apply: morphimGK. Qed.	Lemma	quotientGK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "ker_coset", "morphimGK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_class x A : x \in 'N(H) -> A \subset 'N(H) -> x ^: A / H = coset H x ^: (A / H).	Proof. exact: morphim_class. Qed.	Lemma	quotient_class	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphim_class" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classes_quotient A : A \subset 'N(H) -> classes (A / H) = [set xA / H \| xA in classes A].	Proof. exact: classes_morphim. Qed.	Lemma	classes_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "classes", "classes_morphim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_set1 x : x \in 'N(H) -> coset H @^-1 [set coset H x] = H : x.	Proof. by rewrite -{9}ker_coset; apply: morphpre_set1. Qed.	Lemma	cosetpre_set1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "ker_coset", "morphpre_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x ^ y"

:= (Conj x y) (in custom group_presentation at level 30, right associativity).

Notation

x ^ y

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x ^-1"

:= (Inv x) (in custom group_presentation at level 3).

Notation

x ^-1

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x ^- n"

:= (Inv (Exp x n)) (in custom group_presentation at level 29, n constr at level 28).

Notation

x ^- n

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ ~ x1 , x2 , .. , xn ]"

:= (Comm .. (Comm x1 x2) .. xn) (in custom group_presentation, x1, x2, xn at level 100).

Notation

[ ~ x1 , x2 , .. , xn ]

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x = y"

:= (Eq2 x y) (in custom group_presentation at level 70).

Notation

x = y

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x = y = z"

:= (Eq3 x y z) (in custom group_presentation at level 70, y at next level).

Notation

x = y = z

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Eq3", "next" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"r1 , r2 , .. , rn"

:= (And .. (And r1 r2) .. rn) (in custom group_presentation at level 200).

Notation

r1 , r2 , .. , rn

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "And", "r1", "r2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"( p )"

:= p (in custom group_presentation, p at level 200).

Notation

( p )

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"1"

:= Idx (in custom group_presentation).

Notation

1

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x"

:= x (in custom group_presentation at level 0, x ident).

Notation

x

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x : p"

:= (Generator (fun x => Cast p)) (in custom group_presentation, x ident, p custom group_presentation at level 200).

Notation

x : p

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Cast" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"H \homg 'Grp' p"

:= (hom H p) (at level 70, p at level 0, format "H \homg 'Grp' p") : group_scope.

Notation

H \homg 'Grp' p

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "hom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"H \isog 'Grp' p"

:= (iso H p) (at level 70, p at level 0, format "H \isog 'Grp' p") : group_scope.

Notation

H \isog 'Grp' p

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "iso" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"H \homg 'Grp' ( x : p )"

:= (hom H (fun x => Cast p)) (at level 70, x ident, p custom group_presentation at level 200, format "'[hv' H '/ ' \homg 'Grp' ( x : p ) ']'") : group_scope.

Notation

H \homg 'Grp' ( x : p )

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Cast", "hom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"H \isog 'Grp' ( x : p )"

:= (iso H (fun x => Cast p)) (at level 70, x ident, p custom group_presentation at level 200, format "'[hv' H '/ ' \isog 'Grp' ( x : p ) ']'") : group_scope.

Notation

H \isog 'Grp' ( x : p )

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Cast", "iso" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isoGrp_hom gT (G : {group gT}) p : G \isog Grp p -> G \homg Grp p.

Proof. by move <-; apply: homg_refl. Qed.

Lemma

isoGrp_hom

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Grp", "apply", "gT", "group", "homg", "homg_refl", "isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isoGrpP gT (G : {group gT}) p rT (H : {group rT}) : G \isog Grp p -> reflect (#|H| = #|G| /\ H \homg Grp p) (H \isog G).

Proof. move=> isoGp; apply: (iffP idP) => [isoGH | [oH homHp]]. by rewrite (card_isog isoGH) -isoGp isog_hom. by rewrite isogEcard isoGp homHp /= oH. Qed.

Lemma

isoGrpP

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Grp", "apply", "card_isog", "gT", "group", "homg", "isoGH", "isog", "isogEcard", "isog_hom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homGrp_trans rT gT (H : {set rT}) (G : {group gT}) p : H \homg G -> G \homg Grp p -> H \homg Grp p.

Proof. case/homgP=> h <-{H}; rewrite /hom; move: {p}(p _) => p. have evalG e t: all [in G] e -> eval (map h e) t = h (eval e t). move=> Ge; apply: (@proj2 (eval e t \in G)); elim: t => /=. - move=> i; case: (leqP (size e) i) => [le_e_i | lt_i_e]. by rewrite !nth_default ?size_map ?morph1. by rewrite (nth_...

Lemma

homGrp_trans

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Grp", "all", "allP", "and_rel", "apply", "cycle_subG", "env1", "eq_all_r", "eqxx", "eval", "existsP", "f1", "f2", "gT", "group", "groupJ", "groupM", "groupR", "groupV", "groupX", "hom", "homg", "homgP", "join_subG", "last", "leqP", "map", "map_rev", "mem_nth"...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_homGrp gT rT (G : {group gT}) (H : {group rT}) p : G \isog H -> (G \homg Grp p) = (H \homg Grp p).

Proof. by rewrite isogEhom => /andP[homGH homHG]; apply/idP/idP; apply: homGrp_trans. Qed.

Lemma

eq_homGrp

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Grp", "apply", "gT", "group", "homGrp_trans", "homg", "isog", "isogEhom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isoGrp_trans gT rT (G : {group gT}) (H : {group rT}) p : G \isog H -> H \isog Grp p -> G \isog Grp p.

Proof. by move=> isoGH isoHp kT K; rewrite -isoHp; apply: eq_homgr. Qed.

Lemma

isoGrp_trans

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Grp", "apply", "eq_homgr", "gT", "group", "isoGH", "isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

intro_isoGrp gT (G : {group gT}) p : G \homg Grp p -> (forall rT (H : {group rT}), H \homg Grp p -> H \homg G) -> G \isog Grp p.

Proof. move=> homGp freeG rT H. by apply/idP/idP=> [homHp|]; [apply: homGrp_trans homGp | apply: freeG]. Qed.

Lemma

intro_isoGrp

finite_group

finite_group/presentation.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]

[ "Grp", "apply", "gT", "group", "homGrp_trans", "homg", "isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

H

:= <<A>>.

Notation

H

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_range

:= [pred B in rcosets H 'N(A)].

Definition

coset_range

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "rcosets" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_of : Type

:= Coset { set_of_coset :> GroupSet.sort gT; _ : coset_range set_of_coset }.

Record

coset_of

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_range", "gT", "sort" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_one_proof : coset_range H.

Proof. by apply/rcosetsP; exists (1 : gT); rewrite (group1, mulg1). Qed.

Lemma

coset_one_proof

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset_range", "gT", "group1", "mulg1", "rcosetsP" ]

When A is a group, this is the largest possible quotient 'N(A) / A.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_one

:= Coset coset_one_proof.

Definition

coset_one

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_one_proof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nNH

:= subsetP (norm_gen A).

Let

nNH

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "norm_gen", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_range_mul (B C : coset_of) : coset_range (B * C).

Proof. case: B C => _ /= /rcosetsP[x Nx ->] [_ /= /rcosetsP[y Ny ->]]. by apply/rcosetsP; exists (x * y); rewrite !(groupM, rcoset_mul, nNH). Qed.

Lemma

coset_range_mul

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset_of", "coset_range", "groupM", "nNH", "rcoset_mul", "rcosetsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_mul B C

:= Coset (coset_range_mul B C).

Definition

coset_mul

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_range_mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_range_inv (B : coset_of) : coset_range B^-1.

Proof. case: B => _ /= /rcosetsP[x Nx ->]; rewrite norm_rlcoset ?nNH // invg_lcoset. by apply/rcosetsP; exists x^-1; rewrite ?groupV. Qed.

Lemma

coset_range_inv

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset_of", "coset_range", "groupV", "invg_lcoset", "nNH", "norm_rlcoset", "rcosetsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_inv B

:= Coset (coset_range_inv B).

Definition

coset_inv

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_range_inv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_mulP : associative coset_mul.

Proof. by move=> B C D; apply: val_inj; rewrite /= mulgA. Qed.

Lemma

coset_mulP

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset_mul", "mulgA", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_oneP : left_id coset_one coset_mul.

Proof. case=> B coB; apply: val_inj => /=; case/rcosetsP: coB => x Hx ->{B}. by rewrite mulgA mulGid. Qed.

Lemma

coset_oneP

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset_mul", "coset_one", "mulGid", "mulgA", "rcosetsP", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_invP : left_inverse coset_one coset_inv coset_mul.

Proof. case=> B coB; apply: val_inj => /=; case/rcosetsP: coB => x Hx ->{B}. rewrite invg_rcoset -mulgA (mulgA H) mulGid. by rewrite norm_rlcoset ?nNH // -lcosetM mulVg mul1g. Qed.

Lemma

coset_invP

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset_inv", "coset_mul", "coset_one", "invg_rcoset", "lcosetM", "mul1g", "mulGid", "mulVg", "mulgA", "nNH", "norm_rlcoset", "rcosetsP", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset x : coset_of

:= insubd (1 : coset_of) (H :* x).

Definition

coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_of", "insubd" ]

Projection of the initial group type over the cosets groupType.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_coset_prim x : x \in 'N(A) -> coset x :=: H :* x.

Proof. by move=> Nx; rewrite val_insubd /= mem_rcosets -{1}(mul1g x) mem_mulg. Qed.

Lemma

val_coset_prim

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "mem_mulg", "mem_rcosets", "mul1g", "val_insubd" ]

the case where A is a group.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_morphM : {in 'N(A) &, {morph coset : x y / x * y}}.

Proof. move=> x y Nx Ny; apply: val_inj. by rewrite /= !val_coset_prim ?groupM //= rcoset_mul ?nNH. Qed.

Lemma

coset_morphM

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "groupM", "nNH", "rcoset_mul", "val_coset_prim", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_morphism

:= Morphism coset_morphM.

Canonical

coset_morphism

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_morphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_coset_prim : 'ker coset = 'N_H(A).

Proof. apply/setP=> z; rewrite !in_setI andbC 2!inE -val_eqE /=. case Nz: (z \in 'N(A)); rewrite ?andbF ?val_coset_prim // !andbT. by apply/eqP/idP=> [<-| Az]; rewrite (rcoset_refl, rcoset_id). Qed.

Lemma

ker_coset_prim

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "inE", "in_setI", "ker", "rcoset_id", "rcoset_refl", "setP", "val_coset_prim", "val_eqE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_mem y xbar : y \in xbar -> coset y = xbar.

Proof. case: xbar => /= Hx NHx Hxy; apply: val_inj=> /=. case/rcosetsP: NHx (NHx) Hxy => x Nx -> NHx Hxy. by rewrite val_insubd /= (rcoset_eqP Hxy) NHx. Qed.

Lemma

coset_mem

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "rcoset_eqP", "rcosetsP", "val_inj", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_repr_coset xbar : repr xbar \in xbar.

Proof. by case: xbar => /= _ /rcosetsP[x _ ->]; apply: mem_repr_rcoset. Qed.

Lemma

mem_repr_coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "mem_repr_rcoset", "rcosetsP", "repr" ]

coset is an inverse to repr

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_coset1 : repr (1 : coset_of) = 1.

Proof. exact: repr_group. Qed.

Lemma

repr_coset1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_of", "repr", "repr_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_reprK : cancel (fun xbar => repr xbar) coset.

Proof. by move=> xbar; apply: coset_mem (mem_repr_coset xbar). Qed.

Lemma

coset_reprK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "coset_mem", "mem_repr_coset", "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetP xbar : {x | x \in 'N(A) & xbar = coset x}.

Proof. pose x := repr 'N_xbar(A). have [xbar_x Nx]: x \in xbar /\ x \in 'N(A). apply/setIP; rewrite {}/x; case: xbar => /= _ /rcosetsP[y Ny ->]. by apply: (mem_repr y); rewrite inE rcoset_refl. by exists x; last rewrite (coset_mem xbar_x). Qed.

Lemma

cosetP

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "coset_mem", "inE", "last", "mem_repr", "rcoset_refl", "rcosetsP", "repr", "setIP" ]

guarantee repr xbar \in 'N(A) when A is a group.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_id x : x \in A -> coset x = 1.

Proof. by move=> Ax; apply: coset_mem; apply: mem_gen. Qed.

Lemma

coset_id

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "coset_mem", "mem_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_coset : coset @* 'N(A) = setT.

Proof. by apply/setP=> xbar; case: (cosetP xbar) => x Nx ->; rewrite inE mem_morphim. Qed.

Lemma

im_coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "cosetP", "inE", "mem_morphim", "setP", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_im_coset (C : {set coset_of}) : C \subset coset @* 'N(A).

Proof. by rewrite im_coset subsetT. Qed.

Lemma

sub_im_coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "coset_of", "im_coset", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_proper C D : (coset @*^-1 C \proper coset @*^-1 D) = (C \proper D).

Proof. by rewrite morphpre_proper ?sub_im_coset. Qed.

Lemma

cosetpre_proper

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpre_proper", "proper", "sub_im_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient : {set coset_of}

:= coset @* Q.

Definition

quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "coset_of" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientE : quotient = coset @* Q.

Proof. by []. Qed.

Lemma

quotientE

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A / H"

:= (quotient A H) : group_scope.

Notation

A / H

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_group G A : {group coset_of A}

:= Eval hnf in [group of G / A].

Canonical

quotient_group

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_of", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_coset x : x \in 'N(H) -> coset H x :=: H :* x.

Proof. by move=> Nx; rewrite val_coset_prim // genGid. Qed.

Lemma

val_coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "genGid", "val_coset_prim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_default x : (x \in 'N(H)) = false -> coset H x = 1.

Proof. move=> Nx; apply: val_inj. by rewrite val_insubd /= mem_rcosets /= genGid mulSGid ?normG ?Nx. Qed.

Lemma

coset_default

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "genGid", "mem_rcosets", "mulSGid", "normG", "val_inj", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_norm xbar : xbar \subset 'N(H).

Proof. case: xbar => /= _ /rcosetsP[x Nx ->]. by rewrite genGid mul_subG ?sub1set ?normG. Qed.

Lemma

coset_norm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "genGid", "mul_subG", "normG", "rcosetsP", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_coset : 'ker (coset H) = H.

Proof. by rewrite ker_coset_prim genGid (setIidPl _) ?normG. Qed.

Lemma

ker_coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "genGid", "ker", "ker_coset_prim", "normG", "setIidPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_idr x : x \in 'N(H) -> coset H x = 1 -> x \in H.

Proof. by move=> Nx Hx1; rewrite -ker_coset mem_morphpre //= Hx1 set11. Qed.

Lemma

coset_idr

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "ker_coset", "mem_morphpre", "set11" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_coset_norm xbar : repr xbar \in 'N(H).

Proof. exact: subsetP (coset_norm _) _ (mem_repr_coset _). Qed.

Lemma

repr_coset_norm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_norm", "mem_repr_coset", "repr", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

imset_coset G : coset H @: G = G / H.

Proof. apply/eqP; rewrite eqEsubset andbC imsetS ?subsetIr //=. apply/subsetP=> _ /imsetP[x Gx ->]. by case Nx: (x \in 'N(H)); rewrite ?(coset_default Nx) ?mem_morphim ?group1. Qed.

Lemma

imset_coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "coset_default", "eqEsubset", "group1", "imsetP", "imsetS", "mem_morphim", "subsetIr", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_quotient A : val @: (A / H) = rcosets H 'N_A(H).

Proof. apply/setP=> B; apply/imsetP/rcosetsP=> [[xbar Axbar]|[x /setIP[Ax Nx]]] ->{B}. case/morphimP: Axbar => x Nx Ax ->{xbar}. by exists x; [rewrite inE Ax | rewrite /= val_coset]. by exists (coset H x); [apply/morphimP; exists x | rewrite /= val_coset]. Qed.

Lemma

val_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "imsetP", "inE", "morphimP", "rcosets", "rcosetsP", "setIP", "setP", "val", "val_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_quotient_subnorm A : #|A / H| = #|'N_A(H) : H|.

Proof. by rewrite -(card_imset _ val_inj) val_quotient. Qed.

Lemma

card_quotient_subnorm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "card_imset", "val_inj", "val_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_quotient A : #|A / H| <= #|A|.

Proof. exact: leq_morphim. Qed.

Lemma

leq_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "leq_morphim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_quotient A : H :!=: 1 -> H \subset A -> #|A / H| < #|A|.

Proof. by move=> ntH sHA; rewrite ltn_morphim // ker_coset (setIidPr sHA) proper1G. Qed.

Lemma

ltn_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "ker_coset", "ltn_morphim", "proper1G", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_quotient A : A \subset 'N(H) -> #|A / H| = #|A : H|.

Proof. by move=> nHA; rewrite card_quotient_subnorm (setIidPl nHA). Qed.

Lemma

card_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "card_quotient_subnorm", "setIidPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divg_normal G : H <| G -> #|G| %/ #|H| = #|G / H|.

Proof. by case/andP=> sHG nHG; rewrite divgS ?card_quotient. Qed.

Lemma

divg_normal

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "card_quotient", "divgS", "nHG", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset1 : coset H 1 :=: H.

Proof. by rewrite morph1 /= genGid. Qed.

Lemma

coset1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "genGid", "morph1" ]

Variant of morph1; no specialization for other morph lemmas.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre1 : coset H @*^-1 1 = H.

Proof. by rewrite -kerE ker_coset. Qed.

Lemma

cosetpre1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "kerE", "ker_coset" ]

Variant of kerE.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_quotient : 'N(H) / H = setT.

Proof. exact: im_coset. Qed.

Lemma

im_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "im_coset", "setT" ]

morph[im|pre]Iim are also covered by im_quotient.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientT : setT / H = setT.

Proof. by rewrite -im_quotient; apply: morphimT. Qed.

Lemma

quotientT

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "im_quotient", "morphimT", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientInorm A : 'N_A(H) / H = A / H.

Proof. by rewrite /quotient setIC morphimIdom. Qed.

Lemma

quotientInorm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimIdom", "quotient", "setIC" ]

Variant of morphimIdom.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_setIpre A D : (A :&: coset H @*^-1 D) / H = A / H :&: D.

Proof. exact: morphim_setIpre. Qed.

Lemma

quotient_setIpre

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphim_setIpre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_quotient x G : x \in G -> coset H x \in G / H.

Proof. by move=> Gx; rewrite -imset_coset imset_f. Qed.

Lemma

mem_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "imset_coset", "imset_f" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientS A B : A \subset B -> A / H \subset B / H.

Proof. exact: morphimS. Qed.

Lemma

quotientS

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient0 : set0 / H = set0.

Proof. exact: morphim0. Qed.

Lemma

quotient0

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim0", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_set1 x : x \in 'N(H) -> [set x] / H = [set coset H x].

Proof. exact: morphim_set1. Qed.

Lemma

quotient_set1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphim_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient1 : 1 / H = 1.

Proof. exact: morphim1. Qed.

Lemma

quotient1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientV A : A^-1 / H = (A / H)^-1.

Proof. exact: morphimV. Qed.

Lemma

quotientV

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientMl A B : A \subset 'N(H) -> A * B / H = (A / H) * (B / H).

Proof. exact: morphimMl. Qed.

Lemma

quotientMl

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimMl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientMr A B : B \subset 'N(H) -> A * B / H = (A / H) * (B / H).

Proof. exact: morphimMr. Qed.

Lemma

quotientMr

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimMr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpreM C D : coset H @*^-1 (C * D) = coset H @*^-1 C * coset H @*^-1 D.

Proof. by rewrite morphpreMl ?sub_im_coset. Qed.

Lemma

cosetpreM

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpreMl", "sub_im_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientJ A x : x \in 'N(H) -> A :^ x / H = (A / H) :^ coset H x.

Proof. exact: morphimJ. Qed.

Lemma

quotientJ

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphimJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientU A B : (A :|: B) / H = A / H :|: B / H.

Proof. exact: morphimU. Qed.

Lemma

quotientU

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientI A B : (A :&: B) / H \subset A / H :&: B / H.

Proof. exact: morphimI. Qed.

Lemma

quotientI

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientY A B : A \subset 'N(H) -> B \subset 'N(H) -> (A <*> B) / H = (A / H) <*> (B / H).

Proof. exact: morphimY. Qed.

Lemma

quotientY

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimY" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_homg A : A \subset 'N(H) -> homg (A / H) A.

Proof. exact: morphim_homg. Qed.

Lemma

quotient_homg

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "homg", "morphim_homg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_kerl x y : x \in H -> coset H (x * y) = coset H y.

Proof. move=> Hx; case Ny: (y \in 'N(H)); first by rewrite mkerl ?ker_coset. by rewrite !coset_default ?groupMl // (subsetP (normG H)). Qed.

Lemma

coset_kerl

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "coset_default", "groupMl", "ker_coset", "mkerl", "normG", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset_kerr x y : y \in H -> coset H (x * y) = coset H x.

Proof. move=> Hy; case Nx: (x \in 'N(H)); first by rewrite mkerr ?ker_coset. by rewrite !coset_default ?groupMr // (subsetP (normG H)). Qed.

Lemma

coset_kerr

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "coset_default", "groupMr", "ker_coset", "mkerr", "normG", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_kercosetP x y : x \in 'N(H) -> y \in 'N(H) -> reflect (coset H x = coset H y) (x \in H :* y).

Proof. by rewrite -{6}ker_coset; apply: rcoset_kerP. Qed.

Lemma

rcoset_kercosetP

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "ker_coset", "rcoset_kerP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

kercoset_rcoset x y : x \in 'N(H) -> y \in 'N(H) -> coset H x = coset H y -> exists2 z, z \in H & x = z * y.

Proof. by move=> Nx Ny eqfxy; rewrite -ker_coset; apply: ker_rcoset. Qed.

Lemma

kercoset_rcoset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "ker_coset", "ker_rcoset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientGI G A : H \subset G -> (G :&: A) / H = G / H :&: A / H.

Proof. by rewrite -{1}ker_coset; apply: morphimGI. Qed.

Lemma

quotientGI

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "ker_coset", "morphimGI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientIG A G : H \subset G -> (A :&: G) / H = A / H :&: G / H.

Proof. by rewrite -{1}ker_coset; apply: morphimIG. Qed.

Lemma

quotientIG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "ker_coset", "morphimIG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientD A B : A / H :\: B / H \subset (A :\: B) / H.

Proof. exact: morphimD. Qed.

Lemma

quotientD

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientD1 A : (A / H)^# \subset A^# / H.

Proof. exact: morphimD1. Qed.

Lemma

quotientD1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimD1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientDG A G : H \subset G -> (A :\: G) / H = A / H :\: G / H.

Proof. by rewrite -{1}ker_coset; apply: morphimDG. Qed.

Lemma

quotientDG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "ker_coset", "morphimDG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientK A : A \subset 'N(H) -> coset H @*^-1 (A / H) = H * A.

Proof. by rewrite -{8}ker_coset; apply: morphimK. Qed.

Lemma

quotientK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "ker_coset", "morphimK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientYK G : G \subset 'N(H) -> coset H @*^-1 (G / H) = H <*> G.

Proof. by move=> nHG; rewrite quotientK ?norm_joinEr. Qed.

Lemma

quotientYK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "nHG", "norm_joinEr", "quotientK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientGK G : H <| G -> coset H @*^-1 (G / H) = G.

Proof. by case/andP; rewrite -{1}ker_coset; apply: morphimGK. Qed.

Lemma

quotientGK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "ker_coset", "morphimGK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_class x A : x \in 'N(H) -> A \subset 'N(H) -> x ^: A / H = coset H x ^: (A / H).

Proof. exact: morphim_class. Qed.

Lemma

quotient_class

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphim_class" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classes_quotient A : A \subset 'N(H) -> classes (A / H) = [set xA / H | xA in classes A].

Proof. exact: classes_morphim. Qed.

Lemma

classes_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "classes", "classes_morphim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_set1 x : x \in 'N(H) -> coset H @*^-1 [set coset H x] = H :* x.

Proof. by rewrite -{9}ker_coset; apply: morphpre_set1. Qed.

Lemma

cosetpre_set1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "ker_coset", "morphpre_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d