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
conj_isom x : isom G (G :^ x) (conjgm G x).	Proof. by apply/isomP; rewrite morphim_conj setIid injm_conj. Qed.	Lemma	conj_isom	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "conjgm", "injm_conj", "isom", "isomP", "morphim_conj", "setIid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj_isog x : G \isog G :^ x.	Proof. exact: isom_isog (conj_isom x). Qed.	Lemma	conj_isog	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "conj_isom", "isog", "isom_isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_conjg_im x : x \in 'N(G) -> conjgm G x @* G = G.	Proof. by rewrite morphimEdom; apply: normP. Qed.	Lemma	norm_conjg_im	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "conjgm", "morphimEdom", "normP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_conj_isom x : x \in 'N(G) -> isom G G (conjgm G x).	Proof. by move/norm_conjg_im/restr_isom_to/(_ (conj_isom x))->. Qed.	Lemma	norm_conj_isom	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "conj_isom", "conjgm", "isom", "norm_conjg_im", "restr_isom_to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj_aut x	:= aut (injm_conj _) (norm_conjg_im (subgP (subg _ x))).	Definition	conj_aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "aut", "injm_conj", "norm_conjg_im", "subg", "subgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_conj_autE : {in 'N(G) & G, forall x y, conj_aut x y = y ^ x}.	Proof. by move=> x y nGx Gy; rewrite /= autE //= subgK. Qed.	Lemma	norm_conj_autE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "autE", "conj_aut", "subgK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj_autE : {in G &, forall x y, conj_aut x y = y ^ x}.	Proof. by apply: sub_in11 norm_conj_autE => //; apply: subsetP (normG G). Qed.	Lemma	conj_autE	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "conj_aut", "normG", "norm_conj_autE", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj_aut_morphM : {in 'N(G) &, {morph conj_aut : x y / x * y}}.	Proof. move=> x y nGx nGy; apply/permP=> z /=; rewrite permM. case Gz: (z \in G); last by rewrite !permE /= !Gz. by rewrite !norm_conj_autE // (conjgM, memJ_norm, groupM). Qed.	Lemma	conj_aut_morphM	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "conj_aut", "conjgM", "groupM", "last", "memJ_norm", "norm_conj_autE", "permE", "permM", "permP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj_aut_morphism	:= Morphism conj_aut_morphM.	Canonical	conj_aut_morphism	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "conj_aut_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_conj_aut : 'ker conj_aut = 'C(G).	Proof. apply/setP=> x /[1!inE]; case nGx: (x \in 'N(G)); last first. by symmetry; apply/idP=> cGx; rewrite (subsetP (cent_sub G)) in nGx. rewrite 2!inE /=; apply/eqP/centP=> [cx1 y Gy \| cGx]. by rewrite /commute (conjgC y) -norm_conj_autE // cx1 perm1. apply/permP=> y; case Gy: (y \in G); last by rewrite !permE Gy....	Lemma	ker_conj_aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "centP", "cent_sub", "commute", "conj_aut", "conjgC", "conjgE", "inE", "ker", "last", "mulKg", "norm_conj_autE", "perm1", "permE", "permP", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_conj_aut A : conj_aut @* A \subset Aut G.	Proof. by apply/subsetP=> _ /imsetP[x _ ->]; apply: Aut_aut. Qed.	Lemma	Aut_conj_aut	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut", "Aut_aut", "apply", "conj_aut", "imsetP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
characteristic A B	:= (A \subset B) && [forall f in Aut B, f @: A \subset A].	Definition	characteristic	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charP H G : let fixH (f : {morphism G >-> gT}) := 'injm f -> f @* G = G -> f @* H = H in reflect [/\ H \subset G & forall f, fixH f] (H \char G).	Proof. do [apply: (iffP andP) => -[sHG chHG]; split] => // [f injf Gf\|]. by apply/morphim_fixP; rewrite // -imset_autE ?(forall_inP chHG) ?Aut_aut. apply/forall_inP=> f Af; rewrite -(autmE Af) -morphimEsub //. by rewrite chHG ?injm_autm ?im_autm. Qed.	Lemma	charP	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Aut_aut", "Gf", "apply", "autmE", "char", "forall_inP", "gT", "im_autm", "imset_autE", "injf", "injm_autm", "morphimEsub", "morphim_fixP", "morphism", "sHG", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char1 G : 1 \char G.	Proof. by apply/charP; split=> [\|f _ _]; rewrite (sub1G, morphim1). Qed.	Lemma	char1	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "char", "charP", "morphim1", "split", "sub1G" ]	Characteristic subgroup properties : composition, relational properties	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_refl G : G \char G.	Proof. exact/charP. Qed.	Lemma	char_refl	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "char", "charP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_trans H G K : K \char H -> H \char G -> K \char G.	Proof. case/charP=> sKH chKH; case/charP=> sHG chHG. apply/charP; split=> [\|f injf Gf]; first exact: subset_trans sHG. rewrite -{1}(setIidPr sKH) -(morphim_restrm sHG) chKH //. by rewrite ker_restrm; move/trivgP: injf => ->; apply: subsetIr. by rewrite morphim_restrm setIid chHG. Qed.	Lemma	char_trans	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Gf", "apply", "char", "charP", "injf", "ker_restrm", "morphim_restrm", "sHG", "setIid", "setIidPr", "split", "subsetIr", "subset_trans", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_norms H G : H \char G -> 'N(G) \subset 'N(H).	Proof. case/charP=> sHG chHG; apply/normsP=> x /normP-Nx. have:= chHG [morphism of conjgm G x] => /=. by rewrite !morphimEsub //=; apply; rewrite // injm_conj. Qed.	Lemma	char_norms	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "char", "charP", "conjgm", "injm_conj", "morphimEsub", "morphism", "normP", "normsP", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_sub A B : A \char B -> A \subset B.	Proof. by case/andP. Qed.	Lemma	char_sub	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "char" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_norm_trans H G A : H \char G -> A \subset 'N(G) -> A \subset 'N(H).	Proof. by move/char_norms=> nHnG nGA; apply: subset_trans nHnG. Qed.	Lemma	char_norm_trans	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "char", "char_norms", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_normal_trans H G K : K \char H -> H <\| G -> K <\| G.	Proof. move=> chKH /andP[sHG nHG]. by rewrite /normal (subset_trans (char_sub chKH)) // (char_norm_trans chKH). Qed.	Lemma	char_normal_trans	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "char", "char_norm_trans", "char_sub", "nHG", "normal", "sHG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_normal H G : H \char G -> H <\| G.	Proof. by move/char_normal_trans; apply; apply/andP; rewrite normG. Qed.	Lemma	char_normal	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "apply", "char", "char_normal_trans", "normG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_norm H G : H \char G -> G \subset 'N(H).	Proof. by case/char_normal/andP. Qed.	Lemma	char_norm	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "char", "char_normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charI G H K : H \char G -> K \char G -> H :&: K \char G.	Proof. case/charP=> sHG chHG; case/charP=> _ chKG. apply/charP; split=> [\|f injf Gf]; first by rewrite subIset // sHG. by rewrite morphimGI ?(chHG, chKG) //; apply: subset_trans (sub1G H). Qed.	Lemma	charI	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Gf", "apply", "char", "charP", "injf", "morphimGI", "sHG", "split", "sub1G", "subIset", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charY G H K : H \char G -> K \char G -> H <*> K \char G.	Proof. case/charP=> sHG chHG; case/charP=> sKG chKG. apply/charP; split=> [\|f injf Gf]; first by rewrite gen_subG subUset sHG. by rewrite morphim_gen ?(morphimU, subUset, sHG, chHG, chKG). Qed.	Lemma	charY	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Gf", "apply", "char", "charP", "gen_subG", "injf", "morphimU", "morphim_gen", "sHG", "sKG", "split", "subUset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
charM G H K : H \char G -> K \char G -> H * K \char G.	Proof. move=> chHG chKG; rewrite -norm_joinEl ?charY //. exact: subset_trans (char_sub chHG) (char_norm chKG). Qed.	Lemma	charM	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "char", "charY", "char_norm", "char_sub", "norm_joinEl", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lone_subgroup_char G H : H \subset G -> (forall K, K \subset G -> K \isog H -> K \subset H) -> H \char G.	Proof. move=> sHG Huniq; apply/charP; split=> // f injf Gf; apply/eqP. have{} injf: {in H &, injective f}. by move/injmP: injf; apply: sub_in2; apply/subsetP. have fH: f @* H = f @: H by rewrite /morphim (setIidPr sHG). rewrite eqEcard {2}fH card_in_imset ?{}Huniq //=. by rewrite -{3}Gf morphimS. rewrite isog_sym; ...	Lemma	lone_subgroup_char	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "Gf", "apply", "card_in_imset", "char", "charP", "eqEcard", "fH", "injf", "injmP", "isog", "isogP", "isog_sym", "morphim", "morphimEdom", "morphimS", "morphism", "restrm", "sHG", "setIidPr", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"H \char G"	:= (characteristic H G) : group_scope.	Notation	H \char G	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "characteristic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_char (G H : {group aT}) : G \subset D -> H \char G -> f @* H \char f @* G.	Proof. move=> sGD /charP[sHG charH]. apply/charP; split=> [\|g injg gfG]; first exact: morphimS. have /domP[h [_ ker_h _ im_h]]: 'dom (invm injf \o g \o f) = G. by rewrite /dom /= -(morphpreIim g) (setIidPl _) ?injmK // gfG morphimS. have hH: h @* H = H. apply: charH; first by rewrite ker_h !injm_comp ?injm_invm. ...	Lemma	injm_char	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "aT", "apply", "char", "charP", "dom", "domP", "group", "injf", "injmK", "injm_comp", "injm_invm", "invm", "morphimS", "morphim_comp", "morphim_invm", "morphim_invmE", "morphpreIim", "morphpreK", "sGD", "sHG", "setIidPl", "split", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_injm (G H : {group aT}) : G \subset D -> H \subset D -> (f @* H \char f @* G) = (H \char G).	Proof. move=> sGD sHD; apply/idP/idP; last exact: injm_char. by move/(injm_char (injm_invm injf)); rewrite !morphim_invm ?morphimS // => ->. Qed.	Lemma	char_injm	finite_group	finite_group/automorphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "finset", "fingroup", "perm", "morphism" ]	[ "aT", "apply", "char", "group", "injf", "injm_char", "injm_invm", "last", "morphimS", "morphim_invm", "sGD", "sHD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build G	:= (isStarMonoid.Build G) (only parsing).	Notation	Build	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isMulBaseGroup G	:= (isStarMonoid G) (only parsing).	Notation	isMulBaseGroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build G	:= (StarMonoid_isGroup.Build G) (only parsing).	Notation	Build	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
BaseFinGroup_isGroup G	:= (StarMonoid_isGroup G) (only parsing).	Notation	BaseFinGroup_isGroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
baseFinGroupType	:= finStarMonoidType (only parsing).	Notation	baseFinGroupType	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort	:= (FinStarMonoid.sort) (only parsing).	Notation	sort	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_sort	:= (FinStarMonoid.arg_sort) (only parsing).	Notation	arg_sort	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
on M	:= (FinStarMonoid.on M) (only parsing).	Notation	on	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
copy M N	:= (FinStarMonoid.copy M N) (only parsing).	Notation	copy	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone M N	:= (FinStarMonoid.clone M N) (only parsing).	Notation	clone	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
BaseFinGroup R	:= (FinStarMonoid R) (only parsing).	Notation	BaseFinGroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oneg	:= one (only parsing).	Notation	oneg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg	:= mul (only parsing).	Notation	mulg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg	:= inv (only parsing).	Notation	invg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgn	:= natexp (only parsing).	Notation	expgn	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "natexp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build G	:= (Finite_isGroup.Build G) (only parsing).	Notation	Build	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isMulGroup G	:= (Finite_isGroup G) (only parsing).	Notation	isMulGroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invgK : involutive inv.	Proof. have mulV21 x: x^-1^-1 * 1 = x by rewrite -(mulVg x) mulgA mulVg mul1g. by move=> x; rewrite -[_ ^-1]mulV21 -(mul1g 1) mulgA !mulV21. Qed.	Lemma	invgK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inv", "mul1g", "mulVg", "mulgA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invMg : {morph inv : x y / x * y >-> y * x}.	Proof. have mulgV x: x * x^-1 = 1 by rewrite -{1}[x]invgK mulVg. move=> x y /=; rewrite -[y^-1 * _]mul1g -(mulVg (x * y)) -2!mulgA (mulgA y). by rewrite mulgV mul1g mulgV -(mulgV (x * y)) mulgA mulVg mul1g. Qed.	Lemma	invMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "inv", "invgK", "mul1g", "mulVg", "mulgA", "mulgV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg	:= conjg (only parsing).	Notation	conjg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commg	:= commg (only parsing).	Notation	commg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgA	:= mulgA (only parsing).	Notation	mulgA	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul1g	:= mul1g (only parsing).	Notation	mul1g	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invgK	:= invgK (only parsing).	Notation	invgK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invMg	:= invgM (only parsing).	Notation	invMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "invgM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_inj	:= invg_inj (only parsing).	Notation	invg_inj	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_invg_sym	:= eqg_invLR (only parsing).	Notation	eq_invg_sym	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "eqg_invLR" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg1	:= invg1 (only parsing).	Notation	invg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_invg1	:= invg_eq1 (only parsing).	Notation	eq_invg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "invg_eq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg1	:= mulg1 (only parsing).	Notation	mulg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgnE	:= expgnE (only parsing).	Notation	expgnE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expg0	:= expg0 (only parsing).	Notation	expg0	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expg1	:= expg1 (only parsing).	Notation	expg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expg1n	:= expg1n (only parsing).	Notation	expg1n	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgD	:= expgnDr (only parsing).	Notation	expgD	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "expgnDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgSr	:= expgSr (only parsing).	Notation	expgSr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgM	:= expgnA (only parsing).	Notation	expgM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "expgnA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgAC	:= expgnAC (only parsing).	Notation	expgAC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "expgnAC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commute	:= commute (only parsing).	Notation	commute	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commute_refl	:= commute_refl (only parsing).	Notation	commute_refl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commute_sym	:= commute_sym (only parsing).	Notation	commute_sym	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commute1	:= commute1 (only parsing).	Notation	commute1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commuteM	:= commuteM (only parsing).	Notation	commuteM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commuteX	:= commuteX (only parsing).	Notation	commuteX	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commuteX2	:= commuteX2 (only parsing).	Notation	commuteX2	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgVn	:= expVgn (only parsing).	Notation	expgVn	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "expVgn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgMn	:= expgMn (only parsing).	Notation	expgMn	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulVg	:= mulVg (only parsing).	Notation	mulVg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgV	:= mulgV (only parsing).	Notation	mulgV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulKg	:= mulKg (only parsing).	Notation	mulKg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulKVg	:= mulVKg (only parsing).	Notation	mulKVg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mulVKg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgI	:= mulgI (only parsing).	Notation	mulgI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgK	:= mulgK (only parsing).	Notation	mulgK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgKV	:= mulgVK (only parsing).	Notation	mulgKV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mulgVK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_invg_mul (T : finGroupType) x y : (x^-1 == y :> T) = (x * y == 1).	Proof. by rewrite mulg_eq1 eqg_invLR. Qed.	Lemma	eq_invg_mul	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "eqg_invLR", "mulg_eq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_mulgV1 (T : finGroupType) x y : (x == y) = (x * y^-1 == 1 :> T).	Proof. exact/esym/divg_eq1. Qed.	Lemma	eq_mulgV1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "divg_eq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_mulVg1 (T : finGroupType) x y : (x == y) = (x^-1 * y == 1 :> T).	Proof. by rewrite mulg_eq1 eqg_inv. Qed.	Lemma	eq_mulVg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "eqg_inv", "mulg_eq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commuteV	:= commuteV (only parsing).	Notation	commuteV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgE	:= conjgE (only parsing).	Notation	conjgE	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgC	:= conjgC (only parsing).	Notation	conjgC	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgCV	:= conjgCV (only parsing).	Notation	conjgCV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg1	:= conjg1 (only parsing).	Notation	conjg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj1g	:= conj1g (only parsing).	Notation	conj1g	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjMg	:= conjMg (only parsing).	Notation	conjMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgM	:= conjgM (only parsing).	Notation	conjgM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjVg	:= conjVg (only parsing).	Notation	conjVg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjJg	:= conjJg (only parsing).	Notation	conjJg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjXg	:= conjXg (only parsing).	Notation	conjXg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgK	:= conjgK (only parsing).	Notation	conjgK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgKV	:= conjgKV (only parsing).	Notation	conjgKV	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_inj	:= conjg_inj (only parsing).	Notation	conjg_inj	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_isom x : isom G (G :^ x) (conjgm G x).

Proof. by apply/isomP; rewrite morphim_conj setIid injm_conj. Qed.

Lemma

conj_isom

finite_group

finite_group/automorphism.v

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

[ "apply", "conjgm", "injm_conj", "isom", "isomP", "morphim_conj", "setIid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_isog x : G \isog G :^ x.

Proof. exact: isom_isog (conj_isom x). Qed.

Lemma

conj_isog

finite_group

finite_group/automorphism.v

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

[ "conj_isom", "isog", "isom_isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_conjg_im x : x \in 'N(G) -> conjgm G x @* G = G.

Proof. by rewrite morphimEdom; apply: normP. Qed.

Lemma

norm_conjg_im

finite_group

finite_group/automorphism.v

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

[ "apply", "conjgm", "morphimEdom", "normP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_conj_isom x : x \in 'N(G) -> isom G G (conjgm G x).

Proof. by move/norm_conjg_im/restr_isom_to/(_ (conj_isom x))->. Qed.

Lemma

norm_conj_isom

finite_group

finite_group/automorphism.v

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

[ "conj_isom", "conjgm", "isom", "norm_conjg_im", "restr_isom_to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_aut x

:= aut (injm_conj _) (norm_conjg_im (subgP (subg _ x))).

Definition

conj_aut

finite_group

finite_group/automorphism.v

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

[ "aut", "injm_conj", "norm_conjg_im", "subg", "subgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_conj_autE : {in 'N(G) & G, forall x y, conj_aut x y = y ^ x}.

Proof. by move=> x y nGx Gy; rewrite /= autE //= subgK. Qed.

Lemma

norm_conj_autE

finite_group

finite_group/automorphism.v

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

[ "autE", "conj_aut", "subgK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_autE : {in G &, forall x y, conj_aut x y = y ^ x}.

Proof. by apply: sub_in11 norm_conj_autE => //; apply: subsetP (normG G). Qed.

Lemma

conj_autE

finite_group

finite_group/automorphism.v

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

[ "apply", "conj_aut", "normG", "norm_conj_autE", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_aut_morphM : {in 'N(G) &, {morph conj_aut : x y / x * y}}.

Proof. move=> x y nGx nGy; apply/permP=> z /=; rewrite permM. case Gz: (z \in G); last by rewrite !permE /= !Gz. by rewrite !norm_conj_autE // (conjgM, memJ_norm, groupM). Qed.

Lemma

conj_aut_morphM

finite_group

finite_group/automorphism.v

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

[ "apply", "conj_aut", "conjgM", "groupM", "last", "memJ_norm", "norm_conj_autE", "permE", "permM", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj_aut_morphism

:= Morphism conj_aut_morphM.

Canonical

conj_aut_morphism

finite_group

finite_group/automorphism.v

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

[ "conj_aut_morphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_conj_aut : 'ker conj_aut = 'C(G).

Proof. apply/setP=> x /[1!inE]; case nGx: (x \in 'N(G)); last first. by symmetry; apply/idP=> cGx; rewrite (subsetP (cent_sub G)) in nGx. rewrite 2!inE /=; apply/eqP/centP=> [cx1 y Gy | cGx]. by rewrite /commute (conjgC y) -norm_conj_autE // cx1 perm1. apply/permP=> y; case Gy: (y \in G); last by rewrite !permE Gy....

Lemma

ker_conj_aut

finite_group

finite_group/automorphism.v

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

[ "apply", "centP", "cent_sub", "commute", "conj_aut", "conjgC", "conjgE", "inE", "ker", "last", "mulKg", "norm_conj_autE", "perm1", "permE", "permP", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_conj_aut A : conj_aut @* A \subset Aut G.

Proof. by apply/subsetP=> _ /imsetP[x _ ->]; apply: Aut_aut. Qed.

Lemma

Aut_conj_aut

finite_group

finite_group/automorphism.v

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

[ "Aut", "Aut_aut", "apply", "conj_aut", "imsetP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

characteristic A B

:= (A \subset B) && [forall f in Aut B, f @: A \subset A].

Definition

characteristic

finite_group

finite_group/automorphism.v

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

[ "Aut" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charP H G : let fixH (f : {morphism G >-> gT}) := 'injm f -> f @* G = G -> f @* H = H in reflect [/\ H \subset G & forall f, fixH f] (H \char G).

Proof. do [apply: (iffP andP) => -[sHG chHG]; split] => // [f injf Gf|]. by apply/morphim_fixP; rewrite // -imset_autE ?(forall_inP chHG) ?Aut_aut. apply/forall_inP=> f Af; rewrite -(autmE Af) -morphimEsub //. by rewrite chHG ?injm_autm ?im_autm. Qed.

Lemma

charP

finite_group

finite_group/automorphism.v

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

[ "Aut_aut", "Gf", "apply", "autmE", "char", "forall_inP", "gT", "im_autm", "imset_autE", "injf", "injm_autm", "morphimEsub", "morphim_fixP", "morphism", "sHG", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char1 G : 1 \char G.

Proof. by apply/charP; split=> [|f _ _]; rewrite (sub1G, morphim1). Qed.

Lemma

char1

finite_group

finite_group/automorphism.v

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

[ "apply", "char", "charP", "morphim1", "split", "sub1G" ]

Characteristic subgroup properties : composition, relational properties

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_refl G : G \char G.

Proof. exact/charP. Qed.

Lemma

char_refl

finite_group

finite_group/automorphism.v

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

[ "char", "charP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_trans H G K : K \char H -> H \char G -> K \char G.

Proof. case/charP=> sKH chKH; case/charP=> sHG chHG. apply/charP; split=> [|f injf Gf]; first exact: subset_trans sHG. rewrite -{1}(setIidPr sKH) -(morphim_restrm sHG) chKH //. by rewrite ker_restrm; move/trivgP: injf => ->; apply: subsetIr. by rewrite morphim_restrm setIid chHG. Qed.

Lemma

char_trans

finite_group

finite_group/automorphism.v

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

[ "Gf", "apply", "char", "charP", "injf", "ker_restrm", "morphim_restrm", "sHG", "setIid", "setIidPr", "split", "subsetIr", "subset_trans", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_norms H G : H \char G -> 'N(G) \subset 'N(H).

Proof. case/charP=> sHG chHG; apply/normsP=> x /normP-Nx. have:= chHG [morphism of conjgm G x] => /=. by rewrite !morphimEsub //=; apply; rewrite // injm_conj. Qed.

Lemma

char_norms

finite_group

finite_group/automorphism.v

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

[ "apply", "char", "charP", "conjgm", "injm_conj", "morphimEsub", "morphism", "normP", "normsP", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_sub A B : A \char B -> A \subset B.

Proof. by case/andP. Qed.

Lemma

char_sub

finite_group

finite_group/automorphism.v

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

[ "char" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_norm_trans H G A : H \char G -> A \subset 'N(G) -> A \subset 'N(H).

Proof. by move/char_norms=> nHnG nGA; apply: subset_trans nHnG. Qed.

Lemma

char_norm_trans

finite_group

finite_group/automorphism.v

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

[ "apply", "char", "char_norms", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_normal_trans H G K : K \char H -> H <| G -> K <| G.

Proof. move=> chKH /andP[sHG nHG]. by rewrite /normal (subset_trans (char_sub chKH)) // (char_norm_trans chKH). Qed.

Lemma

char_normal_trans

finite_group

finite_group/automorphism.v

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

[ "char", "char_norm_trans", "char_sub", "nHG", "normal", "sHG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_normal H G : H \char G -> H <| G.

Proof. by move/char_normal_trans; apply; apply/andP; rewrite normG. Qed.

Lemma

char_normal

finite_group

finite_group/automorphism.v

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

[ "apply", "char", "char_normal_trans", "normG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_norm H G : H \char G -> G \subset 'N(H).

Proof. by case/char_normal/andP. Qed.

Lemma

char_norm

finite_group

finite_group/automorphism.v

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

[ "char", "char_normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charI G H K : H \char G -> K \char G -> H :&: K \char G.

Proof. case/charP=> sHG chHG; case/charP=> _ chKG. apply/charP; split=> [|f injf Gf]; first by rewrite subIset // sHG. by rewrite morphimGI ?(chHG, chKG) //; apply: subset_trans (sub1G H). Qed.

Lemma

charI

finite_group

finite_group/automorphism.v

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

[ "Gf", "apply", "char", "charP", "injf", "morphimGI", "sHG", "split", "sub1G", "subIset", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charY G H K : H \char G -> K \char G -> H <*> K \char G.

Proof. case/charP=> sHG chHG; case/charP=> sKG chKG. apply/charP; split=> [|f injf Gf]; first by rewrite gen_subG subUset sHG. by rewrite morphim_gen ?(morphimU, subUset, sHG, chHG, chKG). Qed.

Lemma

charY

finite_group

finite_group/automorphism.v

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

[ "Gf", "apply", "char", "charP", "gen_subG", "injf", "morphimU", "morphim_gen", "sHG", "sKG", "split", "subUset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

charM G H K : H \char G -> K \char G -> H * K \char G.

Proof. move=> chHG chKG; rewrite -norm_joinEl ?charY //. exact: subset_trans (char_sub chHG) (char_norm chKG). Qed.

Lemma

charM

finite_group

finite_group/automorphism.v

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

[ "char", "charY", "char_norm", "char_sub", "norm_joinEl", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lone_subgroup_char G H : H \subset G -> (forall K, K \subset G -> K \isog H -> K \subset H) -> H \char G.

Proof. move=> sHG Huniq; apply/charP; split=> // f injf Gf; apply/eqP. have{} injf: {in H &, injective f}. by move/injmP: injf; apply: sub_in2; apply/subsetP. have fH: f @* H = f @: H by rewrite /morphim (setIidPr sHG). rewrite eqEcard {2}fH card_in_imset ?{}Huniq //=. by rewrite -{3}Gf morphimS. rewrite isog_sym; ...

Lemma

lone_subgroup_char

finite_group

finite_group/automorphism.v

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

[ "Gf", "apply", "card_in_imset", "char", "charP", "eqEcard", "fH", "injf", "injmP", "isog", "isogP", "isog_sym", "morphim", "morphimEdom", "morphimS", "morphism", "restrm", "sHG", "setIidPr", "split", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"H \char G"

:= (characteristic H G) : group_scope.

Notation

H \char G

finite_group

finite_group/automorphism.v

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

[ "characteristic" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_char (G H : {group aT}) : G \subset D -> H \char G -> f @* H \char f @* G.

Proof. move=> sGD /charP[sHG charH]. apply/charP; split=> [|g injg gfG]; first exact: morphimS. have /domP[h [_ ker_h _ im_h]]: 'dom (invm injf \o g \o f) = G. by rewrite /dom /= -(morphpreIim g) (setIidPl _) ?injmK // gfG morphimS. have hH: h @* H = H. apply: charH; first by rewrite ker_h !injm_comp ?injm_invm. ...

Lemma

injm_char

finite_group

finite_group/automorphism.v

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

[ "aT", "apply", "char", "charP", "dom", "domP", "group", "injf", "injmK", "injm_comp", "injm_invm", "invm", "morphimS", "morphim_comp", "morphim_invm", "morphim_invmE", "morphpreIim", "morphpreK", "sGD", "sHG", "setIidPl", "split", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_injm (G H : {group aT}) : G \subset D -> H \subset D -> (f @* H \char f @* G) = (H \char G).

Proof. move=> sGD sHD; apply/idP/idP; last exact: injm_char. by move/(injm_char (injm_invm injf)); rewrite !morphim_invm ?morphimS // => ->. Qed.

Lemma

char_injm

finite_group

finite_group/automorphism.v

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

[ "aT", "apply", "char", "group", "injf", "injm_char", "injm_invm", "last", "morphimS", "morphim_invm", "sGD", "sHD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Build G

:= (isStarMonoid.Build G) (only parsing).

Notation

Build

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isMulBaseGroup G

:= (isStarMonoid G) (only parsing).

Notation

isMulBaseGroup

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Build G

:= (StarMonoid_isGroup.Build G) (only parsing).

Notation

Build

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

BaseFinGroup_isGroup G

:= (StarMonoid_isGroup G) (only parsing).

Notation

BaseFinGroup_isGroup

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

baseFinGroupType

:= finStarMonoidType (only parsing).

Notation

baseFinGroupType

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort

:= (FinStarMonoid.sort) (only parsing).

Notation

sort

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

arg_sort

:= (FinStarMonoid.arg_sort) (only parsing).

Notation

arg_sort

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

on M

:= (FinStarMonoid.on M) (only parsing).

Notation

on

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

copy M N

:= (FinStarMonoid.copy M N) (only parsing).

Notation

copy

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone M N

:= (FinStarMonoid.clone M N) (only parsing).

Notation

clone

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

BaseFinGroup R

:= (FinStarMonoid R) (only parsing).

Notation

BaseFinGroup

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oneg

:= one (only parsing).

Notation

oneg

finite_group

finite_group/fingroup.v

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

[ "one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulg

:= mul (only parsing).

Notation

mulg

finite_group

finite_group/fingroup.v

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

[ "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg

:= inv (only parsing).

Notation

invg

finite_group

finite_group/fingroup.v

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

[ "inv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgn

:= natexp (only parsing).

Notation

expgn

finite_group

finite_group/fingroup.v

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

[ "natexp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Build G

:= (Finite_isGroup.Build G) (only parsing).

Notation

Build

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isMulGroup G

:= (Finite_isGroup G) (only parsing).

Notation

isMulGroup

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invgK : involutive inv.

Proof. have mulV21 x: x^-1^-1 * 1 = x by rewrite -(mulVg x) mulgA mulVg mul1g. by move=> x; rewrite -[_ ^-1]mulV21 -(mul1g 1) mulgA !mulV21. Qed.

Lemma

invgK

finite_group

finite_group/fingroup.v

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

[ "inv", "mul1g", "mulVg", "mulgA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invMg : {morph inv : x y / x * y >-> y * x}.

Proof. have mulgV x: x * x^-1 = 1 by rewrite -{1}[x]invgK mulVg. move=> x y /=; rewrite -[y^-1 * _]mul1g -(mulVg (x * y)) -2!mulgA (mulgA y). by rewrite mulgV mul1g mulgV -(mulgV (x * y)) mulgA mulVg mul1g. Qed.

Lemma

invMg

finite_group

finite_group/fingroup.v

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

[ "inv", "invgK", "mul1g", "mulVg", "mulgA", "mulgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg

:= conjg (only parsing).

Notation

conjg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commg

:= commg (only parsing).

Notation

commg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgA

:= mulgA (only parsing).

Notation

mulgA

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul1g

:= mul1g (only parsing).

Notation

mul1g

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invgK

:= invgK (only parsing).

Notation

invgK

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invMg

:= invgM (only parsing).

Notation

invMg

finite_group

finite_group/fingroup.v

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

[ "invgM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg_inj

:= invg_inj (only parsing).

Notation

invg_inj

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_invg_sym

:= eqg_invLR (only parsing).

Notation

eq_invg_sym

finite_group

finite_group/fingroup.v

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

[ "eqg_invLR" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg1

:= invg1 (only parsing).

Notation

invg1

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_invg1

:= invg_eq1 (only parsing).

Notation

eq_invg1

finite_group

finite_group/fingroup.v

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

[ "invg_eq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulg1

:= mulg1 (only parsing).

Notation

mulg1

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgnE

:= expgnE (only parsing).

Notation

expgnE

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expg0

:= expg0 (only parsing).

Notation

expg0

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expg1

:= expg1 (only parsing).

Notation

expg1

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expg1n

:= expg1n (only parsing).

Notation

expg1n

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgD

:= expgnDr (only parsing).

Notation

expgD

finite_group

finite_group/fingroup.v

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

[ "expgnDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgSr

:= expgSr (only parsing).

Notation

expgSr

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgM

:= expgnA (only parsing).

Notation

expgM

finite_group

finite_group/fingroup.v

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

[ "expgnA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgAC

:= expgnAC (only parsing).

Notation

expgAC

finite_group

finite_group/fingroup.v

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

[ "expgnAC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commute

:= commute (only parsing).

Notation

commute

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commute_refl

:= commute_refl (only parsing).

Notation

commute_refl

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commute_sym

:= commute_sym (only parsing).

Notation

commute_sym

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commute1

:= commute1 (only parsing).

Notation

commute1

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commuteM

:= commuteM (only parsing).

Notation

commuteM

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commuteX

:= commuteX (only parsing).

Notation

commuteX

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commuteX2

:= commuteX2 (only parsing).

Notation

commuteX2

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgVn

:= expVgn (only parsing).

Notation

expgVn

finite_group

finite_group/fingroup.v

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

[ "expVgn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expgMn

:= expgMn (only parsing).

Notation

expgMn

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulVg

:= mulVg (only parsing).

Notation

mulVg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgV

:= mulgV (only parsing).

Notation

mulgV

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulKg

:= mulKg (only parsing).

Notation

mulKg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulKVg

:= mulVKg (only parsing).

Notation

mulKVg

finite_group

finite_group/fingroup.v

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

[ "mulVKg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgI

:= mulgI (only parsing).

Notation

mulgI

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgK

:= mulgK (only parsing).

Notation

mulgK

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgKV

:= mulgVK (only parsing).

Notation

mulgKV

finite_group

finite_group/fingroup.v

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

[ "mulgVK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_invg_mul (T : finGroupType) x y : (x^-1 == y :> T) = (x * y == 1).

Proof. by rewrite mulg_eq1 eqg_invLR. Qed.

Lemma

eq_invg_mul

finite_group

finite_group/fingroup.v

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

[ "eqg_invLR", "mulg_eq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_mulgV1 (T : finGroupType) x y : (x == y) = (x * y^-1 == 1 :> T).

Proof. exact/esym/divg_eq1. Qed.

Lemma

eq_mulgV1

finite_group

finite_group/fingroup.v

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

[ "divg_eq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_mulVg1 (T : finGroupType) x y : (x == y) = (x^-1 * y == 1 :> T).

Proof. by rewrite mulg_eq1 eqg_inv. Qed.

Lemma

eq_mulVg1

finite_group

finite_group/fingroup.v

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

[ "eqg_inv", "mulg_eq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commuteV

:= commuteV (only parsing).

Notation

commuteV

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgE

:= conjgE (only parsing).

Notation

conjgE

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgC

:= conjgC (only parsing).

Notation

conjgC

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgCV

:= conjgCV (only parsing).

Notation

conjgCV

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg1

:= conjg1 (only parsing).

Notation

conjg1

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conj1g

:= conj1g (only parsing).

Notation

conj1g

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjMg

:= conjMg (only parsing).

Notation

conjMg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgM

:= conjgM (only parsing).

Notation

conjgM

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjVg

:= conjVg (only parsing).

Notation

conjVg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjJg

:= conjJg (only parsing).

Notation

conjJg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjXg

:= conjXg (only parsing).

Notation

conjXg

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgK

:= conjgK (only parsing).

Notation

conjgK

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjgKV

:= conjgKV (only parsing).

Notation

conjgKV

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_inj

:= conjg_inj (only parsing).

Notation

conjg_inj

finite_group

finite_group/fingroup.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d