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
sub_isom (A : {set aT}) (C : {set rT}) : A \subset G -> f @* A = C -> 'injm f -> isom A C f.	Proof. move=> sAG; case: (restrmP f sAG) => g [_ _ _ img] <-{C} injf. rewrite /isom -morphimEsub ?morphimDG ?morphim1 //. by rewrite subDset setUC subsetU ?sAG. Qed.	Lemma	sub_isom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "injf", "isom", "morphim1", "morphimDG", "morphimEsub", "restrmP", "sAG", "setUC", "subDset", "subsetU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_isog (A : {set aT}) : A \subset G -> 'injm f -> isog A (f @* A).	Proof. by move=> sAG injf; apply: (isom_isog f sAG); apply: sub_isom. Qed.	Lemma	sub_isog	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "injf", "isog", "isom_isog", "sAG", "sub_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_isom_to (A : {set aT}) (C R : {group rT}) (sAG : A \subset G) : f @* A = C -> isom G R f -> isom A C (restrm sAG f).	Proof. by move=> defC /isomP[inj_f _]; apply: sub_isom. Qed.	Lemma	restr_isom_to	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "group", "inj_f", "isom", "isomP", "restrm", "sAG", "sub_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_isom (A : {group aT}) (R : {group rT}) (sAG : A \subset G) : isom G R f -> isom A (f @* A) (restrm sAG f).	Proof. exact: restr_isom_to. Qed.	Lemma	restr_isom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "group", "isom", "restr_isom_to", "restrm", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x \isog y"	:= (isog x y).	Notation	x \isog y	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idm_isom : isom G G (idm G).	Proof. exact: sub_isom (im_idm G) (injm_idm G). Qed.	Lemma	idm_isom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "idm", "im_idm", "injm_idm", "isom", "sub_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_refl : G \isog G.	Proof. exact: isom_isog idm_isom. Qed.	Lemma	isog_refl	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "idm_isom", "isog", "isom_isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_isog : G \isog H -> #\|G\| = #\|H\|.	Proof. by case/isogP=> f injf <-; apply: isom_card (f) _; apply/isomP. Qed.	Lemma	card_isog	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "injf", "isog", "isogP", "isomP", "isom_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_abelian : G \isog H -> abelian G = abelian H.	Proof. by case/isogP=> f injf <-; rewrite injm_abelian. Qed.	Lemma	isog_abelian	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "abelian", "injf", "injm_abelian", "isog", "isogP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivial_isog : G :=: 1 -> H :=: 1 -> G \isog H.	Proof. move=> -> ->; apply/isogP. exists [morphism of @trivm gT hT 1]; rewrite /= ?morphim1 //. by rewrite ker_trivm; apply: subxx. Qed.	Lemma	trivial_isog	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "gT", "isog", "isogP", "ker_trivm", "morphim1", "morphism", "subxx", "trivm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_eq1 : G \isog H -> (G :==: 1) = (H :==: 1).	Proof. by move=> isoGH; rewrite !trivg_card1 card_isog. Qed.	Lemma	isog_eq1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "card_isog", "isoGH", "isog", "trivg_card1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_sym (f : {morphism G >-> hT}) (isoGH : isom G H f) : isom H G (isom_inv isoGH).	Proof. rewrite sub_isom 1?injm_restrm ?injm_invm // im_restrm. by rewrite -(isom_im isoGH) im_invm. Qed.	Lemma	isom_sym	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "im_invm", "im_restrm", "injm_invm", "injm_restrm", "isoGH", "isom", "isom_im", "isom_inv", "morphism", "sub_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_symr : G \isog H -> H \isog G.	Proof. by case/isog_isom=> f /isom_sym/isom_isog->. Qed.	Lemma	isog_symr	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isog", "isog_isom", "isom_isog", "isom_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_trans : G \isog H -> H \isog K -> G \isog K.	Proof. case/isogP=> f injf <-; case/isogP=> g injg <-. have defG: f @^-1 (f @ G) = G by rewrite morphimGK ?subsetIl. rewrite -morphim_comp -{1 8}defG. by apply/isogP; exists [morphism of g \o f]; rewrite ?injm_comp. Qed.	Lemma	isog_trans	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "defG", "injf", "injm_comp", "isog", "isogP", "morphimGK", "morphim_comp", "morphism", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nclasses_isog : G \isog H -> #\|classes G\| = #\|classes H\|.	Proof. by case/isogP=> f injf <-; rewrite nclasses_injm. Qed.	Lemma	nclasses_isog	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "classes", "injf", "isog", "isogP", "nclasses_injm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_sym : (G \isog H) = (H \isog G).	Proof. by apply/idP/idP; apply: isog_symr. Qed.	Lemma	isog_sym	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "isog", "isog_symr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_transl : G \isog H -> (G \isog K) = (H \isog K).	Proof. by move=> iso; apply/idP/idP; apply: isog_trans; rewrite // -isog_sym. Qed.	Lemma	isog_transl	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "iso", "isog", "isog_sym", "isog_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_transr : G \isog H -> (K \isog G) = (K \isog H).	Proof. by move=> iso; apply/idP/idP; move/isog_trans; apply; rewrite // -isog_sym. Qed.	Lemma	isog_transr	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "iso", "isog", "isog_sym", "isog_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homg rT aT (C : {set rT}) (D : {set aT})	:= [exists (f : {ffun aT -> rT} \| morphic D f), f @: D == C].	Definition	homg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "morphic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homgP rT aT (C : {set rT}) (D : {set aT}) : reflect (exists f : {morphism D >-> rT}, f @* D = C) (homg C D).	Proof. apply: (iffP exists_eq_inP) => [[f fM <-] \| [f <-]]. by exists (morphm_morphism fM); rewrite /morphim /= setIid. exists (finfun f); first by apply/morphicP=> x y Dx Dy; rewrite !ffunE morphM. by rewrite /morphim setIid; apply: eq_imset => x; rewrite ffunE. Qed.	Lemma	homgP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "aT", "apply", "eq_imset", "exists_eq_inP", "fM", "ffunE", "homg", "morphM", "morphicP", "morphim", "morphism", "morphm_morphism", "setIid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_homg aT rT (A D : {set aT}) (f : {morphism D >-> rT}) : A \subset D -> homg (f @* A) A.	Proof. move=> sAD; apply/homgP; exists (restrm_morphism sAD f). by rewrite morphim_restrm setIid. Qed.	Lemma	morphim_homg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "homg", "homgP", "morphim_restrm", "morphism", "restrm_morphism", "sAD", "setIid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_homg rT aT (C : {set rT}) (G : {group aT}) : homg C G -> #\|C\| <= #\|G\|.	Proof. by case/homgP=> f <-; apply: leq_morphim. Qed.	Lemma	leq_homg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "group", "homg", "homgP", "leq_morphim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homg_refl aT (A : {set aT}) : homg A A.	Proof. by apply/homgP; exists (idm_morphism A); rewrite im_idm. Qed.	Lemma	homg_refl	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "homg", "homgP", "idm_morphism", "im_idm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homg_trans aT (B : {set aT}) rT (C : {set rT}) gT (G : {group gT}) : homg C B -> homg B G -> homg C G.	Proof. move=> homCB homBG; case/homgP: homBG homCB => fG <- /homgP[fK <-]. by rewrite -morphim_comp morphim_homg // -sub_morphim_pre. Qed.	Lemma	homg_trans	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "fK", "gT", "group", "homg", "homgP", "morphim_comp", "morphim_homg", "sub_morphim_pre" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isogEcard rT aT (G : {group rT}) (H : {group aT}) : (G \isog H) = (homg G H) && (#\|H\| <= #\|G\|).	Proof. rewrite isog_sym; apply/isogP/andP=> [[f injf <-] \| []]. by rewrite leq_eqVlt eq_sym card_im_injm injf morphim_homg. case/homgP=> f <-; rewrite leq_eqVlt eq_sym card_im_injm. by rewrite ltnNge leq_morphim orbF; exists f. Qed.	Lemma	isogEcard	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "card_im_injm", "eq_sym", "group", "homg", "homgP", "injf", "isog", "isogP", "isog_sym", "leq_eqVlt", "leq_morphim", "ltnNge", "morphim_homg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_hom rT aT (G : {group rT}) (H : {group aT}) : G \isog H -> homg G H.	Proof. by rewrite isogEcard; case/andP. Qed.	Lemma	isog_hom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "group", "homg", "isog", "isogEcard" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isogEhom rT aT (G : {group rT}) (H : {group aT}) : (G \isog H) = homg G H && homg H G.	Proof. apply/idP/andP=> [isoGH \| [homGH homHG]]. by rewrite !isog_hom // isog_sym. by rewrite isogEcard homGH leq_homg. Qed.	Lemma	isogEhom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "group", "homg", "isoGH", "isog", "isogEcard", "isog_hom", "isog_sym", "leq_homg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_homgl gT aT rT (G : {group gT}) (H : {group aT}) (K : {group rT}) : G \isog H -> homg G K = homg H K.	Proof. by rewrite isogEhom => /andP[homGH homHG]; apply/idP/idP; apply: homg_trans. Qed.	Lemma	eq_homgl	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "gT", "group", "homg", "homg_trans", "isog", "isogEhom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_homgr gT rT aT (G : {group gT}) (H : {group rT}) (K : {group aT}) : G \isog H -> homg K G = homg K H.	Proof. rewrite isogEhom => /andP[homGH homHG]. by apply/idP/idP=> homK; apply: homg_trans homK _. Qed.	Lemma	eq_homgr	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "gT", "group", "homg", "homg_trans", "isog", "isogEhom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G \homg H"	:= (homg G H) (at level 70, no associativity) : group_scope.	Notation	G \homg H	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "homg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sgval_morphism	:= Morphism (@sgvalM _ G).	Canonical	sgval_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "sgvalM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_morphism	:= Morphism (@subgM _ G).	Canonical	subg_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "subgM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_sgval : 'injm sgval.	Proof. exact/injmP/(in2W subg_inj). Qed.	Lemma	injm_sgval	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmP", "sgval", "subg_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_subg : 'injm (subg G).	Proof. exact/injmP/(can_in_inj subgK). Qed.	Lemma	injm_subg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmP", "subg", "subgK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_sgval : 'ker sgval = 1.	Proof. exact/trivgP. Qed.	Lemma	ker_sgval	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "sgval", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_subg : 'ker (subg G) = 1.	Proof. exact/trivgP. Qed.	Lemma	ker_subg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "subg", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_subg : subg G @* G = [subg G].	Proof. apply/eqP; rewrite -subTset morphimEdom. by apply/subsetP=> u _; rewrite -(sgvalK u) imset_f ?subgP. Qed.	Lemma	im_subg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "imset_f", "morphimEdom", "sgvalK", "subTset", "subg", "subgP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sgval_sub A : sgval @* A \subset G.	Proof. by apply/subsetP=> x; case/imsetP=> u _ ->; apply: subgP. Qed.	Lemma	sgval_sub	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "imsetP", "sgval", "subgP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sgvalmK A : subg G @* (sgval @* A) = A.	Proof. apply/eqP; rewrite eqEcard !card_injm ?subsetT ?sgval_sub // leqnn andbT. rewrite -morphim_comp; apply/subsetP=> _ /morphimP[v _ Av ->] /=. by rewrite sgvalK. Qed.	Lemma	sgvalmK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "card_injm", "eqEcard", "leqnn", "morphimP", "morphim_comp", "sgval", "sgvalK", "sgval_sub", "subg", "subsetP", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgmK (A : {set gT}) : A \subset G -> sgval @* (subg G @* A) = A.	Proof. move=> sAG; apply/eqP; rewrite eqEcard !card_injm ?subsetT //. rewrite leqnn andbT -morphim_comp morphimE /= morphpreT. by apply/subsetP=> _ /morphimP[v Gv Av ->] /=; rewrite subgK. Qed.	Lemma	subgmK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "card_injm", "eqEcard", "gT", "leqnn", "morphimE", "morphimP", "morphim_comp", "morphpreT", "sAG", "sgval", "subg", "subgK", "subsetP", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_sgval : sgval @* [subg G] = G.	Proof. by rewrite -{2}im_subg subgmK. Qed.	Lemma	im_sgval	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "im_subg", "sgval", "subg", "subgmK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_subg : isom G [subg G] (subg G).	Proof. by apply/isomP; rewrite im_subg. Qed.	Lemma	isom_subg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "im_subg", "isom", "isomP", "subg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_sgval : isom [subg G] G sgval.	Proof. by apply/isomP; rewrite im_sgval. Qed.	Lemma	isom_sgval	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "im_sgval", "isom", "isomP", "sgval", "subg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_subg : isog G [subg G].	Proof. exact: isom_isog isom_subg. Qed.	Lemma	isog_subg	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isog", "isom_isog", "isom_subg", "subg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_type : predArgType	:= Perm (pval : {ffun T -> T}) & injectiveb pval.	Inductive	perm_type	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "injectiveb", "pval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pval p	:= let: Perm f _ := p in f.	Definition	pval	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_of	:= perm_type.	Definition	perm_of	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "perm_type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_proof (f : T -> T) : injective f -> injectiveb (finfun f).	Proof. by move=> f_inj; apply/injectiveP; apply: eq_inj f_inj _ => x; rewrite ffunE. Qed.	Lemma	perm_proof	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "f_inj", "ffunE", "injectiveP", "injectiveb" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'perm' T }"	:= (perm_of T) (format "{ 'perm' T }") : type_scope.	Notation	{ 'perm' T }	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "perm_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''S_' n"	:= {perm 'I_n} (at level 8, n at level 2, format "''S_' n").	Notation	''S_' n	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_unlock	:= Unlockable perm.unlock.	Canonical	perm_unlock	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fun_of_perm_unlock	:= Unlockable fun_of_perm.unlock.	Canonical	fun_of_perm_unlock	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "fun_of_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fun_of_perm : perm_type >-> Funclass.		Coercion	fun_of_perm	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "perm_type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permP s t : s =1 t <-> s = t.	Proof. by split=> [\| -> //]; rewrite unlock => eq_sv; apply/val_inj/ffunP. Qed.	Lemma	permP	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "ffunP", "split", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pvalE s : pval s = s :> (T -> T).	Proof. by rewrite [@fun_of_perm]unlock. Qed.	Lemma	pvalE	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "fun_of_perm", "pval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permE f f_inj : @perm T f f_inj =1 f.	Proof. by move=> x; rewrite -pvalE [@perm]unlock ffunE. Qed.	Lemma	permE	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "f_inj", "ffunE", "pvalE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_inj {s} : injective s.	Proof. by rewrite -!pvalE; apply: (injectiveP _ (valP s)). Qed.	Lemma	perm_inj	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "injectiveP", "pvalE", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_onto s : codom s =i predT.	Proof. by apply/subset_cardP; rewrite ?card_codom ?subset_predT. Qed.	Lemma	perm_onto	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "card_codom", "codom", "subset_cardP", "subset_predT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_one	:= perm (@inj_id T).	Definition	perm_one	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_invK s : cancel (fun x => iinv (perm_onto s x)) s.	Proof. by move=> x /=; rewrite f_iinv. Qed.	Lemma	perm_invK	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "f_iinv", "iinv", "perm_onto" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_inv s	:= perm (can_inj (perm_invK s)).	Definition	perm_inv	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "perm_invK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_mul s t	:= perm (inj_comp (@perm_inj t) (@perm_inj s)).	Definition	perm_mul	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "perm_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_oneP : left_id perm_one perm_mul.	Proof. by move=> s; apply/permP => x; rewrite permE /= permE. Qed.	Lemma	perm_oneP	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "permE", "permP", "perm_mul", "perm_one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_invP : left_inverse perm_one perm_inv perm_mul.	Proof. by move=> s; apply/permP=> x; rewrite !permE /= permE f_iinv. Qed.	Lemma	perm_invP	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "f_iinv", "permE", "permP", "perm_inv", "perm_mul", "perm_one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_mulP : associative perm_mul.	Proof. by move=> s t u; apply/permP=> x; do !rewrite permE /=. Qed.	Lemma	perm_mulP	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "permE", "permP", "perm_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm1 x : (1 : {perm T}) x = x.	Proof. by rewrite permE. Qed.	Lemma	perm1	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "permE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permM s t x : (s * t) x = t (s x).	Proof. by rewrite permE. Qed.	Lemma	permM	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "permE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permK s : cancel s s^-1.	Proof. by move=> x; rewrite -permM mulgV perm1. Qed.	Lemma	permK	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "mulgV", "perm1", "permM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permKV s : cancel s^-1 s.	Proof. by have:= permK s^-1; rewrite invgK. Qed.	Lemma	permKV	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "invgK", "permK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permJ s t x : (s ^ t) (t x) = t (s x).	Proof. by rewrite !permM permK. Qed.	Lemma	permJ	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "permK", "permM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permX s x n : (s ^+ n) x = iter n s x.	Proof. by elim: n => [\|n /= <-]; rewrite ?perm1 // -permM expgSr. Qed.	Lemma	permX	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "expgSr", "iter", "perm1", "permM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permX_fix s x n : s x = x -> (s ^+ n) x = x.	Proof. move=> Hs; elim: n => [\|n IHn]; first by rewrite expg0 perm1. by rewrite expgS permM Hs. Qed.	Lemma	permX_fix	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "expg0", "expgS", "perm1", "permM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_permV s S : s^-1 @: S = s @^-1: S.	Proof. exact: can2_imset_pre (permKV s) (permK s). Qed.	Lemma	im_permV	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "can2_imset_pre", "permK", "permKV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
preim_permV s S : s^-1 @^-1: S = s @: S.	Proof. by rewrite -im_permV invgK. Qed.	Lemma	preim_permV	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "im_permV", "invgK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_on S : pred {perm T}	:= fun s => [pred x \| s x != x] \subset S.	Definition	perm_on	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_closed S s x : perm_on S s -> (s x \in S) = (x \in S).	Proof. move/subsetP=> s_on_S; have [-> // \| nfix_s_x] := eqVneq (s x) x. by rewrite !s_on_S // inE /= ?(inj_eq perm_inj). Qed.	Lemma	perm_closed	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "eqVneq", "inE", "inj_eq", "perm_inj", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_on1 H : perm_on H 1.	Proof. by apply/subsetP=> x; rewrite inE /= perm1 eqxx. Qed.	Lemma	perm_on1	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "eqxx", "inE", "perm1", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_onM H s t : perm_on H s -> perm_on H t -> perm_on H (s * t).	Proof. move/subsetP=> sH /subsetP tH; apply/subsetP => x; rewrite inE /= permM. by have [-> /tH \| /sH] := eqVneq (s x) x. Qed.	Lemma	perm_onM	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "eqVneq", "inE", "permM", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_onV H s : perm_on H s -> perm_on H s^-1.	Proof. move=> /subsetP sH; apply/subsetP => i /[!inE] sVi; apply: sH; rewrite inE. by apply: contra_neq sVi => si_id; rewrite -[in LHS]si_id permK. Qed.	Lemma	perm_onV	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "contra_neq", "inE", "permK", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
out_perm S u x : perm_on S u -> x \notin S -> u x = x.	Proof. by move=> uS; apply: contraNeq (subsetP uS x). Qed.	Lemma	out_perm	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "contraNeq", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_perm_on u S : perm_on S u -> u @: S = S.	Proof. move=> Su; rewrite -preim_permV; apply/setP=> x. by rewrite !inE -(perm_closed _ Su) permKV. Qed.	Lemma	im_perm_on	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "inE", "permKV", "perm_closed", "perm_on", "preim_permV", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_on_id u S : perm_on S u -> #\|S\| <= 1 -> u = 1%g.	Proof. rewrite leq_eqVlt ltnS leqn0 => pSu S10; apply/permP => t; rewrite perm1. case/orP : S10; last first. by move/eqP/cards0_eq => S0; apply: (out_perm pSu); rewrite S0 inE. move=> /cards1P[x Sx]. have [-> \| ntx] := eqVneq t x; last by apply: (out_perm pSu); rewrite Sx inE. by apply/eqP; have := perm_closed x pSu;...	Lemma	perm_on_id	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "S0", "apply", "cards0_eq", "cards1P", "eqVneq", "inE", "last", "leq_eqVlt", "leqn0", "ltnS", "out_perm", "perm1", "permP", "perm_closed", "perm_on" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_onC (S1 S2 : {set T}) (u1 u2 : {perm T}) : perm_on S1 u1 -> perm_on S2 u2 -> [disjoint S1 & S2] -> commute u1 u2.	Proof. move=> pS1 pS2 S12; apply/permP => t; rewrite !permM. case/boolP : (t \in S1) => tS1. have /[!disjoint_subset] /subsetP {}S12 := S12. by rewrite !(out_perm pS2) //; apply: S12; rewrite // perm_closed. case/boolP : (t \in S2) => tS2. have /[1!disjoint_sym] /[!disjoint_subset] /subsetP {}S12 := S12. by rew...	Lemma	perm_onC	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "S1", "S2", "apply", "commute", "disjoint", "disjoint_subset", "disjoint_sym", "out_perm", "permM", "permP", "perm_closed", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
imset_perm1 (S : {set T}) : [set (1 : {perm T}) x \| x in S] = S.	Proof. apply: im_perm_on; exact: perm_on1. Qed.	Lemma	imset_perm1	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "im_perm_on", "perm_on1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tperm_proof x y : involutive [fun z => z with x \|-> y, y \|-> x].	Proof. move=> z /=; case: (z =P x) => [-> \| ne_zx]; first by rewrite eqxx; case: eqP. by case: (z =P y) => [->\| ne_zy]; [rewrite eqxx \| do 2?case: eqP]. Qed.	Lemma	tperm_proof	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "eqxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tperm x y	:= perm (can_inj (tperm_proof x y)).	Definition	tperm	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "tperm_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tperm_spec x y z : T -> Type	:= \| TpermFirst of z = x : tperm_spec x y z y \| TpermSecond of z = y : tperm_spec x y z x \| TpermNone of z <> x & z <> y : tperm_spec x y z z.	Variant	tperm_spec	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermP x y z : tperm_spec x y z (tperm x y z).	Proof. by rewrite permE /=; do 2?[case: eqP => /=]; constructor; auto. Qed.	Lemma	tpermP	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "permE", "tperm", "tperm_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermL x y : tperm x y x = y.	Proof. by case: tpermP. Qed.	Lemma	tpermL	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "tperm", "tpermP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermR x y : tperm x y y = x.	Proof. by case: tpermP. Qed.	Lemma	tpermR	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "tperm", "tpermP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermD x y z : x != z -> y != z -> tperm x y z = z.	Proof. by case: tpermP => // ->; rewrite eqxx. Qed.	Lemma	tpermD	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "eqxx", "tperm", "tpermP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermC x y : tperm x y = tperm y x.	Proof. by apply/permP => z; do 2![case: tpermP => //] => ->. Qed.	Lemma	tpermC	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "permP", "tperm", "tpermP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tperm1 x : tperm x x = 1.	Proof. by apply/permP => z; rewrite perm1; case: tpermP. Qed.	Lemma	tperm1	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "perm1", "permP", "tperm", "tpermP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermK x y : involutive (tperm x y).	Proof. by move=> z; rewrite !permE tperm_proof. Qed.	Lemma	tpermK	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "permE", "tperm", "tperm_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermKg x y : involutive (mul (tperm x y)).	Proof. by move=> s; apply/permP=> z; rewrite !permM tpermK. Qed.	Lemma	tpermKg	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "mul", "permM", "permP", "tperm", "tpermK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermV x y : (tperm x y)^-1 = tperm x y.	Proof. by set t := tperm x y; rewrite -{2}(mulgK t t) -mulgA tpermKg. Qed.	Lemma	tpermV	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "mulgA", "mulgK", "tperm", "tpermKg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tperm2 x y : tperm x y * tperm x y = 1.	Proof. by rewrite -{1}tpermV mulVg. Qed.	Lemma	tperm2	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "mulVg", "tperm", "tpermV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tperm_on x y : perm_on [set x; y] (tperm x y).	Proof. by apply/subsetP => z /[!inE]; case: tpermP => [->\|->\|]; rewrite eqxx // orbT. Qed.	Lemma	tperm_on	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "eqxx", "inE", "perm_on", "subsetP", "tperm", "tpermP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_perm A : #\|perm_on A\| = (#\|A\|)`!.	Proof. pose ffA := {ffun {x \| x \in A} -> T}. rewrite -ffactnn -{2}(card_sig [in A]) /= -card_inj_ffuns_on. pose fT (f : ffA) := [ffun x => oapp f x (insub x)]. pose pfT f := insubd (1 : {perm T}) (fT f). pose fA s : ffA := [ffun u => s (val u)]. rewrite -!sum1dep_card -sum1_card (reindex_onto fA pfT) => [f\|]. case/a...	Lemma	card_perm	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "apply", "card_inj_ffuns_on", "card_sig", "eq_bigl", "eqxx", "fA", "fT", "f_inj", "ffactnn", "ffunE", "ffunP", "forallP", "inE", "injectiveP", "insub", "insubP", "insubd", "insubdK", "last", "out_perm", "perm1", "permP", "perm_closed", "perm_inj", "perm_on", "pval",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
reindex_perm s	:= (reindex_inj (@perm_inj _ s)).	Notation	reindex_perm	finite_group	finite_group/perm.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]	[ "perm_inj", "reindex_inj" ]	Shorthand for using a permutation to reindex a bigop.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_isom (A : {set aT}) (C : {set rT}) : A \subset G -> f @* A = C -> 'injm f -> isom A C f.

Proof. move=> sAG; case: (restrmP f sAG) => g [_ _ _ img] <-{C} injf. rewrite /isom -morphimEsub ?morphimDG ?morphim1 //. by rewrite subDset setUC subsetU ?sAG. Qed.

Lemma

sub_isom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "injf", "isom", "morphim1", "morphimDG", "morphimEsub", "restrmP", "sAG", "setUC", "subDset", "subsetU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_isog (A : {set aT}) : A \subset G -> 'injm f -> isog A (f @* A).

Proof. by move=> sAG injf; apply: (isom_isog f sAG); apply: sub_isom. Qed.

Lemma

sub_isog

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "injf", "isog", "isom_isog", "sAG", "sub_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_isom_to (A : {set aT}) (C R : {group rT}) (sAG : A \subset G) : f @* A = C -> isom G R f -> isom A C (restrm sAG f).

Proof. by move=> defC /isomP[inj_f _]; apply: sub_isom. Qed.

Lemma

restr_isom_to

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "group", "inj_f", "isom", "isomP", "restrm", "sAG", "sub_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_isom (A : {group aT}) (R : {group rT}) (sAG : A \subset G) : isom G R f -> isom A (f @* A) (restrm sAG f).

Proof. exact: restr_isom_to. Qed.

Lemma

restr_isom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "group", "isom", "restr_isom_to", "restrm", "sAG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x \isog y"

:= (isog x y).

Notation

x \isog y

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idm_isom : isom G G (idm G).

Proof. exact: sub_isom (im_idm G) (injm_idm G). Qed.

Lemma

idm_isom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "idm", "im_idm", "injm_idm", "isom", "sub_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_refl : G \isog G.

Proof. exact: isom_isog idm_isom. Qed.

Lemma

isog_refl

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "idm_isom", "isog", "isom_isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_isog : G \isog H -> #|G| = #|H|.

Proof. by case/isogP=> f injf <-; apply: isom_card (f) _; apply/isomP. Qed.

Lemma

card_isog

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "injf", "isog", "isogP", "isomP", "isom_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_abelian : G \isog H -> abelian G = abelian H.

Proof. by case/isogP=> f injf <-; rewrite injm_abelian. Qed.

Lemma

isog_abelian

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "abelian", "injf", "injm_abelian", "isog", "isogP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivial_isog : G :=: 1 -> H :=: 1 -> G \isog H.

Proof. move=> -> ->; apply/isogP. exists [morphism of @trivm gT hT 1]; rewrite /= ?morphim1 //. by rewrite ker_trivm; apply: subxx. Qed.

Lemma

trivial_isog

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "gT", "isog", "isogP", "ker_trivm", "morphim1", "morphism", "subxx", "trivm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_eq1 : G \isog H -> (G :==: 1) = (H :==: 1).

Proof. by move=> isoGH; rewrite !trivg_card1 card_isog. Qed.

Lemma

isog_eq1

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "card_isog", "isoGH", "isog", "trivg_card1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_sym (f : {morphism G >-> hT}) (isoGH : isom G H f) : isom H G (isom_inv isoGH).

Proof. rewrite sub_isom 1?injm_restrm ?injm_invm // im_restrm. by rewrite -(isom_im isoGH) im_invm. Qed.

Lemma

isom_sym

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "im_invm", "im_restrm", "injm_invm", "injm_restrm", "isoGH", "isom", "isom_im", "isom_inv", "morphism", "sub_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_symr : G \isog H -> H \isog G.

Proof. by case/isog_isom=> f /isom_sym/isom_isog->. Qed.

Lemma

isog_symr

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isog", "isog_isom", "isom_isog", "isom_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_trans : G \isog H -> H \isog K -> G \isog K.

Proof. case/isogP=> f injf <-; case/isogP=> g injg <-. have defG: f @*^-1 (f @* G) = G by rewrite morphimGK ?subsetIl. rewrite -morphim_comp -{1 8}defG. by apply/isogP; exists [morphism of g \o f]; rewrite ?injm_comp. Qed.

Lemma

isog_trans

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "defG", "injf", "injm_comp", "isog", "isogP", "morphimGK", "morphim_comp", "morphism", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nclasses_isog : G \isog H -> #|classes G| = #|classes H|.

Proof. by case/isogP=> f injf <-; rewrite nclasses_injm. Qed.

Lemma

nclasses_isog

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "classes", "injf", "isog", "isogP", "nclasses_injm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_sym : (G \isog H) = (H \isog G).

Proof. by apply/idP/idP; apply: isog_symr. Qed.

Lemma

isog_sym

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "isog", "isog_symr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_transl : G \isog H -> (G \isog K) = (H \isog K).

Proof. by move=> iso; apply/idP/idP; apply: isog_trans; rewrite // -isog_sym. Qed.

Lemma

isog_transl

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "iso", "isog", "isog_sym", "isog_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_transr : G \isog H -> (K \isog G) = (K \isog H).

Proof. by move=> iso; apply/idP/idP; move/isog_trans; apply; rewrite // -isog_sym. Qed.

Lemma

isog_transr

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "iso", "isog", "isog_sym", "isog_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homg rT aT (C : {set rT}) (D : {set aT})

:= [exists (f : {ffun aT -> rT} | morphic D f), f @: D == C].

Definition

homg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "morphic" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homgP rT aT (C : {set rT}) (D : {set aT}) : reflect (exists f : {morphism D >-> rT}, f @* D = C) (homg C D).

Proof. apply: (iffP exists_eq_inP) => [[f fM <-] | [f <-]]. by exists (morphm_morphism fM); rewrite /morphim /= setIid. exists (finfun f); first by apply/morphicP=> x y Dx Dy; rewrite !ffunE morphM. by rewrite /morphim setIid; apply: eq_imset => x; rewrite ffunE. Qed.

Lemma

homgP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "Dx", "aT", "apply", "eq_imset", "exists_eq_inP", "fM", "ffunE", "homg", "morphM", "morphicP", "morphim", "morphism", "morphm_morphism", "setIid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_homg aT rT (A D : {set aT}) (f : {morphism D >-> rT}) : A \subset D -> homg (f @* A) A.

Proof. move=> sAD; apply/homgP; exists (restrm_morphism sAD f). by rewrite morphim_restrm setIid. Qed.

Lemma

morphim_homg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "homg", "homgP", "morphim_restrm", "morphism", "restrm_morphism", "sAD", "setIid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_homg rT aT (C : {set rT}) (G : {group aT}) : homg C G -> #|C| <= #|G|.

Proof. by case/homgP=> f <-; apply: leq_morphim. Qed.

Lemma

leq_homg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "group", "homg", "homgP", "leq_morphim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homg_refl aT (A : {set aT}) : homg A A.

Proof. by apply/homgP; exists (idm_morphism A); rewrite im_idm. Qed.

Lemma

homg_refl

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "homg", "homgP", "idm_morphism", "im_idm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homg_trans aT (B : {set aT}) rT (C : {set rT}) gT (G : {group gT}) : homg C B -> homg B G -> homg C G.

Proof. move=> homCB homBG; case/homgP: homBG homCB => fG <- /homgP[fK <-]. by rewrite -morphim_comp morphim_homg // -sub_morphim_pre. Qed.

Lemma

homg_trans

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "fK", "gT", "group", "homg", "homgP", "morphim_comp", "morphim_homg", "sub_morphim_pre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isogEcard rT aT (G : {group rT}) (H : {group aT}) : (G \isog H) = (homg G H) && (#|H| <= #|G|).

Proof. rewrite isog_sym; apply/isogP/andP=> [[f injf <-] | []]. by rewrite leq_eqVlt eq_sym card_im_injm injf morphim_homg. case/homgP=> f <-; rewrite leq_eqVlt eq_sym card_im_injm. by rewrite ltnNge leq_morphim orbF; exists f. Qed.

Lemma

isogEcard

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "card_im_injm", "eq_sym", "group", "homg", "homgP", "injf", "isog", "isogP", "isog_sym", "leq_eqVlt", "leq_morphim", "ltnNge", "morphim_homg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_hom rT aT (G : {group rT}) (H : {group aT}) : G \isog H -> homg G H.

Proof. by rewrite isogEcard; case/andP. Qed.

Lemma

isog_hom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "group", "homg", "isog", "isogEcard" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isogEhom rT aT (G : {group rT}) (H : {group aT}) : (G \isog H) = homg G H && homg H G.

Proof. apply/idP/andP=> [isoGH | [homGH homHG]]. by rewrite !isog_hom // isog_sym. by rewrite isogEcard homGH leq_homg. Qed.

Lemma

isogEhom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "group", "homg", "isoGH", "isog", "isogEcard", "isog_hom", "isog_sym", "leq_homg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_homgl gT aT rT (G : {group gT}) (H : {group aT}) (K : {group rT}) : G \isog H -> homg G K = homg H K.

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

Lemma

eq_homgl

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "gT", "group", "homg", "homg_trans", "isog", "isogEhom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_homgr gT rT aT (G : {group gT}) (H : {group rT}) (K : {group aT}) : G \isog H -> homg K G = homg K H.

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

Lemma

eq_homgr

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "gT", "group", "homg", "homg_trans", "isog", "isogEhom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G \homg H"

:= (homg G H) (at level 70, no associativity) : group_scope.

Notation

G \homg H

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "homg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sgval_morphism

:= Morphism (@sgvalM _ G).

Canonical

sgval_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "sgvalM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_morphism

:= Morphism (@subgM _ G).

Canonical

subg_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "subgM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_sgval : 'injm sgval.

Proof. exact/injmP/(in2W subg_inj). Qed.

Lemma

injm_sgval

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injmP", "sgval", "subg_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_subg : 'injm (subg G).

Proof. exact/injmP/(can_in_inj subgK). Qed.

Lemma

injm_subg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injmP", "subg", "subgK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_sgval : 'ker sgval = 1.

Proof. exact/trivgP. Qed.

Lemma

ker_sgval

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ker", "sgval", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_subg : 'ker (subg G) = 1.

Proof. exact/trivgP. Qed.

Lemma

ker_subg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ker", "subg", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_subg : subg G @* G = [subg G].

Proof. apply/eqP; rewrite -subTset morphimEdom. by apply/subsetP=> u _; rewrite -(sgvalK u) imset_f ?subgP. Qed.

Lemma

im_subg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "imset_f", "morphimEdom", "sgvalK", "subTset", "subg", "subgP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sgval_sub A : sgval @* A \subset G.

Proof. by apply/subsetP=> x; case/imsetP=> u _ ->; apply: subgP. Qed.

Lemma

sgval_sub

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "imsetP", "sgval", "subgP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sgvalmK A : subg G @* (sgval @* A) = A.

Proof. apply/eqP; rewrite eqEcard !card_injm ?subsetT ?sgval_sub // leqnn andbT. rewrite -morphim_comp; apply/subsetP=> _ /morphimP[v _ Av ->] /=. by rewrite sgvalK. Qed.

Lemma

sgvalmK

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "card_injm", "eqEcard", "leqnn", "morphimP", "morphim_comp", "sgval", "sgvalK", "sgval_sub", "subg", "subsetP", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgmK (A : {set gT}) : A \subset G -> sgval @* (subg G @* A) = A.

Proof. move=> sAG; apply/eqP; rewrite eqEcard !card_injm ?subsetT //. rewrite leqnn andbT -morphim_comp morphimE /= morphpreT. by apply/subsetP=> _ /morphimP[v Gv Av ->] /=; rewrite subgK. Qed.

Lemma

subgmK

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "card_injm", "eqEcard", "gT", "leqnn", "morphimE", "morphimP", "morphim_comp", "morphpreT", "sAG", "sgval", "subg", "subgK", "subsetP", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_sgval : sgval @* [subg G] = G.

Proof. by rewrite -{2}im_subg subgmK. Qed.

Lemma

im_sgval

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "im_subg", "sgval", "subg", "subgmK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_subg : isom G [subg G] (subg G).

Proof. by apply/isomP; rewrite im_subg. Qed.

Lemma

isom_subg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "im_subg", "isom", "isomP", "subg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_sgval : isom [subg G] G sgval.

Proof. by apply/isomP; rewrite im_sgval. Qed.

Lemma

isom_sgval

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "im_sgval", "isom", "isomP", "sgval", "subg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_subg : isog G [subg G].

Proof. exact: isom_isog isom_subg. Qed.

Lemma

isog_subg

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isog", "isom_isog", "isom_subg", "subg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_type : predArgType

:= Perm (pval : {ffun T -> T}) & injectiveb pval.

Inductive

perm_type

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "injectiveb", "pval" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pval p

:= let: Perm f _ := p in f.

Definition

pval

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_of

:= perm_type.

Definition

perm_of

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "perm_type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_proof (f : T -> T) : injective f -> injectiveb (finfun f).

Proof. by move=> f_inj; apply/injectiveP; apply: eq_inj f_inj _ => x; rewrite ffunE. Qed.

Lemma

perm_proof

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "f_inj", "ffunE", "injectiveP", "injectiveb" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'perm' T }"

:= (perm_of T) (format "{ 'perm' T }") : type_scope.

Notation

{ 'perm' T }

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "perm_of" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''S_' n"

:= {perm 'I_n} (at level 8, n at level 2, format "''S_' n").

Notation

''S_' n

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_unlock

:= Unlockable perm.unlock.

Canonical

perm_unlock

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fun_of_perm_unlock

:= Unlockable fun_of_perm.unlock.

Canonical

fun_of_perm_unlock

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "fun_of_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fun_of_perm : perm_type >-> Funclass.

Coercion

fun_of_perm

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "perm_type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permP s t : s =1 t <-> s = t.

Proof. by split=> [| -> //]; rewrite unlock => eq_sv; apply/val_inj/ffunP. Qed.

Lemma

permP

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "ffunP", "split", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pvalE s : pval s = s :> (T -> T).

Proof. by rewrite [@fun_of_perm]unlock. Qed.

Lemma

pvalE

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "fun_of_perm", "pval" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permE f f_inj : @perm T f f_inj =1 f.

Proof. by move=> x; rewrite -pvalE [@perm]unlock ffunE. Qed.

Lemma

permE

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "f_inj", "ffunE", "pvalE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_inj {s} : injective s.

Proof. by rewrite -!pvalE; apply: (injectiveP _ (valP s)). Qed.

Lemma

perm_inj

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "injectiveP", "pvalE", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_onto s : codom s =i predT.

Proof. by apply/subset_cardP; rewrite ?card_codom ?subset_predT. Qed.

Lemma

perm_onto

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "card_codom", "codom", "subset_cardP", "subset_predT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_one

:= perm (@inj_id T).

Definition

perm_one

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_invK s : cancel (fun x => iinv (perm_onto s x)) s.

Proof. by move=> x /=; rewrite f_iinv. Qed.

Lemma

perm_invK

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "f_iinv", "iinv", "perm_onto" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_inv s

:= perm (can_inj (perm_invK s)).

Definition

perm_inv

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "perm_invK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_mul s t

:= perm (inj_comp (@perm_inj t) (@perm_inj s)).

Definition

perm_mul

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "perm_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_oneP : left_id perm_one perm_mul.

Proof. by move=> s; apply/permP => x; rewrite permE /= permE. Qed.

Lemma

perm_oneP

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "permE", "permP", "perm_mul", "perm_one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_invP : left_inverse perm_one perm_inv perm_mul.

Proof. by move=> s; apply/permP=> x; rewrite !permE /= permE f_iinv. Qed.

Lemma

perm_invP

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "f_iinv", "permE", "permP", "perm_inv", "perm_mul", "perm_one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_mulP : associative perm_mul.

Proof. by move=> s t u; apply/permP=> x; do !rewrite permE /=. Qed.

Lemma

perm_mulP

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "permE", "permP", "perm_mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm1 x : (1 : {perm T}) x = x.

Proof. by rewrite permE. Qed.

Lemma

perm1

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "permE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permM s t x : (s * t) x = t (s x).

Proof. by rewrite permE. Qed.

Lemma

permM

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "permE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permK s : cancel s s^-1.

Proof. by move=> x; rewrite -permM mulgV perm1. Qed.

Lemma

permK

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "mulgV", "perm1", "permM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permKV s : cancel s^-1 s.

Proof. by have:= permK s^-1; rewrite invgK. Qed.

Lemma

permKV

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "invgK", "permK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permJ s t x : (s ^ t) (t x) = t (s x).

Proof. by rewrite !permM permK. Qed.

Lemma

permJ

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "permK", "permM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permX s x n : (s ^+ n) x = iter n s x.

Proof. by elim: n => [|n /= <-]; rewrite ?perm1 // -permM expgSr. Qed.

Lemma

permX

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "expgSr", "iter", "perm1", "permM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permX_fix s x n : s x = x -> (s ^+ n) x = x.

Proof. move=> Hs; elim: n => [|n IHn]; first by rewrite expg0 perm1. by rewrite expgS permM Hs. Qed.

Lemma

permX_fix

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "expg0", "expgS", "perm1", "permM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_permV s S : s^-1 @: S = s @^-1: S.

Proof. exact: can2_imset_pre (permKV s) (permK s). Qed.

Lemma

im_permV

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "can2_imset_pre", "permK", "permKV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

preim_permV s S : s^-1 @^-1: S = s @: S.

Proof. by rewrite -im_permV invgK. Qed.

Lemma

preim_permV

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "im_permV", "invgK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_on S : pred {perm T}

:= fun s => [pred x | s x != x] \subset S.

Definition

perm_on

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_closed S s x : perm_on S s -> (s x \in S) = (x \in S).

Proof. move/subsetP=> s_on_S; have [-> // | nfix_s_x] := eqVneq (s x) x. by rewrite !s_on_S // inE /= ?(inj_eq perm_inj). Qed.

Lemma

perm_closed

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "eqVneq", "inE", "inj_eq", "perm_inj", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_on1 H : perm_on H 1.

Proof. by apply/subsetP=> x; rewrite inE /= perm1 eqxx. Qed.

Lemma

perm_on1

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "eqxx", "inE", "perm1", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_onM H s t : perm_on H s -> perm_on H t -> perm_on H (s * t).

Proof. move/subsetP=> sH /subsetP tH; apply/subsetP => x; rewrite inE /= permM. by have [-> /tH | /sH] := eqVneq (s x) x. Qed.

Lemma

perm_onM

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "eqVneq", "inE", "permM", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_onV H s : perm_on H s -> perm_on H s^-1.

Proof. move=> /subsetP sH; apply/subsetP => i /[!inE] sVi; apply: sH; rewrite inE. by apply: contra_neq sVi => si_id; rewrite -[in LHS]si_id permK. Qed.

Lemma

perm_onV

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "contra_neq", "inE", "permK", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

out_perm S u x : perm_on S u -> x \notin S -> u x = x.

Proof. by move=> uS; apply: contraNeq (subsetP uS x). Qed.

Lemma

out_perm

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "contraNeq", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_perm_on u S : perm_on S u -> u @: S = S.

Proof. move=> Su; rewrite -preim_permV; apply/setP=> x. by rewrite !inE -(perm_closed _ Su) permKV. Qed.

Lemma

im_perm_on

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "inE", "permKV", "perm_closed", "perm_on", "preim_permV", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_on_id u S : perm_on S u -> #|S| <= 1 -> u = 1%g.

Proof. rewrite leq_eqVlt ltnS leqn0 => pSu S10; apply/permP => t; rewrite perm1. case/orP : S10; last first. by move/eqP/cards0_eq => S0; apply: (out_perm pSu); rewrite S0 inE. move=> /cards1P[x Sx]. have [-> | ntx] := eqVneq t x; last by apply: (out_perm pSu); rewrite Sx inE. by apply/eqP; have := perm_closed x pSu;...

Lemma

perm_on_id

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "S0", "apply", "cards0_eq", "cards1P", "eqVneq", "inE", "last", "leq_eqVlt", "leqn0", "ltnS", "out_perm", "perm1", "permP", "perm_closed", "perm_on" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_onC (S1 S2 : {set T}) (u1 u2 : {perm T}) : perm_on S1 u1 -> perm_on S2 u2 -> [disjoint S1 & S2] -> commute u1 u2.

Proof. move=> pS1 pS2 S12; apply/permP => t; rewrite !permM. case/boolP : (t \in S1) => tS1. have /[!disjoint_subset] /subsetP {}S12 := S12. by rewrite !(out_perm pS2) //; apply: S12; rewrite // perm_closed. case/boolP : (t \in S2) => tS2. have /[1!disjoint_sym] /[!disjoint_subset] /subsetP {}S12 := S12. by rew...

Lemma

perm_onC

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "S1", "S2", "apply", "commute", "disjoint", "disjoint_subset", "disjoint_sym", "out_perm", "permM", "permP", "perm_closed", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

imset_perm1 (S : {set T}) : [set (1 : {perm T}) x | x in S] = S.

Proof. apply: im_perm_on; exact: perm_on1. Qed.

Lemma

imset_perm1

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "im_perm_on", "perm_on1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tperm_proof x y : involutive [fun z => z with x |-> y, y |-> x].

Proof. move=> z /=; case: (z =P x) => [-> | ne_zx]; first by rewrite eqxx; case: eqP. by case: (z =P y) => [->| ne_zy]; [rewrite eqxx | do 2?case: eqP]. Qed.

Lemma

tperm_proof

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "eqxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tperm x y

:= perm (can_inj (tperm_proof x y)).

Definition

tperm

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "tperm_proof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tperm_spec x y z : T -> Type

:= | TpermFirst of z = x : tperm_spec x y z y | TpermSecond of z = y : tperm_spec x y z x | TpermNone of z <> x & z <> y : tperm_spec x y z z.

Variant

tperm_spec

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermP x y z : tperm_spec x y z (tperm x y z).

Proof. by rewrite permE /=; do 2?[case: eqP => /=]; constructor; auto. Qed.

Lemma

tpermP

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "permE", "tperm", "tperm_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermL x y : tperm x y x = y.

Proof. by case: tpermP. Qed.

Lemma

tpermL

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "tperm", "tpermP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermR x y : tperm x y y = x.

Proof. by case: tpermP. Qed.

Lemma

tpermR

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "tperm", "tpermP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermD x y z : x != z -> y != z -> tperm x y z = z.

Proof. by case: tpermP => // ->; rewrite eqxx. Qed.

Lemma

tpermD

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "eqxx", "tperm", "tpermP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermC x y : tperm x y = tperm y x.

Proof. by apply/permP => z; do 2![case: tpermP => //] => ->. Qed.

Lemma

tpermC

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "permP", "tperm", "tpermP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tperm1 x : tperm x x = 1.

Proof. by apply/permP => z; rewrite perm1; case: tpermP. Qed.

Lemma

tperm1

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "perm1", "permP", "tperm", "tpermP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermK x y : involutive (tperm x y).

Proof. by move=> z; rewrite !permE tperm_proof. Qed.

Lemma

tpermK

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "permE", "tperm", "tperm_proof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermKg x y : involutive (mul (tperm x y)).

Proof. by move=> s; apply/permP=> z; rewrite !permM tpermK. Qed.

Lemma

tpermKg

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "mul", "permM", "permP", "tperm", "tpermK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermV x y : (tperm x y)^-1 = tperm x y.

Proof. by set t := tperm x y; rewrite -{2}(mulgK t t) -mulgA tpermKg. Qed.

Lemma

tpermV

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "mulgA", "mulgK", "tperm", "tpermKg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tperm2 x y : tperm x y * tperm x y = 1.

Proof. by rewrite -{1}tpermV mulVg. Qed.

Lemma

tperm2

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "mulVg", "tperm", "tpermV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tperm_on x y : perm_on [set x; y] (tperm x y).

Proof. by apply/subsetP => z /[!inE]; case: tpermP => [->|->|]; rewrite eqxx // orbT. Qed.

Lemma

tperm_on

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "eqxx", "inE", "perm_on", "subsetP", "tperm", "tpermP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_perm A : #|perm_on A| = (#|A|)`!.

Proof. pose ffA := {ffun {x | x \in A} -> T}. rewrite -ffactnn -{2}(card_sig [in A]) /= -card_inj_ffuns_on. pose fT (f : ffA) := [ffun x => oapp f x (insub x)]. pose pfT f := insubd (1 : {perm T}) (fT f). pose fA s : ffA := [ffun u => s (val u)]. rewrite -!sum1dep_card -sum1_card (reindex_onto fA pfT) => [f|]. case/a...

Lemma

card_perm

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "apply", "card_inj_ffuns_on", "card_sig", "eq_bigl", "eqxx", "fA", "fT", "f_inj", "ffactnn", "ffunE", "ffunP", "forallP", "inE", "injectiveP", "insub", "insubP", "insubd", "insubdK", "last", "out_perm", "perm1", "permP", "perm_closed", "perm_inj", "perm_on", "pval",...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

reindex_perm s

:= (reindex_inj (@perm_inj _ s)).

Notation

reindex_perm

finite_group

finite_group/perm.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "binomial", "fingroup", "morphism", "nmodule" ]

[ "perm_inj", "reindex_inj" ]

Shorthand for using a permutation to reindex a bigop.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d