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
inj_tperm (T T' : finType) (f : T -> T') x y z : injective f -> f (tperm x y z) = tperm (f x) (f y) (f z).	Proof. by move=> injf; rewrite !permE /= !(inj_eq injf) !(fun_if f). Qed.	Lemma	inj_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" ]	[ "T'", "inj_eq", "injf", "permE", "tperm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermJ x y s : (tperm x y) ^ s = tperm (s x) (s y).	Proof. by apply/permP => z; rewrite -(permKV s z) permJ; apply/inj_tperm/perm_inj. Qed.	Lemma	tpermJ	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", "inj_tperm", "permJ", "permKV", "permP", "perm_inj", "tperm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tpermJ_tperm x y z : x != z -> y != z -> tperm x z ^ tperm x y = tperm y z.	Proof. by move=> nxz nyz; rewrite tpermJ tpermL [tperm _ _ z]tpermD. Qed.	Lemma	tpermJ_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", "tpermD", "tpermJ", "tpermL" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tuple_permP {T : eqType} {n} {s : seq T} {t : n.-tuple T} : reflect (exists p : 'S_n, s = [tuple tnth t (p i) \| i < n]) (perm_eq s t).	Proof. apply: (iffP idP) => [\|[p ->]]; last first. rewrite /= (map_comp (tnth t)) -{1}(map_tnth_enum t) perm_map //. apply: uniq_perm => [\|\|i]; rewrite ?enum_uniq //. by apply/injectiveP; apply: perm_inj. by rewrite mem_enum -[i](permKV p) image_f. case: n => [\|n] in t *; last have x0 := tnth t ord0. rewrit...	Lemma	tuple_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", "enum_uniq", "eq_from_tnth", "image_f", "injectiveP", "iota_uniq", "last", "map_comp", "map_tnth_enum", "map_uniq", "mem_enum", "mem_iota", "mem_tnth", "mktuple", "ord0", "pP", "permE", "permKV", "perm_eq", "perm_inj", "perm_iotaP", "perm_map", "perm_mem", "per...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aperm (T : finType) x (s : {perm T})	:= s x.	Definition	aperm	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" ]	[]	action.v and hence morphism.v.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbit_unlockable	:= Unlockable porbit.unlock.	Canonical	porbit_unlockable	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
porbits (T : finType) (s : {perm T})	:= porbit s @: T.	Definition	porbits	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
odd_perm (s : perm_type T)	:= odd #\|T\| (+) odd #\|porbits s\|.	Definition	odd_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" ]	[ "odd", "perm_type", "porbits" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
apermE x s : aperm x s = s x.	Proof. by []. Qed.	Lemma	apermE	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" ]	[ "aperm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_porbit s i x : (s ^+ i) x \in porbit s x.	Proof. by rewrite [@porbit]unlock (imset_f (aperm x)) ?mem_cycle. Qed.	Lemma	mem_porbit	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" ]	[ "aperm", "imset_f", "mem_cycle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbit_id s x : x \in porbit s x.	Proof. by rewrite -{1}[x]perm1 (mem_porbit s 0). Qed.	Lemma	porbit_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" ]	[ "mem_porbit", "perm1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_porbit_neq0 s x : #\|porbit s x\| != 0.	Proof. by rewrite -lt0n card_gt0; apply/set0Pn; exists x; exact: porbit_id. Qed.	Lemma	card_porbit_neq0	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_gt0", "lt0n", "porbit_id", "set0Pn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
uniq_traject_porbit s x : uniq (traject s x #\|porbit s x\|).	Proof. case def_n: #\|_\| => // [n]; rewrite looping_uniq. apply: contraL (card_size (traject s x n)) => /loopingP t_sx. rewrite -ltnNge size_traject -def_n ?subset_leq_card // porbit.unlock. by apply/subsetP=> _ /imsetP[_ /cycleP[i ->] ->]; rewrite /aperm permX t_sx. Qed.	Lemma	uniq_traject_porbit	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" ]	[ "aperm", "apply", "card_size", "cycleP", "def_n", "imsetP", "loopingP", "looping_uniq", "ltnNge", "permX", "size_traject", "subsetP", "subset_leq_card", "traject", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbit_traject s x : porbit s x =i traject s x #\|porbit s x\|.	Proof. apply: fsym; apply/subset_cardP. by rewrite (card_uniqP _) ?size_traject ?uniq_traject_porbit. by apply/subsetP=> _ /trajectP[i _ ->]; rewrite -permX mem_porbit. Qed.	Lemma	porbit_traject	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_uniqP", "mem_porbit", "permX", "size_traject", "subsetP", "subset_cardP", "traject", "trajectP", "uniq_traject_porbit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iter_porbit s x : iter #\|porbit s x\| s x = x.	Proof. case def_n: #\|_\| (uniq_traject_porbit s x) => [//\|n] Ut. have: looping s x n.+1. by rewrite -def_n -[looping _ _ _]porbit_traject -permX mem_porbit. rewrite /looping => /trajectP[[\|i] //= lt_i_n /perm_inj eq_i_n_sx]. move: lt_i_n; rewrite ltnS ltn_neqAle andbC => /andP[le_i_n /negP[]]. by rewrite -(nth_uniq x ...	Lemma	iter_porbit	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" ]	[ "def_n", "iter", "looping", "ltnS", "ltn_neqAle", "mem_porbit", "nth_traject", "nth_uniq", "permX", "perm_inj", "porbit_traject", "size_traject", "trajectP", "uniq_traject_porbit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_porbit_mem s x y : (porbit s x == porbit s y) = (x \in porbit s y).	Proof. apply/eqP/idP; first by move<-; exact: porbit_id. rewrite porbit.unlock => /imsetP[si s_si ->]. apply/setP => z; apply/imsetP/imsetP=> [] [sj s_sj ->]. by exists (si * sj); rewrite ?groupM /aperm ?permM. exists (si^-1 * sj); first by rewrite groupM ?groupV. by rewrite /aperm permM permK. Qed.	Lemma	eq_porbit_mem	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" ]	[ "aperm", "apply", "groupM", "groupV", "imsetP", "permK", "permM", "porbit_id", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbit_sym s x y : (x \in porbit s y) = (y \in porbit s x).	Proof. by rewrite -!eq_porbit_mem eq_sym. Qed.	Lemma	porbit_sym	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" ]	[ "eq_porbit_mem", "eq_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbit_perm s i x : porbit s ((s ^+ i) x) = porbit s x.	Proof. by apply/eqP; rewrite eq_porbit_mem mem_porbit. Qed.	Lemma	porbit_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", "eq_porbit_mem", "mem_porbit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbitPmin s x y : y \in porbit s x -> exists2 i, i < #[s] & y = (s ^+ i) x.	Proof. by rewrite porbit.unlock=> /imsetP [z /cyclePmin[ i Hi ->{z}] ->{y}]; exists i. Qed.	Lemma	porbitPmin	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" ]	[ "cyclePmin", "imsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbitP s x y : reflect (exists i, y = (s ^+ i) x) (y \in porbit s x).	Proof. apply (iffP idP) => [/porbitPmin [i _ ->]\| [i ->]]; last exact: mem_porbit. by exists i. Qed.	Lemma	porbitP	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", "last", "mem_porbit", "porbitPmin" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbitV s : porbit s^-1 =1 porbit s.	Proof. move=> x; apply/setP => y; rewrite porbit_sym. by apply/porbitP/porbitP => -[i ->]; exists i; rewrite expgVn ?permK ?permKV. Qed.	Lemma	porbitV	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", "expgVn", "permK", "permKV", "porbitP", "porbit_sym", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbitsV s : porbits s^-1 = porbits s.	Proof. rewrite /porbits; apply/setP => y. by apply/imsetP/imsetP => -[x _ ->{y}]; exists x; rewrite // porbitV. Qed.	Lemma	porbitsV	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", "imsetP", "porbitV", "porbits", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbit_setP s t x : porbit s x =i porbit t x <-> porbit s x = porbit t x.	Proof. by rewrite porbit.unlock; exact: setP. Qed.	Lemma	porbit_setP	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" ]	[ "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbits_mul_tperm s x y : let t := tperm x y in #\|porbits (t * s)\| + (x \notin porbit s y).*2 = #\|porbits s\| + (x != y).	Proof. pose xf a b u := seq.find (pred2 a b) (traject u (u a) #\|porbit u a\|). have xf_size a b u: xf a b u <= #\|porbit u a\|. by rewrite (leq_trans (find_size _ _)) ?size_traject. have lt_xf a b u n : n < xf a b u -> ~~ pred2 a b ((u ^+ n.+1) a). move=> lt_n; apply: contraFN (before_find (u a) lt_n). by rewrite pe...	Lemma	porbits_mul_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" ]	[ "addnA", "addnC", "aperm", "apply", "before_find", "cardsD1", "contraNneq", "eq_porbit_mem", "eq_sym", "eqn_leq", "eqxx", "expgSr", "find", "find_size", "has", "hasP", "has_find", "imsetP", "imset_f", "inE", "iterSr", "last", "leq_trans", "leqnn", "looping", "loopin...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_perm1 : odd_perm 1 = false.	Proof. rewrite /odd_perm card_imset ?addbb // => x y; move/eqP; rewrite eq_porbit_mem. by rewrite porbit.unlock cycle1 imset_set1 /aperm perm1 inE=> /eqP. Qed.	Lemma	odd_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" ]	[ "aperm", "card_imset", "cycle1", "eq_porbit_mem", "imset_set1", "inE", "odd_perm", "perm1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_mul_tperm x y s : odd_perm (tperm x y * s) = (x != y) (+) odd_perm s.	Proof. rewrite addbC -addbA -[~~ _]oddb -oddD -porbits_mul_tperm. by rewrite oddD odd_double addbF. Qed.	Lemma	odd_mul_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" ]	[ "oddD", "odd_double", "odd_perm", "oddb", "porbits_mul_tperm", "tperm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_tperm x y : odd_perm (tperm x y) = (x != y).	Proof. by rewrite -[_ y]mulg1 odd_mul_tperm odd_perm1 addbF. Qed.	Lemma	odd_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" ]	[ "mulg1", "odd_mul_tperm", "odd_perm", "odd_perm1", "tperm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dpair (eT : eqType)	:= [pred t \| t.1 != t.2 :> eT].	Definition	dpair	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
prod_tpermP s : {ts : seq (T * T) \| s = \prod_(t <- ts) tperm t.1 t.2 & all dpair ts}.	Proof. have [n] := ubnP #\|[pred x \| s x != x]\|; elim: n s => // n IHn s /ltnSE-le_s_n. case: (pickP (fun x => s x != x)) => [x s_x \| s_id]; last first. exists nil; rewrite // big_nil; apply/permP=> x. by apply/eqP/idPn; rewrite perm1 s_id. have [\|ts def_s ne_ts] := IHn (tperm x (s^-1 x) * s); last first. exists (...	Lemma	prod_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" ]	[ "all", "apply", "big_cons", "big_nil", "canF_eq", "cardD1", "dpair", "eqVneq", "eq_sym", "eqxx", "inE", "last", "leq_ltn_trans", "ltnSE", "mul1g", "mulgA", "perm1", "permE", "permK", "permKV", "permM", "permP", "pickP", "seq", "subsetP", "subset_leq_card", "tperm"...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_perm_prod ts : all dpair ts -> odd_perm (\prod_(t <- ts) tperm t.1 t.2) = odd (size ts).	Proof. elim: ts => [_\|t ts IHts] /=; first by rewrite big_nil odd_perm1. by case/andP=> dt12 dts; rewrite big_cons odd_mul_tperm dt12 IHts. Qed.	Lemma	odd_perm_prod	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" ]	[ "all", "big_cons", "big_nil", "dpair", "odd", "odd_mul_tperm", "odd_perm", "odd_perm1", "size", "tperm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_permM : {morph odd_perm : s1 s2 / s1 * s2 >-> s1 (+) s2}.	Proof. move=> s1 s2; case: (prod_tpermP s1) => ts1 ->{s1} dts1. case: (prod_tpermP s2) => ts2 ->{s2} dts2. by rewrite -big_cat !odd_perm_prod ?all_cat ?dts1 // size_cat oddD. Qed.	Lemma	odd_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" ]	[ "all_cat", "big_cat", "oddD", "odd_perm", "odd_perm_prod", "prod_tpermP", "s1", "s2", "size_cat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_permV s : odd_perm s^-1 = odd_perm s.	Proof. by rewrite -{2}(mulgK s s) !odd_permM -addbA addKb. Qed.	Lemma	odd_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" ]	[ "mulgK", "odd_perm", "odd_permM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_permJ s1 s2 : odd_perm (s1 ^ s2) = odd_perm s1.	Proof. by rewrite !odd_permM odd_permV addbC addbK. Qed.	Lemma	odd_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" ]	[ "odd_perm", "odd_permM", "odd_permV", "s1", "s2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gen_tperm x : <<[set tperm x y \| y in T]>>%g = [set: {perm T}].	Proof. apply/eqP; rewrite eqEsubset subsetT/=; apply/subsetP => s _. have [ts -> _] := prod_tpermP s; rewrite group_prod// => -[/= y z] _. have [<-\|Nyz] := eqVneq y z; first by rewrite tperm1 group1. have [<-\|Nxz] := eqVneq x z; first by rewrite tpermC mem_gen ?imset_f. by rewrite -(tpermJ_tperm Nxz Nyz) groupJ ?mem_ge...	Lemma	gen_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" ]	[ "apply", "eqEsubset", "eqVneq", "group1", "groupJ", "group_prod", "imset_f", "mem_gen", "prod_tpermP", "subsetP", "subsetT", "tperm", "tperm1", "tpermC", "tpermJ_tperm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_perm : perm_type >-> bool.		Coercion	odd_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
Sym : {set {perm T}}	:= [set s \| perm_on S s].	Definition	Sym	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_on" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Sym_group_set : group_set Sym.	Proof. apply/group_setP; split => [\|s t] /[!inE]; [exact: perm_on1 \| exact: perm_onM]. Qed.	Lemma	Sym_group_set	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" ]	[ "Sym", "apply", "group_set", "group_setP", "inE", "perm_on1", "perm_onM", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Sym_group : {group {perm T}}	:= Group Sym_group_set.	Canonical	Sym_group	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" ]	[ "Sym_group_set", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_Sym : #\|Sym\| = #\|S\|`!.	Proof. by rewrite cardsE /= card_perm. Qed.	Lemma	card_Sym	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" ]	[ "Sym", "card_perm", "cardsE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_Sn : #\|'S_(n)\| = n`!.	Proof. rewrite (eq_card (B := perm_on [set : 'I_n])); last first. by rewrite card_perm /= cardsE /= card_ord. move=> p; rewrite inE unfold_in /perm_on /=. by apply/esym/subsetP => i _; rewrite in_set. Qed.	Lemma	card_Sn	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_ord", "card_perm", "cardsE", "eq_card", "inE", "in_set", "last", "perm_on", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_perm_fun i j s k	:= if unlift i k is Some k' then lift j (s k') else j.	Definition	lift_perm_fun	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" ]	[ "lift", "unlift" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_permK i j s : cancel (lift_perm_fun i j s) (lift_perm_fun j i s^-1).	Proof. rewrite /lift_perm_fun => k. by case: (unliftP i k) => [j'\|] ->; rewrite (liftK, unlift_none) ?permK. Qed.	Lemma	lift_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" ]	[ "liftK", "lift_perm_fun", "permK", "unliftP", "unlift_none" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_perm i j s : 'S_n.+1	:= perm (can_inj (lift_permK i j s)).	Definition	lift_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" ]	[ "lift_permK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_perm_id i j s : lift_perm i j s i = j.	Proof. by rewrite permE /lift_perm_fun unlift_none. Qed.	Lemma	lift_perm_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" ]	[ "lift_perm", "lift_perm_fun", "permE", "unlift_none" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_perm_lift i j s k' : lift_perm i j s (lift i k') = lift j (s k') :> 'I_n.+1.	Proof. by rewrite permE /lift_perm_fun liftK. Qed.	Lemma	lift_perm_lift	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" ]	[ "lift", "liftK", "lift_perm", "lift_perm_fun", "permE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_permM i j k s t : lift_perm i j s * lift_perm j k t = lift_perm i k (s * t).	Proof. apply/permP=> i1; case: (unliftP i i1) => [i2\|] ->{i1}. by rewrite !(permM, lift_perm_lift). by rewrite permM !lift_perm_id. Qed.	Lemma	lift_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" ]	[ "apply", "lift_perm", "lift_perm_id", "lift_perm_lift", "permM", "permP", "unliftP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_perm1 i : lift_perm i i 1 = 1.	Proof. by apply: (mulgI (lift_perm i i 1)); rewrite lift_permM !mulg1. Qed.	Lemma	lift_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", "lift_perm", "lift_permM", "mulg1", "mulgI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_permV i j s : (lift_perm i j s)^-1 = lift_perm j i s^-1.	Proof. by apply/eqP; rewrite eq_invg_mul lift_permM mulgV lift_perm1. Qed.	Lemma	lift_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" ]	[ "apply", "eq_invg_mul", "lift_perm", "lift_perm1", "lift_permM", "mulgV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_lift_perm i j s : lift_perm i j s = odd i (+) odd j (+) s :> bool.	Proof. rewrite -{1}(mul1g s) -(lift_permM _ j) odd_permM. congr (_ (+) _); last first. case: (prod_tpermP s) => ts ->{s} _. elim: ts => [\|t ts IHts] /=; first by rewrite big_nil lift_perm1 !odd_perm1. rewrite big_cons odd_mul_tperm -(lift_permM _ j) odd_permM {}IHts //. congr (_ (+) _); transitivity (tperm (lif...	Lemma	odd_lift_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" ]	[ "add1n", "apply", "big_cons", "big_nil", "bump", "eqSS", "eqVneq", "eq_sym", "inj_eq", "inj_tperm", "invg1", "last", "leqP", "leqW", "leqnn", "lift", "liftK", "lift_inj", "lift_perm", "lift_perm1", "lift_permM", "lift_permV", "lift_perm_id", "lift_perm_lift", "ltnW", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift0_perm s : 'S_n.+1	:= lift_perm 0 0 s.	Definition	lift0_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" ]	[ "lift_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift0_perm0 s : lift0_perm s 0 = 0.	Proof. exact: lift_perm_id. Qed.	Lemma	lift0_perm0	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" ]	[ "lift0_perm", "lift_perm_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift0_perm_lift s k' : lift0_perm s (lift 0 k') = lift (0 : 'I_n.+1) (s k').	Proof. exact: lift_perm_lift. Qed.	Lemma	lift0_perm_lift	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" ]	[ "lift", "lift0_perm", "lift_perm_lift" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift0_permK s : cancel (lift0_perm s) (lift0_perm s^-1).	Proof. by move=> i; rewrite /lift0_perm -lift_permV permK. Qed.	Lemma	lift0_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" ]	[ "lift0_perm", "lift_permV", "permK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift0_perm_eq0 s i : (lift0_perm s i == 0) = (i == 0).	Proof. by rewrite (canF_eq (lift0_permK s)) lift0_perm0. Qed.	Lemma	lift0_perm_eq0	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" ]	[ "canF_eq", "lift0_perm", "lift0_perm0", "lift0_permK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permS0 : all_equal_to (1 : 'S_0).	Proof. by move=> g; apply/permP; case. Qed.	Lemma	permS0	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permS1 : all_equal_to (1 : 'S_1).	Proof. by move=> g; apply/permP => i; rewrite !ord1. Qed.	Lemma	permS1	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", "ord1", "permP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
permS01 n : n <= 1 -> all_equal_to (1 : 'S_n).	Proof. by case: n => [\|[\|]//=] _ g; rewrite (permS0, permS1). Qed.	Lemma	permS01	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" ]	[ "permS0", "permS1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_perm m n (eq_mn : m = n) (s : 'S_m)	:= let: erefl in _ = n := eq_mn return 'S_n in s.	Definition	cast_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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_perm_id n eq_n s : cast_perm eq_n s = s :> 'S_n.	Proof. by apply/permP => i; rewrite /cast_perm /= eq_axiomK. Qed.	Lemma	cast_perm_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" ]	[ "apply", "cast_perm", "eq_axiomK", "permP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ord_permE m n eq_m_n (s : 'S_m) i : @cast_ord m n eq_m_n (s i) = (cast_perm eq_m_n s) (cast_ord eq_m_n i).	Proof. by subst m; rewrite cast_perm_id !cast_ord_id. Qed.	Lemma	cast_ord_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" ]	[ "cast_ord", "cast_ord_id", "cast_perm", "cast_perm_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_permE m n (eq_m_n : m = n) (s : 'S_m) (i : 'I_n) : cast_perm eq_m_n s i = cast_ord eq_m_n (s (cast_ord (esym eq_m_n) i)).	Proof. by rewrite cast_ord_permE cast_ordKV. Qed.	Lemma	cast_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" ]	[ "cast_ord", "cast_ordKV", "cast_ord_permE", "cast_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_perm_comp m n p (eq_m_n : m = n) (eq_n_p : n = p) s : cast_perm eq_n_p (cast_perm eq_m_n s) = cast_perm (etrans eq_m_n eq_n_p) s.	Proof. by case: _ / eq_n_p. Qed.	Lemma	cast_perm_comp	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" ]	[ "cast_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_permK m n eq_m_n : cancel (@cast_perm m n eq_m_n) (cast_perm (esym eq_m_n)).	Proof. by subst m. Qed.	Lemma	cast_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" ]	[ "cast_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_permKV m n eq_m_n : cancel (cast_perm (esym eq_m_n)) (@cast_perm m n eq_m_n).	Proof. by subst m. Qed.	Lemma	cast_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" ]	[ "cast_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_perm_sym m n (eq_m_n : m = n) s t : s = cast_perm eq_m_n t -> t = cast_perm (esym eq_m_n) s.	Proof. by move/(canLR (cast_permK _)). Qed.	Lemma	cast_perm_sym	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" ]	[ "cast_perm", "cast_permK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_perm_inj m n eq_m_n : injective (@cast_perm m n eq_m_n).	Proof. exact: can_inj (cast_permK eq_m_n). Qed.	Lemma	cast_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" ]	[ "cast_perm", "cast_permK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_perm_morphM m n eq_m_n : {morph @cast_perm m n eq_m_n : x y / x * y >-> x * y}.	Proof. by subst m. Qed.	Lemma	cast_perm_morphM	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" ]	[ "cast_perm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_of_cast_perm m n eq_m_n	:= @Morphism _ _ setT (cast_perm eq_m_n) (in2W (@cast_perm_morphM m n eq_m_n)).	Canonical	morph_of_cast_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" ]	[ "cast_perm", "cast_perm_morphM", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_cast_perm m n eq_m_n : isom setT setT (@cast_perm m n eq_m_n).	Proof. case: {n} _ / eq_m_n; apply/isomP; split. exact/injmP/(in2W (@cast_perm_inj _ _ _)). by apply/setP => /= s /[!inE]; apply/imsetP; exists s; rewrite ?inE. Qed.	Lemma	isom_cast_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", "cast_perm", "cast_perm_inj", "imsetP", "inE", "injmP", "isom", "isomP", "setP", "setT", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tact {T n} (s : 'S_n) (t : n.-tuple T) : n.-tuple T	:= [tuple tnth t (s i) \| i < n].	Definition	tact	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" ]	[ "tnth", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tactE {T n} (s : 'S_n) (t : n.-tuple T) : tact s t = [tuple tnth t (s i) \| i < n] :> n.-tuple T.	Proof. exact. Qed.	Lemma	tactE	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" ]	[ "tact", "tnth", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tactP {T : eqType} {n : nat} {r : seq T} {t : n.-tuple T} : reflect (exists s : 'S_n, r = tact s t) (perm_eq r t).	Proof. exact: tuple_permP. Qed.	Lemma	tactP	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" ]	[ "nat", "perm_eq", "seq", "tact", "tuple", "tuple_permP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_tact {T : eqType} {n} (s : 'S_n) (t : n.-tuple T) : perm_eq (tact s t) t.	Proof. by apply/tactP; exists s. Qed.	Lemma	perm_tact	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", "perm_eq", "tact", "tactP", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tact1 {T : eqType} {n} (t : n.-tuple T) : tact 1 t = t :> n.-tuple T.	Proof. apply/eq_from_tnth => i. by rewrite !tnth_map /= permE /= !tnth_ord_tuple. Qed.	Lemma	tact1	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", "eq_from_tnth", "permE", "tact", "tnth_map", "tnth_ord_tuple", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tactM {T : eqType} {n} (s s' : 'S_n) (t : n.-tuple T) : tact (s * s') t = tact s (tact s' t).	Proof. congr tval; apply/eq_from_tnth => i. by rewrite !tnth_map /= permE /= !tnth_ord_tuple. Qed.	Lemma	tactM	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", "eq_from_tnth", "permE", "tact", "tnth_map", "tnth_ord_tuple", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tactK {T : eqType} {n} (s s' : 'S_n) (t : n.-tuple T) : tact s^-1 (tact s t) = t.	Proof. by rewrite -tactM mulVg tact1. Qed.	Lemma	tactK	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", "tact", "tact1", "tactM", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_tact {T n} (s : 'S_n) (t : n.-tuple T) i : tnth (tact s t) i = tnth t (s i).	Proof. by rewrite tactE tnth_mktuple. Qed.	Lemma	tnth_tact	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" ]	[ "tact", "tactE", "tnth", "tnth_mktuple", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tact_lift0 {T n} (s : 'S_n) x (t : n.-tuple T) : tact (lift0_perm s) (x :: t) = x :: tact s t :> n.+1.-tuple T.	Proof. apply/eq_from_tnth => i; rewrite tnth_tact. have [_/[!ord1]->\|k {i}->] := @split_ordP 1 n i. by rewrite !lshift0/= lift0_perm0. by rewrite !rshift1 lift0_perm_lift !tnthS tnth_tact. Qed.	Lemma	tact_lift0	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", "eq_from_tnth", "lift0_perm", "lift0_perm0", "lift0_perm_lift", "lshift0", "ord1", "rshift1", "split_ordP", "tact", "tnthS", "tnth_tact", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tval_tact_lift0 {T n} (s : 'S_n) x (t : n.-tuple T) : tact (lift0_perm s) (x :: t) = x :: tact s t :> seq T.	Proof. by have /(congr1 val) := tact_lift0 s x t. Qed.	Lemma	tval_tact_lift0	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" ]	[ "lift0_perm", "seq", "tact", "tact_lift0", "tuple", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
term	:= \| Cst of nat \| Idx \| Inv of term \| Exp of term & nat \| Mul of term & term \| Conj of term & term \| Comm of term & term.	Inductive	term	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "nat" ]	tuple value type	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eval {gT} e t : gT	:= match t with \| Cst i => nth 1 e i \| Idx => 1 \| Inv t1 => (eval e t1)^-1 \| Exp t1 n => eval e t1 ^+ n \| Mul t1 t2 => eval e t1 * eval e t2 \| Conj t1 t2 => eval e t1 ^ eval e t2 \| Comm t1 t2 => [~ eval e t1, eval e t2] end.	Fixpoint	eval	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "gT", "nth" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
formula	:= Eq2 of term & term \| And of formula & formula.	Inductive	formula	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "And", "term" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Eq1 s	:= Eq2 s Idx.	Definition	Eq1	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Eq3 s1 s2 t	:= And (Eq2 s1 t) (Eq2 s2 t).	Definition	Eq3	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "And", "s1", "s2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rel_type	:= NoRel \| Rel vT of vT & vT.	Inductive	rel_type	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bool_of_rel r	:= if r is Rel vT v1 v2 then v1 == v2 else true.	Definition	bool_of_rel	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bool_of_rel : rel_type >-> bool.		Coercion	bool_of_rel	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "rel_type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
and_rel vT (v1 v2 : vT) r	:= if r is Rel wT w1 w2 then Rel (v1, w1) (v2, w2) else Rel v1 v2.	Definition	and_rel	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rel {gT} (e : seq gT) f r	:= match f with \| Eq2 s t => and_rel (eval e s) (eval e t) r \| And f1 f2 => rel e f1 (rel e f2 r) end.	Fixpoint	rel	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "And", "and_rel", "eval", "f1", "f2", "gT", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
type	:= Generator of term -> type \| Formula of formula.	Inductive	type	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Formula", "formula", "term" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Cast p : type	:= p.	Definition	Cast	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Formula : formula >-> type.		Coercion	Formula	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "formula", "type" ]	syntactic scope cast	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
env gT	:= Env of {set gT} & seq gT.	Inductive	env	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "gT", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
env1 {gT} (x : gT : finType)	:= Env <[x]> [:: x].	Definition	env1	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sat gT vT B n (s : vT -> env gT) p	:= match p with \| Formula f => [exists v, let: Env A e := s v in and_rel A B (rel (rev e) f NoRel)] \| Generator p' => let s' v := let: Env A e := s v.1 in Env (A <*> <[v.2]>) (v.2 :: e) in sat B n.+1 s' (p' (Cst n)) end.	Fixpoint	sat	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "Formula", "and_rel", "env", "gT", "rel", "rev", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hom gT (B : {set gT}) p	:= sat B 1 env1 (p (Cst 0)).	Definition	hom	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "env1", "gT", "sat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iso gT (B : {set gT}) p	:= forall rT (H : {group rT}), (H \homg B) = hom H p.	Definition	iso	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "gT", "group", "hom", "homg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Eq1 : term >-> formula.		Coercion	Eq1	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[ "formula", "term" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x * y"	:= (Mul x y) (in custom group_presentation at level 40, left associativity).	Notation	x * y	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x ^+ n"	:= (Exp x n) (in custom group_presentation at level 29, n constr at level 28).	Notation	x ^+ n	finite_group	finite_group/presentation.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "fintype", "finset", "fingroup", "morphism", "Presentation" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

inj_tperm (T T' : finType) (f : T -> T') x y z : injective f -> f (tperm x y z) = tperm (f x) (f y) (f z).

Proof. by move=> injf; rewrite !permE /= !(inj_eq injf) !(fun_if f). Qed.

Lemma

inj_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" ]

[ "T'", "inj_eq", "injf", "permE", "tperm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermJ x y s : (tperm x y) ^ s = tperm (s x) (s y).

Proof. by apply/permP => z; rewrite -(permKV s z) permJ; apply/inj_tperm/perm_inj. Qed.

Lemma

tpermJ

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", "inj_tperm", "permJ", "permKV", "permP", "perm_inj", "tperm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tpermJ_tperm x y z : x != z -> y != z -> tperm x z ^ tperm x y = tperm y z.

Proof. by move=> nxz nyz; rewrite tpermJ tpermL [tperm _ _ z]tpermD. Qed.

Lemma

tpermJ_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", "tpermD", "tpermJ", "tpermL" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tuple_permP {T : eqType} {n} {s : seq T} {t : n.-tuple T} : reflect (exists p : 'S_n, s = [tuple tnth t (p i) | i < n]) (perm_eq s t).

Proof. apply: (iffP idP) => [|[p ->]]; last first. rewrite /= (map_comp (tnth t)) -{1}(map_tnth_enum t) perm_map //. apply: uniq_perm => [||i]; rewrite ?enum_uniq //. by apply/injectiveP; apply: perm_inj. by rewrite mem_enum -[i](permKV p) image_f. case: n => [|n] in t *; last have x0 := tnth t ord0. rewrit...

Lemma

tuple_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", "enum_uniq", "eq_from_tnth", "image_f", "injectiveP", "iota_uniq", "last", "map_comp", "map_tnth_enum", "map_uniq", "mem_enum", "mem_iota", "mem_tnth", "mktuple", "ord0", "pP", "permE", "permKV", "perm_eq", "perm_inj", "perm_iotaP", "perm_map", "perm_mem", "per...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aperm (T : finType) x (s : {perm T})

:= s x.

Definition

aperm

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" ]

[]

action.v and hence morphism.v.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbit_unlockable

:= Unlockable porbit.unlock.

Canonical

porbit_unlockable

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

porbits (T : finType) (s : {perm T})

:= porbit s @: T.

Definition

porbits

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

odd_perm (s : perm_type T)

:= odd #|T| (+) odd #|porbits s|.

Definition

odd_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" ]

[ "odd", "perm_type", "porbits" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

apermE x s : aperm x s = s x.

Proof. by []. Qed.

Lemma

apermE

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" ]

[ "aperm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_porbit s i x : (s ^+ i) x \in porbit s x.

Proof. by rewrite [@porbit]unlock (imset_f (aperm x)) ?mem_cycle. Qed.

Lemma

mem_porbit

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" ]

[ "aperm", "imset_f", "mem_cycle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbit_id s x : x \in porbit s x.

Proof. by rewrite -{1}[x]perm1 (mem_porbit s 0). Qed.

Lemma

porbit_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" ]

[ "mem_porbit", "perm1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_porbit_neq0 s x : #|porbit s x| != 0.

Proof. by rewrite -lt0n card_gt0; apply/set0Pn; exists x; exact: porbit_id. Qed.

Lemma

card_porbit_neq0

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_gt0", "lt0n", "porbit_id", "set0Pn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

uniq_traject_porbit s x : uniq (traject s x #|porbit s x|).

Proof. case def_n: #|_| => // [n]; rewrite looping_uniq. apply: contraL (card_size (traject s x n)) => /loopingP t_sx. rewrite -ltnNge size_traject -def_n ?subset_leq_card // porbit.unlock. by apply/subsetP=> _ /imsetP[_ /cycleP[i ->] ->]; rewrite /aperm permX t_sx. Qed.

Lemma

uniq_traject_porbit

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" ]

[ "aperm", "apply", "card_size", "cycleP", "def_n", "imsetP", "loopingP", "looping_uniq", "ltnNge", "permX", "size_traject", "subsetP", "subset_leq_card", "traject", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbit_traject s x : porbit s x =i traject s x #|porbit s x|.

Proof. apply: fsym; apply/subset_cardP. by rewrite (card_uniqP _) ?size_traject ?uniq_traject_porbit. by apply/subsetP=> _ /trajectP[i _ ->]; rewrite -permX mem_porbit. Qed.

Lemma

porbit_traject

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_uniqP", "mem_porbit", "permX", "size_traject", "subsetP", "subset_cardP", "traject", "trajectP", "uniq_traject_porbit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iter_porbit s x : iter #|porbit s x| s x = x.

Proof. case def_n: #|_| (uniq_traject_porbit s x) => [//|n] Ut. have: looping s x n.+1. by rewrite -def_n -[looping _ _ _]porbit_traject -permX mem_porbit. rewrite /looping => /trajectP[[|i] //= lt_i_n /perm_inj eq_i_n_sx]. move: lt_i_n; rewrite ltnS ltn_neqAle andbC => /andP[le_i_n /negP[]]. by rewrite -(nth_uniq x ...

Lemma

iter_porbit

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" ]

[ "def_n", "iter", "looping", "ltnS", "ltn_neqAle", "mem_porbit", "nth_traject", "nth_uniq", "permX", "perm_inj", "porbit_traject", "size_traject", "trajectP", "uniq_traject_porbit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_porbit_mem s x y : (porbit s x == porbit s y) = (x \in porbit s y).

Proof. apply/eqP/idP; first by move<-; exact: porbit_id. rewrite porbit.unlock => /imsetP[si s_si ->]. apply/setP => z; apply/imsetP/imsetP=> [] [sj s_sj ->]. by exists (si * sj); rewrite ?groupM /aperm ?permM. exists (si^-1 * sj); first by rewrite groupM ?groupV. by rewrite /aperm permM permK. Qed.

Lemma

eq_porbit_mem

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" ]

[ "aperm", "apply", "groupM", "groupV", "imsetP", "permK", "permM", "porbit_id", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbit_sym s x y : (x \in porbit s y) = (y \in porbit s x).

Proof. by rewrite -!eq_porbit_mem eq_sym. Qed.

Lemma

porbit_sym

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" ]

[ "eq_porbit_mem", "eq_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbit_perm s i x : porbit s ((s ^+ i) x) = porbit s x.

Proof. by apply/eqP; rewrite eq_porbit_mem mem_porbit. Qed.

Lemma

porbit_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", "eq_porbit_mem", "mem_porbit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbitPmin s x y : y \in porbit s x -> exists2 i, i < #[s] & y = (s ^+ i) x.

Proof. by rewrite porbit.unlock=> /imsetP [z /cyclePmin[ i Hi ->{z}] ->{y}]; exists i. Qed.

Lemma

porbitPmin

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" ]

[ "cyclePmin", "imsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbitP s x y : reflect (exists i, y = (s ^+ i) x) (y \in porbit s x).

Proof. apply (iffP idP) => [/porbitPmin [i _ ->]| [i ->]]; last exact: mem_porbit. by exists i. Qed.

Lemma

porbitP

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", "last", "mem_porbit", "porbitPmin" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbitV s : porbit s^-1 =1 porbit s.

Proof. move=> x; apply/setP => y; rewrite porbit_sym. by apply/porbitP/porbitP => -[i ->]; exists i; rewrite expgVn ?permK ?permKV. Qed.

Lemma

porbitV

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", "expgVn", "permK", "permKV", "porbitP", "porbit_sym", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbitsV s : porbits s^-1 = porbits s.

Proof. rewrite /porbits; apply/setP => y. by apply/imsetP/imsetP => -[x _ ->{y}]; exists x; rewrite // porbitV. Qed.

Lemma

porbitsV

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", "imsetP", "porbitV", "porbits", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbit_setP s t x : porbit s x =i porbit t x <-> porbit s x = porbit t x.

Proof. by rewrite porbit.unlock; exact: setP. Qed.

Lemma

porbit_setP

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" ]

[ "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbits_mul_tperm s x y : let t := tperm x y in #|porbits (t * s)| + (x \notin porbit s y).*2 = #|porbits s| + (x != y).

Proof. pose xf a b u := seq.find (pred2 a b) (traject u (u a) #|porbit u a|). have xf_size a b u: xf a b u <= #|porbit u a|. by rewrite (leq_trans (find_size _ _)) ?size_traject. have lt_xf a b u n : n < xf a b u -> ~~ pred2 a b ((u ^+ n.+1) a). move=> lt_n; apply: contraFN (before_find (u a) lt_n). by rewrite pe...

Lemma

porbits_mul_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" ]

[ "addnA", "addnC", "aperm", "apply", "before_find", "cardsD1", "contraNneq", "eq_porbit_mem", "eq_sym", "eqn_leq", "eqxx", "expgSr", "find", "find_size", "has", "hasP", "has_find", "imsetP", "imset_f", "inE", "iterSr", "last", "leq_trans", "leqnn", "looping", "loopin...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_perm1 : odd_perm 1 = false.

Proof. rewrite /odd_perm card_imset ?addbb // => x y; move/eqP; rewrite eq_porbit_mem. by rewrite porbit.unlock cycle1 imset_set1 /aperm perm1 inE=> /eqP. Qed.

Lemma

odd_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" ]

[ "aperm", "card_imset", "cycle1", "eq_porbit_mem", "imset_set1", "inE", "odd_perm", "perm1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_mul_tperm x y s : odd_perm (tperm x y * s) = (x != y) (+) odd_perm s.

Proof. rewrite addbC -addbA -[~~ _]oddb -oddD -porbits_mul_tperm. by rewrite oddD odd_double addbF. Qed.

Lemma

odd_mul_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" ]

[ "oddD", "odd_double", "odd_perm", "oddb", "porbits_mul_tperm", "tperm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_tperm x y : odd_perm (tperm x y) = (x != y).

Proof. by rewrite -[_ y]mulg1 odd_mul_tperm odd_perm1 addbF. Qed.

Lemma

odd_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" ]

[ "mulg1", "odd_mul_tperm", "odd_perm", "odd_perm1", "tperm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dpair (eT : eqType)

:= [pred t | t.1 != t.2 :> eT].

Definition

dpair

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

prod_tpermP s : {ts : seq (T * T) | s = \prod_(t <- ts) tperm t.1 t.2 & all dpair ts}.

Proof. have [n] := ubnP #|[pred x | s x != x]|; elim: n s => // n IHn s /ltnSE-le_s_n. case: (pickP (fun x => s x != x)) => [x s_x | s_id]; last first. exists nil; rewrite // big_nil; apply/permP=> x. by apply/eqP/idPn; rewrite perm1 s_id. have [|ts def_s ne_ts] := IHn (tperm x (s^-1 x) * s); last first. exists (...

Lemma

prod_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" ]

[ "all", "apply", "big_cons", "big_nil", "canF_eq", "cardD1", "dpair", "eqVneq", "eq_sym", "eqxx", "inE", "last", "leq_ltn_trans", "ltnSE", "mul1g", "mulgA", "perm1", "permE", "permK", "permKV", "permM", "permP", "pickP", "seq", "subsetP", "subset_leq_card", "tperm"...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_perm_prod ts : all dpair ts -> odd_perm (\prod_(t <- ts) tperm t.1 t.2) = odd (size ts).

Proof. elim: ts => [_|t ts IHts] /=; first by rewrite big_nil odd_perm1. by case/andP=> dt12 dts; rewrite big_cons odd_mul_tperm dt12 IHts. Qed.

Lemma

odd_perm_prod

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" ]

[ "all", "big_cons", "big_nil", "dpair", "odd", "odd_mul_tperm", "odd_perm", "odd_perm1", "size", "tperm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_permM : {morph odd_perm : s1 s2 / s1 * s2 >-> s1 (+) s2}.

Proof. move=> s1 s2; case: (prod_tpermP s1) => ts1 ->{s1} dts1. case: (prod_tpermP s2) => ts2 ->{s2} dts2. by rewrite -big_cat !odd_perm_prod ?all_cat ?dts1 // size_cat oddD. Qed.

Lemma

odd_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" ]

[ "all_cat", "big_cat", "oddD", "odd_perm", "odd_perm_prod", "prod_tpermP", "s1", "s2", "size_cat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_permV s : odd_perm s^-1 = odd_perm s.

Proof. by rewrite -{2}(mulgK s s) !odd_permM -addbA addKb. Qed.

Lemma

odd_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" ]

[ "mulgK", "odd_perm", "odd_permM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_permJ s1 s2 : odd_perm (s1 ^ s2) = odd_perm s1.

Proof. by rewrite !odd_permM odd_permV addbC addbK. Qed.

Lemma

odd_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" ]

[ "odd_perm", "odd_permM", "odd_permV", "s1", "s2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen_tperm x : <<[set tperm x y | y in T]>>%g = [set: {perm T}].

Proof. apply/eqP; rewrite eqEsubset subsetT/=; apply/subsetP => s _. have [ts -> _] := prod_tpermP s; rewrite group_prod// => -[/= y z] _. have [<-|Nyz] := eqVneq y z; first by rewrite tperm1 group1. have [<-|Nxz] := eqVneq x z; first by rewrite tpermC mem_gen ?imset_f. by rewrite -(tpermJ_tperm Nxz Nyz) groupJ ?mem_ge...

Lemma

gen_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" ]

[ "apply", "eqEsubset", "eqVneq", "group1", "groupJ", "group_prod", "imset_f", "mem_gen", "prod_tpermP", "subsetP", "subsetT", "tperm", "tperm1", "tpermC", "tpermJ_tperm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_perm : perm_type >-> bool.

Coercion

odd_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

Sym : {set {perm T}}

:= [set s | perm_on S s].

Definition

Sym

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_on" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Sym_group_set : group_set Sym.

Proof. apply/group_setP; split => [|s t] /[!inE]; [exact: perm_on1 | exact: perm_onM]. Qed.

Lemma

Sym_group_set

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" ]

[ "Sym", "apply", "group_set", "group_setP", "inE", "perm_on1", "perm_onM", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Sym_group : {group {perm T}}

:= Group Sym_group_set.

Canonical

Sym_group

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" ]

[ "Sym_group_set", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_Sym : #|Sym| = #|S|`!.

Proof. by rewrite cardsE /= card_perm. Qed.

Lemma

card_Sym

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" ]

[ "Sym", "card_perm", "cardsE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_Sn : #|'S_(n)| = n`!.

Proof. rewrite (eq_card (B := perm_on [set : 'I_n])); last first. by rewrite card_perm /= cardsE /= card_ord. move=> p; rewrite inE unfold_in /perm_on /=. by apply/esym/subsetP => i _; rewrite in_set. Qed.

Lemma

card_Sn

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_ord", "card_perm", "cardsE", "eq_card", "inE", "in_set", "last", "perm_on", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_perm_fun i j s k

:= if unlift i k is Some k' then lift j (s k') else j.

Definition

lift_perm_fun

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" ]

[ "lift", "unlift" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_permK i j s : cancel (lift_perm_fun i j s) (lift_perm_fun j i s^-1).

Proof. rewrite /lift_perm_fun => k. by case: (unliftP i k) => [j'|] ->; rewrite (liftK, unlift_none) ?permK. Qed.

Lemma

lift_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" ]

[ "liftK", "lift_perm_fun", "permK", "unliftP", "unlift_none" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_perm i j s : 'S_n.+1

:= perm (can_inj (lift_permK i j s)).

Definition

lift_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" ]

[ "lift_permK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_perm_id i j s : lift_perm i j s i = j.

Proof. by rewrite permE /lift_perm_fun unlift_none. Qed.

Lemma

lift_perm_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" ]

[ "lift_perm", "lift_perm_fun", "permE", "unlift_none" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_perm_lift i j s k' : lift_perm i j s (lift i k') = lift j (s k') :> 'I_n.+1.

Proof. by rewrite permE /lift_perm_fun liftK. Qed.

Lemma

lift_perm_lift

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" ]

[ "lift", "liftK", "lift_perm", "lift_perm_fun", "permE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_permM i j k s t : lift_perm i j s * lift_perm j k t = lift_perm i k (s * t).

Proof. apply/permP=> i1; case: (unliftP i i1) => [i2|] ->{i1}. by rewrite !(permM, lift_perm_lift). by rewrite permM !lift_perm_id. Qed.

Lemma

lift_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" ]

[ "apply", "lift_perm", "lift_perm_id", "lift_perm_lift", "permM", "permP", "unliftP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_perm1 i : lift_perm i i 1 = 1.

Proof. by apply: (mulgI (lift_perm i i 1)); rewrite lift_permM !mulg1. Qed.

Lemma

lift_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", "lift_perm", "lift_permM", "mulg1", "mulgI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_permV i j s : (lift_perm i j s)^-1 = lift_perm j i s^-1.

Proof. by apply/eqP; rewrite eq_invg_mul lift_permM mulgV lift_perm1. Qed.

Lemma

lift_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" ]

[ "apply", "eq_invg_mul", "lift_perm", "lift_perm1", "lift_permM", "mulgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_lift_perm i j s : lift_perm i j s = odd i (+) odd j (+) s :> bool.

Proof. rewrite -{1}(mul1g s) -(lift_permM _ j) odd_permM. congr (_ (+) _); last first. case: (prod_tpermP s) => ts ->{s} _. elim: ts => [|t ts IHts] /=; first by rewrite big_nil lift_perm1 !odd_perm1. rewrite big_cons odd_mul_tperm -(lift_permM _ j) odd_permM {}IHts //. congr (_ (+) _); transitivity (tperm (lif...

Lemma

odd_lift_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" ]

[ "add1n", "apply", "big_cons", "big_nil", "bump", "eqSS", "eqVneq", "eq_sym", "inj_eq", "inj_tperm", "invg1", "last", "leqP", "leqW", "leqnn", "lift", "liftK", "lift_inj", "lift_perm", "lift_perm1", "lift_permM", "lift_permV", "lift_perm_id", "lift_perm_lift", "ltnW", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift0_perm s : 'S_n.+1

:= lift_perm 0 0 s.

Definition

lift0_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" ]

[ "lift_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift0_perm0 s : lift0_perm s 0 = 0.

Proof. exact: lift_perm_id. Qed.

Lemma

lift0_perm0

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" ]

[ "lift0_perm", "lift_perm_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift0_perm_lift s k' : lift0_perm s (lift 0 k') = lift (0 : 'I_n.+1) (s k').

Proof. exact: lift_perm_lift. Qed.

Lemma

lift0_perm_lift

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" ]

[ "lift", "lift0_perm", "lift_perm_lift" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift0_permK s : cancel (lift0_perm s) (lift0_perm s^-1).

Proof. by move=> i; rewrite /lift0_perm -lift_permV permK. Qed.

Lemma

lift0_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" ]

[ "lift0_perm", "lift_permV", "permK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift0_perm_eq0 s i : (lift0_perm s i == 0) = (i == 0).

Proof. by rewrite (canF_eq (lift0_permK s)) lift0_perm0. Qed.

Lemma

lift0_perm_eq0

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" ]

[ "canF_eq", "lift0_perm", "lift0_perm0", "lift0_permK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permS0 : all_equal_to (1 : 'S_0).

Proof. by move=> g; apply/permP; case. Qed.

Lemma

permS0

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permS1 : all_equal_to (1 : 'S_1).

Proof. by move=> g; apply/permP => i; rewrite !ord1. Qed.

Lemma

permS1

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", "ord1", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

permS01 n : n <= 1 -> all_equal_to (1 : 'S_n).

Proof. by case: n => [|[|]//=] _ g; rewrite (permS0, permS1). Qed.

Lemma

permS01

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" ]

[ "permS0", "permS1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_perm m n (eq_mn : m = n) (s : 'S_m)

:= let: erefl in _ = n := eq_mn return 'S_n in s.

Definition

cast_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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_perm_id n eq_n s : cast_perm eq_n s = s :> 'S_n.

Proof. by apply/permP => i; rewrite /cast_perm /= eq_axiomK. Qed.

Lemma

cast_perm_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" ]

[ "apply", "cast_perm", "eq_axiomK", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ord_permE m n eq_m_n (s : 'S_m) i : @cast_ord m n eq_m_n (s i) = (cast_perm eq_m_n s) (cast_ord eq_m_n i).

Proof. by subst m; rewrite cast_perm_id !cast_ord_id. Qed.

Lemma

cast_ord_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" ]

[ "cast_ord", "cast_ord_id", "cast_perm", "cast_perm_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_permE m n (eq_m_n : m = n) (s : 'S_m) (i : 'I_n) : cast_perm eq_m_n s i = cast_ord eq_m_n (s (cast_ord (esym eq_m_n) i)).

Proof. by rewrite cast_ord_permE cast_ordKV. Qed.

Lemma

cast_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" ]

[ "cast_ord", "cast_ordKV", "cast_ord_permE", "cast_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_perm_comp m n p (eq_m_n : m = n) (eq_n_p : n = p) s : cast_perm eq_n_p (cast_perm eq_m_n s) = cast_perm (etrans eq_m_n eq_n_p) s.

Proof. by case: _ / eq_n_p. Qed.

Lemma

cast_perm_comp

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" ]

[ "cast_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_permK m n eq_m_n : cancel (@cast_perm m n eq_m_n) (cast_perm (esym eq_m_n)).

Proof. by subst m. Qed.

Lemma

cast_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" ]

[ "cast_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_permKV m n eq_m_n : cancel (cast_perm (esym eq_m_n)) (@cast_perm m n eq_m_n).

Proof. by subst m. Qed.

Lemma

cast_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" ]

[ "cast_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_perm_sym m n (eq_m_n : m = n) s t : s = cast_perm eq_m_n t -> t = cast_perm (esym eq_m_n) s.

Proof. by move/(canLR (cast_permK _)). Qed.

Lemma

cast_perm_sym

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" ]

[ "cast_perm", "cast_permK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_perm_inj m n eq_m_n : injective (@cast_perm m n eq_m_n).

Proof. exact: can_inj (cast_permK eq_m_n). Qed.

Lemma

cast_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" ]

[ "cast_perm", "cast_permK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_perm_morphM m n eq_m_n : {morph @cast_perm m n eq_m_n : x y / x * y >-> x * y}.

Proof. by subst m. Qed.

Lemma

cast_perm_morphM

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" ]

[ "cast_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_of_cast_perm m n eq_m_n

:= @Morphism _ _ setT (cast_perm eq_m_n) (in2W (@cast_perm_morphM m n eq_m_n)).

Canonical

morph_of_cast_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" ]

[ "cast_perm", "cast_perm_morphM", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_cast_perm m n eq_m_n : isom setT setT (@cast_perm m n eq_m_n).

Proof. case: {n} _ / eq_m_n; apply/isomP; split. exact/injmP/(in2W (@cast_perm_inj _ _ _)). by apply/setP => /= s /[!inE]; apply/imsetP; exists s; rewrite ?inE. Qed.

Lemma

isom_cast_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", "cast_perm", "cast_perm_inj", "imsetP", "inE", "injmP", "isom", "isomP", "setP", "setT", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tact {T n} (s : 'S_n) (t : n.-tuple T) : n.-tuple T

:= [tuple tnth t (s i) | i < n].

Definition

tact

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" ]

[ "tnth", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tactE {T n} (s : 'S_n) (t : n.-tuple T) : tact s t = [tuple tnth t (s i) | i < n] :> n.-tuple T.

Proof. exact. Qed.

Lemma

tactE

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" ]

[ "tact", "tnth", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tactP {T : eqType} {n : nat} {r : seq T} {t : n.-tuple T} : reflect (exists s : 'S_n, r = tact s t) (perm_eq r t).

Proof. exact: tuple_permP. Qed.

Lemma

tactP

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" ]

[ "nat", "perm_eq", "seq", "tact", "tuple", "tuple_permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_tact {T : eqType} {n} (s : 'S_n) (t : n.-tuple T) : perm_eq (tact s t) t.

Proof. by apply/tactP; exists s. Qed.

Lemma

perm_tact

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", "perm_eq", "tact", "tactP", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tact1 {T : eqType} {n} (t : n.-tuple T) : tact 1 t = t :> n.-tuple T.

Proof. apply/eq_from_tnth => i. by rewrite !tnth_map /= permE /= !tnth_ord_tuple. Qed.

Lemma

tact1

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", "eq_from_tnth", "permE", "tact", "tnth_map", "tnth_ord_tuple", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tactM {T : eqType} {n} (s s' : 'S_n) (t : n.-tuple T) : tact (s * s') t = tact s (tact s' t).

Proof. congr tval; apply/eq_from_tnth => i. by rewrite !tnth_map /= permE /= !tnth_ord_tuple. Qed.

Lemma

tactM

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", "eq_from_tnth", "permE", "tact", "tnth_map", "tnth_ord_tuple", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tactK {T : eqType} {n} (s s' : 'S_n) (t : n.-tuple T) : tact s^-1 (tact s t) = t.

Proof. by rewrite -tactM mulVg tact1. Qed.

Lemma

tactK

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", "tact", "tact1", "tactM", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_tact {T n} (s : 'S_n) (t : n.-tuple T) i : tnth (tact s t) i = tnth t (s i).

Proof. by rewrite tactE tnth_mktuple. Qed.

Lemma

tnth_tact

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" ]

[ "tact", "tactE", "tnth", "tnth_mktuple", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tact_lift0 {T n} (s : 'S_n) x (t : n.-tuple T) : tact (lift0_perm s) (x :: t) = x :: tact s t :> n.+1.-tuple T.

Proof. apply/eq_from_tnth => i; rewrite tnth_tact. have [_/[!ord1]->|k {i}->] := @split_ordP 1 n i. by rewrite !lshift0/= lift0_perm0. by rewrite !rshift1 lift0_perm_lift !tnthS tnth_tact. Qed.

Lemma

tact_lift0

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", "eq_from_tnth", "lift0_perm", "lift0_perm0", "lift0_perm_lift", "lshift0", "ord1", "rshift1", "split_ordP", "tact", "tnthS", "tnth_tact", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tval_tact_lift0 {T n} (s : 'S_n) x (t : n.-tuple T) : tact (lift0_perm s) (x :: t) = x :: tact s t :> seq T.

Proof. by have /(congr1 val) := tact_lift0 s x t. Qed.

Lemma

tval_tact_lift0

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" ]

[ "lift0_perm", "seq", "tact", "tact_lift0", "tuple", "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

term

Inductive

term

finite_group

finite_group/presentation.v

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

[ "nat" ]

tuple value type

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eval {gT} e t : gT

Fixpoint

eval

finite_group

finite_group/presentation.v

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

[ "gT", "nth" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

formula

:= Eq2 of term & term | And of formula & formula.

Inductive

formula

finite_group

finite_group/presentation.v

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

[ "And", "term" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Eq1 s

:= Eq2 s Idx.

Definition

Eq1

finite_group

finite_group/presentation.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Eq3 s1 s2 t

:= And (Eq2 s1 t) (Eq2 s2 t).

Definition

Eq3

finite_group

finite_group/presentation.v

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

[ "And", "s1", "s2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rel_type

:= NoRel | Rel vT of vT & vT.

Inductive

rel_type

finite_group

finite_group/presentation.v

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

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bool_of_rel r

:= if r is Rel vT v1 v2 then v1 == v2 else true.

Definition

bool_of_rel

finite_group

finite_group/presentation.v

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

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bool_of_rel : rel_type >-> bool.

Coercion

bool_of_rel

finite_group

finite_group/presentation.v

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

[ "rel_type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

and_rel vT (v1 v2 : vT) r

:= if r is Rel wT w1 w2 then Rel (v1, w1) (v2, w2) else Rel v1 v2.

Definition

and_rel

finite_group

finite_group/presentation.v

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

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rel {gT} (e : seq gT) f r

:= match f with | Eq2 s t => and_rel (eval e s) (eval e t) r | And f1 f2 => rel e f1 (rel e f2 r) end.

Fixpoint

rel

finite_group

finite_group/presentation.v

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

[ "And", "and_rel", "eval", "f1", "f2", "gT", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

type

:= Generator of term -> type | Formula of formula.

Inductive

type

finite_group

finite_group/presentation.v

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

[ "Formula", "formula", "term" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Cast p : type

:= p.

Definition

Cast

finite_group

finite_group/presentation.v

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

[ "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Formula : formula >-> type.

Coercion

Formula

finite_group

finite_group/presentation.v

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

[ "formula", "type" ]

syntactic scope cast

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

env gT

:= Env of {set gT} & seq gT.

Inductive

env

finite_group

finite_group/presentation.v

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

[ "gT", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

env1 {gT} (x : gT : finType)

:= Env <[x]> [:: x].

Definition

env1

finite_group

finite_group/presentation.v

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

[ "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sat gT vT B n (s : vT -> env gT) p

:= match p with | Formula f => [exists v, let: Env A e := s v in and_rel A B (rel (rev e) f NoRel)] | Generator p' => let s' v := let: Env A e := s v.1 in Env (A <*> <[v.2]>) (v.2 :: e) in sat B n.+1 s' (p' (Cst n)) end.

Fixpoint

sat

finite_group

finite_group/presentation.v

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

[ "Formula", "and_rel", "env", "gT", "rel", "rev", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hom gT (B : {set gT}) p

:= sat B 1 env1 (p (Cst 0)).

Definition

hom

finite_group

finite_group/presentation.v

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

[ "env1", "gT", "sat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iso gT (B : {set gT}) p

:= forall rT (H : {group rT}), (H \homg B) = hom H p.

Definition

iso

finite_group

finite_group/presentation.v

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

[ "gT", "group", "hom", "homg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Eq1 : term >-> formula.

Coercion

Eq1

finite_group

finite_group/presentation.v

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

[ "formula", "term" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x * y"

:= (Mul x y) (in custom group_presentation at level 40, left associativity).

Notation

x * y

finite_group

finite_group/presentation.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x ^+ n"

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

Notation

x ^+ n

finite_group

finite_group/presentation.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d