fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
cast_perm_compm 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
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_perm_comp
| |
cast_permKm n eq_m_n :
cancel (@cast_perm m n eq_m_n) (cast_perm (esym eq_m_n)).
Proof. by subst m. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_permK
| |
cast_permKVm n eq_m_n :
cancel (cast_perm (esym eq_m_n)) (@cast_perm m n eq_m_n).
Proof. by subst m. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_permKV
| |
cast_perm_symm 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
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_perm_sym
| |
cast_perm_injm n eq_m_n : injective (@cast_perm m n eq_m_n).
Proof. exact: can_inj (cast_permK eq_m_n). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_perm_inj
| |
cast_perm_morphMm n eq_m_n :
{morph @cast_perm m n eq_m_n : x y / x * y >-> x * y}.
Proof. by subst m. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_perm_morphM
| |
morph_of_cast_permm n eq_m_n :=
@Morphism _ _ setT (cast_perm eq_m_n) (in2W (@cast_perm_morphM m n eq_m_n)).
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
morph_of_cast_perm
| |
isom_cast_permm 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
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
isom_cast_perm
| |
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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
term
| |
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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
eval
| |
formula:= Eq2 of term & term | And of formula & formula.
|
Inductive
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
formula
| |
Eq1s := Eq2 s Idx.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
Eq1
| |
Eq3s1 s2 t := And (Eq2 s1 t) (Eq2 s2 t).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
Eq3
| |
rel_type:= NoRel | Rel vT of vT & vT.
|
Inductive
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
rel_type
| |
bool_of_relr := if r is Rel vT v1 v2 then v1 == v2 else true.
Local Coercion bool_of_rel : rel_type >-> bool.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
bool_of_rel
| |
and_relvT (v1 v2 : vT) r :=
if r is Rel wT w1 w2 then Rel (v1, w1) (v2, w2) else Rel v1 v2.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
and_rel
| |
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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
rel
| |
type:= Generator of term -> type | Formula of formula.
|
Inductive
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
type
| |
Castp : type := p.
Local Coercion Formula : formula >-> type.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
Cast
| |
envgT := Env of {set gT} & seq gT.
|
Inductive
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
env
| |
env1{gT} (x : gT : finType) := Env <[x]> [:: x].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
env1
| |
satgT 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
sat
| |
homgT (B : {set gT}) p := sat B 1 env1 (p (Cst 0)).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
hom
| |
isogT (B : {set gT}) p :=
forall rT (H : {group rT}), (H \homg B) = hom H p.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
iso
| |
bool_of_rel: rel_type >-> bool.
|
Coercion
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
bool_of_rel
| |
Eq1: term >-> formula.
|
Coercion
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
Eq1
| |
Formula: formula >-> type.
Declare Custom Entry group_presentation.
|
Coercion
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
Formula
| |
isoGrp_homgT (G : {group gT}) p : G \isog Grp p -> G \homg Grp p.
Proof. by move <-; apply: homg_refl. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
isoGrp_hom
| |
isoGrpPgT (G : {group gT}) p rT (H : {group rT}) :
G \isog Grp p -> reflect (#|H| = #|G| /\ H \homg Grp p) (H \isog G).
Proof.
move=> isoGp; apply: (iffP idP) => [isoGH | [oH homHp]].
by rewrite (card_isog isoGH) -isoGp isog_hom.
by rewrite isogEcard isoGp homHp /= oH.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
isoGrpP
| |
homGrp_transrT gT (H : {set rT}) (G : {group gT}) p :
H \homg G -> G \homg Grp p -> H \homg Grp p.
Proof.
case/homgP=> h <-{H}; rewrite /hom; move: {p}(p _) => p.
have evalG e t: all [in G] e -> eval (map h e) t = h (eval e t).
move=> Ge; apply: (@proj2 (eval e t \in G)); elim: t => /=.
- move=> i; case: (leqP (size e) i) => [le_e_i | lt_i_e].
by rewrite !nth_default ?size_map ?morph1.
by rewrite (nth_map 1) // [_ \in G](allP Ge) ?mem_nth.
- by rewrite morph1.
- by move=> t [Gt ->]; rewrite groupV morphV.
- by move=> t [Gt ->] n; rewrite groupX ?morphX.
- by move=> t1 [Gt1 ->] t2 [Gt2 ->]; rewrite groupM ?morphM.
- by move=> t1 [Gt1 ->] t2 [Gt2 ->]; rewrite groupJ ?morphJ.
by move=> t1 [Gt1 ->] t2 [Gt2 ->]; rewrite groupR ?morphR.
have and_relE xT x1 x2 r: @and_rel xT x1 x2 r = (x1 == x2) && r :> bool.
by case: r => //=; rewrite andbT.
have rsatG e f: all [in G] e -> rel e f NoRel -> rel (map h e) f NoRel.
move=> Ge; have: NoRel -> NoRel by []; move: NoRel {2 4}NoRel.
elim: f => [x1 x2 | f1 IH1 f2 IH2] r hr IHr; last by apply: IH1; apply: IH2.
by rewrite !and_relE !evalG //; case/andP; move/eqP->; rewrite eqxx.
set s := env1; set vT := gT : finType in s *.
set s' := env1; set vT' := rT : finType in s' *.
have (v): let: Env A e := s v in
A \subset G -> all [in G] e /\ exists v', s' v' = Env (h @* A) (map h e).
- rewrite /= cycle_subG andbT => Gv; rewrite morphim_cycle //.
by split; last exists (h v).
elim: p 1%N vT vT' s s' => /= [p IHp | f] n vT vT
...
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
homGrp_trans
| |
eq_homGrpgT rT (G : {group gT}) (H : {group rT}) p :
G \isog H -> (G \homg Grp p) = (H \homg Grp p).
Proof.
by rewrite isogEhom => /andP[homGH homHG]; apply/idP/idP; apply: homGrp_trans.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
eq_homGrp
| |
isoGrp_transgT rT (G : {group gT}) (H : {group rT}) p :
G \isog H -> H \isog Grp p -> G \isog Grp p.
Proof. by move=> isoGH isoHp kT K; rewrite -isoHp; apply: eq_homgr. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
isoGrp_trans
| |
intro_isoGrpgT (G : {group gT}) p :
G \homg Grp p -> (forall rT (H : {group rT}), H \homg Grp p -> H \homg G) ->
G \isog Grp p.
Proof.
move=> homGp freeG rT H.
by apply/idP/idP=> [homHp|]; [apply: homGrp_trans homGp | apply: freeG].
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq",
"From mathcomp Require Import fintype finset fingroup morphism"
] |
fingroup/presentation.v
|
intro_isoGrp
| |
H:= <<A>>.
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
H
| |
coset_range:= [pred B in rcosets H 'N(A)].
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_range
| |
coset_of: Type :=
Coset { set_of_coset :> GroupSet.sort gT; _ : coset_range set_of_coset }.
HB.instance Definition _ := [isSub for set_of_coset].
#[hnf] HB.instance Definition _ := [Finite of coset_of by <:].
|
Record
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_of
| |
coset_one_proof: coset_range H.
Proof. by apply/rcosetsP; exists (1 : gT); rewrite (group1, mulg1). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_one_proof
| |
coset_one:= Coset coset_one_proof.
Let nNH := subsetP (norm_gen A).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_one
| |
coset_range_mul(B C : coset_of) : coset_range (B * C).
Proof.
case: B C => _ /= /rcosetsP[x Nx ->] [_ /= /rcosetsP[y Ny ->]].
by apply/rcosetsP; exists (x * y); rewrite !(groupM, rcoset_mul, nNH).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_range_mul
| |
coset_mulB C := Coset (coset_range_mul B C).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_mul
| |
coset_range_inv(B : coset_of) : coset_range B^-1.
Proof.
case: B => _ /= /rcosetsP[x Nx ->]; rewrite norm_rlcoset ?nNH // invg_lcoset.
by apply/rcosetsP; exists x^-1; rewrite ?groupV.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_range_inv
| |
coset_invB := Coset (coset_range_inv B).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_inv
| |
coset_mulP: associative coset_mul.
Proof. by move=> B C D; apply: val_inj; rewrite /= mulgA. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_mulP
| |
coset_oneP: left_id coset_one coset_mul.
Proof.
case=> B coB; apply: val_inj => /=; case/rcosetsP: coB => x Hx ->{B}.
by rewrite mulgA mulGid.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_oneP
| |
coset_invP: left_inverse coset_one coset_inv coset_mul.
Proof.
case=> B coB; apply: val_inj => /=; case/rcosetsP: coB => x Hx ->{B}.
rewrite invg_rcoset -mulgA (mulgA H) mulGid.
by rewrite norm_rlcoset ?nNH // -lcosetM mulVg mul1g.
Qed.
HB.instance Definition _ :=
Finite_isGroup.Build coset_of coset_mulP coset_oneP coset_invP.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_invP
| |
cosetx : coset_of := insubd (1 : coset_of) (H :* x).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset
| |
val_coset_primx : x \in 'N(A) -> coset x :=: H :* x.
Proof.
by move=> Nx; rewrite val_insubd /= mem_rcosets -{1}(mul1g x) mem_mulg.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
val_coset_prim
| |
coset_morphM: {in 'N(A) &, {morph coset : x y / x * y}}.
Proof.
move=> x y Nx Ny; apply: val_inj.
by rewrite /= !val_coset_prim ?groupM //= rcoset_mul ?nNH.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_morphM
| |
coset_morphism:= Morphism coset_morphM.
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_morphism
| |
ker_coset_prim: 'ker coset = 'N_H(A).
Proof.
apply/setP=> z; rewrite !in_setI andbC 2!inE -val_eqE /=.
case Nz: (z \in 'N(A)); rewrite ?andbF ?val_coset_prim // !andbT.
by apply/eqP/idP=> [<-| Az]; rewrite (rcoset_refl, rcoset_id).
Qed.
Implicit Type xbar : coset_of.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
ker_coset_prim
| |
coset_memy xbar : y \in xbar -> coset y = xbar.
Proof.
case: xbar => /= Hx NHx Hxy; apply: val_inj=> /=.
case/rcosetsP: NHx (NHx) Hxy => x Nx -> NHx Hxy.
by rewrite val_insubd /= (rcoset_eqP Hxy) NHx.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_mem
| |
mem_repr_cosetxbar : repr xbar \in xbar.
Proof. by case: xbar => /= _ /rcosetsP[x _ ->]; apply: mem_repr_rcoset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
mem_repr_coset
| |
repr_coset1: repr (1 : coset_of) = 1.
Proof. exact: repr_group. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
repr_coset1
| |
coset_reprK: cancel (fun xbar => repr xbar) coset.
Proof. by move=> xbar; apply: coset_mem (mem_repr_coset xbar). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_reprK
| |
cosetPxbar : {x | x \in 'N(A) & xbar = coset x}.
Proof.
pose x := repr 'N_xbar(A).
have [xbar_x Nx]: x \in xbar /\ x \in 'N(A).
apply/setIP; rewrite {}/x; case: xbar => /= _ /rcosetsP[y Ny ->].
by apply: (mem_repr y); rewrite inE rcoset_refl.
by exists x; last rewrite (coset_mem xbar_x).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetP
| |
coset_idx : x \in A -> coset x = 1.
Proof. by move=> Ax; apply: coset_mem; apply: mem_gen. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_id
| |
im_coset: coset @* 'N(A) = setT.
Proof.
by apply/setP=> xbar; case: (cosetP xbar) => x Nx ->; rewrite inE mem_morphim.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
im_coset
| |
sub_im_coset(C : {set coset_of}) : C \subset coset @* 'N(A).
Proof. by rewrite im_coset subsetT. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
sub_im_coset
| |
cosetpre_properC D :
(coset @*^-1 C \proper coset @*^-1 D) = (C \proper D).
Proof. by rewrite morphpre_proper ?sub_im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre_proper
| |
quotient: {set coset_of} := coset @* Q.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient
| |
quotientE: quotient = coset @* Q. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientE
| |
quotient_groupG A : {group coset_of A} :=
Eval hnf in [group of G / A].
Infix "/" := quotient_group : Group_scope.
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_group
| |
val_cosetx : x \in 'N(H) -> coset H x :=: H :* x.
Proof. by move=> Nx; rewrite val_coset_prim // genGid. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
val_coset
| |
coset_defaultx : (x \in 'N(H)) = false -> coset H x = 1.
Proof.
move=> Nx; apply: val_inj.
by rewrite val_insubd /= mem_rcosets /= genGid mulSGid ?normG ?Nx.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_default
| |
coset_normxbar : xbar \subset 'N(H).
Proof.
case: xbar => /= _ /rcosetsP[x Nx ->].
by rewrite genGid mul_subG ?sub1set ?normG.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_norm
| |
ker_coset: 'ker (coset H) = H.
Proof. by rewrite ker_coset_prim genGid (setIidPl _) ?normG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
ker_coset
| |
coset_idrx : x \in 'N(H) -> coset H x = 1 -> x \in H.
Proof. by move=> Nx Hx1; rewrite -ker_coset mem_morphpre //= Hx1 set11. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_idr
| |
repr_coset_normxbar : repr xbar \in 'N(H).
Proof. exact: subsetP (coset_norm _) _ (mem_repr_coset _). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
repr_coset_norm
| |
imset_cosetG : coset H @: G = G / H.
Proof.
apply/eqP; rewrite eqEsubset andbC imsetS ?subsetIr //=.
apply/subsetP=> _ /imsetP[x Gx ->].
by case Nx: (x \in 'N(H)); rewrite ?(coset_default Nx) ?mem_morphim ?group1.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
imset_coset
| |
val_quotientA : val @: (A / H) = rcosets H 'N_A(H).
Proof.
apply/setP=> B; apply/imsetP/rcosetsP=> [[xbar Axbar]|[x /setIP[Ax Nx]]] ->{B}.
case/morphimP: Axbar => x Nx Ax ->{xbar}.
by exists x; [rewrite inE Ax | rewrite /= val_coset].
by exists (coset H x); [apply/morphimP; exists x | rewrite /= val_coset].
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
val_quotient
| |
card_quotient_subnormA : #|A / H| = #|'N_A(H) : H|.
Proof. by rewrite -(card_imset _ val_inj) val_quotient. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
card_quotient_subnorm
| |
leq_quotientA : #|A / H| <= #|A|.
Proof. exact: leq_morphim. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
leq_quotient
| |
ltn_quotientA : H :!=: 1 -> H \subset A -> #|A / H| < #|A|.
Proof.
by move=> ntH sHA; rewrite ltn_morphim // ker_coset (setIidPr sHA) proper1G.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
ltn_quotient
| |
card_quotientA : A \subset 'N(H) -> #|A / H| = #|A : H|.
Proof. by move=> nHA; rewrite card_quotient_subnorm (setIidPl nHA). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
card_quotient
| |
divg_normalG : H <| G -> #|G| %/ #|H| = #|G / H|.
Proof. by case/andP=> sHG nHG; rewrite divgS ?card_quotient. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
divg_normal
| |
coset1: coset H 1 :=: H.
Proof. by rewrite morph1 /= genGid. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset1
| |
cosetpre1: coset H @*^-1 1 = H.
Proof. by rewrite -kerE ker_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpre1
| |
im_quotient: 'N(H) / H = setT.
Proof. exact: im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
im_quotient
| |
quotientT: setT / H = setT.
Proof. by rewrite -im_quotient; apply: morphimT. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientT
| |
quotientInormA : 'N_A(H) / H = A / H.
Proof. by rewrite /quotient setIC morphimIdom. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientInorm
| |
quotient_setIpreA D : (A :&: coset H @*^-1 D) / H = A / H :&: D.
Proof. exact: morphim_setIpre. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_setIpre
| |
mem_quotientx G : x \in G -> coset H x \in G / H.
Proof. by move=> Gx; rewrite -imset_coset imset_f. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
mem_quotient
| |
quotientSA B : A \subset B -> A / H \subset B / H.
Proof. exact: morphimS. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientS
| |
quotient0: set0 / H = set0.
Proof. exact: morphim0. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient0
| |
quotient_set1x : x \in 'N(H) -> [set x] / H = [set coset H x].
Proof. exact: morphim_set1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_set1
| |
quotient1: 1 / H = 1.
Proof. exact: morphim1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient1
| |
quotientVA : A^-1 / H = (A / H)^-1.
Proof. exact: morphimV. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientV
| |
quotientMlA B : A \subset 'N(H) -> A * B / H = (A / H) * (B / H).
Proof. exact: morphimMl. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientMl
| |
quotientMrA B : B \subset 'N(H) -> A * B / H = (A / H) * (B / H).
Proof. exact: morphimMr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientMr
| |
cosetpreMC D : coset H @*^-1 (C * D) = coset H @*^-1 C * coset H @*^-1 D.
Proof. by rewrite morphpreMl ?sub_im_coset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
cosetpreM
| |
quotientJA x : x \in 'N(H) -> A :^ x / H = (A / H) :^ coset H x.
Proof. exact: morphimJ. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientJ
| |
quotientUA B : (A :|: B) / H = A / H :|: B / H.
Proof. exact: morphimU. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientU
| |
quotientIA B : (A :&: B) / H \subset A / H :&: B / H.
Proof. exact: morphimI. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientI
| |
quotientYA B :
A \subset 'N(H) -> B \subset 'N(H) -> (A <*> B) / H = (A / H) <*> (B / H).
Proof. exact: morphimY. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientY
| |
quotient_homgA : A \subset 'N(H) -> homg (A / H) A.
Proof. exact: morphim_homg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotient_homg
| |
coset_kerlx y : x \in H -> coset H (x * y) = coset H y.
Proof.
move=> Hx; case Ny: (y \in 'N(H)); first by rewrite mkerl ?ker_coset.
by rewrite !coset_default ?groupMl // (subsetP (normG H)).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_kerl
| |
coset_kerrx y : y \in H -> coset H (x * y) = coset H x.
Proof.
move=> Hy; case Nx: (x \in 'N(H)); first by rewrite mkerr ?ker_coset.
by rewrite !coset_default ?groupMr // (subsetP (normG H)).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
coset_kerr
| |
rcoset_kercosetPx y :
x \in 'N(H) -> y \in 'N(H) -> reflect (coset H x = coset H y) (x \in H :* y).
Proof. by rewrite -{6}ker_coset; apply: rcoset_kerP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
rcoset_kercosetP
| |
kercoset_rcosetx y :
x \in 'N(H) -> y \in 'N(H) ->
coset H x = coset H y -> exists2 z, z \in H & x = z * y.
Proof. by move=> Nx Ny eqfxy; rewrite -ker_coset; apply: ker_rcoset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
kercoset_rcoset
| |
quotientGIG A : H \subset G -> (G :&: A) / H = G / H :&: A / H.
Proof. by rewrite -{1}ker_coset; apply: morphimGI. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype prime finset fingroup morphism",
"From mathcomp Require Import automorphism"
] |
fingroup/quotient.v
|
quotientGI
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.