Canonical group_set_baseGroupType :=
Eval hnf in BaseFinGroupType (set_type gT) group_set_baseGroupMixin.
Canonical group_set_of_baseGroupType :=
Eval hnf in [baseFinGroupType of {set gT}].
End BaseSetMulDef.
(* Time to open the bag of dirty tricks. When we define groups down below *)
(* as a subtype of {set gT}, we need them to be able to coerce to sets in *)
(* both set-style contexts (x \in G) and monoid-style contexts (G * H),   *)
(* and we need the coercion function to be EXACTLY the structure          *)
(* projection in BOTH cases -- otherwise the canonical unification breaks.*)
(*   Alas, Coq doesn't let us use the same coercion function twice, even  *)
(* when the targets are convertible. Our workaround (ab)uses the module   *)
(* system to declare two different identity coercions on an alias class.  *)
Module GroupSet.
Definition sort (gT : baseFinGroupType) := {set gT}.
End GroupSet.
Identity Coercion GroupSet_of_sort : GroupSet.sort >-> set_of.
Module Type GroupSetBaseGroupSig.
Definition sort gT := group_set_of_baseGroupType gT : Type.
End GroupSetBaseGroupSig.
Module MakeGroupSetBaseGroup (Gset_base : GroupSetBaseGroupSig).
Identity Coercion of_sort : Gset_base.sort >-> FinGroup.arg_sort.
End MakeGroupSetBaseGroup.
Module Export GroupSetBaseGroup := MakeGroupSetBaseGroup GroupSet.
Canonical group_set_eqType gT := Eval hnf in [eqType of GroupSet.sort gT].
Canonical group_set_choiceType gT :=
Eval hnf in [choiceType of GroupSet.sort gT].
Canonical group_set_countType gT := Eval hnf in [countType of GroupSet.sort gT].
Canonical group_set_finType gT := Eval hnf in [finType of GroupSet.sort gT].
Section GroupSetMulDef.
(* Some of these constructs could be defined on a baseFinGroupType. *)
(* We restrict them to proper finGroupType because we only develop  *)
(* the theory for that case.                                        *)
Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Type x y : gT.
Definition lcoset A x := mulg x @: A.
Definition rcoset A x := mulg^~ x @: A.
Definition lcosets A B := lcoset A @: B.
Definition rcosets A B := rcoset A @: B.
Definition indexg B A := #|rcosets A B|.
Definition conjugate A x := conjg^~ x @: A.
Definition conjugates A B := conjugate A @: B.
Definition class x B := conjg x @: B.
Definition classes A := class^~ A @: A.
Definition class_support A B := conjg @2: (A, B).
Definition commg_set A B := commg @2: (A, B).
(* These will only be used later, but are defined here so that we can *)
(* keep all the Notation together.                                    *)
Definition normaliser A := [set x | conjugate A x \subset A].
Definition centraliser A := \bigcap_(x in A) normaliser [set x].
Definition abelian A := A \subset centraliser A.
Definition normal A B := (A \subset B) && (B \subset normaliser A).
(* "normalised" and "centralise[s|d]" are intended to be used with   *)
(* the {in ...} form, as in abelian below.                           *)
Definition normalised A := forall x, conjugate A x = A.
Definition centralises x A := forall y, y \in A -> commute x y.
Definition centralised A := forall x, centralises x A.
End GroupSetMulDef.
Arguments lcoset _ _%g _%g.
Arguments rcoset _ _%g _%g.
Arguments rcosets _ _%g _%g.
Arguments lcosets _ _%g _%g.
Arguments indexg _ _%g _%g.
Arguments conjugate _ _%g _%g.
Arguments conjugates _ _%g _%g.
Arguments class _ _%g _%g.
Arguments classes _ _%g.
Arguments class_support _ _%g _%g.
Arguments commg_set _ _%g _%g.
Arguments normaliser _ _%g.
Arguments centraliser _ _%g.
Arguments abelian _ _%g.
Arguments normal _ _%g _%g.
Arguments normalised _ _%g.
Arguments centralises _ _%g _%g.
Arguments centralised _ _%g.
Notation "[ 1 gT ]" := (1 : {set gT}) : group_scope.
Notation "[ 1 ]" := [1 FinGroup.sort _] : group_scope.
Notation "A ^#" := (A :\ 1) : group_scope.
Notation "x *: A" := ([set x%g] * A) : group_scope.
Notation "A :* x" := (A * [set x%g]) : group_scope.
Notation "A :^ x" := (conjugate A x) : group_scope.
Notation "x ^: B" := (class x B) : group_scope.
Notation "A :^: B" := (conjugates A B) : group_scope.
Notation "#| B : A |" := (indexg B A) : group_scope.
(* No notation for lcoset and rcoset, which are to be used mostly  *)
(* in curried form; x *: B and A :* 1 denote singleton products,   *)
(* so we can use mulgA, mulg1, etc, on, say, A :* 1 * B :* x.      *)
(* No notation for the set commutator generator set commg_set.     *)
Notation "''N' ( A )" := (normaliser A) : group_scope.
Notation "''N_' G ( A )" := (G%g :&: 'N(A)) : group_scope.
Notation "A <| B" := (normal A B) : group_scope.
Notation "''C' ( A )" := (centraliser A) : group_scope.
Notation "''C_' G ( A )" := (G%g :&: 'C(A)) : group_scope.
Notation "''C_' ( G ) ( A )" := 'C_G(A) (only parsing) : group_scope.
Notation "''C' [ x ]" := 'N([set x%g]) : group_scope.
Notation "''C_' G [ x ]" := 'N_G([set x%g]) : group_scope.
Notation "''C_' ( G ) [ x ]" := 'C_G[x] (only parsing) : group_scope.
Prenex Implicits repr lcoset rcoset lcosets rcosets normal.
Prenex Implicits conjugate conjugates class classes class_support.
Prenex Implicits commg_set normalised centralised abelian.
Section BaseSetMulProp.
(* Properties of the purely multiplicative structure. *)
Variable gT : baseFinGroupType.
Implicit Types A B C D : {set gT}.
Implicit Type x y z : gT.
(* Set product. We already have all the pregroup identities, so we *)
(* only need to add the monotonicity rules.                        *)

Lemma mulsgP A B x :
reflect (imset2_spec mulg (mem A) (fun _ => mem B) x) (x \in A * B).