From Corelib Require Export ListDef.
From elpi.apps Require Export tc.
Global Generalizable All Variables.

Elpi TC Solver Override TC.Solver All.

Class ElemOf A B := elem_of: A -> B -> Prop.
Class Elements A C := elements: C -> list A.

Inductive elem_of_list {A} : ElemOf A (list A) :=
  | elem_of_list_here (x : A) l : elem_of x (x :: l)
  | elem_of_list_further (x y : A) l : elem_of x l -> elem_of x (y :: l).
Global Existing Instance elem_of_list.

Inductive NoDup {A} : list A -> Prop :=
  | NoDup_nil_2 : NoDup nil
  | NoDup_cons_2 x l : not (elem_of x l) -> NoDup l -> NoDup (x :: l).

Module A.
  Class FinSet1 A C `{ElemOf A C,Elements A C} : Prop := {
    NoDup_elements (X : C) : NoDup (elements X)
  }.
End A.

Module B.

  Class FinSet2 A C `{ElemOf A C, Elements A C} : Prop := {
    elem_of_elements2 (X : C) x : iff (elem_of x (elements X)) (elem_of x X);
    NoDup_elements2 (X : C) : NoDup (elements X)
  }.

End B.
