(* egalite' *)

Definition Equal_PA_mor (x y : PA_ob) (f g : PA_mor x y) :=
  match f in (PA_mor x' y') return Prop with
  | PA_I1 => 
      (* PA_I1  *)  True
      (* PA_I2  *) 
  | PA_I2 => True
      (* PA_f i *) 
  | PA_f i =>
      match g in (PA_mor x' y') return Prop with
      | PA_I1 => 
          (* PA_I1  *)  False
          (* PA_I2  *) 
      | PA_I2 => False
          (* PA_f j *) 
      | PA_f j => k i =_S k j
      end
  end.

Lemma Equal_PA_mor_Equiv : forall x y : PA_ob, Equivalence (Equal_PA_mor (x:=x) (y:=y)).