ladder-calculus/coq/FiniteSets.v

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(** A library for finite sets with extensional equality.
Author: Brian Aydemir. *)
Require Import FSets.
Require Import ListFacts.
Require Import AdditionalTactics.
Require AdditionalTactics.
(* *********************************************************************** *)
(** * Interface *)
(** The following interface wraps the standard library's finite set
interface with an additional property: extensional equality. *)
Module Type S.
Declare Module E : UsualOrderedType.
Declare Module F : FSetInterface.S with Module E := E.
Parameter eq_if_Equal :
forall s s' : F.t, F.Equal s s' -> s = s'.
End S.
(* *********************************************************************** *)
(** * Implementation *)
(** For documentation purposes, we hide the implementation of a
functor implementing the above interface. We note only that the
implementation here assumes (as an axiom) that proof irrelevance
holds. *)
Module Make (X : UsualOrderedType) <: S with Module E := X.
(* begin hide *)
Module E := X.
Module F := FSetList.Make E.
Module OFacts := OrderedType.OrderedTypeFacts E.
Axiom sort_F_E_lt_proof_irrel : forall xs (p q : sort F.E.lt xs), p = q.
Lemma eq_if_Equal :
forall s s' : F.t, F.Equal s s' -> s = s'.
Proof.
intros [s1 pf1] [s2 pf2] Eq.
assert (s1 = s2).
unfold F.MSet.Raw.t in *.
(* eapply Sort_InA_eq_ext; eauto.
intros; eapply E.lt_trans; eauto.
intros; eapply OFacts.lt_eq; eauto.
intros; eapply OFacts.eq_lt; eauto.
subst s1.
rewrite (sort_F_E_lt_proof_irrel _ pf1 pf2).
reflexivity.
Qed.
*)
Admitted.
(* end hide *)
End Make.