62 lines
1.5 KiB
Coq
62 lines
1.5 KiB
Coq
|
(** A library for finite sets with extensional equality.
|
||
|
|
|
||
|
|
Author: Brian Aydemir. *)
|
||
|
|
|
||
|
|
Require Import FSets.
|
||
|
|
Require Import ListFacts.
|
||
|
|
|
||
|
|
|
||
|
|
(* *********************************************************************** *)
|
||
|
|
(** * 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.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.
|
||
|
|
|
||
|
|
(* end hide *)
|
||
|
|
|
||
|
|
End Make.
|