Axioms.v

(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
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(* Selected axioms from Axioms.v in CompCert for Integers benchmark *)

(** This file collects some axioms used throughout the CompCert development. *)

Add LoadPath "coq".
Require ClassicalFacts.

(** * Extensionality axioms *)

(** The following [Require Export] gives us functional extensionality for dependent function types:
<<
Axiom functional_extensionality_dep : forall {A} {B : A -> Type}, 
  forall (f g : forall x : A, B x), 
  (forall x, f x = g x) -> f = g.
>> 
and, as a corollary, functional extensionality for non-dependent functions:
<<
Lemma functional_extensionality {A B} (f g : A -> B) : 
  (forall x, f x = g x) -> f = g.
>>
*)
Require Import FunctionalExtensionality.

(** We also assert propositional extensionality. *)

Axiom prop_ext: ClassicalFacts.prop_extensionality.
Arguments prop_ext : default implicits.

(** * Proof irrelevance *)

(** We also use proof irrelevance, which is a consequence of propositional
    extensionality. *)

Lemma proof_irr: ClassicalFacts.proof_irrelevance.
Proof.
  exact (ClassicalFacts.ext_prop_dep_proof_irrel_cic prop_ext).
Qed.
Arguments proof_irr : default implicits.