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(* *********************************************************************)
(* *)
(* 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 *)
(* under the terms of the GNU General Public License as published by *)
(* the Free Software Foundation, either version 2 of the License, or *)
(* (at your option) any later version. This file is also distributed *)
(* under the terms of the INRIA Non-Commercial License Agreement. *)
(* *)
(* *********************************************************************)
(** This file collects some axioms used throughout the CompCert development. *)
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 Export FunctionalExtensionality.
(** For compatibility with earlier developments, [extensionality]
is an alias for [functional_extensionality]. *)
Lemma extensionality:
forall (A B: Type) (f g : A -> B), (forall x, f x = g x) -> f = g.
Proof. intros; apply functional_extensionality. auto. Qed.
Implicit Arguments extensionality.
(** We also assert propositional extensionality. *)
Axiom prop_ext: ClassicalFacts.prop_extensionality.
Implicit Arguments prop_ext.
(** * 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.
Implicit Arguments proof_irr.
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