adamc@125: (* Copyright (c) 2008, Adam Chlipala
adamc@125:  * 
adamc@125:  * This work is licensed under a
adamc@125:  * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@125:  * Unported License.
adamc@125:  * The license text is available at:
adamc@125:  *   http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@125:  *)
adamc@125: 
adamc@125: (* Additional axioms not in the Coq standard library *)
adamc@125: 
adamc@173: Set Implicit Arguments.
adamc@173: 
adamc@125: Axiom ext_eq : forall (A : Type) (B : A -> Type)
adamc@125:   (f g : forall x, B x),
adamc@125:   (forall x, f x = g x)
adamc@125:   -> f = g.
adamc@125: 
adamc@173: Theorem ext_eq_Set : forall (A : Set) (B : A -> Set)
adamc@173:   (f g : forall x, B x),
adamc@173:   (forall x, f x = g x)
adamc@173:   -> f = g.
adamc@173:   intros.
adamc@173:   rewrite (ext_eq _ _ _ H); reflexivity.
adamc@173: Qed.
adamc@173: 
adamc@173: Theorem ext_eq_forall : forall (A : Type)
adamc@173:   (f g : A -> Set),
adamc@173:   (forall x, f x = g x)
adamc@173:   -> @eq Type (forall x, f x) (forall x, g x).
adamc@173:   intros.
adamc@173:   rewrite (ext_eq _ _ _ H); trivial.
adamc@173: Qed.
adamc@173: 
adamc@173: Ltac ext_eq := (apply ext_eq || apply ext_eq_Set
adamc@173:   || apply ext_eq_forall); intro.
adamc@190: 
adamc@190: 
adamc@190: Theorem eta : forall (A B : Type) (f : A -> B),
adamc@190:   (fun x => f x) = f.
adamc@190:   intros; ext_eq; trivial.
adamc@190: Qed.