diff options
Diffstat (limited to 'theories/Logic/FunctionalExtensionality.v')
| -rw-r--r-- | theories/Logic/FunctionalExtensionality.v | 32 |
1 files changed, 30 insertions, 2 deletions
diff --git a/theories/Logic/FunctionalExtensionality.v b/theories/Logic/FunctionalExtensionality.v index 7d7792d5..eb50a3aa 100644 --- a/theories/Logic/FunctionalExtensionality.v +++ b/theories/Logic/FunctionalExtensionality.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -19,6 +19,12 @@ Proof. auto. Qed. +Lemma equal_f_dep : forall {A B} {f g : forall (x : A), B x}, + f = g -> forall x, f x = g x. +Proof. +intros A B f g <- H; reflexivity. +Qed. + (** Statements of functional extensionality for simple and dependent functions. *) Axiom functional_extensionality_dep : forall {A} {B : A -> Type}, @@ -31,13 +37,35 @@ Proof. intros ; eauto using @functional_extensionality_dep. Qed. +(** Extensionality of [forall]s follows from functional extensionality. *) +Lemma forall_extensionality {A} {B C : A -> Type} (H : forall x : A, B x = C x) +: (forall x, B x) = (forall x, C x). +Proof. + apply functional_extensionality in H. destruct H. reflexivity. +Defined. + +Lemma forall_extensionalityP {A} {B C : A -> Prop} (H : forall x : A, B x = C x) +: (forall x, B x) = (forall x, C x). +Proof. + apply functional_extensionality in H. destruct H. reflexivity. +Defined. + +Lemma forall_extensionalityS {A} {B C : A -> Set} (H : forall x : A, B x = C x) +: (forall x, B x) = (forall x, C x). +Proof. + apply functional_extensionality in H. destruct H. reflexivity. +Defined. + (** Apply [functional_extensionality], introducing variable x. *) Tactic Notation "extensionality" ident(x) := match goal with [ |- ?X = ?Y ] => (apply (@functional_extensionality _ _ X Y) || - apply (@functional_extensionality_dep _ _ X Y)) ; intro x + apply (@functional_extensionality_dep _ _ X Y) || + apply forall_extensionalityP || + apply forall_extensionalityS || + apply forall_extensionality) ; intro x end. (** Eta expansion follows from extensionality. *) |