diff options
| -rw-r--r-- | theories/Logic/FunctionalExtensionality.v | 24 |
1 files changed, 23 insertions, 1 deletions
diff --git a/theories/Logic/FunctionalExtensionality.v b/theories/Logic/FunctionalExtensionality.v index a5894fc1e5..14d38c7fb7 100644 --- a/theories/Logic/FunctionalExtensionality.v +++ b/theories/Logic/FunctionalExtensionality.v @@ -37,13 +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. *) |