diff options
Diffstat (limited to 'theories/Program/FunctionalExtensionality.v')
| -rw-r--r-- | theories/Program/FunctionalExtensionality.v | 28 |
1 files changed, 17 insertions, 11 deletions
diff --git a/theories/Program/FunctionalExtensionality.v b/theories/Program/FunctionalExtensionality.v index 81bb5390b1..e890261e12 100644 --- a/theories/Program/FunctionalExtensionality.v +++ b/theories/Program/FunctionalExtensionality.v @@ -1,3 +1,4 @@ +(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) @@ -30,17 +31,6 @@ Proof. auto. Qed. -(** Eta expansion *) - -Axiom eta_expansion_dep : forall A (B : A -> Type) (f : forall x : A, B x), - f = fun x => f x. - -Lemma eta_expansion : forall A B (f : A -> B), - f = fun x => f x. -Proof. - intros ; apply eta_expansion_dep. -Qed. - (** Statements of functional equality for simple and dependent functions. *) Axiom fun_extensionality_dep : forall A, forall B : (A -> Type), @@ -63,6 +53,22 @@ Tactic Notation "extensionality" ident(x) := [ |- ?X = ?Y ] => apply (@fun_extensionality _ _ X Y) || apply (@fun_extensionality_dep _ _ X Y) ; intro x end. +(** Eta expansion follows from extensionality. *) + +Lemma eta_expansion_dep : forall A (B : A -> Type) (f : forall x : A, B x), + f = fun x => f x. +Proof. + intros. + extensionality x. + reflexivity. +Qed. + +Lemma eta_expansion : forall A B (f : A -> B), + f = fun x => f x. +Proof. + intros ; apply eta_expansion_dep. +Qed. + (** The two following lemmas allow to unfold a well-founded fixpoint definition without restriction using the functional extensionality axiom. *) |