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Diffstat (limited to 'test-suite/output/FunExt.v')
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diff --git a/test-suite/output/FunExt.v b/test-suite/output/FunExt.v new file mode 100644 index 00000000..7658ce71 --- /dev/null +++ b/test-suite/output/FunExt.v @@ -0,0 +1,168 @@ +Require Import FunctionalExtensionality. + +(* Basic example *) +Goal (forall x y z, x+y+z = z+y+x) -> (fun x y z => z+y+x) = (fun x y z => x+y+z). +intro H. +extensionality in H. +symmetry in H. +assumption. +Qed. + +(* Test rejection of non-equality *) +Goal forall H:(forall A:Prop, A), H=H -> forall H'':True, H''=H''. +intros H H' H''. +Fail extensionality in H. +clear H'. +Fail extensionality in H. +Fail extensionality in H''. +Abort. + +(* Test success on dependent equality *) +Goal forall (p : forall x, S x = x + 1), p = p -> S = fun x => x + 1. +intros p H. +extensionality in p. +assumption. +Qed. + +(* Test dependent functional extensionality *) +Goal forall (P:nat->Type) (Q:forall a, P a -> Type) (f g:forall a (b:P a), Q a b), + (forall x y, f x y = g x y) -> f = g. +intros * H. +extensionality in H. +assumption. +Qed. + +(* Other tests, courtesy of Jason Gross *) + +Goal forall A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c), (forall a b c, f a b c = g a b c) -> f = g. +Proof. + intros A B C D f g H. + extensionality in H. + match type of H with f = g => idtac end. + exact H. +Qed. + +Section test_section. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : forall a b c, f a b c = g a b c). + Goal f = g. + Proof. + extensionality in H. + match type of H with f = g => idtac end. + exact H. + Qed. +End test_section. + +Section test2. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : forall b a c, f a b c = g a b c). + Goal (fun b a c => f a b c) = (fun b a c => g a b c). + Proof. + extensionality in H. + match type of H with (fun b a => f a b) = (fun b' a' => g a' b') => idtac end. + exact H. + Qed. +End test2. + +Section test3. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : forall a c, (fun b => f a b c) = (fun b => g a b c)). + Goal (fun a c b => f a b c) = (fun a c b => g a b c). + Proof. + extensionality in H. + match type of H with (fun a c b => f a b c) = (fun a' c' b' => g a' b' c') => idtac end. + exact H. + Qed. +End test3. + +Section test4. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c -> Type) + (H : forall b, (forall a c d, f a b c d) = (forall a c d, g a b c d)). + Goal (fun b => forall a c d, f a b c d) = (fun b => forall a c d, g a b c d). + Proof. + extensionality in H. + exact H. + Qed. +End test4. + +Section test5. + Goal nat -> True. + Proof. + intro n. + Fail extensionality in n. + constructor. + Qed. +End test5. + +Section test6. + Goal let f := fun A (x : A) => x in let pf := fun A x => @eq_refl _ (f A x) in f = f. + Proof. + intros f pf. + extensionality in pf. + match type of pf with f = f => idtac end. + exact pf. + Qed. +End test6. + +Section test7. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : forall a b c, True -> f a b c = g a b c). + Goal True. + Proof. + extensionality in H. + match type of H with (fun a b c (_ : True) => f a b c) = (fun a' b' c' (_ : True) => g a' b' c') => idtac end. + constructor. + Qed. +End test7. + +Section test8. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : True -> forall a b c, f a b c = g a b c). + Goal True. + Proof. + extensionality in H. + match type of H with (fun (_ : True) => f) = (fun (_ : True) => g) => idtac end. + constructor. + Qed. +End test8. + +Section test9. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : forall b a c, f a b c = g a b c). + Goal (fun b a c => f a b c) = (fun b a c => g a b c). + Proof. + pose H as H'. + extensionality in H. + extensionality in H'. + let T := type of H in let T' := type of H' in constr_eq T T'. + match type of H with (fun b a => f a b) = (fun b' a' => g a' b') => idtac end. + exact H'. + Qed. +End test9. + +Section test10. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : f = g). + Goal True. + Proof. + Fail extensionality in H. + constructor. + Qed. +End test10. + +Section test11. + Context A B C (D : forall a : A, C a -> Type) (f g : forall a : A, B -> forall c : C a, D a c) + (H : forall a b c, f a b c = f a b c). + Goal True. + Proof. + pose H as H'. + pose (eq_refl : H = H') as e. + extensionality in H. + Fail extensionality in H'. + clear e. + extensionality in H'. + let T := type of H in let T' := type of H' in constr_eq T T'. + lazymatch type of H with f = f => idtac end. + constructor. + Qed. +End test11. |