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Require Export Setoid.
Set Implicit Arguments.
Section feq.
Variables A B:Type.
Definition feq (f g: A -> B):=forall a, (f a)=(g a).
Infix "=f":= feq (at level 80, right associativity).
Hint Unfold feq.
Lemma feq_refl: forall f, f =f f.
intuition.
Qed.
Lemma feq_sym: forall f g, f =f g-> g =f f.
intuition.
Qed.
Lemma feq_trans: forall f g h, f =f g-> g =f h -> f =f h.
unfold feq. intuition.
rewrite H.
auto.
Qed.
End feq.
Infix "=f":= feq (at level 80, right associativity).
#[export] Hint Unfold feq. #[export] Hint Resolve feq_refl feq_sym feq_trans.
Parameter K:(nat -> nat)->Prop.
Parameter K_ext:forall a b, (K a)->(a =f b)->(K b).
Add Parametric Relation (A B : Type) : (A -> B) (@feq A B)
reflexivity proved by (@feq_refl A B)
symmetry proved by (@feq_sym A B)
transitivity proved by (@feq_trans A B) as funsetoid.
Add Morphism K with signature (@feq nat nat) ==> iff as K_ext1.
intuition. apply (K_ext H0 H).
intuition. assert (y =f x);auto. apply (K_ext H0 H1).
Qed.
Lemma three:forall n, forall a, (K a)->(a =f (fun m => (a (n+m))))-> (K (fun m
=> (a (n+m)))).
intuition.
setoid_rewrite <- H0.
assumption.
Qed.
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