Section Essais.
(* Parametrized Setoid *)
Parameter K : Type -> Type.
Parameter equiv : forall A : Type, K A -> K A -> Prop.
Parameter equiv_refl : forall (A : Type) (x : K A), equiv x x.
Parameter equiv_sym : forall (A : Type) (x y : K A), equiv x y -> equiv y x.
Parameter equiv_trans : forall (A : Type) (x y z : K A), equiv x y -> equiv y z
-> equiv x z.
(* basic operations *)
Parameter val : forall A : Type, A -> K A.
Parameter bind : forall A B : Type, K A -> (A -> K B) -> K B.
Parameter
bind_compat :
forall (A B : Type) (m1 m2 : K A) (f1 f2 : A -> K B),
equiv m1 m2 ->
(forall x : A, equiv (f1 x) (f2 x)) -> equiv (bind m1 f1) (bind m2 f2).
(* monad axioms *)
Parameter
bind_val_l :
forall (A B : Type) (a : A) (f : A -> K B), equiv (bind (val a) f) (f a).
Parameter
bind_val_r :
forall (A : Type) (m : K A), equiv (bind m (fun a => val a)) m.
Parameter
bind_assoc :
forall (A B C : Type) (m : K A) (f : A -> K B) (g : B -> K C),
equiv (bind (bind m f) g) (bind m (fun a => bind (f a) g)).
Hint Resolve equiv_refl equiv_sym equiv_trans: monad.
Add Parametric Relation A : (K A) (@equiv A)
reflexivity proved by (@equiv_refl A)
symmetry proved by (@equiv_sym A)
transitivity proved by (@equiv_trans A)
as equiv_rel.
Add Parametric Morphism A B : (@bind A B)
with signature (@equiv A) ==> (pointwise_relation A (@equiv B)) ==> (@equiv B)
as bind_mor.

Lemma test:
forall (A B: Type) (m1 m2 m3: K A) (f: A -> A -> K B),
(equiv m1 m2) -> (equiv m2 m3) ->
equiv (bind m1 (fun a => bind m2 (fun a' => f a a')))
(bind m2 (fun a => bind m3 (fun a' => f a a'))).