Record IdiomaticTransformation (F G:Applicative) := idiomaticTransformation
{ transform :> forall a, F a -> G a
; _ : forall a (x:a), transform (pure F x) = pure G x
; _ : forall a b (f : F (a -> b)) x, transform (ap f x) = ap (transform f) (transform x)
}.
Section IdiomaticTransformation.
Variables (F G:Applicative).
Variable (eta : IdiomaticTransformation F G).

(* Specifications: *)
Lemma idiomaticTransform_pure a (x:a) : eta _ (pure F x) = pure G x.
Lemma idiomaticTransform_ap a b (f: F (a -> b)) (x: F a) :
eta _ (ap f x) = ap (eta _ f) (eta _ x).
Lemma idiomaticTransform_map a b (f: a -> b) (x: F a) :
eta _ (map f x) = map f (eta _ x).
Lemma idiomaticTransform_sequenceVector n a (v : Vector.t (F a) n) :
eta _ (sequenceVector v) = sequenceVector (Vector.map (eta _) v).
Lemma idiomaticTransform_traverseVector n a b (f : a -> F b) (v : Vector.t a n) :
eta _ (traverseVector f v) = traverseVector (fun y => eta _ (f y)) v.