Definition duplicateIT i j k : IdiomaticTransformation (KSApplicative i k) 
  (ComposeApplicative (KSApplicative i j) (KSApplicative j k)).Definition dimIT i j : IdiomaticTransformation (KSApplicative i j) NatPlusApplicative.

Lemma pos_ap i j a b (f : KleeneStore i j (a -> b)) (x : KleeneStore i j a) : Vector.to_list (pos (KSap f x)) = app (Vector.to_list (pos f)) (Vector.to_list (pos x)).