Section Lifting.
Variable (sig : Type).
Variable (m n : nat).
(* [P] is the premise of the lifted machine [M@I]. *)
Variable (P : @SpecV sig n).
Variable (I : Vector.t (Fin.t n) m). (* [m<=n] *)
Hypothesis (HI : dupfree I).
(* We want to extract from [P] the premise [P'] for [M] *)
Definition Downlift : @SpecV sig m :=
(select I P).
(* Same specification as in [P] on indices not in [I], but as in [Q] for indices in [I] (lifted).  *)
Definition Frame (Q : @SpecV sig m) : @SpecV sig n := fill I P Q.
End Lifting.

Lemma LiftTapes_Spec_ex (sig : finType) X (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' P Q' Q (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',Downlift P I)) pM (fun y t => exists x:X, tspec (Q' x y,Q x y) t) ->
Triple (tspec (P',P)) (LiftTapes pM I) (fun y t=> exists x, tspec (Q' x y,Frame P I (Q x y)) t ).