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.

(* (* Version with disregarded labels *)
Lemma LiftTapes_Spec' (sig : finType) (F : Type) (m n : nat) (I : Vector.t (Fin.t n) m) (P : Spec sig n) (Q : Spec sig m) (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (Downlift I P)) pM (fun y => tspec Q) ->
Triple (tspec P) (LiftTapes pM  I) (fun _ => tspec (Frame I P Q)).