Section MirrorTape.
Variable (n : nat) (sig : Type).
Definition mirror_tape (t : tape sig) : tape sig :=
match t with
| niltape _ => niltape _
| leftof r rs => rightof r rs
| rightof l ls => leftof l ls
| midtape ls m rs => midtape rs m ls
end.
Definition mirror_move (D : move) : move := match D with | Nmove => Nmove | Lmove => Rmove | Rmove => Lmove end.
(* Realisation plus termination in constant time *)
Definition RealiseIn n (F : Type) (pM : pTM F n) (R : pRel sig F n) (k : nat) :=
forall input, exists outc, loopM (initc (projT1 pM) input) k = Some outc /\ R input ((projT2 pM (cstate outc)), ctapes outc).
Notation "M '⊨c(' k ')' R" := (RealiseIn M R k) (no associativity, at level 45, format "M  '⊨c(' k ')'  R").

Lemma RealiseIn_monotone n (F : Type) (pM : pTM F n) (R R' : pRel sig F n) k k' :
pM ⊨c(k') R' -> k' <= k -> R' <<=2 R -> pM ⊨c(k) R.