Section M.
Variable (sig : finType).
(* Hypothesis (defX: inhabitedC sigX). *)
(* We use the FinType instance of bool, as it has a Case-machine *)
Context `{retr__list : Retract (sigList bool) sig}
`{retr__Pro : Retract Alphabets.sigPro sig}.
Local Instance retr__nat : Retract sigNat sig := ComposeRetract retr__Pro _.
Local Instance retr__bool : Retract bool sig := ComposeRetract retr__list (Retract_sigList_X _).
(* Check _ : codable sig (list bool). *)
(* Check _ : codable sig bool. *)
(* Tapes: 0: compiled and encoded bs (input) 1: result 2: intern (constant for ConcatRepeat [0]) 3: intern (length of bs for concatReepat [1]) *)
(* für step (prepend the bs-dependent symbols) Tapes: 0: bs (input) 1: result 2: head of bs 3: intern (length of bs for concatReepat [1]) *)
Definition M__step : pTM sig^+ (option unit) 3 :=
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;
CaseCom ⇑ ComposeRetract retr__Pro _ @ [|Fin2|];;
If (CaseNat ⇑ _ @ [|Fin2|])
(Return (Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|]
) (Some tt))
(Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;
CaseCom ⇑ ComposeRetract retr__Pro _ @ [|Fin2|];;
Switch (CaseNat ⇑ _ @ [|Fin2|])
(fun b => Cons_constant.M b ⇑ retr__list @[|Fin1|]);;
Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
CaseList _ ⇑ retr__Pro @ [|Fin0;Fin2|];;Reset _ @ [|Fin2|];;
Return Nop (None)
).

Lemma SpecT__step :
{ f : UpToC (fun bs => 1) &
forall (bs res :list bool) (n:nat),
TripleT
≃≃([],[|Contains _ (flat_map enc_bool_perElem bs ++ enc_bool_nil ++ concat (repeat enc_bool_closing n));
Contains _ res;Void|])
(f bs)
M__step
(fun y => ≃≃([y = match bs with [] => Some tt | _ => None end]
,match bs with
| [] => [| Contains _ (concat (repeat enc_bool_closing (n + length bs)));Contains _ res;Void |]
| b::bs' => [|Contains _ (flat_map enc_bool_perElem bs' ++ enc_bool_nil ++ concat (repeat enc_bool_closing n));Contains _ (b::res);Void|]
end)) }.