From Undecidability.Shared.Libs.PSL Require Import FinTypes Vectors.
From Undecidability.L Require Import Datatypes.Lists Tactics.LTactics Util.L_facts.
From Undecidability.TM  Require Import TM_facts ProgrammingTools .
From Undecidability.TM.L.CompilerBoolFuns Require Import Compiler_spec.
From Undecidability.TM Require Import Hoare.
Definition TM_bool_computable_hoare_in {k n Σ} s b (v: Vector.t (list bool) k): SpecV Σ (1+k+n)
:= ([Custom (eq niltape)] ++ Vector.map (fun bs => Custom (eq (encBoolsTM s b bs))) v) ++ Vector.const (Custom (eq niltape)) _.
Definition TM_bool_computable_hoare_out {k n Σ} s b (bs :list bool): SpecV Σ (1+k+n)
:= ([Custom (eq (encBoolsTM s b bs)) ] ++ Vector.const (Custom (fun _ => True)) _)++ Vector.const (Custom (fun _ => True)) _.
Definition TM_bool_computable_hoare {k} (R : Vector.t (list bool) k -> (list bool) -> Prop) :=
exists n : nat, exists Σ : finType, exists s b , s <> b /\
exists M : pTM (Σ^+) unit (1 + k + n),
forall v,
Triple ≃≃([],TM_bool_computable_hoare_in s b v) M (fun y t => exists res, t ≃≃ ([R v res]%list,TM_bool_computable_hoare_out s b res))
/\ forall res, R v res ->
exists k__steps, TripleT ≃≃([],TM_bool_computable_hoare_in s b v) k__steps M (fun y => ≃≃([]%list,TM_bool_computable_hoare_out s b res)).
(* * The tape-order is different than in the implemented machine, so here again with the tape-order implemented: *)
Definition TM_bool_computable_hoare_in' {k n Σ} s b (v: Vector.t (list bool) k): SpecV Σ (1+n+k)
:= Custom (eq niltape) :: Vector.const (Custom (eq niltape)) _ ++ Vector.map (fun bs => Custom (eq (encBoolsTM s b bs))) v.
Definition TM_bool_computable_hoare_out' {n Σ} s b (bs :list bool): SpecV Σ (1+n)
:= Custom (eq (encBoolsTM s b bs)) :: Vector.const (Custom (fun _ => True)) _.
Definition TM_bool_computable_hoare' {k} (R : Vector.t (list bool) k -> (list bool) -> Prop) :=
exists n : nat, exists Σ : finType, exists s b , s <> b /\
exists M : pTM (Σ^+) unit (1 + n + k),
forall v,
Triple ≃≃([],TM_bool_computable_hoare_in' s b v) M (fun y t => exists res, t ≃≃ ([R v res]%list,TM_bool_computable_hoare_out' s b res))
/\ forall res, R v res ->
exists k__steps, TripleT ≃≃([],TM_bool_computable_hoare_in' s b v) k__steps M (fun y => ≃≃([]%list,TM_bool_computable_hoare_out' s b res)).
Local Definition tapeOrderSwap n k:=
(Fin0 :::  tabulate (n := n) (fun x => Fin.R (1 + k) x) ++ (tabulate (n := k) (fun x => Fin.L n (Fin.R 1 x)))).
Local Lemma tapeOrderSwap_dupfree k n : VectorDupfree.dupfree (tapeOrderSwap n k).
Local Set Keyed Unification.

Lemma TM_bool_computable_hoare'_spec k R :
functional R ->
@TM_bool_computable_hoare' k R -> TM_bool_computable R.