(* * Reduction of the Halting Problem of the Heap Machine to the Halting Problem of Turing Machines *)
From Undecidability.TM.L Require Import Alphabets HeapInterpreter.StepTM.
Local Arguments plus : simpl never.
Local Arguments mult : simpl never.
(* Initialise the alphabet of the [Step] Machine *)
Definition sigStep : Type := sigList sigHClos + sigHeap.
Definition retr_heap_step : Retract sigHeap sigStep := _.
Definition retr_closures_step : Retract (sigList sigHClos) sigStep := _.
Definition Loop := While (Step retr_closures_step retr_heap_step).
Local Arguments step_fun : simpl never.
Local Arguments is_halt_state : simpl never.
Local Arguments Step_size : simpl never.
Fixpoint Loop_size (T V : list HClos) (H : Heap) (k : nat) {struct k} : Vector.t (nat->nat) 11 :=
match k with
| 0 => Step_size T V H
| S k' =>
match step_fun (T, V, H) with
| Some (T',V',H') =>
if is_halt_state (T',V',H')
then Step_size T V H >>> Step_size T' V' H'
else Step_size T V H >>> Loop_size T' V' H' k'
| _ => Step_size T V H
end
end.
Definition Loop_Rel : pRel sigStep^+ unit 11 :=
ignoreParam (
fun tin tout =>
forall (T V : list HClos) (H: Heap) (s0 s1 s2 : nat) (sr : Vector.t nat 8),
tin[@Fin0] ≃(;s0) T ->
tin[@Fin1] ≃(;s1) V ->
tin[@Fin2] ≃(;s2) H ->
(forall i : Fin.t 8, isVoid_size tin[@FinR 3 i] sr[@i]) ->
exists T' V' H' (k : nat),
let size := Loop_size T V H k in
steps_k k (T,V,H) (T',V',H') /\
halt_state (T',V',H') /\
match T' with
| nil =>
tout[@Fin0] ≃(; size @> Fin0 s0) @nil HClos /\
tout[@Fin1] ≃(; size @> Fin1 s1) V' /\
tout[@Fin2] ≃(; size @> Fin2 s2) H' /\
(forall i : Fin.t 8, isVoid_size tout[@FinR 3 i] (size @>(FinR 3 i) sr[@i]))
| _ => True
end
).
Fixpoint Loop_steps T V H k :=
match k with
| 0 => Step_steps T V H
| S k' =>
match step_fun (T, V, H) with
| Some (T',V',H') =>
if is_halt_state (T',V',H')
then 1 + Step_steps T V H + Step_steps T' V' H'
else 1 + Step_steps T V H + Loop_steps T' V' H' k'
| None => Step_steps T V H
end
end.
Definition Loop_T : tRel sigStep^+ 11 :=
fun tin i => exists T V H k,
halts_k (T,V,H) k /\
tin[@Fin0] ≃ T /\
tin[@Fin1] ≃ V /\
tin[@Fin2] ≃ H /\
(forall i : Fin.t 8, isVoid tin[@FinR 3 i]) /\
Loop_steps T V H k <= i.
Definition initTapes : state -> tapes sigStep^+ 11 :=
fun '(T,V,H) => initValue _ _ T ::: initValue _ _ V ::: initValue _ _ H ::: Vector.const (initRight _) 8.
(* This vernacular command checks wether we have indeed assumed no axioms. *)
(* Print Assumptions HaltingProblem. *)
(* << Closed under the global context >> *)

Lemma Interpreter_SpecT T V H k V' H' ss:
steps_k k (T,V,H) ([],V',H') -> halt_state ([],V',H') ->
TripleT
≃≃([],
withSpace [|Contains _ T;Contains _ V;Contains _ H;Void;Void;Void;Void;Void;Void;Void;Void|] ss)
(Loop_steps T V H k) Loop
(fun _ =>  ≃≃([],
withSpace [|Contains _ (@nil HClos);Contains _ V';Contains _ H';Void;Void;Void;Void;Void;Void;Void;Void|]
(appSize (Loop_size T V H k) ss))).