(* * Implementation of [ϕ] (aka SplitBody) *)
From Undecidability.TM.L Require Import Alphabets CaseCom.
Local Arguments plus : simpl never.
Local Arguments mult : simpl never.
(* The [JumpTarget] machine only operates on programs. Thus we define [JumpTarget] on the alphabet [sigPro^+]. *)
(* This is the only way we can encode [nat] on [sigPro]: as a variable token. *)
Definition retr_nat_prog : Retract sigNat sigPro := Retract_sigList_X _.
(* append a token to the token list *)
Definition App_ACom (t : ACom) : pTM sigPro^+ unit 2 :=
WriteValue [ACom2Com t] @ [|Fin1|];;
App _.
Definition App_ACom_size (t : ACom) (Q : list Tok) : Vector.t (nat->nat) 2 :=
[| id; WriteValue_size [ACom2Com t] |] >>> App_size Q [ACom2Com t].
Definition App_ACom_Rel (t : ACom) : pRel sigPro^+ unit 2 :=
ignoreParam (
fun tin tout =>
forall (Q : list Tok) (s0 s1:nat),
tin[@Fin0] ≃(;s0) Q ->
isVoid_size tin[@Fin1] s1 ->
tout[@Fin0] ≃(;App_ACom_size t Q @>Fin0 s0) Q ++ [ACom2Com t] /\
isVoid_size tout[@Fin1] (App_ACom_size t Q @>Fin1 s1)
).
Arguments App_ACom_size : simpl never.
Definition App_ACom_steps (Q: Pro) (t: ACom) := 1 + WriteValue_steps (size [ACom2Com t]) + App_steps Q [ACom2Com t].
Definition App_ACom_T (t: ACom) : tRel sigPro^+ 2 :=
fun tin k => exists (Q: list Tok), tin[@Fin0] ≃ Q /\ isVoid tin[@Fin1] /\ App_ACom_steps Q t <= k.
(* Add a singleton list of tokes to [Q] *)
Definition App_Com : pTM sigPro^+ (FinType(EqType unit)) 3 :=
Constr_nil _ @ [|Fin2|];;
Constr_cons _@ [|Fin2; Fin1|];;
App _ @ [|Fin0; Fin2|];;
Reset _ @ [|Fin1|].
Definition App_Com_size (Q : list Tok) (t : Tok) : Vector.t (nat->nat) 3 :=
[| App_size Q [t] @>Fin0; Reset_size t; pred >> Constr_cons_size t >> App_size Q [t] @>Fin1 |].
Definition App_Com_Rel : pRel sigPro^+ (FinType(EqType unit)) 3 :=
ignoreParam (
fun tin tout =>
forall (Q : list Tok) (t : Tok) (s0 s1 s2 : nat),
tin[@Fin0] ≃(;s0) Q ->
tin[@Fin1] ≃(;s1) t ->
isVoid_size tin[@Fin2] s2 ->
tout[@Fin0] ≃(;App_Com_size Q t @>Fin0 s0) Q ++ [t] /\
isVoid_size tout[@Fin1] (App_Com_size Q t @>Fin1 s1) /\
isVoid_size tout[@Fin2] (App_Com_size Q t @>Fin2 s2)
).
Arguments App_Com_size : simpl never.
Definition App_Com_steps (Q: Pro) (t:Tok) :=
3 + Constr_nil_steps + Constr_cons_steps t + App_steps Q [t] + Reset_steps t.
Definition App_Com_T : tRel sigPro^+ 3 :=
fun tin k => exists (Q: list Tok) (t: Tok), tin[@Fin0] ≃ Q /\ tin[@Fin1] ≃ t /\ isVoid tin[@Fin2] /\ App_Com_steps Q t <= k.
Definition JumpTarget_Step : pTM sigPro^+ (option bool) 5 :=
If (CaseList sigCom_fin @ [|Fin0; Fin3|])
(Switch (ChangeAlphabet CaseCom _ @ [|Fin3|])
(fun t : option ACom =>
match t with
| Some retAT =>
If (CaseNat ⇑ retr_nat_prog @ [|Fin2|])
(Return (App_ACom retAT @ [|Fin1; Fin4|]) None) (* continue *)
(Return (ResetEmpty1 _ @ [|Fin2|]) (Some true)) (* return true *)
| Some lamAT =>
Return (Constr_S ⇑ retr_nat_prog @ [|Fin2|];;
App_ACom lamAT @ [|Fin1; Fin4|])
None (* continue *)
| Some appAT =>
Return (App_ACom appAT @ [|Fin1;Fin4|])
None (* continue *)
| None => (* Variable *)
Return (Constr_varT ⇑ _ @ [|Fin3|];;
App_Com @ [|Fin1; Fin3; Fin4|])
None (* continue *)
end))
(Return Nop (Some false)) (* return false *)
.
Definition JumpTarget_Step_size (P Q : Pro) (k : nat) : Vector.t (nat->nat) 5 :=
match P with
| retT :: _ =>
match k with
| S _ =>
[| (*0*) CaseList_size0 retT; (*1*) App_ACom_size retAT Q @>Fin0; (*2*) S;                (*3*) CaseList_size1 retT >> CaseCom_size retT; (*4*) App_ACom_size retAT Q @>Fin1 |]
| 0 =>
[| (*0*) CaseList_size0 retT; (*1*) id;                           (*2*) ResetEmpty1_size; (*3*) CaseList_size1 retT >> CaseCom_size retT; (*4*) id |]
end
| lamT :: _ =>
[| (*0*) CaseList_size0 lamT; (*1*) App_ACom_size lamAT Q @>Fin0; (*2*) pred; (*3*) CaseList_size1 lamT >> CaseCom_size lamT; (*4*) App_ACom_size lamAT Q @>Fin1 |]
| appT :: _ =>
[| (*0*) CaseList_size0 appT; (*1*) App_ACom_size appAT Q @>Fin0; (*2*) id; (*3*) CaseList_size1 retT >> CaseCom_size appT; (*4*) App_ACom_size appAT Q @>Fin1 |]
| varT n :: _ =>
[| (*0*) CaseList_size0 (varT n); (*1*) App_Com_size Q (varT n) @>Fin0; (*2*) id; (*3*) CaseList_size1 (varT n) >> CaseCom_size (varT n) >> pred >> App_Com_size Q (varT n) @>Fin1; (*4*) App_Com_size Q (varT n) @>Fin2 |]
| nil => Vector.const id _
end.
Definition JumpTarget_Step_Rel : pRel sigPro^+ (option bool) 5 :=
fun tin '(yout, tout) =>
forall (P Q : Pro) (k : nat) (s0 s1 s2 s3 s4 : nat),
let size := JumpTarget_Step_size P Q k in
tin[@Fin0] ≃(;s0) P ->
tin[@Fin1] ≃(;s1) Q ->
tin[@Fin2] ≃(;s2) k ->
isVoid_size tin[@Fin3] s3 -> isVoid_size tin[@Fin4] s4 ->
match yout, P with
| _, retT :: P =>
match yout, k with
| Some true, O => (* return true *)
tout[@Fin0] ≃(;size @>Fin0 s0) P /\
tout[@Fin1] ≃(;size @>Fin1 s1) Q /\
isVoid_size tout[@Fin2] (size @>Fin2 s2)
| None, S k' => (* continue *)
tout[@Fin0] ≃(;size @>Fin0 s0) P /\
tout[@Fin1] ≃(;size @>Fin1 s1) Q ++ [retT] /\
tout[@Fin2] ≃(;size @>Fin2 s2) k'
| _, _ => False (* not the case *)
end
| None, lamT :: P => (* continue *)
tout[@Fin0] ≃(;size@>Fin0 s0) P /\
tout[@Fin1] ≃(;size@>Fin1 s1) Q ++ [lamT] /\
tout[@Fin2] ≃(;size@>Fin2 s2) S k
| None, t :: P => (* continue *)
tout[@Fin0] ≃(;size@>Fin0 s0) P /\
tout[@Fin1] ≃(;size@>Fin1 s1) Q ++ [t] /\
tout[@Fin2] ≃(;size@>Fin2 s2) k
| Some false, nil => (* return false *)
True
| _, _ => False (* not the case *)
end /\
isVoid_size tout[@Fin3] (size@>Fin3 s3) /\
isVoid_size tout[@Fin4] (size@>Fin4 s4).
Arguments JumpTarget_Step_size : simpl never.
(* Steps after the [CaseCom], depending on [t] *)
Local Definition JumpTarget_Step_steps_CaseCom (Q: Pro) (k: nat) (t: Tok) :=
match t with
| retT =>
match k with
| S _ => 1 + CaseNat_steps + App_ACom_steps Q retAT
| 0 => 2 + CaseNat_steps + ResetEmpty1_steps
end
| lamT => 1 + Constr_S_steps + App_ACom_steps Q lamAT
| appT => App_ACom_steps Q appAT
| varT n => 1 + Constr_varT_steps + App_Com_steps Q t
end.
(* Steps after the [CaseList] *)
Local Definition JumpTarget_Step_steps_CaseList (P Q : Pro) (k: nat) :=
match P with
| t :: P' => 1 + CaseCom_steps + JumpTarget_Step_steps_CaseCom Q k t
| nil => 0
end.
(* Total steps *)
Definition JumpTarget_Step_steps (P Q: Pro) (k: nat) :=
1 + CaseList_steps P + JumpTarget_Step_steps_CaseList P Q k.
Definition JumpTarget_Step_T : tRel sigPro^+ 5 :=
fun tin steps => (* Warning: I have to use another variable for the steps, since [k] is used. *)
exists (P Q : Pro) (k : nat),
tin[@Fin0] ≃ P /\
tin[@Fin1] ≃ Q /\
tin[@Fin2] ≃ k /\
isVoid tin[@Fin3] /\ isVoid tin[@Fin4] /\
JumpTarget_Step_steps P Q k <= steps.
Fixpoint jumpTarget_k (k:nat) (P:Pro) : nat :=
match P with
| retT :: P' => match k with
| 0 => 0
| S k' => jumpTarget_k k' P'
end
| lamT :: P' => jumpTarget_k (S k) P'
| t :: P'    => jumpTarget_k k P' (* either [varT n] or [appT] *)
| []         => k
end.
Goal forall k P, jumpTarget_k k P <= k + |P|.
Definition JumpTarget_Loop := While JumpTarget_Step.
Fixpoint JumpTarget_Loop_size (P Q : Pro) (k : nat) {struct P} : Vector.t (nat->nat) 5 :=
match P with
| retT :: P' =>
match k with
| O => (* return true *)
JumpTarget_Step_size P Q k
| S k' => (* continue *)
JumpTarget_Step_size P Q k >>> JumpTarget_Loop_size P' (Q ++ [retT]) k'
end
| lamT :: P' => (* continue; increase k *)
JumpTarget_Step_size P Q k >>> JumpTarget_Loop_size P' (Q ++ [lamT]) (S k)
| t :: P' => (* continue *)
JumpTarget_Step_size P Q k >>> JumpTarget_Loop_size P' (Q ++ [t]) k
| nil => (* return false *)
JumpTarget_Step_size P Q k
end.
Definition JumpTarget_Loop_Rel : pRel sigPro^+ bool 5 :=
fun tin '(yout, tout) =>
forall (P Q : Pro) (k : nat) (s0 s1 s2 s3 s4 : nat),
let size := JumpTarget_Loop_size P Q k in
tin[@Fin0] ≃(;s0) P ->
tin[@Fin1] ≃(;s1) Q ->
tin[@Fin2] ≃(;s2) k ->
isVoid_size tin[@Fin3] s3 -> isVoid_size tin[@Fin4] s4 ->
match yout with
| true =>
exists (P' Q' : Pro),
jumpTarget k Q P = Some (Q', P') /\
tout[@Fin0] ≃(;size@>Fin0 s0) P' /\
tout[@Fin1] ≃(;size@>Fin1 s1) Q' /\
isVoid_size tout[@Fin2] (size@>Fin2 s2) /\
isVoid_size tout[@Fin3] (size@>Fin3 s3) /\ isVoid_size tout[@Fin4] (size@>Fin4 s4)
| false =>
jumpTarget k Q P = None
end.
Fixpoint JumpTarget_Loop_steps (P Q: Pro) (k: nat) : nat :=
match P with
| nil => JumpTarget_Step_steps P Q k
| t :: P' =>
match t with
| retT =>
match k with
| S k' => 1 + JumpTarget_Step_steps P Q k + JumpTarget_Loop_steps P' (Q++[t]) k'
| 0 =>        JumpTarget_Step_steps P Q k (* terminal case *)
end
| lamT =>   1 + JumpTarget_Step_steps P Q k + JumpTarget_Loop_steps P' (Q++[t]) (S k)
| appT =>   1 + JumpTarget_Step_steps P Q k + JumpTarget_Loop_steps P' (Q++[t]) k
| varT n => 1 + JumpTarget_Step_steps P Q k + JumpTarget_Loop_steps P' (Q++[t]) k
end
end.
Definition JumpTarget_Loop_T : tRel sigPro^+ 5 :=
fun tin steps =>
exists (P Q : Pro) (k : nat),
tin[@Fin0] ≃ P /\
tin[@Fin1] ≃ Q /\
tin[@Fin2] ≃ k /\
isVoid tin[@Fin3] /\ isVoid tin[@Fin4] /\
JumpTarget_Loop_steps P Q k <= steps.
Definition JumpTarget : pTM sigPro^+ bool 5 :=
Constr_nil _ @ [|Fin1|];;
Constr_O ⇑ _ @ [|Fin2|];;
JumpTarget_Loop.
Definition JumpTarget_size (P : Pro) : Vector.t (nat->nat) 5 :=
[| id; pred; Constr_O_size; id; id |] >>> JumpTarget_Loop_size P nil 0.
Definition JumpTarget_Rel : pRel sigPro^+ bool 5 :=
fun tin '(yout, tout) =>
forall (P : Pro) (s0 s1 : nat) (si : Vector.t nat 3),
tin[@Fin0] ≃(;s0) P ->
isVoid_size tin[@Fin1] s1 ->
(forall i : Fin.t 3, isVoid_size tin[@FinR 2 i : Fin.t 5] si[@i]) ->
match yout with
| true =>
exists (P' Q' : Pro),
jumpTarget 0 nil P = Some (Q', P') /\
tout[@Fin0] ≃(;JumpTarget_size P @>Fin0 s0) P' /\
tout[@Fin1] ≃(;JumpTarget_size P @>Fin1 s1) Q' /\
(forall i : Fin.t 3, isVoid_size tout[@FinR 2 i : Fin.t 5] (JumpTarget_size P @>(FinR 2 i) si[@i]))
| false =>
jumpTarget 0 nil P = None
end.

Lemma JumpTarget_Realise : JumpTarget ⊨ JumpTarget_Rel.