From Undecidability Require Import ProgrammingTools.
From Undecidability Require Import EncodeBinNumbers PosDefinitions PosPointers.
From Undecidability Require Import TM.Basic.Duo.
Local Open Scope positive_scope.
Definition ReadPosSym : pTM sigPos^+ (option bool) 1 :=
CaseChar (fun (s : option sigPos^+) =>
match s with
| Some (inr sigPos_xH) => None
| Some (inr sigPos_xO) => Some false
| Some (inr sigPos_xI) => Some true
| _ => default
end).
Definition ReadPosSym_Rel : pRel sigPos^+ (option bool) 1 :=
fun tin '(yout, tout) =>
(forall (p : positive) (b : bool) (bits : list bool),
atBit tin[@Fin0] p b bits ->
match yout, b with
| Some false, false => tout = tin
| Some true, true => tout = tin
| _, _ => False
end) /\
(forall (p : positive), atHSB tin[@Fin0] p -> tout = tin /\ yout = None).
Lemma ReadPosSym_Sem : ReadPosSym ⊨c(1) ReadPosSym_Rel.
Proof.
eapply RealiseIn_monotone.
{ unfold ReadPosSym. TM_Correct. }
{ reflexivity. }
{ intros tin (yout, tout) (->&->). cbn. split.
- intros p b bits. intros (ls&->). cbn. destruct b; cbn; auto.
- intros p. intros (ls&->). cbn. auto.
}
Qed.
Definition ReadPosSym2 : pTM sigPos^+ (option bool * option bool) 2 :=
CaseChar2 (fun (s1 s2 : option sigPos^+) =>
(match s1 with
| Some (inr sigPos_xH) => None
| Some (inr sigPos_xO) => Some false
| Some (inr sigPos_xI) => Some true
| _ => default
end,
match s2 with
| Some (inr sigPos_xH) => None
| Some (inr sigPos_xO) => Some false
| Some (inr sigPos_xI) => Some true
| _ => default
end)).
Definition ReadPosSym2_Rel : pRel sigPos^+ (option bool * option bool) 2 :=
fun tin '(yout, tout) =>
(forall (p1 : positive) (b1 : bool) (bits1 : list bool) (p2 : positive) (b2 : bool) (bits2 : list bool),
atBit tin[@Fin0] p1 b1 bits1 ->
atBit tin[@Fin1] p2 b2 bits2 ->
match yout, b1, b2 with
| (Some false, Some false), false, false => tout = tin
| (Some false, Some true), false, true => tout = tin
| (Some true, Some false), true, false => tout = tin
| (Some true, Some true), true, true => tout = tin
| _, _, _ => False
end) /\
(forall (p1 : positive) (b1 : bool) (bits1 : list bool) (p2 : positive),
atBit tin[@Fin0] p1 b1 bits1 ->
atHSB tin[@Fin1] p2 ->
match yout, b1 with
| (Some false, None), false => tout = tin
| (Some true, None), true => tout = tin
| _, _ => False
end) /\
(forall (p1 : positive) (p2 : positive) (b2 : bool) (bits2 : list bool),
atHSB tin[@Fin0] p1 ->
atBit tin[@Fin1] p2 b2 bits2 ->
match yout, b2 with
| (None, Some false), false => tout = tin
| (None, Some true), true => tout = tin
| _, _ => False
end) /\
(forall (p1 p2 : positive),
atHSB tin[@Fin0] p1 ->
atHSB tin[@Fin1] p2 ->
tout = tin /\ yout = (None, None)).
Lemma ReadPosSym2_Sem : ReadPosSym2 ⊨c(1) ReadPosSym2_Rel.
Proof.
eapply RealiseIn_monotone.
{ unfold ReadPosSym2. TM_Correct. }
{ reflexivity. }
{ intros tin (yout, tout) (->&->). cbn. split; [ | split; [ | split ] ].
- intros p1 b1 bits1. intros p2 b2 bits2. intros (ls1&->). intros (ls2&->). cbn. destruct b1, b2; cbn; auto.
- intros p1 b1 bits1. intros p2. intros (ls1&->). intros (ls2&->). cbn. destruct b1 ; cbn; auto.
- intros p1. intros p2 b2 bits2. intros (ls1&->). intros (ls2&->). cbn. destruct b2; cbn; auto.
- intros p1. intros p2. intros (ls1&->). intros (l22&->). cbn. auto.
}
Qed.
Definition isxH (s : sigPos) := match s with sigPos_xH => true | _ => false end.
Definition isxH' (s : sigPos^+) := match s with inr s => isxH s | _ => false end.
Definition GoToHSB : pTM sigPos^+ unit 1 := MoveToSymbol_L isxH' id.
Definition GoToHSB_Rel : pRel sigPos^+ unit 1 :=
fun tin '(_, tout) =>
(forall (p : positive) (b : bool) (bits : list bool),
atBit tin[@Fin0] p b bits ->
atHSB tout[@Fin0] (append_bits p (b :: bits))) /\
(forall (p : positive),
atHSB tin[@Fin0] p ->
atHSB tout[@Fin0] p).
Lemma Encode_positive_tl_bits (p : positive) :
forall x, In x (tl (encode_pos p)) -> x = sigPos_xI \/ x = sigPos_xO.
Proof.
induction p; intros; cbn in *.
- rewrite tl_app in H by apply Encode_positive_eq_nil. apply in_app_iff in H as [H|H].
+ now apply IHp.
+ destruct H as [-> | []]. auto.
- rewrite tl_app in H by apply Encode_positive_eq_nil. apply in_app_iff in H as [H|H].
+ now apply IHp.
+ destruct H as [-> | []]. auto.
- auto.
Qed.
Lemma GoToHSB_Realise : GoToHSB ⊨ GoToHSB_Rel.
Proof.
eapply Realise_monotone.
{ unfold GoToHSB. TM_Correct. }
{
intros tin ([], tout) H. cbn in *. rewrite H in *; clear H. split.
- intros p b bits. intros (ls&->). cbn.
pose proof Encode_positive_startsWith_xH p as (str'&Hstr'). setoid_rewrite Hstr'.
cbn. simpl_list. cbn. rewrite MoveToSymbol_L_correct_midtape; cbn; eauto.
+ hnf. exists ls. f_equal. rewrite map_id. rewrite !map_rev. cbn. simpl_list. unfold id.
rewrite encode_append_bits.
rewrite Encode_positive_app_xIO.
simpl_list. cbn.
setoid_rewrite Hstr'. cbn. simpl_list. cbn.
f_equal. f_equal. f_equal. now rewrite map_map.
+ destruct b; auto.
+ intros ? (?&<-&? % in_rev) % in_map_iff.
replace str' with (tl (encode_pos p)) in H0 by now setoid_rewrite Hstr'.
now apply Encode_positive_tl_bits in H0 as [-> | ->].
- intros p. intros (ls&->). rewrite MoveToSymbol_L_Fun_equation. cbn. hnf. eauto.
}
Qed.
Definition GoToLSB_Rel : pRel sigPos^+ unit 1 :=
fun tin '(_, tout) =>
forall (p : positive),
atHSB tin[@Fin0] p ->
match p with
| p' ~ 0 => atLSB tout[@Fin0] p' false
| p' ~ 1 => atLSB tout[@Fin0] p' true
| 1 => atHSB tout[@Fin0] p
end.
Definition GoToLSB : pTM sigPos^+ unit 1 := MoveRight _;; Move L.
Lemma tam_I p :
forall x : boundary + sigPos, x el map inr (tl (encode_pos p ++ [sigPos_xI])) -> isStop x = false.
Proof.
intros x (?&<-&H) % in_map_iff.
rewrite tl_app in H by apply Encode_positive_eq_nil.
apply in_app_iff in H as [H|[<-|[]]].
- now pose proof Encode_positive_tl_bits H as [-> | ->].
- auto.
Qed.
Lemma tam_O p :
forall x : boundary + sigPos, x el map inr (tl (encode_pos p ++ [sigPos_xO])) -> isStop x = false.
Proof.
intros x (?&<-&H) % in_map_iff.
rewrite tl_app in H by apply Encode_positive_eq_nil.
apply in_app_iff in H as [H|[<-|[]]].
- now pose proof Encode_positive_tl_bits H as [-> | ->].
- auto.
Qed.
Lemma GoToLSB_Realise : GoToLSB ⊨ GoToLSB_Rel.
Proof.
eapply Realise_monotone.
{ unfold GoToLSB. unfold MoveRight. TM_Correct. }
{
intros tin ([], tout) H. TMSimp.
intros p. intros (ls&->).
destruct p; cbn in *.
- hnf. exists ls. cbn. rewrite MoveToSymbol_correct_midtape; cbn; auto.
pose proof Encode_positive_startsWith_xH p as (str'&Hstr'). cbn in *.
+ rewrite map_id; unfold id. rewrite !Hstr'; cbn. simpl_list; cbn. f_equal. f_equal. rewrite map_rev. f_equal.
+ apply tam_I.
- hnf. exists ls. cbn. rewrite MoveToSymbol_correct_midtape; cbn; auto.
pose proof Encode_positive_startsWith_xH p as (str'&Hstr'). cbn in *.
+ rewrite map_id; unfold id. rewrite !Hstr'; cbn. simpl_list; cbn. f_equal. f_equal. rewrite map_rev. f_equal.
+ apply tam_O.
- hnf. exists ls. cbn. repeat (rewrite MoveToSymbol_Fun_equation; cbn). reflexivity.
}
Qed.
Definition GoToLSB_start : pTM sigPos^+ unit 1 := Move R;; GoToLSB.
Definition GoToLSB_start_Rel : pRel sigPos^+ unit 1 :=
fun tin '(_, tout) =>
forall (p : positive),
tin[@Fin0] ≃ p ->
match p with
| p' ~ 1 => atLSB tout[@Fin0] p' true
| p' ~ 0 => atLSB tout[@Fin0] p' false
| 1 => atHSB tout[@Fin0] p
end.
Lemma GoToLSB_start_Realise : GoToLSB_start ⊨ GoToLSB_start_Rel.
Proof.
eapply Realise_monotone.
{ unfold GoToLSB_start. TM_Correct. apply GoToLSB_Realise. }
{
intros tin ([], tout) H. intros p Hp. TMSimp.
now apply H0, contains_moveRight_atHSB.
}
Qed.
Definition movedToLeft (t : tape sigPos^+) (p : positive) (b : bool) (bits : list bool) :=
match p with
| p'~1 => atBit t p' true (b :: bits)
| p'~0 => atBit t p' false (b :: bits)
| xH => atHSB t (append_bits p (b :: bits))
end.
Lemma atBit_writeMove_movedToLeft (t : tape sigPos^+) (p : positive) (b b' : bool) (bits : list bool) :
atBit t p b' bits ->
movedToLeft (tape_move (tape_write t (Some (bitToSigPos' b))) L) p b bits.
Proof.
intros (ls&->).
destruct p; cbn in *.
- hnf. exists ls. cbn. simpl_list. cbn. f_equal.
- hnf. exists ls. cbn. simpl_list. cbn. f_equal.
- hnf. exists ls. cbn. simpl_list. cbn. f_equal.
rewrite encode_append_bits. rewrite Encode_positive_app_xIO. cbn. f_equal. f_equal. now rewrite map_map.
Qed.
Definition SetBitAndMoveLeft_Rel (b : bool) : pRel sigPos^+ unit 1 :=
fun tin '(_, tout) =>
forall (p : positive) (b' : bool) (bits : list bool),
atBit tin[@Fin0] p b' bits ->
movedToLeft tout[@Fin0] p b bits.
Definition SetBitAndMoveLeft (b : bool) : pTM sigPos^+ unit 1 := WriteMove (bitToSigPos' b) L.
Lemma SetBitAndMoveLeft_Sem (b : bool) : SetBitAndMoveLeft b ⊨c(1) SetBitAndMoveLeft_Rel b.
Proof.
eapply RealiseIn_monotone.
{ unfold SetBitAndMoveLeft. TM_Correct. }
{ reflexivity. }
{
intros tin ([], tout) H. intros p b' bits. intros.
cbn -[tape_move_left] in H. setoid_rewrite H.
eapply atBit_writeMove_movedToLeft; eauto.
}
Qed.
Definition PushHSB_Rel (b : bool) : pRel sigPos^+ unit 1 :=
fun tin '(_, tout) => forall p, atHSB tin[@Fin0] p -> atHSB tout[@Fin0] (pushHSB p b).
Definition PushHSB (b : bool) : pTM sigPos^+ unit 1 := WriteMove (bitToSigPos' b) L;; WriteMove (inr sigPos_xH) L;; WriteMove (inl START) R.
Definition PushHSB_steps : nat := 5.
Lemma PushHSB_Sem (b : bool) : PushHSB b ⊨c(PushHSB_steps) PushHSB_Rel b.
Proof.
eapply RealiseIn_monotone.
{ unfold PushHSB. TM_Correct. }
{ reflexivity. }
{
intros tin ([], tout) H. TMSimp. intros p (ls&->). hnf; cbn.
pose proof Encode_positive_startsWith_xH p as (str'&Hstr'). repeat setoid_rewrite Hstr'. cbn.
rewrite !encode_pushHSB; cbn.
destruct ls; cbn.
+ exists nil. f_equal. f_equal. f_equal. now setoid_rewrite Hstr'.
+ exists ls. f_equal. f_equal. f_equal. now setoid_rewrite Hstr'.
}
Qed.