Require Import FunInd.
From Undecidability Require Import TM.Util.Prelim.
From Undecidability Require Import TM.Basic.Basic.
From Undecidability Require Import TM.Combinators.Combinators.
From Undecidability.TM.Compound Require Import TMTac Multi MoveToSymbol.
Require Recdef.
Set Default Proof Using "Type".
Section Compare.
Import Recdef.
Variable sig : finType.
Variable stop : sig -> bool.
Definition Compare_Step : pTM sig (option bool) 2 :=
Switch
(CaseChar2 (fun c1 c2 =>
match c1, c2 with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then Some true
else if (stop c1) || (stop c2)
then Some false
else if Dec (c1 = c2)
then None
else Some false
| _, _ => Some false
end))
(fun x : option bool => match x with
| Some b => Return Nop (Some b)
| None => Return (MovePar Rmove Rmove) None
end).
Definition Compare_Step_Rel : pRel sig (option bool) 2 :=
fun tin '(yout, tout) =>
match current tin[@Fin0], current tin[@Fin1] with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then yout = Some true /\ tout = tin
else if (stop c1) || (stop c2)
then yout = Some false /\ tout = tin
else if Dec (c1 = c2)
then yout = None /\ tout[@Fin0] = tape_move_right tin[@Fin0] /\ tout[@Fin1] = tape_move_right tin[@Fin1]
else yout = Some false /\ tout = tin
| _, _ => yout = Some false /\ tout = tin
end.
Lemma Compare_Step_Sem : Compare_Step ⊨c(5) Compare_Step_Rel.
Proof.
eapply RealiseIn_monotone.
{ unfold Compare_Step. TM_Correct. }
{ Unshelve. 4,7: reflexivity. all: lia. }
{ intros tin (yout, tout) H. TMCrush; TMSolve 1. }
Qed.
Definition Compare := While Compare_Step.
Definition Compare_fun_measure (t : tape sig * tape sig) : nat := length (tape_local (fst t)).
Function Compare_fun (t : tape sig * tape sig) {measure Compare_fun_measure t} : bool * (tape sig * tape sig) :=
match (current (fst t)), (current (snd t)) with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then (true, t)
else if (stop c1) || (stop c2)
then (false, t)
else if Dec (c1 = c2)
then Compare_fun (tape_move_right (fst t), tape_move_right (snd t))
else (false, t)
| _, _ => (false, t)
end.
Proof.
intros (t1&t2). intros c1 Hc1 c2 Hc2 HStopC1 HStopC2. cbn in *.
destruct t1; cbn in *; inv Hc1. destruct t2; cbn in *; inv Hc2.
unfold Compare_fun_measure. cbn. simpl_tape. intros. lia.
Qed.
Definition Compare_Rel : pRel sig bool 2 :=
fun tin '(yout, tout) => (yout, (tout[@Fin0], tout[@Fin1])) = Compare_fun (tin[@Fin0], tin[@Fin1]).
Lemma Compare_Realise : Compare ⊨ Compare_Rel.
Proof.
eapply Realise_monotone.
{ unfold Compare. TM_Correct. eapply RealiseIn_Realise. apply Compare_Step_Sem. }
{ apply WhileInduction; intros; cbn in *.
- revert yout HLastStep. TMCrush; intros; rewrite Compare_fun_equation; cbn; TMSolve 1.
all:try rewrite E in *; try rewrite E0 in *;try rewrite E1 in *;try rewrite E2 in *.
all: TMCrush; TMSolve 1.
- revert yout HLastStep. TMCrush; intros. all:TMSimp. all:rewrite HLastStep.
all:symmetry. all:rewrite Compare_fun_equation. all:cbn. all:rewrite E, E0, E1, E2. all:decide (e0=e0) as [ | Tamtam]; [ | now contradiction Tamtam] . all:auto.
}
Qed.
Local Arguments plus : simpl never.
Function Compare_steps (t : tape sig * tape sig) { measure Compare_fun_measure} : nat :=
match (current (fst t)), (current (snd t)) with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then 5
else if (stop c1) || (stop c2)
then 5
else if Dec (c1 = c2)
then 6 + Compare_steps (tape_move_right (fst t), tape_move_right (snd t))
else 5
| _, _ => 5
end.
Proof.
intros (t1&t2). intros c1 Hc1 c2 Hc2 HStopC1 HStopC2. cbn in *.
destruct t1; cbn in *; inv Hc1. destruct t2; cbn in *; inv Hc2.
unfold Compare_fun_measure. cbn. simpl_tape. intros. lia.
Qed.
Definition Compare_T : tRel sig 2 :=
fun tin k => Compare_steps (tin[@Fin0], tin[@Fin1]) <= k.
Lemma Compare_steps_ge t : 5 <= Compare_steps t.
Proof. functional induction Compare_steps t; auto. lia. Qed.
Lemma Compare_TerminatesIn : projT1 Compare ↓ Compare_T.
Proof.
eapply TerminatesIn_monotone.
{ unfold Compare. TM_Correct.
- eapply RealiseIn_Realise. apply Compare_Step_Sem.
- eapply RealiseIn_TerminatesIn. apply Compare_Step_Sem. }
{ apply WhileCoInduction; intros. exists 5. split. reflexivity. intros [ yout | ].
- intros. hnf in HT. TMCrush. all: rewrite <- HT. all: apply Compare_steps_ge.
- intros. hnf in HT. exists (Compare_steps (tape_move tin[@Fin0] Rmove, tape_move tin[@Fin1] Rmove)).
TMCrush.
split.
+ hnf. TMSimp. auto.
+ rewrite Compare_steps_equation in HT. cbn in HT. rewrite E, E0, E1, E2 in HT. rewrite E3 in *. lia.
}
Qed.
End Compare.
Section CompareLists.
Variable X : eqType.
Definition list_comperasion (xs ys : list X) : Prop :=
xs = ys \/
(exists a b l1 l2 l3, a <> b /\ xs = l1 ++ a :: l2 /\ ys = l1 ++ b :: l3) \/
(exists a l1 l2, xs = l1 ++ a :: l2 /\ ys = l1) \/
(exists a l1 l2, xs = l1 /\ ys = l1 ++ a :: l2).
Definition list_comperasion_cons xs ys x :
list_comperasion xs ys ->
list_comperasion (x :: xs) (x :: ys).
Proof.
destruct 1 as [ <- | [ (a&b&l1&l2&l3&H1&H2&H3) | [ (a&l1&l2&H1&H2) | (a&l1&l2&H1&H2) ]]].
- left. reflexivity.
- subst. right; left. exists a, b, (x::l1), l2, l3. auto.
- subst. right. right. left. do 3 eexists. split. 2: reflexivity. cbn. eauto.
- subst. right. right. right. do 3 eexists. split. 1: reflexivity. cbn. eauto.
Qed.
Lemma compare_lists (xs ys : list X) :
list_comperasion xs ys.
Proof.
revert ys. induction xs as [ | x xs IH]; intros; cbn in *.
- destruct ys as [ | y ys].
+ left. reflexivity.
+ right. right. right. do 3 eexists. split. reflexivity. cbn. reflexivity.
- destruct ys as [ | y ys].
+ hnf. right. right. left. do 3 eexists. split. 2: reflexivity. cbn. reflexivity.
+ decide (x = y) as [ <- | HDec].
* now apply list_comperasion_cons.
* hnf. right. left. exists x, y, nil. cbn. do 2 eexists. eauto.
Qed.
End CompareLists.
Local Arguments plus : simpl never.
Local Arguments mult : simpl never.
Section Compare_fun_lemmas.
Variable (X : finType) (stop : X -> bool).
Lemma Compare_correct_eq (str : list X) (s1 s2 : X) rs1 rs2 t :
(forall x, In x str -> stop x = false) ->
stop s1 = true ->
stop s2 = true ->
tape_local (fst t) = str ++ s1 :: rs1 ->
tape_local (snd t) = str ++ s2 :: rs2 ->
Compare_fun stop t =
(true, (midtape (rev str ++ left (fst t)) s1 rs1, midtape (rev str ++ left (snd t)) s2 rs2)).
Proof.
revert str s1 s2 rs1 rs2. functional induction (Compare_fun stop t); intros str s1 s2 rs1 rs2 HStr HS1 HS2 HL1 HL2; destruct t as [t1 t2]; cbn in *;
try rewrite HS1 in *; try rewrite HS2 in *; cbn in *; auto.
- destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1 as ->. apply midtape_tape_local_cons in HL2 as ->. cbn. reflexivity.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2.
rewrite HL1 in *. cbn in *. inv e. rewrite HL2 in *. cbn in *. inv e0.
specialize (HStr c2 ltac:(auto)). rewrite HStr in e1. cbn in *. congruence.
- exfalso. destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0. rewrite HS1, HS2 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0.
specialize (HStr c2 ltac:(auto)). rewrite HStr in *. cbn in *. congruence.
- destruct str as [ | s str']; cbn in *.
+ exfalso. apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0. rewrite HS1 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0.
apply orb_false_iff in e2 as (e2&_).
simpl_tape in IHp. specialize IHp with (4 := eq_refl) (5 := eq_refl) (2 := HS1) (3 := HS2). spec_assert IHp by auto.
simpl_list; cbn; auto.
- exfalso. destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0. rewrite HS1, HS2 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0.
specialize (HStr c2 ltac:(auto)). rewrite HStr in *. cbn in *. congruence.
- exfalso. destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. auto.
Qed.
Lemma Compare_correct_eq_midtape (str : list X) (s1 s2 : X) ls1 rs1 ls2 m rs2 :
(forall x, In x str -> stop x = false) ->
stop m = false ->
stop s1 = true ->
stop s2 = true ->
Compare_fun stop (midtape ls1 m (str ++ s1 :: rs1), midtape ls2 m (str ++ s2 :: rs2)) =
(true, (midtape (rev str ++ m :: ls1) s1 rs1, midtape (rev str ++ m :: ls2) s2 rs2)).
Proof.
intros HStr Hm HS1 HS2.
rewrite Compare_fun_equation; cbn. erewrite Compare_correct_eq with (str := str) (rs1 := rs1) (rs2 := rs2) (s1 := s1) (s2 := s2); eauto.
all: cbn; simpl_tape; auto.
rewrite Hm. cbn. decide (m=m); [ | tauto]. now simpl_tape.
Qed.
Lemma Compare_correct_neq (str1 str2 str3 : list X) (x1 x2 : X) t :
(forall x, In x str1 -> stop x = false) ->
stop x1 = false ->
stop x2 = false ->
x1 <> x2 ->
tape_local (fst t) = str1 ++ x1 :: str2 ->
tape_local (snd t) = str1 ++ x2 :: str3 ->
Compare_fun stop t =
(false, (midtape (rev str1 ++ left (fst t)) x1 str2, midtape (rev str1 ++ left (snd t)) x2 str3)).
Proof.
revert str1 str2 str3 x1 x2. functional induction (Compare_fun stop t); intros str1 str2 str3 x1 x2; intros Hstr1 Hx1 Hx2 Hx12 HT1 HT2; destruct t as [t1 t2]; cbn in *.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx1 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. specialize (Hstr1 s ltac:(eauto)). inv e; inv e0.
rewrite Hstr1 in e1. cbn in *. congruence.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx1 in e2. cbn in *. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. specialize (Hstr1 s ltac:(eauto)).
inv e; inv e0. rewrite Hstr1 in e2. cbn in *. congruence.
- destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. tauto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
simpl_tape in IHp. specialize IHp with (5 := eq_refl) (6 := eq_refl) (2 := Hx1) (3 := Hx2) (4 := Hx12). spec_assert IHp by auto.
simpl_list; cbn; auto.
- destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx1 in e2. cbn in *. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. now contradiction _x.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
Qed.
Lemma Compare_correct_neq_midtape (str1 str2 str3 : list X) (m x1 x2 : X) ls1 ls2 :
(forall x, In x str1 -> stop x = false) ->
stop x1 = false ->
stop x2 = false ->
stop m = false ->
x1 <> x2 ->
Compare_fun stop (midtape ls1 m (str1 ++ x1 :: str2), midtape ls2 m (str1 ++ x2 :: str3)) =
(false, (midtape (rev str1 ++ m :: ls1) x1 str2, midtape (rev str1 ++ m :: ls2) x2 str3)).
Proof.
intros Hstr1 Hx1 Hx2 Hm Hx12.
rewrite Compare_fun_equation; cbn. rewrite Compare_correct_neq with (str1 := str1) (str2 := str2) (str3 := str3) (x1 := x1) (x2 := x2).
all: cbn; simpl_tape; auto.
rewrite Hm. cbn. decide (m=m); [ | tauto]. now simpl_tape.
Qed.
Definition swap (A B : Type) : A*B->B*A := ltac:(intros [b a]; now constructor).
Lemma Compare_correct_swap t :
Compare_fun stop (swap t) = (fst (Compare_fun stop t), swap (snd (Compare_fun stop t))).
Proof.
functional induction (Compare_fun stop t); destruct t as [t1 t2]; cbn in *.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite andb_comm in e1. now rewrite e1.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite andb_comm in e1. rewrite orb_comm in e2. now rewrite e1, e2.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite e1, e2. decide (c1=c1) as [ ? | H]; [ | now contradiction H]. auto.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite andb_comm in e1. rewrite orb_comm in e2. rewrite e1, e2.
decide (c2=c1) as [ <- | H]; [ now contradiction _x | ]. auto.
- rewrite Compare_fun_equation. cbn.
destruct (current t1); auto; destruct (current t2); auto.
Qed.
Lemma Compare_correct_short (str1 str2 rs1 rs2 : list X) (x : X) (s1 s2 : X) t :
(forall x, In x str1 -> stop x = false) ->
stop x = false ->
stop s1 = true ->
stop s2 = true ->
tape_local (fst t) = str1 ++ x :: str2 ++ s1 :: rs1 ->
tape_local (snd t) = str1 ++ s2 :: rs2 ->
Compare_fun stop t =
(false, (midtape (rev str1 ++ left (fst t)) x (str2 ++ s1 :: rs1),
midtape (rev str1 ++ left (snd t)) s2 rs2)).
Proof.
revert str1 str2 rs1 rs2 x s1 s2. functional induction (Compare_fun stop t); intros str1 str2 rs1 rs2 x s1 s2; intros Hstr1 Hx Hs1 Hs2 HT1 HT2; destruct t as [t1 t2]; cbn in *.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx in e1. cbn in e1. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
specialize (Hstr1 c2 ltac:(auto)). rewrite Hstr1 in e1. cbn in e1. congruence.
- destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
specialize (Hstr1 c2 ltac:(auto)). rewrite Hstr1 in e2. cbn in e2. congruence.
- destruct str1 as [ | s str1]; cbn in *.
+ exfalso. apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
rewrite Hs2 in e1. cbn in e1. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
simpl_tape in IHp. specialize IHp with (2 := Hx) (3 := Hs1) (4 := Hs2) (5 := eq_refl) (6 := eq_refl). spec_assert IHp by auto.
simpl_list; cbn; auto.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx in e2. cbn in e2. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
specialize (Hstr1 c2 ltac:(auto)). rewrite Hstr1 in e1. cbn in e1. congruence.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
Qed.
Lemma Compare_correct_short_midtape (str1 str2 : list X) (x s1 s2 : X) ls1 rs1 ls2 m rs2 :
(forall x, In x str1 -> stop x = false) ->
stop m = false ->
stop x = false ->
stop s1 = true ->
stop s2 = true ->
Compare_fun stop (midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1),
midtape ls2 m (str1 ++ s2 :: rs2)) =
(false, (midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1),
midtape (rev str1 ++ m :: ls2) s2 rs2)).
Proof.
intros Hstr1 Hm Hx Hs1 Hs2.
erewrite Compare_correct_short with (str1 := m :: str1) (str2 := str2) (s1 := s1) (s2 := s2); cbn; eauto.
- now simpl_list; cbn.
- intros ? [ <- | H]; auto.
Qed.
Lemma Compare_correct_long_midtape (str1 str2 : list X) (x s1 s2 : X) ls1 rs1 ls2 m rs2 :
(forall x, In x str1 -> stop x = false) ->
stop m = false ->
stop x = false ->
stop s1 = true ->
stop s2 = true ->
Compare_fun stop (midtape ls2 m (str1 ++ s2 :: rs2),
midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1)) =
(false, (midtape (rev str1 ++ m :: ls2) s2 rs2,
midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1))).
Proof.
change ((midtape ls2 m (str1 ++ s2 :: rs2), midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1))) with (swap (midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1), midtape ls2 m (str1 ++ s2 :: rs2))).
change (midtape (rev str1 ++ m :: ls2) s2 rs2, midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1)) with (swap (midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1), midtape (rev str1 ++ m :: ls2) s2 rs2)).
intros. rewrite Compare_correct_swap. cbn. rewrite Compare_correct_short_midtape; eauto.
Qed.
Lemma Compare_steps_correct (str1 str2 : list X) (s1 s2 : X) rs1 rs2 t :
stop s1 = true ->
stop s2 = true ->
tape_local (fst t) = str1 ++ s1 :: rs1 ->
tape_local (snd t) = str2 ++ s2 :: rs2 ->
Compare_steps stop t <= 5 + 6 * max (length str1) (length str2).
Proof.
revert rs1 rs2 str1 str2. functional induction (Compare_steps stop t); intros rs1 rs2 str1 str2; intros Hs1 Hs2 HT1 HT2; destruct t as [t1 t2]; cbn in *.
all: try lia. destruct str1 as [ | s str1], str2 as [ | s' str2]; cbn in *;
apply midtape_tape_local_cons in HT1; apply midtape_tape_local_cons in HT2; rewrite HT1, HT2 in *; cbn in *;
inv e; inv e0.
- exfalso. rewrite Hs1 in e1. cbn in e1. congruence.
- exfalso. rewrite Hs1 in e1. cbn in e1. congruence.
- exfalso. rewrite Hs2 in e2. cbn in e2. congruence.
- simpl_tape in IHn. specialize IHn with (1 := Hs1) (2 := Hs2) (3 := eq_refl) (4 := eq_refl). lia.
Qed.
Lemma Compare_steps_correct_midtape (str1 str2 : list X) (s1 s2 : X) (m : X) ls1 ls2 rs1 rs2 :
stop s1 = true ->
stop s2 = true ->
Compare_steps stop (midtape ls1 m (str1 ++ s1 :: rs1), midtape ls2 m (str2 ++ s2 :: rs2)) <= 11 + 6 * max (length str1) (length str2).
Proof.
intros Hs1 Hs2. rewrite Compare_steps_correct with (str1 := m :: str1) (str2 := m :: str2) (s1 := s1) (s2 := s2); cbn; eauto. lia.
Qed.
Lemma Compare_Move_steps_midtape1 (stop' : X -> bool) (str1 str2 : list X) (s1 s2 : X) (m : X) ls1 ls2 rs1 rs2 :
(forall x, In x str1 -> stop x = false) ->
(forall x, In x str2 -> stop x = false) ->
stop m = false ->
stop s1 = true ->
stop s2 = true ->
stop' m = true ->
(forall x, In x str1 -> stop' x = false) ->
(forall x, In x str2 -> stop' x = false) ->
MoveToSymbol_L_steps stop' id (fst (snd (Compare_fun stop (midtape ls1 m (str1 ++ s1 :: rs1), midtape ls2 m (str2 ++ s2 :: rs2))))) <=
8 + 4 * length str1.
Proof.
intros.
pose proof compare_lists str1 str2 as[ HC | [ (a&b&l1&l2&l3&HC1&HC2&HC3) | [ (a&l1&l2&HC1&HC2) | (a&l1&l2&HC1&HC2) ]]]; subst.
- rewrite Compare_correct_eq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_neq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_short_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_long_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
Qed.
Lemma Compare_Move_steps_midtape2 (stop' : X -> bool) (str1 str2 : list X) (s1 s2 : X) (m : X) ls1 ls2 rs1 rs2 :
(forall x, In x str1 -> stop x = false) ->
(forall x, In x str2 -> stop x = false) ->
stop m = false ->
stop s1 = true ->
stop s2 = true ->
stop' m = true ->
(forall x, In x str1 -> stop' x = false) ->
(forall x, In x str2 -> stop' x = false) ->
MoveToSymbol_L_steps stop' id (snd (snd (Compare_fun stop (midtape ls1 m (str1 ++ s1 :: rs1), midtape ls2 m (str2 ++ s2 :: rs2))))) <=
8 + 4 * length str2.
Proof.
intros.
pose proof compare_lists str1 str2 as[ HC | [ (a&b&l1&l2&l3&HC1&HC2&HC3) | [ (a&l1&l2&HC1&HC2) | (a&l1&l2&HC1&HC2) ]]]; subst.
- rewrite Compare_correct_eq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_neq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_short_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_long_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
Qed.
End Compare_fun_lemmas.
From Undecidability Require Import TM.Util.Prelim.
From Undecidability Require Import TM.Basic.Basic.
From Undecidability Require Import TM.Combinators.Combinators.
From Undecidability.TM.Compound Require Import TMTac Multi MoveToSymbol.
Require Recdef.
Set Default Proof Using "Type".
Section Compare.
Import Recdef.
Variable sig : finType.
Variable stop : sig -> bool.
Definition Compare_Step : pTM sig (option bool) 2 :=
Switch
(CaseChar2 (fun c1 c2 =>
match c1, c2 with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then Some true
else if (stop c1) || (stop c2)
then Some false
else if Dec (c1 = c2)
then None
else Some false
| _, _ => Some false
end))
(fun x : option bool => match x with
| Some b => Return Nop (Some b)
| None => Return (MovePar Rmove Rmove) None
end).
Definition Compare_Step_Rel : pRel sig (option bool) 2 :=
fun tin '(yout, tout) =>
match current tin[@Fin0], current tin[@Fin1] with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then yout = Some true /\ tout = tin
else if (stop c1) || (stop c2)
then yout = Some false /\ tout = tin
else if Dec (c1 = c2)
then yout = None /\ tout[@Fin0] = tape_move_right tin[@Fin0] /\ tout[@Fin1] = tape_move_right tin[@Fin1]
else yout = Some false /\ tout = tin
| _, _ => yout = Some false /\ tout = tin
end.
Lemma Compare_Step_Sem : Compare_Step ⊨c(5) Compare_Step_Rel.
Proof.
eapply RealiseIn_monotone.
{ unfold Compare_Step. TM_Correct. }
{ Unshelve. 4,7: reflexivity. all: lia. }
{ intros tin (yout, tout) H. TMCrush; TMSolve 1. }
Qed.
Definition Compare := While Compare_Step.
Definition Compare_fun_measure (t : tape sig * tape sig) : nat := length (tape_local (fst t)).
Function Compare_fun (t : tape sig * tape sig) {measure Compare_fun_measure t} : bool * (tape sig * tape sig) :=
match (current (fst t)), (current (snd t)) with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then (true, t)
else if (stop c1) || (stop c2)
then (false, t)
else if Dec (c1 = c2)
then Compare_fun (tape_move_right (fst t), tape_move_right (snd t))
else (false, t)
| _, _ => (false, t)
end.
Proof.
intros (t1&t2). intros c1 Hc1 c2 Hc2 HStopC1 HStopC2. cbn in *.
destruct t1; cbn in *; inv Hc1. destruct t2; cbn in *; inv Hc2.
unfold Compare_fun_measure. cbn. simpl_tape. intros. lia.
Qed.
Definition Compare_Rel : pRel sig bool 2 :=
fun tin '(yout, tout) => (yout, (tout[@Fin0], tout[@Fin1])) = Compare_fun (tin[@Fin0], tin[@Fin1]).
Lemma Compare_Realise : Compare ⊨ Compare_Rel.
Proof.
eapply Realise_monotone.
{ unfold Compare. TM_Correct. eapply RealiseIn_Realise. apply Compare_Step_Sem. }
{ apply WhileInduction; intros; cbn in *.
- revert yout HLastStep. TMCrush; intros; rewrite Compare_fun_equation; cbn; TMSolve 1.
all:try rewrite E in *; try rewrite E0 in *;try rewrite E1 in *;try rewrite E2 in *.
all: TMCrush; TMSolve 1.
- revert yout HLastStep. TMCrush; intros. all:TMSimp. all:rewrite HLastStep.
all:symmetry. all:rewrite Compare_fun_equation. all:cbn. all:rewrite E, E0, E1, E2. all:decide (e0=e0) as [ | Tamtam]; [ | now contradiction Tamtam] . all:auto.
}
Qed.
Local Arguments plus : simpl never.
Function Compare_steps (t : tape sig * tape sig) { measure Compare_fun_measure} : nat :=
match (current (fst t)), (current (snd t)) with
| Some c1, Some c2 =>
if (stop c1) && (stop c2)
then 5
else if (stop c1) || (stop c2)
then 5
else if Dec (c1 = c2)
then 6 + Compare_steps (tape_move_right (fst t), tape_move_right (snd t))
else 5
| _, _ => 5
end.
Proof.
intros (t1&t2). intros c1 Hc1 c2 Hc2 HStopC1 HStopC2. cbn in *.
destruct t1; cbn in *; inv Hc1. destruct t2; cbn in *; inv Hc2.
unfold Compare_fun_measure. cbn. simpl_tape. intros. lia.
Qed.
Definition Compare_T : tRel sig 2 :=
fun tin k => Compare_steps (tin[@Fin0], tin[@Fin1]) <= k.
Lemma Compare_steps_ge t : 5 <= Compare_steps t.
Proof. functional induction Compare_steps t; auto. lia. Qed.
Lemma Compare_TerminatesIn : projT1 Compare ↓ Compare_T.
Proof.
eapply TerminatesIn_monotone.
{ unfold Compare. TM_Correct.
- eapply RealiseIn_Realise. apply Compare_Step_Sem.
- eapply RealiseIn_TerminatesIn. apply Compare_Step_Sem. }
{ apply WhileCoInduction; intros. exists 5. split. reflexivity. intros [ yout | ].
- intros. hnf in HT. TMCrush. all: rewrite <- HT. all: apply Compare_steps_ge.
- intros. hnf in HT. exists (Compare_steps (tape_move tin[@Fin0] Rmove, tape_move tin[@Fin1] Rmove)).
TMCrush.
split.
+ hnf. TMSimp. auto.
+ rewrite Compare_steps_equation in HT. cbn in HT. rewrite E, E0, E1, E2 in HT. rewrite E3 in *. lia.
}
Qed.
End Compare.
Section CompareLists.
Variable X : eqType.
Definition list_comperasion (xs ys : list X) : Prop :=
xs = ys \/
(exists a b l1 l2 l3, a <> b /\ xs = l1 ++ a :: l2 /\ ys = l1 ++ b :: l3) \/
(exists a l1 l2, xs = l1 ++ a :: l2 /\ ys = l1) \/
(exists a l1 l2, xs = l1 /\ ys = l1 ++ a :: l2).
Definition list_comperasion_cons xs ys x :
list_comperasion xs ys ->
list_comperasion (x :: xs) (x :: ys).
Proof.
destruct 1 as [ <- | [ (a&b&l1&l2&l3&H1&H2&H3) | [ (a&l1&l2&H1&H2) | (a&l1&l2&H1&H2) ]]].
- left. reflexivity.
- subst. right; left. exists a, b, (x::l1), l2, l3. auto.
- subst. right. right. left. do 3 eexists. split. 2: reflexivity. cbn. eauto.
- subst. right. right. right. do 3 eexists. split. 1: reflexivity. cbn. eauto.
Qed.
Lemma compare_lists (xs ys : list X) :
list_comperasion xs ys.
Proof.
revert ys. induction xs as [ | x xs IH]; intros; cbn in *.
- destruct ys as [ | y ys].
+ left. reflexivity.
+ right. right. right. do 3 eexists. split. reflexivity. cbn. reflexivity.
- destruct ys as [ | y ys].
+ hnf. right. right. left. do 3 eexists. split. 2: reflexivity. cbn. reflexivity.
+ decide (x = y) as [ <- | HDec].
* now apply list_comperasion_cons.
* hnf. right. left. exists x, y, nil. cbn. do 2 eexists. eauto.
Qed.
End CompareLists.
Local Arguments plus : simpl never.
Local Arguments mult : simpl never.
Section Compare_fun_lemmas.
Variable (X : finType) (stop : X -> bool).
Lemma Compare_correct_eq (str : list X) (s1 s2 : X) rs1 rs2 t :
(forall x, In x str -> stop x = false) ->
stop s1 = true ->
stop s2 = true ->
tape_local (fst t) = str ++ s1 :: rs1 ->
tape_local (snd t) = str ++ s2 :: rs2 ->
Compare_fun stop t =
(true, (midtape (rev str ++ left (fst t)) s1 rs1, midtape (rev str ++ left (snd t)) s2 rs2)).
Proof.
revert str s1 s2 rs1 rs2. functional induction (Compare_fun stop t); intros str s1 s2 rs1 rs2 HStr HS1 HS2 HL1 HL2; destruct t as [t1 t2]; cbn in *;
try rewrite HS1 in *; try rewrite HS2 in *; cbn in *; auto.
- destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1 as ->. apply midtape_tape_local_cons in HL2 as ->. cbn. reflexivity.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2.
rewrite HL1 in *. cbn in *. inv e. rewrite HL2 in *. cbn in *. inv e0.
specialize (HStr c2 ltac:(auto)). rewrite HStr in e1. cbn in *. congruence.
- exfalso. destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0. rewrite HS1, HS2 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0.
specialize (HStr c2 ltac:(auto)). rewrite HStr in *. cbn in *. congruence.
- destruct str as [ | s str']; cbn in *.
+ exfalso. apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0. rewrite HS1 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0.
apply orb_false_iff in e2 as (e2&_).
simpl_tape in IHp. specialize IHp with (4 := eq_refl) (5 := eq_refl) (2 := HS1) (3 := HS2). spec_assert IHp by auto.
simpl_list; cbn; auto.
- exfalso. destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0. rewrite HS1, HS2 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. inv e; inv e0.
specialize (HStr c2 ltac:(auto)). rewrite HStr in *. cbn in *. congruence.
- exfalso. destruct str as [ | s str']; cbn in *.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HL1. apply midtape_tape_local_cons in HL2. rewrite HL1, HL2 in *. cbn in *. auto.
Qed.
Lemma Compare_correct_eq_midtape (str : list X) (s1 s2 : X) ls1 rs1 ls2 m rs2 :
(forall x, In x str -> stop x = false) ->
stop m = false ->
stop s1 = true ->
stop s2 = true ->
Compare_fun stop (midtape ls1 m (str ++ s1 :: rs1), midtape ls2 m (str ++ s2 :: rs2)) =
(true, (midtape (rev str ++ m :: ls1) s1 rs1, midtape (rev str ++ m :: ls2) s2 rs2)).
Proof.
intros HStr Hm HS1 HS2.
rewrite Compare_fun_equation; cbn. erewrite Compare_correct_eq with (str := str) (rs1 := rs1) (rs2 := rs2) (s1 := s1) (s2 := s2); eauto.
all: cbn; simpl_tape; auto.
rewrite Hm. cbn. decide (m=m); [ | tauto]. now simpl_tape.
Qed.
Lemma Compare_correct_neq (str1 str2 str3 : list X) (x1 x2 : X) t :
(forall x, In x str1 -> stop x = false) ->
stop x1 = false ->
stop x2 = false ->
x1 <> x2 ->
tape_local (fst t) = str1 ++ x1 :: str2 ->
tape_local (snd t) = str1 ++ x2 :: str3 ->
Compare_fun stop t =
(false, (midtape (rev str1 ++ left (fst t)) x1 str2, midtape (rev str1 ++ left (snd t)) x2 str3)).
Proof.
revert str1 str2 str3 x1 x2. functional induction (Compare_fun stop t); intros str1 str2 str3 x1 x2; intros Hstr1 Hx1 Hx2 Hx12 HT1 HT2; destruct t as [t1 t2]; cbn in *.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx1 in e1. cbn in *. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. specialize (Hstr1 s ltac:(eauto)). inv e; inv e0.
rewrite Hstr1 in e1. cbn in *. congruence.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx1 in e2. cbn in *. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. specialize (Hstr1 s ltac:(eauto)).
inv e; inv e0. rewrite Hstr1 in e2. cbn in *. congruence.
- destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. tauto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
simpl_tape in IHp. specialize IHp with (5 := eq_refl) (6 := eq_refl) (2 := Hx1) (3 := Hx2) (4 := Hx12). spec_assert IHp by auto.
simpl_list; cbn; auto.
- destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx1 in e2. cbn in *. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. now contradiction _x.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
Qed.
Lemma Compare_correct_neq_midtape (str1 str2 str3 : list X) (m x1 x2 : X) ls1 ls2 :
(forall x, In x str1 -> stop x = false) ->
stop x1 = false ->
stop x2 = false ->
stop m = false ->
x1 <> x2 ->
Compare_fun stop (midtape ls1 m (str1 ++ x1 :: str2), midtape ls2 m (str1 ++ x2 :: str3)) =
(false, (midtape (rev str1 ++ m :: ls1) x1 str2, midtape (rev str1 ++ m :: ls2) x2 str3)).
Proof.
intros Hstr1 Hx1 Hx2 Hm Hx12.
rewrite Compare_fun_equation; cbn. rewrite Compare_correct_neq with (str1 := str1) (str2 := str2) (str3 := str3) (x1 := x1) (x2 := x2).
all: cbn; simpl_tape; auto.
rewrite Hm. cbn. decide (m=m); [ | tauto]. now simpl_tape.
Qed.
Definition swap (A B : Type) : A*B->B*A := ltac:(intros [b a]; now constructor).
Lemma Compare_correct_swap t :
Compare_fun stop (swap t) = (fst (Compare_fun stop t), swap (snd (Compare_fun stop t))).
Proof.
functional induction (Compare_fun stop t); destruct t as [t1 t2]; cbn in *.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite andb_comm in e1. now rewrite e1.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite andb_comm in e1. rewrite orb_comm in e2. now rewrite e1, e2.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite e1, e2. decide (c1=c1) as [ ? | H]; [ | now contradiction H]. auto.
- rewrite Compare_fun_equation. cbn. rewrite e, e0.
rewrite andb_comm in e1. rewrite orb_comm in e2. rewrite e1, e2.
decide (c2=c1) as [ <- | H]; [ now contradiction _x | ]. auto.
- rewrite Compare_fun_equation. cbn.
destruct (current t1); auto; destruct (current t2); auto.
Qed.
Lemma Compare_correct_short (str1 str2 rs1 rs2 : list X) (x : X) (s1 s2 : X) t :
(forall x, In x str1 -> stop x = false) ->
stop x = false ->
stop s1 = true ->
stop s2 = true ->
tape_local (fst t) = str1 ++ x :: str2 ++ s1 :: rs1 ->
tape_local (snd t) = str1 ++ s2 :: rs2 ->
Compare_fun stop t =
(false, (midtape (rev str1 ++ left (fst t)) x (str2 ++ s1 :: rs1),
midtape (rev str1 ++ left (snd t)) s2 rs2)).
Proof.
revert str1 str2 rs1 rs2 x s1 s2. functional induction (Compare_fun stop t); intros str1 str2 rs1 rs2 x s1 s2; intros Hstr1 Hx Hs1 Hs2 HT1 HT2; destruct t as [t1 t2]; cbn in *.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx in e1. cbn in e1. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
specialize (Hstr1 c2 ltac:(auto)). rewrite Hstr1 in e1. cbn in e1. congruence.
- destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
specialize (Hstr1 c2 ltac:(auto)). rewrite Hstr1 in e2. cbn in e2. congruence.
- destruct str1 as [ | s str1]; cbn in *.
+ exfalso. apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
rewrite Hs2 in e1. cbn in e1. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
simpl_tape in IHp. specialize IHp with (2 := Hx) (3 := Hs1) (4 := Hs2) (5 := eq_refl) (6 := eq_refl). spec_assert IHp by auto.
simpl_list; cbn; auto.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0. rewrite Hx in e2. cbn in e2. congruence.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. inv e; inv e0.
specialize (Hstr1 c2 ltac:(auto)). rewrite Hstr1 in e1. cbn in e1. congruence.
- exfalso. destruct str1 as [ | s str1]; cbn in *.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
+ apply midtape_tape_local_cons in HT1. apply midtape_tape_local_cons in HT2. rewrite HT1, HT2 in *. cbn in *. auto.
Qed.
Lemma Compare_correct_short_midtape (str1 str2 : list X) (x s1 s2 : X) ls1 rs1 ls2 m rs2 :
(forall x, In x str1 -> stop x = false) ->
stop m = false ->
stop x = false ->
stop s1 = true ->
stop s2 = true ->
Compare_fun stop (midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1),
midtape ls2 m (str1 ++ s2 :: rs2)) =
(false, (midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1),
midtape (rev str1 ++ m :: ls2) s2 rs2)).
Proof.
intros Hstr1 Hm Hx Hs1 Hs2.
erewrite Compare_correct_short with (str1 := m :: str1) (str2 := str2) (s1 := s1) (s2 := s2); cbn; eauto.
- now simpl_list; cbn.
- intros ? [ <- | H]; auto.
Qed.
Lemma Compare_correct_long_midtape (str1 str2 : list X) (x s1 s2 : X) ls1 rs1 ls2 m rs2 :
(forall x, In x str1 -> stop x = false) ->
stop m = false ->
stop x = false ->
stop s1 = true ->
stop s2 = true ->
Compare_fun stop (midtape ls2 m (str1 ++ s2 :: rs2),
midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1)) =
(false, (midtape (rev str1 ++ m :: ls2) s2 rs2,
midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1))).
Proof.
change ((midtape ls2 m (str1 ++ s2 :: rs2), midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1))) with (swap (midtape ls1 m (str1 ++ x :: str2 ++ s1 :: rs1), midtape ls2 m (str1 ++ s2 :: rs2))).
change (midtape (rev str1 ++ m :: ls2) s2 rs2, midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1)) with (swap (midtape (rev str1 ++ m :: ls1) x (str2 ++ s1 :: rs1), midtape (rev str1 ++ m :: ls2) s2 rs2)).
intros. rewrite Compare_correct_swap. cbn. rewrite Compare_correct_short_midtape; eauto.
Qed.
Lemma Compare_steps_correct (str1 str2 : list X) (s1 s2 : X) rs1 rs2 t :
stop s1 = true ->
stop s2 = true ->
tape_local (fst t) = str1 ++ s1 :: rs1 ->
tape_local (snd t) = str2 ++ s2 :: rs2 ->
Compare_steps stop t <= 5 + 6 * max (length str1) (length str2).
Proof.
revert rs1 rs2 str1 str2. functional induction (Compare_steps stop t); intros rs1 rs2 str1 str2; intros Hs1 Hs2 HT1 HT2; destruct t as [t1 t2]; cbn in *.
all: try lia. destruct str1 as [ | s str1], str2 as [ | s' str2]; cbn in *;
apply midtape_tape_local_cons in HT1; apply midtape_tape_local_cons in HT2; rewrite HT1, HT2 in *; cbn in *;
inv e; inv e0.
- exfalso. rewrite Hs1 in e1. cbn in e1. congruence.
- exfalso. rewrite Hs1 in e1. cbn in e1. congruence.
- exfalso. rewrite Hs2 in e2. cbn in e2. congruence.
- simpl_tape in IHn. specialize IHn with (1 := Hs1) (2 := Hs2) (3 := eq_refl) (4 := eq_refl). lia.
Qed.
Lemma Compare_steps_correct_midtape (str1 str2 : list X) (s1 s2 : X) (m : X) ls1 ls2 rs1 rs2 :
stop s1 = true ->
stop s2 = true ->
Compare_steps stop (midtape ls1 m (str1 ++ s1 :: rs1), midtape ls2 m (str2 ++ s2 :: rs2)) <= 11 + 6 * max (length str1) (length str2).
Proof.
intros Hs1 Hs2. rewrite Compare_steps_correct with (str1 := m :: str1) (str2 := m :: str2) (s1 := s1) (s2 := s2); cbn; eauto. lia.
Qed.
Lemma Compare_Move_steps_midtape1 (stop' : X -> bool) (str1 str2 : list X) (s1 s2 : X) (m : X) ls1 ls2 rs1 rs2 :
(forall x, In x str1 -> stop x = false) ->
(forall x, In x str2 -> stop x = false) ->
stop m = false ->
stop s1 = true ->
stop s2 = true ->
stop' m = true ->
(forall x, In x str1 -> stop' x = false) ->
(forall x, In x str2 -> stop' x = false) ->
MoveToSymbol_L_steps stop' id (fst (snd (Compare_fun stop (midtape ls1 m (str1 ++ s1 :: rs1), midtape ls2 m (str2 ++ s2 :: rs2))))) <=
8 + 4 * length str1.
Proof.
intros.
pose proof compare_lists str1 str2 as[ HC | [ (a&b&l1&l2&l3&HC1&HC2&HC3) | [ (a&l1&l2&HC1&HC2) | (a&l1&l2&HC1&HC2) ]]]; subst.
- rewrite Compare_correct_eq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_neq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_short_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_long_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
Qed.
Lemma Compare_Move_steps_midtape2 (stop' : X -> bool) (str1 str2 : list X) (s1 s2 : X) (m : X) ls1 ls2 rs1 rs2 :
(forall x, In x str1 -> stop x = false) ->
(forall x, In x str2 -> stop x = false) ->
stop m = false ->
stop s1 = true ->
stop s2 = true ->
stop' m = true ->
(forall x, In x str1 -> stop' x = false) ->
(forall x, In x str2 -> stop' x = false) ->
MoveToSymbol_L_steps stop' id (snd (snd (Compare_fun stop (midtape ls1 m (str1 ++ s1 :: rs1), midtape ls2 m (str2 ++ s2 :: rs2))))) <=
8 + 4 * length str2.
Proof.
intros.
pose proof compare_lists str1 str2 as[ HC | [ (a&b&l1&l2&l3&HC1&HC2&HC3) | [ (a&l1&l2&HC1&HC2) | (a&l1&l2&HC1&HC2) ]]]; subst.
- rewrite Compare_correct_eq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_neq_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_short_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
- simpl_list; cbn. rewrite Compare_correct_long_midtape; cbn; auto. rewrite MoveToSymbol_L_steps_midtape; auto. simpl_list. lia.
Qed.
End Compare_fun_lemmas.