(* *)
(**************************************************************************)
-include "turing/turing.ma".
+include "turing/multi_universal/moves.ma".
+include "turing/if_multi.ma".
include "turing/inject.ma".
-include "turing/while_multi.ma".
+include "turing/basic_machines.ma".
definition compare_states ≝ initN 3.
*)
definition trans_compare_step ≝
- λi,j.λsig:FinSet.λn.
+ λi,j.λsig:FinSet.λn.λis_endc.
λp:compare_states × (Vector (option sig) (S n)).
let 〈q,a〉 ≝ p in
match pi1 … q with
[ None ⇒ 〈comp2,null_action ? n〉
| Some ai ⇒ match nth j ? a (None ?) with
[ None ⇒ 〈comp2,null_action ? n〉
- | Some aj ⇒ if ai == aj
+ | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj
then 〈comp1,change_vec ? (S n)
(change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
(Some ? 〈aj,R〉) j〉
| S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
definition compare_step ≝
- λi,j,sig,n.
- mk_mTM sig n compare_states (trans_compare_step i j sig n)
+ λi,j,sig,n,is_endc.
+ mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc)
comp0 (λq.q == comp1 ∨ q == comp2).
definition R_comp_step_true ≝
- λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
+ λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
∃x.
+ is_endc x = false ∧
current ? (nth i ? int (niltape ?)) = Some ? x ∧
current ? (nth j ? int (niltape ?)) = Some ? x ∧
outt = change_vec ??
(tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
definition R_comp_step_false ≝
- λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
- (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
+ λi,j:nat.λsig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
current ? (nth i ? int (niltape ?)) = None ? ∨
current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
lemma comp_q0_q2_null :
- ∀i,j,sig,n,v.i < S n → j < S n →
+ ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
(nth i ? (current_chars ?? v) (None ?) = None ? ∨
nth j ? (current_chars ?? v) (None ?) = None ?) →
- step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v)
+ step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
= mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj
+#i #j #sig #n #is_endc #v #Hi #Hj
whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
* #Hcurrent
[ @eq_f2
qed.
lemma comp_q0_q2_neq :
- ∀i,j,sig,n,v.i < S n → j < S n →
- nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?) →
- step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v)
+ ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
+ ((∃x.nth i ? (current_chars ?? v) (None ?) = Some ? x ∧ is_endc x = true) ∨
+ nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) →
+ step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
= mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
+#i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
cases (nth i ?? (None ?)) in ⊢ (???%→?);
[ #Hnth #_ @comp_q0_q2_null // % //
| #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?)))
cases (nth j ?? (None ?)) in ⊢ (???%→?);
[ #Hnth #_ @comp_q0_q2_null // %2 //
- | #aj #Haj #Hneq
- whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
- [ whd in match (trans ????); >Hai >Haj
- whd in ⊢ (??(???%)?); >(\bf ?) // @(not_to_not … Hneq) //
- | whd in match (trans ????); >Hai >Haj
- whd in ⊢ (??(???????(???%))?); >(\bf ?) /2 by not_to_not/
- @tape_move_null_action
-] ]
+ | #aj #Haj *
+ [ * #c * >Hai #Heq #Hendc whd in ⊢ (??%?);
+ >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+ [ whd in match (trans ????); >Hai >Haj destruct (Heq)
+ whd in ⊢ (??(???%)?); >Hendc //
+ | whd in match (trans ????); >Hai >Haj destruct (Heq)
+ whd in ⊢ (??(???????(???%))?); >Hendc @tape_move_null_action
+ ]
+ | #Hneq
+ whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+ [ whd in match (trans ????); >Hai >Haj
+ whd in ⊢ (??(???%)?); cut ((¬is_endc ai∧ai==aj)=false)
+ [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // |#Hcut >Hcut //]
+ | whd in match (trans ????); >Hai >Haj
+ whd in ⊢ (??(???????(???%))?); cut ((¬is_endc ai∧ai==aj)=false)
+ [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) //
+ |#Hcut >Hcut @tape_move_null_action
+ ]
+ ]
+ ]
+ ]
+]
qed.
lemma comp_q0_q1 :
- ∀i,j,sig,n,v,a.i ≠ j → i < S n → j < S n →
- nth i ? (current_chars ?? v) (None ?) = Some ? a →
+ ∀i,j,sig,n,is_endc,v,a.i ≠ j → i < S n → j < S n →
+ nth i ? (current_chars ?? v) (None ?) = Some ? a → is_endc a = false →
nth j ? (current_chars ?? v) (None ?) = Some ? a →
- step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) =
+ step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) =
mk_mconfig ??? comp1
(change_vec ? (S n)
(change_vec ?? v
(tape_move ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i)
(tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) j).
-#i #j #sig #n #v #a #Heq #Hi #Hj #Ha1 #Ha2
+#i #j #sig #n #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
[ whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
+ >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
| whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >(\b ?) //
+ >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) //
change with (change_vec ?????) in ⊢ (??(???????%)?);
<(change_vec_same … v j (niltape ?)) in ⊢ (??%?);
<(change_vec_same … v i (niltape ?)) in ⊢ (??%?);
qed.
lemma sem_comp_step :
- ∀i,j,sig,n.i ≠ j → i < S n → j < S n →
- compare_step i j sig n ⊨
- [ comp1: R_comp_step_true i j sig n,
- R_comp_step_false i j sig n ].
-#i #j #sig #n #Hneq #Hi #Hj #int
+ ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
+ compare_step i j sig n is_endc ⊨
+ [ comp1: R_comp_step_true i j sig n is_endc,
+ R_comp_step_false i j sig n is_endc ].
+#i #j #sig #n #is_endc #Hneq #Hi #Hj #int
lapply (refl ? (current ? (nth i ? int (niltape ?))))
cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
[ #Hcuri %{2} %
[ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%);
@sym_eq @nth_vec_map
| normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % >Ha >Hcurj % % % #H destruct (H) ] ]
- | #b #Hb %{2} cases (true_or_false (a == b)) #Hab
+ | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
+ | #b #Hb %{2}
+ cases (true_or_false (is_endc a)) #Haendc
[ %
- [| % [ %
- [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) //
- [>(\P Hab) <Hb @sym_eq @nth_vec_map
- |<Ha @sym_eq @nth_vec_map ]
- | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) // ]
- | * #H @False_ind @H %
- ] ]
- | %
[| % [ %
[whd in ⊢ (??%?); >comp_q0_q2_neq //
- <(nth_vec_map ?? (current …) i ? int (niltape ?))
- <(nth_vec_map ?? (current …) j ? int (niltape ?)) >Ha >Hb
- @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
+ % %{a} % // <Ha @sym_eq @nth_vec_map
| normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // % % >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+ | #_ % // % % % >Ha %{a} % // ]
+ ]
+ |cases (true_or_false (a == b)) #Hab
+ [ %
+ [| % [ %
+ [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) //
+ [>(\P Hab) <Hb @sym_eq @nth_vec_map
+ |<Ha @sym_eq @nth_vec_map ]
+ | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) % // ]
+ | * #H @False_ind @H %
+ ] ]
+ | %
+ [| % [ %
+ [whd in ⊢ (??%?); >comp_q0_q2_neq //
+ <(nth_vec_map ?? (current …) i ? int (niltape ?))
+ <(nth_vec_map ?? (current …) j ? int (niltape ?)) %2 >Ha >Hb
+ @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
+ | normalize in ⊢ (%→?); #H destruct (H) ]
+ | #_ % // % % %2 >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+ ]
]
]
]
qed.
-definition compare ≝ λi,j,sig,n.
- whileTM … (compare_step i j sig n) comp1.
+definition compare ≝ λi,j,sig,n,is_endc.
+ whileTM … (compare_step i j sig n is_endc) comp1.
definition R_compare ≝
- λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
- (current sig (nth i (tape sig) int (niltape sig))
- ≠current sig (nth j (tape sig) int (niltape sig)) →
- outt = int) ∧
+ λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
+ current ? (nth i ? int (niltape ?)) = None ? ∨
+ current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
(∀ls,x,xs,ci,rs,ls0,cj,rs0.
nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj →
+ nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
+ (is_endc ci = true ∨ ci ≠ cj) →
outt = change_vec ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
(midtape sig (reverse ? xs@x::ls0) cj rs0) j).
-
-lemma wsem_compare : ∀i,j,sig,n.i ≠ j → i < S n → j < S n →
- compare i j sig n ⊫ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj #ta #k #outc #Hloop
-lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) //
+
+lemma wsem_compare : ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
+ compare i j sig n is_endc ⊫ R_compare i j sig n is_endc.
+#i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop
+lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * * [ *
- [ #Hcicj #Houtc %
+[ #tc whd in ⊢ (%→?); * * [ * [ *
+ [* #curi * #Hcuri #Hendi #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj #Hnotendc
+ @False_ind
+ >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H)
+ >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H)
+ ]
+ |#Hcicj #Houtc %
[ #_ @Houtc
| #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj
>Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
- ]
+ ]]
| #Hci #Houtc %
[ #_ @Houtc
| #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
[ #_ @Houtc
| #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #tc #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
+ | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
#IH1 #IH2 %
- [ >Hci >Hcj * #H @False_ind @H %
+ [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H)
+ |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]]
| #ls #c0 #xs #ci #rs #ls0 #cj #rs0 cases xs
- [ #Hnthi #Hnthj #Hcicj >IH1
+ [ #Hnthi #Hnthj #Hnotendc #Hcicj >IH1
[ >Hd @eq_f3 //
[ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
>Hnthi in Hci;normalize #H destruct (H) %
| >(?:c0=x) [ >Hnthj % ]
>Hnthi in Hci;normalize #H destruct (H) % ]
| >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not ??? Hcicj)
- #H destruct (H) % ]
- | #x0 #xs0 #Hnthi #Hnthj #Hcicj
+ >nth_change_vec // >Hnthi >Hnthj normalize
+ cases Hcicj #Hcase
+ [%1 %{ci} % // | %2 %1 %1 @(not_to_not ??? Hcase) #H destruct (H) % ]
+ ]
+ | #x0 #xs0 #Hnthi #Hnthj #Hnotendc #Hcicj
>(IH2 (c0::ls) x0 xs0 ci rs (c0::ls0) cj rs0 … Hcicj)
[ >Hd >change_vec_commute in ⊢ (??%?); //
>change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
@sym_not_eq //
+ | #c1 #Hc1 @Hnotendc @memb_cons @Hc1
| >Hd >nth_change_vec // >Hnthj normalize
>Hnthi in Hci;normalize #H destruct (H) %
| >Hd >nth_change_vec_neq [|@sym_not_eq //] >Hnthi
]]]
qed.
-lemma terminate_compare : ∀i,j,sig,n,t.
+lemma terminate_compare : ∀i,j,sig,n,is_endc,t.
i ≠ j → i < S n → j < S n →
- compare i j sig n ↓ t.
-#i #j #sig #n #t #Hneq #Hi #Hj
+ compare i j sig n is_endc ↓ t.
+#i #j #sig #n #is_endc #t #Hneq #Hi #Hj
@(terminate_while … (sem_comp_step …)) //
<(change_vec_same … t i (niltape ?))
cases (nth i (tape sig) t (niltape ?))
-[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-|2,3: #a0 #al0 % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+[ % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+|2,3: #a0 #al0 % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
- [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
+ [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
#H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 %
- #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+ #t2 * #x0 * * * #Hendcx0 >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
>nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
|#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
normalize in ⊢ (%→?); #H destruct (H) #Hcur
]
qed.
-lemma sem_compare : ∀i,j,sig,n.
+lemma sem_compare : ∀i,j,sig,n,is_endc.
i ≠ j → i < S n → j < S n →
- compare i j sig n ⊨ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize /2/
+ compare i j sig n is_endc ⊨ R_compare i j sig n is_endc.
+#i #j #sig #n #is_endc #Hneq #Hi #Hj @WRealize_to_Realize /2/
+qed.
+
+(*
+ |conf1 $
+ |confin 0/1 confout move
+
+ match machine step ≝
+ compare;
+ if (cur(src) != $)
+ then
+ parmoveL;
+ moveR(dst);
+ else nop
+ *)
+
+definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
+ compare src dst sig n ·
+ (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
+ (single_finalTM ??
+ (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
+ (nop …)
+ tc_true).
+
+definition Rtc_multi_true ≝
+ λalpha,test,n,i.λt1,t2:Vector ? (S n).
+ (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
+
+definition Rtc_multi_false ≝
+ λalpha,test,n,i.λt1,t2:Vector ? (S n).
+ (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
+
+definition R_match_step_false ≝
+ λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ (((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ (* current ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨ *)
+ current sig (nth src (tape sig) int (niltape sig)) = None ? ∨
+ current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨
+ ∃ls,ls0,rs,rs0,x,xs. ∀rsi,rsj,end,c.
+ rs = end::rsi → rs0 = c::rsj →
+ is_endc x = false ∧ is_endc end = true ∧
+ nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src)
+ (midtape sig (reverse ? xs@x::ls0) c rsj) dst.
+
+definition R_match_step_true ≝
+ λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+ ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
+ is_startc s = true →
+ (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
+ (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 →
+ s ≠ s1 →
+ outt = change_vec ?? int
+ (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
+ (∀ls,x,xs,ci,rs,ls0,cj,rs0.
+ nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj →
+ outt = change_vec ?? int
+ (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false).
+
+lemma sem_test_char_multi :
+ ∀alpha,test,n,i.i ≤ n →
+ inject_TM ? (test_char ? test) n i ⊨
+ [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
+#alpha #test #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
+ #Hcur #Htestc #Hnth_i #Hnth_j %
+ [ %{c} % //
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
+ [ @Htestc
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+qed.
+
+axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S. ∃l,tl1,tl2.
+ l1 = l@tl1 ∧ l2 = l@tl2 ∧ ∀a,b,tla,tlb. tl1 = a::tla → tl2 = b::tlb → a≠b.
+
+axiom daemon : ∀X:Prop.X.
+
+lemma sem_match_step :
+ ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
+ match_step src dst sig n is_startc is_endc ⊨
+ [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
+ R_match_step_true src dst sig n is_startc is_endc,
+ R_match_step_false src dst sig n is_endc ].
+#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
+@(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
+ (acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
+ (sem_seq …
+ (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
+ (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
+ (sem_nop …)))
+[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
+ #Htb #s #Hcurta_src #Hstart #Hnotstart %
+ [ #s1 #Hcurta_dst #Hneqss1
+ lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
+ [|@Hcomp1 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
+ #Hcurtc * #te * * #_ #Hte >Hte // whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
+ [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
+ #ls * #rs #Hta_mid >(Htbdst … Hta_mid) >Hta_mid cases rs //
+ | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
+ | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
+ normalize #H destruct (H) // ]
+ |#ls #x #xs #ci #rs #ls0 #cj #rs0 #Htasrc_mid #Htadst_mid #Hcicj
+ lapply (Hcomp2 … Htasrc_mid Htadst_mid Hcicj) -Hcomp2 #Hcomp2
+ cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
+ #H destruct (H)
+ >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs0 (refl ??)) //
+ [| >Hcomp2 >nth_change_vec //
+ | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid
+ cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [@memb_append_l2 >(\P Hc0) @memb_hd
+ |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
+ ]
+ | >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
+ * * #_ #Htbdst #Htbelse %
+ [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj::rs0))
+ [ cases xs //
+ | >nth_change_vec // ]
+ | >nth_change_vec_neq [|@sym_not_eq //]
+ <Htbelse [|@sym_not_eq // ]
+ >nth_change_vec_neq [|@sym_not_eq //]
+ (* STOP. *)
+ cases (decidable_eq_nat i src) #Hisrc
+ [ >Hisrc >nth_change_vec // >Htasrc_mid //
+ | >nth_change_vec_neq [|@sym_not_eq //]
+ <(Htbelse i) [|@sym_not_eq // ]
+ >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ]
+ >nth_change_vec_neq [|@sym_not_eq // ] //
+ ]
+ ]
+ | >Hcomp2 in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // whd in ⊢ (??%?→?);
+ #H destruct (H) cases (is_endc c) in Hcend;
+ normalize #H destruct (H) // ]
+ ]
+|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
+ whd in ⊢ (%→?); #Hout >Hout >Htb whd
+ lapply (current_to_midtape sig (nth src ? intape (niltape ?)))
+ cases (current … (nth src ? intape (niltape ?))) in Hcomp1;
+ [#Hcomp1 #_ %1 % [%1 %2 // | @Hcomp1 %1 %2 %]
+ |#c_src lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
+ cases (current … (nth dst ? intape (niltape ?)))
+ [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %2 %]
+ |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
+ [#Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
+ #ls_dst * #rs_dst #Hmid_dst #_
+ #Hmid_src cases (Hmid_src c_src (refl …)) -Hmid_src
+ #ls_src * #rs_src #Hmid_src %2
+ cases (comp_list … rs_src rs_dst) #xs * #rsi * #rsj * *
+ #Hrs_src #Hrs_dst #Hneq
+ %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs}
+ #rsi0 #rsj0 #end #c #Hend #Hc_dst
+ >Hrs_src in Hmid_src; >Hend #Hmid_src
+ >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
+ lapply(Hcomp2 … Hmid_src Hmid_dst ?)
+ [@(Hneq … Hend Hc_dst)]
+ -Hcomp2 #Hcomp2 <Hcomp2
+ % // % [ %
+ [>Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // #H lapply (H ? (refl …))
+ cases (is_endc end) normalize //
+ |@Hmid_src]
+ |@Hmid_dst]
+ |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
+ %1 %
+ [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
+ [| % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
+ cases (is_endc c_src) //
+ >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
+ |@Hcomp1 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
+ ]
+ ]
+ ]
+ ]
qed.
+definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
+ whileTM … (match_step src dst sig n is_startc is_endc)
+ (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
+
+definition R_match_m ≝
+ λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+ (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ current ? (nth i ? int (niltape ?)) = None ? ∨
+ current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
+ (∀ls,x,xs,ci,rs,ls0,x0,rs0.
+ is_startc x = true → is_endc ci = true →
+ nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+ ∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+ (midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j).
+
+lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+src ≠ dst → src < S n → dst < S n →
+ match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
+#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
+lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc whd in ⊢ (%→?); *
+ [ * * [ *
+ [ * #cur_src * #H1 #H2 #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnthi #Hnthj
+ >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+ ]
+ | #Hci #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+
+
+
+[ #tc whd in ⊢ (%→?); * * [ *
+
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