X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Funistep_aux.ma;h=8033c222627b6123481c324d528aab44652260aa;hb=31790e8f6fa051f710f41e4c17d67701874ba331;hp=ed15669e02edf83710b9ccd458f9431a1b588408;hpb=3a9c3c16e7c7e3a35640a0afa53f044a4f87ed65;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/unistep_aux.ma b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
index ed15669e0..8033c2226 100644
--- a/matita/matita/lib/turing/multi_universal/unistep_aux.ma
+++ b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
@@ -9,59 +9,19 @@
      \ /   GNU General Public License Version 2   
       V_____________________________________________________________*)
 
-include "turing/multi_universal/moves_2.ma".
-include "turing/multi_universal/match.ma".
-include "turing/multi_universal/copy.ma".
+include "turing/auxiliary_machines.ma".
+include "turing/auxiliary_multi_machines.ma".
 include "turing/multi_universal/alphabet.ma".
 include "turing/multi_universal/tuples.ma".
 
-(*
-
-  in.obj : ...  x ...
-                ^
-  in.cfg : ...  ? ? ...
-                    ^
-                
-  out.cfg : ... 1 x ...
-                  ^
-                  
-  ---------------------
-  current (in.obj) = None
-  
-  in.cfg : ...  ? ? ...
-                    ^
-
-  out.cfg : ... 0 0 ...
-                  ^
-                  
-  obj_to_cfg ≝ 
-    move_l(cfg);
-    move_l(cfg);
-    (if (current(in.obj)) == None
-       then write(0,cfg);
-            move_r(cfg);
-            write(0,cfg);
-       else write(1,cfg);
-            move_r(cfg);
-            copy_step(obj,cfg);
-            move_l(obj);)
-    move_to_end_l(cfg);
-    move_r(cfg);
-       
-  
-  cfg_to_obj
-*)
 
 definition obj ≝ (0:DeqNat).
 definition cfg ≝ (1:DeqNat).
 definition prg ≝ (2:DeqNat).
 
 definition obj_to_cfg ≝
-  mmove cfg FSUnialpha 2 L ·
   (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
-    (copy_step obj cfg FSUnialpha 2 ·
-     mmove cfg FSUnialpha 2 L ·
-     mmove obj FSUnialpha 2 L) 
+    (copy_char_N obj cfg FSUnialpha 2)
     (inject_TM ? (write FSUnialpha null) 2 cfg)
      tc_true) ·
   inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
@@ -69,7 +29,7 @@ definition obj_to_cfg ≝
   
 definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
   ∀c,ls.
-  nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] → 
+  nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] → 
   (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso → 
    t2 = change_vec ?? t1 
          (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
@@ -77,22 +37,18 @@ definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
    t2 = change_vec ?? t1
          (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls))) 
            (tail ? (reverse ? (null::ls)))) cfg).
-           
-axiom sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig.
+
+(*
 axiom accRealize_to_Realize :
   ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
   M ⊨ [ acc: Rtrue, Rfalse ] →  M ⊨ Rtrue ∪ Rfalse.
+*)
   
 lemma eq_mk_tape_rightof :
  ∀alpha,a,al.mk_tape alpha (a::al) (None ?) [ ] = rightof ? a al.
 #alpha #a #al %
 qed.
 
-axiom daemon : ∀P:Prop.P.
-
-definition option_cons ≝ λsig.λc:option sig.λl.
-  match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
-
 lemma tape_move_mk_tape_R :
   ∀sig,ls,c,rs.
   (c = None ? → ls = [ ] ∨ rs = [ ]) → 
@@ -103,70 +59,110 @@ lemma tape_move_mk_tape_R :
 normalize //
 qed.
 
+lemma None_or_Some: ∀A.∀a. a =None A ∨ ∃b. a = Some ? b.
+#A * /2/ #a %2 %{a} %
+qed.
+
+lemma not_None_to_Some: ∀A.∀a. a ≠ None A → ∃b. a = Some ? b.
+#A * /2/ * #H @False_ind @H %
+qed. 
+
 lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
-@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
-   (sem_seq ??????
-    (sem_if ??????????
-     (sem_test_null_multi ?? obj ?)
-      (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
-       (sem_seq ?????? (sem_move_multi ? 2 cfg L ?)
-        (sem_move_multi ? 2 obj L ?)))
-      (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
-     (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
-       (sem_move_multi ? 2 cfg R ?)))) //
-#ta #tb *
-#tc * whd in ⊢ (%→?); #Htc *
-#td * *
-[ * #te * * #Hcurtc #Hte
-  * destruct (Hte) #te * *
-  [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte
-    * #tf * whd in ⊢ (%→%→?); #Htf #Htd
-    * #tg * * * whd in ⊢ (%→%→%→%→?); #Htg1 #Htg2 #Htg3 #Htb
-    #c #ls #Hta1 %
-    [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof 
-      whd in match (tape_move ???); #Htc
-      cut (tg = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
-      [@daemon] -Htg1 -Htg2 -Htg3 #Htg destruct (Htg Htf Hte Htd Htc Htb)
-      >change_vec_change_vec >change_vec_change_vec
-      >change_vec_commute // >change_vec_change_vec
-      >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
-      >change_vec_commute // >change_vec_change_vec
-      >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //] 
-      >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
-      >change_vec_commute [|@sym_not_eq //] @eq_f3 //
-      [ >Hta2 cases rso in Hta2; whd in match (tape_move_mono ???);
-        [ #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same
-        | #r1 #rs1 #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same ]
-      | >tape_move_mk_tape_R [| #_ % %] >reverse_cons
-        >nth_change_vec_neq in Hcurtc1; [|@sym_not_eq //] >Hta2
-        normalize in ⊢ (%→?); #H destruct (H) %
-      ]
-    | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
-      >Hta2 #H destruct (H)
+@(sem_seq_app FSUnialpha 2 ????? 
+  (sem_if ??????????
+   (sem_test_null_multi ?? obj ?)
+   (sem_copy_char_N …)
+   (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
+  (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
+    (sem_move_multi ? 2 cfg R ?))) //
+#ta #tout *
+#tb * #Hif * #tc * #HM2 #HM3 #c #ls #Hcfg
+(* Hif *)
+cases Hif -Hif
+[ * #t1 * * #Hcurta #Ht1 destruct (Ht1)
+  lapply (not_None_to_Some … Hcurta) * #curta #Hcurtaeq
+  whd in ⊢ (%→?); #Htb % [2: #Hcur @False_ind /2/]
+  #lso #xo #rso #Hobjta cases HM2 whd in ⊢ (%→?); * #_
+  #HM2 #Heq >Htb in HM2; >nth_change_vec [2: @leb_true_to_le %]
+  >Hcfg >Hcurtaeq #HM2 lapply (HM2 … (refl ??)) -HM2 
+  whd in match (left ??); whd in match (right ??);
+  >reverse_cons #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2 
+  cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
+  [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
+    [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %] 
+     <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+     >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %] 
+    |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
+     >Hobjta in Hcurtaeq; whd in ⊢ (??%?→?); #Htmp destruct(Htmp)
+     >tape_move_mk_tape_R [2: #_ %1 %] %
     ]
-  | * #Hcurtc0 #Hte #_ #_ #c #ls #Hta1 >Hta1 in Htc; >eq_mk_tape_rightof
-    whd in match (tape_move ???); #Htc >Htc in Hcurtc0; *
-    [ >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
-      #Hcurtc #Hcurtc0 >Hcurtc0 in Hcurtc; * #H @False_ind @H %
-    | >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) ]
+  |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %] 
+   >nth_change_vec_neq [2:@eqb_false_to_not_eq %] 
+   <(Heq 2) [2:@eqb_false_to_not_eq %] >Htb
+   >nth_change_vec_neq [%|2:@eqb_false_to_not_eq %] 
+  ]
+| * #t1 * * #Hcurta #Ht1 destruct (Ht1)
+  * whd in ⊢ (%→?); #Htb lapply (Htb … Hcfg) -Htb #Htb
+  #Htbeq % 
+    [#lso #xo #rso #Hmid @False_ind >Hmid in Hcurta;
+     whd in ⊢ (??%?→?); #Htmp destruct (Htmp)]
+  #_ cases HM2 whd in ⊢ (%→?); * #_
+  #HM2 #Heq >Htb in HM2; #HM2 lapply (HM2 … (refl ??)) -HM2 
+  #Htc >HM3 @(eq_vec … (niltape ?)) #i #lei2 
+  cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
+  [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
+    [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+     <(Heq 0) [2:@eqb_false_to_not_eq %] >Htb
+     <(Htbeq 0) [2:@eqb_false_to_not_eq %] % 
+    |#Hi >Hi >nth_change_vec // >nth_change_vec // >Htc
+     >tape_move_mk_tape_R [2: #_ %1 %] >reverse_cons % 
+    ]
+  |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %] 
+   >nth_change_vec_neq [2:@eqb_false_to_not_eq %] 
+   <(Heq 2) [2:@eqb_false_to_not_eq %] 
+   <(Htbeq 2) [%|@eqb_false_to_not_eq %] 
   ]
-| * #te * * #Hcurtc #Hte
-  * whd in ⊢ (%→%→?); #Htd1 #Htd2
-  * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb
-  #c #ls #Hta1 %
-  [ #lso #x #rso #Hta2 >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] 
-    >Hta2 normalize in ⊢ (%→?); #H destruct (H)
-  | #_ >Hta1 in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
-    destruct (Hte) cut (td = change_vec ?? tc (midtape ? ls null []) cfg)
-    [@daemon] -Htd1 -Htd2 #Htd
-    -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
-    [@daemon] -Htf2 -Htf3 #Htf destruct (Htf Htd Htc Htb)
-    >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
-    >change_vec_change_vec >change_vec_change_vec >nth_change_vec //
-    >reverse_cons >tape_move_mk_tape_R /2/ ]
 ]
 qed.
 
+(* another semantics for obj_to_cfg *)
+definition low_char' ≝ λc.
+  match c with
+  [ None ⇒ null 
+  | Some b ⇒ if (is_bit b) then b else null
+  ].
+  
+lemma low_char_option : ∀s.
+  low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
+* //
+qed.
+
+definition R_obj_to_cfg1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+  ∀c,ls.
+  nth cfg ? t1 (niltape ?) = midtape ? ls c [ ] → 
+  let x ≝ current ? (nth obj ? t1 (niltape ?)) in
+  (∀b. x= Some ? b → is_bit b = true) →
+  t2 = change_vec ?? t1
+         (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (low_char' x::ls))) 
+           (tail ? (reverse ? (low_char' x::ls)))) cfg.
+
+lemma sem_obj_to_cfg1: obj_to_cfg ⊨ R_obj_to_cfg1.
+@(Realize_to_Realize … sem_obj_to_cfg) #t1 #t2 #Hsem
+#c #ls #Hcfg lapply(Hsem c ls Hcfg) * #HSome #HNone #Hb 
+cases (None_or_Some ? (current ? (nth obj ? t1 (niltape ?)))) 
+  [#Hcur >Hcur @HNone @Hcur
+  |* #b #Hb1 >Hb1
+   cut (low_char' (Some ? b) = b)  [whd in ⊢ (??%?); >(Hb b Hb1) %] #Hlow >Hlow
+   lapply(current_to_midtape … Hb1) * #lsobj * #rsobj #Hmid
+   @(HSome … Hmid)
+  ]
+qed.
+    
+(* test_null_char *)
 definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
 
 definition R_test_null_char_true ≝ λt1,t2.
@@ -190,83 +186,25 @@ lemma sem_test_null_char :
   | <Houtc % ] ]
 qed.
 
-definition copy_char_states ≝ initN 3.
-
-definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
-definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
-
-definition trans_copy_char ≝ 
- λsrc,dst.λsig:FinSet.λn.
- λp:copy_char_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ 〈cc1,change_vec ? (S n) 
-           (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
-           (〈nth src ? a (None ?),R〉) dst〉
- | S _ ⇒ 〈cc1,null_action ? n〉 ].
-
-definition copy_char ≝ 
-  λsrc,dst,sig,n.
-  mk_mTM sig n copy_char_states (trans_copy_char src dst sig n) 
-    cc0 (λq.q == cc1).
-
-definition R_copy_char ≝ 
-  λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
-  outt = change_vec ?? 
-         (change_vec ?? int
-          (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
-          (tape_move_mono ? (nth dst ? int (niltape ?)) 
-           〈current ? (nth src ? int (niltape ?)), R〉) dst.
-
-lemma copy_char_q0_q1 :
-  ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n → 
-  step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
-    mk_mconfig ??? cc1 
-     (change_vec ? (S n) 
-       (change_vec ?? v
-         (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
-            (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
-#src #dst #sig #n #v #Heq #Hsrc #Hdst
-whd in ⊢ (??%?);
-<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
-<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
->tape_move_multi_def @eq_f2 //
->pmap_change >pmap_change <tape_move_multi_def
->tape_move_null_action @eq_f2 // @eq_f2
-[ >change_vec_same %
-| >change_vec_same >change_vec_same // ]
-qed.
-
-lemma sem_copy_char:
-  ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
-  copy_char src dst sig n ⊨ R_copy_char src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #int
-%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
-qed.
-
 definition cfg_to_obj ≝
   mmove cfg FSUnialpha 2 L ·
   (ifTM ?? (inject_TM ? test_null_char 2 cfg)
     (nop ? 2)
-    (copy_char cfg obj FSUnialpha 2 ·
-     mmove cfg FSUnialpha 2 L ·
-     mmove obj FSUnialpha 2 L) 
-     tc_true) ·
+    (copy_char_N cfg obj FSUnialpha 2)
+    tc_true).
+(* ·
   inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
-  mmove cfg FSUnialpha 2 R.
+  mmove cfg FSUnialpha 2 R. *)
   
 definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
   ∀c,ls.
   nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] → 
-  (c = null → 
-   t2 = change_vec ?? t1
-         (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (c::ls))) 
-           (tail ? (reverse ? (c::ls)))) cfg) ∧
+  (c = null → t2 = change_vec ?? t1 (midtape ? ls c [ ]) cfg) ∧
   (c ≠ null → 
    t2 = change_vec ??
           (change_vec ?? t1
              (midtape ? (left ? (nth obj ? t1 (niltape ?))) c (right ? (nth obj ? t1 (niltape ?)))) obj)
-          (mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg).
+          (midtape ? ls c [ ]) cfg).
           
 lemma tape_move_mk_tape_L :
   ∀sig,ls,c,rs.
@@ -280,65 +218,37 @@ qed.
 
 lemma sem_cfg_to_obj : cfg_to_obj ⊨  R_cfg_to_obj.
 @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?)
-  (sem_seq ??????
-   (sem_if ??????????
-    (acc_sem_inject ?????? cfg ? sem_test_null_char)
-    (sem_nop …)
-    (sem_seq ?????? (sem_copy_char …)
-     (sem_seq ?????? (sem_move_multi ? 2 cfg L ?) (sem_move_multi ? 2 obj L ?))))
-   (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
-    (sem_move_multi ? 2 cfg R ?)))) // [@sym_not_eq //]
+  (sem_if ??????????
+   (acc_sem_inject ?????? cfg ? sem_test_null_char)
+   (sem_nop …)
+   (sem_copy_char_N …)))
+// [@sym_not_eq //]
 #ta #tb *
 #tc * whd in ⊢ (%→?); #Htc *
-#td * *
-[ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htd destruct (Htd)
-  * #tf * * * #Htf1 #Htf2 #Htf3
-  whd in ⊢ (%→?); #Htb
+[ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htb destruct (Htb)
   #c #ls #Hta %
   [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
-    cut (te = tc) [@daemon] -Hte1 -Hte2 #Hte
-    cut (tf = change_vec ? 3 te (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
-    [@daemon] -Htf1 -Htf2 -Htf3 #Htf
-    destruct (Htf Hte Htc Htb)
-    >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
-    >nth_change_vec // >tape_move_mk_tape_R [| #_ % % ] 
-    >reverse_cons %
-  | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
-    >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?); 
-    #H destruct (H) @False_ind cases Hc /2/ ]
-  * #tf * *
+    cut (te = tc)
+    [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
+      -Hte1 -Hte2 #Hte //
+    | #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
+      >Htc in Hcurtc; >nth_change_vec // normalize in ⊢ (%→?); 
+      #H destruct (H) @False_ind cases Hc /2/ ]
 | * #te * * * #Hcurtc #Hte1 #Hte2
-  * #tf * whd in ⊢ (%→?); #Htf
-  * #tg * whd in ⊢ (%→%→?); #Htg #Htd
-  * #th * * * #Hth1 #Hth2 #Hth3
-  whd in ⊢ (%→?); #Htb 
+  whd in ⊢ (%→?); #Htb
   #c #ls #Hta % #Hc
-  [ >Htc in Hcurtc; >Hta >nth_change_vec // >tape_move_mk_tape_L //
-    >Hc normalize in ⊢ (%→?); * #H @False_ind /2/
-  | cut (te = tc) [ @daemon ] -Hte1 -Hte2 #Hte
-    cut (th = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
-    [@daemon] -Hth1 -Hth2 -Hth3 #Hth
-    destruct (Hth Hte Hta Htb Htd Htg Htc Htf) 
-    >change_vec_change_vec >change_vec_change_vec
-    >change_vec_commute // >change_vec_change_vec
-    >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
-    >change_vec_commute // >change_vec_change_vec    
-    >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //] 
-    >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
-    >change_vec_commute [|@sym_not_eq //]
-    @eq_f3 //
-    [ >Hta >tape_move_mk_tape_L // >nth_change_vec // whd in match (current ??);
-      @eq_f2 // cases (nth obj ? ta (niltape ?))
-      [| #r0 #rs0 | #l0 #ls0 | #ls0 #c0 #rs0 ] try %
-      cases rs0 //
-    | >reverse_cons >tape_move_mk_tape_R // #_ % % ]
+  [ >Htc in Hcurtc; >Hta >nth_change_vec // 
+    normalize in ⊢ (%→?); * #H @False_ind /2/
+  | cut (te = tc)
+    [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
+      >change_vec_same // ] -Hte1 -Hte2 #Hte destruct (Hte)
+    >Hta in Htc; whd in match (tape_move ???); #Htc
+    >Htc in Htb; >nth_change_vec // 
+    >nth_change_vec_neq [2:@eqb_false_to_not_eq //] >Hta 
+    #Htb @Htb
   ]
-]
 qed.
 
-(* macchina che muove il nastro obj a destra o sinistra a seconda del valore
-   del current di prg, che codifica la direzione in cui ci muoviamo *)
-
 definition char_to_move ≝ λc.match c with
   [ bit b ⇒ if b then R else L
   | _ ⇒ N].
@@ -346,7 +256,58 @@ definition char_to_move ≝ λc.match c with
 definition char_to_bit_option ≝ λc.match c with
   [ bit b ⇒ Some ? (bit b)
   | _ ⇒ None ?]. 
- 
+
+definition R_cfg_to_obj1 ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+  ∀c,ls.
+  nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] → 
+  c ≠ bar →
+  let new_obj ≝ 
+      tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c) in
+    t2 = change_vec ??
+          (change_vec ?? t1
+            (tape_write ? (nth obj ? t1 (niltape ?)) (char_to_bit_option c)) obj)
+          (midtape ? ls c [ ]) cfg.
+
+lemma sem_cfg_to_obj1: cfg_to_obj ⊨  R_cfg_to_obj1.
+@(Realize_to_Realize … sem_cfg_to_obj) #t1 #t2 #H #c #ls #Hcfg #Hbar
+cases (H c ls Hcfg) cases (true_or_false (c==null)) #Hc
+  [#Ht2 #_ >(Ht2 (\P Hc)) -Ht2 @(eq_vec … (niltape ?))
+   #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
+    [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
+      [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
+       >nth_change_vec_neq [2:@eqb_false_to_not_eq %] 
+       >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
+       >nth_change_vec // >(\P Hc) % 
+      |#Hi >Hi >nth_change_vec //
+      ] 
+    |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
+    ]
+  |#_ #Ht2 >(Ht2 (\Pf Hc)) -Ht2 @(eq_vec … (niltape ?))
+   #i #lei2 cases (le_to_or_lt_eq … (le_S_S_to_le …lei2))
+    [#lei1 cases (le_to_or_lt_eq … (le_S_S_to_le …lei1))
+      [#lei0 lapply(le_n_O_to_eq … (le_S_S_to_le …lei0)) #eqi <eqi
+       >nth_change_vec_neq [2:@eqb_false_to_not_eq %] 
+       >nth_change_vec_neq in ⊢ (???%); [2:@eqb_false_to_not_eq %]
+       >nth_change_vec // >nth_change_vec // 
+       lapply (\bf Hbar) lapply Hc elim c //
+        [whd in ⊢ (??%?→?); #H destruct (H)
+        |#_ whd in ⊢ (??%?→?); #H destruct (H)
+        ]
+      |#Hi >Hi >nth_change_vec //
+      ] 
+    |#Hi >Hi >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
+     >nth_change_vec_neq [2:@eqb_false_to_not_eq %] %
+    ]
+  ]
+qed.
+       
+
+(* macchina che muove il nastro obj a destra o sinistra a seconda del valore
+   del current di prg, che codifica la direzione in cui ci muoviamo *)
+      
 definition tape_move_obj : mTM FSUnialpha 2 ≝ 
   ifTM ?? 
    (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
@@ -379,17 +340,24 @@ lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
 cases HR -HR
 [ * #tb * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htb1 #Htb2
   whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
-  [ >Hcurta_prg #H destruct (H) >(?:tb = ta) [|@daemon] %
+  [ >Hcurta_prg #H destruct (H) >(?:tb = ta) 
+    [| lapply (eq_vec_change_vec ??????? Htb1 Htb2)
+       >change_vec_same // ] %
   | >Hcurta_prg #H destruct (H) destruct (Hc) ]
   | >Hcurta_prg >Hc * #H @False_ind /2/ ]
-| * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta) [@daemon] -Htb1 -Htb2
-  #Htb destruct (Htb) *
+| * #tb * * * #Hnotfalse #Htb1 #Htb2 cut (tb = ta) 
+  [ lapply (eq_vec_change_vec ??????? Htb1 Htb2)
+     >change_vec_same // ] -Htb1 -Htb2 #Htb destruct (Htb) *
   [ * #tc * * * * #c * #Hcurta_prg #Hc lapply (\P Hc) -Hc #Hc #Htc1 #Htc2
     whd in ⊢ (%→%); #Houtc >Houtc -Houtc % [ %
     [ >Hcurta_prg #H destruct (H) destruct (Hc)
-    | >Hcurta_prg #H destruct (H) >(?:tc = ta) [|@daemon] % ]
+    | >Hcurta_prg #H destruct (H) >(?:tc = ta) 
+      [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
+        >change_vec_same // ] % ]
     | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
-  | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta) [@daemon] -Htc1 -Htc2 
+  | * #tc * * * #Hnottrue #Htc1 #Htc2 cut (tc = ta) 
+    [ lapply (eq_vec_change_vec ??????? Htc1 Htc2)
+      >change_vec_same // ] -Htc1 -Htc2 
     #Htc destruct (Htc) whd in ⊢ (%→?); #Houtc % [ %
     [ #Hcurta_prg lapply (\Pf (Hnotfalse ? Hcurta_prg)) * #H @False_ind /2/
     | #Hcurta_prg lapply (\Pf (Hnottrue ? Hcurta_prg)) * #H @False_ind /2/ ]
@@ -413,204 +381,141 @@ lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
 ]
 qed.
 
-definition restart_tape ≝ λi. 
-  inject_TM ? (move_to_end FSUnialpha L) 2 i ·
-  mmove i FSUnialpha 2 R. 
-
-definition unistep ≝ 
-  match_m cfg prg FSUnialpha 2 · 
-  restart_tape cfg · copy prg cfg FSUnialpha 2 ·
-  cfg_to_obj · tape_move_obj · restart_tape prg · obj_to_cfg.
-
-(*
-definition legal_tape ≝ λn,l,h,t.
-  ∃state,char,table.
-  nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
-  is_config n (bar::state@[char]) →  
-  nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
-  bar::table = table_TM n l h → *)
-
-definition list_of_tape ≝ λsig,t. 
-  left sig t@option_cons ? (current ? t) (right ? t).
-
-definition low_char' ≝ λc.
-  match c with
-  [ None ⇒ null 
-  | Some b ⇒ if (is_bit b) then b else null
-  ].
-  
-lemma low_char_option : ∀s.
-  low_char' (option_map FinBool FSUnialpha bit s) = low_char s.
-* //
-qed.
-
-definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
-  ∀state,char,table.
-  (* cfg *)
-  nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) →
-  is_config n (bar::state@[char]) →  
-  (* prg *)
-  nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
-  bar::table = table_TM n l h →
-  (* obj *)
-  only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
-  let conf ≝ (bar::state@[char]) in
-  (∃ll,lr.bar::table = ll@conf@lr) →
-(*
-    ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧ 
-    mem ? t l ∧  *)
-    ∀nstate,nchar,m,t. 
-    tuple_encoding n h t = (conf@nstate@[nchar;m])→ 
-    mem ? t l →
-    let new_obj ≝ 
-     tape_move_mono ? (nth obj ? t1 (niltape ?)) 
-       〈char_to_bit_option nchar,char_to_move m〉 in
-    let next_char ≝ low_char' (current ? new_obj) in
-    t2 = 
-      change_vec ??
-        (change_vec ?? t1 (midtape ? [ ] bar (nstate@[next_char])) cfg)
-        new_obj obj.
-
-definition tape_map ≝ λA,B:FinSet.λf:A→B.λt.
-  mk_tape B (map ?? f (left ? t)) 
-    (option_map ?? f (current ? t)) 
-    (map ?? f (right ? t)).
-    
-lemma map_list_of_tape: ∀A,B,f,t.
-  list_of_tape B (tape_map ?? f t) = map ?? f (list_of_tape A t).
-#A #B #f * // normalize // #ls #c #rs <map_append %
+(************** list of tape ******************)
+definition list_of_tape ≝ λsig.λt:tape sig.
+  reverse ? (left ? t)@option_cons ? (current ? t) (right ? t).
+
+lemma list_of_midtape: ∀sig,ls,c,rs.
+  list_of_tape sig (midtape ? ls c rs) = reverse ? ls@c::rs.
+// qed-.
+
+lemma list_of_rightof: ∀sig,ls,c.
+  list_of_tape sig (rightof ? c ls) = reverse ? (c::ls).
+#sig #ls #c <(append_nil ? (reverse ? (c::ls)))
+// qed-.
+
+lemma list_of_tape_move: ∀sig,t,m.
+  list_of_tape sig t = list_of_tape sig (tape_move ? t m).
+#sig #t * // cases t //
+ [(* rightof, move L *) #a #l >list_of_midtape 
+  >append_cons <reverse_single <reverse_append %
+ |(* midtape, move L *) * // 
+  #a #ls #c #rs >list_of_midtape >list_of_midtape
+  >reverse_cons >associative_append %
+ |(* midtape, move R *) #ls #c * 
+   [>list_of_midtape >list_of_rightof >reverse_cons %
+   |#a #rs >list_of_midtape >list_of_midtape >reverse_cons 
+    >associative_append %
+   ]
+ ]
 qed.
 
-lemma low_char_current : ∀t.
-  low_char' (current FSUnialpha (tape_map FinBool FSUnialpha bit t))
-  = low_char (current FinBool t).
-* // qed.
-
-definition low_tapes: ∀M:normalTM.∀c:nconfig (no_states M).Vector ? 3 ≝ 
-λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
-  [tape_map ?? bit (ctape ?? c);
-   midtape ? [ ] bar 
-    ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]);
-   midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
-  ].
-
-lemma obj_low_tapes: ∀M,c.
-  nth obj ? (low_tapes M c) (niltape ?) = tape_map ?? bit (ctape ?? c).
-// qed.
-
-lemma cfg_low_tapes: ∀M,c.
-  nth cfg ? (low_tapes M c) (niltape ?) = 
-  midtape ? [ ] bar 
-    ((bits_of_state ? (nhalt M) (cstate ?? c))@[low_char (current ? (ctape ?? c))]).
-// qed.
-
-lemma prg_low_tapes: ∀M,c.
-  nth prg ? (low_tapes M c) (niltape ?) = 
-  midtape ? [ ] bar (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))).
-// qed.
-
-(* commutation lemma for write *)
-lemma map_write: ∀t,cout.
- tape_write ? (tape_map FinBool ? bit t) (char_to_bit_option (low_char cout))
-  = tape_map ?? bit (tape_write ? t cout).
-#t * // #b whd in match (char_to_bit_option ?);
-whd in ⊢ (??%%); @eq_f3 [elim t // | // | elim t //]
+lemma list_of_tape_write: ∀sig,cond,t,c. 
+(∀b. c = Some ? b → cond b =true) →
+(∀x. mem ? x (list_of_tape ? t) → cond x =true ) →
+∀x. mem ? x (list_of_tape sig (tape_write ? t c)) → cond x =true.
+#sig #cond #t #c #Hc #Htape #x lapply Hc cases c 
+  [(* c is None *) #_ whd in match (tape_write ???); @Htape
+  |#b #Hb lapply (Hb … (refl ??)) -Hb #Hb
+   whd in match (tape_write ???); >list_of_midtape
+   #Hx cases(mem_append ???? Hx) -Hx
+    [#Hx @Htape @mem_append_l1 @Hx
+    |* [//] 
+     #Hx @Htape @mem_append_l2 cases (current sig t)
+      [@Hx | #c1 %2 @Hx]
+    ]
+  ]
 qed.
-
-(* commutation lemma for moves *)
-lemma map_move: ∀t,m.
- tape_move ? (tape_map FinBool ? bit t) (char_to_move (low_mv m))
-  = tape_map ?? bit (tape_move ? t m).
-#t * // whd in match (char_to_move ?);
-  [cases t // * // | cases t // #ls #a * //]
+   
+lemma current_in_list: ∀sig,t,b. 
+  current sig t = Some ? b → mem ? b (list_of_tape sig t).
+#sig #t #b cases t
+  [whd in ⊢ (??%?→?); #Htmp destruct
+  |#l #b whd in ⊢ (??%?→?); #Htmp destruct
+  |#l #b whd in ⊢ (??%?→?); #Htmp destruct
+  |#ls #c #rs whd in ⊢ (??%?→?); #Htmp destruct
+   >list_of_midtape @mem_append_l2 % %
+  ]
 qed.
   
-(* commutation lemma for actions *)
-lemma map_action: ∀t,cout,m.
- tape_move ? (tape_write ? (tape_map FinBool ? bit t)
-    (char_to_bit_option (low_char cout))) (char_to_move (low_mv m)) 
- = tape_map ?? bit (tape_move ? (tape_write ? t cout) m).
-#t #cout #m >map_write >map_move % 
-qed. 
-
-lemma map_move_mono: ∀t,cout,m.
- tape_move_mono ? (tape_map FinBool ? bit t)
-  〈char_to_bit_option (low_char cout), char_to_move (low_mv m)〉
- = tape_map ?? bit (tape_move_mono ? t 〈cout,m〉).
-@map_action
-qed. 
-
-definition R_unistep_high ≝ λM:normalTM.λt1,t2.
-∀c:nconfig (no_states M).
-  t1 = low_tapes M c → 
-  t2 = low_tapes M (step ? M c). 
-
-lemma R_unistep_equiv : ∀M,t1,t2. 
-  R_unistep (no_states M) (graph_enum ?? (ntrans M)) (nhalt M) t1 t2 →
-  R_unistep_high M t1 t2.
-#M #t1 #t2 #H whd whd in match (nconfig ?); #c #Ht1
-lapply (initial_bar ? (nhalt M) (graph_enum ?? (ntrans M)) (nTM_nog ?)) #Htable
-(* tup = current tuple *)
-cut (∃t.t = 〈〈cstate … c,current ? (ctape … c)〉, 
-             ntrans M 〈cstate … c,current ? (ctape … c)〉〉) [% //] * #tup #Htup
-(* tup is in the graph *)
-cut (mem ? tup (graph_enum ?? (ntrans M)))
-  [@memb_to_mem >Htup @(graph_enum_complete … (ntrans M)) %] #Hingraph
-(* tupe target = 〈qout,cout,m〉 *)
-lapply (decomp_target ? (ntrans M 〈cstate … c,current ? (ctape … c)〉))
-* #qout * #cout * #m #Htg >Htg in Htup; #Htup
-(* new config *)
-cut (step FinBool M c = mk_config ?? qout (tape_move ? (tape_write ? (ctape … c) cout) m))
-  [>(config_expand … c) whd in ⊢ (??%?); (* >Htg ?? why not?? *)
-   cut (trans ? M 〈cstate  … c, current ? (ctape … c)〉 = 〈qout,cout,m〉) [<Htg %] #Heq1 
-   >Heq1 %] #Hstep
-(* new state *)
-cut (cstate ?? (step FinBool M c) = qout) [>Hstep %] #Hnew_state
-(* new tape *)
-cut (ctape ?? (step FinBool M c) = tape_move ? (tape_write ? (ctape … c) cout) m)
-  [>Hstep %] #Hnew_tape
-lapply(H (bits_of_state ? (nhalt M) (cstate ?? c)) 
-         (low_char (current ? (ctape ?? c)))
-         (tail ? (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)))
-         ??????)
-[<Htable
- lapply(list_to_table … (nhalt M) …Hingraph) * #ll * #lr #Htable1 %{ll} 
- %{(((bits_of_state ? (nhalt M) qout)@[low_char cout;low_mv m])@lr)} 
- >Htable1 @eq_f <associative_append @eq_f2 // >Htup
- whd in ⊢ (??%?); @eq_f >associative_append %
-|>Ht1 >obj_low_tapes >map_list_of_tape elim (list_of_tape ??) 
-  [#b @False_ind | #b #tl #Hind #a * [#Ha >Ha //| @Hind]]
-|@sym_eq @Htable
-|>Ht1 %
-|%{(bits_of_state ? (nhalt M) (cstate ?? c))} %{(low_char (current ? (ctape ?? c)))}
- % [% [% [// | cases (current ??) normalize [|#b] % #Hd destruct (Hd)]
-      |>length_map whd in match (length ??); @eq_f //]
-   |//]
-|>Ht1 >cfg_low_tapes //] -H #H 
-lapply(H (bits_of_state … (nhalt M) qout) (low_char … cout) 
-         (low_mv … m) tup ? Hingraph)
-  [>Htup whd in ⊢ (??%?); @eq_f >associative_append %] -H
-#Ht2 @(eq_vec ? 3 ?? (niltape ?) ?) >Ht2 #i #Hi 
-cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
-  [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
-    [cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
-      [@False_ind /2/
-      |>Hi >obj_low_tapes >nth_change_vec //
-       >Ht1 >obj_low_tapes >Hstep @map_action 
-      ]
-    |>Hi >cfg_low_tapes >nth_change_vec_neq 
-      [|% whd in ⊢ (??%?→?);  #H destruct (H)]
-     >nth_change_vec // >Hnew_state @eq_f @eq_f >Hnew_tape 
-     @eq_f2 [|2:%] >Ht1 >obj_low_tapes >map_move_mono >low_char_current %
-    ]
-  |(* program tapes do not change *)
-   >Hi >prg_low_tapes 
-   >nth_change_vec_neq [|% whd in ⊢ (??%?→?);  #H destruct (H)]
-   >nth_change_vec_neq [|% whd in ⊢ (??%?→?);  #H destruct (H)]
-   >Ht1 >prg_low_tapes //
-  ]
-qed. 
+definition restart_tape ≝ λi,n. 
+  mmove i FSUnialpha n L ·
+  inject_TM ? (move_to_end FSUnialpha L) n i ·
+  mmove i FSUnialpha n R.
   
-
-    
-         
\ No newline at end of file
+definition R_restart_tape ≝ λi,n.λint,outt:Vector (tape FSUnialpha) (S n).
+   ∀t.t = nth i ? int (niltape ?) → 
+   outt = change_vec ?? int 
+    (mk_tape ? [ ] (option_hd ? (list_of_tape ? t)) (tail ? (list_of_tape ? t))) i.
+
+lemma sem_restart_tape : ∀i,n.i < S n → restart_tape i n ⊨ R_restart_tape i n.
+#i #n #Hleq
+@(sem_seq_app ??????? (sem_move_multi ? n i L ?)
+  (sem_seq ?????? (sem_inject ???? i ? (sem_move_to_end_l ?))
+   (sem_move_multi ? n i R ?))) [1,2,3:@le_S_S_to_le //]
+#ta #tb * #tc * whd in ⊢ (%→?); #Htc
+* #td * * * #Htd1 #Htd2 #Htd3 
+whd in ⊢ (%→?); #Htb *
+[ #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+  cut (td = tc) 
+  [ <(change_vec_same … tc … i … (niltape ?))
+    @(eq_vec_change_vec … (niltape ?))
+    [ @Htd1 >Htc >nth_change_vec //
+    | @Htd3 ] ]
+  (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
+  #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
+  #Htb >Htb %
+| #r0 #rs0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+  cut (td = tc)
+  [ <(change_vec_same … tc … i … (niltape ?))
+    @(eq_vec_change_vec … (niltape ?))
+    [ @Htd1 >Htc >nth_change_vec //
+    | @Htd3 ] ]
+  (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
+  #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
+  #Htb >Htb %
+| #l0 #ls0 #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+  cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@[l0])) i)
+  [ <(change_vec_same … tc … i … (niltape ?))
+    @(eq_vec_change_vec … (niltape ?))
+    [ @Htd2 >Htc >nth_change_vec //
+    | #j #Hij >nth_change_vec_neq // @Htd3 // ]]
+  #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
+  >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
+  cases (reverse ? ls0)
+  [ %
+  | #l1 #ls1 >reverse_cons
+     >(?: list_of_tape ? (rightof ? l0 (reverse ? ls1@[l1])) =
+          l1::ls1@[l0])
+     [|change with (reverse ??@?) in ⊢ (??%?);
+       whd in match (left ??); >reverse_cons >reverse_append 
+       whd in ⊢ (??%?); @eq_f >reverse_reverse normalize >append_nil % ] % ]
+| * 
+  [ #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+    cut (td = tc) 
+    [ <(change_vec_same … tc … i … (niltape ?))
+    @(eq_vec_change_vec … (niltape ?))
+    [ @Htd1 >Htc >nth_change_vec //
+    | @Htd3 ] ]
+    (* >Htc in Htd1; >nth_change_vec // *) -Htd1 -Htd2 -Htd3
+    #Htd >Htd in Htb; >Htc >change_vec_change_vec >nth_change_vec //
+    #Htb >Htb %
+  | #l0 #ls0 #c #rs #Hta_i <Hta_i in Htc; whd in ⊢ (???(????%?)→?); #Htc
+    cut (td = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls0@l0::c::rs)) i)
+    [ @(eq_vec_change_vec … (niltape ?))
+      [ @Htd2 >Htc >nth_change_vec //
+      | @Htd3 ] ]
+    #Htd >Htd in Htb; >Htc >change_vec_change_vec >change_vec_change_vec
+    >nth_change_vec // #Htb >Htb <(reverse_reverse ? ls0) in ⊢ (???%);
+    cases (reverse ? ls0)
+    [ %
+    | #l1 #ls1 >reverse_cons
+      >(?: list_of_tape ? (midtape ? (l0::reverse ? ls1@[l1]) c rs) =
+            l1::ls1@l0::c::rs)
+      [|change with (reverse ??@?) in ⊢ (??%?);
+        whd in match (left ??); >reverse_cons >reverse_append 
+        whd in ⊢ (??%?); @eq_f >reverse_reverse normalize
+        >associative_append % ] % ]
+  ]
+]
+qed.