X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Funistep_aux.ma;h=4f37564c0ea459d8aa8365813dec2084917e13e5;hb=3b280742956d9d62aab53dd81704c250e3befe91;hp=06427b3bb638889731844709de706c61c39f0e50;hpb=e461bb10382fe2ea65bd37dd56be46c6aec67db7;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 06427b3bb..4f37564c0 100644
--- a/matita/matita/lib/turing/multi_universal/unistep_aux.ma
+++ b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
@@ -13,6 +13,7 @@ include "turing/multi_universal/moves_2.ma".
 include "turing/multi_universal/match.ma".
 include "turing/multi_universal/copy.ma".
 include "turing/multi_universal/alphabet.ma".
+include "turing/multi_universal/tuples.ma".
 
 (*
 
@@ -51,160 +52,585 @@ include "turing/multi_universal/alphabet.ma".
   cfg_to_obj
 *)
 
-definition obj ≝ 0.
-definition cfg ≝ 1.
-definition prg ≝ 2.
+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 obj ≝ (0:DeqNat).
+definition cfg ≝ (1:DeqNat).
+definition prg ≝ (2:DeqNat).
 
 definition obj_to_cfg ≝
-  mmove cfg FSUnialpha 2 L ·
   mmove cfg FSUnialpha 2 L ·
   (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
-    (inject_TM ? (write FSUnialpha (bit false)) 2 cfg ·
-     inject_TM ? (move_r FSUnialpha) 2 cfg ·
-     inject_TM ? (write FSUnialpha (bit false)) 2 cfg)
-    (inject_TM ? (write FSUnialpha (bit true)) 2 cfg ·
-     inject_TM ? (move_r FSUnialpha) 2 cfg ·
-     copy_step obj cfg FSUnialpha 2) tc_true ·
-     inject_TM ? (move_l FSUnialpha) 2 cfg) ·
+    (copy_char obj cfg FSUnialpha 2 ·
+     mmove cfg FSUnialpha 2 L ·
+     mmove obj FSUnialpha 2 L) 
+    (inject_TM ? (write FSUnialpha null) 2 cfg)
+     tc_true) ·
   inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
-  inject_TM ? (move_r FSUnialpha) 2 cfg.
+  mmove cfg FSUnialpha 2 R.
   
 definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
-  ∀c,opt,ls.
-  nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::opt::ls) (None ?) [ ] → 
+  ∀c,ls.
+  nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] → 
   (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso → 
    t2 = change_vec ?? t1 
-         (mk_tape ? [ ] (option_hd ? (reverse ? (c::opt::ls))) (tail ? (reverse ? (c::opt::ls)))) cfg) ∧
+         (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧
   (current ? (nth obj ? t1 (niltape ?)) = None ? → 
    t2 = change_vec ?? t1
-         (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (bit false::bit false::ls))) 
-           (tail ? (reverse ? (bit false :: bit false::ls)))) cfg).
+         (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.
 
-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_app ?? ????? (sem_move_multi ? 2 cfg L ?)
-   (sem_seq_app ???????    
-    (sem_seq_app ???????
-     (sem_if ? 2 ????????
-      (sem_test_null_multi ?? obj ?)
-      (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false)))
-       (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_r ?)) 
-        (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?) ?)
-      ?)
-     ??) ??) ?) ?)
-[|||||||||||||||| @     
-        
-        ??) ??) ??) ?) ?)
-       ?) ?) ?) ?)
-
-
-@(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?) ??)
-[||
-@(sem_seq_app ?? ????? (sem_move_multi ? 2 cfg L ?) ??)
-[|| @sem_seq_app
-[|| @sem_seq_app
-[|| @(sem_if ? 2 ???????? (sem_test_null_multi ?? obj ?))
-[|||@(sem_seq_app ??????? (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?)
-[||@(sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_r ?)) 
-       (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?)
-[||      
+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 = [ ]) → 
+  tape_move ? (mk_tape sig ls c rs) R =
+  mk_tape ? (option_cons ? c ls) (option_hd ? rs) (tail ? rs).
+#sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
+[| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ] 
+normalize //
+qed.
+
+lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
+  nth i ? v2 d = t → 
+  (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) → 
+  v2 = change_vec ?? v1 t i.
+#sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
+#i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
+[ >Hii0 >nth_change_vec //
+| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
+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_app ?? ????? (sem_move_multi ? 2 cfg L ?)
-   (sem_seq_app ???????
-    (sem_if ? 2 ????????
+   (sem_seq ??????
+    (sem_if ??????????
      (sem_test_null_multi ?? obj ?)
-     (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false)))     
-      (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_r ?)) 
-       (sem_inject ???? cfg ? (sem_write FSUnialpha (bit false))) ?) ?)
-       ?)
-    (sem_seq_app ??????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
-     (sem_inject ???? cfg ? (sem_move_r ?)) ?) ?) ?) ?)
-      
-
-lemma wsem_copy : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
-  copy src dst sig n ⊫ R_copy src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
-lapply (sem_while … (sem_copy_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ whd in ⊢ (%→?); * #Hnone #Hout %
-  [#_ @Hout
-  |#ls #x #x0 #rs #ls0 #rs0 #Hsrc1 #Hdst1 @False_ind cases Hnone
-    [>Hsrc1 normalize #H destruct (H) | >Hdst1 normalize #H destruct (H)]
-  ]
-|#tc #td * #x * #y * * #Hcx #Hcy #Htd #Hstar #IH #He lapply (IH He) -IH *
- #IH1 #IH2 %
-  [* [>Hcx #H destruct (H) | >Hcy #H destruct (H)]
-  |#ls #x' #y' #rs #ls0 #rs0 #Hnth_src #Hnth_dst
-   >Hnth_src in Hcx; whd in ⊢ (??%?→?); #H destruct (H)
-   >Hnth_dst in Hcy; whd in ⊢ (??%?→?); #H destruct (H)
-   >Hnth_src in Htd; >Hnth_dst -Hnth_src -Hnth_dst
-   cases rs
-    [(* the source tape is empty after the move *)
-     #Htd lapply (IH1 ?) 
-      [%1 >Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] >nth_change_vec //]
-     #Hout (* whd in match (tape_move ???); *) %1 %{([])} %{rs0} % 
-      [% [// | // ] 
-      |whd in match (reverse ??); whd in match (reverse ??);
-       >Hout >Htd @eq_f2 // cases rs0 //
-      ]
-    |#c1 #tl1 cases rs0
-      [(* the dst tape is empty after the move *)
-       #Htd lapply (IH1 ?) [%2 >Htd >nth_change_vec //] 
-       #Hout (* whd in match (tape_move ???); *) %2 %{[ ]} %{(c1::tl1)} % 
-        [% [// | // ] 
-        |whd in match (reverse ??); whd in match (reverse ??);
-         >Hout >Htd @eq_f2 // 
-        ]
-      |#c2 #tl2 whd in match (tape_move_mono ???); whd in match (tape_move_mono ???);
-       #Htd
-       cut (nth src (tape sig) td (niltape sig)=midtape sig (x::ls) c1 tl1)
-         [>Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] @nth_change_vec //]
-       #Hsrc_td
-       cut (nth dst (tape sig) td (niltape sig)=midtape sig (x::ls0) c2 tl2)
-         [>Htd @nth_change_vec //]
-       #Hdst_td cases (IH2 … Hsrc_td Hdst_td) -Hsrc_td -Hdst_td
-        [* #rs01 * #rs02 * * #H1 #H2 #H3 %1
-         %{(c2::rs01)} %{rs02} % [% [@eq_f //|normalize @eq_f @H2]]
-         >Htd in H3; >change_vec_commute // >change_vec_change_vec
-         >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec 
-         #H >reverse_cons >associative_append >associative_append @H 
-        |* #rs11 * #rs12 * * #H1 #H2 #H3 %2
-         %{(c1::rs11)} %{rs12} % [% [@eq_f //|normalize @eq_f @H2]]
-         >Htd in H3; >change_vec_commute // >change_vec_change_vec
-         >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec 
-         #H >reverse_cons >associative_append >associative_append @H 
-        ]
+      (sem_seq ?????? (sem_copy_char …)
+       (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 ⊢ (%→?); #Hte
+  cut (∃x.current ? (nth obj ? tc (niltape ?)) = Some ? x)
+  [ cases (current ? (nth obj ? tc (niltape ?))) in Hcurtc;
+    [ * #H @False_ind /2/ | #x #_ %{x} % ] ] * #x #Hcurtc'
+(*  [ 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)
+      [ lapply (eq_vec_change_vec ??????? (Htg2 ls x [ ] ?) Htg3) //
+        >Htd >nth_change_vec_neq // >Htf >nth_change_vec //
+        >Hte >Hcurtc' >nth_change_vec // >Htc >nth_change_vec // ] 
+      -Htg1 -Htg2 -Htg3 #Htg destruct 
+      >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 Hcurtc'; [|@sym_not_eq //] >Hta2
+        normalize in ⊢ (%→?); #H destruct (H) %
       ]
+    | #Hta2 >Htc in Hcurtc'; >nth_change_vec_neq [| @sym_not_eq //]
+      >Hta2 #H destruct (H)
     ]
+| * #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)
+    [ lapply (eq_vec_change_vec ??????? (Htd1 ls c [ ] ?) Htd2) // 
+      >Htc >nth_change_vec // ] -Htd1 -Htd2 #Htd
+    -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg)
+    [ lapply (eq_vec_change_vec ??????? (Htf2 ls null [ ] ?) Htf3) //
+      >Htd >nth_change_vec // ] -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.
+
+definition test_null_char ≝ test_char FSUnialpha (λc.c == null).
+
+definition R_test_null_char_true ≝ λt1,t2.
+  current FSUnialpha t1 = Some ? null ∧ t1 = t2.
+  
+definition R_test_null_char_false ≝ λt1,t2.
+  current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2.
+  
+lemma sem_test_null_char :
+  test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false].
+#t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue
+#Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull)
+  -Hcnull #H destruct (H) #Houtc1 %
+  [ @Hcurt1 | <Houtc1 % ] ]
+| #Houtc cases (Hfalse ?) [| @Houtc] #Hc #Houtc %
+  [ % #Hcurt1 >Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??)) 
+    >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ]
+    #H destruct (H)
+  | <Houtc % ] ]
+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) ·
+  inject_TM ? (move_to_end FSUnialpha L) 2 cfg ·
+  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 ??
+          (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).
+          
+lemma tape_move_mk_tape_L :
+  ∀sig,ls,c,rs.
+  (c = None ? → ls = [ ] ∨ rs = [ ]) → 
+  tape_move ? (mk_tape sig ls c rs) L =
+  mk_tape ? (tail ? ls) (option_hd ? ls) (option_cons ? c rs).
+#sig * [ * [ * | #c * ] | #l0 #ls0 * [ *
+[| #r0 #rs0 #H @False_ind cases (H (refl ??)) #H1 destruct (H1) ] | #c * ] ] 
+normalize //
+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 //]
+#ta #tb *
+#tc * whd in ⊢ (%→?); #Htc *
+#td * *
+[ * #te * * * #Hcurtc #Hte1 #Hte2 whd in ⊢ (%→?); #Htd destruct (Htd)
+  * #tf * * * #Htf1 #Htf2 #Htf3
+  whd in ⊢ (%→?); #Htb
+  #c #ls #Hta %
+  [ #Hc >Hta in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc
+    cut (te = tc)
+    [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2) >change_vec_same // ]
+    -Hte1 -Hte2 #Hte
+    cut (tf = change_vec ? 3 te (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
+    [ lapply (eq_vec_change_vec ??????? (Htf2 ls c [ ] ?) Htf3) //
+      >Hte >Htc >nth_change_vec // ] -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 * *
+| * #te * * * #Hcurtc #Hte1 #Hte2
+  * #tf * whd in ⊢ (%→?); #Htf
+  * #tg * whd in ⊢ (%→%→?); #Htg #Htd
+  * #th * * * #Hth1 #Hth2 #Hth3
+  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)
+    [ lapply (eq_vec_change_vec ??????? (sym_eq … Hte1) Hte2)
+      >change_vec_same // ] -Hte1 -Hte2 #Hte
+    cut (th = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[c])) cfg)
+    [ lapply (eq_vec_change_vec ??????? (Hth2 ls c [ ] ?) Hth3) //
+      >Htd >nth_change_vec_neq // >Htg >nth_change_vec //
+      >Htf >nth_change_vec_neq // >nth_change_vec // 
+      >Hte >Htc >nth_change_vec // >Hta // ] -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 // #_ % % ]
   ]
+]
 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].
+
+definition char_to_bit_option ≝ λc.match c with
+  [ bit b ⇒ Some ? (bit b)
+  | _ ⇒ None ?]. 
  
-lemma terminate_copy :  ∀src,dst,sig,n,t.
-  src ≠ dst → src < S n → dst < S n → copy src dst sig n ↓ t.
-#src #dst #sig #n #t #Hneq #Hsrc #Hdts
-@(terminate_while … (sem_copy_step …)) //
-<(change_vec_same … t src (niltape ?))
-cases (nth src (tape sig) t (niltape ?))
-[ % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-|2,3: #a0 #al0 % #t1 * #x * #y * * >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 * #y * * >nth_change_vec // normalize in ⊢ (%→?);
-   #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 % 
-   #t2 * #x0 * #y0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
-   >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
-  |#r0 #rs0 #IH #t #ls #c % #t1 * #x * #y * * >nth_change_vec //
-   normalize in ⊢ (%→?); #H destruct (H) #Hcur
-   >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+definition tape_move_obj : mTM FSUnialpha 2 ≝ 
+  ifTM ?? 
+   (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
+   (mmove obj FSUnialpha 2 L)
+   (ifTM ?? 
+    (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg)
+    (mmove obj FSUnialpha 2 R)
+    (nop ??)
+    tc_true)
+   tc_true.
+
+definition R_tape_move_obj' ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+  (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit false) → 
+   t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) L) obj) ∧
+  (current ? (nth prg ? t1 (niltape ?)) = Some ? (bit true) → 
+   t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) R) obj) ∧
+  (current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit false) →
+   current ? (nth prg ? t1 (niltape ?)) ≠ Some ? (bit true) →  
+   t2 = t1).
+   
+lemma sem_tape_move_obj' : tape_move_obj ⊨ R_tape_move_obj'.
+#ta cases (sem_if ??????????
+  (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit false)))
+  (sem_move_multi ? 2 obj L ?)
+  (sem_if ??????????
+   (acc_sem_inject ?????? prg ? (sem_test_char ? (λc:FSUnialpha.c == bit true)))
+   (sem_move_multi ? 2 obj R ?)
+   (sem_nop …)) ta) //
+#i * #outc * #Hloop #HR %{i} %{outc} % [@Hloop] -i
+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) 
+    [| 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) 
+  [ 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) 
+      [| lapply (eq_vec_change_vec ??????? Htc1 Htc2)
+        >change_vec_same // ] % ]
+    | >Hcurta_prg >Hc #_ * #H @False_ind /2/ ]
+  | * #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/ ]
+    | #_ #_ @Houtc ]
   ]
 ]
 qed.
 
-lemma sem_copy : ∀src,dst,sig,n.
-  src ≠ dst → src < S n → dst < S n → 
-  copy src dst sig n ⊨ R_copy src dst sig n.
-#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize [/2/| @wsem_copy // ]
+definition R_tape_move_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
+  ∀c. current ? (nth prg ? t1 (niltape ?)) = Some ? c → 
+  t2 = change_vec ?? t1 (tape_move ? (nth obj ? t1 (niltape ?)) (char_to_move c)) obj.
+
+lemma sem_tape_move_obj : tape_move_obj ⊨ R_tape_move_obj.
+@(Realize_to_Realize … sem_tape_move_obj')
+#ta #tb * * #Htb1 #Htb2 #Htb3 * [ *
+[ @Htb2 | @Htb1 ] 
+| #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
+  >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
+| #Hcurta_prg change with (nth obj ? ta (niltape ?)) in match (tape_move ???);
+  >change_vec_same @Htb3 >Hcurta_prg % #H destruct (H)
+]
+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 %
+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 //]
+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 * //]
+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.