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=4a7e252d94a8a853634364df458f4a3ca904a368;hpb=d774b8f9c73e2497fb953e7feb4bc1840a464564;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 4a7e252d9..4f37564c0 100644
--- a/matita/matita/lib/turing/multi_universal/unistep_aux.ma
+++ b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
@@ -52,6 +52,60 @@ include "turing/multi_universal/tuples.ma".
   cfg_to_obj
 *)
 
+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).
@@ -59,7 +113,7 @@ 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 ·
+    (copy_char obj cfg FSUnialpha 2 ·
      mmove cfg FSUnialpha 2 L ·
      mmove obj FSUnialpha 2 L) 
     (inject_TM ? (write FSUnialpha null) 2 cfg)
@@ -78,7 +132,6 @@ definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
          (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.
@@ -88,8 +141,6 @@ lemma eq_mk_tape_rightof :
 #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 ].
 
@@ -103,12 +154,22 @@ lemma tape_move_mk_tape_R :
 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 ??????
     (sem_if ??????????
      (sem_test_null_multi ?? obj ?)
-      (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
+      (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)))
@@ -118,15 +179,21 @@ lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
 #tc * whd in ⊢ (%→?); #Htc *
 #td * *
 [ * #te * * #Hcurtc #Hte
-  * destruct (Hte) #te * *
-  [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #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)
-      [@daemon] -Htg1 -Htg2 -Htg3 #Htg destruct (Htg Htf Hte Htd Htc Htb)
+      [ 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
@@ -138,18 +205,12 @@ lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
         [ #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
+        >nth_change_vec_neq in Hcurtc'; [|@sym_not_eq //] >Hta2
         normalize in ⊢ (%→?); #H destruct (H) %
       ]
-    | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
+    | #Hta2 >Htc in Hcurtc'; >nth_change_vec_neq [| @sym_not_eq //]
       >Hta2 #H destruct (H)
     ]
-  | * #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) ]
-  ]
 | * #te * * #Hcurtc #Hte
   * whd in ⊢ (%→%→?); #Htd1 #Htd2
   * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb
@@ -158,9 +219,11 @@ lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
     >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
+    [ 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)
-    [@daemon] -Htf2 -Htf3 #Htf destruct (Htf Htd Htc Htb)
+    [ 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/ ]
@@ -194,7 +257,7 @@ definition cfg_to_obj ≝
   mmove cfg FSUnialpha 2 L ·
   (ifTM ?? (inject_TM ? test_null_char 2 cfg)
     (nop ? 2)
-    (copy_step cfg obj FSUnialpha 2 ·
+    (copy_char cfg obj FSUnialpha 2 ·
      mmove cfg FSUnialpha 2 L ·
      mmove obj FSUnialpha 2 L) 
      tc_true) ·
@@ -213,58 +276,82 @@ definition R_cfg_to_obj ≝ λt1,t2:Vector (tape FSUnialpha) 3.
           (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.
 
-axiom 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 ??????????
-     (sem_test_null_multi ?? obj ?)
-      (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
-       (sem_move_multi ? 2 cfg L ?))
-      (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
-     (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?))
-       (sem_move_multi ? 2 cfg R ?)))) //
+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 #Hte
-  * destruct (Hte) #te * *
-  [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte #Htd
-    * #tf * * * whd in ⊢ (%→%→%→%→?); #Htf1 #Htf2 #Htf3 #Htb
-    #c #ls #Hta1 %
-    [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof 
-      whd in match (tape_move ???); #Htc
-      cut (tf = change_vec ?? tc (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
-      [@daemon] -Htf1 -Htf2 -Htf3 #Htf destruct (Htf Hte Htd Htc Htb)
-      >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec
-      >nth_change_vec // >tape_move_mk_tape_R
-      @daemon
-    | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
-      >Hta2 #H destruct (H)
-    ]
-  | * #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) ]
-  ]
-| * #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)
+[ * #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
-    >change_vec_change_vec >change_vec_change_vec >nth_change_vec //
-    >reverse_cons >tape_move_mk_tape_R /2/ ]
+    >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 *)
@@ -272,7 +359,11 @@ qed.
 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 ?]. 
+ 
 definition tape_move_obj : mTM FSUnialpha 2 ≝ 
   ifTM ?? 
    (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg)
@@ -284,6 +375,68 @@ definition tape_move_obj : mTM FSUnialpha 2 ≝
     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.
+
+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. 
@@ -310,6 +463,11 @@ definition low_char' ≝ λc.
   | 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 *)
@@ -321,12 +479,16 @@ definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
   (* obj *)
   only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
   let conf ≝ (bar::state@[char]) in
-  (∃ll,lr.bar::table = ll@conf@lr) → 
+  (∃ll,lr.bar::table = ll@conf@lr) →
+(*
     ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧ 
-    mem ? t l ∧
+    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 ?)) 
-       〈Some ? nchar,char_to_move m〉 in
+       〈char_to_bit_option nchar,char_to_move m〉 in
     let next_char ≝ low_char' (current ? new_obj) in
     t2 = 
       change_vec ??
@@ -337,17 +499,138 @@ 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.
 
-definition low_tapes ≝ λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
+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));
-   midtape ? [ ] bar (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M))
+   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)))
   ].
 
-definition R_unistep_high ≝ λM:normalTM.λc:nconfig (no_states M).λt1,t2.
+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). 
 
-  
-  
- 
\ No newline at end of file
+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.