X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Funistep.ma;h=ee594814189e3f790076129f75a4d39b4dad9556;hb=d6b8021e8c83eb19033cad0aeaeebf95b327e78a;hp=2e4e1e253d2425dd7a237b6930f1f03628d8d773;hpb=3026dfea8eae158433ed13df5156a733fa926794;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/unistep.ma b/matita/matita/lib/turing/multi_universal/unistep.ma
index 2e4e1e253..ee5948141 100644
--- a/matita/matita/lib/turing/multi_universal/unistep.ma
+++ b/matita/matita/lib/turing/multi_universal/unistep.ma
@@ -11,6 +11,7 @@
 
 
 include "turing/multi_universal/unistep_aux.ma".
+include "turing/multi_universal/match.ma".
 
 definition exec_move ≝ 
   cfg_to_obj · tape_move_obj · restart_tape prg 2 · obj_to_cfg.
@@ -101,6 +102,13 @@ definition legal_tape ≝ λn,l,h,t.
   is_config n (bar::state@[char]) →  
   nth prg ? t1 (niltape ?) = midtape ? [ ] bar table →
   bar::table = table_TM n l h → *)
+  
+definition deterministic_tuples ≝ λn,h,l.
+ ∀t1,t2,conf,out1,out2.
+  is_config n conf →
+  mem ? t1 l → mem ? t2 l →
+  tuple_encoding n h t1 = conf@out1 →
+    tuple_encoding n h t2 = conf@out2 → out1 = out2.
 
 definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
   ∀state,char,table.
@@ -112,6 +120,8 @@ definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3.
   bar::table = table_TM n l h →
   (* obj *)
   only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) →
+  (* deterministic tuples *)
+  deterministic_tuples n h l →
   let conf ≝ (bar::state@[char]) in
   (∃ll,lr.bar::table = ll@conf@lr) →
 (*
@@ -148,12 +158,29 @@ definition R_copy_strict ≝
             (tail sig rs2)) src)
             (mk_tape sig (reverse sig rs1@x::ls0) (None sig) []) dst)).
 
-axiom sem_copy_strict : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n → 
+lemma sem_copy_strict : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n → 
   copy src dst sig n ⊨ R_copy_strict src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst @(Realize_to_Realize … (sem_copy …)) //
+#ta #tb * #Htb1 #Htb2 % [ @Htb1 ]
+#ls #x #x0 #rs #ls0 #rs0 #Htamid_src #Htamid_dst #Hlen
+cases (Htb2 … Htamid_src Htamid_dst) -Htb1 -Htb2
+[ * #rs1 * #rs2 * * #Hrs0 #Heq #Htb <Heq in Hlen; >Hrs0 >length_append
+  >(plus_n_O (|rs1|)) #Hlen cut (|rs2| ≤ 0) [ /2 by le_plus_to_le/ ]
+  #Hlenrs2 cut (rs2 = [ ]) 
+  [ cases rs2 in Hlenrs2; // #r3 #rs3 normalize #H @False_ind cases (not_le_Sn_O (|rs3|)) /2/ ]
+  #Hrs2 destruct (Hrs2) >append_nil in Hrs0; #Hrs0 destruct (Hrs0) -Hlenrs2 -Hlen
+  <plus_n_O <plus_n_O %{rs} %{[ ]} >append_nil % [ % // ] @Htb
+| * #rs1 * #rs2 #H %{rs1} %{rs2} @H ]
+qed.
 
-axiom daemon : ∀P:Prop.P.
+lemma config_to_len : ∀n,b,q,c.is_config n (b::q@[c]) → |q| = S n.
+#n #b #q #c * #q0 * #cin * * * #_ #_ #Hq0 #H >(?:q = q0) //
+lapply (cons_injective_r ????? H) #H1 @(append_l1_injective … H1)
+lapply (eq_f … (length ?) … H) normalize >length_append >length_append
+<plus_n_Sm <plus_n_Sm <plus_n_O <plus_n_O #Hlen destruct (Hlen) //
+qed.
 
-lemma sem_unistep : ∀n,l,h.unistep ⊨ R_unistep n l h.
+lemma sem_unistep_low : ∀n,l,h.unistep ⊨ R_unistep n l h.
 #n #l #h
 @(sem_seq_app ??????? (sem_match_m cfg prg FSUnialpha 2 ???)
   (sem_seq ?????? (sem_restart_tape ???)
@@ -162,15 +189,15 @@ lemma sem_unistep : ∀n,l,h.unistep ⊨ R_unistep n l h.
      (sem_exec_move …)))))
   /2 by le_n,sym_not_eq/
 #ta #tb #HR #state #char #table #Hta_cfg #Hcfg #Hta_prg #Htable
-#Hbits_obj #Htotaltable
+#Hbits_obj #Hdeterm #Hmatching
 #nstate #nchar #m #t #Htuple #Hmatch
 cases HR -HR #tc * whd in ⊢ (%→?); 
 >Hta_cfg #H cases (H ?? (refl ??)) -H 
 (* prg starts with a bar, so it's not empty *) #_
 >Hta_prg #H lapply (H ??? (refl ??)) -H *
-[| cases Htotaltable #ll * #lr #H >H 
+[| cases Hmatching #ll * #lr #H >H 
    #Hfalse @False_ind cases (Hfalse ll lr) #H1 @H1 //]
-* #ll * #lr * #Hintable -Htotaltable #Htc
+* #ll * #lr * #Hintable  #Htc
 * #td * whd in ⊢ (%→?); >Htc
 >nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
 #Htd lapply (Htd ? (refl ??)) -Htd
@@ -183,61 +210,79 @@ whd in ⊢ (???(???(????%?)??)→?); whd in match (tail ??); #Htd
 (* move cfg to R *)
 * #te * whd in ⊢ (%→?); >Htd
 >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
->nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+>nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec [2:@leb_true_to_le %]
+cut (∃ll1.ll@[bar] = bar::ll1) 
+[ cases ll in Hintable; [ #_ %{[ ]} % ] 
+  #llhd #lltl normalize #H destruct (H) %{(lltl@[bar])} % ] * #ll1 #Hll1
 >Htable in Hintable; #Hintable #Hte
 (* copy *)
-lapply (cfg_in_table_to_tuple ???? Hcfg ?? Hintable)
-* #newconfig * #m0 * #lr0 * * #Hlr destruct (Hlr) #Hnewcfg #Hm0
+lapply (table_to_list ???? Hcfg ?? Hintable) * #out * #lr0 * #t0
+* * #Hlr #Htuple_t0 #mem_t0 
+cut (out = nstate@[nchar;m])
+  [@(Hdeterm … Hcfg mem_t0 Hmatch Htuple_t0 Htuple)] #Hout
+>(append_cons ? nchar) in Htuple; #Htuple
+lapply (tuple_to_config ?????? Hcfg … Htuple) #newconfig
 cut (∃fo,so.state = fo::so ∧ |so| = n)
-[ @daemon ] * #fo * #so * #Hstate_exp #Hsolen
-cut (∃fn,sn,cn.newconfig = fn::sn@[cn] ∧ |sn| = n)
-[ @daemon ] * #fn * #sn * #cn * #Hnewstate_exp #Hsnlen
+[ lapply (config_to_len … Hcfg) cases state [ normalize #H destruct (H) ]
+  #fo #so normalize #H destruct (H) %{fo} %{so} % // ]
+* #fo * #so * #Hstate_exp #Hsolen
+cut (∃fn,sn.nstate = fn::sn  ∧ |sn| = n)
+[ lapply (config_to_len … newconfig) cases nstate [ normalize #H destruct (H) ]
+  #fn #sn normalize #H destruct (H) %{fn} %{sn} % // ]
+* #fn * #sn * #Hnewstate_exp #Hsnlen
+>Hstate_exp >Hout in Hlr; >Hnewstate_exp whd in ⊢ (???%→?); #Hlr
 * #tf * * #_ >Hte >(nth_change_vec ?????? lt_prg)
 >nth_change_vec_neq [|@sym_not_eq //] >(nth_change_vec ?????? lt_cfg)
->Hstate_exp >Hnewstate_exp
+>Hstate_exp >Hnewstate_exp >Hlr
 whd in match (mk_tape ????); whd in match (tape_move ???);
 #Htf cases (Htf ?????? (refl ??) (refl ??) ?) -Htf
 [| whd in match (tail ??); >length_append >length_append 
-   >Hsolen >length_append >Hsnlen //]
+   >Hsolen >length_append >Hsnlen normalize //]
 #rs1 * #rs2 whd in match (tail ??); * *
 #Hrs1rs2 #Hrs1len
 >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
 >change_vec_change_vec >append_nil #Htf 
 (* exec_move *)
-cut ((sn@[cn])=rs1)
+>append_cons in Hrs1rs2; >associative_append #Hrs1rs2
+cut ((sn@[nchar])=rs1)
   [@(append_l1_injective ?????? Hrs1rs2) >Hrs1len 
    >length_append >length_append normalize >Hsolen >Hsnlen % ] #Hrs1  
-cut (m0::lr0=rs2) [@(append_l2_injective ?????? Hrs1rs2) >Hrs1 % ] #Hrs2 
-cut (∃ll1. ll@[bar] = bar::ll1) 
-  [@daemon (* ll is at the begininng of a table *)] * #ll1 #Hll
+cut (m::lr0=rs2) [@(append_l2_injective ?????? Hrs1rs2) >Hrs1 % ] #Hrs2 
 whd in ⊢ (%→?); >Htf >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
 >nth_change_vec [2:@leb_true_to_le %] 
 >nth_change_vec [2:@leb_true_to_le %]
 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
 >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
 >append_cons <Hrs1 >reverse_append >reverse_single 
-<Hrs2 >(append_cons … bar ll) >Hll >reverse_cons
+<Hrs2 >(append_cons … bar ll) >Hll1 >reverse_cons
 #sem_exec_move 
 lapply
-  (sem_exec_move ? m0 ?
-    (([cn]@reverse FSUnialpha sn)
+  (sem_exec_move ? m ?
+    (([nchar]@reverse FSUnialpha sn)
           @fn::reverse FSUnialpha (ll1@(fo::so)@[char])) lr0 … (refl ??) ????) 
-  [@Hm0 
-  |@daemon (* OK *)
+  [ cut (tuple_TM (S n) (tuple_encoding n h t)) // >Htuple
+    * #qin * #cin * #qout * #cout * #mv * * * * #_ #Hmv #_ #_
+    normalize >(?: bar::qin@cin::qout@[cout;mv] = (bar::qin@cin::qout@[cout])@[mv])
+    [| normalize >associative_append normalize >associative_append % ] 
+    >(?: bar::(state@[char])@(nstate@[nchar])@[m] = (bar::(state@[char])@(nstate@[nchar]))@[m]) 
+    [|normalize >associative_append >associative_append >associative_append >associative_append >associative_append % ] 
+    #Heq lapply (append_l2_injective_r ?????? Heq) // #H destruct (H) //
+  | cases newconfig #qout * #cout * * * #_ #Hcout #_ #H destruct (H) -H
+    lapply (append_l2_injective_r ?????? e0) // #H destruct (H) @Hcout 
   |@Hbits_obj
   |whd in ⊢ (??%?); >associative_append >associative_append 
-   >associative_append %
+   >associative_append  %
   |#Htb >Htb @(eq_vec … (niltape ?)) (* tape by tape *) #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
        (* obj tape *)
-       >nth_change_vec [2:@leb_true_to_le %] 
-       (* dimostrare che la tabella e' univoca
-          da cui m = m0 e nchar = cn *)
+       >nth_change_vec [2:@leb_true_to_le %] % 
       |(* cfg tape *) #eqi >eqi
        >nth_change_vec_neq [2:@eqb_false_to_not_eq %]
-       >nth_change_vec [2:@leb_true_to_le %] 
+       >nth_change_vec [2:@leb_true_to_le %]  
+       whd in ⊢ (??%?); cut (∀A.∀l:list A.[]@l = l) [//] #Hnil >Hnil
+       >reverse_append >reverse_single >reverse_reverse %
        (* idem *)
       ]
     |(* prg tape *) #eqi >eqi
@@ -247,61 +292,33 @@ lapply
       >Htable >Hintable >reverse_append >reverse_cons
       >reverse_reverse >reverse_cons >reverse_reverse
       >associative_append >associative_append >associative_append
-      >(append_cons ? bar ll) >Hll @eq_f @eq_f <Hstate_exp @eq_f
-      >Hnewstate_exp %
+      >(append_cons ? bar ll) >Hll1 @eq_f @eq_f <Hstate_exp @eq_f
+      >Hnewstate_exp >Hlr normalize >associative_append >associative_append %
     ]
   ]
-  
-  
 qed.
 
-
-
-  cut ((mk_tape FSUnialpha []
-     (option_hd FSUnialpha
-      (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar])))
-     (tail FSUnialpha
-      (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar])))) =
-      midtape ? [ ] bar (fn::sn@[cn;m0]))
-  [cut (reverse FSUnialpha (m0::[]@reverse FSUnialpha (sn@[cn])@[fn; bar]) =
-          bar::fn::sn@[cn;m0])
-    [>reverse_cons whd in ⊢ (??(??(??%)?)?); >reverse_append >reverse_reverse
-     >append_cons in ⊢ (???%); % ] #Hrev >Hrev % ] #Hmk_tape >Hmk_tape -Hmk_tape
-   >change_vec_commute
-
-  >reverse_append >reverse_append
-  check reverse_cons
-  <reverse_cons >reverse_cons
-   -Htg #Htg 
-
->(change_vec_commute ????? cfg prg) [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 %]
-lapply (append_l1_injective ?????? Hrs1rs2)
-[ >Hsnlen >Hrs1len >length_append >length_append >length_append >length_append
-  normalize >Hsolen >Hsnlen % ]
-#Hrs1 <Hrs1 >reverse_append #Htg cases (Htg ?? (refl ??)) -Htg
-cases m0
-  [#mv #_ #Htg #_
-
-   
-      
-      
-
-[ * 
- 
-  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 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)).
-    
+
+(* da spostare *)
+lemma map_reverse: ∀A,B,f,l. 
+  map ?? f (reverse A l) = reverse B (map ?? f l).
+#A #B #f #l elim l //
+#a #l1 #Hind >reverse_cons >reverse_cons <map_append @eq_f2 //
+qed.
+
 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 %
+#A #B #f * // 
+  [#a #l normalize >rev_append_def >rev_append_def >append_nil
+   >append_nil <map_append <map_reverse @eq_f2 //
+  |#rs #a #ls normalize >rev_append_def >rev_append_def
+   >append_nil >append_nil <map_append normalize 
+   @eq_f2 //
+  ]
 qed.
 
 lemma low_char_current : ∀t.
@@ -395,12 +412,14 @@ cut (ctape ?? (step FinBool M c) = tape_move ? (tape_write ? (ctape … c) cout)
 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 %
+|#tx #ty #conf #outx #outy #isconf #memx #memy #tuplex #tupley
+ @(deterministic M … (refl ??) memx memy isconf tuplex tupley)
 |>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
@@ -432,4 +451,8 @@ cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
    >nth_change_vec_neq [|% whd in ⊢ (??%?→?);  #H destruct (H)]
    >Ht1 >prg_low_tapes //
   ]
-qed. 
+qed.
+
+lemma sem_unistep : ∀M.unistep ⊨ R_unistep_high M.
+#M @(Realize_to_Realize … (R_unistep_equiv …)) //
+qed.