X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Funiversal.ma;h=3ed3015bec83c38932ce359c20faa37218c31815;hb=31cb2f0b374657eb5acb95708443e2c1b8481891;hp=d35283a6f61159bde56b61884209f3b7d41690c9;hpb=05458990020a373abef8df4590d131c96824eac7;p=helm.git

diff --git a/matita/matita/lib/turing/universal/universal.ma b/matita/matita/lib/turing/universal/universal.ma
index d35283a6f..3ed3015be 100644
--- a/matita/matita/lib/turing/universal/universal.ma
+++ b/matita/matita/lib/turing/universal/universal.ma
@@ -10,29 +10,10 @@
       V_____________________________________________________________*)
 
 
-include "turing/universal/trans_to_tuples.ma".
+include "turing/universal/uni_step.ma".
 
 (* definition zero : ∀n.initN n ≝ λn.mk_Sig ?? 0 (le_O_n n). *)
 
-record normalTM : Type[0] ≝ 
-{ no_states : nat;
-  pos_no_states : (0 < no_states); 
-  ntrans : trans_source no_states → trans_target no_states;
-  nhalt : initN no_states → bool
-}.
-
-definition normalTM_to_TM ≝ λM:normalTM.
-  mk_TM FinBool (initN (no_states M)) 
-   (ntrans M) (mk_Sig ?? 0 (pos_no_states M)) (nhalt M).
-
-coercion normalTM_to_TM.
-
-definition nconfig ≝ λn. config FinBool (initN n).
-
-(* 
-definition normalTM ≝ λn,t,h. 
-  λk:0<n.mk_TM FinBool (initN n) t (mk_Sig ?? 0 k) h. *)
-
 definition low_config: ∀M:normalTM.nconfig (no_states M) → tape STape ≝ 
 λM:normalTM.λc.
   let n ≝ no_states M in
@@ -40,15 +21,108 @@ definition low_config: ∀M:normalTM.nconfig (no_states M) → tape STape ≝
   let t ≝ntrans M in 
   let q ≝ cstate … c in
   let q_low ≝  m_bits_of_state n h q in 
-  let current_low ≝
-    match current … (ctape … c) with
-    [ None ⇒ 〈null,false〉
-    | Some b ⇒ 〈bit b,false〉] in
+  let current_low ≝ match current … (ctape … c) with [ None ⇒ null | Some b ⇒ bit b] in
   let low_left ≝ map … (λb.〈bit b,false〉) (left … (ctape …c)) in
   let low_right ≝ map … (λb.〈bit b,false〉) (right … (ctape …c)) in
-  let table ≝ flatten ? (tuples_of_pairs n h (graph_enum ?? t)) in
-  mk_tape STape low_left (Some ? 〈grid,false〉) 
-    (q_low@current_low::〈grid,false〉::table@〈grid,false〉::low_right).
+  let table ≝ flatten ? (tuples_list n h (graph_enum ?? t)) in
+  let right ≝ q_low@〈current_low,false〉::〈grid,false〉::table@〈grid,false〉::low_right in
+  mk_tape STape (〈grid,false〉::low_left) (option_hd … right) (tail … right).
+  
+lemma low_config_eq: ∀t,M,c. t = low_config M c → 
+  ∃low_left,low_right,table,q_low_hd,q_low_tl,c_low.
+  low_left = map … (λb.〈bit b,false〉) (left … (ctape …c)) ∧
+  low_right = map … (λb.〈bit b,false〉) (right … (ctape …c)) ∧
+  table = flatten ? (tuples_list (no_states M) (nhalt M) (graph_enum ?? (ntrans M))) ∧
+  〈q_low_hd,false〉::q_low_tl = m_bits_of_state (no_states M) (nhalt M) (cstate … c) ∧
+  c_low =  match current … (ctape … c) with [ None ⇒ null| Some b ⇒ bit b] ∧
+  t = midtape STape (〈grid,false〉::low_left) 〈q_low_hd,false〉 (q_low_tl@〈c_low,false〉::〈grid,false〉::table@〈grid,false〉::low_right).
+#t #M #c #eqt
+  @(ex_intro … (map … (λb.〈bit b,false〉) (left … (ctape …c))))
+  @(ex_intro … (map … (λb.〈bit b,false〉) (right … (ctape …c))))
+  @(ex_intro … (flatten ? (tuples_list (no_states M) (nhalt M) (graph_enum ?? (ntrans M)))))
+  @(ex_intro … (\fst (hd ? (m_bits_of_state (no_states M) (nhalt M) (cstate … c)) 〈bit true,false〉)))
+  @(ex_intro … (tail ? (m_bits_of_state (no_states M) (nhalt M) (cstate … c))))
+  @(ex_intro … (match current … (ctape … c) with [ None ⇒ null| Some b ⇒ bit b]))
+% [% [% [% [% // | // ] | // ] | // ] | >eqt //]
+qed.
+
+let rec to_bool_list (l: list (unialpha×bool)) ≝ 
+  match l with
+  [ nil ⇒ nil ?
+  | cons a tl ⇒ 
+    match \fst a with 
+    [bit b ⇒ b::to_bool_list tl
+    |_ ⇒ nil ?
+    ]
+  ].
+
+definition high_c ≝ λc:unialpha×bool.
+  match \fst c with
+  [ null ⇒ None ?
+  | bit b ⇒ Some ? b 
+  | _ ⇒ None ?].
+
+definition high_tape ≝ λls,c,rs.
+  mk_tape FinBool (to_bool_list ls) (high_c c) (to_bool_list rs).
+
+lemma high_tape_eq : ∀ls,c,rs. high_tape ls c rs =
+  mk_tape FinBool (to_bool_list ls) (high_c c) (to_bool_list rs).
+// qed.
+
+definition high_tape_from_tape ≝ λt:tape STape.
+  match t with
+  [niltape ⇒ niltape ?
+  |leftof a l ⇒ match \fst a with 
+     [bit b ⇒ leftof ? b (to_bool_list l)
+     |_ ⇒ niltape ?
+     ]
+  |rightof a r ⇒ match \fst a with 
+     [bit b ⇒ rightof ? b (to_bool_list r)
+     |_ ⇒ niltape ?
+     ]
+  |midtape l c r ⇒ high_tape l c r
+  ].
+
+lemma high_tape_of_lift : ∀ls,c,rs. legal_tape ls c rs →
+  high_tape ls c rs =
+    high_tape_from_tape (lift_tape ls c rs).
+#ls * #c #b #rs * #H cases c // 
+>high_tape_eq  
+* [ * [#H @False_ind /2/
+      | #Heq >Heq cases rs // * #a #b1 #tl 
+        whd in match (lift_tape ???); cases a // 
+      ]
+  |#Heq >Heq cases ls // * #a #b1 #tl 
+        whd in match (lift_tape ???); cases a //
+  ]
+qed.
+
+lemma bool_embedding: ∀l.
+  to_bool_list (map ?? (λb.〈bit b,false〉) l) = l.
+#l elim l // #a #tl #Hind normalize @eq_f @Hind
+qed.
+
+lemma current_embedding: ∀c.
+  high_c (〈match c with [None ⇒ null | Some b ⇒ bit b],false〉) = c.
+  * normalize // qed.
+
+lemma tape_embedding: ∀ls,c,rs.
+ high_tape 
+   (map ?? (λb.〈bit b,false〉) ls) 
+   (〈match c with [None ⇒ null | Some b ⇒ bit b],false〉)
+   (map ?? (λb.〈bit b,false〉) rs) = mk_tape ? ls c rs.
+#ls #c #rs >high_tape_eq >bool_embedding >bool_embedding
+>current_embedding %
+qed.
+
+definition high_move ≝ λc,mv.
+  match c with 
+  [ bit b ⇒ Some ? 〈b,move_of_unialpha mv〉
+  | _ ⇒ None ?
+  ].
+
+definition map_move ≝ 
+  λc,mv.match c with [ null ⇒ None ? | _ ⇒ Some ? 〈c,false,move_of_unialpha mv〉 ].
 
 definition low_step_R_true ≝ λt1,t2.
   ∀M:normalTM.
@@ -56,41 +130,223 @@ definition low_step_R_true ≝ λt1,t2.
     t1 = low_config M c →
     halt ? M (cstate … c) = false ∧
       t2 = low_config M (step ? M c).
+
+definition low_tape_aux : ∀M:normalTM.tape FinBool → tape STape ≝ 
+λM:normalTM.λt.
+  let current_low ≝ match current … t with 
+    [ None ⇒ None ? | Some b ⇒ Some ? 〈bit b,false〉] in
+  let low_left ≝ map … (λb.〈bit b,false〉) (left … t) in
+  let low_right ≝ map … (λb.〈bit b,false〉) (right … t) in
+  mk_tape STape low_left current_low low_right. 
+
+lemma left_of_low_tape: ∀M,t. 
+  left ? (low_tape_aux M t) = map … (λb.〈bit b,false〉) (left … t).
+#M * //
+qed. 
+
+lemma right_of_low_tape: ∀M,t. 
+  right ? (low_tape_aux M t) = map … (λb.〈bit b,false〉) (right … t).
+#M * //
+qed. 
+
+definition low_move ≝ λaction:option (bool × move).
+  match action with
+  [None ⇒ None ?
+  |Some act ⇒ Some ? (〈〈bit (\fst act),false〉,\snd act〉)].
+
+(* simulation lemma *)
+lemma low_tape_move : ∀M,action,t.
+  tape_move STape (low_tape_aux M t) (low_move action) =
+  low_tape_aux M (tape_move FinBool t action). 
+#M * // (* None *)
+* #b #mv #t cases mv cases t //
+  [#ls #c #rs cases ls //|#ls #c #rs cases rs //]
+qed.
+
+lemma left_of_lift: ∀ls,c,rs. left ? (lift_tape ls c rs) = ls.
+#ls * #c #b #rs cases c // cases ls // cases rs //
+qed.
+
+lemma right_of_lift: ∀ls,c,rs. legal_tape ls c rs →
+  right ? (lift_tape ls c rs) = rs.
+#ls * #c #b #rs * #_ cases c // cases ls cases rs // #a #tll #b #tlr
+#H @False_ind cases H [* [#H1 /2/ |#H1 destruct] |#H1 destruct]
+qed.
+
+
+lemma current_of_lift: ∀ls,c,b,rs. legal_tape ls 〈c,b〉 rs →
+  current STape (lift_tape ls 〈c,b〉 rs) =
+    match c with [null ⇒ None ? | _ ⇒ Some ? 〈c,b〉].
+#ls #c #b #rs cases c // whd in ⊢ (%→?); * #_ 
+* [* [#Hnull @False_ind /2/ | #Hls >Hls whd in ⊢ (??%%); cases rs //]
+  |#Hrs >Hrs whd in ⊢ (??%%); cases ls //]
+qed.
+
+lemma current_of_lift_None: ∀ls,c,b,rs. legal_tape ls 〈c,b〉 rs →
+  current STape (lift_tape ls 〈c,b〉 rs) = None ? →
+    c = null.
+#ls #c #b #rs #Hlegal >(current_of_lift … Hlegal) cases c normalize  
+  [#b #H destruct |// |3,4,5:#H destruct ]
+qed.
+
+lemma current_of_lift_Some: ∀ls,c,c1,rs. legal_tape ls c rs →
+  current STape (lift_tape ls c rs) = Some ? c1 →
+    c = c1.
+#ls * #c #cb #b #rs #Hlegal >(current_of_lift … Hlegal) cases c normalize 
+ [#b1 #H destruct // |#H destruct |3,4,5:#H destruct //]
+qed.
+
+lemma current_of_low_None: ∀M,t. current FinBool t = None ? → 
+  current STape (low_tape_aux M t) = None ?.
+#M #t cases t // #l #b #r whd in ⊢ ((??%?)→?); #H destruct
+qed.
+  
+lemma current_of_low_Some: ∀M,t,b. current FinBool t = Some ? b → 
+  current STape (low_tape_aux M t) = Some ? 〈bit b,false〉.
+#M #t cases t 
+  [#b whd in ⊢ ((??%?)→?); #H destruct
+  |#b #l #b1 whd in ⊢ ((??%?)→?); #H destruct
+  |#b #l #b1 whd in ⊢ ((??%?)→?); #H destruct
+  |#c #c1 #l #r whd in ⊢ ((??%?)→?); #H destruct %
+  ]
+qed.
+
+lemma current_of_low:∀M,tape,ls,c,rs. legal_tape ls c rs → 
+  lift_tape ls c rs = low_tape_aux M tape →
+  c = 〈match current … tape  with 
+       [ None ⇒ null | Some b ⇒ bit b], false〉.
+#M #tape #ls * #c #cb #rs #Hlegal #Hlift  
+cut (current ? (lift_tape ls 〈c,cb〉 rs) = current ? (low_tape_aux M tape))
+  [@eq_f @Hlift] -Hlift #Hlift
+cut (current … tape = None ? ∨ ∃b.current … tape = Some ? b)
+  [cases (current … tape) [%1 // | #b1 %2 /2/ ]] *  
+  [#Hcurrent >Hcurrent normalize
+   >(current_of_low_None …Hcurrent) in Hlift; #Hlift 
+   >(current_of_lift_None … Hlegal Hlift) 
+   @eq_f cases Hlegal * * #Hmarks #_ #_ #_ @(Hmarks 〈c,cb〉) @memb_hd
+  |* #b #Hcurrent >Hcurrent normalize
+   >(current_of_low_Some …Hcurrent) in Hlift; #Hlift 
+   @(current_of_lift_Some … Hlegal Hlift) 
+  ]
+qed.
+
+(*
+lemma current_of_low:∀M,tape,ls,c,rs. legal_tape ls c rs → 
+  lift_tape ls c rs = low_tape_aux M tape →
+  c = 〈match current … tape  with 
+       [ None ⇒ null | Some b ⇒ bit b], false〉.
+#M #tape #ls * #c #cb #rs * * #_ #H cases (orb_true_l … H)
+  [cases c [2,3,4,5: whd in ⊢ ((??%?)→?); #Hfalse destruct]
+   #b #_ #_ cases tape 
+    [whd in ⊢ ((??%%)→?); #H destruct
+    |#a #l whd in ⊢ ((??%%)→?); #H destruct 
+    |#a #l whd in ⊢ ((??%%)→?); #H destruct 
+    |#a #l #r whd in ⊢ ((??%%)→?); #H destruct //
+    ]
+  |cases c 
+    [#b whd in ⊢ ((??%?)→?); #Hfalse destruct
+    |3,4,5:whd in ⊢ ((??%?)→?); #Hfalse destruct]
+   #_ * [* [#Habs @False_ind /2/
+           |#Hls >Hls whd in ⊢ ((??%%)→?); *)
+          
     
+(* sufficent conditions to have a low_level_config *)
+lemma is_low_config: ∀ls,c,rs,M,s,tape,qhd,q_tl,table.
+legal_tape ls c rs →
+table = flatten ? (tuples_list (no_states M) (nhalt M) (graph_enum ?? (ntrans M))) →
+lift_tape ls c rs = low_tape_aux M tape →
+〈qhd,false〉::q_tl = m_bits_of_state (no_states M) (nhalt M) s →
+midtape STape (〈grid,false〉::ls) 
+  〈qhd,false〉 
+  (q_tl@c::〈grid,false〉::table@〈grid,false〉::rs) = 
+  low_config M (mk_config ?? s tape).
+#ls #c #rs #M #s #tape #qhd #q_tl #table #Hlegal #Htable
+#Hlift #Hstate whd in match (low_config ??); <Hstate 
+@eq_f3 
+  [@eq_f <(left_of_lift ls c rs) >Hlift //
+  | cut (∀A.∀a,b:A.∀l1,l2. a::l1 = b::l2 → a=b)
+    [#A #a #b #l1 #l2 #H destruct (H) %] #Hcut
+   @(Hcut …Hstate)
+  |@eq_f <(current_of_low … Hlegal Hlift) @eq_f @eq_f <Htable @eq_f @eq_f
+   <(right_of_lift ls c rs Hlegal) >Hlift @right_of_low_tape
+  ]
+qed.
+
+lemma unistep_true_to_low_step: ∀t1,t2.
+  R_uni_step_true t1 t2 → low_step_R_true t1 t2.
+#t1 #t2 (* whd in ⊢ (%→%); *) #Huni_step * #n #posn #t #h * #qin #tape #eqt1
+cases (low_config_eq … eqt1) 
+#low_left * #low_right * #table * #q_low_hd * #q_low_tl * #current_low
+***** #Hlow_left #Hlow_right #Htable #Hq_low #Hcurrent_low #Ht1
+letin trg ≝ (t 〈qin,current ? tape〉)
+letin qout_low ≝ (m_bits_of_state n h (\fst trg))
+letin qout_low_hd ≝ (hd ? qout_low 〈bit true,false〉)
+letin qout_low_tl ≝ (tail ? qout_low)
+letin low_act ≝ (low_action (\snd (t 〈qin,current ? tape〉)))
+letin low_cout ≝ (\fst low_act)
+letin low_m ≝ (\snd low_act)
+lapply (Huni_step n table q_low_hd (\fst qout_low_hd) 
+       current_low low_cout low_left low_right q_low_tl qout_low_tl low_m … Ht1) 
+  [@daemon
+  |>Htable
+   @(trans_to_match n h t 〈qin,current ? tape〉 … (refl …))
+   >Hq_low  >Hcurrent_low whd in match (mk_tuple ?????);
+   >(eq_pair_fst_snd … (t …)) whd in ⊢ (??%?);
+   >(eq_pair_fst_snd … (low_action …)) %
+  |//
+  |@daemon
+  ]
+-Ht1 #Huni_step lapply (Huni_step ? (refl …)) -Huni_step *
+#q_low_head_false * #ls1 * #rs1 * #c2 * * 
+#Ht2 #Hlift #Hlegal %
+  [whd in ⊢ (??%?); >q_low_head_false in Hq_low; 
+   whd in ⊢ ((???%)→?); generalize in match (h qin);
+   #x #H destruct (H) %
+  |>Ht2 whd in match (step FinBool ??); 
+   whd in match (trans ???); 
+   >(eq_pair_fst_snd … (t ?))
+   @is_low_config // >Hlift
+   <low_tape_move @eq_f2
+    [>Hlow_left >Hlow_right >Hcurrent_low whd in ⊢ (??%%); 
+     cases (current …tape) [%] #b whd in ⊢ (??%%); %
+    |whd in match low_cout; whd in match low_m; whd in match low_act; 
+     generalize in match (\snd (t ?)); * [%] * #b #mv
+     whd in  ⊢ (??(?(???%)?)%); cases mv % 
+    ]
+  ]
+qed.
+
 definition low_step_R_false ≝ λt1,t2.
   ∀M:normalTM.
   ∀c: nconfig (no_states M).  
     t1 = low_config M c → halt ? M (cstate … c) = true  ∧ t1 = t2.
 
+lemma unistep_false_to_low_step: ∀t1,t2.
+  R_uni_step_false t1 t2 → low_step_R_false t1 t2.
+#t1 #t2 (* whd in ⊢ (%→%); *) #Huni_step * #n #posn #t #h * #qin #tape #eqt1
+cases (low_config_eq … eqt1) #low_left * #low_right * #table * #q_low_hd * #q_low_tl * #current_low
+***** #_ #_ #_ #Hqin #_ #Ht1 whd in match (halt ???);
+cases (Huni_step (h qin) ?) [/2/] >Ht1 whd in ⊢ (??%?); @eq_f
+normalize in Hqin; destruct (Hqin) %
+qed.
+
 definition low_R ≝ λM,qstart,R,t1,t2.
     ∀tape1. t1 = low_config M (mk_config ?? qstart tape1) → 
     ∃q,tape2.R tape1 tape2 ∧
     halt ? M q = true ∧ t2 = low_config M (mk_config ?? q tape2).
 
-definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
-∃i,outc.
-  loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (mk_config ?? q t1) = Some ? outc ∧ 
-  t2 = (ctape ?? outc).
-
-(*
-definition universal_R ≝ λM,R,t1,t2. 
-    Realize ? M R →    
-    ∀tape1,tape2.
-    R tape1 tape 2 ∧
-    t1 = low_config M (initc ? M tape1) ∧
-    ∃q.halt ? M q = true → t2 = low_config M (mk_config ?? q tape2).*)
-
-axiom uni_step: TM STape.
-axiom us_acc: states ? uni_step.
-
-axiom sem_uni_step: accRealize ? uni_step us_acc low_step_R_true low_step_R_false.
+lemma sem_uni_step1: 
+  uni_step ⊨ [us_acc: low_step_R_true, low_step_R_false].
+@(acc_Realize_to_acc_Realize … sem_uni_step) 
+  [@unistep_true_to_low_step | @unistep_false_to_low_step ]
+qed. 
 
 definition universalTM ≝ whileTM ? uni_step us_acc.
 
 theorem sem_universal: ∀M:normalTM. ∀qstart.
   WRealize ? universalTM (low_R M qstart (R_TM FinBool M qstart)).
 #M #q #intape #i #outc #Hloop
-lapply (sem_while … sem_uni_step intape i outc Hloop)
+lapply (sem_while … sem_uni_step1 intape i outc Hloop)
   [@daemon] -Hloop 
 * #ta * #Hstar generalize in match q; -q 
 @(star_ind_l ??????? Hstar)
@@ -108,14 +364,14 @@ lapply (sem_while … sem_uni_step intape i outc Hloop)
    #q #Htd #tape1 #Htb 
    lapply (IH (\fst (trans ? M 〈q,current ? tape1〉)) Htd) -IH 
    #IH cases (Htc … Htb); -Htc #Hhaltq 
-   whd in match (step ???); >(eq_pair_fst_snd ?? (trans ???)) 
+   whd in match (step FinBool ??); >(eq_pair_fst_snd ?? (trans ???)) 
    #Htc change with (mk_config ????) in Htc:(???(??%)); 
    cases (IH ? Htc) #q1 * #tape2 * * #HRTM #Hhaltq1 #Houtc
    @(ex_intro … q1) @(ex_intro … tape2) % 
     [%
       [cases HRTM #k * #outc1 * #Hloop #Houtc1
        @(ex_intro … (S k)) @(ex_intro … outc1) % 
-        [>loop_S_false [2://] whd in match (step ???); 
+        [>loop_S_false [2://] whd in match (step FinBool ??); 
          >(eq_pair_fst_snd ?? (trans ???)) @Hloop
         |@Houtc1
         ]
@@ -125,15 +381,6 @@ lapply (sem_while … sem_uni_step intape i outc Hloop)
   ]
 qed.
 
-lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2. 
-  WRealize sig M R → R_TM ? M (start ? M) t1 t2 → R t1 t2.
-#sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
-#Hloop #Ht2 >Ht2 @(HMR … Hloop)
-qed.
-
-axiom WRealize_to_WRealize: ∀sig,M,R1,R2.
-  (∀t1,t2.R1 t1 t2 → R2 t1 t2) → WRealize sig M R1 → WRealize ? M R2.
-
 theorem sem_universal2: ∀M:normalTM. ∀R.
   WRealize ? M R → WRealize ? universalTM (low_R M (start ? M) R).
 #M #R #HMR lapply (sem_universal … M (start ? M)) @WRealize_to_WRealize