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=dae251c38e392d180110a5e3d93522333747296c;hp=f5767e2cfaf77b0f05bb9640b3d43a8978171639;hpb=76a993b80bb33d1075f84c55637ca1897644b16a;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 f5767e2cf..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/ ]
@@ -190,60 +253,6 @@ 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)
@@ -296,9 +305,12 @@ lemma sem_cfg_to_obj : cfg_to_obj ⊨  R_cfg_to_obj.
   whd in ⊢ (%→?); #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 (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)
-    [@daemon] -Htf1 -Htf2 -Htf3 #Htf
+    [ 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 [| #_ % % ] 
@@ -315,9 +327,14 @@ lemma sem_cfg_to_obj : cfg_to_obj ⊨  R_cfg_to_obj.
   #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 (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)
-    [@daemon] -Hth1 -Hth2 -Hth3 #Hth
+    [ 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
@@ -379,17 +396,24 @@ 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/ ]
@@ -486,13 +510,14 @@ lemma low_char_current : ∀t.
   = low_char (current FinBool t).
 * // qed.
 
-definition low_tapes ≝ λM:normalTM.λc:nconfig (no_states M).Vector_of_list ?
+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.
@@ -539,14 +564,15 @@ lemma map_move_mono: ∀t,cout,m.
 @map_action
 qed. 
 
-definition R_unistep_high ≝ λM:normalTM.λc:nconfig (no_states M).λt1,t2.
+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,c,t1,t2. 
+lemma R_unistep_equiv : ∀M,t1,t2. 
   R_unistep (no_states M) (graph_enum ?? (ntrans M)) (nhalt M) t1 t2 →
-  R_unistep_high M c t1 t2.
-#M #c #t1 #t2 #H #Ht1
+  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)〉, 
@@ -588,7 +614,7 @@ lapply(H (bits_of_state ? (nhalt M) (cstate ?? c))
 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 >Ht2 @(eq_vec ? 3 … (niltape ?)) #i #Hi 
+#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
@@ -608,7 +634,3 @@ cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
    >Ht1 >prg_low_tapes //
   ]
 qed. 
-   
-
-    
-         
\ No newline at end of file