X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Funi_step.ma;h=fc663ff2c92114aa363c1a57738b2b1696087769;hb=66d22e9bc8ecc624e93e3e142676045d511ed9b0;hp=5c48d9272c2c011b678cc9666b696dae15bf737b;hpb=ceb81586cd493164f9c980c4f97ed0b4dbc6f545;p=helm.git

diff --git a/matita/matita/lib/turing/universal/uni_step.ma b/matita/matita/lib/turing/universal/uni_step.ma
index 5c48d9272..fc663ff2c 100644
--- a/matita/matita/lib/turing/universal/uni_step.ma
+++ b/matita/matita/lib/turing/universal/uni_step.ma
@@ -15,6 +15,8 @@
 *)
 
 include "turing/universal/copy.ma".
+include "turing/universal/move_tape.ma".
+include "turing/universal/match_machines.ma".
 
 (*
 
@@ -55,26 +57,23 @@ if is_true(current) (* current state is final *)
 *)
 
 definition init_match ≝ 
-  seq ? (mark ?) 
-    (seq ? (adv_to_mark_r ? (λc:STape.is_grid (\fst c)))
-      (seq ? (move_r ?) 
-        (seq ? (mark ?)
-          (seq ? (move_l ?) 
-            (adv_to_mark_l ? (is_marked ?)))))).
-            
+  mark ? · adv_to_mark_r ? (λc:STape.is_grid (\fst c)) · move_r ? · 
+    move_r ? · mark ? · move_l ? · adv_to_mark_l ? (is_marked ?).
+             
 definition R_init_match ≝ λt1,t2.
   ∀ls,l,rs,c,d. no_grids (〈c,false〉::l) → no_marks l → 
-  t1 = midtape STape ls 〈c,false〉 (l@〈grid,false〉::〈d,false〉::rs) →
-  t2 = midtape STape ls 〈c,true〉 (l@〈grid,false〉::〈d,true〉::rs).
+  t1 = midtape STape ls 〈c,false〉 (l@〈grid,false〉::〈bar,false〉::〈d,false〉::rs) →
+  t2 = midtape STape ls 〈c,true〉 (l@〈grid,false〉::〈bar,false〉::〈d,true〉::rs).
   
 lemma sem_init_match : Realize ? init_match R_init_match.
 #intape 
 cases (sem_seq ????? (sem_mark ?)
        (sem_seq ????? (sem_adv_to_mark_r ? (λc:STape.is_grid (\fst c)))
         (sem_seq ????? (sem_move_r ?)
-         (sem_seq ????? (sem_mark ?)
-          (sem_seq ????? (sem_move_l ?)
-           (sem_adv_to_mark_l ? (is_marked ?)))))) intape)
+         (sem_seq ????? (sem_move_r ?)
+          (sem_seq ????? (sem_mark ?)
+           (sem_seq ????? (sem_move_l ?)
+            (sem_adv_to_mark_l ? (is_marked ?))))))) intape)
 #k * #outc * #Hloop #HR 
 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
 #ls #l #rs #c #d #Hnogrids #Hnomarks #Hintape
@@ -83,49 +82,493 @@ cases HR -HR
 * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb -Hta 
   [* #Hgridc @False_ind @(absurd … Hgridc) @eqnot_to_noteq 
    @(Hnogrids 〈c,false〉) @memb_hd ]
-* #Hgrdic #Htb lapply (Htb l 〈grid,false〉 (〈d,false〉::rs) (refl …) (refl …) ?) 
+* #Hgrdic #Htb lapply (Htb l 〈grid,false〉 (〈bar,false〉::〈d,false〉::rs) (refl …) (refl …) ?) 
   [#x #membl @Hnogrids @memb_cons @membl] -Htb #Htb
 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb #Htc
 * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd -Htc #Htd
 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd #Hte
-whd in ⊢ (%→?); #Htf cases (Htf … Hte) -Htf -Hte 
+* #tf * whd in ⊢ (%→?); #Htf lapply (Htf … Hte) -Htf -Hte #Htf
+whd in ⊢ (%→?); #Htg cases (Htg … Htf) -Htg -Htf
   [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
-* #_ #Htf lapply (Htf (reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?) 
-  [#x #membl @Hnomarks @daemon] -Htf #Htf >Htf >reverse_reverse %
+* #_ #Htg lapply (Htg (〈grid,false〉::reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?) 
+  [#x #membl @Hnomarks @daemon] -Htg #Htg >Htg >reverse_cons >reverse_reverse
+   >associative_append %
+qed.
+
+(* init_copy 
+
+           init_current_on_match; (* no marks in current *)
+           move_r;
+           adv_to_mark_r;
+           adv_mark_r;
+
+*)
+
+definition init_copy ≝ 
+  init_current_on_match · move_r ? · 
+    adv_to_mark_r ? (is_marked ?) · adv_mark_r ?.
+
+definition R_init_copy ≝ λt1,t2.
+  ∀l1,l2,c,ls,d,rs. 
+  no_marks l1 → no_grids l1 → 
+  no_marks l2 → is_grid c = false → 
+  t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉 (l2@〈comma,true〉::〈d,false〉::rs) → 
+  t2 = midtape STape (〈comma,false〉::(reverse ? l2)@〈grid,false〉::l1@〈c,true〉::〈grid,false〉::ls) 〈d,true〉 rs.
+
+lemma list_last: ∀A.∀l:list A.
+  l = [ ] ∨ ∃a,l1. l = l1@[a].
+#A #l <(reverse_reverse ? l) cases (reverse A l)
+  [%1 //
+  |#a #l1 %2 @(ex_intro ?? a) @(ex_intro ?? (reverse ? l1)) //
+  ]
+qed.
+   
+lemma sem_init_copy : Realize ? init_copy R_init_copy.
+#intape 
+cases (sem_seq ????? sem_init_current_on_match
+        (sem_seq ????? (sem_move_r ?)
+          (sem_seq ????? (sem_adv_to_mark_r ? (is_marked ?))
+            (sem_adv_mark_r ?))) intape)
+#k * #outc * #Hloop #HR 
+@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#l1 #l2 #c #ls #d #rs #Hl1marks #Hl1grids #Hl2marks #Hc #Hintape
+cases HR -HR
+#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hl1grids Hc Hintape) -Hta -Hintape #Hta
+* #tb * whd in ⊢ (%→?); #Htb lapply (Htb  … Hta) -Htb -Hta
+generalize in match Hl1marks; -Hl1marks cases (list_last ? l1) 
+  [#eql1 >eql1 #Hl1marks whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?); #Htb
+   * #tc * whd in ⊢ (%→?); #Htc lapply (Htc  … Htb) -Htc -Htb *
+    [* whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)]
+   * #_ #Htc lapply (Htc … (refl …) (refl …) ?)
+    [#x #membx @Hl2marks @membx]
+   #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+   >Houtc %
+  |* #c1 * #tl #eql1 >eql1 #Hl1marks >reverse_append >reverse_single 
+   whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?);
+   >associative_append whd in ⊢ ((???(????%))→?); #Htb
+   * #tc * whd in ⊢ (%→?); #Htc lapply (Htc  … Htb) -Htc -Htb *
+    [* >Hl1marks [#Htemp destruct (Htemp)] @memb_append_l2 @memb_hd]
+   * #_ >append_cons <associative_append #Htc lapply (Htc … (refl …) (refl …) ?)
+    [#x #membx cases (memb_append … membx) -membx #membx
+      [cases (memb_append … membx) -membx #membx
+        [@Hl1marks @memb_append_l1 @daemon
+        |>(memb_single … membx) %
+        ]
+      |@Hl2marks @membx
+      ]]
+  #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+  >Houtc >reverse_append >reverse_append >reverse_single 
+  >reverse_reverse >associative_append >associative_append 
+  >associative_append %
 qed.
+  
+(* OLD 
+definition init_copy ≝ 
+  seq ? (adv_mark_r ?) 
+    (seq ? init_current_on_match
+      (seq ? (move_r ?) 
+        (adv_to_mark_r ? (is_marked ?)))).
 
-#Htb lapply (Htb ??? (refl …) (refl …))
+definition R_init_copy ≝ λt1,t2.
+  ∀l1,l2,c,l3,d,rs. 
+  no_marks l1 → no_grids l1 → 
+  no_marks l2 → no_grids l2 → is_grid c = false → is_grid d =false →
+  t1 = midtape STape (l1@〈grid,false〉::l2@〈c,false〉::〈grid,false〉::l3) 〈comma,true〉 (〈d,false〉::rs) → 
+  t2 = midtape STape (〈comma,false〉::l1@〈grid,false〉::l2@〈c,true〉::〈grid,false〉::l3) 〈d,true〉 rs.
 
-lapply (Htb l 〈grid,false〉 (〈d,false〉::rs) (refl … ))
+lemma list_last: ∀A.∀l:list A.
+  l = [ ] ∨ ∃a,l1. l = l1@[a].
+#A #l <(reverse_reverse ? l) cases (reverse A l)
+  [%1 //
+  |#a #l1 %2 @(ex_intro ?? a) @(ex_intro ?? (reverse ? l1)) //
+  ]
+qed.
    
- 
-* #tc * whd in ⊢ (%→?); #Htc 
-* #td * whd in ⊢ (%→%→?); #Htd #Houtc
-#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
-cases (Hta … Hintape) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
--Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
--Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (Hc) ] 
--Htc * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
--Htc #Htc lapply (Htd … Htc) -Htd
->reverse_append >reverse_cons 
->reverse_cons in Hc0; cases (reverse … l2)
-[ normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
-  #Htd >(Houtc … Htd) %
-| * #c2 #b2 #tl2 normalize in ⊢ (%→?);
-  #Hc0 #Htd >(Houtc … Htd)
-  whd in ⊢ (???%); destruct (Hc0)
-  >associative_append >associative_append %
+lemma sem_init_copy : Realize ? init_copy R_init_copy.
+#intape 
+cases (sem_seq ????? (sem_adv_mark_r ?)
+       (sem_seq ????? sem_init_current_on_match
+        (sem_seq ????? (sem_move_r ?)
+         (sem_adv_to_mark_r ? (is_marked ?)))) intape)
+#k * #outc * #Hloop #HR 
+@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#l1 #l2 #c #l3 #d #rs #Hl1marks #Hl1grids #Hl2marks #Hl2grids #Hc #Hd #Hintape
+cases HR -HR
+#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape #Hta
+* #tb * whd in ⊢ (%→?); 
+>append_cons #Htb lapply (Htb (〈comma,false〉::l1) l2 c … Hta) 
+  [@Hd |@Hc |@Hl2grids 
+   |#x #membx cases (orb_true_l … membx) -membx #membx 
+     [>(\P membx) // | @Hl1grids @membx]
+  ] -Htb #Htb
+* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
+>reverse_append >reverse_cons cases (list_last ? l2)
+  [#Hl2 >Hl2 >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
+   whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd -Htc
+    [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
+   * #_ #Htf lapply (Htf … (refl …) (refl …) ?) 
+    [#x >reverse_cons #membx cases (memb_append … membx) -membx #membx
+      [@Hl1marks @daemon |>(memb_single … membx) //] 
+    -Htf
+    |#Htf >Htf >reverse_reverse >associative_append %
+    ]
+  |* #a * #l21 #Heq >Heq >reverse_append >reverse_single 
+   >associative_append >associative_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
+   whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd -Htc
+    [* >Hl2marks [#Habs destruct (Habs) |>Heq @memb_append_l2 @memb_hd]]
+   * #_ <associative_append <associative_append #Htf lapply (Htf … (refl …) (refl …) ?) 
+    [#x >reverse_cons #membx cases (memb_append … membx) -membx #membx
+      [cases (memb_append … membx) -membx #membx
+        [@Hl2marks >Heq @memb_append_l1 @daemon
+        |>(memb_single … membx) //]
+      |cases (memb_append … membx) -membx #membx
+        [@Hl1marks @daemon |>(memb_single … membx) //]
+      ]
+    | #Htf >Htf >reverse_append >reverse_reverse
+      >reverse_append >reverse_reverse >associative_append 
+      >reverse_single >associative_append >associative_append 
+      >associative_append % 
+    ]
+  ]
+qed. *)
+
+definition exec_action ≝ 
+  init_copy · copy · move_r … · move_tape.
+
+definition map_move ≝ 
+  λc,mv.match c with [ null ⇒ None ? | _ ⇒ Some ? 〈c,false,move_of_unialpha mv〉 ].
+
+(* - aggiungere a legal_tape le condizioni
+       only_bits ls, rs; bit_or_null c
+   - ci vuole un lemma che dimostri 
+       bit_or_null c1 = true    bit_or_null mv = true
+       mv ≠ null → c1 ≠ null
+     dal fatto che c1 e mv sono contenuti nella table
+ *)
+definition R_exec_action ≝ λt1,t2.
+  ∀n,curconfig,ls,rs,c0,c1,s0,s1,table1,newconfig,mv,table2.
+  table_TM n (table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2) → 
+  no_marks curconfig → only_bits (curconfig@[〈s0,false〉]) → 
+  only_bits (〈s1,false〉::newconfig) → bit_or_null c1 = true → 
+  |curconfig| = |newconfig| → 
+  legal_tape ls 〈c0,false〉 rs →
+  t1 = midtape STape (〈c0,false〉::curconfig@〈s0,false〉::〈grid,false〉::ls) 〈grid,false〉 
+    (table1@〈comma,true〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) → 
+  ∀t1'.t1' = lift_tape ls 〈c0,false〉 rs → 
+  ∃ls1,rs1,c2.
+  t2 = midtape STape ls1 〈grid,false〉 
+    (〈s1,false〉::newconfig@〈c2,false〉::〈grid,false〉::
+     table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs1) ∧   
+  lift_tape ls1 〈c2,false〉 rs1 = 
+  tape_move STape t1' (map_move c1 mv) ∧ legal_tape ls1 〈c2,false〉 rs1.
+  
+(* move the following 2 lemmata to mono.ma *)
+lemma tape_move_left_eq :
+  ∀A.∀t:tape A.∀c.
+  tape_move ? t (Some ? 〈c,L〉) = 
+  tape_move_left ? (left ? t) c (right ? t).
+//
+qed.
+
+lemma tape_move_right_eq :
+  ∀A.∀t:tape A.∀c.
+  tape_move ? t (Some ? 〈c,R〉) = 
+  tape_move_right ? (left ? t) c (right ? t).
+//
+qed.
+
+lemma lift_tape_not_null : 
+ ∀ls,c,bc,rs.c ≠ null → lift_tape ls 〈c,bc〉 rs = midtape ? ls 〈c,bc〉 rs.
+#ls #c #bc #rs cases c //
+#Hfalse @False_ind /2/
+qed.
+
+lemma merge_char_not_null :
+  ∀c1,c2.c1 ≠ null → merge_char c1 c2 ≠ null.
+#c1 #c2 @not_to_not cases c2
+[ #c1' normalize #Hfalse destruct (Hfalse)
+| normalize //
+| *: normalize #Hfalse destruct (Hfalse)
 ]
 qed.
 
-definition match_tuple_step ≝ 
-  ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c))) 
-   (single_finalTM ? 
-     (seq ? compare
-      (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
-        (nop ?)
-        (seq ? mark_next_tuple 
-           (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
-             (mark ?) (seq ? (move_l ?) init_current) tc_true)) tc_true)))
+lemma merge_char_null : ∀c.merge_char null c = c.
+* //
+qed.
+
+lemma merge_char_cases : ∀c1,c2.merge_char c1 c2 = c1 ∨ merge_char c1 c2 = c2.
+#c1 *
+[ #c1' %2 %
+| % %
+| *: %2 % ]
+qed.
+
+(* lemma merge_char_c_bit :
+  ∀c1,c2.is_bit c2 = true → merge_char c1 c2 = c2.
+#c1 *
+[ #c2' #_ %
+|*: normalize #Hfalse destruct (Hfalse) ]
+qed.
+
+lemma merge_char_c_bit :
+  ∀c1,c2.is_null c2 = true → merge_char c1 c2 = c1.
+#c1 *
+[ #c2' #_ %
+|*: normalize #Hfalse destruct (Hfalse) ]
+qed.
+
+*)
+
+lemma sem_exec_action : Realize ? exec_action R_exec_action.
+#intape
+cases (sem_seq … sem_init_copy
+        (sem_seq … sem_copy
+          (sem_seq … (sem_move_r …) sem_move_tape )) intape)
+#k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop ] -Hloop
+#n #curconfig #ls #rs #c0 #c1 #s0 #s1 #table1 #newconfig #mv #table2
+#Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hc1 #Hlen #Htape #Hintape #t1' #Ht1'
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta
+lapply (Hta (〈c0,false〉::curconfig) table1 s0 ls s1
+        (newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) … Hintape) -Hta
+[ (*Hcurconfig2*) @daemon
+| (*Htable*) @daemon
+| (*bit_or_null c0 = true *) @daemon
+| (*Hcurconfig1*) @daemon
+| #Hta * #tb * whd in ⊢ (%→?); #Htb
+  lapply (Htb (〈grid,false〉::ls) s0 s1 c0 c1 (〈mv,false〉::table2@〈grid,false〉::rs) newconfig (〈comma,false〉::reverse ? table1) curconfig Hta ????????) -Htb
+  [9:|*:(* bit_or_null c0,c1; |curconfig| = |newconfig|*) @daemon ]
+  #Htb * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc whd in ⊢(???(??%%%)→?);#Htc
+  whd in ⊢ (%→?); #Houtc whd in Htc:(???%); whd in Htc:(???(??%%%));
+  lapply (Houtc rs n 
+    (〈comma,false〉::〈c1,false〉::reverse ? newconfig@〈s1,false〉::〈comma,false〉::reverse ? table1)
+    mv table2 (merge_char c0 c1) (reverse ? newconfig@[〈s1,false〉]) ls ????????)
+  [3: cases Htape -Htape * * #Hnomarks #Hbits #Hc0 #Hlsrs % [ % [ %
+    [ #x #Hx cases (orb_true_l … Hx) #Hx'
+      [ >(\P Hx') %
+      | @Hnomarks @memb_cons // ]
+    | @Hbits ]
+    | cases (merge_char_cases c0 c1) #Hmerge >Hmerge // ]
+    | cases (true_or_false (c0 == null)) #Hc0'
+      [ cases Hlsrs -Hlsrs 
+        [ *
+          [ >(\P Hc0') * #Hfalse @False_ind /2/
+          | #Hlsnil % %2 // ]
+        | #Hrsnil %2 // ] 
+      | % % @merge_char_not_null @(\Pf Hc0') ] ]
+  |4:>Htc @(eq_f3 … (midtape ?))
+    [ @eq_f @eq_f >associative_append >associative_append %
+    | %
+    | % ]
+  | %
+  || >reverse_cons >reverse_cons >reverse_append >reverse_reverse 
+     >reverse_cons >reverse_cons >reverse_reverse
+     >associative_append >associative_append >associative_append
+     >associative_append >associative_append
+     @Htable
+  | (* well formedness of table *) @daemon
+  | (* Hnewconfig *) @daemon
+  | (* bit_or_null mv = true (well formedness of table) *) @daemon
+  | -Houtc * #ls1 * #rs1 * #newc * #Hnewtapelegal * #Houtc *
+    [ *
+      [ * #Hmv #Htapemove
+        @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc)
+        %
+        [ %
+          [ >Houtc -Houtc >reverse_append
+            >reverse_reverse >reverse_single @eq_f
+            >reverse_cons >reverse_cons >reverse_append >reverse_cons
+            >reverse_cons >reverse_reverse >reverse_reverse
+            >associative_append >associative_append
+            >associative_append >associative_append
+            >associative_append >associative_append %
+          | >Hmv >Ht1' >Htapemove 
+            (* mv = bit false -→ c1 = bit ? *)
+            cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
+            >Hc1 >tape_move_left_eq >(legal_tape_left … Htape) 
+            >(legal_tape_right … Htape) %
+          ]
+        | //
+        ]
+      | * #Hmv #Htapemove 
+        @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
+        [ %
+          [ >Houtc -Houtc >reverse_append
+            >reverse_reverse >reverse_single @eq_f
+            >reverse_cons >reverse_cons >reverse_append >reverse_cons
+            >reverse_cons >reverse_reverse >reverse_reverse
+            >associative_append >associative_append
+            >associative_append >associative_append
+            >associative_append >associative_append %
+          |>Hmv >Ht1' >Htapemove 
+            cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
+            >Hc1 >tape_move_right_eq >(legal_tape_left … Htape) 
+            >(legal_tape_right … Htape) %
+          ]
+        | //
+        ]
+      ]
+    | * * * #Hmv #Hlseq #Hrseq #Hnewc 
+      @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
+      [ %
+        [ >Houtc -Houtc >reverse_append
+          >reverse_reverse >reverse_single @eq_f
+          >reverse_cons >reverse_cons >reverse_append >reverse_cons
+          >reverse_cons >reverse_reverse >reverse_reverse
+          >associative_append >associative_append
+          >associative_append >associative_append
+          >associative_append >associative_append %
+        |>Hmv >Ht1' cases c1 in Hnewc;
+          [ #c1' whd in ⊢ (??%?→?);#Hnewc <Hnewc
+            >Hlseq >Hrseq whd in ⊢ (??%%);
+            >(legal_tape_left … Htape) >(legal_tape_right … Htape) %
+          | whd in ⊢ (??%?→?); #Hnewc >Hnewc >Hlseq >Hrseq %
+          |*: whd in ⊢ (??%?→?);#Hnewc <Hnewc
+            >Hlseq >Hrseq whd in ⊢ (??%%);
+            >(legal_tape_left … Htape) >(legal_tape_right … Htape) %
+          ]
+        ]
+      | //
+      ]
+    ]
+  ]
+]
+qed.
+
+(*
+if is_false(current) (* current state is not final *)
+   then init_match;
+    match_tuple;
+    if is_marked(current) = false (* match ok *)
+      then 
+           exec_action
+           move_r;
+      else sink;
+   else nop;
+*)
+
+definition uni_step ≝ 
+  ifTM ? (test_char STape (λc.\fst c == bit false))
+    (single_finalTM ? 
+      (init_match · match_tuple ·
+        (ifTM ? (test_char ? (λc.¬is_marked ? c))
+          (exec_action · move_r …)
+          (nop ?) tc_true)))
     (nop ?) tc_true.
-          
\ No newline at end of file
+
+definition R_uni_step_true ≝ λt1,t2.
+  ∀n,table,s0,s1,c0,c1,ls,rs,curconfig,newconfig,mv.
+  0 < |table| → table_TM (S n) table → 
+  match_in_table (S n) (〈s0,false〉::curconfig) 〈c0,false〉
+    (〈s1,false〉::newconfig) 〈c1,false〉 〈mv,false〉 table → 
+  legal_tape ls 〈c0,false〉 rs → 
+  t1 = midtape STape (〈grid,false〉::ls) 〈s0,false〉 
+    (curconfig@〈c0,false〉::〈grid,false〉::table@〈grid,false〉::rs) → 
+  ∀t1'.t1' = lift_tape ls 〈c0,false〉 rs → 
+  s0 = bit false ∧
+  ∃ls1,rs1,c2.
+  (t2 = midtape STape (〈grid,false〉::ls1) 〈s1,false〉 
+    (newconfig@〈c2,false〉::〈grid,false〉::table@〈grid,false〉::rs1) ∧
+   lift_tape ls1 〈c2,false〉 rs1 = 
+   tape_move STape t1' (map_move c1 mv) ∧ legal_tape ls1 〈c2,false〉 rs1).
+   
+definition R_uni_step_false ≝ λt1,t2.
+  ∀b. current STape t1 = Some ? 〈bit b,false〉 → b = true ∧ t2 = t1.
+  
+axiom sem_match_tuple : Realize ? match_tuple R_match_tuple.
+
+definition us_acc : states ? uni_step ≝ (inr … (inl … (inr … start_nop))).
+
+lemma sem_uni_step :
+  accRealize ? uni_step us_acc
+     R_uni_step_true R_uni_step_false. 
+@(acc_sem_if_app STape … (sem_test_char ? (λc:STape.\fst c == bit false))
+   (sem_seq … sem_init_match      
+     (sem_seq … sem_match_tuple        
+       (sem_if … (* ????????? (sem_test_char …  (λc.¬is_marked FSUnialpha c)) *)
+          (sem_seq … sem_exec_action (sem_move_r …))
+          (sem_nop …))))
+   (sem_nop …)
+   …)
+[@sem_test_char||]
+[ #intape #outtape 
+  #ta whd in ⊢ (%→?); #Hta #HR
+  #n #fulltable #s0 #s1 #c0 #c1 #ls #rs #curconfig #newconfig #mv
+  #Htable_len cut (∃t0,table. fulltable =〈bar,false〉::〈t0,false〉::table) [(* 0 < |table| *) @daemon]
+  * #t0 * #table #Hfulltable >Hfulltable -fulltable 
+  #Htable #Hmatch #Htape #Hintape #t1' #Ht1'
+  >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta 
+  #Hs0 lapply (\P Hs0) -Hs0 #Hs0 #Hta % //
+  cases HR -HR 
+  #tb * whd in ⊢ (%→?); #Htb
+  lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) s0 t0 ???)
+  [ >Hta >associative_append %
+  | @daemon
+  | @daemon
+  | -Hta -Htb #Htb * 
+    #tc * whd in ⊢ (%→?); #Htc cases (Htc … Htable … Htb) -Htb -Htc
+    [| * #Hcurrent #Hfalse @False_ind
+      (* absurd by Hmatch *) @daemon
+    | >Hs0 %
+    | (* Htable (con lemma) *) @daemon
+    | (* Hmatch *) @daemon
+    | (* Htable *) @daemon
+    | (* Htable, Hmatch → |config| = n 
+       necessaria modifica in R_match_tuple, le dimensioni non corrispondono
+      *) @daemon
+    ]
+    * #table1 * #newc * #mv1 * #table2 * #Htableeq #Htc *
+    [ * #td * whd in ⊢ (%→?); >Htc -Htc #Htd
+      cases (Htd ? (refl ??)) #_ -Htd
+      cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
+      >Hnewc #Htd cut (mv1 = 〈mv,false〉)
+      [@daemon] #Hmv1
+      * #te * whd in ⊢ (%→?); #Hte
+      cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈s0,false〉::〈grid,false〉::ls) 
+              〈grid,false〉
+              ((table1@〈bar,false〉::〈s0,false〉::curconfig@[〈c0,false〉])@〈comma,true〉::〈s1,false〉::
+               newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs))
+      [ >Htd @eq_f3 //
+        [ >reverse_append >reverse_single %
+        | >associative_append >associative_append normalize
+          >associative_append >associative_append >Hmv1 % 
+        ]
+      ]
+      -Htd #Htd lapply (Hte … (S n) … Htd … Ht1') -Htd -Hte
+      [ //
+      | (*|curconfig| = |newconfig|*) @daemon
+      | (* Htable → bit_or_null c1 = true *) @daemon
+      | (* only_bits (〈s1,false〉::newconfig) *) @daemon
+      | (* only_bits (curconfig@[〈s0,false〉]) *) @daemon
+      | (* no_marks (reverse ? curconfig) *) @daemon
+      | >Hmv1 in Htableeq; >Hnewc 
+        >associative_append >associative_append normalize
+        >associative_append >associative_append
+        #Htableeq <Htableeq // ]
+      * #ls1 * #rs1 * #c2 * * #Hte #Hliftte #Hlegalte
+      whd in ⊢ (%→?); #Houttape lapply (Houttape … Hte) -Houttape #Houttape
+      whd in Houttape:(???%); whd in Houttape:(???(??%%%));
+      @ex_intro [| @(ex_intro ?? rs1) @ex_intro [| % [ % 
+      [ >Houttape @eq_f @eq_f @eq_f @eq_f
+        change with ((〈bar,false〉::〈t0,false〉::table)@?) in ⊢ (???%);
+        >Htableeq >associative_append >associative_append 
+        >associative_append normalize >associative_append
+        >associative_append normalize >Hnewc <Hmv1
+        >associative_append normalize >associative_append
+        >Hmv1 % 
+      | @Hliftte
+      ]
+     | //
+     ]
+    ]
+   ] 
+  | * #td * whd in ⊢ (%→%→?); >Htc #Htd
+    cases (Htd ? (refl ??)) normalize in ⊢ (%→?);
+    #Hfalse destruct (Hfalse)
+  ]
+ ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #Ht1 #Ht2
+  #b #Hb cases (Ht1 ? Hb) #Hb' #Ht3 >Ht2 % //
+  cases b in Hb'; normalize #H1 //
+]
+qed.
\ No newline at end of file