X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Funi_step.ma;h=eabab3e838c66dbac33a0da3b8e42eae2f708d5b;hb=df4cfc76ab059f6b3d5daf324712ad27ec281088;hp=d24dc07ba9f3bdbf2e8f4e6d463414f9109b8824;hpb=1695dd8b1861176954666067243c02020e1dcea5;p=helm.git

diff --git a/matita/matita/lib/turing/universal/uni_step.ma b/matita/matita/lib/turing/universal/uni_step.ma
index d24dc07ba..eabab3e83 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".
 
 (*
 
@@ -55,46 +57,44 @@ 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
 cases HR -HR
-#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape #Hta
-* #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb -Hta 
+#ta * whd in ⊢ (%→?); * #Hta #_ lapply (Hta … Hintape) -Hta -Hintape #Hta
+* #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 
-  [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
-* #_ #Htf lapply (Htf (reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?) 
-  [#x #membl @Hnomarks @daemon] -Htf #Htf >Htf >reverse_reverse %
+* #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
+* #tf * whd in ⊢ (%→?); * #_ #Htf lapply (Htf … Hte) -Htf -Hte #Htf
+whd in ⊢ (%→?); * #_ #Htg cases (Htg … Htf) -Htg -Htf
+#_ #Htg cases (Htg (refl …)) -Htg #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 *)
@@ -105,10 +105,8 @@ qed.
 *)
 
 definition init_copy ≝ 
-  seq ? init_current_on_match
-    (seq ? (move_r ?) 
-      (seq ? (adv_to_mark_r ? (is_marked ?))
-        (adv_mark_r ?))).
+  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. 
@@ -136,21 +134,22 @@ cases (sem_seq ????? sem_init_current_on_match
 #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
+* #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 *
+   * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc  … Htb) -Htc -Htb *
     [* whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)]
-   * #_ #Htc lapply (Htc … (refl …) (refl …) ?)
+   * * #_ #Htc #_ lapply (Htc … (refl …) (refl …) ?)
     [#x #membx @Hl2marks @membx]
-   #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+   #Htc whd in ⊢ (%→?); * #Houtc #_ cases (Houtc (reverse ? l2@〈grid,false〉::〈c,true〉::〈grid,false〉::ls) comma)
+   -Houtc #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 *
+   * #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 …) ?)
+   * * #_ >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
@@ -158,7 +157,9 @@ generalize in match Hl1marks; -Hl1marks cases (list_last ? l1)
         ]
       |@Hl2marks @membx
       ]]
-  #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+  -Htc #Htc #_ whd in ⊢ (%→?); * #Houtc #_ cases (Houtc (reverse (FinProd FSUnialpha FinBool) ((reverse STape tl@[〈grid,false〉])@l2)
+     @c1::〈c,true〉::〈grid,false〉::ls) comma)
+  -Houtc #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc #_
   >Houtc >reverse_append >reverse_append >reverse_single 
   >reverse_reverse >associative_append >associative_append 
   >associative_append %
@@ -234,134 +235,544 @@ cases HR -HR
   ]
 qed. *)
 
-include "turing/universal/move_tape.ma".
+definition exec_action ≝ 
+  init_copy · copy · move_r … · move_tape.
 
-definition exec_move ≝ 
-  seq ? (adv_to_mark_r … (is_marked ?))
-    (seq ? init_copy
-      (seq ? copy
-        (seq ? (move_r …)
-          (seq ? move_tape (move_r …))))).
-
-definition lift_tape ≝ λls,c,rs.
-  let 〈c0,b〉 ≝ c in
-  let c' ≝ match c0 with
-  [ null ⇒ None ?
-  | _ ⇒ Some ? c ]
-  in
-  mk_tape STape ls c' rs.
+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 Pre_exec_action ≝ λt1.
+  ∃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).
+
+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.
   
-definition sim_current_of_tape ≝ λt.
-  match current STape t with
-  [ None ⇒ 〈null,false〉
-  | Some c0 ⇒ c0 ].
+(* 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.
+
+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.
 
-(*
- t1 =  ls # cs c # table # rs  
- 
- let simt ≝ lift_tape ls c rs in
- let simt' ≝ move_left simt' in
- 
- t2 = left simt'# cs (sim_current_of_tape simt') # table # right simt'
-*)
-          
-(*
-definition R_move
-
-definition R_exec_move ≝ λt1,t2.
-  ∀ls,current,table1,newcurrent,table2,rs.
-  t1 = midtape STape (current@〈grid,false〉::ls) 〈grid,false〉
-       (table1@〈comma,true〉::newcurrent@〈comma,false〉::move::table2@
-        〈grid,false〉::rs) → 
-  table_TM (table1@〈comma,false〉::newcurrent@〈comma,false〉::move::table2) →
-  t2 = midtape
 *)
 
-(*
+(*lemma GRealize_to_Realize : 
+  ∀sig,M,R.GRealize sig M (λx.True) R → Realize sig M R.
+#sig #M #R #HR #t @HR //
+qed.*)
 
-step :
+lemma sem_exec_action : GRealize ? exec_action Pre_exec_action R_exec_action.
+@(sem_seq_app_guarded … (Realize_to_GRealize … sem_init_copy) ???)
+[3:@(sem_seq_app_guarded … sem_copy)
+ [3:@(sem_seq_app_guarded … (Realize_to_GRealize … (sem_move_r STape)))
+  [3:@(Realize_to_GRealize … (sem_move_tape …))
+  |@(λt.True)
+  |4://
+  |5:@sub_reflexive]
+ |@(λt.True)
+ |4://
+ |5:@sub_reflexive]
+|4:#t1 #t2 
+   * #n * #curconfig * #ls * #rs * #c0 * #c1 * #s0 * #s1 * #table1 * #newconfig 
+   * #mv * #table2 * * * * * * *
+   #Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hc1 #Hlen #Htape #Ht1
+   whd in ⊢ (%→?); >Ht1 #HR
+   lapply (HR (〈c0,false〉::curconfig) table1 s0 ls s1 
+              (newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) ???? (refl ??))
+   [(*Hcurconfig2*) @daemon
+   |(*by Htable *) @daemon
+   |(*Hcurconfig2*) @daemon
+   |(*Hcurconfig1*) @daemon ]
+   -HR #Ht2 whd
+   %{(〈grid,false〉::ls)} %{s0} %{s1} %{c0} %{c1} %{(〈mv,false〉::table2@〈grid,false〉::rs)}
+   %{newconfig} %{(〈comma,false〉::reverse ? table1)} %{curconfig} >Ht2
+   % [ % [ % [ % [ % [ % [ % [ % 
+   [ %
+   |(*by Htabel*) @daemon ]
+   |(*by Htable*) @daemon ]
+   |// ]
+   |>Hlen % ]
+   |@Hcurconfig2 ]
+   |@Hnewconfig ]
+   |cases Htape * * // ]
+   | // ] 
+   |1,2:skip]
+#ta #outtape #HR #n #curconfig #ls #rs #c0 #c1 #s0 #s1 #table1 #newconfig #mv
+#table2 #Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hbitnullc1 #Hlen #Htape
+#Hta #ta' #Hta'
+cases HR -HR #tb * whd in ⊢ (%→?); #Htb 
+lapply (Htb (〈c0,false〉::curconfig) table1 s0 ls s1 
+            (newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs)
+            ???? Hta) -Htb
+[ (*Hcurconfig2*) @daemon
+| (*Htable*) @daemon
+| (* by Htape bit_or_null c0 = true, moreover Hcurconfig2 *) @daemon
+| (*Hcurconfig1*) @daemon ]
+#Htb * #tc * whd in ⊢ (%→?); #Htc
+lapply (Htc (〈grid,false〉::ls) s0 s1 c0 c1 (〈mv,false〉::table2@〈grid,false〉::rs) newconfig (〈comma,false〉::reverse ? table1) curconfig Htb ????????) -Htc
+[9:|*:(* bit_or_null c0,c1; |curconfig| = |newconfig|*) @daemon ]
+#Htc * #td * whd in ⊢ (%→?); * #_ #Htd lapply (Htd … Htc) -Htd whd in ⊢(???(??%%%)→?);#Htd
+whd in ⊢ (%→?); #Houtc whd in Htd:(???%); whd in Htd:(???(??%%%));
+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:>Htd @(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 >Hta' >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 >Hta' >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 >Hta' 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_true(current) (* current state is final *)
-   then nop
-   else 
-   init_match;
-   match_tuple;
-   if is_marked(current) = false (* match ok *)
-      then exec_move; 
+(*
+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 mk_tuple ≝ λc,newc,mv.
-  c @ 〈comma,false〉:: newc @ 〈comma,false〉 :: [〈mv,false〉].
-
-inductive match_in_table (c,newc:list STape) (mv:unialpha) : list STape → Prop ≝ 
-| mit_hd : 
-   ∀tb.
-   match_in_table c newc mv (mk_tuple c newc mv@〈bar,false〉::tb)
-| mit_tl :
-   ∀c0,newc0,mv0,tb.
-   match_in_table c newc mv tb → 
-   match_in_table c newc mv (mk_tuple c0 newc0 mv0@〈bar,false〉::tb).
-
-definition move_of_unialpha ≝ 
-  λc.match c with
-  [ bit x ⇒ match x with [ true ⇒ R | false ⇒ L ]
-  | _ ⇒ N ].
-
-definition R_uni_step ≝ λt1,t2.
-  ∀n,table,c,c1,ls,rs,curs,curc,news,newc,mv.
-  table_TM n table → 
-  match_in_table (〈c,false〉::curs@[〈curc,false〉]) 
-    (〈c1,false〉::news@[〈newc,false〉]) mv table → 
-  t1 = midtape STape (〈grid,false〉::ls) 〈c,false〉 
-    (curs@〈curc,false〉::〈grid,false〉::table@〈grid,false〉::rs) → 
-  ∀t1',ls1,rs1.t1' = lift_tape ls 〈curc,false〉 rs → 
-  (t2 = midtape STape (〈grid,false〉::ls1) 〈c1,false〉 
-    (news@〈newc,false〉::〈grid,false〉::table@〈grid,false〉::rs1) ∧
-   lift_tape ls1 〈newc,false〉 rs1 = 
-   tape_move STape t1' (Some ? 〈〈newc,false〉,move_of_unialpha mv〉)).
-
-definition no_nulls ≝ 
- λl:list STape.∀x.memb ? x l = true → is_null (\fst x) = false.
- 
-definition R_move_tape_r_abstract ≝ λt1,t2.
-  ∀rs,n,table,curc,curconfig,ls.
-  bit_or_null curc = true → only_bits_or_nulls curconfig → table_TM n table → 
-  t1 = midtape STape (table@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls) 
-         〈grid,false〉 rs →
-  no_nulls rs → 
-  ∀t1'.t1' = lift_tape ls 〈curc,false〉 rs → 
-  ∃ls1,rs1,newc.
-  (t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@newc::
-    〈grid,false〉::reverse ? table@〈grid,false〉::rs1) ∧
-   lift_tape ls1 newc rs1 = 
-   tape_move_right STape ls 〈curc,false〉 rs).
+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.
+
+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).
    
-lemma lift_tape_not_null :
-  ∀ls,c,rs. is_null (\fst c) = false → 
-  lift_tape ls c rs = mk_tape STape ls (Some ? c) rs.
-#ls * #c0 #bc0 #rs cases c0
-[|normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
-//
+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))).
+
+definition Pre_uni_step ≝ λt1.
+  ∃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).
+    
+lemma sem_uni_step :
+  accGRealize ? uni_step us_acc Pre_uni_step
+     R_uni_step_true R_uni_step_false. 
+@daemon (* this no longer works: TODO *) (*
+@(acc_sem_if_app_guarded STape … (sem_test_char ? (λc:STape.\fst c == bit false)) 
+  ? (test_char_inv …) (sem_nop …) …)
+[| @(sem_seq_app_guarded … (Realize_to_GRealize … sem_init_match) ???)
+  [ 5: @sub_reflexive
+  | 3: @(sem_seq_app_guarded … sem_match_tuple 
+         (Realize_to_GRealize … (sem_if ????????? (sem_test_char …  (λc.¬is_marked FSUnialpha c))
+          (sem_seq … sem_exec_action (sem_move_r …))
+          (sem_nop …))))
+     [@(λx.True)
+     |//
+     |@sub_reflexive]
+  ||| #t1 #t2 * #n * #table * #s0 * #s1 * #c0 * #c1 * #ls * #rs * #curconfig 
+      * #newconfig * #mv * * * *
+      #Hlen1 #Htable #Hmatch #Hlegal #Ht1
+      whd in ⊢ (%→?);
+      cut (∃tup,table0.table = tup@table0 ∧ tuple_TM (S n) tup)
+      [@daemon]
+      * #tup * #table0 * #Htableeq * #qin * #cin * #qout * #cout * #mv0
+      * * * * * * * * * *
+      #Hqinnomarks #_ #Hqinbits #_ #_ #_ #_ #_ #Hqinlen #_ #Htupeq
+      cut (∃d,qin0.qin = 〈d,false〉::qin0)
+      [ lapply Hqinlen lapply Hqinnomarks -Hqinlen -Hqinnomarks cases qin
+        [ #_ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) 
+        | * #d #bd #qin0 #Hqinnomarks #_ %{d} %{qin0}
+          >(?:bd=false) [%]
+          @(Hqinnomarks 〈d,bd〉) @memb_hd ] ]
+      * #d * #qin0 #Hqineq
+      #Ht2 
+      lapply (Ht2 (〈grid,false〉::ls) (curconfig@[〈c0,false〉])
+               (qin0@〈cin,false〉::〈comma,false〉::qout@〈cout,false〉::〈comma,false〉::〈mv0,false〉::table0@〈grid,false〉::rs) s0 d ???)
+      [ >Ht1 @eq_f >associative_append @eq_f @eq_f @eq_f
+        >Htableeq >Htupeq >associative_append whd in ⊢ (??%?);
+        @eq_f >Hqineq >associative_append @eq_f whd in ⊢ (??%?);
+        @eq_f whd in ⊢ (??%?); @eq_f
+        >associative_append %
+      | @daemon
+      | @daemon
+      ]
+      #Ht2 % [| % [| % [| % [ @Ht2 ]
+        %2
+        (* ls0 = ls
+           c = s0
+           l1 = curconfig@[〈c0,false〉]
+           l2 = [〈bar,false〉]
+           c10 = d
+           l3 = qin0@[〈cin,false〉]
+           l4 = qout@〈cout,false〉::〈comma,false〉::〈mv0,false〉::table0
+           rs00 = rs
+           n0 = S n ?
+         *)
+        %{ls} %{s0} %{(curconfig@[〈c0,false〉])}
+        %{([〈bar,false〉])} %{d} %{(qin0@[〈cin,false〉])}
+        %{(qout@〈cout,false〉::〈comma,false〉::〈mv0,false〉::table0)}
+        %{rs} %{n} @daemon (* TODO *)
+      ]
+     ]
+    ]
+   ]
+ | #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; * * * #c #bc *
+  whd in ⊢ (??%?→?); #HSome destruct (HSome) #Hc #Hta % [@(\P Hc)]
+  cases HR -HR 
+  #tb * whd in ⊢ (%→?); #Htb
+  lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) c 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
+    | >(\P Hc) %
+    | (* 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 * * #c2 * whd in ⊢ (??%?→?); #Hc2 destruct (Hc2) 
+      #_ #Htd
+      cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
+      >Hnewc cut (mv1 = 〈mv,false〉)
+      [@daemon] #Hmv1
+      * #te * whd in ⊢ (%→?); #Hte
+      cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈c,false〉::〈grid,false〉::ls) 
+              〈grid,false〉
+              ((table1@〈bar,false〉::〈c,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 >Hmv1 >Hnewc @eq_f @eq_f @eq_f @eq_f @eq_f @eq_f
+          whd in ⊢ (??%?); >associative_append % 
+        ]
+      ]
+      -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
+    lapply (Htd ? (refl ??)) normalize in ⊢ (%→?);
+    #Hfalse destruct (Hfalse)
+  ]
+ ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #Ht1 #Ht2
+  #b #Hb >Hb in Ht1; * #Hc #Ht1 lapply (Hc ? (refl ??)) -Hc #Hb' %
+  // cases b in Hb'; normalize #H1 //
+]
+*)
 qed.
- 
-lemma mtr_concrete_to_abstract :
-  ∀t1,t2.R_move_tape_r t1 t2 → R_move_tape_r_abstract t1 t2.
-#t1 #t2 whd in ⊢(%→?); #Hconcrete
-#rs #n #table #curc #curconfig #ls #Hcurc #Hcurconfig #Htable #Ht1
-#Hrsnonulls #t1' #Ht1'
-cases (Hconcrete … Htable Ht1) //
-[ * #Hrs #Ht2
-  @(ex_intro ?? (〈curc,false〉::ls)) @(ex_intro ?? [])
-  @(ex_intro ?? 〈null,false〉) %
-  [ >Ht2 %
-  | >Hrs % ]
-| * #r0 * #rs0 * #Hrs #Ht2 
-  @(ex_intro ?? (〈curc,false〉::ls)) @(ex_intro ?? rs0)
-  @(ex_intro ?? r0) %
-  [ >Ht2 %
-  | >Hrs >lift_tape_not_null
-    [ %
-    | @Hrsnonulls >Hrs @memb_hd ] ]
-qed.
\ No newline at end of file
+
+(*
+@(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; * * * #c #bc *
+  whd in ⊢ (??%?→?); #HSome destruct (HSome) #Hc #Hta % [@(\P Hc)]
+  cases HR -HR 
+  #tb * whd in ⊢ (%→?); #Htb
+  lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) c 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
+    | >(\P Hc) %
+    | (* 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 * * #c2 * whd in ⊢ (??%?→?); #Hc2 destruct (Hc2) 
+      #_ #Htd
+      cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
+      >Hnewc cut (mv1 = 〈mv,false〉)
+      [@daemon] #Hmv1
+      * #te * whd in ⊢ (%→?); #Hte
+      cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈c,false〉::〈grid,false〉::ls) 
+              〈grid,false〉
+              ((table1@〈bar,false〉::〈c,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 >Hmv1 >Hnewc @eq_f @eq_f @eq_f @eq_f @eq_f @eq_f
+          whd in ⊢ (??%?); >associative_append % 
+        ]
+      ]
+      -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
+    lapply (Htd ? (refl ??)) normalize in ⊢ (%→?);
+    #Hfalse destruct (Hfalse)
+  ]
+ ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #Ht1 #Ht2
+  #b #Hb >Hb in Ht1; * #Hc #Ht1 lapply (Hc ? (refl ??)) -Hc #Hb' %
+  // cases b in Hb'; normalize #H1 //
+]
+qed.
+*)
\ No newline at end of file