X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Fcopy.ma;h=b2fcf29f11e72c744719e499d0363f2b79754637;hb=77bf99a9ac05a61573d621d576e269870668f076;hp=5775eab05df5b3b578d9dfb08f456050cc67c582;hpb=1c878b2e2fcdc1d61c9604741f2c8f22e8cfee8a;p=helm.git

diff --git a/matita/matita/lib/turing/universal/copy.ma b/matita/matita/lib/turing/universal/copy.ma
index 5775eab05..b2fcf29f1 100644
--- a/matita/matita/lib/turing/universal/copy.ma
+++ b/matita/matita/lib/turing/universal/copy.ma
@@ -10,26 +10,7 @@
       V_____________________________________________________________*)
 
 
-(* COMPARE BIT
-
-*)
-
-include "turing/universal/tuples.ma".
-
-definition write_states ≝ initN 2.
-
-definition write ≝ λalpha,c.
-  mk_TM alpha write_states
-  (λp.let 〈q,a〉 ≝ p in
-    match q with 
-    [ O ⇒ 〈1,Some ? 〈c,N〉〉
-    | S _ ⇒ 〈1,None ?〉 ])
-  O (λx.x == 1).
-  
-definition R_write ≝ λalpha,c,t1,t2.
-  ∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs.
-  
-axiom sem_write : ∀alpha,c.Realize ? (write alpha c) (R_write alpha c).
+include "turing/universal/marks.ma".
 
 definition copy_step_subcase ≝
   λalpha,c,elseM.ifTM ? (test_char ? (λx.x == 〈c,true〉))
@@ -44,17 +25,62 @@ definition copy_step_subcase ≝
 
 definition R_copy_step_subcase ≝ 
   λalpha,c,RelseM,t1,t2.
-    ∀ls,x,rs.t1 = midtape (FinProd … alpha FinBool) ls 〈x,true〉 rs → 
-    (x = c ∧
-     ∀a,l1,x0,a0,l2,l3. (∀c.memb ? c l1 = true → is_marked ? c = false) → 
-     ls = l1@〈a0,false〉::〈x0,true〉::l2 → 
-     rs = 〈a,false〉::l3 → 
-     t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
+    ∀a,l1,x0,a0,l2,x,l3.
+    t1 = midtape (FinProd … alpha FinBool) (l1@〈a0,false〉::〈x0,true〉::l2) 
+         〈x,true〉 (〈a,false〉::l3) → 
+    (∀c.memb ? c l1 = true → is_marked ? c = false) →          
+    (x = c ∧ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
     (x ≠ c ∧ RelseM t1 t2).
     
-axiom sem_copy_step_subcase : 
-  ∀alpha,c,elseM,RelseM.
+lemma sem_copy_step_subcase : 
+  ∀alpha,c,elseM,RelseM. Realize ? elseM RelseM → 
   Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM).
+#alpha #c #elseM #RelseM #HelseM #intape
+cases (sem_if ? (test_char ? (λx. x == 〈c,true〉)) ?????? tc_true (sem_test_char ? (λx.x == 〈c,true〉))
+        (sem_seq ????? (sem_adv_mark_r alpha)
+          (sem_seq ????? (sem_move_l …)
+            (sem_seq ????? (sem_adv_to_mark_l … (is_marked alpha))
+              (sem_seq ????? (sem_write ? 〈c,false〉)
+                (sem_seq ????? (sem_move_r …)
+                  (sem_seq ????? (sem_mark …)
+                    (sem_seq ????? (sem_move_r …) (sem_adv_to_mark_r … (is_marked alpha)))))))))
+        HelseM intape)
+#k * #outc * #Hloop #HR %{k} %{outc} % [@Hloop] -Hloop
+#a #l1 #x0 #a0 #l2 #x #l3 #Hintape #Hl1marks cases HR -HR
+[ * #ta * whd in ⊢ (%→?); >Hintape * * #c0 * whd in ⊢ (??%?→?); #Hx #Hc #Hta
+  * #tb * whd in ⊢ (%→?); * #Htb cases (Htb (l1@〈a0,false〉::〈x0,true〉::l2) x) -Htb
+  #Htb lapply (Htb … Hta) -Htb #Htb #_ #_
+  * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc #Htc
+  * #td * whd in ⊢ (%→?); * #_ #Htd cases (Htd … Htc) -Htd #_ #Htd cases (Htd (refl ??))
+  -Htd #Htd lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ???) //
+  [#x1 #Hx1 cases (memb_append … Hx1) [ @Hl1marks | #Hsingle >(memb_single … Hsingle) % ]
+  |whd in ⊢ (??%?); // ]
+  -Htd #Htd #_
+  * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd -Htc #Hte
+  * #tf * whd in ⊢ (%→?); * #_ #Htf lapply (Htf … Hte) -Hte -Htf >reverse_append #Htf
+  * #tg * whd in ⊢ (%→?); * #Htg #_ lapply (Htg … Htf) -Htf -Htg >reverse_single #Htg
+  * #th * whd in ⊢ (%→?); * #_ #Hth lapply (Hth … Htg) -Htg -Hth
+  generalize in match Hl1marks; -Hl1marks @(list_elim_left … l1)
+  [ #Hl1marks #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc
+    [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+    * * #_ #Houtc #_ lapply (Houtc [] ?? (refl ??) (refl ??) Hl1marks) -Houtc
+    #Houtc % >(\P Hc) in Hx; #Hx destruct (Hx) % // @Houtc
+  | -l1 #c1 #l1 #_ #Hl1marks >reverse_append >reverse_single
+    #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc
+    [ * >Hl1marks [ #Hfalse destruct (Hfalse) ] @memb_append_l2 @memb_hd ]
+    * * #_ #Houtc lapply (Houtc (reverse ? l1@[〈x,false〉]) 〈a,true〉 l3 ? (refl ??) ?) -Houtc
+    [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1   
+      [@Hl1marks @memb_append_l1 <(reverse_reverse … l1) @memb_reverse @Hx1 
+      |>(memb_single … Hx1) % ]
+    | normalize >associative_append % ] 
+    #Houtc #_ % destruct (Hx) lapply (\P Hc) -Hc #Hc destruct (Hc) % //
+    >Houtc >reverse_append >reverse_reverse >associative_append >associative_append % ]
+| * #ta * whd in ⊢ (%→?); >Hintape * #Hxc #Hta #Helse %2 %
+  [| <Hta @Helse ]
+  % #Hxc0 >Hxc0 in Hxc; #Hxc lapply (Hxc 〈c,true〉 (refl …)) #Hfalse
+  cases (\Pf Hfalse) #Hfalse0 @Hfalse0 %
+]
+qed.
     
 (*
 if current = 0,tt
@@ -75,439 +101,722 @@ else if current = 1,tt
         mark;
         move_r;
         advance_to_mark_r;
-else nop
+else if current = null 
+   then advance_mark_r;
+        move_l;
+        advance_to_mark_l
+        adv_mark_r;
+        move_r;
+        advance_to_mark_r
 *)
 
+definition nocopy_subcase ≝
+  ifTM STape (test_char ? (λx:STape.x == 〈null,true〉))
+    (seq ? (adv_mark_r …)
+      (seq ? (move_l …)
+        (seq ? (adv_to_mark_l … (is_marked ?))
+          (seq ? (adv_mark_r …)
+            (seq ? (move_r …) (adv_to_mark_r … (is_marked ?)))))))
+    (nop ?) tc_true.
+
+definition R_nocopy_subcase ≝ 
+  λt1,t2.
+    ∀a,l1,x0,a0,l2,x,l3.
+    t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2) 
+         〈x,true〉 (〈a,false〉::l3) → 
+    (∀c.memb ? c l1 = true → is_marked ? c = false) →          
+    (x = null ∧
+     t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨
+    (x ≠ null ∧ t2 = t1).
+    
+lemma sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase.
+#intape
+cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true
+        (sem_test_char ? (λx:STape.x == 〈null,true〉))
+          (sem_seq … (sem_adv_mark_r …)
+            (sem_seq … (sem_move_l …)
+              (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
+                (sem_seq … (sem_adv_mark_r …)
+                  (sem_seq … (sem_move_r …) 
+                    (sem_adv_to_mark_r … (is_marked ?))))))) (sem_nop ?) intape)
+#k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc)  % [@Hloop] -Hloop
+#a #l1 #x0 #a0 #l2 #x #l3 #Hintape #Hl1marks cases HR -HR
+[ * #ta * whd in ⊢ (%→?); >Hintape * * #c * whd in ⊢ (??%?→?); #Hc destruct (Hc) #Hx #Hta
+  * #tb * whd in ⊢ (%→?); * #Htb #_ cases (Htb (l1@〈a0,false〉::〈x0,true〉::l2) x) -Htb #Htb #_ lapply (Htb … Hta) -Hta -Htb #Htb
+  * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc #Htc
+  * #td * whd in ⊢ (%→?); * #_ #Htd cases (Htd … Htc) -Htd #_ #Htd cases (Htd (refl …)) -Htd #Htd #_
+  lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ? (refl …) ?)
+  [#x1 #Hx1 cases (memb_append … Hx1) [@Hl1marks| -Hx1 #Hx1 >(memb_single … Hx1) % ]
+  |>associative_append % ] -Htd >reverse_append in ⊢ (???%→?); >associative_append in ⊢ (???%→?); #Htd
+  * #te * whd in ⊢ (%→?); * #Hte cases (Hte l2 x0) -Hte #Hte #_ #_ lapply (Hte … Htd) -Hte -Htd -Htc #Hte
+  * #tf * whd in ⊢ (%→?); * #_ #Htf lapply (Htf … Hte) -Hte -Htf
+  generalize in match Hl1marks; -Hl1marks @(list_elim_left … l1)
+  [ #Hl1marks #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc
+    [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+    * * #_ #Houtc lapply (Houtc [] ?? (refl ??) (refl ??) Hl1marks) -Houtc
+    #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) #_ % % [%] @Houtc
+  | -l1 #c1 #l1 #_ #Hl1marks >reverse_append >reverse_single
+    #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc
+    [ * >Hl1marks [ #Hfalse destruct (Hfalse) ] @memb_append_l2 @memb_hd ]
+    * * #Hc1 #Houtc #_ lapply (Houtc (reverse ? l1@[〈x,false〉]) 〈a,true〉 l3 ? (refl ??) ?) -Houtc
+    [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1
+      [@Hl1marks @memb_append_l1 <(reverse_reverse … l1) @memb_reverse @Hx1 
+      |>(memb_single … Hx1) % ]
+    | normalize >associative_append % ] 
+    #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) % % [%] >Houtc
+    >reverse_append >reverse_reverse >associative_append >associative_append % ]
+| * #ta * whd in ⊢ (%→?); >Hintape * #Hxc
+  cut (x ≠ null) [ % #Hx cases (\Pf (Hxc ? (refl …))) #Hfalse @Hfalse >Hx % ] -Hxc #Hxnull
+  #Hta whd in ⊢ (%→?); #Houtc %2 % // <Hta @Houtc ]
+qed.
+
 definition copy_step ≝
-  ifTM ? (test_char STape (λc.is_bit (\fst c)))
+  ifTM ? (test_char STape (λc.bit_or_null (\fst c)))
   (single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
-    (copy_step_subcase FSUnialpha (bit true) (nop ?))))
+    (copy_step_subcase FSUnialpha (bit true) nocopy_subcase)))
   (nop ?)
   tc_true.
   
 definition R_copy_step_true ≝ 
   λt1,t2.
-    ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls 〈c,true〉 rs → 
-    ∃x. c = bit x ∧
-    (∀a,l1,c0,a0,l2,l3. (∀y.memb ? y l1 = true → is_marked ? y = false) → 
-     ls = l1@〈a0,false〉::〈c0,true〉::l2 → 
-     rs = 〈a,false〉::l3 → 
-     t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3).
-
+    ∀ls,c,rs. t1 = midtape STape ls 〈c,true〉 rs → 
+    bit_or_null c = true ∧
+    (∀a,l1,x0,a0,l2,l3.
+     ls = (l1@〈a0,false〉::〈x0,true〉::l2) → 
+     rs = (〈a,false〉::l3) → 
+     no_marks l1 →          
+     ((∃x. c = bit x ∧
+      t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3) ∨
+      (c = null ∧
+      t2 = midtape ? (〈null,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3))).
+     
 definition R_copy_step_false ≝ 
   λt1,t2.
    ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs → 
-   is_bit (\fst c) = false ∧ t2 = t1.
-
-axiom sem_comp_step : 
-  accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false.
+   bit_or_null (\fst c) = false ∧ t2 = t1.
    
-definition copy ≝ whileTM ? copy_step (inr … (inl … (inr … 0))).
-
-definition R_copy ≝ λt1,t2.
-  ∀ls,c,rs.t1 = midtape ? ls 〈c,true〉 rs → 
-  (∀l1,d,l2,l3,l4.
-   〈c,false〉::rs = l1@〈d,false〉::l2 → only_bits l1 → is_bit d = false → 
-   ls = l3@l4@〈c0,true〉::l5 → |l4| = |l1@[〈d,false〉]|
-  
-  
+lemma sem_copy_step : 
+  accRealize ? copy_step (inr … (inl … (inr … start_nop))) R_copy_step_true R_copy_step_false.
+#intape
+@(acc_sem_if_app … (sem_test_char ? (λc:STape.bit_or_null (\fst c)))  …
+    (sem_copy_step_subcase FSUnialpha (bit false) …
+       (sem_copy_step_subcase FSUnialpha (bit true) … (sem_nocopy_subcase …)))
+          (sem_nop …))
+[ #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1 >Ht1 in H1;
+  * * #c0 * whd in ⊢ (??%?→?); #Hc0 destruct (Hc0) #Hc #Ht3 % //
+  #a #l1 #x0 #a0 #l2 #l3 #Hls #Hrs #Hl1marks >Hls in Ht3; >Hrs #Ht3
+  cases (H2 … Ht3 ?)
+  [ * #Hc' #Ht2 % %{false} % // <Hc' @Ht2
+  | * #Hnotfalse whd in ⊢ (%→?); #Ht2 cases (Ht2 … Ht3 ?) -Ht2
+    [ * #Hc' #Ht2 % %{true} % // <Hc' @Ht2
+    |  * #Hnottrue whd in ⊢ (%→?); #Ht2 cases (Ht2 … Ht3 ?) -Ht2
+      [ * #Hc' #Ht2 %2 <Hc' % // @Ht2
+      | * #Hnotnull @False_ind
+        generalize in match Hnotnull;generalize in match Hnottrue;generalize in match Hnotfalse;
+        cases c in Hc; normalize
+        [ * [ #_ #_ * #Hfalse #_ | #_ * #Hfalse #_ #_ ]
+        | #_ #_ #_ * #Hfalse
+        |*: #Hfalse destruct (Hfalse) ] @Hfalse %
+      | @Hl1marks ]
+    | @Hl1marks ]
+  | @Hl1marks ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1
+  >Ht1 in H1; * #Hcur #Ht3 % // @Hcur % ]
+qed.
 
-axiom no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
 (*
-l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
-   ^                               ^
-
-if current (* x *) = #
-   then 
-   else if x = 0
-      then move_right; ----
-           adv_to_mark_r;
-           if current (* x0 *) = 0
-              then advance_mark ----
-                   adv_to_mark_l;
-                   advance_mark
-              else STOP
-      else x = 1 (* analogo *)
-
+1) il primo carattere è marcato
+2) l'ultimo carattere è l'unico che può essere null, gli altri sono bit
+3) il terminatore non è né bit, né null
 *)
-
-
-(*
-   MARK NEXT TUPLE machine
-   (partially axiomatized)
    
-   marks the first character after the first bar (rightwards)
- *)
- 
-definition bar_or_grid ≝ λc:STape.is_bar (\fst c) ∨ is_grid (\fst c).
-
-definition mark_next_tuple ≝ 
-  seq ? (adv_to_mark_r ? bar_or_grid)
-     (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
-       (move_right_and_mark ?) (nop ?) 1).
+definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … start_nop))).
+
+let rec merge_config (l1,l2:list STape) ≝ 
+  match l1 with
+  [ nil ⇒ nil ?
+  | cons p1 l1' ⇒ match l2 with
+    [ nil ⇒ nil ? 
+    | cons p2 l2' ⇒ 
+           let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in
+           match c2 with
+           [ null ⇒ p1
+           | _ ⇒ p2 ] :: merge_config l1' l2' ] ].
+           
+lemma merge_config_append :
+ ∀l1,l2,l3,l4.|l1| = |l2| → 
+ merge_config (l1@l3) (l2@l4) = merge_config l1 l2@merge_config l3 l4.
+#l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen)
+[normalize //
+| #t1 #t2 * #c1 #b1 * #c2 #b2 #IH whd in ⊢ (??%%); >IH % ]
+qed.
 
-definition R_mark_next_tuple ≝ 
-  λt1,t2.
-    ∀ls,c,rs1,rs2.
-    (* c non può essere un separatore ... speriamo *)
-    t1 = midtape STape ls c (rs1@〈grid,false〉::rs2) → 
-    no_marks rs1 → no_grids rs1 → bar_or_grid c = false → 
-    (∃rs3,rs4,d,b.rs1 = rs3 @ 〈bar,false〉 :: rs4 ∧
-      no_bars rs3 ∧
-      Some ? 〈d,b〉 = option_hd ? (rs4@〈grid,false〉::rs2) ∧
-      t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2)))
-    ∨
-    (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
-     
-axiom tech_split :
-  ∀A:DeqSet.∀f,l.
-   (∀x.memb A x l = true → f x = false) ∨
-   (∃l1,c,l2.f c = true ∧ l = l1@c::l2 ∧ ∀x.memb ? x l1 = true → f x = false).
-(*#A #f #l elim l
-[ % #x normalize #Hfalse *)
+definition R_copy0 ≝ λt1,t2.
+  ∀ls,c,c0,rs,l1,l3,l4.
+  t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) → 
+  no_marks l1 → no_marks (l3@l4) → |l1| = |l4| → 
+  ∀l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] → only_bits_or_nulls l1' → 
+  ∀l4',bg.l4@[〈c0,false〉] = 〈grid,bg〉::l4' → only_bits_or_nulls l4' → 
+  (c = comma ∧ t2 = t1) ∨
+  (c ≠ comma ∧ 
+    t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉::
+                  merge_config l4' (reverse ? l1')@ls) 
+     〈comma,true〉 rs).
      
-theorem sem_mark_next_tuple :
-  Realize ? mark_next_tuple R_mark_next_tuple.
-#intape 
-lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
-         (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) 1) ????)
-[@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip]
-| //
-|||#Hif cases (Hif intape) -Hif
-   #j * #outc * #Hloop * #ta * #Hleft #Hright
-   @(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
-   -Hloop
-   #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
-   cases (Hleft … Hrs)
-   [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
-   | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1)
-     [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
-       [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
-       | %
-       | -Hta #Hta cases Hright
-         [ * #tb * whd in ⊢ (%→?); #Hcurrent
-           @False_ind cases (Hcurrent 〈grid,false〉 ?)
-           [ normalize #Hfalse destruct (Hfalse)
-           | >Hta % ]
-         | * #tb * whd in ⊢ (%→?); #Hcurrent
-           cases (Hcurrent 〈grid,false〉 ?)
-           [  #_ #Htb whd in ⊢ (%→?); #Houtc
-             %2 %
-             [ @H1
-             | >Houtc >Htb >Hta % ]
-           | >Hta % ]
-         ]
-       ]
-    | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
-      % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
-     lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
-     [ #x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
-       #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
-       >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
-     | whd in ⊢ (??%?); >Hc0 %
-     | >Hsplit >associative_append % ] -Hta #Hta
-       cases Hright
-       [ * #tb * whd in ⊢ (%→?); #Hta'
-         whd in ⊢ (%→?); #Htb
-         cases (Hta' c0 ?)
-         [ #_ #Htb' >Htb' in Htb; #Htb
-           generalize in match Hsplit; -Hsplit
-           cases rs4 in Hta;
-           [ #Hta #Hsplit >(Htb … Hta)
-             >(?:c0 = 〈bar,false〉)
-             [ @(ex_intro ?? grid) @(ex_intro ?? false)
-               % [ % [ % 
-               [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ] 
-               | (* Hc0 *) @daemon ]
-           | #r5 #rs5 >(eq_pair_fst_snd … r5)
-             #Hta #Hsplit >(Htb … Hta)
-             >(?:c0 = 〈bar,false〉)
-             [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
-               % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
-                     | % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
-             | * #tb * whd in ⊢ (%→?); #Hta'
-               whd in ⊢ (%→?); #Htb
-               cases (Hta' c0 ?)
-               [ #Hfalse @False_ind >Hfalse in Hc0;
-                 #Hc0 destruct (Hc0)
-               | >Hta % ]
-]]]]
+lemma inj_append_singleton_l1 :
+  ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → l1 = l2.
+#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
+>reverse_append >reverse_append normalize #H1 destruct
+lapply (eq_f … (reverse ?) … e0) >reverse_reverse >reverse_reverse //
 qed.
 
-definition init_current ≝ 
-  seq ? (adv_to_mark_l ? (is_marked ?))
-    (seq ? (clear_mark ?)
-       (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
-          (seq ? (move_r ?) (mark ?)))).
-          
-definition R_init_current ≝ λt1,t2.
-  ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false → 
-  Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) → 
-  t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs → 
-  t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
-        ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
+lemma inj_append_singleton_l2 :
+  ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → a1 = a2.
+#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
+>reverse_append >reverse_append normalize #H1 destruct %
+qed.
 
-lemma sem_init_current : Realize ? init_current R_init_current.
-#intape 
-cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
-        (sem_seq ????? (sem_clear_mark ?)
-           (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
-             (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
-#k * #outc * #Hloop #HR 
-@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta 
-* #tb * whd in ⊢ (%→?); #Htb 
-* #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 %
-]
+axiom daemon : ∀P:Prop.P.
+
+lemma list_cases2_full : 
+  ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:Prop.
+  length ? l1 = length ? l2 →
+  (l1 = [] → l2 = [] → P) → 
+  (∀hd1,hd2,tl1,tl2.l1 = hd1::tl1 → l2 = hd2::tl2 → P) → P.
+#T1 #T2 #l1 #l2 #P #Hl @(list_ind2 … Hl)
+[ #Pnil #Pcons @Pnil //
+| #tl1 #tl2 #hd1 #hd2 #IH1 #IH2 #Hp @Hp // ]
 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)))
-    (nop ?) tc_true.
+lemma wsem_copy0 : WRealize ? copy0 R_copy0.
+#intape #k #outc #Hloop 
+lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
+* #ta * #Hstar @(star_ind_l ??????? Hstar)
+[ whd in ⊢ (%→?); #Hleft
+  #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+  #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
+  cases (Hleft … Htb) -Hleft #Hc #Houtc % %
+  [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
+    [ normalize #Hl1 #c1 destruct (Hl1) %
+    | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
+      #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ] 
+      >Hc #Hfalse destruct ]
+  | @Houtc ]
+| #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
+  lapply (Hind Htd) -Hind #Hind
+  #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+  #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
+  cases (Htc … Htb) -Htc #Hcbitnull #Htc
+  % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ]
+  cut (|l1| = |reverse ? l4|) [>length_reverse @Hlen] #Hlen1
+  @(list_cases2_full … Hlen1)
+  [ (* case l1 = [] is discriminated because l1 contains at least comma *)
+    #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize
+    [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull)
+    | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0;
+      [ normalize #Hfalse1 destruct (Hfalse1)
+      | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ]
+  | (* case c::l1 = c::a::l1'' *)
+    * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons
+    lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons
+    cut (ba = false) 
+    [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?)
+      [ @memb_hd | normalize // ] ] #Hba
+    cut (ba0 = false) 
+    [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?)
+      [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0
+    >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons
+    >Hl4cons in Htc; >Hl1cons #Htc
+    lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?)
+    [ #x #Hx @Hl3l4nomarks >Hl4cons <associative_append
+      @memb_append_l1 @Hx
+    | >associative_append >associative_append %
+    | -Htc
+      cut (∃la.l1' = 〈c,false〉::la)
+      [ >Hl1cons in Hl1; cases l1'
+        [normalize #Hfalse destruct (Hfalse)
+        | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ]
+      * #la #Hla
+      cut (∃lb.l4' = lb@[〈c0,false〉])
+      [ >Hl4cons in Hl4;
+        @(list_elim_left … l4')
+        [ #Heq lapply (eq_f … (length ?) … Heq)
+          >length_append >length_append 
+          >commutative_plus normalize >commutative_plus normalize
+          #Hfalse destruct
+        | #a1 #tl #_ #Heq 
+          >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq)
+          @ex_intro //
+      ] ] * #lb #Hlb
+      cut (|lb| = |reverse ? la|) 
+      [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1
+        >(?:l1 = la@[〈comma,bv〉]) in Hlen;
+        [|normalize in Hl1; destruct (Hl1) %]
+        >(?:l4 = 〈grid,bg〉::lb)
+        [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ]
+        >length_append >commutative_plus >length_reverse
+        normalize #Hlalb destruct (Hlalb) //
+      ] #Hlen2
+      *
+      (* by hyp on the first iteration step, 
+         we consider whether c = bit x or c = null *)
+      (* c = bit x *)
+      [ * #x * #Hx #Htc 
+        lapply (Hind (〈bit x,false〉::ls) a a0 rs l1'' 
+                (〈bit x,false〉::l3) (reverse ? l4'') ????) 
+        [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus 
+          normalize #Hlen destruct (Hlen) //
+        | #x0 #Hx0 cases (orb_true_l … Hx0)
+          [ #Hx0eq >(\P Hx0eq) %
+          | -Hx0 #Hx0 @Hl3l4nomarks >Hl4cons
+            <associative_append @memb_append_l1 // ]
+        | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+        | >Htc >associative_append % 
+        | -Hind 
+          <Hl1cons <Hl4cons #Hind lapply (Hind la bv ?? lb bg ??)
+          [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+          | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+            @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+          | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+          | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+            destruct (Hl1) // ] -Hind
+          (* by IH, we proceed by cases, whether a = comma 
+             (consequently several lists = []) or not *)          
+          *
+          [ * #Ha #Houtc1
+           cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
+           [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+           >Hl1cons in Hl1; >Hla
+           >Houtc1 >Htc #Hl1
+           >Hl4cons in Hl4; >Hlb #Hl4
+           >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx
+           cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+           normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+          | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+            >Hla >reverse_cons >associative_append @eq_f
+            >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+            >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %            
+          ]
+       ]
+    | (* c = null *)
+      * #Hc #Htc 
+      lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????)
+      [  >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize
+         #Hlen destruct (Hlen) @e0
+      | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0
+        [ >(memb_single … Hx0) %
+        | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0
+          [ @memb_append_l1 //
+          | @memb_append_l2 >Hl4cons @memb_append_l1 // ]
+        ]
+      | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+      | >Htc @eq_f3 // >associative_append % ] -Hind <Hl1cons <Hl4cons #Hind
+        lapply (Hind la bv ?? lb bg ??)
+          [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+          | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+            @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+          | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+          | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+            destruct (Hl1) // ] -Hind *
+          (* by IH, we proceed by cases, whether a = comma 
+             (consequently several lists = []) or not *)          
+          [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla
+           >Houtc1 >Htc #Hl1
+           >Hl4cons in Hl4; >Hlb #Hl4
+           cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = []) 
+           [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+           >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hc
+           cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+           normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+          | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+            >Hla >reverse_cons >associative_append @eq_f
+            >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+            >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
+          ]
+       ]
+]]]
+qed.
 
-definition R_match_tuple_step_true ≝ λt1,t2.
-  ∀ls,c,l1,l2,c1,l3,l4,rs,n.
-  is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true →
-  only_bits l3 → n = |l1| → |l1| = |l3| →
-  table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) → 
-  t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉 
-         (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs) → 
-  (* facciamo match *)
-  (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
-  t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
-        (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs))
-  ∨
-  (* non facciamo match e marchiamo la prossima tupla *)
-  ((〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
-   ∃c2,l5,l6,l7.l4 = l5@〈bar,false〉::〈c2,false〉::l6@〈comma,false〉::l7 ∧
-   (* condizioni su l5 l6 l7 *)
-   t2 = midtape STape (〈grid,false〉::ls) 〈c,true〉 
-         (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::
-          l5@〈bar,false〉::〈c2,true〉::l6@〈comma,false〉::l7))
-  ∨  
-  (* non facciamo match e non c'è una prossima tupla:
-     non specifichiamo condizioni sul nastro di output, perché
-     non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
-  (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).  
+definition Pre_copy0 ≝ λt1.
+  ∃ls,c,c0,rs,l1,l3,l4.
+  t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) ∧
+  no_marks l1 ∧ no_marks (l3@l4) ∧ |l1| = |l4| ∧
+  (∃l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] ∧ only_bits_or_nulls l1') ∧
+  (∃l4',bg.l4@[〈c0,false〉] = 〈grid,bg〉::l4' ∧ only_bits_or_nulls l4').
+
+definition Pre_copy ≝ λt1.
+  ∃ls,s0,s1,c0,c1,rs,l1,l3,l4.
+  t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) ∧
+  no_marks l1 ∧ no_marks l3 ∧ no_marks l4 ∧ |l1| = |l4| ∧ 
+  only_bits (l4@[〈s0,false〉]) ∧ only_bits (〈s1,false〉::l1) ∧ 
+  bit_or_null c0 = true ∧ bit_or_null c1 = true.
   
-definition R_match_tuple_step_false ≝ λt1,t2.
-  ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
-  
-include alias "basics/logic.ma". 
-
-(*
-lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
-  ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →   
-    f x1 x2 x3 x4 = f y1 y2 y3 y4.
-//
-qed-. *)
+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 some_option_hd: ∀A.∀l:list A.∀a.∃b.
-  Some ? b = option_hd ? (l@[a]) .
-#A #l #a cases l normalize /2/
+lemma terminate_copy0 : ∀t.Pre_copy0 t → Terminate ? copy0 t.
+#t #HPre
+@(terminate_while_guarded ??? 
+   Pre_copy0 … 
+   (acc_Realize_to_acc_GRealize ??? Pre_copy0 … sem_copy_step)
+   … HPre) [%]
+  [ -HPre -t #t1 #t2 * #ls * #c * #c0 * #rs * #l1 * #l3 * #l4
+    * * * * * #Ht1 #Hl1nomarks #Hl3l4nomarks #Hlen
+    * #l1' * #bv * #Hl1 #Hbitsnullsl1' * #l4' * #bg * #Hl4 #Hbitsnullsl4'
+    #HR cases (HR … Ht1) -HR #Hbitnullc
+    cut (∃d1,l1''.l1 = 〈d1,false〉::l1'')
+    [ lapply Hl1nomarks cases l1 in Hl1;
+      [ whd in ⊢ (???%→?); #Hl1
+        lapply (append_l2_injective_r ? [] l1' [〈c,false〉] [〈comma,bv〉] (refl ??) Hl1)
+        #Hc destruct (Hc) normalize in Hbitnullc; destruct (Hbitnullc) 
+      | * #d #bd #l1'' #_ #Hl1nomarks >(?:bd = false) [| @(Hl1nomarks 〈d,bd〉) @memb_hd ] /3/ ] ]
+    * #d1 * #l1'' #Hl1''
+    cut (∃d2,l4''.l4 = l4''@[〈d2,false〉])
+    [ lapply Hl4 cases (list_last ? l4)
+      [ #Hl4' >Hl4' in Hlen; >Hl1'' normalize in ⊢ (%→?); #HFalse destruct (HFalse)
+      | * * #d2 #bd2 * #l4'' #Hl4'' >Hl4'' in Hl3l4nomarks; #Hl3l4nomarks #_ 
+        <(?:bd2 = false) [| @(Hl3l4nomarks 〈d2,bd2〉) @memb_append_l2 @memb_append_l2 @memb_hd ]
+        /3/ ] ]
+    * #d2 * #l4'' #Hl4'' >Hl1'' >Hl4''
+    cut (∃l1''',bv0.〈d1,false〉::l1''=l1'''@[〈comma,bv0〉]∧only_bits_or_nulls l1''')
+    [ <Hl1'' cases (list_last ? l1) in Hl1'';
+        [ #Hl1'' >Hl1''#HFalse destruct(HFalse)
+        | * #a * #l1''' #Hl1''' >Hl1''' in Hl1; #Hl1 #_
+          lapply (append_l2_injective_r ? (〈c,false〉::l1''') l1' [a] [〈comma,bv〉] (refl …) Hl1)
+          #Ha destruct (Ha) %{l1'''} %{bv} % //
+          #x #Hx @Hbitsnullsl1'
+          lapply (append_l1_injective_r ? (〈c,false〉::l1''') l1' [〈comma,bv〉] [〈comma,bv〉] (refl …) Hl1)
+          #H <H @memb_cons @Hx ] ]
+    cut (∃l4''',bg0.l4''@[〈d2,false〉]=〈grid,bg0〉::l4'''∧only_bits_or_nulls l4''')
+    [ <Hl4'' cases (list_last ? l4') in Hl4;
+        [ #Hl4' >Hl4' >Hl4'' cases l4''
+          [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+          | #y #yl normalize in ⊢ (%→?); #H1 destruct (H1) cases yl in e0;
+            [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+            | #z #zl normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] ]
+        | * #a * #l4''' #Hl4''' >Hl4''' #Hl4
+          lapply (append_l1_injective_r ? l4 (〈grid,bg〉::l4''') [〈c0,false〉] [a] (refl …) Hl4)
+          -Hl4 #Hl4 >Hl4 %{l4'''} %{bg} % //
+          #x #Hx @Hbitsnullsl4' >Hl4''' @memb_append_l1 @Hx ] ]
+    #Hl4''' #Hl1'''
+    #HR cases (HR d1 (l3@l4'') c0 d2 ls (l1''@rs) ? ? ?)
+    [3: >associative_append >associative_append %
+    |4: %
+    |5: #x #Hx @Hl3l4nomarks cases (memb_append … Hx)
+      [ @memb_append_l1
+      | >Hl4'' -Hx #Hx @memb_append_l2 @memb_append_l1 @Hx ]
+    | * #x1 * #Hc #Ht2 whd >Ht2
+      %{(〈bit x1,false〉::ls)} %{d1} %{d2} %{rs} %{l1''} %{(〈bit x1,false〉::l3)} %{l4''}
+      % [ % [ % [ % [ % 
+      [ >associative_append %
+      | #x #Hx @Hl1nomarks >Hl1'' @memb_cons @Hx ]
+      | #x #Hx cases (orb_true_l … Hx) -Hx #Hx
+        [ >(\P Hx) %
+        | @Hl3l4nomarks cases (memb_append … Hx)
+          [ @memb_append_l1
+          | -Hx #Hx >Hl4'' @memb_append_l2 @memb_append_l1 @Hx ] ] ]
+      | >Hl1'' in Hlen; >Hl4'' >length_append >commutative_plus
+        normalize in ⊢ (%→?); #Hlen destruct (Hlen) @e0 ]
+      | @Hl1''' ]
+      | @Hl4''' ]
+   | * #Hc #Ht2 whd >Ht2
+      %{(〈c0,false〉::ls)} %{d1} %{d2} %{rs} %{l1''} %{(〈null,false〉::l3)} %{l4''}
+      % [ % [ % [ % [ % 
+      [ >associative_append %
+      | #x #Hx @Hl1nomarks >Hl1'' @memb_cons @Hx ]
+      | #x #Hx cases (orb_true_l … Hx) -Hx #Hx
+        [ >(\P Hx) %
+        | @Hl3l4nomarks cases (memb_append … Hx)
+          [ @memb_append_l1
+          | -Hx #Hx >Hl4'' @memb_append_l2 @memb_append_l1 @Hx ] ] ]
+      | >Hl1'' in Hlen; >Hl4'' >length_append >commutative_plus
+        normalize in ⊢ (%→?); #Hlen destruct (Hlen) @e0 ]
+      | @Hl1''' ] 
+      | @Hl4''' ]
+    ]
+ |cases HPre -HPre #ls * #c * #c0 * #rs * #l1 * #l3 * #l4
+  * * * * * #Ht #Hl1nomarks #Hl3l4nomarks #Hlen
+  * #l1' * #bv * #Hl1 #Hbitsnullsl1'
+  * #l4' * #bg * #Hl4 #Hbitsnullsl4'
+  >Ht lapply Hbitsnullsl1' lapply Hl1 lapply l1' lapply Hl3l4nomarks 
+  lapply Hl1nomarks lapply l3 lapply c0 lapply c lapply ls 
+  -Hbitsnullsl1' -Hl1 -l1' -Hl3l4nomarks -Hl1nomarks -l3 -Hl4 -c0 -c -ls
+  <(reverse_reverse ? l4) <(length_reverse ? l4) in Hlen; #Hlen
+  @(list_ind2 … Hlen)
+  [ #ls #c #c0 #l3 #_ #_ #l1' #Hl1 #Hbitsnullsl1' cases l1' in Hl1;
+    [| #x * [| #x0 #xs ] normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+    normalize in ⊢ (%→?); #Hl1' destruct (Hl1') % #t1 whd in ⊢ (%→?);
+    #HR cases (HR … (refl …)) whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
+  | #xs #ys * #x #bx * #y #by #IH
+    #ls #c #c0 #l3 #Hl1nomarks #Hl3l4nomarks #l1' #Hl1 
+    #Hbitsnullsl1' %
+    #t1 whd in ⊢ (%→?); #HR cases (HR … (refl …)) -HR
+    #Hbitnullc #HR
+    lapply (Hl1nomarks 〈x,bx〉 (memb_hd …)) normalize in ⊢ (%→?); #Hbx destruct (Hbx)
+    (*cut (∃d2,l4''.〈y,by〉::ys = l4''@[〈d2,false〉])
+    [ cases (list_last ? ys) in Hl3l4nomarks;
+      [ #Hl4' #Hl3l4nomarks >Hl4' >(?:by = false) [%{y} %{([])} %]
+        @(Hl3l4nomarks 〈y,by〉) @memb_append_l2 @memb_hd
+      | * * #d2 #bd2 * #l4'' #Hl4'' >Hl4'' #Hl3l4nomarks 
+        <(?:bd2 = false) [| @(Hl3l4nomarks 〈d2,bd2〉) @memb_append_l2 @memb_cons @memb_append_l2 @memb_hd ]
+        %{d2} %{(〈y,by〉::l4'')} % ] ]
+    * #a0 * #l4'' #Hl4'' >Hl4'' in HR; #HR *)
+    cases (HR x (l3@reverse ? ys) c0 y ls (xs@rs) ? (refl …) ?)
+    [3: >reverse_cons >associative_append >associative_append
+        >(?:by = false) [|@(Hl3l4nomarks 〈y,by〉) @memb_append_l2 >reverse_cons @memb_append_l2 @memb_hd ] %
+    |4: #x #Hx @Hl3l4nomarks >reverse_cons <associative_append @memb_append_l1 @Hx
+    | * #x1 * #Hc #Ht1 >Ht1 
+      <(?: (〈bit x1,false〉::l3)@reverse (FSUnialpha×bool) ys@〈y,true〉::〈bit x1,false〉::ls =
+            〈bit x1,false〉::(l3@reverse (FSUnialpha×bool) ys)@〈y,true〉::〈bit x1,false〉::ls)
+     [cut (∃l1''.〈x,false〉::xs = l1''@[〈comma,bv〉] ∧ only_bits_or_nulls l1'')
+      [lapply Hbitsnullsl1' lapply Hl1 -Hl1 -Hbitsnullsl1' cases l1'
+       [normalize in ⊢(%→?); #Hfalse destruct (Hfalse)
+       |#x2 #l1'' normalize in ⊢ (%→?); #Heq #Hbitsnullsl1' destruct (Heq)
+        %{l1''} % [@e0]
+        #x #Hx @Hbitsnullsl1' @memb_cons @Hx ] ]
+      * #l1'' * #Hl1'' #Hbitsnullsl1''
+      @(IH (〈bit x1,false〉::ls) x y (〈bit x1,false〉::l3) ?? l1'' Hl1'' Hbitsnullsl1'')
+      [#x #Hx @Hl1nomarks @memb_cons @Hx
+      |#x #Hx cases (orb_true_l … Hx) -Hx #Hx
+       [>(\P Hx) %
+       |cases (memb_append … Hx) -Hx #Hx @Hl3l4nomarks
+        [@memb_append_l1 @Hx
+        |@memb_append_l2 >reverse_cons @memb_append_l1 @Hx ] ] ]
+     |>associative_append % ]
+    | * #Hc #Ht1 >Ht1 
+      <(?: (〈null,false〉::l3)@reverse (FSUnialpha×bool) ys@〈y,true〉::〈c0,false〉::ls =
+            〈null,false〉::(l3@reverse (FSUnialpha×bool) ys)@〈y,true〉::〈c0,false〉::ls)
+     [cut (∃l1''.〈x,false〉::xs = l1''@[〈comma,bv〉] ∧ only_bits_or_nulls l1'')
+      [lapply Hbitsnullsl1' lapply Hl1 -Hl1 -Hbitsnullsl1' cases l1'
+       [normalize in ⊢(%→?); #Hfalse destruct (Hfalse)
+       |#x2 #l1'' normalize in ⊢ (%→?); #Heq #Hbitsnullsl1' destruct (Heq)
+        %{l1''} % [@e0]
+        #x #Hx @Hbitsnullsl1' @memb_cons @Hx ] ]
+      * #l1'' * #Hl1'' #Hbitsnullsl1''
+      @(IH (〈c0,false〉::ls) x y (〈null,false〉::l3) ?? l1'' Hl1'' Hbitsnullsl1'')
+      [#x #Hx @Hl1nomarks @memb_cons @Hx
+      |#x #Hx cases (orb_true_l … Hx) -Hx #Hx
+       [>(\P Hx) %
+       |cases (memb_append … Hx) -Hx #Hx @Hl3l4nomarks
+        [@memb_append_l1 @Hx
+        |@memb_append_l2 >reverse_cons @memb_append_l1 @Hx ] ] ]
+    |>associative_append % ]
+]]]
 qed.
 
-lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false.
-* // normalize #H destruct
+definition merge_char ≝ λc1,c2.
+  match c2 with
+  [ null ⇒ c1
+  | _ ⇒ c2 ].
+  
+lemma merge_cons : 
+  ∀c1,c2,conf1,conf2.
+  merge_config (〈c1,false〉::conf1) (〈c2,false〉::conf2) = 
+    〈merge_char c1 c2,false〉::merge_config conf1 conf2.
+#c1 #c2 #conf1 #conf2 normalize @eq_f2 //
+cases c2 /2/
 qed.
 
-lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
-* // normalize #H destruct
+lemma merge_bits : ∀l1,l2.|l1| = |l2| → only_bits l2 → merge_config l1 l2 = l2.
+#l1 #l2 #Hlen @(list_ind2 … Hlen) //
+#tl1 #tl2 #hd1 #hd2 #IH
+>(eq_pair_fst_snd … hd1) >(eq_pair_fst_snd … hd2) #Hbits
+change with (cons ???) in ⊢ (??%?); @eq_f2
+[ cases (\fst hd2) in Hbits; 
+  [ #b #_ %
+  |*: #Hfalse lapply (Hfalse … (memb_hd …)) normalize #Hfalse1 destruct (Hfalse1) ]
+| @IH #x #Hx @Hbits @memb_cons // ]
 qed.
 
-axiom sem_match_tuple_step: 
-    accRealize ? match_tuple_step (inr … (inl … (inr … 0))) 
-    R_match_tuple_step_true R_match_tuple_step_false.
-(* @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
-  (sem_seq … sem_compare
-    (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
-      (sem_nop …)
-        (sem_seq … sem_mark_next_tuple 
-           (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
-             (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
-  (sem_nop ?) …)
-[(* is_grid: termination case *)
- 2:#t1 #t2 #t3 whd in ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1
-  cases (H c ?) [2: >Ht1 %] #Hgrid #Heq %
-    [@injective_notb @Hgrid | <Heq @H1]
-|#tapea #tapeout #tapeb whd in ⊢ (%→?); #Htapea
- * #tapec * #Hcompare #Hor 
- #ls #c #l1 #l2 #c1 #l3 #l4 #rs #n #Hc #Hl1 #Hl2 #Hc1 #Hl3 #eqn
- #eqlen #Htable #Htapea1 cases (Htapea 〈c,true〉 ?) >Htapea1 [2:%]
- #notgridc -Htapea -Htapea1 -tapea #Htapeb  
- cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare 
- cases (Hcompare c c1 l1 l3 (l2@[〈bar,false〉]) (l4@〈grid,false〉::rs) eqlen … (refl …) Hc ?)  
- -Hcompare 
-   [* #Htemp destruct (Htemp) #Htapec %1 % [%]
-    >Htapec in Hor; -Htapec *
-     [2: * #t3 * whd in ⊢ (%→?); #H @False_ind
-      cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
-     |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped *
-      #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped >associative_append
-      %
-     ]
-   |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec 
-    cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3) 
-      [>H2 >H3 elim la
-        [@(not_to_not …H1) normalize #H destruct % 
-        |#x #tl @not_to_not normalize #H destruct // 
-        ]
-      ] #Hnoteq %2
-    cut (is_bit d' = true) 
-      [cases la in H3;
-        [normalize in ⊢ (%→?); #H destruct //
-        |#x #tl #H @(Hl3 〈d',false〉)
-         normalize in H; destruct @memb_append_l2 @memb_hd
-        ] 
-      ] #Hd'
-    >Htapec in Hor; -Htapec *
-     [* #taped * whd in ⊢ (%→?); #H @False_ind
-      cases (H … (refl …)) >Hd' #Htemp destruct (Htemp)
-     |* #taped * whd in ⊢ (%→?); #H cases (H … (refl …)) -H #_
-      #Htaped * #tapee * whd in ⊢ (%→?); #Htapee  
-      <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
-      lapply (Htapee … Htaped ???) -Htaped -Htapee 
-       [whd in ⊢ (??%?); >(bit_not_grid … Hd') >(bit_not_bar … Hd') %
-       |#x #Hx cases (memb_append … Hx) 
-         [-Hx #Hx @bit_not_grid @Hl3 cases la in H3; normalize 
-           [#H3 destruct (H3) @Hx | #y #tl #H3 destruct (H3) 
-            @memb_append_l2 @memb_cons @Hx ]
-         |-Hx #Hx @(no_grids_in_table … Htable) 
-          @memb_cons @memb_append_l2 @Hx
-         ]
-       |@daemon (* TODO *)
-       |* 
-         [* #rs3 * * (* we proceed by cases on rs4 *) 
-           [* #d * #b * * * #Heq1 #Hnobars
-            whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
-            #Htapee * 
-             [* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef 
-              cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
-              whd in ⊢ (%→?); #H lapply (H … ???? (refl …)) #Htapeout
-              %1 %
-              [ //| @daemon]
-              | >Htapeout %
-              ]
-           |* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef
-            cases (Htapef … (refl …)) whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
-           ]
-         |* #d2 #b2 #rs3' * #d  * #b * * * #Heq1 #Hnobars
-          cut (is_grid d2 = false) [@daemon (* ??? *)] #Hd2
-          whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
-           [* #tapef * whd in ⊢ (%→?); #Htapef 
-            cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp) 
-           |* #tapef * whd in ⊢ (%→?); #Htapef 
-            cases (Htapef … (refl …)) #_ -Htapef #Htapef
-            * #tapeg >Htapef -Htapef * whd in ⊢ (%→?); 
-            #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H  #Htapeg
-            >Htapeg -Htapeg whd in ⊢ (%→?); #Htapeout
-            %1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
-            * #c00 #b00 #Hoption
-            lapply 
-             (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb))
-             c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout
-              [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
-               >associative_append 
-               generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls); #l
-               whd in ⊢ (???(???%)); >associative_append >associative_append 
-               % 
-              |@daemon 
-              |@daemon
-              |@daemon
-              |@daemon
-              |@daemon
-              ]
-           ]
-         ]
-       |* #Hnobars #Htapee >Htapee -Htapee *
-         [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
-          cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
-          whd in ⊢ (%→?); #Htapeout %2
-          >(Htapeout … (refl …)) %
-           [ % 
-             [ @daemon 
-             | @daemon
-             ]
-           | %
-           ] 
-         |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
-          cases (Htapef … (refl …)) -Htapef 
-          whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp) 
-         ]
-       |
-           
-           
-      
-      
-   
-  
-  ????? (refl …) Hc ?) -Hcompare 
- #Hcompare 
-  is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true →
-  only_bits l3 → n = |l2| → |l2| = |l3| →
-  table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) →#x
+lemma merge_config_c_nil : 
+  ∀c.merge_config c [] = [].
+#c cases c normalize //
+qed.
 
-  #intape
-cases 
-  (acc_sem_if … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) 
-    (sem_seq … sem_compare
-      (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
-        (sem_nop …)
-        (sem_seq … sem_mark_next_tuple 
-           (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
-             (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
-    (sem_nop ?) intape)
-#k * #outc * * #Hloop #H1 #H2
-@(ex_intro ?? k) @(ex_intro ?? outc) %
-[ % [@Hloop ] ] -Hloop
-  *)
+lemma reverse_merge_config :
+  ∀c1,c2.|c1| = |c2| → reverse ? (merge_config c1 c2) = 
+    merge_config (reverse ? c1) (reverse ? c2).        
+#c1 #c2 <(length_reverse ? c1) <(length_reverse ? c2) #Hlen
+<(reverse_reverse ? c1) in ⊢ (??%?); <(reverse_reverse ? c2) in ⊢ (??%?);
+generalize in match Hlen; @(list_ind2 … Hlen) -Hlen //
+#tl1 #tl2 #hd1 #hd2 #IH whd in ⊢ (??%%→?); #Hlen destruct (Hlen) -Hlen
+<(length_reverse ? tl1) in e0; <(length_reverse ? tl2) #Hlen
+>reverse_cons >reverse_cons >(merge_config_append ???? Hlen)
+>reverse_append >(eq_pair_fst_snd ?? hd1) >(eq_pair_fst_snd ?? hd2)
+whd in ⊢ (??%%); @eq_f2 // @IH //
+qed.
 
-(* 
-  MATCH TUPLE
+definition copy
+≝ 
+  seq STape copy0 (seq ? (move_l …) (seq ? (adv_to_mark_l … (is_marked ?))
+   (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …))))).
 
-  scrolls through the tuples in the transition table until one matching the
-  current configuration is found
+(*
+   s0, s1 = caratteri di testa dello stato
+   c0 = carattere corrente del nastro oggetto
+   c1 = carattere in scrittura sul nastro oggetto
+   
+   questa dimostrazione sfrutta il fatto che 
+   merge_config (l0@[c0]) (l1@[c1]) = l1@[merge_char c0 c1] 
+   se l0 e l1 non contengono null
 *)
 
-definition match_tuple ≝  whileTM ? match_tuple_step (inr … (inl … (inr … 0))).
+definition R_copy ≝ λt1,t2.
+  ∀ls,s0,s1,c0,c1,rs,l1,l3,l4.
+  t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) → 
+  no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| → 
+  only_bits (l4@[〈s0,false〉]) → only_bits (〈s1,false〉::l1) → 
+  bit_or_null c0 = true → bit_or_null c1 = true → 
+  t2 = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l3@〈grid,false〉::
+                      〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
+       〈comma,false〉 rs.
+       
+lemma sem_copy0 : GRealize ? copy0 Pre_copy0 R_copy0.
+@WRealize_to_GRealize
+[ @terminate_copy0
+| @wsem_copy0 ]
+qed.
 
-definition R_match_tuple ≝ λt1,t2.
-  ∀ls,c,l1,c1,l2,rs,n.
-  is_bit c = true → only_bits l1 → is_bit c1 = true → n = |l1| →
-  table_TM (S n) (〈c1,true〉::l2) → 
-  t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉 
-         (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) → 
-  (* facciamo match *)
-  (∃l3,newc,mv,l4.
-   〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧
-   t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
-        (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv@l4@〈grid,false〉::rs))
-  ∨
-  (* non facciamo match su nessuna tupla;
-     non specifichiamo condizioni sul nastro di output, perché
-     non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
-  (current ? t2 = Some ? 〈grid,true〉 ∧
-   ∀l3,newc,mv,l4.
-   〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4).  
+definition option_cons ≝ λA.λa:option A.λl.
+  match a with
+  [ None ⇒ l
+  | Some a' ⇒ a'::l ].
+  
+lemma sem_copy : GRealize ? copy Pre_copy R_copy.
+@(GRealize_to_GRealize_2 ?? Pre_copy0 ? R_copy) //
+(* preamble: Pre_copy0 implies Pre_copy *)
+[ #t1 * #ls * #s0 * #s1 * #c0 * #c1 * #rs * #l1 * #l3 * #l4 * * * * * * * *
+  #Ht1 #Hl1nomarks #Hl3nomarks #Hl4nomarks #Hlen #Hbitsl4 #Hbitsl1 
+  #Hbitnullc0 #Hbitnullc1 whd
+  %{ls} %{s1} %{s0} %{rs} %{(l1@[〈c1,false〉;〈comma,false〉])}
+  %{l3} %{(〈grid,false〉::〈c0,false〉::l4)}
+  % [ % [ % [ % [ % 
+  [ >Ht1 -Ht1 @eq_f >associative_append %
+  | #x #Hx cases (memb_append … Hx) -Hx #Hx
+    [ @(Hl1nomarks ? Hx)
+    | cases (orb_true_l … Hx) -Hx #Hx 
+      [ >(\P Hx) %
+      | >(memb_single … Hx) % ] ] ]
+  | #x #Hx cases (memb_append … Hx) -Hx #Hx
+    [ @(Hl3nomarks ? Hx)
+    | cases (orb_true_l … Hx) -Hx #Hx
+      [ >(\P Hx) %
+      | cases (orb_true_l … Hx) -Hx #Hx
+        [ >(\P Hx) %
+        | @(Hl4nomarks ? Hx) ]]]]
+  |>length_append >Hlen >commutative_plus % ]
+  | %{(〈s1,false〉::l1@[〈c1,false〉])} %{false} %
+    [ @eq_f >associative_append %
+    | #x #Hx cases (orb_true_l … Hx) -Hx #Hx
+      [ >(\P Hx) @is_bit_to_bit_or_null @(Hbitsl1 〈s1,false〉) @memb_hd
+      | cases (memb_append … Hx) -Hx #Hx
+        [ @is_bit_to_bit_or_null @(Hbitsl1 〈\fst x,\snd x〉) @memb_cons <eq_pair_fst_snd @Hx
+        | >(memb_single… Hx) // ]
+      ]
+    ] ]
+  | %{(〈c0,false〉::l4@[〈s0,false〉])} %{false} %
+    [ %
+    | #x #Hx cases (orb_true_l … Hx) -Hx #Hx
+      [ >(\P Hx) //
+      | cases (memb_append … Hx) -Hx #Hx
+        [ @is_bit_to_bit_or_null @(Hbitsl4 〈\fst x,\snd x〉) @memb_append_l1 <eq_pair_fst_snd @Hx
+        | >(memb_single… Hx) @is_bit_to_bit_or_null @(Hbitsl4 〈s0,false〉) @memb_append_l2 @memb_hd ]
+      ]
+] ] ]
+(*whd in ⊢ (%→%)
+change with (?·?) in match copy;*)
+@(sem_seq_app_guarded ???? (λx.True) ??? sem_copy0)
+[2:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_move_l (FinProd FSUnialpha FinBool))))
+  [@(λx.True)
+  |4://
+  |5:@sub_reflexive
+  |3:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_adv_to_mark_l … (is_marked FSUnialpha))))
+    [@(λx.True)
+    |4://
+    |5:@sub_reflexive
+    |3:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_clear_mark ?)))
+      [@(λx.True)
+      |4://
+      |5:@sub_reflexive
+      |3:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_adv_to_mark_r ? (is_marked ?))))
+        [@(λx.True)
+        |4://
+        |5:@sub_reflexive
+        |3:@(Realize_to_GRealize … (sem_clear_mark ?)) ] ] ] ]
+|3: //
+|1:skip]
+#intape #outtape #HR
+#ls #s0 #s1 #c0 #c1 #rs #l1 #l2 #l3 #Hintape #Hl1marks #Hl2marks #Hl3marks #Hlen
+#Hbits1 #Hbits2 #Hc0bits #Hc1bits
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta 
+cut (ta = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l2@〈grid,true〉::
+                      〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
+       〈comma,true〉 rs)
+[lapply (Hta ls s1 s0 rs (l1@[〈c1,false〉;〈comma,false〉]) l2 (〈grid,false〉::〈c0,false〉::l3) ?)
+  [>associative_append in ⊢ (???(????%)); normalize in ⊢ (???(??%%%)); @Hintape ]
+ -Hta #Hta cases (Hta ??? (〈s1,false〉::l1@[〈c1,false〉]) false ? ? ?? (refl ??) ?)
+  [3: #x #Hx cases (memb_append … Hx) -Hx #Hx
+    [ @Hl1marks //
+    | cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) % | >(memb_single … Hx) % ]] 
+  |4: #x #Hx cases (memb_append … Hx) -Hx #Hx
+    [ @Hl2marks //
+    | cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) % | cases (orb_true_l … Hx) [-Hx #Hx >(\P Hx) % | @Hl3marks ] ] ]
+  |5: >length_append normalize >Hlen >commutative_plus %
+  |6: normalize >associative_append %
+  |7: #x #Hx cases (memb_append ?? (〈s1,false〉::l1) … Hx) -Hx #Hx
+    [ whd in ⊢ (??%?); >(Hbits2 … Hx) %
+    | >(memb_single … Hx) // ]
+  |8: #x #Hx cases (memb_append … Hx) -Hx #Hx
+    [ cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) // | whd in ⊢ (??%?); >Hbits1 // @memb_append_l1 // ]
+    | >(memb_single … Hx) whd in ⊢ (??%?); >(Hbits1 〈s0,false〉) // @memb_append_l2 @memb_hd ]
+  | * #Hs1 @False_ind >Hs1 in Hbits2; #Hbits2 lapply (Hbits2 〈comma,false〉 ?) //
+    normalize #Hfalse destruct (Hfalse)
+  | * #Hs1 #Ht2 >Ht2 >reverse_cons >reverse_append >reverse_single @eq_f3 //
+    >merge_cons >merge_bits
+    [2: #x #Hx @Hbits2 cases (memb_append STape ? (reverse ? l1) ? Hx) -Hx #Hx
+      [@daemon | >(memb_single … Hx) @memb_hd ]
+    |3: >length_append >length_append >length_reverse >Hlen % ]
+    normalize >associative_append normalize >associative_append %
+  ]
+] -Hta #Hta * #tb * * #_ #Htb
+lapply (Htb … Hta) -Htb #Htb change with (midtape ????) in Htb:(???%);
+* #tc * * #_ #Htc 
+cases (Htc … Htb)
+(* [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ] *)
+#_ #Htc cases (Htc (refl ??)) -Htc #Htc #_
+lapply (Htc (reverse ? l1@〈s1,false〉::l2) 〈grid,true〉
+          (〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)???)
+[ #x #Hx cases (memb_append … Hx) -Hx #Hx
+  [ @Hl1marks @daemon
+  | cases (orb_true_l … Hx) -Hx #Hx
+    [ >(\P Hx) % | @(Hl2marks … Hx) ] ]
+| %
+| whd in ⊢ (??%?); >associative_append % ] -Htc #Htc
+* #td * whd in ⊢ (%→?); * #_ #Htd lapply (Htd … Htc) -Htd #Htd
+* #te * whd in ⊢ (%→?); * #_ #Hte cases (Hte … Htd) -Hte -Htd
+[ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
+* * #_ #Hte #_ 
+lapply (Hte (reverse ? (reverse ? l1@〈s1,false〉::l2)@[〈c1,false〉])
+         〈comma,true〉 rs ? (refl ??) ?) -Hte
+[ >reverse_append >reverse_cons >reverse_reverse #x #Hx
+  cases (memb_append … Hx) -Hx #Hx
+  [ cases (memb_append … Hx) -Hx #Hx
+    [ cases (memb_append … Hx) -Hx #Hx
+      [ @daemon 
+      | lapply (memb_single … Hx) -Hx #Hx >Hx % ]
+    | @(Hl1marks … Hx) ]
+  | lapply (memb_single … Hx) -Hx #Hx >Hx % ]
+| >reverse_append >reverse_reverse >reverse_cons
+  >associative_append >associative_append >associative_append
+  >associative_append >associative_append % ]
+#Hte * #_ #Houtc lapply (Houtc … Hte) -Houtc #Houtc >Houtc
+@eq_f3 //
+>reverse_append >reverse_append >reverse_single >reverse_cons
+>reverse_append >reverse_append >reverse_reverse >reverse_reverse
+>reverse_single >associative_append >associative_append %
+qed.
\ No newline at end of file