X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=f16190491a202810abf3f2e5fb192b2d477a2884;hb=302d83e414e27210fbfbc1de4d213786d9580e23;hp=b153ef8c7c5929f318a0206d623e571b133a9f32;hpb=315610badd512e271f6e99011721a3b4d3e316fc;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma
index b153ef8c7..f16190491 100644
--- a/matita/matita/lib/turing/multi_universal/match.ma
+++ b/matita/matita/lib/turing/multi_universal/match.ma
@@ -12,9 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "turing/turing.ma".
+include "turing/multi_universal/moves.ma".
+include "turing/if_multi.ma".
 include "turing/inject.ma".
-include "turing/while_multi.ma".
+include "turing/basic_machines.ma".
 
 definition compare_states ≝ initN 3.
 
@@ -32,7 +33,7 @@ definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)
 *)
 
 definition trans_compare_step ≝ 
- λi,j.λsig:FinSet.λn.
+ λi,j.λsig:FinSet.λn.λis_endc.
  λp:compare_states × (Vector (option sig) (S n)).
  let 〈q,a〉 ≝ p in
  match pi1 … q with
@@ -40,7 +41,7 @@ definition trans_compare_step ≝
    [ None ⇒ 〈comp2,null_action ? n〉
    | Some ai ⇒ match nth j ? a (None ?) with 
      [ None ⇒ 〈comp2,null_action ? n〉
-     | Some aj ⇒ if ai == aj 
+     | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj 
          then 〈comp1,change_vec ? (S n) 
                       (change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
                         (Some ? 〈aj,R〉) j〉
@@ -51,13 +52,14 @@ definition trans_compare_step ≝
    | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
 
 definition compare_step ≝ 
-  λi,j,sig,n.
-  mk_mTM sig n compare_states (trans_compare_step i j sig n) 
+  λi,j,sig,n,is_endc.
+  mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc) 
     comp0 (λq.q == comp1 ∨ q == comp2).
 
 definition R_comp_step_true ≝ 
-  λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
+  λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
   ∃x.
+   is_endc x = false ∧
    current ? (nth i ? int (niltape ?)) = Some ? x ∧
    current ? (nth j ? int (niltape ?)) = Some ? x ∧
    outt = change_vec ?? 
@@ -66,18 +68,19 @@ definition R_comp_step_true ≝
             (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
 
 definition R_comp_step_false ≝ 
-  λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
-  (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
+  λi,j:nat.λsig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+   ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+   current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
    current ? (nth i ? int (niltape ?)) = None ? ∨
    current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
 
 lemma comp_q0_q2_null :
-  ∀i,j,sig,n,v.i < S n → j < S n → 
+  ∀i,j,sig,n,is_endc,v.i < S n → j < S n → 
   (nth i ? (current_chars ?? v) (None ?) = None ? ∨
    nth j ? (current_chars ?? v) (None ?) = None ?) → 
-  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) 
+  step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) 
   = mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj
+#i #j #sig #n #is_endc #v #Hi #Hj
 whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
 * #Hcurrent
 [ @eq_f2
@@ -90,42 +93,57 @@ whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%
 qed.
 
 lemma comp_q0_q2_neq :
-  ∀i,j,sig,n,v.i < S n → j < S n → 
-  nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?) → 
-  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) 
+  ∀i,j,sig,n,is_endc,v.i < S n → j < S n → 
+  ((∃x.nth i ? (current_chars ?? v) (None ?) = Some ? x ∧ is_endc x = true) ∨ 
+    nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) → 
+  step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) 
   = mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
+#i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
 cases (nth i ?? (None ?)) in ⊢ (???%→?);
 [ #Hnth #_ @comp_q0_q2_null // % //
 | #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?)))
   cases (nth j ?? (None ?)) in ⊢ (???%→?);
   [ #Hnth #_ @comp_q0_q2_null // %2 //
-  | #aj #Haj #Hneq
-    whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-    [ whd in match (trans ????); >Hai >Haj
-      whd in ⊢ (??(???%)?); >(\bf ?) // @(not_to_not … Hneq) //
-    | whd in match (trans ????); >Hai >Haj
-      whd in ⊢ (??(???????(???%))?); >(\bf ?) /2 by not_to_not/
-      @tape_move_null_action
-] ]
+  | #aj #Haj *
+    [ * #c * >Hai #Heq #Hendc whd in ⊢ (??%?); 
+      >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+      [ whd in match (trans ????); >Hai >Haj destruct (Heq) 
+        whd in ⊢ (??(???%)?); >Hendc // 
+      | whd in match (trans ????); >Hai >Haj destruct (Heq) 
+        whd in ⊢ (??(???????(???%))?); >Hendc @tape_move_null_action
+      ]
+    | #Hneq
+      whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+      [ whd in match (trans ????); >Hai >Haj
+        whd in ⊢ (??(???%)?); cut ((¬is_endc ai∧ai==aj)=false)
+        [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // |#Hcut >Hcut //]
+        | whd in match (trans ????); >Hai >Haj
+          whd in ⊢ (??(???????(???%))?); cut ((¬is_endc ai∧ai==aj)=false)
+          [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // 
+          |#Hcut >Hcut @tape_move_null_action
+          ]
+        ]
+      ]
+    ]
+]
 qed.
 
 lemma comp_q0_q1 :
-  ∀i,j,sig,n,v,a.i ≠ j → i < S n → j < S n → 
-  nth i ? (current_chars ?? v) (None ?) = Some ? a →
+  ∀i,j,sig,n,is_endc,v,a.i ≠ j → i < S n → j < S n → 
+  nth i ? (current_chars ?? v) (None ?) = Some ? a → is_endc a = false →
   nth j ? (current_chars ?? v) (None ?) = Some ? a → 
-  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) =
+  step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) =
     mk_mconfig ??? comp1 
      (change_vec ? (S n) 
        (change_vec ?? v
          (tape_move ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i)
        (tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) j).
-#i #j #sig #n #v #a #Heq #Hi #Hj #Ha1 #Ha2
+#i #j #sig #n #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2
 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
 [ whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
+  >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
 | whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >(\b ?) //
+  >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) //
   change with (change_vec ?????) in ⊢ (??(???????%)?);
   <(change_vec_same … v j (niltape ?)) in ⊢ (??%?);
   <(change_vec_same … v i (niltape ?)) in ⊢ (??%?);
@@ -135,11 +153,11 @@ whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
 qed.
 
 lemma sem_comp_step :
-  ∀i,j,sig,n.i ≠ j → i < S n → j < S n → 
-  compare_step i j sig n ⊨ 
-    [ comp1: R_comp_step_true i j sig n, 
-             R_comp_step_false i j sig n ].
-#i #j #sig #n #Hneq #Hi #Hj #int
+  ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n → 
+  compare_step i j sig n is_endc ⊨ 
+    [ comp1: R_comp_step_true i j sig n is_endc, 
+             R_comp_step_false i j sig n is_endc ].
+#i #j #sig #n #is_endc #Hneq #Hi #Hj #int
 lapply (refl ? (current ? (nth i ? int (niltape ?))))
 cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
 [ #Hcuri %{2} %
@@ -155,103 +173,163 @@ cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
        [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%); 
          @sym_eq @nth_vec_map
        | normalize in ⊢ (%→?); #H destruct (H) ]
-       | #_ % >Ha >Hcurj % % % #H destruct (H) ] ]
-  | #b #Hb %{2} cases (true_or_false (a == b)) #Hab
+       | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
+  | #b #Hb %{2} 
+   cases (true_or_false (is_endc a)) #Haendc
     [ %
-      [| % [ % 
-        [whd in ⊢  (??%?);  >(comp_q0_q1 … a Hneq Hi Hj) //
-          [>(\P Hab) <Hb @sym_eq @nth_vec_map
-          |<Ha @sym_eq @nth_vec_map ]
-        | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) // ]
-        | * #H @False_ind @H %
-      ] ]
-    | %
       [| % [ % 
         [whd in ⊢  (??%?);  >comp_q0_q2_neq //
-         <(nth_vec_map ?? (current …) i ? int (niltape ?))
-         <(nth_vec_map ?? (current …) j ? int (niltape ?)) >Ha >Hb
-         @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
+         % %{a} % // <Ha @sym_eq @nth_vec_map
         | normalize in ⊢ (%→?); #H destruct (H) ]
-      | #_ % // % % >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+      | #_ % // % % % >Ha %{a} % // ]
+      ]
+    |cases (true_or_false (a == b)) #Hab
+      [ %
+        [| % [ % 
+          [whd in ⊢  (??%?);  >(comp_q0_q1 … a Hneq Hi Hj) //
+            [>(\P Hab) <Hb @sym_eq @nth_vec_map
+            |<Ha @sym_eq @nth_vec_map ]
+          | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) % // ]
+          | * #H @False_ind @H %
+        ] ]
+      | %
+        [| % [ % 
+          [whd in ⊢  (??%?);  >comp_q0_q2_neq //
+           <(nth_vec_map ?? (current …) i ? int (niltape ?))
+           <(nth_vec_map ?? (current …) j ? int (niltape ?)) %2 >Ha >Hb
+           @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
+          | normalize in ⊢ (%→?); #H destruct (H) ]
+        | #_ % // % % %2 >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+      ]
     ]
   ]
 ]
 qed.
 
-definition compare ≝ λi,j,sig,n.
-  whileTM … (compare_step i j sig n) comp1.
+definition compare ≝ λi,j,sig,n,is_endc.
+  whileTM … (compare_step i j sig n is_endc) comp1.
 
 definition R_compare ≝ 
-  λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
-  (current sig (nth i (tape sig) int (niltape sig))
-   ≠current sig (nth j (tape sig) int (niltape sig)) → 
-   outt = int) ∧  
-  (∀ls,x,xs,ci,rs,ls0,cj,rs0. 
+  λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+  ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+   (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
+    current ? (nth i ? int (niltape ?)) = None ? ∨
+    current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
+  (∀ls,x,xs,ci,rs,ls0,rs0. 
     nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
-    nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj → 
+    nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
+    (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → 
+    (rs0 = [ ] ∧
+     outt = change_vec ?? 
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+           (mk_tape sig (reverse ? xs@x::ls0) (None ?) []) j) ∨
+    ∃cj,rs1.rs0 = cj::rs1 ∧
+    ((is_endc ci = true ∨ ci ≠ cj) → 
     outt = change_vec ?? 
            (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
-           (midtape sig (reverse ? xs@x::ls0) cj rs0) j).
-
-lemma wsem_compare : ∀i,j,sig,n.i ≠ j → i < S n → j < S n → 
-  compare i j sig n ⊫ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj #ta #k #outc #Hloop
-lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) //
+           (midtape sig (reverse ? xs@x::ls0) cj rs1) j)).
+          
+lemma wsem_compare : ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n → 
+  compare i j sig n is_endc ⊫ R_compare i j sig n is_endc.
+#i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop
+lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * * [ *
-  [ #Hcicj #Houtc %
+[ #tc whd in ⊢ (%→?); * * [ * [ *
+  [* #curi * #Hcuri #Hendi #Houtc %
+    [ #_ @Houtc  
+    | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj #Hnotendc 
+      @False_ind
+      >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H)
+      >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H)
+    ]
+  |#Hcicj #Houtc % 
     [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj
+    | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj
       >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
-    ]
+    ]]
   | #Hci #Houtc %
     [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
+    | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi >Hnthi in Hci;
       normalize in ⊢ (%→?); #H destruct (H) ] ]
   | #Hcj #Houtc %
     [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
+    | #ls #x #xs #ci #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj;
       normalize in ⊢ (%→?); #H destruct (H) ] ]
-  | #tc #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
+  | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
     #IH1 #IH2 %
-    [ >Hci >Hcj * #H @False_ind @H %
-    | #ls #c0 #xs #ci #rs #ls0 #cj #rs0 cases xs
-      [ #Hnthi #Hnthj #Hcicj >IH1 
-        [ >Hd @eq_f3 // 
+    [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H) 
+    |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]] 
+    | #ls #c0 #xs #ci #rs #ls0 #rs0 cases xs
+      [ #Hnthi #Hnthj #Hnotendc cases rs0 in Hnthj;
+        [ #Hnthj % % // >IH1
+          [ >Hd @eq_f3 //
+            [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
+              >Hnthi in Hci;normalize #H destruct (H) %
+            | >(?:c0=x) [ >Hnthj % ]
+              >Hnthi in Hci;normalize #H destruct (H) % ]
+          | >Hd %2 %2 >nth_change_vec // >Hnthj % ]
+        | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // *
+          [ #Hendci >IH1
+            [ >Hd @eq_f3 // 
+              [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
+             >Hnthi in Hci;normalize #H destruct (H) %
+            | >(?:c0=x) [ >Hnthj % ]
+            >Hnthi in Hci;normalize #H destruct (H) % ]
+        | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
+          >nth_change_vec // >Hnthi >Hnthj normalize % %{ci} % //
+        ]
+      |#Hcir1 >IH1
+        [>Hd @eq_f3 // 
           [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
             >Hnthi in Hci;normalize #H destruct (H) %
           | >(?:c0=x) [ >Hnthj % ]
             >Hnthi in Hci;normalize #H destruct (H) % ]
-        | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
-          >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not ??? Hcicj)
+        | >Hd %2 % % >nth_change_vec //
+          >nth_change_vec_neq [|@sym_not_eq //]
+          >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not … Hcir1)
           #H destruct (H) % ]
-      | #x0 #xs0 #Hnthi #Hnthj #Hcicj
-        >(IH2 (c0::ls) x0 xs0 ci rs (c0::ls0) cj rs0 … Hcicj)
-        [ >Hd >change_vec_commute in ⊢ (??%?); //
-          >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
+      ]
+    ]
+  |#x0 #xs0 #Hnthi #Hnthj #Hnotendc 
+   cut (c0 = x) [ >Hnthi in Hci; normalize #H destruct (H) // ]
+   #Hcut destruct (Hcut) cases rs0 in Hnthj;
+    [ #Hnthj % % // 
+      cases (IH2 (x::ls) x0 xs0 ci rs (x::ls0) [ ] ???) -IH2
+      [ * #_ #IH2 >IH2 >Hd >change_vec_commute in ⊢ (??%?); //
+        >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
+        @sym_not_eq //
+      | * #cj * #rs1 * #H destruct (H)
+      | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+        >Hnthi %
+      | >Hd >nth_change_vec // >Hnthj %
+      | #c0 #Hc0 @Hnotendc @memb_cons @Hc0 ]
+    | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // #Hcir1
+      cases(IH2 (x::ls) x0 xs0 ci rs (x::ls0) (r1::rs1) ???)
+      [ * #H destruct (H)
+      | * #r1' * #rs1' * #H destruct (H) #Hc1r1 >Hc1r1 //
+        >Hd >change_vec_commute in ⊢ (??%?); //
+        >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
           @sym_not_eq //
-        | >Hd >nth_change_vec // >Hnthj normalize
-          >Hnthi in Hci;normalize #H destruct (H) %
-        | >Hd >nth_change_vec_neq [|@sym_not_eq //] >Hnthi
-          >nth_change_vec // normalize
-          >Hnthi in Hci;normalize #H destruct (H) %
-        ]
-]]]
+      | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+        >Hnthi //
+      | >Hd >nth_change_vec // >Hnthi >Hnthj %
+      | #c0 #Hc0 @Hnotendc @memb_cons @Hc0
+]]]]]
 qed.      
  
-lemma terminate_compare :  ∀i,j,sig,n,t.
+lemma terminate_compare :  ∀i,j,sig,n,is_endc,t.
   i ≠ j → i < S n → j < S n → 
-  compare i j sig n  ↓ t.
-#i #j #sig #n #t #Hneq #Hi #Hj
+  compare i j sig n is_endc ↓ t.
+#i #j #sig #n #is_endc #t #Hneq #Hi #Hj
 @(terminate_while … (sem_comp_step …)) //
 <(change_vec_same … t i (niltape ?))
 cases (nth i (tape sig) t (niltape ?))
-[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct 
-|2,3: #a0 #al0 % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+[ % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct 
+|2,3: #a0 #al0 % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
 | #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
-  [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
+  [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
    #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % 
-   #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+   #t2 * #x0 * * * #Hendcx0 >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
    >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
   |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
    normalize in ⊢ (%→?); #H destruct (H) #Hcur
@@ -260,9 +338,582 @@ cases (nth i (tape sig) t (niltape ?))
 ]
 qed.
 
-lemma sem_compare : ∀i,j,sig,n.
+lemma sem_compare : ∀i,j,sig,n,is_endc.
   i ≠ j → i < S n → j < S n → 
-  compare i j sig n ⊨ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize /2/
+  compare i j sig n is_endc ⊨ R_compare i j sig n is_endc.
+#i #j #sig #n #is_endc #Hneq #Hi #Hj @WRealize_to_Realize /2/
+qed.
+
+(*
+   |conf1   $
+   |confin 0/1 confout move
+
+  match machine step ≝
+    compare;
+    if (cur(src) != $)
+      then
+        parmoveL;
+        moveR(dst);
+      else nop
+ *)
+
+definition Rtc_multi_true ≝ 
+  λalpha,test,n,i.λt1,t2:Vector ? (S n).
+   (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
+   
+definition Rtc_multi_false ≝ 
+  λalpha,test,n,i.λt1,t2:Vector ? (S n).
+    (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
+
+lemma sem_test_char_multi :
+  ∀alpha,test,n,i.i ≤ n → 
+  inject_TM ? (test_char ? test) n i ⊨ 
+  [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
+#alpha #test #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
+  #Hcur #Htestc #Hnth_i #Hnth_j %
+  [ %{c} % //
+  | @(eq_vec … (niltape ?)) #i0 #Hi0
+    cases (decidable_eq_nat i0 i) #Hi0i
+    [ >Hi0i @Hnth_i
+    | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
+  [ @Htestc
+  | @(eq_vec … (niltape ?)) #i0 #Hi0
+    cases (decidable_eq_nat i0 i) #Hi0i
+    [ >Hi0i @Hnth_i
+    | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+qed.
+
+axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2. 
+  l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
+  ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
+  
+axiom daemon : ∀X:Prop.X.
+
+(*
+definition R_match_step_false ≝ 
+  λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+  ∀ls,x,xs,end,rs.
+  nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
+  (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
+   ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
+   (∃ls0,rs0. 
+    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+    ∀rsj,c. 
+    rs0 = c::rsj →
+    outt = change_vec ??
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
+           (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
+
+definition R_match_step_true ≝ 
+  λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+  ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s → 
+  is_startc s = true → 
+  (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
+  (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
+  (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →  
+   outt = change_vec ?? int 
+          (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧  
+  (∀ls,x,xs,ci,rs,ls0,rs0. 
+    nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
+    (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → 
+    (∀cj,rs1.rs0 = cj::rs1 → ci ≠ cj →
+     (outt = change_vec ?? int 
+           (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)) ∧
+    (rs0 = [ ] →
+     outt = change_vec ??
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
+           (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)).
+           
+lemma sem_match_step :
+  ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n → 
+  match_step src dst sig n is_startc is_endc ⊨ 
+    [ inr ?? (inr ?? (inl … (inr ?? start_nop))) : 
+      R_match_step_true src dst sig n is_startc is_endc, 
+      R_match_step_false src dst sig n is_endc ].
+#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst 
+@(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
+    (acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
+      (sem_seq … 
+        (sem_parmoveL ???? is_startc Hneq Hsrc Hdst) 
+        (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
+      (sem_nop …)))
+[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
+ #Htb #s #Hcurta_src #Hstart #Hnotstart % [ %
+ [#Hdst_none @daemon 
+ | #s1 #Hcurta_dst #Hneqss1
+   lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta) 
+   [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
+   #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //] 
+   whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
+   [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+     [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
+       #ls * #rs #Hta_mid >(Htbdst … Hta_mid) >Hta_mid cases rs //
+     | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
+   | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
+     normalize #H destruct (H) // ]
+   ]
+ |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc 
+  cases rs00 in Htadst_mid;
+   [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ]
+     #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2 
+     [2: * #x0 * #rs1 * #H destruct (H) ]
+    * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src; 
+    normalize in ⊢ (%→?); #H destruct (H)  
+    >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
+    >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
+     @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+      [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
+      |@sym_eq @Htbelse @sym_not_eq //
+      ] 
+    |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ]
+     #cj #rs1 #H destruct (H) #Hcicj
+     cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
+     * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
+     lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc
+     cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
+     #H destruct (H)
+     >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??)) //
+     [| >Htc >nth_change_vec //
+     | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid 
+      cases (orb_true_l … Hc0) -Hc0 #Hc0
+      [@memb_append_l2 >(\P Hc0) @memb_hd
+      |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
+      ]
+    | >Htc >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
+    * * #_ #Htbdst #Htbelse %
+    [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+      [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj'::rs0'))
+        [ cases xs //
+        | >nth_change_vec // ]
+      | >nth_change_vec_neq [|@sym_not_eq //]
+        <Htbelse [|@sym_not_eq // ]
+        >nth_change_vec_neq [|@sym_not_eq //]
+        cases (decidable_eq_nat i src) #Hisrc
+        [ >Hisrc >nth_change_vec // >Htasrc_mid //
+        | >nth_change_vec_neq [|@sym_not_eq //]
+          <(Htbelse i) [|@sym_not_eq // ]
+          >Htc >nth_change_vec_neq [|@sym_not_eq // ]
+          >nth_change_vec_neq [|@sym_not_eq // ] //
+        ]
+       ] 
+    | >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] 
+      >nth_change_vec // whd in ⊢ (??%?→?); 
+      #H destruct (H) cases (is_endc c) in Hcend;
+      normalize #H destruct (H) // ]
+    ]
+  ]
+|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb 
+ whd in ⊢ (%→?); #Hout >Hout >Htb whd
+ #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
+ lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
+ cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
+  [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
+  |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
+    [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
+     #ls_dst * #rs_dst #Hmid_dst %2
+     cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * 
+     #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst
+     cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
+     #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
+     lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?) 
+     [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+       [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //]
+     | *
+       [ * #Hrsj #Hta %
+         [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+           #Hc lapply (Hc ? (refl ??)) #Hendr1
+           cut (xs = xs1)
+           [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
+             -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
+             [ * normalize in ⊢ (%→?); //
+               #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
+               lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
+             | #x2 #xs2 #IH *
+               [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
+                 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
+                 normalize in ⊢ (%→?); #H destruct (H)
+               | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
+                 #Hnotendc #Hnotendcxs1 @eq_f @IH
+                 [ @(cons_injective_r … Heq)
+                 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
+                   [ >(\P Hc0) @memb_hd
+                   | @memb_cons @memb_cons // ]
+                 | #c #Hc @Hnotendcxs1 @memb_cons // ]
+               ]
+             ]
+           | #Hxsxs1 >Hmid_dst >Hxsxs1 % ]
+         | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
+       | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?) 
+         [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
+         -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] 
+         >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
+         (* lemmatize this proof *) cut (xs = xs1)
+         [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
+           -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
+           [ * normalize in ⊢ (%→?); //
+             #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
+             lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
+           | #x2 #xs2 #IH *
+             [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
+               >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
+               normalize in ⊢ (%→?); #H destruct (H)
+             | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
+               #Hnotendc #Hnotendcxs1 @eq_f @IH
+               [ @(cons_injective_r … Heq)
+               | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
+                 [ >(\P Hc0) @memb_hd
+                 | @memb_cons @memb_cons // ]
+               | #c #Hc @Hnotendcxs1 @memb_cons // ]
+             ]
+           ]
+         | #Hxsxs1 >Hmid_dst >Hxsxs1 % //
+           #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
+           @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
+           #Hendr1 destruct (Hendr1) % ]
+       ]
+     ]
+   (* STOP *)
+   |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst 
+    @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
+    @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape 
+    >Hintape in Hc; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc 
+    >(Hnotend c_src) // normalize #H destruct (H)   
+   ]
+  ]
+]
+qed. 
+*)
+
+definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
+  compare src dst sig n is_endc ·
+    (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
+      (ifTM ?? (inject_TM ? (test_null ?) n src)
+        (single_finalTM ??
+          (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
+        (nop …) tc_true)
+      (nop …)
+      tc_true).
+      
+definition R_match_step_false ≝ 
+  λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+  ∀ls,x,xs,end,rs.
+  nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
+  (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
+   ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
+   (∃ls0,rs0. 
+    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+    ∀rsj,c. 
+    rs0 = c::rsj →
+    outt = change_vec ??
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
+           (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
+
+definition R_match_step_true ≝ 
+  λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+  ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s → 
+  is_startc s = true → 
+  (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
+  current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
+  (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →  
+   outt = change_vec ?? int 
+          (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧  
+  (∀ls,x,xs,ci,rs,ls0,rs0. 
+    nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
+    (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → 
+    (∀cj,rs1.rs0 = cj::rs1 → ci ≠ cj →
+     (outt = change_vec ?? int 
+           (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)) ∧
+    (rs0 = [ ] →
+     outt = change_vec ??
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
+           (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)).
+           
+lemma sem_match_step :
+  ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n → 
+  match_step src dst sig n is_startc is_endc ⊨ 
+    [ inr ?? (inr ?? (inl … (inr ?? (inr ?? start_nop)))) : 
+      R_match_step_true src dst sig n is_startc is_endc, 
+      R_match_step_false src dst sig n is_endc ].
+#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst 
+(* test_null versione multi? *)
+@(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
+    (acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
+      (acc_sem_if ? n … (sem_test_null sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
+      
+      sem_seq … 
+        (sem_parmoveL ???? is_startc Hneq Hsrc Hdst) 
+        (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
+      (sem_nop …)))
+[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
+ #Htb #s #Hcurta_src #Hstart #Hnotstart % [ %
+ [#Hdst_none @daemon 
+ | #s1 #Hcurta_dst #Hneqss1
+   lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta) 
+   [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
+   #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //] 
+   whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
+   [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+     [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
+       #ls * #rs #Hta_mid >(Htbdst … Hta_mid) >Hta_mid cases rs //
+     | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
+   | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
+     normalize #H destruct (H) // ]
+   ]
+ |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc 
+  cases rs00 in Htadst_mid;
+   [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ]
+     #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2 
+     [2: * #x0 * #rs1 * #H destruct (H) ]
+    * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src; 
+    normalize in ⊢ (%→?); #H destruct (H)  
+    >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
+    >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
+     @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+      [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
+      |@sym_eq @Htbelse @sym_not_eq //
+      ] 
+    |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ]
+     #cj #rs1 #H destruct (H) #Hcicj
+     cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
+     * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
+     lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc
+     cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
+     #H destruct (H)
+     >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??)) //
+     [| >Htc >nth_change_vec //
+     | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid 
+      cases (orb_true_l … Hc0) -Hc0 #Hc0
+      [@memb_append_l2 >(\P Hc0) @memb_hd
+      |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
+      ]
+    | >Htc >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
+    * * #_ #Htbdst #Htbelse %
+    [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
+      [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj'::rs0'))
+        [ cases xs //
+        | >nth_change_vec // ]
+      | >nth_change_vec_neq [|@sym_not_eq //]
+        <Htbelse [|@sym_not_eq // ]
+        >nth_change_vec_neq [|@sym_not_eq //]
+        cases (decidable_eq_nat i src) #Hisrc
+        [ >Hisrc >nth_change_vec // >Htasrc_mid //
+        | >nth_change_vec_neq [|@sym_not_eq //]
+          <(Htbelse i) [|@sym_not_eq // ]
+          >Htc >nth_change_vec_neq [|@sym_not_eq // ]
+          >nth_change_vec_neq [|@sym_not_eq // ] //
+        ]
+       ] 
+    | >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] 
+      >nth_change_vec // whd in ⊢ (??%?→?); 
+      #H destruct (H) cases (is_endc c) in Hcend;
+      normalize #H destruct (H) // ]
+    ]
+  ]
+|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb 
+ whd in ⊢ (%→?); #Hout >Hout >Htb whd
+ #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
+ lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
+ cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
+  [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
+  |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
+    [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
+     #ls_dst * #rs_dst #Hmid_dst %2
+     cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * 
+     #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst
+     cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
+     #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
+     lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?) 
+     [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+       [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //]
+     | *
+       [ * #Hrsj #Hta %
+         [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+           #Hc lapply (Hc ? (refl ??)) #Hendr1
+           cut (xs = xs1)
+           [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
+             -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
+             [ * normalize in ⊢ (%→?); //
+               #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
+               lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
+             | #x2 #xs2 #IH *
+               [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
+                 >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
+                 normalize in ⊢ (%→?); #H destruct (H)
+               | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
+                 #Hnotendc #Hnotendcxs1 @eq_f @IH
+                 [ @(cons_injective_r … Heq)
+                 | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
+                   [ >(\P Hc0) @memb_hd
+                   | @memb_cons @memb_cons // ]
+                 | #c #Hc @Hnotendcxs1 @memb_cons // ]
+               ]
+             ]
+           | #Hxsxs1 >Hmid_dst >Hxsxs1 % ]
+         | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
+       | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?) 
+         [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
+         -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] 
+         >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
+         (* lemmatize this proof *) cut (xs = xs1)
+         [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
+           -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
+           [ * normalize in ⊢ (%→?); //
+             #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
+             lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
+           | #x2 #xs2 #IH *
+             [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
+               >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
+               normalize in ⊢ (%→?); #H destruct (H)
+             | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
+               #Hnotendc #Hnotendcxs1 @eq_f @IH
+               [ @(cons_injective_r … Heq)
+               | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
+                 [ >(\P Hc0) @memb_hd
+                 | @memb_cons @memb_cons // ]
+               | #c #Hc @Hnotendcxs1 @memb_cons // ]
+             ]
+           ]
+         | #Hxsxs1 >Hmid_dst >Hxsxs1 % //
+           #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
+           @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
+           #Hendr1 destruct (Hendr1) % ]
+       ]
+     ]
+   (* STOP *)
+   |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst 
+    @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
+    @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape 
+    >Hintape in Hc; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc 
+    >(Hnotend c_src) // normalize #H destruct (H)   
+   ]
+  ]
+]
+qed. 
+
+definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
+  whileTM … (match_step src dst sig n is_startc is_endc) 
+    (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
+
+definition R_match_m ≝ 
+  λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+(*  (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
+  ∀ls,x,xs,end,rs.
+  nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
+  (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
+  (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
+  (is_startc x = true →
+   (∀ls0,x0,rs0.
+    nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+    (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
+     ∀cj,l2.l1=cj::l2 →
+     outt = change_vec ?? 
+            (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
+            (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
+    ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
+
+(*
+definition R_match_m ≝ 
+  λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+  (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+    current ? (nth i ? int (niltape ?)) = None ? ∨
+    current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
+  (∀ls,x,xs,ci,rs,ls0,x0,rs0.
+    (∀x. is_startc x ≠ is_endc x) → 
+    is_startc x = true → is_endc ci = true → 
+    (∀z. memb ? z (x::xs) = true → is_endc x = false) →
+    nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+    nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+    (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
+     ∀cj,l2.l1=cj::l2 →
+     outt = change_vec ?? 
+            (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+            (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
+    ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
+*)
+
+(*
+axiom sub_list_dec: ∀A.∀l,ls:list A. 
+  ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
+*)
+
+lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+src ≠ dst → src < S n → dst < S n → 
+  match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
+#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
+lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
+  cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse 
+  [(* current dest = None *) * #Hcur_dst #Houtc %
+    [#_ >Houtc //
+    |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst; 
+     normalize in ⊢ (%→?); #H destruct (H)
+    ]
+  |* #ls0 * #rs0 * #Hmid_dst #HFalse %
+    [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
+    | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
+     %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil 
+     >reverse_cons >associative_append @(HFalse ?? Hnotnil)
+    ]
+  ]
+|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+ #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend 
+ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); 
+  [#Hmid_dst % 
+    [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
+     cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon]
+     * #Htb #_ #_ >Htb in IH; // #IH  
+     cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend)
+     #Hcur_outc #_ @Hcur_outc //
+    |#ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?); 
+     #H destruct (H)
+    ]
+  | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
+    #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
+    #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
+    cases (Htrue x (refl …) Hstart ?) -Htrue
+    [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
+    cases (true_or_false (x==c)) #eqx
+    [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
+      #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
+      cases tl1 in Hxs; 
+      [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
+      #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
+      [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
+       #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
+      |#Hcomp lapply (Htrue ls x x1 ci tl1 ls0 tl2 ???)
+       [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
+         [ @Hnotend >(\P Hc0) @memb_hd
+         | @Hnotendx1 // ]
+       | >Hmid_dst >Hrs0 >(\P eqx) %
+       | >Hxs %
+       | *  cases tl2 in Hrs0;
+         [ >append_nil #Hrs0 #_ #Htb whd in IH;
+           lapply (IH ls x x1 ci tl1 ? Hstart ??)
+           [
+           |
+           | >Htb // >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+          
+        >Hrs0 in Hmid_dst; #Hmid_dst
+       cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
+       whd in IH;
+       cases(IH ls x xs end rs ? Hstart Hnotend Hend)
+       [* #H1 #H2 >Htb in H1; >nth_change_vec // 
+        >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)] 
+        #_ %2 @daemon (* si dimostra *)
+       |@daemon
+       |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
+       ] 
+    ]
+  ]
+]
 qed.