X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fcompare.ma;h=8ad92d1e8b8f299e8908c63c36424cc3e397c24f;hb=a1dd5e64f21738a3f9ee1f635affeb9033e90954;hp=2a26abb3e73aa8d94c1d18d3131827e4d2cac19c;hpb=cdcfe9f97936f02dab1970ebf3911940bf0a4e29;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/compare.ma b/matita/matita/lib/turing/multi_universal/compare.ma
index 2a26abb3e..8ad92d1e8 100644
--- a/matita/matita/lib/turing/multi_universal/compare.ma
+++ b/matita/matita/lib/turing/multi_universal/compare.ma
@@ -33,18 +33,18 @@ definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)
 *)
 
 definition trans_compare_step ≝ 
- λi,j.λsig:FinSet.λn.λis_endc.
+ λi,j.λsig:FinSet.λn.
  λp:compare_states × (Vector (option sig) (S n)).
  let 〈q,a〉 ≝ p in
  match pi1 … q with
  [ O ⇒ match nth i ? a (None ?) with
-   [ None ⇒ 〈comp2,null_action ? n〉
+   [ None ⇒ 〈comp2,null_action sig n〉
    | Some ai ⇒ match nth j ? a (None ?) with 
      [ None ⇒ 〈comp2,null_action ? n〉
-     | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj 
+     | Some aj ⇒ if ai == aj 
          then 〈comp1,change_vec ? (S n) 
-                      (change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
-                        (Some ? 〈aj,R〉) j〉
+                      (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) i)
+                        (〈None ?,R〉) j〉
          else 〈comp2,null_action ? n〉 ]
    ]
  | S q ⇒ match q with 
@@ -52,284 +52,251 @@ definition trans_compare_step ≝
    | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
 
 definition compare_step ≝ 
-  λi,j,sig,n,is_endc.
-  mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc) 
+  λi,j,sig,n.
+  mk_mTM sig n compare_states (trans_compare_step i j sig n) 
     comp0 (λq.q == comp1 ∨ q == comp2).
 
 definition R_comp_step_true ≝ 
-  λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+  λi,j,sig,n.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 ?? 
             (change_vec ?? int
-              (tape_move ? (nth i ? int (niltape ?)) (Some ? 〈x,R〉)) i)
-            (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
+              (tape_move_right ? (nth i ? int (niltape ?))) i)
+            (tape_move_right ? (nth j ? int (niltape ?))) j.
 
 definition R_comp_step_false ≝ 
-  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.
+  λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
+   (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,is_endc,v.i < S n → j < S n → 
+  ∀i,j,sig,n,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 is_endc) (mk_mconfig ??? comp0 v) 
+  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) 
   = mk_mconfig ??? comp2 v.
-#i #j #sig #n #is_endc #v #Hi #Hj
+#i #j #sig #n #v #Hi #Hj
 whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
 * #Hcurrent
 [ @eq_f2
   [ whd in ⊢ (??(???%)?); >Hcurrent %
-  | whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ]
+  | whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ]
 | @eq_f2
   [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) //
-  | whd in ⊢ (??(???????(???%))?); >Hcurrent
+  | whd in ⊢ (??(????(???%))?); >Hcurrent
     cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ]
 qed.
 
 lemma comp_q0_q2_neq :
-  ∀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) 
+  ∀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) 
   = mk_mconfig ??? comp2 v.
-#i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
+#i #j #sig #n #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 *
-    [ * #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
-          ]
-        ]
+  | #aj #Haj * #Hneq
+    whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+    [ whd in match (trans ????); >Hai >Haj
+      whd in ⊢ (??(???%)?); cut ((ai==aj)=false)
+      [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq
+       >Hai >Haj //
+      | #Haiaj >Haiaj % ]
+    | whd in match (trans ????); >Hai >Haj
+      whd in ⊢ (??(????(???%))?); cut ((ai==aj)=false)
+      [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq
+       >Hai >Haj //
+      |#Hcut >Hcut @tape_move_null_action
       ]
     ]
+  ]
 ]
 qed.
 
 lemma comp_q0_q1 :
-  ∀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 →
+  ∀i,j,sig,n,v,a.i ≠ j → i < S n → j < S n → 
+  nth i ? (current_chars ?? v) (None ?) = Some ? a →
   nth j ? (current_chars ?? v) (None ?) = Some ? a → 
-  step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) =
+  step sig n (compare_step i j sig n) (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 #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2
+         (tape_move_right ? (nth i ? v (niltape ?))) i)
+       (tape_move_right ? (nth j ? v (niltape ?))) j).
+#i #j #sig #n #v #a #Heq #Hi #Hj #Ha1 #Ha2
 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
 [ whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
+  >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
 | whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) //
-  change with (change_vec ?????) in ⊢ (??(???????%)?);
+  >Ha1 >Ha2 whd in ⊢ (??(????(???%))?); >(\b ?) //
+  change with (change_vec ?????) in ⊢ (??(????%)?);
   <(change_vec_same … v j (niltape ?)) in ⊢ (??%?);
   <(change_vec_same … v i (niltape ?)) in ⊢ (??%?);
-  >pmap_change >pmap_change >tape_move_null_action
-  @eq_f2 // @eq_f2 // >nth_change_vec_neq //
+  >tape_move_multi_def 
+  >pmap_change >pmap_change <tape_move_multi_def
+  >tape_move_null_action
+  @eq_f2 // >nth_change_vec_neq //
 ]
 qed.
 
 lemma sem_comp_step :
-  ∀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
+  ∀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
 lapply (refl ? (current ? (nth i ? int (niltape ?))))
 cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
 [ #Hcuri %{2} %
   [| % [ %
-    [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ % <Hcuri in ⊢ (???%); 
-      @sym_eq @nth_vec_map
+    [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ 
     | normalize in ⊢ (%→?); #H destruct (H) ]
   | #_ % // % %2 // ] ]
 | #a #Ha lapply (refl ? (current ? (nth j ? int (niltape ?))))
   cases (current ? (nth j ? int (niltape ?))) in ⊢ (???%→?);
   [ #Hcurj %{2} %
     [| % [ %
-       [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%); 
-         @sym_eq @nth_vec_map
+       [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2
        | normalize in ⊢ (%→?); #H destruct (H) ]
-       | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
-  | #b #Hb %{2} 
-   cases (true_or_false (is_endc a)) #Haendc
+       | #_ % // >Ha >Hcurj % % % #H destruct (H) ] ]
+  | #b #Hb %{2} cases (true_or_false (a == b)) #Hab
     [ %
+      [| % [ % 
+        [whd in ⊢  (??%?);  >(comp_q0_q1 … a Hneq Hi Hj) //
+         >(\P Hab) <Hb @sym_eq @nth_vec_map
+        | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) // ]
+        | * #H @False_ind @H %
+      ] ]
+    | %
       [| % [ % 
         [whd in ⊢  (??%?);  >comp_q0_q2_neq //
-         % %{a} % // <Ha @sym_eq @nth_vec_map
+         <(nth_vec_map ?? (current …) i ? int (niltape ?))
+         <(nth_vec_map ?? (current …) j ? int (niltape ?)) >Ha >Hb
+         @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
         | normalize in ⊢ (%→?); #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) % ] ]
-      ]
+      | #_ % // % % >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
     ]
   ]
 ]
 qed.
 
-definition compare ≝ λi,j,sig,n,is_endc.
-  whileTM … (compare_step i j sig n is_endc) comp1.
+definition compare ≝ λi,j,sig,n.
+  whileTM … (compare_step i j sig n) comp1.
 
+(*    (∃rs'.rs = rs0@rs' ∧ current ? (nth j ? outt (niltape ?)) = None ?) ∨
+    (∃rs0'.rs0 = rs@rs0' ∧ 
+     outt = change_vec ?? 
+            (change_vec ?? int  
+              (mk_tape sig (reverse sig rs@x::ls) (None sig) []) i)
+            (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs0')
+            (tail sig rs0')) j) ∨
+    (∃xs,ci,cj,rs',rs0'.ci ≠ cj ∧ rs = xs@ci::rs' ∧ rs0 = xs@cj::rs0' ∧
+     outt = change_vec ?? 
+            (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs') i)
+            (midtape sig (reverse ? xs@x::ls0) cj rs0') j)).*)
 definition R_compare ≝ 
-  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 ?)) ∨
+  λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
+  ((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@rs0) →
-    (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → 
-    (rs0 = [ ] ∧
+  (∀ls,x,rs,ls0,rs0. 
+(*    nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → *)
+    nth i ? int (niltape ?) = midtape sig ls x rs →
+    nth j ? int (niltape ?) = midtape sig ls0 x rs0 →
+    (∃rs'.rs = rs0@rs' ∧ 
+     outt = change_vec ?? 
+            (change_vec ?? int  
+              (mk_tape sig (reverse sig rs0@x::ls) (option_hd sig rs') (tail ? rs')) i)
+            (mk_tape sig (reverse sig rs0@x::ls0) (None ?) [ ]) j) ∨
+    (∃rs0'.rs0 = rs@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 rs1) j)).
+            (change_vec ?? int  
+              (mk_tape sig (reverse sig rs@x::ls) (None sig) []) i)
+            (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs0')
+            (tail sig rs0')) j) ∨
+    (∃xs,ci,cj,rs',rs0'.ci ≠ cj ∧ rs = xs@ci::rs' ∧ rs0 = xs@cj::rs0' ∧
+     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,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) //
+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) //
 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ 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 #rs0 #Hnthi #Hnthj
-      >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
-    ]]
-  | #Hci #Houtc %
-    [ #_ @Houtc
-    | #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 #rs0 #_ #Hnthj >Hnthj in Hcj;
-      normalize in ⊢ (%→?); #H destruct (H) ] ]
-  | #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
-    #IH1 #IH2 %
-    [ >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 %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 #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_neq [|@sym_not_eq //] >nth_change_vec //
-        >Hnthi //
-      | >Hd >nth_change_vec // >Hnthi >Hnthj %
-      | #c0 #Hc0 @Hnotendc @memb_cons @Hc0
-]]]]]
-qed.      
+[ whd in ⊢ (%→?); * * [ *
+ [ #Hcicj #Houtc % 
+   [ #_ @Houtc
+   | #ls #x #rs #ls0 #rs0 #Hnthi #Hnthj
+     >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+   ]
+ | #Hci #Houtc %
+   [ #_ @Houtc
+   | #ls #x #rs #ls0 #rs0 #Hnthi >Hnthi in Hci;
+     normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+  [ #_ @Houtc
+  | #ls #x #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj;
+    normalize in ⊢ (%→?); #H destruct (H) ] ]
+| #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
+  #IH1 #IH2 %
+  [ >Hci >Hcj * [ * 
+    [ * #H @False_ind @H % | #H destruct (H)] | #H destruct (H)] 
+  | #ls #c0 #rs #ls0 #rs0 cases rs
+    [ -IH2 #Hnthi #Hnthj % %2 %{rs0} % [%]
+      >Hnthi in Hd; #Hd >Hd in IH1; #IH1 >IH1
+      [| % %2 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // % ]
+      >Hnthj cases rs0 [| #r1 #rs1 ] %
+    | #r1 #rs1 #Hnthi cases rs0
+      [ -IH2 #Hnthj % % %{(r1::rs1)} % [%]
+        >Hnthj in Hd; #Hd >Hd in IH1; #IH1 >IH1
+        [| %2 >nth_change_vec // ]
+        >Hnthi >Hnthj %
+      | #r2 #rs2 #Hnthj lapply IH2; >Hd in IH1; >Hnthi >Hnthj
+        >nth_change_vec //
+        >nth_change_vec_neq [| @sym_not_eq // ] >nth_change_vec //
+        cases (true_or_false (r1 == r2)) #Hr1r2
+        [ >(\P Hr1r2) #_ #IH2 cases (IH2 … (refl ??) (refl ??)) [ *
+          [ * #rs' * #Hrs1 #Hcurout_j % % %{rs'}
+            >Hrs1 % 
+            [ % 
+            | >Hcurout_j >change_vec_commute // >change_vec_change_vec
+              >change_vec_commute // @sym_not_eq // ]
+          | * #rs0' * #Hrs2 #Hcurout_i % %2 %{rs0'}
+            >Hrs2 >Hcurout_i % //
+            >change_vec_commute // >change_vec_change_vec
+            >change_vec_commute [|@sym_not_eq//] >change_vec_change_vec
+            >reverse_cons >associative_append >associative_append % ]
+          | * #xs * #ci * #cj * #rs' * #rs0' * * * #Hcicj #Hrs1 #Hrs2 
+            >change_vec_commute // >change_vec_change_vec 
+            >change_vec_commute [| @sym_not_eq ] // >change_vec_change_vec 
+            #Houtc %2 %{(r2::xs)} %{ci} %{cj} %{rs'} %{rs0'}
+            % [ % [ % [ // | >Hrs1 // ] | >Hrs2 // ] 
+              | >reverse_cons >associative_append >associative_append >Houtc % ] ]
+        | lapply (\Pf Hr1r2) -Hr1r2 #Hr1r2 #IH1 #_ %2
+          >IH1 [| % % normalize @(not_to_not … Hr1r2) #H destruct (H) % ]
+          %{[]} %{r1} %{r2} %{rs1} %{rs2} % [ % [ % /2/ | % ] | % ] ]]]]]
+qed.
  
-lemma terminate_compare :  ∀i,j,sig,n,is_endc,t.
+lemma terminate_compare :  ∀i,j,sig,n,t.
   i ≠ j → i < S n → j < S n → 
-  compare i j sig n is_endc ↓ t.
-#i #j #sig #n #is_endc #t #Hneq #Hi #Hj
+  compare i j sig n ↓ t.
+#i #j #sig #n #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 * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
-   #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % 
-   #t2 * #x0 * * * #Hendcx0 >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+  [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
+   #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 % 
+   #t2 * #x0 * * >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
@@ -338,8 +305,9 @@ cases (nth i (tape sig) t (niltape ?))
 ]
 qed.
 
-lemma sem_compare : ∀i,j,sig,n,is_endc.
+lemma sem_compare : ∀i,j,sig,n.
   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 @WRealize_to_Realize /2/
+  compare i j sig n ⊨ R_compare i j sig n.
+#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize 
+  [/2/| @wsem_compare // ]
 qed.