X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Fmarks.ma;h=829f56dc57ea7c1a2f80eef71df3a2d60a399ce4;hb=31cb2f0b374657eb5acb95708443e2c1b8481891;hp=4b66e6af524fe89362fdc19d455a3ac80b37aa53;hpb=35e0d9c8c27601b93413f4e9cbf1558404f24a41;p=helm.git

diff --git a/matita/matita/lib/turing/universal/marks.ma b/matita/matita/lib/turing/universal/marks.ma
index 4b66e6af5..829f56dc5 100644
--- a/matita/matita/lib/turing/universal/marks.ma
+++ b/matita/matita/lib/turing/universal/marks.ma
@@ -14,10 +14,9 @@
 
 *)
 
-include "turing/while_machine.ma".
 include "turing/if_machine.ma".
+include "turing/basic_machines.ma".
 include "turing/universal/alphabet.ma".
-include "turing/universal/tests.ma".
 
 (* ADVANCE TO MARK (right)
 
@@ -30,17 +29,21 @@ include "turing/universal/tests.ma".
 
 definition atm_states ≝ initN 3.
 
+definition atm0 : atm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition atm1 : atm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition atm2 : atm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
 definition atmr_step ≝ 
   λalpha:FinSet.λtest:alpha→bool.
   mk_TM alpha atm_states
   (λp.let 〈q,a〉 ≝ p in
    match a with
-   [ None ⇒ 〈1, None ?〉
+   [ None ⇒ 〈atm1, None ?〉
    | Some a' ⇒ 
      match test a' with
-     [ true ⇒ 〈1,None ?〉
-     | false ⇒ 〈2,Some ? 〈a',R〉〉 ]])
-  O (λx.notb (x == 0)).
+     [ true ⇒ 〈atm1,None ?〉
+     | false ⇒ 〈atm2,Some ? 〈a',R〉〉 ]])
+  atm0 (λx.notb (x == atm0)).
 
 definition Ratmr_step_true ≝ 
   λalpha,test,t1,t2.
@@ -57,43 +60,45 @@ definition Ratmr_step_false ≝
 lemma atmr_q0_q1 :
   ∀alpha,test,ls,a0,rs. test a0 = true → 
   step alpha (atmr_step alpha test)
-    (mk_config ?? 0 (midtape … ls a0 rs)) =
-  mk_config alpha (states ? (atmr_step alpha test)) 1
+    (mk_config ?? atm0 (midtape … ls a0 rs)) =
+  mk_config alpha (states ? (atmr_step alpha test)) atm1
     (midtape … ls a0 rs).
-#alpha #test #ls #a0 #ts #Htest normalize >Htest %
+#alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest %
 qed.
      
 lemma atmr_q0_q2 :
   ∀alpha,test,ls,a0,rs. test a0 = false → 
   step alpha (atmr_step alpha test)
-    (mk_config ?? 0 (midtape … ls a0 rs)) =
-  mk_config alpha (states ? (atmr_step alpha test)) 2
+    (mk_config ?? atm0 (midtape … ls a0 rs)) =
+  mk_config alpha (states ? (atmr_step alpha test)) atm2
     (mk_tape … (a0::ls) (option_hd ? rs) (tail ? rs)).
-#alpha #test #ls #a0 #ts #Htest normalize >Htest cases ts //
+#alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest cases ts //
 qed.
 
 lemma sem_atmr_step :
   ∀alpha,test.
   accRealize alpha (atmr_step alpha test) 
-    2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
+    atm2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
 #alpha #test *
 [ @(ex_intro ?? 2)
-  @(ex_intro ?? (mk_config ?? 1 (niltape ?))) %
-  [ % // #Hfalse destruct | #_ % // % % ]
-| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? 1 (leftof ? a al)))
-  % [ % // #Hfalse destruct | #_ % // % % ]
-| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? 1 (rightof ? a al)))
-  % [ % // #Hfalse destruct | #_ % // % % ]
+  @(ex_intro ?? (mk_config ?? atm1 (niltape ?))) %
+  [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (leftof ? a al)))
+  % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (rightof ? a al)))
+  % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
 | #ls #c #rs @(ex_intro ?? 2)
   cases (true_or_false (test c)) #Htest
-  [ @(ex_intro ?? (mk_config ?? 1 ?))
+  [ @(ex_intro ?? (mk_config ?? atm1 ?))
     [| % 
       [ % 
         [ whd in ⊢ (??%?); >atmr_q0_q1 //
-        | #Hfalse destruct ]
+        | whd in ⊢ ((??%%)→?); #Hfalse destruct ]
       | #_ % // %2 @(ex_intro ?? c) % // ]
     ]
-  | @(ex_intro ?? (mk_config ?? 2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
+  | @(ex_intro ?? (mk_config ?? atm2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
     % 
     [ %
       [ whd in ⊢ (??%?); >atmr_q0_q2 //
@@ -116,7 +121,7 @@ definition R_adv_to_mark_r ≝ λalpha,test,t1,t2.
      t2 = midtape ? (reverse ? rs1@c::ls) b rs2))).
      
 definition adv_to_mark_r ≝ 
-  λalpha,test.whileTM alpha (atmr_step alpha test) 2.
+  λalpha,test.whileTM alpha (atmr_step alpha test) atm2.
 
 lemma wsem_adv_to_mark_r :
   ∀alpha,test.
@@ -188,15 +193,18 @@ qed.
  
 definition mark_states ≝ initN 2.
 
+definition ms0 : mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition ms1 : mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
+
 definition mark ≝ 
   λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mark_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈1,Some ? 〈〈a'',true〉,N〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
+    [ None ⇒ 〈ms1,None ?〉
+    | Some a' ⇒ match (pi1 … q) with
+      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈ms1,Some ? 〈〈a'',true〉,N〉〉
+      | S q ⇒ 〈ms1,None ?〉 ] ])
+  ms0 (λq.q == ms1).
   
 definition R_mark ≝ λalpha,t1,t2.
   ∀ls,c,b,rs.
@@ -216,77 +224,6 @@ lemma sem_mark :
   @ex_intro [| % [ % | #ls0 #c0 #b0 #rs0 #H1 destruct (H1) % ] ] ]
 qed.
 
-(* MOVE RIGHT 
-
-   moves the head one step to the right
-
-*)
-
-definition move_states ≝ initN 2.
-
-definition move_r ≝ 
-  λalpha:FinSet.mk_TM alpha move_states
-  (λp.let 〈q,a〉 ≝ p in
-    match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ 〈1,Some ? 〈a',R〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
-  
-definition R_move_r ≝ λalpha,t1,t2.
-  ∀ls,c,rs.
-  t1 = midtape alpha ls c rs → 
-  t2 = mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs).
-    
-lemma sem_move_r :
-  ∀alpha.Realize ? (move_r alpha) (R_move_r alpha).
-#alpha #intape @(ex_intro ?? 2) cases intape
-[ @ex_intro
-  [| % [ % | #ls #c #rs #Hfalse destruct ] ]
-|#a #al @ex_intro
-  [| % [ % | #ls #c #rs #Hfalse destruct ] ]
-|#a #al @ex_intro
-  [| % [ % | #ls #c #rs #Hfalse destruct ] ]
-| #ls #c #rs
-  @ex_intro [| % [ % | #ls0 #c0 #rs0 #H1 destruct (H1)
-  cases rs0 // ] ] ]
-qed.
-
-(* MOVE LEFT
-
-   moves the head one step to the right
-
-*)
-
-definition move_l ≝ 
-  λalpha:FinSet.mk_TM alpha move_states
-  (λp.let 〈q,a〉 ≝ p in
-    match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ 〈1,Some ? 〈a',L〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
-  
-definition R_move_l ≝ λalpha,t1,t2.
-  ∀ls,c,rs.
-  t1 = midtape alpha ls c rs → 
-  t2 = mk_tape ? (tail ? ls) (option_hd ? ls) (c::rs).
-    
-lemma sem_move_l :
-  ∀alpha.Realize ? (move_l alpha) (R_move_l alpha).
-#alpha #intape @(ex_intro ?? 2) cases intape
-[ @ex_intro
-  [| % [ % | #ls #c #rs #Hfalse destruct ] ]
-|#a #al @ex_intro
-  [| % [ % | #ls #c #rs #Hfalse destruct ] ]
-|#a #al @ex_intro
-  [| % [ % | #ls #c #rs #Hfalse destruct ] ]
-| #ls #c #rs
-  @ex_intro [| % [ % | #ls0 #c0 #rs0 #H1 destruct (H1)
-  cases ls0 // ] ] ]
-qed.
 
 (* MOVE RIGHT AND MARK machine
 
@@ -297,18 +234,22 @@ qed.
  
 definition mrm_states ≝ initN 3.
 
+definition mrm0 : mrm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition mrm1 : mrm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition mrm2 : mrm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
 definition move_right_and_mark ≝ 
   λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mrm_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈2,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ 〈1,Some ? 〈a',R〉〉
+    [ None ⇒ 〈mrm2,None ?〉
+    | Some a' ⇒ match pi1 … q with
+      [ O ⇒ 〈mrm1,Some ? 〈a',R〉〉
       | S q ⇒ match q with
         [ O ⇒ let 〈a'',b〉 ≝ a' in
-              〈2,Some ? 〈〈a'',true〉,N〉〉
-        | S _ ⇒ 〈2,None ?〉 ] ] ])
-  O (λq.q == 2).
+              〈mrm2,Some ? 〈〈a'',true〉,N〉〉
+        | S _ ⇒ 〈mrm2,None ?〉 ] ] ])
+  mrm0 (λq.q == mrm2).
   
 definition R_move_right_and_mark ≝ λalpha,t1,t2.
   ∀ls,c,d,b,rs.
@@ -337,15 +278,19 @@ qed.
  
 definition clear_mark_states ≝ initN 3.
 
+definition clear0 : clear_mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition clear1 : clear_mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition claer2 : clear_mark_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
 definition clear_mark ≝ 
   λalpha:FinSet.mk_TM (FinProd … alpha FinBool) clear_mark_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈1,Some ? 〈〈a'',false〉,N〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
+    [ None ⇒ 〈clear1,None ?〉
+    | Some a' ⇒ match pi1 … q with
+      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈clear1,Some ? 〈〈a'',false〉,N〉〉
+      | S q ⇒ 〈clear1,None ?〉 ] ])
+  clear0 (λq.q == clear1).
   
 definition R_clear_mark ≝ λalpha,t1,t2.
   ∀ls,c,b,rs.
@@ -498,10 +443,6 @@ qed.
         ^
 *)
 
-definition is_marked ≝ 
-  λalpha.λp:FinProd … alpha FinBool.
-  let 〈x,b〉 ≝ p in b.
-
 definition adv_both_marks ≝ 
   λalpha.seq ? (adv_mark_r alpha)
     (seq ? (move_l ?)
@@ -650,7 +591,7 @@ cases (sem_if ? (test_char ? (λx.x == c)) … tc_true
     | * #Hx0 #Houtc %2
       % [ <(\P Hx) in Hx0; #Hx0 lapply (\Pf Hx0) @not_to_not #Hx' >Hx' %
         | >Houtc % ]
-    | (* members of lists are invariant under reverse *) @daemon ]
+    | #x #membx @Hl1 <(reverse_reverse …l1) @memb_reverse @membx ]
   | %2 % [ @(\Pf Hc) ]
     >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta
     >Hx in Hc;#Hc destruct (Hc) ]
@@ -678,7 +619,8 @@ definition comp_step ≝
   ifTM ? (test_char ? (is_marked ?))
   (single_finalTM … (comp_step_subcase FSUnialpha 〈bit false,true〉
     (comp_step_subcase FSUnialpha 〈bit true,true〉
-      (clear_mark …))))
+      (comp_step_subcase FSUnialpha 〈null,true〉
+        (clear_mark …)))))
   (nop ?)
   tc_true.
   
@@ -686,7 +628,7 @@ definition R_comp_step_true ≝
   λt1,t2.
     ∀l0,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) l0 c rs → 
     ∃c'. c = 〈c',true〉 ∧
-    ((is_bit c' = true ∧
+    ((bit_or_null c' = true ∧
      ∀a,l1,c0,a0,l2.
       rs = 〈a,false〉::l1@〈c0,true〉::〈a0,false〉::l2 → 
       (∀c.memb ? c l1 = true → is_marked ? c = false) → 
@@ -695,7 +637,7 @@ definition R_comp_step_true ≝
       (c0 ≠ c' ∧
        t2 = midtape (FinProd … FSUnialpha FinBool) 
         (reverse ? l1@〈a,false〉::〈c',true〉::l0) 〈c0,false〉 (〈a0,false〉::l2))) ∨
-     (is_bit c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
+     (bit_or_null c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
 
 definition R_comp_step_false ≝ 
   λt1,t2.
@@ -703,13 +645,14 @@ definition R_comp_step_false ≝
    is_marked ? c = false ∧ t2 = t1.
    
 lemma sem_comp_step : 
-  accRealize ? comp_step (inr … (inl … (inr … 0))) 
+  accRealize ? comp_step (inr … (inl … (inr … start_nop))) 
     R_comp_step_true R_comp_step_false.
 #intape
 cases (acc_sem_if … (sem_test_char ? (is_marked ?))
         (sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
           (sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ?? 
-            (sem_clear_mark …)))
+            (sem_comp_step_subcase FSUnialpha 〈null,true〉 ?? 
+              (sem_clear_mark …))))
         (sem_nop …) intape)
 #k * #outc * * #Hloop #H1 #H2
 @(ex_intro ?? k) @(ex_intro ?? outc) %
@@ -737,13 +680,24 @@ cases (acc_sem_if … (sem_test_char ? (is_marked ?))
         [ @sym_not_eq //
         | @Houtc ]
       ]
-    | * #Hc' whd in ⊢ (%→?); #Helse2 %2 %
-      [ generalize in match Hc'; generalize in match Hc;
-        cases c'
-        [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
-            | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
-        |*: #_ #_ % ]
-      | @(Helse2 … Hta)
+    | * #Hc' #Helse2 cases (Helse2 … Hta)
+      [ * #Hc'' #H1 % % [destruct (Hc'') % ]
+        #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
+        cases (H1 … Hl1 Hrs)
+        [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
+        | * #Hneq #Houtc %2 %
+          [ @sym_not_eq //
+          | @Houtc ]
+        ]
+      | * #Hc'' whd in ⊢ (%→?); #Helse3 %2 %
+        [ generalize in match Hc''; generalize in match Hc'; generalize in match Hc;
+          cases c'
+          [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
+              | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
+          | #_ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
+          |*: #_ #_ #_ % ]
+        | @(Helse3 … Hta)
+        ]
       ]
     ]
   ]
@@ -756,7 +710,7 @@ cases (acc_sem_if … (sem_test_char ? (is_marked ?))
 qed.
 
 definition compare ≝ 
-  whileTM ? comp_step (inr … (inl … (inr … 0))).
+  whileTM ? comp_step (inr … (inl … (inr … start_nop))).
 
 (*
 definition R_compare :=
@@ -807,16 +761,16 @@ RIFIUTO: c ≠ d
 definition R_compare :=
   λt1,t2.
   ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs → 
-  (∀c'.is_bit c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
+  (∀c'.bit_or_null c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
   (∀c'. c = 〈c',false〉 → t2 = t1) ∧
   ∀b,b0,bs,b0s,l1,l2.
   |bs| = |b0s| → 
-  (∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → is_bit (\fst c) = true) → 
-  (∀c.memb (FinProd … FSUnialpha FinBool) c b0s = true → is_bit (\fst c) = true) → 
+  (∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → bit_or_null (\fst c) = true) → 
+  (∀c.memb (FinProd … FSUnialpha FinBool) c b0s = true → bit_or_null (\fst c) = true) → 
   (∀c.memb ? c bs = true → is_marked ? c = false) → 
   (∀c.memb ? c b0s = true → is_marked ? c = false) → 
   (∀c.memb ? c l1 = true → is_marked ? c = false) → 
-  c = 〈b,true〉 → is_bit b = true → 
+  c = 〈b,true〉 → bit_or_null b = true → 
   rs = bs@〈grid,false〉::l1@〈b0,true〉::b0s@〈comma,false〉::l2 → 
   (〈b,true〉::bs = 〈b0,true〉::b0s ∧
    t2 = midtape ? (reverse ? bs@〈b,false〉::ls)
@@ -832,32 +786,6 @@ definition R_compare :=
                     reverse ? la@ls)
                     〈d',false〉 (lc@〈comma,false〉::l2)).
                     
-lemma list_ind2 : 
-  ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:list T1 → list T2 → Prop.
-  length ? l1 = length ? l2 →
-  (P [] []) → 
-  (∀tl1,tl2,hd1,hd2. P tl1 tl2 → P (hd1::tl1) (hd2::tl2)) → 
-  P l1 l2.
-#T1 #T2 #l1 #l2 #P #Hl #Pnil #Pcons
-generalize in match Hl; generalize in match l2;
-elim l1
-[#l2 cases l2 // normalize #t2 #tl2 #H destruct
-|#t1 #tl1 #IH #l2 cases l2
-   [normalize #H destruct
-   |#t2 #tl2 #H @Pcons @IH normalize in H; destruct // ]
-]
-qed.
-
-lemma list_cases_2 : 
-  ∀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.
-
 lemma wsem_compare : WRealize ? compare R_compare.
 #t #i #outc #Hloop
 lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
@@ -873,7 +801,7 @@ lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
   ]
 | #tapea #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
   whd in Hleft; #ls #c #rs #Htapea cases (Hleft … Htapea) -Hleft
-  #c' * #Hc >Hc cases (true_or_false (is_bit c')) #Hc'
+  #c' * #Hc >Hc cases (true_or_false (bit_or_null c')) #Hc'
   [2: * 
     [ * >Hc' #H @False_ind destruct (H)
     | * #_ #Htapeb cases (IH … Htapeb) * #_ #H #_ %
@@ -893,7 +821,7 @@ lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
     |#b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1
      #Heq destruct (Heq) #_ #Hrs cases Hleft -Hleft
       [2: * >Hc' #Hfalse @False_ind destruct ] * #_
-       @(list_cases_2 … Hlen)
+       @(list_cases2 … Hlen)
        [ #Hbs #Hb0s generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
        -Hrs #Hrs normalize in Hrs; #Hleft cases (Hleft ????? Hrs ?) -Hleft
          [ * #Heqb #Htapeb cases (IH … Htapeb) -IH * #IH #_ #_
@@ -912,7 +840,7 @@ lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
          | @Hl1 ]
       | * #b' #bitb' * #b0' #bitb0' #bs' #b0s' #Hbs #Hb0s 
         generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
-        cut (is_bit b' = true ∧ is_bit b0' = true ∧ 
+        cut (bit_or_null b' = true ∧ bit_or_null b0' = true ∧ 
              bitb' = false ∧ bitb0' = false)
         [ % [ % [ % [ >Hbs in Hbs1; #Hbs1 @(Hbs1 〈b',bitb'〉) @memb_hd
             | >Hb0s in Hb0s1; #Hb0s1 @(Hb0s1 〈b0',bitb0'〉) @memb_hd ]
@@ -970,5 +898,5 @@ lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
          ]
 ]]]]]
 qed.       
-           
+
 axiom sem_compare : Realize ? compare R_compare.
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