Hints sui DeqSets

diff --git a/matita/matita/lib/re/lang.ma b/matita/matita/lib/re/lang.ma
index a0a5a938d1785d58bf1422016caad88e5089c7f7..c1b894a7d787edc592a5f6d29622a4acd104a6ad 100644 (file)
--- a/matita/matita/lib/re/lang.ma
+++ b/matita/matita/lib/re/lang.ma
@@ -17,6 +17,7 @@
 include "arithmetics/nat.ma".
 include "basics/lists/list.ma".
 include "basics/sets.ma".
+include "basics/deqsets.ma".
 
 definition word ≝ λS:DeqSet.list S.
 
@@ -58,6 +59,24 @@ lemma distr_cat_r: ∀S.∀A,B,C:word S →Prop.
   [* #w1 * #w2 * * #eqw * /6/ |* * #w1 * #w2 * * /6/] 
 qed.
 
+(* derivative *)
+
+definition deriv ≝ λS.λA:word S → Prop.λa,w. A (a::w).
+
+lemma deriv_middot: ∀S,A,B,a. ¬ A ϵ → 
+  deriv S (A·B) a =1 (deriv S A a) · B.
+#S #A #B #a #noteps #w normalize %
+  [* #w1 cases w1 
+    [* #w2 * * #_ #Aeps @False_ind /2/
+    |#b #w2 * #w3 * * whd in ⊢ ((??%?)→?); #H destruct
+     #H #H1 @(ex_intro … w2) @(ex_intro … w3) % // % //
+    ]
+  |* #w1 * #w2 * * #H #H1 #H2 @(ex_intro … (a::w1))
+   @(ex_intro … w2) % // % normalize //
+  ]
+qed. 
+
+(* star properties *)
 lemma espilon_in_star: ∀S.∀A:word S → Prop.
   A^* ϵ.
 #S #A @(ex_intro … [ ]) normalize /2/

diff --git a/matita/matita/lib/re/moves.ma b/matita/matita/lib/re/moves.ma
index 4967bf0596ddf7ec35fecebd29d8460761247e2e..c260a6d403dd34c60cc0697a2cf64568a1654b36 100644 (file)
--- a/matita/matita/lib/re/moves.ma
+++ b/matita/matita/lib/re/moves.ma
@@ -64,38 +64,21 @@ theorem move_ok:
   |normalize /2/
   |normalize /2/
   |normalize #x #w cases (true_or_false (a==x)) #H >H normalize
-    [>(proj1 … (eqb_true …) H) % 
-      [* // #bot @False_ind //| #H1 destruct /2/]
-    |% [#bot @False_ind // 
-       | #H1 destruct @(absurd ((a==a)=true))
-         [>(proj2 … (eqb_true …) (refl …)) // | /2/] 
-       ]
+    [>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
+    |% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
     ]
   |#i1 #i2 #HI1 #HI2 #w >(sem_cat S i1 i2) >move_cat
    @iff_trans[|@sem_odot] >same_kernel >sem_cat_w
-   @iff_trans[||@(iff_or_l … (HI2 w))] @iff_or_r %
-    [* #w1 * #w2 * * #eqw #w1in #w2in @(ex_intro … (a::w1))
-     @(ex_intro … w2) % // % normalize // cases (HI1 w1) /2/
-    |* #w1 * #w2 * cases w1
-      [* #_ #H @False_ind /2/
-      |#x #w3 * #eqaw normalize in eqaw; destruct #w3in #w2in 
-      @(ex_intro … w3) @(ex_intro … w2) % // % // cases (HI1 w3) /2/
-      ]
-    ]
+   @iff_trans[||@(iff_or_l … (HI2 w))] @iff_or_r 
+   @iff_trans[||@iff_sym @deriv_middot //]
+   @cat_ext_l @HI1
   |#i1 #i2 #HI1 #HI2 #w >(sem_plus S i1 i2) >move_plus >sem_plus_w 
    @iff_trans[|@sem_oplus] 
    @iff_trans[|@iff_or_l [|@HI2]| @iff_or_r //]
   |#i1 #HI1 #w >move_star 
-   @iff_trans[|@sem_ostar] >same_kernel >sem_star_w %
-    [* #w1 * #w2 * * #eqw #w1in #w2in 
-     @(ex_intro … (a::w1)) @(ex_intro … w2) % // % normalize //
-     cases (HI1 w1 ) /2/
-    |* #w1 * #w2 * cases w1
-      [* #_ #H @False_ind /2/
-      |#x #w3 * #eqaw normalize in eqaw; destruct #w3in #w2in 
-       @(ex_intro … w3) @(ex_intro … w2) % // % // cases (HI1 w3) /2/
-      ]
-    ]
+   @iff_trans[|@sem_ostar] >same_kernel >sem_star_w 
+   @iff_trans[||@iff_sym @deriv_middot //]
+   @cat_ext_l @HI1
   ]
 qed.
     
@@ -165,18 +148,6 @@ coinductive equiv (S:DeqSet) : pre S → pre S → Prop ≝
      equiv S e1 e2.
 *)
 
-definition beqb ≝ λb1,b2.
-  match b1 with
-  [ true ⇒ b2
-  | false ⇒ notb b2
-  ].
-
-lemma beqb_ok: ∀b1,b2. iff (beqb b1 b2 = true) (b1 = b2).
-#b1 #b2 cases b1 cases b2 normalize /2/
-qed.
-
-definition Bin ≝ mk_DeqSet bool beqb beqb_ok. 
-
 let rec beqitem S (i1,i2: pitem S) on i1 ≝ 
   match i1 with
   [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
@@ -192,27 +163,53 @@ let rec beqitem S (i1,i2: pitem S) on i1 ≝
   | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
   ].
 
-axiom beqitem_ok: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2). 
+lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2). 
+#S #i1 elim i1
+  [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+  |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+  |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+    [>(\P H) // | @(\b (refl …))]
+  |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+    [>(\P H) // | @(\b (refl …))]
+  |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+   normalize #H destruct 
+    [cases (true_or_false (beqitem S i11 i21)) #H1
+      [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @False_ind /2/
+      ]
+    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    ]
+  |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+   normalize #H destruct 
+    [cases (true_or_false (beqitem S i11 i21)) #H1
+      [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @False_ind /2/
+      ]
+    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    ]
+  |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
+   normalize #H destruct 
+    [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
+  ]
+qed. 
 
 definition DeqItem ≝ λS.
-  mk_DeqSet (pitem S) (beqitem S) (beqitem_ok S).
-
-definition beqpre ≝ λS:DeqSet.λe1,e2:pre S. 
-  beqitem S (\fst e1) (\fst e2) ∧ beqb (\snd e1) (\snd e2).
-  
-definition beqpairs ≝ λS:DeqSet.λp1,p2:(pre S)×(pre S). 
-  beqpre S (\fst p1) (\fst p2) ∧ beqpre S (\snd p1) (\snd p2).
+  mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
   
-axiom beqpairs_ok: ∀S,p1,p2. iff (beqpairs S p1 p2 = true) (p1 = p2). 
-
-definition space ≝ λS.mk_DeqSet ((pre S)×(pre S)) (beqpairs S) (beqpairs_ok S).
-
-(* (sons S l p) computes all sons of p relative to characters in l *)
+unification hint  0 ≔ S; 
+    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢ 
+    pitem S ≡ carr X.
+    
+unification hint  0 ≔ S,i1,i2; 
+    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢ 
+    beqitem S i1 i2 ≡ eqb X i1 i2.
 
-definition sons ≝ λS:DeqSet.λl:list S.λp:space S. 
+definition sons ≝ λS:DeqSet.λl:list S.λp:(pre S)×(pre S). 
  map ?? (λa.〈move S a (\fst (\fst p)),move S a (\fst (\snd p))〉) l.
 
-lemma memb_sons: ∀S,l,p,q. memb (space S) p (sons S l q) = true →
+lemma memb_sons: ∀S,l.∀p,q:(pre S)×(pre S). memb ? p (sons ? l q) = true →
   ∃a.(move ? a (\fst (\fst q)) = \fst p ∧
       move ? a (\fst (\snd q)) = \snd p).
 #S #l elim l [#p #q normalize in ⊢ (%→?); #abs @False_ind /2/] 
@@ -222,7 +219,7 @@ lemma memb_sons: ∀S,l,p,q. memb (space S) p (sons S l q) = true →
   ]
 qed.
 
-let rec bisim S l n (frontier,visited: list (space S)) on n ≝
+let rec bisim S l n (frontier,visited: list ?) on n ≝
   match n with 
   [ O ⇒ 〈false,visited〉 (* assert false *)
   | S m ⇒ 
@@ -236,7 +233,7 @@ let rec bisim S l n (frontier,visited: list (space S)) on n ≝
     ]
   ].
   
-lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list (space S).
+lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
   bisim S l n frontier visited =
   match n with 
   [ O ⇒ 〈false,visited〉 (* assert false *)
@@ -252,39 +249,39 @@ lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list (space S).
   ].
 #S #l #n cases n // qed.
   
-lemma bisim_never: ∀S,l.∀frontier,visited: list (space S).
+lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
   bisim S l O frontier visited = 〈false,visited〉.
 #frontier #visited >unfold_bisim // 
 qed.
 
-lemma bisim_end: ∀Sig,l,m.∀visited: list (space Sig).
+lemma bisim_end: ∀Sig,l,m.∀visited: list ?.
   bisim Sig l (S m) [] visited = 〈true,visited〉.
 #n #visisted >unfold_bisim // 
 qed.
 
-lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list (space Sig).
+lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
 beqb (\snd (\fst p)) (\snd (\snd p)) = true →
   bisim Sig l (S m) (p::frontier) visited = 
-  bisim Sig l m (unique_append ? (filter ? (λx.notb(memb (space Sig) x (p::visited))) 
+  bisim Sig l m (unique_append ? (filter ? (λx.notb(memb ? x (p::visited))) 
     (sons Sig l p)) frontier) (p::visited).
 #Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test // 
 qed.
 
-lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list (space Sig).
+lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
 beqb (\snd (\fst p)) (\snd (\snd p)) = false →
   bisim Sig l (S m) (p::frontier) visited = 〈false,visited〉.
 #Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test // 
 qed.
  
-definition visited_inv ≝ λS.λe1,e2:pre S.λvisited: list (space S).
+definition visited_inv ≝ λS.λe1,e2:pre S.λvisited: list ?.
 uniqueb ? visited = true ∧  
   ∀p. memb ? p visited = true → 
    (∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p)) ∧ 
    (beqb (\snd (\fst p)) (\snd (\snd p)) = true).
   
-definition frontier_inv ≝ λS.λfrontier,visited: list (space S).
+definition frontier_inv ≝ λS.λfrontier,visited.
 uniqueb ? frontier = true ∧ 
-∀p. memb ? p frontier = true →  
+∀p:(pre S)×(pre S). memb ? p frontier = true →  
   memb ? p visited = false ∧
   ∃p1.((memb ? p1 visited = true) ∧
    (∃a. move ? a (\fst (\fst p1)) = \fst p ∧ 
@@ -339,91 +336,98 @@ let rec pitem_enum S (i:re S) on i ≝
   | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
   | k i ⇒ map ?? (pk S) (pitem_enum S i)
   ].
-
-(* axiom pitem_enum_complete: ∀S:DeqSet.∀i: pitem S.
-  memb ((pitem S)×(pitem S)) i (pitem_enum ? (forget ? i)) = true. *)
-(*
-#i elim i
-  [//
-  |//
-  |* //
-  |* // 
-  |#i1 #i2 #Hind1 #Hind2 @memb_compose //
-  |#i1 #i2 #Hind1 #Hind2 @memb_compose //
-  |
-*)
+  
+lemma pitem_enum_complete : ∀S.∀i:pitem S.
+  memb (DeqItem S) i (pitem_enum S (|i|)) = true.
+#S #i elim i 
+  [1,2://
+  |3,4:#c normalize >(\b (refl … c)) //
+  |5,6:#i1 #i2 #Hind1 #Hind2 @(memb_compose (DeqItem S) (DeqItem S)) //
+  |#i #Hind @(memb_map (DeqItem S)) //
+  ]
+qed.
 
 definition pre_enum ≝ λS.λi:re S.
   compose ??? (λi,b.〈i,b〉) (pitem_enum S i) [true;false].
+  
+lemma pre_enum_complete : ∀S.∀e:pre S.
+  memb ? e (pre_enum S (|\fst e|)) = true.
+#S * #i #b @(memb_compose (DeqItem S) DeqBool ? (λi,b.〈i,b〉))
+// cases b normalize //
+qed.
  
 definition space_enum ≝ λS.λi1,i2:re S.
-  compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i1).
+  compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i2).
 
-axiom space_enum_complete : ∀S.∀e1,e2: pre S.
-  memb (space S) 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true.
+lemma space_enum_complete : ∀S.∀e1,e2: pre S.
+  memb ? 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true.
+#S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉))
+// qed.
+
+definition visited_inv_1 ≝ λS.λe1,e2:pre S.λvisited: list ?.
+uniqueb ? visited = true ∧  
+  ∀p. memb ? p visited = true → 
+    ∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p). 
    
 lemma bisim_ok1: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} → 
- ∀l,n.∀frontier,visited:list (space S).
+ ∀l,n.∀frontier,visited:list (*(space S) *) ((pre S)×(pre S)).
  |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
- visited_inv S e1 e2 visited →  frontier_inv S frontier visited →
+ visited_inv_1 S e1 e2 visited →  frontier_inv S frontier visited →
  \fst (bisim S l n frontier visited) = true.
 #Sig #e1 #e2 #same #l #n elim n 
   [#frontier #visited #abs * #unique #H @False_ind @(absurd … abs)
    @le_to_not_lt @sublist_length // * #e11 #e21 #membp 
    cut ((|\fst e11| = |\fst e1|) ∧ (|\fst e21| = |\fst e2|))
    [|* #H1 #H2 <H1 <H2 @space_enum_complete]
-   cases (H … membp) * #w * #we1 #we2 #_
-   <we1 <we2 % //    
+   cases (H … membp) #w * #we1 #we2 <we1 <we2 % >same_kernel_moves //    
   |#m #HI * [#visited #vinv #finv >bisim_end //]
    #p #front_tl #visited #Hn * #u_visited #vinv * #u_frontier #finv
    cases (finv p (memb_hd …)) #Hp * #p2 * #visited_p2
-   * #a * #movea1 #movea2 
+   * #a * #movea1 #movea2
    cut (∃w.(moves Sig w e1 = \fst p) ∧ (moves Sig w e2 = \snd p))
-     [cases (vinv … visited_p2) -vinv * #w1 * #mw1 #mw2 #_
-      @(ex_intro … (w1@[a])) /2/] 
+     [cases (vinv … visited_p2) -vinv #w1 * #mw1 #mw2 
+      @(ex_intro … (w1@[a])) % //] 
    -movea2 -movea1 -a -visited_p2 -p2 #reachp
    cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true)
      [cases reachp #w * #move_e1 #move_e2 <move_e1 <move_e2
-      @(proj2 … (beqb_ok … )) @(proj1 … (equiv_sem … )) @same 
-     |#ptest >(bisim_step_true … ptest) @HI -HI
-       [<plus_n_Sm //
-       |% [whd in ⊢ (??%?); >Hp whd in ⊢ (??%?); //]
-        #p1 #H (cases (orb_true_l … H))
-         [#eqp <(proj1 … (eqb_true (space Sig) ? p1) eqp) % // 
+      @(\b ?) @(proj1 … (equiv_sem … )) @same] #ptest 
+   >(bisim_step_true … ptest) @HI -HI 
+     [<plus_n_Sm //
+     |% [whd in ⊢ (??%?); >Hp whd in ⊢ (??%?); //]
+       #p1 #H (cases (orb_true_l … H))
+         [#eqp <(\P eqp) // 
          |#visited_p1 @(vinv … visited_p1)
          ]
-       |whd % [@unique_append_unique @(andb_true_r … u_frontier)]
-        @unique_append_elim #q #H
-         [% 
-           [@notb_eq_true_l @(filter_true … H) 
-           |@(ex_intro … p) % // 
-            @(memb_sons … (memb_filter_memb … H))
+     |whd % [@unique_append_unique @(andb_true_r … u_frontier)]
+      @unique_append_elim #q #H
+       [% 
+         [@notb_eq_true_l @(filter_true … H) 
+         |@(ex_intro … p) % [@memb_hd|@(memb_sons … (memb_filter_memb … H))]
+         ]
+       |cases (finv q ?) [|@memb_cons //]
+        #nvq * #p1 * #Hp1 #reach %
+         [cut ((p==q) = false) [|#Hpq whd in ⊢ (??%?); >Hpq @nvq]
+          cases (andb_true … u_frontier) #notp #_ 
+          @(not_memb_to_not_eq … H) @notb_eq_true_l @notp 
+         |cases (proj2 … (finv q ?)) 
+           [#p1 *  #Hp1 #reach @(ex_intro … p1) % // @memb_cons //
+           |@memb_cons //
            ]
-         |cases (finv q ?) [|@memb_cons //]
-          #nvq * #p1 * #Hp1 #reach %
-           [cut ((p==q) = false) [|#Hpq whd in ⊢ (??%?); >Hpq @nvq]
-            cases (andb_true … u_frontier) #notp #_ 
-            @(not_memb_to_not_eq … H) @notb_eq_true_l @notp 
-           |cases (proj2 … (finv q ?)) 
-             [#p1 *  #Hp1 #reach @(ex_intro … p1) % // @memb_cons //
-             |@memb_cons //
-             ]
-          ]
-        ]  
-      ]
+        ]
+      ]  
     ]
   ]
 qed.
 
-definition all_true ≝ λS.λl.∀p. memb (space S) p l = true → 
+definition all_true ≝ λS.λl.∀p:(pre S) × (pre S). memb ? p l = true → 
   (beqb (\snd (\fst p)) (\snd (\snd p)) = true).
 
-definition sub_sons ≝ λS,l,l1,l2.∀x,a. 
-memb (space S) x l1 = true → memb S a l = true →
-  memb (space S) 〈move ? a (\fst (\fst x)), move ? a (\fst (\snd x))〉 l2 = true.
+definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S).∀a:S. 
+memb ? x l1 = true → memb S a l = true →
+  memb ? 〈move ? a (\fst (\fst x)), move ? a (\fst (\snd x))〉 l2 = true.
 
 lemma reachable_bisim: 
- ∀S,l,n.∀frontier,visited,visited_res:list (space S).
+ ∀S,l,n.∀frontier,visited,visited_res:list ?.
  all_true S visited →
  sub_sons S l visited (frontier@visited) →
  bisim S l n frontier visited = 〈true,visited_res〉 → 
@@ -435,7 +439,7 @@ lemma reachable_bisim:
   |#m #Hind * 
     [(* case empty frontier *)
      -Hind #vis #vis_res #allv #H normalize in  ⊢ (%→?);
-     #H1 destruct % // % // #p /2/ 
+     #H1 destruct % // % // #p /2 by / 
     |#hd cases (true_or_false (beqb (\snd (\fst hd)) (\snd (\snd hd))))
       [|(* case head of the frontier is non ok (absurd) *)
        #H #tl #vis #vis_res #allv >(bisim_step_false … H) #_ #H1 destruct]
@@ -443,8 +447,7 @@ lemma reachable_bisim:
      #H #tl #visited #visited_res #allv >(bisim_step_true … H)
      (* new_visited = hd::visited are all ok *)
      cut (all_true S (hd::visited)) 
-      [#p #H cases (orb_true_l … H) 
-        [#eqp <(proj1 … (eqb_true …) eqp) // |@allv]]
+      [#p #H1 cases (orb_true_l … H1) [#eqp <(\P eqp) @H |@allv]]
      (* we now exploit the induction hypothesis *)
      #allh #subH #bisim cases (Hind … allh … bisim) -Hind
       [* #H1 #H2 #H3 % // % // #p #H4 @H2 @memb_cons //]
@@ -456,7 +459,7 @@ lemma reachable_bisim:
        (* xa is the son of x w.r.t. a; we must distinguish the case xa 
         was already visited form the case xa is new *)
        letin xa ≝ 〈move S a (\fst (\fst x)), move S a (\fst (\snd x))〉
-       cases (true_or_false … (memb (space S) xa (x::visited)))
+       cases (true_or_false … (memb ? xa (x::visited)))
         [(* xa visited - trivial *) #membxa @memb_append_l2 //
         |(* xa new *) #membxa @memb_append_l1 @sublist_unique_append_l1 @memb_filter_l
           [>membxa //
@@ -593,7 +596,7 @@ qed.
 lemma bisim_char: ∀S.∀e1,e2:pre S.
 (∀w.(beqb (\snd (moves S w e1)) (\snd (moves ? w e2))) = true) → 
 \sem{e1}=1\sem{e2}.
-#S #e1 #e2 #H @(proj2 … (equiv_sem …)) #w @(proj1 …(beqb_ok …)) @H
+#S #e1 #e2 #H @(proj2 … (equiv_sem …)) #w @(\P ?) @H
 qed.
 
 lemma bisim_ok2: ∀S.∀e1,e2:pre S.
@@ -620,16 +623,17 @@ cut (sub_sons S rsig [〈e1,e2〉] (frontier@[〈e1,e2〉]))
     ]
   ] #init
 cases (reachable_bisim … allH init … H) * #H1 #H2 #H3
-cut (∀w.sublist ? w (occ S e1 e2)→∀p.memb (space S) p visited_res = true → 
-  memb (space S) 〈moves ? w (\fst p), moves ? w (\snd p)〉 visited_res = true)
-  [#w elim w [//] 
+cut (∀w.sublist ? w (occ S e1 e2)→∀p.memb ? p visited_res = true → 
+  memb ? 〈moves ? w (\fst p), moves ? w (\snd p)〉 visited_res = true)
+  [#w elim w  [#_ #p #H4 >moves_empty >moves_empty <eq_pair_fst_snd //] 
    #a #w1 #Hind #Hsub * #e11 #e21 #visp >moves_cons >moves_cons 
    @(Hind ? 〈?,?〉) [#x #H4 @Hsub @memb_cons //] 
-   @(H1 〈?,?〉) // @Hsub @memb_hd] #all_reach
+   @(H1 〈?,?〉) [@visp| @Hsub @memb_hd]] #all_reach
 @bisim_char @occ_enough
 #w #Hsub @(H3 〈?,?〉) @(all_reach w Hsub 〈?,?〉) @H2 //
 qed.
   
+(*
 definition tt ≝ ps Bin true.
 definition ff ≝ ps Bin false.
 definition eps ≝ pe Bin.
@@ -639,7 +643,7 @@ definition exp2 ≝  ff · (eps + tt).
 definition exp3 ≝ move Bin true (\fst (•exp1)).
 definition exp4 ≝ move Bin true (\fst (•exp2)).
 definition exp5 ≝ move Bin false (\fst (•exp1)).
-definition exp6 ≝ move Bin false (\fst (•exp2)).
+definition exp6 ≝ move Bin false (\fst (•exp2)). *)