From 18166618ab9029ed213ff3de556b5a9cba8673a8 Mon Sep 17 00:00:00 2001
From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Tue, 6 Mar 2012 11:44:26 +0000
Subject: [PATCH] chapter 9 and 10

---
 weblib/tutorial/chapter10.ma | 326 +++++++++++++++++++++++++++++++++++
 weblib/tutorial/chapter9.ma  | 324 ----------------------------------
 2 files changed, 326 insertions(+), 324 deletions(-)
 create mode 100644 weblib/tutorial/chapter10.ma

diff --git a/weblib/tutorial/chapter10.ma b/weblib/tutorial/chapter10.ma
new file mode 100644
index 000000000..f52e80570
--- /dev/null
+++ b/weblib/tutorial/chapter10.ma
@@ -0,0 +1,326 @@
+include "cahpter9.ma".
+
+(* bisimulation *)
+definition cofinal ≝ λS.λp:(pre S)×(pre S). 
+  \snd (\fst p) = \snd (\snd p).
+  
+theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S. 
+  \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
+#S #e1 #e2 % 
+[#same_sem #w 
+  cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2)) 
+    [* * // * #H1 #H2 [@sym_eq @H1 //| @H2 //]]
+  #Hcut @Hcut @iff_trans [|@decidable_sem] 
+  @iff_trans [|@same_sem] @iff_sym @decidable_sem
+|#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
+qed.
+
+definition occ ≝ λS.λe1,e2:pre S. 
+  unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
+
+lemma occ_enough: ∀S.∀e1,e2:pre S.
+(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
+ →∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
+#S #e1 #e2 #H #w
+cases (decidable_sublist S w (occ S e1 e2)) [@H] -H #H
+ >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //]
+ >to_pit [2: @(not_to_not … H) #H1 #a #memba  @sublist_unique_append_l2 @H1 //]
+ //
+qed.
+
+lemma equiv_sem_occ: ∀S.∀e1,e2:pre S.
+(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
+→ \sem{e1}=1\sem{e2}.
+#S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H 
+qed.
+
+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:(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/] 
+#a #tl #Hind #p #q #H cases (orb_true_l … H) -H
+  [#H @(ex_intro … a) >(\P H) /2/ |#H @Hind @H]
+qed.
+
+definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S.
+  ∀p:(pre S)×(pre S). memb ? p l = true → cofinal ? p ∧ (sublist ? (sons ? alpha p) l).
+
+lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S. 
+  is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}.
+#S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ 
+#w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?))
+lapply Hsub @(list_elim_left … w) [//]
+#a #w1 #Hind #Hsub >moves_left >moves_left @(proj2 …(Hbisim …(Hind ?)))
+  [#x #Hx @Hsub @memb_append_l1 //
+  |cut (memb S a (occ S e1 e2) = true) [@Hsub @memb_append_l2 //] #occa 
+   @(memb_map … occa)
+  ]
+qed.
+
+(* the algorithm *)
+let rec bisim S l n (frontier,visited: list ?) on n ≝
+  match n with 
+  [ O ⇒ 〈false,visited〉 (* assert false *)
+  | S m ⇒ 
+    match frontier with
+    [ nil ⇒ 〈true,visited〉
+    | cons hd tl ⇒
+      if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
+        bisim S l m (unique_append ? (filter ? (λx.notb (memb ? x (hd::visited))) 
+        (sons S l hd)) tl) (hd::visited)
+      else 〈false,visited〉
+    ]
+  ].
+  
+lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
+  bisim S l n frontier visited =
+  match n with 
+  [ O ⇒ 〈false,visited〉 (* assert false *)
+  | S m ⇒ 
+    match frontier with
+    [ nil ⇒ 〈true,visited〉
+    | cons hd tl ⇒
+      if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
+        bisim S l m (unique_append ? (filter ? (λx.notb(memb ? x (hd::visited))) 
+          (sons S l hd)) tl) (hd::visited)
+      else 〈false,visited〉
+    ]
+  ].
+#S #l #n cases n // qed.
+  
+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 ?.
+  bisim Sig l (S m) [] visited = 〈true,visited〉.
+#n #visisted >unfold_bisim // 
+qed.
+
+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 ? 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 ?.
+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.
+
+lemma notb_eq_true_l: ∀b. notb b = true → b = false.
+#b cases b normalize //
+qed.
+
+let rec pitem_enum S (i:re S) on i ≝
+  match i with
+  [ z ⇒ [pz S]
+  | e ⇒ [pe S]
+  | s y ⇒ [ps S y; pp S y]
+  | o i1 i2 ⇒ compose ??? (po S) (pitem_enum S i1) (pitem_enum S i2)
+  | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
+  | k i ⇒ map ?? (pk S) (pitem_enum S i)
+  ].
+  
+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 i2).
+
+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 all_reachable ≝ λS.λe1,e2:pre S.λl: list ?.
+uniqueb ? l = true ∧ 
+  ∀p. memb ? p l = true → 
+    ∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p). 
+
+definition disjoint ≝ λS:DeqSet.λl1,l2.
+  ∀p:S. memb S p l1 = true →  memb S p l2 = false.
+        
+lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} → 
+ ∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
+ |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
+ all_reachable S e1 e2 visited →  
+ all_reachable S e1 e2 frontier →
+ disjoint ? 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 % >same_kernel_moves //    
+  |#m #HI * [#visited #vinv #finv >bisim_end //]
+   #p #front_tl #visited #Hn * #u_visited #r_visited * #u_frontier #r_frontier 
+   #disjoint
+   cut (∃w.(moves ? w e1 = \fst p) ∧ (moves ? w e2 = \snd p)) 
+    [@(r_frontier … (memb_hd … ))] #rp
+   cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true)
+    [cases rp #w * #fstp #sndp <fstp <sndp @(\b ?) 
+     @(proj1 … (equiv_sem … )) @same] #ptest 
+   >(bisim_step_true … ptest) @HI -HI 
+     [<plus_n_Sm //
+     |% [whd in ⊢ (??%?); >(disjoint … (memb_hd …)) whd in ⊢ (??%?); //
+        |#p1 #H (cases (orb_true_l … H)) [#eqp >(\P eqp) // |@r_visited]
+        ]
+     |whd % [@unique_append_unique @(andb_true_r … u_frontier)]
+      @unique_append_elim #q #H
+       [cases (memb_sons … (memb_filter_memb … H)) -H
+        #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(ex_intro … (w1@[a]))
+        >moves_left >moves_left >mw1 >mw2 >m1 >m2 % // 
+       |@r_frontier @memb_cons //
+       ]
+     |@unique_append_elim #q #H
+       [@injective_notb @(filter_true … H)
+       |cut ((q==p) = false) 
+         [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @memb_cons //]
+        cases (andb_true … u_frontier) #notp #_ @(\bf ?) 
+        @(not_to_not … not_eq_true_false) #eqqp <notp <eqqp >H //
+       ]
+     ]
+   ]  
+qed.     
+
+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:(pre S) × (pre S). 
+memb ? x l1 = true → sublist ? (sons ? l x) l2. 
+
+lemma bisim_complete: 
+ ∀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〉 →
+ is_bisim S visited_res l ∧ sublist ? (frontier@visited) visited_res. 
+#S #l #n elim n
+  [#fron #vis #vis_res #_ #_ >bisim_never #H destruct
+  |#m #Hind * 
+    [(* case empty frontier *)
+     -Hind #vis #vis_res #allv #H normalize in  ⊢ (%→?);
+     #H1 destruct % #p 
+      [#membp % [@(\P ?) @allv //| @H //]|#H1 @H1]
+    |#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]
+     (* frontier = hd:: tl and hd is ok *)
+     #H #tl #visited #visited_res #allv >(bisim_step_true … H)
+     (* new_visited = hd::visited are all ok *)
+     cut (all_true S (hd::visited)) 
+      [#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) -bisim -Hind
+      [#H1 #H2 % // #p #membp @H2 -H2 cases (memb_append … membp) -membp #membp
+        [cases (orb_true_l … membp) -membp #membp
+          [@memb_append_l2 >(\P membp) @memb_hd
+          |@memb_append_l1 @sublist_unique_append_l2 // 
+          ]
+        |@memb_append_l2 @memb_cons //
+        ] 
+      |(* the only thing left to prove is the sub_sons invariant *)  
+     #x #membx cases (orb_true_l … membx)
+      [(* case x = hd *) 
+       #eqhdx <(\P eqhdx) #xa #membxa
+       (* xa is a son of x; we must distinguish the case xa 
+        was already visited form the case xa is new *)
+       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 //|//]
+        ]
+      |(* case x in visited *)
+       #H1 #xa #membxa cases (memb_append … (subH x … H1 … membxa))  
+        [#H2 (cases (orb_true_l … H2)) 
+          [#H3 @memb_append_l2 <(\P H3) @memb_hd
+          |#H3 @memb_append_l1 @sublist_unique_append_l2 @H3
+          ]
+        |#H2 @memb_append_l2 @memb_cons @H2
+        ]
+      ]
+    ]
+  ]
+qed.
+
+definition equiv ≝ λSig.λre1,re2:re Sig. 
+  let e1 ≝ •(blank ? re1) in
+  let e2 ≝ •(blank ? re2) in
+  let n ≝ S (length ? (space_enum Sig (|\fst e1|) (|\fst e2|))) in
+  let sig ≝ (occ Sig e1 e2) in
+  (bisim ? sig n [〈e1,e2〉] []).
+
+theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig.
+   \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}.
+#Sig #re1 #re2 %
+  [#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding]
+   cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉)
+     [<H //] #Hcut
+   cases (bisim_complete … Hcut) 
+     [2,3: #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/] 
+   #Hbisim #Hsub @(bisim_to_sem … Hbisim) 
+   @Hsub @memb_hd
+  |#H @(bisim_correct ? (•(blank ? re1)) (•(blank ? re2))) 
+    [@eqP_trans [|@re_embedding] @eqP_trans [|@H] @eqP_sym @re_embedding
+    |// 
+    |% // #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/  
+    |% // #p #H >(memb_single … H) @(ex_intro … ϵ) /2/
+    |#p #_ normalize //
+    ]
+  ]
+qed.
+
+lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
+#n #m % [@eqbnat_true_to_eq | @eq_to_eqbnat_true]
+qed.
+
+definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true.
+
+definition a ≝ s DeqNat O.
+definition b ≝ s DeqNat (S O).
+definition c ≝ s DeqNat (S (S O)).
+
+definition exp1 ≝ ((a·b)^*·a).
+definition exp2 ≝ a·(b·a)^*.
+definition exp4 ≝ (b·a)^*.
+
+definition exp6 ≝ a·(a ·a ·b^* + b^* ).
+definition exp7 ≝ a · a^* · b^*.
+
+definition exp8 ≝ a·a·a·a·a·a·a·a·(a^* ).
+definition exp9 ≝ (a·a·a + a·a·a·a·a)^*.
+
+example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true.
+normalize // qed.
+
+
+
+
+
+
+
diff --git a/weblib/tutorial/chapter9.ma b/weblib/tutorial/chapter9.ma
index dc7474e21..c7585df80 100644
--- a/weblib/tutorial/chapter9.ma
+++ b/weblib/tutorial/chapter9.ma
@@ -159,327 +159,3 @@ lemma to_pit: ∀S,w,e. ¬ sublist S w (occur S (|\fst e|)) →
   ]
 qed.
 
-(* bisimulation *)
-definition cofinal ≝ λS.λp:(pre S)×(pre S). 
-  \snd (\fst p) = \snd (\snd p).
-  
-theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S. 
-  \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
-#S #e1 #e2 % 
-[#same_sem #w 
-  cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2)) 
-    [* * // * #H1 #H2 [@sym_eq @H1 //| @H2 //]]
-  #Hcut @Hcut @iff_trans [|@decidable_sem] 
-  @iff_trans [|@same_sem] @iff_sym @decidable_sem
-|#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
-qed.
-
-definition occ ≝ λS.λe1,e2:pre S. 
-  unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
-
-lemma occ_enough: ∀S.∀e1,e2:pre S.
-(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
- →∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
-#S #e1 #e2 #H #w
-cases (decidable_sublist S w (occ S e1 e2)) [@H] -H #H
- >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //]
- >to_pit [2: @(not_to_not … H) #H1 #a #memba  @sublist_unique_append_l2 @H1 //]
- //
-qed.
-
-lemma equiv_sem_occ: ∀S.∀e1,e2:pre S.
-(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
-→ \sem{e1}=1\sem{e2}.
-#S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H 
-qed.
-
-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:(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/] 
-#a #tl #Hind #p #q #H cases (orb_true_l … H) -H
-  [#H @(ex_intro … a) >(\P H) /2/ |#H @Hind @H]
-qed.
-
-definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S.
-  ∀p:(pre S)×(pre S). memb ? p l = true → cofinal ? p ∧ (sublist ? (sons ? alpha p) l).
-
-lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S. 
-  is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}.
-#S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ 
-#w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?))
-lapply Hsub @(list_elim_left … w) [//]
-#a #w1 #Hind #Hsub >moves_left >moves_left @(proj2 …(Hbisim …(Hind ?)))
-  [#x #Hx @Hsub @memb_append_l1 //
-  |cut (memb S a (occ S e1 e2) = true) [@Hsub @memb_append_l2 //] #occa 
-   @(memb_map … occa)
-  ]
-qed.
-
-(* the algorithm *)
-let rec bisim S l n (frontier,visited: list ?) on n ≝
-  match n with 
-  [ O ⇒ 〈false,visited〉 (* assert false *)
-  | S m ⇒ 
-    match frontier with
-    [ nil ⇒ 〈true,visited〉
-    | cons hd tl ⇒
-      if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
-        bisim S l m (unique_append ? (filter ? (λx.notb (memb ? x (hd::visited))) 
-        (sons S l hd)) tl) (hd::visited)
-      else 〈false,visited〉
-    ]
-  ].
-  
-lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
-  bisim S l n frontier visited =
-  match n with 
-  [ O ⇒ 〈false,visited〉 (* assert false *)
-  | S m ⇒ 
-    match frontier with
-    [ nil ⇒ 〈true,visited〉
-    | cons hd tl ⇒
-      if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
-        bisim S l m (unique_append ? (filter ? (λx.notb(memb ? x (hd::visited))) 
-          (sons S l hd)) tl) (hd::visited)
-      else 〈false,visited〉
-    ]
-  ].
-#S #l #n cases n // qed.
-  
-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 ?.
-  bisim Sig l (S m) [] visited = 〈true,visited〉.
-#n #visisted >unfold_bisim // 
-qed.
-
-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 ? 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 ?.
-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.
-
-lemma notb_eq_true_l: ∀b. notb b = true → b = false.
-#b cases b normalize //
-qed.
-
-let rec pitem_enum S (i:re S) on i ≝
-  match i with
-  [ z ⇒ [pz S]
-  | e ⇒ [pe S]
-  | s y ⇒ [ps S y; pp S y]
-  | o i1 i2 ⇒ compose ??? (po S) (pitem_enum S i1) (pitem_enum S i2)
-  | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
-  | k i ⇒ map ?? (pk S) (pitem_enum S i)
-  ].
-  
-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 i2).
-
-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 all_reachable ≝ λS.λe1,e2:pre S.λl: list ?.
-uniqueb ? l = true ∧ 
-  ∀p. memb ? p l = true → 
-    ∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p). 
-
-definition disjoint ≝ λS:DeqSet.λl1,l2.
-  ∀p:S. memb S p l1 = true →  memb S p l2 = false.
-        
-lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} → 
- ∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
- |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
- all_reachable S e1 e2 visited →  
- all_reachable S e1 e2 frontier →
- disjoint ? 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 % >same_kernel_moves //    
-  |#m #HI * [#visited #vinv #finv >bisim_end //]
-   #p #front_tl #visited #Hn * #u_visited #r_visited * #u_frontier #r_frontier 
-   #disjoint
-   cut (∃w.(moves ? w e1 = \fst p) ∧ (moves ? w e2 = \snd p)) 
-    [@(r_frontier … (memb_hd … ))] #rp
-   cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true)
-    [cases rp #w * #fstp #sndp <fstp <sndp @(\b ?) 
-     @(proj1 … (equiv_sem … )) @same] #ptest 
-   >(bisim_step_true … ptest) @HI -HI 
-     [<plus_n_Sm //
-     |% [whd in ⊢ (??%?); >(disjoint … (memb_hd …)) whd in ⊢ (??%?); //
-        |#p1 #H (cases (orb_true_l … H)) [#eqp >(\P eqp) // |@r_visited]
-        ]
-     |whd % [@unique_append_unique @(andb_true_r … u_frontier)]
-      @unique_append_elim #q #H
-       [cases (memb_sons … (memb_filter_memb … H)) -H
-        #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(ex_intro … (w1@[a]))
-        >moves_left >moves_left >mw1 >mw2 >m1 >m2 % // 
-       |@r_frontier @memb_cons //
-       ]
-     |@unique_append_elim #q #H
-       [@injective_notb @(filter_true … H)
-       |cut ((q==p) = false) 
-         [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @memb_cons //]
-        cases (andb_true … u_frontier) #notp #_ @(\bf ?) 
-        @(not_to_not … not_eq_true_false) #eqqp <notp <eqqp >H //
-       ]
-     ]
-   ]  
-qed.     
-
-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:(pre S) × (pre S). 
-memb ? x l1 = true → sublist ? (sons ? l x) l2. 
-
-lemma bisim_complete: 
- ∀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〉 →
- is_bisim S visited_res l ∧ sublist ? (frontier@visited) visited_res. 
-#S #l #n elim n
-  [#fron #vis #vis_res #_ #_ >bisim_never #H destruct
-  |#m #Hind * 
-    [(* case empty frontier *)
-     -Hind #vis #vis_res #allv #H normalize in  ⊢ (%→?);
-     #H1 destruct % #p 
-      [#membp % [@(\P ?) @allv //| @H //]|#H1 @H1]
-    |#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]
-     (* frontier = hd:: tl and hd is ok *)
-     #H #tl #visited #visited_res #allv >(bisim_step_true … H)
-     (* new_visited = hd::visited are all ok *)
-     cut (all_true S (hd::visited)) 
-      [#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) -bisim -Hind
-      [#H1 #H2 % // #p #membp @H2 -H2 cases (memb_append … membp) -membp #membp
-        [cases (orb_true_l … membp) -membp #membp
-          [@memb_append_l2 >(\P membp) @memb_hd
-          |@memb_append_l1 @sublist_unique_append_l2 // 
-          ]
-        |@memb_append_l2 @memb_cons //
-        ] 
-      |(* the only thing left to prove is the sub_sons invariant *)  
-     #x #membx cases (orb_true_l … membx)
-      [(* case x = hd *) 
-       #eqhdx <(\P eqhdx) #xa #membxa
-       (* xa is a son of x; we must distinguish the case xa 
-        was already visited form the case xa is new *)
-       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 //|//]
-        ]
-      |(* case x in visited *)
-       #H1 #xa #membxa cases (memb_append … (subH x … H1 … membxa))  
-        [#H2 (cases (orb_true_l … H2)) 
-          [#H3 @memb_append_l2 <(\P H3) @memb_hd
-          |#H3 @memb_append_l1 @sublist_unique_append_l2 @H3
-          ]
-        |#H2 @memb_append_l2 @memb_cons @H2
-        ]
-      ]
-    ]
-  ]
-qed.
-
-definition equiv ≝ λSig.λre1,re2:re Sig. 
-  let e1 ≝ •(blank ? re1) in
-  let e2 ≝ •(blank ? re2) in
-  let n ≝ S (length ? (space_enum Sig (|\fst e1|) (|\fst e2|))) in
-  let sig ≝ (occ Sig e1 e2) in
-  (bisim ? sig n [〈e1,e2〉] []).
-
-theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig.
-   \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}.
-#Sig #re1 #re2 %
-  [#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding]
-   cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉)
-     [<H //] #Hcut
-   cases (bisim_complete … Hcut) 
-     [2,3: #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/] 
-   #Hbisim #Hsub @(bisim_to_sem … Hbisim) 
-   @Hsub @memb_hd
-  |#H @(bisim_correct ? (•(blank ? re1)) (•(blank ? re2))) 
-    [@eqP_trans [|@re_embedding] @eqP_trans [|@H] @eqP_sym @re_embedding
-    |// 
-    |% // #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/  
-    |% // #p #H >(memb_single … H) @(ex_intro … ϵ) /2/
-    |#p #_ normalize //
-    ]
-  ]
-qed.
-
-lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
-#n #m % [@eqbnat_true_to_eq | @eq_to_eqbnat_true]
-qed.
-
-definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true.
-
-definition a ≝ s DeqNat O.
-definition b ≝ s DeqNat (S O).
-definition c ≝ s DeqNat (S (S O)).
-
-definition exp1 ≝ ((a·b)^*·a).
-definition exp2 ≝ a·(b·a)^*.
-definition exp4 ≝ (b·a)^*.
-
-definition exp6 ≝ a·(a ·a ·b^* + b^* ).
-definition exp7 ≝ a · a^* · b^*.
-
-definition exp8 ≝ a·a·a·a·a·a·a·a·(a^* ).
-definition exp9 ≝ (a·a·a + a·a·a·a·a)^*.
-
-example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true.
-normalize // qed.
-
-
-
-
-
-
-
-- 
2.39.5