From c65aeeb22bdc7bf289e5db108c358b199773c857 Mon Sep 17 00:00:00 2001
From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Fri, 2 Mar 2012 09:02:13 +0000
Subject: [PATCH] Splitted chapter 7

---
 weblib/tutorial/chapter7.ma | 304 +---------------
 weblib/tutorial/chapter8.ma | 671 +++++++++++++-----------------------
 weblib/tutorial/chapter9.ma | 485 ++++++++++++++++++++++++++
 3 files changed, 733 insertions(+), 727 deletions(-)
 create mode 100644 weblib/tutorial/chapter9.ma

diff --git a/weblib/tutorial/chapter7.ma b/weblib/tutorial/chapter7.ma
index da90566e8..b9a35a8ad 100644
--- a/weblib/tutorial/chapter7.ma
+++ b/weblib/tutorial/chapter7.ma
@@ -1,4 +1,5 @@
-(* We shall apply all the previous machinery to the study of regular languages 
+(* <h1>Regular Expressions</h1>
+We shall apply all the previous machinery to the study of regular languages 
 and the constructions of the associated finite automata. *)
 
 include "tutorial/chapter6.ma".
@@ -46,7 +47,8 @@ lemma rsem_star : ∀S.∀r: a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1
 // qed.
 
 
-(* We now introduce pointed regular expressions, that are the main tool we shall 
+(* <h2>Pointed Regular expressions </h2>
+We now introduce pointed regular expressions, that are the main tool we shall 
 use for the construction of the automaton. 
 A pointed regular expression is just a regular expression internally labelled 
 with some additional points. Intuitively, points mark the positions inside the 
@@ -128,7 +130,8 @@ lemma erase_plus : ∀S.∀i1,i2:pitem S.
 lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
 // qed.
 
-(* Items and pres are very concrete datatypes: they can be effectively compared, 
+(* <h2>Comparing items and pres<h2>
+Items and pres are very concrete datatypes: they can be effectively compared, 
 and enumerated. In particular, we can define a boolean equality beqitem and a proof
 beqitem_true that it refects propositional equality, enriching the set (pitem S)
 to a DeqSet. *)
@@ -195,7 +198,8 @@ unification hint  0 ≔ S,i1,i2;
 (* ---------------------------------------- *) ⊢ 
     beqitem S i1 i2 ≡ eqb X i1 i2.
 
-(* The intuitive semantic of a point is to mark the position where
+(* <h2>Semantics of pointed regular expression<h2>
+The intuitive semantic of a point is to mark the position where
 we should start reading the regular expression. The language associated
 to a pre is the union of the languages associated with its points. *)
 
@@ -295,295 +299,3 @@ lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
   ]
 qed.
 
-definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
-notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
-interpretation "oplus" 'oplus a b = (lo ? a b).
-
-lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
-// qed.
-
-definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
-  match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
- 
-notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
-interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
-
-lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. 
-  A = B → A =1 B. 
-#S #A #B #H >H /2/ qed.
-
-lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
-  \sem{i ◃ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
-#S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //] 
->sem_pre_true >sem_cat >sem_pre_true /2/ 
-qed.
- 
-definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
-  match e1 with 
-  [ mk_Prod i1 b1 ⇒ match b1 with 
-    [ true ⇒ (i1 ◃ (bcast ? i2)) 
-    | false ⇒ 〈i1 · i2,false〉
-    ]
-  ].
-
-notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
-interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
-
-notation "•" non associative with precedence 60 for @{eclose ?}.
-
-let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
- match i with
-  [ pz ⇒ 〈 `∅, false 〉
-  | pe ⇒ 〈 ϵ,  true 〉
-  | ps x ⇒ 〈 `.x, false〉
-  | pp x ⇒ 〈 `.x, false 〉
-  | po i1 i2 ⇒ •i1 ⊕ •i2
-  | pc i1 i2 ⇒ •i1 ▹ i2
-  | pk i ⇒ 〈(\fst (•i))^*,true〉].
-  
-notation "• x" non associative with precedence 60 for @{'eclose $x}.
-interpretation "eclose" 'eclose x = (eclose ? x).
-
-lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 + i2) = •i1 ⊕ •i2.
-// qed.
-
-lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 · i2) = •i1 ▹ i2.
-// qed.
-
-lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
-  •i^* = 〈(\fst(•i))^*,true〉.
-// qed.
-
-definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
-  match e with 
-  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
-  
-definition preclose ≝ λS. lift S (eclose S). 
-interpretation "preclose" 'eclose x = (preclose ? x).
-
-(* theorem 16: 2 *)
-lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
-  \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}. 
-#S * #i1 #b1 * #i2 #b2 #w %
-  [cases b1 cases b2 normalize /2/ * /3/ * /3/
-  |cases b1 cases b2 normalize /2/ * /3/ * /3/
-  ]
-qed.
-
-lemma odot_true : 
-  ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
-// qed.
-
-lemma odot_true_bis : 
-  ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
-#S #i1 #i2 normalize cases (•i2) // qed.
-
-lemma odot_false: 
-  ∀S.∀i1,i2:pitem S.
-  〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
-// qed.
-
-lemma LcatE : ∀S.∀e1,e2:pitem S.
-  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
-// qed.
-
-lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
-#S #i elim i // 
-  [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
-    cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
-  | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
-    cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //  
-  | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
-  ]
-qed.
-
-(*
-lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
-  \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
-/2/ qed.
-*)
-
-(* theorem 16: 1 → 3 *)
-lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
-   \sem{•i2} =1  \sem{i2} ∪ \sem{|i2|} →
-   \sem{e1 ▹ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
-#S * #i1 #b1 #i2 cases b1
-  [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
-  |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
-   >erase_bull @eqP_trans [|@(eqP_union_l … H)]
-    @eqP_trans [|@eqP_union_l[|@union_comm ]]
-    @eqP_trans [|@eqP_sym @union_assoc ] /3/ 
-  ]
-qed.
-  
-lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
- \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
-#S #e #i #A #seme
-@eqP_trans [|@minus_eps_pre]
-@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
-@eqP_trans [||@distribute_substract] 
-@eqP_substract_r //
-qed.
-
-(* theorem 16: 1 *)
-theorem sem_bull: ∀S:DeqSet. ∀i:pitem S.  \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
-#S #e elim e 
-  [#w normalize % [/2/ | * //]
-  |/2/ 
-  |#x normalize #w % [ /2/ | * [@False_ind | //]]
-  |#x normalize #w % [ /2/ | * // ] 
-  |#i1 #i2 #IH1 #IH2 >eclose_dot
-   @eqP_trans [|@odot_dot_aux //] >sem_cat 
-   @eqP_trans
-     [|@eqP_union_r
-       [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
-   @eqP_trans [|@union_assoc]
-   @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_union_l //
-  |#i1 #i2 #IH1 #IH2 >eclose_plus
-   @eqP_trans [|@sem_oplus] >sem_plus >erase_plus 
-   @eqP_trans [|@(eqP_union_l … IH2)]
-   @eqP_trans [|@eqP_sym @union_assoc]
-   @eqP_trans [||@union_assoc] @eqP_union_r
-   @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_trans [||@eqP_union_l [|@union_comm]]
-   @eqP_trans [||@union_assoc] /2/
-  |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
-   @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
-   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
-   @eqP_trans [|@union_assoc] @eqP_union_l >erase_star 
-   @eqP_sym @star_fix_eps 
-  ]
-qed.
-
-(* blank item *)
-let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
- match i with
-  [ z ⇒ `∅
-  | e ⇒ ϵ
-  | s y ⇒ `y
-  | o e1 e2 ⇒ (blank S e1) + (blank S e2) 
-  | c e1 e2 ⇒ (blank S e1) · (blank S e2)
-  | k e ⇒ (blank S e)^* ].
-  
-lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
-#S #e elim e normalize //
-qed.
-
-lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
-#S #e elim e 
-  [1,2:@eq_to_ex_eq // 
-  |#s @eq_to_ex_eq //
-  |#e1 #e2 #Hind1 #Hind2 >sem_cat 
-   @eqP_trans [||@(union_empty_r … ∅)] 
-   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
-   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
-  |#e1 #e2 #Hind1 #Hind2 >sem_plus 
-   @eqP_trans [||@(union_empty_r … ∅)] 
-   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
-  |#e #Hind >sem_star
-   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
-  ]
-qed.
-   
-theorem re_embedding: ∀S.∀e:re S. 
-  \sem{•(blank S e)} =1 \sem{e}.
-#S #e @eqP_trans [|@sem_bull] >forget_blank 
-@eqP_trans [|@eqP_union_r [|@sem_blank]]
-@eqP_trans [|@union_comm] @union_empty_r.
-qed.
-
-(* lefted operations *)
-definition lifted_cat ≝ λS:DeqSet.λe:pre S. 
-  lift S (pre_concat_l S eclose e).
-
-notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
-
-interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
-
-lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b. 
-  〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
-#S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) // 
-qed.
-
-lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
-  〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
-// 
-qed.
-  
-lemma erase_odot:∀S.∀e1,e2:pre S.
-  |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
-#S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot //  
-qed.
-
-definition lk ≝ λS:DeqSet.λe:pre S.
-  match e with 
-  [ mk_Prod i1 b1 ⇒
-    match b1 with 
-    [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
-    |false ⇒ 〈i1^*,false〉
-    ]
-  ]. 
-
-(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
-interpretation "lk" 'lk a = (lk ? a).
-notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
-
-
-lemma ostar_true: ∀S.∀i:pitem S.
-  〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
-// qed.
-
-lemma ostar_false: ∀S.∀i:pitem S.
-  〈i,false〉^⊛ = 〈i^*, false〉.
-// qed.
-  
-lemma erase_ostar: ∀S.∀e:pre S.
-  |\fst (e^⊛)| = |\fst e|^*.
-#S * #i * // qed.
-
-lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. 
-  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
-#S #e1 #i 
-cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
-#H >H cases (e1 ▹ i) #i1 #b1 cases b1 
-  [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_union_l /2/ 
-  |/2/
-  ]
-qed.
-
-lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. 
-  e1 ⊙ 〈i,false〉 = e1 ▹ i.
-#S #e1 #i  
-cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
-cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
-qed.
-
-lemma sem_odot: 
-  ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
-#S #e1 * #i2 * 
-  [>sem_pre_true 
-   @eqP_trans [|@sem_odot_true]
-   @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
-  |>sem_pre_false >eq_odot_false @odot_dot_aux //
-  ]
-qed.
-
-(* theorem 16: 4 *)      
-theorem sem_ostar: ∀S.∀e:pre S. 
-  \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
-#S * #i #b cases b
-  [>sem_pre_true >sem_pre_true >sem_star >erase_bull
-   @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
-   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
-   @eqP_trans [||@eqP_sym @distr_cat_r]
-   @eqP_trans [|@union_assoc] @eqP_union_l
-   @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps 
-  |>sem_pre_false >sem_pre_false >sem_star /2/
-  ]
-qed.
\ No newline at end of file
diff --git a/weblib/tutorial/chapter8.ma b/weblib/tutorial/chapter8.ma
index dc7474e21..7aac6fabc 100644
--- a/weblib/tutorial/chapter8.ma
+++ b/weblib/tutorial/chapter8.ma
@@ -1,485 +1,294 @@
-include "re.ma".
-include "basics/listb.ma".
+include "tutorial/chapter7.ma".
 
-let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
- match E with
+definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
+notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
+interpretation "oplus" 'oplus a b = (lo ? a b).
+
+lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
+// qed.
+
+definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
+  match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
+ 
+notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
+interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
+
+lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. 
+  A = B → A =1 B. 
+#S #A #B #H >H /2/ qed.
+
+lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
+  \sem{i ◃ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
+#S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //] 
+>sem_pre_true >sem_cat >sem_pre_true /2/ 
+qed.
+ 
+definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
+  match e1 with 
+  [ mk_Prod i1 b1 ⇒ match b1 with 
+    [ true ⇒ (i1 ◃ (bcast ? i2)) 
+    | false ⇒ 〈i1 · i2,false〉
+    ]
+  ].
+
+notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
+interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
+
+notation "•" non associative with precedence 60 for @{eclose ?}.
+
+let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
+ match i with
   [ pz ⇒ 〈 `∅, false 〉
-  | pe ⇒ 〈 ϵ, false 〉
-  | ps y ⇒ 〈 `y, false 〉
-  | pp y ⇒ 〈 `y, x == y 〉
-  | po e1 e2 ⇒ (move ? x e1) ⊕ (move ? x e2) 
-  | pc e1 e2 ⇒ (move ? x e1) ⊙ (move ? x e2)
-  | pk e ⇒ (move ? x e)^⊛ ].
+  | pe ⇒ 〈 ϵ,  true 〉
+  | ps x ⇒ 〈 `.x, false〉
+  | pp x ⇒ 〈 `.x, false 〉
+  | po i1 i2 ⇒ •i1 ⊕ •i2
+  | pc i1 i2 ⇒ •i1 ▹ i2
+  | pk i ⇒ 〈(\fst (•i))^*,true〉].
   
-lemma move_plus: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
-  move S x (i1 + i2) = (move ? x i1) ⊕ (move ? x i2).
-// qed.
+notation "• x" non associative with precedence 60 for @{'eclose $x}.
+interpretation "eclose" 'eclose x = (eclose ? x).
 
-lemma move_cat: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
-  move S x (i1 · i2) = (move ? x i1) ⊙ (move ? x i2).
+lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
+  •(i1 + i2) = •i1 ⊕ •i2.
 // qed.
 
-lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
-  move S x i^* = (move ? x i)^⊛.
+lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
+  •(i1 · i2) = •i1 ▹ i2.
 // qed.
 
-definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
-
-lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b. 
-  pmove ? x 〈i,b〉 = move ? x i.
+lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
+  •i^* = 〈(\fst(•i))^*,true〉.
 // qed.
 
-lemma eq_to_eq_hd: ∀A.∀l1,l2:list A.∀a,b. 
-  a::l1 = b::l2 → a = b.
-#A #l1 #l2 #a #b #H destruct //
-qed. 
-
-lemma same_kernel: ∀S:DeqSet.∀a:S.∀i:pitem S.
-  |\fst (move ? a i)| = |i|.
-#S #a #i elim i //
-  [#i1 #i2 #H1 #H2 >move_cat >erase_odot //
-  |#i1 #i2 #H1 #H2 >move_plus whd in ⊢ (??%%); // 
+definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
+  match e with 
+  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
+  
+definition preclose ≝ λS. lift S (eclose S). 
+interpretation "preclose" 'eclose x = (preclose ? x).
+
+(* theorem 16: 2 *)
+lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
+  \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}. 
+#S * #i1 #b1 * #i2 #b2 #w %
+  [cases b1 cases b2 normalize /2/ * /3/ * /3/
+  |cases b1 cases b2 normalize /2/ * /3/ * /3/
   ]
 qed.
 
-theorem move_ok:
- ∀S:DeqSet.∀a:S.∀i:pitem S.∀w: word S. 
-   \sem{move ? a i} w ↔ \sem{i} (a::w).
-#S #a #i elim i 
-  [normalize /2/
-  |normalize /2/
-  |normalize /2/
-  |normalize #x #w cases (true_or_false (a==x)) #H >H normalize
-    [>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
-    |% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
-    ]
-  |#i1 #i2 #HI1 #HI2 #w >move_cat
-   @iff_trans[|@sem_odot] >same_kernel >sem_cat_w
-   @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 
-   @iff_trans[||@iff_sym @deriv_middot //]
-   @cat_ext_l @HI1
-  ]
-qed.
-    
-notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
-let rec moves (S : DeqSet) w e on w : pre S ≝
- match w with
-  [ nil ⇒ e
-  | cons x w' ⇒ w' ↦* (move S x (\fst e))]. 
-
-lemma moves_empty: ∀S:DeqSet.∀e:pre S. 
-  moves ? [ ] e = e.
+lemma odot_true : 
+  ∀S.∀i1,i2:pitem S.
+  〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
 // qed.
 
-lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S. 
-  moves ? (a::w)  e = moves ? w (move S a (\fst e)).
+lemma odot_true_bis : 
+  ∀S.∀i1,i2:pitem S.
+  〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+#S #i1 #i2 normalize cases (•i2) // qed.
+
+lemma odot_false: 
+  ∀S.∀i1,i2:pitem S.
+  〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
 // qed.
 
-lemma moves_left : ∀S,a,w,e. 
-  moves S (w@[a]) e = move S a (\fst (moves S w e)). 
-#S #a #w elim w // #x #tl #Hind #e >moves_cons >moves_cons //
-qed.
+lemma LcatE : ∀S.∀e1,e2:pitem S.
+  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
+// qed.
 
-lemma not_epsilon_sem: ∀S:DeqSet.∀a:S.∀w: word S. ∀e:pre S. 
-  iff ((a::w) ∈ e) ((a::w) ∈ \fst e).
-#S #a #w * #i #b cases b normalize 
-  [% /2/ * // #H destruct |% normalize /2/]
+lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
+#S #i elim i // 
+  [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
+    cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
+  | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
+    cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //  
+  | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
+  ]
 qed.
 
-lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
-  |\fst (moves ? w e)| = |\fst e|.
-#S #w elim w //
+(*
+lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
+  \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
+/2/ qed.
+*)
+
+(* theorem 16: 1 → 3 *)
+lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
+   \sem{•i2} =1  \sem{i2} ∪ \sem{|i2|} →
+   \sem{e1 ▹ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+#S * #i1 #b1 #i2 cases b1
+  [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
+  |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
+   >erase_bull @eqP_trans [|@(eqP_union_l … H)]
+    @eqP_trans [|@eqP_union_l[|@union_comm ]]
+    @eqP_trans [|@eqP_sym @union_assoc ] /3/ 
+  ]
 qed.
-
-theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S. 
-   (\snd (moves ? w e) = true) ↔ \sem{e} w.
-#S #w elim w 
- [* #i #b >moves_empty cases b % /2/
- |#a #w1 #Hind #e >moves_cons
-  @iff_trans [||@iff_sym @not_epsilon_sem]
-  @iff_trans [||@move_ok] @Hind
- ]
+  
+lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
+ \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
+#S #e #i #A #seme
+@eqP_trans [|@minus_eps_pre]
+@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
+@eqP_trans [||@distribute_substract] 
+@eqP_substract_r //
 qed.
 
-(************************ pit state ***************************)
-definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉. 
-
-let rec occur (S: DeqSet) (i: re S) on i ≝  
-  match i with
-  [ z ⇒ [ ]
-  | e ⇒ [ ]
-  | s y ⇒ [y]
-  | o e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2) 
-  | c e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2) 
-  | k e ⇒ occur S e].
-
-lemma not_occur_to_pit: ∀S,a.∀i:pitem S. memb S a (occur S (|i|)) ≠ true →
-  move S a i  = pit_pre S i.
-#S #a #i elim i //
-  [#x normalize cases (a==x) normalize // #H @False_ind /2/
-  |#i1 #i2 #Hind1 #Hind2 #H >move_cat 
-   >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
-   >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
-  |#i1 #i2 #Hind1 #Hind2 #H >move_plus 
-   >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
-   >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
-  |#i #Hind #H >move_star >Hind // 
+(* theorem 16: 1 *)
+theorem sem_bull: ∀S:DeqSet. ∀i:pitem S.  \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
+#S #e elim e 
+  [#w normalize % [/2/ | * //]
+  |/2/ 
+  |#x normalize #w % [ /2/ | * [@False_ind | //]]
+  |#x normalize #w % [ /2/ | * // ] 
+  |#i1 #i2 #IH1 #IH2 >eclose_dot
+   @eqP_trans [|@odot_dot_aux //] >sem_cat 
+   @eqP_trans
+     [|@eqP_union_r
+       [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
+   @eqP_trans [|@union_assoc]
+   @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_union_l //
+  |#i1 #i2 #IH1 #IH2 >eclose_plus
+   @eqP_trans [|@sem_oplus] >sem_plus >erase_plus 
+   @eqP_trans [|@(eqP_union_l … IH2)]
+   @eqP_trans [|@eqP_sym @union_assoc]
+   @eqP_trans [||@union_assoc] @eqP_union_r
+   @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_trans [||@eqP_union_l [|@union_comm]]
+   @eqP_trans [||@union_assoc] /2/
+  |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
+   @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
+   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+   @eqP_trans [|@union_assoc] @eqP_union_l >erase_star 
+   @eqP_sym @star_fix_eps 
   ]
 qed.
 
-lemma move_pit: ∀S,a,i. move S a (\fst (pit_pre S i)) = pit_pre S i.
-#S #a #i elim i //
-  [#i1 #i2 #Hind1 #Hind2 >move_cat >Hind1 >Hind2 // 
-  |#i1 #i2 #Hind1 #Hind2 >move_plus >Hind1 >Hind2 // 
-  |#i #Hind >move_star >Hind //
-  ]
-qed. 
+(* blank item *)
+let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
+ match i with
+  [ z ⇒ `∅
+  | e ⇒ ϵ
+  | s y ⇒ `y
+  | o e1 e2 ⇒ (blank S e1) + (blank S e2) 
+  | c e1 e2 ⇒ (blank S e1) · (blank S e2)
+  | k e ⇒ (blank S e)^* ].
+  
+lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
+#S #e elim e normalize //
+qed.
 
-lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
-#S #w #i elim w // 
-qed. 
- 
-lemma to_pit: ∀S,w,e. ¬ sublist S w (occur S (|\fst e|)) →
- moves S w e = pit_pre S (\fst e).
-#S #w elim w
-  [#e * #H @False_ind @H normalize #a #abs @False_ind /2/
-  |#a #tl #Hind #e #H cases (true_or_false (memb S a (occur S (|\fst e|))))
-    [#Htrue >moves_cons whd in ⊢ (???%); <(same_kernel … a) 
-     @Hind >same_kernel @(not_to_not … H) #H1 #b #memb cases (orb_true_l … memb)
-      [#H2 >(\P H2) // |#H2 @H1 //]
-    |#Hfalse >moves_cons >not_occur_to_pit // >Hfalse /2/ 
-    ]
+lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
+#S #e elim e 
+  [1,2:@eq_to_ex_eq // 
+  |#s @eq_to_ex_eq //
+  |#e1 #e2 #Hind1 #Hind2 >sem_cat 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
+  |#e1 #e2 #Hind1 #Hind2 >sem_plus 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
+  |#e #Hind >sem_star
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
   ]
 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]
+   
+theorem re_embedding: ∀S.∀e:re S. 
+  \sem{•(blank S e)} =1 \sem{e}.
+#S #e @eqP_trans [|@sem_bull] >forget_blank 
+@eqP_trans [|@eqP_union_r [|@sem_blank]]
+@eqP_trans [|@union_comm] @union_empty_r.
 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.
+(* lefted operations *)
+definition lifted_cat ≝ λS:DeqSet.λe:pre S. 
+  lift S (pre_concat_l S eclose e).
 
-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.
+notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
 
-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.
+interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
 
-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]
+lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b. 
+  〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
+#S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) // 
 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)
-  ]
+lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
+  〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
+// 
 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 // 
+lemma erase_odot:∀S.∀e1,e2:pre S.
+  |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
+#S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot //  
 qed.
 
-lemma bisim_end: ∀Sig,l,m.∀visited: list ?.
-  bisim Sig l (S m) [] visited = 〈true,visited〉.
-#n #visisted >unfold_bisim // 
-qed.
+definition lk ≝ λS:DeqSet.λe:pre S.
+  match e with 
+  [ mk_Prod i1 b1 ⇒
+    match b1 with 
+    [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
+    |false ⇒ 〈i1^*,false〉
+    ]
+  ]. 
 
-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.
+(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
+interpretation "lk" 'lk a = (lk ? a).
+notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
 
-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.
+lemma ostar_true: ∀S.∀i:pitem S.
+  〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
+// 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 ostar_false: ∀S.∀i:pitem S.
+  〈i,false〉^⊛ = 〈i^*, false〉.
+// qed.
   
-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)) //
+lemma erase_ostar: ∀S.∀e:pre S.
+  |\fst (e^⊛)| = |\fst e|^*.
+#S * #i * // qed.
+
+lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. 
+  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
+#S #e1 #i 
+cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
+#H >H cases (e1 ▹ i) #i1 #b1 cases b1 
+  [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_union_l /2/ 
+  |/2/
   ]
 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 //
+lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. 
+  e1 ⊙ 〈i,false〉 = e1 ▹ i.
+#S #e1 #i  
+cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
+cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
 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
-        ]
-      ]
-    ]
+lemma sem_odot: 
+  ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
+#S #e1 * #i2 * 
+  [>sem_pre_true 
+   @eqP_trans [|@sem_odot_true]
+   @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
+  |>sem_pre_false >eq_odot_false @odot_dot_aux //
   ]
 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 //
-    ]
+(* theorem 16: 4 *)      
+theorem sem_ostar: ∀S.∀e:pre S. 
+  \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
+#S * #i #b cases b
+  [>sem_pre_true >sem_pre_true >sem_star >erase_bull
+   @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
+   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+   @eqP_trans [||@eqP_sym @distr_cat_r]
+   @eqP_trans [|@union_assoc] @eqP_union_l
+   @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps 
+  |>sem_pre_false >sem_pre_false >sem_star /2/
   ]
-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.
-
-
-
-
-
-
-
+qed.
\ No newline at end of file
diff --git a/weblib/tutorial/chapter9.ma b/weblib/tutorial/chapter9.ma
new file mode 100644
index 000000000..dc7474e21
--- /dev/null
+++ b/weblib/tutorial/chapter9.ma
@@ -0,0 +1,485 @@
+include "re.ma".
+include "basics/listb.ma".
+
+let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
+ match E with
+  [ pz ⇒ 〈 `∅, false 〉
+  | pe ⇒ 〈 ϵ, false 〉
+  | ps y ⇒ 〈 `y, false 〉
+  | pp y ⇒ 〈 `y, x == y 〉
+  | po e1 e2 ⇒ (move ? x e1) ⊕ (move ? x e2) 
+  | pc e1 e2 ⇒ (move ? x e1) ⊙ (move ? x e2)
+  | pk e ⇒ (move ? x e)^⊛ ].
+  
+lemma move_plus: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
+  move S x (i1 + i2) = (move ? x i1) ⊕ (move ? x i2).
+// qed.
+
+lemma move_cat: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
+  move S x (i1 · i2) = (move ? x i1) ⊙ (move ? x i2).
+// qed.
+
+lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
+  move S x i^* = (move ? x i)^⊛.
+// qed.
+
+definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
+
+lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b. 
+  pmove ? x 〈i,b〉 = move ? x i.
+// qed.
+
+lemma eq_to_eq_hd: ∀A.∀l1,l2:list A.∀a,b. 
+  a::l1 = b::l2 → a = b.
+#A #l1 #l2 #a #b #H destruct //
+qed. 
+
+lemma same_kernel: ∀S:DeqSet.∀a:S.∀i:pitem S.
+  |\fst (move ? a i)| = |i|.
+#S #a #i elim i //
+  [#i1 #i2 #H1 #H2 >move_cat >erase_odot //
+  |#i1 #i2 #H1 #H2 >move_plus whd in ⊢ (??%%); // 
+  ]
+qed.
+
+theorem move_ok:
+ ∀S:DeqSet.∀a:S.∀i:pitem S.∀w: word S. 
+   \sem{move ? a i} w ↔ \sem{i} (a::w).
+#S #a #i elim i 
+  [normalize /2/
+  |normalize /2/
+  |normalize /2/
+  |normalize #x #w cases (true_or_false (a==x)) #H >H normalize
+    [>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
+    |% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
+    ]
+  |#i1 #i2 #HI1 #HI2 #w >move_cat
+   @iff_trans[|@sem_odot] >same_kernel >sem_cat_w
+   @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 
+   @iff_trans[||@iff_sym @deriv_middot //]
+   @cat_ext_l @HI1
+  ]
+qed.
+    
+notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
+let rec moves (S : DeqSet) w e on w : pre S ≝
+ match w with
+  [ nil ⇒ e
+  | cons x w' ⇒ w' ↦* (move S x (\fst e))]. 
+
+lemma moves_empty: ∀S:DeqSet.∀e:pre S. 
+  moves ? [ ] e = e.
+// qed.
+
+lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S. 
+  moves ? (a::w)  e = moves ? w (move S a (\fst e)).
+// qed.
+
+lemma moves_left : ∀S,a,w,e. 
+  moves S (w@[a]) e = move S a (\fst (moves S w e)). 
+#S #a #w elim w // #x #tl #Hind #e >moves_cons >moves_cons //
+qed.
+
+lemma not_epsilon_sem: ∀S:DeqSet.∀a:S.∀w: word S. ∀e:pre S. 
+  iff ((a::w) ∈ e) ((a::w) ∈ \fst e).
+#S #a #w * #i #b cases b normalize 
+  [% /2/ * // #H destruct |% normalize /2/]
+qed.
+
+lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
+  |\fst (moves ? w e)| = |\fst e|.
+#S #w elim w //
+qed.
+
+theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S. 
+   (\snd (moves ? w e) = true) ↔ \sem{e} w.
+#S #w elim w 
+ [* #i #b >moves_empty cases b % /2/
+ |#a #w1 #Hind #e >moves_cons
+  @iff_trans [||@iff_sym @not_epsilon_sem]
+  @iff_trans [||@move_ok] @Hind
+ ]
+qed.
+
+(************************ pit state ***************************)
+definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉. 
+
+let rec occur (S: DeqSet) (i: re S) on i ≝  
+  match i with
+  [ z ⇒ [ ]
+  | e ⇒ [ ]
+  | s y ⇒ [y]
+  | o e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2) 
+  | c e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2) 
+  | k e ⇒ occur S e].
+
+lemma not_occur_to_pit: ∀S,a.∀i:pitem S. memb S a (occur S (|i|)) ≠ true →
+  move S a i  = pit_pre S i.
+#S #a #i elim i //
+  [#x normalize cases (a==x) normalize // #H @False_ind /2/
+  |#i1 #i2 #Hind1 #Hind2 #H >move_cat 
+   >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
+   >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
+  |#i1 #i2 #Hind1 #Hind2 #H >move_plus 
+   >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
+   >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
+  |#i #Hind #H >move_star >Hind // 
+  ]
+qed.
+
+lemma move_pit: ∀S,a,i. move S a (\fst (pit_pre S i)) = pit_pre S i.
+#S #a #i elim i //
+  [#i1 #i2 #Hind1 #Hind2 >move_cat >Hind1 >Hind2 // 
+  |#i1 #i2 #Hind1 #Hind2 >move_plus >Hind1 >Hind2 // 
+  |#i #Hind >move_star >Hind //
+  ]
+qed. 
+
+lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
+#S #w #i elim w // 
+qed. 
+ 
+lemma to_pit: ∀S,w,e. ¬ sublist S w (occur S (|\fst e|)) →
+ moves S w e = pit_pre S (\fst e).
+#S #w elim w
+  [#e * #H @False_ind @H normalize #a #abs @False_ind /2/
+  |#a #tl #Hind #e #H cases (true_or_false (memb S a (occur S (|\fst e|))))
+    [#Htrue >moves_cons whd in ⊢ (???%); <(same_kernel … a) 
+     @Hind >same_kernel @(not_to_not … H) #H1 #b #memb cases (orb_true_l … memb)
+      [#H2 >(\P H2) // |#H2 @H1 //]
+    |#Hfalse >moves_cons >not_occur_to_pit // >Hfalse /2/ 
+    ]
+  ]
+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.
+
+
+
+
+
+
+
-- 
2.39.2