commit by user andrea

diff --git a/weblib/basics/list.ma b/weblib/basics/list.ma
index 4e44ce298396d089a0fb97543c26cf94b770fa1e..d6c0661208c15e31c7b10684657a3f98bbd0060c 100644 (file)
--- a/weblib/basics/list.ma
+++ b/weblib/basics/list.ma
@@ -9,7 +9,7 @@
      \ /   GNU General Public License Version 2   
       V_______________________________________________________________ *)
 
-include "arithmetics/nat.ma".\ 5span class="error" title="disambiguation error"\ 6\ 5/span\ 6
+include "arithmetics/nat.ma".
 
 inductive list (A:Type[0]) : Type[0] :=
   | nil: list A
@@ -237,4 +237,4 @@ theorem fold_sum: ∀A,B. ∀I,J:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1
     \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i∈(I\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6J)} (f i).
 #A #B #I #J #nil #op #f (elim I) normalize 
   [>\ 5a href="cic:/matita/basics/list/nill.fix(0,2,2)"\ 6nill\ 5/a\ 6 //|#a #tl #Hind <\ 5a href="cic:/matita/basics/list/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 //]
-qed.
\ No newline at end of file
+qed.

diff --git a/weblib/tutorial/chapter10.ma b/weblib/tutorial/chapter10.ma
index 95e3a55c121f51e6d4e870933c552901846fd0a8..8fe459cd2ef9f99f4a8d9183a879be3112ef9401 100644 (file)
--- a/weblib/tutorial/chapter10.ma
+++ b/weblib/tutorial/chapter10.ma
@@ -3,9 +3,8 @@
 
 include "tutorial/chapter9.ma".
 
-(* We say that two pres \langle i_1,b_1\rangle$ and
-$\langle i_1,b_1\rangle$ are {\em cofinal} if and only if
-$b_1 = b_2$. *)
+(* We say that two pres 〈i_1,b_1〉 and 〈i_1,b_1〉 are {\em cofinal} if and 
+only if b_1 = b_2. *)
 
 definition cofinal ≝ λS.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). 
   \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
@@ -15,7 +14,7 @@ e1 and e2 are equivalent iff for any word w the states reachable
 through w are cofinal. *)
 
 theorem equiv_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. 
-  \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2} \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 ∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉.
+  \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2} \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 ∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉.
 #S #e1 #e2 % 
 [#same_sem #w 
   cut (∀b1,b2. \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) (b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b2)) 
@@ -34,7 +33,7 @@ definition occ ≝ λS.λe1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)
   \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|)) (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)).
 
 lemma occ_enough: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
-(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
+(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
  →∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉.
 #S #e1 #e2 #H #w
 cases (\ 5a href="cic:/matita/tutorial/chapter5/decidable_sublist.def(6)"\ 6decidable_sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2)) [@H] -H #H
@@ -47,7 +46,7 @@ qed.
 occurring the given regular expressions. *)
 
 lemma equiv_sem_occ: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
-(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
+(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
 → \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
 #S #e1 #e2 #H @(\ 5a href="cic:/matita/basics/logic/proj2.def(2)"\ 6proj2\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter10/equiv_sem.def(16)"\ 6equiv_sem\ 5/a\ 6 …)) @\ 5a href="cic:/matita/tutorial/chapter10/occ_enough.def(11)"\ 6occ_enough\ 5/a\ 6 #w @H 
 qed.
@@ -60,7 +59,7 @@ w.r.t. moves, and all its members are cofinal.
 *)
 
 definition sons ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). 
- \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (λa.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)),\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p))〉) l.
+ \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (λa.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)),\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p))〉) l.
 
 lemma memb_sons: ∀S,l.∀p,q:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? l q) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
   \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6a.(\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 q)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
@@ -76,7 +75,7 @@ definition is_bisim ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,
 (* Using lemma equiv_sem_occ it is easy to prove the following result: *)
 
 lemma bisim_to_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. 
-  \ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S l (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+  \ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S l (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ?\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
 #S #l #e1 #e2 #Hbisim #Hmemb @\ 5a href="cic:/matita/tutorial/chapter10/equiv_sem_occ.def(17)"\ 6equiv_sem_occ\ 5/a\ 6 
 #w #Hsub @(\ 5a href="cic:/matita/basics/logic/proj1.def(2)"\ 6proj1\ 5/a\ 6 … (Hbisim \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e2〉 ?))
 lapply Hsub @(\ 5a href="cic:/matita/basics/list/list_elim_left.def(10)"\ 6list_elim_left\ 5/a\ 6 … w) [//]
@@ -114,15 +113,15 @@ Here is the extremely simple algorithm: *)
 
 let rec bisim S l n (frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?) on n ≝
   match n with 
-  [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
+  [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
   | S m ⇒ 
     match frontier with
-    [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
+    [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
     | cons hd tl ⇒
       if \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 hd)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 hd)) then
         bisim S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited))) 
         (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)
-      else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
+      else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
     ]
   ].
 
@@ -137,26 +136,26 @@ case and in some relevant instances *)
 lemma unfold_bisim: ∀S,l,n.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
   \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
   match n with 
-  [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
+  [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
   | S m ⇒ 
     match frontier with
-    [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
+    [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
     | cons hd tl ⇒
       if \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 hd)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 hd)) then
         \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited))) 
           (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)
-      else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
+      else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
     ]
   ].
 #S #l #n cases n // qed.
 
 lemma bisim_never: ∀S,l.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
-  \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
+  \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
 #frontier #visited >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 // 
 qed.
 
 lemma bisim_end: ∀Sig,l,m.∀visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
-  \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉.
+  \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉.
 #n #visisted >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 // 
 qed.
 
@@ -170,7 +169,7 @@ qed.
 
 lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
 \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
-  \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
+  \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
 #Sig #l #m #p #frontier #visited #test >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 whd in ⊢ (??%?); >test // 
 qed.
 
@@ -183,7 +182,7 @@ qed. *)
 
 let rec pitem_enum S (i:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i ≝
   match i with
-  [ z ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"\ 6\ 5/span\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
+  [ z ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
   | e ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,2,1)"\ 6pe\ 5/a\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
   | s y ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,3,1)"\ 6ps\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:(\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,4,1)"\ 6pp\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
   | o i1 i2 ⇒ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,6,1)"\ 6po\ 5/a\ 6 S) (pitem_enum S i1) (pitem_enum S i2)
@@ -202,23 +201,23 @@ lemma pitem_enum_complete : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pit
 qed.
 
 definition pre_enum ≝ λS.λi:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
-  \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉) (\ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S i) (\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
+  \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉) ( \ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S i) (\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
   
 lemma pre_enum_complete : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
   \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? e (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-#S * #i #b @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) \ 5a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6 ? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
+#S * #i #b @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) \ 5a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6 ? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
 // cases b normalize //
 qed.
  
-definition space_enum ≝ λS.λi1,i2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
-  \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λe1,e2.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i1) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i2).
+definition space_enum ≝ λS.λi1,i2: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
+  \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λe1,e2.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉) ( \ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i1) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i2).
 
 lemma space_enum_complete : ∀S.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
-  \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 (\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-#S #e1 #e2 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 … (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
+  \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 ( \ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S #e1 #e2 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 … (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
 // qed.
 
-definition all_reachable ≝ λS.λe1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+definition all_reachable ≝ λS.λe1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
 \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 ? l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 
   ∀p. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → 
     \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w.(\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p). 
@@ -228,8 +227,8 @@ definition disjoint ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,
         
 (* We are ready to prove that bisim is correct; we use the invariant 
 that at each call of bisim the two lists visited and frontier only contain 
-nodes reachable from \langle e_1,e_2\rangle, hence it is absurd to suppose 
-to meet a pair which is not cofinal. *)
+nodes reachable from 〈e_1,e_2〉, hence it is absurd to suppose to meet a pair 
+which is not cofinal. *)
 
 lemma bisim_correct: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2} → 
  ∀l,n.∀frontier,visited:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ((\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)).
@@ -265,7 +264,7 @@ lemma bisim_correct: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.de
        |@r_frontier @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //
        ]
      |@\ 5a href="cic:/matita/tutorial/chapter5/unique_append_elim.def(7)"\ 6unique_append_elim\ 5/a\ 6 #q #H
-       [@\ 5a href="cic:/matita/basics/bool/injective_notb.def(4)"\ 6injective_notb\ 5/a\ 6 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_true.def(5)"\ 6memb_filter_true\ 5/a\ 6 … H\ 5span class="error" title="error location"\ 6\ 5/span\ 6)
+       [@\ 5a href="cic:/matita/basics/bool/injective_notb.def(4)"\ 6injective_notb\ 5/a\ 6 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_true.def(5)"\ 6memb_filter_true\ 5/a\ 6 … H)
        |cut ((q\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=p) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) 
          [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //]
         cases (\ 5a href="cic:/matita/basics/bool/andb_true.def(5)"\ 6andb_true\ 5/a\ 6 … u_frontier) #notp #_ @(\bf ?) 
@@ -274,7 +273,7 @@ lemma bisim_correct: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.de
      ]
    ]  
 qed.  
-   
+
 (* For completeness, we use the invariant that all the nodes in visited are cofinal, 
 and the sons of visited are either in visited or in the frontier; since
 at the end frontier is empty, visited is hence a bisimulation. *)
@@ -289,7 +288,7 @@ lemma bisim_complete:
  ∀S,l,n.∀frontier,visited,visited_res:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
  \ 5a href="cic:/matita/tutorial/chapter10/all_true.def(8)"\ 6all_true\ 5/a\ 6 S visited →
  \ 5a href="cic:/matita/tutorial/chapter10/sub_sons.def(8)"\ 6sub_sons\ 5/a\ 6 S l visited (frontier\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6visited) →
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited_res〉 →
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited_res〉 →
  \ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S visited_res l \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 ? (frontier\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6visited) visited_res. 
 #S #l #n elim n
   [#fron #vis #vis_res #_ #_ >\ 5a href="cic:/matita/tutorial/chapter10/bisim_never.def(10)"\ 6bisim_never\ 5/a\ 6 #H destruct
@@ -354,7 +353,7 @@ theorem euqiv_sem : ∀Sig.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/re.in
    \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? e1 e2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
 #Sig #re1 #re2 %
   [#H @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"\ 6eqP_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6] @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6]
-   cut (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2)〉)
+   cut (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2)〉)
      [<H //] #Hcut
    cases (\ 5a href="cic:/matita/tutorial/chapter10/bisim_complete.def(11)"\ 6bisim_complete\ 5/a\ 6 … Hcut) 
      [2,3: #p whd in ⊢ ((??%?)→?); #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]