X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter9.ma;h=ba759d47cf34a8c20c42c5db01f8d8e42f86cdfc;hb=ccf5878f2a2ec7f952f140e162391708a740517b;hp=b3f46ae04282a96988fde5cd17d334c7fa48234c;hpb=cee37c7a5aa40d1b6d978a5e1aa3a2d91cdb7a63;p=helm.git

diff --git a/weblib/tutorial/chapter9.ma b/weblib/tutorial/chapter9.ma
index b3f46ae04..ba759d47c 100644
--- a/weblib/tutorial/chapter9.ma
+++ b/weblib/tutorial/chapter9.ma
@@ -11,30 +11,30 @@ lifted operators of the previous section:
 
 include "tutorial/chapter8.ma".
 
-let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
+let rec move (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (x:S) (E: a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) on E : a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a 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)^⊛ ].
+  [ pz ⇒ 〈a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"pz/a S, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | pe ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem epsilon" href="cic:/fakeuri.def(1)"ϵ/a, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | ps y ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem ps" href="cic:/fakeuri.def(1)"`/ay, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | pp y ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem ps" href="cic:/fakeuri.def(1)"`/ay, x a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a y a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | po e1 e2 ⇒ (move ? x e1) a title="oplus" href="cic:/fakeuri.def(1)"⊕/a (move ? x e2) 
+  | pc e1 e2 ⇒ (move ? x e1) a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a (move ? x e2)
+  | pk e ⇒ (move ? x e)a title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a ].
   
-lemma move_plus: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
-  move S x (i1 + i2) = (move ? x i1) ⊕ (move ? x i2).
+lemma move_plus: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀x:S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S x (i1 a title="pitem or" href="cic:/fakeuri.def(1)"+/a i2) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? x i1) a title="oplus" href="cic:/fakeuri.def(1)"⊕/a (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? 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).
+lemma move_cat: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀x:S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S x (i1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? x i1) a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? x i2).
 // qed.
 
-lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
-  move S x i^* = (move ? x i)^⊛.
+lemma move_star: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀x:S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S x ia title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? x i)a title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a.
 // qed.
 
 (*
-Example. Let us consider the item                      
+bExample/b. Let us consider the item                      
   
                                (•a + ϵ)((•b)*•a + •b)b
 
@@ -43,147 +43,208 @@ For a, we have two possible positions (all other points gets erased); the innerm
 point stops in front of the final b, while the other one broadcast inside (b^*a + b)b, 
 so
  
-      move((•a + ϵ)((•b)*•a + •b)b,a) = 〈(a + ϵ)((•b)^*•a + •b)•b, false〉
+      move((•a + ϵ)((•b)*•a + •b)b,a) = 〈(a + ϵ)((•b)^*•a + •b)•b, false〉
 
 For b, we have two positions too. The innermost point stops in front of the final b too, 
 while the other point reaches the end of b* and must go back through b*a:  
     
-      move((•a + ϵ)((•b)*•a + •b)b ,b) = 〈(a +  ϵ)((•b)*•a + b)•b, false〉
+      move((•a + ϵ)((•b)*•a + •b)b ,b) = 〈(a +  ϵ)((•b)*•a + b)•b, false〉
 
 *)
 
-definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
+definition pmove ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λx:S.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? x (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e).
 
-lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b. 
-  pmove ? x 〈i,b〉 = move ? x i.
+lemma pmove_def : ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀x:S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀b. 
+  a href="cic:/matita/tutorial/chapter9/pmove.def(7)"pmove/a ? x a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? x i.
 // qed.
 
-lemma eq_to_eq_hd: ∀A.∀l1,l2:list A.∀a,b. 
-  a::l1 = b::l2 → a = b.
+lemma eq_to_eq_hd: ∀A.∀l1,l2:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a A.∀a,b. 
+  aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/al1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a ba title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/al2 → a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/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|.
+(* Obviously, a move does not change the carrier of the item, as one can easily 
+prove by induction on the item. *)
+
+lemma same_kernel: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a:S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? a i)a title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/a.
 #S #a #i elim i //
-  [#i1 #i2 #H1 #H2 >move_cat >erase_odot //
-  |#i1 #i2 #H1 #H2 >move_plus whd in ⊢ (??%%); // 
+  [#i1 #i2 #H1 #H2 >a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"move_cat/a >a href="cic:/matita/tutorial/chapter8/erase_odot.def(7)"erase_odot/a //
+  |#i1 #i2 #H1 #H2 >a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"move_plus/a whd in ⊢ (??%%); // 
   ]
 qed.
 
+(* Here is our first, major result, stating the correctness of the
+move operation. The proof is a simple induction on i. *)
+
 theorem move_ok:
- ∀S:DeqSet.∀a:S.∀i:pitem S.∀w: word S. 
-   \sem{move ? a i} w ↔ \sem{i} (a::w).
+ ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a:S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀w: a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S. 
+   a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? a ia title="in_prl" href="cic:/fakeuri.def(1)"}/a w a title="iff" href="cic:/fakeuri.def(1)"↔/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a (aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aw).
 #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 //]
+  [normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/
+  |normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/
+  |normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/
+  |normalize #x #w cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ax)) #H >H normalize
+    [>(\P H) % [* // #bot @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a //| #H1 destruct /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/]
+    |% [@a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a |#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
+  |#i1 #i2 #HI1 #HI2 #w >a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"move_cat/a
+   @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a[|@a href="cic:/matita/tutorial/chapter8/sem_odot.def(13)"sem_odot/a] >a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"same_kernel/a >a href="cic:/matita/tutorial/chapter7/sem_cat_w.def(8)"sem_cat_w/a
+   @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a[||@(a href="cic:/matita/basics/logic/iff_or_l.def(2)"iff_or_l/a … (HI2 w))] @a href="cic:/matita/basics/logic/iff_or_r.def(2)"iff_or_r/a 
+   @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a[||@a href="cic:/matita/basics/logic/iff_sym.def(2)"iff_sym/a @a href="cic:/matita/tutorial/chapter6/deriv_middot.def(5)"deriv_middot/a //]
+   @a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a @HI1
+  |#i1 #i2 #HI1 #HI2 #w >(a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"sem_plus/a S i1 i2) >a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"move_plus/a >a href="cic:/matita/tutorial/chapter7/sem_plus_w.def(8)"sem_plus_w/a 
+   @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a[|@a href="cic:/matita/tutorial/chapter8/sem_oplus.def(9)"sem_oplus/a] 
+   @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a[|@a href="cic:/matita/basics/logic/iff_or_l.def(2)"iff_or_l/a [|@HI2]| @a href="cic:/matita/basics/logic/iff_or_r.def(2)"iff_or_r/a //]
+  |#i1 #HI1 #w >a href="cic:/matita/tutorial/chapter9/move_star.def(7)"move_star/a 
+   @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a[|@a href="cic:/matita/tutorial/chapter8/sem_ostar.def(13)"sem_ostar/a] >a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"same_kernel/a >a href="cic:/matita/tutorial/chapter7/sem_star_w.def(8)"sem_star_w/a 
+   @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a[||@a href="cic:/matita/basics/logic/iff_sym.def(2)"iff_sym/a @a href="cic:/matita/tutorial/chapter6/deriv_middot.def(5)"deriv_middot/a //]
+   @a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a @HI1
   ]
 qed.
     
+(* The move operation is generalized to strings in the obvious way. *)
+
 notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
-let rec moves (S : DeqSet) w e on w : pre S ≝
+
+let rec moves (S : a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) w e on w : a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S ≝
  match w with
   [ nil ⇒ e
-  | cons x w' ⇒ w' ↦* (move S x (\fst e))]. 
+  | cons x w' ⇒ w' ↦* (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S x (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e))]. 
 
-lemma moves_empty: ∀S:DeqSet.∀e:pre S. 
-  moves ? [ ] e = e.
+lemma moves_empty: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
+  a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? a title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a e.
 // qed.
 
-lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S. 
-  moves ? (a::w)  e = moves ? w (move S a (\fst e)).
+lemma moves_cons: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a:S.∀w.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
+  a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? (aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aw)  e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w (a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a 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 //
+  a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S (wa title="append" href="cic:/fakeuri.def(1)"@/a(aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a)) e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e)). 
+#S #a #w elim w // #x #tl #Hind #e >a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"moves_cons/a >a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"moves_cons/a //
 qed.
 
-lemma not_epsilon_sem: ∀S:DeqSet.∀a:S.∀w: word S. ∀e:pre S. 
-  iff ((a::w) ∈ e) ((a::w) ∈ \fst e).
+lemma not_epsilon_sem: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a:S.∀w: a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S. ∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
+  a href="cic:/matita/basics/logic/iff.def(1)"iff/a ((aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aw) a title="in_prl mem" href="cic:/fakeuri.def(1)"∈/a e) ((aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aw) a title="in_pl mem" href="cic:/fakeuri.def(1)"∈/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e).
 #S #a #w * #i #b cases b normalize 
-  [% /2/ * // #H destruct |% normalize /2/]
+  [% /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a/span/span/ * // #H destruct |% normalize /span class="autotactic"2span class="autotrace" trace /span/span/]
 qed.
 
-lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
-  |\fst (moves ? w e)| = |\fst e|.
+lemma same_kernel_moves: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀w.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.
+  a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e)a title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea title="forget" href="cic:/fakeuri.def(1)"|/a.
 #S #w elim w //
 qed.
 
-theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S. 
-   (\snd (moves ? w e) = true) ↔ \sem{e} w.
+theorem decidable_sem: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀w: a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S. ∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
+   (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) a title="iff" href="cic:/fakeuri.def(1)"↔/a a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_prl" href="cic:/fakeuri.def(1)"}/a 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
+ [* #i #b >a href="cic:/matita/tutorial/chapter9/moves_empty.def(8)"moves_empty/a cases b % /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter7/true_to_epsilon.def(9)"true_to_epsilon/a/span/span/ #H @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
+ |#a #w1 #Hind #e >a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"moves_cons/a
+  @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a [||@a href="cic:/matita/basics/logic/iff_sym.def(2)"iff_sym/a @a href="cic:/matita/tutorial/chapter9/not_epsilon_sem.def(9)"not_epsilon_sem/a]
+  @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a [||@a href="cic:/matita/tutorial/chapter9/move_ok.def(14)"move_ok/a] @Hind
  ]
 qed.
 
-(************************ pit state ***************************)
-definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉. 
+(* It is now clear that we can build a DFA D_e for e by taking pre as states, 
+and move as transition function; the initial state is •(e) and a state 〈i,b〉 is 
+final if and only if b is true. The fact that states in D_e are finite is obvious: 
+in fact, their cardinality is at most 2^{n+1} where n is the number of symbols in 
+e. This is one of the advantages of pointed regular expressions w.r.t. derivatives, 
+whose finite nature only holds after a suitable quotient.
+
+Let us discuss a couple of examples.
+
+bExample/b. 
+Below is the DFA associated with the regular expression (ac+bc)*.
+
+img src="http://www.cs.unibo.it/~asperti/FIGURES/acUbc.gif" alt="DFA for (ac+bc)"  
+
+The graphical description of the automaton is the traditional one, with nodes for 
+states and labelled arcs for transitions. Unreachable states are not shown.
+Final states are emphasized by a double circle: since a state 〈e,b〉 is final if and 
+only if b is true, we may just label nodes with the item.
+The automaton is not minimal: it is easy to see that the two states corresponding to 
+the items (a•c +bc)* and (ac+b•c)* are equivalent (a way to prove it is to observe 
+that they define the same language!). In fact, an important property of pres e is that 
+each state has a clear semantics, given in terms of the specification e and not of the 
+behaviour of the automaton. As a consequence, the construction of the automaton is not 
+only direct, but also extremely intuitive and locally verifiable. 
+
+Let us consider a more complex case.
+
+bExample/b. 
+Starting form the regular expression (a+ϵ)(b*a + b)b, we obtain the following automaton.
 
-let rec occur (S: DeqSet) (i: re S) on i ≝  
+img src="http://www.cs.unibo.it/~asperti/FIGURES/automaton.gif" alt="DFA for (a+ϵ)(b*a + b)b" 
+
+Remarkably, this DFA is minimal, testifying the small number of states produced by our 
+technique (the pair of states 6-8 and 7-9 differ for the fact that 6 and 7 
+are final, while 8 and 9 are not). 
+
+
+h2Move to pit/h2. 
+
+We conclude this chapter with a few properties of the move opertions in relation
+with the pit state. *)
+
+definition pit_pre ≝ λS.λi.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/a), a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. 
+
+(* The following function compute the list of characters occurring in a given
+item i. *)
+
+let rec occur (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (i: a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a 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) 
+  [ z ⇒ a title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/a
+  | e ⇒ a title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/a
+  | s y ⇒ ya title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a
+  | o e1 e2 ⇒ a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (occur S e1) (occur S e2) 
+  | c e1 e2 ⇒ a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (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.
+(* If a symbol a does not occur in i, then move(i,a) gets to the
+pit state. *)
+
+lemma not_occur_to_pit: ∀S,a.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"occur/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/a)) a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a →
+  a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a i  a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"pit_pre/a 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 // 
+  [#x normalize cases (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ax) normalize // #H @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
+  |#i1 #i2 #Hind1 #Hind2 #H >a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"move_cat/a 
+   >Hind1 [2:@(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … H) #H1 @a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"sublist_unique_append_l1/a //]
+   >Hind2 [2:@(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … H) #H1 @a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"sublist_unique_append_l2/a //] //
+  |#i1 #i2 #Hind1 #Hind2 #H >a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"move_plus/a 
+   >Hind1 [2:@(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … H) #H1 @a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"sublist_unique_append_l1/a //]
+   >Hind2 [2:@(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … H) #H1 @a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"sublist_unique_append_l2/a //] //
+  |#i #Hind #H >a href="cic:/matita/tutorial/chapter9/move_star.def(7)"move_star/a >Hind // 
   ]
 qed.
 
-lemma move_pit: ∀S,a,i. move S a (\fst (pit_pre S i)) = pit_pre S i.
+(* We cannot escape form the pit state. *)
+
+lemma move_pit: ∀S,a,i. a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"pit_pre/a S i)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"pit_pre/a 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 //
+  [#i1 #i2 #Hind1 #Hind2 >a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"move_cat/a >Hind1 >Hind2 // 
+  |#i1 #i2 #Hind1 #Hind2 >a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"move_plus/a >Hind1 >Hind2 // 
+  |#i #Hind >a href="cic:/matita/tutorial/chapter9/move_star.def(7)"move_star/a >Hind //
   ]
 qed. 
 
-lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
+lemma moves_pit: ∀S,w,i. a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w (a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"pit_pre/a S i) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"pit_pre/a 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).
+(* If any character in w does not occur in i, then moves(i,w) gets
+to the pit state. *)
+
+lemma to_pit: ∀S,w,e. a title="logical not" href="cic:/fakeuri.def(1)"¬/a a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S w (a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"occur/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea title="forget" href="cic:/fakeuri.def(1)"|/a)) →
+ a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"pit_pre/a S (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a 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)
+  [#e * #H @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a @H normalize #a #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
+  |#a #tl #Hind #e #H cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"occur/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea title="forget" href="cic:/fakeuri.def(1)"|/a))))
+    [#Htrue >a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"moves_cons/a whd in ⊢ (???%); <(a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"same_kernel/a … a) 
+     @Hind >a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"same_kernel/a @(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … H) #H1 #b #memb cases (a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a … memb)
       [#H2 >(\P H2) // |#H2 @H1 //]
-    |#Hfalse >moves_cons >not_occur_to_pit // >Hfalse /2/ 
+    |#Hfalse >a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"moves_cons/a >a href="cic:/matita/tutorial/chapter9/not_occur_to_pit.def(8)"not_occur_to_pit/a // >Hfalse /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/bool/eqnot_to_noteq.def(4)"eqnot_to_noteq/a/span/span/ 
     ]
   ]
 qed.
\ No newline at end of file