From 0485e1e1845d054ab1a570c78cd5b6f2992d6909 Mon Sep 17 00:00:00 2001
From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Tue, 6 Mar 2012 12:27:17 +0000
Subject: [PATCH] commit by user andrea

---
 weblib/tutorial/chapter9.ma | 130 ++++++++++++++++++++++++------------
 1 file changed, 87 insertions(+), 43 deletions(-)

diff --git a/weblib/tutorial/chapter9.ma b/weblib/tutorial/chapter9.ma
index b3f46ae04..6126269d8 100644
--- a/weblib/tutorial/chapter9.ma
+++ b/weblib/tutorial/chapter9.ma
@@ -11,26 +11,26 @@ 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 title="Pair construction" href="cic:/fakeuri.def(1)"〈/a 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 〉
+  | 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 〉
+  | 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 〉
+  | 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)"=/a= y 〉
+  | 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)"^/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)"^/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)"^/a⊛.
 // qed.
 
 (*
@@ -52,52 +52,61 @@ while the other point reaches the end of b* and must go back through b*a:
 
 *)
 
-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,b〉 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)":/a:l1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a ba title="cons" href="cic:/fakeuri.def(1)":/a:l2 → 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="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/ai|.
 #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 i} w a title="iff" href="cic:/fakeuri.def(1)"↔/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i} (aa title="cons" href="cic:/fakeuri.def(1)":/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 //]
+  [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)"=/a=x)) #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 ≝
  match w with
   [ nil ⇒ e
@@ -137,6 +146,41 @@ theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S.
  ]
 qed.
 
+(* 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.
+Below is the DFA associated with the regular expression (ac+bc)*.
+
+\includegraphics[width=.8\textwidth]{acUbc.pdf}
+\caption{DFA for $(ac+bc)^*$\label{acUbc}}
+
+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.
+Starting form the regular expression (a+ϵ)(b*a + b)b, we obtain the following automaton.
+
+\includegraphics[width=.8\textwidth]{automaton.pdf}
+\caption{DFA for $(a+\epsilon)(b^*a + b)b$\label{automaton}}
+
+Remarkably, this DFA is minimal, testifying the small number of states produced by our 
+technique (the pair ofstates $6-8$ and $7-9$ differ for the fact that $6$ and $7$ 
+are final, while $8$ and $9$ are not). 
+*) 
+
 (************************ pit state ***************************)
 definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉. 
 
-- 
2.39.2