// qed.
(*
-Example. Let us consider the item
+\ 5b\ 6Example\ 5/b\ 6. Let us consider the item
(•a + ϵ)((•b)*•a + •b)b
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 : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) w e on w : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
match w with
[ nil ⇒ e
- | cons x w' ⇒ w' ↦* (move S x (\fst e))].
+ | cons x w' ⇒ w' ↦* (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e))].
-lemma moves_empty: ∀S:DeqSet.∀e:pre S.
- moves ? [ ] e = e.
+lemma moves_empty: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 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:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w) e \ 5a title="leibnitz's equality" 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 (\ 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 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 //
+ \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S (w\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6a]) e \ 5a title="leibnitz's equality" 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 href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e)).
+#S #a #w elim w // #x #tl #Hind #e >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 //
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:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w) \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e) ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w) \ 5a title="in_pl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
#S #a #w * #i #b cases b normalize
- [% /2/ * // #H destruct |% normalize /2/]
+ [% /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * // #H destruct |% normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/]
qed.
-lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
- |\fst (moves ? w e)| = |\fst e|.
+lemma same_kernel_moves: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 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 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e)| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 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|.
#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:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 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 href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w 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) \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{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
+ [* #i #b >\ 5a href="cic:/matita/tutorial/chapter9/moves_empty.def(8)"\ 6moves_empty\ 5/a\ 6 cases b % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter7/true_to_epsilon.def(9)"\ 6true_to_epsilon\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/\ 5span class="error" title="error location"\ 6\ 5/span\ 6 #H @\ 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/
+ |#a #w1 #Hind #e >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6
+ @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"\ 6iff_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter9/not_epsilon_sem.def(9)"\ 6not_epsilon_sem\ 5/a\ 6]
+ @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter9/move_ok.def(14)"\ 6move_ok\ 5/a\ 6] @Hind
]
qed.
whose finite nature only holds after a suitable quotient.
Let us discuss a couple of examples.
+
+\ 5b\ 6Example\ 5/b\ 6.
Below is the DFA associated with the regular expression (ac+bc)*.
-\includegraphics[width=.8\textwidth]{acUbc.pdf}
-\caption{DFA for $(ac+bc)^*$\label{acUbc}}
+\ 5img src="http://www.cs.unibo.it/~asperti/FIGURES/acUbc.gif" alt="DFA for (ac+bc)"\ 6
The graphical description of the automaton is the traditional one, with nodes for
states and labelled arcs for transitions. Unreachable states are not shown.
only direct, but also extremely intuitive and locally verifiable.
Let us consider a more complex case.
+
+\ 5b\ 6Example\ 5/b\ 6.
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}}
+\ 5img src="http://www.cs.unibo.it/~asperti/FIGURES/automaton.gif" alt="DFA for (a+ϵ)(b*a + b)b"\ 6
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).
-*)
+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).
+
+
+\ 5h2\ 6Move to pit\ 5/h2\ 6.
+
+We conclude this chapter with a few properties of the move opertions in relation
+with the pit state. *).
-(************************ pit state ***************************)
definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉.
+(* The following function compute if a symbol in S occurs in a given
+item i *).
+
let rec occur (S: DeqSet) (i: re S) on i ≝
match i with
[ z ⇒ [ ]
| c e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2)
| k e ⇒ occur S e].
+(* 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:pitem S. memb S a (occur S (|i|)) ≠ true →
move S a i = pit_pre S i.
#S #a #i elim i //
]
qed.
+(* We cannot escape form the pit state. *).
+
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 //
#S #w #i elim w //
qed.
+(* 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. ¬ sublist S w (occur S (|\fst e|)) →
moves S w e = pit_pre S (\fst e).
#S #w elim w