include "tutorial/chapter8.ma".
-let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
+let rec move (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (E: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on E : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 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 ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 S, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
+ | pe ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
+ | ps y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
+ | pp y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= y 〉
+ | po e1 e2 ⇒ (move ? x e1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (move ? x e2)
+ | pc e1 e2 ⇒ (move ? x e1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (move ? x e2)
+ | pk e ⇒ (move ? x e)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛ ].
-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:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2) \ 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 ? x i1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? 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:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2) \ 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 ? x i1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i2).
// qed.
-lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
- move S x i^* = (move ? x i)^⊛.
+lemma move_star: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 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 ? x i)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛.
// qed.
(*
*)
-definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
+definition pmove ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λx:S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
-lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b.
- pmove ? x 〈i,b〉 = move ? x i.
+lemma pmove_def : ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
+ \ 5a href="cic:/matita/tutorial/chapter9/pmove.def(7)"\ 6pmove\ 5/a\ 6 ? x \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉 \ 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 ? 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:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a,b.
+ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l2 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 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:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 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/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i)| \ 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\ 6i|.
#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 >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter8/erase_odot.def(7)"\ 6erase_odot\ 5/a\ 6 //
+ |#i1 #i2 #H1 #H2 >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 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:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i} w \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6: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 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |normalize #x #w cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x)) #H >H normalize
+ [>(\P H) % [* // #bot @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 //| #H1 destruct /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
+ |% [@\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 |#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 >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6
+ @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6[|@\ 5a href="cic:/matita/tutorial/chapter8/sem_odot.def(13)"\ 6sem_odot\ 5/a\ 6] >\ 5a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"\ 6same_kernel\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_cat_w.def(8)"\ 6sem_cat_w\ 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_or_l.def(2)"\ 6iff_or_l\ 5/a\ 6 … (HI2 w))] @\ 5a href="cic:/matita/basics/logic/iff_or_r.def(2)"\ 6iff_or_r\ 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/chapter6/deriv_middot.def(5)"\ 6deriv_middot\ 5/a\ 6 //]
+ @\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"\ 6cat_ext_l\ 5/a\ 6 @HI1
+ |#i1 #i2 #HI1 #HI2 #w >(\ 5a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"\ 6sem_plus\ 5/a\ 6 S i1 i2) >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_plus_w.def(8)"\ 6sem_plus_w\ 5/a\ 6
+ @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6[|@\ 5a href="cic:/matita/tutorial/chapter8/sem_oplus.def(9)"\ 6sem_oplus\ 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_or_l.def(2)"\ 6iff_or_l\ 5/a\ 6 [|@HI2]| @\ 5a href="cic:/matita/basics/logic/iff_or_r.def(2)"\ 6iff_or_r\ 5/a\ 6 //]
+ |#i1 #HI1 #w >\ 5a href="cic:/matita/tutorial/chapter9/move_star.def(7)"\ 6move_star\ 5/a\ 6
+ @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6[|@\ 5a href="cic:/matita/tutorial/chapter8/sem_ostar.def(13)"\ 6sem_ostar\ 5/a\ 6] >\ 5a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"\ 6same_kernel\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_star_w.def(8)"\ 6sem_star_w\ 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/chapter6/deriv_middot.def(5)"\ 6deriv_middot\ 5/a\ 6 //]
+ @\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"\ 6cat_ext_l\ 5/a\ 6 @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
]
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〉.
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