From 0485e1e1845d054ab1a570c78cd5b6f2992d6909 Mon Sep 17 00:00:00 2001 From: matitaweb 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