X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter8.ma;h=0e3293c1e79186894105304f7d6d3b9eefe45e25;hb=0bcf2dc1a27e38cb6cd3d44eb838d652926841e0;hp=dc7474e21c42c2cfcc951ede34655b4699d1fe0c;hpb=df8e8b840c36a1a789ec7bebc47c3cc8aca4f663;p=helm.git diff --git a/weblib/tutorial/chapter8.ma b/weblib/tutorial/chapter8.ma index dc7474e21..0e3293c1e 100644 --- a/weblib/tutorial/chapter8.ma +++ b/weblib/tutorial/chapter8.ma @@ -1,485 +1,374 @@ -include "re.ma". -include "basics/listb.ma". - -let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre 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)^⊛ ]. - -lemma move_plus: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S. - move S x (i1 + i2) = (move ? x i1) ⊕ (move ? x i2). +(* +h1Broadcasting points/h1 +Intuitively, a regular expression e must be understood as a pointed expression with a single +point in front of it. Since however we only allow points before symbols, we must broadcast +this initial point inside e traversing all nullable subexpressions, that essentially corresponds +to the ϵ-closure operation on automata. We use the notation •(_) to denote such an operation; +its definition is the expected one: let us start discussing an example. + +bExample/b +Let us broadcast a point inside (a + ϵ)(b*a + b)b. We start working in parallel on the +first occurrence of a (where the point stops), and on ϵ that gets traversed. We have hence +reached the end of a + ϵ and we must pursue broadcasting inside (b*a + b)b. Again, we work in +parallel on the two additive subterms b^*a and b; the first point is allowed to both enter the +star, and to traverse it, stopping in front of a; the second point just stops in front of b. +No point reached that end of b^*a + b hence no further propagation is possible. In conclusion: + •((a + ϵ)(b^*a + b)b) = 〈(•a + ϵ)((•b)^*•a + •b)b, false〉 +*) + +include "tutorial/chapter7.ma". + +(* Broadcasting a point inside an item generates a pre, since the point could possibly reach +the end of the expression. +Broadcasting inside a i1+i2 amounts to broadcast in parallel inside i1 and i2. +If we define + 〈i1,b1〉 ⊕ 〈i2,b2〉 = 〈i1 + i2, b1∨ b2〉 +then, we just have •(i1+i2) = •(i1)⊕ •(i2). +*) + +img class="anchor" src="icons/tick.png" id="lo"definition lo ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λa,b:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a a a title="pitem or" href="cic:/fakeuri.def(1)"+/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a b,a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a a a title="boolean or" href="cic:/fakeuri.def(1)"∨/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. +notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}. +interpretation "oplus" 'oplus a b = (lo ? a b). + +img class="anchor" src="icons/tick.png" id="lo_def"lemma lo_def: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀b1,b2. a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,b1a title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="oplus" href="cic:/fakeuri.def(1)"⊕/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai2,b2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1a title="pitem or" href="cic:/fakeuri.def(1)"+/ai2,b1a title="boolean or" href="cic:/fakeuri.def(1)"∨/ab2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. // qed. -lemma move_cat: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S. - move S x (i1 · i2) = (move ? x i1) ⊙ (move ? x i2). -// qed. +(* +Concatenation is a bit more complex. In order to broadcast a point inside i1 · i2 +we should start broadcasting it inside i1 and then proceed into i2 if and only if a +point reached the end of i1. This suggests to define •(i1 · i2) as •(i1) ▹ i2, where +e ▹ i is a general operation of concatenation between a pre and an item, defined by +cases on the boolean in e: + + 〈i1,true〉 ▹ i2 = i1 ◃ •(i_2) + 〈i1,false〉 ▹ i2 = i1 · i2 +In turn, ◃ says how to concatenate an item with a pre, that is however extremely simple: + i1 ◃ 〈i1,b〉 = 〈i_1 · i2, b〉 +Let us come to the formalized definitions: +*) + +img class="anchor" src="icons/tick.png" id="pre_concat_r"definition pre_concat_r ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λi:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + match e with [ mk_Prod i1 b ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i1, ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a]. + +notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}. +interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e). -lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S. - move S x i^* = (move ? x i)^⊛. -// qed. +img class="anchor" src="icons/tick.png" id="eq_to_ex_eq"lemma eq_to_ex_eq: ∀S.∀A,B:a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S → Prop. + A a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a B → A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B. +#S #A #B #H >H #x % // qed. -definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e). +(* The behaviour of ◃ is summarized by the following, easy lemma: *) -lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b. - pmove ? x 〈i,b〉 = move ? x i. -// qed. +img class="anchor" src="icons/tick.png" id="sem_pre_concat_r"lemma sem_pre_concat_r : ∀S,i.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{i a title="pre_concat_r" href="cic:/fakeuri.def(1)"◃/a ea title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/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)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a a title="union" 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. +#S #i * #i1 #b1 cases b1 [2: @a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a //] +>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a /span class="autotactic"2span class="autotrace" trace /span/span/ +qed. + +(* The definition of $•(-)$ (eclose) and ▹ (pre_concat_l) are mutually recursive. +In this situation, a viable alternative that is usually simpler to reason about, +is to abstract one of the two functions with respect to the other. In particular +we abstract pre_concat_l with respect to an input bcast function from items to +pres. *) + +img class="anchor" src="icons/tick.png" id="pre_concat_l"definition pre_concat_l ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λbcast:∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S → a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λe1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λi2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + match e1 with + [ mk_Prod i1 b1 ⇒ match b1 with + [ true ⇒ (i1 a title="pre_concat_r" href="cic:/fakeuri.def(1)"◃/a (bcast ? i2)) + | false ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a + ] + ]. -lemma eq_to_eq_hd: ∀A.∀l1,l2:list A.∀a,b. - a::l1 = b::l2 → a = b. -#A #l1 #l2 #a #b #H destruct // -qed. +notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}. +interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b). -lemma same_kernel: ∀S:DeqSet.∀a:S.∀i:pitem S. - |\fst (move ? a i)| = |i|. -#S #a #i elim i // - [#i1 #i2 #H1 #H2 >move_cat >erase_odot // - |#i1 #i2 #H1 #H2 >move_plus whd in ⊢ (??%%); // - ] -qed. +(* We are ready to give the formal definition of the broadcasting operation. *) -theorem move_ok: - ∀S:DeqSet.∀a:S.∀i:pitem S.∀w: word S. - \sem{move ? a i} w ↔ \sem{i} (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 //] - ] - |#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 - ] -qed. - -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 - | cons x w' ⇒ w' ↦* (move S x (\fst e))]. - -lemma moves_empty: ∀S:DeqSet.∀e:pre S. - moves ? [ ] e = e. +notation "•" non associative with precedence 60 for @{eclose ?}. + +img class="anchor" src="icons/tick.png" id="eclose"let rec eclose (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (i: a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) on i : a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S ≝ + match i with + [ pz ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"pz/a ?, 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(0,1,0)"true/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a + | ps x ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem pp" href="cic:/fakeuri.def(1)"`/aa title="pitem pp" href="cic:/fakeuri.def(1)"./ax, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a + | pp x ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem pp" href="cic:/fakeuri.def(1)"`/aa title="pitem pp" href="cic:/fakeuri.def(1)"./ax, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a + | po i1 i2 ⇒ •i1 a title="oplus" href="cic:/fakeuri.def(1)"⊕/a •i2 + | pc i1 i2 ⇒ •i1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2 + | pk i ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (•i))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a]. + +notation "• x" non associative with precedence 60 for @{'eclose $x}. +interpretation "eclose" 'eclose x = (eclose ? x). + +(* Here are a few simple properties of ▹ and •(-) *) + +img class="anchor" src="icons/tick.png" id="pcl_true"lemma pcl_true : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a i1 a title="pre_concat_r" href="cic:/fakeuri.def(1)"◃/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2). // qed. -lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S. - moves ? (a::w) e = moves ? w (move S a (\fst e)). +img class="anchor" src="icons/tick.png" id="pcl_true_bis"lemma pcl_true_bis : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2)a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. +#S #i1 #i2 normalize cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2) // qed. + +img class="anchor" src="icons/tick.png" id="pcl_false"lemma pcl_false: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. // 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 // -qed. +img class="anchor" src="icons/tick.png" id="eclose_plus"lemma eclose_plus: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="eclose" href="cic:/fakeuri.def(1)"•/a(i1 a title="pitem or" href="cic:/fakeuri.def(1)"+/a i2) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="eclose" href="cic:/fakeuri.def(1)"•/ai1 a title="oplus" href="cic:/fakeuri.def(1)"⊕/a a title="eclose" href="cic:/fakeuri.def(1)"•/ai2. +// qed. -lemma not_epsilon_sem: ∀S:DeqSet.∀a:S.∀w: word S. ∀e:pre S. - iff ((a::w) ∈ e) ((a::w) ∈ \fst e). -#S #a #w * #i #b cases b normalize - [% /2/ * // #H destruct |% normalize /2/] -qed. +img class="anchor" src="icons/tick.png" id="eclose_dot"lemma eclose_dot: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="eclose" href="cic:/fakeuri.def(1)"•/a(i1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="eclose" href="cic:/fakeuri.def(1)"•/ai1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2. +// qed. -lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S. - |\fst (moves ? w e)| = |\fst e|. -#S #w elim w // -qed. +img class="anchor" src="icons/tick.png" id="eclose_star"lemma eclose_star: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="eclose" href="cic:/fakeuri.def(1)"•/aia 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 title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a(a title="eclose" href="cic:/fakeuri.def(1)"•/ai))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. +// qed. -theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S. - (\snd (moves ? w e) = true) ↔ \sem{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 - ] -qed. +(* The definition of •(-) (eclose) can then be lifted from items to pres +in the obvious way. *) -(************************ pit state ***************************) -definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉. - -let rec occur (S: DeqSet) (i: re 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) - | 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. -#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 // +img class="anchor" src="icons/tick.png" id="lift"definition lift ≝ λS.λf:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S →a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + match e with + [ mk_Prod i b ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (f i), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (f i) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a]. + +img class="anchor" src="icons/tick.png" id="preclose"definition preclose ≝ λS. a href="cic:/matita/tutorial/chapter8/lift.def(2)"lift/a S (a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"eclose/a S). +interpretation "preclose" 'eclose x = (preclose ? x). + +(* Obviously, broadcasting does not change the carrier of the item, +as it is easily proved by structural induction. *) + +img class="anchor" src="icons/tick.png" id="erase_bull"lemma erase_bull : ∀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 title="eclose" href="cic:/fakeuri.def(1)"•/ai)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 #i elim i // + [ #i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"erase_dot/a a href="cic:/matita/tutorial/chapter8/eclose_dot.def(5)"eclose_dot/a + cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai1) #i11 #b1 cases b1 // a href="cic:/matita/tutorial/chapter8/pcl_true_bis.def(5)"pcl_true_bis/a // + | #i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter8/eclose_plus.def(5)"eclose_plus/a >(a href="cic:/matita/tutorial/chapter7/erase_plus.def(4)"erase_plus/a … i1) a href="cic:/matita/tutorial/chapter8/eclose_star.def(5)"eclose_star/a >(a href="cic:/matita/tutorial/chapter7/erase_star.def(4)"erase_star/a … i) move_cat >Hind1 >Hind2 // - |#i1 #i2 #Hind1 #Hind2 >move_plus >Hind1 >Hind2 // - |#i #Hind >move_star >Hind // - ] -qed. +(* We are now ready to state the main semantic properties of ⊕, ◃ and •(-): -lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre 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). -#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) - [#H2 >(\P H2) // |#H2 @H1 //] - |#Hfalse >moves_cons >not_occur_to_pit // >Hfalse /2/ - ] +sem_oplus: \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2} +sem_pcl: \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2} +sem_bullet \sem{•i} =1 \sem{i} ∪ \sem{|i|} + +The proof of sem_oplus is straightforward. *) + +img class="anchor" src="icons/tick.png" id="sem_oplus"lemma sem_oplus: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="oplus" href="cic:/fakeuri.def(1)"⊕/a e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a. +#S * #i1 #b1 * #i2 #b2 #w % + [cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace /span/span/ * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ + |cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace /span/span/ * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ ] qed. -(* bisimulation *) -definition cofinal ≝ λS.λp:(pre S)×(pre S). - \snd (\fst p) = \snd (\snd p). - -theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S. - \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉. -#S #e1 #e2 % -[#same_sem #w - cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2)) - [* * // * #H1 #H2 [@sym_eq @H1 //| @H2 //]] - #Hcut @Hcut @iff_trans [|@decidable_sem] - @iff_trans [|@same_sem] @iff_sym @decidable_sem -|#H #w1 @iff_trans [||@decidable_sem] to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //] - >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l2 @H1 //] - // -qed. +sem_pcl_aux: + \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} → + \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}. -lemma equiv_sem_occ: ∀S.∀e1,e2:pre S. -(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉) -→ \sem{e1}=1\sem{e2}. -#S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H -qed. +Then, using the previous result, we prove sem_bullet by induction +on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *) -definition sons ≝ λS:DeqSet.λl:list S.λp:(pre S)×(pre S). - map ?? (λa.〈move S a (\fst (\fst p)),move S a (\fst (\snd p))〉) l. +img class="anchor" src="icons/tick.png" id="LcatE"lemma LcatE : ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a e2a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/ae2a title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_pl" href="cic:/fakeuri.def(1)"}/a. +// qed. -lemma memb_sons: ∀S,l.∀p,q:(pre S)×(pre S). memb ? p (sons ? l q) = true → - ∃a.(move ? a (\fst (\fst q)) = \fst p ∧ - move ? a (\fst (\snd q)) = \snd p). -#S #l elim l [#p #q normalize in ⊢ (%→?); #abs @False_ind /2/] -#a #tl #Hind #p #q #H cases (orb_true_l … H) -H - [#H @(ex_intro … a) >(\P H) /2/ |#H @Hind @H] +img class="anchor" src="icons/tick.png" id="sem_pcl_aux"lemma sem_pcl_aux : ∀S.∀e1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{a title="eclose" href="cic:/fakeuri.def(1)"•/ai2a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/ai2a title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a → + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/ai2a title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a. +#S * #i1 #b1 #i2 cases b1 + [2:#th >a href="cic:/matita/tutorial/chapter8/pcl_false.def(5)"pcl_false/a >a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/ + |#H >a href="cic:/matita/tutorial/chapter8/pcl_true.def(5)"pcl_true/a >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a @(a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a … (a href="cic:/matita/tutorial/chapter8/sem_pre_concat_r.def(10)"sem_pre_concat_r/a …)) + >a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"erase_bull/a @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@(a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a … H)] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a[|@a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"union_comm/a ]] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a ] /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a, a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a/span/span/ + ] +qed. + +img class="anchor" src="icons/tick.png" id="minus_eps_pre_aux"lemma minus_eps_pre_aux: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀A. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a A → a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a (A a title="substraction" href="cic:/fakeuri.def(1)"-/a a title="singleton" href="cic:/fakeuri.def(1)"{/aa title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/aa title="singleton" href="cic:/fakeuri.def(1)"}/a). +#S #e #i #A #seme +@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter7/minus_eps_pre.def(10)"minus_eps_pre/a] +@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter7/minus_eps_item.def(9)"minus_eps_item/a]] +@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/distribute_substract.def(3)"distribute_substract/a] +@a href="cic:/matita/tutorial/chapter4/eqP_substract_r.def(3)"eqP_substract_r/a // qed. -definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S. - ∀p:(pre S)×(pre S). memb ? p l = true → cofinal ? p ∧ (sublist ? (sons ? alpha p) l). - -lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S. - is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}. -#S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ -#w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?)) -lapply Hsub @(list_elim_left … w) [//] -#a #w1 #Hind #Hsub >moves_left >moves_left @(proj2 …(Hbisim …(Hind ?))) - [#x #Hx @Hsub @memb_append_l1 // - |cut (memb S a (occ S e1 e2) = true) [@Hsub @memb_append_l2 //] #occa - @(memb_map … occa) +img class="anchor" src="icons/tick.png" id="sem_bull"theorem sem_bull: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a. ∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{a title="eclose" href="cic:/fakeuri.def(1)"•/aia title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a. +#S #e elim e + [#w normalize % [/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | * //] + |/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/ + |#x normalize #w % [ /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 | //]] + |#x normalize #w % [ /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | * // ] + |#i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter8/eclose_dot.def(5)"eclose_dot/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"sem_pcl_aux/a //] >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a + [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a + [|@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@(a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a … IH1)] @a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a]] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] + @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a // + |#i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter8/eclose_plus.def(5)"eclose_plus/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_oplus.def(9)"sem_oplus/a] >a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"sem_plus/a >a href="cic:/matita/tutorial/chapter7/erase_plus.def(4)"erase_plus/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@(a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a … IH2)] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a [|@a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"union_comm/a]] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a/span/span/ + |#i #H >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a >a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"erase_bull/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a [|@a href="cic:/matita/tutorial/chapter8/minus_eps_pre_aux.def(11)"minus_eps_pre_aux/a //]]] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a]] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a >a href="cic:/matita/tutorial/chapter7/erase_star.def(4)"erase_star/a + @a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/star_fix_eps.def(7)"star_fix_eps/a ] qed. -(* the algorithm *) -let rec bisim S l n (frontier,visited: list ?) on n ≝ - match n with - [ O ⇒ 〈false,visited〉 (* assert false *) - | S m ⇒ - match frontier with - [ nil ⇒ 〈true,visited〉 - | cons hd tl ⇒ - if beqb (\snd (\fst hd)) (\snd (\snd hd)) then - bisim S l m (unique_append ? (filter ? (λx.notb (memb ? x (hd::visited))) - (sons S l hd)) tl) (hd::visited) - else 〈false,visited〉 - ] - ]. - -lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?. - bisim S l n frontier visited = - match n with - [ O ⇒ 〈false,visited〉 (* assert false *) - | S m ⇒ - match frontier with - [ nil ⇒ 〈true,visited〉 - | cons hd tl ⇒ - if beqb (\snd (\fst hd)) (\snd (\snd hd)) then - bisim S l m (unique_append ? (filter ? (λx.notb(memb ? x (hd::visited))) - (sons S l hd)) tl) (hd::visited) - else 〈false,visited〉 - ] - ]. -#S #l #n cases n // qed. +(* +h2Blank item/h2 + +As a corollary of theorem sem_bullet, given a regular expression e, we can easily +find an item with the same semantics of $e$: it is enough to get an item (blank e) +having e as carrier and no point, and then broadcast a point in it. The semantics of +(blank e) is obviously the empty language: from the point of view of the automaton, +it corresponds with the pit state. *) + +img class="anchor" src="icons/tick.png" id="blank"let rec blank (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 :a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S ≝ + match i with + [ z ⇒ a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"pz/a ? + | e ⇒ a title="pitem epsilon" href="cic:/fakeuri.def(1)"ϵ/a + | s y ⇒ a title="pitem ps" href="cic:/fakeuri.def(1)"`/ay + | o e1 e2 ⇒ (blank S e1) a title="pitem or" href="cic:/fakeuri.def(1)"+/a (blank S e2) + | c e1 e2 ⇒ (blank S e1) a title="pitem cat" href="cic:/fakeuri.def(1)"·/a (blank S e2) + | k e ⇒ (blank S e)a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a ]. -lemma bisim_never: ∀S,l.∀frontier,visited: list ?. - bisim S l O frontier visited = 〈false,visited〉. -#frontier #visited >unfold_bisim // +img class="anchor" src="icons/tick.png" id="forget_blank"lemma forget_blank: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S.a title="forget" href="cic:/fakeuri.def(1)"|/aa href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a S ea title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a e. +#S #e elim e normalize // qed. -lemma bisim_end: ∀Sig,l,m.∀visited: list ?. - bisim Sig l (S m) [] visited = 〈true,visited〉. -#n #visisted >unfold_bisim // +img class="anchor" src="icons/tick.png" id="sem_blank"lemma sem_blank: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S.a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a S ea title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="empty set" href="cic:/fakeuri.def(1)"∅/a. +#S #e elim e + [1,2:@a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a // + |#s @a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a // + |#e1 #e2 #Hind1 #Hind2 >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"union_empty_r/a … a title="empty set" href="cic:/fakeuri.def(1)"∅/a)] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a[|@Hind2]] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter6/cat_empty_l.def(5)"cat_empty_l/a … ?)] @a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a @Hind1 + |#e1 #e2 #Hind1 #Hind2 >a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"sem_plus/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"union_empty_r/a … a title="empty set" href="cic:/fakeuri.def(1)"∅/a)] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a[|@Hind2]] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a @Hind1 + |#e #Hind >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter6/cat_empty_l.def(5)"cat_empty_l/a … ?)] @a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a @Hind + ] qed. - -lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list ?. -beqb (\snd (\fst p)) (\snd (\snd p)) = true → - bisim Sig l (S m) (p::frontier) visited = - bisim Sig l m (unique_append ? (filter ? (λx.notb(memb ? x (p::visited))) - (sons Sig l p)) frontier) (p::visited). -#Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test // + +img class="anchor" src="icons/tick.png" id="re_embedding"theorem re_embedding: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{a title="eclose" href="cic:/fakeuri.def(1)"•/a(a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a S e)a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_l" href="cic:/fakeuri.def(1)"}/a. +#S #e @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_bull.def(12)"sem_bull/a] >a href="cic:/matita/tutorial/chapter8/forget_blank.def(4)"forget_blank/a +@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter8/sem_blank.def(9)"sem_blank/a]] +@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"union_comm/a] @a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"union_empty_r/a. qed. -lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list ?. -beqb (\snd (\fst p)) (\snd (\snd p)) = false → - bisim Sig l (S m) (p::frontier) visited = 〈false,visited〉. -#Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test // -qed. +(* +h2Lifted Operators/h2 -lemma notb_eq_true_l: ∀b. notb b = true → b = false. -#b cases b normalize // -qed. +Plus and bullet have been already lifted from items to pres. We can now +do a similar job for concatenation ⊙ and Kleene's star ⊛.*) -let rec pitem_enum S (i:re S) on i ≝ - match i with - [ z ⇒ [pz S] - | e ⇒ [pe S] - | s y ⇒ [ps S y; pp S y] - | o i1 i2 ⇒ compose ??? (po S) (pitem_enum S i1) (pitem_enum S i2) - | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2) - | k i ⇒ map ?? (pk S) (pitem_enum S i) - ]. - -lemma pitem_enum_complete : ∀S.∀i:pitem S. - memb (DeqItem S) i (pitem_enum S (|i|)) = true. -#S #i elim i - [1,2:// - |3,4:#c normalize >(\b (refl … c)) // - |5,6:#i1 #i2 #Hind1 #Hind2 @(memb_compose (DeqItem S) (DeqItem S)) // - |#i #Hind @(memb_map (DeqItem S)) // - ] -qed. +img class="anchor" src="icons/tick.png" id="lifted_cat"definition lifted_cat ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a href="cic:/matita/tutorial/chapter8/lift.def(2)"lift/a S (a href="cic:/matita/tutorial/chapter8/pre_concat_l.def(3)"pre_concat_l/a S a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"eclose/a e). -definition pre_enum ≝ λS.λi:re S. - compose ??? (λi,b.〈i,b〉) (pitem_enum S i) [true;false]. - -lemma pre_enum_complete : ∀S.∀e:pre S. - memb ? e (pre_enum S (|\fst e|)) = true. -#S * #i #b @(memb_compose (DeqItem S) DeqBool ? (λi,b.〈i,b〉)) -// cases b normalize // -qed. - -definition space_enum ≝ λS.λi1,i2:re S. - compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i2). +notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}. -lemma space_enum_complete : ∀S.∀e1,e2: pre S. - memb ? 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true. -#S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉)) -// qed. +interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2). -definition all_reachable ≝ λS.λe1,e2:pre S.λl: list ?. -uniqueb ? l = true ∧ - ∀p. memb ? p l = true → - ∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p). - -definition disjoint ≝ λS:DeqSet.λl1,l2. - ∀p:S. memb S p l1 = true → memb S p l2 = false. - -lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} → - ∀l,n.∀frontier,visited:list ((pre S)×(pre S)). - |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→ - all_reachable S e1 e2 visited → - all_reachable S e1 e2 frontier → - disjoint ? frontier visited → - \fst (bisim S l n frontier visited) = true. -#Sig #e1 #e2 #same #l #n elim n - [#frontier #visited #abs * #unique #H @False_ind @(absurd … abs) - @le_to_not_lt @sublist_length // * #e11 #e21 #membp - cut ((|\fst e11| = |\fst e1|) ∧ (|\fst e21| = |\fst e2|)) - [|* #H1 #H2

same_kernel_moves // - |#m #HI * [#visited #vinv #finv >bisim_end //] - #p #front_tl #visited #Hn * #u_visited #r_visited * #u_frontier #r_frontier - #disjoint - cut (∃w.(moves ? w e1 = \fst p) ∧ (moves ? w e2 = \snd p)) - [@(r_frontier … (memb_hd … ))] #rp - cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true) - [cases rp #w * #fstp #sndp (bisim_step_true … ptest) @HI -HI - [(disjoint … (memb_hd …)) whd in ⊢ (??%?); // - |#p1 #H (cases (orb_true_l … H)) [#eqp >(\P eqp) // |@r_visited] - ] - |whd % [@unique_append_unique @(andb_true_r … u_frontier)] - @unique_append_elim #q #H - [cases (memb_sons … (memb_filter_memb … H)) -H - #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(ex_intro … (w1@[a])) - >moves_left >moves_left >mw1 >mw2 >m1 >m2 % // - |@r_frontier @memb_cons // - ] - |@unique_append_elim #q #H - [@injective_notb @(filter_true … H) - |cut ((q==p) = false) - [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @memb_cons //] - cases (andb_true … u_frontier) #notp #_ @(\bf ?) - @(not_to_not … not_eq_true_false) #eqqp H // - ] - ] - ] -qed. - -definition all_true ≝ λS.λl.∀p:(pre S) × (pre S). memb ? p l = true → - (beqb (\snd (\fst p)) (\snd (\snd p)) = true). - -definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S). -memb ? x l1 = true → sublist ? (sons ? l x) l2. - -lemma bisim_complete: - ∀S,l,n.∀frontier,visited,visited_res:list ?. - all_true S visited → - sub_sons S l visited (frontier@visited) → - bisim S l n frontier visited = 〈true,visited_res〉 → - is_bisim S visited_res l ∧ sublist ? (frontier@visited) visited_res. -#S #l #n elim n - [#fron #vis #vis_res #_ #_ >bisim_never #H destruct - |#m #Hind * - [(* case empty frontier *) - -Hind #vis #vis_res #allv #H normalize in ⊢ (%→?); - #H1 destruct % #p - [#membp % [@(\P ?) @allv //| @H //]|#H1 @H1] - |#hd cases (true_or_false (beqb (\snd (\fst hd)) (\snd (\snd hd)))) - [|(* case head of the frontier is non ok (absurd) *) - #H #tl #vis #vis_res #allv >(bisim_step_false … H) #_ #H1 destruct] - (* frontier = hd:: tl and hd is ok *) - #H #tl #visited #visited_res #allv >(bisim_step_true … H) - (* new_visited = hd::visited are all ok *) - cut (all_true S (hd::visited)) - [#p #H1 cases (orb_true_l … H1) [#eqp >(\P eqp) @H |@allv]] - (* we now exploit the induction hypothesis *) - #allh #subH #bisim cases (Hind … allh … bisim) -bisim -Hind - [#H1 #H2 % // #p #membp @H2 -H2 cases (memb_append … membp) -membp #membp - [cases (orb_true_l … membp) -membp #membp - [@memb_append_l2 >(\P membp) @memb_hd - |@memb_append_l1 @sublist_unique_append_l2 // - ] - |@memb_append_l2 @memb_cons // - ] - |(* the only thing left to prove is the sub_sons invariant *) - #x #membx cases (orb_true_l … membx) - [(* case x = hd *) - #eqhdx <(\P eqhdx) #xa #membxa - (* xa is a son of x; we must distinguish the case xa - was already visited form the case xa is new *) - cases (true_or_false … (memb ? xa (x::visited))) - [(* xa visited - trivial *) #membxa @memb_append_l2 // - |(* xa new *) #membxa @memb_append_l1 @sublist_unique_append_l1 @memb_filter_l - [>membxa //|//] - ] - |(* case x in visited *) - #H1 #xa #membxa cases (memb_append … (subH x … H1 … membxa)) - [#H2 (cases (orb_true_l … H2)) - [#H3 @memb_append_l2 <(\P H3) @memb_hd - |#H3 @memb_append_l1 @sublist_unique_append_l2 @H3 - ] - |#H2 @memb_append_l2 @memb_cons @H2 - ] - ] - ] - ] +img class="anchor" src="icons/tick.png" id="odot_true_b"lemma odot_true_b : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀b. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai2,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2)),a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. +#S #i1 #i2 #b normalize in ⊢ (??%?); cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2) // qed. -definition equiv ≝ λSig.λre1,re2:re Sig. - let e1 ≝ •(blank ? re1) in - let e2 ≝ •(blank ? re2) in - let n ≝ S (length ? (space_enum Sig (|\fst e1|) (|\fst e2|))) in - let sig ≝ (occ Sig e1 e2) in - (bisim ? sig n [〈e1,e2〉] []). - -theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig. - \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}. -#Sig #re1 #re2 % - [#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding] - cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉) - [(memb_single … H) @(ex_intro … ϵ) /2/ - |#p #_ normalize // - ] - ] +img class="anchor" src="icons/tick.png" id="odot_false_b"lemma odot_false_b : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀b. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai2,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2 ,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. +// qed. - -lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m. -#n #m % [@eqbnat_true_to_eq | @eq_to_eqbnat_true] + +img class="anchor" src="icons/tick.png" id="erase_odot"lemma erase_odot:∀S.∀e1,e2: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 (e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a e2)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 e1a title="forget" href="cic:/fakeuri.def(1)"|/a a title="re cat" 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 e2a title="forget" href="cic:/fakeuri.def(1)"|/a). +#S * #i1 * * #i2 #b2 // >a href="cic:/matita/tutorial/chapter8/odot_true_b.def(6)"odot_true_b/a >a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"erase_dot/a // qed. -definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true. +(* Let us come to the star operation: *) -definition a ≝ s DeqNat O. -definition b ≝ s DeqNat (S O). -definition c ≝ s DeqNat (S (S O)). - -definition exp1 ≝ ((a·b)^*·a). -definition exp2 ≝ a·(b·a)^*. -definition exp4 ≝ (b·a)^*. - -definition exp6 ≝ a·(a ·a ·b^* + b^* ). -definition exp7 ≝ a · a^* · b^*. - -definition exp8 ≝ a·a·a·a·a·a·a·a·(a^* ). -definition exp9 ≝ (a·a·a + a·a·a·a·a)^*. - -example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true. -normalize // qed. +img class="anchor" src="icons/tick.png" id="lk"definition lk ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + match e with + [ mk_Prod i1 b1 ⇒ + match b1 with + [true ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"eclose/a ? i1))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a, a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a + |false ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" 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 + ] + ]. +(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*) +interpretation "lk" 'lk a = (lk ? a). +notation "a^⊛" non associative with precedence 90 for @{'lk $a}. +img class="anchor" src="icons/tick.png" id="ostar_true"lemma ostar_true: ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a, a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. +// qed. +img class="anchor" src="icons/tick.png" id="ostar_false"lemma ostar_false: ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aia title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" 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. +// qed. + +img class="anchor" src="icons/tick.png" id="erase_ostar"lemma erase_ostar: ∀S.∀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 (ea title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a)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)"|/aa title="re star" href="cic:/fakeuri.def(1)"^/aa title="re star" href="cic:/fakeuri.def(1)"*/a. +#S * #i * // qed. + +img class="anchor" src="icons/tick.png" id="sem_odot_true"lemma sem_odot_true: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a ia title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="singleton" href="cic:/fakeuri.def(1)"{/a a title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/a a title="singleton" href="cic:/fakeuri.def(1)"}/a. +#S #e1 #i +cut (e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a(e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) [//] +#H >H cases (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i) #i1 #b1 cases b1 + [>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] + @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a/span/span/ + |/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/ + ] +qed. +img class="anchor" src="icons/tick.png" id="eq_odot_false"lemma eq_odot_false: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i. + e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i. +#S #e1 #i +cut (e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a(e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i) a title="boolean or" 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) [//] +cases (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i) #i1 #b1 cases b1 #H @H +qed. +(* We conclude this section with the proof of the main semantic properties +of ⊙ and ⊛. *) +img class="anchor" src="icons/tick.png" id="sem_odot"lemma sem_odot: + ∀S.∀e1,e2: a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/aa title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a. +#S #e1 * #i2 * + [>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_odot_true.def(10)"sem_odot_true/a] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a @a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"sem_pcl_aux/a // + |>a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter8/eq_odot_false.def(6)"eq_odot_false/a @a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"sem_pcl_aux/a // + ] +qed. +img class="anchor" src="icons/tick.png" id="sem_ostar"theorem sem_ostar: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/aa title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/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)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/aa title="star lang" href="cic:/fakeuri.def(1)"^/aa title="star lang" href="cic:/fakeuri.def(1)"*/a. +#S * #i #b cases b + [>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a >a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"erase_bull/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a[|@a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a [|@a href="cic:/matita/tutorial/chapter8/minus_eps_pre_aux.def(11)"minus_eps_pre_aux/a //]]] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a]] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a] + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a + @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/epsilon_cat_l.def(5)"epsilon_cat_l/a] @a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/star_fix_eps.def(7)"star_fix_eps/a + |>a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/ + ] +qed. \ No newline at end of file