2 \ 5h1
\ 6Broadcasting points
\ 5/h1
\ 6
3 Intuitively, a regular expression e must be understood as a pointed expression with a single
4 point in front of it. Since however we only allow points before symbols, we must broadcast
5 this initial point inside e traversing all nullable subexpressions, that essentially corresponds
6 to the ϵ-closure operation on automata. We use the notation •(_) to denote such an operation;
7 its definition is the expected one: let us start discussing an example.
10 Let us broadcast a point inside (a + ϵ)(b*a + b)b. We start working in parallel on the
11 first occurrence of a (where the point stops), and on ϵ that gets traversed. We have hence
12 reached the end of a + ϵ and we must pursue broadcasting inside (b*a + b)b. Again, we work in
13 parallel on the two additive subterms b^*a and b; the first point is allowed to both enter the
14 star, and to traverse it, stopping in front of a; the second point just stops in front of b.
15 No point reached that end of b^*a + b hence no further propagation is possible. In conclusion:
16 •((a + ϵ)(b^*a + b)b) = 〈(•a + ϵ)((•b)^*•a + •b)b, false〉
19 include "tutorial/chapter7.ma".
21 (* Broadcasting a point inside an item generates a pre, since the point could possibly reach
22 the end of the expression.
23 Broadcasting inside a i1+i2 amounts to broadcast in parallel inside i1 and i2.
25 〈i1,b1〉 ⊕ 〈i2,b2〉 = 〈i1 + i2, b1∨ b2〉
26 then, we just have •(i1+i2) = •(i1)⊕ •(i2).
29 include "tutorial/chapter7.ma".
31 definition lo ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λa,b:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 a
\ 5a title="pitem or" href="cic:/fakeuri.def(1)"
\ 6+
\ 5/a
\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 b,
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6 a
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6 b〉.
32 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
33 interpretation "oplus" 'oplus a b = (lo ? a b).
35 lemma lo_def: ∀S.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.∀b1,b2.
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1,b1〉
\ 5a title="oplus" href="cic:/fakeuri.def(1)"
\ 6⊕
\ 5/a
\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i2,b2〉
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1
\ 5a title="pitem or" href="cic:/fakeuri.def(1)"
\ 6+
\ 5/a
\ 6i2,b1
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6b2〉.
39 Concatenation is a bit more complex. In order to broadcast a point inside i1 · i2
40 we should start broadcasting it inside i1 and then proceed into i2 if and only if a
41 point reached the end of i1. This suggests to define •(i1 · i2) as •(i1) ▹ i2, where
42 e ▹ i is a general operation of concatenation between a pre and an item, defined by
43 cases on the boolean in e:
45 〈i1,true〉 ▹ i2 = i1 ◃ •(i_2)
46 〈i1,false〉 ▹ i2 = i1 · i2
47 In turn, ◃ says how to concatenate an item with a pre, that is however extremely simple:
48 i1 ◃ 〈i1,b〉 = 〈i_1 · i2, b〉
49 Let us come to the formalized definitions:
52 definition pre_concat_r ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λi:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.λe:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
53 match e with [ mk_Prod i1 b ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 i1, b〉].
55 notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
56 interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
58 lemma eq_to_ex_eq: ∀S.∀A,B:
\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"
\ 6word
\ 5/a
\ 6 S → Prop.
59 A
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 B → A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 B.
60 #S #A #B #H >H #x % // qed.
62 (* The behaviour of ◃ is summarized by the following, easy lemma: *)
64 lemma sem_pre_concat_r : ∀S,i.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
65 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i
\ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"
\ 6◃
\ 5/a
\ 6 e}
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i}
\ 5a title="cat lang" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 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|}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e}.
66 #S #i * #i1 #b1 cases b1 [2: @
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6 //]
67 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"
\ 6sem_cat
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5/span
\ 6\ 5/span
\ 6/
70 (* The definition of $•(-)$ (eclose) and ▹ (pre_concat_l) are mutually recursive.
71 In this situation, a viable alternative that is usually simpler to reason about,
72 is to abstract one of the two functions with respect to the other. In particular
73 we abstract pre_concat_l with respect to an input bcast function from items to
76 definition pre_concat_l ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λbcast:∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S →
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.λe1:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.λi2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
78 [ mk_Prod i1 b1 ⇒ match b1 with
79 [ true ⇒ (i1
\ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"
\ 6◃
\ 5/a
\ 6 (bcast ? i2))
80 | false ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 i2,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
84 notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
85 interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
87 (* We are ready to give the formal definition of the broadcasting operation. *)
89 notation "•" non associative with precedence 60 for @{eclose ?}.
91 let rec eclose (S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6) (i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S) on i :
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S ≝
93 [ 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 ?,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6 〉
94 | 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(0,1,0)"
\ 6true
\ 5/a
\ 6 〉
95 | ps x ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"
\ 6`
\ 5/a
\ 6.x,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
96 | pp x ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"
\ 6`
\ 5/a
\ 6.x,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6 〉
97 | po i1 i2 ⇒ •i1
\ 5a title="oplus" href="cic:/fakeuri.def(1)"
\ 6⊕
\ 5/a
\ 6 •i2
98 | pc i1 i2 ⇒ •i1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i2
99 | pk i ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6(
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (•i))
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉].
101 notation "• x" non associative with precedence 60 for @{'eclose $x}.
102 interpretation "eclose" 'eclose x = (eclose ? x).
104 (* Here are a few simple properties of ▹ and •(-) *)
108 〈i1,true〉 ▸ i2 = i1 ◂ (•i2).
113 〈i1,true〉 ▸ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
114 #S #i1 #i2 normalize cases (•i2) // qed.
118 〈i1,false〉 ▸ i2 = 〈i1 · i2, false〉.
121 lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
122 •(i1 + i2) = •i1 ⊕ •i2.
125 lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
126 •(i1 · i2) = •i1 ▹ i2.
129 lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
130 •i^* = 〈(\fst(•i))^*,true〉.
133 (* The definition of •(-) (eclose) can then be lifted from items to pres
134 in the obvious way. *)
136 definition lift ≝ λS.λf:pitem S →pre S.λe:pre S.
138 [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
140 definition preclose ≝ λS. lift S (eclose S).
141 interpretation "preclose" 'eclose x = (preclose ? x).
143 (* Obviously, broadcasting does not change the carrier of the item,
144 as it is easily proved by structural induction. *)
146 lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
148 [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
149 cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
150 | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
151 cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //
152 | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
156 (* We are now ready to state the main semantic properties of $\oplus,
157 \triangleleft$ and $\bullet(-)$:
159 \begin{lstlisting}[language=grafite]
160 lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
161 \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
162 lemma sem_pcl : ∀S.∀e1:pre S.∀i2:pitem S.
163 \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
164 theorem sem_bullet: ∀S:DeqSet. ∀i:pitem S.
165 \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
167 The proof of \verb+sem_oplus+ is straightforward. *)
169 lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
170 \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
171 #S * #i1 #b1 * #i2 #b2 #w %
172 [cases b1 cases b2 normalize /2/ * /3/ * /3/
173 |cases b1 cases b2 normalize /2/ * /3/ * /3/
177 (* For the others, we proceed as follow: we first prove the following
178 auxiliary lemma, that assumes sem_bullet:
180 lemma sem_pcl_aux: ∀S.∀e1:pre S.∀i2:pitem S.
181 \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
182 \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
184 Then, using the previous result, we prove sem_bullet by induction
185 on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *)
187 lemma LcatE : ∀S.∀e1,e2:pitem S.
188 \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
191 lemma sem_pcl_aux : ∀S.∀e1:pre S.∀i2:pitem S.
192 \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
193 \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
194 #S * #i1 #b1 #i2 cases b1
195 [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
196 |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
197 >erase_bull @eqP_trans [|@(eqP_union_l … H)]
198 @eqP_trans [|@eqP_union_l[|@union_comm ]]
199 @eqP_trans [|@eqP_sym @union_assoc ] /3/
203 lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A.
204 \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
206 @eqP_trans [|@minus_eps_pre]
207 @eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
208 @eqP_trans [||@distribute_substract]
213 theorem sem_bull: ∀S:DeqSet. ∀i:pitem S. \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
215 [#w normalize % [/2/ | * //]
217 |#x normalize #w % [ /2/ | * [@False_ind | //]]
218 |#x normalize #w % [ /2/ | * // ]
219 |#i1 #i2 #IH1 #IH2 >eclose_dot
220 @eqP_trans [|@odot_dot_aux //] >sem_cat
223 [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
224 @eqP_trans [|@union_assoc]
225 @eqP_trans [||@eqP_sym @union_assoc]
227 |#i1 #i2 #IH1 #IH2 >eclose_plus
228 @eqP_trans [|@sem_oplus] >sem_plus >erase_plus
229 @eqP_trans [|@(eqP_union_l … IH2)]
230 @eqP_trans [|@eqP_sym @union_assoc]
231 @eqP_trans [||@union_assoc] @eqP_union_r
232 @eqP_trans [||@eqP_sym @union_assoc]
233 @eqP_trans [||@eqP_union_l [|@union_comm]]
234 @eqP_trans [||@union_assoc] /2/
235 |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
236 @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
237 @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
238 @eqP_trans [|@union_assoc] @eqP_union_l >erase_star
239 @eqP_sym @star_fix_eps
244 let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
249 | o e1 e2 ⇒ (blank S e1) + (blank S e2)
250 | c e1 e2 ⇒ (blank S e1) · (blank S e2)
251 | k e ⇒ (blank S e)^* ].
253 lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
254 #S #e elim e normalize //
257 lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
261 |#e1 #e2 #Hind1 #Hind2 >sem_cat
262 @eqP_trans [||@(union_empty_r … ∅)]
263 @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
264 @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
265 |#e1 #e2 #Hind1 #Hind2 >sem_plus
266 @eqP_trans [||@(union_empty_r … ∅)]
267 @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
269 @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
273 theorem re_embedding: ∀S.∀e:re S.
274 \sem{•(blank S e)} =1 \sem{e}.
275 #S #e @eqP_trans [|@sem_bull] >forget_blank
276 @eqP_trans [|@eqP_union_r [|@sem_blank]]
277 @eqP_trans [|@union_comm] @union_empty_r.
280 (* lefted operations *)
281 definition lifted_cat ≝ λS:DeqSet.λe:pre S.
282 lift S (pre_concat_l S eclose e).
284 notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
286 interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
288 lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b.
289 〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
290 #S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) //
293 lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
294 〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
298 lemma erase_odot:∀S.∀e1,e2:pre S.
299 |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
300 #S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot //
303 definition lk ≝ λS:DeqSet.λe:pre S.
307 [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
308 |false ⇒ 〈i1^*,false〉
312 (* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
313 interpretation "lk" 'lk a = (lk ? a).
314 notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
317 lemma ostar_true: ∀S.∀i:pitem S.
318 〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
321 lemma ostar_false: ∀S.∀i:pitem S.
322 〈i,false〉^⊛ = 〈i^*, false〉.
325 lemma erase_ostar: ∀S.∀e:pre S.
326 |\fst (e^⊛)| = |\fst e|^*.
329 lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
330 \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
332 cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
333 #H >H cases (e1 ▹ i) #i1 #b1 cases b1
334 [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
340 lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i.
341 e1 ⊙ 〈i,false〉 = e1 ▹ i.
343 cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
344 cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
348 ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
351 @eqP_trans [|@sem_odot_true]
352 @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
353 |>sem_pre_false >eq_odot_false @odot_dot_aux //
358 theorem sem_ostar: ∀S.∀e:pre S.
359 \sem{e^⊛} =1 \sem{e} · \sem{|\fst e|}^*.
361 [>sem_pre_true >sem_pre_true >sem_star >erase_bull
362 @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
363 @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
364 @eqP_trans [||@eqP_sym @distr_cat_r]
365 @eqP_trans [|@union_assoc] @eqP_union_l
366 @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps
367 |>sem_pre_false >sem_pre_false >sem_star /2/