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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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17 inductive re (S: DeqSet) : Type[0] ≝
21 | c: re S → re S → re S
22 | o: re S → re S → re S
25 interpretation "re epsilon" 'epsilon = (e ?).
26 interpretation "re or" 'plus a b = (o ? a b).
27 interpretation "re cat" 'middot a b = (c ? a b).
28 interpretation "re star" 'star a = (k ? a).
30 notation < "a" non associative with precedence 90 for @{ 'ps $a}.
31 notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
32 interpretation "atom" 'ps a = (s ? a).
34 notation "`∅" non associative with precedence 90 for @{ 'empty }.
35 interpretation "empty" 'empty = (z ?).
37 let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝
42 | c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
43 | o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
44 | k r1 ⇒ (in_l ? r1) ^*].
46 notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
47 interpretation "in_l" 'in_l E = (in_l ? E).
48 interpretation "in_l mem" 'mem w l = (in_l ? l w).
50 lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
55 inductive pitem (S: DeqSet) : Type[0] ≝
60 | pc: pitem S → pitem S → pitem S
61 | po: pitem S → pitem S → pitem S
62 | pk: pitem S → pitem S.
64 definition pre ≝ λS.pitem S × bool.
66 interpretation "pitem star" 'star a = (pk ? a).
67 interpretation "pitem or" 'plus a b = (po ? a b).
68 interpretation "pitem cat" 'middot a b = (pc ? a b).
69 notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
70 notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
71 interpretation "pitem pp" 'pp a = (pp ? a).
72 interpretation "pitem ps" 'ps a = (ps ? a).
73 interpretation "pitem epsilon" 'epsilon = (pe ?).
74 interpretation "pitem empty" 'empty = (pz ?).
76 let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
82 | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
83 | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
84 | pk E ⇒ (forget ? E)^* ].
86 (* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
87 interpretation "forget" 'norm a = (forget ? a).
89 lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
92 lemma erase_plus : ∀S.∀i1,i2:pitem S.
93 |i1 + i2| = |i1| + |i2|.
96 lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
99 (* boolean equality *)
100 let rec beqitem S (i1,i2: pitem S) on i1 ≝
102 [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
103 | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
104 | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
105 | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
106 | po i11 i12 ⇒ match i2 with
107 [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
109 | pc i11 i12 ⇒ match i2 with
110 [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
112 | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
115 lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
117 [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
118 |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
119 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
120 [>(\P H) // | @(\b (refl …))]
121 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
122 [>(\P H) // | @(\b (refl …))]
123 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
124 normalize #H destruct
125 [cases (true_or_false (beqitem S i11 i21)) #H1
126 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
127 |>H1 in H; normalize #abs @False_ind /2/
129 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
131 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
132 normalize #H destruct
133 [cases (true_or_false (beqitem S i11 i21)) #H1
134 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
135 |>H1 in H; normalize #abs @False_ind /2/
137 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
139 |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
140 normalize #H destruct
141 [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
145 definition DeqItem ≝ λS.
146 mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
148 unification hint 0 ≔ S;
149 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
150 (* ---------------------------------------- *) ⊢
153 unification hint 0 ≔ S,i1,i2;
154 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
155 (* ---------------------------------------- *) ⊢
156 beqitem S i1 i2 ≡ eqb X i1 i2.
160 let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
166 | pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
167 | po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
168 | pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
170 interpretation "in_pl" 'in_l E = (in_pl ? E).
171 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
173 definition in_prl ≝ λS : DeqSet.λp:pre S.
174 if (\snd p) then \sem{\fst p} ∪ {ϵ} else \sem{\fst p}.
176 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
177 interpretation "in_prl" 'in_l E = (in_prl ? E).
179 lemma sem_pre_true : ∀S.∀i:pitem S.
180 \sem{〈i,true〉} = \sem{i} ∪ {ϵ}.
183 lemma sem_pre_false : ∀S.∀i:pitem S.
184 \sem{〈i,false〉} = \sem{i}.
187 lemma sem_cat: ∀S.∀i1,i2:pitem S.
188 \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
191 lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
192 \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
195 lemma sem_plus: ∀S.∀i1,i2:pitem S.
196 \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
199 lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w.
200 \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
203 lemma sem_star : ∀S.∀i:pitem S.
204 \sem{i^*} = \sem{i} · \sem{|i|}^*.
207 lemma sem_star_w : ∀S.∀i:pitem S.∀w.
208 \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
211 lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
212 #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
214 lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ (ϵ ∈ e).
215 #S #e elim e normalize /2/
216 [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
217 >(append_eq_nil …H…) /2/
218 |#r1 #r2 #n1 #n2 % * /2/
219 |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
224 lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
225 #S * #i #b cases b // normalize #H @False_ind /2/
228 lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → ϵ ∈ e.
229 #S * #i #b #btrue normalize in btrue; >btrue %2 //
232 lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
234 [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
239 lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
241 [>sem_pre_true normalize in ⊢ (??%?); #w %
242 [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
243 |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
247 definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
248 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
249 interpretation "oplus" 'oplus a b = (lo ? a b).
251 lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
254 definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
255 match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
257 notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
258 interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
260 lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop.
262 #S #A #B #H >H /2/ qed.
264 lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
265 \sem{i ◃ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
266 #S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //]
267 >sem_pre_true >sem_cat >sem_pre_true /2/
270 definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
272 [ mk_Prod i1 b1 ⇒ match b1 with
273 [ true ⇒ (i1 ◃ (bcast ? i2))
274 | false ⇒ 〈i1 · i2,false〉
278 notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
279 interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
281 notation "•" non associative with precedence 60 for @{eclose ?}.
283 let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
287 | ps x ⇒ 〈 `.x, false〉
288 | pp x ⇒ 〈 `.x, false 〉
289 | po i1 i2 ⇒ •i1 ⊕ •i2
290 | pc i1 i2 ⇒ •i1 ▹ i2
291 | pk i ⇒ 〈(\fst (•i))^*,true〉].
293 notation "• x" non associative with precedence 60 for @{'eclose $x}.
294 interpretation "eclose" 'eclose x = (eclose ? x).
296 lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
297 •(i1 + i2) = •i1 ⊕ •i2.
300 lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
301 •(i1 · i2) = •i1 ▹ i2.
304 lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
305 •i^* = 〈(\fst(•i))^*,true〉.
308 definition lift ≝ λS.λf:pitem S →pre S.λe:pre S.
310 [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
312 definition preclose ≝ λS. lift S (eclose S).
313 interpretation "preclose" 'eclose x = (preclose ? x).
316 lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
317 \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
318 #S * #i1 #b1 * #i2 #b2 #w %
319 [cases b1 cases b2 normalize /2/ * /3/ * /3/
320 |cases b1 cases b2 normalize /2/ * /3/ * /3/
326 〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
329 lemma odot_true_bis :
331 〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
332 #S #i1 #i2 normalize cases (•i2) // qed.
336 〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
339 lemma LcatE : ∀S.∀e1,e2:pitem S.
340 \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
343 lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
345 [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
346 cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
347 | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
348 cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //
349 | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
354 lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
355 \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
359 (* theorem 16: 1 → 3 *)
360 lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
361 \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
362 \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
363 #S * #i1 #b1 #i2 cases b1
364 [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
365 |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
366 >erase_bull @eqP_trans [|@(eqP_union_l … H)]
367 @eqP_trans [|@eqP_union_l[|@union_comm ]]
368 @eqP_trans [|@eqP_sym @union_assoc ] /3/
372 lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A.
373 \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
375 @eqP_trans [|@minus_eps_pre]
376 @eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
377 @eqP_trans [||@distribute_substract]
382 theorem sem_bull: ∀S:DeqSet. ∀i:pitem S. \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
384 [#w normalize % [/2/ | * //]
386 |#x normalize #w % [ /2/ | * [@False_ind | //]]
387 |#x normalize #w % [ /2/ | * // ]
388 |#i1 #i2 #IH1 #IH2 >eclose_dot
389 @eqP_trans [|@odot_dot_aux //] >sem_cat
392 [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
393 @eqP_trans [|@union_assoc]
394 @eqP_trans [||@eqP_sym @union_assoc]
396 |#i1 #i2 #IH1 #IH2 >eclose_plus
397 @eqP_trans [|@sem_oplus] >sem_plus >erase_plus
398 @eqP_trans [|@(eqP_union_l … IH2)]
399 @eqP_trans [|@eqP_sym @union_assoc]
400 @eqP_trans [||@union_assoc] @eqP_union_r
401 @eqP_trans [||@eqP_sym @union_assoc]
402 @eqP_trans [||@eqP_union_l [|@union_comm]]
403 @eqP_trans [||@union_assoc] /2/
404 |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
405 @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
406 @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
407 @eqP_trans [|@union_assoc] @eqP_union_l >erase_star
408 @eqP_sym @star_fix_eps
413 let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
418 | o e1 e2 ⇒ (blank S e1) + (blank S e2)
419 | c e1 e2 ⇒ (blank S e1) · (blank S e2)
420 | k e ⇒ (blank S e)^* ].
422 lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
423 #S #e elim e normalize //
426 lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
430 |#e1 #e2 #Hind1 #Hind2 >sem_cat
431 @eqP_trans [||@(union_empty_r … ∅)]
432 @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
433 @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
434 |#e1 #e2 #Hind1 #Hind2 >sem_plus
435 @eqP_trans [||@(union_empty_r … ∅)]
436 @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
438 @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
442 theorem re_embedding: ∀S.∀e:re S.
443 \sem{•(blank S e)} =1 \sem{e}.
444 #S #e @eqP_trans [|@sem_bull] >forget_blank
445 @eqP_trans [|@eqP_union_r [|@sem_blank]]
446 @eqP_trans [|@union_comm] @union_empty_r.
449 (* lefted operations *)
450 definition lifted_cat ≝ λS:DeqSet.λe:pre S.
451 lift S (pre_concat_l S eclose e).
453 notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
455 interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
457 lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b.
458 〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
459 #S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) //
462 lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
463 〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
467 lemma erase_odot:∀S.∀e1,e2:pre S.
468 |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
469 #S * #i1 * * #i2 #b2 // >odot_true_b //
472 definition lk ≝ λS:DeqSet.λe:pre S.
476 [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
477 |false ⇒ 〈i1^*,false〉
481 (* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
482 interpretation "lk" 'lk a = (lk ? a).
483 notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
486 lemma ostar_true: ∀S.∀i:pitem S.
487 〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
490 lemma ostar_false: ∀S.∀i:pitem S.
491 〈i,false〉^⊛ = 〈i^*, false〉.
494 lemma erase_ostar: ∀S.∀e:pre S.
495 |\fst (e^⊛)| = |\fst e|^*.
498 lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
499 \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
501 cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
502 #H >H cases (e1 ▹ i) #i1 #b1 cases b1
503 [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
509 lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i.
510 e1 ⊙ 〈i,false〉 = e1 ▹ i.
512 cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
513 cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
517 ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
520 @eqP_trans [|@sem_odot_true]
521 @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
522 |>sem_pre_false >eq_odot_false @odot_dot_aux //
527 theorem sem_ostar: ∀S.∀e:pre S.
528 \sem{e^⊛} =1 \sem{e} · \sem{|\fst e|}^*.
530 [>sem_pre_true >sem_pre_true >sem_star >erase_bull
531 @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
532 @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
533 @eqP_trans [||@eqP_sym @distr_cat_r]
534 @eqP_trans [|@union_assoc] @eqP_union_l
535 @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps
536 |>sem_pre_false >sem_pre_false >sem_star /2/