1 (**************************************************************************)
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 *)
13 (**************************************************************************)
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 let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
95 | pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
96 | po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
97 | pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
99 interpretation "in_pl" 'in_l E = (in_pl ? E).
100 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
102 definition in_prl ≝ λS : DeqSet.λp:pre S.
103 if (\snd p) then \sem{\fst p} ∪ {ϵ} else \sem{\fst p}.
105 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
106 interpretation "in_prl" 'in_l E = (in_prl ? E).
108 lemma sem_pre_true : ∀S.∀i:pitem S.
109 \sem{〈i,true〉} = \sem{i} ∪ {ϵ}.
112 lemma sem_pre_false : ∀S.∀i:pitem S.
113 \sem{〈i,false〉} = \sem{i}.
116 lemma sem_cat: ∀S.∀i1,i2:pitem S.
117 \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
120 lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
121 \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
124 lemma sem_plus: ∀S.∀i1,i2:pitem S.
125 \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
128 lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w.
129 \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
132 lemma sem_star : ∀S.∀i:pitem S.
133 \sem{i^*} = \sem{i} · \sem{|i|}^*.
136 lemma sem_star_w : ∀S.∀i:pitem S.∀w.
137 \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
140 lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
141 #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
143 lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ (ϵ ∈ e).
144 #S #e elim e normalize /2/
145 [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
146 >(append_eq_nil …H…) /2/
147 |#r1 #r2 #n1 #n2 % * /2/
148 |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
153 lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
154 #S * #i #b cases b // normalize #H @False_ind /2/
157 lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → ϵ ∈ e.
158 #S * #i #b #btrue normalize in btrue; >btrue %2 //
161 definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
162 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
163 interpretation "oplus" 'oplus a b = (lo ? a b).
165 lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
168 definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
169 match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
171 notation "i ◂ e" left associative with precedence 60 for @{'ltrif $i $e}.
172 interpretation "pre_concat_r" 'ltrif i e = (pre_concat_r ? i e).
174 lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop.
176 #S #A #B #H >H /2/ qed.
178 lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
179 \sem{i ◂ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
180 #S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //]
181 >sem_pre_true >sem_cat >sem_pre_true /2/
184 definition lc ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
186 [ mk_Prod i1 b1 ⇒ match b1 with
187 [ true ⇒ (i1 ◂ (bcast ? i2))
188 | false ⇒ 〈i1 · i2,false〉
192 definition lift ≝ λS.λf:pitem S →pre S.λe:pre S.
194 [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
196 notation "a ▸ b" left associative with precedence 60 for @{'lc eclose $a $b}.
197 interpretation "lc" 'lc op a b = (lc ? op a b).
199 definition lk ≝ λS:DeqSet.λbcast:∀S:DeqSet.∀E:pitem S.pre S.λe:pre S.
203 [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
204 |false ⇒ 〈i1^*,false〉
208 (* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.*)
209 interpretation "lk" 'lk op a = (lk ? op a).
210 notation "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
212 notation "•" non associative with precedence 60 for @{eclose ?}.
214 let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
218 | ps x ⇒ 〈 `.x, false〉
219 | pp x ⇒ 〈 `.x, false 〉
220 | po i1 i2 ⇒ •i1 ⊕ •i2
221 | pc i1 i2 ⇒ •i1 ▸ i2
222 | pk i ⇒ 〈(\fst (•i))^*,true〉].
224 notation "• x" non associative with precedence 60 for @{'eclose $x}.
225 interpretation "eclose" 'eclose x = (eclose ? x).
227 lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
228 •(i1 + i2) = •i1 ⊕ •i2.
231 lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
232 •(i1 · i2) = •i1 ▸ i2.
235 lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
236 •i^* = 〈(\fst(•i))^*,true〉.
239 definition reclose ≝ λS. lift S (eclose S).
240 interpretation "reclose" 'eclose x = (reclose ? x).
243 lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
244 \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
245 #S * #i1 #b1 * #i2 #b2 #w %
246 [cases b1 cases b2 normalize /2/ * /3/ * /3/
247 |cases b1 cases b2 normalize /2/ * /3/ * /3/
253 〈i1,true〉 ▸ i2 = i1 ◂ (•i2).
256 lemma odot_true_bis :
258 〈i1,true〉 ▸ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
259 #S #i1 #i2 normalize cases (•i2) // qed.
263 〈i1,false〉 ▸ i2 = 〈i1 · i2, false〉.
266 lemma LcatE : ∀S.∀e1,e2:pitem S.
267 \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
270 lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
273 lemma erase_plus : ∀S.∀i1,i2:pitem S.
274 |i1 + i2| = |i1| + |i2|.
277 lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
280 lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
282 [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
283 cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
284 | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
285 cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //
286 | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
290 lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
291 \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
294 (* theorem 16: 1 → 3 *)
295 lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
296 \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
297 \sem{e1 ▸ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
298 #S * #i1 #b1 #i2 cases b1
299 [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
300 |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
301 >erase_bull @eqP_trans [|@(eqP_union_l … H)]
302 @eqP_trans [|@eqP_union_l[|@union_comm ]]
303 @eqP_trans [|@eqP_sym @union_assoc ] /3/
307 lemma sem_fst: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
309 [>sem_pre_true normalize in ⊢ (??%?); #w %
310 [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
311 |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
315 lemma item_eps: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
317 [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
322 lemma sem_fst_aux: ∀S.∀e:pre S.∀i:pitem S.∀A.
323 \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
325 @eqP_trans [|@sem_fst]
326 @eqP_trans [||@eqP_union_r [|@eqP_sym @item_eps]]
327 @eqP_trans [||@distribute_substract]
332 theorem sem_bull: ∀S:DeqSet. ∀e:pitem S. \sem{•e} =1 \sem{e} ∪ \sem{|e|}.
334 [#w normalize % [/2/ | * //]
336 |#x normalize #w % [ /2/ | * [@False_ind | //]]
337 |#x normalize #w % [ /2/ | * // ]
338 |#i1 #i2 #IH1 #IH2 >eclose_dot
339 @eqP_trans [|@odot_dot_aux //] >sem_cat
342 [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
343 @eqP_trans [|@union_assoc]
344 @eqP_trans [||@eqP_sym @union_assoc]
346 |#i1 #i2 #IH1 #IH2 >eclose_plus
347 @eqP_trans [|@sem_oplus] >sem_plus >erase_plus
348 @eqP_trans [|@(eqP_union_l … IH2)]
349 @eqP_trans [|@eqP_sym @union_assoc]
350 @eqP_trans [||@union_assoc] @eqP_union_r
351 @eqP_trans [||@eqP_sym @union_assoc]
352 @eqP_trans [||@eqP_union_l [|@union_comm]]
353 @eqP_trans [||@union_assoc] /2/
354 |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
355 @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@sem_fst_aux //]]]
356 @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
357 @eqP_trans [|@union_assoc] @eqP_union_l >erase_star
358 @eqP_sym @star_fix_eps
362 definition lifted_cat ≝ λS:DeqSet.λe:pre S.
363 lift S (lc S eclose e).
365 notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
367 interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
369 lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b.
370 〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
371 #S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) //
374 lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
375 〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
379 lemma erase_odot:∀S.∀e1,e2:pre S.
380 |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
381 #S * #i1 * * #i2 #b2 // >odot_true_b //
384 lemma ostar_true: ∀S.∀i:pitem S.
385 〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
388 lemma ostar_false: ∀S.∀i:pitem S.
389 〈i,false〉^⊛ = 〈i^*, false〉.
392 lemma erase_ostar: ∀S.∀e:pre S.
393 |\fst (e^⊛)| = |\fst e|^*.
396 lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
397 \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▸ i} ∪ { [ ] }.
399 cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ true〉) [//]
400 #H >H cases (e1 ▸ i) #i1 #b1 cases b1
401 [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
407 lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i.
408 e1 ⊙ 〈i,false〉 = e1 ▸ i.
410 cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ false〉) [//]
411 cases (e1 ▸ i) #i1 #b1 cases b1 #H @H
415 ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
418 @eqP_trans [|@sem_odot_true]
419 @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
420 |>sem_pre_false >eq_odot_false @odot_dot_aux //
425 theorem sem_ostar: ∀S.∀e:pre S.
426 \sem{e^⊛} =1 \sem{e} · \sem{|\fst e|}^*.
428 [>sem_pre_true >sem_pre_true >sem_star >erase_bull
429 @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@sem_fst_aux //]]]
430 @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
431 @eqP_trans [||@eqP_sym @distr_cat_r]
432 @eqP_trans [|@union_assoc] @eqP_union_l
433 @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps
434 |>sem_pre_false >sem_pre_false >sem_star /2/