3 (**************************************************************************)
6 (* ||A|| A project by Andrea Asperti *)
8 (* ||I|| Developers: *)
9 (* ||T|| The HELM team. *)
10 (* ||A|| http://helm.cs.unibo.it *)
12 (* \ / This file is distributed under the terms of the *)
13 (* v GNU General Public License Version 2 *)
15 (**************************************************************************)
17 include "arithmetics/nat.ma".
18 include "basics/list.ma".
20 interpretation "iff" 'iff a b = (iff a b).
22 record Alpha : Type[1] ≝ { carr :> Type[0];
23 eqb: carr → carr → bool;
24 eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y)
27 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
28 interpretation "eqb" 'eqb a b = (eqb ? a b).
30 definition word ≝ λS:Alpha.list S.
32 inductive re (S: Alpha) : Type[0] ≝
36 | c: re S → re S → re S
37 | o: re S → re S → re S
40 (* notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.*)
41 notation "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
42 interpretation "star" 'pk a = (k ? a).
43 interpretation "or" 'plus a b = (o ? a b).
45 notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
46 interpretation "cat" 'pc a b = (c ? a b).
48 (* to get rid of \middot
49 ncoercion c : ∀S:Alpha.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
52 notation < "a" non associative with precedence 90 for @{ 'ps $a}.
53 notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
54 interpretation "atom" 'ps a = (s ? a).
56 notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
57 interpretation "epsilon" 'epsilon = (e ?).
59 notation "∅" non associative with precedence 90 for @{ 'empty }.
60 interpretation "empty" 'empty = (z ?).
62 let rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝
63 match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
65 let rec conjunct (S : Alpha) (l : list (word S)) (r : word S → Prop) on l: Prop ≝
66 match l with [ nil ⇒ ? | cons w tl ⇒ r w ∧ conjunct ? tl r ].
69 definition empty_lang ≝ λS.λw:word S.False.
70 notation "{}" non associative with precedence 90 for @{'empty_lang}.
71 interpretation "empty lang" 'empty_lang = (empty_lang ?).
73 definition sing_lang ≝ λS.λx,w:word S.x=w.
74 (* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.*)
75 interpretation "sing lang" 'singl x = (sing_lang ? x).
77 definition union : ∀S,l1,l2,w.Prop ≝ λS.λl1,l2.λw:word S.l1 w ∨ l2 w.
78 interpretation "union lang" 'union a b = (union ? a b).
80 definition cat : ∀S,l1,l2,w.Prop ≝
81 λS.λl1,l2.λw:word S.∃w1,w2.w1 @ w2 = w ∧ l1 w1 ∧ l2 w2.
82 interpretation "cat lang" 'pc a b = (cat ? a b).
84 definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
85 interpretation "star lang" 'pk l = (star ? l).
87 let rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝
92 | c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
93 | o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
94 | k r1 ⇒ (in_l ? r1) ^*].
96 notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
97 interpretation "in_l" 'in_l E = (in_l ? E).
98 interpretation "in_l mem" 'mem w l = (in_l ? l w).
100 notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
101 interpretation "orb" 'orb a b = (orb a b).
103 definition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
104 notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
105 notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
106 interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
108 inductive pitem (S: Alpha) : Type[0] ≝
113 | pc: pitem S → pitem S → pitem S
114 | po: pitem S → pitem S → pitem S
115 | pk: pitem S → pitem S.
117 definition pre ≝ λS.pitem S × bool.
119 interpretation "pstar" 'pk a = (pk ? a).
120 interpretation "por" 'plus a b = (po ? a b).
121 interpretation "pcat" 'pc a b = (pc ? a b).
122 notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
123 notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
124 interpretation "ppatom" 'pp a = (pp ? a).
125 (* to get rid of \middot
126 ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?.
128 interpretation "patom" 'ps a = (ps ? a).
129 interpretation "pepsilon" 'epsilon = (pe ?).
130 interpretation "pempty" 'empty = (pz ?).
132 let rec forget (S: Alpha) (l : pitem S) on l: re S ≝
138 | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
139 | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
140 | pk E ⇒ (forget ? E)^* ].
142 (* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
143 interpretation "forget" 'norm a = (forget ? a).
146 let rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝
152 | pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
153 | po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
154 | pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
156 interpretation "in_pl" 'in_l E = (in_pl ? E).
157 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
159 definition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}.
161 interpretation "epsilon" 'epsilon = (epsilon ?).
162 notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
163 interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
165 definition in_prl ≝ λS : Alpha.λp:pre S.
166 if (\snd p) then \sem{\fst p} ∪ { ([ ] : word S) } else \sem{\fst p}.
168 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
169 interpretation "in_prl" 'in_l E = (in_prl ? E).
171 lemma sem_pre_true : ∀S.∀i:pitem S.
172 \sem{〈i,true〉} = \sem{i} ∪ { ([ ] : word S) }.
175 lemma sem_pre_false : ∀S.∀i:pitem S.
176 \sem{〈i,false〉} = \sem{i}.
179 lemma sem_cat: ∀S.∀i1,i2:pitem S.
180 \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
183 lemma sem_plus: ∀S.∀i1,i2:pitem S.
184 \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
187 lemma sem_star : ∀S.∀i:pitem S.
188 \sem{i^*} = \sem{i} · \sem{|i|}^*.
191 lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
192 #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
194 lemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ([ ] ∈ e).
195 #S #e elim e normalize /2/
196 [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
197 >(append_eq_nil …H…) /2/
198 |#r1 #r2 #n1 #n2 % * /2/
199 |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
204 lemma epsilon_to_true : ∀S.∀e:pre S. [ ] ∈ e → \snd e = true.
205 #S * #i #b cases b // normalize #H @False_ind /2/
208 lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → [ ] ∈ e.
209 #S * #i #b #btrue normalize in btrue >btrue %2 //
212 definition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
213 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
214 interpretation "oplus" 'oplus a b = (lo ? a b).
216 lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1||b2〉.
219 definition pre_concat_r ≝ λS:Alpha.λi:pitem S.λe:pre S.
220 match e with [ pair i1 b ⇒ 〈i · i1, b〉].
222 notation "i ▸ e" left associative with precedence 60 for @{'trir $i $e}.
223 interpretation "pre_concat_r" 'trir i e = (pre_concat_r ? i e).
225 definition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
226 notation "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
227 interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
229 lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop.
231 #S #A #B #H >H /2/ qed.
233 lemma ext_eq_trans: ∀S.∀A,B,C:word S → Prop.
234 A =1 B → B =1 C → A =1 C.
235 #S #A #B #C #eqAB #eqBC #w cases (eqAB w) cases (eqBC w) /4/
238 lemma union_assoc: ∀S.∀A,B,C:word S → Prop.
239 A ∪ B ∪ C =1 A ∪ (B ∪ C).
240 #S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
243 lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
244 \sem{i ▸ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
245 #S #i * #i1 #b1 cases b1 /2/
246 >sem_pre_true >sem_cat >sem_pre_true /2/
249 definition lc ≝ λS:Alpha.λbcast:∀S:Alpha.pitem S → pre S.λe1:pre S.λi2:pitem S.
251 [ pair i1 b1 ⇒ match b1 with
252 [ true ⇒ (i1 ▸ (bcast ? i2))
253 | false ⇒ 〈i1 · i2,false〉
257 definition lift ≝ λf:∀S.pitem S →pre S.λS.λe:pre S.
259 [ pair i b ⇒ 〈\fst (f S i), \snd (f S i) || b〉].
261 notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
262 interpretation "lc" 'lc op a b = (lc ? op a b).
263 notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}.
265 definition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λe:pre S.
269 [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
274 (* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.*)
275 interpretation "lk" 'lk op a = (lk ? op a).
276 notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
278 notation > "•" non associative with precedence 60 for @{eclose ?}.
280 let rec eclose (S: Alpha) (i: pitem S) on i : pre S ≝
284 | ps x ⇒ 〈 `.x, false〉
285 | pp x ⇒ 〈 `.x, false 〉
286 | po i1 i2 ⇒ •i1 ⊕ •i2
287 | pc i1 i2 ⇒ •i1 ⊙ i2
288 | pk i ⇒ 〈(\fst(•i))^*,true〉].
291 notation "• x" non associative with precedence 60 for @{'eclose $x}.
292 interpretation "eclose" 'eclose x = (eclose ? x).
294 lemma eclose_plus: ∀S:Alpha.∀i1,i2:pitem S.
295 •(i1 + i2) = •i1 ⊕ •i2.
298 lemma eclose_dot: ∀S:Alpha.∀i1,i2:pitem S.
299 •(i1 · i2) = •i1 ⊙ i2.
302 lemma eclose_star: ∀S:Alpha.∀i:pitem S.
303 •i^* = 〈(\fst(•i))^*,true〉.
306 definition reclose ≝ lift eclose.
307 interpretation "reclose" 'eclose x = (reclose ? x).
309 lemma epsilon_or : ∀S:Alpha.∀b1,b2. epsilon S (b1 || b2) =1 ϵ b1 ∪ ϵ b2.
310 #S #b1 #b2 #w % cases b1 cases b2 normalize /2/ * /2/ * ;
314 lemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
315 #S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
317 nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
318 #S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.*)
322 lemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.
323 \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
324 #S * #i1 #b1 * #i2 #b2 #w %
325 [normalize * [* /3/ | cases b1 cases b2 normalize /3/ ]
326 |normalize * * /3/ cases b1 cases b2 normalize /3/ *]
331 〈i1,true〉 ⊙ i2 = i1 ▸ (•i2).
334 lemma odot_true_bis :
336 〈i1,true〉 ⊙ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
337 #S #i1 #i2 normalize cases (•i2) // qed.
341 〈i1,false〉 ⊙ i2 = 〈i1 · i2, false〉.
344 lemma LcatE : ∀S.∀e1,e2:pitem S.
345 \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
349 nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r).
350 #S p q r; napply extP; #w; nnormalize; @;
351 ##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
352 ##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
355 nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
356 #S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.*)
358 lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
361 lemma erase_plus : ∀S.∀i1,i2:pitem S.
362 |i1 + i2| = |i1| + |i2|.
365 lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
369 definition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
370 interpretation "substract" 'minus a b = (substract ? a b).
372 nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
373 #S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
375 nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
376 #S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
378 nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
379 #S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
381 nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
382 #S a b w; nnormalize; *; //; nqed. *)
384 lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
386 [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
387 cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
388 | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
389 cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //
390 | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
394 axiom eq_ext_sym: ∀S.∀A,B:word S →Prop.
397 axiom union_ext_l: ∀S.∀A,B,C:word S →Prop.
398 A =1 C → A ∪ B =1 C ∪ B.
400 axiom union_ext_r: ∀S.∀A,B,C:word S →Prop.
401 B =1 C → A ∪ B =1 A ∪ C.
403 axiom union_comm : ∀S.∀A,B:word S →Prop.
406 lemma distr_cat_r: ∀S.∀A,B,C:word S →Prop.
407 (A ∪ B) · C =1 A · C ∪ B · C.
409 [* #w1 * #w2 * * #eqw * /6/ |* * #w1 * #w2 * * /6/]
412 (* this kind of results are pretty bad for automation;
413 better not index them *)
414 lemma epsilon_cat_r: ∀S.∀A:word S →Prop.
417 [* #w1 * #w2 * * #eqw #inw1 normalize #eqw2 <eqw //
418 |#inA @(ex_intro … w) @(ex_intro … [ ]) /3/
422 lemma epsilon_cat_l: ∀S.∀A:word S →Prop.
425 [* #w1 * #w2 * * #eqw normalize #eqw2 <eqw <eqw2 //
426 |#inA @(ex_intro … [ ]) @(ex_intro … w) /3/
431 lemma distr_cat_r_eps: ∀S.∀A,C:word S →Prop.
432 (A ∪ { [ ] }) · C =1 A · C ∪ C.
433 #S #A #C @ext_eq_trans [|@distr_cat_r |@union_ext_r @epsilon_cat_l]
436 (* axiom eplison_cut_l: ∀S.∀A:word S →Prop.
439 axiom eplison_cut_r: ∀S.∀A:word S →Prop.
443 lemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
444 #S p; ncases p; //; nqed.
446 nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p.
447 #S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
448 *; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
451 (* theorem 16: 1 → 3 *)
452 lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
453 \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
454 \sem{e1 ⊙ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
455 #S * #i1 #b1 #i2 cases b1
456 [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
457 |#H >odot_true >sem_pre_true @(ext_eq_trans … (sem_pre_concat_r …))
462 [|@union_ext_r [|@union_comm ]
463 |@ext_eq_trans (* /3 by eq_ext_sym, union_ext_l/; *)
464 [|@eq_ext_sym @union_assoc
467 by eq_ext_sym, union_ext_l; @union_ext_l /3
469 /3 width=5 by eq_ext_sym, union_ext_r/ *)
476 (* nlemma sub_dot_star :
477 ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
478 #S X b; napply extP; #w; @;
479 ##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
480 *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
481 @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
482 @; //; napply (subW … sube);
483 ##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
484 #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
485 ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
486 @; ncases b in H1; #H1;
487 ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
488 nrewrite > (associative_append ? w' w1 w2);
489 nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
490 ##| ncases w' in Pw';
491 ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
492 ##| #x xs Px; @(x::xs); @(w1@w2);
493 nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
495 ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
496 nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
497 ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
498 nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
500 ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //]
501 @; //; @; //; @; *;##]##]##]
505 alias symbol "pc" (instance 13) = "cat lang".
506 alias symbol "in_pl" (instance 23) = "in_pl".
507 alias symbol "in_pl" (instance 5) = "in_pl".
508 alias symbol "eclose" (instance 21) = "eclose".
509 ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 |e|.
511 ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
512 ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
514 nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
515 nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
516 nrewrite > (IH1 …); nrewrite > (cup_dotD …);
517 nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
518 nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
519 nrewrite < (erase_dot …); nrewrite < (cupA …); //;
521 nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
522 nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
523 nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
524 nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …);
525 nrewrite < (erase_plus …); //.
526 ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
527 nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
528 nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
529 nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
530 nrewrite > (erase_bull…e);
531 nrewrite > (erase_star …);
532 nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[##2:
533 nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH;
534 ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH;
535 nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
536 ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
537 nrewrite > (cup_dotD…); nrewrite > (cupA…);
538 nrewrite > (?: ((?·?)∪{[]} = 𝐋 |e^*|)); //;
539 nchange in match (𝐋 |e^*|) with ((𝐋 |e|)^* ); napply sub_dot_star;##]
544 ∀S.∀e1,e2: pre S. 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
545 #S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
547 nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} = 𝐋 e^*.
548 #S e; napply extP; #w; nnormalize; @;
549 ##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2;
550 *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
551 nrewrite < defw; nrewrite < defw2; @; //; @;//;
552 ##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
553 #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
557 nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
558 #S e; @[]; /2/; nqed.
560 nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
561 #S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
563 nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
564 #S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
566 nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
567 ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
568 #S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
569 ##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
573 nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 |\fst e|)^*.
574 #S p; ncases p; #e b; ncases b;
575 ##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
576 nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
577 nchange in ⊢ (??%?) with (?∪?);
578 nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
579 nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b' )); ##[##2:
580 nlapply (bull_cup ? e); #bc;
581 nchange in match (𝐋\p (•e)) in bc with (?∪?);
582 nchange in match b' in bc with b';
583 ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
584 nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
585 nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
586 nrewrite > (sub_dot_star…);
587 nchange in match (𝐋\p 〈?,?〉) with (?∪?);
588 nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;
589 ##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
591 nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 |e|^* );
592 nrewrite < (cup0 ? (𝐋\p e)); //;##]
595 nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝
600 | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
601 | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
602 | k e1 ⇒ pk ? (pre_of_re ? e1)].
604 nlemma notFalse : ¬False. @; //; nqed.
606 nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
607 #S A; nnormalize; napply extP; #w; @; ##[##2: *]
608 *; #w1; *; #w2; *; *; //; nqed.
610 nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
611 #S e; nelim e; ##[##1,2,3: //]
612 ##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
613 nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
614 ##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
615 nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
616 ##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
617 nrewrite > H1; napply dot0; ##]
620 nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e.
622 ##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
623 nrewrite < H1; nrewrite < H2; //
624 ##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
625 nrewrite < H1; nrewrite < H2; //
626 ##| #e1 H1; nchange in match (𝐋 (e1^* )) with ((𝐋 e1)^* );
631 nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
632 #S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
633 nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
636 nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
637 #S f g H; nrewrite > H; //; nqed.
640 ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|.
642 ##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
643 nrewrite > defsnde; #H;
644 nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
648 notation > "\move term 90 x term 90 E"
649 non associative with precedence 60 for @{move ? $x $E}.
650 nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
654 | ps y ⇒ 〈 `y, false 〉
655 | pp y ⇒ 〈 `y, x == y 〉
656 | po e1 e2 ⇒ \move x e1 ⊕ \move x e2
657 | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
658 | pk e ⇒ (\move x e)^⊛ ].
659 notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
660 notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
661 interpretation "move" 'move x E = (move ? x E).
663 ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
664 interpretation "rmove" 'move x E = (rmove ? x E).
666 nlemma XXz : ∀S:Alpha.∀w:word S. w ∈ ∅ → False.
667 #S w abs; ninversion abs; #; ndestruct;
671 nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
672 #S w abs; ninversion abs; #; ndestruct;
675 nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False.
676 #S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
681 ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
682 (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
683 #S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
684 ncases e1 in H; ncases e2;
685 ##[##1: *; ##[*; nnormalize; #; ndestruct]
686 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
687 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
688 ##|##2: *; ##[*; nnormalize; #; ndestruct]
689 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
690 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
691 ##| #r; *; ##[ *; nnormalize; #; ndestruct]
692 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
693 ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
694 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
695 ##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
696 #H; ninversion H; nnormalize; #; ndestruct;
697 ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
698 ##| #r1 r2; *; ##[ *; #defw]
703 ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E.
704 #S E; ncases E; #r b; nelim r;
706 ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
707 #H; ninversion H; #; ndestruct;
708 ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
709 #H; ninversion H; #; ndestruct;##]
710 ##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
711 *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
712 ##|#a c w; @; nnormalize;
713 ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##]
714 #H; ninversion H; #; ndestruct;
715 ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
716 #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
717 ##|#r1 r2 H1 H2 a w; @;
718 ##[ #H; ncases (in_move_cat … H);
719 ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
720 ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good;
721 nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
730 notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
731 nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
734 | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
736 ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
738 ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
740 ∀E1,E2: bool × (pre S).
742 (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
745 ndefinition NAT: decidable.
749 include "hints_declaration.ma".
751 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
752 unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
754 ninductive unit: Type[0] ≝ I: unit.
756 nlet corec foo_nop (b: bool):
758 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
759 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
761 [ nnormalize in ⊢ (??%%); napply (foo_nop false)
763 [ nnormalize in ⊢ (??%%); napply (foo_nop false)
764 | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
768 nlet corec foo (a: unit):
770 (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
771 (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
776 [ nnormalize in ⊢ (??%%);
777 nnormalize in foo: (? → ??%%);
779 [ nnormalize in ⊢ (??%%); napply foo_nop
781 [ nnormalize in ⊢ (??%%);
783 ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
784 ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
789 ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
790 ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
791 ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
794 nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
795 nnormalize in match test3;
800 (**********************************************************)
802 ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
803 der_z: der S a (z S) (z S)
804 | der_e: der S a (e S) (z S)
805 | der_s1: der S a (s S a) (e ?)
806 | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
807 | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
808 der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
809 | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
810 der S a (c ? e1 e2) (c ? e1' e2)
811 | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
812 der S a (o ? e1 e2) (o ? e1' e2').
814 nlemma eq_rect_CProp0_r:
815 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
816 #A; #a; #x; #p; ncases p; #P; #H; nassumption.
819 nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
821 naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
822 (* #S; #r1; #r2; #w; nelim r1
824 | #H1; #H2; napply (in_c ? []); //
825 | (* tutti casi assurdi *) *)
827 ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
828 in_l_empty1: ∀E.in_l S [] E → in_l' S [] E
829 | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
831 ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
833 (in_l S [] E1 → in_l S [] E2) →
834 (in_l S [] E2 → in_l S [] E1) →
835 (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
838 (* serve il lemma dopo? *)
839 ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
840 #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
842 | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
844 (* IL VICEVERSA NON VALE *)
845 naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
846 (* #S; #w; #E; #H; nelim H
848 | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
852 ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
853 #S; #a; #E; #E'; #w; #H; nelim H
854 [##1,2: #H1; ninversion H1
855 [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
856 |##2,9: #X; #Y; #K; ncases (?:False); /2/
857 |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
858 |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
859 |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
860 |##6,13: #x; #y; #K; ncases (?:False); /2/
861 |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
862 ##| #H1; ninversion H1
864 | #X; #Y; #K; ncases (?:False); /2/
865 | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
866 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
867 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
868 | #x; #y; #K; ncases (?:False); /2/
869 | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
870 ##| #H1; #H2; #H3; ninversion H3
871 [ #_; #K; ncases (?:False); /2/
872 | #X; #Y; #K; ncases (?:False); /2/
873 | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
874 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
875 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
876 | #x; #y; #K; ncases (?:False); /2/
877 | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
878 ##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;