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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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16 include "basics/lists/listb.ma".
18 let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
22 | ps y ⇒ 〈 `y, false 〉
23 | pp y ⇒ 〈 `y, x == y 〉
24 | po e1 e2 ⇒ (move ? x e1) ⊕ (move ? x e2)
25 | pc e1 e2 ⇒ (move ? x e1) ⊙ (move ? x e2)
26 | pk e ⇒ (move ? x e)^⊛ ].
28 lemma move_plus: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
29 move S x (i1 + i2) = (move ? x i1) ⊕ (move ? x i2).
32 lemma move_cat: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
33 move S x (i1 · i2) = (move ? x i1) ⊙ (move ? x i2).
36 lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
37 move S x i^* = (move ? x i)^⊛.
40 definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
42 lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b.
43 pmove ? x 〈i,b〉 = move ? x i.
46 lemma eq_to_eq_hd: ∀A.∀l1,l2:list A.∀a,b.
47 a::l1 = b::l2 → a = b.
48 #A #l1 #l2 #a #b #H destruct //
51 lemma same_kernel: ∀S:DeqSet.∀a:S.∀i:pitem S.
52 |\fst (move ? a i)| = |i|.
54 [#i1 #i2 >move_cat #H1 #H2 whd in ⊢ (???%); <H1 <H2 //
55 |#i1 #i2 >move_plus #H1 #H2 whd in ⊢ (???%); <H1 <H2 //
60 ∀S:DeqSet.∀a:S.∀i:pitem S.∀w: word S.
61 \sem{move ? a i} w ↔ \sem{i} (a::w).
66 |normalize #x #w cases (true_or_false (a==x)) #H >H normalize
67 [>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
68 |% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
70 |#i1 #i2 #HI1 #HI2 #w >(sem_cat S i1 i2) >move_cat
71 @iff_trans[|@sem_odot] >same_kernel >sem_cat_w
72 @iff_trans[||@(iff_or_l … (HI2 w))] @iff_or_r
73 @iff_trans[||@iff_sym @deriv_middot //]
75 |#i1 #i2 #HI1 #HI2 #w >(sem_plus S i1 i2) >move_plus >sem_plus_w
76 @iff_trans[|@sem_oplus]
77 @iff_trans[|@iff_or_l [|@HI2]| @iff_or_r //]
78 |#i1 #HI1 #w >move_star
79 @iff_trans[|@sem_ostar] >same_kernel >sem_star_w
80 @iff_trans[||@iff_sym @deriv_middot //]
85 notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
86 let rec moves (S : DeqSet) w e on w : pre S ≝
89 | cons x w' ⇒ w' ↦* (move S x (\fst e))].
91 lemma moves_empty: ∀S:DeqSet.∀e:pre S.
95 lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S.
96 moves ? (a::w) e = moves ? w (move S a (\fst e)).
99 lemma not_epsilon_sem: ∀S:DeqSet.∀a:S.∀w: word S. ∀e:pre S.
100 iff ((a::w) ∈ e) ((a::w) ∈ \fst e).
101 #S #a #w * #i #b cases b normalize
102 [% /2/ * // #H destruct |% normalize /2/]
105 lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
106 |\fst (moves ? w e)| = |\fst e|.
110 theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S.
111 (\snd (moves ? w e) = true) ↔ \sem{e} w.
113 [* #i #b >moves_empty cases b % /2/
114 |#a #w1 #Hind #e >moves_cons
115 @iff_trans [||@iff_sym @not_epsilon_sem]
116 @iff_trans [||@move_ok] @Hind
120 lemma not_true_to_false: ∀b.b≠true → b =false.
121 #b * cases b // #H @False_ind /2/
124 theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S.
125 iff (\sem{e1} =1 \sem{e2}) (∀w.\snd (moves ? w e1) = \snd (moves ? w e2)).
128 cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2))
129 [* * // * #H1 #H2 [@sym_eq @H1 //| @H2 //]]
130 #Hcut @Hcut @iff_trans [|@decidable_sem]
131 @iff_trans [|@same_sem] @iff_sym @decidable_sem
132 |#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
135 lemma moves_left : ∀S,a,w,e.
136 moves S (w@[a]) e = move S a (\fst (moves S w e)).
137 #S #a #w elim w // #x #tl #Hind #e >moves_cons >moves_cons //
140 definition in_moves ≝ λS:DeqSet.λw.λe:pre S. \snd(w ↦* e).
143 coinductive equiv (S:DeqSet) : pre S → pre S → Prop ≝
147 (∀x. equiv S (move ? x (\fst e1)) (move ? x (\fst e2))) →
151 let rec beqitem S (i1,i2: pitem S) on i1 ≝
153 [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
154 | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
155 | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
156 | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
157 | po i11 i12 ⇒ match i2 with
158 [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
160 | pc i11 i12 ⇒ match i2 with
161 [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
163 | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
166 lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
168 [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
169 |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
170 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
171 [>(\P H) // | @(\b (refl …))]
172 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
173 [>(\P H) // | @(\b (refl …))]
174 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
175 normalize #H destruct
176 [cases (true_or_false (beqitem S i11 i21)) #H1
177 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
178 |>H1 in H; normalize #abs @False_ind /2/
180 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
182 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
183 normalize #H destruct
184 [cases (true_or_false (beqitem S i11 i21)) #H1
185 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
186 |>H1 in H; normalize #abs @False_ind /2/
188 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
190 |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
191 normalize #H destruct
192 [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
196 definition DeqItem ≝ λS.
197 mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
199 unification hint 0 ≔ S;
200 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
201 (* ---------------------------------------- *) ⊢
204 unification hint 0 ≔ S,i1,i2;
205 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
206 (* ---------------------------------------- *) ⊢
207 beqitem S i1 i2 ≡ eqb X i1 i2.
209 definition sons ≝ λS:DeqSet.λl:list S.λp:(pre S)×(pre S).
210 map ?? (λa.〈move S a (\fst (\fst p)),move S a (\fst (\snd p))〉) l.
212 lemma memb_sons: ∀S,l.∀p,q:(pre S)×(pre S). memb ? p (sons ? l q) = true →
213 ∃a.(move ? a (\fst (\fst q)) = \fst p ∧
214 move ? a (\fst (\snd q)) = \snd p).
215 #S #l elim l [#p #q normalize in ⊢ (%→?); #abs @False_ind /2/]
216 #a #tl #Hind #p #q #H cases (orb_true_l … H) -H
217 [#H @(ex_intro … a) <(proj1 … (eqb_true …)H) /2/
222 let rec bisim S l n (frontier,visited: list ?) on n ≝
224 [ O ⇒ 〈false,visited〉 (* assert false *)
227 [ nil ⇒ 〈true,visited〉
229 if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
230 bisim S l m (unique_append ? (filter ? (λx.notb (memb ? x (hd::visited)))
231 (sons S l hd)) tl) (hd::visited)
236 lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
237 bisim S l n frontier visited =
239 [ O ⇒ 〈false,visited〉 (* assert false *)
242 [ nil ⇒ 〈true,visited〉
244 if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
245 bisim S l m (unique_append ? (filter ? (λx.notb(memb ? x (hd::visited)))
246 (sons S l hd)) tl) (hd::visited)
250 #S #l #n cases n // qed.
252 lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
253 bisim S l O frontier visited = 〈false,visited〉.
254 #frontier #visited >unfold_bisim //
257 lemma bisim_end: ∀Sig,l,m.∀visited: list ?.
258 bisim Sig l (S m) [] visited = 〈true,visited〉.
259 #n #visisted >unfold_bisim //
262 lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
263 beqb (\snd (\fst p)) (\snd (\snd p)) = true →
264 bisim Sig l (S m) (p::frontier) visited =
265 bisim Sig l m (unique_append ? (filter ? (λx.notb(memb ? x (p::visited)))
266 (sons Sig l p)) frontier) (p::visited).
267 #Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
270 lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
271 beqb (\snd (\fst p)) (\snd (\snd p)) = false →
272 bisim Sig l (S m) (p::frontier) visited = 〈false,visited〉.
273 #Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
276 definition visited_inv ≝ λS.λe1,e2:pre S.λvisited: list ?.
277 uniqueb ? visited = true ∧
278 ∀p. memb ? p visited = true →
279 (∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p)) ∧
280 (beqb (\snd (\fst p)) (\snd (\snd p)) = true).
282 definition frontier_inv ≝ λS.λfrontier,visited.
283 uniqueb ? frontier = true ∧
284 ∀p:(pre S)×(pre S). memb ? p frontier = true →
285 memb ? p visited = false ∧
286 ∃p1.((memb ? p1 visited = true) ∧
287 (∃a. move ? a (\fst (\fst p1)) = \fst p ∧
288 move ? a (\fst (\snd p1)) = \snd p)).
290 (* lemma andb_true: ∀b1,b2:bool.
291 (b1 ∧ b2) = true → (b1 = true) ∧ (b2 = true).
292 #b1 #b2 cases b1 normalize #H [>H /2/ |@False_ind /2/].
295 lemma andb_true_r: ∀b1,b2:bool.
296 (b1 = true) ∧ (b2 = true) → (b1 ∧ b2) = true.
297 #b1 #b2 cases b1 normalize * //
300 lemma notb_eq_true_l: ∀b. notb b = true → b = false.
301 #b cases b normalize //
305 lemma notb_eq_true_r: ∀b. b = false → notb b = true.
306 #b cases b normalize //
309 lemma notb_eq_false_l:∀b. notb b = false → b = true.
310 #b cases b normalize //
313 lemma notb_eq_false_r:∀b. b = true → notb b = false.
314 #b cases b normalize //
317 (* include "arithmetics/exp.ma". *)
319 let rec pos S (i:re S) on i ≝
324 | o i1 i2 ⇒ pos S i1 + pos S i2
325 | c i1 i2 ⇒ pos S i1 + pos S i2
330 let rec pitem_enum S (i:re S) on i ≝
334 | s y ⇒ [ps S y; pp S y]
335 | o i1 i2 ⇒ compose ??? (po S) (pitem_enum S i1) (pitem_enum S i2)
336 | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
337 | k i ⇒ map ?? (pk S) (pitem_enum S i)
340 lemma pitem_enum_complete : ∀S.∀i:pitem S.
341 memb (DeqItem S) i (pitem_enum S (|i|)) = true.
344 |3,4:#c normalize >(\b (refl … c)) //
345 |5,6:#i1 #i2 #Hind1 #Hind2 @(memb_compose (DeqItem S) (DeqItem S)) //
346 |#i #Hind @(memb_map (DeqItem S)) //
350 definition pre_enum ≝ λS.λi:re S.
351 compose ??? (λi,b.〈i,b〉) (pitem_enum S i) [true;false].
353 lemma pre_enum_complete : ∀S.∀e:pre S.
354 memb ? e (pre_enum S (|\fst e|)) = true.
355 #S * #i #b @(memb_compose (DeqItem S) DeqBool ? (λi,b.〈i,b〉))
356 // cases b normalize //
359 definition space_enum ≝ λS.λi1,i2:re S.
360 compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i2).
362 lemma space_enum_complete : ∀S.∀e1,e2: pre S.
363 memb ? 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true.
364 #S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉))
367 definition visited_inv_1 ≝ λS.λe1,e2:pre S.λvisited: list ?.
368 uniqueb ? visited = true ∧
369 ∀p. memb ? p visited = true →
370 ∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p).
372 lemma bisim_ok1: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} →
373 ∀l,n.∀frontier,visited:list (*(space S) *) ((pre S)×(pre S)).
374 |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
375 visited_inv_1 S e1 e2 visited → frontier_inv S frontier visited →
376 \fst (bisim S l n frontier visited) = true.
377 #Sig #e1 #e2 #same #l #n elim n
378 [#frontier #visited #abs * #unique #H @False_ind @(absurd … abs)
379 @le_to_not_lt @sublist_length // * #e11 #e21 #membp
380 cut ((|\fst e11| = |\fst e1|) ∧ (|\fst e21| = |\fst e2|))
381 [|* #H1 #H2 <H1 <H2 @space_enum_complete]
382 cases (H … membp) #w * #we1 #we2 <we1 <we2 % >same_kernel_moves //
383 |#m #HI * [#visited #vinv #finv >bisim_end //]
384 #p #front_tl #visited #Hn * #u_visited #vinv * #u_frontier #finv
385 cases (finv p (memb_hd …)) #Hp * #p2 * #visited_p2
386 * #a * #movea1 #movea2
387 cut (∃w.(moves Sig w e1 = \fst p) ∧ (moves Sig w e2 = \snd p))
388 [cases (vinv … visited_p2) -vinv #w1 * #mw1 #mw2
389 @(ex_intro … (w1@[a])) % //]
390 -movea2 -movea1 -a -visited_p2 -p2 #reachp
391 cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true)
392 [cases reachp #w * #move_e1 #move_e2 <move_e1 <move_e2
393 @(\b ?) @(proj1 … (equiv_sem … )) @same] #ptest
394 >(bisim_step_true … ptest) @HI -HI
396 |% [whd in ⊢ (??%?); >Hp whd in ⊢ (??%?); //]
397 #p1 #H (cases (orb_true_l … H))
399 |#visited_p1 @(vinv … visited_p1)
401 |whd % [@unique_append_unique @(andb_true_r … u_frontier)]
402 @unique_append_elim #q #H
404 [@notb_eq_true_l @(filter_true … H)
405 |@(ex_intro … p) % [@memb_hd|@(memb_sons … (memb_filter_memb … H))]
407 |cases (finv q ?) [|@memb_cons //]
408 #nvq * #p1 * #Hp1 #reach %
409 [cut ((p==q) = false) [|#Hpq whd in ⊢ (??%?); >Hpq @nvq]
410 cases (andb_true … u_frontier) #notp #_
411 @(not_memb_to_not_eq … H) @notb_eq_true_l @notp
412 |cases (proj2 … (finv q ?))
413 [#p1 * #Hp1 #reach @(ex_intro … p1) % // @memb_cons //
422 definition all_true ≝ λS.λl.∀p:(pre S) × (pre S). memb ? p l = true →
423 (beqb (\snd (\fst p)) (\snd (\snd p)) = true).
425 definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S).∀a:S.
426 memb ? x l1 = true → memb S a l = true →
427 memb ? 〈move ? a (\fst (\fst x)), move ? a (\fst (\snd x))〉 l2 = true.
429 lemma reachable_bisim:
430 ∀S,l,n.∀frontier,visited,visited_res:list ?.
432 sub_sons S l visited (frontier@visited) →
433 bisim S l n frontier visited = 〈true,visited_res〉 →
434 (sub_sons S l visited_res visited_res ∧
435 sublist ? visited visited_res ∧
436 all_true S visited_res).
438 [#fron #vis #vis_res #_ #_ >bisim_never #H destruct
440 [(* case empty frontier *)
441 -Hind #vis #vis_res #allv #H normalize in ⊢ (%→?);
442 #H1 destruct % // % // #p /2 by /
443 |#hd cases (true_or_false (beqb (\snd (\fst hd)) (\snd (\snd hd))))
444 [|(* case head of the frontier is non ok (absurd) *)
445 #H #tl #vis #vis_res #allv >(bisim_step_false … H) #_ #H1 destruct]
446 (* frontier = hd:: tl and hd is ok *)
447 #H #tl #visited #visited_res #allv >(bisim_step_true … H)
448 (* new_visited = hd::visited are all ok *)
449 cut (all_true S (hd::visited))
450 [#p #H1 cases (orb_true_l … H1) [#eqp <(\P eqp) @H |@allv]]
451 (* we now exploit the induction hypothesis *)
452 #allh #subH #bisim cases (Hind … allh … bisim) -Hind
453 [* #H1 #H2 #H3 % // % // #p #H4 @H2 @memb_cons //]
454 (* the only thing left to prove is the sub_sons invariant *)
456 cases (orb_true_l … membx)
458 #eqhdx >(proj1 … (eqb_true …) eqhdx)
459 (* xa is the son of x w.r.t. a; we must distinguish the case xa
460 was already visited form the case xa is new *)
461 letin xa ≝ 〈move S a (\fst (\fst x)), move S a (\fst (\snd x))〉
462 cases (true_or_false … (memb ? xa (x::visited)))
463 [(* xa visited - trivial *) #membxa @memb_append_l2 //
464 |(* xa new *) #membxa @memb_append_l1 @sublist_unique_append_l1 @memb_filter_l
466 |(* this can be probably improved *)
467 generalize in match memba; -memba elim l
468 [whd in ⊢ (??%?→?); #abs @False_ind /2/
469 |#b #others #Hind #memba cases (orb_true_l … memba) #H
470 [>(proj1 … (eqb_true …) H) @memb_hd
476 |(* case x in visited *)
477 #H1 letin xa ≝ 〈move S a (\fst (\fst x)), move S a (\fst (\snd x))〉
478 cases (memb_append … (subH x a H1 memba))
479 [#H2 (cases (orb_true_l … H2))
480 [#H3 @memb_append_l2 >(proj1 … (eqb_true …) H3) @memb_hd
481 |#H3 @memb_append_l1 @sublist_unique_append_l2 @H3
483 |#H2 @memb_append_l2 @memb_cons @H2
491 let rec blank_item (S: DeqSet) (i: re S) on i :pitem S ≝
496 | o e1 e2 ⇒ (blank_item S e1) + (blank_item S e2)
497 | c e1 e2 ⇒ (blank_item S e1) · (blank_item S e2)
498 | k e ⇒ (blank_item S e)^* ].
500 definition pit_pre ≝ λS.λi.〈blank_item S (|i|), false〉.
502 let rec occur (S: DeqSet) (i: re S) on i ≝
507 | o e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2)
508 | c e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2)
511 axiom memb_single: ∀S,a,x. memb S a [x] = true → a = x.
513 axiom tech: ∀b. b ≠ true → b = false.
514 axiom tech2: ∀b. b = false → b ≠ true.
516 lemma not_occur_to_pit: ∀S,a.∀i:pitem S. memb S a (occur S (|i|)) = false →
517 move S a i = pit_pre S i.
519 [#x cases (true_or_false (a==x))
520 [#H >(proj1 …(eqb_true …) H) whd in ⊢ ((??%?)→?);
521 >(proj2 …(eqb_true …) (refl …)) whd in ⊢ ((??%?)→?); #abs @False_ind /2/
524 |#i1 #i2 #Hind1 #Hind2 #H >move_cat >Hind1 [2:@tech
525 @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l1 //]
526 >Hind2 [2:@tech @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l2 //]
528 |#i1 #i2 #Hind1 #Hind2 #H >move_plus >Hind1 [2:@tech
529 @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l1 //]
530 >Hind2 [2:@tech @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l2 //]
532 |#i #Hind #H >move_star >Hind // @H
536 lemma move_pit: ∀S,a,i. move S a (\fst (pit_pre S i)) = pit_pre S i.
538 [#i1 #i2 #Hind1 #Hind2 >move_cat >Hind1 >Hind2 //
539 |#i1 #i2 #Hind1 #Hind2 >move_plus >Hind1 >Hind2 //
540 |#i #Hind >move_star >Hind //
544 lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
545 #S #w #i elim w // #a #tl >moves_cons //
548 lemma to_pit: ∀S,w,e. ¬ sublist S w (occur S (|\fst e|)) →
549 moves S w e = pit_pre S (\fst e).
552 #e * #H @False_ind @H normalize #a #abs @False_ind /2/
553 |#a #tl #Hind #e #H cases (true_or_false (memb S a (occur S (|\fst e|))))
554 [#Htrue >moves_cons whd in ⊢ (???%); <(same_kernel … a)
555 @Hind >same_kernel @(not_to_not … H) #H1 #b #memb cases (orb_true_l … memb)
556 [#H2 <(proj1 … (eqb_true …) H2) // |#H2 @H1 //]
557 |#Hfalse >moves_cons >not_occur_to_pit //
562 definition occ ≝ λS.λe1,e2:pre S.
563 unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
565 (* definition occS ≝ λS:DeqSet.λoccur.
566 PSig S (λx.memb S x occur = true). *)
568 lemma occ_enough: ∀S.∀e1,e2:pre S.
569 (∀w.(sublist S w (occ S e1 e2))→
570 (beqb (\snd (moves S w e1)) (\snd (moves ? w e2))) = true) \to
571 ∀w.(beqb (\snd (moves S w e1)) (\snd (moves ? w e2))) = true.
573 cut (sublist S w (occ S e1 e2) ∨ ¬(sublist S w (occ S e1 e2)))
575 [%1 #a normalize in ⊢ (%→?); #abs @False_ind /2/
577 [cases (true_or_false (memb S a (occ S e1 e2))) #memba
578 [%1 whd #x #membx cases (orb_true_l … membx)
579 [#eqax <(proj1 … (eqb_true …) eqax) //
582 |%2 @(not_to_not … (tech2 … memba)) #H1 @H1 @memb_hd
584 |%2 @(not_to_not … subtl) #H1 #x #H2 @H1 @memb_cons //
589 [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //]
591 [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l2 @H1 //]
596 lemma bisim_char: ∀S.∀e1,e2:pre S.
597 (∀w.(beqb (\snd (moves S w e1)) (\snd (moves ? w e2))) = true) →
599 #S #e1 #e2 #H @(proj2 … (equiv_sem …)) #w @(\P ?) @H
602 lemma bisim_ok2: ∀S.∀e1,e2:pre S.
603 (beqb (\snd e1) (\snd e2) = true) → ∀n.
604 \fst (bisim S (occ S e1 e2) n (sons S (occ S e1 e2) 〈e1,e2〉) [〈e1,e2〉]) = true →
607 letin rsig ≝ (occ S e1 e2)
608 letin frontier ≝ (sons S rsig 〈e1,e2〉)
609 letin visited_res ≝ (\snd (bisim S rsig n frontier [〈e1,e2〉]))
611 cut (bisim S rsig n frontier [〈e1,e2〉] = 〈true,visited_res〉)
612 [<bisim_true <eq_pair_fst_snd //] #H
613 cut (all_true S [〈e1,e2〉])
614 [#p #Hp cases (orb_true_l … Hp)
615 [#eqp <(proj1 … (eqb_true …) eqp) //
616 | whd in ⊢ ((??%?)→?); #abs @False_ind /2/
618 cut (sub_sons S rsig [〈e1,e2〉] (frontier@[〈e1,e2〉]))
619 [#x #a #H1 cases (orb_true_l … H1)
620 [#eqx <(proj1 … (eqb_true …) eqx) #H2 @memb_append_l1
621 whd in ⊢ (??(???%)?); @(memb_map … H2)
622 |whd in ⊢ ((??%?)→?); #abs @False_ind /2/
625 cases (reachable_bisim … allH init … H) * #H1 #H2 #H3
626 cut (∀w.sublist ? w (occ S e1 e2)→∀p.memb ? p visited_res = true →
627 memb ? 〈moves ? w (\fst p), moves ? w (\snd p)〉 visited_res = true)
628 [#w elim w [#_ #p #H4 >moves_empty >moves_empty <eq_pair_fst_snd //]
629 #a #w1 #Hind #Hsub * #e11 #e21 #visp >moves_cons >moves_cons
630 @(Hind ? 〈?,?〉) [#x #H4 @Hsub @memb_cons //]
631 @(H1 〈?,?〉) [@visp| @Hsub @memb_hd]] #all_reach
632 @bisim_char @occ_enough
633 #w #Hsub @(H3 〈?,?〉) @(all_reach w Hsub 〈?,?〉) @H2 //
637 definition tt ≝ ps Bin true.
638 definition ff ≝ ps Bin false.
639 definition eps ≝ pe Bin.
640 definition exp1 ≝ (ff + tt · ff).
641 definition exp2 ≝ ff · (eps + tt).
643 definition exp3 ≝ move Bin true (\fst (•exp1)).
644 definition exp4 ≝ move Bin true (\fst (•exp2)).
645 definition exp5 ≝ move Bin false (\fst (•exp1)).
646 definition exp6 ≝ move Bin false (\fst (•exp2)). *)