4 definition cofinal ≝ λS.λp:(pre S)×(pre S).
5 \snd (\fst p) = \snd (\snd p).
7 theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S.
8 \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
11 cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2))
12 [* * // * #H1 #H2 [@sym_eq @H1 //| @H2 //]]
13 #Hcut @Hcut @iff_trans [|@decidable_sem]
14 @iff_trans [|@same_sem] @iff_sym @decidable_sem
15 |#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
18 definition occ ≝ λS.λe1,e2:pre S.
19 unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
21 lemma occ_enough: ∀S.∀e1,e2:pre S.
22 (∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
23 →∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
25 cases (decidable_sublist S w (occ S e1 e2)) [@H] -H #H
26 >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //]
27 >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l2 @H1 //]
31 lemma equiv_sem_occ: ∀S.∀e1,e2:pre S.
32 (∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
34 #S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H
37 definition sons ≝ λS:DeqSet.λl:list S.λp:(pre S)×(pre S).
38 map ?? (λa.〈move S a (\fst (\fst p)),move S a (\fst (\snd p))〉) l.
40 lemma memb_sons: ∀S,l.∀p,q:(pre S)×(pre S). memb ? p (sons ? l q) = true →
41 ∃a.(move ? a (\fst (\fst q)) = \fst p ∧
42 move ? a (\fst (\snd q)) = \snd p).
43 #S #l elim l [#p #q normalize in ⊢ (%→?); #abs @False_ind /2/]
44 #a #tl #Hind #p #q #H cases (orb_true_l … H) -H
45 [#H @(ex_intro … a) >(\P H) /2/ |#H @Hind @H]
48 definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S.
49 ∀p:(pre S)×(pre S). memb ? p l = true → cofinal ? p ∧ (sublist ? (sons ? alpha p) l).
51 lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S.
52 is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}.
53 #S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ
54 #w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?))
55 lapply Hsub @(list_elim_left … w) [//]
56 #a #w1 #Hind #Hsub >moves_left >moves_left @(proj2 …(Hbisim …(Hind ?)))
57 [#x #Hx @Hsub @memb_append_l1 //
58 |cut (memb S a (occ S e1 e2) = true) [@Hsub @memb_append_l2 //] #occa
64 let rec bisim S l n (frontier,visited: list ?) on n ≝
66 [ O ⇒ 〈false,visited〉 (* assert false *)
69 [ nil ⇒ 〈true,visited〉
71 if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
72 bisim S l m (unique_append ? (filter ? (λx.notb (memb ? x (hd::visited)))
73 (sons S l hd)) tl) (hd::visited)
78 lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
79 bisim S l n frontier visited =
81 [ O ⇒ 〈false,visited〉 (* assert false *)
84 [ nil ⇒ 〈true,visited〉
86 if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
87 bisim S l m (unique_append ? (filter ? (λx.notb(memb ? x (hd::visited)))
88 (sons S l hd)) tl) (hd::visited)
92 #S #l #n cases n // qed.
94 lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
95 bisim S l O frontier visited = 〈false,visited〉.
96 #frontier #visited >unfold_bisim //
99 lemma bisim_end: ∀Sig,l,m.∀visited: list ?.
100 bisim Sig l (S m) [] visited = 〈true,visited〉.
101 #n #visisted >unfold_bisim //
104 lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
105 beqb (\snd (\fst p)) (\snd (\snd p)) = true →
106 bisim Sig l (S m) (p::frontier) visited =
107 bisim Sig l m (unique_append ? (filter ? (λx.notb(memb ? x (p::visited)))
108 (sons Sig l p)) frontier) (p::visited).
109 #Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
112 lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
113 beqb (\snd (\fst p)) (\snd (\snd p)) = false →
114 bisim Sig l (S m) (p::frontier) visited = 〈false,visited〉.
115 #Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
118 lemma notb_eq_true_l: ∀b. notb b = true → b = false.
119 #b cases b normalize //
122 let rec pitem_enum S (i:re S) on i ≝
126 | s y ⇒ [ps S y; pp S y]
127 | o i1 i2 ⇒ compose ??? (po S) (pitem_enum S i1) (pitem_enum S i2)
128 | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
129 | k i ⇒ map ?? (pk S) (pitem_enum S i)
132 lemma pitem_enum_complete : ∀S.∀i:pitem S.
133 memb (DeqItem S) i (pitem_enum S (|i|)) = true.
136 |3,4:#c normalize >(\b (refl … c)) //
137 |5,6:#i1 #i2 #Hind1 #Hind2 @(memb_compose (DeqItem S) (DeqItem S)) //
138 |#i #Hind @(memb_map (DeqItem S)) //
142 definition pre_enum ≝ λS.λi:re S.
143 compose ??? (λi,b.〈i,b〉) (pitem_enum S i) [true;false].
145 lemma pre_enum_complete : ∀S.∀e:pre S.
146 memb ? e (pre_enum S (|\fst e|)) = true.
147 #S * #i #b @(memb_compose (DeqItem S) DeqBool ? (λi,b.〈i,b〉))
148 // cases b normalize //
151 definition space_enum ≝ λS.λi1,i2:re S.
152 compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i2).
154 lemma space_enum_complete : ∀S.∀e1,e2: pre S.
155 memb ? 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true.
156 #S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉))
159 definition all_reachable ≝ λS.λe1,e2:pre S.λl: list ?.
161 ∀p. memb ? p l = true →
162 ∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p).
164 definition disjoint ≝ λS:DeqSet.λl1,l2.
165 ∀p:S. memb S p l1 = true → memb S p l2 = false.
167 lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} →
168 ∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
169 |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
170 all_reachable S e1 e2 visited →
171 all_reachable S e1 e2 frontier →
172 disjoint ? frontier visited →
173 \fst (bisim S l n frontier visited) = true.
174 #Sig #e1 #e2 #same #l #n elim n
175 [#frontier #visited #abs * #unique #H @False_ind @(absurd … abs)
176 @le_to_not_lt @sublist_length // * #e11 #e21 #membp
177 cut ((|\fst e11| = |\fst e1|) ∧ (|\fst e21| = |\fst e2|))
178 [|* #H1 #H2 <H1 <H2 @space_enum_complete]
179 cases (H … membp) #w * #we1 #we2 <we1 <we2 % >same_kernel_moves //
180 |#m #HI * [#visited #vinv #finv >bisim_end //]
181 #p #front_tl #visited #Hn * #u_visited #r_visited * #u_frontier #r_frontier
183 cut (∃w.(moves ? w e1 = \fst p) ∧ (moves ? w e2 = \snd p))
184 [@(r_frontier … (memb_hd … ))] #rp
185 cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true)
186 [cases rp #w * #fstp #sndp <fstp <sndp @(\b ?)
187 @(proj1 … (equiv_sem … )) @same] #ptest
188 >(bisim_step_true … ptest) @HI -HI
190 |% [whd in ⊢ (??%?); >(disjoint … (memb_hd …)) whd in ⊢ (??%?); //
191 |#p1 #H (cases (orb_true_l … H)) [#eqp >(\P eqp) // |@r_visited]
193 |whd % [@unique_append_unique @(andb_true_r … u_frontier)]
194 @unique_append_elim #q #H
195 [cases (memb_sons … (memb_filter_memb … H)) -H
196 #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(ex_intro … (w1@[a]))
197 >moves_left >moves_left >mw1 >mw2 >m1 >m2 % //
198 |@r_frontier @memb_cons //
200 |@unique_append_elim #q #H
201 [@injective_notb @(filter_true … H)
202 |cut ((q==p) = false)
203 [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @memb_cons //]
204 cases (andb_true … u_frontier) #notp #_ @(\bf ?)
205 @(not_to_not … not_eq_true_false) #eqqp <notp <eqqp >H //
211 definition all_true ≝ λS.λl.∀p:(pre S) × (pre S). memb ? p l = true →
212 (beqb (\snd (\fst p)) (\snd (\snd p)) = true).
214 definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S).
215 memb ? x l1 = true → sublist ? (sons ? l x) l2.
217 lemma bisim_complete:
218 ∀S,l,n.∀frontier,visited,visited_res:list ?.
220 sub_sons S l visited (frontier@visited) →
221 bisim S l n frontier visited = 〈true,visited_res〉 →
222 is_bisim S visited_res l ∧ sublist ? (frontier@visited) visited_res.
224 [#fron #vis #vis_res #_ #_ >bisim_never #H destruct
226 [(* case empty frontier *)
227 -Hind #vis #vis_res #allv #H normalize in ⊢ (%→?);
229 [#membp % [@(\P ?) @allv //| @H //]|#H1 @H1]
230 |#hd cases (true_or_false (beqb (\snd (\fst hd)) (\snd (\snd hd))))
231 [|(* case head of the frontier is non ok (absurd) *)
232 #H #tl #vis #vis_res #allv >(bisim_step_false … H) #_ #H1 destruct]
233 (* frontier = hd:: tl and hd is ok *)
234 #H #tl #visited #visited_res #allv >(bisim_step_true … H)
235 (* new_visited = hd::visited are all ok *)
236 cut (all_true S (hd::visited))
237 [#p #H1 cases (orb_true_l … H1) [#eqp >(\P eqp) @H |@allv]]
238 (* we now exploit the induction hypothesis *)
239 #allh #subH #bisim cases (Hind … allh … bisim) -bisim -Hind
240 [#H1 #H2 % // #p #membp @H2 -H2 cases (memb_append … membp) -membp #membp
241 [cases (orb_true_l … membp) -membp #membp
242 [@memb_append_l2 >(\P membp) @memb_hd
243 |@memb_append_l1 @sublist_unique_append_l2 //
245 |@memb_append_l2 @memb_cons //
247 |(* the only thing left to prove is the sub_sons invariant *)
248 #x #membx cases (orb_true_l … membx)
250 #eqhdx <(\P eqhdx) #xa #membxa
251 (* xa is a son of x; we must distinguish the case xa
252 was already visited form the case xa is new *)
253 cases (true_or_false … (memb ? xa (x::visited)))
254 [(* xa visited - trivial *) #membxa @memb_append_l2 //
255 |(* xa new *) #membxa @memb_append_l1 @sublist_unique_append_l1 @memb_filter_l
258 |(* case x in visited *)
259 #H1 #xa #membxa cases (memb_append … (subH x … H1 … membxa))
260 [#H2 (cases (orb_true_l … H2))
261 [#H3 @memb_append_l2 <(\P H3) @memb_hd
262 |#H3 @memb_append_l1 @sublist_unique_append_l2 @H3
264 |#H2 @memb_append_l2 @memb_cons @H2
271 definition equiv ≝ λSig.λre1,re2:re Sig.
272 let e1 ≝ •(blank ? re1) in
273 let e2 ≝ •(blank ? re2) in
274 let n ≝ S (length ? (space_enum Sig (|\fst e1|) (|\fst e2|))) in
275 let sig ≝ (occ Sig e1 e2) in
276 (bisim ? sig n [〈e1,e2〉] []).
278 theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig.
279 \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}.
281 [#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding]
282 cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉)
284 cases (bisim_complete … Hcut)
285 [2,3: #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/]
286 #Hbisim #Hsub @(bisim_to_sem … Hbisim)
288 |#H @(bisim_correct ? (•(blank ? re1)) (•(blank ? re2)))
289 [@eqP_trans [|@re_embedding] @eqP_trans [|@H] @eqP_sym @re_embedding
291 |% // #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/
292 |% // #p #H >(memb_single … H) @(ex_intro … ϵ) /2/
298 lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
299 #n #m % [@eqbnat_true_to_eq | @eq_to_eqbnat_true]
302 definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true.
304 definition a ≝ s DeqNat O.
305 definition b ≝ s DeqNat (S O).
306 definition c ≝ s DeqNat (S (S O)).
308 definition exp1 ≝ ((a·b)^*·a).
309 definition exp2 ≝ a·(b·a)^*.
310 definition exp4 ≝ (b·a)^*.
312 definition exp6 ≝ a·(a ·a ·b^* + b^* ).
313 definition exp7 ≝ a · a^* · b^*.
315 definition exp8 ≝ a·a·a·a·a·a·a·a·(a^* ).
316 definition exp9 ≝ (a·a·a + a·a·a·a·a)^*.
318 example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true.