[>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
|% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
]
- |#i1 #i2 #HI1 #HI2 #w >(sem_cat S i1 i2) >move_cat
+ |#i1 #i2 #HI1 #HI2 #w >move_cat
@iff_trans[|@sem_odot] >same_kernel >sem_cat_w
@iff_trans[||@(iff_or_l … (HI2 w))] @iff_or_r
@iff_trans[||@iff_sym @deriv_middot //]
let rec moves (S : DeqSet) w e on w : pre S ≝
match w with
[ nil ⇒ e
- | cons x w' ⇒ w' ↦* (move S x (\fst e))].
+ | cons x w' ⇒ w' ↦* (move S x (\fst e))].
lemma moves_empty: ∀S:DeqSet.∀e:pre S.
moves ? [ ] e = e.
moves ? (a::w) e = moves ? w (move S a (\fst e)).
// qed.
+lemma moves_left : ∀S,a,w,e.
+ moves S (w@[a]) e = move S a (\fst (moves S w e)).
+#S #a #w elim w // #x #tl #Hind #e >moves_cons >moves_cons //
+qed.
+
lemma not_epsilon_sem: ∀S:DeqSet.∀a:S.∀w: word S. ∀e:pre S.
iff ((a::w) ∈ e) ((a::w) ∈ \fst e).
#S #a #w * #i #b cases b normalize
qed.
theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S.
- (\snd (moves ? w e) = true) ↔ \sem{e} w.
+ (\snd (moves ? w e) = true) ↔ \sem{e} w.
#S #w elim w
[* #i #b >moves_empty cases b % /2/
|#a #w1 #Hind #e >moves_cons
]
qed.
-lemma not_true_to_false: ∀b.b≠true → b =false.
+(* lemma not_true_to_false: ∀b.b≠true → b =false.
#b * cases b // #H @False_ind /2/
+qed. *)
+
+(************************ pit state ***************************)
+definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉.
+
+let rec occur (S: DeqSet) (i: re S) on i ≝
+ match i with
+ [ z ⇒ [ ]
+ | e ⇒ [ ]
+ | s y ⇒ [y]
+ | o e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2)
+ | c e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2)
+ | k e ⇒ occur S e].
+
+lemma not_occur_to_pit: ∀S,a.∀i:pitem S. memb S a (occur S (|i|)) ≠ true →
+ move S a i = pit_pre S i.
+#S #a #i elim i //
+ [#x normalize cases (a==x) normalize // #H @False_ind /2/
+ |#i1 #i2 #Hind1 #Hind2 #H >move_cat
+ >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
+ >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
+ |#i1 #i2 #Hind1 #Hind2 #H >move_plus
+ >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
+ >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
+ |#i #Hind #H >move_star >Hind //
+ ]
+qed.
+
+lemma move_pit: ∀S,a,i. move S a (\fst (pit_pre S i)) = pit_pre S i.
+#S #a #i elim i //
+ [#i1 #i2 #Hind1 #Hind2 >move_cat >Hind1 >Hind2 //
+ |#i1 #i2 #Hind1 #Hind2 >move_plus >Hind1 >Hind2 //
+ |#i #Hind >move_star >Hind //
+ ]
qed.
+lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
+#S #w #i elim w // #a #tl >moves_cons //
+qed.
+
+lemma to_pit: ∀S,w,e. ¬ sublist S w (occur S (|\fst e|)) →
+ moves S w e = pit_pre S (\fst e).
+#S #w elim w
+ [#e * #H @False_ind @H normalize #a #abs @False_ind /2/
+ |#a #tl #Hind #e #H cases (true_or_false (memb S a (occur S (|\fst e|))))
+ [#Htrue >moves_cons whd in ⊢ (???%); <(same_kernel … a)
+ @Hind >same_kernel @(not_to_not … H) #H1 #b #memb cases (orb_true_l … memb)
+ [#H2 >(\P H2) // |#H2 @H1 //]
+ |#Hfalse >moves_cons >not_occur_to_pit // >Hfalse /2/
+ ]
+ ]
+qed.
+
+(* bisimulation *)
+definition cofinal ≝ λS.λp:(pre S)×(pre S).
+ \snd (\fst p) = \snd (\snd p).
+
theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S.
- iff (\sem{e1} =1 \sem{e2}) (∀w.\snd (moves ? w e1) = \snd (moves ? w e2)).
+ \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
#S #e1 #e2 %
[#same_sem #w
cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2))
|#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
qed.
-lemma moves_left : ∀S,a,w,e.
- moves S (w@[a]) e = move S a (\fst (moves S w e)).
-#S #a #w elim w // #x #tl #Hind #e >moves_cons >moves_cons //
-qed.
-
-definition in_moves ≝ λS:DeqSet.λw.λe:pre S. \snd(w ↦* e).
-
-(*
-coinductive equiv (S:DeqSet) : pre S → pre S → Prop ≝
- mk_equiv:
- ∀e1,e2: pre S.
- \snd e1 = \snd e2 →
- (∀x. equiv S (move ? x (\fst e1)) (move ? x (\fst e2))) →
- equiv S e1 e2.
-*)
-
-let rec beqitem S (i1,i2: pitem S) on i1 ≝
- match i1 with
- [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
- | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
- | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
- | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
- | po i11 i12 ⇒ match i2 with
- [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
- | _ ⇒ false]
- | pc i11 i12 ⇒ match i2 with
- [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
- | _ ⇒ false]
- | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
- ].
+definition occ ≝ λS.λe1,e2:pre S.
+ unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
-lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
-#S #i1 elim i1
- [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
- |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
- |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
- [>(\P H) // | @(\b (refl …))]
- |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
- [>(\P H) // | @(\b (refl …))]
- |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
- normalize #H destruct
- [cases (true_or_false (beqitem S i11 i21)) #H1
- [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
- |>H1 in H; normalize #abs @False_ind /2/
- ]
- |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
- ]
- |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
- normalize #H destruct
- [cases (true_or_false (beqitem S i11 i21)) #H1
- [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
- |>H1 in H; normalize #abs @False_ind /2/
- ]
- |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
- ]
- |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
- normalize #H destruct
- [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
- ]
-qed.
+lemma occ_enough: ∀S.∀e1,e2:pre S.
+(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
+ →∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
+#S #e1 #e2 #H #w
+cases (decidable_sublist S w (occ S e1 e2)) [@H] -H #H
+ >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //]
+ >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l2 @H1 //]
+ //
+qed.
-definition DeqItem ≝ λS.
- mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
-
-unification hint 0 ≔ S;
- X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
-(* ---------------------------------------- *) ⊢
- pitem S ≡ carr X.
-
-unification hint 0 ≔ S,i1,i2;
- X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
-(* ---------------------------------------- *) ⊢
- beqitem S i1 i2 ≡ eqb X i1 i2.
+lemma equiv_sem_occ: ∀S.∀e1,e2:pre S.
+(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
+→ \sem{e1}=1\sem{e2}.
+#S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H
+qed.
definition sons ≝ λS:DeqSet.λl:list S.λp:(pre S)×(pre S).
map ?? (λa.〈move S a (\fst (\fst p)),move S a (\fst (\snd p))〉) l.
move ? a (\fst (\snd q)) = \snd p).
#S #l elim l [#p #q normalize in ⊢ (%→?); #abs @False_ind /2/]
#a #tl #Hind #p #q #H cases (orb_true_l … H) -H
- [#H @(ex_intro … a) <(proj1 … (eqb_true …)H) /2/
- |#H @Hind @H
+ [#H @(ex_intro … a) >(\P H) /2/ |#H @Hind @H]
+qed.
+
+definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S.
+ ∀p:(pre S)×(pre S). memb ? p l = true → cofinal ? p ∧ (sublist ? (sons ? alpha p) l).
+
+lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S.
+ is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}.
+#S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ
+#w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?))
+lapply Hsub @(list_elim_left … w) [//]
+#a #w1 #Hind #Hsub >moves_left >moves_left @(proj2 …(Hbisim …(Hind ?)))
+ [#x #Hx @Hsub @memb_append_l1 //
+ |cut (memb S a (occ S e1 e2) = true) [@Hsub @memb_append_l2 //] #occa
+ @(memb_map … occa)
]
qed.
+(* the algorithm *)
let rec bisim S l n (frontier,visited: list ?) on n ≝
match n with
[ O ⇒ 〈false,visited〉 (* assert false *)
bisim Sig l (S m) (p::frontier) visited = 〈false,visited〉.
#Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
qed.
-
-definition visited_inv ≝ λS.λe1,e2:pre S.λvisited: list ?.
-uniqueb ? visited = true ∧
- ∀p. memb ? p visited = true →
- (∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p)) ∧
- (beqb (\snd (\fst p)) (\snd (\snd p)) = true).
-
-definition frontier_inv ≝ λS.λfrontier,visited.
-uniqueb ? frontier = true ∧
-∀p:(pre S)×(pre S). memb ? p frontier = true →
- memb ? p visited = false ∧
- ∃p1.((memb ? p1 visited = true) ∧
- (∃a. move ? a (\fst (\fst p1)) = \fst p ∧
- move ? a (\fst (\snd p1)) = \snd p)).
-
-(* lemma andb_true: ∀b1,b2:bool.
- (b1 ∧ b2) = true → (b1 = true) ∧ (b2 = true).
-#b1 #b2 cases b1 normalize #H [>H /2/ |@False_ind /2/].
-qed.
-
-lemma andb_true_r: ∀b1,b2:bool.
- (b1 = true) ∧ (b2 = true) → (b1 ∧ b2) = true.
-#b1 #b2 cases b1 normalize * //
-qed. *)
lemma notb_eq_true_l: ∀b. notb b = true → b = false.
#b cases b normalize //
qed.
-(*
-lemma notb_eq_true_r: ∀b. b = false → notb b = true.
-#b cases b normalize //
-qed.
-
-lemma notb_eq_false_l:∀b. notb b = false → b = true.
-#b cases b normalize //
-qed.
-
-lemma notb_eq_false_r:∀b. b = true → notb b = false.
-#b cases b normalize //
-qed. *)
-
-(* include "arithmetics/exp.ma". *)
-
-let rec pos S (i:re S) on i ≝
- match i with
- [ z ⇒ 0
- | e ⇒ 0
- | s y ⇒ 1
- | o i1 i2 ⇒ pos S i1 + pos S i2
- | c i1 i2 ⇒ pos S i1 + pos S i2
- | k i ⇒ pos S i
- ].
-
-
let rec pitem_enum S (i:re S) on i ≝
match i with
[ z ⇒ [pz S]
#S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉))
// qed.
-definition visited_inv_1 ≝ λS.λe1,e2:pre S.λvisited: list ?.
-uniqueb ? visited = true ∧
- ∀p. memb ? p visited = true →
+definition all_reachable ≝ λS.λe1,e2:pre S.λl: list ?.
+uniqueb ? l = true ∧
+ ∀p. memb ? p l = true →
∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p).
-
-lemma bisim_ok1: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} →
- ∀l,n.∀frontier,visited:list (*(space S) *) ((pre S)×(pre S)).
+
+definition disjoint ≝ λS:DeqSet.λl1,l2.
+ ∀p:S. memb S p l1 = true → memb S p l2 = false.
+
+lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} →
+ ∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
|space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
- visited_inv_1 S e1 e2 visited → frontier_inv S frontier visited →
+ all_reachable S e1 e2 visited →
+ all_reachable S e1 e2 frontier →
+ disjoint ? frontier visited →
\fst (bisim S l n frontier visited) = true.
#Sig #e1 #e2 #same #l #n elim n
[#frontier #visited #abs * #unique #H @False_ind @(absurd … abs)
[|* #H1 #H2 <H1 <H2 @space_enum_complete]
cases (H … membp) #w * #we1 #we2 <we1 <we2 % >same_kernel_moves //
|#m #HI * [#visited #vinv #finv >bisim_end //]
- #p #front_tl #visited #Hn * #u_visited #vinv * #u_frontier #finv
- cases (finv p (memb_hd …)) #Hp * #p2 * #visited_p2
- * #a * #movea1 #movea2
- cut (∃w.(moves Sig w e1 = \fst p) ∧ (moves Sig w e2 = \snd p))
- [cases (vinv … visited_p2) -vinv #w1 * #mw1 #mw2
- @(ex_intro … (w1@[a])) % //]
- -movea2 -movea1 -a -visited_p2 -p2 #reachp
+ #p #front_tl #visited #Hn * #u_visited #r_visited * #u_frontier #r_frontier
+ #disjoint
+ cut (∃w.(moves ? w e1 = \fst p) ∧ (moves ? w e2 = \snd p))
+ [@(r_frontier … (memb_hd … ))] #rp
cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true)
- [cases reachp #w * #move_e1 #move_e2 <move_e1 <move_e2
- @(\b ?) @(proj1 … (equiv_sem … )) @same] #ptest
+ [cases rp #w * #fstp #sndp <fstp <sndp @(\b ?)
+ @(proj1 … (equiv_sem … )) @same] #ptest
>(bisim_step_true … ptest) @HI -HI
[<plus_n_Sm //
- |% [whd in ⊢ (??%?); >Hp whd in ⊢ (??%?); //]
- #p1 #H (cases (orb_true_l … H))
- [#eqp <(\P eqp) //
- |#visited_p1 @(vinv … visited_p1)
- ]
+ |% [whd in ⊢ (??%?); >(disjoint … (memb_hd …)) whd in ⊢ (??%?); //
+ |#p1 #H (cases (orb_true_l … H)) [#eqp >(\P eqp) // |@r_visited]
+ ]
|whd % [@unique_append_unique @(andb_true_r … u_frontier)]
@unique_append_elim #q #H
- [%
- [@notb_eq_true_l @(filter_true … H)
- |@(ex_intro … p) % [@memb_hd|@(memb_sons … (memb_filter_memb … H))]
- ]
- |cases (finv q ?) [|@memb_cons //]
- #nvq * #p1 * #Hp1 #reach %
- [cut ((p==q) = false) [|#Hpq whd in ⊢ (??%?); >Hpq @nvq]
- cases (andb_true … u_frontier) #notp #_
- @(not_memb_to_not_eq … H) @notb_eq_true_l @notp
- |cases (proj2 … (finv q ?))
- [#p1 * #Hp1 #reach @(ex_intro … p1) % // @memb_cons //
- |@memb_cons //
- ]
- ]
- ]
- ]
- ]
-qed.
+ [cases (memb_sons … (memb_filter_memb … H)) -H
+ #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(ex_intro … (w1@[a]))
+ >moves_left >moves_left >mw1 >mw2 >m1 >m2 % //
+ |@r_frontier @memb_cons //
+ ]
+ |@unique_append_elim #q #H
+ [@injective_notb @(filter_true … H)
+ |cut ((q==p) = false)
+ [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @memb_cons //]
+ cases (andb_true … u_frontier) #notp #_ @(\bf ?)
+ @(not_to_not … not_eq_true_false) #eqqp <notp <eqqp >H //
+ ]
+ ]
+ ]
+qed.
definition all_true ≝ λS.λl.∀p:(pre S) × (pre S). memb ? p l = true →
(beqb (\snd (\fst p)) (\snd (\snd p)) = true).
-definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S).∀a:S.
-memb ? x l1 = true → memb S a l = true →
- memb ? 〈move ? a (\fst (\fst x)), move ? a (\fst (\snd x))〉 l2 = true.
+definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S).
+memb ? x l1 = true → sublist ? (sons ? l x) l2.
-lemma reachable_bisim:
+lemma bisim_complete:
∀S,l,n.∀frontier,visited,visited_res:list ?.
all_true S visited →
sub_sons S l visited (frontier@visited) →
- bisim S l n frontier visited = 〈true,visited_res〉 →
- (sub_sons S l visited_res visited_res ∧
- sublist ? visited visited_res ∧
- all_true S visited_res).
+ bisim S l n frontier visited = 〈true,visited_res〉 →
+ is_bisim S visited_res l ∧ sublist ? (frontier@visited) visited_res.
#S #l #n elim n
[#fron #vis #vis_res #_ #_ >bisim_never #H destruct
|#m #Hind *
[(* case empty frontier *)
-Hind #vis #vis_res #allv #H normalize in ⊢ (%→?);
- #H1 destruct % // % // #p /2 by /
+ #H1 destruct % #p
+ [#membp % [@(\P ?) @allv //| @H //]|#H1 @H1]
|#hd cases (true_or_false (beqb (\snd (\fst hd)) (\snd (\snd hd))))
[|(* case head of the frontier is non ok (absurd) *)
#H #tl #vis #vis_res #allv >(bisim_step_false … H) #_ #H1 destruct]
#H #tl #visited #visited_res #allv >(bisim_step_true … H)
(* new_visited = hd::visited are all ok *)
cut (all_true S (hd::visited))
- [#p #H1 cases (orb_true_l … H1) [#eqp <(\P eqp) @H |@allv]]
+ [#p #H1 cases (orb_true_l … H1) [#eqp >(\P eqp) @H |@allv]]
(* we now exploit the induction hypothesis *)
- #allh #subH #bisim cases (Hind … allh … bisim) -Hind
- [* #H1 #H2 #H3 % // % // #p #H4 @H2 @memb_cons //]
- (* the only thing left to prove is the sub_sons invariant *)
- #x #a #membx #memba
- cases (orb_true_l … membx)
+ #allh #subH #bisim cases (Hind … allh … bisim) -bisim -Hind
+ [#H1 #H2 % // #p #membp @H2 -H2 cases (memb_append … membp) -membp #membp
+ [cases (orb_true_l … membp) -membp #membp
+ [@memb_append_l2 >(\P membp) @memb_hd
+ |@memb_append_l1 @sublist_unique_append_l2 //
+ ]
+ |@memb_append_l2 @memb_cons //
+ ]
+ |(* the only thing left to prove is the sub_sons invariant *)
+ #x #membx cases (orb_true_l … membx)
[(* case x = hd *)
- #eqhdx >(proj1 … (eqb_true …) eqhdx)
- (* xa is the son of x w.r.t. a; we must distinguish the case xa
+ #eqhdx <(\P eqhdx) #xa #membxa
+ (* xa is a son of x; we must distinguish the case xa
was already visited form the case xa is new *)
- letin xa ≝ 〈move S a (\fst (\fst x)), move S a (\fst (\snd x))〉
cases (true_or_false … (memb ? xa (x::visited)))
[(* xa visited - trivial *) #membxa @memb_append_l2 //
|(* xa new *) #membxa @memb_append_l1 @sublist_unique_append_l1 @memb_filter_l
- [>membxa //
- |(* this can be probably improved *)
- generalize in match memba; -memba elim l
- [whd in ⊢ (??%?→?); #abs @False_ind /2/
- |#b #others #Hind #memba cases (orb_true_l … memba) #H
- [>(proj1 … (eqb_true …) H) @memb_hd
- |@memb_cons @Hind //
- ]
- ]
- ]
+ [>membxa //|//]
]
|(* case x in visited *)
- #H1 letin xa ≝ 〈move S a (\fst (\fst x)), move S a (\fst (\snd x))〉
- cases (memb_append … (subH x a H1 memba))
+ #H1 #xa #membxa cases (memb_append … (subH x … H1 … membxa))
[#H2 (cases (orb_true_l … H2))
- [#H3 @memb_append_l2 >(proj1 … (eqb_true …) H3) @memb_hd
+ [#H3 @memb_append_l2 <(\P H3) @memb_hd
|#H3 @memb_append_l1 @sublist_unique_append_l2 @H3
]
|#H2 @memb_append_l2 @memb_cons @H2
]
qed.
-(* pit state *)
-let rec blank_item (S: DeqSet) (i: re S) on i :pitem S ≝
- match i with
- [ z ⇒ `∅
- | e ⇒ ϵ
- | s y ⇒ `y
- | o e1 e2 ⇒ (blank_item S e1) + (blank_item S e2)
- | c e1 e2 ⇒ (blank_item S e1) · (blank_item S e2)
- | k e ⇒ (blank_item S e)^* ].
-
-definition pit_pre ≝ λS.λi.〈blank_item S (|i|), false〉.
-
-let rec occur (S: DeqSet) (i: re S) on i ≝
- match i with
- [ z ⇒ [ ]
- | e ⇒ [ ]
- | s y ⇒ [y]
- | o e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2)
- | c e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2)
- | k e ⇒ occur S e].
-
-axiom memb_single: ∀S,a,x. memb S a [x] = true → a = x.
-
-axiom tech: ∀b. b ≠ true → b = false.
-axiom tech2: ∀b. b = false → b ≠ true.
-
-lemma not_occur_to_pit: ∀S,a.∀i:pitem S. memb S a (occur S (|i|)) = false →
- move S a i = pit_pre S i.
-#S #a #i elim i //
- [#x cases (true_or_false (a==x))
- [#H >(proj1 …(eqb_true …) H) whd in ⊢ ((??%?)→?);
- >(proj2 …(eqb_true …) (refl …)) whd in ⊢ ((??%?)→?); #abs @False_ind /2/
- |#H normalize >H //
+definition equiv ≝ λSig.λre1,re2:re Sig.
+ let e1 ≝ •(blank ? re1) in
+ let e2 ≝ •(blank ? re2) in
+ let n ≝ S (length ? (space_enum Sig (|\fst e1|) (|\fst e2|))) in
+ let sig ≝ (occ Sig e1 e2) in
+ (bisim ? sig n [〈e1,e2〉] []).
+
+theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig.
+ \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}.
+#Sig #re1 #re2 %
+ [#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding]
+ cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉)
+ [<H //] #Hcut
+ cases (bisim_complete … Hcut)
+ [2,3: #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/]
+ #Hbisim #Hsub @(bisim_to_sem … Hbisim)
+ @Hsub @memb_hd
+ |#H @(bisim_correct ? (•(blank ? re1)) (•(blank ? re2)))
+ [@eqP_trans [|@re_embedding] @eqP_trans [|@H] @eqP_sym @re_embedding
+ |//
+ |% // #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/
+ |% // #p #H >(memb_single … H) @(ex_intro … ϵ) /2/
+ |#p #_ normalize //
]
- |#i1 #i2 #Hind1 #Hind2 #H >move_cat >Hind1 [2:@tech
- @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l1 //]
- >Hind2 [2:@tech @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l2 //]
- //
- |#i1 #i2 #Hind1 #Hind2 #H >move_plus >Hind1 [2:@tech
- @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l1 //]
- >Hind2 [2:@tech @(not_to_not … (tech2 … H)) #H1 @sublist_unique_append_l2 //]
- //
- |#i #Hind #H >move_star >Hind // @H
]
qed.
-lemma move_pit: ∀S,a,i. move S a (\fst (pit_pre S i)) = pit_pre S i.
-#S #a #i elim i //
- [#i1 #i2 #Hind1 #Hind2 >move_cat >Hind1 >Hind2 //
- |#i1 #i2 #Hind1 #Hind2 >move_plus >Hind1 >Hind2 //
- |#i #Hind >move_star >Hind //
- ]
-qed.
+definition eqbnat ≝ λn,m:nat. eqb n m.
-lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
-#S #w #i elim w // #a #tl >moves_cons //
-qed.
-
-lemma to_pit: ∀S,w,e. ¬ sublist S w (occur S (|\fst e|)) →
- moves S w e = pit_pre S (\fst e).
-#S #w elim w
- [(* orribile *)
- #e * #H @False_ind @H normalize #a #abs @False_ind /2/
- |#a #tl #Hind #e #H cases (true_or_false (memb S a (occur S (|\fst e|))))
- [#Htrue >moves_cons whd in ⊢ (???%); <(same_kernel … a)
- @Hind >same_kernel @(not_to_not … H) #H1 #b #memb cases (orb_true_l … memb)
- [#H2 <(proj1 … (eqb_true …) H2) // |#H2 @H1 //]
- |#Hfalse >moves_cons >not_occur_to_pit //
- ]
- ]
+lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
+#n #m % [@eqb_true_to_eq | @eq_to_eqb_true]
qed.
-
-definition occ ≝ λS.λe1,e2:pre S.
- unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
-(* definition occS ≝ λS:DeqSet.λoccur.
- PSig S (λx.memb S x occur = true). *)
+definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true.
+
+definition a ≝ s DeqNat 0.
+definition b ≝ s DeqNat 1.
+definition c ≝ s DeqNat 2.
+
+definition exp1 ≝ ((a·b)^*·a).
+definition exp2 ≝ a·(b·a)^*.
+definition exp4 ≝ (b·a)^*.
+
+definition exp6 ≝ a·(a ·a ·b^* + b^* ).
+definition exp7 ≝ a · a^* · b^*.
+
+definition exp8 ≝ a·a·a·a·a·a·a·a·(a^* ).
+definition exp9 ≝ (a·a·a + a·a·a·a·a)^*.
+
+example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true.
+normalize // qed.
-lemma occ_enough: ∀S.∀e1,e2:pre S.
-(∀w.(sublist S w (occ S e1 e2))→
- (beqb (\snd (moves S w e1)) (\snd (moves ? w e2))) = true) \to
-∀w.(beqb (\snd (moves S w e1)) (\snd (moves ? w e2))) = true.
-#S #e1 #e2 #H #w
-cut (sublist S w (occ S e1 e2) ∨ ¬(sublist S w (occ S e1 e2)))
-[elim w
- [%1 #a normalize in ⊢ (%→?); #abs @False_ind /2/
- |#a #tl * #subtl
- [cases (true_or_false (memb S a (occ S e1 e2))) #memba
- [%1 whd #x #membx cases (orb_true_l … membx)
- [#eqax <(proj1 … (eqb_true …) eqax) //
- |@subtl
- ]
- |%2 @(not_to_not … (tech2 … memba)) #H1 @H1 @memb_hd
- ]
- |%2 @(not_to_not … subtl) #H1 #x #H2 @H1 @memb_cons //
- ]
- ]
-|* [@H]
- #H >to_pit
- [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //]
- >to_pit
- [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l2 @H1 //]
- //
-]
-qed.
-lemma bisim_char: ∀S.∀e1,e2:pre S.
-(∀w.(beqb (\snd (moves S w e1)) (\snd (moves ? w e2))) = true) →
-\sem{e1}=1\sem{e2}.
-#S #e1 #e2 #H @(proj2 … (equiv_sem …)) #w @(\P ?) @H
-qed.
-lemma bisim_ok2: ∀S.∀e1,e2:pre S.
- (beqb (\snd e1) (\snd e2) = true) → ∀n.
- \fst (bisim S (occ S e1 e2) n (sons S (occ S e1 e2) 〈e1,e2〉) [〈e1,e2〉]) = true →
- \sem{e1}=1\sem{e2}.
-#S #e1 #e2 #Hnil #n
-letin rsig ≝ (occ S e1 e2)
-letin frontier ≝ (sons S rsig 〈e1,e2〉)
-letin visited_res ≝ (\snd (bisim S rsig n frontier [〈e1,e2〉]))
-#bisim_true
-cut (bisim S rsig n frontier [〈e1,e2〉] = 〈true,visited_res〉)
- [<bisim_true <eq_pair_fst_snd //] #H
-cut (all_true S [〈e1,e2〉])
- [#p #Hp cases (orb_true_l … Hp)
- [#eqp <(proj1 … (eqb_true …) eqp) //
- | whd in ⊢ ((??%?)→?); #abs @False_ind /2/
- ]] #allH
-cut (sub_sons S rsig [〈e1,e2〉] (frontier@[〈e1,e2〉]))
- [#x #a #H1 cases (orb_true_l … H1)
- [#eqx <(proj1 … (eqb_true …) eqx) #H2 @memb_append_l1
- whd in ⊢ (??(???%)?); @(memb_map … H2)
- |whd in ⊢ ((??%?)→?); #abs @False_ind /2/
- ]
- ] #init
-cases (reachable_bisim … allH init … H) * #H1 #H2 #H3
-cut (∀w.sublist ? w (occ S e1 e2)→∀p.memb ? p visited_res = true →
- memb ? 〈moves ? w (\fst p), moves ? w (\snd p)〉 visited_res = true)
- [#w elim w [#_ #p #H4 >moves_empty >moves_empty <eq_pair_fst_snd //]
- #a #w1 #Hind #Hsub * #e11 #e21 #visp >moves_cons >moves_cons
- @(Hind ? 〈?,?〉) [#x #H4 @Hsub @memb_cons //]
- @(H1 〈?,?〉) [@visp| @Hsub @memb_hd]] #all_reach
-@bisim_char @occ_enough
-#w #Hsub @(H3 〈?,?〉) @(all_reach w Hsub 〈?,?〉) @H2 //
-qed.
-
-(*
-definition tt ≝ ps Bin true.
-definition ff ≝ ps Bin false.
-definition eps ≝ pe Bin.
-definition exp1 ≝ (ff + tt · ff).
-definition exp2 ≝ ff · (eps + tt).
-
-definition exp3 ≝ move Bin true (\fst (•exp1)).
-definition exp4 ≝ move Bin true (\fst (•exp2)).
-definition exp5 ≝ move Bin false (\fst (•exp1)).
-definition exp6 ≝ move Bin false (\fst (•exp2)). *)
| pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
| po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
| pk E ⇒ (forget ? E)^* ].
-
+
(* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
interpretation "forget" 'norm a = (forget ? a).
+lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
+// qed.
+
+lemma erase_plus : ∀S.∀i1,i2:pitem S.
+ |i1 + i2| = |i1| + |i2|.
+// qed.
+
+lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
+// qed.
+
+(* boolean equality *)
+let rec beqitem S (i1,i2: pitem S) on i1 ≝
+ match i1 with
+ [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
+ | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
+ | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
+ | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
+ | po i11 i12 ⇒ match i2 with
+ [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+ | _ ⇒ false]
+ | pc i11 i12 ⇒ match i2 with
+ [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+ | _ ⇒ false]
+ | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
+ ].
+
+lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
+#S #i1 elim i1
+ [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+ |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+ |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+ [>(\P H) // | @(\b (refl …))]
+ |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+ [>(\P H) // | @(\b (refl …))]
+ |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+ normalize #H destruct
+ [cases (true_or_false (beqitem S i11 i21)) #H1
+ [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+ |>H1 in H; normalize #abs @False_ind /2/
+ ]
+ |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+ ]
+ |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+ normalize #H destruct
+ [cases (true_or_false (beqitem S i11 i21)) #H1
+ [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+ |>H1 in H; normalize #abs @False_ind /2/
+ ]
+ |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+ ]
+ |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
+ normalize #H destruct
+ [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
+ ]
+qed.
+
+definition DeqItem ≝ λS.
+ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
+
+unification hint 0 ≔ S;
+ X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢
+ pitem S ≡ carr X.
+
+unification hint 0 ≔ S,i1,i2;
+ X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢
+ beqitem S i1 i2 ≡ eqb X i1 i2.
+
+(* semantics *)
+
let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
match r with
[ pz ⇒ ∅
#S * #i #b #btrue normalize in btrue; >btrue %2 //
qed.
+lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
+#S #i #w %
+ [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
+ |* //
+ ]
+qed.
+
+lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
+#S * #i *
+ [>sem_pre_true normalize in ⊢ (??%?); #w %
+ [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
+ |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
+ ]
+qed.
+
definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
interpretation "oplus" 'oplus a b = (lo ? a b).
definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
-notation "i â\97\82 e" left associative with precedence 60 for @{'ltrif $i $e}.
-interpretation "pre_concat_r" 'ltrif i e = (pre_concat_r ? i e).
+notation "i â\97\83 e" left associative with precedence 60 for @{'lhd $i $e}.
+interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop.
A = B → A =1 B.
#S #A #B #H >H /2/ qed.
lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
- \sem{i â\97\82 e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
+ \sem{i â\97\83 e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
#S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //]
>sem_pre_true >sem_cat >sem_pre_true /2/
qed.
-definition lc ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
+definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
match e1 with
[ mk_Prod i1 b1 ⇒ match b1 with
- [ true â\87\92 (i1 â\97\82 (bcast ? i2))
+ [ true â\87\92 (i1 â\97\83 (bcast ? i2))
| false ⇒ 〈i1 · i2,false〉
]
].
-
-definition lift ≝ λS.λf:pitem S →pre S.λe:pre S.
- match e with
- [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
-notation "a ▸ b" left associative with precedence 60 for @{'lc eclose $a $b}.
-interpretation "lc" 'lc op a b = (lc ? op a b).
-
-definition lk ≝ λS:DeqSet.λbcast:∀S:DeqSet.∀E:pitem S.pre S.λe:pre S.
- match e with
- [ mk_Prod i1 b1 ⇒
- match b1 with
- [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
- |false ⇒ 〈i1^*,false〉
- ]
- ].
-
-(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.*)
-interpretation "lk" 'lk op a = (lk ? op a).
-notation "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
+notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
+interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
notation "•" non associative with precedence 60 for @{eclose ?}.
| ps x ⇒ 〈 `.x, false〉
| pp x ⇒ 〈 `.x, false 〉
| po i1 i2 ⇒ •i1 ⊕ •i2
- | pc i1 i2 â\87\92 â\80¢i1 â\96¸ i2
+ | pc i1 i2 â\87\92 â\80¢i1 â\96¹ i2
| pk i ⇒ 〈(\fst (•i))^*,true〉].
notation "• x" non associative with precedence 60 for @{'eclose $x}.
// qed.
lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
- â\80¢(i1 · i2) = â\80¢i1 â\96¸ i2.
+ â\80¢(i1 · i2) = â\80¢i1 â\96¹ i2.
// qed.
lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
•i^* = 〈(\fst(•i))^*,true〉.
// qed.
-definition reclose ≝ λS. lift S (eclose S).
-interpretation "reclose" 'eclose x = (reclose ? x).
+definition lift ≝ λS.λf:pitem S →pre S.λe:pre S.
+ match e with
+ [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
+
+definition preclose ≝ λS. lift S (eclose S).
+interpretation "preclose" 'eclose x = (preclose ? x).
(* theorem 16: 2 *)
lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
lemma odot_true :
∀S.∀i1,i2:pitem S.
- â\8c©i1,trueâ\8cª â\96¸ i2 = i1 â\97\82 (•i2).
+ â\8c©i1,trueâ\8cª â\96¹ i2 = i1 â\97\83 (•i2).
// qed.
lemma odot_true_bis :
∀S.∀i1,i2:pitem S.
- â\8c©i1,trueâ\8cª â\96¸ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+ â\8c©i1,trueâ\8cª â\96¹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
#S #i1 #i2 normalize cases (•i2) // qed.
lemma odot_false:
∀S.∀i1,i2:pitem S.
- â\8c©i1,falseâ\8cª â\96¸ i2 = 〈i1 · i2, false〉.
+ â\8c©i1,falseâ\8cª â\96¹ i2 = 〈i1 · i2, false〉.
// qed.
lemma LcatE : ∀S.∀e1,e2:pitem S.
\sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
// qed.
-lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
-// qed.
-
-lemma erase_plus : ∀S.∀i1,i2:pitem S.
- |i1 + i2| = |i1| + |i2|.
-// qed.
-
-lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
-// qed.
-
lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
#S #i elim i //
[ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
| #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
]
qed.
-
+
+(*
lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
\sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
/2/ qed.
+*)
(* theorem 16: 1 → 3 *)
lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
\sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
- \sem{e1 â\96¸ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+ \sem{e1 â\96¹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
#S * #i1 #b1 #i2 cases b1
[2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
|#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
@eqP_trans [|@eqP_sym @union_assoc ] /3/
]
qed.
-
-lemma sem_fst: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
-#S * #i *
- [>sem_pre_true normalize in ⊢ (??%?); #w %
- [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
- |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
- ]
-qed.
-
-lemma item_eps: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
-#S #i #w %
- [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
- |* //
- ]
-qed.
-lemma sem_fst_aux: ∀S.∀e:pre S.∀i:pitem S.∀A.
+lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A.
\sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
#S #e #i #A #seme
-@eqP_trans [|@sem_fst]
-@eqP_trans [||@eqP_union_r [|@eqP_sym @item_eps]]
+@eqP_trans [|@minus_eps_pre]
+@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
@eqP_trans [||@distribute_substract]
@eqP_substract_r //
qed.
(* theorem 16: 1 *)
-theorem sem_bull: ∀S:DeqSet. ∀e:pitem S. \sem{•e} =1 \sem{e} ∪ \sem{|e|}.
+theorem sem_bull: ∀S:DeqSet. ∀i:pitem S. \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
#S #e elim e
[#w normalize % [/2/ | * //]
|/2/
@eqP_trans [||@eqP_union_l [|@union_comm]]
@eqP_trans [||@union_assoc] /2/
|#i #H >sem_pre_true >sem_star >erase_bull >sem_star
- @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@sem_fst_aux //]]]
+ @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
@eqP_trans [|@eqP_union_r [|@distr_cat_r]]
@eqP_trans [|@union_assoc] @eqP_union_l >erase_star
@eqP_sym @star_fix_eps
]
qed.
+(* blank item *)
+let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
+ match i with
+ [ z ⇒ `∅
+ | e ⇒ ϵ
+ | s y ⇒ `y
+ | o e1 e2 ⇒ (blank S e1) + (blank S e2)
+ | c e1 e2 ⇒ (blank S e1) · (blank S e2)
+ | k e ⇒ (blank S e)^* ].
+
+lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
+#S #e elim e normalize //
+qed.
+
+lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
+#S #e elim e
+ [1,2:@eq_to_ex_eq //
+ |#s @eq_to_ex_eq //
+ |#e1 #e2 #Hind1 #Hind2 >sem_cat
+ @eqP_trans [||@(union_empty_r … ∅)]
+ @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
+ @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
+ |#e1 #e2 #Hind1 #Hind2 >sem_plus
+ @eqP_trans [||@(union_empty_r … ∅)]
+ @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
+ |#e #Hind >sem_star
+ @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
+ ]
+qed.
+
+theorem re_embedding: ∀S.∀e:re S.
+ \sem{•(blank S e)} =1 \sem{e}.
+#S #e @eqP_trans [|@sem_bull] >forget_blank
+@eqP_trans [|@eqP_union_r [|@sem_blank]]
+@eqP_trans [|@union_comm] @union_empty_r.
+qed.
+
+(* lefted operations *)
definition lifted_cat ≝ λS:DeqSet.λe:pre S.
- lift S (lc S eclose e).
+ lift S (pre_concat_l S eclose e).
notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
#S * #i1 * * #i2 #b2 // >odot_true_b //
qed.
+definition lk ≝ λS:DeqSet.λe:pre S.
+ match e with
+ [ mk_Prod i1 b1 ⇒
+ match b1 with
+ [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
+ |false ⇒ 〈i1^*,false〉
+ ]
+ ].
+
+(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
+interpretation "lk" 'lk a = (lk ? a).
+notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
+
+
lemma ostar_true: ∀S.∀i:pitem S.
〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
// qed.
#S * #i * // qed.
lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
- \sem{e1 â\8a\99 â\8c©i,trueâ\8cª} =1 \sem{e1 â\96¸ i} ∪ { [ ] }.
+ \sem{e1 â\8a\99 â\8c©i,trueâ\8cª} =1 \sem{e1 â\96¹ i} ∪ { [ ] }.
#S #e1 #i
-cut (e1 â\8a\99 â\8c©i,trueâ\8cª = â\8c©\fst (e1 â\96¸ i), \snd(e1 â\96¸ i) ∨ true〉) [//]
-#H >H cases (e1 â\96¸ i) #i1 #b1 cases b1
+cut (e1 â\8a\99 â\8c©i,trueâ\8cª = â\8c©\fst (e1 â\96¹ i), \snd(e1 â\96¹ i) ∨ true〉) [//]
+#H >H cases (e1 â\96¹ i) #i1 #b1 cases b1
[>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
@eqP_union_l /2/
|/2/
qed.
lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i.
- e1 â\8a\99 â\8c©i,falseâ\8cª = e1 â\96¸ i.
+ e1 â\8a\99 â\8c©i,falseâ\8cª = e1 â\96¹ i.
#S #e1 #i
-cut (e1 â\8a\99 â\8c©i,falseâ\8cª = â\8c©\fst (e1 â\96¸ i), \snd(e1 â\96¸ i) ∨ false〉) [//]
-cases (e1 â\96¸ i) #i1 #b1 cases b1 #H @H
+cut (e1 â\8a\99 â\8c©i,falseâ\8cª = â\8c©\fst (e1 â\96¹ i), \snd(e1 â\96¹ i) ∨ false〉) [//]
+cases (e1 â\96¹ i) #i1 #b1 cases b1 #H @H
qed.
lemma sem_odot:
\sem{e^⊛} =1 \sem{e} · \sem{|\fst e|}^*.
#S * #i #b cases b
[>sem_pre_true >sem_pre_true >sem_star >erase_bull
- @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@sem_fst_aux //]]]
+ @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
@eqP_trans [|@eqP_union_r [|@distr_cat_r]]
@eqP_trans [||@eqP_sym @distr_cat_r]
@eqP_trans [|@union_assoc] @eqP_union_l
|>sem_pre_false >sem_pre_false >sem_star /2/
]
qed.
-
+
projects / helm.git / commitdiff