include "re/re.ma".
include "basics/lists/listb.ma".
+(*
+Moves
+
+We now define the move operation, that corresponds to the advancement of the
+state in response to the processing of an input character a. The intuition is
+clear: we have to look at points inside $e$ preceding the given character a,
+let the point traverse the character, and broadcast it. All other points must
+be removed.
+
+We can give a particularly elegant definition in terms of the
+lifted operators of the previous section:
+*)
+
let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
match E with
- [ pz ⇒ 〈 `∅, false 〉
- | pe ⇒ 〈 ϵ, false 〉
- | ps y ⇒ 〈 `y, false 〉
- | pp y ⇒ 〈 `y, x == y 〉
+ [ pz ⇒ 〈 pz ?, false 〉
+ | pe ⇒ 〈 pe ? , false 〉
+ | ps y ⇒ 〈 ps ? y, false 〉
+ | pp y ⇒ 〈 ps ? y, x == y 〉
| po e1 e2 ⇒ (move ? x e1) ⊕ (move ? x e2)
| pc e1 e2 ⇒ (move ? x e1) ⊙ (move ? x e2)
| pk e ⇒ (move ? x e)^⊛ ].
|normalize /2/
|normalize /2/
|normalize #x #w cases (true_or_false (a==x)) #H >H normalize
- [>(proj1 … (eqb_true …) H) %
- [* // #bot @False_ind //| #H1 destruct /2/]
- |% [#bot @False_ind //
- | #H1 destruct @(absurd ((a==a)=true))
- [>(proj2 … (eqb_true …) (refl …)) // | /2/]
- ]
- ]
- |#i1 #i2 #HI1 #HI2 #w >(sem_cat S i1 i2) >move_cat
- @iff_trans[|@sem_odot] >same_kernel >sem_cat_w
- @iff_trans[||@(iff_or_l … (HI2 w))] @iff_or_r %
- [* #w1 * #w2 * * #eqw #w1in #w2in @(ex_intro … (a::w1))
- @(ex_intro … w2) % // % normalize // cases (HI1 w1) /2/
- |* #w1 * #w2 * cases w1
- [* #_ #H @False_ind /2/
- |#x #w3 * #eqaw normalize in eqaw; destruct #w3in #w2in
- @(ex_intro … w3) @(ex_intro … w2) % // % // cases (HI1 w3) /2/
- ]
+ [>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
+ |% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
]
+ |#i1 #i2 #HI1 #HI2 #w
+ (* lhs = w∈\sem{move S a (i1·i2)} *)
+ >move_cat
+ (* lhs = w∈\sem{move S a i1}⊙\sem{move S a i2} *)
+ @iff_trans[|@sem_odot] >same_kernel
+ (* lhs = w∈\sem{move S a i1}·\sem{|i2|} ∨ a∈\sem{move S a i2} *)
+ (* now we work on the rhs, that is
+ rhs = a::w1∈\sem{i1·i2} *)
+ >sem_cat_w
+ (* rhs = a::w1∈\sem{i1}\sem{|i2|} ∨ a::w∈\sem{i2} *)
+ @iff_trans[||@(iff_or_l … (HI2 w))]
+ (* rhs = a::w1∈\sem{i1}\sem{|i2|} ∨ w∈\sem{move S a i2} *)
+ @iff_or_r
+ (* we are left to prove that
+ w∈\sem{move S a i1}·\sem{|i2|} ↔ a::w∈\sem{i1}\sem{|i2|}
+ we use deriv_middot on the rhs *)
+ @iff_trans[||@iff_sym @deriv_middot //]
+ (* w∈\sem{move S a i1}·\sem{|i2|} ↔ w∈(deriv S \sem{i1} a) · \sem{|i2|} *)
+ @cat_ext_l @HI1
|#i1 #i2 #HI1 #HI2 #w >(sem_plus S i1 i2) >move_plus >sem_plus_w
@iff_trans[|@sem_oplus]
@iff_trans[|@iff_or_l [|@HI2]| @iff_or_r //]
|#i1 #HI1 #w >move_star
- @iff_trans[|@sem_ostar] >same_kernel >sem_star_w %
- [* #w1 * #w2 * * #eqw #w1in #w2in
- @(ex_intro … (a::w1)) @(ex_intro … w2) % // % normalize //
- cases (HI1 w1 ) /2/
- |* #w1 * #w2 * cases w1
- [* #_ #H @False_ind /2/
- |#x #w3 * #eqaw normalize in eqaw; destruct #w3in #w2in
- @(ex_intro … w3) @(ex_intro … w2) % // % // cases (HI1 w3) /2/
- ]
- ]
+ @iff_trans[|@sem_ostar] >same_kernel >sem_star_w
+ @iff_trans[||@iff_sym @deriv_middot //]
+ @cat_ext_l @HI1
]
qed.
-notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
+notation > "x ↦* E" non associative with precedence 65 for @{moves ? $x $E}.
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.
-#b * cases b // #H @False_ind /2/
+(************************ 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).
+
+(* As a corollary of decidable_sem, we have that two expressions
+e1 and e2 are equivalent iff for any word w the states reachable
+through w are cofinal. *)
+
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} ≐ \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).
+(* This does not directly imply decidability: we have no bound over the
+length of w; moreover, so far, we made no assumption over the cardinality
+of S. Instead of requiring S to be finite, we may restrict the analysis
+to characters occurring in the given pres. *)
-(*
-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.
-*)
-
-definition beqb ≝ λb1,b2.
- match b1 with
- [ true ⇒ b2
- | false ⇒ notb b2
- ].
+definition occ ≝ λS.λe1,e2:pre S.
+ unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
-lemma beqb_ok: ∀b1,b2. iff (beqb b1 b2 = true) (b1 = b2).
-#b1 #b2 cases b1 cases b2 normalize /2/
+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 Bin ≝ mk_DeqSet bool beqb beqb_ok.
-
-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]
- ].
+(* The following is a stronger version of equiv_sem, relative to characters
+occurring the given regular expressions. *)
-axiom beqitem_ok: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
-
-definition DeqItem ≝ λS.
- mk_DeqSet (pitem S) (beqitem S) (beqitem_ok S).
-
-definition beqpre ≝ λS:DeqSet.λe1,e2:pre S.
- beqitem S (\fst e1) (\fst e2) ∧ beqb (\snd e1) (\snd e2).
-
-definition beqpairs ≝ λS:DeqSet.λp1,p2:(pre S)×(pre S).
- beqpre S (\fst p1) (\fst p2) ∧ beqpre S (\snd p1) (\snd p2).
-
-axiom beqpairs_ok: ∀S,p1,p2. iff (beqpairs S p1 p2 = true) (p1 = p2).
+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}≐\sem{e2}.
+#S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H
+qed.
-definition space ≝ λS.mk_DeqSet ((pre S)×(pre S)) (beqpairs S) (beqpairs_ok S).
+(*
+We say that a list of pairs of pres is a bisimulation if it is closed
+w.r.t. moves, and all its members are cofinal.
+*)
-(* (sons S l p) computes all sons of p relative to characters in l *)
+(* the sons of p w.r.t a list of symbols l are all states reachable from p
+with a move in l *)
-definition sons ≝ λS:DeqSet.λl:list S.λp:space S.
+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.
-lemma memb_sons: ∀S,l,p,q. memb (space S) p (sons S l q) = true →
+lemma memb_sons: ∀S,l.∀p,q:(pre S)×(pre S). memb ? p (sons ? l q) = true →
∃a.(move ? a (\fst (\fst q)) = \fst p ∧
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).
+
+(* Using lemma equiv_sem_occ it is easy to prove the following result: *)
+
+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}≐\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.
-let rec bisim S l n (frontier,visited: list (space S)) on n ≝
+(* This is already an interesting result: checking if l is a bisimulation
+is decidable, hence we could generate l with some untrusted piece of code
+and then run a (boolean version of) is_bisim to check that it is actually
+a bisimulation.
+However, in order to prove that equivalence of regular expressions
+is decidable we must prove that we can always effectively build such a list
+(or find a counterexample).
+The idea is that the list we are interested in is just the set of
+all pair of pres reachable from the initial pair via some
+sequence of moves.
+
+The algorithm for computing reachable nodes in graph is a very
+traditional one. We split nodes in two disjoint lists: a list of
+visited nodes and a frontier, composed by all nodes connected
+to a node in visited but not visited already. At each step we select a node
+a from the frontier, compute its sons, add a to the set of
+visited nodes and the (not already visited) sons to the frontier.
+
+Instead of fist computing reachable nodes and then performing the
+bisimilarity test we can directly integrate it in the algorithm:
+the set of visited nodes is closed by construction w.r.t. reachability,
+so we have just to check cofinality for any node we add to visited.
+
+Here is the extremely simple algorithm: *)
+
+let rec bisim S l n (frontier,visited: list ?) on n ≝
match n with
[ O ⇒ 〈false,visited〉 (* assert false *)
| S m ⇒
]
].
-lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list (space S).
+lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
bisim S l n frontier visited =
match n with
[ O ⇒ 〈false,visited〉 (* assert false *)
]
].
#S #l #n cases n // qed.
-
-lemma bisim_never: ∀S,l.∀frontier,visited: list (space S).
+
+(* The integer n is an upper bound to the number of recursive call,
+equal to the dimension of the graph. It returns a pair composed
+by a boolean and a the set of visited nodes; the boolean is true
+if and only if all visited nodes are cofinal.
+
+The following results explicitly state the behaviour of bisim is the general
+case and in some relevant instances *)
+
+lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
bisim S l O frontier visited = 〈false,visited〉.
#frontier #visited >unfold_bisim //
qed.
-lemma bisim_end: ∀Sig,l,m.∀visited: list (space Sig).
+lemma bisim_end: ∀Sig,l,m.∀visited: list ?.
bisim Sig l (S m) [] visited = 〈true,visited〉.
#n #visisted >unfold_bisim //
qed.
-lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list (space Sig).
+lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
beqb (\snd (\fst p)) (\snd (\snd p)) = true →
bisim Sig l (S m) (p::frontier) visited =
- bisim Sig l m (unique_append ? (filter ? (λx.notb(memb (space Sig) x (p::visited)))
+ bisim Sig l m (unique_append ? (filter ? (λx.notb(memb ? x (p::visited)))
(sons Sig l p)) frontier) (p::visited).
#Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
qed.
-lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list (space Sig).
+lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
beqb (\snd (\fst p)) (\snd (\snd p)) = 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 (space S).
-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: list (space S).
-uniqueb ? frontier = true ∧
-∀p. 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
- ].
+(* In order to prove termination of bisim we must be able to effectively
+enumerate all possible pres: *)
-
let rec pitem_enum S (i:re S) on i ≝
match i with
[ z ⇒ [pz S]
| c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
| k i ⇒ map ?? (pk S) (pitem_enum S i)
].
-
-(* axiom pitem_enum_complete: ∀S:DeqSet.∀i: pitem S.
- memb ((pitem S)×(pitem S)) i (pitem_enum ? (forget ? i)) = true. *)
-(*
-#i elim i
- [//
- |//
- |* //
- |* //
- |#i1 #i2 #Hind1 #Hind2 @memb_compose //
- |#i1 #i2 #Hind1 #Hind2 @memb_compose //
- |
-*)
+
+lemma pitem_enum_complete : ∀S.∀i:pitem S.
+ memb (DeqItem S) i (pitem_enum S (|i|)) = true.
+#S #i elim i
+ [1,2://
+ |3,4:#c normalize >(\b (refl … c)) //
+ |5,6:#i1 #i2 #Hind1 #Hind2 @(memb_compose (DeqItem S) (DeqItem S)) //
+ |#i #Hind @(memb_map (DeqItem S)) //
+ ]
+qed.
definition pre_enum ≝ λS.λi:re S.
compose ??? (λi,b.〈i,b〉) (pitem_enum S i) [true;false].
+
+lemma pre_enum_complete : ∀S.∀e:pre S.
+ memb ? e (pre_enum S (|\fst e|)) = true.
+#S * #i #b @(memb_compose (DeqItem S) DeqBool ? (λi,b.〈i,b〉))
+// cases b normalize //
+qed.
definition space_enum ≝ λS.λi1,i2:re S.
- compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i1).
+ compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i2).
-axiom space_enum_complete : ∀S.∀e1,e2: pre S.
- memb (space S) 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true.
-
-lemma bisim_ok1: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} →
- ∀l,n.∀frontier,visited:list (space S).
+lemma space_enum_complete : ∀S.∀e1,e2: pre S.
+ memb ? 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true.
+#S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉))
+// qed.
+
+(* We are ready to prove that bisim is correct; we use the invariant
+that at each call of bisim the two lists visited and frontier only contain
+nodes reachable from \langle e_1,e_2\rangle, hence it is absurd to suppose
+to meet a pair which is not cofinal. *)
+
+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).
+
+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}≐\sem{e2} →
+ ∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
|space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
- visited_inv 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)
@le_to_not_lt @sublist_length // * #e11 #e21 #membp
cut ((|\fst e11| = |\fst e1|) ∧ (|\fst e21| = |\fst e2|))
[|* #H1 #H2 <H1 <H2 @space_enum_complete]
- cases (H … membp) * #w * #we1 #we2 #_
- <we1 <we2 % //
+ 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])) /2/]
- -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
- @(proj2 … (beqb_ok … )) @(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 <(proj1 … (eqb_true (space Sig) ? p1) eqp) % //
- |#visited_p1 @(vinv … visited_p1)
- ]
- |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_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 rp #w * #fstp #sndp <fstp <sndp @(\b ?)
+ @(proj1 … (equiv_sem … )) @same] #ptest
+ >(bisim_step_true … ptest) @HI -HI
+ [<plus_n_Sm //
+ |% [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
+ [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. memb (space S) p l = true →
+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,a.
-memb (space S) x l1 = true → memb S a l = true →
- memb (space S) 〈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.
+
+(* For completeness, we use the invariant that all the nodes in visited are cofinal,
+and the sons of visited are either in visited or in the frontier; since
+at the end frontier is empty, visited is hence a bisimulation. *)
-lemma reachable_bisim:
- ∀S,l,n.∀frontier,visited,visited_res:list (space S).
+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/
+ #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 #H cases (orb_true_l … H)
- [#eqp <(proj1 … (eqb_true …) eqp) // |@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 (space S) xa (x::visited)))
+ 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} ≐ \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.
-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.
+definition a ≝ s DeqNat 0.
+definition b ≝ s DeqNat 1.
+definition c ≝ s DeqNat 2.
-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 @(proj1 …(beqb_ok …)) @H
-qed.
+definition exp1 ≝ ((a·b)^*·a).
+definition exp2 ≝ a·(b·a)^*.
+definition exp4 ≝ (b·a)^*.
-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 (space S) p visited_res = true →
- memb (space S) 〈moves ? w (\fst p), moves ? w (\snd p)〉 visited_res = true)
- [#w elim w [//]
- #a #w1 #Hind #Hsub * #e11 #e21 #visp >moves_cons >moves_cons
- @(Hind ? 〈?,?〉) [#x #H4 @Hsub @memb_cons //]
- @(H1 〈?,?〉) // @Hsub @memb_hd] #all_reach
-@bisim_char @occ_enough
-#w #Hsub @(H3 〈?,?〉) @(all_reach w Hsub 〈?,?〉) @H2 //
+definition exp5 ≝ (a·(a·(a·b)^*·b)^*·b)^*.
+
+example
+ moves1: \snd (moves DeqNat [0;1;0] (•(blank ? exp2))) = true.
+normalize //
qed.
+
+example
+ moves2: \snd (moves DeqNat [0;1;0;0;0] (•(blank ? exp2))) = false.
+normalize // qed.
+
+example
+ moves3: \snd (moves DeqNat [0;0;0;1;0;1;1;0;1;1] (•(blank ? exp5))) = true.
+normalize // qed.
+
+example
+ moves4: \snd (moves DeqNat [0;0;0;1;0;1;1;0;1;1;1;0] (•(blank ? exp5))) = false.
+normalize // qed.
+
+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.
+
+definition exp10 ≝ a·a·a·a·a·a·a·a·a·a·a·a·(a^* ).
+definition exp11 ≝ (a·a·a·a·a + a·a·a·a·a·a·a)^*.
+
+example ex2 : \fst (equiv ? (exp10+exp11) exp11) = false.
+normalize // qed.
+
+definition exp12 ≝
+ (a·a·a·a·a·a·a·a)·(a·a·a·a·a·a·a·a)·(a·a·a·a·a·a·a·a)·(a^* ).
-definition tt ≝ ps Bin true.
-definition ff ≝ ps Bin false.
-definition eps ≝ pe Bin.
-definition exp1 ≝ (ff + tt · ff).
-definition exp2 ≝ ff · (eps + tt).
+example ex3 : \fst (equiv ? (exp12+exp11) exp11) = true.
+normalize // qed.
+
+let rec raw (n:nat) ≝
+ match n with
+ [ O ⇒ a
+ | S p ⇒ a · (raw p)
+ ].
+
+let rec alln (n:nat) ≝
+ match n with
+ [O ⇒ ϵ
+ |S m ⇒ raw m + alln m
+ ].
+
+definition testA ≝ λx,y,z,b.
+ let e1 ≝ raw x in
+ let e2 ≝ raw y in
+ let e3 ≝ (raw z) · a^* in
+ let e4 ≝ (e1 + e2)^* in
+ \fst (equiv ? (e3+e4) e4) = b.
+
+example ex4 : testA 2 4 7 true.
+normalize // qed.
+
+example ex5 : testA 3 4 10 false.
+normalize // qed.
+
+example ex6 : testA 3 4 11 true.
+normalize // qed.
+
+example ex7 : testA 4 5 18 false.
+normalize // qed.
+
+example ex8 : testA 4 5 19 true.
+normalize // qed.
+
+example ex9 : testA 4 6 22 false.
+normalize // qed.
+
+example ex10 : testA 4 6 23 true.
+normalize // qed.
+
+definition testB ≝ λn,b.
+ \fst (equiv ? ((alln n)·((raw n)^* )) a^* ) = b.
+
+example ex11 : testB 6 true.
+normalize // qed.
+
+example ex12 : testB 8 true.
+normalize // qed.
-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)).
+example ex13 : testB 10 true.
+normalize // qed.
+example ex14 : testB 12 true.
+normalize // qed.
+example ex15 : testB 14 true.
+normalize // qed.
+example ex16 : testB 16 true.
+normalize // qed.
+example ex17 : testB 18 true.
+normalize // qed.
projects / helm.git / blobdiff