(*
-Regular Expressions Equivalence
+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:
*)
include "tutorial/chapter9.ma".
-(* We say that two pres 〈i_1,b_1〉 and 〈i_1,b_1〉 are {\em cofinal} if and
-only if b_1 = b_2. *)
-
-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.
- \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))
- [* * // * #H1 #H2 [@sym_eq @H1 //| @H2 //]]
- #Hcut @Hcut @iff_trans [|@decidable_sem]
- @iff_trans [|@same_sem] @iff_sym @decidable_sem
-|#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
-qed.
+let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
+ match E with
+ [ pz ⇒ 〈 pz S, false 〉
+ | pe ⇒ 〈 ϵ, false 〉
+ | ps y ⇒ 〈 `y, false 〉
+ | pp y ⇒ 〈 `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)^⊛ ].
+
+lemma move_plus: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
+ move S x (i1 + i2) = (move ? x i1) ⊕ (move ? x i2).
+// qed.
-(* 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. *)
-
-definition occ ≝ λS.λe1,e2:pre S.
- unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
-
-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.
+lemma move_cat: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
+ move S x (i1 · i2) = (move ? x i1) ⊙ (move ? x i2).
+// qed.
-(* The following is a stronger version of equiv_sem, relative to characters
-occurring the given regular expressions. *)
+lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
+ move S x i^* = (move ? x i)^⊛.
+// qed.
-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.
+(*
+Example. Let us consider the item
+
+ (•a + ϵ)((•b)*•a + •b)b
-(*
-Bisimulations
+and the two moves w.r.t. the characters a and b.
+For a, we have two possible positions (all other points gets erased); the innermost
+point stops in front of the final b, while the other one broadcast inside (b^*a + b)b,
+so
+
+ move((•a + ϵ)((•b)*•a + •b)b,a) = 〈(a + ϵ)((•b)^*•a + •b)•b, false〉
+For b, we have two positions too. The innermost point stops in front of the final b too,
+while the other point reaches the end of b* and must go back through b*a:
+
+ move((•a + ϵ)((•b)*•a + •b)b ,b) = 〈(a + ϵ)((•b)*•a + b)•b, false〉
-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.
*)
-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.
+definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
-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) >(\P H) /2/ |#H @Hind @H]
-qed.
+lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b.
+ pmove ? x 〈i,b〉 = move ? x i.
+// 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 eq_to_eq_hd: ∀A.∀l1,l2:list A.∀a,b.
+ a::l1 = b::l2 → a = b.
+#A #l1 #l2 #a #b #H destruct //
+qed.
-(* Using lemma equiv_sem_occ it is easy to prove the following result: *)
+(* Obviously, a move does not change the carrier of the item, as one can easily
+prove by induction on the item. *)
-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)
+lemma same_kernel: ∀S:DeqSet.∀a:S.∀i:pitem S.
+ |\fst (move ? a i)| = |i|.
+#S #a #i elim i //
+ [#i1 #i2 #H1 #H2 >move_cat >erase_odot //
+ |#i1 #i2 #H1 #H2 >move_plus whd in ⊢ (??%%); //
]
qed.
-(* 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 ⇒
- match frontier with
- [ nil ⇒ 〈true,visited〉
- | cons hd tl ⇒
- if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
- bisim S l m (unique_append ? (filter ? (λx.notb (memb ? x (hd::visited)))
- (sons S l hd)) tl) (hd::visited)
- else 〈false,visited〉
+(* Here is our first, major result, stating the correctness of the
+move operation. The proof is a simple induction on i. *)
+
+theorem move_ok:
+ ∀S:DeqSet.∀a:S.∀i:pitem S.∀w: word S.
+ \sem{move ? a i} w ↔ \sem{i} (a::w).
+#S #a #i elim i
+ [normalize /2/
+ |normalize /2/
+ |normalize /2/
+ |normalize #x #w cases (true_or_false (a==x)) #H >H normalize
+ [>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
+ |% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
]
- ].
-
-(* 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 unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
- bisim S l n frontier visited =
- match n with
- [ O ⇒ 〈false,visited〉 (* assert false *)
- | S m ⇒
- match frontier with
- [ nil ⇒ 〈true,visited〉
- | cons hd tl ⇒
- if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
- bisim S l m (unique_append ? (filter ? (λx.notb(memb ? x (hd::visited)))
- (sons S l hd)) tl) (hd::visited)
- else 〈false,visited〉
- ]
- ].
-#S #l #n cases n // qed.
+ |#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 //]
+ @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
+ @iff_trans[||@iff_sym @deriv_middot //]
+ @cat_ext_l @HI1
+ ]
+qed.
+
+(* The move operation is generalized to strings in the obvious way. *)
+
+notation > "x ↦* E" non associative with precedence 60 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))].
+
+lemma moves_empty: ∀S:DeqSet.∀e:pre S.
+ moves ? [ ] e = e.
+// qed.
+
+lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S.
+ moves ? (a::w) e = moves ? w (move S a (\fst e)).
+// qed.
-lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
- bisim S l O frontier visited = 〈false,visited〉.
-#frontier #visited >unfold_bisim //
+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 bisim_end: ∀Sig,l,m.∀visited: list ?.
- bisim Sig l (S m) [] visited = 〈true,visited〉.
-#n #visisted >unfold_bisim //
+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
+ [% /2/ * // #H destruct |% normalize /2/]
qed.
-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 ? x (p::visited)))
- (sons Sig l p)) frontier) (p::visited).
-#Sig #l #m #p #frontier #visited #test >unfold_bisim whd in ⊢ (??%?); >test //
+lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
+ |\fst (moves ? w e)| = |\fst e|.
+#S #w elim w //
qed.
-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 whd in ⊢ (??%?); >test //
+theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S.
+ (\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
+ @iff_trans [||@iff_sym @not_epsilon_sem]
+ @iff_trans [||@move_ok] @Hind
+ ]
qed.
-(* In order to prove termination of bisim we must be able to effectively
-enumerate all possible pres: *)
-
-(* lemma notb_eq_true_l: ∀b. notb b = true → b = false.
-#b cases b normalize //
-qed. *)
+(* It is now clear that we can build a DFA D_e for e by taking pre as states,
+and move as transition function; the initial state is •(e) and a state 〈i,b〉 is
+final if and only if b is true. The fact that states in D_e are finite is obvious:
+in fact, their cardinality is at most 2^{n+1} where n is the number of symbols in
+e. This is one of the advantages of pointed regular expressions w.r.t. derivatives,
+whose finite nature only holds after a suitable quotient.
-let rec pitem_enum S (i:re S) on i ≝
- match i with
- [ z ⇒ (pz S)::[]
- | e ⇒ (pe S)::[]
- | s y ⇒ (ps S y)::(pp S y)::[]
- | o i1 i2 ⇒ compose ??? (po S) (pitem_enum S i1) (pitem_enum S i2)
- | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
- | k i ⇒ map ?? (pk S) (pitem_enum S i)
- ].
-
-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.
+Let us discuss a couple of examples.
-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 i2).
+Example.
+Below is the DFA associated with the regular expression (ac+bc)*.
-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.
+DFA for (ac+bc)
-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.
-
-(* 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 〈e_1,e_2〉, hence it is absurd to suppose to meet a pair
-which is not cofinal. *)
-
-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|→
- 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 % >same_kernel_moves //
- |#m #HI * [#visited #vinv #finv >bisim_end //]
- #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 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 @(memb_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.
-
-(* 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. *)
-
-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).
-memb ? x l1 = true → sublist ? (sons ? l x) l2.
-
-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〉 →
- 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
- [#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]
- (* frontier = hd:: tl and hd is ok *)
- #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]]
- (* we now exploit the induction hypothesis *)
- #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 <(\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 *)
- 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 //|//]
- ]
- |(* case x in visited *)
- #H1 #xa #membxa cases (memb_append … (subH x … H1 … membxa))
- [#H2 (cases (orb_true_l … H2))
- [#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.
+The graphical description of the automaton is the traditional one, with nodes for
+states and labelled arcs for transitions. Unreachable states are not shown.
+Final states are emphasized by a double circle: since a state 〈e,b〉 is final if and
+only if b is true, we may just label nodes with the item.
+The automaton is not minimal: it is easy to see that the two states corresponding to
+the items (a•c +bc)* and (ac+b•c)* are equivalent (a way to prove it is to observe
+that they define the same language!). In fact, an important property of pres e is that
+each state has a clear semantics, given in terms of the specification e and not of the
+behaviour of the automaton. As a consequence, the construction of the automaton is not
+only direct, but also extremely intuitive and locally verifiable.
-(* We can now give the definition of the equivalence algorithm, and
-prove that two expressions are equivalente if and only if they define
-the same language. *)
-
-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 //
- ]
- ]
-qed.
+Let us consider a more complex case.
-definition eqbnat ≝ λn,m:nat. eqb n m.
+Example.
+Starting form the regular expression (a+ϵ)(b*a + b)b, we obtain the following automaton.
+
+DFA for (a+ϵ)(b*a + b)b
+
+Remarkably, this DFA is minimal, testifying the small number of states produced by our
+technique (the pair of states 6-8 and 7-9 differ for the fact that 6 and 7
+are final, while 8 and 9 are not).
-lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
-#n #m % [@eqb_true_to_eq | @eq_to_eqb_true]
-qed.
-definition DeqNat ≝ mk_DeqSet nat eqb eqbnat_true.
+Move to pit
+.
-definition a ≝ s DeqNat O.
-definition b ≝ s DeqNat (S O).
-definition c ≝ s DeqNat (S (S O)).
+We conclude this chapter with a few properties of the move opertions in relation
+with the pit state. *)
-definition exp1 ≝ ((a·b)^*·a).
-definition exp2 ≝ a·(b·a)^*.
-definition exp4 ≝ (b·a)^*.
+definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉.
+
+(* The following function compute the list of characters occurring in a given
+item i. *)
+
+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].
+
+(* If a symbol a does not occur in i, then move(i,a) gets to the
+pit state. *)
+
+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.
-definition exp6 ≝ a·(a ·a ·b^* + b^* ).
-definition exp7 ≝ a · a^* · b^*.
+(* We cannot escape form the pit state. *)
-definition exp8 ≝ a·a·a·a·a·a·a·a·(a^* ).
-definition exp9 ≝ (a·a·a + a·a·a·a·a)^*.
+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.
-example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true.
-normalize // qed.
\ No newline at end of file
+lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
+#S #w #i elim w //
+qed.
+
+(* If any character in w does not occur in i, then moves(i,w) gets
+to the pit state. *)
+
+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.
\ No newline at end of file
projects / helm.git / blobdiff