-(* We shall apply all the previous machinery to the study of regular languages
+(* <h1>Regular Expressions</h1>
+We shall apply all the previous machinery to the study of regular languages
and the constructions of the associated finite automata. *)
include "tutorial/chapter6.ma".
// qed.
-(* We now introduce pointed regular expressions, that are the main tool we shall
+(* <h2>Pointed Regular expressions </h2>
+We now introduce pointed regular expressions, that are the main tool we shall
use for the construction of the automaton.
A pointed regular expression is just a regular expression internally labelled
with some additional points. Intuitively, points mark the positions inside the
lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
// qed.
-(* Items and pres are very concrete datatypes: they can be effectively compared,
+(* <h2>Comparing items and pres<h2>
+Items and pres are very concrete datatypes: they can be effectively compared,
and enumerated. In particular, we can define a boolean equality beqitem and a proof
beqitem_true that it refects propositional equality, enriching the set (pitem S)
to a DeqSet. *)
(* ---------------------------------------- *) ⊢
beqitem S i1 i2 ≡ eqb X i1 i2.
-(* The intuitive semantic of a point is to mark the position where
+(* <h2>Semantics of pointed regular expression<h2>
+The intuitive semantic of a point is to mark the position where
we should start reading the regular expression. The language associated
to a pre is the union of the languages associated with its points. *)
]
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).
-
-lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
-// qed.
-
-definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
- match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
-
-notation "i ◃ 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 ◃ 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 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 ⇒ (i1 ◃ (bcast ? i2))
- | false ⇒ 〈i1 · i2,false〉
- ]
- ].
-
-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 ?}.
-
-let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
- match i with
- [ pz ⇒ 〈 `∅, false 〉
- | pe ⇒ 〈 ϵ, true 〉
- | ps x ⇒ 〈 `.x, false〉
- | pp x ⇒ 〈 `.x, false 〉
- | po i1 i2 ⇒ •i1 ⊕ •i2
- | pc i1 i2 ⇒ •i1 ▹ i2
- | pk i ⇒ 〈(\fst (•i))^*,true〉].
-
-notation "• x" non associative with precedence 60 for @{'eclose $x}.
-interpretation "eclose" 'eclose x = (eclose ? x).
-
-lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
- •(i1 + i2) = •i1 ⊕ •i2.
-// qed.
-
-lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
- •(i1 · i2) = •i1 ▹ i2.
-// qed.
-
-lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
- •i^* = 〈(\fst(•i))^*,true〉.
-// qed.
-
-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.
- \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
-#S * #i1 #b1 * #i2 #b2 #w %
- [cases b1 cases b2 normalize /2/ * /3/ * /3/
- |cases b1 cases b2 normalize /2/ * /3/ * /3/
- ]
-qed.
-
-lemma odot_true :
- ∀S.∀i1,i2:pitem S.
- 〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
-// qed.
-
-lemma odot_true_bis :
- ∀S.∀i1,i2:pitem S.
- 〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
-#S #i1 #i2 normalize cases (•i2) // qed.
-
-lemma odot_false:
- ∀S.∀i1,i2:pitem S.
- 〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
-// qed.
-
-lemma LcatE : ∀S.∀e1,e2:pitem S.
- \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
-// qed.
-
-lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
-#S #i elim i //
- [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
- cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
- | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
- cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //
- | #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 ▹ 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 …))
- >erase_bull @eqP_trans [|@(eqP_union_l … H)]
- @eqP_trans [|@eqP_union_l[|@union_comm ]]
- @eqP_trans [|@eqP_sym @union_assoc ] /3/
- ]
-qed.
-
-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 [|@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. ∀i:pitem S. \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
-#S #e elim e
- [#w normalize % [/2/ | * //]
- |/2/
- |#x normalize #w % [ /2/ | * [@False_ind | //]]
- |#x normalize #w % [ /2/ | * // ]
- |#i1 #i2 #IH1 #IH2 >eclose_dot
- @eqP_trans [|@odot_dot_aux //] >sem_cat
- @eqP_trans
- [|@eqP_union_r
- [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
- @eqP_trans [|@union_assoc]
- @eqP_trans [||@eqP_sym @union_assoc]
- @eqP_union_l //
- |#i1 #i2 #IH1 #IH2 >eclose_plus
- @eqP_trans [|@sem_oplus] >sem_plus >erase_plus
- @eqP_trans [|@(eqP_union_l … IH2)]
- @eqP_trans [|@eqP_sym @union_assoc]
- @eqP_trans [||@union_assoc] @eqP_union_r
- @eqP_trans [||@eqP_sym @union_assoc]
- @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 [|@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 (pre_concat_l S eclose e).
-
-notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
-
-interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
-
-lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b.
- 〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
-#S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) //
-qed.
-
-lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
- 〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
-//
-qed.
-
-lemma erase_odot:∀S.∀e1,e2:pre S.
- |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
-#S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot //
-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.
-
-lemma ostar_false: ∀S.∀i:pitem S.
- 〈i,false〉^⊛ = 〈i^*, false〉.
-// qed.
-
-lemma erase_ostar: ∀S.∀e:pre S.
- |\fst (e^⊛)| = |\fst e|^*.
-#S * #i * // qed.
-
-lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
- \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
-#S #e1 #i
-cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
-#H >H cases (e1 ▹ 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 ⊙ 〈i,false〉 = e1 ▹ i.
-#S #e1 #i
-cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
-cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
-qed.
-
-lemma sem_odot:
- ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
-#S #e1 * #i2 *
- [>sem_pre_true
- @eqP_trans [|@sem_odot_true]
- @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
- |>sem_pre_false >eq_odot_false @odot_dot_aux //
- ]
-qed.
-
-(* theorem 16: 4 *)
-theorem sem_ostar: ∀S.∀e:pre S.
- \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 [|@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
- @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps
- |>sem_pre_false >sem_pre_false >sem_star /2/
- ]
-qed.
\ No newline at end of file
-include "re.ma".
-include "basics/listb.ma".
+include "tutorial/chapter7.ma".
-let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
- match E with
+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).
+
+lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
+// qed.
+
+definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
+ match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
+
+notation "i ◃ 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 ◃ 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 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 ⇒ (i1 ◃ (bcast ? i2))
+ | false ⇒ 〈i1 · i2,false〉
+ ]
+ ].
+
+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 ?}.
+
+let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
+ match i with
[ pz ⇒ 〈 `∅, 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)^⊛ ].
+ | pe ⇒ 〈 ϵ, true 〉
+ | ps x ⇒ 〈 `.x, false〉
+ | pp x ⇒ 〈 `.x, false 〉
+ | po i1 i2 ⇒ •i1 ⊕ •i2
+ | pc i1 i2 ⇒ •i1 ▹ i2
+ | pk i ⇒ 〈(\fst (•i))^*,true〉].
-lemma move_plus: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
- move S x (i1 + i2) = (move ? x i1) ⊕ (move ? x i2).
-// qed.
+notation "• x" non associative with precedence 60 for @{'eclose $x}.
+interpretation "eclose" 'eclose x = (eclose ? x).
-lemma move_cat: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
- move S x (i1 · i2) = (move ? x i1) ⊙ (move ? x i2).
+lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
+ •(i1 + i2) = •i1 ⊕ •i2.
// qed.
-lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
- move S x i^* = (move ? x i)^⊛.
+lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
+ •(i1 · i2) = •i1 ▹ i2.
// qed.
-definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
-
-lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b.
- pmove ? x 〈i,b〉 = move ? x i.
+lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
+ •i^* = 〈(\fst(•i))^*,true〉.
// qed.
-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.
-
-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 ⊢ (??%%); //
+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.
+ \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
+#S * #i1 #b1 * #i2 #b2 #w %
+ [cases b1 cases b2 normalize /2/ * /3/ * /3/
+ |cases b1 cases b2 normalize /2/ * /3/ * /3/
]
qed.
-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 //]
- ]
- |#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.
-
-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.
+lemma odot_true :
+ ∀S.∀i1,i2:pitem S.
+ 〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
// qed.
-lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S.
- moves ? (a::w) e = moves ? w (move S a (\fst e)).
+lemma odot_true_bis :
+ ∀S.∀i1,i2:pitem S.
+ 〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+#S #i1 #i2 normalize cases (•i2) // qed.
+
+lemma odot_false:
+ ∀S.∀i1,i2:pitem S.
+ 〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
// 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 LcatE : ∀S.∀e1,e2:pitem S.
+ \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
+// 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
- [% /2/ * // #H destruct |% normalize /2/]
+lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
+#S #i elim i //
+ [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
+ cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
+ | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
+ cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //
+ | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
+ ]
qed.
-lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
- |\fst (moves ? w e)| = |\fst e|.
-#S #w elim w //
+(*
+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 ▹ 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 …))
+ >erase_bull @eqP_trans [|@(eqP_union_l … H)]
+ @eqP_trans [|@eqP_union_l[|@union_comm ]]
+ @eqP_trans [|@eqP_sym @union_assoc ] /3/
+ ]
qed.
-
-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
- ]
+
+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 [|@minus_eps_pre]
+@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
+@eqP_trans [||@distribute_substract]
+@eqP_substract_r //
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 //
+(* theorem 16: 1 *)
+theorem sem_bull: ∀S:DeqSet. ∀i:pitem S. \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
+#S #e elim e
+ [#w normalize % [/2/ | * //]
+ |/2/
+ |#x normalize #w % [ /2/ | * [@False_ind | //]]
+ |#x normalize #w % [ /2/ | * // ]
+ |#i1 #i2 #IH1 #IH2 >eclose_dot
+ @eqP_trans [|@odot_dot_aux //] >sem_cat
+ @eqP_trans
+ [|@eqP_union_r
+ [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
+ @eqP_trans [|@union_assoc]
+ @eqP_trans [||@eqP_sym @union_assoc]
+ @eqP_union_l //
+ |#i1 #i2 #IH1 #IH2 >eclose_plus
+ @eqP_trans [|@sem_oplus] >sem_plus >erase_plus
+ @eqP_trans [|@(eqP_union_l … IH2)]
+ @eqP_trans [|@eqP_sym @union_assoc]
+ @eqP_trans [||@union_assoc] @eqP_union_r
+ @eqP_trans [||@eqP_sym @union_assoc]
+ @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 [|@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.
-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.
+(* 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 moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
-#S #w #i elim w //
-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/
- ]
+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.
-
-(* bisimulation *)
-definition cofinal ≝ λS.λp:(pre S)×(pre S).
- \snd (\fst p) = \snd (\snd p).
-
-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]
+
+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.
-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.
+(* lefted operations *)
+definition lifted_cat ≝ λS:DeqSet.λe:pre S.
+ lift S (pre_concat_l S eclose e).
-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.
+notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
-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.
+interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
-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]
+lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b.
+ 〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
+#S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) //
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)
- ]
+lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
+ 〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
+//
qed.
-
-(* the 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〉
- ]
- ].
-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.
-
-lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
- bisim S l O frontier visited = 〈false,visited〉.
-#frontier #visited >unfold_bisim //
+lemma erase_odot:∀S.∀e1,e2:pre S.
+ |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
+#S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot //
qed.
-lemma bisim_end: ∀Sig,l,m.∀visited: list ?.
- bisim Sig l (S m) [] visited = 〈true,visited〉.
-#n #visisted >unfold_bisim //
-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〉
+ ]
+ ].
-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 normalize nodelta >test //
-qed.
+(* 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 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.
-lemma notb_eq_true_l: ∀b. notb b = true → b = false.
-#b cases b normalize //
-qed.
+lemma ostar_true: ∀S.∀i:pitem S.
+ 〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
+// qed.
-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 ostar_false: ∀S.∀i:pitem S.
+ 〈i,false〉^⊛ = 〈i^*, false〉.
+// qed.
-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)) //
+lemma erase_ostar: ∀S.∀e:pre S.
+ |\fst (e^⊛)| = |\fst e|^*.
+#S * #i * // qed.
+
+lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
+ \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
+#S #e1 #i
+cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
+#H >H cases (e1 ▹ i) #i1 #b1 cases b1
+ [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
+ @eqP_union_l /2/
+ |/2/
]
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 //
+lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i.
+ e1 ⊙ 〈i,false〉 = e1 ▹ i.
+#S #e1 #i
+cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
+cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
qed.
-
-definition space_enum ≝ λS.λi1,i2:re S.
- compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i2).
-
-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.
-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}=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 @(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).
-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
- ]
- ]
- ]
+lemma sem_odot:
+ ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
+#S #e1 * #i2 *
+ [>sem_pre_true
+ @eqP_trans [|@sem_odot_true]
+ @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
+ |>sem_pre_false >eq_odot_false @odot_dot_aux //
]
qed.
-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 //
- ]
+(* theorem 16: 4 *)
+theorem sem_ostar: ∀S.∀e:pre S.
+ \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 [|@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
+ @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps
+ |>sem_pre_false >sem_pre_false >sem_star /2/
]
-qed.
-
-lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
-#n #m % [@eqbnat_true_to_eq | @eq_to_eqbnat_true]
-qed.
-
-definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true.
-
-definition a ≝ s DeqNat O.
-definition b ≝ s DeqNat (S O).
-definition c ≝ s DeqNat (S (S O)).
-
-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.
-
-
-
-
-
-
-
+qed.
\ No newline at end of file
--- /dev/null
+include "re.ma".
+include "basics/listb.ma".
+
+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 〉
+ | 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.
+
+lemma move_cat: ∀S:DeqSet.∀x:S.∀i1,i2:pitem S.
+ move S x (i1 · i2) = (move ? x i1) ⊙ (move ? x i2).
+// qed.
+
+lemma move_star: ∀S:DeqSet.∀x:S.∀i:pitem S.
+ move S x i^* = (move ? x i)^⊛.
+// qed.
+
+definition pmove ≝ λS:DeqSet.λx:S.λe:pre S. move ? x (\fst e).
+
+lemma pmove_def : ∀S:DeqSet.∀x:S.∀i:pitem S.∀b.
+ pmove ? x 〈i,b〉 = move ? x i.
+// qed.
+
+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.
+
+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.
+
+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 //]
+ ]
+ |#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.
+
+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 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
+ [% /2/ * // #H destruct |% normalize /2/]
+qed.
+
+lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
+ |\fst (moves ? w e)| = |\fst e|.
+#S #w elim w //
+qed.
+
+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.
+
+(************************ 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 //
+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.
+ \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.
+
+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 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.
+
+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.
+
+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 *)
+ | 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〉
+ ]
+ ].
+
+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.
+
+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 ?.
+ bisim Sig l (S m) [] visited = 〈true,visited〉.
+#n #visisted >unfold_bisim //
+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 normalize nodelta >test //
+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 normalize nodelta >test //
+qed.
+
+lemma notb_eq_true_l: ∀b. notb b = true → b = false.
+#b cases b normalize //
+qed.
+
+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.
+
+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).
+
+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.
+
+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}=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 @(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).
+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.
+
+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.
+
+lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
+#n #m % [@eqbnat_true_to_eq | @eq_to_eqbnat_true]
+qed.
+
+definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true.
+
+definition a ≝ s DeqNat O.
+definition b ≝ s DeqNat (S O).
+definition c ≝ s DeqNat (S (S O)).
+
+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.
+
+
+
+
+
+
+
projects / helm.git / commitdiff