-(*
-Broadcasting points
-
-Intuitively, a regular expression e must be understood as a pointed expression with a single
-point in front of it. Since however we only allow points before symbols, we must broadcast
-this initial point inside e traversing all nullable subexpressions, that essentially corresponds
-to the ϵ-closure operation on automata. We use the notation •(_) to denote such an operation;
-its definition is the expected one: let us start discussing an example.
-
-Example
-Let us broadcast a point inside (a + ϵ)(b*a + b)b. We start working in parallel on the
-first occurrence of a (where the point stops), and on ϵ that gets traversed. We have hence
-reached the end of a + ϵ and we must pursue broadcasting inside (b*a + b)b. Again, we work in
-parallel on the two additive subterms b^*a and b; the first point is allowed to both enter the
-star, and to traverse it, stopping in front of a; the second point just stops in front of b.
-No point reached that end of b^*a + b hence no further propagation is possible. In conclusion:
- •((a + ϵ)(b^*a + b)b) = 〈(•a + ϵ)((•b)^*•a + •b)b, false〉
-*)
+(*
+Regular Expressions
+
+We shall apply all the previous machinery to the study of regular languages
+and the constructions of the associated finite automata. *)
include "tutorial/chapter7.ma".
-(* Broadcasting a point inside an item generates a pre, since the point could possibly reach
-the end of the expression.
-Broadcasting inside a i1+i2 amounts to broadcast in parallel inside i1 and i2.
-If we define
- 〈i1,b1〉 ⊕ 〈i2,b2〉 = 〈i1 + i2, b1∨ b2〉
-then, we just have •(i1+i2) = •(i1)⊕ •(i2).
-*)
+(* The type re of regular expressions over an alphabet $S$ is the smallest
+collection of objects generated by the following constructors: *)
-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).
+inductive re (S: DeqSet) : Type[0] ≝
+ z: re S (* empty: ∅ *)
+ | e: re S (* epsilon: ϵ *)
+ | s: S → re S (* symbol: a *)
+ | c: re S → re S → re S (* concatenation: e1 · e2 *)
+ | o: re S → re S → re S (* plus: e1 + e2 *)
+ | k: re S → re S. (* kleene's star: e* *)
-lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
-// qed.
+interpretation "re epsilon" 'epsilon = (e ?).
+interpretation "re or" 'plus a b = (o ? a b).
+interpretation "re cat" 'middot a b = (c ? a b).
+interpretation "re star" 'star a = (k ? a).
-(*
-Concatenation is a bit more complex. In order to broadcast a point inside i1 · i2
-we should start broadcasting it inside i1 and then proceed into i2 if and only if a
-point reached the end of i1. This suggests to define •(i1 · i2) as •(i1) ▹ i2, where
-e ▹ i is a general operation of concatenation between a pre and an item, defined by
-cases on the boolean in e:
-
- 〈i1,true〉 ▹ i2 = i1 ◃ •(i_2)
- 〈i1,false〉 ▹ i2 = i1 · i2
-In turn, ◃ says how to concatenate an item with a pre, that is however extremely simple:
- i1 ◃ 〈i1,b〉 = 〈i_1 · i2, b〉
-Let us come to the formalized definitions:
-*)
+notation < "a" non associative with precedence 90 for @{ 'ps $a}.
+notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
+interpretation "atom" 'ps a = (s ? a).
-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).
+notation "`∅" non associative with precedence 90 for @{ 'empty }.
+interpretation "empty" 'empty = (z ?).
-lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop.
- A = B → A =1 B.
-#S #A #B #H >H #x % // qed.
+(* The language sem{e} associated with the regular expression e is inductively
+defined by the following function: *)
-(* The behaviour of ◃ is summarized by the following, easy lemma: *)
+let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝
+match r with
+[ z ⇒ ∅
+| e ⇒ {ϵ}
+| s x ⇒ { (x::[]) }
+| c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
+| o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
+| k r1 ⇒ (in_l ? r1) ^*].
-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.
-
-(* The definition of $•(-)$ (eclose) and ▹ (pre_concat_l) are mutually recursive.
-In this situation, a viable alternative that is usually simpler to reason about,
-is to abstract one of the two functions with respect to the other. In particular
-we abstract pre_concat_l with respect to an input bcast function from items to
-pres. *)
-
-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).
-
-(* We are ready to give the formal definition of the broadcasting operation. *)
-
-notation "•" non associative with precedence 60 for @{eclose ?}.
-let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
- match i with
- [ pz ⇒ 〈 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).
-
-(* Here are a few simple properties of ▹ and •(-) *)
+notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
+interpretation "in_l" 'in_l E = (in_l ? E).
+interpretation "in_l mem" 'mem w l = (in_l ? l w).
-lemma pcl_true : ∀S.∀i1,i2:pitem S.
- 〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
+lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
// qed.
-lemma pcl_true_bis : ∀S.∀i1,i2:pitem S.
- 〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
-#S #i1 #i2 normalize cases (•i2) // qed.
-lemma pcl_false: ∀S.∀i1,i2:pitem S.
- 〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
-// qed.
+(*
+Pointed Regular expressions
+
+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
+regular expression which have been reached after reading some prefix of
+the input string, or better the positions where the processing of the remaining
+string has to be started. Each pointed expression for $e$ represents a state of
+the {\em deterministic} automaton associated with $e$; since we obviously have
+only a finite number of possible labellings, the number of states of the automaton
+is finite.
+
+Pointed regular expressions provide the tool for an algebraic revisitation of
+McNaughton and Yamada's algorithm for position automata, making the proof of its
+correctness, that is far from trivial, particularly clear and simple. In particular,
+pointed expressions offer an appealing alternative to Brzozowski's derivatives,
+avoiding their weakest point, namely the fact of being forced to quotient derivatives
+w.r.t. a suitable notion of equivalence in order to get a finite number of states
+(that is not essential for recognizing strings, but is crucial for comparing regular
+expressions).
+
+Our main data structure is the notion of pointed item, that is meant whose purpose
+is to encode a set of positions inside a regular expression.
+The idea of formalizing pointers inside a data type by means of a labelled version
+of the data type itself is probably one of the first, major lessons learned in the
+formalization of the metatheory of programming languages. For our purposes, it is
+enough to mark positions preceding individual characters, so we shall have two kinds
+of characters •a (pp a) and a (ps a) according to the case a is pointed or not. *)
+
+inductive pitem (S: DeqSet) : Type[0] ≝
+ pz: pitem S (* empty *)
+ | pe: pitem S (* epsilon *)
+ | ps: S → pitem S (* symbol *)
+ | pp: S → pitem S (* pointed sysmbol *)
+ | pc: pitem S → pitem S → pitem S (* concatenation *)
+ | po: pitem S → pitem S → pitem S (* plus *)
+ | pk: pitem S → pitem S. (* kleene's star *)
+
+(* A pointed regular expression (pre) is just a pointed item with an additional
+boolean, that must be understood as the possibility to have a trailing point at
+the end of the expression. As we shall see, pointed regular expressions can be
+understood as states of a DFA, and the boolean indicates if
+the state is final or not. *)
+
+definition pre ≝ λS.pitem S × bool.
+
+interpretation "pitem star" 'star a = (pk ? a).
+interpretation "pitem or" 'plus a b = (po ? a b).
+interpretation "pitem cat" 'middot a b = (pc ? a b).
+notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
+notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
+interpretation "pitem pp" 'pp a = (pp ? a).
+interpretation "pitem ps" 'ps a = (ps ? a).
+interpretation "pitem epsilon" 'epsilon = (pe ?).
+interpretation "pitem empty" 'empty = (pz ?).
+
+(* The carrier $|i|$ of an item i is the regular expression obtained from i by
+removing all the points. Similarly, the carrier of a pointed regular expression
+is the carrier of its item. *)
+
+let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
+ match l with
+ [ pz ⇒ z ? (* `∅ *)
+ | pe ⇒ ϵ
+ | ps x ⇒ `x
+ | pp x ⇒ `x
+ | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
+ | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
+ | pk E ⇒ (forget ? E)^* ].
+
+(* Already in the library
+notation "| term 19 C |" with precedence 70 for @{ 'card $C }.
+*)
+interpretation "forget" 'card a = (forget ? a).
-lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
- •(i1 + i2) = •i1 ⊕ •i2.
+lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
// qed.
-lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
- •(i1 · i2) = •i1 ▹ i2.
+lemma erase_plus : ∀S.∀i1,i2:pitem S.
+ |i1 + i2| = |i1| + |i2|.
// qed.
-lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
- •i^* = 〈(\fst(•i))^*,true〉.
+lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
// qed.
-(* The definition of •(-) (eclose) can then be lifted from items to pres
-in the obvious way. *)
+(*
+Comparing items and pres
+
+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. *)
+
+let rec beqitem S (i1,i2: pitem S) on i1 ≝
+ match i1 with
+ [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
+ | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
+ | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
+ | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
+ | po i11 i12 ⇒ match i2 with
+ [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+ | _ ⇒ false]
+ | pc i11 i12 ⇒ match i2 with
+ [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+ | _ ⇒ false]
+ | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
+ ].
-definition 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).
-
-(* Obviously, broadcasting does not change the carrier of the item,
-as it is easily proved by structural induction. *)
-
-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 >pcl_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) //
+lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
+#S #i1 elim i1
+ [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+ |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+ |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+ [>(\P H) // | @(\b (refl …))]
+ |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+ [>(\P H) // | @(\b (refl …))]
+ |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+ normalize #H destruct
+ [cases (true_or_false (beqitem S i11 i21)) #H1
+ [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+ |>H1 in H; normalize #abs @False_ind /2/
+ ]
+ |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+ ]
+ |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+ normalize #H destruct
+ [cases (true_or_false (beqitem S i11 i21)) #H1
+ [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+ |>H1 in H; normalize #abs @False_ind /2/
+ ]
+ |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+ ]
+ |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
+ normalize #H destruct
+ [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
]
-qed.
-
-(* We are now ready to state the main semantic properties of ⊕, ◃ and •(-):
-
-sem_oplus: \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}
-sem_pcl: \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}
-sem_bullet \sem{•i} =1 \sem{i} ∪ \sem{|i|}
-
-The proof of sem_oplus is straightforward. *)
+qed.
+
+definition DeqItem ≝ λS.
+ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
+
+(* We also add a couple of unification hints to allow the type inference system
+to look at (pitem S) as the carrier of a DeqSet, and at beqitem as if it was the
+equality function of a DeqSet. *)
+
+unification hint 0 ≔ S;
+ X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢
+ pitem S ≡ carr X.
+
+unification hint 0 ≔ S,i1,i2;
+ X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢
+ beqitem S i1 i2 ≡ eqb X i1 i2.
+
+(*
+Semantics of pointed regular expressions
+
+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. *)
+
+let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
+match r with
+[ pz ⇒ ∅
+| pe ⇒ ∅
+| ps _ ⇒ ∅
+| pp x ⇒ { (x::[]) }
+| pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
+| po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
+| pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
+
+interpretation "in_pl" 'in_l E = (in_pl ? E).
+interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
+
+definition in_prl ≝ λS : DeqSet.λp:pre S.
+ if (snd ?? p) then \sem{fst ?? p} ∪ {ϵ} else \sem{fst ?? p}.
+
+interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
+interpretation "in_prl" 'in_l E = (in_prl ? E).
-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.
+(* The following, trivial lemmas are only meant for rewriting purposes. *)
-(* For the others, we proceed as follow: we first prove the following
-auxiliary lemma, that assumes sem_bullet:
+lemma sem_pre_true : ∀S.∀i:pitem S.
+ \sem{〈i,true〉} = \sem{i} ∪ {ϵ}.
+// qed.
-sem_pcl_aux:
- \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
- \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+lemma sem_pre_false : ∀S.∀i:pitem S.
+ \sem{〈i,false〉} = \sem{i}.
+// qed.
-Then, using the previous result, we prove sem_bullet by induction
-on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *)
+lemma sem_cat: ∀S.∀i1,i2:pitem S.
+ \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
+// qed.
-lemma LcatE : ∀S.∀e1,e2:pitem S.
- \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
+lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
+ \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
// qed.
-lemma sem_pcl_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 >pcl_false >sem_pre_false >sem_pre_false >sem_cat /2/
- |#H >pcl_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.
+lemma sem_plus: ∀S.∀i1,i2:pitem S.
+ \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
+// qed.
-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 [|@sem_pcl_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 sem_plus_w: ∀S.∀i1,i2:pitem S.∀w.
+ \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
+// qed.
-(*
-Blank item
-
+lemma sem_star : ∀S.∀i:pitem S.
+ \sem{i^*} = \sem{i} · \sem{|i|}^*.
+// qed.
-As a corollary of theorem sem_bullet, given a regular expression e, we can easily
-find an item with the same semantics of $e$: it is enough to get an item (blank e)
-having e as carrier and no point, and then broadcast a point in it. The semantics of
-(blank e) is obviously the empty language: from the point of view of the automaton,
-it corresponds with the pit state. *)
-
-let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
- match i with
- [ z ⇒ pz ?
- | 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_star_w : ∀S.∀i:pitem S.∀w.
+ \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
+// 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
+(* Below are a few, simple, semantic properties of items. In particular:
+- not_epsilon_item : ∀S:DeqSet.∀i:pitem S. ¬ (\sem{i} ϵ).
+- epsilon_pre : ∀S.∀e:pre S. (\sem{i} ϵ) ↔ (\snd e = true).
+- minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \sem{i}-{[ ]}.
+- minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} ≐ \sem{e}-{[ ]}.
+The first property is proved by a simple induction on $i$; the other
+results are easy corollaries. We need an auxiliary lemma first. *)
+
+lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
+#S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
+
+lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ (ϵ ∈ e).
+#S #e elim e normalize
+ [1,2,3:/2/
+ |#x % #H destruct
+ |#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
+ >(append_eq_nil …H…) /2/
+ |#r1 #r2 #n1 #n2 % * /2/
+ |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
]
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.
-
-(*
-Lifted Operators
-
-
-Plus and bullet have been already lifted from items to pres. We can now
-do a similar job for concatenation ⊙ and Kleene's star ⊛.*)
-
-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) //
+lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → snd … e = true.
+#S * #i #b cases b // normalize #H @False_ind /2/
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 //
+lemma true_to_epsilon : ∀S.∀e:pre S. snd … e = true → ϵ ∈ e.
+#S * #i #b #btrue normalize in btrue; >btrue %2 //
qed.
-(* Let us come to the star operation: *)
-
-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/
+lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \sem{i}-{ϵ}.
+#S #i #w %
+ [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
+ |* //
]
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.
-
-(* We conclude this section with the proof of the main semantic properties
-of ⊙ and ⊛. *)
-
-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 @sem_pcl_aux //
- |>sem_pre_false >eq_odot_false @sem_pcl_aux //
+lemma minus_eps_pre: ∀S.∀e:pre S. \sem{fst ?? e} ≐ \sem{e}-{ϵ}.
+#S * #i *
+ [>sem_pre_true normalize in ⊢ (??%?); #w %
+ [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
+ |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
]
qed.
-
-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
projects / helm.git / blobdiff