-include "lang.ma".
+(* <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. *)
-inductive re (S: DeqSet) : Type[0] ≝
- z: re S
- | e: re S
- | s: S → re S
- | c: re S → re S → re S
- | o: re S → re S → re S
- | k: re S → re S.
+include "tutorial/chapter6.ma".
+
+(* The type re of regular expressions over an alphabet $S$ is the smallest
+collection of objects generated by the following constructors: *)
+
+inductive re (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) : 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* *)
interpretation "re epsilon" 'epsilon = (e ?).
interpretation "re or" 'plus a b = (o ? a b).
notation "`∅" non associative with precedence 90 for @{ 'empty }.
interpretation "empty" 'empty = (z ?).
-let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝
+(* The language sem{e} associated with the regular expression e is inductively
+defined by the following function: *)
+
+let rec in_l (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6\ 5span class="error" title="Parse error: RPAREN expected after [term] (in [arg])"\ 6\ 5/span\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '≝' expected (in [let_defs])"\ 6\ 5/span\ 6 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) ^*].
+[ z ⇒ \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '|' or SYMBOL ']' expected (in [term])"\ 6\ 5/span\ 6
+| e ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}
+| s x ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5span class="error" title="Parse error: [ident] or [term level 19] expected after [sym{] (in [term])"\ 6\ 5/span\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym[] (in [term])"\ 6\ 5/span\ 6x]}
+| c r1 r2 ⇒ (in_l ? r1) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (in_l ? r2)
+| o r1 r2 ⇒ (in_l ? r1) \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (in_l ? r2)
+| k r1 ⇒ (in_l ? r1) \ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*].
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 rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
+lemma rsem_star : ∀S.∀r: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S. \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{r\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{r}\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
// qed.
-(* pointed items *)
-inductive pitem (S: DeqSet) : Type[0] ≝
- pz: pitem S
- | pe: pitem S
- | ps: S → pitem S
- | pp: S → pitem S
- | pc: pitem S → pitem S → pitem S
- | po: pitem S → pitem S → pitem S
- | pk: pitem S → pitem S.
+(* <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
+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: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) : 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 *)
-definition pre ≝ λS.pitem S × bool.
+(* 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.\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
interpretation "pitem star" 'star a = (pk ? a).
interpretation "pitem or" 'plus a b = (po ? a b).
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 ⇒ `∅
lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
// qed.
-(* boolean equality *)
+(* <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. *)
+
let rec beqitem S (i1,i2: pitem S) on i1 ≝
match i1 with
[ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
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)
(* ---------------------------------------- *) ⊢
(* ---------------------------------------- *) ⊢
beqitem S i1 i2 ≡ eqb X i1 i2.
-(* semantics *)
+(* <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. *)
let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
match r with
interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
interpretation "in_prl" 'in_l E = (in_prl ? E).
+(* The following, trivial lemmas are only meant for rewriting purposes. *)
+
lemma sem_pre_true : ∀S.∀i:pitem S.
\sem{〈i,true〉} = \sem{i} ∪ {ϵ}.
// qed.
\sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
// qed.
+(* 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} =1 \sem{i}-{[ ]}.
+- minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \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.
]
qed.
-(* lemma 12 *)
lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
#S * #i #b cases b // normalize #H @False_ind /2/
qed.
]
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.
-
projects / helm.git / blobdiff