4 In this chapter we shall apply our notion of DeqSet, and the list operations
5 defined in Chapter 4 to formal languages. A formal language is an arbitrary set
6 of words over a given alphabet, that we shall represent as a predicate over words.
8 include "tutorial/chapter6.ma".
10 (* A word (or string) over an alphabet S is just a list of elements of S.*)
11 definition word ≝ λS:DeqSet.list S.
13 (* For any alphabet there is only one word of length 0, the empty word, which is
16 notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
17 interpretation "epsilon" 'epsilon = (nil ?).
19 (* The operation that consists in appending two words to form a new word, whose
20 length is the sum of the lengths of the original words is called concatenation.
21 String concatenation is just the append operation over lists, hence there is no
22 point to define it. Similarly, many of its properties, such as the fact that
23 concatenating a word with the empty word gives the original word, follow by
24 general results over lists.
28 Operations over languages
30 Languages inherit all the basic operations for sets, namely union, intersection,
31 complementation, substraction, and so on. In addition, we may define some new
32 operations induced by string concatenation, and in particular the concatenation
33 A · B of two languages A and B, the so called Kleene's star A* of A and the
34 derivative of a language A w.r.t. a given character a. *)
36 definition cat : ∀S,l1,l2,w.Prop ≝
37 λS.λl1,l2.λw:word S.∃w1,w2.w1 @ w2 = w ∧ l1 w1 ∧ l2 w2.
39 notation "a · b" non associative with precedence 60 for @{ 'middot $a $b}.
41 interpretation "cat lang" 'middot a b = (cat ? a b).
43 (* Given a language l, the Kleene's star of l, denoted by l*, is the set of
44 finite-length strings that can be generated by concatenating arbitrary strings of
45 l. In other words, w belongs to l* is and only if there exists a list of strings
46 w1,w2,...wk all belonging to l, such that l = w1w2...wk.
47 We need to define the latter operations. The following flatten function takes in
48 input a list of words and concatenates them together. *)
50 (* Already in the library
51 let rec flatten (S : DeqSet) (l : list (word S)) on l : word S ≝
52 match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
55 (* Given a list of words l and a language r, (conjunct l r) is true if and only
56 if all words in l are in r, that is for every w in l, r w holds. *)
58 let rec conjunct (S : DeqSet) (l : list (word S)) (r : word S → Prop) on l: Prop ≝
59 match l with [ nil ⇒ True | cons w tl ⇒ r w ∧ conjunct ? tl r ].
61 (* We are ready to give the formal definition of the Kleene's star of l:
62 a word w belongs to l* is and only if there exists a list of strings
63 lw such that (conjunct lw l) and l = flatten lw. *)
65 definition flatten ≝ λA.foldr (list A) (list A) (append A) [].
67 definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
69 notation "a ^ *" non associative with precedence 90 for @{ 'star $a}.
70 interpretation "star lang" 'star l = (star ? l).
72 (* The derivative of a language A with respect to a character a is the set of
73 all strings w such that aw is in A. *)
75 definition deriv ≝ λS.λA:word S → Prop.λa,w. A (a::w).
80 Equality between languages is just the usual extensional equality between
81 sets. The operation of concatenation behaves well with respect to this equality. *)
83 lemma cat_ext_l: ∀S.∀A,B,C:word S →Prop.
84 A ≐ C → A · B ≐ C · B.
85 #S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
89 lemma cat_ext_r: ∀S.∀A,B,C:word S →Prop.
90 B ≐ C → A · B ≐ A · C.
91 #S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
95 (* Concatenating a language with the empty language results in the
97 lemma cat_empty_l: ∀S.∀A:word S→Prop. ∅ · A ≐ ∅.
98 #S #A #w % [|*] * #w1 * #w2 * * #_ *
101 (* Concatenating a language l with the singleton language containing the
102 empty string, results in the language l; that is {ϵ} is a left and right
103 unit with respect to concatenation. *)
105 lemma epsilon_cat_r: ∀S.∀A:word S →Prop.
108 [* #w1 * #w2 * * #eqw #inw1 normalize #eqw2 <eqw //
109 |#inA @(ex_intro … w) @(ex_intro … [ ]) /3/
113 lemma epsilon_cat_l: ∀S.∀A:word S →Prop.
116 [* #w1 * #w2 * * #eqw normalize #eqw2 <eqw <eqw2 //
117 |#inA @(ex_intro … ϵ) @(ex_intro … w) /3/
121 (* Concatenation is distributive w.r.t. union. *)
123 lemma distr_cat_r: ∀S.∀A,B,C:word S →Prop.
124 (A ∪ B) · C ≐ A · C ∪ B · C.
126 [* #w1 * #w2 * * #eqw * /6/ |* * #w1 * #w2 * * /6/]
129 lemma distr_cat_r_eps: ∀S.∀A,C:word S →Prop.
130 (A ∪ {ϵ}) · C ≐ A · C ∪ C.
131 #S #A #C @eqP_trans [|@distr_cat_r |@eqP_union_l @epsilon_cat_l]
134 (* The following is a major property of derivatives *)
136 lemma deriv_middot: ∀S,A,B,a. ¬ A ϵ → deriv S (A·B) a ≐ (deriv S A a) · B.
137 #S #A #B #a #noteps #w normalize %
139 [* #w2 * * #_ #Aeps @False_ind /2/
140 |#b #w2 * #w3 * * whd in ⊢ ((??%?)→?); #H destruct
141 #H #H1 @(ex_intro … w2) @(ex_intro … w3) % // % //
143 |* #w1 * #w2 * * #H #H1 #H2 @(ex_intro … (a::w1))
144 @(ex_intro … w2) % // % normalize //
149 Main Properties of Kleene's star
151 We conclude this section with some important properties of Kleene's
152 star that will be used in the following chapters. *)
154 lemma espilon_in_star: ∀S.∀A:word S → Prop.
156 #S #A @(ex_intro … [ ]) normalize /2/
159 lemma cat_to_star:∀S.∀A:word S → Prop.
160 ∀w1,w2. A w1 → A^* w2 → A^* (w1@w2).
161 #S #A #w1 #w2 #Aw * #l * #H #H1 @(ex_intro … (w1::l))
162 % normalize destruct /2/
165 lemma fix_star: ∀S.∀A:word S → Prop.
168 [* #l generalize in match w; -w cases l [normalize #w * /2/]
169 #w1 #tl #w * whd in ⊢ ((??%?)→?); #eqw whd in ⊢ (%→?); *
170 #w1A #cw1 %1 @(ex_intro … w1) @(ex_intro … (flatten S tl))
171 % destruct /2/ whd @(ex_intro … tl) /2/
172 |* [2: whd in ⊢ (%→?); #eqw <eqw //]
173 * #w1 * #w2 * * #eqw <eqw @cat_to_star
177 lemma star_fix_eps : ∀S.∀A:word S → Prop.
178 A^* ≐ (A - {ϵ}) · A^* ∪ {ϵ}.
181 [* whd in ⊢ ((??%?)→?); #eqw #_ %2 <eqw //
182 |* [#tl #Hind * #H * #_ #H2 @Hind % [@H | //]
183 |#a #w1 #tl #Hind * whd in ⊢ ((??%?)→?); #H1 * #H2 #H3 %1
184 @(ex_intro … (a::w1)) @(ex_intro … (flatten S tl)) %
185 [% [@H1 | normalize % [@H2 | % #H4 destruct (H4)] ]
186 |whd @(ex_intro … tl) /2/]
189 |* [* #w1 * #w2 * * #eqw * #H1 #_ <eqw @cat_to_star //
190 | whd in ⊢ (%→?); #H <H //
195 lemma star_epsilon: ∀S:DeqSet.∀A:word S → Prop.