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/chapter5.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}.
40 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 let rec flatten (S : DeqSet) (l : list (word S)) on l : word S ≝
51 match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
53 (* Given a list of words l and a language r, (conjunct l r) is true if and only if
54 all words in l are in r, that is for every w in l, r w holds. *)
56 let rec conjunct (S : DeqSet) (l : list (word S)) (r : word S → Prop) on l: Prop ≝
57 match l with [ nil ⇒ True | cons w tl ⇒ r w ∧ conjunct ? tl r ].
59 (* We are ready to give the formal definition of the Kleene's star of l:
60 a word w belongs to l* is and only if there exists a list of strings
61 lw such that (conjunct lw l) and l = flatten lw. *)
63 definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
64 notation "a ^ *" non associative with precedence 90 for @{ 'star $a}.
65 interpretation "star lang" 'star l = (star ? l).
67 (* The derivative of a language A with respect to a character a is the set of
68 all strings w such that aw is in A. *)
70 definition deriv ≝ λS.λA:word S → Prop.λa,w. A (a::w).
75 Equality between languages is just the usual extensional equality between
76 sets. The operation of concatenation behaves well with respect to this equality. *)
78 lemma cat_ext_l: ∀S.∀A,B,C:word S →Prop.
79 A =1 C → A · B =1 C · B.
80 #S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
84 lemma cat_ext_r: ∀S.∀A,B,C:word S →Prop.
85 B =1 C → A · B =1 A · C.
86 #S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
90 (* Concatenating a language with the empty language results in the
92 lemma cat_empty_l: ∀S.∀A:word S→Prop. ∅ · A =1 ∅.
93 #S #A #w % [|*] * #w1 * #w2 * * #_ *
96 (* Concatenating a language l with the singleton language containing the
97 empty string, results in the language l; that is {ϵ} is a left and right
98 unit with respect to concatenation. *)
100 lemma epsilon_cat_r: ∀S.∀A:word S →Prop.
103 [* #w1 * #w2 * * #eqw #inw1 normalize #eqw2 <eqw //
104 |#inA @(ex_intro … w) @(ex_intro … [ ]) /3/
108 lemma epsilon_cat_l: ∀S.∀A:word S →Prop.
111 [* #w1 * #w2 * * #eqw normalize #eqw2 <eqw <eqw2 //
112 |#inA @(ex_intro … ϵ) @(ex_intro … w) /3/
116 (* Concatenation is distributive w.r.t. union. *)
118 lemma distr_cat_r: ∀S.∀A,B,C:word S →Prop.
119 (A ∪ B) · C =1 A · C ∪ B · C.
121 [* #w1 * #w2 * * #eqw * /6/ |* * #w1 * #w2 * * /6/]
124 lemma distr_cat_r_eps: ∀S.∀A,C:word S →Prop.
125 (A ∪ {ϵ}) · C =1 A · C ∪ C.
126 #S #A #C @eqP_trans [|@distr_cat_r |@eqP_union_l @epsilon_cat_l]
129 (* The following is a major property of derivatives *)
131 lemma deriv_middot: ∀S,A,B,a. ¬ A ϵ → deriv S (A·B) a =1 (deriv S A a) · B.
132 #S #A #B #a #noteps #w normalize %
134 [* #w2 * * #_ #Aeps @False_ind /2/
135 |#b #w2 * #w3 * * whd in ⊢ ((??%?)→?); #H destruct
136 #H #H1 @(ex_intro … w2) @(ex_intro … w3) % // % //
138 |* #w1 * #w2 * * #H #H1 #H2 @(ex_intro … (a::w1))
139 @(ex_intro … w2) % // % normalize //
144 Main Properties of Kleene's star
146 We conclude this section with some important properties of Kleene's
147 star that will be used in the following chapters. *)
149 lemma espilon_in_star: ∀S.∀A:word S → Prop.
151 #S #A @(ex_intro … [ ]) normalize /2/
154 lemma cat_to_star:∀S.∀A:word S → Prop.
155 ∀w1,w2. A w1 → A^* w2 → A^* (w1@w2).
156 #S #A #w1 #w2 #Aw * #l * #H #H1 @(ex_intro … (w1::l))
160 lemma fix_star: ∀S.∀A:word S → Prop.
161 A^* =1 A · A^* ∪ {ϵ}.
163 [* #l generalize in match w; -w cases l [normalize #w * /2/]
164 #w1 #tl #w * whd in ⊢ ((??%?)→?); #eqw whd in ⊢ (%→?); *
165 #w1A #cw1 %1 @(ex_intro … w1) @(ex_intro … (flatten S tl))
166 % /2/ whd @(ex_intro … tl) /2/
167 |* [2: whd in ⊢ (%→?); #eqw <eqw //]
168 * #w1 * #w2 * * #eqw <eqw @cat_to_star
172 lemma star_fix_eps : ∀S.∀A:word S → Prop.
173 A^* =1 (A - {ϵ}) · A^* ∪ {ϵ}.
176 [* whd in ⊢ ((??%?)→?); #eqw #_ %2 <eqw //
177 |* [#tl #Hind * #H * #_ #H2 @Hind % [@H | //]
178 |#a #w1 #tl #Hind * whd in ⊢ ((??%?)→?); #H1 * #H2 #H3 %1
179 @(ex_intro … (a::w1)) @(ex_intro … (flatten S tl)) %
180 [% [@H1 | normalize % /2/] |whd @(ex_intro … tl) /2/]
183 |* [* #w1 * #w2 * * #eqw * #H1 #_ <eqw @cat_to_star //
184 | whd in ⊢ (%→?); #H <H //
189 lemma star_epsilon: ∀S:DeqSet.∀A:word S → Prop.