1 (* Title: HOL/Lex/NAe_of_RegExp.thy |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow |
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4 Copyright 1998 TUM |
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5 |
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6 Conversion of regular expressions |
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7 into nondeterministic automata with epsilon transitions |
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8 *) |
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9 |
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10 NAe_of_RegExp = NAe + RegExp + |
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11 |
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12 types 'a r2nae = ('a,bool list)nae |
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13 |
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14 syntax "##" :: 'a => 'a list set => 'a list set (infixr 65) |
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15 translations "x ## S" == "op # x `` S" |
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16 |
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17 constdefs |
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18 atom :: 'a => 'a r2nae |
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19 "atom a == ([True], |
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20 %b s. if s=[True] & b=Some a then {[False]} else {}, |
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21 %s. s=[False])" |
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22 |
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23 union :: 'a r2nae => 'a r2nae => 'a r2nae |
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24 "union == %(ql,dl,fl)(qr,dr,fr). |
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25 ([], |
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26 %a s. case s of |
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27 [] => if a=None then {True#ql,False#qr} else {} |
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28 | left#s => if left then True ## dl a s |
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29 else False ## dr a s, |
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30 %s. case s of [] => False | left#s => if left then fl s else fr s)" |
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31 |
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32 conc :: 'a r2nae => 'a r2nae => 'a r2nae |
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33 "conc == %(ql,dl,fl)(qr,dr,fr). |
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34 (True#ql, |
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35 %a s. case s of |
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36 [] => {} |
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37 | left#s => if left then (True ## dl a s) Un |
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38 (if fl s & a=None then {False#qr} else {}) |
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39 else False ## dr a s, |
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40 %s. case s of [] => False | left#s => ~left & fr s)" |
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41 |
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42 star :: 'a r2nae => 'a r2nae |
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43 "star == %(q,d,f). |
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44 ([], |
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45 %a s. case s of |
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46 [] => if a=None then {True#q} else {} |
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47 | left#s => if left then (True ## d a s) Un |
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48 (if f s & a=None then {True#q} else {}) |
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49 else {}, |
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50 %s. case s of [] => True | left#s => left & f s)" |
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51 |
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52 consts r2n :: 'a rexp => 'a r2nae |
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53 primrec r2n rexp |
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54 "r2n Empty = ([], %a s. {}, %s. False)" |
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55 "r2n(Atom a) = atom a" |
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56 "r2n(Union el er) = union(r2n el)(r2n er)" |
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57 "r2n(Conc el er) = conc(r2n el)(r2n er)" |
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58 "r2n(Star e) = star(r2n e)" |
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59 |
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60 end |
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