(* Title: HOL/Lex/RegExp2NA.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1998 TUM
Conversion of regular expressions *directly*
into nondeterministic automata *without* epsilon transitions
*)
RegExp2NA = RegExp + NA +
types 'a bitsNA = ('a,bool list)na
syntax "##" :: 'a => 'a list set => 'a list set (infixr 65)
translations "x ## S" == "Cons x `` S"
constdefs
atom :: 'a => 'a bitsNA
"atom a == ([True],
%b s. if s=[True] & b=a then {[False]} else {},
%s. s=[False])"
union :: 'a bitsNA => 'a bitsNA => 'a bitsNA
"union == %(ql,dl,fl)(qr,dr,fr).
([],
%a s. case s of
[] => (True ## dl a ql) Un (False ## dr a qr)
| left#s => if left then True ## dl a s
else False ## dr a s,
%s. case s of [] => (fl ql | fr qr)
| left#s => if left then fl s else fr s)"
conc :: 'a bitsNA => 'a bitsNA => 'a bitsNA
"conc == %(ql,dl,fl)(qr,dr,fr).
(True#ql,
%a s. case s of
[] => {}
| left#s => if left then (True ## dl a s) Un
(if fl s then False ## dr a qr else {})
else False ## dr a s,
%s. case s of [] => False | left#s => left & fl s & fr qr | ~left & fr s)"
epsilon :: 'a bitsNA
"epsilon == ([],%a s. {}, %s. s=[])"
plus :: 'a bitsNA => 'a bitsNA
"plus == %(q,d,f). (q, %a s. d a s Un (if f s then d a q else {}), f)"
star :: 'a bitsNA => 'a bitsNA
"star A == union epsilon (plus A)"
consts rexp2na :: 'a rexp => 'a bitsNA
primrec
"rexp2na Empty = ([], %a s. {}, %s. False)"
"rexp2na(Atom a) = atom a"
"rexp2na(Union r s) = union (rexp2na r) (rexp2na s)"
"rexp2na(Conc r s) = conc (rexp2na r) (rexp2na s)"
"rexp2na(Star r) = star (rexp2na r)"
end