(* Title: HOL/Lex/AutoChopper.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1995 TUM
auto_chopper turns an automaton into a chopper. Tricky, because primrec.
is_auto_chopper requires its argument to produce longest_prefix_choppers
wrt the language accepted by the automaton.
Main result: auto_chopper satisfies the is_auto_chopper specification.
WARNING: auto_chopper is exponential(?)
if the recursive calls in the penultimate argument are evaluated eagerly.
A more efficient version is defined in AutoChopper1.
But both versions are far too specific. Better development in Scanner.thy and
its antecedents.
*)
AutoChopper = Prefix + DA + Chopper +
constdefs
is_auto_chopper :: (('a,'s)da => 'a chopper) => bool
"is_auto_chopper(chopperf) ==
!A. is_longest_prefix_chopper(accepts A)(chopperf A)"
consts
acc :: "[('a,'s)da, 's, 'a list list*'a list, 'a list, 'a list, 'a list]
=> 'a list list * 'a list"
primrec
"acc A s r ps [] zs = (if ps=[] then r else (ps#fst(r),snd(r)))"
"acc A s r ps (x#xs) zs =
(let t = next A x s
in if fin A t
then acc A t (acc A (start A) ([],xs) [] xs [])
(zs@[x]) xs (zs@[x])
else acc A t r ps xs (zs@[x]))"
constdefs
auto_chopper :: ('a,'s)da => 'a chopper
"auto_chopper A xs == acc A (start A) ([],xs) [] xs []"
(* acc_prefix is an auxiliary notion for the proof *)
constdefs
acc_prefix :: ('a,'s)da => 'a list => 's => bool
"acc_prefix A xs s == ? us. us <= xs & us~=[] & fin A (delta A us s)"
end