(* Title: HOL/ex/PropLog.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
Inductive definition of propositional logic.
*)
PropLog = Finite +
datatype
'a pl = false | var 'a ("#_" [1000]) | "->" ('a pl) ('a pl) (infixr 90)
consts
thms :: 'a pl set => 'a pl set
"|-" :: ['a pl set, 'a pl] => bool (infixl 50)
"|=" :: ['a pl set, 'a pl] => bool (infixl 50)
eval2 :: ['a pl, 'a set] => bool
eval :: ['a set, 'a pl] => bool ("_[_]" [100,0] 100)
hyps :: ['a pl, 'a set] => 'a pl set
translations
"H |- p" == "p : thms(H)"
inductive "thms(H)"
intrs
H "p:H ==> H |- p"
K "H |- p->q->p"
S "H |- (p->q->r) -> (p->q) -> p->r"
DN "H |- ((p->false) -> false) -> p"
MP "[| H |- p->q; H |- p |] ==> H |- q"
defs
sat_def "H |= p == (!tt. (!q:H. tt[q]) --> tt[p])"
eval_def "tt[p] == eval2 p tt"
primrec eval2 pl
"eval2(false) = (%x.False)"
"eval2(#v) = (%tt.v:tt)"
"eval2(p->q) = (%tt.eval2 p tt-->eval2 q tt)"
primrec hyps pl
"hyps(false) = (%tt.{})"
"hyps(#v) = (%tt.{if v:tt then #v else #v->false})"
"hyps(p->q) = (%tt.hyps p tt Un hyps q tt)"
end