(* Title: HOL/ex/PropLog.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
Inductive definition of propositional logic.
*)
PropLog = Main +
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)
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]])"
primrec
"tt[[false]] = False"
"tt[[#v]] = (v:tt)"
eval_imp "tt[[p->q]] = (tt[[p]] --> tt[[q]])"
primrec
"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