(* Title: HOL/ex/prop-log
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Inductive definition of propositional logic.
*)
PL = Finite +
types pl 1
arities pl :: (term)term
consts
false :: "'a pl"
"->" :: "['a pl,'a pl] => 'a pl" (infixr 90)
var :: "'a => 'a pl" ("#_")
pl_rec :: "['a pl,'a => 'b, 'b, ['b,'b] => 'b] => 'b"
axK,axS,axDN:: "'a pl set"
ruleMP,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"
rules
(** Proof theory for propositional logic **)
axK_def "axK == {x . ? p q. x = p->q->p}"
axS_def "axS == {x . ? p q r. x = (p->q->r) -> (p->q) -> p->r}"
axDN_def "axDN == {x . ? p. x = ((p->false) -> false) -> p}"
(*the use of subsets simplifies the proof of monotonicity*)
ruleMP_def "ruleMP(X) == {q. ? p:X. p->q : X}"
thms_def
"thms(H) == lfp(%X. H Un axK Un axS Un axDN Un ruleMP(X))"
conseq_def "H |- p == p : thms(H)"
sat_def "H |= p == (!tt. (!q:H. tt[q]) --> tt[p])"
pl_rec_var "pl_rec(#v,f,y,z) = f(v)"
pl_rec_false "pl_rec(false,f,y,z) = y"
pl_rec_imp "pl_rec(p->q,f,y,g) = g(pl_rec(p,f,y,g),pl_rec(q,f,y,g))"
eval_def "tt[p] == pl_rec(p, %v.v:tt, False, op -->)"
hyps_def
"hyps(p,tt) == pl_rec(p, %a. {if(a:tt, #a, (#a)->false)}, {}, op Un)"
var_inject "(#v = #w) ==> v = w"
imp_inject "[| (p -> q) = (p' -> q'); [| p = p'; q = q' |] ==> R |] ==> R"
var_neq_imp "(#v = (p -> q)) ==> R"
pl_ind "[| P(false); !!v. P(#v); !!p q. P(p)-->P(q)-->P(p->q)|] ==> !t.P(t)"
end