--- a/ex/PL.thy Sat Aug 13 16:33:53 1994 +0200
+++ b/ex/PL.thy Sat Aug 13 16:34:30 1994 +0200
@@ -14,34 +14,36 @@
ruleMP,thms :: "'a pl set => 'a pl set"
"|-" :: "['a pl set, 'a pl] => bool" (infixl 50)
"|=" :: "['a pl set, 'a pl] => bool" (infixl 50)
- pl_rec :: "['a pl,'a => 'b, 'b, ['b,'b] => 'b] => 'b"
+ eval2 :: "['a pl, 'a set] => bool"
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}"
+ 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}"
+ (*the use of subsets simplifies the proof of monotonicity*)
+ ruleMP_def "ruleMP(X) == {q. ? p:X. p->q : X}"
- thms_def
+ thms_def
"thms(H) == lfp(%X. H Un axK Un axS Un axDN Un ruleMP(X))"
- conseq_def "H |- p == p : thms(H)"
+ conseq_def "H |- p == p : thms(H)"
- sat_def "H |= p == (!tt. (!q:H. tt[q]) --> tt[p])"
+ 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] == eval2(p,tt)"
+primrec eval2 pl
+ eval2_false "eval2(false) = (%x.False)"
+ eval2_var "eval2(#v) = (%tt.v:tt)"
+ eval2_imp "eval2(p->q) = (%tt.eval2(p,tt)-->eval2(q,tt))"
- 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)"
+primrec hyps pl
+ hyps_false "hyps(false) = (%tt.{})"
+ hyps_var "hyps(#v) = (%tt.{if(v:tt, #v, #v->false)})"
+ hyps_imp "hyps(p->q) = (%tt.hyps(p,tt) Un hyps(q,tt))"
end