ex/PropLog.thy

-(*  Title: 	HOL/ex/pl.thy
+(*  Title: 	HOL/ex/PropLog.thy
     ID:         $Id$
     Author: 	Tobias Nipkow
     Copyright   1994  TU Muenchen
 
 Inductive definition of propositional logic.

   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"
 
-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])"
-
+defs
+  sat_def  "H |= p  ==  (!tt. (!q:H. tt[q]) --> tt[p])"
   eval_def "tt[p] == eval2(p,tt)"
 
 primrec eval2 pl
   eval2_false "eval2(false) = (%x.False)"
   eval2_var   "eval2(#v) = (%tt.v:tt)"