ex/PropLog.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/PropLog.thy	Tue Aug 30 10:05:46 1994 +0200
@@ -0,0 +1,61 @@
+(*  Title: 	HOL/ex/pl.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"
+
+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])"
+
+  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))"
+
+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