ex/PropLog.thy

Updated PL to PropLog using Larrys ind. defs.

(*  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