ex/PL.thy

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(*  Title: 	HOL/ex/pl.thy
    ID:         $Id$
    Author: 	Tobias Nipkow
    Copyright   1994  TU Muenchen

Inductive definition of propositional logic.
*)

PL = Finite +
datatype
    'a pl = false | var ('a) ("#_" [1000]) | "->" ('a pl,'a pl) (infixr 90)
consts
    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)
    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}"

  (*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