Theory Modal0

(*  Title:      Sequents/Modal0.thy
    Author:     Martin Coen
    Copyright   1991  University of Cambridge
*)

theory Modal0
imports LK0
begin

ML_file modal.ML

consts
  box           :: "oo"       ("[]_" [50] 50)
  dia           :: "oo"       ("<>_" [50] 50)
  Lstar         :: "two_seqi"
  Rstar         :: "two_seqi"

syntax
  "_Lstar"      :: "two_seqe"   ("(_)|L>(_)" [6,6] 5)
  "_Rstar"      :: "two_seqe"   ("(_)|R>(_)" [6,6] 5)

ML 
  fun star_tr c [s1, s2] = Syntax.const c $ seq_tr s1 $ seq_tr s2;
  fun star_tr' c [s1, s2] = Syntax.const c $ seq_tr' s1 $ seq_tr' s2;


parse_translation 
 [(syntax_const‹_Lstar›, K (star_tr const_syntaxLstar)),
  (syntax_const‹_Rstar›, K (star_tr const_syntaxRstar))]


print_translation 
 [(const_syntaxLstar, K (star_tr' syntax_const‹_Lstar›)),
  (const_syntaxRstar, K (star_tr' syntax_const‹_Rstar›))]


definition strimp :: "[o,o]o"   (infixr "--<" 25)
  where "P --< Q == [](P  Q)"

definition streqv :: "[o,o]o"   (infixr ">-<" 25)
  where "P >-< Q == (P --< Q)  (Q --< P)"


lemmas rewrite_rls = strimp_def streqv_def

lemma iffR: "$H,P  $E,Q,$F;  $H,Q  $E,P,$F  $H  $E, P  Q, $F"
  apply (unfold iff_def)
  apply (assumption | rule conjR impR)+
  done

lemma iffL: "$H,$G  $E,P,Q;  $H,Q,P,$G  $E   $H, P  Q, $G  $E"
  apply (unfold iff_def)
  apply (assumption | rule conjL impL basic)+
  done

lemmas safe_rls = basic conjL conjR disjL disjR impL impR notL notR iffL iffR
  and unsafe_rls = allR exL
  and bound_rls = allL exR

end