(* Title: Sequents/Modal0.thy
Author: Martin Coen
Copyright 1991 University of Cambridge
*)
theory Modal0
imports LK0
begin
ML_file \<open>modal.ML\<close>
consts
box :: "o\<Rightarrow>o" ("[]_" [50] 50)
dia :: "o\<Rightarrow>o" ("<>_" [50] 50)
Lstar :: "two_seqi"
Rstar :: "two_seqi"
syntax
"_Lstar" :: "two_seqe" ("(_)|L>(_)" [6,6] 5)
"_Rstar" :: "two_seqe" ("(_)|R>(_)" [6,6] 5)
ML \<open>
fun star_tr c [s1, s2] = Const(c, dummyT) $ seq_tr s1 $ seq_tr s2;
fun star_tr' c [s1, s2] = Const(c, dummyT) $ seq_tr' s1 $ seq_tr' s2;
\<close>
parse_translation \<open>
[(\<^syntax_const>\<open>_Lstar\<close>, K (star_tr \<^const_syntax>\<open>Lstar\<close>)),
(\<^syntax_const>\<open>_Rstar\<close>, K (star_tr \<^const_syntax>\<open>Rstar\<close>))]
\<close>
print_translation \<open>
[(\<^const_syntax>\<open>Lstar\<close>, K (star_tr' \<^syntax_const>\<open>_Lstar\<close>)),
(\<^const_syntax>\<open>Rstar\<close>, K (star_tr' \<^syntax_const>\<open>_Rstar\<close>))]
\<close>
definition strimp :: "[o,o]\<Rightarrow>o" (infixr "--<" 25)
where "P --< Q == [](P \<longrightarrow> Q)"
definition streqv :: "[o,o]\<Rightarrow>o" (infixr ">-<" 25)
where "P >-< Q == (P --< Q) \<and> (Q --< P)"
lemmas rewrite_rls = strimp_def streqv_def
lemma iffR: "\<lbrakk>$H,P \<turnstile> $E,Q,$F; $H,Q \<turnstile> $E,P,$F\<rbrakk> \<Longrightarrow> $H \<turnstile> $E, P \<longleftrightarrow> Q, $F"
apply (unfold iff_def)
apply (assumption | rule conjR impR)+
done
lemma iffL: "\<lbrakk>$H,$G \<turnstile> $E,P,Q; $H,Q,P,$G \<turnstile> $E \<rbrakk> \<Longrightarrow> $H, P \<longleftrightarrow> Q, $G \<turnstile> $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