theory Modal0
imports LK0
begin
ML_file "modal.ML"
consts
box :: "o=>o" ("[]_" [50] 50)
dia :: "o=>o" ("<>_" [50] 50)
strimp :: "[o,o]=>o" (infixr "--<" 25)
streqv :: "[o,o]=>o" (infixr ">-<" 25)
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] = Const(c, dummyT) $ seq_tr s1 $ seq_tr s2;
fun star_tr' c [s1, s2] = Const(c, dummyT) $ seq_tr' s1 $ seq_tr' s2;
*}
parse_translation {*
[(@{syntax_const "_Lstar"}, star_tr @{const_syntax Lstar}),
(@{syntax_const "_Rstar"}, star_tr @{const_syntax Rstar})]
*}
print_translation {*
[(@{const_syntax Lstar}, star_tr' @{syntax_const "_Lstar"}),
(@{const_syntax Rstar}, star_tr' @{syntax_const "_Rstar"})]
*}
defs
strimp_def: "P --< Q == [](P --> Q)"
streqv_def: "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