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Definition 5.1
![$\widetilde{M_0}:= \{ \Gamma \mid \Gamma \mbox{ maximal
$\mathit{ILM_0}$ -consis...
...orall
\mathrm{A} \in \Gamma \ \ Prop( \mathrm{A})
\subset
Prop( \mathcal{A}) \}$](img292.gif)
.
Prop(
B) is the set of propositional
variables
occurring in
B.
The definitions of
and
are precisely
as before.
In perfect analogy with the case of
we have the following
two lemmas:
Lemma 5.2
Let
![$ x \in \widetilde{M_0} $](img293.gif)
and
![$ \neg ( A \rhd B ) \in x \cap R $](img294.gif)
.
There exists
y such that
![$ x \prec_{\mbox{\footnotesize {B}}} y $](img250.gif)
and
![$ A \in y $](img181.gif)
.
Moreover,
y can be chosen to contain a
``maximal amount'' of
![$\Box$](img11.gif)
-formulas.
The M0-axiom is essentially used only in the next
lemma.
Lemma 5.4
Consider
![$ w \prec_{\mbox{\footnotesize {B}}} x \prec y $](img299.gif)
,
all in
![$\widetilde{M_0}$](img300.gif)
,
such that
![$ C \rhd D \in w$](img301.gif)
,
and
![$ C \in y $](img183.gif)
.
Then there exists
![$z \ _B \! \succ x $](img302.gif)
with both
![$D \in z$](img303.gif)
and
![$ x \subset_{\Box} z$](img304.gif)
.
This
z can be chosen to
be maximal w.r.t. the
![$\Box$](img11.gif)
-inclusion.
Joost Joosten
2000-02-07