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George Pirlea
Iris
Commits
7b92a6d7
Commit
7b92a6d7
authored
Oct 06, 2016
by
Ralf Jung
Browse files
docs: update semantic entailment
parent
8bfac1ad
Changes
1
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docs/model.tex
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7b92a6d7
...
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@@ -98,20 +98,22 @@ We can now define \emph{semantic} logical entailment.
\typedsection
{
Interpretation of entailment
}{
\Sem
{
\vctx
\mid
\pfctx
\proves
\prop
}
:
\mProp
}
\[
\Sem
{
\vctx
\mid
\p
fctx
\proves
\prop
}
\eqdef
\Sem
{
\vctx
\mid
\p
rop
\proves
\prop
B
}
\eqdef
\begin
{
aligned
}
[
t
]
\MoveEqLeft
\forall
n
\in
\mathbb
{
N
}
.
\;
\forall
\rs
\in
\textdom
{
Res
}
.
\;
\forall
\gamma
\in
\Sem
{
\vctx
}
,
\;
\\
&
\bigl
(
\All
\propB
\in
\pfctx
.
n
\in
\Sem
{
\vctx
\proves
\prop
B
:
\Prop
}_
\gamma
(
\rs
)
\bigr
)
\Ra
n
\in
\Sem
{
\vctx
\proves
\prop
:
\Prop
}_
\gamma
(
\rs
)
n
\in
\Sem
{
\vctx
\proves
\prop
:
\Prop
}_
\gamma
(
\rs
)
\Ra
n
\in
\Sem
{
\vctx
\proves
\prop
B
:
\Prop
}_
\gamma
(
\rs
)
\end
{
aligned
}
\]
The soundness statement of the logic reads
\[
\vctx
\mid
\pfctx
\proves
\prop
\Ra
\Sem
{
\vctx
\mid
\pfctx
\proves
\prop
}
\]
The following theorem connects syntactic and semantic entailment:
\[
\vctx
\mid
\prop
\proves
\propB
\Ra
\Sem
{
\vctx
\mid
\prop
\proves
\propB
}
\]
It now becomes trivial to show soundness of the logic.
%%% Local Variables:
%%% mode: latex
...
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