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Ralf Jung
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\section{Extensions of the Base Logic}
In this section we discuss some additional constructions that we define within and on top of the base logic.
Ralf Jung
committed
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These are not ``extensions'' in the sense that they change the proof power of the logic, they just form useful derived principles.
\subsection{Derived rules about base connectives}
We collect here some important and frequently used derived proof rules.
\begin{mathparpagebreakable}
\infer{}
{\prop \Ra \propB \proves \prop \wand \propB}
\infer{}
{\prop * \Exists\var.\propB \provesIff \Exists\var. \prop * \propB}
\infer{}
{\prop * \All\var.\propB \proves \All\var. \prop * \propB}
\infer{}
{\always(\prop*\propB) \provesIff \always\prop * \always\propB}
\infer{}
{\always(\prop \Ra \propB) \proves \always\prop \Ra \always\propB}
\infer{}
{\always(\prop \wand \propB) \proves \always\prop \wand \always\propB}
\infer{}
{\always(\prop \wand \propB) \provesIff \always(\prop \Ra \propB)}
\infer{}
{\later(\prop \Ra \propB) \proves \later\prop \Ra \later\propB}
\infer{}
{\later(\prop \wand \propB) \proves \later\prop \wand \later\propB}
\infer{}
{\prop \proves \later\prop}
\end{mathparpagebreakable}
\subsection{Persistent assertions}
We call an assertion $\prop$ \emph{persistent} if $\prop \proves \always\prop$.
These are assertions that ``don't own anything'', so we can (and will) treat them like ``normal'' intuitionistic assertions.
Of course, $\always\prop$ is persistent for any $\prop$.
Furthermore, by the proof rules given in \Sref{sec:proof-rules}, $\TRUE$, $\FALSE$, $t = t'$ as well as $\ownGhost\gname{\mcore\melt}$ and $\mval(\melt)$ are persistent.
Persistence is preserved by conjunction, disjunction, separating conjunction as well as universal and existential quantification and $\later$.
\subsection{Timeless assertions and except-0}
One of the troubles of working in a step-indexed logic is the ``later'' modality $\later$.
It turns out that we can somewhat mitigate this trouble by working below the following \emph{except-0} modality:
\[ \diamond \prop \eqdef \later\FALSE \lor \Prop \]
This modality is useful because there is a class of assertions which we call \emph{timeless} assertions, for which we have
\[ \timeless{\prop} \eqdef \later\prop \proves \diamond\prop \]
In other words, when working below the except-0 modality, we can \emph{strip away} the later from timeless assertions.
The following ruels can be derived about except-0:
\begin{mathpar}
\inferH{ex0-mono}
{\prop \proves \propB}
{\diamond\prop \proves \diamond\propB}
\axiomH{ex0-intro}
{\prop \proves \diamond\prop}
\axiomH{ex0-idem}
{\diamond\diamond\prop \proves \diamond\prop}
\begin{array}[c]{rMcMl}
\diamond{(\prop * \propB)} &\provesIff& \diamond\prop * \diamond\propB \\
\diamond{(\prop \land \propB)} &\provesIff& \diamond\prop \land \diamond\propB \\
\diamond{(\prop \lor \propB)} &\provesIff& \diamond\prop \lor \diamond\propB
\end{array}
\begin{array}[c]{rMcMl}
\diamond{\All x. \prop} &\provesIff& \All x. \diamond{\prop} \\
\diamond{\Exists x. \prop} &\provesIff& \Exists x. \diamond{\prop} \\
\diamond\always{\prop} &\provesIff& \always\diamond{\prop} \\
\diamond\later\prop &\proves& \later{\prop}
\end{array}
\end{mathpar}
The following rules identify the class of timeless assertions:
\begin{mathparpagebreakable}
\infer
{\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \land \propB}}
\infer
{\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \lor \propB}}
\infer
{\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop * \propB}}
\infer
{\vctx \proves \timeless{\prop}}
{\vctx \proves \timeless{\always\prop}}
\infer
{\vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \Ra \propB}}
\infer
{\vctx \proves \timeless{\propB}}
{\vctx \proves \timeless{\prop \wand \propB}}
\infer
{\vctx,\var:\type \proves \timeless{\prop}}
{\vctx \proves \timeless{\All\var:\type.\prop}}
\infer
{\vctx,\var:\type \proves \timeless{\prop}}
{\vctx \proves \timeless{\Exists\var:\type.\prop}}
\axiom{\timeless{\TRUE}}
\axiom{\timeless{\FALSE}}
\infer
{\text{$\term$ or $\term'$ is a discrete COFE element}}
{\timeless{\term =_\type \term'}}
\infer
{\text{$\melt$ is a discrete COFE element}}
{\timeless{\ownM\melt}}
\infer
{\text{$\melt$ is an element of a discrete CMRA}}
{\timeless{\mval(\melt)}}
\end{mathparpagebreakable}