ghost-state.tex

\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.
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}

  \infer{}
  {\TRUE \proves \plainly\TRUE}
\end{mathparpagebreakable}

Noteworthy here is the fact that $\prop \proves \later\prop$ can be derived from Löb induction, and $\TRUE \proves \plainly\TRUE$ can be derived via $\plainly$ commuting with universal quantification ranging over the empty type $0$.

\subsection{Persistent Propositions}
We call a proposition $\prop$ \emph{persistent} if $\prop \proves \always\prop$.
These are propositions that ``do not own anything'', so we can (and will) treat them like ``normal'' intuitionistic propositions.

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 Propositions 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 propositions which we call \emph{timeless} propositions, 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 propositions.

The following rules 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 propositions:
\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 OFE element}}
{\timeless{\term =_\type \term'}}

\infer
{\text{$\melt$ is a discrete OFE element}}
{\timeless{\ownM\melt}}

\infer
{\text{$\melt$ is an element of a discrete camera}}
{\timeless{\mval(\melt)}}
\end{mathparpagebreakable}



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