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 $0$.

Noteworthy here is the fact that Löb induction can be derived from $\later$-introduction and the fact that we can take fixed-points of functions where the recursive occurrences are below $\later$.%

Furthermore, $\TRUE\proves\plainly\TRUE$ can be derived via $\plainly$ commuting with universal quantification ranging over the empty type $0$.

To derive that existential quantifiers commute with separating conjunction requires an intermediate step using a magic wand: From $P *\exists x, Q \vdash\Exists x. P * Q$ we can deduce $\Exists x. Q \vdash P \wand\Exists x. P * Q$ and then proceed via $\exists$-elimination.

\subsection{Persistent Propositions}

We call a proposition $\prop$\emph{persistent} if $\prop\proves\always\prop$.