Commit d4aa9dd9 authored by Ralf Jung's avatar Ralf Jung

on unique representatives

parent 1df177bd
......@@ -31,6 +31,23 @@ You can think of uniform predicates as monotone, step-indexed predicates over a
$\UPred(-)$ is a locally non-expansive functor from $\CMRAs$ to $\COFEs$.
It is worth noting that the above quotient admits canonical
representatives. More precisely, one can show that every
equivalence class contains exactly one element $P_0$ such that:
\[ \All n, \melt. (\mval(\melt) \nincl{n} P_0(\melt)) \Ra n \in P_0(\melt) \tagH{UPred-canonical} \]
Intuitively, this says that $P_0$ trivially holds whenever the resource is invalid.
Starting from any element $P$, one can find this canonical
representative by choosing $P_0(\melt) := \setComp{n}{n \in \mval(\melt) \Ra n \in P(\melt)}$.
Hence, as an alternative definition of $\UPred$, we could use the set
of canonical representatives. This alternative definition would
save us from using a quotient. However, the definitions of the various
connectives would get more complicated, because we have to make sure
they all verify \ruleref{UPred-canonical}, which sometimes requires some adjustments. We
would moreover need to prove one more property for every logical
connective.
\clearpage
\section{RA and CMRA constructions}
......
......@@ -24,10 +24,11 @@ Set Default Proof Using "Type".
It is worth noting that this equivalence relation admits canonical
representatives. More precisely, one can show that every
equivalence class contains exactly one element P0 such that:
∀ x, (✓ x → P(x)) → P(x) (2)
(Again, this assertion has to be understood in sProp). Starting
from an element P of a given class, one can build this canonical
representative by chosing:
∀ x, (✓ x → P0(x)) → P0(x) (2)
(Again, this assertion has to be understood in sProp). Intuitively,
this says that P0 trivially holds whenever the resource is invalid.
Starting from any element P, one can find this canonical
representative by choosing:
P0(x) := ✓ x → P(x) (3)
Hence, as an alternative definition of uPred, we could use the set
......
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