### on unique representatives

parent 1df177bd
 ... @@ -31,6 +31,23 @@ You can think of uniform predicates as monotone, step-indexed predicates over a ... @@ -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 $\COFEs$. $\UPred(-)$ 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 \clearpage \section{RA and CMRA constructions} \section{RA and CMRA constructions} ... ...
 ... @@ -24,10 +24,11 @@ Set Default Proof Using "Type". ... @@ -24,10 +24,11 @@ Set Default Proof Using "Type". It is worth noting that this equivalence relation admits canonical It is worth noting that this equivalence relation admits canonical representatives. More precisely, one can show that every representatives. More precisely, one can show that every equivalence class contains exactly one element P0 such that: equivalence class contains exactly one element P0 such that: ∀ x, (✓ x → P(x)) → P(x) (2) ∀ x, (✓ x → P0(x)) → P0(x) (2) (Again, this assertion has to be understood in sProp). Starting (Again, this assertion has to be understood in sProp). Intuitively, from an element P of a given class, one can build this canonical this says that P0 trivially holds whenever the resource is invalid. representative by chosing: Starting from any element P, one can find this canonical representative by choosing: P0(x) := ✓ x → P(x) (3) P0(x) := ✓ x → P(x) (3) Hence, as an alternative definition of uPred, we could use the set Hence, as an alternative definition of uPred, we could use the set ... ...
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