Commit cd3a1805 by Ralf Jung

### docs: notation for nonexp functions

parent d7cae999
 ... ... @@ -25,7 +25,7 @@ \end{defn} \begin{defn} A function $f : \cofe \to \cofeB$ between two COFEs is \emph{non-expansive} if A function $f : \cofe \to \cofeB$ between two COFEs is \emph{non-expansive} (written $f : \cofe \nfn \cofeB$) if $\All n, x \in \cofe, y \in \cofe. x \nequiv{n} y \Ra f(x) \nequiv{n} f(y)$ It is \emph{contractive} if $\All n, x \in \cofe, y \in \cofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(x)$ ... ... @@ -49,8 +49,9 @@ Note that $\COFEs$ is cartesian closed. \subsection{CMRA} \begin{defn} A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \mcore{-}: \monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid, (\mdiv) : \monoid \times \monoid \to \monoid)$ satisfying A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \mcore{-}: \monoid \nfn \monoid, (\mtimes) : \monoid \times \monoid \nfn \monoid, (\mdiv) : \monoid \times \monoid \nfn \monoid)$ satisfying \begin{align*} \All n, \melt, \meltB.& \melt \nequiv{n} \meltB \land \melt\in\mval_n \Ra \meltB\in\mval_n \tagH{cmra-valid-nonexp} \\ \All n, m.& n \geq m \Ra V_n \subseteq V_m \tagH{cmra-valid-mono} \\ \All \melt, \meltB, \meltC.& (\melt \mtimes \meltB) \mtimes \meltC = \melt \mtimes (\meltB \mtimes \meltC) \tagH{cmra-assoc} \\ \All \melt, \meltB.& \melt \mtimes \meltB = \meltB \mtimes \melt \tagH{cmra-comm} \\ ... ... @@ -143,7 +144,7 @@ Note that every RA is a discrete CMRA, by picking the discrete COFE for the equi Furthermore, discrete CMRAs can be turned into RAs by ignoring their COFE structure, as well as the step-index of $\mval$. \begin{defn} A function $f : \monoid_1 \to \monoid_2$ between two CMRAs is \emph{monotone} if it satisfies the following conditions: A function $f : \monoid_1 \to \monoid_2$ between two CMRAs is \emph{monotone} (written $f : \monoid_1 \monra \monoid_2$) if it satisfies the following conditions: \begin{enumerate}[itemsep=0pt] \item $f$ is non-expansive \item $f$ preserves validity: \\ ... ...
 ... ... @@ -52,6 +52,7 @@ \newcommand{\Ra}{\Rightarrow} \newcommand{\Lra}{\Leftrightarrow} \newcommand\monra{\xrightarrow{\kern-0.25ex\textrm{mon}\kern-0.25ex}} \newcommand\nfn{\xrightarrow{\kern-0.15ex\textrm{n}\kern-0.05ex}} \newcommand{\eqdef}{\triangleq} \newcommand{\bnfdef}{\vcentcolon\vcentcolon=} ... ...
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