Commit 25118e82 authored by Ralf Jung's avatar Ralf Jung
Browse files

complete description of COFEs

parent 87e74b01
Pipeline #192 passed with stage
......@@ -2,13 +2,17 @@
Given some set $T$ and an indexed family $({\nequiv{n}} \subseteq T \times T)_{n \in \mathbb{N}}$ of equivalence relations, a \emph{chain} is a function $c : \mathbb{N} \to T$ such that $\All n, m. n < m \Ra c (m) \nequiv{n} c (n+1)$.
A COFE is a tuple $(T, (\nequiv{n})_{n \in \mathbb{N}}, c : (\mathbb{N} \to T) \to T)$ satisfying
A \emph{complete ordered family of equivalences} (COFE) is a tuple $(T, ({\nequiv{n}} \subseteq T \times T)_{n \in \mathbb{N}}, \lim : \chain(T) \to T)$ satisfying
\All n. (\nequiv{n}) ~& \text{is an equivalence relation} \tagH{cofe-equiv} \\
\All n, m.& n \geq m \Ra (\nequiv{n}) \subseteq (\nequiv{m}) \tagH{cofe-mono} \\
\All x, y.& x = y \Lra (\All n. x \nequiv{n} y) \tagH{cofe-limit} \\
\All n, X.& c(X) \nequiv{n} X(n+1) \tagH{cofe-compl}
\All n, c.& \lim(c) \nequiv{n} c(n+1) \tagH{cofe-compl}
......@@ -17,7 +21,7 @@
A CMRA is a tuple $(\monoid, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \munit: \monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid, (\mdiv) : \monoid \times \monoid \to \monoid)$ satisfying
A \emph{CMRA} is a tuple $(\monoid, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \munit: \monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid, (\mdiv) : \monoid \times \monoid \to \monoid)$ satisfying
\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} \\
......@@ -176,10 +176,9 @@
\newcommand{\pset}[1]{\wp(#1)} % Powerset
\newcommand{\WHEN}{\textrm{when }}
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment