[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"P74OPoi4kv":3},"# Domain Theory in Lean\n\nThis library defined basic concepts from domain theory, following [Abramsky and Jung 1995](https://achimjungbham.github.io/pub/papers/handy1.pdf).\nDomain Theory is useful for defining denotational semantics of programming languages, particularly probabilistic languages.\n\nAlthough mathlib aleady defines a `CompletePartialOrder` class, that class is painful to use because it extends `SupSet` and `OrderBot`.\n`SupSet` is problematic because it requires one to define a supremum of any arbitrary set, whereas in many cases the supremum only exists for a directed set.\n`OrderBot` is problematic because some DCPOs do not have a least element, if they do have a least element, then they are called *pointed* DCPOs.\nMoreover, directed sets must be nonempty, which is not imposed in mathlib's `CompletePartialOrder` class.\nBy contrast, this library more closely follows the established theory.\nThe implementation is closer to mathlib's `OmegaCompletePartialOrder` class, which only requires giving a supremum of an omega chain rather than an arbitrary set.",1785080796541]