---
title: "Implementing and Verifying \"Static Program Analysis\" in Agda, Part 2: Combining Lattices"
series: "Static Program Analysis in Agda"
date: 2024-04-13T14:23:03-07:01
draft: true
---

In the previous post, I wrote about how lattices arise when tracking, comparing
and combining static information about programs. I then showed two simple lattices:
the natural numbers, and the (parameterized) "above-below" lattice, which
modified an arbitrary set with "bottom" and "top" elements (\(\bot\) and \(\top\)
respectively). One instance of the "above-below" lattice was the sign lattice,
which could be used to reason about the signs (positive, negative, or zero)
of variables in a program.

At the end of that post, I introduced a source of complexity: the "full"
lattices that we want to use for the program analysis aren't signs or numbers,
but maps of states and variables to lattices-based states. The full lattice
for sign analysis might something in the form:

{{< latex >}}
\text{Info} \triangleq \text{ProgramStates} \to (\text{Variables} \to \text{Sign})
{{< /latex >}}

Thus, we have to compare and find least upper bounds (e.g.) of not just
signs, but maps! Proving the various lattice laws for signs was not too
challenging, but for for a two-level map like \(\text{info}\) above, we'd
need to do a lot more work. We need tools to build up such complicated lattices!

The way to do this, it turns out, is by using simpler lattices as building blocks.