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+---
+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.