Since March 2022, MLIR has a Presburger library, FPL, that provides native support for a full set of polyhedral analysis operators. This functionality has already been deployed in the loop fusion pass in the Affine dialect and has also been used to enable better dependence analysis in CIRCT. In this tutorial, we demonstrate several case studies showing how to use FPL’s Presburger arithmetic functionality in MLIR. Reasoning precisely about sets of integers enables accurate analytical cache models and brings powerful transformations to loop optimizers for ML and HPC, formal verification, and hardware design. Despite many efforts to use Presburger arithmetic in LLVM, their use has thus far been confined to optional extensions like Polly due to the need for external Presburger libraries (e.g., isl) that were not part of the core compiler toolchain. In the course of developing FPL we worked closely with the LLVM community to make FPL and Presburger arithmetic available in the MLIR upstream repositories. In this talk, we give a detailed walkthrough and demonstrate how they can be used. Our objective is to overcome the longstanding ‘vendor-lock-in’ and inflexibility of polyhedral toolchains by working with the LLVM community to provide targeted analyses that enhance the native components of the LLVM ecosystem.