thanks for these useful info and doc
Just as a info/update on this thread, MLIR now have “controlable schedule” concept that allows a separate schedule to control things on a IR. As a example, in their presentation, Halide and TVM are cited. Probably now, with this, is more easy (or at least it would be) and straightforward to represent TVM’s schedule part and the TIR parts with MLIR. This “controlable part” (or schedule) is a dialect itself, but one controlling elements of a IR.
I leave this here, just as a pure info, but I assume many of you are already being aware.
The presentation: https://youtu.be/P4gUj3QtH_Y
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The shared focus between Poly and TVM lies in their use of integer and integer set analysis. I see a valuable opportunity for both MLIR and TVM to enhance and benefit from each other in this area. Essentially, the central concept here involves leveraging integer set analysis, which could be referred to as polyhedral or hypercube analysis. 
May be this will help you more https://mlir.llvm.org/docs/Tutorials/transform-details/Ch0/
Currently there is an intial support (optional) for MLIR Presburger (Polly) right in TVM:
- [Arith] Add tvm::arith::PresburgerSetNode to work with Presburger Set in MLIR by multiverstack-intellif · Pull Request #14690 · apache/tvm · GitHub
- [RFC][Tracking Issue] Introduce PresburgerSet · Issue #14006 · apache/tvm · GitHub
As for the IR expressivness part, there is a example (out on github) for a TVM relay dialect parser done in MLIR:
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