Lax Lean Archive

Proof of `Splitting by minimal representatives preserves coarsening`

groundedproofs/Lax185574Proofs/SplittingMonotone.lean · lax-185574

What this proof establishes

no assumptions

lem✓Lax185574.SplittingMonotone

Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.

Read the Lean proof on GitHub

In the paper

  • page 35 of this submission's paper

Description

As in the paper. If đ’«iâ‰Œđ’«jđ’«á”ą ≌ đ’«â±Œ then a vertex that is minimal in its part of đ’«jđ’«â±Œ is minimal in its part of đ’«iđ’«á”ą, so Tj⊆TiTⱌ ⊆ Tᔹ. Two vertices in the same part of 𝒬iđ’Źá”ą lie in the same part of đ’«iđ’«á”ą, hence of đ’«jđ’«â±Œ. They have the same neighbors in Tiâˆ–đ’«i(u)Tᔹ ∖ đ’«á”ą(u), which contains Tjâˆ–đ’«j(u)Tⱌ ∖ đ’«â±Œ(u), so they lie in the same part of 𝒬jđ’Źâ±Œ.