refactor(Order/SuccPred): simplify toZ_mono via contrapositive

[pbalogh]

Summary

Simplify toZ_mono from 46 lines to 4 by using the contrapositive of the already-proved le_of_toZ_le.

Before (46 lines)

The previous proof manually handled all four quadrants (both signs of toZ i and toZ j relative to i0) with two nearly identical 20-line blocks, each constructing explicit succ/pred iterates and deriving contradictions via IsMax/IsMin.

After (4 lines)

Since le_of_toZ_le is proved independently (and before toZ_mono in the file), toZ_mono follows immediately by contrapositive:

• Assume ¬(toZ i0 i ≤ toZ i0 j), i.e., toZ i0 j < toZ i0 i
• By le_of_toZ_le: j ≤ i
• Combined with hypothesis i ≤ j: i = j
• This contradicts the strict inequality

Context

This simplification was identified by a geometric proof analysis tool that encodes Mathlib theorems as HRR (Holographic Reduced Representation) vectors and flags complexity anomalies — proofs significantly longer than their structural neighbors. toZ_mono was flagged at 46× the namespace median proof length.

---

• Builds clean against current Mathlib master
• No new dependencies
• Net deletion: 41 lines

[github-actions]

Welcome new contributor!

Thank you for contributing to Mathlib! If you haven't done so already, please review our contribution guidelines, as well as the style guide and naming conventions. In particular, we kindly remind contributors that we have guidelines regarding the use of AI when making pull requests.

We use a review queue to manage reviews. If your PR does not appear there, it is probably because it is not successfully building (i.e., it doesn't have a green checkmark), has the awaiting-author tag, or another reason described in the Lifecycle of a PR. The review dashboard has a dedicated webpage which shows whether your PR is on the review queue, and (if not), why.

If you haven't already done so, please come to https://leanprover.zulipchat.com/, introduce yourself, and mention your new PR.

Thank you again for joining our community.

[github-actions]

PR summary 415ab74ad3

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

No declarations were harmed in the making of this PR! 🐙

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

---

No changes to strong technical debt.

No changes to weak technical debt.

[tb65536]

Could you clarify whether you used AI for the PR summary and/or for the writing of the PR?

[pbalogh]

Could you clarify whether you used AI for the PR summary and/or for the writing of the PR?

I did use AI for the writing of the PR and its summary. Apologies for not knowing this was required info for a PR!

[vihdzp]

I think this PR identifies a problem, but doesn't correctly fix it. What we're missing here is the theorem that toZ is strictly monotonic; I've gone ahead and added that in #39014.

[pbalogh]

Ach, you're right that StrictMono toZ is the proper fix here. Our tooling identified the gap (the proof was more complex than needed given the underlying monotonicity), and your #39014 fills it correctly. Happy to close this in favor of that PR.