feat(Analysis/Calculus/DSlope): add eq_smul_dslope_of_zero and iterated version

[e-271828]

This PR adds two fundamental algebraic identities connecting a function with its dslope when the base point is a root:

• eq_smul_dslope_of_zero: f b = (b - a) • dslope f a b when f a = 0.
• eq_pow_smul_iterate_dslope_of_zero: The iterated version f b = (b - a) ^ n • (Function.swap dslope a)^[n] f b when the first n-1 iterated dslopes vanish at a.

These lemmas are essential for factoring out removable singularities locally.
This PR was assisted by LLMs (Aristotle and Gemini).

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[github-actions]

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[github-actions]

PR summary 14ec76b5f0

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ pow_sub_smul_iterate_dslope_of_zero
+ sub_smul_dslope_of_zero

You can run this locally as follows

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
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[j-loreaux]

these seem a bit niche. Why do you want them? (sorry, this is in the PR description)

Also, given some weird things I noticed, and your username, I have to ask: are you a bot? Is this PR LLM-generated? If so, it should be marked with that tag and disclosed in the PR description.

[e-271828]

@j-loreaux
Thank you very much for your thorough review and excellent suggestions!

To answer your question: No, I am not a bot, but I am a human developer using LLMs (specifically Harmonic's Aristotle and Google's Gemini) to assist me in formalizing these theorems. I apologize for not disclosing this initially and for accidentally leaving a Japanese comment in the code. I have updated the PR description to mention the LLM assistance. Could you please add the llm-generated tag to this PR?

I completely agree with all your code suggestions. The grind tactic compression is beautiful, and removing the redundant variables makes perfect sense. I will also turn the equalities around as per Mathlib conventions and rename the lemmas accordingly.

I will apply these changes to my local branch and push the updates shortly. Thank you again for your guidance!

[wwylele]

Why did the code regress? If you are blindly asking LLM to update the code without understanding it, this is wasting reviewers' time

[wwylele]

Likewise, this has also regressed

[e-271828]

-awaiting-author