feat(Analysis/Calculus): define absolutely monotone functions

[mrdouglasny]

Summary

• Define AbsolutelyMonotoneOn f s for functions f : ℝ → ℝ that are smooth on s with all iterated derivatives within s nonneg
• Prove closure under add, smul, mul
• Show exp, cosh, constants, and powers are absolutely monotone on appropriate domains

This is the first part of a split of #37879. Follow-up PRs will add:

• Bernstein backward direction (nonneg coefficients → absolutely monotone)
• Bernstein forward direction (absolutely monotone → analytic at 0)
• Matrix applications (Schur product theorem for power series, entrywise exp)

Test plan

• CI passes (build + lint)
• Verify AbsolutelyMonotoneOn structure and examples type-check

🤖 Generated with Claude Code

Co-Authored-By: Claude Opus 4.6 (1M context) noreply@anthropic.com

[github-actions]

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[github-actions]

PR summary 2ff88851d5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.Calculus.AbsolutelyMonotone (new file)	1802

---

Declarations diff

+ AbsolutelyMonotoneOn
+ add
+ mul
+ of_contDiff
+ smul

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.

---

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.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[vihdzp]

Suggested change

	on `s` and all its iterated derivatives within `s` are nonneg on `s`.
	on `s` and all its iterated derivatives within `s` are nonnegative on `s`.

[vihdzp]

This seems redundant to me, since it follows immediately from 0 ≤ iteratedDerivWithin n f s x.

[vihdzp]

I think these examples should go in separate files. This definition shouldn't require Real.cosh in order to be imported, for instance.

[vihdzp]

Suggested change

	· exact le_refl 0
	· rfl

[wwylele]

Could you format the code to fill up the 100-character limit per line?
(Why is this a recurring problem from Claude code? This should be trivial to solve, either by human or AI)