feat(Data/ENNReal, Probability/Independence): division decomposition and self-independence

[jdhart81]

Two general-purpose lemmas extracted from a formal verification of "The Intelligence Bound" (Hart 2025).

---

New lemmas

• ENNReal.div_eq_div_mul_div: Division decomposition a / c = a / b * (b / c) when b ≠ 0 and b ≠ ⊤. Added to Mathlib.Data.ENNReal.Inv.

• IndepFun.of_self: If Y is independent of itself under a probability measure, then X and Y are independent. Self-independence implies Y is a.e. constant. Added to Mathlib.Probability.Independence.Basic.

Context

These lemmas arose in formalizing information-theoretic results where:

• The ENNReal division identity is needed for decomposing rate expressions (limsup of ratios)
• The self-independence result bridges entropy = 0 and measure-theoretic independence

References

• Formal verification: https://github.com/jdhart81/intelligence-bound-lean
• Paper: "The Intelligence Bound" (Hart 2025)

[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.

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.

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 b73ba7707a

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ IndepFun.of_self
+ div_eq_div_mul_div

You can run this locally as follows

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

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

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

---

Increase in tech debt: (relative, absolute) = (1.00, 0.01)

Current number	Change	Type
140	1	flexible linter exceptions

Current commit 0dcaf7fd5b
Reference commit b73ba7707a

You can run this locally as

./scripts/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).