[Merged by Bors] - feat(Analysis/Complex/Exponential): add new bounds on exponential

[b-mehta]

We add new upper bounds on Real.exp in terms of 2x/(2+x), and use these to move some bounds on log higher in mathlib.
We also add a lemma for the common bound (1+1/n)^n <= e, though this is a special-case of one_sub_div_pow_le_exp_neg (immediately before), but is added for convenience and discoverability, as this is a "well-known" bound.

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[github-actions]

PR summary 3a0c2e492f

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Analysis.Complex.Exponential	1332	1335	+3 (+0.23%)
Mathlib.Analysis.SpecialFunctions.Log.Basic	1749	1751	+2 (+0.11%)
Mathlib.Analysis.SpecialFunctions.Log.Deriv	1959	1961	+2 (+0.10%)
Mathlib.Analysis.Complex.ExponentialBounds	1960	1962	+2 (+0.10%)

Import changes for all files

Files	Import difference
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Mathlib.Probability.Distributions.Binomial Mathlib.Probability.Distributions.Cauchy Mathlib.Probability.Distributions.Exponential Mathlib.Probability.Distributions.Fernique Mathlib.Probability.Distributions.Gamma Mathlib.Probability.Distributions.Gaussian.Basic Mathlib.Probability.Distributions.Gaussian.CharFun Mathlib.Probability.Distributions.Gaussian.Fernique Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Def Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence Mathlib.Probability.Distributions.Gaussian.Multivariate Mathlib.Probability.Distributions.Gaussian.Real Mathlib.Probability.Distributions.Geometric Mathlib.Probability.Distributions.Pareto 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Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic Mathlib.Tactic.NormNum.Irrational Mathlib.Tactic.NormNum.LegendreSymbol Mathlib.Topology.Algebra.PontryaginDual Mathlib.Topology.Algebra.Valued.LocallyCompact Mathlib.Topology.Algebra.Valued.NormedValued Mathlib.Topology.CWComplex.Abstract.Basic Mathlib.Topology.Category.TopCat.Sphere Mathlib.Topology.Compactification.OnePoint.Sphere Mathlib.Topology.EMetricSpace.PairReduction Mathlib.Topology.MetricSpace.GromovHausdorff Mathlib.Topology.MetricSpace.HausdorffDimension Mathlib.Topology.MetricSpace.HolderNorm Mathlib.Topology.MetricSpace.Holder Mathlib.Topology.MetricSpace.Kuratowski Mathlib.Topology.MetricSpace.Snowflaking	2
4 files Mathlib.Analysis.Complex.Exponential Mathlib.Analysis.SpecialFunctions.Exponential Mathlib.Analysis.SpecialFunctions.Log.ERealExp Mathlib.Combinatorics.Derangements.Exponential	3

---

Declarations diff

+ exp_le_two_add_div_two_sub
+ exp_lt_two_add_div_two_sub
+ one_add_inv_pow_le_exp

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.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[j-loreaux]

I have one concern, but if you're not worried about it you can merge anyway. If you are, I guess you can add the goal explicitly.

bors d+

[j-loreaux]

I realize why you're not filling in the RHS here, because it's ugly. Are you not worried about maintainability? e.g., something breaks here and then the user has to go look up what to write down because Lean can't infer the goal?

[b-mehta]

In general I am worried, but in this case I'm a lot less worried, since this can't break so badly - grw and apply are doing rigid things, and the x := and n := are making it explicit exactly what we're grewriting with

[mathlib-bors]

✌️ b-mehta can now approve this pull request until 2026-06-23 22:27 UTC (in 2 weeks). To approve and merge, reply with bors r+. More detailed instructions are available here.

⚠️ This delegation only covers changes within Archive/**, Counterexamples/**, docs/**, DownstreamTest/**, Mathlib/**, MathlibTest/**, widget/**, Archive.lean, Counterexamples.lean, docs.lean, Mathlib.lean; an author commit touching anything else will revoke it. Bors also revokes it if a later push changes too many files for it to check the full list — even if it stays within scope.

[b-mehta]

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Post-Build Step
• ci (staging) / Lint style
• ci (staging) / Build
• ci (staging) / Post-CI job