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Add influences of Boolean functions and the KKL theorem - #83

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formalize/kkl-theorem
Nov 2, 2026
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tukamilano merged 1 commit into
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formalize/kkl-theorem

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Formalize the influence lower bounds on the uniform hypercube {±1}^n:

  • Influence/Martingale.lean: finite-average expectation, variance, entropy and Dirichlet form; the Doob martingale condFirst and its increments; orthogonal decompositions Var f = ∑ E[Δᵢ²] and 𝓔(f) = ∑ 𝓔(Δᵢ); the Bernoulli log-Sobolev inequality restated as Ent(f²) ≤ 2𝓔(f).
  • Influence/Basic.lean: influences, Lemma 9.3, Theorem 9.4 (via the log-sum inequality), the bound 4 Var f · exp(-I/(2 Var f)) ≤ ∑ Iᵢ², Theorem 9.5 (∑ Iᵢ² ≥ Var² log² n / n) and the KKL corollary exists_influence_ge.

Formalize the influence lower bounds on the uniform hypercube {±1}^n:

- Influence/Martingale.lean: finite-average expectation, variance, entropy
  and Dirichlet form; the Doob martingale `condFirst` and its increments;
  orthogonal decompositions Var f = ∑ E[Δᵢ²] and 𝓔(f) = ∑ 𝓔(Δᵢ); the
  Bernoulli log-Sobolev inequality restated as Ent(f²) ≤ 2𝓔(f).
- Influence/Basic.lean: influences, Lemma 9.3, Theorem 9.4 (via the
  log-sum inequality), the bound 4 Var f · exp(-I/(2 Var f)) ≤ ∑ Iᵢ²,
  Theorem 9.5 (∑ Iᵢ² ≥ Var² log² n / n) and the KKL corollary
  `exists_influence_ge`.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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tukamilano merged commit f529cd2 into main Nov 2, 2026
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