Blueprint geometric lemma exppos#44
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Align blueprint structure with Lean code by extracting the claim that 𝔼[f(Z₁,...,Zᵣ)] ≥ 0 into a separate lemma (lem:expPos). This was previously stated inline in the geometric lemma proof, following the original paper's structure. The Lean code has this as a separate lemma `expPos`, so we extract it to match. The proof remains as stated in the paper: follows from the moments lemma and linearity of expectation, using the fact that all Taylor coefficients of f are non-negative. This is the first step toward addressing issue #8.
Extract the integral calculation ∫₀^∞ (1/(2√x)) e^(-√x) dx = 1 as a separate lemma in the Integration theory subsection. This was previously just mentioned inline in the geometric lemma proof. The Lean code has this as `integral_exp_neg_sqrt` in Integrals.lean, and the blueprint was referencing a non-existent lem:integral-exp-decay in the uses clause. The proof is straightforward by substitution u = √x. This continues alignment work before tackling issue #8.
Extract two key technical lemmas that were inline in the geometric lemma proof: 1. lem:exp-ineq (exp_ineq_ENN): Shows 1 - Pr(E) is bounded by an expectation, derived by splitting f into E and E^c parts and applying the special function bounds. 2. lem:lintegral-sum-bound: Bounds the expectation via integral manipulation and exponential decay, using the fundamental theorem of calculus and Tonelli's theorem. The geometric lemma proof is now much cleaner, simply referencing these helper lemmas. This matches the Lean structure where these are separate lemmas in GeometricLemma.lean. This completes the structural alignment before tackling issue #8.
Expand the proof of lem:taylor-nonneg to show explicitly how f
has a Taylor expansion with non-negative coefficients.
The proof:
1. Expands each term x_j ∏_{i≠j} (2 + cosh√x_i) using the definition
cosh√x = ∑ₙ xⁿ/(2n)!
2. Shows how products expand via independent choices at each index
3. Introduces multi-index notation α = (α₁,...,αᵣ) for coefficients
4. Proves coefficient formula: c_{j,α} = 2^{...} ∏ 1/(2αᵢ)!
5. Verifies all c_{j,α} > 0 since factorials and powers of 2 are positive
6. Includes concrete example for r=2 to illustrate
This detailed expansion is needed for lem:expPos to explicitly
identify the ℓᵢ exponents used in the moments lemma.
Addresses part of issue #8.
Expand the proof of lem:expPos to explicitly show how the moments lemma is applied, addressing issue #8's request for explicit ℓᵢ values. The proof: 1. Uses the Taylor expansion from lem:taylor-nonneg 2. Substitutes Zᵢ = 3r·⟨σᵢ(U),σᵢ(U')⟩ 3. Applies linearity of expectation to distribute over the sum 4. For each multi-index α, explicitly applies lem:moments with ℓ₁ = α₁, ..., ℓᵣ = αᵣ 5. Shows each term is non-negative as a product of: - Non-negative coefficient c_{j,α} ≥ 0 - Positive constant (3r)^{|α|} > 0 - Non-negative expectation from moments lemma 6. Concludes that the sum of non-negative terms is non-negative This completes the detailed proof requested in issue #8. Closes #8.
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