Research proof index

A dependent-random-choice corollary. Let $A$ be nonempty, let $0<\varepsilon\le 1$ and $\delta>0$, and let $p$ be a nonzero even integer satisfying $\varepsilon^{-1}\log(2/\delta)\le p$. Then there are sets $A_1,A_2$ such that the normalized difference distribution $\mu_{A_1}\mathbin{\circleddash}\mu_{A_2}$ assigns mass at least $1-\delta$ to the source's sifted set $s_{p,\varepsilon}(A)$. Both sets retain explicit density: $\operatorname{dens}(A_i)\ge \tfrac14\operatorname{dens}(A)^{2p}$ for $i=1,2$.

Generic derivative bound for σ_k q-series on compact sets. Uses σ_k(n) ≤ n^(k+1) (sigma_bound) and iter_deriv_comp_bound3 for exponential decay.

Generic summability for n^a * σ_b(n) * exp(2πinz) series. Uses σ_b(n) ≤ n^(b+1) (sigma_bound) and a33 (a+b+1) for exponential summability.

If every value x in the support of gen satisfies Pr[= a | f x] ≥ 1 - δ, then the overall probability satisfies Pr[= a | gen >>= f] ≥ 1 - δ. This reduces a "for all keys" completeness statement to per-key bounds.

Phase B (freshness-drop) bound. The CMA advantage is bounded above by the success probability of the same experiment with the freshness check dropped. Both unforgeableExp and unforgeableExpNoFresh factor as runtime.evalDist (joint >>= ...) sharing the same prefix joint. The hypothesis h_pull packages the runtime-specific factoring step that...

Failing companion to simulateQ_optionT_forIn_yield_pure_some: if each loop body, under simulateQ, resolves to pure (some (ForInStep.yield init)) when its per-element condition cond a holds and to pure none otherwise, and some element of the list fails its condition, then the whole OptionT-monadic forIn resolves to pure none (the failure...