Research proof index

For $0\le a\le b$ and a differentiable function $f$ whose derivative is continuous on $[a,b]$, $\int_a^b f'(t)(\lfloor t\rfloor+1)\,dt=(b+\tfrac12)f(b)-(a+\tfrac12)f(a)-\int_a^b f(t)\,dt-\int_a^b f'(t)B_1(t)\,dt,$ where $B_1(t)=t-\lfloor t\rfloor-\tfrac12$.

The statement that for a measure μ of dimension d, and a function f : M → G of dimension (CarriesDimension.d G * d⁻¹) (where CarriesDimension.d G is the dimension associated with terms of type G), then ∫ x, f x ∂μ has the correct dimension, namely CarriesDimension.d G. In other words, the function: ``` fun (μ : DimSet (MeasureTheory.Measu...

If f vanishes a.e. on U and g is supported in U, then ∫ f • g ∂μU = 0.

If f i converges to g in Lᵖ(U), then for any test function ψ ∈ C_c^∞(U), the integrals ∫ f i * ψ ∂μU converge to ∫ g * ψ ∂μU.

Under the empty env alphabet (EnvAction.empty Unit) and the trivial env scheduler (always processTick), runStepsAsync reduces to ProcessOver.runSteps with the env state pinned to () and a constant processTick trace. This is the operational core of the sync-recovery story: it factors the async engine into the synchronous ProcessOver.runSteps...

inductively to each such matrix, we conclude that, for each r_{ϑ-1} ∈ R*, d^{2^{ϑ-1}}((1-r_{ϑ-1}) · U₀ + r_{ϑ-1} · U₁, C^{2^{ϑ-1}}) ≤ e -/ lemma correlatedAgreement_of_mem_R_star_tensor {ϑ_pred : ℕ} (ih : δ_ε_multilinearCorrelatedAgreement_Nat (F := F) (A := A) (ι := ι) (C := MC) (ϑ := ϑ_pred) (e := e) (ε := ε)) (u : Fin (2 ^ (ϑ_pred + 1)) → (Word A ι...