Research proof index

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...

The spatial-derivative in terms of the derivative of functions between manifolds with the manifold structure Space.manifoldStructure d.

A power series which is differentiable everywhere, and in the limit as n → ∞ tends to ‖x‖. -/ def normPowerSeries {d} : ℕ → Space d → ℝ := fun n x => √(‖x‖ ^ 2 + 1/(n + 1)) lemma normPowerSeries_eq (n : ℕ) : normPowerSeries (d := d) n = fun x => √(‖x‖ ^ 2 + 1/(n + 1)) := rfl lemma normPowerSeries_eq_rpow {d} (n : ℕ) : normPowerSeries (d := d) n = fun...

The distributional divergence of the radial field x ↦ ‖x‖ ^ (-d) • x (i.e. x / ‖x‖ ^ d) equals d * volume (Metric.ball 0 1) , the surface area of the unit sphere S^{d-1} , times the Dirac delta at the origin. This is the Gauss-law identity underlying the fundamental solution of the Laplacian: away from 0 the field is divergence-free, and all of...

A power series which is differentiable everywhere, and in the limit as n → ∞ tends to ‖x‖. -/ def normPowerSeries {d} : ℕ → Space d → ℝ := fun n x => √(‖x‖ ^ 2 + 1/(n + 1)) lemma normPowerSeries_eq (n : ℕ) : normPowerSeries (d := d) n = fun x => √(‖x‖ ^ 2 + 1/(n + 1)) := rfl lemma normPowerSeries_eq_rpow {d} (n : ℕ) : normPowerSeries (d := d) n = fun...

The distributional Laplacian of ‖x‖ ^ (2 - d) is (2 - d) * d * volume (Metric.ball 0 1) times the Dirac delta at the origin. For d ≥ 3 this ‖x‖ ^ (2 - d) is the (singular) fundamental solution of the Laplacian, and for d = 1 it is ‖x‖. When d = 2 the exponent vanishes, so the identity collapses to the trivial Δᵈ 1 = 0; the genuine two-dime...