Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Anti Der Pos

antiDerPos

sphere packing Fourier analysis modular forms derivative

Plain-language statement

If $F$ is a modular form where $F(it)$ is positive for sufficiently large $t$ (i.e. constant term is positive) and the derivative is positive, then $F$ is also positive.

Exact Lean statement

theorem antiDerPos {F : ℍ → ℂ} (hFderiv : MDiff F)
    (hFepos : ResToImagAxis.EventuallyPos F) (hDF : ResToImagAxis.Pos (D F)) :
    ResToImagAxis.Pos F

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem antiDerPos {F : ℍ → ℂ} (hFderiv : MDiff F)    (hFepos : ResToImagAxis.EventuallyPos F) (hDF : ResToImagAxis.Pos (D F)) :    ResToImagAxis.Pos F := by  obtain ⟨hF_real, t₀, ht₀_pos, hF_pos⟩ := hFepos  obtain ⟨-, hDF_pos⟩ := hDF  let g := fun t => (F.resToImagAxis t).re  have hg : ∀ t, 0 < t → HasDerivAt g (-2 * π * (ResToImagAxis (D F) t).re) t :=    fun t ht => hasDerivAt_resToImagAxis_re hFderiv ht  have hn : ∀ t ∈ Set.Ioi (0 : ℝ), deriv g t < 0 := fun t (ht : 0 < t) => by    rw [(hg t ht).deriv]    have ht' : 0 < (ResToImagAxis (D F) t).re := hDF_pos t ht    nlinarith [Real.pi_pos]  have gpos := fun t ht =>    StrictAntiOn.eventuallyPos_Ioi (strictAntiOn_of_deriv_neg (convex_Ioi 0)    (fun x hx => (hg x hx).continuousAt.continuousWithinAt)      (by simpa [interior_Ioi] using hn)) ht₀_pos hF_pos t ht  exact ⟨hF_real, gpos⟩

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/Derivative.lean:833-849

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Anti Serre Der Pos

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Plain-language statement

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Person-level attribution pending.

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closedBall_center_subset_upperHalfPlane

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Closed ball centered at z with radius z.im/2 is contained in the upper half plane.

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cuspForm_rpow_mul_resToImagAxis_tendsto_zero

Plain-language statement

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Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

View proof record