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Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Scale density

SpherePacking.scale_density

sphere packingFourier analysismodular forms

Plain-language statement

Density of a scaled packing.

Exact Lean statement

lemma scale_density {d : ℕ} (hd : 0 < d) (S : SpherePacking d) {c : ℝ} (hc : 0 < c) :
    (S.scale hc).density = S.density

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma scale_density {d : ℕ} (hd : 0 < d) (S : SpherePacking d) {c : ℝ} (hc : 0 < c) :    (S.scale hc).density = S.density := by  simp only [density, limsup, limsSup, eventually_map, eventually_atTop]  apply le_antisymm  -- The following are almost identical. Can we condense the proof?  · simp only [sInf_le_iff, le_sInf_iff, Set.mem_setOf_eq, lowerBounds]    intro x hx y hy    rcases hx with ⟨a, ha⟩    apply hy    use c * a    intro b' hb'    rw [scale_finiteDensity' hd S hc]    apply ha    exact (le_div_iff₀' hc).mpr hb'  · simp only [sInf_le_iff, le_sInf_iff, Set.mem_setOf_eq, lowerBounds]    intro x hx y hy    rcases hx with ⟨a, ha⟩    apply hy    use a / c    intro b' hb'    rw [← scale_finiteDensity hd S hc]    apply ha    exact (div_le_iff₀' hc).mp hb'
Project
Sphere Packing in Dimension 8
License
Apache-2.0
Commit
acfc6204e65a
Source
SpherePacking/Basic/SpherePacking.lean:298-320

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Related declarations

Project-declaredLean 4.31.0

Anti Der Pos

antiDerPos

Plain-language statement

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

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Project-declaredLean 4.31.0

Anti Serre Der Pos

antiSerreDerPos

Plain-language statement

Let F:H→CF : \mathbb{H} \to \mathbb{C} be a holomorphic function where F(it)F(it) is real for all t>0t > 0. Assume that Serre derivative ∂kF\partial_k F is positive on the imaginary axis. If F(it)F(it) is positive for sufficiently large tt, then F(it)F(it) is positive for all t>0t > 0.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

View proof record