Plain-language statement
The error terms satisfy f₂ + f₄ = f₃ (from Jacobi identity)
Exact Lean statement
lemma f₂_add_f₄_eq_f₃ : f₂ + f₄ = f₃
Formal artifact
Lean source
lemma f₂_add_f₄_eq_f₃ : f₂ + f₄ = f₃ := by ext z; simp only [Pi.add_apply, f₂, f₃, f₄] -- Key relation: serre_D 2 H₂ z + serre_D 2 H₄ z = serre_D 2 H₃ z (via Jacobi identity) have h_serre : serre_D 2 H₂ z + serre_D 2 H₄ z = serre_D 2 H₃ z := by have h := congrFun (serre_D_add (2 : ℤ) H₂ H₄ H₂_SIF_MDifferentiable H₄_SIF_MDifferentiable) z simp only [Pi.add_apply] at h rw [jacobi_identity] at h exact h.symm calc serre_D 2 H₂ z - 1/6 * (H₂ z * (H₂ z + 2 * H₄ z)) + (serre_D 2 H₄ z + 1/6 * (H₄ z * (2 * H₂ z + H₄ z))) = (serre_D 2 H₂ z + serre_D 2 H₄ z) + (1/6 * (H₄ z * (2 * H₂ z + H₄ z)) - 1/6 * (H₂ z * (H₂ z + 2 * H₄ z))) := by ring _ = serre_D 2 H₃ z + (1/6 * (H₄ z * (2 * H₂ z + H₄ z)) - 1/6 * (H₂ z * (H₂ z + 2 * H₄ z))) := by rw [h_serre] _ = serre_D 2 H₃ z - 1/6 * (H₂ z ^ 2 - H₄ z ^ 2) := by ring- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/JacobiTheta/Derivative.lean:115-129
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Closed Ball center subset upper Half Plane
closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.