Closed-form derivatives, Christoffel symbols and explicit Ricci expression of the Kerr metric (generated)

Definition
KerrBL_Kerr_ClosedForms
by He Wang · Sep 9, 2026 · Mathlib 0df444a (Lean v4.33.1)
coordinate-geometrygeneral-relativitykerr-metrickerrbl-missionricci-flatness
This bundle contains machine-generated closed forms in the five atoms $(r,s,c,S,D)$ , which stand for $(r,\sin\theta,\cos\theta,\Sigma,\Delta)$ but are treated as independent real variables:
dgKerr M a l i j r s c S D: the claimed value of $\partial_l g_{ij}$ ;
GammaKerr M a i j k r s c S D: the claimed value of $\Gamma^i_{jk}$ , and GammaKerrpt its evaluation at a point $x$ with the atoms instantiated consistently ( $S=\Sigma(x)$ , $D=\Delta(x)$ , $s=\sin x_2$ , $c=\cos x_2$ );
dGammaKerr M a l i j k r s c S D: the claimed value of $\partial_l\Gamma^i_{jk}$ ;
RicciKerr M a b d r s c S D: the Ricci formula of KerrBL.ricci with the generic derivatives replaced by dGammaKerr and the generic $\Gamma$ by GammaKerr:
\mathrm{RicciKerr}_{bd}=\sum_i\Big(\mathrm d\Gamma^{i}_{i;bd}-\mathrm d\Gamma^{i}_{d;bi}+\sum_j\big(\Gamma^i_{ij}\Gamma^j_{bd}-\Gamma^i_{dj}\Gamma^j_{bi}\big)\Big),\qquad \mathrm d\Gamma^{i}_{l;jk}:=\texttt{dGammaKerr}\,l\,i\,j\,k .
Each definition is a case split on the indices with rational-function bodies, followed by rfl accessor lemmas.
The closed forms were produced by SymPy from the metric of KerrBL_Kerr_Metric and are not trusted. Layer II of the mission proves from the generic definitions that each of them is the true derivative (hdgKerr_all, hdchrKerr_all), the true Christoffel symbol (chrKerr_all), and that the generic Ricci tensor equals RicciKerr (ricci_bridge_Kerr). A wrong closed form would make those bridge theorems unprovable, never a false theorem provable. RicciKerr is the full formula, not $0$ ; its vanishing is proved in Layer III.
Formalization Note Every exponent is written ^(k:ℕ). The bundle is large (about 120 kB) because every case of the $4^3$ and $4^4$ index tables is spelled out; it compiles in about 7 s.
Definition code
import Definitions.Def_KerrBL_Kerr_Metric
set_option linter.unusedVariables false

namespace KerrBL

/-- Closed-form partial derivatives ∂_l g_{ij} (generated). -/
noncomputable def dgKerr (M a : ℝ) : Fin 4 → Fin 4 → Fin 4 → ℝ → ℝ → ℝ → ℝ → ℝ → ℝ
  | 1, 0, 0 => fun r s c S D => (2*M*S - 4*M*r^(2:ℕ))/(S^(2:ℕ))
  | 1, 0, 3 => fun r s c S D => (((-2))*M*a*S*s^(2:ℕ) + 4*M*a*r^(2:ℕ)*s^(2:ℕ))/(S^(2:ℕ))
  | 1, 1, 1 => fun r s c S D => (2*r*D + 2*M*S - 2*r*S)/(D^(2:ℕ))
  | 1, 2, 2 => fun r s c S D => (2*r)/(1)
  | 1, 3, 0 => fun r s c S D => (((-2))*M*a*S*s^(2:ℕ) + 4*M*a*r^(2:ℕ)*s^(2:ℕ))/(S^(2:ℕ))
  | 1, 3, 3 => fun r s c S D => (2*r*s^(2:ℕ)*S^(2:ℕ) + 2*M*S*a^(2:ℕ)*s^(4:ℕ) - 4*M*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ))/(S^(2:ℕ))
  | 2, 0, 0 => fun r s c S D => (4*M*r*s*c*a^(2:ℕ))/(S^(2:ℕ))
  | 2, 0, 3 => fun r s c S D => (((-4))*M*r*c*a^(3:ℕ)*s^(3:ℕ) - 4*M*a*r*s*c*S)/(S^(2:ℕ))
  | 2, 1, 1 => fun r s c S D => (((-2))*s*c*a^(2:ℕ))/(D)
  | 2, 2, 2 => fun r s c S D => (((-2))*s*c*a^(2:ℕ))/(1)
  | 2, 3, 0 => fun r s c S D => (((-4))*M*r*c*a^(3:ℕ)*s^(3:ℕ) - 4*M*a*r*s*c*S)/(S^(2:ℕ))
  | 2, 3, 3 => fun r s c S D => (2*s*c*a^(2:ℕ)*S^(2:ℕ) + 2*s*c*r^(2:ℕ)*S^(2:ℕ) + 4*M*r*c*a^(4:ℕ)*s^(5:ℕ) + 8*M*r*c*S*a^(2:ℕ)*s^(3:ℕ))/(S^(2:ℕ))
  | _, _, _ => fun _ _ _ _ _ => 0

/-- Closed-form Christoffel symbols Γ^i_{jk} (generated). -/
noncomputable def GammaKerr (M a : ℝ) : Fin 4 → Fin 4 → Fin 4 → ℝ → ℝ → ℝ → ℝ → ℝ → ℝ
  | 0, 0, 1 => fun r s c S D => (2*M*r^(6:ℕ) - M*S*a^(4:ℕ) - M*S*r^(4:ℕ) + 4*M*a^(2:ℕ)*r^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ) - 2*M*S*a^(2:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ))
  | 0, 0, 2 => fun r s c S D => (((-2))*M*s*c*a^(2:ℕ)*r^(5:ℕ) - 4*M*s*c*a^(4:ℕ)*r^(3:ℕ) - 2*M*r*s*c*a^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(3:ℕ) + 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ))/(D*S^(3:ℕ))
  | 0, 1, 0 => fun r s c S D => (2*M*r^(6:ℕ) - M*S*a^(4:ℕ) - M*S*r^(4:ℕ) + 4*M*a^(2:ℕ)*r^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ) - 2*M*S*a^(2:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ))
  | 0, 1, 3 => fun r s c S D => (M*S*a^(5:ℕ)*s^(2:ℕ) - 2*M*a*r^(6:ℕ)*s^(2:ℕ) - 4*M*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 2*M*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - M*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*D*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + M*a*S*r^(4:ℕ)*s^(2:ℕ) + 2*M*S*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*S*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ))/(D*S^(3:ℕ))
  | 0, 2, 0 => fun r s c S D => (((-2))*M*s*c*a^(2:ℕ)*r^(5:ℕ) - 4*M*s*c*a^(4:ℕ)*r^(3:ℕ) - 2*M*r*s*c*a^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(3:ℕ) + 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ))/(D*S^(3:ℕ))
  | 0, 2, 3 => fun r s c S D => (2*M*c*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 4*M*c*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*r*c*a^(7:ℕ)*s^(3:ℕ) - 4*c*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 2*M*r*c*D*a^(5:ℕ)*s^(5:ℕ) + 2*M*a*s*c*S*r^(5:ℕ) + 4*M*s*c*S*a^(3:ℕ)*r^(3:ℕ) + 2*M*r*s*c*S*a^(5:ℕ) - 2*M*a*s*c*r^(3:ℕ)*S^(2:ℕ) - 2*M*r*s*c*a^(3:ℕ)*S^(2:ℕ) - 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 2*M*r*c*S*D*a^(3:ℕ)*s^(3:ℕ))/(D*S^(3:ℕ))
  | 0, 3, 1 => fun r s c S D => (M*S*a^(5:ℕ)*s^(2:ℕ) - 2*M*a*r^(6:ℕ)*s^(2:ℕ) - 4*M*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 2*M*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - M*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*D*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + M*a*S*r^(4:ℕ)*s^(2:ℕ) + 2*M*S*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*S*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ))/(D*S^(3:ℕ))
  | 0, 3, 2 => fun r s c S D => (2*M*c*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 4*M*c*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*r*c*a^(7:ℕ)*s^(3:ℕ) - 4*c*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 2*M*r*c*D*a^(5:ℕ)*s^(5:ℕ) + 2*M*a*s*c*S*r^(5:ℕ) + 4*M*s*c*S*a^(3:ℕ)*r^(3:ℕ) + 2*M*r*s*c*S*a^(5:ℕ) - 2*M*a*s*c*r^(3:ℕ)*S^(2:ℕ) - 2*M*r*s*c*a^(3:ℕ)*S^(2:ℕ) - 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 2*M*r*c*S*D*a^(3:ℕ)*s^(3:ℕ))/(D*S^(3:ℕ))
  | 1, 0, 0 => fun r s c S D => (-(M*S*D) + 2*M*D*r^(2:ℕ))/(S^(3:ℕ))
  | 1, 0, 3 => fun r s c S D => (M*a*S*D*s^(2:ℕ) - 2*M*a*D*r^(2:ℕ)*s^(2:ℕ))/(S^(3:ℕ))
  | 1, 1, 1 => fun r s c S D => (r*D + M*S - r*S)/(S*D)
  | 1, 1, 2 => fun r s c S D => (-(s*c*a^(2:ℕ)))/(S)
  | 1, 2, 1 => fun r s c S D => (-(s*c*a^(2:ℕ)))/(S)
  | 1, 2, 2 => fun r s c S D => (-(r*D))/(S)
  | 1, 3, 0 => fun r s c S D => (M*a*S*D*s^(2:ℕ) - 2*M*a*D*r^(2:ℕ)*s^(2:ℕ))/(S^(3:ℕ))
  | 1, 3, 3 => fun r s c S D => (-(r*D*s^(2:ℕ)*S^(2:ℕ)) - M*S*D*a^(2:ℕ)*s^(4:ℕ) + 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ))/(S^(3:ℕ))
  | 2, 0, 0 => fun r s c S D => (((-2))*M*r*s*c*a^(2:ℕ))/(S^(3:ℕ))
  | 2, 0, 3 => fun r s c S D => (2*M*r*c*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*r*s*c*S)/(S^(3:ℕ))
  | 2, 1, 1 => fun r s c S D => (s*c*a^(2:ℕ))/(S*D)
  | 2, 1, 2 => fun r s c S D => (r)/(S)
  | 2, 2, 1 => fun r s c S D => (r)/(S)
  | 2, 2, 2 => fun r s c S D => (-(s*c*a^(2:ℕ)))/(S)
  | 2, 3, 0 => fun r s c S D => (2*M*r*c*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*r*s*c*S)/(S^(3:ℕ))
  | 2, 3, 3 => fun r s c S D => (-(s*c*a^(2:ℕ)*S^(2:ℕ)) - s*c*r^(2:ℕ)*S^(2:ℕ) - 2*M*r*c*a^(4:ℕ)*s^(5:ℕ) - 4*M*r*c*S*a^(2:ℕ)*s^(3:ℕ))/(S^(3:ℕ))
  | 3, 0, 1 => fun r s c S D => (4*a*M^(2:ℕ)*r^(3:ℕ) - M*a*S*D + 2*M*a*D*r^(2:ℕ) + M*S*a^(3:ℕ)*s^(2:ℕ) - 2*M*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*a*r*S*M^(2:ℕ))/(D*S^(3:ℕ))
  | 3, 0, 2 => fun r s c S D => (2*M*r*c*a^(5:ℕ)*s^(4:ℕ) - 4*c*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r*c*S*D - 2*M*r*c*D*a^(3:ℕ)*s^(2:ℕ) + 2*M*r*c*S*a^(3:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ))
  | 3, 1, 0 => fun r s c S D => (4*a*M^(2:ℕ)*r^(3:ℕ) - M*a*S*D + 2*M*a*D*r^(2:ℕ) + M*S*a^(3:ℕ)*s^(2:ℕ) - 2*M*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*a*r*S*M^(2:ℕ))/(D*S^(3:ℕ))
  | 3, 1, 3 => fun r s c S D => (r*D*S^(2:ℕ) - M*S*a^(4:ℕ)*s^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) - r*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ))
  | 3, 2, 0 => fun r s c S D => (2*M*r*c*a^(5:ℕ)*s^(4:ℕ) - 4*c*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r*c*S*D - 2*M*r*c*D*a^(3:ℕ)*s^(2:ℕ) + 2*M*r*c*S*a^(3:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ))
  | 3, 2, 3 => fun r s c S D => (c*D*a^(2:ℕ)*S^(2:ℕ) + c*D*r^(2:ℕ)*S^(2:ℕ) - c*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*c*a^(6:ℕ)*s^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(4:ℕ) - 4*M*r*c*S*a^(4:ℕ)*s^(4:ℕ) + 4*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M*r*c*S*D*a^(2:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ))
  | 3, 3, 1 => fun r s c S D => (r*D*S^(2:ℕ) - M*S*a^(4:ℕ)*s^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) - r*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ))
  | 3, 3, 2 => fun r s c S D => (c*D*a^(2:ℕ)*S^(2:ℕ) + c*D*r^(2:ℕ)*S^(2:ℕ) - c*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*c*a^(6:ℕ)*s^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(4:ℕ) - 4*M*r*c*S*a^(4:ℕ)*s^(4:ℕ) + 4*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M*r*c*S*D*a^(2:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ))
  | _, _, _ => fun _ _ _ _ _ => 0

/-- Closed-form Γ as a function on coordinate points. -/
noncomputable def GammaKerrpt (M a : ℝ) (i j k : Fin 4) (x : Pt) : ℝ :=
  GammaKerr M a i j k (x 1) (Real.sin (x 2)) (Real.cos (x 2)) (Sig a (x 1) (Real.cos (x 2))) (Del M a (x 1))

/-- Closed-form partial derivatives ∂_l Γ^i_{jk} (generated). -/
noncomputable def dGammaKerr (M a : ℝ) : Fin 4 → Fin 4 → Fin 4 → Fin 4 → ℝ → ℝ → ℝ → ℝ → ℝ → ℝ
  | 1, 0, 0, 1 => fun r s c S D => (((-12))*M*D*r^(7:ℕ) - 4*M*S*r^(7:ℕ) + 2*M*r^(5:ℕ)*S^(2:ℕ) + 4*S*M^(2:ℕ)*r^(6:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*S^(2:ℕ) - 2*M^(2:ℕ)*r^(4:ℕ)*S^(2:ℕ) + 16*M*S*D*r^(5:ℕ) - 4*M*D*r^(3:ℕ)*S^(2:ℕ) - 24*M*D*a^(2:ℕ)*r^(5:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ) - 8*M*S*a^(2:ℕ)*r^(5:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ) + 4*M*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 2*M*r*a^(4:ℕ)*S^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 24*M*S*D*a^(2:ℕ)*r^(3:ℕ) + 8*M*r*S*D*a^(4:ℕ) - 4*M*r*D*a^(2:ℕ)*S^(2:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 0, 0, 2 => fun r s c S D => (((-2))*M*s*c*S*D*a^(6:ℕ) + 12*M*s*c*D*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*D*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*D*a^(6:ℕ)*r^(2:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(3:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ) + 4*M*s*c*S*a^(2:ℕ)*r^(6:ℕ) + 8*M*s*c*S*a^(4:ℕ)*r^(4:ℕ) + 4*M*s*c*S*a^(6:ℕ)*r^(2:ℕ) - 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(5:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 4*r*s*c*S*M^(2:ℕ)*a^(6:ℕ) - 8*s*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 8*s*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 10*M*s*c*S*D*a^(2:ℕ)*r^(4:ℕ) - 12*M*s*c*S*D*a^(4:ℕ)*r^(2:ℕ) - 16*s*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 8*r*s*c*D*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 0, 1, 0 => fun r s c S D => (((-12))*M*D*r^(7:ℕ) - 4*M*S*r^(7:ℕ) + 2*M*r^(5:ℕ)*S^(2:ℕ) + 4*S*M^(2:ℕ)*r^(6:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*S^(2:ℕ) - 2*M^(2:ℕ)*r^(4:ℕ)*S^(2:ℕ) + 16*M*S*D*r^(5:ℕ) - 4*M*D*r^(3:ℕ)*S^(2:ℕ) - 24*M*D*a^(2:ℕ)*r^(5:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ) - 8*M*S*a^(2:ℕ)*r^(5:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ) + 4*M*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 2*M*r*a^(4:ℕ)*S^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 24*M*S*D*a^(2:ℕ)*r^(3:ℕ) + 8*M*r*S*D*a^(4:ℕ) - 4*M*r*D*a^(2:ℕ)*S^(2:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 0, 1, 3 => fun r s c S D => (2*M^(2:ℕ)*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(7:ℕ)*s^(2:ℕ) + 24*M*D*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 12*M*D*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*D*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*D*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 12*M*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*a*S*r^(7:ℕ)*s^(2:ℕ) + 8*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 4*a*S*M^(2:ℕ)*r^(6:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*a*M^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 4*a*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(3:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*M^(3:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 16*M*a*S*D*r^(5:ℕ)*s^(2:ℕ) - 24*M*S*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(5:ℕ)*s^(2:ℕ) + 8*M*a*D*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*r*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a*r*D*s^(2:ℕ)*S^(3:ℕ) + 20*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*M*r*S*a^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 0, 2, 0 => fun r s c S D => (((-2))*M*s*c*S*D*a^(6:ℕ) + 12*M*s*c*D*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*D*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*D*a^(6:ℕ)*r^(2:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(3:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ) + 4*M*s*c*S*a^(2:ℕ)*r^(6:ℕ) + 8*M*s*c*S*a^(4:ℕ)*r^(4:ℕ) + 4*M*s*c*S*a^(6:ℕ)*r^(2:ℕ) - 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(5:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 4*r*s*c*S*M^(2:ℕ)*a^(6:ℕ) - 8*s*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 8*s*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 10*M*s*c*S*D*a^(2:ℕ)*r^(4:ℕ) - 12*M*s*c*S*D*a^(4:ℕ)*r^(2:ℕ) - 16*s*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 8*r*s*c*D*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 0, 2, 3 => fun r s c S D => (2*M*c*S*D*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(5:ℕ)*S^(2:ℕ) - 2*M*s*c*D*a^(3:ℕ)*S^(3:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*D*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*D*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*D*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 2*M*c*S*a^(5:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 2*M*c*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 8*M*c*S*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*c*S*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*r^(6:ℕ)*S^(2:ℕ) - 8*M*s*c*a^(3:ℕ)*r^(4:ℕ)*S^(2:ℕ) - 4*M*s*c*a^(5:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 4*M*a*s*c*r^(4:ℕ)*S^(3:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(3:ℕ) + 4*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(7:ℕ)*s^(3:ℕ) + 4*a*s*c*M^(2:ℕ)*r^(5:ℕ)*S^(2:ℕ) + 8*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 16*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(5:ℕ)*S^(2:ℕ) - 4*a*s*c*M^(2:ℕ)*r^(3:ℕ)*S^(3:ℕ) - 4*r*s*c*M^(2:ℕ)*a^(3:ℕ)*S^(3:ℕ) - 8*c*S*M^(3:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 16*c*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 8*M*a*s*c*S*D*r^(6:ℕ) - 16*M*s*c*S*D*a^(3:ℕ)*r^(4:ℕ) + 10*M*c*S*D*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 8*M*s*c*S*D*a^(5:ℕ)*r^(2:ℕ) + 12*M*c*S*D*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 14*M*a*s*c*D*r^(4:ℕ)*S^(2:ℕ) + 16*M*s*c*D*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 6*M*a*s*c*D*r^(2:ℕ)*S^(3:ℕ) + 32*c*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 16*r*c*D*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 0, 3, 1 => fun r s c S D => (2*M^(2:ℕ)*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(7:ℕ)*s^(2:ℕ) + 24*M*D*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 12*M*D*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*D*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*D*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 12*M*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*a*S*r^(7:ℕ)*s^(2:ℕ) + 8*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 4*a*S*M^(2:ℕ)*r^(6:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*a*M^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 4*a*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(3:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*M^(3:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 16*M*a*S*D*r^(5:ℕ)*s^(2:ℕ) - 24*M*S*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(5:ℕ)*s^(2:ℕ) + 8*M*a*D*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*r*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a*r*D*s^(2:ℕ)*S^(3:ℕ) + 20*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*M*r*S*a^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 0, 3, 2 => fun r s c S D => (2*M*c*S*D*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(5:ℕ)*S^(2:ℕ) - 2*M*s*c*D*a^(3:ℕ)*S^(3:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*D*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*D*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*D*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 2*M*c*S*a^(5:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 2*M*c*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 8*M*c*S*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*c*S*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*r^(6:ℕ)*S^(2:ℕ) - 8*M*s*c*a^(3:ℕ)*r^(4:ℕ)*S^(2:ℕ) - 4*M*s*c*a^(5:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 4*M*a*s*c*r^(4:ℕ)*S^(3:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(3:ℕ) + 4*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(7:ℕ)*s^(3:ℕ) + 4*a*s*c*M^(2:ℕ)*r^(5:ℕ)*S^(2:ℕ) + 8*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 16*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(5:ℕ)*S^(2:ℕ) - 4*a*s*c*M^(2:ℕ)*r^(3:ℕ)*S^(3:ℕ) - 4*r*s*c*M^(2:ℕ)*a^(3:ℕ)*S^(3:ℕ) - 8*c*S*M^(3:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 16*c*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 8*M*a*s*c*S*D*r^(6:ℕ) - 16*M*s*c*S*D*a^(3:ℕ)*r^(4:ℕ) + 10*M*c*S*D*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 8*M*s*c*S*D*a^(5:ℕ)*r^(2:ℕ) + 12*M*c*S*D*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 14*M*a*s*c*D*r^(4:ℕ)*S^(2:ℕ) + 16*M*s*c*D*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 6*M*a*s*c*D*r^(2:ℕ)*S^(3:ℕ) + 32*c*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 16*r*c*D*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 1, 0, 0 => fun r s c S D => (2*M^(2:ℕ)*S^(2:ℕ) - 12*M*D*r^(3:ℕ) + 4*M*S*r^(3:ℕ) - 2*M*r*S^(2:ℕ) - 4*S*M^(2:ℕ)*r^(2:ℕ) + 8*M*r*S*D)/(S^(4:ℕ))
  | 1, 1, 0, 3 => fun r s c S D => (((-2))*a*M^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(3:ℕ)*s^(2:ℕ) - 4*M*a*S*r^(3:ℕ)*s^(2:ℕ) + 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 4*a*S*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*a*r*S*D*s^(2:ℕ))/(S^(4:ℕ))
  | 1, 1, 1, 1 => fun r s c S D => (-(D*S^(2:ℕ)) + S*D^(2:ℕ) - 2*r^(2:ℕ)*D^(2:ℕ) + 2*M^(2:ℕ)*S^(2:ℕ) + 2*r^(2:ℕ)*S^(2:ℕ) - 4*M*r*S^(2:ℕ))/(S^(2:ℕ)*D^(2:ℕ))
  | 1, 1, 1, 2 => fun r s c S D => (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ))
  | 1, 1, 2, 1 => fun r s c S D => (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ))
  | 1, 1, 2, 2 => fun r s c S D => (-(S*D) + 2*D*r^(2:ℕ) - 2*S*r^(2:ℕ) + 2*M*r*S)/(S^(2:ℕ))
  | 1, 1, 3, 0 => fun r s c S D => (((-2))*a*M^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(3:ℕ)*s^(2:ℕ) - 4*M*a*S*r^(3:ℕ)*s^(2:ℕ) + 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 4*a*S*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*a*r*S*D*s^(2:ℕ))/(S^(4:ℕ))
  | 1, 1, 3, 3 => fun r s c S D => (-(D*s^(2:ℕ)*S^(3:ℕ)) - 2*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*D*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*s^(2:ℕ)*S^(3:ℕ) + 2*M^(2:ℕ)*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*D*a^(2:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 4*M*S*a^(2:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 2*M*r*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*M*r*S*D*a^(2:ℕ)*s^(4:ℕ))/(S^(4:ℕ))
  | 1, 2, 0, 0 => fun r s c S D => (((-2))*M*s*c*S*a^(2:ℕ) + 12*M*s*c*a^(2:ℕ)*r^(2:ℕ))/(S^(4:ℕ))
  | 1, 2, 0, 3 => fun r s c S D => (2*M*c*S*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*M*a*s*c*S*r^(2:ℕ))/(S^(4:ℕ))
  | 1, 2, 1, 1 => fun r s c S D => (((-2))*r*s*c*D*a^(2:ℕ) + 2*M*s*c*S*a^(2:ℕ) - 2*r*s*c*S*a^(2:ℕ))/(S^(2:ℕ)*D^(2:ℕ))
  | 1, 2, 1, 2 => fun r s c S D => (S - 2*r^(2:ℕ))/(S^(2:ℕ))
  | 1, 2, 2, 1 => fun r s c S D => (S - 2*r^(2:ℕ))/(S^(2:ℕ))
  | 1, 2, 2, 2 => fun r s c S D => (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ))
  | 1, 2, 3, 0 => fun r s c S D => (2*M*c*S*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*M*a*s*c*S*r^(2:ℕ))/(S^(4:ℕ))
  | 1, 2, 3, 3 => fun r s c S D => (2*s*c*r^(3:ℕ)*S^(2:ℕ) - 2*r*s*c*S^(3:ℕ) - 2*M*c*S*a^(4:ℕ)*s^(5:ℕ) - 4*M*c*a^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(5:ℕ) + 2*r*s*c*a^(2:ℕ)*S^(2:ℕ) + 16*M*c*S*a^(2:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(S^(4:ℕ))
  | 1, 3, 0, 1 => fun r s c S D => (((-2))*a*D*M^(2:ℕ)*S^(2:ℕ) - 24*a*D*M^(2:ℕ)*r^(4:ℕ) - 12*M*a*r^(3:ℕ)*D^(2:ℕ) - 8*a*S*M^(2:ℕ)*r^(4:ℕ) + 4*a*M^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 2*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*a*S*M^(3:ℕ)*r^(3:ℕ) - 4*a*r*M^(3:ℕ)*S^(2:ℕ) + 12*M*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 20*a*S*D*M^(2:ℕ)*r^(2:ℕ) + 8*M*a*r*S*D^(2:ℕ) + 4*M*S*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(3:ℕ)*s^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 3, 0, 2 => fun r s c S D => (((-2))*M*a*c*S^(2:ℕ)*D^(2:ℕ) + 2*M*c*S*D*a^(5:ℕ)*s^(4:ℕ) + 2*M*c*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 24*c*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*M*a*c*S*r^(2:ℕ)*D^(2:ℕ) - 2*M*c*S*a^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(4:ℕ) + 4*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*c*S*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ))
  | 1, 3, 1, 0 => fun r s c S D => (((-2))*a*D*M^(2:ℕ)*S^(2:ℕ) - 24*a*D*M^(2:ℕ)*r^(4:ℕ) - 12*M*a*r^(3:ℕ)*D^(2:ℕ) - 8*a*S*M^(2:ℕ)*r^(4:ℕ) + 4*a*M^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 2*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*a*S*M^(3:ℕ)*r^(3:ℕ) - 4*a*r*M^(3:ℕ)*S^(2:ℕ) + 12*M*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 20*a*S*D*M^(2:ℕ)*r^(2:ℕ) + 8*M*a*r*S*D^(2:ℕ) + 4*M*S*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(3:ℕ)*s^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 3, 1, 3 => fun r s c S D => (S^(3:ℕ)*D^(2:ℕ) - 2*r^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 2*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*M*r*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*r*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 3, 2, 0 => fun r s c S D => (((-2))*M*a*c*S^(2:ℕ)*D^(2:ℕ) + 2*M*c*S*D*a^(5:ℕ)*s^(4:ℕ) + 2*M*c*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 24*c*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*M*a*c*S*r^(2:ℕ)*D^(2:ℕ) - 2*M*c*S*a^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(4:ℕ) + 4*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*c*S*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ))
  | 1, 3, 2, 3 => fun r s c S D => (((-2))*c*r^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 2*r*c*S^(3:ℕ)*D^(2:ℕ) - 2*r*c*a^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 2*M*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*c*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*r*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*c*S*D*a^(6:ℕ)*s^(6:ℕ) - 4*M*c*D*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*c*D*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*r*c*D*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*c*D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*S*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*c*S*M^(2:ℕ)*a^(6:ℕ)*s^(6:ℕ) - 8*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*r*c*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 16*M*c*S*D*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 16*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ) + 8*r*c*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 16*M*c*S*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ))
  | 1, 3, 3, 1 => fun r s c S D => (S^(3:ℕ)*D^(2:ℕ) - 2*r^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 2*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*M*r*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*r*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ))
  | 1, 3, 3, 2 => fun r s c S D => (((-2))*c*r^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 2*r*c*S^(3:ℕ)*D^(2:ℕ) - 2*r*c*a^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 2*M*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*c*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*r*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*c*S*D*a^(6:ℕ)*s^(6:ℕ) - 4*M*c*D*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*c*D*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*r*c*D*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*c*D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*S*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*c*S*M^(2:ℕ)*a^(6:ℕ)*s^(6:ℕ) - 8*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*r*c*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 16*M*c*S*D*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 16*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ) + 8*r*c*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 16*M*c*S*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ))
  | 2, 0, 0, 1 => fun r s c S D => (((-4))*M*s*c*S*a^(6:ℕ) + 12*M*s*c*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*a^(6:ℕ)*r^(2:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(2:ℕ)*r^(4:ℕ) - 8*M*s*c*S*a^(4:ℕ)*r^(2:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 0, 0, 2 => fun r s c S D => (((-2))*M*S*a^(2:ℕ)*r^(5:ℕ)*c^(2:ℕ) + 2*M*S*a^(2:ℕ)*r^(5:ℕ)*s^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*c^(2:ℕ) + 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*S*a^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(2:ℕ) - 12*M*a^(4:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 24*M*a^(6:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 6*M*r*S*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 0, 1, 0 => fun r s c S D => (((-4))*M*s*c*S*a^(6:ℕ) + 12*M*s*c*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*a^(6:ℕ)*r^(2:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(2:ℕ)*r^(4:ℕ) - 8*M*s*c*S*a^(4:ℕ)*r^(2:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 0, 1, 3 => fun r s c S D => (4*M*c*S*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(5:ℕ)*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 4*M*c*S*D*a^(5:ℕ)*s^(5:ℕ) - 4*M*c*D*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 4*M*a*s*c*S*r^(6:ℕ) - 8*M*s*c*S*a^(3:ℕ)*r^(4:ℕ) + 4*M*c*S*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(5:ℕ)*r^(2:ℕ) + 8*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*a*s*c*r^(4:ℕ)*S^(2:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 4*M*a*s*c*r^(2:ℕ)*S^(3:ℕ) + 16*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 8*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(D*S^(4:ℕ))
  | 2, 0, 2, 0 => fun r s c S D => (((-2))*M*S*a^(2:ℕ)*r^(5:ℕ)*c^(2:ℕ) + 2*M*S*a^(2:ℕ)*r^(5:ℕ)*s^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*c^(2:ℕ) + 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*S*a^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(2:ℕ) - 12*M*a^(4:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 24*M*a^(6:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 6*M*r*S*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 0, 2, 3 => fun r s c S D => (((-2))*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(4:ℕ) - 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 2*M*r*S*a^(7:ℕ)*s^(4:ℕ) + 2*M*a*r^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a^(3:ℕ)*r^(3:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*a^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*r*a^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 12*M*a^(5:ℕ)*r^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 24*M*a^(7:ℕ)*r^(3:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 12*M*r*a^(9:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*M^(2:ℕ)*a^(7:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(5:ℕ)*s^(6:ℕ) + 2*M*r*D*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 14*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 28*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(7:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*r*a^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 52*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 24*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 18*M*r*S*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 6*M*r*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 0, 3, 1 => fun r s c S D => (4*M*c*S*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(5:ℕ)*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 4*M*c*S*D*a^(5:ℕ)*s^(5:ℕ) - 4*M*c*D*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 4*M*a*s*c*S*r^(6:ℕ) - 8*M*s*c*S*a^(3:ℕ)*r^(4:ℕ) + 4*M*c*S*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(5:ℕ)*r^(2:ℕ) + 8*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*a*s*c*r^(4:ℕ)*S^(2:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 4*M*a*s*c*r^(2:ℕ)*S^(3:ℕ) + 16*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 8*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(D*S^(4:ℕ))
  | 2, 0, 3, 2 => fun r s c S D => (((-2))*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(4:ℕ) - 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 2*M*r*S*a^(7:ℕ)*s^(4:ℕ) + 2*M*a*r^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a^(3:ℕ)*r^(3:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*a^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*r*a^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 12*M*a^(5:ℕ)*r^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 24*M*a^(7:ℕ)*r^(3:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 12*M*r*a^(9:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*M^(2:ℕ)*a^(7:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(5:ℕ)*s^(6:ℕ) + 2*M*r*D*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 14*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 28*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(7:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*r*a^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 52*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 24*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 18*M*r*S*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 6*M*r*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 1, 0, 0 => fun r s c S D => (((-4))*M*s*c*S*D*a^(2:ℕ) + 12*M*s*c*D*a^(2:ℕ)*r^(2:ℕ))/(S^(4:ℕ))
  | 2, 1, 0, 3 => fun r s c S D => (4*M*c*S*D*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*D*S^(2:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*S*D*r^(2:ℕ))/(S^(4:ℕ))
  | 2, 1, 1, 1 => fun r s c S D => (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ))
  | 2, 1, 1, 2 => fun r s c S D => (-(S*a^(2:ℕ)*c^(2:ℕ)) + S*a^(2:ℕ)*s^(2:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(2:ℕ))
  | 2, 1, 2, 1 => fun r s c S D => (-(S*a^(2:ℕ)*c^(2:ℕ)) + S*a^(2:ℕ)*s^(2:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(2:ℕ))
  | 2, 1, 2, 2 => fun r s c S D => (((-2))*r*s*c*D*a^(2:ℕ))/(S^(2:ℕ))
  | 2, 1, 3, 0 => fun r s c S D => (4*M*c*S*D*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*D*S^(2:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*S*D*r^(2:ℕ))/(S^(4:ℕ))
  | 2, 1, 3, 3 => fun r s c S D => (((-2))*r*s*c*D*S^(3:ℕ) - 4*M*c*S*D*a^(4:ℕ)*s^(5:ℕ) - 4*M*c*D*a^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 2*r*c*D*a^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*D*a^(2:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(S^(4:ℕ))
  | 2, 2, 0, 0 => fun r s c S D => (((-2))*M*r*S*a^(2:ℕ)*c^(2:ℕ) + 2*M*r*S*a^(2:ℕ)*s^(2:ℕ) - 12*M*r*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(4:ℕ))
  | 2, 2, 0, 3 => fun r s c S D => (((-2))*M*r*S*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 12*M*r*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(4:ℕ))
  | 2, 2, 1, 1 => fun r s c S D => (S*a^(2:ℕ)*c^(2:ℕ) - S*a^(2:ℕ)*s^(2:ℕ) + 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*S^(2:ℕ))
  | 2, 2, 1, 2 => fun r s c S D => (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ))
  | 2, 2, 2, 1 => fun r s c S D => (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ))
  | 2, 2, 2, 2 => fun r s c S D => (-(S*a^(2:ℕ)*c^(2:ℕ)) + S*a^(2:ℕ)*s^(2:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(2:ℕ))
  | 2, 2, 3, 0 => fun r s c S D => (((-2))*M*r*S*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 12*M*r*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(4:ℕ))
  | 2, 2, 3, 3 => fun r s c S D => (-(a^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ)) + a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - r^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) + r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 2*M*r*S*a^(4:ℕ)*s^(6:ℕ) + 4*M*r*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*r*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 2*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 26*M*r*S*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 12*M*r*a^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(S^(4:ℕ))
  | 2, 3, 0, 1 => fun r s c S D => (4*M*c*S*a^(5:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(3:ℕ)*S^(2:ℕ) - 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ) - 4*M*s*c*S*D*a^(3:ℕ) + 12*M*s*c*D*a^(3:ℕ)*r^(2:ℕ) - 4*M*s*c*S*a^(3:ℕ)*r^(2:ℕ) - 8*r*s*c*S*M^(2:ℕ)*a^(3:ℕ))/(D*S^(4:ℕ))
  | 2, 3, 0, 2 => fun r s c S D => (((-2))*M*r*S*a^(5:ℕ)*s^(6:ℕ) - 2*M*r*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 24*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*D*c^(2:ℕ)*S^(2:ℕ) + 2*M*a*r*D*s^(2:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 10*M*r*S*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ))
  | 2, 3, 1, 0 => fun r s c S D => (4*M*c*S*a^(5:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(3:ℕ)*S^(2:ℕ) - 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ) - 4*M*s*c*S*D*a^(3:ℕ) + 12*M*s*c*D*a^(3:ℕ)*r^(2:ℕ) - 4*M*s*c*S*a^(3:ℕ)*r^(2:ℕ) - 8*r*s*c*S*M^(2:ℕ)*a^(3:ℕ))/(D*S^(4:ℕ))
  | 2, 3, 1, 3 => fun r s c S D => (((-4))*M*c*S*a^(6:ℕ)*s^(5:ℕ) - 4*M*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*a^(6:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 2*r*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 2*r*s*c*a^(2:ℕ)*S^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*r*s*c*D*a^(2:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 3, 2, 0 => fun r s c S D => (((-2))*M*r*S*a^(5:ℕ)*s^(6:ℕ) - 2*M*r*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 24*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*D*c^(2:ℕ)*S^(2:ℕ) + 2*M*a*r*D*s^(2:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 10*M*r*S*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ))
  | 2, 3, 2, 3 => fun r s c S D => (a^(4:ℕ)*s^(4:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(3:ℕ) - a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(8:ℕ) + 4*M*r*a^(4:ℕ)*s^(6:ℕ)*S^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(8:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 2*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) - a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(6:ℕ) - 4*M*r*D*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 26*M*r*S*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 12*M*r*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 22*M*r*S*D*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M*r*D*a^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ))
  | 2, 3, 3, 1 => fun r s c S D => (((-4))*M*c*S*a^(6:ℕ)*s^(5:ℕ) - 4*M*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*a^(6:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 2*r*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 2*r*s*c*a^(2:ℕ)*S^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*r*s*c*D*a^(2:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ))
  | 2, 3, 3, 2 => fun r s c S D => (a^(4:ℕ)*s^(4:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(3:ℕ) - a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(8:ℕ) + 4*M*r*a^(4:ℕ)*s^(6:ℕ)*S^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(8:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 2*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) - a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(6:ℕ) - 4*M*r*D*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 26*M*r*S*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 12*M*r*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 22*M*r*S*D*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M*r*D*a^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ))
  | _, _, _, _ => fun _ _ _ _ _ => 0

/-- Explicit Ricci expression: the `ricci` formula with Γ, ∂Γ replaced by their closed forms. -/
noncomputable def RicciKerr (M a : ℝ) (b d : Fin 4) (r s c S D : ℝ) : ℝ :=
  ∑ i : Fin 4, (dGammaKerr M a i i b d r s c S D - dGammaKerr M a d i b i r s c S D
    + ∑ j : Fin 4, (GammaKerr M a i i j r s c S D * GammaKerr M a j b d r s c S D - GammaKerr M a i d j r s c S D * GammaKerr M a j b i r s c S D))

theorem dgKerr_000 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 0 0 r s c S D = 0 := rfl
theorem dgKerr_001 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 0 1 r s c S D = 0 := rfl
theorem dgKerr_002 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 0 2 r s c S D = 0 := rfl
theorem dgKerr_003 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 0 3 r s c S D = 0 := rfl
theorem dgKerr_010 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 1 0 r s c S D = 0 := rfl
theorem dgKerr_011 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 1 1 r s c S D = 0 := rfl
theorem dgKerr_012 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 1 2 r s c S D = 0 := rfl
theorem dgKerr_013 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 1 3 r s c S D = 0 := rfl
theorem dgKerr_020 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 2 0 r s c S D = 0 := rfl
theorem dgKerr_021 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 2 1 r s c S D = 0 := rfl
theorem dgKerr_022 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 2 2 r s c S D = 0 := rfl
theorem dgKerr_023 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 2 3 r s c S D = 0 := rfl
theorem dgKerr_030 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 3 0 r s c S D = 0 := rfl
theorem dgKerr_031 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 3 1 r s c S D = 0 := rfl
theorem dgKerr_032 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 3 2 r s c S D = 0 := rfl
theorem dgKerr_033 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 0 3 3 r s c S D = 0 := rfl
theorem dgKerr_100 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 0 0 r s c S D = (2*M*S - 4*M*r^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dgKerr_101 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 0 1 r s c S D = 0 := rfl
theorem dgKerr_102 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 0 2 r s c S D = 0 := rfl
theorem dgKerr_103 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 0 3 r s c S D = (((-2))*M*a*S*s^(2:ℕ) + 4*M*a*r^(2:ℕ)*s^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dgKerr_110 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 1 0 r s c S D = 0 := rfl
theorem dgKerr_111 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 1 1 r s c S D = (2*r*D + 2*M*S - 2*r*S)/(D^(2:ℕ)) := rfl
theorem dgKerr_112 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 1 2 r s c S D = 0 := rfl
theorem dgKerr_113 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 1 3 r s c S D = 0 := rfl
theorem dgKerr_120 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 2 0 r s c S D = 0 := rfl
theorem dgKerr_121 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 2 1 r s c S D = 0 := rfl
theorem dgKerr_122 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 2 2 r s c S D = (2*r)/(1) := rfl
theorem dgKerr_123 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 2 3 r s c S D = 0 := rfl
theorem dgKerr_130 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 3 0 r s c S D = (((-2))*M*a*S*s^(2:ℕ) + 4*M*a*r^(2:ℕ)*s^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dgKerr_131 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 3 1 r s c S D = 0 := rfl
theorem dgKerr_132 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 3 2 r s c S D = 0 := rfl
theorem dgKerr_133 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 1 3 3 r s c S D = (2*r*s^(2:ℕ)*S^(2:ℕ) + 2*M*S*a^(2:ℕ)*s^(4:ℕ) - 4*M*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ))/(S^(2:ℕ)) := rfl
theorem dgKerr_200 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 0 0 r s c S D = (4*M*r*s*c*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dgKerr_201 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 0 1 r s c S D = 0 := rfl
theorem dgKerr_202 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 0 2 r s c S D = 0 := rfl
theorem dgKerr_203 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 0 3 r s c S D = (((-4))*M*r*c*a^(3:ℕ)*s^(3:ℕ) - 4*M*a*r*s*c*S)/(S^(2:ℕ)) := rfl
theorem dgKerr_210 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 1 0 r s c S D = 0 := rfl
theorem dgKerr_211 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 1 1 r s c S D = (((-2))*s*c*a^(2:ℕ))/(D) := rfl
theorem dgKerr_212 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 1 2 r s c S D = 0 := rfl
theorem dgKerr_213 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 1 3 r s c S D = 0 := rfl
theorem dgKerr_220 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 2 0 r s c S D = 0 := rfl
theorem dgKerr_221 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 2 1 r s c S D = 0 := rfl
theorem dgKerr_222 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 2 2 r s c S D = (((-2))*s*c*a^(2:ℕ))/(1) := rfl
theorem dgKerr_223 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 2 3 r s c S D = 0 := rfl
theorem dgKerr_230 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 3 0 r s c S D = (((-4))*M*r*c*a^(3:ℕ)*s^(3:ℕ) - 4*M*a*r*s*c*S)/(S^(2:ℕ)) := rfl
theorem dgKerr_231 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 3 1 r s c S D = 0 := rfl
theorem dgKerr_232 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 3 2 r s c S D = 0 := rfl
theorem dgKerr_233 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 2 3 3 r s c S D = (2*s*c*a^(2:ℕ)*S^(2:ℕ) + 2*s*c*r^(2:ℕ)*S^(2:ℕ) + 4*M*r*c*a^(4:ℕ)*s^(5:ℕ) + 8*M*r*c*S*a^(2:ℕ)*s^(3:ℕ))/(S^(2:ℕ)) := rfl
theorem dgKerr_300 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 0 0 r s c S D = 0 := rfl
theorem dgKerr_301 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 0 1 r s c S D = 0 := rfl
theorem dgKerr_302 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 0 2 r s c S D = 0 := rfl
theorem dgKerr_303 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 0 3 r s c S D = 0 := rfl
theorem dgKerr_310 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 1 0 r s c S D = 0 := rfl
theorem dgKerr_311 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 1 1 r s c S D = 0 := rfl
theorem dgKerr_312 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 1 2 r s c S D = 0 := rfl
theorem dgKerr_313 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 1 3 r s c S D = 0 := rfl
theorem dgKerr_320 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 2 0 r s c S D = 0 := rfl
theorem dgKerr_321 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 2 1 r s c S D = 0 := rfl
theorem dgKerr_322 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 2 2 r s c S D = 0 := rfl
theorem dgKerr_323 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 2 3 r s c S D = 0 := rfl
theorem dgKerr_330 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 3 0 r s c S D = 0 := rfl
theorem dgKerr_331 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 3 1 r s c S D = 0 := rfl
theorem dgKerr_332 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 3 2 r s c S D = 0 := rfl
theorem dgKerr_333 (M a : ℝ) (r s c S D : ℝ) : dgKerr M a 3 3 3 r s c S D = 0 := rfl
theorem GammaKerr_000 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 0 0 r s c S D = 0 := rfl
theorem GammaKerr_001 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 0 1 r s c S D = (2*M*r^(6:ℕ) - M*S*a^(4:ℕ) - M*S*r^(4:ℕ) + 4*M*a^(2:ℕ)*r^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ) - 2*M*S*a^(2:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_002 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 0 2 r s c S D = (((-2))*M*s*c*a^(2:ℕ)*r^(5:ℕ) - 4*M*s*c*a^(4:ℕ)*r^(3:ℕ) - 2*M*r*s*c*a^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(3:ℕ) + 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_003 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 0 3 r s c S D = 0 := rfl
theorem GammaKerr_010 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 1 0 r s c S D = (2*M*r^(6:ℕ) - M*S*a^(4:ℕ) - M*S*r^(4:ℕ) + 4*M*a^(2:ℕ)*r^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ) - 2*M*S*a^(2:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_011 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 1 1 r s c S D = 0 := rfl
theorem GammaKerr_012 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 1 2 r s c S D = 0 := rfl
theorem GammaKerr_013 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 1 3 r s c S D = (M*S*a^(5:ℕ)*s^(2:ℕ) - 2*M*a*r^(6:ℕ)*s^(2:ℕ) - 4*M*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 2*M*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - M*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*D*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + M*a*S*r^(4:ℕ)*s^(2:ℕ) + 2*M*S*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*S*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_020 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 2 0 r s c S D = (((-2))*M*s*c*a^(2:ℕ)*r^(5:ℕ) - 4*M*s*c*a^(4:ℕ)*r^(3:ℕ) - 2*M*r*s*c*a^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(3:ℕ) + 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_021 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 2 1 r s c S D = 0 := rfl
theorem GammaKerr_022 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 2 2 r s c S D = 0 := rfl
theorem GammaKerr_023 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 2 3 r s c S D = (2*M*c*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 4*M*c*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*r*c*a^(7:ℕ)*s^(3:ℕ) - 4*c*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 2*M*r*c*D*a^(5:ℕ)*s^(5:ℕ) + 2*M*a*s*c*S*r^(5:ℕ) + 4*M*s*c*S*a^(3:ℕ)*r^(3:ℕ) + 2*M*r*s*c*S*a^(5:ℕ) - 2*M*a*s*c*r^(3:ℕ)*S^(2:ℕ) - 2*M*r*s*c*a^(3:ℕ)*S^(2:ℕ) - 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 2*M*r*c*S*D*a^(3:ℕ)*s^(3:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_030 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 3 0 r s c S D = 0 := rfl
theorem GammaKerr_031 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 3 1 r s c S D = (M*S*a^(5:ℕ)*s^(2:ℕ) - 2*M*a*r^(6:ℕ)*s^(2:ℕ) - 4*M*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 2*M*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - M*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*D*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + M*a*S*r^(4:ℕ)*s^(2:ℕ) + 2*M*S*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*S*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_032 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 3 2 r s c S D = (2*M*c*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 4*M*c*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*r*c*a^(7:ℕ)*s^(3:ℕ) - 4*c*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 2*M*r*c*D*a^(5:ℕ)*s^(5:ℕ) + 2*M*a*s*c*S*r^(5:ℕ) + 4*M*s*c*S*a^(3:ℕ)*r^(3:ℕ) + 2*M*r*s*c*S*a^(5:ℕ) - 2*M*a*s*c*r^(3:ℕ)*S^(2:ℕ) - 2*M*r*s*c*a^(3:ℕ)*S^(2:ℕ) - 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 2*M*r*c*S*D*a^(3:ℕ)*s^(3:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_033 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 0 3 3 r s c S D = 0 := rfl
theorem GammaKerr_100 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 0 0 r s c S D = (-(M*S*D) + 2*M*D*r^(2:ℕ))/(S^(3:ℕ)) := rfl
theorem GammaKerr_101 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 0 1 r s c S D = 0 := rfl
theorem GammaKerr_102 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 0 2 r s c S D = 0 := rfl
theorem GammaKerr_103 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 0 3 r s c S D = (M*a*S*D*s^(2:ℕ) - 2*M*a*D*r^(2:ℕ)*s^(2:ℕ))/(S^(3:ℕ)) := rfl
theorem GammaKerr_110 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 1 0 r s c S D = 0 := rfl
theorem GammaKerr_111 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 1 1 r s c S D = (r*D + M*S - r*S)/(S*D) := rfl
theorem GammaKerr_112 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 1 2 r s c S D = (-(s*c*a^(2:ℕ)))/(S) := rfl
theorem GammaKerr_113 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 1 3 r s c S D = 0 := rfl
theorem GammaKerr_120 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 2 0 r s c S D = 0 := rfl
theorem GammaKerr_121 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 2 1 r s c S D = (-(s*c*a^(2:ℕ)))/(S) := rfl
theorem GammaKerr_122 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 2 2 r s c S D = (-(r*D))/(S) := rfl
theorem GammaKerr_123 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 2 3 r s c S D = 0 := rfl
theorem GammaKerr_130 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 3 0 r s c S D = (M*a*S*D*s^(2:ℕ) - 2*M*a*D*r^(2:ℕ)*s^(2:ℕ))/(S^(3:ℕ)) := rfl
theorem GammaKerr_131 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 3 1 r s c S D = 0 := rfl
theorem GammaKerr_132 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 3 2 r s c S D = 0 := rfl
theorem GammaKerr_133 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 1 3 3 r s c S D = (-(r*D*s^(2:ℕ)*S^(2:ℕ)) - M*S*D*a^(2:ℕ)*s^(4:ℕ) + 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ))/(S^(3:ℕ)) := rfl
theorem GammaKerr_200 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 0 0 r s c S D = (((-2))*M*r*s*c*a^(2:ℕ))/(S^(3:ℕ)) := rfl
theorem GammaKerr_201 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 0 1 r s c S D = 0 := rfl
theorem GammaKerr_202 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 0 2 r s c S D = 0 := rfl
theorem GammaKerr_203 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 0 3 r s c S D = (2*M*r*c*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*r*s*c*S)/(S^(3:ℕ)) := rfl
theorem GammaKerr_210 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 1 0 r s c S D = 0 := rfl
theorem GammaKerr_211 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 1 1 r s c S D = (s*c*a^(2:ℕ))/(S*D) := rfl
theorem GammaKerr_212 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 1 2 r s c S D = (r)/(S) := rfl
theorem GammaKerr_213 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 1 3 r s c S D = 0 := rfl
theorem GammaKerr_220 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 2 0 r s c S D = 0 := rfl
theorem GammaKerr_221 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 2 1 r s c S D = (r)/(S) := rfl
theorem GammaKerr_222 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 2 2 r s c S D = (-(s*c*a^(2:ℕ)))/(S) := rfl
theorem GammaKerr_223 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 2 3 r s c S D = 0 := rfl
theorem GammaKerr_230 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 3 0 r s c S D = (2*M*r*c*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*r*s*c*S)/(S^(3:ℕ)) := rfl
theorem GammaKerr_231 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 3 1 r s c S D = 0 := rfl
theorem GammaKerr_232 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 3 2 r s c S D = 0 := rfl
theorem GammaKerr_233 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 2 3 3 r s c S D = (-(s*c*a^(2:ℕ)*S^(2:ℕ)) - s*c*r^(2:ℕ)*S^(2:ℕ) - 2*M*r*c*a^(4:ℕ)*s^(5:ℕ) - 4*M*r*c*S*a^(2:ℕ)*s^(3:ℕ))/(S^(3:ℕ)) := rfl
theorem GammaKerr_300 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 0 0 r s c S D = 0 := rfl
theorem GammaKerr_301 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 0 1 r s c S D = (4*a*M^(2:ℕ)*r^(3:ℕ) - M*a*S*D + 2*M*a*D*r^(2:ℕ) + M*S*a^(3:ℕ)*s^(2:ℕ) - 2*M*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*a*r*S*M^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_302 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 0 2 r s c S D = (2*M*r*c*a^(5:ℕ)*s^(4:ℕ) - 4*c*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r*c*S*D - 2*M*r*c*D*a^(3:ℕ)*s^(2:ℕ) + 2*M*r*c*S*a^(3:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ)) := rfl
theorem GammaKerr_303 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 0 3 r s c S D = 0 := rfl
theorem GammaKerr_310 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 1 0 r s c S D = (4*a*M^(2:ℕ)*r^(3:ℕ) - M*a*S*D + 2*M*a*D*r^(2:ℕ) + M*S*a^(3:ℕ)*s^(2:ℕ) - 2*M*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*a*r*S*M^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_311 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 1 1 r s c S D = 0 := rfl
theorem GammaKerr_312 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 1 2 r s c S D = 0 := rfl
theorem GammaKerr_313 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 1 3 r s c S D = (r*D*S^(2:ℕ) - M*S*a^(4:ℕ)*s^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) - r*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_320 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 2 0 r s c S D = (2*M*r*c*a^(5:ℕ)*s^(4:ℕ) - 4*c*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 2*M*a*r*c*S*D - 2*M*r*c*D*a^(3:ℕ)*s^(2:ℕ) + 2*M*r*c*S*a^(3:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ)) := rfl
theorem GammaKerr_321 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 2 1 r s c S D = 0 := rfl
theorem GammaKerr_322 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 2 2 r s c S D = 0 := rfl
theorem GammaKerr_323 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 2 3 r s c S D = (c*D*a^(2:ℕ)*S^(2:ℕ) + c*D*r^(2:ℕ)*S^(2:ℕ) - c*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*c*a^(6:ℕ)*s^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(4:ℕ) - 4*M*r*c*S*a^(4:ℕ)*s^(4:ℕ) + 4*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M*r*c*S*D*a^(2:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ)) := rfl
theorem GammaKerr_330 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 3 0 r s c S D = 0 := rfl
theorem GammaKerr_331 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 3 1 r s c S D = (r*D*S^(2:ℕ) - M*S*a^(4:ℕ)*s^(4:ℕ) + 2*M*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) - r*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + M*S*D*a^(2:ℕ)*s^(2:ℕ) - 2*M*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*r*S*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ))/(D*S^(3:ℕ)) := rfl
theorem GammaKerr_332 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 3 2 r s c S D = (c*D*a^(2:ℕ)*S^(2:ℕ) + c*D*r^(2:ℕ)*S^(2:ℕ) - c*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*c*a^(6:ℕ)*s^(6:ℕ) + 4*c*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*c*D*a^(4:ℕ)*s^(4:ℕ) - 4*M*r*c*S*a^(4:ℕ)*s^(4:ℕ) + 4*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 4*M*r*c*S*D*a^(2:ℕ)*s^(2:ℕ))/(s*D*S^(3:ℕ)) := rfl
theorem GammaKerr_333 (M a : ℝ) (r s c S D : ℝ) : GammaKerr M a 3 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_0000 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_0001 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_0002 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_0003 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_0010 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_0011 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_0012 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_0013 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_0020 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_0021 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_0022 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_0023 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_0030 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_0031 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_0032 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_0033 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 0 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_0100 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_0101 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_0102 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_0103 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_0110 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_0111 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_0112 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_0113 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_0120 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_0121 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_0122 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_0123 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_0130 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_0131 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_0132 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_0133 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 1 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_0200 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_0201 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_0202 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_0203 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_0210 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_0211 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_0212 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_0213 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_0220 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_0221 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_0222 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_0223 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_0230 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_0231 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_0232 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_0233 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 2 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_0300 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_0301 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_0302 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_0303 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_0310 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_0311 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_0312 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_0313 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_0320 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_0321 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_0322 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_0323 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_0330 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_0331 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_0332 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_0333 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 0 3 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_1000 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_1001 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 0 1 r s c S D = (((-12))*M*D*r^(7:ℕ) - 4*M*S*r^(7:ℕ) + 2*M*r^(5:ℕ)*S^(2:ℕ) + 4*S*M^(2:ℕ)*r^(6:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*S^(2:ℕ) - 2*M^(2:ℕ)*r^(4:ℕ)*S^(2:ℕ) + 16*M*S*D*r^(5:ℕ) - 4*M*D*r^(3:ℕ)*S^(2:ℕ) - 24*M*D*a^(2:ℕ)*r^(5:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ) - 8*M*S*a^(2:ℕ)*r^(5:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ) + 4*M*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 2*M*r*a^(4:ℕ)*S^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 24*M*S*D*a^(2:ℕ)*r^(3:ℕ) + 8*M*r*S*D*a^(4:ℕ) - 4*M*r*D*a^(2:ℕ)*S^(2:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1002 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 0 2 r s c S D = (((-2))*M*s*c*S*D*a^(6:ℕ) + 12*M*s*c*D*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*D*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*D*a^(6:ℕ)*r^(2:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(3:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ) + 4*M*s*c*S*a^(2:ℕ)*r^(6:ℕ) + 8*M*s*c*S*a^(4:ℕ)*r^(4:ℕ) + 4*M*s*c*S*a^(6:ℕ)*r^(2:ℕ) - 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(5:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 4*r*s*c*S*M^(2:ℕ)*a^(6:ℕ) - 8*s*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 8*s*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 10*M*s*c*S*D*a^(2:ℕ)*r^(4:ℕ) - 12*M*s*c*S*D*a^(4:ℕ)*r^(2:ℕ) - 16*s*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 8*r*s*c*D*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1003 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_1010 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 1 0 r s c S D = (((-12))*M*D*r^(7:ℕ) - 4*M*S*r^(7:ℕ) + 2*M*r^(5:ℕ)*S^(2:ℕ) + 4*S*M^(2:ℕ)*r^(6:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*S^(2:ℕ) - 2*M^(2:ℕ)*r^(4:ℕ)*S^(2:ℕ) + 16*M*S*D*r^(5:ℕ) - 4*M*D*r^(3:ℕ)*S^(2:ℕ) - 24*M*D*a^(2:ℕ)*r^(5:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ) - 8*M*S*a^(2:ℕ)*r^(5:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ) + 4*M*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 2*M*r*a^(4:ℕ)*S^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 24*M*S*D*a^(2:ℕ)*r^(3:ℕ) + 8*M*r*S*D*a^(4:ℕ) - 4*M*r*D*a^(2:ℕ)*S^(2:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1011 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_1012 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_1013 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 1 3 r s c S D = (2*M^(2:ℕ)*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(7:ℕ)*s^(2:ℕ) + 24*M*D*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 12*M*D*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*D*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*D*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 12*M*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*a*S*r^(7:ℕ)*s^(2:ℕ) + 8*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 4*a*S*M^(2:ℕ)*r^(6:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*a*M^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 4*a*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(3:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*M^(3:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 16*M*a*S*D*r^(5:ℕ)*s^(2:ℕ) - 24*M*S*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(5:ℕ)*s^(2:ℕ) + 8*M*a*D*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*r*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a*r*D*s^(2:ℕ)*S^(3:ℕ) + 20*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*M*r*S*a^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1020 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 2 0 r s c S D = (((-2))*M*s*c*S*D*a^(6:ℕ) + 12*M*s*c*D*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*D*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*D*a^(6:ℕ)*r^(2:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(3:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ) + 4*M*s*c*S*a^(2:ℕ)*r^(6:ℕ) + 8*M*s*c*S*a^(4:ℕ)*r^(4:ℕ) + 4*M*s*c*S*a^(6:ℕ)*r^(2:ℕ) - 4*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(5:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 4*r*s*c*S*M^(2:ℕ)*a^(6:ℕ) - 8*s*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 8*s*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 10*M*s*c*S*D*a^(2:ℕ)*r^(4:ℕ) - 12*M*s*c*S*D*a^(4:ℕ)*r^(2:ℕ) - 16*s*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 8*r*s*c*D*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1021 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_1022 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_1023 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 2 3 r s c S D = (2*M*c*S*D*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(5:ℕ)*S^(2:ℕ) - 2*M*s*c*D*a^(3:ℕ)*S^(3:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*D*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*D*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*D*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 2*M*c*S*a^(5:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 2*M*c*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 8*M*c*S*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*c*S*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*r^(6:ℕ)*S^(2:ℕ) - 8*M*s*c*a^(3:ℕ)*r^(4:ℕ)*S^(2:ℕ) - 4*M*s*c*a^(5:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 4*M*a*s*c*r^(4:ℕ)*S^(3:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(3:ℕ) + 4*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(7:ℕ)*s^(3:ℕ) + 4*a*s*c*M^(2:ℕ)*r^(5:ℕ)*S^(2:ℕ) + 8*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 16*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(5:ℕ)*S^(2:ℕ) - 4*a*s*c*M^(2:ℕ)*r^(3:ℕ)*S^(3:ℕ) - 4*r*s*c*M^(2:ℕ)*a^(3:ℕ)*S^(3:ℕ) - 8*c*S*M^(3:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 16*c*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 8*M*a*s*c*S*D*r^(6:ℕ) - 16*M*s*c*S*D*a^(3:ℕ)*r^(4:ℕ) + 10*M*c*S*D*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 8*M*s*c*S*D*a^(5:ℕ)*r^(2:ℕ) + 12*M*c*S*D*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 14*M*a*s*c*D*r^(4:ℕ)*S^(2:ℕ) + 16*M*s*c*D*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 6*M*a*s*c*D*r^(2:ℕ)*S^(3:ℕ) + 32*c*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 16*r*c*D*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1030 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_1031 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 3 1 r s c S D = (2*M^(2:ℕ)*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(7:ℕ)*s^(2:ℕ) + 24*M*D*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 12*M*D*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*D*M^(2:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*D*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 12*M*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*a*S*r^(7:ℕ)*s^(2:ℕ) + 8*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ) + 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 4*a*S*M^(2:ℕ)*r^(6:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(2:ℕ) - 8*S*M^(2:ℕ)*a^(3:ℕ)*r^(4:ℕ)*s^(4:ℕ) - 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(2:ℕ) + 2*a*M^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 4*a*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(3:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*M^(3:ℕ)*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 16*M*a*S*D*r^(5:ℕ)*s^(2:ℕ) - 24*M*S*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(5:ℕ)*s^(2:ℕ) + 8*M*a*D*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*r*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*M*a*r*D*s^(2:ℕ)*S^(3:ℕ) + 20*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*M*r*S*a^(3:ℕ)*s^(4:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1032 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 3 2 r s c S D = (2*M*c*S*D*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(5:ℕ)*S^(2:ℕ) - 2*M*s*c*D*a^(3:ℕ)*S^(3:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*D*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*D*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*D*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 2*M*c*S*a^(5:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 2*M*c*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 8*M*c*S*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*c*S*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*r^(6:ℕ)*S^(2:ℕ) - 8*M*s*c*a^(3:ℕ)*r^(4:ℕ)*S^(2:ℕ) - 4*M*s*c*a^(5:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 4*M*a*s*c*r^(4:ℕ)*S^(3:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(3:ℕ) + 4*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(5:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 8*c*S*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(7:ℕ)*s^(3:ℕ) + 4*a*s*c*M^(2:ℕ)*r^(5:ℕ)*S^(2:ℕ) + 8*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*S^(2:ℕ) + 16*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(5:ℕ)*S^(2:ℕ) - 4*a*s*c*M^(2:ℕ)*r^(3:ℕ)*S^(3:ℕ) - 4*r*s*c*M^(2:ℕ)*a^(3:ℕ)*S^(3:ℕ) - 8*c*S*M^(3:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 16*c*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 8*M*a*s*c*S*D*r^(6:ℕ) - 16*M*s*c*S*D*a^(3:ℕ)*r^(4:ℕ) + 10*M*c*S*D*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 8*M*s*c*S*D*a^(5:ℕ)*r^(2:ℕ) + 12*M*c*S*D*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 14*M*a*s*c*D*r^(4:ℕ)*S^(2:ℕ) + 16*M*s*c*D*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 6*M*a*s*c*D*r^(2:ℕ)*S^(3:ℕ) + 32*c*S*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 16*r*c*D*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1033 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 0 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_1100 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 0 0 r s c S D = (2*M^(2:ℕ)*S^(2:ℕ) - 12*M*D*r^(3:ℕ) + 4*M*S*r^(3:ℕ) - 2*M*r*S^(2:ℕ) - 4*S*M^(2:ℕ)*r^(2:ℕ) + 8*M*r*S*D)/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1101 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_1102 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_1103 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 0 3 r s c S D = (((-2))*a*M^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(3:ℕ)*s^(2:ℕ) - 4*M*a*S*r^(3:ℕ)*s^(2:ℕ) + 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 4*a*S*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*a*r*S*D*s^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1110 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_1111 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 1 1 r s c S D = (-(D*S^(2:ℕ)) + S*D^(2:ℕ) - 2*r^(2:ℕ)*D^(2:ℕ) + 2*M^(2:ℕ)*S^(2:ℕ) + 2*r^(2:ℕ)*S^(2:ℕ) - 4*M*r*S^(2:ℕ))/(S^(2:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1112 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 1 2 r s c S D = (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_1113 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_1120 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_1121 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 2 1 r s c S D = (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_1122 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 2 2 r s c S D = (-(S*D) + 2*D*r^(2:ℕ) - 2*S*r^(2:ℕ) + 2*M*r*S)/(S^(2:ℕ)) := rfl
theorem dGammaKerr_1123 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_1130 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 3 0 r s c S D = (((-2))*a*M^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a*D*r^(3:ℕ)*s^(2:ℕ) - 4*M*a*S*r^(3:ℕ)*s^(2:ℕ) + 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 4*a*S*M^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*a*r*S*D*s^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1131 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_1132 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_1133 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 1 3 3 r s c S D = (-(D*s^(2:ℕ)*S^(3:ℕ)) - 2*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*D*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*s^(2:ℕ)*S^(3:ℕ) + 2*M^(2:ℕ)*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*D*a^(2:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 4*M*S*a^(2:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 2*M*r*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*M*r*S*D*a^(2:ℕ)*s^(4:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1200 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 0 0 r s c S D = (((-2))*M*s*c*S*a^(2:ℕ) + 12*M*s*c*a^(2:ℕ)*r^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1201 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_1202 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_1203 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 0 3 r s c S D = (2*M*c*S*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*M*a*s*c*S*r^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1210 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_1211 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 1 1 r s c S D = (((-2))*r*s*c*D*a^(2:ℕ) + 2*M*s*c*S*a^(2:ℕ) - 2*r*s*c*S*a^(2:ℕ))/(S^(2:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1212 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 1 2 r s c S D = (S - 2*r^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_1213 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_1220 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_1221 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 2 1 r s c S D = (S - 2*r^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_1222 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 2 2 r s c S D = (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_1223 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_1230 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 3 0 r s c S D = (2*M*c*S*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*M*a*s*c*S*r^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1231 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_1232 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_1233 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 2 3 3 r s c S D = (2*s*c*r^(3:ℕ)*S^(2:ℕ) - 2*r*s*c*S^(3:ℕ) - 2*M*c*S*a^(4:ℕ)*s^(5:ℕ) - 4*M*c*a^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(5:ℕ) + 2*r*s*c*a^(2:ℕ)*S^(2:ℕ) + 16*M*c*S*a^(2:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_1300 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_1301 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 0 1 r s c S D = (((-2))*a*D*M^(2:ℕ)*S^(2:ℕ) - 24*a*D*M^(2:ℕ)*r^(4:ℕ) - 12*M*a*r^(3:ℕ)*D^(2:ℕ) - 8*a*S*M^(2:ℕ)*r^(4:ℕ) + 4*a*M^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 2*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*a*S*M^(3:ℕ)*r^(3:ℕ) - 4*a*r*M^(3:ℕ)*S^(2:ℕ) + 12*M*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 20*a*S*D*M^(2:ℕ)*r^(2:ℕ) + 8*M*a*r*S*D^(2:ℕ) + 4*M*S*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(3:ℕ)*s^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1302 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 0 2 r s c S D = (((-2))*M*a*c*S^(2:ℕ)*D^(2:ℕ) + 2*M*c*S*D*a^(5:ℕ)*s^(4:ℕ) + 2*M*c*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 24*c*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*M*a*c*S*r^(2:ℕ)*D^(2:ℕ) - 2*M*c*S*a^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(4:ℕ) + 4*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*c*S*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1303 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_1310 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 1 0 r s c S D = (((-2))*a*D*M^(2:ℕ)*S^(2:ℕ) - 24*a*D*M^(2:ℕ)*r^(4:ℕ) - 12*M*a*r^(3:ℕ)*D^(2:ℕ) - 8*a*S*M^(2:ℕ)*r^(4:ℕ) + 4*a*M^(2:ℕ)*r^(2:ℕ)*S^(2:ℕ) + 2*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*a*S*M^(3:ℕ)*r^(3:ℕ) - 4*a*r*M^(3:ℕ)*S^(2:ℕ) + 12*M*D*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 20*a*S*D*M^(2:ℕ)*r^(2:ℕ) + 8*M*a*r*S*D^(2:ℕ) + 4*M*S*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*D*a^(3:ℕ)*s^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1311 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_1312 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_1313 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 1 3 r s c S D = (S^(3:ℕ)*D^(2:ℕ) - 2*r^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 2*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*M*r*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*r*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1320 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 2 0 r s c S D = (((-2))*M*a*c*S^(2:ℕ)*D^(2:ℕ) + 2*M*c*S*D*a^(5:ℕ)*s^(4:ℕ) + 2*M*c*D*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 24*c*D*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*M*a*c*S*r^(2:ℕ)*D^(2:ℕ) - 2*M*c*S*a^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) + 12*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(4:ℕ) + 4*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*c*S*M^(3:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*r*c*S*D*M^(2:ℕ)*a^(3:ℕ)*s^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1321 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_1322 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_1323 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 2 3 r s c S D = (((-2))*c*r^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 2*r*c*S^(3:ℕ)*D^(2:ℕ) - 2*r*c*a^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 2*M*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*c*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*r*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*c*S*D*a^(6:ℕ)*s^(6:ℕ) - 4*M*c*D*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*c*D*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*r*c*D*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*c*D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*S*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*c*S*M^(2:ℕ)*a^(6:ℕ)*s^(6:ℕ) - 8*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*r*c*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 16*M*c*S*D*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 16*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ) + 8*r*c*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 16*M*c*S*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1330 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_1331 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 3 1 r s c S D = (S^(3:ℕ)*D^(2:ℕ) - 2*r^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 2*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 12*M*D*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*D*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 12*M*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*D^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*M*r*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*r*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 8*S*M^(2:ℕ)*a^(2:ℕ)*r^(4:ℕ)*s^(2:ℕ) + 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*S*M^(3:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 4*r*M^(3:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 8*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) - 20*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ) - 8*M*r*S*a^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1332 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 3 2 r s c S D = (((-2))*c*r^(3:ℕ)*S^(2:ℕ)*D^(2:ℕ) + 2*r*c*S^(3:ℕ)*D^(2:ℕ) - 2*r*c*a^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 2*M*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*c*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*r*c*a^(4:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*c*S*D*a^(6:ℕ)*s^(6:ℕ) - 4*M*c*D*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 24*c*D*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) + 2*c*D*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*r*c*D*a^(4:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*r*c*D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 2*M*c*S*a^(4:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ)*D^(2:ℕ) - 12*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*D^(2:ℕ) + 4*M*c*S*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M*c*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 2*M*c*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 8*c*S*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 4*r*c*S*M^(2:ℕ)*a^(6:ℕ)*s^(6:ℕ) - 8*c*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 8*r*c*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 8*c*S*M^(3:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 8*c*M^(3:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 16*M*c*S*D*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 16*c*S*D*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ)*s^(2:ℕ) + 8*r*c*S*D*M^(2:ℕ)*a^(4:ℕ)*s^(4:ℕ) + 8*r*c*D*M^(2:ℕ)*a^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 16*M*c*S*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*D^(2:ℕ))/(s*S^(4:ℕ)*D^(2:ℕ)) := rfl
theorem dGammaKerr_1333 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 1 3 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_2000 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_2001 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 0 1 r s c S D = (((-4))*M*s*c*S*a^(6:ℕ) + 12*M*s*c*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*a^(6:ℕ)*r^(2:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(2:ℕ)*r^(4:ℕ) - 8*M*s*c*S*a^(4:ℕ)*r^(2:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2002 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 0 2 r s c S D = (((-2))*M*S*a^(2:ℕ)*r^(5:ℕ)*c^(2:ℕ) + 2*M*S*a^(2:ℕ)*r^(5:ℕ)*s^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*c^(2:ℕ) + 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*S*a^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(2:ℕ) - 12*M*a^(4:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 24*M*a^(6:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 6*M*r*S*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2003 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_2010 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 1 0 r s c S D = (((-4))*M*s*c*S*a^(6:ℕ) + 12*M*s*c*a^(2:ℕ)*r^(6:ℕ) + 24*M*s*c*a^(4:ℕ)*r^(4:ℕ) + 12*M*s*c*a^(6:ℕ)*r^(2:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(2:ℕ)*r^(4:ℕ) - 8*M*s*c*S*a^(4:ℕ)*r^(2:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2011 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_2012 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_2013 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 1 3 r s c S D = (4*M*c*S*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(5:ℕ)*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 4*M*c*S*D*a^(5:ℕ)*s^(5:ℕ) - 4*M*c*D*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 4*M*a*s*c*S*r^(6:ℕ) - 8*M*s*c*S*a^(3:ℕ)*r^(4:ℕ) + 4*M*c*S*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(5:ℕ)*r^(2:ℕ) + 8*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*a*s*c*r^(4:ℕ)*S^(2:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 4*M*a*s*c*r^(2:ℕ)*S^(3:ℕ) + 16*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 8*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2020 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 2 0 r s c S D = (((-2))*M*S*a^(2:ℕ)*r^(5:ℕ)*c^(2:ℕ) + 2*M*S*a^(2:ℕ)*r^(5:ℕ)*s^(2:ℕ) - 4*M*S*a^(4:ℕ)*r^(3:ℕ)*c^(2:ℕ) + 4*M*S*a^(4:ℕ)*r^(3:ℕ)*s^(2:ℕ) - 2*M*r*S*a^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(2:ℕ) - 12*M*a^(4:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 24*M*a^(6:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(4:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 6*M*r*S*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2021 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_2022 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_2023 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 2 3 r s c S D = (((-2))*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(4:ℕ) - 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 2*M*r*S*a^(7:ℕ)*s^(4:ℕ) + 2*M*a*r^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a^(3:ℕ)*r^(3:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*a^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*r*a^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 12*M*a^(5:ℕ)*r^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 24*M*a^(7:ℕ)*r^(3:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 12*M*r*a^(9:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*M^(2:ℕ)*a^(7:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(5:ℕ)*s^(6:ℕ) + 2*M*r*D*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 14*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 28*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(7:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*r*a^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 52*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 24*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 18*M*r*S*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 6*M*r*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2030 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_2031 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 3 1 r s c S D = (4*M*c*S*a^(7:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(5:ℕ)*S^(2:ℕ) - 12*M*c*a^(3:ℕ)*r^(6:ℕ)*s^(3:ℕ) - 24*M*c*a^(5:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 12*M*c*a^(7:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*c*M^(2:ℕ)*a^(5:ℕ)*r^(3:ℕ)*s^(5:ℕ) - 4*M*c*S*D*a^(5:ℕ)*s^(5:ℕ) - 4*M*c*D*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(5:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 4*M*a*s*c*S*r^(6:ℕ) - 8*M*s*c*S*a^(3:ℕ)*r^(4:ℕ) + 4*M*c*S*a^(3:ℕ)*r^(4:ℕ)*s^(3:ℕ) - 4*M*s*c*S*a^(5:ℕ)*r^(2:ℕ) + 8*M*c*S*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*M*a*s*c*r^(4:ℕ)*S^(2:ℕ) + 4*M*s*c*a^(3:ℕ)*r^(2:ℕ)*S^(2:ℕ) - 4*M*c*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 4*M*a*s*c*r^(2:ℕ)*S^(3:ℕ) + 16*c*S*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 8*r*c*S*M^(2:ℕ)*a^(5:ℕ)*s^(5:ℕ) - 8*r*c*M^(2:ℕ)*a^(3:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2032 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 3 2 r s c S D = (((-2))*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(4:ℕ) - 4*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(4:ℕ) - 2*M*r*S*a^(7:ℕ)*s^(4:ℕ) + 2*M*a*r^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 4*M*a^(3:ℕ)*r^(3:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*S^(2:ℕ) + 2*M*r*a^(5:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 2*M*r*a^(5:ℕ)*s^(2:ℕ)*S^(2:ℕ) - 2*M*a*r^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*a*r^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*M*r*a^(3:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*S^(3:ℕ) + 12*M*a^(5:ℕ)*r^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 24*M*a^(7:ℕ)*r^(3:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 12*M*r*a^(9:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(6:ℕ) + 8*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 24*M^(2:ℕ)*a^(7:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(5:ℕ)*s^(6:ℕ) + 2*M*r*D*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 14*M*S*a^(3:ℕ)*r^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 28*M*S*a^(5:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(7:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 4*M*a^(3:ℕ)*r^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*M*r*a^(5:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 52*S*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 24*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 18*M*r*S*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 6*M*r*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2033 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 0 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_2100 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 0 0 r s c S D = (((-4))*M*s*c*S*D*a^(2:ℕ) + 12*M*s*c*D*a^(2:ℕ)*r^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2101 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_2102 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_2103 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 0 3 r s c S D = (4*M*c*S*D*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*D*S^(2:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*S*D*r^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2110 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_2111 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 1 1 r s c S D = (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_2112 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 1 2 r s c S D = (-(S*a^(2:ℕ)*c^(2:ℕ)) + S*a^(2:ℕ)*s^(2:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_2113 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_2120 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_2121 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 2 1 r s c S D = (-(S*a^(2:ℕ)*c^(2:ℕ)) + S*a^(2:ℕ)*s^(2:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_2122 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 2 2 r s c S D = (((-2))*r*s*c*D*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_2123 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_2130 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 3 0 r s c S D = (4*M*c*S*D*a^(3:ℕ)*s^(3:ℕ) + 2*M*a*s*c*D*S^(2:ℕ) - 12*M*c*D*a^(3:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 4*M*a*s*c*S*D*r^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2131 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_2132 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_2133 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 1 3 3 r s c S D = (((-2))*r*s*c*D*S^(3:ℕ) - 4*M*c*S*D*a^(4:ℕ)*s^(5:ℕ) - 4*M*c*D*a^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 2*r*c*D*a^(2:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 8*M*c*S*D*a^(2:ℕ)*r^(2:ℕ)*s^(3:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2200 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 0 0 r s c S D = (((-2))*M*r*S*a^(2:ℕ)*c^(2:ℕ) + 2*M*r*S*a^(2:ℕ)*s^(2:ℕ) - 12*M*r*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2201 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_2202 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_2203 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 0 3 r s c S D = (((-2))*M*r*S*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 12*M*r*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2210 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_2211 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 1 1 r s c S D = (S*a^(2:ℕ)*c^(2:ℕ) - S*a^(2:ℕ)*s^(2:ℕ) + 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*S^(2:ℕ)) := rfl
theorem dGammaKerr_2212 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 1 2 r s c S D = (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_2213 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_2220 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_2221 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 2 1 r s c S D = (2*r*s*c*a^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_2222 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 2 2 r s c S D = (-(S*a^(2:ℕ)*c^(2:ℕ)) + S*a^(2:ℕ)*s^(2:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(2:ℕ)) := rfl
theorem dGammaKerr_2223 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_2230 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 3 0 r s c S D = (((-2))*M*r*S*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*c^(2:ℕ)*S^(2:ℕ) - 2*M*a*r*s^(2:ℕ)*S^(2:ℕ) + 12*M*r*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2231 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_2232 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_2233 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 2 3 3 r s c S D = (-(a^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ)) + a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - r^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) + r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 2*M*r*S*a^(4:ℕ)*s^(6:ℕ) + 4*M*r*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) - 12*M*r*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 2*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 26*M*r*S*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ) - 12*M*r*a^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(S^(4:ℕ)) := rfl
theorem dGammaKerr_2300 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_2301 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 0 1 r s c S D = (4*M*c*S*a^(5:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(3:ℕ)*S^(2:ℕ) - 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ) - 4*M*s*c*S*D*a^(3:ℕ) + 12*M*s*c*D*a^(3:ℕ)*r^(2:ℕ) - 4*M*s*c*S*a^(3:ℕ)*r^(2:ℕ) - 8*r*s*c*S*M^(2:ℕ)*a^(3:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2302 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 0 2 r s c S D = (((-2))*M*r*S*a^(5:ℕ)*s^(6:ℕ) - 2*M*r*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 24*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*D*c^(2:ℕ)*S^(2:ℕ) + 2*M*a*r*D*s^(2:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 10*M*r*S*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ)) := rfl
theorem dGammaKerr_2303 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_2310 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 1 0 r s c S D = (4*M*c*S*a^(5:ℕ)*s^(3:ℕ) + 2*M*s*c*a^(3:ℕ)*S^(2:ℕ) - 12*M*c*a^(5:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 24*s*c*M^(2:ℕ)*a^(3:ℕ)*r^(3:ℕ) - 4*M*s*c*S*D*a^(3:ℕ) + 12*M*s*c*D*a^(3:ℕ)*r^(2:ℕ) - 4*M*s*c*S*a^(3:ℕ)*r^(2:ℕ) - 8*r*s*c*S*M^(2:ℕ)*a^(3:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2311 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_2312 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_2313 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 1 3 r s c S D = (((-4))*M*c*S*a^(6:ℕ)*s^(5:ℕ) - 4*M*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*a^(6:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 2*r*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 2*r*s*c*a^(2:ℕ)*S^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*r*s*c*D*a^(2:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2320 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 2 0 r s c S D = (((-2))*M*r*S*a^(5:ℕ)*s^(6:ℕ) - 2*M*r*a^(3:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*a^(7:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(4:ℕ) - 24*M^(2:ℕ)*a^(5:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*S*D*a^(3:ℕ)*s^(4:ℕ) + 2*M*a*r*D*c^(2:ℕ)*S^(2:ℕ) + 2*M*a*r*D*s^(2:ℕ)*S^(2:ℕ) - 12*M*r*D*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 14*M*r*S*a^(5:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 2*M*r*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 4*S*M^(2:ℕ)*a^(3:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ) - 10*M*r*S*D*a^(3:ℕ)*s^(2:ℕ)*c^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ)) := rfl
theorem dGammaKerr_2321 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_2322 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_2323 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 2 3 r s c S D = (a^(4:ℕ)*s^(4:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(3:ℕ) - a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(8:ℕ) + 4*M*r*a^(4:ℕ)*s^(6:ℕ)*S^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(8:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 2*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) - a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(6:ℕ) - 4*M*r*D*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 26*M*r*S*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 12*M*r*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 22*M*r*S*D*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M*r*D*a^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ)) := rfl
theorem dGammaKerr_2330 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_2331 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 3 1 r s c S D = (((-4))*M*c*S*a^(6:ℕ)*s^(5:ℕ) - 4*M*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) + 12*M*c*a^(6:ℕ)*r^(2:ℕ)*s^(5:ℕ) - 24*c*M^(2:ℕ)*a^(4:ℕ)*r^(3:ℕ)*s^(3:ℕ) - 2*r*c*a^(4:ℕ)*s^(3:ℕ)*S^(2:ℕ) - 2*r*s*c*a^(2:ℕ)*S^(3:ℕ) + 4*M*c*S*D*a^(4:ℕ)*s^(3:ℕ) + 2*M*s*c*D*a^(2:ℕ)*S^(2:ℕ) - 12*M*c*D*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) + 2*r*s*c*D*a^(2:ℕ)*S^(2:ℕ) + 8*M*c*S*a^(4:ℕ)*r^(2:ℕ)*s^(3:ℕ) - 8*s*c*S*M^(2:ℕ)*a^(2:ℕ)*r^(3:ℕ) + 8*r*c*S*M^(2:ℕ)*a^(4:ℕ)*s^(3:ℕ) + 4*r*s*c*M^(2:ℕ)*a^(2:ℕ)*S^(2:ℕ) - 4*M*s*c*S*D*a^(2:ℕ)*r^(2:ℕ))/(D*S^(4:ℕ)) := rfl
theorem dGammaKerr_2332 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 3 2 r s c S D = (a^(4:ℕ)*s^(4:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*a^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - D*r^(2:ℕ)*s^(2:ℕ)*S^(3:ℕ) - 2*a^(6:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(3:ℕ) - a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) + 2*D*a^(4:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 2*M*r*S*a^(6:ℕ)*s^(8:ℕ) + 4*M*r*a^(4:ℕ)*s^(6:ℕ)*S^(2:ℕ) - 12*M*r*a^(8:ℕ)*s^(8:ℕ)*c^(2:ℕ) - 4*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(6:ℕ) - 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 24*M^(2:ℕ)*a^(6:ℕ)*r^(2:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 2*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) - a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(3:ℕ) - 2*M*r*S*D*a^(4:ℕ)*s^(6:ℕ) - 4*M*r*D*a^(2:ℕ)*s^(4:ℕ)*S^(2:ℕ) + 12*M*r*D*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) + 2*D*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) - 26*M*r*S*a^(6:ℕ)*s^(6:ℕ)*c^(2:ℕ) - 12*M*r*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 28*S*M^(2:ℕ)*a^(4:ℕ)*r^(2:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M^(2:ℕ)*a^(2:ℕ)*r^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ) + 22*M*r*S*D*a^(4:ℕ)*s^(4:ℕ)*c^(2:ℕ) + 4*M*r*D*a^(2:ℕ)*s^(2:ℕ)*c^(2:ℕ)*S^(2:ℕ))/(D*s^(2:ℕ)*S^(4:ℕ)) := rfl
theorem dGammaKerr_2333 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 2 3 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_3000 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_3001 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_3002 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_3003 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_3010 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_3011 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_3012 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_3013 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_3020 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_3021 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_3022 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_3023 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_3030 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_3031 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_3032 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_3033 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 0 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_3100 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_3101 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_3102 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_3103 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_3110 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_3111 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_3112 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_3113 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_3120 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_3121 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_3122 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_3123 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_3130 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_3131 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_3132 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_3133 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 1 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_3200 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_3201 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_3202 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_3203 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_3210 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_3211 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_3212 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_3213 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_3220 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_3221 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_3222 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_3223 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_3230 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_3231 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_3232 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_3233 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 2 3 3 r s c S D = 0 := rfl
theorem dGammaKerr_3300 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 0 0 r s c S D = 0 := rfl
theorem dGammaKerr_3301 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 0 1 r s c S D = 0 := rfl
theorem dGammaKerr_3302 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 0 2 r s c S D = 0 := rfl
theorem dGammaKerr_3303 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 0 3 r s c S D = 0 := rfl
theorem dGammaKerr_3310 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 1 0 r s c S D = 0 := rfl
theorem dGammaKerr_3311 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 1 1 r s c S D = 0 := rfl
theorem dGammaKerr_3312 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 1 2 r s c S D = 0 := rfl
theorem dGammaKerr_3313 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 1 3 r s c S D = 0 := rfl
theorem dGammaKerr_3320 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 2 0 r s c S D = 0 := rfl
theorem dGammaKerr_3321 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 2 1 r s c S D = 0 := rfl
theorem dGammaKerr_3322 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 2 2 r s c S D = 0 := rfl
theorem dGammaKerr_3323 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 2 3 r s c S D = 0 := rfl
theorem dGammaKerr_3330 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 3 0 r s c S D = 0 := rfl
theorem dGammaKerr_3331 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 3 1 r s c S D = 0 := rfl
theorem dGammaKerr_3332 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 3 2 r s c S D = 0 := rfl
theorem dGammaKerr_3333 (M a : ℝ) (r s c S D : ℝ) : dGammaKerr M a 3 3 3 3 r s c S D = 0 := rfl

end KerrBL
Source
R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963) 237-238, https://doi.org/10.1103/PhysRevLett.11.237; R. H. Boyer and R. W. Lindquist, Maximal analytic extension of the Kerr metric, J. Math. Phys. 8 (1967) 265-281, https://doi.org/10.1063/1.1705193, Sec. 2 (Boyer-Lindquist form of the Kerr line element); metric components transcribed token-for-token from the project certificate EinsteinSolver/certificate/kerr/metric.json (sha256 d729883d95fd7d3cf84d9c971c6725f847155562cc4e88660535b8d0bd0be336); design record LEAN/kerr-formalization/mission/DESIGN.md, definition bundle D3 (generated by mission_emit.py; certified by the Layer II bridge theorems)
Human review
Endorsed by Shuze Chen · Sep 14, 2026
Endorsed by He Wang · Sep 14, 2026
Confirmed by the mission captain (proposal self-audit).
View graph