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theorem proved tactic proof

lorentzian_signature

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formal statement (Lean)

 162theorem lorentzian_signature :
 163    (Finset.univ.filter (fun i : Fin 4 => η i i < 0)).card = temporal_dim ∧
 164    (Finset.univ.filter (fun i : Fin 4 => 0 < η i i)).card = spatial_dim :=

proof body

Tactic-mode proof.

 165  ⟨negative_eigenvalue_count, positive_eigenvalue_count⟩
 166
 167/-- The trace of the metric: Tr(η) = −1 + 1 + 1 + 1 = 2. -/
 168theorem η_trace : ∑ i : Fin 4, η i i = 2 := by
 169  simp only [Fin.sum_univ_four]; rw [η_00, η_11, η_22, η_33]; norm_num
 170
 171/-- The determinant of the metric: det(η) = −1. -/
 172theorem η_det : ∏ i : Fin 4, η i i = -1 := by
 173  simp only [Fin.prod_univ_four]; rw [η_00, η_11, η_22, η_33]; norm_num
 174
 175/-- Negative determinant confirms Lorentzian signature. -/

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