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theorem proved tactic proof

born_rule_normalized

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formal statement (Lean)

 162theorem born_rule_normalized (C₁ C₂ : ℝ) (α₁ α₂ : ℂ)
 163  (h₁ : Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₁‖ ^ 2)
 164  (h₂ : Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₂‖ ^ 2) :
 165  ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2 = 1 := by

proof body

Tactic-mode proof.

 166  have hdenom : Real.exp (-C₁) + Real.exp (-C₂) ≠ 0 :=
 167    (add_pos (Real.exp_pos _) (Real.exp_pos _)).ne'
 168  calc ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2
 169      = Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) +
 170        Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) := by rw [← h₁, ← h₂]
 171      _ = (Real.exp (-C₁) + Real.exp (-C₂)) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
 172        simpa using
 173          (add_div (Real.exp (-C₁)) (Real.exp (-C₂)) (Real.exp (-C₁) + Real.exp (-C₂))).symm
 174      _ = 1 := div_self hdenom
 175
 176end Measurement
 177end IndisputableMonolith

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