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theorem proved tactic proof

born_rule_normalized

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formal statement (Lean)

  15theorem born_rule_normalized (C₁ C₂ : ℝ) (α₁ α₂ : ℂ)
  16  (h₁ : Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₁‖ ^ 2)
  17  (h₂ : Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₂‖ ^ 2) :
  18  ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2 = 1 := by

proof body

Tactic-mode proof.

  19  have hdenom : Real.exp (-C₁) + Real.exp (-C₂) ≠ 0 :=
  20    (add_pos (Real.exp_pos _) (Real.exp_pos _)).ne'
  21  calc ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2
  22      = Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) +
  23        Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
  24          rw [← h₁, ← h₂]
  25      _ = (Real.exp (-C₁) + Real.exp (-C₂)) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
  26          rw [div_add_div_same]
  27      _ = 1 := div_self hdenom
  28
  29end Measurement
  30end IndisputableMonolith

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