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theorem proved tactic proof

phi_in_observed_range

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formal statement (Lean)

 119theorem phi_in_observed_range : 0.5 < φ ∧ φ < 5 := by

proof body

Tactic-mode proof.

 120  constructor
 121  · -- 0.5 < φ: Since φ = (1 + √5)/2 and √5 > 0, we have φ > 0.5
 122    unfold φ Constants.phi
 123    have h_sqrt5_pos : 0 < Real.sqrt 5 := Real.sqrt_pos.mpr (by norm_num : (5 : ℝ) > 0)
 124    linarith
 125  · -- φ < 5: Since φ = (1 + √5)/2 and √5 < 3, we have φ < 2 < 5
 126    unfold φ Constants.phi
 127    have h_sqrt5_lt_3 : Real.sqrt 5 < 3 := by
 128      rw [show (3 : ℝ) = Real.sqrt 9 by norm_num]
 129      exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
 130    linarith
 131
 132/-- **THEOREM (RIGOROUS)**: φ is strictly between 1 and 2. -/

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