phi_lt_two

formal source

  70  linarith
  71
  72/-- φ < 2. -/
  73theorem phi_lt_two : φ < 2 := by
  74  simp only [φ]
  75  have h5 : Real.sqrt 5 < 3 := by
  76    have h9 : (5 : ℝ) < 9 := by norm_num
  77    have hsqrt9 : Real.sqrt 9 = 3 := by
  78      rw [show (9 : ℝ) = 3^2 by norm_num, Real.sqrt_sq (by norm_num : (3 : ℝ) ≥ 0)]
  79    calc Real.sqrt 5 < Real.sqrt 9 := Real.sqrt_lt_sqrt (by norm_num) h9
  80      _ = 3 := hsqrt9
  81  linarith
  82
  83/-- φ > 1.618. -/
  84theorem phi_gt_onePointSixOneEight : φ > (1.618 : ℝ) := by
  85  simp only [φ]
  86  have h5 : Real.sqrt 5 > (2.236 : ℝ) := by
  87    have h : (2.236 : ℝ)^2 < 5 := by norm_num
  88    rw [← Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2.236)]
  89    exact Real.sqrt_lt_sqrt (by norm_num) h
  90  linarith
  91
  92/-- φ < 1.619. -/
  93theorem phi_lt_onePointSixOneNine : φ < (1.619 : ℝ) := by
  94  simp only [φ]
  95  have h5 : Real.sqrt 5 < (2.238 : ℝ) := by
  96    have h : (5 : ℝ) < (2.238 : ℝ)^2 := by norm_num
  97    rw [← Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2.238)]
  98    exact Real.sqrt_lt_sqrt (by norm_num) h
  99  linarith
 100
 101/-- φ < 1.8. -/
 102theorem phi_lt_onePointEight : φ < (1.8 : ℝ) :=
 103  lt_trans phi_lt_onePointSixOneNine (by norm_num)