prove_upper_bound_le

formal source

  39    (h_contains : I.contains x) (h_lo : b ≤ I.lo) : (b : ℝ) ≤ x :=
  40  I.lo_ge_implies_contains_ge h_lo h_contains
  41
  42theorem prove_upper_bound_le {I : Interval} {x : ℝ} {b : ℚ}
  43    (h_contains : I.contains x) (h_hi : I.hi ≤ b) : x ≤ (b : ℝ) :=
  44  I.hi_le_implies_contains_le h_hi h_contains
  45
  46/-- Prove that φ is in its interval -/
  47theorem phi_interval_contains :
  48    phiInterval.contains ((1 + Real.sqrt 5) / 2) := phi_in_phiInterval
  49
  50/-- Prove that log(φ) is in its interval -/
  51theorem log_phi_interval_contains :
  52    logPhiInterval.contains (Real.log ((1 + Real.sqrt 5) / 2)) := log_phi_in_interval
  53
  54/-- Example: Prove log(φ) > 0.48 (using interval lo = 481/1000 = 0.481) -/
  55theorem log_phi_gt_048 : (0.48 : ℝ) < Real.log ((1 + Real.sqrt 5) / 2) := by
  56  have h := log_phi_in_interval
  57  -- logPhiInterval.lo = 481/1000 > 0.48
  58  have h1 : (0.48 : ℝ) < (481 / 1000 : ℚ) := by norm_num
  59  exact lt_of_lt_of_le h1 h.1
  60
  61/-- Example: Prove log(φ) < 0.49 (using interval hi = 482/1000 = 0.482) -/
  62theorem log_phi_lt_049 : Real.log ((1 + Real.sqrt 5) / 2) < (0.49 : ℝ) := by
  63  have h := log_phi_in_interval
  64  -- logPhiInterval.hi = 482/1000 < 0.49
  65  have h1 : ((482 / 1000 : ℚ) : ℝ) < (0.49 : ℝ) := by norm_num
  66  exact lt_of_le_of_lt h.2 h1
  67
  68/-- Example: Prove φ > 1.61 (using interval lo = 1618/1000) -/
  69theorem phi_gt_161 : (1.61 : ℝ) < (1 + Real.sqrt 5) / 2 := by
  70  have h := phi_in_phiInterval
  71  -- phiInterval.lo = 1618/1000 > 1.61
  72  have h1 : (1.61 : ℝ) < (1618 / 1000 : ℚ) := by norm_num