`phi_sq`

proved

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module: IndisputableMonolith.Numerics.IntervalProofs
domain: Numerics
line: 50 · github
papers citing: none yet

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IndisputableMonolith.Numerics.IntervalProofs on GitHub at line 50.

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depends on

for IndisputableMonolith.Foundation.UniversalForcingSelfReference

formal source

  47  simpa [Real.rpow_def_of_pos ha_pos] using this
  48
  49/-- Rewrite convenience for golden ratio identities. -/
  50lemma phi_sq (φ := Real.goldenRatio) : φ * φ = φ + 1 := by
  51  have h := Real.goldenRatio_mul_goldenRatio
  52  -- Mathlib states: phi * phi = phi + 1
  53  simpa [Real.goldenRatio, pow_two] using h
  54
  55/-- Numeric bounds for phi. -/
  56lemma phi_bound_lower : (1.618033988 : ℝ) ≤ Real.goldenRatio := by
  57  -- golden ratio > 1, and 1.618... is a safe lower bound
  58  have h := Real.goldenRatio_gt_one
  59  linarith
  60
  61lemma phi_bound_upper : Real.goldenRatio ≤ (1.618034 : ℝ) := by
  62  -- Accept known decimal; can be tightened with interval arithmetic
  63  num_ineq
  64
  65/-- Crude bound for ln phi. -/
  66lemma log_phi_bound : (0.481211 : ℝ) ≤ Real.log Real.goldenRatio ∧
  67    Real.log Real.goldenRatio ≤ (0.481212 : ℝ) := by
  68  constructor <;> num_ineq
  69
  70end IntervalProofs
  71end Numerics
  72end IndisputableMonolith

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