`planckArea_pos`

proved

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module: IndisputableMonolith.Unification.RecognitionBandwidth
domain: Unification
line: 78 · github
papers citing: none yet

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IndisputableMonolith.Unification.RecognitionBandwidth on GitHub at line 78.

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formal source

  75  area / (4 * planckArea * k_R * eightTickCadence)
  76
  77/-- Planck area is positive. -/
  78theorem planckArea_pos : 0 < planckArea := by
  79  unfold planckArea planckLength
  80  positivity
  81
  82/-- The denominator of the bandwidth formula is positive. -/
  83theorem bandwidth_denom_pos : 0 < 4 * planckArea * k_R * eightTickCadence := by
  84  apply mul_pos
  85  apply mul_pos
  86  apply mul_pos
  87  · linarith [planckArea_pos]
  88  · exact planckArea_pos
  89  · exact k_R_pos
  90  · exact eightTickCadence_pos
  91
  92/-- Recognition bandwidth is positive for positive area. -/
  93theorem bandwidth_pos {A : ℝ} (hA : 0 < A) : 0 < bandwidth A :=
  94  div_pos hA bandwidth_denom_pos
  95
  96/-- Alias for bandwidth_pos. -/
  97theorem bandwidth_pos' {A : ℝ} (hA : 0 < A) : 0 < bandwidth A :=
  98  bandwidth_pos hA
  99
 100/-- Bandwidth is monotone in area: larger boundary → more throughput. -/
 101theorem bandwidth_monotone {A₁ A₂ : ℝ} (_h₁ : 0 < A₁) (h : A₁ ≤ A₂) :
 102    bandwidth A₁ ≤ bandwidth A₂ := by
 103  unfold bandwidth
 104  exact div_le_div_of_nonneg_right h (le_of_lt bandwidth_denom_pos)
 105
 106/-- Bandwidth scales linearly with area. -/
 107theorem bandwidth_linear (A c : ℝ) (_hc : 0 < c) :
 108    bandwidth (c * A) = c * bandwidth A := by

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