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theorem proved tactic proof

affine_log_collapses_to_gap

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formal statement (Lean)

 180theorem affine_log_collapses_to_gap
 181    {a c : ℝ}
 182    (h0 : gapAffineLogR a phi c 0 = 0)
 183    (h1 : gapAffineLogR a phi c 1 = 1) :
 184    ∀ Z : ℤ, gapAffineLog a phi c Z = RSBridge.gap Z := by

proof body

Tactic-mode proof.

 185  have hc : c = 0 := zero_normalization_forces_offset h0
 186  have ha : a = 1 / Real.log phi := unit_step_forces_log_scale h0 h1
 187  intro Z
 188  unfold gapAffineLog gapAffineLogR RSBridge.gap
 189  calc
 190    a * Real.log (1 + (Z : ℝ) / phi) + c
 191        = (1 / Real.log phi) * Real.log (1 + (Z : ℝ) / phi) := by
 192            simp [ha, hc]
 193    _ = Real.log (1 + (Z : ℝ) / phi) / Real.log phi := by
 194          simp [div_eq_mul_inv, mul_comm]
 195
 196/-- Three-point calibration gives direct collapse to the canonical gap. -/

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