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theorem proved tactic proof

two_pow_neg22_in_interval

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formal statement (Lean)

 207theorem two_pow_neg22_in_interval : two_pow_neg22_interval.contains ((2 : ℝ) ^ (-22 : ℤ)) := by

proof body

Tactic-mode proof.

 208  simp only [Interval.contains, two_pow_neg22_interval]
 209  -- 2^(-22) = 1/2^22 = 1/4194304
 210  have h : (2 : ℝ) ^ (-22 : ℤ) = 1 / 4194304 := by
 211    have h2 : (2 : ℝ) ^ (22 : ℤ) = 4194304 := by norm_num
 212    have h3 : (2 : ℝ) ^ (-22 : ℤ) = ((2 : ℝ) ^ (22 : ℤ))⁻¹ := by
 213      rw [zpow_neg]
 214    rw [h3, h2]
 215    norm_num
 216  rw [h]
 217  constructor
 218  · -- 238/1000000000 ≤ 1/4194304
 219    -- 238 * 4194304 ≤ 1000000000
 220    -- 998223552 ≤ 1000000000 ✓
 221    norm_num
 222  · -- 1/4194304 ≤ 239/1000000000
 223    -- 1000000000 ≤ 239 * 4194304
 224    -- 1000000000 ≤ 1002438656 ✓
 225    norm_num
 226
 227/-! ## Monotonicity Lemmas for φ^x -/
 228
 229/-- φ > 1, so φ^x is strictly increasing in x -/

depends on (10)

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