| import OAI.Combinatorics.EditApproximation.Geometry.Parameters |
| import OAI.Combinatorics.EditApproximation.Definitions |
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| namespace OAI |
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| /-! A computable fixed-accuracy cutoff. The algorithm tests the small-log |
| conditions directly; it does not construct the double exponential bound. -/ |
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| namespace EditApproximation |
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| def accuracyCutoffExponent (epsilon : ℚ) : ℕ := max (2 ^ 1000) ⌈10 / epsilon⌉₊ |
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| def accuracyCutoff (epsilon : ℚ) : ℕ := 2 ^ (2 ^ accuracyCutoffExponent epsilon) |
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| theorem largeAccuracyInput_pos (N : ℕ) (epsilon : ℚ) (h : largeAccuracyInput N epsilon) : 0 < N := by |
| have htwo : 2 ≤ (2 : ℕ) ^ 1000 := by |
| simpa only [pow_one] using pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) (by decide : 1 ≤ 1000) |
| have hell : 2 ≤ inputSmallLog N := htwo.trans h.1 |
| by_contra hn |
| have hzero : N = 0 := Nat.eq_zero_of_not_pos hn |
| rw [hzero] at hell |
| norm_num [inputSmallLog, smallLogExponent, heightExponent] at hell |
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| theorem heightExponent_above_double_power (N exponent : ℕ) (h : 2 ^ (2 ^ exponent) ≤ N) : |
| exponent ≤ heightExponent N := by |
| have hfirst := Nat.clog_mono_right 2 (h.trans (Nat.le_add_right N 2)) |
| rw [Nat.clog_pow _ _ (by decide : 1 < (2 : ℕ))] at hfirst |
| have hsecond := Nat.clog_mono_right 2 hfirst |
| simpa only [Nat.clog_pow _ _ (by decide : 1 < (2 : ℕ)), heightExponent] using hsecond |
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| theorem accuracyCutoff_sufficient (N : ℕ) (epsilon : ℚ) (hepsilon : 0 < epsilon) |
| (hN : accuracyCutoff epsilon ≤ N) : largeAccuracyInput N epsilon := by |
| have hpositive : 0 < N := (Nat.two_pow_pos _).trans_le hN |
| have he : accuracyCutoffExponent epsilon ≤ inputSmallLog N := |
| (heightExponent_above_double_power N (accuracyCutoffExponent epsilon) hN).trans |
| (inputSmallLog_bounds N hpositive).1 |
| constructor |
| · exact (le_max_left _ _).trans he |
| · have hc : (10 / epsilon : ℚ) ≤ (inputSmallLog N : ℚ) := |
| (Nat.le_ceil _).trans (Nat.cast_le.mpr ((le_max_right _ _).trans he)) |
| exact (div_le_iff₀ hepsilon).mp hc |
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| theorem largeAccuracyInput_real (N : ℕ) (epsilon : ℚ) (h : largeAccuracyInput N epsilon) : |
| 2 ^ 1000 ≤ inputSmallLog N ∧ 10 ≤ (inputSmallLog N : ℝ) * (epsilon : ℝ) := |
| ⟨h.1, by exact_mod_cast h.2⟩ |
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| end EditApproximation |
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| end OAI |
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