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e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | import OAI.Combinatorics.EditApproximation.Geometry.Parameters
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! A computable fixed-accuracy cutoff. The algorithm tests the small-log
conditions directly; it does not construct the double exponential bound. -/
namespace EditApproximation
def accuracyCutoffExponent (epsilon : ℚ) : ℕ := max (2 ^ 1000) ⌈10 / epsilon⌉₊
def accuracyCutoff (epsilon : ℚ) : ℕ := 2 ^ (2 ^ accuracyCutoffExponent epsilon)
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
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
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
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⟩
end EditApproximation
end OAI
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