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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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import OAI.Combinatorics.EditApproximation.Geometry.GroupNumerical
import OAI.Combinatorics.EditApproximation.Geometry.BandGroupSampling
namespace OAI
/-! Computed parameters discharge the group scale-count and H^-4 tail bounds. -/
namespace EditApproximation
theorem computed_group_scale_count (N eb : ℕ) (hN : 0 < N) (hell : 8 ≤ inputSmallLog N) :
(groupDyadicScales (integerParameters N).M (2 ^ eb) (inputSmallLog N)).length ≤
2 * inputSmallLog N := by
have hlogs := (inputSmallLog_bounds N hN).1
have hsmall : smallLogExponent N ≤ inputSmallLog N :=
(smallLogExponent N).lt_two_pow_self.le
have hdiv := Nat.div_le_div_right (c := 20) hlogs
change (groupDyadicScales (2 ^ (heightExponent N / 20)) (2 ^ eb)
((2 ^ smallLogExponent N : ℕ) : ℚ)).length ≤ _
rw [Nat.cast_pow, Nat.cast_ofNat, groupDyadicScales_length_dyadic]
omega
theorem computed_height_exp_upper (N : ℕ) (hN : 0 < N) :
(inputHeight N : ℝ) ≤ Real.exp (inputSmallLog N) := by
change ((2 ^ heightExponent N : ℕ) : ℝ) ≤ _
rw [Nat.cast_pow, Nat.cast_ofNat]
have htwo : (2 : ℝ) ≤ Real.exp 1 := by
simpa only [one_add_one_eq_two] using Real.add_one_le_exp (1 : ℝ)
have hpower := pow_le_pow_left₀ (by norm_num : (0 : ℝ) ≤ 2) htwo (heightExponent N)
rw [← Real.exp_nat_mul, mul_one] at hpower
have hlogs : (heightExponent N : ℝ) ≤ inputSmallLog N := by
exact_mod_cast (inputSmallLog_bounds N hN).1
exact hpower.trans (Real.exp_le_exp.mpr hlogs)
theorem computed_group_member_exp_bound (N : ℕ) (hN : 0 < N)
(hell : 1764 ≤ inputSmallLog N) :
(((integerParameters N).P + 1 : ℕ) : ℝ) ^ 2 ≤
4 * Real.exp (2 * (inputSmallLog N : ℝ) ^ 2) := by
let ell := inputSmallLog N
let p := coarseExponent ell
have hp := coarseExponent_shift_square ell hell
have hp0 : (0 : ℝ) ≤ p := Nat.cast_nonneg _
have hpell : (p : ℝ) + 20 ≤ ell := by
change ((p : ℝ) + 21) ^ 2 ≤ ell at hp
nlinarith only [hp, hp0]
have hell0 : (0 : ℝ) ≤ ell := Nat.cast_nonneg _
have hP : ((integerParameters N).P : ℝ) ≤ Real.exp ((ell : ℝ) ^ 2) := by
change ((inputHeight N ^ (p + 20) : ℕ) : ℝ) ≤ _
rw [Nat.cast_pow]
have h := pow_le_pow_left₀ (Nat.cast_nonneg (inputHeight N))
(computed_height_exp_upper N hN) (p + 20)
rw [← Real.exp_nat_mul] at h
have hexponent : ((p + 20 : ℕ) : ℝ) * ell ≤ (ell : ℝ) ^ 2 := by
push_cast
nlinarith only [hpell, hell0]
exact h.trans (Real.exp_le_exp.mpr hexponent)
have hPone : (1 : ℝ) ≤ (integerParameters N).P := by
have hNat : 1 ≤ (integerParameters N).P :=
one_le_pow₀ (Nat.one_le_two_pow (n := heightExponent N))
exact_mod_cast hNat
have hplus : (((integerParameters N).P + 1 : ℕ) : ℝ) ≤ 2 * Real.exp ((ell : ℝ) ^ 2) := by
push_cast
linarith only [hP, hPone]
have hsquare := pow_le_pow_left₀ (Nat.cast_nonneg ((integerParameters N).P + 1)) hplus 2
rw [mul_pow, ← Real.exp_nat_mul] at hsquare
norm_num only [Nat.cast_ofNat] at hsquare
exact hsquare
theorem computed_group_union_failure (N eb members : ℕ) (hN : 0 < N)
(hell : 1764 ≤ inputSmallLog N)
(hmembers : members ≤ ((integerParameters N).P + 1) ^ 2) :
let ell := inputSmallLog N
let T := (groupDyadicScales (integerParameters N).M (2 ^ eb) ell).length
(members : ℝ) * T *
(2 * Real.exp (-((ell : ℝ) ^ 80 * (1 / (2 * (ell : ℝ) ^ 21)) ^ 2 /
(512 * ell + (4 / 3 : ℝ) * (1 / (2 * (ell : ℝ) ^ 21)))))) ≤
1 / (inputHeight N : ℝ) ^ 4 := by
dsimp only
let ell := inputSmallLog N
let T := (groupDyadicScales (integerParameters N).M (2 ^ eb) ell).length
have hellR : (8 : ℝ) ≤ ell := by exact_mod_cast (show 8 ≤ ell from by omega)
have hcount : (members : ℝ) ≤ 4 * Real.exp (2 * (ell : ℝ) ^ 2) := by
have hcast : (members : ℝ) ≤ (((integerParameters N).P + 1 : ℕ) : ℝ) ^ 2 := by
exact_mod_cast hmembers
exact hcast.trans (computed_group_member_exp_bound N hN hell)
have hscale : (T : ℝ) ≤ 2 * ell := by
exact_mod_cast computed_group_scale_count N eb hN (by omega)
have hnum := group_union_failure_numeric hellR
(by exact_mod_cast Nat.two_pow_pos (heightExponent N) : (0 : ℝ) < inputHeight N)
(computed_height_exp_upper N hN) (Nat.cast_nonneg members) hcount (Nat.cast_nonneg T) hscale
have hexponent := group_normalized_exponent (show (1 : ℝ) ≤ ell from by linarith only [hellR])
have hexp : Real.exp (-((ell : ℝ) ^ 80 * (1 / (2 * (ell : ℝ) ^ 21)) ^ 2 /
(512 * ell + (4 / 3 : ℝ) * (1 / (2 * (ell : ℝ) ^ 21))))) ≤
Real.exp (-((ell : ℝ) ^ 37 / 3000)) :=
Real.exp_le_exp.mpr (neg_le_neg hexponent)
exact (mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hexp (by norm_num : (0 : ℝ) ≤ 2))
(mul_nonneg (Nat.cast_nonneg members) (Nat.cast_nonneg T))).trans hnum
end EditApproximation
end OAI
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