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a2ee19a | 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
lemma zero_cap_coefficient {d : ℕ} (hd : 32 ≤ d) :
(3*4^d:ℝ) ≤ (20:ℝ)^(((d:ℝ)-1)/2) := by
have hs : (9:ℝ)*16^d ≤ 20^(d-1) := by
obtain ⟨k,rfl⟩ := Nat.exists_eq_add_of_le hd
have he : 32+k-1 = 31+k := by omega
rw [he, pow_add, pow_add]
have hb : (9:ℝ)*16^32 ≤ 20^31 := by norm_num
calc
_ = ((9:ℝ)*16^32)*16^k := by ring
_ ≤ 20^31*20^k := mul_le_mul hb (pow_le_pow_left₀ (by norm_num) (by norm_num : (16:ℝ) ≤ 20) k)
(by positivity) (by positivity)
apply (sq_le_sq₀ (by positivity) (by positivity)).mp
have hl : (3*4^d:ℝ)^2 = 9*16^d := by
rw [mul_pow, ← pow_mul, Nat.mul_comm d 2, pow_mul]
norm_num
have hr : ((20:ℝ)^(((d:ℝ)-1)/2))^2 = 20^(d-1) := by
rw [← Real.rpow_natCast, ← Real.rpow_mul (by norm_num : (0:ℝ) ≤ 20)]
rw [← Real.rpow_natCast]
congr 1
rw [Nat.cast_sub (by omega), Nat.cast_one]
ring
rwa [hl,hr]
lemma zero_cap_precise {d : ℕ} (hd : 32 ≤ d) {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1/10) :
(3*4^d:ℝ)*ε^(d-1) ≤ (2*ε)^(((d:ℝ)-1)/2) := by
let a := ((d:ℝ)-1)/2
have ha : 0 ≤ a := by
dsimp [a]
have : (32:ℝ) ≤ d := by exact_mod_cast hd
linarith
have he : ε^(d-1) = ε^a*ε^a := by
rw [← Real.rpow_add hε, ← Real.rpow_natCast]
congr 1
dsimp [a]
rw [Nat.cast_sub (by omega), Nat.cast_one]
ring
have hsmall : (10*ε)^a ≤ 1 := Real.rpow_le_one (by positivity) (by linarith) ha
calc
_ ≤ (20:ℝ)^a * ε^(d-1) := mul_le_mul_of_nonneg_right (zero_cap_coefficient hd) (by positivity)
_ = (2*ε)^a*(10*ε)^a := by
rw [he, Real.mul_rpow (by norm_num : (0:ℝ) ≤ 2) hε.le,
Real.mul_rpow (by norm_num : (0:ℝ) ≤ 10) hε.le]
have h20 := Real.mul_rpow (z := a) (by norm_num : (0:ℝ) ≤ 2) (by norm_num : (0:ℝ) ≤ 10)
norm_num only [show (2:ℝ)*10=20 by norm_num] at h20
rw [h20]
ring
_ ≤ (2*ε)^a * 1 := mul_le_mul_of_nonneg_left hsmall (by positivity)
_ = _ := mul_one _
end MemoryPrecision
end
end OAI
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