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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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.CubeContinuation
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 cube_memory_ratio {d M : ℕ} (hd : 16 ≤ d) (hM : M ≤ d^2) :
(M:ℝ) / (((d/2)/4:ℕ):ℝ) ≤ 72 * (d/2:ℕ) := by
have hd3 : d ≤ 3*(d/2) := by omega
have hn8 : d/2 ≤ 8*((d/2)/4) := by omega
have hm : M ≤ (72*(d/2))*((d/2)/4) := by
calc
M ≤ d^2 := hM
_ ≤ (3*(d/2))^2 := Nat.pow_le_pow_left hd3 2
_ = 9*(d/2)*(d/2) := by ring
_ ≤ 9*(d/2)*(8*((d/2)/4)) := Nat.mul_le_mul_left _ hn8
_ = _ := by ring
apply (div_le_iff₀ (by exact_mod_cast (show 0 < (d/2)/4 by omega))).2
exact_mod_cast hm
lemma cube_memory_factor {d M : ℕ} (hd : 16 ≤ d) (hM : M ≤ d^2) :
((2^M:ℕ):ℝ)^(1/(((d/2)/4:ℕ):ℝ)) ≤ ((2:ℝ)^72)^(d/2) := by
have he : ((2^M:ℕ):ℝ)^(1/(((d/2)/4:ℕ):ℝ)) =
(2:ℝ)^((M:ℝ)/(((d/2)/4:ℕ):ℝ)) := by
rw [Nat.cast_pow, Nat.cast_ofNat, ← Real.rpow_natCast,
← Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2)]
congr 1
ring
rw [he, ← pow_mul, ← Real.rpow_natCast]
apply Real.rpow_le_rpow_of_exponent_le (by norm_num : (1:ℝ) ≤ 2)
simpa using cube_memory_ratio hd hM
lemma cube_stopping_factor {d : ℕ} (hd : 16 ≤ d) (T : ℕ) :
(T+1:ℝ) ≤ ((2:ℝ)^(d/2))^BlockCount T ((d/2)/4) := by
have hT := blockCount_mul_ge T (by omega : 0 < (d/2)/4)
have hp : (d/2)/4 ≤ d/2 := Nat.div_le_self _ _
have hTn : T ≤ (d/2) * BlockCount T ((d/2)/4) := by nlinarith
have htwo : (T+1:ℕ) ≤ 2^T := Nat.lt_two_pow_self
have hpow : (2:ℝ)^T ≤ 2^((d/2) * BlockCount T ((d/2)/4)) :=
pow_le_pow_right₀ (by norm_num) hTn
rw [pow_mul] at hpow
exact (by exact_mod_cast htwo : (T+1:ℝ) ≤ (2:ℝ)^T).trans hpow
def CubeSuccessConstant : ℝ := max 12 (2*CubeStepConstant*2^72)
lemma cubeSuccessConstant_twelve : 12 ≤ CubeSuccessConstant := le_max_left _ _
lemma cubeSuccessConstant_one : 1 ≤ CubeSuccessConstant :=
(by norm_num : (1:ℝ) ≤ 12).trans cubeSuccessConstant_twelve
lemma cubeSuccessConstant_exp : Real.exp 1 ≤ CubeSuccessConstant :=
Real.exp_one_lt_three.le.trans ((by norm_num : (3:ℝ) ≤ 12).trans cubeSuccessConstant_twelve)
lemma cubeSuccessConstant_pos : 0 < CubeSuccessConstant :=
lt_of_lt_of_le zero_lt_one cubeSuccessConstant_one
lemma cube_whole_factor {d M : ℕ} (hd : 16 ≤ d) (hM : M ≤ d^2) (T : ℕ) :
(T+1:ℝ) * ((12:ℝ)^(d/2) * CubeStep d (2^M) ^ BlockCount T ((d/2)/4)) ≤
CubeSuccessConstant^((d/2)*(BlockCount T ((d/2)/4)+1)) := by
have hC := cubeStepConstant_pos
have hD := cubeSuccessConstant_pos
have hS := cubeStep_pos d (show 1 ≤ 2^M by exact one_le_pow₀ (by norm_num))
have hmem := cube_memory_factor hd hM
have hstep : (2:ℝ)^(d/2) * CubeStep d (2^M) ≤ CubeSuccessConstant^(d/2) := by
calc
_ ≤ 2^(d/2) * (CubeStepConstant^(d/2) * ((2:ℝ)^72)^(d/2)) :=
mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hmem (by positivity)) (by positivity)
_ = (2*CubeStepConstant*2^72)^(d/2) := by simp only [mul_pow]; ring
_ ≤ _ := pow_le_pow_left₀ (by positivity) (le_max_right _ _) _
calc
_ = 12^(d/2) * ((T+1:ℝ) * CubeStep d (2^M) ^ BlockCount T ((d/2)/4)) := by ring
_ ≤ 12^(d/2) * (((2:ℝ)^(d/2))^BlockCount T ((d/2)/4) *
CubeStep d (2^M) ^ BlockCount T ((d/2)/4)) :=
mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right (cube_stopping_factor hd T) (by positivity))
(by positivity)
_ = 12^(d/2) * ((2:ℝ)^(d/2) * CubeStep d (2^M))^BlockCount T ((d/2)/4) := by rw [mul_pow]
_ ≤ CubeSuccessConstant^(d/2) * (CubeSuccessConstant^(d/2))^BlockCount T ((d/2)/4) :=
mul_le_mul (pow_le_pow_left₀ (by norm_num) cubeSuccessConstant_twelve _)
(pow_le_pow_left₀ (by positivity) hstep _) (by positivity) (by positivity)
_ = _ := by rw [pow_mul, pow_succ]; ring
end MemoryPrecision
end
end OAI
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