| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.RealContinuation |
|
|
| 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 |
|
|
| def SuccessConstant : ℝ := 6*StepConstant |
| lemma successConstant_one : 1 ≤ SuccessConstant := by |
| have h := stepConstant_one |
| dsimp [SuccessConstant] |
| linarith |
|
|
| lemma stopping_factor_bound {d : ℕ} (hd : 16 ≤ d) (T : ℕ) : |
| (T+1:ℝ) ≤ ((2:ℝ)^d)^BlockCount T (d/8) := by |
| have hT := blockCount_mul_ge T (by omega : 0 < d/8) |
| have hp : d/8 ≤ d := Nat.div_le_self _ _ |
| have hTd : T ≤ d * BlockCount T (d/8) := by nlinarith |
| have htwo : (T+1:ℕ) ≤ 2^T := Nat.lt_two_pow_self |
| have hpow : (2:ℝ)^T ≤ 2^(d * BlockCount T (d/8)) := |
| pow_le_pow_right₀ (by norm_num) hTd |
| rw [pow_mul] at hpow |
| exact (by exact_mod_cast htwo : (T+1:ℝ) ≤ (2:ℝ)^T).trans hpow |
|
|
| lemma stopping_step_bound {d N K : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N) (hNK : N ≤ K) |
| (T : ℕ) : |
| (T+1:ℝ) * RealStep d N ^ BlockCount T (d/8) ≤ |
| ((2*StepConstant)^d * (K:ℝ)^(1/((d/8:ℕ):ℝ))) ^ BlockCount T (d/8) := by |
| have hSp := stepConstant_pos |
| have hR := realStep_pos d hN |
| have hn0 : (0:ℝ) ≤ N := by positivity |
| have hroot : (N:ℝ)^(1/((d/8:ℕ):ℝ)) ≤ (K:ℝ)^(1/((d/8:ℕ):ℝ)) := |
| Real.rpow_le_rpow hn0 (by exact_mod_cast hNK) (by positivity) |
| calc |
| _ ≤ ((2:ℝ)^d)^BlockCount T (d/8) * RealStep d N ^ BlockCount T (d/8) := |
| mul_le_mul_of_nonneg_right (stopping_factor_bound hd T) (by positivity) |
| _ = ((2*StepConstant)^d * (N:ℝ)^(1/((d/8:ℕ):ℝ))) ^ BlockCount T (d/8) := by |
| rw [← mul_pow] |
| congr 1 |
| simp only [RealStep, mul_pow] |
| ring |
| _ ≤ _ := pow_le_pow_left₀ (by positivity) |
| (mul_le_mul_of_nonneg_left hroot (by positivity)) _ |
|
|
| lemma stopping_step_bound_three {d N K T : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N) |
| (hNK : N ≤ K) (hT : 0 < T) : |
| 3 * ((T+1:ℝ) * RealStep d N ^ BlockCount T (d/8)) ≤ |
| (SuccessConstant^d * (K:ℝ)^(1/((d/8:ℕ):ℝ))) ^ BlockCount T (d/8) := by |
| have hSp := stepConstant_pos |
| have hR := realStep_pos d hN |
| have hb := blockCount_pos hT (by omega : 0 < d/8) |
| have h3 : (3:ℝ) ≤ ((3:ℝ)^d)^BlockCount T (d/8) := by |
| rw [← pow_mul] |
| have hdb : 1 ≤ d * BlockCount T (d/8) := Nat.mul_pos (by omega) hb |
| simpa using pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 3) hdb |
| have hroot0 : 0 ≤ (K:ℝ)^(1/((d/8:ℕ):ℝ)) := by positivity |
| calc |
| _ ≤ ((3:ℝ)^d)^BlockCount T (d/8) * |
| ((2*StepConstant)^d * (K:ℝ)^(1/((d/8:ℕ):ℝ))) ^ BlockCount T (d/8) := |
| mul_le_mul h3 (stopping_step_bound hd hN hNK T) |
| (by positivity) (by positivity) |
| _ = _ := by |
| rw [← mul_pow] |
| congr 1 |
| rw [← mul_assoc, ← mul_pow] |
| congr 2 |
| dsimp [SuccessConstant] |
| ring |
|
|
| lemma cap_constant_le {d : ℕ} (hd : 1 ≤ d) : (3*4^d:ℝ) ≤ 12^d := by |
| have h3 : (3:ℝ) ≤ 3^d := by |
| simpa using pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 3) hd |
| calc |
| _ ≤ 3^d*4^d := mul_le_mul_of_nonneg_right h3 (by positivity) |
| _ = _ := by rw [← mul_pow]; norm_num |
|
|
| lemma terminal_angular_zero_bound {d : ℕ} (hd : 16 ≤ d) {ε : ℝ} |
| (hε : 0 < ε) (hε1 : ε ≤ 1/10) : |
| (3*4^d:ℝ)*ε^(d-1) ≤ 4^d*ε^(((d:ℝ)-1)/2) := by |
| have hd' : (16:ℝ) ≤ d := by exact_mod_cast hd |
| have hα : (1:ℝ) ≤ ((d:ℝ)-1)/2 := by linarith |
| have hεa : ε^(((d:ℝ)-1)/2) ≤ ε := by |
| simpa using Real.rpow_le_rpow_of_exponent_ge hε (by linarith : ε ≤ 1) hα |
| have hn : ((d-1:ℕ):ℝ) = (d:ℝ)-1 := by rw [Nat.cast_sub (by omega), Nat.cast_one] |
| have he : ε^(d-1) = ε^(((d:ℝ)-1)/2) * ε^(((d:ℝ)-1)/2) := by |
| rw [← Real.rpow_add hε, ← Real.rpow_natCast, hn] |
| congr 1 |
| ring |
| rw [he] |
| have haux : 3*ε^(((d:ℝ)-1)/2) ≤ 1 := by linarith |
| calc |
| _ = (4^d*ε^(((d:ℝ)-1)/2)) * (3*ε^(((d:ℝ)-1)/2)) := by ring |
| _ ≤ (4^d*ε^(((d:ℝ)-1)/2)) * 1 := mul_le_mul_of_nonneg_left haux (by positivity) |
| _ = _ := mul_one _ |
| |
| end MemoryPrecision |
| |
| |
| end |
| |
| end OAI |
| |
