| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.KernelEllipsoid |
| import OAI.Probability.MemoryPrecision.StreamArithmetic |
| import OAI.Probability.MemoryPrecision.StreamZero |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
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| noncomputable section |
| universe u v |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision.Kernel |
| open MemoryPrecision Policy |
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| def SourceStreamStatement : Prop := |
| ∃ C : ℝ, 0 < C ∧ ∃ d₀ : ℕ, 0 < d₀ ∧ |
| ∀ d M T : ℕ, d₀ ≤ d → ∀ ε : ℝ, 0 < ε → ε ≤ 1/10 → |
| ∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ → |
| ∀ L : Ω → MemoryPrecision.Kernel.Policy d M T, |
| MemoryPrecision.Kernel.Policy.Admissible L ρ (MemoryPrecision.Kernel.UniformSphere d) → |
| MemoryPrecision.Kernel.Policy.Success L ρ (MemoryPrecision.Kernel.UniformSphere d) ε ≤ min 1 (ENNReal.ofReal |
| ((2*ε*Real.exp (C*(1+(M:ℝ)/(d:ℝ)^2)*(MemoryPrecision.Kernel.BlockCount T ((d-1)/8):ℝ)))^(((d:ℝ)-1)/2))) |
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| lemma source_stream_explicit : |
| ∀ d M T : ℕ, 32 ≤ d → ∀ ε : ℝ, 0 < ε → ε ≤ 1/10 → |
| ∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ → |
| ∀ L : Ω → Policy d M T, Admissible L ρ (UniformSphere d) → |
| Success L ρ (UniformSphere d) ε ≤ min 1 (ENNReal.ofReal |
| ((2*ε*Real.exp (StreamConstant*(1+(M:ℝ)/(d:ℝ)^2)*(BlockCount T ((d-1)/8):ℝ)))^(((d:ℝ)-1)/2))) := by |
| intro d M T hd ε hε hε1 Ω _ ρ hρ L hL |
| have : IsProbabilityMeasure ρ := hρ |
| let : NeZero d := ⟨by omega⟩ |
| refine le_min (Policy.success_le_one L ρ (UniformSphere d) ε) ?_ |
| apply (Policy.success_le_chord L ρ (UniformSphere d) ε).trans |
| by_cases hT : T = 0 |
| · subst T |
| simp only [BlockCount, Nat.ceil_zero, zero_div, Nat.cast_zero, mul_zero, Real.exp_zero, mul_one] |
| exact (Policy.chord_zero_bound L ρ hL (by omega) hε).trans |
| (ENNReal.ofReal_le_ofReal (zero_cap_precise hd hε hε1)) |
| · apply (chord_half_sphere_bound L ρ hL (by omega) hε hε1).trans |
| apply ENNReal.ofReal_le_ofReal |
| let a := ((d:ℝ)-1)/2 |
| let X := StreamConstant*(1+(M:ℝ)/(d:ℝ)^2)*(BlockCount T ((d-1)/8):ℝ) |
| have ha : 0 ≤ a := by |
| dsimp [a] |
| have : (32:ℝ) ≤ d := by exact_mod_cast hd |
| linarith |
| have he : Real.exp (a*X) = (Real.exp X)^a := by rw [← Real.exp_mul, mul_comm] |
| have h2 : ε^a ≤ (2*ε)^a := Real.rpow_le_rpow hε.le (by linarith) ha |
| calc |
| _ = ε^a * ((4:ℝ)^d * (SuccessConstant^d * |
| (((d/8+2)*2^M:ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^BlockCount T (d/8)) := by dsimp [a, BlockCount, MemoryPrecision.BlockCount]; ring |
| _ ≤ ε^a * Real.exp (a*X) := by |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| simpa only [a, X, BlockCount, MemoryPrecision.BlockCount, mul_assoc] using stream_full_factor hd (by omega : 0 < T) |
| _ ≤ (2*ε)^a * Real.exp (a*X) := mul_le_mul_of_nonneg_right h2 (by positivity) |
| _ = (2*ε*Real.exp X)^a := by |
| rw [he] |
| exact (Real.mul_rpow (by positivity : (0:ℝ) ≤ 2*ε) (Real.exp_pos X).le).symm |
| theorem source_stream_success : SourceStreamStatement := |
| ⟨StreamConstant, streamConstant_pos, 32, by norm_num, source_stream_explicit⟩ |
| end MemoryPrecision.Kernel |
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| end |
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| end OAI |
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