import Mathlib import OAI.Probability.SKGap.Localization.DimensionDecay namespace OAI section open scoped BigOperators open scoped BigOperators open scoped BigOperators open scoped BigOperators open scoped BigOperators open scoped BigOperators NNReal open MeasureTheory ProbabilityTheory open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal open MeasureTheory ProbabilityTheory open scoped BigOperators NNReal ENNReal open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal ENNReal open MeasureTheory ProbabilityTheory open scoped BigOperators Matrix Matrix.Norms.Elementwise open scoped BigOperators open MeasureTheory ProbabilityTheory open scoped BigOperators Matrix Matrix.Norms.Elementwise open scoped BigOperators open scoped BigOperators NNReal ENNReal open MeasureTheory Metric Set open scoped BigOperators NNReal ENNReal open MeasureTheory ProbabilityTheory Filter Set open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator open MeasureTheory ProbabilityTheory Filter Set open scoped BigOperators Matrix.Norms.L2Operator open MeasureTheory ProbabilityTheory Filter Set open scoped BigOperators Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Set open MeasureTheory ProbabilityTheory Filter open scoped BigOperators ENNReal NNReal open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal ENNReal Matrix open MeasureTheory ProbabilityTheory Filter open scoped BigOperators ENNReal NNReal open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal ENNReal open scoped BigOperators open MeasureTheory ProbabilityTheory open scoped BigOperators Matrix Matrix.Norms.Elementwise NNReal ENNReal open scoped BigOperators open Filter Topology open MeasureTheory ProbabilityTheory Filter open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open Matrix open scoped NNReal ENNReal BigOperators Topology Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal ENNReal Topology open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal ENNReal Topology open MeasureTheory ProbabilityTheory Filter open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal Topology BigOperators open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal Topology BigOperators open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal Topology BigOperators open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal Topology BigOperators open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Set Matrix open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Matrix open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise open Filter MeasureTheory ProbabilityTheory open scoped Topology NNReal ENNReal open Filter MeasureTheory ProbabilityTheory open scoped Topology NNReal ENNReal open MeasureTheory Filter open scoped Topology NNReal ENNReal open MeasureTheory Filter ProbabilityTheory open scoped Topology NNReal ENNReal open MeasureTheory Filter open scoped Topology open MeasureTheory Filter ProbabilityTheory open scoped Topology NNReal ENNReal open MeasureTheory Filter ProbabilityTheory open scoped Topology NNReal ENNReal open MeasureTheory Filter ProbabilityTheory open scoped Topology NNReal ENNReal open MeasureTheory Filter ProbabilityTheory open scoped Topology NNReal ENNReal open MeasureTheory Filter ProbabilityTheory ContinuousLinearMap open scoped Topology NNReal ENNReal open Filter MeasureTheory ProbabilityTheory open scoped Topology NNReal ENNReal open MeasureTheory Filter open scoped BigOperators Topology open MeasureTheory Filter open scoped BigOperators Topology open MeasureTheory Filter open scoped BigOperators Topology open MeasureTheory Filter open scoped BigOperators Topology namespace SKGapCutoff lemma dimensionDecay_pos (a : ℝ) (n : ℕ) : 0 < dimensionDecay a n := Real.exp_pos _ lemma dimensionDecay_le_one {a : ℝ} (ha : 0 ≤ a) {n : ℕ} (hn : 0 < n) : dimensionDecay a n ≤ 1 := by apply Real.exp_le_one_iff.mpr have hlog := Real.log_nonneg (show (1:ℝ) ≤ n by exact_mod_cast hn) nlinarith lemma dimensionDecay_mul (a b : ℝ) (n : ℕ) : dimensionDecay a n * dimensionDecay b n = dimensionDecay (a+b) n := by unfold dimensionDecay rw [← Real.exp_add] congr 1 ring lemma nat_mul_dimensionDecay {n : ℕ} (hn : 0 < n) (a : ℝ) : (n:ℝ)*dimensionDecay a n = dimensionDecay (a-1) n := by simpa only [show 1+(a-1) = a by ring, dimensionDecay] using mul_exp_log_identity hn (a-1) lemma nat_sq_mul_dimensionDecay {n : ℕ} (hn : 0 < n) (a : ℝ) : (n:ℝ)^2*dimensionDecay a n = dimensionDecay (a-2) n := by rw [pow_two, mul_assoc, nat_mul_dimensionDecay hn, nat_mul_dimensionDecay hn] congr 1 ring lemma exp_cutoffTime_eq_dimensionDecay {rate : ℝ} (hrate : rate ≠ 0) (ρ c : ℝ) (n : ℕ) : Real.exp (-2*ρ*(c*cutoffTime rate n)) = dimensionDecay (c*ρ/rate) n := by unfold cutoffTime dimensionDecay congr 1 field_simp [hrate] lemma dimensionDecay_mul_cutoffTime_tendsto {q rate : ℝ} (hq : 0 < q) : Tendsto (fun n => dimensionDecay q n*cutoffTime rate n) atTop (𝓝 0) := by have hh := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 q hq).comp (Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop) simp only [Function.comp_def, Real.rpow_one] at hh have hh' := hh.div_const (2*rate) convert hh' using 1 · funext n unfold dimensionDecay cutoffTime ring · simp lemma linear_mean_error_tendsto {C α q : ℝ} (hα : 1 < α) (hq : 1 < q) : Tendsto (fun n => 2*Real.sqrt (C*dimensionDecay (α-1) n+C*dimensionDecay (q-1) n)) atTop (𝓝 0) := by have hh := ((dimensionDecay_tendsto (α-1) (by linarith)).const_mul C).add ((dimensionDecay_tendsto (q-1) (by linarith)).const_mul C) simpa using hh.sqrt.const_mul 2 lemma upper_polynomial_error_tendsto {C M θ m α δ q : ℝ} (hexp : 1+2*δ < α) (hα : 1 < α) (htail : 2 < θ*δ) (hq : 1 < q) : Tendsto (fun n => Real.sqrt (C/4*dimensionDecay (α-1) n + C/4*dimensionDecay (α-1-2*δ) n + 2*C*M*Real.exp (θ*m)*dimensionDecay (θ*δ-2) n + C*dimensionDecay (q-1) n)) atTop (𝓝 0) := by have h1 := (dimensionDecay_tendsto (α-1) (by linarith)).const_mul (C/4) have h2 := (dimensionDecay_tendsto (α-1-2*δ) (by linarith)).const_mul (C/4) have h3 := (dimensionDecay_tendsto (θ*δ-2) (by linarith)).const_mul (2*C*M*Real.exp (θ*m)) have h4 := (dimensionDecay_tendsto (q-1) (by linarith)).const_mul C simpa using ((h1.add h2).add h3 |>.add h4).sqrt end SKGapCutoff open MeasureTheory Filter open scoped BigOperators Topology end end OAI