| import Mathlib.Analysis.Complex.Basic |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Topology.MetricSpace.Pseudo.Defs |
|
|
| namespace OAI |
|
|
| namespace Generator |
|
|
| /-! Scalar error budgets and lower bounds for perturbation correlation. -/ |
|
|
| theorem exists_correlation_parameter {s : ℝ} (hs : 0 < s) : |
| ∃ α : ℝ, 0 < α ∧ α < 1 / 4 ∧ α < s / 32 := by |
| refine ⟨min (1 / 8) (s / 64), lt_min (by norm_num) (by positivity), ?_, ?_⟩ |
| · exact lt_of_le_of_lt (min_le_left _ _) (by norm_num) |
| · exact lt_of_le_of_lt (min_le_right _ _) (by linarith) |
|
|
| /-- The strict nonlinear error budget in the manuscript, with its original constants. -/ |
| theorem correlation_error_budget {s α : ℝ} (hs : 0 < s) (hα : 0 < α) |
| (hquarter : α < 1 / 4) (hsmall : α < s / 32) : |
| s * (4 * α ^ 2 / (1 - 2 * α)) < α * s ^ 2 / 4 := by |
| have hden : 0 < 1 - 2 * α := by linarith |
| have hnum : 0 < 4 * α ^ 2 := by positivity |
| have hquot : 4 * α ^ 2 / (1 - 2 * α) < 8 * α ^ 2 := by |
| apply (div_lt_iff₀ hden).mpr |
| nlinarith [mul_pos hnum (show 0 < 1 - 4 * α by linarith)] |
| have hfirst := mul_lt_mul_of_pos_left hquot hs |
| have hsecond : 8 * s * α ^ 2 < α * s ^ 2 / 4 := by |
| have hp : 0 < α * s := mul_pos hα hs |
| nlinarith [mul_pos hp (show 0 < s - 32 * α by linarith)] |
| nlinarith |
|
|
| /-- Triangle-inequality step converting the correlation error into a lower bound. -/ |
| theorem correlation_lower_bound {s α δ : ℝ} (hs : 0 < s) (hα : 0 < α) |
| (hquarter : α < 1 / 4) (hsmall : α < s / 32) {c : ℂ} |
| (hc : ‖c - ((α * s ^ 2 / 2 : ℝ) : ℂ)‖ ≤ |
| s * (4 * α ^ 2 / (1 - 2 * α)) + δ) : |
| α * s ^ 2 / 4 - δ ≤ ‖c‖ := by |
| have he := correlation_error_budget hs hα hquarter hsmall |
| have hcenter : 0 ≤ α * s ^ 2 / 2 := by positivity |
| have ht := norm_sub_le c (c - ((α * s ^ 2 / 2 : ℝ) : ℂ)) |
| rw [sub_sub_cancel, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hcenter] at ht |
| linarith |
|
|
| /-! The numerical and topological contradiction for a continuous correlated sequence. -/ |
| open Filter Topology |
|
|
| theorem no_continuous_correlated_sequence {X : Type*} [TopologicalSpace X] |
| (F : X → ℝ) {x : X} (hF : ContinuousAt F x) (hzero : F x = 0) |
| (u : ℕ → X) (hu : Tendsto u atTop (𝓝 x)) |
| (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) (c : ℕ → ℂ) |
| {s α : ℝ} (hs : 0 < s) (hα : 0 < α) (hq : α < 1 / 4) (hsmall : α < s / 32) |
| (hc : ∀ n, ‖c n - ((α * s ^ 2 / 2 : ℝ) : ℂ)‖ ≤ |
| s * (4 * α ^ 2 / (1 - 2 * α)) + δ n) |
| (hproj : ∀ n, ‖c n‖ ≤ F (u n)) : False := by |
| have hFu : Tendsto (fun n => F (u n)) atTop (𝓝 0) := by |
| simpa only [hzero, Function.comp_def] using hF.tendsto.comp hu |
| have hpos : 0 < α * s ^ 2 / 8 := by positivity |
| have hnearF := Metric.tendsto_nhds.mp hFu (α * s ^ 2 / 8) hpos |
| have hnearδ := Metric.tendsto_nhds.mp hδ (α * s ^ 2 / 8) hpos |
| obtain ⟨n, hnF, hnδ⟩ := (hnearF.and hnearδ).exists |
| rw [Real.dist_eq, sub_zero] at hnF hnδ |
| have hlow := correlation_lower_bound hs hα hq hsmall (hc n) |
| have hupp := hproj n |
| have hFlt := (abs_lt.mp hnF).2 |
| have hδlt := (abs_lt.mp hnδ).2 |
| linarith |
|
|
| end Generator |
|
|
| end OAI |
|
|
