openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /FarDepthRegimeEventually.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.FarSourceScaleRegime | |
| namespace OAI | |
| /-! Recursive policies and quantitative depth estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4FullFar | |
| open Filter Real | |
| open scoped Topology | |
| theorem far_depth_regime_eventuallyND {σ : ℝ} (hσ : 0<σ) (D : ℝ) : | |
| ∀ᶠ R : ℝ in atTop, ∀ (N L k : ℕ) (a v : ℝ), R=(2:ℝ)^N → | |
| a≤k → v-a≤N → | |
| (k:ℝ)≤v-2*(max L (Nat.ceil (10*σ*N)):ℕ)+D → | |
| (L:ℝ)≤(v-k)/2+σ*N ∧ 10*σ*N≤v-k ∧ v-k≤2*N := by | |
| filter_upwards [eventually_ge_atTop ((2:ℝ)^(D/σ))] with R hR | |
| intro N L k a v hRN hak hv hk | |
| have hDN : D≤σ*N := by | |
| rw [hRN,← Real.rpow_natCast (2:ℝ) N] at hR | |
| have hh := (Real.rpow_le_rpow_left_iff (by norm_num : (1:ℝ)<2)).mp hR | |
| exact (div_le_iff₀ hσ).mp hh |>.trans_eq (mul_comm _ _) | |
| have hL : (L:ℝ)≤(max L (Nat.ceil (10*σ*N)):ℕ) := by exact_mod_cast Nat.le_max_left L _ | |
| have hm : 10*σ*N≤(max L (Nat.ceil (10*σ*N)):ℕ) := | |
| (Nat.le_ceil (10*σ*(N:ℝ))).trans (by exact_mod_cast Nat.le_max_right L (Nat.ceil (10*σ*N))) | |
| have hnonneg : 0≤σ*(N:ℝ) := mul_nonneg hσ.le (Nat.cast_nonneg N) | |
| have hN : (0:ℝ)≤N := Nat.cast_nonneg N | |
| exact ⟨by linarith,by linarith,by linarith⟩ | |
| theorem far_source_point_regime_eventuallyND {σ : ℝ} (hσ : 0<σ) (hσ₁ : σ≤1) (D : ℝ) : | |
| ∀ᶠ R : ℝ in atTop, ∀ (N L k : ℕ) (a v dist : ℝ), R=(2:ℝ)^N → | |
| a≤k → v-a≤N → | |
| (k:ℝ)≤v-2*(max L (Nat.ceil (10*σ*N)):ℕ)+D → | |
| (2:ℝ)^(-(k:ℝ))/4≤dist → | |
| 1≤ shellEffectiveScaleND L R σ ∧ shellEffectiveScaleND L R σ≤R^2 ∧ | |
| R^σ≤(2:ℝ)^v*dist ∧ (shellEffectiveScaleND L R σ)^2/((2:ℝ)^v*dist)≤R^(-σ) := by | |
| filter_upwards [far_depth_regime_eventuallyND hσ D, | |
| far_source_scale_regime_eventuallyND hσ hσ₁ (by norm_num : (0:ℝ)<1/4)] with R hdepth hscale | |
| intro N L k a v dist hRN hak hv hk hdist | |
| obtain ⟨hl,hqlo,hqup⟩ := hdepth N L k a v hRN hak hv hk | |
| obtain ⟨hl',hqlo',hqup'⟩ := far_source_exponentsND hl hqlo hqup | |
| rw [← hRN] at hl' hqlo' hqup' | |
| have hΛ : (1/4:ℝ)*(2:ℝ)^(v-k)≤(2:ℝ)^v*dist := by | |
| calc | |
| _ = (2:ℝ)^v*((2:ℝ)^(-(k:ℝ))/4) := by | |
| rw [Real.rpow_sub (by norm_num : (0:ℝ)<2),Real.rpow_neg (by norm_num : (0:ℝ)≤2)] | |
| ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left hdist (by positivity) | |
| obtain ⟨h₁,h₂,h₃,h₄,_⟩ := hscale L (v-k) ((2:ℝ)^v*dist) hl' hqlo' hqup' hΛ | |
| exact ⟨h₁,h₂,h₃,h₄⟩ | |
| end Attempt4FullFar | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |