| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.GridOccupation |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BoundedContinuousFunction |
| namespace DirectionalTransience |
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| def heightTubeEndpointEvent (Y : ℕ → ℝ) (H : ℕ) (L R : ℝ) : Prop := |
| (∀ j ≤ H, |Y j| ≤ L) ∧ |Y H| ≤ R |
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| lemma measurableSet_heightTubeEndpointEvent {Ω : Type*} [MeasurableSpace Ω] |
| (Y : ℕ → Ω → ℝ) (hY : ∀ j, Measurable (Y j)) (H : ℕ) (L R : ℝ) : |
| MeasurableSet {x | heightTubeEndpointEvent (fun j => Y j x) H L R} := by |
| change MeasurableSet ({x | ∀ j ≤ H, |Y j x| ≤ L} ∩ {x | |Y H x| ≤ R}) |
| apply MeasurableSet.inter |
| · simp only [Set.ofPred_forall] |
| exact MeasurableSet.iInter fun j => MeasurableSet.iInter fun _ => |
| measurableSet_le (hY j).abs measurable_const |
| · exact measurableSet_le (hY H).abs measurable_const |
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| lemma heightPolygon_window_implies_tube (Y : ℕ → ℝ) (r n : ℝ) (hr : 0 < r) (hn : 0 < n) |
| (B L ρ δ : ℝ) (hBL : B ≤ L) (τ : unitInterval) |
| (hg : ‖heightPolygon Y r n 2‖ < B) |
| (he : ∀ t : unitInterval, dist t τ ≤ δ → |heightPolygon Y r n 2 t| < ρ) |
| (ht : dist (layerGridPoint (2*n) ⌊n⌋₊) τ ≤ δ) : |
| heightTubeEndpointEvent Y ⌊n⌋₊ (L*r) (ρ*r) := by |
| have h2n : 0 < 2*n := by positivity |
| have hj (j : ℕ) (h : j ≤ ⌊n⌋₊) : (j:ℝ) ≤ 2*n := by |
| have h' := (Nat.cast_le (α := ℝ)).mpr h |
| have hfloor := Nat.floor_le hn.le |
| linarith |
| have heval (j : ℕ) (h : j ≤ ⌊n⌋₊) : heightPolygon Y r n 2 (layerGridPoint (2*n) j)=Y j/r := by |
| apply heightPolygon_grid _ _ _ _ h2n.le |
| rw [layerGridPoint_coe h2n (hj j h),mul_div_cancel₀ _ h2n.ne'] |
| constructor |
| · intro j h |
| have hh := (heightPolygon Y r n 2).norm_coe_le_norm (layerGridPoint (2*n) j) |
| rw [heval j h,Real.norm_eq_abs,abs_div,abs_of_pos hr] at hh |
| exact ((div_le_iff₀ hr).mp (hh.trans (hg.le.trans hBL))) |
| · have hh := he _ ht |
| rw [heval ⌊n⌋₊ le_rfl,abs_div,abs_of_pos hr] at hh |
| exact ((div_lt_iff₀ hr).mp hh).le |
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| lemma half_layer_grid_tendsto (n : ℕ → ℝ) (hn : Tendsto n atTop atTop) : |
| Tendsto (fun i => layerGridPoint (2*n i) ⌊n i⌋₊) atTop |
| (𝓝 (⟨(1:ℝ)/2,by norm_num,by norm_num⟩ : unitInterval)) := by |
| apply layerGridPoint_tendsto |
| have hh := ((tendsto_nat_floor_mul_div_atTop (by norm_num : (0:ℝ)≤1)).comp hn).div_const 2 |
| convert hh using 1 |
| simp only [one_mul,Function.comp_def] |
| funext i |
| ring |
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| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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