| import Mathlib |
| import OAI.Probability.Ballisticity.Geometry.HeightIncrement |
| import OAI.Probability.Ballisticity.Estimates.GridLinear |
|
|
| namespace OAI |
|
|
| section |
| section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| noncomputable def heightPolygon (F : ℕ → ℝ) (r n T : ℝ) : C(unitInterval,ℝ) := |
| ContinuousMap.const _ (F 0/r) + scaledPolygon (heightIncrement F) r n T |
|
|
| lemma measurable_heightPolygon {Ω : Type*} [MeasurableSpace Ω] |
| (F : ℕ → Ω → ℝ) (hF : ∀ k, Measurable (F k)) (r n T : ℝ) : |
| Measurable (fun ω => heightPolygon (fun k => F k ω) r n T) := by |
| exact (ContinuousMap.continuous_const'.measurable.comp ((hF 0).div_const _)).add |
| (measurable_scaledPolygon _ (measurable_heightIncrement F hF) _ _ _) |
| |
| lemma heightPolygon_formula (F : ℕ → ℝ) (r n T : ℝ) (h : 0 ≤ T*n) (t : unitInterval) : |
| heightPolygon F r n T t = |
| (F ⌊T*n*(t:ℝ)⌋₊ + (T*n*(t:ℝ)-⌊T*n*(t:ℝ)⌋₊)* |
| (F (⌊T*n*(t:ℝ)⌋₊+1)-F ⌊T*n*(t:ℝ)⌋₊))/r := by |
| have ht : 0 ≤ T*n*(t:ℝ) := mul_nonneg h t.2.1 |
| have hN : ⌊T*n*(t:ℝ)⌋₊ < ⌊T*n⌋₊+1 := |
| Nat.lt_succ_of_le (Nat.floor_mono (mul_le_of_le_one_right h t.2.2)) |
| simp only [heightPolygon,ContinuousMap.add_apply,ContinuousMap.const_apply,scaledPolygon_apply, |
| linearPolygon_formula _ _ ht hN,realPartialSum_heightIncrement,heightIncrement] |
| ring |
| |
| lemma heightPolygon_grid (F : ℕ → ℝ) (r n T : ℝ) (h : 0 ≤ T*n) |
| (j : ℕ) (t : unitInterval) (ht : T*n*(t:ℝ) = j) : |
| heightPolygon F r n T t = F j/r := by |
| rw [heightPolygon_formula F r n T h,ht] |
| simp |
| |
| lemma realPartialSum_sub (x y : ℕ → ℝ) (j : ℕ) : |
| realPartialSum (fun k => x k-y k) j = realPartialSum x j-realPartialSum y j := by |
| simp [realPartialSum,Finset.sum_sub_distrib] |
| |
| lemma scaledPolygon_sub (x y : ℕ → ℝ) (r n T : ℝ) : |
| scaledPolygon (fun k => x k-y k) r n T = scaledPolygon x r n T-scaledPolygon y r n T := by |
| ext t |
| simp only [scaledPolygon_apply,ContinuousMap.sub_apply,linearPolygon,mul_sub,Finset.sum_sub_distrib,sub_div] |
| |
| lemma heightPolygon_sub (F G : ℕ → ℝ) (r n T : ℝ) : |
| heightPolygon (fun k => F k-G k) r n T = heightPolygon F r n T-heightPolygon G r n T := by |
| have he : heightIncrement (fun k => F k-G k) = fun k => heightIncrement F k-heightIncrement G k := by |
| funext k |
| simp only [heightIncrement] |
| ring |
| ext t |
| simp only [heightPolygon,he,scaledPolygon_sub,sub_div,ContinuousMap.add_apply, |
| ContinuousMap.sub_apply,ContinuousMap.const_apply] |
| ring |
| |
| lemma heightPolygon_segment (F : ℕ → ℝ) (r n T : ℝ) (h : 0 ≤ T*n) |
| (j : ℕ) (t : unitInterval) (hjt : (j:ℝ) ≤ T*n*(t:ℝ)) (htj : T*n*(t:ℝ) ≤ (j:ℝ)+1) : |
| heightPolygon F r n T t = |
| (F j+(T*n*(t:ℝ)-j)*(F (j+1)-F j))/r := by |
| by_cases he : T*n*(t:ℝ) = (j:ℝ)+1 |
| · rw [heightPolygon_grid F r n T h (j+1) t (by simpa using he),he] |
| congr 1 |
| ring |
| · have hj : ⌊T*n*(t:ℝ)⌋₊ = j := Nat.floor_eq_iff (mul_nonneg h t.2.1) |>.mpr ⟨hjt,lt_of_le_of_ne htj he⟩ |
| rw [heightPolygon_formula F r n T h,hj] |
| |
| lemma heightPolygon_gridLinear (F : ℕ → ℝ) (r n T : ℝ) (h : 0 ≤ T*n) : |
| GridLinear (T*n) (heightPolygon F r n T) := by |
| intro j u v t hu hv htu htv |
| have hjt : (j:ℝ) ≤ T*n*(t:ℝ) := hu ▸ mul_le_mul_of_nonneg_left htu h |
| have htj : T*n*(t:ℝ) ≤ (j:ℝ)+1 := hv ▸ mul_le_mul_of_nonneg_left htv h |
| rw [heightPolygon_segment F r n T h j t hjt htj, |
| heightPolygon_grid F r n T h j u hu, |
| heightPolygon_grid F r n T h (j+1) v (by simpa using hv)] |
| ring |
| |
| lemma heightPolygon_restrict (F : ℕ → ℝ) (r n T S : ℝ) (hn : 0 ≤ n) |
| (hT : 0 ≤ T) (hS : 0 ≤ S) (a : unitInterval) (ha : S*(a:ℝ) = T) : |
| pathRestrictScale a (heightPolygon F r n S) = heightPolygon F r n T := by |
| ext t |
| rw [pathRestrictScale_apply,heightPolygon_formula F r n S (mul_nonneg hS hn), |
| heightPolygon_formula F r n T (mul_nonneg hT hn)] |
| have he : S*n*((a*t:unitInterval):ℝ) = T*n*(t:ℝ) := by |
| change S*n*((a:ℝ)*(t:ℝ)) = T*n*(t:ℝ) |
| rw [← ha] |
| ring |
| rw [he] |
| |
| end DirectionalTransience |
| |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| |
| end |
| end |
| |
| end OAI |
| |
