openAI-math-dataset / lean /OAI /Probability /Ballisticity /Geometry /HeightPolygonTranslateApply.lean
| import Mathlib | |
| import OAI.Probability.Ballisticity.Geometry.HeightPolygon | |
| 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 | |
| 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 BoundedContinuousFunction | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction | |
| 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 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 Classical | |
| namespace DirectionalTransience | |
| lemma heightPolygon_translate_apply (F : ℕ → ℝ) (r n T : ℝ) (hn : 0 ≤ n) (hT : 0 ≤ T) | |
| (H : ℕ) (c : ℝ) (t u : unitInterval) | |
| (hu : (T+1)*n*(u:ℝ) = T*n*(t:ℝ)+(H:ℝ)) : | |
| heightPolygon (fun k => F (k+H)+c) r n T t = | |
| heightPolygon F r n (T+1) u+c/r := by | |
| let j := ⌊T*n*(t:ℝ)⌋₊ | |
| have h0 : 0 ≤ T*n*(t:ℝ) := mul_nonneg (mul_nonneg hT hn) t.2.1 | |
| have hj0 : (j:ℝ) ≤ T*n*(t:ℝ) := Nat.floor_le h0 | |
| have hj1 : T*n*(t:ℝ) ≤ (j:ℝ)+1 := (Nat.lt_floor_add_one _).le | |
| rw [heightPolygon_formula _ _ _ _ (mul_nonneg hT hn), | |
| heightPolygon_segment F r n (T+1) (mul_nonneg (by linarith) hn) (j+H) u | |
| (by rw [hu]; push_cast; linarith) | |
| (by rw [hu]; push_cast; linarith),hu] | |
| simp only [Nat.cast_add,Nat.add_assoc,Nat.add_comm 1 H] | |
| dsimp only [j] | |
| ring | |
| lemma heightPolygon_translate_norm_le (F : ℕ → ℝ) (r n T : ℝ) | |
| (hn : 0 < n) (hT : 0 ≤ T) (H : ℕ) (hH : (H:ℝ) ≤ n) (c ε δ : ℝ) | |
| (hδ : (H:ℝ)/n ≤ δ) | |
| (hmod : heightPolygon F r n (T+1) ∉ ContinuousOscillation ε δ) : | |
| ‖heightPolygon (fun k => F (k+H)+c) r n T-heightPolygon F r n T‖ ≤ ε+|c/r| := by | |
| have hL : 0 < T+1 := by linarith | |
| have hLn : 0 < (T+1)*n := mul_pos hL hn | |
| apply (ContinuousMap.norm_le_of_nonempty _).mpr | |
| intro t | |
| let s : unitInterval := ⟨T/(T+1)*(t:ℝ),mul_nonneg (div_nonneg hT hL.le) t.2.1, | |
| (mul_le_of_le_one_right (div_nonneg hT hL.le) t.2.2).trans | |
| ((div_le_one hL).mpr (by linarith))⟩ | |
| let u : unitInterval := ⟨(T*n*(t:ℝ)+(H:ℝ))/((T+1)*n), | |
| div_nonneg (add_nonneg (mul_nonneg (mul_nonneg hT hn.le) t.2.1) (Nat.cast_nonneg _)) hLn.le, | |
| (div_le_one hLn).mpr (by nlinarith [mul_le_of_le_one_right (mul_nonneg hT hn.le) t.2.2])⟩ | |
| have hs : (T+1)*n*(s:ℝ) = T*n*(t:ℝ) := by dsimp [s]; field_simp | |
| have hu : (T+1)*n*(u:ℝ) = T*n*(t:ℝ)+(H:ℝ) := by dsimp [u]; field_simp | |
| have hdist : dist u s ≤ δ := by | |
| have he : (u:ℝ)-(s:ℝ) = (H:ℝ)/((T+1)*n) := by dsimp [u,s]; field_simp; ring | |
| rw [Subtype.dist_eq,Real.dist_eq,he,abs_of_nonneg (div_nonneg (Nat.cast_nonneg _) hLn.le)] | |
| exact (div_le_div_of_nonneg_left (Nat.cast_nonneg H) hn (by nlinarith)).trans hδ | |
| have hm : |heightPolygon F r n (T+1) u-heightPolygon F r n (T+1) s| ≤ ε := | |
| le_of_not_gt (fun h => hmod ⟨u,s,hdist,h⟩) | |
| have hsval := heightPolygon_translate_apply F r n T hn.le hT 0 0 t s (by simpa using hs) | |
| simp only [add_zero,zero_div] at hsval | |
| simp only [ContinuousMap.sub_apply,Real.norm_eq_abs] | |
| rw [heightPolygon_translate_apply F r n T hn.le hT H c t u hu,hsval] | |
| calc | |
| |heightPolygon F r n (T+1) u+c/r-heightPolygon F r n (T+1) s| | |
| = |(heightPolygon F r n (T+1) u-heightPolygon F r n (T+1) s)+c/r| := by ring_nf | |
| _ ≤ |heightPolygon F r n (T+1) u-heightPolygon F r n (T+1) s|+|c/r| := abs_add_le _ _ | |
| _ ≤ ε+|c/r| := by gcongr | |
| end DirectionalTransience | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| end | |
| end | |
| end OAI | |