openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /CompactIntegral.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.Model | |
| namespace OAI | |
| noncomputable section | |
| namespace VelocityDetection.CompactIntegral | |
| open scoped BigOperators Topology ContDiff | |
| open Set Function Filter | |
| open MeasureTheory | |
| theorem hasDerivAt_integral {n : ℕ} {F : ℝ × Coord n → ℝ} | |
| (hF : ContDiff ℝ 1 F) (t : ℝ) {K : Set (Coord n)} (hK : IsCompact K) | |
| (hoff : ∀ s ∈ Icc (t - 1) (t + 1), ∀ X ∉ K, F (s, X) = 0) : | |
| HasDerivAt (fun s => ∫ X, F (s, X)) | |
| (∫ X, fderiv ℝ F (t, X) (1, 0)) t := by | |
| let F' : ℝ → Coord n → ℝ := fun s X => fderiv ℝ F (s, X) (1, 0) | |
| have hd (s : ℝ) (X : Coord n) : HasDerivAt (fun r => F (r, X)) (F' s X) s := by | |
| have hpair : HasDerivAt (fun r : ℝ => (r, X)) (1, 0) s := | |
| (hasDerivAt_id s).prodMk (hasDerivAt_const s X) | |
| exact ((hF.differentiable (by norm_num) (s, X)).hasFDerivAt.comp_hasDerivAt s hpair) | |
| have hFc : Continuous F := hF.continuous | |
| have hF'c : Continuous (fun q : ℝ × Coord n => F' q.1 q.2) := | |
| (hF.continuous_fderiv (by norm_num)).clm_apply continuous_const | |
| have hF'supp {s : ℝ} (hs : s ∈ Ioo (t - 1) (t + 1)) {X : Coord n} (hX : X ∉ K) : | |
| F' s X = 0 := by | |
| have he : (fun r => F (r, X)) =ᶠ[𝓝 s] (fun _ => (0 : ℝ)) := by | |
| filter_upwards [Icc_mem_nhds hs.1 hs.2] with r hr | |
| exact hoff r hr X hX | |
| exact (hd s X).unique ((hasDerivAt_const s 0).congr_of_eventuallyEq he) | |
| have hFcpt : HasCompactSupport (fun X => F (t, X)) := by | |
| apply HasCompactSupport.of_support_subset_isCompact hK | |
| intro X hX | |
| by_contra hnot | |
| exact hX (hoff t (by constructor <;> linarith) X hnot) | |
| have hFcX (s : ℝ) : Continuous (fun X => F (s, X)) := | |
| hFc.comp (continuous_const.prodMk continuous_id) | |
| have hF'cX : Continuous (F' t) := hF'c.comp (continuous_const.prodMk continuous_id) | |
| obtain ⟨B, hB⟩ := ((isCompact_Icc (a := t - 1) (b := t + 1)).prod hK).exists_bound_of_continuousOn hF'c.continuousOn | |
| let bound : Coord n → ℝ := K.indicator (fun _ => B) | |
| have hboundint : Integrable bound := by | |
| exact (integrable_indicator_iff hK.measurableSet).2 | |
| (integrableOn_const (C := B) (hK.measure_lt_top (μ := volume)).ne) | |
| have hb : ∀ X, ∀ s ∈ Ioo (t - 1) (t + 1), ‖F' s X‖ ≤ bound X := by | |
| intro X s hs | |
| by_cases hX : X ∈ K | |
| · simpa only [bound, indicator_of_mem hX] using hB (s, X) ⟨⟨hs.1.le, hs.2.le⟩, hX⟩ | |
| · simp only [hF'supp hs hX, norm_zero, bound, indicator_of_notMem hX, le_refl] | |
| exact (hasDerivAt_integral_of_dominated_loc_of_deriv_le | |
| (Ioo_mem_nhds (by linarith) (by linarith)) | |
| (Filter.Eventually.of_forall (fun s => (hFcX s).aestronglyMeasurable)) | |
| ((hFcX t).integrable_of_hasCompactSupport hFcpt) hF'cX.aestronglyMeasurable | |
| (Filter.Eventually.of_forall hb) hboundint | |
| (Filter.Eventually.of_forall (fun X s _ => hd s X))).2 | |
| theorem hasDerivAt_moving_capture {n : ℕ} {φ : Coord n → ℝ} | |
| {ρ : ℝ → Coord n → ℝ} {c : ℝ → Coord n} | |
| (hφ : ContDiff ℝ 1 φ) (hφc : HasCompactSupport φ) | |
| (hρ : ContDiff ℝ 1 (uncurry ρ)) (hc : ContDiff ℝ 1 c) (t : ℝ) : | |
| HasDerivAt (fun s => ∫ X, φ (X - c s) * ρ s X) | |
| (∫ X, φ (X - c t) * deriv (fun s => ρ s X) t - | |
| fderiv ℝ φ (X - c t) (deriv c t) * ρ t X) t := by | |
| let F : ℝ × Coord n → ℝ := fun q => φ (q.2 - c q.1) * ρ q.1 q.2 | |
| have hF : ContDiff ℝ 1 F := | |
| (hφ.comp (contDiff_snd.sub (hc.comp contDiff_fst))).mul hρ | |
| let K : Set (Coord n) := (fun q : ℝ × Coord n => c q.1 + q.2) '' | |
| (Icc (t - 1) (t + 1) ×ˢ tsupport φ) | |
| have hK : IsCompact K := | |
| (isCompact_Icc.prod hφc).image ((hc.continuous.comp continuous_fst).add continuous_snd) | |
| have hoff : ∀ s ∈ Icc (t - 1) (t + 1), ∀ X ∉ K, F (s, X) = 0 := by | |
| intro s hs X hX | |
| have hz : φ (X - c s) = 0 := by | |
| by_contra hnz | |
| apply hX | |
| refine ⟨(s, X - c s), ⟨hs, subset_closure hnz⟩, ?_⟩ | |
| dsimp only | |
| abel | |
| simp only [F, hz, zero_mul] | |
| have hi := hasDerivAt_integral hF t hK hoff | |
| have hdF (X : Coord n) : fderiv ℝ F (t, X) (1, 0) = | |
| φ (X - c t) * deriv (fun s => ρ s X) t - | |
| fderiv ℝ φ (X - c t) (deriv c t) * ρ t X := by | |
| have hdiff : HasFDerivAt F (fderiv ℝ F (t, X)) (t, X) := | |
| (hF.differentiable (by norm_num) (t, X)).hasFDerivAt | |
| have hline : HasDerivAt (fun s => F (s, X)) (fderiv ℝ F (t, X) (1, 0)) t := by | |
| simpa only [Function.comp_def, id_eq] using | |
| hdiff.comp_hasDerivAt t ((hasDerivAt_id t).prodMk (hasDerivAt_const t X)) | |
| have hρX : ContDiff ℝ 1 (fun s => ρ s X) := hρ.comp (contDiff_id.prodMk contDiff_const) | |
| have harg := (hasDerivAt_const t X).sub ((hc.differentiable (by norm_num) t).hasDerivAt) | |
| have hφX := (hφ.differentiable (by norm_num) (X - c t)).hasFDerivAt.comp_hasDerivAt t harg | |
| have hprod := hφX.mul ((hρX.differentiable (by norm_num) t).hasDerivAt) | |
| have hval := hline.unique hprod | |
| simp only [zero_sub, map_neg, Function.comp_apply, Pi.sub_apply] at hval | |
| rw [hval] | |
| ring | |
| have heq : (∫ X, fderiv ℝ F (t, X) (1, 0)) = | |
| ∫ X, φ (X - c t) * deriv (fun s => ρ s X) t - | |
| fderiv ℝ φ (X - c t) (deriv c t) * ρ t X := | |
| integral_congr_ae (Filter.Eventually.of_forall hdF) | |
| rw [heq] at hi | |
| exact hi | |
| end VelocityDetection.CompactIntegral | |
| end | |
| end OAI | |