| import OAI.Geometry.IsometricImmersion.Flows.FlowParameterContinuity |
| import Mathlib.Analysis.Calculus.Deriv.Slope |
|
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| namespace OAI |
|
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| noncomputable section |
| open Set Metric Filter Function MeasureTheory |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
|
|
| theorem hasDerivAt_of_exact_secant {f g : ℝ → ℝ} {s : ℝ} |
| (hg : ContinuousAt g s) |
| (he : ∀ᶠ u in 𝓝 s, f u - f s = (u - s) * g u) : |
| HasDerivAt f (g s) s := by |
| rw [hasDerivAt_iff_tendsto_slope] |
| have hslope : slope f s =ᶠ[𝓝[≠] s] g := by |
| filter_upwards [he.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with u hu hne |
| have hne' : u ≠ s := by simpa only [mem_compl_iff, mem_singleton_iff] using hne |
| rw [slope_def_field, hu] |
| exact mul_div_cancel_left₀ (g u) (sub_ne_zero.mpr hne') |
| exact (hg.mono_left nhdsWithin_le_nhds).congr' hslope.symm |
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| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
|
|
| theorem flow_difference_formula |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) |
| (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) |
| {s0 s1 t : ℝ} (hs0 : s0 ∈ Icc (-2 : ℝ) 2) (hs1 : s1 ∈ Icc (-2 : ℝ) 2) |
| (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| Y s1 t - Y s0 t = (s1 - s0) * |
| Real.exp (-(∫ u in 0..t, flowAverageCoefficient q Y s0 s1 u)) := by |
| have hcoeff := flowAverageCoefficient_continuous hq hU hSU hY hrange |
| have ha : ContinuousOn (flowAverageCoefficient q Y s0 s1) (Icc (-2 : ℝ) 2) := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| exact hcoeff.comp |
| ((continuous_const (y := (⟨s0, hs0⟩ : CapInterval))).prodMk |
| ((continuous_const (y := (⟨s1, hs1⟩ : CapInterval))).prodMk continuous_id)) |
| have hz (u : ℝ) (hu : u ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun v => Y s1 v - Y s0 v) |
| (-flowAverageCoefficient q Y s0 s1 u * (Y s1 u - Y s0 u)) u := by |
| exact (flow_difference_hasDerivWithinAt hq hU hSU ⟨hu.1.le, hu.2.le⟩ |
| (hrange s0 hs0 u ⟨hu.1.le, hu.2.le⟩) (hrange s1 hs1 u ⟨hu.1.le, hu.2.le⟩) |
| (hode s0 hs0 u ⟨hu.1.le, hu.2.le⟩) (hode s1 hs1 u ⟨hu.1.le, hu.2.le⟩)).hasDerivAt |
| (Icc_mem_nhds hu.1 hu.2) |
| have he := scalar_linear_evolution_formula (show (0 : ℝ) < 2 by norm_num) |
| (ha.mono Ioo_subset_Icc_self) hz ht |
| simpa only [hstart s0 hs0, hstart s1 hs1] using he |
|
|
| theorem cap_flow_hasDerivAt_initial |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) |
| (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun u => Y u t) |
| (Real.exp (-(∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))))) s := by |
| let G (u : ℝ) := Real.exp (-(∫ r in 0..t, flowAverageCoefficient q Y s u r)) |
| have hG : ContinuousOn G (Icc (-2 : ℝ) 2) := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| exact (((flowAverageIntegral_continuous hq hU hSU hY hrange |
| ⟨ht.1.le, ht.2.le⟩).comp |
| ((continuous_const (y := (⟨s, ⟨hs.1.le, hs.2.le⟩⟩ : CapInterval))).prodMk continuous_id)).neg).rexp |
| have he : ∀ᶠ u in 𝓝 s, Y u t - Y s t = (u - s) * G u := by |
| filter_upwards [Icc_mem_nhds hs.1 hs.2] with u hu |
| exact flow_difference_formula hq hU hSU hY hrange hstart hode ⟨hs.1.le, hs.2.le⟩ hu ht |
| have hd := hasDerivAt_of_exact_secant (hG.continuousAt (Icc_mem_nhds hs.1 hs.2)) he |
| have hdiag : G s = Real.exp (-(∫ r in 0..t, |
| coordPartial 1 q (coordinatePoint r (Y s r)))) := by |
| simp only [G, flowAverageCoefficient, verticalAverageDerivative_diagonal] |
| rw [hdiag] at hd |
| exact hd |
|
|
| theorem cap_flow_initial_deriv_pos |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) |
| (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| 0 < deriv (fun u => Y u t) s := by |
| rw [(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv] |
| exact Real.exp_pos _ |
|
|
| theorem exists_cap_flow_first_variation |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) : |
| ∃ Y : ℝ → ℝ → ℝ, |
| ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2) ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, ContDiffOn ℝ ∞ (Y s) (Icc (-2 : ℝ) 2)) ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| |Y s t - s| ≤ (1 : ℝ) / 50) ∧ |
| (∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, |
| HasDerivAt (fun u => Y u t) |
| (Real.exp (-(∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))))) s ∧ |
| 0 < deriv (fun u => Y u t) s) := by |
| obtain ⟨Y, hY, htime, hstart, hode, hdisp, hdisp', hrange, hLip⟩ := |
| exists_cap_flow hq hU hSU hsmall |
| have hrange' (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) (t : ℝ) |
| (ht : t ∈ Icc (-2 : ℝ) 2) : Y s t ∈ Icc (-3 : ℝ) 3 := |
| abs_le.mp ((hrange s hs t ht).1.trans (by norm_num)) |
| refine ⟨Y, hY, htime, hstart, hode, hdisp', ?_⟩ |
| intro s hs t ht |
| exact ⟨cap_flow_hasDerivAt_initial hq hU hSU hY hrange' hstart hode hs ht, |
| cap_flow_initial_deriv_pos hq hU hSU hY hrange' hstart hode hs ht⟩ |
|
|
| end SmoothLocal.Flow |
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| end |
|
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| end OAI |
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