| import OAI.Geometry.IsometricImmersion.Flows.FlowTimeVariation |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory Function |
| open scoped ContDiff Topology Interval |
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| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
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| private theorem abs_transport_bound {a b c M : ℝ} (hM : 0 ≤ M) |
| (ha : |a| ≤ M) (hb : |b| ≤ M) (hc : |c| ≤ 1) : |
| |a - c * b| ≤ 2 * M := by |
| calc |
| _ ≤ |a| + |c| * |b| := by simpa only [abs_mul] using abs_sub a (c * b) |
| _ ≤ M + 1 * M := add_le_add ha (by |
| simpa only [mul_comm] using mul_le_mul hb hc (abs_nonneg _) hM) |
| _ = 2 * M := by ring |
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| private theorem abs_second_initial_coefficient_bound {a b c d M : ℝ} (hM : 0 ≤ M) |
| (ha : |a| ≤ M) (hb : |b| ≤ M) (hc : |c| ≤ 1) (hd : |d| ≤ M) : |
| |-a + b * c + d^2| ≤ 2 * M + M^2 := by |
| have hbase : |-a + b * c| ≤ 2 * M := by |
| calc |
| _ = |-(a - c * b)| := by congr 1; ring |
| _ = |a - c * b| := abs_neg _ |
| _ ≤ 2 * M := abs_transport_bound hM ha hb hc |
| have hsq : |d^2| ≤ M^2 := by |
| rw [abs_pow] |
| exact pow_le_pow_left₀ (abs_nonneg _) hd 2 |
| exact (abs_add_le _ _).trans (add_le_add hbase hsq) |
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| private theorem abs_third_time_coefficient_bound {a00 a10 a01 a11 a0 a1 c M : ℝ} |
| (hM : 0 ≤ M) (hc : |c| ≤ 1) |
| (h00 : |a00| ≤ M) (h10 : |a10| ≤ M) (h01 : |a01| ≤ M) |
| (h11 : |a11| ≤ M) (h0 : |a0| ≤ M) (h1 : |a1| ≤ M) : |
| |-a00 + c * a10 + (a01 - c * a11) * c + a1 * (a0 - c * a1)| ≤ |
| 4 * M + 2 * M^2 := by |
| have ha : |-a00 + c * a10| ≤ 2 * M := by |
| calc |
| _ = |-(a00 - c * a10)| := by congr 1; ring |
| _ = |a00 - c * a10| := abs_neg _ |
| _ ≤ 2 * M := abs_transport_bound hM h00 h10 hc |
| have hb := abs_transport_bound hM h01 h11 hc |
| have hd := abs_transport_bound hM h0 h1 hc |
| calc |
| _ ≤ |-a00 + c * a10| + |a01 - c * a11| * |c| + |a1| * |a0 - c * a1| := by |
| simpa only [abs_mul] using (abs_add_le (-a00 + c * a10 + (a01 - c * a11) * c) |
| (a1 * (a0 - c * a1))).trans |
| (add_le_add (abs_add_le (-a00 + c * a10) ((a01 - c * a11) * c)) le_rfl) |
| _ ≤ 2 * M + (2 * M) * 1 + M * (2 * M) := |
| add_le_add (add_le_add ha (mul_le_mul hb hc (abs_nonneg _) (by positivity))) |
| (mul_le_mul h1 hd (abs_nonneg _) hM) |
| _ = 4 * M + 2 * M^2 := by ring |
|
|
| section ActualFlow |
| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
| variable (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| variable (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| variable (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| variable (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) |
| variable (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) |
| include hq hU hSU hY hrange hstart hode |
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| theorem cap_flow_time_bound |
| (hq0 : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| |timeFlowDerivative Y (t, s)| ≤ (1 : ℝ) / 100 := by |
| rw [timeFlowDerivative_eq hq hU hSU hY hrange hstart hode hs ht, abs_neg] |
| exact hq0 _ (flowCoordinateMap_mem_modelSquare (p := (t, s)) hq hU hSU hY hrange hstart hode ⟨ht, hs⟩) |
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| theorem cap_flow_time_second_bound {M : ℝ} (hM : 0 ≤ M) |
| (hq0 : ∀ p ∈ modelSquare, |q p| ≤ 1) |
| (hq1 : ∀ i : Fin 2, ∀ p ∈ modelSquare, |coordPartial i q p| ≤ M) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| |timeSecondFlowDerivative Y (t, s)| ≤ 2 * M := by |
| rw [timeSecondFlowDerivative_eq hq hU hSU hY hrange hstart hode hs ht] |
| have hp := flowCoordinateMap_mem_modelSquare (p := (t, s)) hq hU hSU hY hrange hstart hode ⟨ht, hs⟩ |
| have hh := abs_transport_bound hM (hq1 0 _ hp) (hq1 1 _ hp) (hq0 _ hp) |
| calc |
| _ = |-(coordPartial 0 q (coordinatePoint t (Y s t)) - |
| q (coordinatePoint t (Y s t)) * coordPartial 1 q (coordinatePoint t (Y s t)))| := by |
| congr 1; ring |
| _ = |coordPartial 0 q (coordinatePoint t (Y s t)) - |
| q (coordinatePoint t (Y s t)) * coordPartial 1 q (coordinatePoint t (Y s t))| := abs_neg _ |
| _ ≤ 2 * M := hh |
|
|
| theorem cap_flow_mixed_bound {M : ℝ} (hM : 0 ≤ M) |
| (hq1 : ∀ i : Fin 2, ∀ p ∈ modelSquare, |coordPartial i q p| ≤ M) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| |mixedFlowDerivative Y (t, s)| ≤ M * Real.exp (2 * M) := by |
| rw [mixedFlowDerivative_eq hq hU hSU hY hrange hstart hode hs ht, abs_mul, abs_neg, initialFlowDerivative, |
| abs_of_pos (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht)] |
| exact mul_le_mul (hq1 1 _ |
| (flowCoordinateMap_mem_modelSquare (p := (t, s)) hq hU hSU hY hrange hstart hode ⟨ht, hs⟩)) |
| (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM (hq1 1) hs ht).2 |
| (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht).le hM |
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| theorem cap_flow_mixed_initial_bound {M : ℝ} (hM : 0 ≤ M) |
| (hq1 : ∀ i : Fin 2, ∀ p ∈ modelSquare, |coordPartial i q p| ≤ M) |
| (hq2 : ∀ i j : Fin 2, ∀ p ∈ modelSquare, |coordPartial i (coordPartial j q) p| ≤ M) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| |deriv (fun u => mixedFlowDerivative Y (t, u)) s| ≤ |
| (M + 2 * M^2) * Real.exp (4 * M) := by |
| rw [(cap_flow_mixed_initial_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv] |
| have hp := flowCoordinateMap_mem_modelSquare (p := (t, s)) hq hU hSU hY hrange hstart hode ⟨ht, hs⟩ |
| have hV := (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM (hq1 1) hs ht).2 |
| have hVpos := cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht |
| have hVsq : |(initialFlowDerivative Y (t, s))^2| ≤ Real.exp (4 * M) := by |
| rw [abs_of_nonneg (sq_nonneg _)] |
| calc |
| _ ≤ (Real.exp (2 * M))^2 := pow_le_pow_left₀ hVpos.le hV 2 |
| _ = Real.exp (4 * M) := by rw [pow_two, ← Real.exp_add]; congr 1; ring |
| have hW := cap_flow_initial_second_bound hq hU hSU hY hrange hstart hode hM (hq1 1) (hq2 1 1) hs ht |
| calc |
| _ ≤ |coordPartial 1 (coordPartial 1 q) (coordinatePoint t (Y s t))| * |
| |(initialFlowDerivative Y (t, s))^2| + |
| |coordPartial 1 q (coordinatePoint t (Y s t))| * |initialSecondFlowDerivative Y (t, s)| := by |
| calc |
| _ ≤ |-coordPartial 1 (coordPartial 1 q) (coordinatePoint t (Y s t)) * |
| (initialFlowDerivative Y (t, s))^2| + |
| |coordPartial 1 q (coordinatePoint t (Y s t)) * initialSecondFlowDerivative Y (t, s)| := |
| abs_sub _ _ |
| _ = _ := by rw [abs_mul, abs_mul, abs_neg] |
| _ ≤ M * Real.exp (4 * M) + M * (2 * M * Real.exp (4 * M)) := |
| add_le_add (mul_le_mul (hq2 1 1 _ hp) hVsq (abs_nonneg _) hM) |
| (mul_le_mul (hq1 1 _ hp) hW (abs_nonneg _) hM) |
| _ = (M + 2 * M^2) * Real.exp (4 * M) := by ring |
|
|
| theorem cap_flow_time_second_initial_bound {M : ℝ} (hM : 0 ≤ M) |
| (hq0 : ∀ p ∈ modelSquare, |q p| ≤ 1) |
| (hq1 : ∀ i : Fin 2, ∀ p ∈ modelSquare, |coordPartial i q p| ≤ M) |
| (hq2 : ∀ i j : Fin 2, ∀ p ∈ modelSquare, |coordPartial i (coordPartial j q) p| ≤ M) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| |deriv (fun u => timeSecondFlowDerivative Y (t, u)) s| ≤ |
| (2 * M + M^2) * Real.exp (2 * M) := by |
| rw [(cap_flow_time_second_initial_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv, |
| abs_mul, initialFlowDerivative, abs_of_pos (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht)] |
| have hp := flowCoordinateMap_mem_modelSquare (p := (t, s)) hq hU hSU hY hrange hstart hode ⟨ht, hs⟩ |
| exact mul_le_mul |
| (abs_second_initial_coefficient_bound hM (hq2 1 0 _ hp) (hq2 1 1 _ hp) (hq0 _ hp) (hq1 1 _ hp)) |
| (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM (hq1 1) hs ht).2 |
| (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht).le (by positivity) |
|
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| theorem cap_flow_time_third_bound {M : ℝ} (hM : 0 ≤ M) |
| (hq0 : ∀ p ∈ modelSquare, |q p| ≤ 1) |
| (hq1 : ∀ i : Fin 2, ∀ p ∈ modelSquare, |coordPartial i q p| ≤ M) |
| (hq2 : ∀ i j : Fin 2, ∀ p ∈ modelSquare, |coordPartial i (coordPartial j q) p| ≤ M) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| |deriv (fun u => timeSecondFlowDerivative Y (u, s)) t| ≤ 4 * M + 2 * M^2 := by |
| rw [(cap_flow_time_third_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv] |
| have hp := flowCoordinateMap_mem_modelSquare (p := (t, s)) hq hU hSU hY hrange hstart hode ⟨ht, hs⟩ |
| exact abs_third_time_coefficient_bound hM (hq0 _ hp) |
| (hq2 0 0 _ hp) (hq2 1 0 _ hp) (hq2 0 1 _ hp) (hq2 1 1 _ hp) |
| (hq1 0 _ hp) (hq1 1 _ hp) |
|
|
| end ActualFlow |
| end SmoothLocal.Flow |
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| end |
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| end OAI |
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