| import OAI.Geometry.IsometricImmersion.Immersions.HeightCoefficientClosedBounds |
| import OAI.Geometry.IsometricImmersion.Caps.CapReciprocalBounds |
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|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
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|
| theorem capPullback_bound_one {u : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {A F : ℝ} |
| (hY : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2)) |
| (hu : ContDiffOn ℝ ∞ u U) (hU : IsOpen U) |
| (hmap : MapsTo (capChart Y) capChartDomain U) |
| (hmapS : MapsTo (capChart Y) capChartDomain modelSquare) |
| (hA : 0 ≤ A) (hF : 0 ≤ F) |
| (huB : CoordinateBound u modelSquare 1 A) |
| (hFB : CoordinateBound (capFlowHeight Y) capChartDomain 3 F) : |
| CoordinateBound (capPullback Y u) capChartDomain 1 (A * (1 + F)) := by |
| intro ds hds p hp |
| have hv : |u (capChart Y p)| ≤ A := huB [] (by norm_num) _ (hmapS hp) |
| have hdu (i : Fin 2) : |coordPartial i u (capChart Y p)| ≤ A := huB [i] (by norm_num) _ (hmapS hp) |
| have hdf (i : Fin 2) : |coordPartial i (capFlowHeight Y) p| ≤ F := hFB [i] (by norm_num) p hp |
| cases ds with |
| | nil => exact hv.trans (by nlinarith [mul_nonneg hA hF]) |
| | cons i ds => |
| cases ds with |
| | nil => |
| change |coordPartial i (capPullback Y u) p| ≤ A * (1 + F) |
| rw [capPullback_partial hY hp ((hu.contDiffAt (hU.mem_nhds (hmap hp))).differentiableAt (by simp)) i] |
| have h0 : |if i = 0 then coordPartial 0 u (capChart Y p) else 0| ≤ A := by |
| split_ifs |
| · exact hdu 0 |
| · simpa only [abs_zero] using hA |
| have h1 : |coordPartial 1 u (capChart Y p) * coordPartial i (capFlowHeight Y) p| ≤ A * F := by |
| rw [abs_mul] |
| exact mul_le_mul (hdu 1) (hdf i) (abs_nonneg _) hA |
| exact ((abs_add_le _ _).trans (add_le_add h0 h1)).trans (le_of_eq (by ring)) |
| | cons _ ds => simp only [List.length_cons] at hds; omega |
|
|
| theorem heightOriginalB_bound_one {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {P Q : ℝ} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hz : ContDiffOn ℝ ∞ z U) (hne : ∀ p ∈ U, covHessian g z p 1 1 ≠ 0) |
| (hPB : CoordinateBound (heightPFirst g z 0) modelSquare 1 P) |
| (hqB : CoordinateBound (hessianQuotient g z) modelSquare 3 Q) (ell : ℕ) : |
| CoordinateBound (heightOriginalB g z ell) modelSquare 1 (P + 2 * (ell : ℝ) * Q) := by |
| have hI : interior modelSquare ⊆ U := interior_subset.trans hSU |
| have hP := heightPFirst_contDiffOn hg hU hz hne 0 |
| have hq := hessianQuotient_contDiffOn hg hU hz hne |
| have hqy := partial_contDiffOn hq hU 1 |
| have hqyB : CoordinateBound (coordPartial 1 (hessianQuotient g z)) modelSquare 1 Q := |
| (hqB.mono (by norm_num : 2 ≤ 3) le_rfl).partial_bound 1 |
| have ht := CoordinateBound.const_mul (hqy.mono hI) isOpen_interior |
| (hqyB.restrict_domain interior_subset) (2 * (ell : ℝ)) |
| have hb := CoordinateBound.sub (hP.neg.mono hI) (contDiffOn_const.mul (hqy.mono hI)) isOpen_interior |
| (CoordinateBound.neg (hP.mono hI) isOpen_interior (hPB.restrict_domain interior_subset)) ht |
| have hc : |2 * (ell : ℝ)| = 2 * (ell : ℝ) := abs_of_nonneg (by positivity) |
| rw [hc] at hb |
| exact CoordinateBound.of_modelSquare_interior (heightOriginalB_contDiffOn hg hU hz hne ell) hU hSU hb |
|
|
| theorem capChartRho_lower_from_q |
| {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {M : ℝ} |
| (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) |
| (hM : 0 ≤ M) (hq1 : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M) |
| {p : Coord} (hp : p ∈ capChartDomain) : 1 / Real.exp (2 * M) ≤ capChartRho Y p := by |
| have hs := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode |
| have hpos := cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hp.2 hp.1 |
| have hupper := (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hq1 hp.2 hp.1).2 |
| unfold capChartRho |
| rw [capFlowHeight_partial_one_eq hs hp] |
| apply (le_div_iff₀ hpos).mpr |
| calc |
| _ ≤ (1 / Real.exp (2 * M)) * Real.exp (2 * M) := mul_le_mul_of_nonneg_left hupper (by positivity) |
| _ = 1 := one_div_mul_cancel (Real.exp_ne_zero _) |
|
|
| theorem heightChartRemainder_bound_of_low_coefficients |
| (g : MetricField) (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (p : Coord) (ell : ℕ) |
| {DE DG SG lowerG RL : ℝ} (hDE : 0 ≤ DE) (hDG : 0 ≤ DG) (hSG : 0 ≤ SG) |
| (hGL : 0 < lowerG) (hRL : 0 < RL) |
| (herror : |curvatureDriftError g z (capChart Y p)| ≤ DE) |
| (hGy : |coordPartial 1 (darbouxG g z) (capChart Y p)| ≤ DG) |
| (hG : lowerG ≤ darbouxG g z (capChart Y p)) |
| (hrho : RL ≤ capChartRho Y p) (hsigma : |capChartSigma Y p| ≤ SG) : |
| |heightChartRemainder g z Y ell p| ≤ |
| (DE + DG) / (lowerG * RL) + (2 * (ell : ℝ) + 1) * SG / RL^2 := by |
| have hGpos : 0 < darbouxG g z (capChart Y p) := hGL.trans_le hG |
| have hrhopos : 0 < capChartRho Y p := hRL.trans_le hrho |
| have hden : lowerG * RL ≤ |darbouxG g z (capChart Y p) * capChartRho Y p| := by |
| rw [abs_of_pos (mul_pos hGpos hrhopos)] |
| exact mul_le_mul hG hrho hRL.le hGpos.le |
| have hden2 : RL^2 ≤ |(capChartRho Y p)^2| := by |
| rw [abs_pow, abs_of_pos hrhopos] |
| exact pow_le_pow_left₀ hRL.le hrho 2 |
| have hnum := (abs_sub _ _).trans (add_le_add herror hGy) |
| have hfirst := LowQuotient.abs_div_bound (add_nonneg hDE hDG) hnum (mul_pos hGL hRL) hden |
| have hnum2 : |(2 * (ell : ℝ) + 1) * capChartSigma Y p| ≤ (2 * (ell : ℝ) + 1) * SG := by |
| rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ 2 * (ell : ℝ) + 1)] |
| exact mul_le_mul_of_nonneg_left hsigma (by positivity) |
| have hsecond := LowQuotient.abs_div_bound (by positivity) hnum2 (pow_pos hRL 2) hden2 |
| exact (abs_sub _ _).trans (add_le_add hfirst hsecond) |
|
|
| end SmoothLocal.Flow |
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| end |
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| end OAI |
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