openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Coordinates /BoundedClassSlabSmallness.lean
| import OAI.Geometry.IsometricImmersion.Estimates.BoundedClassQLow | |
| import OAI.Geometry.IsometricImmersion.Pulses.PulseFlowDistance | |
| import OAI.Geometry.IsometricImmersion.Calculus.ClosedSquareJetBounds | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped ContDiff Topology Matrix | |
| namespace SmoothLocal.Perturbation | |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow | |
| def boundedClassQuotientBudget (G d : ℝ) (M : ℕ) : ℝ := | |
| heightQuotientJetBound G (M : ℝ) d (1 / (M : ℝ)) | |
| def boundedClassSlabRadius (G d kappa q0 : ℝ) (M : ℕ) : ℝ := | |
| 1 / (100 * boundedClassWidth kappa M * (1 + boundedClassQuotientBudget G d M) * | |
| (1 + (1 + |q0|) * boundedClassWidth kappa M)) | |
| theorem boundedClassQuotientBudget_nonneg {G d : ℝ} {M : ℕ} | |
| (hG : 0 ≤ G) (hd : 0 < d) (hM : 0 < M) : 0 ≤ boundedClassQuotientBudget G d M := | |
| heightQuotientJetBound_nonneg hG (Nat.cast_nonneg M) hd | |
| (one_div_pos.mpr (Nat.cast_pos.mpr hM)) | |
| theorem boundedClassSlabRadius_pos {G d : ℝ} (kappa q0 : ℝ) {M : ℕ} | |
| (hG : 0 ≤ G) (hd : 0 < d) (hM : 0 < M) : 0 < boundedClassSlabRadius G d kappa q0 M := by | |
| have hL := boundedClassWidth_pos kappa M | |
| have hQ := boundedClassQuotientBudget_nonneg hG hd hM | |
| unfold boundedClassSlabRadius | |
| positivity | |
| theorem bounded_class_quotient_change_from_origin | |
| {g : MetricField} {z : Coord → ℝ} {M : ℕ} (h : BoundedAdmissibleHeight g M z) | |
| {G d : ℝ} (hG : 0 ≤ G) (hd : 0 < d) | |
| (hgB : ∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare, | |
| ‖iteratedFDeriv ℝ k (fun r => g r i j) p‖ ≤ G) | |
| (hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|) | |
| {p : Coord} (hp : p ∈ modelSquare) : | |
| |hessianQuotient g z p - hessianQuotient g z 0| ≤ | |
| boundedClassQuotientBudget G d M * (|p 0| + |p 1|) := by | |
| obtain ⟨hM, hadm, hjet, hHb, _⟩ := h | |
| obtain ⟨U, hU, hSU, hg, hz, _, _, _, _⟩ := hadm | |
| have hgCoord : ∀ i j, CoordinateBound (fun p => g p i j) modelSquare 4 G := by | |
| intro i j ds hds p hp | |
| exact (norm_iteratedCoordPartial_le_jet (hg.1 i j) hU ds (hSU hp)).trans | |
| (hgB i j ds.length hds p hp) | |
| have hzCoord : CoordinateBound z modelSquare 5 (M : ℝ) := by | |
| intro ds hds p hp | |
| exact (norm_iteratedCoordPartial_le_jet hz hU ds (hSU hp)).trans | |
| (hjet ds.length (by omega) p hp) | |
| obtain ⟨V, hV, hSV, _, _, hqV, hqB⟩ := | |
| exists_open_quotient_domain_with_closed_low_bounds hg hU hSU hz hG (Nat.cast_nonneg M) hd | |
| (one_div_pos.mpr (Nat.cast_pos.mpr hM)) hgCoord hzCoord hdet (fun p hp => (hHb p hp).1) | |
| apply quotient_change_from_origin hqV hV hSV (boundedClassQuotientBudget_nonneg hG hd hM) ?_ hp | |
| intro p hp i | |
| exact hqB [i] (by norm_num) p hp | |
| theorem inverseShear_coordinate_sum_le {q0 L r : ℝ} {p : Coord} | |
| (_hr : 0 ≤ r) (hx : |p 0| ≤ L * r) (hy : |p 1| ≤ r) : | |
| |inverseShearCoordinates q0 p 0| + |inverseShearCoordinates q0 p 1| ≤ | |
| r * (1 + (1 + |q0|) * L) := by | |
| change |p 0| + |p 1 - q0 * p 0| ≤ _ | |
| calc | |
| _ ≤ |p 0| + (|p 1| + |q0 * p 0|) := add_le_add le_rfl (abs_sub _ _) | |
| _ ≤ L * r + (r + |q0| * (L * r)) := by | |
| rw [abs_mul] | |
| exact add_le_add hx (add_le_add hy (mul_le_mul_of_nonneg_left hx (abs_nonneg _))) | |
| _ = _ := by ring | |
| theorem bounded_class_slab_quotient_small | |
| {g : MetricField} {z : Coord → ℝ} {M : ℕ} (h : BoundedAdmissibleHeight g M z) | |
| {G d kappa q0 r : ℝ} (hG : 0 ≤ G) (hd : 0 < d) (hr : 0 ≤ r) | |
| (hrsmall : r ≤ boundedClassSlabRadius G d kappa q0 M) | |
| (hgB : ∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare, | |
| ‖iteratedFDeriv ℝ k (fun a => g a i j) p‖ ≤ G) | |
| (hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|) | |
| (hcenter : |hessianQuotient g z 0 - q0| ≤ 1 / (100 * boundedClassWidth kappa M)) | |
| {p : Coord} (hp : inverseShearCoordinates q0 p ∈ modelSquare) | |
| (hx : |p 0| ≤ boundedClassWidth kappa M * r) (hy : |p 1| ≤ r) : | |
| |hessianQuotient g z (inverseShearCoordinates q0 p) - q0| ≤ | |
| 1 / (10 * boundedClassWidth kappa M) := by | |
| let L := boundedClassWidth kappa M | |
| let Q := boundedClassQuotientBudget G d M | |
| let D := 1 + (1 + |q0|) * L | |
| have hL : 0 < L := boundedClassWidth_pos kappa M | |
| have hQ : 0 ≤ Q := boundedClassQuotientBudget_nonneg hG hd h.1 | |
| have hD : 0 < D := by dsimp [D]; positivity | |
| have hbudget : Q * (r * D) ≤ 1 / (100 * L) := by | |
| calc | |
| _ ≤ (1 + Q) * (r * D) := mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hr hD.le) | |
| _ ≤ (1 + Q) * (boundedClassSlabRadius G d kappa q0 M * D) := | |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hrsmall hD.le) (by positivity) | |
| _ = _ := by | |
| change (1 + Q) * (1 / (100 * L * (1 + Q) * D) * D) = 1 / (100 * L) | |
| field_simp [hL.ne', hD.ne', ne_of_gt (show 0 < 1 + Q by positivity)] | |
| have hchange := bounded_class_quotient_change_from_origin h hG hd hgB hdet hp | |
| have hdist := inverseShear_coordinate_sum_le (q0 := q0) hr hx hy | |
| have hchange' : |hessianQuotient g z (inverseShearCoordinates q0 p) - hessianQuotient g z 0| ≤ | |
| 1 / (100 * L) := hchange.trans ((mul_le_mul_of_nonneg_left hdist hQ).trans hbudget) | |
| calc | |
| _ ≤ |hessianQuotient g z (inverseShearCoordinates q0 p) - hessianQuotient g z 0| + | |
| |hessianQuotient g z 0 - q0| := abs_sub_le _ _ _ | |
| _ ≤ 1 / (100 * L) + 1 / (100 * L) := add_le_add hchange' hcenter | |
| _ ≤ 1 / (10 * L) := by | |
| have hInv : 0 < 1 / L := one_div_pos.mpr hL | |
| calc | |
| _ = (1 / 50 : ℝ) * (1 / L) := by | |
| field_simp [hL.ne'] | |
| ring | |
| _ ≤ (1 / 10 : ℝ) * (1 / L) := | |
| mul_le_mul_of_nonneg_right (by norm_num) hInv.le | |
| _ = _ := by field_simp [hL.ne'] | |
| theorem exists_bounded_class_slab_QLowBounds | |
| (G R kappa d q0 : ℝ) (M : ℕ) (hG : 0 ≤ G) (hR : 0 ≤ R) | |
| (hkappa : 0 < kappa) (hd : 0 < d) (hM : 0 < M) : | |
| ∃ MQ : ℝ, 0 ≤ MQ ∧ ∀ (g : MetricField) (z : Coord → ℝ) (S : Set Coord) (r : ℝ), | |
| BoundedAdmissibleHeight g M z → 0 ≤ r → r ≤ boundedClassSlabRadius G d kappa q0 M → | |
| (∀ p ∈ S, inverseShearCoordinates q0 p ∈ modelSquare) → | |
| (∀ p ∈ S, |p 0| ≤ R ∧ |p 1| ≤ R) → | |
| (∀ p ∈ S, |p 0| ≤ boundedClassWidth kappa M * r ∧ |p 1| ≤ r) → | |
| (∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare, | |
| ‖iteratedFDeriv ℝ k (fun a => g a i j) p‖ ≤ G) → | |
| (∀ p ∈ modelSquare, d ≤ |(g p).det|) → | |
| (∀ p ∈ S, gaussianCurvature g (inverseShearCoordinates q0 p) ≤ -kappa / 2) → | |
| |hessianQuotient g z 0 - q0| ≤ 1 / (100 * boundedClassWidth kappa M) → | |
| QLowBounds (metricInShearCoordinates g q0) (heightInShearCoordinates z q0) S | |
| (shearedQPositiveFloor G (M : ℝ) d q0 (M : ℝ)) MQ (boundedClassWidth kappa M / 4) := by | |
| obtain ⟨MQ, hMQ, hlow⟩ := exists_bounded_class_sheared_QLowBounds G R kappa d q0 M hG hR hkappa hd hM | |
| refine ⟨MQ, hMQ, ?_⟩ | |
| intro g z S r hclass hr hrsmall hSp hcoords haxes hgB hdet hK hcenter | |
| exact hlow g z S hclass hSp hcoords | |
| (fun i j k hk p hp => hgB i j k hk _ (hSp p hp)) | |
| (fun p hp => hdet _ (hSp p hp)) hK | |
| (fun p hp => bounded_class_slab_quotient_small hclass hG hd hr hrsmall hgB hdet hcenter | |
| (hSp p hp) (haxes p hp).1 (haxes p hp).2) | |
| end SmoothLocal.Perturbation | |
| end | |
| end OAI | |