openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Coordinates /BoundedPatchSlabGeometry.lean
| import OAI.Geometry.IsometricImmersion.Caps.BoundedClassCapRadius | |
| import OAI.Geometry.IsometricImmersion.Metrics.MetricPatchOpen | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped ContDiff Matrix | |
| namespace SmoothLocal.Perturbation | |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow | |
| open SmoothLocal.ODE SmoothLocal.Hyperbolic | |
| theorem inverseShear_slab_mem_centralBox {L r q0 : ℝ} {p : Coord} | |
| (hL : 1 ≤ L) (hr : 0 ≤ r) (hLr : L * r ≤ 1 / 20) (hq0 : |q0| ≤ 1 / 20) | |
| (hx : |p 0| ≤ L * r) (hy : |p 1| ≤ r) : | |
| inverseShearCoordinates q0 p ∈ centralBox := by | |
| have hrL : r ≤ L * r := by nlinarith | |
| have hrsmall : r ≤ 1 / 20 := hrL.trans hLr | |
| have hxsmall : |p 0| ≤ 1 / 20 := hx.trans hLr | |
| have hprod : |q0 * p 0| ≤ (1 / 20 : ℝ) * (1 / 20) := by | |
| rw [abs_mul] | |
| exact mul_le_mul hq0 hxsmall (abs_nonneg _) (by norm_num) | |
| have hysmall : |p 1 - q0 * p 0| ≤ (1 / 20 : ℝ) + (1 / 20) * (1 / 20) := | |
| (abs_sub _ _).trans (add_le_add (hy.trans hrsmall) hprod) | |
| change (fun _ => -(1 / 5 : ℝ)) ≤ inverseShearCoordinates q0 p ∧ | |
| inverseShearCoordinates q0 p ≤ (fun _ => (1 / 5 : ℝ)) | |
| constructor | |
| · intro i | |
| fin_cases i | |
| · change -(1 / 5 : ℝ) ≤ p 0 | |
| linarith [(abs_le.mp hxsmall).1] | |
| · change -(1 / 5 : ℝ) ≤ p 1 - q0 * p 0 | |
| linarith [(abs_le.mp hysmall).1] | |
| · intro i | |
| fin_cases i | |
| · change p 0 ≤ (1 / 5 : ℝ) | |
| linarith [(abs_le.mp hxsmall).2] | |
| · change p 1 - q0 * p 0 ≤ (1 / 5 : ℝ) | |
| linarith [(abs_le.mp hysmall).2] | |
| theorem shrinking_class_slab_axis_bounds {kappa r b : ℝ} {M : ℕ} {p : Coord} | |
| (hr : 0 ≤ r) (hb : b ≤ 0) | |
| (hp : p ∈ shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r) | |
| (-r) b (boundedClassWidth kappa M / 4)) : | |
| |p 0| ≤ boundedClassWidth kappa M * r ∧ |p 1| ≤ r := by | |
| have ht := hp.1 | |
| have hx := hp.2 | |
| have hspeed : 0 ≤ boundedClassWidth kappa M / 4 := | |
| div_nonneg (boundedClassWidth_pos kappa M).le (by norm_num) | |
| have hmove : 0 ≤ (boundedClassWidth kappa M / 4) * (p 1 - (-r)) := | |
| mul_nonneg hspeed (sub_nonneg.mpr ht.1) | |
| unfold inwardLeft inwardRight at hx | |
| exact ⟨abs_le.mpr ⟨by linarith [hx.1], by linarith [hx.2]⟩, | |
| abs_le.mpr ⟨ht.1, by linarith [ht.2]⟩⟩ | |
| theorem actual_patch_curvature_on_class_slab {g0 : MetricField} {kappa q0 r b : ℝ} {M : ℕ} | |
| (eta : metricPatchSet g0 kappa) (hr : 0 ≤ r) (hb : b ≤ 0) | |
| (hLr : boundedClassWidth kappa M * r ≤ 1 / 20) (hq0 : |q0| ≤ 1 / 20) | |
| {p : Coord} | |
| (hp : p ∈ shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r) | |
| (-r) b (boundedClassWidth kappa M / 4)) : | |
| inverseShearCoordinates q0 p ∈ centralBox ∧ | |
| gaussianCurvature (perturbedMetric g0 eta.val) (inverseShearCoordinates q0 p) < -kappa / 2 := by | |
| obtain ⟨hx, hy⟩ := shrinking_class_slab_axis_bounds hr hb hp | |
| have hc := inverseShear_slab_mem_centralBox | |
| ((by norm_num : (1 : ℝ) ≤ 100).trans (boundedClassWidth_ge_hundred kappa M)) hr hLr hq0 hx hy | |
| exact ⟨hc, eta.property.2 _ hc⟩ | |
| theorem exists_bounded_patch_actual_slab_QLowBounds | |
| (G kappa d q0 : ℝ) (M : ℕ) (hG : 0 ≤ G) | |
| (hkappa : 0 < kappa) (hd : 0 < d) (hM : 0 < M) (hq0 : |q0| ≤ 1 / 20) : | |
| ∃ MQ : ℝ, 0 ≤ MQ ∧ ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa) | |
| (z : Coord → ℝ) (r b : ℝ), | |
| BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z → 0 ≤ r → b ≤ 0 → | |
| boundedClassWidth kappa M * r ≤ 1 / 20 → | |
| heightQuotientJetBound G (M : ℝ) d (1 / (M : ℝ)) * | |
| (r + 107 * (boundedClassWidth kappa M * r) / 100) ≤ | |
| 9 / (100 * boundedClassWidth kappa M) → | |
| (∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare, | |
| ‖iteratedFDeriv ℝ k (fun a => perturbedMetric g0 eta.val a i j) p‖ ≤ G) → | |
| (∀ p ∈ modelSquare, d ≤ |(perturbedMetric g0 eta.val p).det|) → | |
| |hessianQuotient (perturbedMetric g0 eta.val) z 0 - q0| ≤ | |
| 1 / (100 * boundedClassWidth kappa M) → | |
| QLowBounds (metricInShearCoordinates (perturbedMetric g0 eta.val) q0) | |
| (heightInShearCoordinates z q0) | |
| (shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r) | |
| (-r) b (boundedClassWidth kappa M / 4)) | |
| (shearedQPositiveFloor G (M : ℝ) d q0 (M : ℝ)) MQ (boundedClassWidth kappa M / 4) := by | |
| obtain ⟨MQ, hMQ, hlow⟩ := exists_bounded_class_cap_radius_QLowBounds | |
| G 1 kappa d q0 M hG (by norm_num) hkappa hd hM hq0 | |
| refine ⟨MQ, hMQ, ?_⟩ | |
| intro g0 eta z r b hclass hr hb hLr hrsmall hgB hdet hcenter | |
| let S := shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r) | |
| (-r) b (boundedClassWidth kappa M / 4) | |
| have haxes (p : Coord) (hp : p ∈ S) := shrinking_class_slab_axis_bounds hr hb hp | |
| have hcentral (p : Coord) (hp : p ∈ S) := actual_patch_curvature_on_class_slab eta hr hb hLr hq0 hp | |
| have hrsmall' : r ≤ 1 / 20 := by | |
| have hL : 1 ≤ boundedClassWidth kappa M := | |
| (by norm_num : (1 : ℝ) ≤ 100).trans (boundedClassWidth_ge_hundred kappa M) | |
| have hh : r ≤ boundedClassWidth kappa M * r := by nlinarith | |
| exact hh.trans hLr | |
| exact hlow (perturbedMetric g0 eta.val) z S r hclass hr hrsmall | |
| (fun p hp => centralBox_subset_modelSquare (hcentral p hp).1) | |
| (fun p hp => ⟨by linarith [(haxes p hp).1], by linarith [(haxes p hp).2]⟩) | |
| haxes hgB hdet (fun p hp => (hcentral p hp).2.le) hcenter | |
| end SmoothLocal.Perturbation | |
| end | |
| end OAI | |