openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Darboux /QSpatialFiniteInduction.lean
| import OAI.Geometry.IsometricImmersion.Darboux.QSpatialInductionStep | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped ContDiff Topology NNReal | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic | |
| theorem exists_Q_spatial_induction_offset | |
| (G R Z s0 M speed xl xr T lengthFloor : ℝ) (A : ℝ≥0) | |
| (hG : 0 ≤ G) (hR : 0 ≤ R) (hZ : 0 ≤ Z) (hs0 : 0 < s0) (hM : 0 ≤ M) | |
| (hspeed : 0 ≤ speed) (hT : 0 ≤ T) (hlen : 0 < lengthFloor) | |
| (hwidth : lengthFloor ≤ xr - xl - 2 * speed * T) | |
| {d c : ℝ} (hd : 0 < d) (hc : 0 < c) : | |
| ∀ k : ℕ, ∃ H : ℝ≥0, | |
| ∀ (g : MetricField) (z : Coord → ℝ) (radius lo hi a b : ℝ), | |
| SmoothPositiveOn g (coordinateRectangle radius lo hi) → | |
| ContDiffOn ℝ ∞ z (coordinateRectangle radius lo hi) → | |
| (∀ p ∈ coordinateRectangle radius lo hi, | |
| (covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p) → | |
| (∀ p ∈ coordinateRectangle radius lo hi, covHessian g z p 0 0 ≠ 0) → | |
| 0 < radius → a ≤ b → b - a ≤ T → | |
| -radius < xl → xr < radius → a ∈ Ioo lo hi → b ∈ Ioo lo hi → | |
| QLowBounds g z (shrinkingSlab xl xr a b speed) s0 M speed → | |
| (∀ p ∈ shrinkingSlab xl xr a b speed, |p 0| ≤ R ∧ |p 1| ≤ R) → | |
| (∀ i j, CoordinateBound (fun p => g p i j) (shrinkingSlab xl xr a b speed) (k + 9) G) → | |
| CoordinateBound z (shrinkingSlab xl xr a b speed) 8 Z → | |
| (∀ p ∈ shrinkingSlab xl xr a b speed, d ≤ |(g p).det|) → | |
| (∀ p ∈ shrinkingSlab xl xr a b speed, c ≤ |covHessian g z p 0 0|) → | |
| (∀ ds : List (Fin 2), ds.length ≤ k + 8 → ∀ x ∈ Icc xl xr, | |
| |iteratedCoordPartial ds z (coordinatePoint x a)| ≤ (A : ℝ)) → | |
| ∀ t ∈ Icc a b, SpatialSliceL2Bound (spatialFirstJet z) t | |
| (inwardLeft xl a speed t) (inwardRight xr a speed t) (k + 7) H := by | |
| intro k | |
| induction k with | |
| | zero => | |
| refine ⟨firstJetSliceBudget Z (xr - xl), ?_⟩ | |
| intro g z radius lo hi a b hg hz hD hxx hradius hab habT hxl hxr ha hb | |
| hLow hcoords hgB hzB hdet hden hcut t ht | |
| apply coordinateC8_to_spatialFirstJet_L2_seven hzB hZ | |
| (fun x hx => point_mem_shrinkingSlab ht hx) | |
| unfold inwardLeft inwardRight | |
| have hh : 0 ≤ speed * (t - a) := mul_nonneg hspeed (sub_nonneg.mpr ht.1) | |
| linarith | |
| | succ k ih => | |
| obtain ⟨Hprev, hprev⟩ := ih | |
| obtain ⟨Hnew, hle, hstep⟩ := exists_Q_spatial_induction_step | |
| G R Z s0 M speed xl xr T lengthFloor Hprev A hG hR hs0 hM hspeed hT hlen hwidth | |
| hd hc (k + 5) (by omega) | |
| refine ⟨Hnew, ?_⟩ | |
| intro g z radius lo hi a b hg hz hD hxx hradius hab habT hxl hxr ha hb | |
| hLow hcoords hgB hzB hdet hden hcut t ht | |
| have hold : ∀ v ∈ Icc a b, SpatialSliceL2Bound (spatialFirstJet z) v | |
| (inwardLeft xl a speed v) (inwardRight xr a speed v) (k + 7) Hprev := | |
| hprev g z radius lo hi a b hg hz hD hxx hradius hab habT hxl hxr ha hb | |
| hLow hcoords (fun i j => (hgB i j).mono (by omega) le_rfl) hzB hdet hden | |
| (fun ds hds x hx => hcut ds (by omega) x hx) | |
| have hnew := hstep g z radius lo hi a b hg hz hD hxx hradius hab habT hxl hxr ha hb | |
| hLow hcoords (fun i j => (hgB i j).mono (by omega) le_rfl) hzB hdet hden | |
| (fun ds hds x hx => hcut ds (by omega) x hx) | |
| (by simpa only [Nat.add_assoc] using hold) t ht | |
| simpa only [Nat.succ_eq_add_one, Nat.add_assoc] using hnew | |
| theorem exists_finite_Q_spatial_bound | |
| (G R Z s0 M speed xl xr T lengthFloor : ℝ) (A : ℝ≥0) | |
| (hG : 0 ≤ G) (hR : 0 ≤ R) (hZ : 0 ≤ Z) (hs0 : 0 < s0) (hM : 0 ≤ M) | |
| (hspeed : 0 ≤ speed) (hT : 0 ≤ T) (hlen : 0 < lengthFloor) | |
| (hwidth : lengthFloor ≤ xr - xl - 2 * speed * T) | |
| {d c : ℝ} (hd : 0 < d) (hc : 0 < c) (N : ℕ) : | |
| ∃ H : ℝ≥0, | |
| ∀ (g : MetricField) (z : Coord → ℝ) (radius lo hi a b : ℝ), | |
| SmoothPositiveOn g (coordinateRectangle radius lo hi) → | |
| ContDiffOn ℝ ∞ z (coordinateRectangle radius lo hi) → | |
| (∀ p ∈ coordinateRectangle radius lo hi, | |
| (covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p) → | |
| (∀ p ∈ coordinateRectangle radius lo hi, covHessian g z p 0 0 ≠ 0) → | |
| 0 < radius → a ≤ b → b - a ≤ T → | |
| -radius < xl → xr < radius → a ∈ Ioo lo hi → b ∈ Ioo lo hi → | |
| QLowBounds g z (shrinkingSlab xl xr a b speed) s0 M speed → | |
| (∀ p ∈ shrinkingSlab xl xr a b speed, |p 0| ≤ R ∧ |p 1| ≤ R) → | |
| (∀ i j, CoordinateBound (fun p => g p i j) (shrinkingSlab xl xr a b speed) (max 9 (N + 2)) G) → | |
| CoordinateBound z (shrinkingSlab xl xr a b speed) 8 Z → | |
| (∀ p ∈ shrinkingSlab xl xr a b speed, d ≤ |(g p).det|) → | |
| (∀ p ∈ shrinkingSlab xl xr a b speed, c ≤ |covHessian g z p 0 0|) → | |
| (∀ ds : List (Fin 2), ds.length ≤ max 8 (N + 1) → ∀ x ∈ Icc xl xr, | |
| |iteratedCoordPartial ds z (coordinatePoint x a)| ≤ (A : ℝ)) → | |
| ∀ t ∈ Icc a b, SpatialSliceL2Bound (spatialFirstJet z) t | |
| (inwardLeft xl a speed t) (inwardRight xr a speed t) N H := by | |
| obtain ⟨H, hH⟩ := exists_Q_spatial_induction_offset | |
| G R Z s0 M speed xl xr T lengthFloor A hG hR hZ hs0 hM hspeed hT hlen hwidth hd hc (N - 7) | |
| refine ⟨H, ?_⟩ | |
| intro g z radius lo hi a b hg hz hD hxx hradius hab habT hxl hxr ha hb | |
| hLow hcoords hgB hzB hdet hden hcut t ht | |
| have hmetricOrder : N - 7 + 9 ≤ max 9 (N + 2) := by omega | |
| have hcutOrder : N - 7 + 8 ≤ max 8 (N + 1) := by omega | |
| have hb := hH g z radius lo hi a b hg hz hD hxx hradius hab habT hxl hxr ha hb | |
| hLow hcoords (fun i j => (hgB i j).mono hmetricOrder le_rfl) hzB hdet hden | |
| (fun ds hds x hx => hcut ds (hds.trans hcutOrder) x hx) t ht | |
| exact SpatialSliceL2Bound.mono_order_budget hb (by omega) le_rfl | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |