openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Calculus /ShearedMetricFiniteJets.lean
| import OAI.Geometry.IsometricImmersion.Pulses.OffPulseMetricBounds | |
| import OAI.Geometry.IsometricImmersion.Calculus.ShearMetricJets | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped ContDiff Topology BigOperators Matrix | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE SmoothLocal.Weighted | |
| open SmoothLocal.HighEquation SmoothLocal.Hyperbolic | |
| theorem exists_uniform_off_pulse_sheared_metric_bounds | |
| {gStar : MetricField} {V : Set Coord} | |
| (hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V) | |
| {q0 L r : ℝ} (hL : 0 < L) (hr : 0 < r) (hrhalf : r < 1/2) | |
| (hLr : L*r ≤ 1/20) (hq0 : |q0| ≤ 1/20) (m : ℕ) : | |
| ∃ B : ℝ, 0 ≤ B ∧ ∀ N : ℕ, ∀ delta : ℝ, 0 < delta → | |
| ∀ tau : ℕ, 1 ≤ tau → m ≤ tau → | |
| ∀ (gTau : MetricField) (U : Set Coord), SmoothPositiveOn gTau U → IsOpen U → modelSquare ⊆ U → | |
| (∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare, | |
| ‖iteratedFDeriv ℝ k (fun q => gTau q i j- | |
| testMetric gStar q0 (L*r/16) N delta (tau : ℝ) q i j) p‖ ≤ metricApproximationAccuracy tau) → | |
| ∀ i j : Fin 2, CoordinateBound (fun p => metricInShearCoordinates gTau q0 p i j) | |
| {p | |p 0| ≤ L*r ∧ p 1 ∈ Icc (-r) (-delta/(tau : ℝ))} m B := by | |
| obtain ⟨B,hB,hjets⟩ := exists_uniform_off_pulse_metric_jets hgStar hV hSV | |
| hL hr hrhalf hLr hq0 m | |
| refine ⟨shearedMetricCoordinateBudget B q0 m, | |
| shearedMetricCoordinateBudget_nonneg q0 m hB,?_⟩ | |
| intro N delta hd tau htau hmtau gTau U hgTau hU hSU happrox | |
| let T : Set Coord := {p | |p 0| ≤ L*r ∧ p 1 ∈ Icc (-r) (-delta/(tau : ℝ))} | |
| let S : Set Coord := inverseShearCoordinates q0 '' T | |
| have htr : (0 : ℝ) < tau := zero_lt_one.trans_le (by exact_mod_cast htau) | |
| have ht0 : -delta/(tau : ℝ) < 0 := div_neg_of_neg_of_pos (neg_neg_of_pos hd) htr | |
| have hpS (p : Coord) (hp : p ∈ T) : inverseShearCoordinates q0 p ∈ modelSquare := by | |
| apply sectionFourClosedSlab_mem_shearedModelSquare hr hrhalf hLr hq0 | |
| exact ⟨hp.1,abs_le.mpr ⟨hp.2.1,(hp.2.2.trans ht0.le).trans hr.le⟩⟩ | |
| have hsub : S ⊆ U := by | |
| rintro y ⟨p,hp,rfl⟩ | |
| exact hSU (hpS p hp) | |
| have hb (i j : Fin 2) : CoordinateBound (fun p => gTau p i j) S m B := by | |
| apply coordinateBound_of_frechet_bounds (hgTau.1 i j) hU hsub | |
| intro k hk y hy | |
| obtain ⟨p,hp,rfl⟩ := hy | |
| have he : ![p 0,p 1] = p := by ext a; fin_cases a <;> rfl | |
| simpa only [he] using hjets N delta hd tau htau hmtau gTau U hgTau hU hSU happrox | |
| (p 0) (p 1) hp.1 hp.2 i j k hk | |
| have hs := metric_coordinate_bound_inverse_shear hgTau hU hsub hB hb q0 | |
| intro i j ds hds p hp | |
| exact hs i j ds hds p ⟨p,hp,rfl⟩ | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |