openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Pulses /QReferenceForcingMargins.lean
| import OAI.Geometry.IsometricImmersion.Darboux.QSameStripMargins | |
| import OAI.Geometry.IsometricImmersion.Pulses.ForcingCoefficientSign | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.Pulse | |
| theorem actual_det_upper_of_entry_bound (g : MetricField) (p : Coord) {G : ℝ} | |
| (hG : 0 ≤ G) (hg : ∀ i j, |g p i j| ≤ G) : (g p).det ≤ 2*G^2 := by | |
| rw [Matrix.det_fin_two] | |
| calc | |
| g p 0 0*g p 1 1-g p 0 1*g p 1 0 ≤ | |
| |g p 0 0*g p 1 1|+|g p 0 1*g p 1 0| := by | |
| linarith [le_abs_self (g p 0 0*g p 1 1), | |
| neg_le_abs (g p 0 1*g p 1 0)] | |
| _ = |g p 0 0| *|g p 1 1|+|g p 0 1| *|g p 1 0| := by rw [abs_mul,abs_mul] | |
| _ ≤ G*G+G*G := add_le_add | |
| (mul_le_mul (hg 0 0) (hg 1 1) (abs_nonneg _) hG) | |
| (mul_le_mul (hg 0 1) (hg 1 0) (abs_nonneg _) hG) | |
| _ = _ := by ring | |
| theorem exists_uniform_reference_forcing_margins (G R M M0 : ℝ) | |
| {d nu c eta kappa : ℝ} (hG : 0 ≤ G) (hR : 0 ≤ R) | |
| (hM : 0 ≤ M) (hM0 : 0 ≤ M0) | |
| (hd : 0 < d) (hnu : 0 < nu) (hc : 0 < c) (heta : 0 < eta) (hkappa : 0 < kappa) : | |
| ∃ epsilon A : ℝ, 0 < epsilon ∧ 1 ≤ A ∧ | |
| ∀ (g g0 : MetricField) (z z0 : Coord → ℝ) (U : Set Coord), | |
| SmoothPositiveOn g U → SmoothPositiveOn g0 U → IsOpen U → | |
| ContDiffOn ℝ ∞ z U → ContDiffOn ℝ ∞ z0 U → | |
| ∀ left right a b : ℝ, a ≤ b → | |
| closedRectangle left right a b ⊆ U → | |
| (∀ p ∈ closedRectangle left right a b, ‖p‖ ≤ R) → | |
| (∀ i j k, k ≤ 2 → ∀ p ∈ closedRectangle left right a b, | |
| ‖iteratedFDeriv ℝ k (fun x => g x i j) p‖ ≤ G) → | |
| (∀ i j k, k ≤ 2 → ∀ p ∈ closedRectangle left right a b, | |
| ‖iteratedFDeriv ℝ k (fun x => g0 x i j) p‖ ≤ G) → | |
| CoordinateBound z (closedRectangle left right a b) 3 M → | |
| CoordinateBound z0 (closedRectangle left right a b) 3 M0 → | |
| (∀ p ∈ closedRectangle left right a b, d ≤ |(g p).det|) → | |
| (∀ p ∈ closedRectangle left right a b, d ≤ |(g0 p).det|) → | |
| (∀ i j k, k ≤ 2 → ∀ p ∈ closedRectangle left right a b, | |
| ‖iteratedFDeriv ℝ k (fun x => g0 x i j - g x i j) p‖ ≤ epsilon) → | |
| (M + M0) * (b - a) ≤ epsilon → | |
| (∀ x ∈ Ioo left right, z0 (boxPoint x a) = z (boxPoint x a)) → | |
| (∀ x ∈ Ioo left right, | |
| coordPartial 1 z0 (boxPoint x a) = coordPartial 1 z (boxPoint x a)) → | |
| (∀ p ∈ closedRectangle left right a b, nu ≤ |covHessian g z p 0 0|) → | |
| (∀ p ∈ closedRectangle left right a b, c ≤ |covHessian g z p 1 1|) → | |
| (∀ p ∈ closedRectangle left right a b, eta ≤ heightEnergy g z p) → | |
| (∀ p ∈ closedRectangle left right a b, gaussianCurvature g p ≤ -kappa) → | |
| (∀ p ∈ closedRectangle left right a b, | |
| (covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p) → | |
| (∀ p ∈ closedRectangle left right a b, | |
| (hessianQuotient g z p)^2 + gaussianCurvature g p * darbouxG g z p < 0) → | |
| ∀ l r : ℝ, left < l → r < right → | |
| ∀ p ∈ closedRectangle l r a b, | |
| eta/2 ≤ heightEnergy g0 z p ∧ | |
| nu/2 ≤ |covHessian g0 z p 0 0| ∧ | |
| |covHessian g0 z p 0 0| ≤ A ∧ | |
| (g0 p).det ≤ 2*G^2 ∧ | |
| forcingFloor (eta/2) (2*G^2) A ≤ |forcingCoefficient g0 z p| := by | |
| obtain ⟨epsilon,A,hepsilon,hA,hmargin⟩ := | |
| exists_uniform_Q_same_strip_margins G R M M0 hG hR hM hM0 hd hnu hc heta hkappa | |
| refine ⟨epsilon,A,hepsilon,hA,?_⟩ | |
| intro g g0 z z0 U hg hg0 hU hz hz0 left right a b hab hbox hpoint | |
| hgB hg0B hzB hz0B hdet hdet0 hmetric hwidth hvalue hvelocity | |
| hxx hyy henergy hK hD htime l r hleft hright p hp | |
| have hp0 : p 0 ∈ Ioo left right := | |
| ⟨hleft.trans_le hp.1.1, hp.1.2.trans_lt hright⟩ | |
| have hpointEq : boxPoint (p 0) (p 1) = p := by | |
| ext i | |
| fin_cases i <;> simp [boxPoint] | |
| have hpbox : p ∈ closedRectangle left right a b := | |
| ⟨⟨hp0.1.le,hp0.2.le⟩,hp.2⟩ | |
| have hm := ((hmargin g g0 z z0 U hg hg0 hU hz hz0 left right a b hab hbox hpoint | |
| hgB hg0B hzB hz0B hdet hdet0 hmetric hwidth hvalue hvelocity | |
| hxx hyy henergy hK hD htime (p 0) hp0 (p 1) hp.2).1 1 (by norm_num)) | |
| simp only [qHeightJetSegment,stateSegment_one,hpointEq, | |
| stateQDenominator_qSolutionJet] at hm | |
| have hE : eta/2 ≤ heightEnergy g0 z p := by | |
| have hs := hm.2.2.1 | |
| unfold stateEnergy at hs | |
| rw [statePoint_qSolutionJet,stateGradient_qSolutionJet, | |
| jetEnergy_at_height hg0 (hbox hpbox) z] at hs | |
| exact hs | |
| have hentry (i j : Fin 2) : |g0 p i j| ≤ G := by | |
| have hh := hg0B i j 0 (by norm_num) p hpbox | |
| simpa only [norm_iteratedFDeriv_zero,Real.norm_eq_abs] using hh | |
| have hdetup := actual_det_upper_of_entry_bound g0 p hG hentry | |
| exact ⟨hE,hm.1,hm.2.1,hdetup, | |
| forcingCoefficient_floor_of_reference_margins hg0 (hbox hpbox) | |
| (half_pos heta) (half_pos hnu) hE hdetup hm.1 hm.2.1⟩ | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |