Datasets:

roseking
/

openAI-math-dataset

	import OAI.Geometry.IsometricImmersion.Taylor.TopChainExtraction
	import OAI.Geometry.IsometricImmersion.Estimates.LargestBlockLinearization
	import OAI.Geometry.IsometricImmersion.Immersions.HeightCoefficientRegularity

	namespace OAI

	noncomputable section
	open Set Function Filter
	open scoped ContDiff Topology BigOperators Matrix

	namespace SmoothLocal.HighEquation
	open SmoothLocal.Geometry

	theorem second_fderiv_comp_constant_direction
	{F : DarbouxState → ℝ} {J : Coord → DarbouxState} {p : Coord}
	(hF : ContDiffAt ℝ ∞ F (J p)) (hJ : DifferentiableAt ℝ J p) (v : DarbouxState) :
	iteratedFDeriv ℝ 2 F (J p) ![coordPartial 1 J p, v] =
	coordPartial 1 (fun r => fderiv ℝ F (J r) v) p := by
	have hdF : DifferentiableAt ℝ (fderiv ℝ F) (J p) :=
	(hF.fderiv_right (m := 1) (WithTop.coe_le_coe.mpr le_top)).differentiableAt (by norm_num)
	have hc : DifferentiableAt ℝ (fun r => fderiv ℝ F (J r)) p := hdF.comp p hJ
	rw [iteratedFDeriv_two_apply]
	simp only [Matrix.cons_val_zero, Matrix.cons_val_one]
	unfold coordPartial
	have hcomp : fderiv ℝ (fun r => fderiv ℝ F (J r)) p =
	(fderiv ℝ (fderiv ℝ F) (J p)).comp (fderiv ℝ J p) := by
	convert fderiv_comp p hdF hJ using 1
	rfl
	rw [fderiv_clm_apply hc (differentiableAt_const (c := v)), hcomp]
	simp

	theorem sixVariableP_second_vertical_direction
	{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
	(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
	(hyy : ∀ r ∈ U, covHessian g z r 1 1 ≠ 0) (hp : p ∈ U)
	(v : DarbouxState) (hv0 : v 0 = 0) (hv1 : v 1 = 0) :
	iteratedFDeriv ℝ 2 (sixVariableP g) (solutionJet z p)
	![coordPartial 1 (solutionJet z) p, v] =
	coordPartial 1 (heightPFirst g z 0) p * v 2 +
	coordPartial 1 (heightPFirst g z 1) p * v 3 +
	2 * coordPartial 1 (hessianQuotient g z) p * v 4 -
	coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
	gaussianCurvature g r * darbouxG g z r) p * v 5 := by
	rw [second_fderiv_comp_constant_direction
	(sixVariableP_contDiffAt_solutionJet hg hU hp (hyy _ hp))
	(((solutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp))]
	have heq : (fun r => fderiv ℝ (sixVariableP g) (solutionJet z r) v) =ᶠ[𝓝 p]
	(fun r => heightPFirst g z 0 r * v 2 + heightPFirst g z 1 r * v 3 +
	2 * hessianQuotient g z r * v 4 -
	((hessianQuotient g z r)^2 + gaussianCurvature g r * darbouxG g z r) * v 5) := by
	filter_upwards [hU.mem_nhds hp] with r hr
	exact sixVariableP_vertical_direction hg hU hr (hyy _ hr) v hv0 hv1
	have h0 := ((heightPFirst_contDiffOn hg hU hz hyy 0).contDiffAt
	(hU.mem_nhds hp)).differentiableAt (by simp)
	have h1 := ((heightPFirst_contDiffOn hg hU hz hyy 1).contDiffAt
	(hU.mem_nhds hp)).differentiableAt (by simp)
	have hq := ((hessianQuotient_contDiffOn hg hU hz hyy).contDiffAt
	(hU.mem_nhds hp)).differentiableAt (by simp)
	have hK := ((gaussianCurvature_contDiffOn hg hU).contDiffAt
	(hU.mem_nhds hp)).differentiableAt (by simp)
	have hG := ((darbouxG_contDiffOn hg hU hz hyy).contDiffAt
	(hU.mem_nhds hp)).differentiableAt (by simp)
	have ha : DifferentiableAt ℝ (fun r => (hessianQuotient g z r)^2 +
	gaussianCurvature g r * darbouxG g z r) p := (hq.pow 2).add (hK.mul hG)
	have hd := (((h0.hasFDerivAt.mul_const (v 2)).add
	(h1.hasFDerivAt.mul_const (v 3))).add
	((hq.hasFDerivAt.const_mul 2).mul_const (v 4))).sub (ha.hasFDerivAt.mul_const (v 5))
	simp only [Pi.add_def, Pi.sub_def] at hd
	unfold coordPartial
	rw [heq.fderiv_eq]
	change fderiv ℝ (fun r => heightPFirst g z 0 r * v 2 + heightPFirst g z 1 r * v 3 +
	2 * hessianQuotient g z r * v 4 -
	((hessianQuotient g z r)^2 + gaussianCurvature g r * darbouxG g z r) * v 5) p
	(Pi.single 1 1) = _
	rw [hd.fderiv]
	simp only [add_apply, sub_apply, smul_apply, smul_eq_mul]
	ring

	def firstJetPairRemainder (g : MetricField) (z : Coord → ℝ) (n : ℕ) (p : Coord) : ℝ :=
	coordPartial 1 (heightPFirst g z 0) p * coordPartial 0 (verticalJet z n) p +
	coordPartial 1 (heightPFirst g z 1) p * coordPartial 1 (verticalJet z n) p

	def secondJetCorrection (g : MetricField) (z u : Coord → ℝ) (p : Coord) : ℝ :=
	2 * coordPartial 1 (hessianQuotient g z) p * coordPartial 0 u p -
	coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
	gaussianCurvature g r * darbouxG g z r) p * coordPartial 1 u p

	theorem actualP_pair_extract_secondJet
	{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {x y : ℝ}
	(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
	(hyy : ∀ p ∈ U, covHessian g z p 1 1 ≠ 0)
	(hp : (![x, y] : Coord) ∈ U) (n : ℕ) (hn : 2 ≤ n) :
	chainTerm (sixVariableP g) (solutionJetCurve z x) (ChainWord.pair 1 n) y =
	secondJetCorrection g z (verticalJet z (n + 1)) ![x, y] +
	firstJetPairRemainder g z n ![x, y] := by
	have hfirst : iteratedDeriv 1 (solutionJetCurve z x) y =
	coordPartial 1 (solutionJet z) ![x, y] :=
	verticalJet_slice hU (solutionJet_contDiffOn hU hz) x 1 y hp
	rw [chainTerm_pair_eq, hfirst, solutionJetCurve_iteratedDeriv hU hz hp n hn]
	change iteratedFDeriv ℝ 2 (sixVariableP g) (solutionJet z ![x, y]) _ = _
	rw [sixVariableP_second_vertical_direction hg hU hz hyy hp _ (by rfl) (by rfl)]
	change
	coordPartial 1 (heightPFirst g z 0) ![x, y] * coordPartial 0 (verticalJet z n) ![x, y] +
	coordPartial 1 (heightPFirst g z 1) ![x, y] * coordPartial 1 (verticalJet z n) ![x, y] +
	2 * coordPartial 1 (hessianQuotient g z) ![x, y] *
	coordPartial 0 (coordPartial 1 (verticalJet z n)) ![x, y] -
	coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
	gaussianCurvature g r * darbouxG g z r) ![x, y] *
	coordPartial 1 (coordPartial 1 (verticalJet z n)) ![x, y] = _
	unfold secondJetCorrection firstJetPairRemainder
	change _ =
	(2 * coordPartial 1 (hessianQuotient g z) ![x, y] *
	coordPartial 0 (coordPartial 1 (verticalJet z n)) ![x, y] -
	coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
	gaussianCurvature g r * darbouxG g z r) ![x, y] *
	coordPartial 1 (coordPartial 1 (verticalJet z n)) ![x, y]) + _
	ring

	end SmoothLocal.HighEquation

	end

	end OAI