openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Darboux /SixVariableLinearization.lean
| import OAI.Geometry.IsometricImmersion.Calculus.VerticalSliceJets | |
| import OAI.Geometry.IsometricImmersion.Darboux.ActualDriftEquation | |
| import Mathlib.Analysis.Calculus.Deriv.Pi | |
| namespace OAI | |
| noncomputable section | |
| open Set Function | |
| open scoped ContDiff Matrix | |
| namespace SmoothLocal.HighEquation | |
| open SmoothLocal.Geometry | |
| theorem fderiv_axis_eq_deriv_update {n : ℕ} {f : (Fin n → ℝ) → ℝ} {w : Fin n → ℝ} | |
| (hf : DifferentiableAt ℝ f w) (i : Fin n) : | |
| fderiv ℝ f w (Pi.single i 1) = deriv (fun t => f (Function.update w i t)) (w i) := by | |
| have hd : HasFDerivAt f (fderiv ℝ f w) (Function.update w i (w i)) := by | |
| simpa using hf.hasFDerivAt | |
| exact (hd.comp_hasDerivAt (w i) (hasDerivAt_update w i (w i))).deriv.symm | |
| def gradientStateIndex (i : Fin 2) : Fin 6 := if i = 0 then 2 else 3 | |
| theorem sixVariableP_update_gradient (g : MetricField) (w : DarbouxState) (i : Fin 2) (t : ℝ) : | |
| sixVariableP g (Function.update w (gradientStateIndex i) t) = | |
| solvedDarboux g (statePoint w) (w 4) (w 5) (Function.update (stateGradient w) i t) := by | |
| have hp : statePoint (Function.update w (gradientStateIndex i) t) = statePoint w := by | |
| ext j | |
| fin_cases i <;> fin_cases j <;> simp [gradientStateIndex, statePoint] | |
| have hv : stateGradient (Function.update w (gradientStateIndex i) t) = | |
| Function.update (stateGradient w) i t := by | |
| ext j | |
| fin_cases i <;> fin_cases j <;> simp [gradientStateIndex, stateGradient] | |
| unfold sixVariableP | |
| rw [hp, hv] | |
| fin_cases i <;> simp [gradientStateIndex] | |
| theorem sixVariableP_update_mixed (g : MetricField) (w : DarbouxState) (t : ℝ) : | |
| sixVariableP g (Function.update w 4 t) = | |
| solvedDarboux g (statePoint w) t (w 5) (stateGradient w) := by | |
| have hp : statePoint (Function.update w 4 t) = statePoint w := by | |
| ext j | |
| fin_cases j <;> simp [statePoint] | |
| have hv : stateGradient (Function.update w 4 t) = stateGradient w := by | |
| ext j | |
| fin_cases j <;> simp [stateGradient] | |
| unfold sixVariableP | |
| rw [hp, hv] | |
| simp | |
| theorem sixVariableP_update_yy (g : MetricField) (w : DarbouxState) (t : ℝ) : | |
| sixVariableP g (Function.update w 5 t) = | |
| solvedDarboux g (statePoint w) (w 4) t (stateGradient w) := by | |
| have hp : statePoint (Function.update w 5 t) = statePoint w := by | |
| ext j | |
| fin_cases j <;> simp [statePoint] | |
| have hv : stateGradient (Function.update w 5 t) = stateGradient w := by | |
| ext j | |
| fin_cases j <;> simp [stateGradient] | |
| unfold sixVariableP | |
| rw [hp, hv] | |
| simp | |
| theorem solvedDarboux_first_variables_differentiableAt (g : MetricField) (p u : Coord) (b c : ℝ) | |
| (hyy : jetYY g p c u ≠ 0) : DifferentiableAt ℝ (solvedDarboux g p b c) u := by | |
| have hd := (jetConnection_differentiableAt g p u 0 0).add | |
| ((jetNumerator_differentiableAt g p u b).mul ((jetYY_differentiableAt g p u c).inv hyy)) | |
| exact hd | |
| theorem sixVariableP_gradient_axis | |
| {g : MetricField} {U : Set Coord} {w : DarbouxState} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hw : w ∈ darbouxStateDomain g U) (i : Fin 2) : | |
| fderiv ℝ (sixVariableP g) w (Pi.single (gradientStateIndex i) 1) = | |
| coordPartial i (solvedDarboux g (statePoint w) (w 4) (w 5)) (stateGradient w) := by | |
| have hP : DifferentiableAt ℝ (sixVariableP g) w := | |
| ((sixVariableP_contDiffOn hg hU).contDiffAt | |
| ((darbouxStateDomain_isOpen hg hU).mem_nhds hw)).differentiableAt (by simp) | |
| rw [fderiv_axis_eq_deriv_update hP] | |
| have heq : (fun t => sixVariableP g (Function.update w (gradientStateIndex i) t)) = | |
| (fun t => solvedDarboux g (statePoint w) (w 4) (w 5) (Function.update (stateGradient w) i t)) := | |
| funext (sixVariableP_update_gradient g w i) | |
| rw [heq] | |
| have hi : w (gradientStateIndex i) = stateGradient w i := by | |
| fin_cases i <;> simp [gradientStateIndex, stateGradient] | |
| rw [hi] | |
| exact (fderiv_axis_eq_deriv_update | |
| (solvedDarboux_first_variables_differentiableAt g (statePoint w) (stateGradient w) (w 4) (w 5) hw.2) i).symm | |
| theorem sixVariableP_gradient_axis_at_height | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U) | |
| (hyy : covHessian g z p 1 1 ≠ 0) (i : Fin 2) : | |
| fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single (gradientStateIndex i) 1) = | |
| heightPFirst g z i p := by | |
| rw [sixVariableP_gradient_axis hg hU (solutionJet_mem_domain g z hp hyy) i, | |
| statePoint_solutionJet, stateGradient_solutionJet] | |
| rfl | |
| theorem sixVariableP_mixed_axis_at_height | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U) | |
| (hyy : covHessian g z p 1 1 ≠ 0) : | |
| fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 4 1) = 2 * hessianQuotient g z p := by | |
| have hP := (sixVariableP_contDiffAt_solutionJet hg hU hp hyy).differentiableAt (by simp) | |
| rw [fderiv_axis_eq_deriv_update hP] | |
| have heq : (fun t => sixVariableP g (Function.update (solutionJet z p) 4 t)) = | |
| (fun t => solvedDarboux g p t (coordPartial 1 (coordPartial 1 z) p) (fun i => coordPartial i z p)) := by | |
| funext t | |
| rw [sixVariableP_update_mixed, statePoint_solutionJet, stateGradient_solutionJet] | |
| rfl | |
| rw [heq] | |
| change deriv (fun t => solvedDarboux g p t (coordPartial 1 (coordPartial 1 z) p) | |
| (fun i => coordPartial i z p)) (coordPartial 0 (coordPartial 1 z) p) = _ | |
| rw [solvedDarboux_deriv_b g p _ _ _ hyy, jetMixed_at_height, jetYY_at_height] | |
| rfl | |
| theorem sixVariableP_yy_axis_at_height | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U) | |
| (hyy : covHessian g z p 1 1 ≠ 0) : | |
| fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 5 1) = | |
| -((hessianQuotient g z p)^2 + gaussianCurvature g p * darbouxG g z p) := by | |
| have hP := (sixVariableP_contDiffAt_solutionJet hg hU hp hyy).differentiableAt (by simp) | |
| rw [fderiv_axis_eq_deriv_update hP] | |
| have heq : (fun t => sixVariableP g (Function.update (solutionJet z p) 5 t)) = | |
| (fun t => solvedDarboux g p (coordPartial 0 (coordPartial 1 z) p) t (fun i => coordPartial i z p)) := by | |
| funext t | |
| rw [sixVariableP_update_yy, statePoint_solutionJet, stateGradient_solutionJet] | |
| rfl | |
| rw [heq] | |
| change deriv (fun t => solvedDarboux g p (coordPartial 0 (coordPartial 1 z) p) t | |
| (fun i => coordPartial i z p)) (coordPartial 1 (coordPartial 1 z) p) = _ | |
| rw [solvedDarboux_deriv_c g p _ _ _ hyy, jetMixed_at_height, jetYY_at_height, | |
| jetEnergy_at_height hg hp z] | |
| unfold hessianQuotient darbouxG | |
| ring | |
| theorem sixVariableP_vertical_direction | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} | |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U) | |
| (hyy : covHessian g z p 1 1 ≠ 0) (v : DarbouxState) (hv0 : v 0 = 0) (hv1 : v 1 = 0) : | |
| fderiv ℝ (sixVariableP g) (solutionJet z p) v = | |
| heightPFirst g z 0 p * v 2 + heightPFirst g z 1 p * v 3 + | |
| 2 * hessianQuotient g z p * v 4 - | |
| ((hessianQuotient g z p)^2 + gaussianCurvature g p * darbouxG g z p) * v 5 := by | |
| have he : v = v 2 • (Pi.single 2 (1 : ℝ) : DarbouxState) + | |
| v 3 • Pi.single 3 1 + v 4 • Pi.single 4 1 + v 5 • Pi.single 5 1 := by | |
| ext i | |
| fin_cases i <;> simp [hv0, hv1] | |
| have h0 : fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 2 1) = heightPFirst g z 0 p := by | |
| simpa only [gradientStateIndex, ↓reduceIte] using sixVariableP_gradient_axis_at_height hg hU hp hyy 0 | |
| have h1 : fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 3 1) = heightPFirst g z 1 p := by | |
| simpa only [gradientStateIndex, show (1 : Fin 2) ≠ 0 by decide, ↓reduceIte] using | |
| sixVariableP_gradient_axis_at_height hg hU hp hyy 1 | |
| have hl := congrArg (fderiv ℝ (sixVariableP g) (solutionJet z p)) he | |
| simp only [map_add, map_smul, h0, h1, sixVariableP_mixed_axis_at_height hg hU hp hyy, | |
| sixVariableP_yy_axis_at_height hg hU hp hyy, smul_eq_mul] at hl | |
| rw [hl] | |
| ring | |
| end SmoothLocal.HighEquation | |
| end | |
| end OAI | |