| import OAI.Analysis.HyperbolicCones.KernelSmooth | |
| import OAI.Analysis.HyperbolicCones.TangentBasic | |
| namespace OAI | |
| noncomputable section | |
| open scoped ContDiff Topology RealInnerProductSpace Matrix.Norms.L2Operator | |
| open Set Filter | |
| namespace Paper256 | |
| theorem kernel_projection_tangent_derivative (c : ℕ) (hC : 0 < c) | |
| (E : Sym 4 →ₗ[ℝ] Sym c) : | |
| ∃ (v : Vec 4) (T : Vec 4 →ₗ[ℝ] Mat c ℝ), | |
| ‖v‖ = 1 ∧ v 0 ≠ 0 ∧ ∀ h : Vec 4, inner ℝ v h = 0 → | |
| HasDerivAt (fun s : ℝ => kernelProjection | |
| (E (1 - outerSym (tangentCurve v h s)) : Mat c ℝ)) (T h) 0 := by | |
| obtain ⟨v, U, F, hv, hv0, hU, hvU, hF, heq⟩ := | |
| kernel_projection_smooth_local_extension c hC E | |
| have hd : DifferentiableAt ℝ F v := | |
| (hF.contDiffAt (hU.mem_nhds hvU)).differentiableAt (by norm_num) | |
| refine ⟨v, (fderiv ℝ F v).toLinearMap, hv, hv0, ?_⟩ | |
| intro h hh | |
| have hc := tangentCurve_derivative v h | |
| have hcomp : HasDerivAt (fun s => F (tangentCurve v h s)) ((fderiv ℝ F v) h) 0 := by | |
| simpa only [Function.comp_apply] using! | |
| hd.hasFDerivAt.comp_hasDerivAt_of_eq 0 hc (by simp [tangentCurve]) | |
| apply hcomp.congr_of_eventuallyEq | |
| have hmem : ∀ᶠ s in 𝓝 (0 : ℝ), tangentCurve v h s ∈ U := by | |
| apply hc.continuousAt.preimage_mem_nhds | |
| simpa [tangentCurve] using hU.mem_nhds hvU | |
| filter_upwards [hmem] with s hs | |
| exact (heq _ hs (tangentCurve_unit v h hv hh s)).symm | |
| end Paper256 | |
| end | |
| end OAI | |