openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Geometry /AnalyticInverseChart.lean
| import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff | |
| import Mathlib.Topology.OpenPartialHomeomorph.Composition | |
| namespace OAI | |
| /-! # Restricting an analytic inverse chart to its analytic domain -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open Set Filter | |
| open scoped ContDiff Topology | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] | |
| theorem exists_analytic_inverse_chart {f : E → E} | |
| (hf : ContDiffAt ℝ ω f 0) (h0 : f 0 = 0) | |
| (hd : HasFDerivAt f (ContinuousLinearMap.id ℝ E) 0) : | |
| ∃ q : OpenPartialHomeomorph E E, | |
| (0 : E) ∈ q.source ∧ q 0 = 0 ∧ (q : E → E) = f ∧ | |
| ContDiffOn ℝ ω q.symm q.target := by | |
| let d := ContinuousLinearEquiv.refl ℝ E | |
| let q : OpenPartialHomeomorph E E := hf.toOpenPartialHomeomorph f (f' := d) hd (by simp) | |
| have hq0 : (0 : E) ∈ q.source := hf.mem_toOpenPartialHomeomorph_source (f' := d) hd (by simp) | |
| have hi : ContDiffAt ℝ ω q.symm 0 := by | |
| change ContDiffAt ℝ ω (hf.localInverse (f' := d) hd (by simp)) 0 | |
| simpa only [h0] using hf.to_localInverse (f' := d) hd (by simp) | |
| obtain ⟨s,hs,hsi⟩ := hi.contDiffOn le_rfl (by intro _; rfl) | |
| obtain ⟨u,hus,hu,hu0⟩ := mem_nhds_iff.mp hs | |
| let r := OpenPartialHomeomorph.trans q (OpenPartialHomeomorph.ofSet u hu) | |
| refine ⟨r, ?_, ?_, rfl, ?_⟩ | |
| · refine ⟨hq0, ?_⟩ | |
| change f 0 ∈ u | |
| rwa [h0] | |
| · exact h0 | |
| · change ContDiffOn ℝ ω (fun x => q.symm x) _ | |
| apply hsi.mono | |
| intro x hx | |
| exact hus hx.1 | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |