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6aac220 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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
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