lean/OAI/Analysis/C0Absorption/LocalInverse.lean

import OAI.Analysis.C0Absorption.Operators

namespace OAI

open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty
open NormedSpace

namespace C0Absorption
noncomputable section
open Set Filter Topology
open scoped NNReal BigOperators

section LocalOnto
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
  [NormedAddCommGroup F] [NormedSpace ℝ F]

theorem range_mem_nhds_of_approx {f : E → F} {a : E} (S : E ≃L[ℝ] F)
    (s : Set E) (hs : s∈nhds a) (c : ℝ≥0)
    (hc : c<‖(S.symm : F →L[ℝ] E)‖₊⁻¹) (ha : ApproximatesLinearOn f (S : E →L[ℝ] F) s c) :
    Set.range f∈nhds (f a) := by
  have hh := ha.image_mem_nhds S.toNonlinearRightInverse hs (Or.inr hc)
  exact mem_of_superset hh (Set.image_subset_range f s)

theorem surjective_of_punctured_local_onto {E F : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
    [NormedAddCommGroup F] [NormedSpace ℝ F]
    [Nontrivial E]
    (f : E → F) {c C : ℝ≥0} (hf : AntilipschitzWith c f) (hLip : LipschitzWith C f)
    (hzero : f 0=0) (hconn : IsPreconnected ({0}ᶜ : Set F))
    (hlocal : ∀ x≠0, Set.range f∈nhds (f x)) : Function.Surjective f := by
  let K : Set F := {0}ᶜ
  let A : Set K := Subtype.val ⁻¹' Set.range f
  let : PreconnectedSpace K := Subtype.preconnectedSpace hconn
  have hclosed : IsClosed A := (hf.isClosed_range hLip.uniformContinuous).preimage continuous_subtype_val
  have hopen : IsOpen A := by
    rw [isOpen_iff_mem_nhds]
    intro y hy
    obtain ⟨x,hx⟩ := hy
    have hx0 : x≠0 := by
      intro hx0
      have he : y.val=0 := by rw [← hx,hx0,hzero]
      exact y.property (by simpa only [Set.mem_singleton_iff] using he)
    have hn : Set.range f∈nhds y.val := by rw [← hx]; exact hlocal x hx0
    exact continuous_subtype_val.continuousAt hn
  have hne : A.Nonempty := by
    obtain ⟨x,hx⟩ := exists_ne (0 : E)
    have hx' : f x≠0 := by intro he; exact hx (hf.injective (he.trans hzero.symm))
    refine ⟨⟨f x,?_⟩,x,rfl⟩
    simpa only [K,Set.mem_compl_iff,Set.mem_singleton_iff] using hx'
  have heq : A=Set.univ := IsClopen.eq_univ ⟨hclosed,hopen⟩ hne
  intro y
  by_cases hy : y=0
  · exact ⟨0,hzero.trans hy.symm⟩
  · let yy : K := ⟨y,by simpa only [K,Set.mem_compl_iff,Set.mem_singleton_iff] using hy⟩
    have hh : yy∈A := by rw [heq]; trivial
    exact hh

end LocalOnto
end
end C0Absorption

end OAI