import Mathlib

namespace OAI

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namespace VertexCover.AffineSelector
open Filter MeasureTheory
open scoped Topology

variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
variable {ι : Type*} [Fintype ι]

def Regular (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ) (x : E) : Prop :=
  ∀ i j, a i x + b i = a j x + b j → a i = a j

theorem locally_affine (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ)
    {f : E → ℝ} (hf : Continuous f)
    (hselect : ∀ y, ∃ i, f y = a i y + b i) {x : E}
    (hx : Regular a b x) :
    ∃ i, f =ᶠ[𝓝 x] (fun y => a i y + b i) := by
  obtain ⟨i, hi⟩ := hselect x
  refine ⟨i, ?_⟩
  have hj : ∀ j, ∀ᶠ y in 𝓝 x,
      f y = a j y + b j → a j y + b j = a i y + b i := by
    intro j
    by_cases h : f x = a j x + b j
    · have ha : a i = a j := hx i j (hi.symm.trans h)
      have hb : b i = b j := by rw [ha] at hi; linarith
      exact Filter.Eventually.of_forall (fun y _ => by rw [ha, hb])
    · have hn : f x - (a j x + b j) ≠ 0 := sub_ne_zero.mpr h
      have he := (hf.continuousAt.sub ((a j).continuous.continuousAt.add_const (b j))).eventually_ne hn
      filter_upwards [he] with y hy
      intro heq
      exact (hy (sub_eq_zero.mpr heq)).elim
  filter_upwards [Filter.eventually_all.mpr hj] with y hy
  obtain ⟨j, hj⟩ := hselect y
  exact hj.trans (hy j hj)

theorem hasFDerivAt (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ)
    {f : E → ℝ} (hf : Continuous f)
    (hselect : ∀ y, ∃ i, f y = a i y + b i) {x : E}
    (hx : Regular a b x) : ∃ i, HasFDerivAt f (a i) x := by
  obtain ⟨i, hi⟩ := locally_affine a b hf hselect hx
  exact ⟨i, ((a i).hasFDerivAt.add_const (b i)).congr_of_eventuallyEq hi⟩

variable [MeasurableSpace E] [BorelSpace E]

theorem regular_measurableSet (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ) :
    MeasurableSet {x | Regular a b x} := by
  classical
  simp only [Regular, Set.ofPred_forall]
  apply MeasurableSet.iInter; intro i
  apply MeasurableSet.iInter; intro j
  by_cases h : a i = a j
  · simp [h]
  · simp only [h, imp_false]
    exact (measurableSet_eq_fun ((a i).continuous.measurable.add_const (b i))
      ((a j).continuous.measurable.add_const (b j))).compl

theorem affine_tie_null [FiniteDimensional ℝ E] (μ : Measure E) [μ.IsAddHaarMeasure]
    (a a' : E →L[ℝ] ℝ) (b b' : ℝ) (ha : a ≠ a') :
    μ {x | a x + b = a' x + b'} = 0 := by
  let S : AffineSubspace ℝ E := AffineSubspace.comap
    (a-a').toLinearMap.toAffineMap (affineSpan ℝ ({b'-b} : Set ℝ))
  have hmem : ∀ x, x ∈ S ↔ a x + b = a' x + b' := by
    intro x
    simp only [S, AffineSubspace.mem_comap, AffineSubspace.mem_affineSpan_singleton]
    change a x - a' x = b'-b ↔ _
    constructor <;> intro h <;> linarith
  have hproper : S ≠ ⊤ := by
    intro htop
    have hall : ∀ x, a x + b = a' x + b' := fun x =>
      (hmem x).mp (by rw [htop]; exact AffineSubspace.mem_top ℝ E x)
    have hz : b = b' := by simpa using hall 0
    apply ha
    ext x
    linarith [hall x]
  have heq : (S : Set E) = {x | a x + b = a' x + b'} := Set.ext hmem
  rw [← heq]
  exact Measure.addHaar_affineSubspace μ S hproper

theorem ae_regular [FiniteDimensional ℝ E] (μ : Measure E) [μ.IsAddHaarMeasure]
    (a : ι → E →L[ℝ] ℝ) (b : ι → ℝ) : ∀ᵐ x ∂μ, Regular a b x := by
  classical
  apply Filter.eventually_all.mpr
  intro i
  apply Filter.eventually_all.mpr
  intro j
  by_cases hij : a i = a j
  · exact Filter.Eventually.of_forall (fun _ _ => hij)
  · have hne : ∀ᵐ x ∂μ, a i x + b i ≠ a j x + b j := by
      rw [ae_iff]
      simpa only [not_not] using affine_tie_null μ (a i) (a j) (b i) (b j) hij
    filter_upwards [hne] with x hx
    exact fun h => (hx h).elim

end VertexCover.AffineSelector


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end OAI