| import Mathlib |
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| 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 |
| 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 < |
| have hproper : S ≠ ⊤ := by |
| intro htop |
| have hall : ∀ x, a x + b = a' x + b' := fun x => |
| (hmem x).mp (by rw [htop] |
| 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 |
| end |
| end |
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| end |
| |
| end OAI |
| |
