| import OAI.Combinatorics.HalvingLines.Calculus |
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| namespace OAI |
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| namespace PlanarHalving |
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| namespace RankLimit |
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| open Filter MeasureTheory Set Metric |
| open scoped Topology |
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| abbrev RatPoint := ℚ × ℚ |
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| def ratPoint (q : RatPoint) : Point := ((q.1 : ℝ), (q.2 : ℝ)) |
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| noncomputable def lowerGrid (g : RatPoint → unitInterval) (x : Point) : unitInterval := |
| ⨆ q : RatPoint, if (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2 then g q else ⊥ |
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| lemma lowerGrid_monotone (g : RatPoint → unitInterval) : Monotone (lowerGrid g) := by |
| intro x y hxy |
| apply iSup_le |
| intro q |
| by_cases hq : (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2 |
| · have hqy : (q.1 : ℝ) < y.1 ∧ (q.2 : ℝ) < y.2 := |
| ⟨hq.1.trans_le hxy.1, hq.2.trans_le hxy.2⟩ |
| simpa only [lowerGrid, ite_eq_left hq, ite_eq_left hqy] using |
| (le_iSup (fun r : RatPoint => |
| if (r.1 : ℝ) < y.1 ∧ (r.2 : ℝ) < y.2 then g r else ⊥) q) |
| · simp only [ite_eq_right hq, bot_le] |
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| lemma lowerGrid_measurable (g : RatPoint → unitInterval) : Measurable (lowerGrid g) := by |
| apply Measurable.iSup |
| intro q |
| exact Measurable.ite |
| ((isOpen_lt continuous_const continuous_fst).inter |
| (isOpen_lt continuous_const continuous_snd)).measurableSet |
| measurable_const measurable_const |
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| lemma grid_le_lowerGrid {g : RatPoint → unitInterval} {q : RatPoint} {x : Point} |
| (hq : (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2) : g q ≤ lowerGrid g x := by |
| simpa only [lowerGrid, ite_eq_left hq] using |
| (le_iSup (fun r : RatPoint => |
| if (r.1 : ℝ) < x.1 ∧ (r.2 : ℝ) < x.2 then g r else ⊥) q) |
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| theorem moving_tendsto_of_grid {f : ℕ → Point → unitInterval} |
| (hf : ∀ n, Monotone (f n)) {g : RatPoint → unitInterval} |
| (hg : ∀ q, Tendsto (fun n => f n (ratPoint q)) atTop (𝓝 (g q))) |
| {x : Point} (hc : ContinuousAt (lowerGrid g) x) {y : ℕ → Point} |
| (hy : Tendsto y atTop (𝓝 x)) : |
| Tendsto (fun n => f n (y n)) atTop (𝓝 (lowerGrid g x)) := by |
| rw [tendsto_order] |
| constructor |
| · intro a ha |
| obtain ⟨q, hq⟩ := lt_iSup_iff.mp ha |
| by_cases hqx : (q.1 : ℝ) < x.1 ∧ (q.2 : ℝ) < x.2 |
| · simp only [ite_eq_left hqx] at hq |
| filter_upwards [(tendsto_order.mp (hg q)).1 a hq, |
| (tendsto_order.mp ((continuous_fst.tendsto x).comp hy)).1 _ hqx.1, |
| (tendsto_order.mp ((continuous_snd.tendsto x).comp hy)).1 _ hqx.2] with n hn hn₁ hn₂ |
| exact hn.trans_le (hf n ⟨hn₁.le, hn₂.le⟩) |
| · simp only [ite_eq_right hqx, not_lt_bot] at hq |
| · intro a ha |
| have hn := hc.preimage_mem_nhds (isOpen_Iio.mem_nhds ha) |
| obtain ⟨δ, hδ, hδU⟩ := Metric.mem_nhds_iff.mp hn |
| let b : Point := x + (δ / 2, δ / 2) |
| have hb : b ∈ ball x δ := by |
| rw [Metric.mem_ball, dist_eq_norm] |
| simp only [b, add_sub_cancel_left, Prod.norm_def, Real.norm_eq_abs, |
| abs_of_pos (half_pos hδ), max_self] |
| linarith only [hδ] |
| have hba : lowerGrid g b < a := hδU hb |
| obtain ⟨r₁, hxr₁, hr₁b⟩ := exists_rat_btwn |
| (show x.1 < b.1 by dsimp [b]; linarith only [hδ]) |
| obtain ⟨r₂, hxr₂, hr₂b⟩ := exists_rat_btwn |
| (show x.2 < b.2 by dsimp [b]; linarith only [hδ]) |
| let r : RatPoint := (r₁, r₂) |
| have hra : g r < a := (grid_le_lowerGrid ⟨hr₁b, hr₂b⟩).trans_lt hba |
| filter_upwards [(tendsto_order.mp (hg r)).2 a hra, |
| (tendsto_order.mp ((continuous_fst.tendsto x).comp hy)).2 _ hxr₁, |
| (tendsto_order.mp ((continuous_snd.tendsto x).comp hy)).2 _ hxr₂] with n hn hn₁ hn₂ |
| exact (hf n ⟨hn₁.le, hn₂.le⟩).trans_lt hn |
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| theorem monotone_grid_subsequence {ι : Type*} [Countable ι] |
| (f : ℕ → ι → Point → unitInterval) (hf : ∀ n i, Monotone (f n i)) : |
| ∃ (φ : ℕ → ℕ) (F : ι → Point → unitInterval), StrictMono φ ∧ |
| (∀ i, Monotone (F i) ∧ Measurable (F i)) ∧ |
| (∀ i x, ContinuousAt (F i) x → ∀ y : ℕ → Point, |
| Tendsto y atTop (𝓝 x) → |
| Tendsto (fun n => f (φ n) i (y n)) atTop (𝓝 (F i x))) := by |
| obtain ⟨g, φ, hφ, hg⟩ := CompactSpace.tendsto_subseq |
| (fun n i q => f n i (ratPoint q)) |
| refine ⟨φ, fun i => lowerGrid (g i), hφ, |
| fun i => ⟨lowerGrid_monotone _, lowerGrid_measurable _⟩, ?_⟩ |
| intro i x hx y hy |
| apply moving_tendsto_of_grid (fun n => hf (φ n) i) _ hx hy |
| intro q |
| exact ((tendsto_pi_nhds.mp hg) i).apply_nhds q |
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| end RankLimit |
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| namespace Transversality |
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| open MeasureTheory Set |
| open scoped MeasureTheory ENNReal |
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| theorem critical_projection_null {Z : Type*} [MeasurableSpace Z] |
| (σ : Measure Z) (coordinate : Z → Point) (hcoordinate : Measurable coordinate) |
| (hdom : σ.map coordinate ≪ volume) (p : Z → Point) (F : Point → Point) |
| (S : Set Z) (hidentity : ∀ z ∈ S, coordinate z = F (p z)) |
| (hdiff : ∀ z ∈ S, DifferentiableAt ℝ F (p z)) |
| (hsing : ∀ z ∈ S, (fderiv ℝ F (p z)).det = 0) : σ S = 0 := by |
| obtain ⟨N, hsub, hN, hnull⟩ := exists_measurable_superset_of_null (critical_image_null F) |
| apply measure_mono_null (t := coordinate ⁻¹' N) |
| · intro z hz |
| change coordinate z ∈ N |
| rw [hidentity z hz] |
| exact hsub ⟨p z, ⟨hdiff z hz, hsing z hz⟩, rfl⟩ |
| · rw [← Measure.map_apply hcoordinate hN] |
| exact hdom hnull |
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| theorem independent_gradients_ae {Z ι : Type*} [MeasurableSpace Z] [Countable ι] |
| (ν : Measure Z) (h : Z → ℝ≥0∞) (hh : Measurable h) |
| (σ : ι → Measure Z) (hν : ν.withDensity h ≪ Measure.sum σ) |
| (coordinate : ι → Z → Point) (hcoordinate : ∀ i, Measurable (coordinate i)) |
| (hdom : ∀ i, (σ i).map (coordinate i) ≪ volume) |
| (p : Z → Point) (F : ι → Point → Point) (S : Set Z) |
| (hidentity : ∀ i z, z ∈ S → coordinate i z = F i (p z)) |
| (hdiff : ∀ i z, z ∈ S → DifferentiableAt ℝ (F i) (p z)) : |
| ∀ᵐ z ∂ν, z ∈ S → h z ≠ 0 → ∃ i, (fderiv ℝ (F i) (p z)).det ≠ 0 := by |
| have hbad : (ν.withDensity h) {z | z ∈ S ∧ ∀ i, (fderiv ℝ (F i) (p z)).det = 0} = 0 := by |
| apply hν |
| rw [Measure.sum_apply_eq_zero] |
| intro i |
| apply critical_projection_null (σ i) (coordinate i) (hcoordinate i) (hdom i) p |
| (F i) _ |
| · intro z hz |
| exact hidentity i z hz.1 |
| · intro z hz |
| exact hdiff i z hz.1 |
| · intro z hz |
| exact hz.2 i |
| have hgood : ∀ᵐ z ∂ν.withDensity h, |
| z ∈ S → ∃ i, (fderiv ℝ (F i) (p z)).det ≠ 0 := by |
| apply ae_iff.mpr |
| simpa only [Classical.not_imp, not_exists, not_not] using hbad |
| have hg := (ae_withDensity_iff hh).mp hgood |
| filter_upwards [hg] with z hz hzS hzh |
| exact hz hzh hzS |
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| end Transversality |
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| end PlanarHalving |
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| end OAI |
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