| import OAI.Combinatorics.HalvingLines.Derivatives |
|
|
| namespace OAI |
|
|
| open Filter Topology |
|
|
| namespace PlanarHalving.TimeWindows |
| open Set Filter MeasureTheory |
| open scoped unitInterval Topology |
|
|
| noncomputable def lower (J : Window) : I := UniformPartition.boundary (2^J.1) J.2.castSucc |
| noncomputable def upper (J : Window) : I := UniformPartition.boundary (2^J.1) J.2.succ |
|
|
| lemma lower_lt_upper (J : Window) : lower J < upper J := by |
| change (J.2 : ℝ)/(2^J.1 : ℕ) < ((J.2.val+1 : ℕ) : ℝ)/(2^J.1 : ℕ) |
| apply div_lt_div_of_pos_right _ (by positivity) |
| simp |
|
|
| lemma upper_sub_lower (J : Window) : (upper J : ℝ) - (lower J : ℝ) = 1/(2:ℝ)^J.1 := by |
| simp only [upper, lower, UniformPartition.boundary, Fin.val_succ, Fin.val_castSucc, |
| Nat.cast_add, Nat.cast_one, Nat.cast_pow, Nat.cast_ofNat] |
| ring |
|
|
| lemma Ioo_subset_carrier (J : Window) : Ioo (lower J) (upper J) ⊆ carrier J := by |
| intro t ht |
| unfold carrier UniformPartition.cell |
| split_ifs |
| · exact ⟨ht.1.le, ht.2.le⟩ |
| · exact ⟨ht.1.le, ht.2⟩ |
|
|
| lemma carrier_subset_Icc (J : Window) : carrier J ⊆ Icc (lower J) (upper J) := by |
| intro t ht |
| unfold carrier UniformPartition.cell at ht |
| split_ifs at ht |
| · exact ht |
| · exact ⟨ht.1, ht.2.le⟩ |
|
|
| lemma closure_carrier (J : Window) : closure (carrier J) = Icc (lower J) (upper J) := by |
| unfold carrier UniformPartition.cell |
| split_ifs |
| · exact isClosed_Icc.closure_eq |
| · exact closure_Ico (lower_lt_upper J).ne |
|
|
| lemma endpoints_of_subset {J K : Window} (h : carrier J ⊆ carrier K) : |
| lower K ≤ lower J ∧ upper J ≤ upper K := by |
| have hh := closure_mono h |
| rw [closure_carrier, closure_carrier] at hh |
| exact ⟨(hh ⟨le_rfl,(lower_lt_upper J).le⟩).1, |
| (hh ⟨(lower_lt_upper J).le,le_rfl⟩).2⟩ |
|
|
| lemma before_or_before {A B : Window} (h : Disjoint (carrier A) (carrier B)) : |
| Before A B ∨ Before B A := by |
| by_contra hn |
| have hn' : upper B > lower A ∧ upper A > lower B := by |
| simpa only [Before, lower, upper, not_or, not_le, and_comm] using hn |
| have hlt : max (lower A) (lower B) < min (upper A) (upper B) := by |
| exact max_lt_iff.mpr ⟨lt_min (lower_lt_upper A) hn'.1, |
| lt_min hn'.2 (lower_lt_upper B)⟩ |
| obtain ⟨t,ht1,ht2⟩ := exists_between hlt |
| exact Set.disjoint_left.mp h (Ioo_subset_carrier A ⟨(le_max_left _ _).trans_lt ht1, |
| ht2.trans_le (min_le_left _ _)⟩) |
| (Ioo_subset_carrier B ⟨(le_max_right _ _).trans_lt ht1,ht2.trans_le (min_le_right _ _)⟩) |
|
|
| lemma before_of_separated {K H C A : Window} (hKH : upper K < lower H) |
| (hC : carrier C ⊆ carrier K) (hA : carrier A ⊆ carrier H) : Before C A := by |
| exact ((endpoints_of_subset hC).2.trans hKH.le).trans (endpoints_of_subset hA).1 |
|
|
| lemma ae_avoid_boundaries (μ : Measure I) [NullSingletonClass μ] : |
| ∀ᵐ t ∂μ, ∀ m (i : Fin (2^m+1)), t ≠ UniformPartition.boundary (2^m) i := by |
| rw [ae_all_iff] |
| intro m |
| rw [ae_all_iff] |
| intro i |
| exact μ.ae_ne _ |
|
|
| lemma exists_separated_positive (μ : Measure I) [IsProbabilityMeasure μ] |
| [NullSingletonClass μ] : |
| ∃ K H : Window, (0 : ℝ) < lower K ∧ upper K < lower H ∧ (upper H : ℝ) < 1 ∧ |
| μ (carrier K) ≠ 0 ∧ μ (carrier H) ≠ 0 := by |
| have he0 : ∀ᵐ t ∂μ, t ≠ (0 : I) := μ.ae_ne _ |
| have he1 : ∀ᵐ t ∂μ, t ≠ (1 : I) := μ.ae_ne _ |
| have hes : ∀ᵐ t ∂μ, t ∈ μ.support := μ.support_mem_ae |
| obtain ⟨x,hxs,hxb,hx0,hx1⟩ := |
| (hes.and ((ae_avoid_boundaries μ).and (he0.and he1))).exists |
| have hne : ∀ᵐ t ∂μ, t ≠ x := μ.ae_ne _ |
| obtain ⟨y,hys,hyb,hy0,hy1,hyx⟩ := |
| (hes.and ((ae_avoid_boundaries μ).and (he0.and (he1.and hne)))).exists |
| have base (x y : I) (hxs : x ∈ μ.support) (hys : y ∈ μ.support) |
| (hxb : ∀ m (i : Fin (2^m+1)), x ≠ UniformPartition.boundary (2^m) i) |
| (hyb : ∀ m (i : Fin (2^m+1)), y ≠ UniformPartition.boundary (2^m) i) |
| (hx0 : x ≠ 0) (hy1 : y ≠ 1) (hxy : x < y) : |
| ∃ K H : Window, (0 : ℝ) < lower K ∧ upper K < lower H ∧ (upper H : ℝ) < 1 ∧ |
| μ (carrier K) ≠ 0 ∧ μ (carrier H) ≠ 0 := by |
| have hxpos : 0 < (x : ℝ) := lt_of_le_of_ne x.property.1 (by |
| intro hh; exact hx0 (Subtype.ext hh.symm)) |
| have hylt : (y : ℝ) < 1 := lt_of_le_of_ne y.property.2 (by |
| intro hh; exact hy1 (Subtype.ext hh)) |
| have hxy' : (x : ℝ) < (y : ℝ) := hxy |
| have hmesh : Tendsto (fun m : ℕ => 1/(2:ℝ)^m) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop (tendsto_pow_atTop_atTop_of_one_lt (by norm_num)) |
| have he := hmesh.eventually (gt_mem_nhds (show 0 < min (x : ℝ) (min (1-(y:ℝ)) (((y:ℝ)-x)/2)) by positivity)) |
| obtain ⟨m,hm⟩ := he.exists |
| have hm1 := (lt_min_iff.mp hm).1 |
| have hm2 := (lt_min_iff.mp (lt_min_iff.mp hm).2).1 |
| have hm3 := (lt_min_iff.mp (lt_min_iff.mp hm).2).2 |
| obtain ⟨i,hi⟩ := UniformPartition.mem_some_cell (2^m) x |
| obtain ⟨j,hj⟩ := UniformPartition.mem_some_cell (2^m) y |
| let K : Window := ⟨m,i⟩ |
| let H : Window := ⟨m,j⟩ |
| have hk := carrier_subset_Icc K hi |
| have hh := carrier_subset_Icc H hj |
| have hki : x ∈ Ioo (lower K) (upper K) := |
| ⟨lt_of_le_of_ne hk.1 (Ne.symm (hxb m i.castSucc)), lt_of_le_of_ne hk.2 (hxb m i.succ)⟩ |
| have hhi : y ∈ Ioo (lower H) (upper H) := |
| ⟨lt_of_le_of_ne hh.1 (Ne.symm (hyb m j.castSucc)), lt_of_le_of_ne hh.2 (hyb m j.succ)⟩ |
| have kw := upper_sub_lower K |
| have hw := upper_sub_lower H |
| have hk1 : (lower K : ℝ) ≤ x := hk.1 |
| have hk2 : (x : ℝ) ≤ upper K := hk.2 |
| have hh1 : (lower H : ℝ) ≤ y := hh.1 |
| have hh2 : (y : ℝ) ≤ upper H := hh.2 |
| refine ⟨K,H,?_,?_,?_,?_,?_⟩ |
| · dsimp [K] at kw |
| linarith |
| · change (upper K : ℝ) < lower H |
| dsimp [K] at kw |
| dsimp [H] at hw |
| linarith |
| · dsimp [H] at hw |
| linarith |
| · exact ne_of_gt ((measure_mono (Ioo_subset_carrier K)).trans_lt' ((Measure.mem_support_iff_forall x).mp hxs _ (Ioo_mem_nhds hki.1 hki.2))) |
| · exact ne_of_gt ((measure_mono (Ioo_subset_carrier H)).trans_lt' ((Measure.mem_support_iff_forall y).mp hys _ (Ioo_mem_nhds hhi.1 hhi.2))) |
| rcases lt_or_gt_of_ne hyx with h | h |
| · exact base y x hys hxs hyb hxb hy0 hx1 h |
| · exact base x y hxs hys hxb hyb hx0 hy1 h |
|
|
| end PlanarHalving.TimeWindows |
| namespace PlanarHalving.Batches |
| open Set MeasureTheory TimeWindows Limits |
| variable {n R k a : ℕ} {P : Fin n → Point} |
| lemma System.windowCrossings_comm (S : |
| |
| |
| |
| |
| |
| |
| |
| |
| System n R k a P) [NeZero R] |
| (A B : Window) {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X] |
| (z : Point → X) : S.windowCrossMeasure A B z = S.windowCrossMeasure B A z := by |
| apply FiniteMeasure.eq_of_forall_apply_eq |
| intro U hU |
| apply NNReal.coe_injective |
| rw [S.windowCrossMeasure_apply A B z hU,S.windowCrossMeasure_apply B A z hU] |
| rw [System.crossingsIn,System.crossingsIn,S.windowCrossings_comm A B] |
| end PlanarHalving.Batches |
|
|
| namespace PlanarHalving.RankLimits |
| open Set Filter MeasureTheory Batches TimeWindows Limits |
| open scoped unitInterval Topology |
|
|
| lemma Converges.crossings_comm {F : ℕ → Configuration} {L : Data} (h : Converges F L) |
| (A B : Window) : L.crossings A B = L.crossings B A := by |
| apply tendsto_nhds_unique (h.crossings A B) |
| convert h.crossings B A using 1 |
| funext n |
| exact (F n).system.windowCrossMeasure_comm A B _ |
|
|
| lemma unitPair_quasiMeasurePreserving : |
| Measure.QuasiMeasurePreserving (fun p : I × I => ((p.1 : ℝ),(p.2 : ℝ))) volume volume := by |
| have hi : Measure.QuasiMeasurePreserving ((↑) : I → ℝ) volume volume := |
| unitInterval.measurePreserving_coe.quasiMeasurePreserving.mono_right (Measure.absolutelyContinuous_of_le (Measure.restrict_le_self)) |
| exact QuasiMeasurePreserving.prodMap hi hi |
|
|
| lemma signedPair_measurePreserving (b : Bool) : MeasurePreserving (signedPair b) volume volume := by |
| cases b |
| · exact (Measure.measurePreserving_neg (volume : Measure ℝ)).prod (MeasurePreserving.id _) |
| · exact (MeasurePreserving.id _).prod (Measure.measurePreserving_neg (volume : Measure ℝ)) |
|
|
| lemma Converges.crossing_real_projection {F : ℕ → Configuration} {L : Data} (h : Converges F L) |
| (hR : Tendsto (fun n => (F n).R) atTop atTop) |
| (hM : Tendsto (fun n => (F n).a*(F n).n) atTop atTop) |
| {A B : Window} (hAB : Before A B) : |
| (L.crossings A B : Measure Labels).map (fun z => ((z A : ℝ),(z B : ℝ))) ≪ volume := by |
| have hp : Measure.QuasiMeasurePreserving (fun z : Labels => (z A,z B)) |
| (L.crossings A B : Measure Labels) volume := |
| ⟨(continuous_apply A |>.prodMk (continuous_apply B)).measurable, |
| Measure.absolutelyContinuous_of_le (h.crossing_projection hR hM hAB)⟩ |
| exact (unitPair_quasiMeasurePreserving.comp hp).absolutelyContinuous |
|
|
| lemma Converges.crossing_signed_projection {F : ℕ → Configuration} {L : Data} (h : Converges F L) |
| (hR : Tendsto (fun n => (F n).R) atTop atTop) |
| (hM : Tendsto (fun n => (F n).a*(F n).n) atTop atTop) |
| (b : Bool) {A B : Window} (hAB : Before A B) : |
| (L.crossings A B : Measure Labels).map (pairCoordinates b A B) ≪ volume := by |
| have hp : Measure.QuasiMeasurePreserving (fun z : Labels => ((z A : ℝ),(z B : ℝ))) |
| (L.crossings A B : Measure Labels) volume := |
| ⟨by fun_prop,h.crossing_real_projection hR hM hAB⟩ |
| exact ((signedPair_measurePreserving b).quasiMeasurePreserving.comp hp).absolutelyContinuous |
|
|
| lemma Converges.window_withDensity {F : ℕ → Configuration} {L : Data} (h : Converges F L) |
| (hR : Tendsto (fun n => (F n).R) atTop atTop) (J : Window) : |
| (L.edgeWindow J : Measure Labels) = (L.edge : Measure Labels).withDensity |
| (fun z => L.conditional z (carrier J)) := by |
| ext U hU |
| rw [withDensity_apply _ hU, Data.edge_toMeasure, Data.conditional, |
| Measure.setLIntegral_condKernel_eq_measure_prod hU (measurable_carrier J)] |
| rw [Data.edgeWindow,FiniteMeasure.toMeasure_map,Measure.map_apply measurable_fst hU, |
| h.restriction_eq hR,Measure.restrict_apply (hU.preimage measurable_fst)] |
| rfl |
|
|
| lemma Converges.edge_probability {F : ℕ → Configuration} {L : Data} (h : Converges F L) : |
| IsProbabilityMeasure (L.edge : Measure Labels) := by |
| constructor |
| rw [Data.edge,FiniteMeasure.toMeasure_map,Measure.map_apply measurable_fst MeasurableSet.univ, |
| preimage_univ,← FiniteMeasure.ennreal_mass,h.edgeTime_mass] |
| rfl |
|
|
| end PlanarHalving.RankLimits |
|
|
| namespace PlanarHalving.RankLimit |
| open Filter Set Metric MeasureTheory |
| open scoped Topology |
|
|
| lemma dense_strict_brackets {S : Set Point} (hS : Dense S) {p : Point} {U : Set Point} |
| (hU : U ∈ 𝓝 p) : |
| (∃ l ∈ S ∩ U, l.1 < p.1 ∧ l.2 < p.2) ∧ |
| (∃ u ∈ S ∩ U, p.1 < u.1 ∧ p.2 < u.2) := by |
| obtain ⟨r,hr,hrU⟩ := Metric.mem_nhds_iff.mp hU |
| have hb : ‖(r/2,r/2)‖ < r := by |
| simp only [Prod.norm_def,Real.norm_eq_abs,abs_of_pos (half_pos hr),max_self] |
| linarith only [hr] |
| constructor |
| · obtain ⟨l,hlS,hl⟩ := hS.exists_mem_open |
| ((isOpen_ball.inter (isOpen_lt continuous_fst continuous_const)).inter |
| (isOpen_lt continuous_snd continuous_const)) |
| (show ((ball p r ∩ {x : Point | x.1 < p.1}) ∩ {x : Point | x.2 < p.2}).Nonempty from |
| ⟨p-(r/2,r/2),⟨⟨by simpa [Metric.mem_ball,dist_eq_norm] using hb,by dsimp; linarith⟩,by dsimp; linarith⟩⟩) |
| exact ⟨l,⟨hlS,hrU hl.1.1⟩,hl.1.2,hl.2⟩ |
| · obtain ⟨u,huS,hu⟩ := hS.exists_mem_open |
| ((isOpen_ball.inter (isOpen_lt continuous_const continuous_fst)).inter |
| (isOpen_lt continuous_const continuous_snd)) |
| (show ((ball p r ∩ {x : Point | p.1 < x.1}) ∩ {x : Point | p.2 < x.2}).Nonempty from |
| ⟨p+(r/2,r/2),⟨⟨by simpa [Metric.mem_ball,dist_eq_norm] using hb,by dsimp; linarith⟩,by dsimp; linarith⟩⟩) |
| exact ⟨u,⟨huS,hrU hu.1.1⟩,hu.1.2,hu.2⟩ |
|
|
| lemma monotone_eventually_uniform_approx {f : ℕ → Point → ℝ} {G g : Point → ℝ} |
| (hf : ∀ n, Monotone (f n)) {S : Set Point} (hS : Dense S) |
| (hlim : ∀ x ∈ S, Tendsto (fun n => f n x) atTop (𝓝 (G x))) |
| (hg : Continuous g) {K U : Set Point} (hK : IsCompact K) (hU : IsOpen U) |
| (hKU : K ⊆ U) {ε : ℝ} (hε : 0 < ε) |
| (happrox : ∀ x ∈ U, dist (G x) (g x) < ε/4) : |
| ∀ᶠ n in atTop, ∀ x ∈ K, dist (f n x) (g x) < ε := by |
| have he : ∀ᶠ q : ℕ × Point in atTop ×ˢ 𝓝ˢ K, dist (f q.1 q.2) (g q.2) < ε := by |
| apply hK.mem_prod_nhdsSet_of_forall |
| intro p hp |
| let V := U ∩ {x | dist (g x) (g p) < ε/4} |
| have hV : V ∈ 𝓝 p := inter_mem (hU.mem_nhds (hKU hp)) |
| ((hg.tendsto p).eventually (Metric.ball_mem_nhds _ (by positivity))) |
| obtain ⟨⟨l,hl,hlp⟩,⟨u,hu,hpu⟩⟩ := dense_strict_brackets hS hV |
| have hll := (hlim l hl.1).eventually (Metric.ball_mem_nhds _ (show 0 < ε/4 by positivity)) |
| have hlu := (hlim u hu.1).eventually (Metric.ball_mem_nhds _ (show 0 < ε/4 by positivity)) |
| have hne : ∀ᶠ x in 𝓝 p, l.1 < x.1 ∧ l.2 < x.2 ∧ x.1 < u.1 ∧ x.2 < u.2 ∧ |
| dist (g x) (g p) < ε/4 := by |
| filter_upwards [(continuous_fst.tendsto p).eventually (Ioi_mem_nhds hlp.1), |
| (continuous_snd.tendsto p).eventually (Ioi_mem_nhds hlp.2), |
| (continuous_fst.tendsto p).eventually (Iio_mem_nhds hpu.1), |
| (continuous_snd.tendsto p).eventually (Iio_mem_nhds hpu.2), |
| (hg.tendsto p).eventually (Metric.ball_mem_nhds _ (show 0 < ε/4 by positivity))] |
| with x hx1 hx2 hx3 hx4 hx5 |
| exact ⟨hx1,hx2,hx3,hx4,hx5⟩ |
| filter_upwards [(hll.and hlu).prod_mk hne] with q hq |
| rcases hq with ⟨⟨hnl,hnu⟩,hx1,hx2,hx3,hx4,hxg⟩ |
| have hlo := hf q.1 (show l ≤ q.2 from ⟨hx1.le,hx2.le⟩) |
| have hup := hf q.1 (show q.2 ≤ u from ⟨hx3.le,hx4.le⟩) |
| have hdl := happrox l hl.2.1 |
| have hdu := happrox u hu.2.1 |
| have hlg : dist (g l) (g p) < ε/4 := hl.2.2 |
| have hug : dist (g u) (g p) < ε/4 := hu.2.2 |
| rw [Real.dist_eq,abs_lt] at hnl hnu |
| rw [Real.dist_eq,abs_lt] at hdl hdu hlg hug hxg |
| change dist (f q.1 q.2) (g q.2) < ε |
| rw [Real.dist_eq,abs_lt] |
| constructor <;> linarith |
| have hen := he.curry |
| exact hen.mono fun _ hn => hn.self_of_nhdsSet |
|
|
| lemma firstOrder_eventually_uniform {f : ℕ → Point → ℝ} {G : Point → ℝ} |
| (hf : ∀ n, Monotone (f n)) {S : Set Point} (hS : Dense S) |
| (hlim : ∀ x ∈ S, Tendsto (fun n => f n x) atTop (𝓝 (G x))) |
| {p : Point} {D : Point →L[ℝ] ℝ} (hd : HasFDerivAt G D p) |
| {R ε : ℝ} (hR : 0 ≤ R) (hε : 0 < ε) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ h : ℝ, 0 < h → h < δ → |
| ∀ᶠ n in atTop, ∀ x ∈ closedBall p (R*h), |
| dist (f n x) (G p + D (x-p)) < ε*h := by |
| let η := ε/(8*(R+1)) |
| have hη : 0 < η := div_pos hε (by positivity) |
| have he := hd.isLittleO.def hη |
| obtain ⟨r,hr,hrb⟩ := Metric.mem_nhds_iff.mp he |
| refine ⟨r/(R+1),div_pos hr (by positivity),?_⟩ |
| intro h hh hhr |
| have hhr' : (R+1)*h < r := by |
| have ht := (lt_div_iff₀ (show 0 < R+1 by positivity)).mp hhr |
| nlinarith only [ht] |
| apply monotone_eventually_uniform_approx hf hS hlim |
| (continuous_const.add (D.continuous.comp (continuous_id.sub continuous_const))) |
| (isCompact_closedBall p (R*h)) isOpen_ball (U:=ball p ((R+1)*h)) |
| (ε:=ε*h) |
| · intro x hx |
| exact (Metric.mem_closedBall.mp hx).trans_lt (by nlinarith only [hh]) |
| · exact mul_pos hε hh |
| · intro x hx |
| have hxnorm : ‖x-p‖ < (R+1)*h := by |
| simpa only [Metric.mem_ball,dist_eq_norm] using hx |
| have hxb : x ∈ ball p r := by |
| simpa only [Metric.mem_ball,dist_eq_norm] using hxnorm.trans hhr' |
| have hxlin := hrb hxb |
| calc |
| dist (G x) (G p + D (x-p)) = ‖G x-G p-D (x-p)‖ := by |
| rw [dist_eq_norm, sub_add_eq_sub_sub] |
| _ ≤ η*‖x-p‖ := hxlin |
| _ < η*((R+1)*h) := mul_lt_mul_of_pos_left hxnorm hη |
| _ = ε*h/8 := by |
| dsimp [η] |
| field_simp |
| _ < ε*h/4 := by nlinarith [mul_pos hε hh] |
|
|
| lemma firstOrder_of_joint {f : ℕ → Point → ℝ} {G : Point → ℝ} |
| (hf : ∀ n, Monotone (f n)) |
| (hlim : ∀ p, ContinuousAt G p → Tendsto (fun q : ℕ × Point => f q.1 q.2) |
| (atTop ×ˢ 𝓝 p) (𝓝 (G p))) |
| (hdiff : ∀ᵐ p ∂(volume : Measure Point), DifferentiableAt ℝ G p) |
| {p : Point} {D : Point →L[ℝ] ℝ} (hd : HasFDerivAt G D p) |
| {R ε : ℝ} (hR : 0 ≤ R) (hε : 0 < ε) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ h : ℝ, 0 < h → h < δ → |
| ∀ᶠ n in atTop, ∀ x ∈ closedBall p (R*h), |
| dist (f n x) (G p + D (x-p)) < ε*h := by |
| apply firstOrder_eventually_uniform hf |
| (Measure.dense_of_ae (μ := volume) (hdiff.mono fun _ hx => hx.continuousAt)) |
| _ hd hR hε |
| intro x hx |
| exact (hlim x hx).comp (tendsto_id.prodMk tendsto_const_nhds) |
|
|
| end PlanarHalving.RankLimit |
|
|
| namespace PlanarHalving.RankLimits |
| open Set Filter MeasureTheory Batches TimeWindows Limits |
| open scoped Topology unitInterval ENNReal |
|
|
| def SeparatedBefore (K H : Window) : Prop := |
| (0 : ℝ) < lower K ∧ upper K < lower H ∧ (upper H : ℝ) < 1 |
|
|
| lemma Data.exists_positive_windows (L : Data) [IsProbabilityMeasure (L.edge : Measure Labels)] |
| (hna : ∀ᵐ z ∂(L.edge : Measure Labels), NullSingletonClass (L.conditional z)) : |
| ∃ K H : Window, SeparatedBefore K H ∧ |
| (L.edge : Measure Labels) {z | L.conditional z (carrier K) ≠ 0 ∧ |
| L.conditional z (carrier H) ≠ 0} ≠ 0 := by |
| by_contra hn |
| have hz : ∀ K H : Window, SeparatedBefore K H → |
| (L.edge : Measure Labels) {z | L.conditional z (carrier K) ≠ 0 ∧ |
| L.conditional z (carrier H) ≠ 0} = 0 := by |
| simpa only [not_exists, not_and, not_not] using hn |
| have he : ∀ᵐ z ∂(L.edge : Measure Labels), ∀ K H : Window, SeparatedBefore K H → |
| ¬(L.conditional z (carrier K) ≠ 0 ∧ L.conditional z (carrier H) ≠ 0) := by |
| rw [ae_all_iff] |
| intro K |
| rw [ae_all_iff] |
| intro H |
| by_cases hs : SeparatedBefore K H |
| · exact (ae_iff.mpr (by simpa only [not_not] using hz K H hs)).mono fun _ hh _ => hh |
| · exact Filter.Eventually.of_forall fun _ hh => (hs hh).elim |
| obtain ⟨z,hz,hzna⟩ := (he.and hna).exists |
| let := hzna |
| obtain ⟨K,H,hK,hKH,hH,hk,hh⟩ := exists_separated_positive (L.conditional z) |
| exact hz K H ⟨hK,hKH,hH⟩ ⟨hk,hh⟩ |
|
|
| theorem Converges.positive_part {F : ℕ → Configuration} {L : Data} (h : Converges F L) |
| (E : ∀ n, (F n).SweepEmbedding) (hR : Tendsto (fun n => (F n).R) atTop atTop) |
| {C : ℝ} (hC : 0 ≤ C) |
| (hcoeff : ∀ n, 4608*((F n).a : ℝ)^2*(F n).n^4/(((F n).R : ℝ)*(F n).a^2)^3 ≤ C) |
| (herror : Tendsto (fun n => 13824*((F n).n : ℝ)^3/(((F n).R : ℝ)*(F n).a^2)^3) atTop (𝓝 0)) |
| (hshort : Tendsto (fun n => ((F n).n : ℝ)/((F n).R*(F n).a)) atTop (𝓝 0)) : |
| ∃ (K H A B : Window) (T : Set Labels), SeparatedBefore K H ∧ |
| carrier A ⊆ carrier H ∧ carrier B ⊆ carrier H ∧ Before A B ∧ |
| MeasurableSet T ∧ (L.edge : Measure Labels) T ≠ 0 ∧ |
| (∀ z ∈ T, L.conditional z (carrier K) ≠ 0 ∧ L.conditional z (carrier H) ≠ 0) ∧ |
| (L.edge : Measure Labels).restrict T ≪ (L.crossings A B : Measure Labels) ∧ |
| (L.edge : Measure Labels).restrict T ≪ (L.points : Measure Labels) := by |
| let := h.edge_probability |
| obtain ⟨K,H,hsep,hpos⟩ := L.exists_positive_windows (h.conditional_nullSingleton E hR hC hcoeff herror) |
| let U : Set Labels := {z | L.conditional z (carrier K) ≠ 0 ∧ L.conditional z (carrier H) ≠ 0} |
| have hU : MeasurableSet U := |
| ((L.conditional.measurable_coe (measurable_carrier K)) (measurableSet_singleton 0)).compl.inter |
| ((L.conditional.measurable_coe (measurable_carrier H)) (measurableSet_singleton 0)).compl |
| have hac (J : Window) (hJ : ∀ z ∈ U, L.conditional z (carrier J) ≠ 0) : |
| (L.edge : Measure Labels).restrict U ≪ (L.edgeWindow J : Measure Labels) := by |
| rw [h.window_withDensity hR J] |
| exact restrict_ac_withDensity _ _ (L.conditional.measurable_coe (measurable_carrier J)) hU hJ |
| have hacH := (hac H (fun _ hz => hz.2)).trans (h.edge_crossing_ac E hR hC hcoeff herror hshort H) |
| obtain ⟨AB,V,hV,hVp,hVac⟩ := positive_single_component |
| ((L.edge : Measure Labels).restrict U) |
| (fun JH : CrossingPairs H => (L.crossings JH.val.1 JH.val.2 : Measure Labels)) hacH |
| (by simpa only [Measure.restrict_apply_univ] using hpos) |
| let T := V ∩ U |
| have hTeq : ((L.edge : Measure Labels).restrict U).restrict V = |
| (L.edge : Measure Labels).restrict T := Measure.restrict_restrict hV |
| have hTpos : (L.edge : Measure Labels) T ≠ 0 := by |
| simpa only [Measure.restrict_apply hV] using hVp |
| have hTpts : (L.edge : Measure Labels).restrict T ≪ (L.points : Measure Labels) := by |
| exact (Measure.absolutelyContinuous_of_le (Measure.restrict_mono_set _ inter_subset_right)).trans |
| ((hac K (fun _ hz => hz.1)).trans (h.window_absolutelyContinuous hC hcoeff herror K)) |
| rw [hTeq] at hVac |
| rcases before_or_before AB.property.2.2 with hAB | hBA |
| · exact ⟨K,H,AB.val.1,AB.val.2,T,hsep,AB.property.1,AB.property.2.1,hAB, |
| hV.inter hU,hTpos,fun _ hz => hz.2,hVac,hTpts⟩ |
| · rw [h.crossings_comm] at hVac |
| exact ⟨K,H,AB.val.2,AB.val.1,T,hsep,AB.property.2.1,AB.property.1,hBA, |
| hV.inter hU,hTpos,fun _ hz => hz.2,hVac,hTpts⟩ |
|
|
| end PlanarHalving.RankLimits |
|
|
| end OAI |
|
|
