| import OAI.Combinatorics.HalvingLines.ActivityRetention |
|
|
| namespace OAI |
|
|
| open Filter Topology |
|
|
| namespace PlanarHalving.RankLimits |
| open Set Filter MeasureTheory Metric Batches TimeWindows Zoom |
| open scoped Topology unitInterval |
|
|
| noncomputable def commonDirection : Fin 3 → Point := ![(1,0),(1,1),(0,1)] |
|
|
| lemma commonDirection_independent (i j : Fin 3) (h : i≠j) : |
| detPair (commonDirection i) (commonDirection j)≠0 := by |
| fin_cases i <;> fin_cases j <;> norm_num [commonDirection,detPair] at * |
|
|
| lemma common_frame (F : ℕ → Configuration) {K : Window} (q : CommonChoice K) |
| (h12 : ∀ n, (F n).timeSlope (q.time 0)<(F n).timeSlope (q.time 1)) |
| (h23 : ∀ n, (F n).timeSlope (q.time 1)<(F n).timeSlope (q.time 2)) : |
| ∃ E : ℕ → Point ≃L[ℝ] Point, |
| (∀ n i, detPair ((E n).symm (1,(F n).timeSlope (q.time i))) (commonDirection i)=0) ∧ |
| ∀ n, ∃ a b l u : ℝ, 0<a ∧ 0<b ∧ l<u ∧ |
| E n (1,0)=a • (1,l) ∧ E n (0,1)=b • (1,u) ∧ |
| ∀ i, ((F n).system.timeWindow q.val.2.2).slope i∈Icc l u := by |
| choose E a b ha hb h1 h2 h3 using fun n => slope_frame (h12 n) (h23 n) |
| refine ⟨E,?_,?_⟩ |
| · intro n i |
| have h1' : (E n).symm (1,(F n).timeSlope (q.time 0))=(a n)⁻¹ • (1,0) := by |
| apply (E n).injective |
| rw [(E n).apply_symm_apply,map_smul,h1,smul_smul,inv_mul_cancel₀ (ha n).ne',one_smul] |
| have h2' : (E n).symm (1,(F n).timeSlope (q.time 2))=(b n)⁻¹ • (0,1) := by |
| apply (E n).injective |
| rw [(E n).apply_symm_apply,map_smul,h2,smul_smul,inv_mul_cancel₀ (hb n).ne',one_smul] |
| have h3' : (E n).symm (1,(F n).timeSlope (q.time 1))=(1,1) := by |
| rw [←h3,(E n).symm_apply_apply] |
| fin_cases i |
| · change detPair ((E n).symm (1,(F n).timeSlope (q.time 0))) (1,0)=0 |
| rw [h1']; simp [detPair] |
| · change detPair ((E n).symm (1,(F n).timeSlope (q.time 1))) (1,1)=0 |
| rw [h3']; simp [detPair] |
| · change detPair ((E n).symm (1,(F n).timeSlope (q.time 2))) (0,1)=0 |
| rw [h2']; simp [detPair] |
| · intro n |
| refine ⟨a n,b n,_,_,ha n,hb n,(h12 n).trans (h23 n),h1 n,h2 n,?_⟩ |
| intro i |
| constructor |
| · exact ((F n).timeSlope_before (J:=q.val.2.2) q.property.2.2.2.1.le).trans |
| (((F n).system.timeWindow q.val.2.2).increasing.monotone (Fin.zero_le i)) |
| · apply (((F n).system.timeWindow q.val.2.2).increasing.monotone (Fin.le_last i)).trans |
| exact (F n).timeSlope_after (J:=q.val.2.2) |
| (q.property.2.2.2.2.trans (q.property.2.1.trans (lower_lt_upper q.val.2.1))).le |
|
|
| noncomputable def commonChart (F : Configuration) {A B : Window} (hAB : Before A B) |
| (hA : 0<length F.R A) (hB : 0<length F.R B) (E : Point ≃L[ℝ] Point) : |
| OpenPartialHomeomorph Point Point := |
| E.toHomeomorph.transOpenPartialHomeomorph (F.rankChart hAB hA hB) |
|
|
| @[simp] lemma commonChart_symm (F : Configuration) {A B : Window} (hAB : Before A B) |
| (hA : 0<length F.R A) (hB : 0<length F.R B) (E : Point ≃L[ℝ] Point) (p : Point) : |
| (commonChart F hAB hA hB E).symm p=E.symm ((F.rankChart hAB hA hB).symm p) := rfl |
|
|
| lemma commonChart_source (F : Configuration) {A B : Window} (hAB : Before A B) |
| (hA : 0<length F.R A) (hB : 0<length F.R B) (E : Point ≃L[ℝ] Point) : |
| (commonChart F hAB hA hB E).source=univ := by |
| simp [commonChart,F.rankChart_source] |
|
|
| lemma commonChart_target (F : Configuration) {A B : Window} (hAB : Before A B) |
| (hA : 0<length F.R A) (hB : 0<length F.R B) (E : Point ≃L[ℝ] Point) : |
| (commonChart F hAB hA hB E).target=Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0 := by |
| simp [commonChart,F.rankChart_target] |
|
|
| end PlanarHalving.RankLimits |
|
|
| namespace PlanarHalving.RankLimits |
| open Set Filter MeasureTheory Metric Batches TimeWindows Zoom |
| open scoped Topology unitInterval |
|
|
| lemma Converges.radial_ae {F : ℕ → Configuration} {L : Data} (h : Converges F L) |
| (emb : ∀ n, (F n).SweepEmbedding) (hR : Tendsto (fun n => (F n).R) atTop atTop) |
| (hM : Tendsto (fun n => (F n).a*(F n).n) 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 A B : Window} (hAB : Before A B) (hKA : Before K A) (hKB : Before K B) |
| (hA : ∀ n, 0<length (F n).R A) (hB : ∀ n, 0<length (F n).R B) |
| (G : Subwindows K → Point → I) |
| (hjoint : ∀ i p, ContinuousAt (G i) p → Tendsto |
| (fun q : ℕ × Point => (F q.1).extendedRank true A B i.val q.2) |
| (atTop ×ˢ 𝓝 p) (𝓝 (G i p))) |
| (hGaed : ∀ i, ∀ᵐ p ∂(volume : Measure Point), DifferentiableAt ℝ (fun x => (G i x:ℝ)) p) |
| (hid : ∀ i z, z∈(L.points:Measure Labels).support → |
| ContinuousAt (G i) (pairCoordinates true A B z) → z i.val=G i (pairCoordinates true A B z)) |
| (ν : Measure Labels) (he : ν≪(L.edge:Measure Labels)) (hp : ν≪(L.points:Measure Labels)) |
| (hproj : ν.map (pairCoordinates true A B)≪volume) |
| (q : CommonChoice K) |
| (hpos : ∀ᵐ z ∂ν, L.conditional z (carrier K)≠0 ∧ z∈commonSet L q) |
| (E : ℕ → Point ≃L[ℝ] Point) |
| (hE : ∀ n i, detPair ((E n).symm (1,(F n).timeSlope (q.time i))) (commonDirection i)=0) : |
| ∀ᵐ z ∂ν, pairCoordinates true A B z∈Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0 ∧ |
| ∃ C₀ : ℝ, 0<C₀ ∧ ∀ T : ℝ, 1≤T → ∀ᶠ r in 𝓝[>] (0:ℝ), ∀ᶠ n in atTop, |
| RadialCompare (commonChart (F n) hAB (hA n) (hB n) (E n)).symm |
| (pairCoordinates true A B z) r C₀ T := by |
| have hreg := h.rank_regular_ae ν hp hproj G hGaed hid |
| have hind := he.ae_le (h.rank_independent_ae (A:=A) (B:=B) emb hR hM hC hcoeff herror hshort G) |
| have hna := he.ae_le (h.conditional_nullSingleton emb hR hC hcoeff herror) |
| filter_upwards [hreg,hind,hna,hpos,pairCoordinates_interior_ae hproj] with z hz hi hn hpos hzp |
| let := hn |
| refine ⟨hzp,?_⟩ |
| have h := pointwise_radial F hR hAB hKA hKB hA hB G hjoint hGaed hzp |
| (fun i => (hz i).1) (L.conditional z) hpos.1 (fun J hJ => by |
| obtain ⟨i,j,hi,hj,_,hij⟩ := hi hz J hJ |
| exact ⟨i,j,hi,hj,hij⟩) q.time q.time_mem |
| (q.positive_gaps (L.conditional z) hpos.2.1 hpos.2.2.1) |
| (fun n => (E n).symm) commonDirection commonDirection_independent hE |
| exact h |
|
|
| end PlanarHalving.RankLimits |
|
|
| namespace PlanarHalving.RankLimits |
| open Set Filter MeasureTheory Metric Batches TimeWindows Zoom Spatial Enhancement Boundary |
| open scoped Topology unitInterval |
|
|
| lemma normalized_commonChart (F : Configuration) {A B : Window} (hAB : Before A B) |
| (hA : 0<length F.R A) (hB : 0<length F.R B) (E : Point ≃L[ℝ] Point) |
| (p : Point) {U : ℝ} (hU : U≠0) (x : Point) : |
| (normalizeChart (commonChart F hAB hA hB E) p U hU).symm x= |
| physical E ((F.rankChart hAB hA hB).symm p) U⁻¹ ((F.rankChart hAB hA hB).symm x) := by |
| simp only [normalizeChart_symm,commonChart_symm,physical,map_sub] |
|
|
| lemma enhancement_points_ae (F : Configuration) (ψ : Point → Point) (H : Space → Prop) |
| (hH : ∀ p, H (Enhancement.label F ψ p)) : |
| ∀ᵐ z ∂(Enhancement.points F ψ : Measure Space), H z := by |
| apply Measure.ae_smul_measure |
| rw [ae_finsetSum_measure_iff] |
| intro i _ |
| simpa only [ae_dirac_eq,Filter.eventually_pure,Function.comp_apply] using hH (F.point i) |
|
|
| lemma normalized_points_graph (F : Configuration) {A B : Window} (hAB : Before A B) |
| (hA : 0<length F.R A) (hB : 0<length F.R B) (E : Point ≃L[ℝ] Point) |
| (p₀ : Point) {U : ℝ} (hU : U≠0) : |
| let o := (F.rankChart hAB hA hB).symm p₀ |
| let ψ := physical E o U⁻¹ |
| ∀ᵐ z ∂(Enhancement.points F ψ : Measure Space), z.2=encode |
| ((normalizeChart (commonChart F hAB hA hB E) p₀ U hU).symm |
| (pairCoordinates true A B z.1)) := by |
| dsimp only |
| apply enhancement_points_ae |
| intro p |
| change encode (physical E _ _ p)=encode ((normalizeChart (commonChart F hAB hA hB E) p₀ U hU).symm |
| (pairCoordinates true A B (F.system.labels F.generic p))) |
| rw [normalized_commonChart,←F.rankChart_apply hAB hA hB p] |
| rw [(F.rankChart hAB hA hB).left_inv (mem_univ p)] |
|
|
| lemma support_graph_subseq {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| {μs : ℕ → FiniteMeasure (X × Cube)} {μ : FiniteMeasure (X × Cube)} |
| {τ : ℕ → ℕ} (hτ : StrictMono τ) (hμ : Tendsto (μs∘τ) atTop (𝓝 μ)) |
| (π : X → Point) (hπ : Continuous π) (f : ℕ → Point → Point) |
| (hid : ∀ n, ∀ᵐ z ∂(μs n:Measure (X×Cube)), z.2=encode (f n (π z.1))) |
| (G : Set X) (l : (π '' G) → Point) |
| (hjoint : ∀ p : (π '' G), Tendsto (fun q : ℕ × Point => f q.1 q.2) |
| (atTop ×ˢ 𝓝 (p:Point)) (𝓝 (l p))) : |
| ∀ z∈(μ:Measure (X×Cube)).support, ∀ hz : z.1∈G, |
| z.2=encode (l ⟨π z.1,mem_image_of_mem π hz⟩) := by |
| intro z hz hG |
| apply support_spatial_graph hμ π hπ (f∘τ) |
| (Filter.Eventually.of_forall (fun n => hid (τ n))) G l _ hz hG |
| intro p |
| exact (hjoint p).comp (hτ.tendsto_atTop.prodMap tendsto_id) |
|
|
| lemma Converges.spatial_impossible {F : ℕ → Configuration} {L : Data} (hF : Converges F L) |
| (hR : Tendsto (fun n => (F n).R) atTop atTop) (emb : ∀ n, (F n).SweepEmbedding) |
| {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)) |
| (J : Window) {A B : Window} (hAB : Before A B) |
| (hA : ∀ n, 0<length (F n).R A) (hB : ∀ n, 0<length (F n).R B) |
| (E : ℕ → Point ≃L[ℝ] Point) |
| (hframe : ∀ n, ∃ a b l u : ℝ, 0<a ∧ 0<b ∧ l<u ∧ |
| E n (1,0)=a • (1,l) ∧ E n (0,1)=b • (1,u) ∧ |
| ∀ i, ((F n).system.timeWindow J).slope i∈Icc l u) |
| (T : Set Labels) (hT : MeasurableSet T) (hTp : (L.edgeWindow J:Measure Labels) T≠0) |
| (hac : ((L.edgeWindow J:Measure Labels).restrict T).map (pairCoordinates true A B)≪volume) |
| (hrad : ∀ᵐ z ∂(L.edgeWindow J:Measure Labels).restrict T, |
| pairCoordinates true A B z∈Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0 ∧ |
| ∃ C₀ : ℝ, 0<C₀ ∧ ∀ Q : ℝ, 1≤Q → ∀ᶠ r in 𝓝[>] (0:ℝ), ∀ᶠ n in atTop, |
| RadialCompare (commonChart (F n) hAB (hA n) (hB n) (E n)).symm |
| (pairCoordinates true A B z) r C₀ Q) : False := by |
| let π := pairCoordinates true A B |
| let f := fun n => commonChart (F n) hAB (hA n) (hB n) (E n) |
| obtain ⟨G,hG,hGT,hGp,m,hdensity,p₀,U,hU,φ,l,c,hc,hφ,hl,hjoint,hsep⟩ := |
| positive_spatial_witness (L.points:Measure Labels) (L.edgeWindow J:Measure Labels) |
| T hT hTp π (continuous_pairCoordinates true A B).measurable hac f |
| (fun n => commonChart_source _ _ _ _ _) (fun n => commonChart_target _ _ _ _ _) hrad |
| let o := fun n => ((F n).rankChart hAB (hA n) (hB n)).symm p₀ |
| let ψ := fun n => physical (E n) (o n) (U n)⁻¹ |
| obtain ⟨ρ,M,hρ,hM,hprojection⟩ := Enhancement.subsequence (hF.subseq hφ) (fun n => ψ (φ n)) |
| have hgraph : ∀ z∈(M .point:Measure Space).support, ∀ hz : z.1∈G, |
| z.2=encode (l ⟨π z.1,mem_image_of_mem π hz⟩) := |
| support_graph_subseq (μs:=fun n => Enhancement.points (F (φ n)) (ψ (φ n))) hρ |
| (hM .point) π (continuous_pairCoordinates true A B) |
| (fun n => (normalizeChart (f (φ n)) p₀ (U (φ n)) (hU (φ n)).ne').symm) |
| (fun n => normalized_points_graph (F (φ n)) hAB (hA _) (hB _) (E _) p₀ (hU _).ne') |
| G l hjoint |
| have hmap : (M .point:Measure Space).map Prod.fst=(L.points:Measure Labels) := by |
| rw [← FiniteMeasure.toMeasure_map] |
| exact congrArg FiniteMeasure.toMeasure (hprojection .point) |
| obtain ⟨hH,hbounded,D,hD,hdisk⟩ := support_lift_disk (M .point:Measure Space) G hG π |
| (continuous_pairCoordinates true A B).measurable l hc (by rwa [hmap]) hgraph hl hsep |
| have hz := enhanced_boundary_zero ((hF.subseq hφ).subseq hρ) |
| ((hR.comp hφ.tendsto_atTop).comp hρ.tendsto_atTop) (fun n => emb (φ (ρ n))) |
| hC (fun n => hcoeff (φ (ρ n))) ((herror.comp hφ.tendsto_atTop).comp hρ.tendsto_atTop) |
| J (fun n => E (φ (ρ n))) (fun n => o (φ (ρ n))) (fun n => (U (φ (ρ n)))⁻¹) |
| (fun n => inv_pos.mpr (hU _)) (fun n => hframe (φ (ρ n))) hM hprojection G hG |
| (fun z hz => ⟨hbounded z hz,by rw [hgraph z hz.2 hz.1,decode_encode]⟩) hD hdisk |
| have hz' : L.edgeWindow J G=0 := NNReal.coe_injective hz |
| apply hGp |
| rw [← FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure,hz',ENNReal.coe_zero] |
|
|
| end PlanarHalving.RankLimits |
| namespace PlanarHalving.RankLimits |
| open Set Filter MeasureTheory Metric Batches TimeWindows Zoom Spatial Enhancement Boundary |
| open scoped Topology unitInterval |
|
|
| lemma common_finite_eventually (F : ℕ → Configuration) |
| (hR : Tendsto (fun n => (F n).R) atTop atTop) |
| (A B : Window) {K : Window} (q : CommonChoice K) : |
| ∀ᶠ n in atTop, 0<length (F n).R A ∧ 0<length (F n).R B ∧ |
| (F n).timeSlope (q.time 0)<(F n).timeSlope (q.time 1) ∧ |
| (F n).timeSlope (q.time 1)<(F n).timeSlope (q.time 2) := by |
| filter_upwards [eventually_length_pos _ hR A,eventually_length_pos _ hR B, |
| timeSlope_eventually_lt F hR (q.time_strictMono (by decide : (0:Fin 3)<1)), |
| timeSlope_eventually_lt F hR (q.time_strictMono (by decide : (1:Fin 3)<2))] |
| with n hA hB h12 h23 |
| exact ⟨hA,hB,h12,h23⟩ |
|
|
| lemma joint_shift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
| (f : ℕ → X → Y) {p : X} {y : Y} |
| (h : Tendsto (fun z : ℕ × X => f z.1 z.2) (atTop ×ˢ 𝓝 p) (𝓝 y)) (N : ℕ) : |
| Tendsto (fun z : ℕ × X => f (z.1+N) z.2) (atTop ×ˢ 𝓝 p) (𝓝 y) := |
| h.comp ((tendsto_add_atTop_nat N).prodMap tendsto_id) |
|
|
| lemma Converges.analytic_ready {F : ℕ → Configuration} {L : Data} (hF : Converges F L) |
| (hR : Tendsto (fun n => (F n).R) atTop atTop) |
| {K A B : Window} (hAB : Before A B) (hKA : Before K A) (hKB : Before K B) |
| (q : CommonChoice K) : |
| ∃ (φ : ℕ → ℕ) (G : Subwindows K → Point → I), StrictMono φ ∧ |
| (∀ n, 0<length (F (φ n)).R A ∧ 0<length (F (φ n)).R B ∧ |
| (F (φ n)).timeSlope (q.time 0)<(F (φ n)).timeSlope (q.time 1) ∧ |
| (F (φ n)).timeSlope (q.time 1)<(F (φ n)).timeSlope (q.time 2)) ∧ |
| (∀ i, ∀ᵐ p ∂(volume:Measure Point), DifferentiableAt ℝ (fun x => (G i x:ℝ)) p) ∧ |
| (∀ i z, z∈(L.points:Measure Labels).support → |
| ContinuousAt (G i) (pairCoordinates true A B z) → z i.val=G i (pairCoordinates true A B z)) ∧ |
| ∀ i p, ContinuousAt (G i) p → Tendsto |
| (fun z : ℕ × Point => (F (φ z.1)).extendedRank true A B i.val z.2) |
| (atTop ×ˢ 𝓝 p) (𝓝 (G i p)) := by |
| have hC (i : Subwindows K) : Before i.val A ∧ Before i.val B := |
| ⟨(endpoints_of_subset i.property).2.trans hKA,(endpoints_of_subset i.property).2.trans hKB⟩ |
| obtain ⟨φ,G,hφ,_,hGaed,hid,hjoint⟩ := hF.rank_analytic hR true hAB |
| (fun i : Subwindows K => i.val) (fun i => hC i) |
| have hR' : Tendsto (fun n => (F (φ n)).R) atTop atTop := hR.comp hφ.tendsto_atTop |
| have he := common_finite_eventually (F∘φ) hR' A B q |
| obtain ⟨N,hN⟩ := eventually_atTop.mp he |
| refine ⟨fun n => φ (n+N),G,(fun _ _ h => hφ (Nat.add_lt_add_right h N)), |
| (fun n => hN (n+N) (Nat.le_add_left N n)),hGaed,hid,?_⟩ |
| intro i p hp |
| exact joint_shift (fun n => (F (φ n)).extendedRank true A B i.val) (hjoint i p hp) N |
|
|
| end PlanarHalving.RankLimits |
|
|
| namespace PlanarHalving.RankLimits |
| open Set Filter MeasureTheory Metric Batches TimeWindows Zoom Spatial Enhancement Boundary |
| open scoped Topology unitInterval |
|
|
| lemma Converges.impossible {F : ℕ → Configuration} {L : Data} (hF : Converges F L) |
| (emb : ∀ n, (F n).SweepEmbedding) (hR : Tendsto (fun n => (F n).R) atTop atTop) |
| (hM : Tendsto (fun n => (F n).a*(F n).n) 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)) : False := by |
| obtain ⟨K,H,A,B,T,hKH,hAH,hBH,hAB,hT,hTp,hpos,hcross,hpts⟩ := |
| hF.positive_part emb hR hC hcoeff herror hshort |
| obtain ⟨q,V,hV,hVT,hcommon,hVp,hVle⟩ := hF.common_positive hR K |
| (hF.conditional_nullSingleton emb hR hC hcoeff herror) hT hTp (fun z hz => (hpos z hz).1) |
| let ν := (L.edgeWindow q.val.2.2:Measure Labels).restrict V |
| have he : ν≪(L.edge:Measure Labels) := |
| Measure.absolutelyContinuous_of_le (hVle.trans Measure.restrict_le_self) |
| have hp : ν≪(L.points:Measure Labels) := (Measure.absolutelyContinuous_of_le hVle).trans hpts |
| have hproj : ν.map (pairCoordinates true A B)≪volume := |
| (((Measure.absolutelyContinuous_of_le hVle).trans hcross).map |
| (continuous_pairCoordinates true A B).measurable).trans (hF.crossing_signed_projection hR hM true hAB) |
| have hgood : ∀ᵐ z ∂ν, L.conditional z (carrier K)≠0 ∧ z∈commonSet L q := by |
| filter_upwards [ae_restrict_mem hV] with z hz |
| exact ⟨(hpos z (hVT hz)).1,hcommon z hz⟩ |
| have hKA : Before K A := hKH.2.1.le.trans (endpoints_of_subset hAH).1 |
| have hKB : Before K B := hKH.2.1.le.trans (endpoints_of_subset hBH).1 |
| obtain ⟨φ,G,hφ,hfinite,hGaed,hid,hjoint⟩ := hF.analytic_ready hR hAB hKA hKB q |
| have hF' : Converges (F∘φ) L := hF.subseq hφ |
| have hR' : Tendsto (fun n => (F (φ n)).R) atTop atTop := hR.comp hφ.tendsto_atTop |
| obtain ⟨E,hE,hframe⟩ := common_frame (F∘φ) q |
| (fun n => (hfinite n).2.2.1) (fun n => (hfinite n).2.2.2) |
| exact hF'.spatial_impossible hR' (fun n => emb (φ n)) hC (fun n => hcoeff (φ n)) |
| (herror.comp hφ.tendsto_atTop) q.val.2.2 hAB (fun n => (hfinite n).1) |
| (fun n => (hfinite n).2.1) E hframe V hV hVp hproj |
| (hF'.radial_ae (fun n => emb (φ n)) hR' (hM.comp hφ.tendsto_atTop) hC |
| (fun n => hcoeff (φ n)) (herror.comp hφ.tendsto_atTop) (hshort.comp hφ.tendsto_atTop) |
| hAB hKA hKB (fun n => (hfinite n).1) (fun n => (hfinite n).2.1) |
| G hjoint hGaed hid ν he hp hproj q hgood E hE) |
|
|
| end PlanarHalving.RankLimits |
| namespace PlanarHalving.RankLimits |
| open Filter |
| open scoped Topology |
|
|
| lemma normalized_sequence_impossible (F : ℕ → Configuration) |
| (emb : ∀ n, (F n).SweepEmbedding) {c : ℝ} (hc : 0<c) |
| (hnear : ∀ n, c*((F n).n:ℝ)^4≤((F n).R:ℝ)^3*((F n).a:ℝ)^4) |
| (hq : Tendsto (fun n => ((F n).n:ℝ)/(F n).a) atTop atTop) : False := by |
| obtain ⟨φ,L,hφ,hF⟩ := subsequence F |
| have hnear' : ∀ n, c*((F (φ n)).n:ℝ)^4≤((F (φ n)).R:ℝ)^3*((F (φ n)).a:ℝ)^4 := |
| fun n => hnear (φ n) |
| have hq' := hq.comp hφ.tendsto_atTop |
| have hs := normalization_sizes_atTop hc hnear' hq' |
| exact hF.impossible (fun n => emb (φ n)) hs.2.2 hs.2.1 (by positivity : 0≤4608/c) |
| (fun n => (F (φ n)).normalization_coefficient hc (hnear' n)) |
| (sparse_error_tendsto_zero hc hnear' hq') (short_ratio_tendsto_zero hc hnear' hq') |
|
|
| theorem uniform_batch_saving {c : ℝ} (hc : 0<c) : |
| ∃ Q : ℝ, 0<Q ∧ ∀ F : Configuration, F.SweepEmbedding → Q≤(F.n:ℝ)/F.a → |
| (F.R:ℝ)^3*(F.a:ℝ)^4≤c*(F.n:ℝ)^4 := by |
| by_contra h |
| push Not at h |
| have hex (m : ℕ) : ∃ F : Configuration, Nonempty F.SweepEmbedding ∧ |
| (m:ℝ)+1≤(F.n:ℝ)/F.a ∧ c*(F.n:ℝ)^4≤(F.R:ℝ)^3*(F.a:ℝ)^4 := by |
| obtain ⟨F,emb,hq,hnear⟩ := h ((m:ℝ)+1) (by positivity) |
| exact ⟨F,⟨emb⟩,hq,hnear.le⟩ |
| choose F he hq hn using hex |
| have hlim : Tendsto (fun m => ((F m).n:ℝ)/(F m).a) atTop atTop := by |
| apply tendsto_atTop_mono hq |
| exact tendsto_atTop_add_const_right _ 1 tendsto_natCast_atTop_atTop |
| exact normalized_sequence_impossible F (fun n => (he n).some) hc hn hlim |
|
|
| end PlanarHalving.RankLimits |
| namespace PlanarHalving.Sweep |
| open Set RankLimits |
|
|
| noncomputable def greedyConfiguration {n k : ℕ} {P : Fin n → Point} |
| (hP : Generic P) (hk : k≤n) (a : ℕ) (ha : 0<a) (hn : 0<n) |
| (hR : 0<((states P hP hk).greedy 0 a ha).count) : Configuration where |
| n := n |
| R := ((states P hP hk).greedy 0 a ha).count |
| k := k |
| a := a |
| n_pos := hn |
| R_pos := hR |
| a_pos := ha |
| point := P |
| generic := hP |
| system := (states P hP hk).greedySystem a ha |
|
|
| noncomputable def greedyConfiguration_embedding {n k : ℕ} {P : Fin n → Point} |
| (hP : Generic P) (hk : k≤n) (a : ℕ) (ha : 0<a) (hn : 0<n) |
| (hR : 0<((states P hP hk).greedy 0 a ha).count) : |
| (greedyConfiguration hP hk a ha hn hR).SweepEmbedding where |
| rank_le := hk |
| index := ((states P hP hk).greedy 0 a ha).index |
| increasing := ((states P hP hk).greedy 0 a ha).increasing |
| marks := fun _ => rfl |
|
|
| lemma distance_saving {c : ℝ} (hc : 0<c) : |
| ∃ Q : ℝ, 0<Q ∧ ∀ {n k : ℕ} (P : Fin n → Point) (hP : Generic P) (hk : k≤n), |
| ∀ {d : ℕ}, 0<d → Q≤(n:ℝ)/d → |
| ∀ C : Finset (Fin ((traversal (criticalSlopes P)).size+1) × Fin ((traversal (criticalSlopes P)).size+1)), |
| StepDisjoint C → (∀ p∈C, p.1≤p.2) → |
| (∀ p∈C, d≤(states P hP hk).distance p.1 p.2) → |
| (C.card:ℝ)^3*(d:ℝ)^4≤16*c*(n:ℝ)^4 := by |
| obtain ⟨Q,hQ,hbatch⟩ := uniform_batch_saving hc |
| refine ⟨Q,hQ,?_⟩ |
| intro n k P hP hk d hd hq C hC hord hdist |
| let S := states P hP hk |
| let a := (d+1)/2 |
| have ha : 0<a := by dsimp [a];omega |
| have had : a≤d := by dsimp [a];omega |
| let R := (S.greedy 0 a ha).count |
| have hcount : C.card≤R := S.greedy_packing hd C hC hord hdist |
| by_cases hR : 0<R |
| · have hn : 0<n := by |
| by_contra hn |
| have hn0 : n=0 := by omega |
| have hq0 : (n:ℝ)/d=0 := by simp [hn0] |
| rw [hq0] at hq |
| exact (not_le_of_gt hQ) hq |
| let F := greedyConfiguration hP hk a ha hn hR |
| have hbd : (R:ℝ)^3*(a:ℝ)^4≤c*(n:ℝ)^4 := |
| hbatch F (greedyConfiguration_embedding hP hk a ha hn hR) |
| (hq.trans (div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast ha) (by exact_mod_cast had))) |
| have hda : (d:ℝ)≤2*(a:ℝ) := by exact_mod_cast (show d≤2*a by dsimp [a];omega) |
| calc |
| _ ≤ (R:ℝ)^3*(2*(a:ℝ))^4 := by gcongr |
| _ = 16*((R:ℝ)^3*(a:ℝ)^4) := by ring |
| _ ≤ 16*(c*(n:ℝ)^4) := mul_le_mul_of_nonneg_left hbd (by norm_num) |
| _ = _ := by ring |
| · have hz : C.card=0 := by omega |
| simp only [hz,Nat.cast_zero,zero_pow (by omega : (3:ℕ)≠0),zero_mul] |
| positivity |
|
|
| end PlanarHalving.Sweep |
| namespace PlanarHalving.Sweep |
| open Set |
|
|
| theorem activity_saving_cubic {δ : ℝ} (hδ : 0<δ) : |
| ∃ Q : ℝ, 0<Q ∧ ∀ {n k : ℕ} (P : Fin n → Point) (hP : Generic P) (hk : k≤n), |
| ∀ {b : ℕ}, 0<b → Q≤(n:ℝ)/b → |
| ∀ C : Finset (Fin ((traversal (criticalSlopes P)).size+1) × Fin ((traversal (criticalSlopes P)).size+1)), |
| StepDisjoint C → (horder : ∀ p∈C, p.1≤p.2) → |
| (∀ p, ∀ hp : p∈C, b≤((states P hP hk).intervalActivity p.1 p.2 (horder p hp)).card) → |
| (C.card:ℝ)^3*(b:ℝ)^4≤524288*δ*(n:ℝ)^4 := by |
| obtain ⟨Q,hQ,hdist⟩ := distance_saving hδ |
| refine ⟨Q,hQ,?_⟩ |
| intro n k P hP hk b hb hq C hC horder hactivity |
| classical |
| let S := states P hP hk |
| by_cases hb4 : 4≤b |
| · let c := b/4 |
| have hc : 0<c := by dsimp [c]; omega |
| have hbLe : (b : ℝ)≤8*(c : ℝ) := by exact_mod_cast (show b≤8*c by dsimp [c]; omega) |
| let A := C.filter (fun p => ∃ hk' : k+c≤n, c≤(states P hP hk').distance p.1 p.2) |
| let B := C\A |
| have hA : A⊆C := Finset.filter_subset _ _ |
| have hB : B⊆C := Finset.sdiff_subset |
| have hbound (D : Finset _) (hD : D⊆C) (r : ℕ) |
| (hd : ∀ p∈D, ∃ hr : r≤n, c≤(states P hP hr).distance p.1 p.2) : |
| (D.card : ℝ)^3*(c : ℝ)^4≤16*δ*(n : ℝ)^4 := by |
| by_cases hne : D.Nonempty |
| · obtain ⟨p,hp⟩ := hne |
| obtain ⟨hr,_⟩ := hd p hp |
| apply hdist P hP hr hc (hq.trans (div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast hc) (by exact_mod_cast (show c≤b by dsimp [c];omega)))) D |
| · exact fun p hp q hq hpq => hC p (hD hp) q (hD hq) hpq |
| · exact fun p hp => horder p (hD hp) |
| · intro p hp |
| obtain ⟨hr',hh⟩ := hd p hp |
| exact hh |
| · simp only [Finset.not_nonempty_iff_eq_empty.mp hne,Finset.card_empty,Nat.cast_zero,zero_pow (by omega : (3:ℕ)≠0),zero_mul] |
| positivity |
| have hAb := hbound A hA (k+c) (fun p hp => (Finset.mem_filter.mp hp).2) |
| have hBb := hbound B hB (k-c) (fun p hp => by |
| have hpC := hB hp |
| have hh := states_intervalActivity_distance P hP hk (horder p hpC) |
| ((show 4*c≤b from Nat.mul_div_le b 4).trans (hactivity p hpC)) |
| rcases hh with hh | ⟨hr,_,hh⟩ |
| · exact False.elim ((Finset.mem_sdiff.mp hp).2 (Finset.mem_filter.mpr ⟨hpC,hh⟩)) |
| · exact ⟨hr,hh⟩) |
| have hcard : (C.card : ℝ)=(A.card : ℝ)+(B.card : ℝ) := by |
| exact_mod_cast (by simpa only [B,Nat.add_comm] using (Finset.card_sdiff_add_card_eq_card hA).symm : C.card=A.card+B.card) |
| have hsum := mul_le_mul_of_nonneg_right (cube_add_le (show (0:ℝ)≤A.card by positivity) (show (0:ℝ)≤B.card by positivity)) (show (0:ℝ)≤(c : ℝ)^4 by positivity) |
| rw [← hcard] at hsum |
| have hcub : (C.card : ℝ)^3*(c : ℝ)^4≤128*δ*(n : ℝ)^4 := by nlinarith [hAb,hBb] |
| calc |
| _ ≤ (C.card : ℝ)^3*(8*(c : ℝ))^4 := by gcongr |
| _ = 4096*((C.card : ℝ)^3*(c : ℝ)^4) := by ring |
| _ ≤ 4096*(128*δ*(n : ℝ)^4) := mul_le_mul_of_nonneg_left hcub (by norm_num) |
| _ ≤ _ := by nlinarith [mul_nonneg hδ.le (sq_nonneg ((n : ℝ)^2))] |
| · have hd := hdist P hP hk (by omega : 0<1) (by simpa using hq.trans (div_le_self (by positivity) (by exact_mod_cast hb))) C hC horder |
| (fun p hp => S.intervalActivity_pos_distance (horder p hp) (lt_of_lt_of_le hb (hactivity p hp))) |
| norm_num at hd |
| have hbLe : (b : ℝ)≤3 := by exact_mod_cast (by omega : b≤3) |
| calc |
| _ ≤ (C.card : ℝ)^3*3^4 := by gcongr |
| _ ≤ 16*δ*(n : ℝ)^4*3^4 := mul_le_mul_of_nonneg_right hd (by norm_num) |
| _ ≤ _ := by nlinarith [mul_nonneg hδ.le (sq_nonneg ((n : ℝ)^2))] |
|
|
| end PlanarHalving.Sweep |
|
|
| end OAI |
|
|
