import OAI.Combinatorics.HalvingLines.ThinStrips namespace OAI open Filter Topology namespace PlanarHalving.Boundary open Set Filter MeasureTheory Batches TimeWindows RankLimits Enhancement Spatial open scoped Topology unitInterval NNReal ENNReal Classical lemma finite_window_params (F : Configuration) (K : Window) (E : Point ≃L[ℝ] Point) {A B l u : ℝ} (hA : 0 < A) (hB : 0 < B) (hlu : l < u) (h1 : E (1,0)=A • (1,l)) (h2 : E (0,1)=B • (1,u)) (hspan : ∀ i, (F.system.timeWindow K).slope i ∈ Icc l u) (o : Point) {r : ℝ} (hr : 0 < r) : ∃ p : Subwindow K → CompactParams, (∀ J, (vertices F (physical E o r) J.val (Prod.snd ⁻¹' (encode '' awayStrips (p J).2.1.val (p J).2.2.1.val (fun i => ((p J).2.2.2 i).val))):ℝ) ≤ 2*(p J).1.val) ∧ (∀ D : Finset (Subwindow K), Set.PairwiseDisjoint (↑D : Set (Subwindow K)) (fun J => carrier J.val) → (∑ J∈D, (p J).1.val) ≤ 4 ∧ (∑ J∈D, (p J).2.2.1.val) ≤ 1) := by classical let S := F.system.timeWindow K let pnt := fun v => physical E o r (F.point v) let j : Fin F.a := ⟨0,F.a_pos⟩ let w : Fin (length F.R K) → ℝ := fun i => S.boundaryCount pnt j i let t : Fin (length F.R K+1) → ℝ := fun i => frameParameter A B l u (S.slope i) let s : Subwindow K → ℕ := fun J => start F.R J.val-start F.R K let m : Subwindow K → ℕ := fun J => length F.R J.val have hsm (J : Subwindow K) : s J+m J ≤ length F.R K := relative_end_le F.R J.property let b : Subwindow K → ℝ := fun J => (∑ i∈batchRange _ (s J) (m J), w i)/F.n let ti : Subwindow K → ℝ := fun J => t ⟨s J,by have := hsm J; omega⟩ let theta : Subwindow K → ℝ := fun J => t ⟨s J+m J,by have := hsm J; omega⟩-ti J have hw (i) : 0 ≤ w i := Nat.cast_nonneg _ have ht : Monotone t := by intro i i' hii' apply frameParameter_monotone hA hB hlu (hspan _) (hspan _) exact S.increasing.monotone hii' have htI (i) : t i ∈ Icc (0:ℝ) 1 := frameParameter_mem hA hB hlu (hspan _) have hcost : (∑ i, w i) ≤ 4*(F.n:ℝ) := by have hh := S.boundary_budget pnt j (fun v => (actual_tokens_monotone S E hA hB hlu h1 h2 hspan o hr j v).1) (fun v => (actual_tokens_monotone S E hA hB hlu h1 h2 hspan o hr j v).2) dsimp only [w] exact_mod_cast hh have hpos : 0 < (F.n:ℝ) := Nat.cast_pos.mpr F.n_pos have hb (J) : b J ∈ Icc (0:ℝ) 4 := by constructor · exact div_nonneg (Finset.sum_nonneg (fun i _ => hw i)) hpos.le · apply (div_le_iff₀ hpos).mpr exact (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun i _ _ => hw i)).trans hcost have hti (J) : ti J ∈ Icc (0:ℝ) 1 := htI _ have htheta (J) : theta J ∈ Icc (0:ℝ) 1 := by constructor · exact sub_nonneg.mpr (ht (by simp [Fin.le_def])) · have hh := (htI (⟨s J+m J,by have := hsm J; omega⟩)).2 have hh' := (hti J).1 dsimp [theta]; linarith obtain ⟨c,hc,hfinite⟩ : ∃ c : Subwindow K → Fin 3 → ℝ, (∀ J h, c J h ∈ Icc (-4:ℝ) 4) ∧ ∀ J, (((S.window (s J) (m J) (hsm J)).vertices.filter fun v => pnt v∈awayStrips (ti J) (theta J) (c J)).card) ≤ 2*∑ i : Fin (m J), S.boundaryCount pnt j ⟨s J+i.val,by have := hsm J; omega⟩ := by choose c hc hb using fun J => S.window_boundary E hA hB hlu h1 h2 hspan o hr j (s J) (m J) (hsm J) refine ⟨c,hc,fun J => ?_⟩ convert hb J using 1 congr 1 apply Finset.filter_congr intro v _ rfl let p : Subwindow K → CompactParams := fun J => (⟨b J,hb J⟩,⟨ti J,hti J⟩,⟨theta J,htheta J⟩,fun i => ⟨c J i,hc J i⟩) refine ⟨p,?_,?_⟩ · intro J change (vertices F (physical E o r) J.val (Prod.snd ⁻¹' (encode '' awayStrips (ti J) (theta J) (c J))):ℝ) ≤ 2*b J rw [vertices,System.vertexMeasure_apply _ _ ((isOpenMap_encode _ (isOpen_awayStrips _ _ _)).measurableSet.preimage measurable_snd)] have he : ((F.system.timeWindow J.val).vertices.filter fun v => label F (physical E o r) (F.point v)∈Prod.snd ⁻¹' (encode '' awayStrips (ti J) (theta J) (c J))) = ((S.window (s J) (m J) (hsm J)).vertices.filter fun v => pnt v∈awayStrips (ti J) (theta J) (c J)) := by rw [show (S.window (s J) (m J) (hsm J)).vertices=(F.system.timeWindow J.val).vertices from F.system.relative_vertices J.property] apply Finset.filter_congr intro v _ exact encode_injective.mem_set_image rw [he] apply (div_le_iff₀ hpos).mpr have hh : (((S.window (s J) (m J) (hsm J)).vertices.filter fun v => pnt v∈awayStrips (ti J) (theta J) (c J)).card:ℝ) ≤ 2*∑ i∈batchRange _ (s J) (m J), w i := by rw [sum_batchRange (hsm J)] dsimp only [w] exact_mod_cast hfinite J calc _ ≤ 2*∑ i∈batchRange _ (s J) (m J), w i := hh _ = 2*b J*(F.n:ℝ) := by dsimp [b] field_simp · intro D hD have hd : Set.PairwiseDisjoint (↑D : Set (Subwindow K)) (fun J => batchRange (length F.R K) (s J) (m J)) := by intro J hJ H hH hJH exact relative_range_disjoint F.R J H (hD hJ hH hJH) constructor · change (∑ J∈D, b J) ≤ 4 simp only [b,← Finset.sum_div] apply (div_le_iff₀ hpos).mpr exact (sum_disjoint_ranges D s m hd w hw).trans hcost · change (∑ J∈D, theta J) ≤ 1 have he (J : Subwindow K) : theta J=∑ i∈batchRange _ (s J) (m J), (t i.succ-t i.castSucc) := (parameter_range_sum (hsm J) t).symm simp_rw [he] exact parameter_budget D s m hd t ht htI end PlanarHalving.Boundary namespace PlanarHalving.RankLimits open Set TimeWindows Boundary open scoped Classical lemma localPartition_subset {K J : Window} {m : ℕ} (h : J∈localPartition K m) : carrier J⊆carrier K := (Finset.mem_filter.mp h).2 noncomputable def subPartition (K : Window) (m : ℕ) : Finset (Subwindow K) := (localPartition K m).attach.image (fun J => (⟨J.val,localPartition_subset J.property⟩ : Subwindow K)) lemma mem_subPartition {K : Window} {m : ℕ} {J : Subwindow K} : J∈subPartition K m ↔ J.val∈localPartition K m := by unfold subPartition constructor · intro h obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp h exact j.property · intro h exact Finset.mem_image.mpr ⟨⟨J.val,h⟩,Finset.mem_attach _ _,Subtype.ext rfl⟩ lemma subPartition_map_val (K : Window) (m : ℕ) : (subPartition K m).image Subtype.val=localPartition K m := by ext J constructor · intro h obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp h exact mem_subPartition.mp hj · intro h exact Finset.mem_image.mpr ⟨⟨J,localPartition_subset h⟩,mem_subPartition.mpr h,rfl⟩ lemma subPartition_sum {M : Type*} [AddCommMonoid M] (K : Window) (m : ℕ) (f : Window → M) : ∑ J∈subPartition K m, f J.val=∑ J∈localPartition K m, f J := by rw [← subPartition_map_val K m] rw [Finset.sum_image (fun _ _ _ _ h => Subtype.ext h)] lemma subPartition_disjoint (K : Window) (m : ℕ) : Set.PairwiseDisjoint (↑(subPartition K m):Set (Subwindow K)) (fun J => carrier J.val) := by intro J hJ H hH hJH exact localPartition_disjoint K m (mem_subPartition.mp hJ) (mem_subPartition.mp hH) (fun he => hJH (Subtype.ext he)) lemma subPartition_card (K : Window) (m : ℕ) : (subPartition K m).card ≤ 2^(K.1+m) := by have he : (subPartition K m).card=(localPartition K m).card := by rw [← subPartition_map_val K m,Finset.card_image_of_injective _ Subtype.val_injective] rw [he] apply (Finset.card_le_card (Finset.filter_subset _ _)).trans exact (Finset.card_image_le).trans (by simp) lemma subPartition_tail {F : ℕ → Configuration} {L : Data} (h : Converges F L) (E : ∀ n, (F n).SweepEmbedding) (K : Window) (m : ℕ) {u : ℝ} (hu : 0 < u) : (((subPartition K m).filter fun J => u < ((L.vertices J.val).mass:ℝ)).card:ℝ) ≤ 10000*u^(-(4:ℝ)/3) := by have hh := h.activity_tail E (K.1+m) hu rw [← filter_timePartition (K.1+m) (fun J => u < ((L.vertices J).mass:ℝ))] at hh apply hh.trans' apply Nat.cast_le.mpr apply Finset.card_le_card_of_injOn Subtype.val · intro J hJ obtain ⟨hJ,hJu⟩ := Finset.mem_filter.mp hJ exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp (mem_subPartition.mp hJ)).1,hJu⟩ · exact Subtype.val_injective.injOn end PlanarHalving.RankLimits namespace PlanarHalving.Enhancement open Set Filter MeasureTheory Batches TimeWindows RankLimits Limits Spatial open scoped Topology unitInterval NNReal ENNReal Classical lemma limit_vertices_le {F : ℕ → Configuration} {ψ : ℕ → Point → Point} {M : Index → FiniteMeasure Space} (hM : ∀ i, Tendsto (fun n => measures (F n) (ψ n) i) atTop (𝓝 (M i))) (J : Window) : (M (.vertex J) : Measure Space) ≤ (M .point : Measure Space) := measure_le_of_weak_limits (hM (.vertex J)) (hM .point) (fun n => vertices_le_points (F n) (ψ n) J) lemma limit_edge_ac {F : ℕ → Configuration} {ψ : ℕ → Point → Point} {M : Index → FiniteMeasure Space} (hM : ∀ i, Tendsto (fun n => measures (F n) (ψ n) i) atTop (𝓝 (M i))) {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) : (M (.edge J) : Measure Space) ≪ (M .point : Measure Space) := by apply Measure.AbsolutelyContinuous.trans (μ₂ := (M (.vertex J) : Measure Space)) _ (Measure.absolutelyContinuous_of_le (limit_vertices_le hM J)) intro U hU let V := toMeasurable (M (.vertex J) : Measure Space) U have hc := local_cubic hM hC hcoeff herror J (measurableSet_toMeasurable (M (.vertex J) : Measure Space) U) have hV : M (.vertex J) V=0 := by apply ENNReal.coe_eq_zero.mp rw [FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure,measure_toMeasurable] exact hU change (M (.edge J) V:ℝ)^3 ≤ C*((M (.vertex J)).mass:ℝ)^3*(M (.vertex J) V:ℝ) at hc rw [hV,NNReal.coe_zero,mul_zero] at hc have hz : M (.edge J) V=0 := by apply NNReal.coe_injective exact (pow_eq_zero_iff (by norm_num : (3:ℕ)≠0)).mp (le_antisymm hc (pow_nonneg (NNReal.coe_nonneg _) _)) apply measure_mono_null (subset_toMeasurable _ _) rw [← FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure,hz,ENNReal.coe_zero] lemma lifted_mass (μ ν : FiniteMeasure Space) (ρ : FiniteMeasure Labels) (hac : (ν : Measure Space) ≪ (μ : Measure Space)) (hmap : ν.map Prod.fst=ρ) {G : Set Labels} (hG : MeasurableSet G) : (ν (Prod.fst ⁻¹' G ∩ (μ : Measure Space).support):ℝ)=(ρ G:ℝ) := by have hr := congrArg (fun m : Measure Space => m univ) (support_lift_restrict (μ:Measure Space) (ν:Measure Space) hac (Prod.fst ⁻¹' G)) simp only [Measure.restrict_apply_univ] at hr have he := congrArg (fun m : FiniteMeasure Labels => (m:Measure Labels) G) hmap rw [FiniteMeasure.toMeasure_map,Measure.map_apply measurable_fst hG] at he change ((ν:Measure Space) _).toReal=((ρ:Measure Labels) G).toReal rw [hr,he] lemma projected_mass (ν : FiniteMeasure Space) (ρ : FiniteMeasure Labels) (hmap : ν.map Prod.fst=ρ) : ν.mass=ρ.mass := by rw [← hmap] change ν univ=(ν.map Prod.fst) univ rw [FiniteMeasure.map_apply _ measurable_fst MeasurableSet.univ,preimage_univ] end PlanarHalving.Enhancement namespace PlanarHalving.Boundary open Set Filter Metric MeasureTheory Batches TimeWindows RankLimits Enhancement Spatial open scoped Topology unitInterval NNReal ENNReal Classical lemma enhanced_boundary_zero {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)) (K : Window) (E : ℕ → Point ≃L[ℝ] Point) (o : ℕ → Point) (r : ℕ → ℝ) (hr : ∀ n, 0 measures (F n) (physical (E n) (o n) (r n)) i) atTop (𝓝 (M i))) (hprojection : ∀ i, (M i).map Prod.fst=old L i) (G : Set Labels) (hG : MeasurableSet G) (hpos : ∀ z∈Prod.fst ⁻¹' G ∩ (M .point : Measure Space).support, ‖decode z.2‖≤1 ∧ encode (decode z.2)=z.2) {D : ℝ} (hD : 0≤D) (hdisk : ∀ x : Point, ∀ t : ℝ, 0 finite_window_params (F n) K (E n) (ha n) (hb n) (hlu n) (h1 n) (h2 n) (hspan n) (o n) (hr n) choose p hp hbudget using hparams obtain ⟨φ,b,t,θ,c,hφ,hbounds,hb,ht,hθ,hc⟩ := params_subsequence p have haway (J : Subwindow K) : (M (.vertex J.val) (Prod.snd ⁻¹' (encode '' awayStrips (t J) (θ J) (c J))):ℝ)≤2*b J := by apply moving_strips_limit ((hM (.vertex J.val)).comp hφ.tendsto_atTop) (ht J) (hθ J) (hc J) (hb J) exact Filter.Eventually.of_forall (fun n => hp (φ n) J) have hactivity (J : Subwindow K) : (M (.vertex J.val) H:ℝ)≤ 2*b J+3*(13*D*(1+3^((32:ℝ)/17))*36^((32:ℝ)/17-1))*θ J^((32:ℝ)/17-1) := by exact limiting_activity_bound (M .point) (M (.vertex J.val)) H hH (limit_vertices_le hM J.val) hpos hD (by norm_num) (hbounds J).2.2.1.1 (hbounds J).1.1 (c J) hdisk (haway J) have hbudgets (m : ℕ) : (∑ J∈subPartition K m, b J)≤4 ∧ (∑ J∈subPartition K m, θ J)≤1 := by constructor · exact finite_budget_limit (subPartition K m) _ b hb (Filter.Eventually.of_forall (fun n => (hbudget (φ n) (subPartition K m) (subPartition_disjoint K m)).1)) · exact finite_budget_limit (subPartition K m) _ θ hθ (Filter.Eventually.of_forall (fun n => (hbudget (φ n) (subPartition K m) (subPartition_disjoint K m)).2)) have hlam (J : Window) : (M (.edge J) H:ℝ)=(L.edgeWindow J G:ℝ) := lifted_mass (M .point) (M (.edge J)) (L.edgeWindow J) (limit_edge_ac hM hC hcoeff herror J) (hprojection (.edge J)) hG have hvertex (J : Window) : (M (.vertex J)).mass=(L.vertices J).mass := projected_mass _ _ (hprojection (.vertex J)) let q : ℕ → ℝ := fun m => (2:ℝ)^(K.1+m) have hq : ∀ m, 0 by positivity have hqtop : Tendsto q atTop atTop := (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ)<2)).comp (tendsto_atTop_mono (fun n => Nat.le_add_left n K.1) tendsto_id) apply mass_eq_zero (subPartition K) (fun _ J => (M (.edge J.val) H:ℝ)) (fun _ J => (M (.vertex J.val) H:ℝ)) (fun _ J => ((L.vertices J.val).mass:ℝ)) (fun _ J => b J) (fun _ J => θ J) q (A:=C) (H:=10000) (B:=3*(13*D*(1+3^((32:ℝ)/17))*36^((32:ℝ)/17-1))) (B₀:=4) (T:=1) (α:=(32:ℝ)/17-1) (NNReal.coe_nonneg _) hC (by positivity) (by norm_num) (by norm_num) (by norm_num) hq hqtop · intro m simpa only [q,Nat.cast_pow,Nat.cast_ofNat] using (Nat.cast_le (α:=ℝ).mpr (subPartition_card K m)) · intro m J _ exact NNReal.coe_nonneg _ · intro m J hJ rw [hlam] have hd := mem_timePartition.mp (Finset.mem_filter.mp (mem_subPartition.mp hJ)).1 apply (NNReal.coe_le_coe.mpr ((L.edgeWindow J.val).apply_le_mass G)).trans_eq rw [hF.edgeWindow_mass hR,hd] simp only [q,Nat.cast_pow,Nat.cast_ofNat] · intro m J _; exact NNReal.coe_nonneg _ · intro m J _; exact NNReal.coe_nonneg _ · intro m J _; exact (hbounds J).2.2.1.1 · intro m J _ have hh := Enhancement.local_cubic hM hC hcoeff herror J.val hH rwa [hvertex] at hh · intro m J _; exact hactivity J · intro m; exact (hbudgets m).1 · intro m; exact (hbudgets m).2 · intro m u hu; exact subPartition_tail hF emb K m hu · intro m simp_rw [hlam] rw [subPartition_sum K m (fun J => (L.edgeWindow J G:ℝ)),hF.localPartition_sum hR K m hG] end PlanarHalving.Boundary namespace PlanarHalving.RankLimits open Set Filter MeasureTheory Batches TimeWindows Limits open scoped Topology unitInterval ENNReal lemma Converges.window_le_edge {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) := by rw [Data.edgeWindow,Data.edge,FiniteMeasure.toMeasure_map,FiniteMeasure.toMeasure_map, h.restriction_eq hR] exact Measure.map_mono Measure.restrict_le_self measurable_fst noncomputable def commonSet (L : Data) {K : Window} (q : CommonChoice K) : Set Labels := {z | L.conditional z (carrier q.val.1)≠0 ∧ L.conditional z (carrier q.val.2.1)≠0 ∧ L.conditional z (carrier q.val.2.2)≠0} lemma commonSet_measurable (L : Data) {K : Window} (q : CommonChoice K) : MeasurableSet (commonSet L q) := by have hm (J : Window) : MeasurableSet {z | L.conditional z (carrier J)≠0} := ((L.conditional.measurable_coe (measurable_carrier J)) (measurableSet_singleton 0)).compl exact (hm q.val.1).inter ((hm q.val.2.1).inter (hm q.val.2.2)) lemma Converges.common_positive {F : ℕ → Configuration} {L : Data} (h : Converges F L) (hR : Tendsto (fun n => (F n).R) atTop atTop) (K : Window) (hna : ∀ᵐ z ∂(L.edge : Measure Labels), NullSingletonClass (L.conditional z)) {T : Set Labels} (hT : MeasurableSet T) (hTp : (L.edge : Measure Labels) T≠0) (hK : ∀ z∈T, L.conditional z (carrier K)≠0) : ∃ (q : CommonChoice K) (E : Set Labels), MeasurableSet E ∧ E⊆T ∧ (∀ z∈E, z∈commonSet L q) ∧ (L.edgeWindow q.val.2.2 : Measure Labels) E≠0 ∧ (L.edgeWindow q.val.2.2 : Measure Labels).restrict E ≤ (L.edge : Measure Labels).restrict T := by have hcover : ∀ᵐ z ∂(L.edge : Measure Labels).restrict T, ∃ q : CommonChoice K, z∈commonSet L q := by filter_upwards [ae_restrict_of_ae hna,ae_restrict_mem hT] with z hz hzt let := hz exact exists_commonChoice (L.conditional z) K (hK z hzt) obtain ⟨q,hq⟩ := positive_countable_selection (L.edge : Measure Labels) T hT hTp (commonSet L) hcover let E := T∩commonSet L q have hE : MeasurableSet E := hT.inter (commonSet_measurable L q) have hac : (L.edge : Measure Labels).restrict E ≪ (L.edgeWindow q.val.2.2 : Measure Labels) := by rw [h.window_withDensity hR] exact restrict_ac_withDensity _ _ (L.conditional.measurable_coe (measurable_carrier _)) hE (fun z hz => hz.2.2.2) refine ⟨q,E,hE,inter_subset_left,fun _ hz => hz.2,?_,?_⟩ · intro he have hzero := hac he rw [Measure.restrict_apply hE,inter_self] at hzero exact hq hzero · exact Measure.restrict_mono inter_subset_left (h.window_le_edge hR _) end PlanarHalving.RankLimits namespace PlanarHalving.RankLimits open Set Filter MeasureTheory Batches TimeWindows open scoped Topology unitInterval ENNReal lemma coordinate_lines_null (x : ℝ) : (volume : Measure Point) {p | p.1=x}=0 ∧ (volume : Measure Point) {p | p.2=x}=0 := by have h1 : {p : Point | p.1=x}=({x}:Set ℝ)×ˢuniv := by ext p; simp have h2 : {p : Point | p.2=x}=(univ:Set ℝ)×ˢ{x} := by ext p; simp rw [h1,h2,Measure.volume_eq_prod,Measure.prod_prod,Measure.prod_prod] simp lemma pairCoordinates_interior_ae {ν : Measure Labels} {A B : Window} (hproj : ν.map (pairCoordinates true A B) ≪ volume) : ∀ᵐ z ∂ν, pairCoordinates true A B z∈Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0 := by have hq : Measure.QuasiMeasurePreserving (pairCoordinates true A B) ν volume := ⟨(continuous_pairCoordinates true A B).measurable,hproj⟩ have ha (x : ℝ) : ∀ᵐ p ∂(volume : Measure Point), p.1≠x := by simpa only [ae_iff,not_not] using (coordinate_lines_null x).1 have hb (x : ℝ) : ∀ᵐ p ∂(volume : Measure Point), p.2≠x := by simpa only [ae_iff,not_not] using (coordinate_lines_null x).2 filter_upwards [hq.ae (ha 0),hq.ae (ha 1),hq.ae (hb (-1)),hq.ae (hb 0)] with z h1 h2 h3 h4 dsimp [pairCoordinates,signedPair] at h1 h2 h3 h4 ⊢ have hzA := (z A).property have hzB := (z B).property exact ⟨⟨lt_of_le_of_ne hzA.1 (Ne.symm h1),lt_of_le_of_ne hzA.2 h2⟩, ⟨lt_of_le_of_ne (by linarith [hzB.2]) (Ne.symm h3),lt_of_le_of_ne (by linarith [hzB.1]) h4⟩⟩ end PlanarHalving.RankLimits end OAI