import OAI.Combinatorics.HalvingLines.LocalZoom namespace OAI open Filter Topology namespace PlanarHalving.Zoom open Set Filter Metric open scoped Topology lemma detPair_zero_parallel {u v : Point} (hv : v ≠ 0) (h : detPair u v=0) : ∃ a : ℝ, u=a • v := by by_cases h₁ : v.1=0 · have h₂ : v.2≠0 := by intro hh; exact hv (Prod.ext h₁ hh) refine ⟨u.2/v.2,Prod.ext ?_ ?_⟩ · change u.1=u.2/v.2*v.1 rw [h₁,mul_zero] have he : u.1*v.2=0 := by simpa [detPair,h₁] using h exact (mul_eq_zero.mp he).resolve_right h₂ · change u.2=u.2/v.2*v.2 field_simp · refine ⟨u.1/v.1,Prod.ext ?_ ?_⟩ · change u.1=u.1/v.1*v.1 field_simp · change u.2=u.1/v.1*v.2 rw [div_mul_eq_mul_div] apply (eq_div_iff h₁).mpr dsimp [detPair] at h nlinarith lemma map_eq_column (A : Point →L[ℝ] Point) (x : Point) : A x=x.1 • A (1,0)+x.2 • A (0,1) := by have hx : x=x.1 • ((1,0):Point)+x.2 • ((0,1):Point) := by ext <;> simp conv_lhs => rw [hx,map_add,map_smul,map_smul] lemma detPair_map (A : Point →L[ℝ] Point) (u v : Point) : detPair (A u) (A v)=detPair (A (1,0)) (A (0,1))*detPair u v := by rw [map_eq_column A u,map_eq_column A v] dsimp [detPair]; ring lemma map_injective_of_det (A : Point →L[ℝ] Point) (h : detPair (A (1,0)) (A (0,1)) ≠ 0) : Function.Injective A := by have he : A= (columnEquiv (A (1,0)) (A (0,1)) h).toContinuousLinearMap := by apply ContinuousLinearMap.ext intro x exact map_eq_column A x rw [he] exact (columnEquiv _ _ h).injective lemma map_zero_of_two (A : Point →L[ℝ] Point) {u v : Point} (h : detPair u v ≠ 0) (hu : A u=0) (hv : A v=0) : A=0 := by apply ContinuousLinearMap.ext intro x obtain ⟨a,b,he⟩ := plane_span u v h x rw [he,map_add,map_smul,map_smul,hu,hv,smul_zero,smul_zero,add_zero] rfl lemma three_lines_injective (A : Point →L[ℝ] Point) (hA : A≠0) (u v : Fin 3 → Point) (hu : ∀ i j, i≠j → detPair (u i) (u j) ≠ 0) (hv : ∀ i j, i≠j → detPair (v i) (v j) ≠ 0) (hline : ∀ i, detPair (A (u i)) (v i)=0) : Function.Injective A := by have hv0 (i : Fin 3) : v i≠0 := by obtain ⟨j,hj⟩ := exists_ne i intro he have hh := hv i j hj.symm simp only [he,detPair,Prod.fst_zero,Prod.snd_zero,zero_mul,sub_self,ne_eq,not_true_eq_false] at hh choose c hc using fun i => detPair_zero_parallel (hv0 i) (hline i) apply map_injective_of_det intro hdet have hp (i j : Fin 3) (hij : i≠j) : c i=0 ∨ c j=0 := by have hh := detPair_map A (u i) (u j) rw [hdet,zero_mul,hc,hc,detPair_smul] at hh have hz : c i*c j=0 := (mul_eq_zero.mp hh).resolve_right (hv i j hij) exact mul_eq_zero.mp hz have hz (i j : Fin 3) (hij : i≠j) (hi : c i=0) (hj : c j=0) : False := by apply hA apply map_zero_of_two A (hu i j hij) · rw [hc,hi,zero_smul] · rw [hc,hj,zero_smul] rcases hp 0 1 (by decide) with h0 | h1 · rcases hp 1 2 (by decide) with h1 | h2 · exact hz 0 1 (by decide) h0 h1 · exact hz 0 2 (by decide) h0 h2 · rcases hp 0 2 (by decide) with h0 | h2 · exact hz 0 1 (by decide) h0 h1 · exact hz 1 2 (by decide) h1 h2 lemma three_lines_uniform_conorm (v : Fin 3 → Point) (hv : ∀ i j, i≠j → detPair (v i) (v j) ≠ 0) (δ : ℝ) (hδ : 0<δ) : ∃ ε : ℝ, 0<ε ∧ ∀ (A : Point →L[ℝ] Point), ‖A‖=1 → ∀ u : Fin 3 → Point, (∀ i, ‖u i‖=1) → (∀ i j, i≠j → δ ≤ |detPair (u i) (u j)|) → (∀ i, detPair (A (u i)) (v i)=0) → ∀ y : Point, ‖y‖=1 → ε ≤ ‖A y‖ := by by_contra! hn have hex (n : ℕ) : ∃ (A : Point →L[ℝ] Point) (u : Fin 3 → Point) (y : Point), ‖A‖=1 ∧ (∀ i, ‖u i‖=1) ∧ (∀ i j, i≠j → δ ≤ |detPair (u i) (u j)|) ∧ (∀ i, detPair (A (u i)) (v i)=0) ∧ ‖y‖=1 ∧ ‖A y‖ < 1/(n+1:ℝ) := by obtain ⟨A,hA,u,hu,hsep,hl,y,hy,hsmall⟩ := hn (1/(n+1:ℝ)) (by positivity) exact ⟨A,u,y,hA,hu,hsep,hl,hy,hsmall⟩ choose A u y hA hu hsep hl hy hsmall using hex let Z := ((Point →L[ℝ] Point) × (Fin 3 → Point)) × Point let z : ℕ → Z := fun n => ((A n,u n),y n) have hz (n : ℕ) : z n∈closedBall (0:Z) 1 := by rw [mem_closedBall_zero_iff] change max (max ‖A n‖ ‖u n‖) ‖y n‖ ≤ 1 rw [hA,hy,max_le_iff,max_le_iff] refine ⟨⟨le_rfl,?_⟩,le_rfl⟩ exact (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr (fun i => (hu n i).le) obtain ⟨z₀,hz₀,ρ,hρ,hlim⟩ := (isCompact_closedBall (0:Z) 1).tendsto_subseq hz let A₀ := z₀.1.1 let u₀ := z₀.1.2 let y₀ := z₀.2 have hAL : Tendsto (fun n => A (ρ n)) atTop (𝓝 A₀) := by exact ((continuous_fst.comp continuous_fst).tendsto z₀).comp hlim have huL : Tendsto (fun n => u (ρ n)) atTop (𝓝 u₀) := by exact ((continuous_snd.comp continuous_fst).tendsto z₀).comp hlim have hyL : Tendsto (fun n => y (ρ n)) atTop (𝓝 y₀) := by exact (continuous_snd.tendsto z₀).comp hlim have hAn : ‖A₀‖=1 := tendsto_nhds_unique hAL.norm (by simp only [hA]; exact tendsto_const_nhds) have hu_n (i : Fin 3) : ‖u₀ i‖=1 := tendsto_nhds_unique ((tendsto_pi_nhds.mp huL i).norm) (by simp only [hu]; exact tendsto_const_nhds) have hyn : ‖y₀‖=1 := tendsto_nhds_unique hyL.norm (by simp only [hy]; exact tendsto_const_nhds) have hsep₀ (i j : Fin 3) (hij : i≠j) : detPair (u₀ i) (u₀ j) ≠ 0 := by have hi := tendsto_pi_nhds.mp huL i have hj := tendsto_pi_nhds.mp huL j have hd : Tendsto (fun n => |detPair (u (ρ n) i) (u (ρ n) j)|) atTop (𝓝 |detPair (u₀ i) (u₀ j)|) := ((hi.fst_nhds.mul hj.snd_nhds).sub (hi.snd_nhds.mul hj.fst_nhds)).abs have hb := ge_of_tendsto hd (Eventually.of_forall (fun n => hsep (ρ n) i j hij)) exact abs_pos.mp (hδ.trans_le hb) have hl₀ (i : Fin 3) : detPair (A₀ (u₀ i)) (v i)=0 := by have ht := ((continuous_fst.clm_apply continuous_snd).tendsto (A₀,u₀ i)).comp (hAL.prodMk_nhds (tendsto_pi_nhds.mp huL i)) have hd : Tendsto (fun n => detPair (A (ρ n) (u (ρ n) i)) (v i)) atTop (𝓝 (detPair (A₀ (u₀ i)) (v i))) := (ht.fst_nhds.mul_const _).sub (ht.snd_nhds.mul_const _) exact tendsto_nhds_unique hd (by simp only [hl]; exact tendsto_const_nhds) have hInj := three_lines_injective A₀ (by intro he; rw [he,norm_zero] at hAn; norm_num at hAn) u₀ v hsep₀ hv hl₀ have hzero : ‖A₀ y₀‖=0 := by have ht := (((continuous_fst.clm_apply continuous_snd).tendsto (A₀,y₀)).comp (hAL.prodMk_nhds hyL)).norm have hnlim : Tendsto (fun n : ℕ => 1/(n+1:ℝ)) atTop (𝓝 0) := by simpa using tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ) have hbound : Tendsto (fun n => ‖A (ρ n) (y (ρ n))‖) atTop (𝓝 0) := squeeze_zero (fun n => norm_nonneg _) (fun n => (hsmall (ρ n)).le) (hnlim.comp hρ.tendsto_atTop) exact tendsto_nhds_unique ht hbound have hye : y₀=0 := hInj ((norm_eq_zero.mp hzero).trans (map_zero A₀).symm) rw [hye,norm_zero] at hyn norm_num at hyn end PlanarHalving.Zoom namespace PlanarHalving.Zoom open Set Filter Metric open scoped Topology lemma detPair_equiv_ne_zero (B : Point ≃L[ℝ] Point) {u v : Point} (h : detPair u v≠0) : detPair (B u) (B v)≠0 := by intro he have ht := detPair_map B.symm.toContinuousLinearMap (B u) (B v) apply h simpa only [ContinuousLinearEquiv.coe_coe,B.symm_apply_apply,he,mul_zero] using ht lemma signed_simplex_ne_zero (z : ℝ) : RankLimits.reflectFirst (Directions.simplex z)≠0 := by intro he have h1 := congrArg Prod.fst he have h2 := congrArg Prod.snd he dsimp [RankLimits.reflectFirst,Directions.simplex] at h1 h2 linarith lemma unit_simplex_parallel_det_ne_zero (B : Point ≃L[ℝ] Point) {u v : Point} (hu : ‖u‖=1) (hv : ‖v‖=1) {z w : ℝ} (hzw : z≠w) (huz : detPair u (B (RankLimits.reflectFirst (Directions.simplex z)))=0) (hvw : detPair v (B (RankLimits.reflectFirst (Directions.simplex w)))=0) : detPair u v≠0 := by have hnon (r : ℝ) : B (RankLimits.reflectFirst (Directions.simplex r))≠0 := by intro he exact signed_simplex_ne_zero r (B.injective (he.trans (map_zero B).symm)) obtain ⟨a,ha⟩ := detPair_zero_parallel (hnon z) huz obtain ⟨b,hb⟩ := detPair_zero_parallel (hnon w) hvw have ha0 : a≠0 := by intro he; rw [he,zero_smul] at ha; rw [ha,norm_zero] at hu; norm_num at hu have hb0 : b≠0 := by intro he; rw [he,zero_smul] at hb; rw [hb,norm_zero] at hv; norm_num at hv rw [ha,hb,detPair_smul] apply mul_ne_zero (mul_ne_zero ha0 hb0) apply detPair_equiv_ne_zero have ht : detPair (RankLimits.reflectFirst (Directions.simplex z)) (RankLimits.reflectFirst (Directions.simplex w))=w-z := by dsimp [detPair,RankLimits.reflectFirst,Directions.simplex]; ring rw [ht] exact sub_ne_zero.mpr hzw.symm lemma compact_allowed_separation (B : Point ≃L[ℝ] Point) (S T : Set ℝ) (hS : IsClosed S) (hT : IsClosed T) (hSb : S ⊆ Icc (0:ℝ) 1) (hTb : T ⊆ Icc (0:ℝ) 1) (hST : Disjoint S T) : ∃ δ : ℝ, 0<δ ∧ ∀ u v : Point, ‖u‖=1 → ‖v‖=1 → (∃ z∈S, detPair u (B (RankLimits.reflectFirst (Directions.simplex z)))=0) → (∃ z∈T, detPair v (B (RankLimits.reflectFirst (Directions.simplex z)))=0) → δ ≤ |detPair u v| := by let dir : ℝ → Point := fun z => B (RankLimits.reflectFirst (Directions.simplex z)) have hdir : Continuous dir := by apply B.continuous.comp apply RankLimits.reflectFirst.continuous.comp exact (continuous_id.prodMk (continuous_const.sub continuous_id)) let K : Set (Point × ℝ) := {x | ‖x.1‖=1 ∧ x.2∈S ∧ detPair x.1 (dir x.2)=0} let L : Set (Point × ℝ) := {x | ‖x.1‖=1 ∧ x.2∈T ∧ detPair x.1 (dir x.2)=0} have hdet : Continuous (fun x : Point × ℝ => detPair x.1 (dir x.2)) := by exact ((continuous_fst.fst.mul ((hdir.comp continuous_snd).snd)).sub (continuous_fst.snd.mul ((hdir.comp continuous_snd).fst))) have hKC : IsClosed K := (isClosed_eq continuous_fst.norm continuous_const).inter ((hS.preimage continuous_snd).inter (isClosed_eq hdet continuous_const)) have hLC : IsClosed L := (isClosed_eq continuous_fst.norm continuous_const).inter ((hT.preimage continuous_snd).inter (isClosed_eq hdet continuous_const)) have hK : IsCompact K := ((isCompact_closedBall (0:Point) 1).prod isCompact_Icc).of_isClosed_subset hKC (by intro x hx; exact ⟨by simp [hx.1],hSb hx.2.1⟩) have hL : IsCompact L := ((isCompact_closedBall (0:Point) 1).prod isCompact_Icc).of_isClosed_subset hLC (by intro x hx; exact ⟨by simp [hx.1],hTb hx.2.1⟩) let f : (Point × ℝ) × (Point × ℝ) → ℝ := fun x => |detPair x.1.1 x.2.1| have hf : Continuous f := by exact ((continuous_fst.fst.fst.mul continuous_snd.fst.snd).sub (continuous_fst.fst.snd.mul continuous_snd.fst.fst)).abs have hpos (x) (hx : x∈K ×ˢ L) : 0 (h hij).elim⟩ · obtain ⟨δ,hδ,hd⟩ := compact_allowed_separation B (S ij.1) (S ij.2) (hS ij.1) (hS ij.2) (hSb ij.1) (hSb ij.2) (hdis _ _ hij) exact ⟨δ,hδ,fun u v hu hv hz hw _ => hd u v hu hv hz hw⟩ choose δ hδ hd using hex obtain ⟨ij,_,hmin⟩ := isCompact_univ.exists_isMinOn (s:= (univ:Set (Fin 3 × Fin 3))) (by simp) (continuous_of_discreteTopology (f:=δ)).continuousOn refine ⟨δ ij,hδ ij,?_⟩ intro u hu hz i j hij exact (hmin (show (i,j)∈(univ:Set (Fin 3 × Fin 3)) from mem_univ _)).trans (hd (i,j) (u i) (u j) (hu i) (hu j) (hz i) (hz j) hij) lemma eventually_separated_of_allowed_clusters (B : Point ≃L[ℝ] Point) (S : Fin 3 → Set ℝ) (hS : ∀ i, IsClosed (S i)) (hSb : ∀ i, S i ⊆ Icc (0:ℝ) 1) (hdis : ∀ i j, i≠j → Disjoint (S i) (S j)) : ∃ δ : ℝ, 0<δ ∧ ∀ (u : ℕ → Fin 3 → Point), (∀ n i, ‖u n i‖=1) → (∀ ρ : ℕ → ℕ, StrictMono ρ → ∀ v : Fin 3 → Point, Tendsto (u ∘ ρ) atTop (𝓝 v) → ∀ i, ∃ z∈S i, detPair (v i) (B (RankLimits.reflectFirst (Directions.simplex z)))=0) → ∀ᶠ n in atTop, ∀ i j, i≠j → δ ≤ |detPair (u n i) (u n j)| := by obtain ⟨δ,hδ,hsep⟩ := triple_allowed_separation B S hS hSb hdis refine ⟨δ/2,by positivity,?_⟩ intro u hu hcluster by_contra hn obtain ⟨ρ,hρ,hbad⟩ := extraction_of_frequently_atTop (not_eventually.mp hn) have hb (n : ℕ) : u (ρ n)∈closedBall (0:Fin 3 → Point) 1 := by rw [mem_closedBall_zero_iff] exact (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤1)).mpr (fun i => (hu _ i).le) obtain ⟨v,_,τ,hτ,hlim⟩ := (isCompact_closedBall (0:Fin 3 → Point) 1).tendsto_subseq hb have hv (i : Fin 3) : ‖v i‖=1 := tendsto_nhds_unique ((tendsto_pi_nhds.mp hlim i).norm) (by simp only [Function.comp_def,hu]; exact tendsto_const_nhds) have hd := hsep v hv (hcluster (ρ ∘ τ) (hρ.comp hτ) v hlim) have he : ∀ᶠ n in atTop, ∀ i j, i≠j → δ/2 ≤ |detPair (u (ρ (τ n)) i) (u (ρ (τ n)) j)| := by apply eventually_all.mpr intro i apply eventually_all.mpr intro j by_cases hij : i=j · exact Eventually.of_forall (fun _ h => (h hij).elim) · have hi := tendsto_pi_nhds.mp hlim i have hj := tendsto_pi_nhds.mp hlim j have ht : Tendsto (fun n => |detPair (u (ρ (τ n)) i) (u (ρ (τ n)) j)|) atTop (𝓝 |detPair (v i) (v j)|) := ((hi.fst_nhds.mul hj.snd_nhds).sub (hi.snd_nhds.mul hj.fst_nhds)).abs have hlt : δ/2 < |detPair (v i) (v j)| := by linarith [hd i j hij] exact (ht.eventually (lt_mem_nhds hlt)).mono (fun _ hh _ => hh.le) obtain ⟨n,hn⟩ := he.exists exact hbad (τ n) hn end PlanarHalving.Zoom namespace PlanarHalving.RankLimits open Set Filter MeasureTheory Metric Batches TimeWindows RankCoordinates Zoom open scoped Topology unitInterval lemma affine_rank_forward_limit {ι : Type*} (F : ℕ → Configuration) {A B : Window} (hAB : Before A B) (hA : ∀ n, 0 < length (F n).R A) (hB : ∀ n, 0 < length (F n).R B) (C : ι → Window) (hCB : ∀ i, Before (C i) A ∧ Before (C i) B) (hC : ∀ i, ∀ᶠ n in atTop, 0 < length (F n).R (C i)) (D : ι → Point →L[ℝ] ℝ) (c : ι → ℝ) (p : Point) (h q : ℕ → ℝ) (hh : ∀ n, 0 affineZoom ((F n).rankChart hAB (hA n) (hB n)) (E n) B' p (x₀ n) (q n) (h n) (hq n).ne' (hh n).ne' (x n)) atTop (𝓝 a)) (hb : Tendsto (fun n => affineZoom ((F n).rankChart hAB (hA n) (hB n)) (E n) B' p (x₀ n) (q n) (h n) (hq n).ne' (hh n).ne' (y n)) atTop (𝓝 b)) (hj : ∀ i R, 0 ≤ R → TendstoUniformlyOn (fun n u => (((F n).extendedRank true A B (C i) (p+h n • u) : ℝ)-c i)/h n) (D i) atTop (closedBall 0 R)) : ∃ z∈Directions.allowed reflectFirst D (fun i => (lower (C i):ℝ)) (fun i => (upper (C i):ℝ)) t, detPair (b-a) (B' (reflectFirst (Directions.simplex z)))=0 := by let X := fun n => x₀ n+q n • E n (x n) let Y := fun n => x₀ n+q n • E n (y n) have hXY (n : ℕ) : Y n=X n+(q n*s n) • (1,(F n).timeSlope t) := by have he := hE n have hy : E n (y n)=E n (x n)+s n • (1,(F n).timeSlope t) := by rw [map_sub] at he; exact eq_add_of_sub_eq' he dsimp only [X,Y] rw [hy,smul_add,smul_smul,add_assoc] have hxa : Tendsto (fun n => (h n)⁻¹ • ((F n).rankChart hAB (hA n) (hB n) (X n)-p)) atTop (𝓝 (B'.symm a)) := by simpa only [Function.comp_def,affineZoom_apply,B'.symm_apply_apply,X] using (B'.symm.continuous.tendsto a).comp ha have hyb : Tendsto (fun n => (h n)⁻¹ • ((F n).rankChart hAB (hA n) (hB n) (Y n)-p)) atTop (𝓝 (B'.symm b)) := by simpa only [Function.comp_def,affineZoom_apply,B'.symm_apply_apply,Y] using (B'.symm.continuous.tendsto b).comp hb have hne : B'.symm b-B'.symm a≠0 := by intro he; apply hab have hg := congrArg B' he simpa only [map_sub,B'.apply_symm_apply,map_zero] using hg obtain ⟨r,_,z,hz,he⟩ := actual_forward_limit F hAB hA hB C hC D c p h hh t htA htB X Y (fun n => q n*s n) (fun n => mul_nonneg (hq n).le (hs n)) hXY (B'.symm a) (B'.symm b) hxa hyb (fun i => jet_at_physical F hAB (hCB i) hA hB (hC i) (D i) (c i) p h hh (hj i) X _ hxa) (fun i => jet_at_physical F hAB (hCB i) hA hB (hC i) (D i) (c i) p h hh (hj i) Y _ hyb) hne refine ⟨z,hz,?_⟩ have ht := congrArg B' he simp only [map_sub,map_smul,B'.apply_symm_apply] at ht rw [ht] dsimp [detPair]; ring end PlanarHalving.RankLimits namespace PlanarHalving.RankLimits open Set Filter MeasureTheory Metric Batches TimeWindows RankCoordinates Zoom open scoped Topology unitInterval noncomputable def forwardUnit (F : Configuration) (E : Point ≃L[ℝ] Point) (t : I) : Point := ‖E.symm (1,F.timeSlope t)‖⁻¹ • E.symm (1,F.timeSlope t) lemma forwardUnit_norm (F : Configuration) (E : Point ≃L[ℝ] Point) (t : I) : ‖forwardUnit F E t‖=1 := by apply norm_smul_inv_norm intro he have hh := congrArg E he simp at hh lemma forwardUnit_map (F : Configuration) (E : Point ≃L[ℝ] Point) (t : I) : E (forwardUnit F E t)=‖E.symm (1,F.timeSlope t)‖⁻¹ • (1,F.timeSlope t) := by simp only [forwardUnit,map_smul,E.apply_symm_apply] lemma actual_forwardUnit_clusters {ι : Type*} (F : ℕ → Configuration) {A B : Window} (hAB : Before A B) (hA : ∀ n, 0 < length (F n).R A) (hB : ∀ n, 0 < length (F n).R B) (C : ι → Window) (hCB : ∀ i, Before (C i) A ∧ Before (C i) B) (hC : ∀ i, ∀ᶠ n in atTop, 0 < length (F n).R (C i)) (D : ι → Point →L[ℝ] ℝ) (c : ι → ℝ) (p : Point) (hp : p ∈ Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0) (h q : ℕ → ℝ) (hh : ∀ n, 0 (((F n).extendedRank true A B (C i) (p+h n • u) : ℝ)-c i)/h n) (D i) atTop (closedBall 0 R)) (hunif : TendstoUniformlyOn (fun n => affineZoom ((F n).rankChart hAB (hA n) (hB n)) (E n) B' p (((F n).rankChart hAB (hA n) (hB n)).symm p) (q n) (h n) (hq n).ne' (hh n).ne') (fun x => sgn • x) atTop (closedBall 0 1)) (ρ : ℕ → ℕ) (hρ : StrictMono ρ) (v : Point) (hvlim : Tendsto (fun n => forwardUnit (F (ρ n)) (E (ρ n)) t) atTop (𝓝 v)) : ∃ z∈Directions.allowed reflectFirst D (fun i => (lower (C i):ℝ)) (fun i => (upper (C i):ℝ)) t, detPair v (B' (reflectFirst (Directions.simplex z)))=0 := by let f := fun n => (F n).rankChart hAB (hA n) (hB n) let x₀ := fun n => (f n).symm p let Z := fun n => affineZoom (f n) (E n) B' p (x₀ n) (q n) (h n) (hq n).ne' (hh n).ne' have hpt (n : ℕ) : p∈(f n).target := by rw [(F n).rankChart_target]; exact hp have hz (n : ℕ) : Z n 0=0 := affineZoom_zero (f n) (E n) B' (hpt n) _ _ _ _ have hvn : ‖v‖=1 := tendsto_nhds_unique hvlim.norm (by simp only [forwardUnit_norm]; exact tendsto_const_nhds) have hsgn0 : sgn≠0 := by rcases hsgn with rfl | rfl <;> norm_num have hne : sgn • v-0≠0 := by rw [sub_zero] exact smul_ne_zero hsgn0 (by intro he; rw [he,norm_zero] at hvn; norm_num at hvn) have hvwithin : Tendsto (fun n => forwardUnit (F (ρ n)) (E (ρ n)) t) atTop (𝓝[closedBall (0:Point) 1] v) := tendsto_nhdsWithin_iff.mpr ⟨hvlim,Eventually.of_forall (fun n => by simp [forwardUnit_norm])⟩ have hyl : Tendsto (fun n => Z (ρ n) (forwardUnit (F (ρ n)) (E (ρ n)) t)) atTop (𝓝 (sgn • v)) := (hunif.seq_tendstoUniformlyOn ρ hρ.tendsto_atTop).tendsto_comp ((continuous_const : Continuous (fun _ : Point => sgn)).smul continuous_id).continuousWithinAt hvwithin obtain ⟨z,hz',hdet⟩ := affine_rank_forward_limit (F ∘ ρ) hAB (fun n => hA (ρ n)) (fun n => hB (ρ n)) C hCB (fun i => hρ.tendsto_atTop.eventually (hC i)) D c p (h ∘ ρ) (q ∘ ρ) (fun n => hh (ρ n)) (fun n => hq (ρ n)) (E ∘ ρ) B' (x₀ ∘ ρ) t htA htB (fun _ => 0) (fun n => forwardUnit (F (ρ n)) (E (ρ n)) t) (fun n => ‖(E (ρ n)).symm (1,(F (ρ n)).timeSlope t)‖⁻¹) (fun n => inv_nonneg.mpr (norm_nonneg _)) (fun n => by rw [sub_zero]; exact forwardUnit_map _ _ _) 0 (sgn • v) hne (by change Tendsto (fun n => Z (ρ n) 0) atTop (𝓝 0); simp only [hz]; exact tendsto_const_nhds) hyl (fun i R hR => (hj i R hR).seq_tendstoUniformlyOn ρ hρ.tendsto_atTop) refine ⟨z,hz',?_⟩ rw [sub_zero] at hdet have he : detPair (sgn • v) (B' (reflectFirst (Directions.simplex z)))= sgn*detPair v (B' (reflectFirst (Directions.simplex z))) := by dsimp [detPair]; ring rw [he] at hdet exact (mul_eq_zero.mp hdet).resolve_left hsgn0 end PlanarHalving.RankLimits end OAI