import OAI.Combinatorics.CompleteCrossing.Plane import OAI.Combinatorics.Crossing.ChordOrientations import OAI.Combinatorics.Crossing.RadialBoundaryPorts import OAI.Combinatorics.Crossing.SimilarityCoordinates import Mathlib.Analysis.Complex.Arg import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan import Mathlib.Tactic namespace OAI /-! Spoke angles, circle angles, scaling, and coordinate frames. -/ section noncomputable section open scoped Classical namespace Paper170 def complexPoint (a : Plane) : ℂ := ⟨a.1, a.2⟩ lemma complexPoint_injective : Function.Injective complexPoint := by intro a b h exact Prod.ext (congrArg Complex.re h) (congrArg Complex.im h) @[simp] lemma complexPoint_zero : complexPoint 0 = 0 := rfl @[simp] lemma complexPoint_smul (r : ℝ) (a : Plane) : complexPoint (r • a) = (r : ℂ) * complexPoint a := by apply Complex.ext <;> simp [complexPoint] lemma complexPoint_ne_zero {a : Plane} (ha : a ≠ 0) : complexPoint a ≠ 0 := by exact fun h => ha (complexPoint_injective h) lemma spokeAngle_coe (cut : ℝ) (a : Plane) (ha : a ≠ 0) : (spokeAngle cut a : Real.Angle) = (Complex.arg (complexPoint a) : Real.Angle) - cut := by change (Complex.arg (Complex.exp (-(cut : ℂ) * Complex.I) * complexPoint a) : Real.Angle) = _ rw [Complex.arg_mul_coe_angle (Complex.exp_ne_zero _) (complexPoint_ne_zero ha)] simp [Complex.arg_exp, sub_eq_add_neg, add_comm] lemma finite_spoke_cut {ι : Type*} [Fintype ι] (a : ι → Plane) (ha : ∀ i, a i ≠ 0) : ∃ cut : ℝ, ∀ i, spokeAngle cut (a i) ≠ Real.pi := by have hreal (t : ℝ) : (Real.arctan t : Real.Angle).toReal = Real.arctan t := by apply Real.Angle.toReal_coe_eq_self_iff.mpr rcases Real.arctan_mem_Ioo t with ⟨hl, hu⟩ constructor <;> linarith [Real.pi_pos] have hinj : Function.Injective (fun t : ℝ => (Real.arctan t : Real.Angle)) := by intro t u h apply Real.arctan_injective simpa only [hreal] using congrArg Real.Angle.toReal h let := Infinite.of_injective _ hinj let bad : ι → Real.Angle := fun i => (Complex.arg (complexPoint (a i)) : Real.Angle) - Real.pi obtain ⟨θ, hθ⟩ := (Set.finite_range bad).exists_notMem refine ⟨θ.toReal, ?_⟩ intro i hi have h := spokeAngle_coe θ.toReal (a i) (ha i) rw [hi, Real.Angle.coe_toReal] at h apply hθ refine ⟨i, ?_⟩ apply sub_eq_iff_eq_add.mpr simpa [add_comm] using (eq_sub_iff_add_eq.mp h).symm lemma spokeAngle_eq_smul (cut : ℝ) (a b : Plane) (ha : a ≠ 0) (hb : b ≠ 0) (h : spokeAngle cut a = spokeAngle cut b) : ∃ r : ℝ, 0 < r ∧ r • a = b := by let q := Complex.exp (-(cut : ℂ) * Complex.I) have hq : q ≠ 0 := Complex.exp_ne_zero _ have ha' : q * complexPoint a ≠ 0 := mul_ne_zero hq (complexPoint_ne_zero ha) have hb' : q * complexPoint b ≠ 0 := mul_ne_zero hq (complexPoint_ne_zero hb) let r : ℝ := ‖q * complexPoint b‖ / ‖q * complexPoint a‖ have hr : 0 < r := div_pos (norm_pos_iff.mpr hb') (norm_pos_iff.mpr ha') refine ⟨r, hr, complexPoint_injective ?_⟩ rw [complexPoint_smul] apply mul_left_cancel₀ hq have heq := (Complex.arg_eq_arg_iff ha' hb').mp h rw [← Complex.ofReal_div] at heq change ((r : ℂ) * (q * complexPoint a)) = q * complexPoint b at heq calc q * ((r : ℂ) * complexPoint a) = (r : ℂ) * (q * complexPoint a) := by ring _ = q * complexPoint b := heq end Paper170 end end section noncomputable section namespace Paper170 lemma spokeAngle_smul (cut r : ℝ) (hr : 0 < r) (a : Plane) : spokeAngle cut (r • a) = spokeAngle cut a := by change Complex.arg (Complex.exp (-(cut : ℂ) * Complex.I) * complexPoint (r • a)) = _ rw [complexPoint_smul, mul_left_comm, Complex.arg_real_mul _ hr] rfl end Paper170 end end section noncomputable section open scoped Classical namespace Paper170 lemma circlePoint_complex (t : ℝ) : complexPoint (Zarankiewicz.circlePoint t) = (Real.cos (2 * Real.arctan t) : ℂ) + (Real.sin (2 * Real.arctan t) : ℂ) * Complex.I := by have hden := Zarankiewicz.circle_denominator_pos t have hsqrt := Real.sq_sqrt hden.le have hsqrt0 : Real.sqrt (1 + t ^ 2) ≠ 0 := ne_of_gt (Real.sqrt_pos.mpr hden) apply Complex.ext · simp only [Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, zero_mul, sub_zero, add_zero] rw [Real.cos_two_mul, Real.cos_sq_arctan] dsimp [complexPoint, Zarankiewicz.circlePoint] field_simp ring · simp only [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, mul_one, zero_add, add_zero] rw [Real.sin_two_mul, Real.sin_arctan, Real.cos_arctan] dsimp [complexPoint, Zarankiewicz.circlePoint] field_simp rw [hsqrt] lemma circlePoint_arg (t : ℝ) : Complex.arg (complexPoint (Zarankiewicz.circlePoint t)) = 2 * Real.arctan t := by rw [circlePoint_complex] rw [Complex.ofReal_cos, Complex.ofReal_sin] apply Complex.arg_cos_add_sin_mul_I rcases Real.arctan_mem_Ioo t with ⟨hl, hu⟩ constructor <;> linarith lemma circlePoint_arg_lt (a b : ℝ) : Complex.arg (complexPoint (Zarankiewicz.circlePoint a)) < Complex.arg (complexPoint (Zarankiewicz.circlePoint b)) ↔ a < b := by rw [circlePoint_arg, circlePoint_arg, mul_lt_mul_iff_right₀ (by norm_num : (0 : ℝ) < 2)] exact Real.arctan_strictMono.lt_iff_lt def angleRotation (cut : ℝ) (a : Plane) : Plane := Zarankiewicz.similarityPoint 0 ((Complex.exp (-(cut : ℂ) * Complex.I)).re, (Complex.exp (-(cut : ℂ) * Complex.I)).im) a lemma angleRotation_complex (cut : ℝ) (a : Plane) : complexPoint (angleRotation cut a) = Complex.exp (-(cut : ℂ) * Complex.I) * complexPoint a := by apply Complex.ext <;> simp [complexPoint, angleRotation, Zarankiewicz.similarityPoint, add_comm] lemma angleRotation_unit (cut : ℝ) (a : Plane) (ha : squaredLength a = 1) : squaredLength (angleRotation cut a) = 1 := by have he : Complex.normSq (Complex.exp (-(cut : ℂ) * Complex.I)) = 1 := by rw [← Complex.sq_norm] simp [Complex.norm_exp] have hform (p : Plane) : Complex.normSq (complexPoint p) = squaredLength p := by simp [Complex.normSq_apply, complexPoint, squaredLength, pow_two] have h : Complex.normSq (complexPoint a) = 1 := by rw [hform, ha] rw [← hform] rw [angleRotation_complex, map_mul, he, h, mul_one] lemma angleRotation_injective (cut : ℝ) : Function.Injective (angleRotation cut) := by intro a b h apply complexPoint_injective apply mul_left_cancel₀ (Complex.exp_ne_zero (-(cut : ℂ) * Complex.I)) simpa only [angleRotation_complex] using congrArg complexPoint h lemma angleRotation_sign (cut : ℝ) (a b c d : Plane) : segmentIntersection (angleRotation cut a) (angleRotation cut b) (angleRotation cut c) (angleRotation cut d) = segmentIntersection a b c d := by simp only [segmentIntersection_eq_segmentCrossingSign] apply Zarankiewicz.segmentCrossingSign_similarity intro h apply Complex.exp_ne_zero (-(cut : ℂ) * Complex.I) exact Complex.ext (congrArg Prod.fst h) (congrArg Prod.snd h) lemma angleRotation_parameter (cut : ℝ) (a : Plane) (ha : squaredLength a = 1) (hc : spokeAngle cut a ≠ Real.pi) : ∃ t, Zarankiewicz.circlePoint t = angleRotation cut a := by have hu := angleRotation_unit cut a ha have hx : (angleRotation cut a).1 ≠ -1 := by intro h have hy : (angleRotation cut a).2 = 0 := by dsimp [squaredLength] at hu rw [h] at hu nlinarith [sq_nonneg (angleRotation cut a).2] apply hc change Complex.arg (Complex.exp (-(cut : ℂ) * Complex.I) * complexPoint a) = _ rw [← angleRotation_complex] exact Complex.arg_eq_pi_iff.mpr ⟨by simpa only [complexPoint, h] using (by norm_num : (-1:ℝ)<0), hy⟩ exact ⟨(angleRotation cut a).2 / (1 + (angleRotation cut a).1), Zarankiewicz.circlePoint_surjective_away_cut _ hu hx⟩ theorem signed_chord_formula (p : Fin 4 → Plane) (hp : Function.Injective p) (hc : ∀ i, squaredLength (p i) = 1) (cut : ℝ) (hcut : ∀ i, spokeAngle cut (p i) ≠ Real.pi) : segmentIntersection (p 0) (p 1) (p 2) (p 3) = (1 / 2 : ℝ) * (circleComparison cut (p 0) (p 2) - circleComparison cut (p 0) (p 3) - circleComparison cut (p 1) (p 2) + circleComparison cut (p 1) (p 3)) := by choose t ht using fun i => angleRotation_parameter cut (p i) (hc i) (hcut i) have hi : Function.Injective t := by intro i j hij apply hp apply angleRotation_injective cut rw [← ht i, ← ht j, hij] have hs (i j : Fin 4) : circleComparison cut (p i) (p j) = Zarankiewicz.orderSign (t i) (t j) := by have he (k : Fin 4) : spokeAngle cut (p k) = Complex.arg (complexPoint (Zarankiewicz.circlePoint (t k))) := by rw [ht, angleRotation_complex] rfl simp only [circleComparison, he, circlePoint_arg_lt, Zarankiewicz.orderSign] rw [← angleRotation_sign cut, ← ht 0, ← ht 1, ← ht 2, ← ht 3, segmentIntersection_eq_segmentCrossingSign, Zarankiewicz.circle_chord_pairing _ _ _ _ (hi.ne (by decide)) (hi.ne (by decide)) (hi.ne (by decide)) (hi.ne (by decide)) (hi.ne (by decide)) (hi.ne (by decide))] simp only [hs] ring end Paper170 end end section noncomputable section open Set open scoped Classical namespace Paper170 variable {V E : Type} {G : Graph V E} lemma outwardSpoke_ne_zero (N : NormalizedDrawing G) (v : V) (e : E) : outwardSpoke N v e ≠ 0 := by unfold outwardSpoke split_ifs · exact (N.polygonal e).direction_ne_zero (N.edge_injective e) 0 (N.polygonal e).positive · have h := (N.polygonal e).direction_ne_zero (N.edge_injective e) ((N.polygonal e).pieces - 1) (by have := (N.polygonal e).positive; omega) have hi : (N.polygonal e).pieces - 1 + 1 = (N.polygonal e).pieces := by have := (N.polygonal e).positive omega rw [hi] at h exact sub_ne_zero.mpr (Ne.symm (sub_ne_zero.mp h)) lemma outwardSpoke_interior (N : NormalizedDrawing G) (v : V) (e : G.IncidentEdge v) (t : ℝ) (ht : 0 < t ∧ t < 1) : N.vertex v + t • outwardSpoke N v e.val ∈ edgeInterior N.toContinuousDrawing e.val := by by_cases hs : G.src e.val = v · have h := (N.polygonal e.val).segment_mem_interior N.toContinuousDrawing e.val ⟨0, (N.polygonal e.val).positive⟩ t ht have hp : (N.polygonal e.val).vertices 0 = N.vertex v := by simpa [hs] using (N.polygonal e.val).vertices_first convert h using 1 rw [outwardSpoke, ite_eq_left hs, hp] simp only [segmentPoint] module · have hd : G.dst e.val = v := e.property.resolve_left hs have hi : (N.polygonal e.val).pieces - 1 + 1 = (N.polygonal e.val).pieces := by have := (N.polygonal e.val).positive omega have h := (N.polygonal e.val).segment_mem_interior N.toContinuousDrawing e.val ⟨(N.polygonal e.val).pieces - 1, by have := (N.polygonal e.val).positive; omega⟩ (1-t) ⟨by linarith, by linarith⟩ have hp : (N.polygonal e.val).vertices (N.polygonal e.val).pieces = N.vertex v := by simpa [hd] using (N.polygonal e.val).vertices_last simp only [hi, hp] at h convert h using 1 rw [outwardSpoke, ite_eq_right hs, hp] simp only [segmentPoint] module lemma spokeAngle_injective (N : NormalizedDrawing G) (v : V) (cut : ℝ) : Function.Injective (fun e : G.IncidentEdge v => spokeAngle cut (outwardSpoke N v e.val)) := by intro e f h by_contra hne have hef : e.val ≠ f.val := fun h => hne (Subtype.ext h) have hadj : G.src e.val = G.src f.val ∨ G.src e.val = G.dst f.val ∨ G.dst e.val = G.src f.val ∨ G.dst e.val = G.dst f.val := by rcases e.property with he | he <;> rcases f.property with hf | hf <;> grind obtain ⟨r, hr, heq⟩ := spokeAngle_eq_smul cut (outwardSpoke N v e.val) (outwardSpoke N v f.val) (outwardSpoke_ne_zero N v e.val) (outwardSpoke_ne_zero N v f.val) h have hden : 0 < 1+r := by linarith have ht : 0 < 1/(1+r) ∧ 1/(1+r) < 1 := by constructor · exact div_pos zero_lt_one hden · exact (div_lt_one hden).mpr (by linarith) have hu : 0 < r/(1+r) ∧ r/(1+r) < 1 := by constructor · exact div_pos hr hden · exact (div_lt_one hden).mpr (by linarith) have hx := outwardSpoke_interior N v e (r/(1+r)) hu have hy := outwardSpoke_interior N v f (1/(1+r)) ht have hxy : N.vertex v + (1/(1+r)) • outwardSpoke N v f.val = N.vertex v + (r/(1+r)) • outwardSpoke N v e.val := by rw [← heq, smul_smul] congr 2 ring rw [hxy] at hy exact Set.disjoint_left.mp (N.no_adjacent e.val f.val hef hadj) hx hy theorem spoke_orders_exist [Fintype V] [Fintype E] (G : Graph V E) (N : NormalizedDrawing G) : Nonempty (SpokeOrders N) := by have hc (v : V) : ∃ cut : ℝ, ∀ e : G.IncidentEdge v, spokeAngle cut (outwardSpoke N v e.val) ≠ Real.pi := finite_spoke_cut _ (fun e => outwardSpoke_ne_zero N v e.val) choose cut hcut using hc exact ⟨{ cut := cut order := fun v => LinearOrder.lift' _ (spokeAngle_injective N v (cut v)) cut_avoids := hcut compatible := fun _ _ _ => Iff.rfl }⟩ end Paper170 end end section noncomputable section open Zarankiewicz namespace Paper170 theorem complexPoint_similarity (o u x : Plane) : complexPoint (similarityPoint o u x)=complexPoint o+complexPoint u*complexPoint x := by apply Complex.ext <;> simp [complexPoint,similarityPoint,add_comm] def angleCoordinateFrame (cut radius : ℝ) : Plane := radius • (((Complex.exp (-(cut:ℂ)*Complex.I))⁻¹).re, ((Complex.exp (-(cut:ℂ)*Complex.I))⁻¹).im) theorem complexPoint_angleCoordinateFrame (cut radius : ℝ) : complexPoint (angleCoordinateFrame cut radius)= (radius:ℂ)*(Complex.exp (-(cut:ℂ)*Complex.I))⁻¹ := by rw [angleCoordinateFrame,complexPoint_smul] rfl theorem angleCoordinateFrame_length (cut radius : ℝ) : squaredLength (angleCoordinateFrame cut radius)=radius^2 := by have hq : Complex.normSq (Complex.exp (-(cut:ℂ)*Complex.I))=1 := by rw [← Complex.sq_norm] simp [Complex.norm_exp] have hnorm (p : Plane) : squaredLength p=Complex.normSq (complexPoint p) := by simp [complexPoint,squaredLength,Complex.normSq_apply,pow_two] rw [hnorm,complexPoint_angleCoordinateFrame,map_mul,map_inv₀,hq,inv_one,mul_one] simp [Complex.normSq_apply,pow_two] theorem angleCoordinateFrame_ne_zero (cut radius : ℝ) (hr : 0