| import Mathlib |
| import OAI.Geometry.HyperbolicGroups.ConeData |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped Topology ContinuousMap |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
|
|
| namespace HyperbolicObstruction |
| namespace CATZero |
| variable {X : Type*} [MetricSpace X] |
|
|
| /-- The midpoint part of square rigidity, retaining just the metric data |
| needed later. -/ |
| theorem square_mid (h : CATZero X) {p q r s : X} |
| (hpq : dist p q ^ 2 = 1) (hqr : dist q r ^ 2 = 1) |
| (hrs : dist r s ^ 2 = 1) (hsp : dist s p ^ 2 = 1) |
| (hpr : dist p r ^ 2 = 2) (hqs : dist q s ^ 2 = 2) : |
| mid h p r = mid h q s := by |
| have hq := quadrilateral h p q r s |
| rw [hpq, hqr, hrs, hsp, hpr, hqs] at hq |
| apply dist_eq_zero.mp |
| nlinarith [dist_nonneg (x := mid h p r) (y := mid h q s)] |
|
|
| end CATZero |
| namespace TangentCone |
| variable {X : Type*} [MetricSpace X] (h : CATZero X) |
|
|
| /-- A log-isometric base gives every radial CN equality, directly from the |
| constructed cone metric law and uniqueness of segments. -/ |
| theorem cn_eq_of_log_dist (o x y : X) |
| (he : dist (log h o x) (log h o y) = dist x y) {t : ℝ} |
| (ht : t ∈ Set.Icc (0 : ℝ) 1) : |
| dist o (CATZero.segment h x y t) ^ 2 = |
| (1 - t) * dist o x ^ 2 + t * dist o y ^ 2 - t * (1 - t) * dist x y ^ 2 := by |
| have hmap := CATZero.map_segment_of_dist h (catZero h o) (log h o) |
| (dist_log_le h o) x y he ht |
| have hn := (coneData h o).normSq_segment (catZero h o) (log h o x) (log h o y) ht |
| rw [← hmap, normSq_log, normSq_log, normSq_log, he] at hn |
| exact hn |
|
|
| /-- The fan midpoint identity, derived from a metric square and the central |
| flatness of its line diagonal. -/ |
| theorem fan_identity (c x y a b : X) |
| (hcx : dist c x ^ 2 = 3 / 4) (hcy : dist c y ^ 2 = 3 / 4) |
| (hca : dist c a ^ 2 = 3 / 4) (hcb : dist c b ^ 2 = 3 / 4) |
| (hxa : dist x a ^ 2 = 1) (hay : dist a y ^ 2 = 1) |
| (hyb : dist y b ^ 2 = 1) (hbx : dist b x ^ 2 = 1) |
| (hxy : dist x y ^ 2 = 2) (hab : dist a b ^ 2 = 2) |
| (hflat : dist c (CATZero.mid h a b) ^ 2 = 1 / 4) : |
| (coneData h c).pairing (log h c x) (log h c y) = -1 / 4 ∧ |
| (coneData h c).add (catZero h c) (log h c x) (log h c y) = |
| (coneData h c).add (catZero h c) (log h c a) (log h c b) := by |
| have hm := CATZero.square_mid h hxa hay hyb hbx hxy hab |
| have heab : dist c (CATZero.mid h a b) ^ 2 = |
| (dist c a ^ 2 + dist c b ^ 2) / 2 - dist a b ^ 2 / 4 := by |
| rw [hca, hcb, hab, hflat] |
| norm_num |
| have hexy : dist c (CATZero.mid h x y) ^ 2 = |
| (dist c x ^ 2 + dist c y ^ 2) / 2 - dist x y ^ 2 / 4 := by |
| rw [hm, hflat, hcx, hcy, hxy] |
| norm_num |
| constructor |
| · unfold ConeData.pairing |
| rw [normSq_log, normSq_log, log_dist_of_midpoint_equality h c x y hexy, hcx, hcy, hxy] |
| norm_num |
| · unfold ConeData.add |
| rw [← log_mid_of_midpoint_equality h c x y hexy, |
| ← log_mid_of_midpoint_equality h c a b heab, hm] |
|
|
| /-- Source exceptional-incidence identity. Its hypotheses are entirely metric, |
| including the fixed-fraction test; no flat triangle or cone identity is assumed. -/ |
| theorem exceptional_identity (c p q z k : X) |
| (hcp : dist c p ^ 2 = 3 / 4) (hcq : dist c q ^ 2 = 3 / 4) |
| (hcz : dist c z ^ 2 = 3 / 4) (hck : dist c k ^ 2 = 3 / 4) |
| (hpq : dist p q ^ 2 = 2) (hpz : dist p z ^ 2 = 2) (hqz : dist q z ^ 2 = 2) |
| (hcj : dist c (CATZero.mid h p q) ^ 2 = 1 / 4) |
| (hkj : dist k (CATZero.mid h p q) ^ 2 = 1 / 2) (hkz : dist k z ^ 2 = 1) |
| (hfrac : ∀ t ∈ Set.Ioc (0 : ℝ) 1, |
| t ^ 2 * (1 / 2 + 1) ≤ dist (CATZero.segment h k (CATZero.mid h p q) t) |
| (CATZero.segment h k z t) ^ 2) : |
| log h c k = (coneData h c).add (catZero h c) |
| ((coneData h c).add (catZero h c) (log h c p) (log h c q)) (log h c z) := by |
| let j := CATZero.mid h p q |
| let b := CATZero.segment h j z (1 / 3) |
| have hjz : dist j z ^ 2 = 3 / 2 := by |
| have hlo := hfrac 1 (by norm_num) |
| rw [(CATZero.segment_spec h k j).2.1, (CATZero.segment_spec h k z).2.1] at hlo |
| have hhi := CATZero.mid_cn h p q z |
| rw [dist_comm z p, dist_comm z q, hpz, hqz, hpq, dist_comm z j] at hhi |
| norm_num at hlo |
| linarith |
| have hlogk : dist (log h k j) (log h k z) = dist j z := by |
| apply log_dist_of_right_fractional h k j z |
| · rw [hjz, hkj, hkz] |
| norm_num |
| · intro t ht |
| rw [hkj, hkz] |
| exact hfrac t ht |
| have hkb : dist k b ^ 2 = 1 / 3 := by |
| have he := cn_eq_of_log_dist h k j z hlogk (t := 1 / 3) (by norm_num) |
| rw [hkj, hkz, hjz] at he |
| dsimp only [b] |
| linarith |
| have hcb : dist c b ^ 2 ≤ 1 / 12 := by |
| have he := CATZero.segment_cn h j z c (t := 1 / 3) (by norm_num) |
| rw [hcj, hcz, hjz] at he |
| exact le_trans he (by norm_num) |
| have hkbl : dist k b = 2 / 3 * dist c k := by |
| nlinarith [dist_nonneg (x := k) (y := b), dist_nonneg (x := c) (y := k)] |
| have hcbl : dist c b = 1 / 3 * dist c k := by |
| have htri := dist_triangle c b k |
| rw [dist_comm b k, hkbl] at htri |
| nlinarith [dist_nonneg (x := c) (y := b), dist_nonneg (x := c) (y := k)] |
| have hb : b = CATZero.segment h c k (1 / 3) := by |
| apply CATZero.eq_segment_of_dist h (by norm_num) |
| · exact hcbl |
| · rw [dist_comm, hkbl] |
| ring |
| have hec : dist c b ^ 2 = |
| (1 - 1 / 3) * dist c j ^ 2 + (1 / 3) * dist c z ^ 2 - |
| (1 / 3) * (1 - 1 / 3) * dist j z ^ 2 := by |
| rw [hcbl, hcj, hcz, hjz, mul_pow, hck] |
| norm_num |
| have hlogc := log_dist_of_interior_equality h c j z (t := 1 / 3) (by norm_num) hec |
| have hmap := CATZero.map_segment_of_dist h (catZero h c) (log h c) (dist_log_le h c) |
| j z hlogc (t := 1 / 3) (by norm_num) |
| have hlogb : log h c b = (coneData h c).scale (1 / 3) (log h c k) := by |
| rw [hb, log_radial h c k (by norm_num)] |
| have hsum : log h c k = (coneData h c).add (catZero h c) |
| ((coneData h c).scale 2 (log h c j)) (log h c z) := by |
| have he := congrArg ((coneData h c).scale 3) hmap |
| change (coneData h c).scale 3 (log h c b) = _ at he |
| rw [hlogb, (coneData h c).scale_mul (by norm_num) (by norm_num)] at he |
| norm_num only [show (3 : ℝ) * (1 / 3) = 1 by norm_num, (coneData h c).scale_one] at he |
| rw [(coneData h c).segment_eq_add_scale (catZero h c) _ _ (by norm_num), |
| (coneData h c).scale_add (catZero h c) (by norm_num), |
| (coneData h c).scale_mul (by norm_num) (by norm_num), |
| (coneData h c).scale_mul (by norm_num) (by norm_num)] at he |
| norm_num at he |
| simpa only [(coneData h c).scale_one] using he |
| have hepq : dist c j ^ 2 = (dist c p ^ 2 + dist c q ^ 2) / 2 - dist p q ^ 2 / 4 := by |
| rw [hcj, hcp, hcq, hpq] |
| norm_num |
| rw [hsum, log_mid_of_midpoint_equality h c p q hepq] |
| rfl |
|
|
| end TangentCone |
| end HyperbolicObstruction |
|
|
| noncomputable section |
| namespace HyperbolicObstruction |
|
|
| /-- Squared metric data for the unit square, in cyclic order. -/ |
| structure MetricSquare {X : Type*} [MetricSpace X] (p q r s : X) : Prop where |
| side₀ : dist p q ^ 2 = 1 |
| side₁ : dist q r ^ 2 = 1 |
| side₂ : dist r s ^ 2 = 1 |
| side₃ : dist s p ^ 2 = 1 |
| diagonal₀ : dist p r ^ 2 = 2 |
| diagonal₁ : dist q s ^ 2 = 2 |
|
|
| theorem MetricSquare.mid_eq {X : Type*} [MetricSpace X] {p q r s : X} |
| (hs : MetricSquare p q r s) (h : CATZero X) : CATZero.mid h p r = CATZero.mid h q s := |
| CATZero.square_mid h hs.side₀ hs.side₁ hs.side₂ hs.side₃ hs.diagonal₀ hs.diagonal₁ |
|
|
| /-- Full incidences among the six lines AB, AC, AD, BC, BD, CD at A,B,C,D. |
| These are exactly the doubled fans of the source. -/ |
| def fullLines : Fin 4 → Finset (Fin 6) := |
| ![{0, 1, 2}, {0, 3, 4}, {1, 3, 5}, {2, 4}] |
|
|
| /-- The specific subconfiguration used by the source obstruction. This is |
| metric input to a contradiction, not an assumed existence assertion. All its |
| conditions are consequences asserted by source Proposition `configuration`; |
| the existence-from-actions direction remains a separate obligation. -/ |
| structure FiniteObstructionData {X : Type*} [MetricSpace X] (h : CATZero X) where |
| c : X |
| x : Fin 4 → X |
| y : Fin 4 → X |
| line : Fin 6 → X |
| radiusX : ∀ i, dist c (x i) ^ 2 = 3 / 4 |
| radiusY : ∀ i, dist c (y i) ^ 2 = 3 / 4 |
| radiusLine : ∀ i, dist c (line i) ^ 2 = 3 / 4 |
| lineDist : ∀ a b, a ≠ b → dist (line a) (line b) ^ 2 = 2 |
| lineFlat : ∀ a b, a ≠ b → dist c (CATZero.mid h (line a) (line b)) ^ 2 = 1 / 4 |
| square : ∀ i a b, a ∈ fullLines i → b ∈ fullLines i → a ≠ b → |
| MetricSquare (x i) (line a) (y i) (line b) |
| twoStepX : dist (x 2) (x 3) ^ 2 = 2 |
| twoStepY : dist (y 2) (x 3) ^ 2 = 2 |
| loose : dist (line 5) (x 3) ^ 2 = 1 |
| missing : 3 ≤ dist (y 3) (line 5) ^ 2 |
| fractional : ∀ t ∈ Set.Ioc (0 : ℝ) 1, |
| t ^ 2 * (1 / 2 + 1) ≤ |
| dist (CATZero.segment h (line 5) (CATZero.mid h (x 2) (y 2)) t) |
| (CATZero.segment h (line 5) (x 3) t) ^ 2 |
|
|
| /-- The complete finite CAT(0) metric obstruction from Section 6. The proof |
| constructs the completed tangent cone and all rigidity identities internally. -/ |
| theorem no_finite_obstruction {X : Type*} [MetricSpace X] (h : CATZero X) |
| (D : FiniteObstructionData h) : False := by |
| let T := TangentCone.coneData h D.c |
| let H := TangentCone.catZero h D.c |
| let L : Fin 6 → TangentCone h D.c := fun i => TangentCone.log h D.c (D.line i) |
| let U : Fin 4 → TangentCone h D.c := fun i => TangentCone.log h D.c (D.x i) |
| let V : Fin 4 → TangentCone h D.c := fun i => TangentCone.log h D.c (D.y i) |
| let M : Fin 4 → TangentCone h D.c := fun i => T.add H (U i) (V i) |
| have hnL (i) : T.normSq (L i) = 3 / 4 := by |
| exact (TangentCone.normSq_log h D.c (D.line i)).trans (D.radiusLine i) |
| have hnU (i) : T.normSq (U i) = 3 / 4 := by |
| exact (TangentCone.normSq_log h D.c (D.x i)).trans (D.radiusX i) |
| have hnV (i) : T.normSq (V i) = 3 / 4 := by |
| exact (TangentCone.normSq_log h D.c (D.y i)).trans (D.radiusY i) |
| have hpL (a b) (hne : a ≠ b) : T.pairing (L a) (L b) = -1 / 4 := by |
| have he : dist D.c (CATZero.mid h (D.line a) (D.line b)) ^ 2 = |
| (dist D.c (D.line a) ^ 2 + dist D.c (D.line b) ^ 2) / 2 - |
| dist (D.line a) (D.line b) ^ 2 / 4 := by |
| rw [D.lineFlat a b hne, D.radiusLine a, D.radiusLine b, D.lineDist a b hne] |
| norm_num |
| have hd := TangentCone.log_dist_of_midpoint_equality h D.c (D.line a) (D.line b) he |
| change dist (L a) (L b) = _ at hd |
| rw [ConeData.pairing, hnL, hnL, hd, D.lineDist a b hne] |
| norm_num |
| have hfan (i a b) (ha : a ∈ fullLines i) (hb : b ∈ fullLines i) (hne : a ≠ b) : |
| T.pairing (U i) (V i) = -1 / 4 ∧ M i = T.add H (L a) (L b) := by |
| have hs := D.square i a b ha hb hne |
| exact TangentCone.fan_identity h D.c (D.x i) (D.y i) (D.line a) (D.line b) |
| (D.radiusX i) (D.radiusY i) (D.radiusLine a) (D.radiusLine b) |
| hs.side₀ hs.side₁ hs.side₂ hs.side₃ hs.diagonal₀ hs.diagonal₁ (D.lineFlat a b hne) |
| have hA₁ := hfan 0 0 1 (by decide) (by decide) (by decide) |
| have hA₂ := hfan 0 1 2 (by decide) (by decide) (by decide) |
| have hB₁ := hfan 1 0 3 (by decide) (by decide) (by decide) |
| have hB₂ := hfan 1 3 4 (by decide) (by decide) (by decide) |
| have hC := hfan 2 1 3 (by decide) (by decide) (by decide) |
| have hD := hfan 3 2 4 (by decide) (by decide) (by decide) |
| have hnM₂ : T.normSq (M 2) = 1 := by |
| change T.normSq (T.add H (U 2) (V 2)) = 1 |
| rw [T.normSq_add, hnU, hnV, hC.1] |
| norm_num |
| have hnM₃ : T.normSq (M 3) = 1 := by |
| change T.normSq (T.add H (U 3) (V 3)) = 1 |
| rw [T.normSq_add, hnU, hnV, hD.1] |
| norm_num |
| have hmy : T.pairing (M 3) (V 3) = 1 / 2 := by |
| rw [T.pairing_comm] |
| change T.pairing (V 3) (T.add H (U 3) (V 3)) = _ |
| rw [T.pairing_add_right, hD.1, hnV] |
| norm_num |
| have hmx : T.pairing (M 3) (U 3) = 1 / 2 := by |
| rw [T.pairing_comm] |
| change T.pairing (U 3) (T.add H (U 3) (V 3)) = _ |
| rw [T.pairing_add_left, hD.1, hnU] |
| norm_num |
| have hmid : D.c = CATZero.mid h (D.y 3) (D.line 5) := by |
| have hc := CATZero.mid_cn h (D.y 3) (D.line 5) D.c |
| rw [D.radiusY, D.radiusLine] at hc |
| apply dist_eq_zero.mp |
| nlinarith [D.missing, dist_nonneg (x := D.c) (y := CATZero.mid h (D.y 3) (D.line 5))] |
| have hop : T.Opposite (V 3) (L 5) := by |
| have he := TangentCone.opposite_mid_logs h (D.y 3) (D.line 5) |
| rw [← hmid] at he |
| exact he |
| have hsC := D.square 2 5 1 (by decide) (by decide) (by decide) |
| have hcj : dist D.c (CATZero.mid h (D.x 2) (D.y 2)) ^ 2 = 1 / 4 := by |
| rw [hsC.mid_eq h] |
| exact D.lineFlat 5 1 (by decide) |
| have hkj : dist (D.line 5) (CATZero.mid h (D.x 2) (D.y 2)) ^ 2 = 1 / 2 := by |
| rw [hsC.mid_eq h, CATZero.dist_left_mid, div_pow, D.lineDist 5 1 (by decide)] |
| norm_num |
| have hex : L 5 = T.add H (M 2) (U 3) := |
| TangentCone.exceptional_identity h D.c (D.x 2) (D.y 2) (D.x 3) (D.line 5) |
| (D.radiusX 2) (D.radiusY 2) (D.radiusX 3) (D.radiusLine 5) |
| hsC.diagonal₀ D.twoStepX D.twoStepY hcj hkj D.loose D.fractional |
| have hopex : T.Opposite (M 2) (M 3) := |
| T.exceptional_antipodes H hnM₂ hnM₃ hmy hmx hop hex |
| apply T.five_line_fans_contradiction H (fun i : Fin 5 => L i.castSucc) |
| · intro i |
| exact hnL i.castSucc |
| · intro i j hij |
| exact hpL i.castSucc j.castSucc (fun he => hij (Fin.ext (congrArg (fun k : Fin 6 => k.val) he))) |
| · exact hA₁.2.symm.trans hA₂.2 |
| · exact hB₁.2.symm.trans hB₂.2 |
| · change T.Opposite (T.add H (L 1) (L 3)) (T.add H (L 2) (L 4)) |
| rw [← hC.2, ← hD.2] |
| exact hopex |
|
|
| end HyperbolicObstruction |
|
|
| noncomputable section |
| namespace HyperbolicObstruction |
| namespace CATZero |
| open Filter Topology |
| variable {X : Type*} [MetricSpace X] |
|
|
| /-- The elementary endpoint estimate behind first variation: fixing one |
| radial endpoint changes a comparison product by at most a first-order error. |
| It follows from the triangle inequality and convexity of the real square. -/ |
| theorem radialPairing_endpoint_gap (h : CATZero X) (o x y : X) {a b : ℝ} |
| (ha : a ∈ Set.Ioc (0 : ℝ) 1) (hb : b ∈ Set.Ioc (0 : ℝ) 1) : |
| 2 * b * (radialPairing h o x y a b - radialPairing h o x y a 1) ≤ |
| a * dist o x ^ 2 := by |
| let u := dist (segment h o x a) (segment h o y b) |
| let v := dist (segment h o x a) y |
| have htri : v ≤ u + (1 - b) * dist o y := by |
| have he := dist_triangle (segment h o x a) (segment h o y b) y |
| rw [segment_right h o y ⟨hb.1.le, hb.2⟩] at he |
| exact he |
| have huv : v ^ 2 ≤ (u + (1 - b) * dist o y) ^ 2 := by |
| have hv : 0 ≤ v := dist_nonneg |
| have hu : 0 ≤ u := dist_nonneg |
| nlinarith [mul_nonneg (sub_nonneg.mpr hb.2) (dist_nonneg (x := o) (y := y))] |
| have hs : b * v ^ 2 ≤ u ^ 2 + b * (1 - b) * dist o y ^ 2 := by |
| nlinarith [mul_le_mul_of_nonneg_left huv hb.1.le, |
| mul_nonneg (sub_nonneg.mpr hb.2) (sq_nonneg (u - b * dist o y))] |
| have he₁ : 2 * a * radialPairing h o x y a 1 = |
| a ^ 2 * dist o x ^ 2 + dist o y ^ 2 - v ^ 2 := by |
| dsimp [radialPairing, v] |
| rw [(segment_spec h o y).2.1] |
| field_simp [ha.1.ne', hb.1.ne'] |
| have he₂ : 2 * a * b * radialPairing h o x y a b = |
| a ^ 2 * dist o x ^ 2 + b ^ 2 * dist o y ^ 2 - u ^ 2 := by |
| dsimp [radialPairing, u] |
| field_simp [ha.1.ne', hb.1.ne'] |
| apply (mul_le_mul_iff_of_pos_left ha.1).mp |
| have he₁' := congrArg (fun z : ℝ => b * z) he₁ |
| nlinarith [mul_nonneg hb.1.le (mul_nonneg (sq_nonneg a) (sq_nonneg (dist o x)))] |
| |
| /-- First-variation comparison product along a quadratic radial parameter. |
| This suffices for the minimizer argument and avoids any assumed angle theorem. -/ |
| theorem radialPairing_endpoint_tendsto (h : CATZero X) (o x y : X) : |
| Tendsto (fun t : ℝ => radialPairing h o x y (t ^ 2) 1) (𝓝[>] 0) |
| (𝓝 (tangentPairing h o x y)) := by |
| have ht : Tendsto (fun t : ℝ => t) (𝓝[>] 0) (𝓝 0) := nhdsWithin_le_nhds |
| have htp : ∀ᶠ t : ℝ in 𝓝[>] 0, 0 < t := eventually_mem_nhdsWithin |
| have hlim := radialPairing_tendsto h o x y (by simpa using ht.pow 2) ht |
| (htp.mono fun _ hh => sq_pos_of_pos hh) htp |
| have hlo : Tendsto (fun t : ℝ => radialPairing h o x y (t ^ 2) t - |
| t * dist o x ^ 2 / 2) (𝓝[>] 0) (𝓝 (tangentPairing h o x y)) := by |
| simpa using hlim.sub ((ht.mul_const (dist o x ^ 2)).div_const 2) |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hlo tendsto_const_nhds |
| · filter_upwards [htp, ht.eventually (ge_mem_nhds (by norm_num : (0 : ℝ) < 1))] |
| with t ht0 ht1 |
| have hts : t ^ 2 ∈ Set.Ioc (0 : ℝ) 1 := ⟨sq_pos_of_pos ht0, by nlinarith⟩ |
| have he := radialPairing_endpoint_gap h o x y hts ⟨ht0, ht1⟩ |
| apply (mul_le_mul_iff_of_pos_left ht0).mp |
| nlinarith |
| · filter_upwards [htp, ht.eventually (ge_mem_nhds (by norm_num : (0 : ℝ) < 1))] |
| with t ht0 ht1 |
| exact radialPairing_le_tangentPairing h o x y ⟨sq_pos_of_pos ht0, by nlinarith⟩ |
| (by norm_num) |
|
|
| end CATZero |
| end HyperbolicObstruction |
|
|
| noncomputable section |
| namespace HyperbolicObstruction |
| namespace CATZero |
| open Filter Topology |
| variable {X : Type*} [MetricSpace X] {ι : Type*} [Fintype ι] |
|
|
| /-- A finite squared-distance minimizer has nonpositive directional product |
| sum. The proof uses the actual comparison-product limit, not first variation |
| as an imported hypothesis. -/ |
| theorem tangentPairing_sum_nonpos_of_minimum (h : CATZero X) (o : X) (y : ι → X) |
| (hmin : ∀ x : X, (∑ i, dist o (y i) ^ 2) ≤ ∑ i, dist x (y i) ^ 2) (z : X) : |
| (∑ i, tangentPairing h o z (y i)) ≤ 0 := by |
| have hbound (a : ℝ) (ha : a ∈ Set.Ioc (0 : ℝ) 1) : |
| (∑ i, radialPairing h o z (y i) a 1) ≤ |
| a * (Fintype.card ι : ℝ) * dist o z ^ 2 / 2 := by |
| have he (i : ι) : dist (segment h o z a) (y i) ^ 2 = |
| a ^ 2 * dist o z ^ 2 + dist o (y i) ^ 2 - |
| 2 * a * radialPairing h o z (y i) a 1 := by |
| dsimp [radialPairing] |
| rw [(segment_spec h o (y i)).2.1] |
| field_simp [ha.1.ne'] |
| ring |
| have hes := Finset.sum_congr (s₁ := Finset.univ) (s₂ := Finset.univ) rfl |
| (fun i _ => he i) |
| simp only [Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_const, |
| Finset.card_univ, nsmul_eq_mul, ← Finset.mul_sum] at hes |
| have hm := hmin (segment h o z a) |
| apply (mul_le_mul_iff_of_pos_left ha.1).mp |
| nlinarith |
| have ht : Tendsto (fun t : ℝ => t) (𝓝[>] 0) (𝓝 0) := nhdsWithin_le_nhds |
| have htp : ∀ᶠ t : ℝ in 𝓝[>] 0, 0 < t := eventually_mem_nhdsWithin |
| have hlim := tendsto_finsetSum Finset.univ |
| (fun i _ => radialPairing_endpoint_tendsto h o z (y i)) |
| have hr : Tendsto (fun t : ℝ => t ^ 2 * (Fintype.card ι : ℝ) * dist o z ^ 2 / 2) |
| (𝓝[>] 0) (𝓝 0) := by |
| simpa using (((ht.pow 2).mul_const (Fintype.card ι : ℝ)).mul_const |
| (dist o z ^ 2)).div_const 2 |
| apply le_of_tendsto_of_tendsto hlim hr |
| filter_upwards [htp, ht.eventually (ge_mem_nhds (by norm_num : (0 : ℝ) < 1))] |
| with t ht0 ht1 |
| exact hbound (t ^ 2) ⟨sq_pos_of_pos ht0, by nlinarith⟩ |
| |
| end CATZero |
| end HyperbolicObstruction |
| |
| noncomputable section |
| namespace HyperbolicObstruction |
| namespace TangentCone |
| open UniformSpace |
| variable {X : Type*} [MetricSpace X] (h : CATZero X) (o : X) |
| |
| theorem pairing_ray_log (p : TangentRay h o) (y : X) : |
| (coneData h o).pairing (p : TangentCone h o) (log h o y) = |
| (p.weight : ℝ) * CATZero.tangentPairing h o p.point y := by |
| unfold ConeData.pairing ConeData.normSq |
| change (dist (vertex h o) (p : TangentCone h o) ^ 2 + |
| dist (vertex h o) (log h o y) ^ 2 - dist (p : TangentCone h o) (log h o y) ^ 2) / 2 = _ |
| rw [dist_vertex_log, vertex, log, Completion.dist_eq, Completion.dist_eq, |
| TangentRay.dist_vertex, TangentRay.dist_sq] |
| dsimp only [TangentRay.log, NNReal.coe_one] |
| ring |
| |
| /-- The tangent barycenter inequality at an actual finite squared-distance |
| minimizer, extended to the completed cone by density. -/ |
| theorem pairing_sum_nonpos_of_minimum {ι : Type*} [Fintype ι] (y : ι → X) |
| (hmin : ∀ x : X, (∑ i, dist o (y i) ^ 2) ≤ ∑ i, dist x (y i) ^ 2) |
| (v : TangentCone h o) : (∑ i, (coneData h o).pairing v (log h o (y i))) ≤ 0 := by |
| induction v using Completion.induction_on with |
| | hp => |
| apply isClosed_le _ continuous_const |
| unfold ConeData.pairing ConeData.normSq |
| fun_prop |
| | ih p => |
| simp_rw [pairing_ray_log] |
| rw [← Finset.mul_sum] |
| exact mul_nonpos_of_nonneg_of_nonpos p.weight.coe_nonneg |
| (CATZero.tangentPairing_sum_nonpos_of_minimum h o y hmin p.point) |
| |
| /-- Variance form of the zero tangent barycenter. -/ |
| theorem variance_of_minimum {ι : Type*} [Fintype ι] (y : ι → X) |
| (hmin : ∀ x : X, (∑ i, dist o (y i) ^ 2) ≤ ∑ i, dist x (y i) ^ 2) |
| (v : TangentCone h o) : |
| (Fintype.card ι : ℝ) * (coneData h o).normSq v + (∑ i, dist o (y i) ^ 2) ≤ |
| ∑ i, dist v (log h o (y i)) ^ 2 := by |
| have hs := pairing_sum_nonpos_of_minimum h o y hmin v |
| simp_rw [(coneData h o).dist_sq, normSq_log] |
| simp only [Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_const, |
| Finset.card_univ, nsmul_eq_mul, ← Finset.mul_sum] |
| linarith |
| |
| end TangentCone |
| end HyperbolicObstruction |
| |
| noncomputable section |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
