import Mathlib
import OAI.Geometry.HyperbolicGroups.ConeData

namespace OAI

section
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open scoped Topology ContinuousMap
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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