Theorem: Besicovitch.TauPackage.color_lt
File: Mathlib/MeasureTheory/Covering/Besicovitch.lean

tactic: induction' i using Ordinal.induction with i IH
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
⊢ p.color i < N
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
⊢ p.color i < N

tactic: let A : Set ℕ :=
  ⋃ (j : { j // j < i })
    (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩
      closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
    {p.color j}
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
⊢ p.color i < N
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
⊢ p.color i < N

tactic: have color_i : p.color i = sInf (univ \ A) := by rw [color]
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
⊢ p.color i < N
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
⊢ p.color i < N

tactic: rw [color_i]
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
⊢ p.color i < N
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
⊢ sInf (univ \ A) < N

tactic: have N_mem : N ∈ univ \ A := by
  simp only [A, not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff,
    mem_closedBall, not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]
  intro j ji _
  exact (IH j ji (ji.trans hi)).ne'
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
⊢ sInf (univ \ A) < N
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
⊢ sInf (univ \ A) < N

tactic: intro Inf_eq_N
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
⊢ sInf (univ \ A) ≠ N
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
⊢ False

tactic: have :
  ∀ k, k < N → ∃ j, j < i ∧
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩
      closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧ k = p.color j := by
  intro k hk
  rw [← Inf_eq_N] at hk
  have : k ∈ A := by
    simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using
      Nat.not_mem_of_lt_sInf hk
  simp only [mem_iUnion, mem_singleton_iff, exists_prop, Subtype.exists, exists_and_right,
    and_assoc] at this
  simpa only [A, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists,
    Subtype.coe_mk]
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
⊢ False
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
this :
  ∀ k < N,
    ∃ j < i,
      (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color j
⊢ False

tactic: choose! g hg using this
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
this :
  ∀ k < N,
    ∃ j < i,
      (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color j
⊢ False
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
⊢ False

tactic: let G : ℕ → Ordinal := fun n => if n = N then i else g n
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
⊢ False
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
⊢ False

tactic: have fGn :
    ∀ n, n ≤ N →
      p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n)) := by
  intro n hn
  have :
    p.index (G n) =
      Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by
    rw [index]; rfl
  rw [this]
  have : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by
    simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using
      not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)
  exact Classical.epsilon_spec this
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
⊢ False
state_after: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
⊢ False

tactic: exact hN.false sc
state_before: case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
sc : SatelliteConfig α N p.τ :=
  { c := fun k => p.c (p.index (G ↑k)), r := fun k => p.r (p.index (G ↑k)), rpos := ⋯, h := ⋯, hlast := ⋯, inter := ⋯ }
⊢ False
state_after: no goals

tactic: rw [color]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
⊢ p.color i = sInf (univ \ A)
state_after: no goals

tactic: simp only [A, not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff,
  mem_closedBall, not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
⊢ N ∈ univ \ A
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
⊢ ∀ x < i,
    (closedBall (p.c (p.index x)) (p.r (p.index x)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty →
      ¬N = p.color x

tactic: intro j ji _
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
⊢ ∀ x < i,
    (closedBall (p.c (p.index x)) (p.r (p.index x)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty →
      ¬N = p.color x
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
j : Ordinal.{u}
ji : j < i
a✝ : (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty
⊢ ¬N = p.color j

tactic: exact (IH j ji (ji.trans hi)).ne'
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
j : Ordinal.{u}
ji : j < i
a✝ : (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty
⊢ ¬N = p.color j
state_after: no goals

tactic: rcases (csInf_le (OrderBot.bddBelow (univ \ A)) N_mem).lt_or_eq with (H | H)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
this : sInf (univ \ A) ≠ N
⊢ sInf (univ \ A) < N
state_after: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
this : sInf (univ \ A) ≠ N
H : sInf (univ \ A) < N
⊢ sInf (univ \ A) < N

case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
this : sInf (univ \ A) ≠ N
H : sInf (univ \ A) = N
⊢ sInf (univ \ A) < N

tactic: exact H
state_before: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
this : sInf (univ \ A) ≠ N
H : sInf (univ \ A) < N
⊢ sInf (univ \ A) < N
state_after: no goals

tactic: exact (this H).elim
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
this : sInf (univ \ A) ≠ N
H : sInf (univ \ A) = N
⊢ sInf (univ \ A) < N
state_after: no goals

tactic: intro k hk
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
⊢ ∀ k < N,
    ∃ j < i,
      (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color j
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < N
⊢ ∃ j < i,
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
      k = p.color j

tactic: rw [← Inf_eq_N] at hk
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < N
⊢ ∃ j < i,
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
      k = p.color j
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < sInf (univ \ A)
⊢ ∃ j < i,
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
      k = p.color j

tactic: have : k ∈ A := by
  simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using
    Nat.not_mem_of_lt_sInf hk
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < sInf (univ \ A)
⊢ ∃ j < i,
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
      k = p.color j
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < sInf (univ \ A)
this : k ∈ A
⊢ ∃ j < i,
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
      k = p.color j

tactic: simpa only [A, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists,
  Subtype.coe_mk]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < sInf (univ \ A)
this : k ∈ A
⊢ ∃ j < i,
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
      k = p.color j
state_after: no goals

tactic: simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using
  Nat.not_mem_of_lt_sInf hk
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < sInf (univ \ A)
⊢ k ∈ A
state_after: no goals

tactic: intro n hn
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
⊢ ∀ n ≤ N, p.color (G n) = n
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
n : ℕ
hn : n ≤ N
⊢ p.color (G n) = n

tactic: rcases hn.eq_or_lt with (rfl | H)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
n : ℕ
hn : n ≤ N
⊢ p.color (G n) = n
state_after: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
hn : n ≤ n
⊢ p.color (G n) = n

case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
n : ℕ
hn : n ≤ N
H : n < N
⊢ p.color (G n) = n

tactic: simp only [G]
state_before: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
hn : n ≤ n
⊢ p.color (G n) = n
state_after: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
hn : n ≤ n
⊢ p.color (if True then i else g n) = n

tactic: simp only [color_i, Inf_eq_N, if_true, eq_self_iff_true]
state_before: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
hn : n ≤ n
⊢ p.color (if True then i else g n) = n
state_after: no goals

tactic: simp only [G]
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
n : ℕ
hn : n ≤ N
H : n < N
⊢ p.color (G n) = n
state_after: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
n : ℕ
hn : n ≤ N
H : n < N
⊢ p.color (if n = N then i else g n) = n

tactic: simp only [H.ne, (hg n H).right.right.symm, if_false]
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
n : ℕ
hn : n ≤ N
H : n < N
⊢ p.color (if n = N then i else g n) = n
state_after: no goals

tactic: intro n hn
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
⊢ ∀ n ≤ N, G n < p.lastStep
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
⊢ G n < p.lastStep

tactic: rcases hn.eq_or_lt with (rfl | H)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
⊢ G n < p.lastStep
state_after: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
color_G : ∀ n_1 ≤ n, p.color (G n_1) = n_1
hn : n ≤ n
⊢ G n < p.lastStep

case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
H : n < N
⊢ G n < p.lastStep

tactic: simp only [G]
state_before: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
color_G : ∀ n_1 ≤ n, p.color (G n_1) = n_1
hn : n ≤ n
⊢ G n < p.lastStep
state_after: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
color_G : ∀ n_1 ≤ n, p.color (G n_1) = n_1
hn : n ≤ n
⊢ (if True then i else g n) < p.lastStep

tactic: simp only [hi, if_true, eq_self_iff_true]
state_before: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
g : ℕ → Ordinal.{u}
n : ℕ
hN : IsEmpty (SatelliteConfig α n p.τ)
IH : ∀ k < i, k < p.lastStep → p.color k < n
N_mem : n ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = n
hg :
  ∀ k < n,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1
color_G : ∀ n_1 ≤ n, p.color (G n_1) = n_1
hn : n ≤ n
⊢ (if True then i else g n) < p.lastStep
state_after: no goals

tactic: simp only [G]
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
H : n < N
⊢ G n < p.lastStep
state_after: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
H : n < N
⊢ (if n = N then i else g n) < p.lastStep

tactic: simp only [H.ne, (hg n H).left.trans hi, if_false]
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
H : n < N
⊢ (if n = N then i else g n) < p.lastStep
state_after: no goals

tactic: intro n hn
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
⊢ ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
⊢ p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))

tactic: have :
  p.index (G n) =
    Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by
  rw [index]; rfl
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
⊢ p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))

tactic: rw [this]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t) ∉ p.iUnionUpTo (G n) ∧
    p.R (G n) ≤ p.τ * p.r (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)

tactic: have : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by
  simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using
    not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t) ∉ p.iUnionUpTo (G n) ∧
    p.R (G n) ≤ p.τ * p.r (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this✝ : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
this : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t) ∉ p.iUnionUpTo (G n) ∧
    p.R (G n) ≤ p.τ * p.r (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)

tactic: exact Classical.epsilon_spec this
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this✝ : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
this : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t) ∉ p.iUnionUpTo (G n) ∧
    p.R (G n) ≤ p.τ * p.r (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)
state_after: no goals

tactic: rw [index]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
⊢ p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
⊢ (Classical.epsilon fun b => p.c b ∉ ⋃ j, ball (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∧ ⨆ b, p.r ↑b ≤ p.τ * p.r b) =
    Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t

tactic: rfl
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
⊢ (Classical.epsilon fun b => p.c b ∉ ⋃ j, ball (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∧ ⨆ b, p.r ↑b ≤ p.τ * p.r b) =
    Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
state_after: no goals

tactic: simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using
  not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
state_after: no goals

tactic: intro a b G_lt
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
⊢ ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
    p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))

tactic: have ha : (a : ℕ) ≤ N := Nat.lt_succ_iff.1 a.2
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
    p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
    p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))

tactic: have hb : (b : ℕ) ≤ N := Nat.lt_succ_iff.1 b.2
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
    p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
    p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))

tactic: constructor
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
    p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
state_after: case left
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b)))

case right
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))

tactic: have := (fGn b hb).1
state_before: case left
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b)))
state_after: case left
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
this : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑b)
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b)))

tactic: simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,
  Subtype.exists, Subtype.coe_mk] at this
state_before: case left
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
this : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑b)
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b)))
state_after: case left
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
this : ∀ x < G ↑b, p.c (p.index (G ↑b)) ∉ ball (p.c (p.index x)) (p.r (p.index x))
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b)))

tactic: simpa only [dist_comm, mem_ball, not_lt] using this (G a) G_lt
state_before: case left
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
this : ∀ x < G ↑b, p.c (p.index (G ↑b)) ∉ ball (p.c (p.index x)) (p.r (p.index x))
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b)))
state_after: no goals

tactic: apply le_trans _ (fGn a ha).2
state_before: case right
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑b)) ≤ p.R (G ↑a)

tactic: have B : p.c (p.index (G b)) ∉ p.iUnionUpTo (G a) := by
  intro H; exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.r (p.index (G ↑b)) ≤ p.R (G ↑a)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
⊢ p.r (p.index (G ↑b)) ≤ p.R (G ↑a)

tactic: let b' : { t // p.c t ∉ p.iUnionUpTo (G a) } := ⟨p.index (G b), B⟩
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
⊢ p.r (p.index (G ↑b)) ≤ p.R (G ↑a)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
⊢ p.r (p.index (G ↑b)) ≤ p.R (G ↑a)

tactic: apply @le_ciSup _ _ _ (fun t : { t // p.c t ∉ p.iUnionUpTo (G a) } => p.r t) _ b'
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
⊢ p.r (p.index (G ↑b)) ≤ p.R (G ↑a)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
⊢ BddAbove (range fun t => p.r ↑t)

tactic: refine ⟨p.r_bound, fun t ht => ?_⟩
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
⊢ BddAbove (range fun t => p.r ↑t)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
ht : t ∈ range fun t => p.r ↑t
⊢ t ≤ p.r_bound

tactic: simp only [exists_prop, mem_range, Subtype.exists, Subtype.coe_mk] at ht
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
ht : t ∈ range fun t => p.r ↑t
⊢ t ≤ p.r_bound
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
ht : ∃ a_1, p.c a_1 ∉ p.iUnionUpTo (G ↑a) ∧ p.r a_1 = t
⊢ t ≤ p.r_bound

tactic: rcases ht with ⟨u, hu⟩
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
ht : ∃ a_1, p.c a_1 ∉ p.iUnionUpTo (G ↑a) ∧ p.r a_1 = t
⊢ t ≤ p.r_bound
state_after: case intro
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
u : β
hu : p.c u ∉ p.iUnionUpTo (G ↑a) ∧ p.r u = t
⊢ t ≤ p.r_bound

tactic: rw [← hu.2]
state_before: case intro
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
u : β
hu : p.c u ∉ p.iUnionUpTo (G ↑a) ∧ p.r u = t
⊢ t ≤ p.r_bound
state_after: case intro
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
u : β
hu : p.c u ∉ p.iUnionUpTo (G ↑a) ∧ p.r u = t
⊢ p.r u ≤ p.r_bound

tactic: exact p.r_le _
state_before: case intro
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
B : p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
b' : { t // p.c t ∉ p.iUnionUpTo (G ↑a) } := ⟨p.index (G ↑b), B⟩
t : ℝ
u : β
hu : p.c u ∉ p.iUnionUpTo (G ↑a) ∧ p.r u = t
⊢ p.r u ≤ p.r_bound
state_after: no goals

tactic: intro H
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
⊢ p.c (p.index (G ↑b)) ∉ p.iUnionUpTo (G ↑a)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
H : p.c (p.index (G ↑b)) ∈ p.iUnionUpTo (G ↑a)
⊢ False

tactic: exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
ha : ↑a ≤ N
hb : ↑b ≤ N
H : p.c (p.index (G ↑b)) ∈ p.iUnionUpTo (G ↑a)
⊢ False
state_after: no goals

tactic: intro a b a_ne_b
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
⊢ Pairwise fun i j =>
    (fun k => p.r (p.index (G ↑k))) i ≤ dist ((fun k => p.c (p.index (G ↑k))) i) ((fun k => p.c (p.index (G ↑k))) j) ∧
        (fun k => p.r (p.index (G ↑k))) j ≤ p.τ * (fun k => p.r (p.index (G ↑k))) i ∨
      (fun k => p.r (p.index (G ↑k))) j ≤ dist ((fun k => p.c (p.index (G ↑k))) j) ((fun k => p.c (p.index (G ↑k))) i) ∧
        (fun k => p.r (p.index (G ↑k))) i ≤ p.τ * (fun k => p.r (p.index (G ↑k))) j
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
⊢ (fun k => p.r (p.index (G ↑k))) a ≤ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
      (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
    (fun k => p.r (p.index (G ↑k))) b ≤ dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
      (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b

tactic: wlog G_le : G a ≤ G b generalizing a b
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
⊢ (fun k => p.r (p.index (G ↑k))) a ≤ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
      (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
    (fun k => p.r (p.index (G ↑k))) b ≤ dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
      (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b
state_after: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
this :
  ∀ ⦃a b : Fin N.succ⦄,
    a ≠ b →
      G ↑a ≤ G ↑b →
        (fun k => p.r (p.index (G ↑k))) a ≤
              dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
            (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
          (fun k => p.r (p.index (G ↑k))) b ≤
              dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
            (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b
G_le : ¬G ↑a ≤ G ↑b
⊢ (fun k => p.r (p.index (G ↑k))) a ≤ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
      (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
    (fun k => p.r (p.index (G ↑k))) b ≤ dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
      (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b

α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
⊢ (fun k => p.r (p.index (G ↑k))) a ≤ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
      (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
    (fun k => p.r (p.index (G ↑k))) b ≤ dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
      (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b

tactic: exact Or.inl (Gab a b G_lt)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
G_lt : G ↑a < G ↑b
⊢ (fun k => p.r (p.index (G ↑k))) a ≤ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
      (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
    (fun k => p.r (p.index (G ↑k))) b ≤ dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
      (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b
state_after: no goals

tactic: exact (this a_ne_b.symm (le_of_not_le G_le)).symm
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
this :
  ∀ ⦃a b : Fin N.succ⦄,
    a ≠ b →
      G ↑a ≤ G ↑b →
        (fun k => p.r (p.index (G ↑k))) a ≤
              dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
            (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
          (fun k => p.r (p.index (G ↑k))) b ≤
              dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
            (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b
G_le : ¬G ↑a ≤ G ↑b
⊢ (fun k => p.r (p.index (G ↑k))) a ≤ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) b) ∧
      (fun k => p.r (p.index (G ↑k))) b ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a ∨
    (fun k => p.r (p.index (G ↑k))) b ≤ dist ((fun k => p.c (p.index (G ↑k))) b) ((fun k => p.c (p.index (G ↑k))) a) ∧
      (fun k => p.r (p.index (G ↑k))) a ≤ p.τ * (fun k => p.r (p.index (G ↑k))) b
state_after: no goals

tactic: rcases G_le.lt_or_eq with (H | H)
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
⊢ G ↑a < G ↑b
state_after: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a < G ↑b
⊢ G ↑a < G ↑b

case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a = G ↑b
⊢ G ↑a < G ↑b

tactic: have A : (a : ℕ) ≠ b := Fin.val_injective.ne a_ne_b
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a = G ↑b
⊢ G ↑a < G ↑b
state_after: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A✝ : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A✝)
N_mem : N ∈ univ \ A✝
Inf_eq_N : sInf (univ \ A✝) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a = G ↑b
A : ↑a ≠ ↑b
⊢ G ↑a < G ↑b

tactic: rw [← color_G a (Nat.lt_succ_iff.1 a.2), ← color_G b (Nat.lt_succ_iff.1 b.2), H] at A
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A✝ : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A✝)
N_mem : N ∈ univ \ A✝
Inf_eq_N : sInf (univ \ A✝) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a = G ↑b
A : ↑a ≠ ↑b
⊢ G ↑a < G ↑b
state_after: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A✝ : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A✝)
N_mem : N ∈ univ \ A✝
Inf_eq_N : sInf (univ \ A✝) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a = G ↑b
A : p.color (G ↑b) ≠ p.color (G ↑b)
⊢ G ↑a < G ↑b

tactic: exact (A rfl).elim
state_before: case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A✝ : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A✝)
N_mem : N ∈ univ \ A✝
Inf_eq_N : sInf (univ \ A✝) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a = G ↑b
A : p.color (G ↑b) ≠ p.color (G ↑b)
⊢ G ↑a < G ↑b
state_after: no goals

tactic: exact H
state_before: case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G ↑a ≤ G ↑b
H : G ↑a < G ↑b
⊢ G ↑a < G ↑b
state_after: no goals

tactic: intro a ha
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
⊢ ∀ i < last N,
    (fun k => p.r (p.index (G ↑k))) i ≤
        dist ((fun k => p.c (p.index (G ↑k))) i) ((fun k => p.c (p.index (G ↑k))) (last N)) ∧
      (fun k => p.r (p.index (G ↑k))) (last N) ≤ p.τ * (fun k => p.r (p.index (G ↑k))) i
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
⊢ (fun k => p.r (p.index (G ↑k))) a ≤
      dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ∧
    (fun k => p.r (p.index (G ↑k))) (last N) ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a

tactic: have I : (a : ℕ) < N := ha
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
⊢ (fun k => p.r (p.index (G ↑k))) a ≤
      dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ∧
    (fun k => p.r (p.index (G ↑k))) (last N) ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ (fun k => p.r (p.index (G ↑k))) a ≤
      dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ∧
    (fun k => p.r (p.index (G ↑k))) (last N) ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a

tactic: have : G a < G (Fin.last N) := by dsimp; simp [G, I.ne, (hg a I).1]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ (fun k => p.r (p.index (G ↑k))) a ≤
      dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ∧
    (fun k => p.r (p.index (G ↑k))) (last N) ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
this : G ↑a < G ↑(last N)
⊢ (fun k => p.r (p.index (G ↑k))) a ≤
      dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ∧
    (fun k => p.r (p.index (G ↑k))) (last N) ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a

tactic: exact Gab _ _ this
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
this : G ↑a < G ↑(last N)
⊢ (fun k => p.r (p.index (G ↑k))) a ≤
      dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ∧
    (fun k => p.r (p.index (G ↑k))) (last N) ≤ p.τ * (fun k => p.r (p.index (G ↑k))) a
state_after: no goals

tactic: dsimp
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ G ↑a < G ↑(last N)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ G ↑a < G N

tactic: simp [G, I.ne, (hg a I).1]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ G ↑a < G N
state_after: no goals

tactic: intro a ha
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
⊢ ∀ i < last N,
    dist ((fun k => p.c (p.index (G ↑k))) i) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
      (fun k => p.r (p.index (G ↑k))) i + (fun k => p.r (p.index (G ↑k))) (last N)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)

tactic: have I : (a : ℕ) < N := ha
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)

tactic: have J : G (Fin.last N) = i := by dsimp; simp only [G, if_true, eq_self_iff_true]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
J : G ↑(last N) = i
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)

tactic: have K : G a = g a := by dsimp [G]; simp [I.ne, (hg a I).1]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
J : G ↑(last N) = i
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
J : G ↑(last N) = i
K : G ↑a = g ↑a
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)

tactic: convert dist_le_add_of_nonempty_closedBall_inter_closedBall (hg _ I).2.1
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
J : G ↑(last N) = i
K : G ↑a = g ↑a
⊢ dist ((fun k => p.c (p.index (G ↑k))) a) ((fun k => p.c (p.index (G ↑k))) (last N)) ≤
    (fun k => p.r (p.index (G ↑k))) a + (fun k => p.r (p.index (G ↑k))) (last N)
state_after: no goals

tactic: dsimp
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ G ↑(last N) = i
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ G N = i

tactic: simp only [G, if_true, eq_self_iff_true]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
⊢ G N = i
state_after: no goals

tactic: dsimp [G]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
J : G ↑(last N) = i
⊢ G ↑a = g ↑a
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
J : G ↑(last N) = i
⊢ (if ↑a = N then i else g ↑a) = g ↑a

tactic: simp [I.ne, (hg a I).1]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Gab :
  ∀ (a b : Fin N.succ),
    G ↑a < G ↑b →
      p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
        p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
a : Fin (N + 1)
ha : a < last N
I : ↑a < N
J : G ↑(last N) = i
⊢ (if ↑a = N then i else g ↑a) = g ↑a
state_after: no goals