-- Mathlib/MeasureTheory/Covering/Besicovitch.lean
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N

Proof state:
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]

Result:
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

Proof state:
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

Tactic:
exact (this a_ne_b.symm (le_of_not_le G_le)).symm

Result:
no goals

Proof state:
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, 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]

Result:
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