l3lab/ntp-mathlib huggingface dataset:
/-- Every point is covered by chosen balls, before `p.lastStep`. -/
theorem mem_iUnionUpTo_lastStep (x : β) : p.c x ∈ p.iUnionUpTo p.lastStep := by
  have A : ∀ z : β, p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep := by
    have : p.lastStep ∈ {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} :=
      csInf_mem p.lastStep_nonempty
    simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem]
  by_contra h
  rcases A x with (H | H); · exact h H
  have Rpos : 0 < p.R p.lastStep := by
    apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H
  have B : p.τ⁻¹ * p.R p.lastStep < p.R p.lastStep := by
    conv_rhs => rw [← one_mul (p.R p.lastStep)]
    exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one
  obtain ⟨y, hy1, hy2⟩ :
      ∃ y : β, p.c y ∉ p.iUnionUpTo p.lastStep ∧ p.τ⁻¹ * p.R p.lastStep < p.r y := by
    have := exists_lt_of_lt_csSup ?_ B
    · simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists,
      Subtype.coe_mk]
    rw [← image_univ, nonempty_image_iff]
    exact ⟨⟨_, h⟩, mem_univ _⟩
  rcases A y with (Hy | Hy)
  · 

Proof state:
case inr.intro.intro.inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x : β
A :
  ∀ (z : β),
    BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨
      p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p)
h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p)
H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p)
Rpos : 0 < R p (lastStep p)
B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p)
y : β
hy1 : BallPackage.c p.toBallPackage y ∉ iUnionUpTo p (lastStep p)
hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y
Hy : BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p)
⊢ False

Next tactic:
exact hy1 Hy
/-- Every point is covered by chosen balls, before `p.lastStep`. -/
theorem mem_iUnionUpTo_lastStep (x : β) : p.c x ∈ p.iUnionUpTo p.lastStep := by
  have A : ∀ z : β, p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep := by
    have : p.lastStep ∈ {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} :=
      csInf_mem p.lastStep_nonempty
    simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem]
  by_contra h
  rcases A x with (H | H); · exact h H
  have Rpos : 0 < p.R p.lastStep := by
    apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H
  have B : p.τ⁻¹ * p.R p.lastStep < p.R p.lastStep := by
    conv_rhs => rw [← one_mul (p.R p.lastStep)]
    exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one
  obtain ⟨y, hy1, hy2⟩ :
      ∃ y : β, p.c y ∉ p.iUnionUpTo p.lastStep ∧ p.τ⁻¹ * p.R p.lastStep < p.r y := by
    have := exists_lt_of_lt_csSup ?_ B
    · simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists,
      Subtype.coe_mk]
    rw [← image_univ, nonempty_image_iff]
    exact ⟨⟨_, h⟩, mem_univ _⟩
  rcases A y with (Hy | Hy)
  · exact hy1 Hy
  · 

Proof state:
case inr.intro.intro.inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x : β
A :
  ∀ (z : β),
    BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨
      p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p)
h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p)
H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p)
Rpos : 0 < R p (lastStep p)
B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p)
y : β
hy1 : BallPackage.c p.toBallPackage y ∉ iUnionUpTo p (lastStep p)
hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y
Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p)
⊢ False

Next tactic:
rw [← div_eq_inv_mul] at hy2
/-- Every point is covered by chosen balls, before `p.lastStep`. -/
theorem mem_iUnionUpTo_lastStep (x : β) : p.c x ∈ p.iUnionUpTo p.lastStep := by
  have A : ∀ z : β, p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep := by
    have : p.lastStep ∈ {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} :=
      csInf_mem p.lastStep_nonempty
    simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem]
  by_contra h
  rcases A x with (H | H); · exact h H
  have Rpos : 0 < p.R p.lastStep := by
    apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H
  have B : p.τ⁻¹ * p.R p.lastStep < p.R p.lastStep := by
    conv_rhs => rw [← one_mul (p.R p.lastStep)]
    exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one
  obtain ⟨y, hy1, hy2⟩ :
      ∃ y : β, p.c y ∉ p.iUnionUpTo p.lastStep ∧ p.τ⁻¹ * p.R p.lastStep < p.r y := by
    have := exists_lt_of_lt_csSup ?_ B
    · simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists,
      Subtype.coe_mk]
    rw [← image_univ, nonempty_image_iff]
    exact ⟨⟨_, h⟩, mem_univ _⟩
  rcases A y with (Hy | Hy)
  · exact hy1 Hy
  · rw [← div_eq_inv_mul] at hy2
    

Proof state:
case inr.intro.intro.inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x : β
A :
  ∀ (z : β),
    BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨
      p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p)
h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p)
H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p)
Rpos : 0 < R p (lastStep p)
B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p)
y : β
hy1 : BallPackage.c p.toBallPackage y ∉ iUnionUpTo p (lastStep p)
hy2 : R p (lastStep p) / p.τ < BallPackage.r p.toBallPackage y
Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p)
⊢ False

Next tactic:
have := (div_le_iff' (_root_.zero_lt_one.trans p.one_lt_tau)).1 hy2.le
/-- Every point is covered by chosen balls, before `p.lastStep`. -/
theorem mem_iUnionUpTo_lastStep (x : β) : p.c x ∈ p.iUnionUpTo p.lastStep := by
  have A : ∀ z : β, p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep := by
    have : p.lastStep ∈ {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} :=
      csInf_mem p.lastStep_nonempty
    simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem]
  by_contra h
  rcases A x with (H | H); · exact h H
  have Rpos : 0 < p.R p.lastStep := by
    apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H
  have B : p.τ⁻¹ * p.R p.lastStep < p.R p.lastStep := by
    conv_rhs => rw [← one_mul (p.R p.lastStep)]
    exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one
  obtain ⟨y, hy1, hy2⟩ :
      ∃ y : β, p.c y ∉ p.iUnionUpTo p.lastStep ∧ p.τ⁻¹ * p.R p.lastStep < p.r y := by
    have := exists_lt_of_lt_csSup ?_ B
    · simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists,
      Subtype.coe_mk]
    rw [← image_univ, nonempty_image_iff]
    exact ⟨⟨_, h⟩, mem_univ _⟩
  rcases A y with (Hy | Hy)
  · exact hy1 Hy
  · rw [← div_eq_inv_mul] at hy2
    have := (div_le_iff' (_root_.zero_lt_one.trans p.one_lt_tau)).1 hy2.le
    

Proof state:
case inr.intro.intro.inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x : β
A :
  ∀ (z : β),
    BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨
      p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p)
h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p)
H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p)
Rpos : 0 < R p (lastStep p)
B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p)
y : β
hy1 : BallPackage.c p.toBallPackage y ∉ iUnionUpTo p (lastStep p)
hy2 : R p (lastStep p) / p.τ < BallPackage.r p.toBallPackage y
Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p)
this : R p (lastStep p) ≤ p.τ * BallPackage.r p.toBallPackage y
⊢ False

Next tactic:
exact lt_irrefl _ (Hy.trans_le this)
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < lastStep p
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
⊢ color p i < N

Next tactic:
induction' i using Ordinal.induction with i IH
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  

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 < lastStep p → color p k < N
hi : i < lastStep p
⊢ color p i < N

Next 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}
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
⊢ color p i < N

Next tactic:
have color_i : p.color i = sInf (univ \ A) := by rw [color]
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  have color_i : p.color i = sInf (univ \ A) := by 

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
⊢ color p i = sInf (univ \ A)

Next tactic:
rw [color]
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  have color_i : p.color i = sInf (univ \ A) := by rw [color]
  

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
color_i : color p i = sInf (univ \ A)
⊢ color p i < N

Next tactic:
rw [color_i]
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  have color_i : p.color i = sInf (univ \ A) := by rw [color]
  rw [color_i]
  

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
color_i : color p i = sInf (univ \ A)
⊢ sInf (univ \ A) < N

Next tactic:
have N_mem : N ∈ univ \ A := by
    simp only [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'
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  have color_i : p.color i = sInf (univ \ A) := by rw [color]
  rw [color_i]
  have N_mem : N ∈ univ \ A := by
    

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
color_i : color p i = sInf (univ \ A)
⊢ N ∈ univ \ A

Next tactic:
simp only [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]
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  have color_i : p.color i = sInf (univ \ A) := by rw [color]
  rw [color_i]
  have N_mem : N ∈ univ \ A := by
    simp only [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]
    

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
color_i : color p i = sInf (univ \ A)
⊢ ∀ x < i,
    Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) →
      ¬N = color p x

Next tactic:
intro j ji _
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  have color_i : p.color i = sInf (univ \ A) := by rw [color]
  rw [color_i]
  have N_mem : N ∈ univ \ A := by
    simp only [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 _
    

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
color_i : color p i = sInf (univ \ A)
j : Ordinal.{u}
ji : j < i
a✝ :
  Set.Nonempty
    (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩
      closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))
⊢ ¬N = color p j

Next tactic:
exact (IH j ji (ji.trans hi)).ne'
/-- If there are no configurations of satellites with `N+1` points, one never uses more than `N`
distinct families in the Besicovitch inductive construction. -/
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by
  /- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
    Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
    (there is such a ball, otherwise one would have used the color `k` and not `N`).
    Then this family of `N+1` balls forms a satellite configuration, which is forbidden by
    the assumption `hN`. -/
  induction' i using Ordinal.induction with i IH
  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}
  have color_i : p.color i = sInf (univ \ A) := by rw [color]
  rw [color_i]
  have N_mem : N ∈ univ \ A := by
    simp only [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'
  

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 < lastStep p → color p k < N
hi : i < lastStep p
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      Set.Nonempty
        (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
          closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),
      {color p ↑j}
color_i : color p i = sInf (univ \ A)
N_mem : N ∈ univ \ A
⊢ sInf (univ \ A) < N

Next tactic:
suffices sInf (univ \ A) ≠ N by
    rcases (csInf_le (OrderBot.bddBelow (univ \ A)) N_mem).lt_or_eq with (H | H)
    · exact H
    · exact (this H).elim