awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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:
(this H).elim

Premises used: False.elim
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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 (this H).elim

Premises used: False.elim
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
ind : sInf (univ \ A) ≠ N
⊢ sInf (univ \ A) < N

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
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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
⊢ k < sInf (univ \ A)

Tactic:
rw [← Inf_eq_N] at hk
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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

Tactic:
simpa only [true_and, mem_univ, Classical.not_not, mem_diff] using
        Nat.notMem_of_lt_sInf hk

Premises used: Nat.notMem_of_lt_sInf, Classical.not_not, Set.mem_diff, Set.mem_univ, true_and
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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

Tactic:
have : k ∈ A := by
      simpa only [true_and, mem_univ, Classical.not_not, mem_diff] using
        Nat.notMem_of_lt_sInf hk
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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]

Premises used: Subtype.coe_mk, Set.mem_iUnion, Subtype.exists, exists_prop, Set.mem_singleton_iff, Metric.mem_closedBall
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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

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.notMem_of_lt_sInf hk
    simp only [] at this
    simpa only [A, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists,
      Subtype.coe_mk]
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
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
ind :
  ∀ (g : (k : 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
⊢ False

Tactic:
choose! g hg using this
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : ℕ) ↦ if n = N then i else g n
n : ℕ
hn : n ≤ N
inl :
  ∀ (hN : IsEmpty.{u_1 + 1} (SatelliteConfig α n p.τ)) (IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < 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)),
    let G : ℕ → Ordinal.{u} := fun (n_1 : ℕ) ↦ if n_1 = n then i else g n_1;
    ∀ (hn : n ≤ n), p.color (G n) = n
inr : ∀ (H : n < N), p.color (G n) = n
⊢ p.color (G n) = n

Tactic:
rcases hn.eq_or_lt with (rfl | H)

Premises used: LE.le.eq_or_lt, rfl
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
g : (k : ℕ) → Ordinal.{u}
n : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α n p.τ)
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < 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 : ℕ) ↦ if n_1 = n then i else g n_1
hn : n ≤ n
inl : p.color (if True then i else g n) = n
⊢ p.color (G n) = n

Tactic:
simp only [G]
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
g : (k : ℕ) → Ordinal.{u}
n : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α n p.τ)
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < 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 : ℕ) ↦ 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]

Premises used: if_true
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : ℕ) ↦ if n = N then i else g n
n : ℕ
hn : n ≤ N
H : n < N
inr : p.color (if n = N then i else g n) = n
⊢ p.color (G n) = n

Tactic:
simp only [G]
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : 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]

Premises used: if_false, And.right, Eq.symm, LT.lt.ne
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : ℕ) ↦ if n = N then i else g n
⊢ ∀ n ≤ N, p.color (G n) = n

Tactic:
have color_G : ∀ n, n ≤ N → p.color (G n) = n := by
    intro n hn
    rcases hn.eq_or_lt with (rfl | H)
    · simp only [G]; simp only [color_i, Inf_eq_N, if_true]
    · simp only [G]; simp only [H.ne, (hg n H).right.right.symm, if_false]
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : ℕ) ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
inl :
  ∀ (hN : IsEmpty.{u_1 + 1} (SatelliteConfig α n p.τ)) (IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < 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)),
    let G : ℕ → Ordinal.{u} := fun (n_1 : ℕ) ↦ 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
inr : ∀ (H : n < N), G n < p.lastStep
⊢ G n < p.lastStep

Tactic:
rcases hn.eq_or_lt with (rfl | H)

Premises used: LE.le.eq_or_lt, rfl
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
g : (k : ℕ) → Ordinal.{u}
n : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α n p.τ)
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < 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 : ℕ) ↦ 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
inl : (if True then i else g n) < p.lastStep
⊢ G n < p.lastStep

Tactic:
simp only [G]
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
g : (k : ℕ) → Ordinal.{u}
n : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α n p.τ)
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < 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 : ℕ) ↦ 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]

Premises used: if_true
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : 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
inr : (if n = N then i else g n) < p.lastStep
⊢ G n < p.lastStep

Tactic:
simp only [G]
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : 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]

Premises used: if_false, And.left, LT.lt.trans, LT.lt.ne
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : ℕ) ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
⊢ ∀ n ≤ N, G n < p.lastStep

Tactic:
have G_lt_last : ∀ n, n ≤ N → G n < p.lastStep := by
    intro n hn
    rcases hn.eq_or_lt with (rfl | H)
    · simp only [G]; simp only [hi, if_true]
    · simp only [G]; simp only [H.ne, (hg n H).left.trans hi, if_false]
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
inst_1 : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, ∀ (hi : y < p.lastStep), p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < i }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : (k : 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 : 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
rw :
  Eq.{u + 1} (α := β)
    (Classical.epsilon fun (b : β) ↦
      p.c b ∉
          ⋃ (j : { j : Ordinal.{u} // j < G n }),
            ball (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∧
        ⨆ (b :
            { b : β //
              p.c b ∉
                ⋃ (j : { j : Ordinal.{u} // j < G n }),
                  ball (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) }),
            p.r (↑b : β) ≤
          p.τ * p.r b)
    (Classical.epsilon fun (t : β) ↦ p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)
⊢ p.index (G n) = Classical.epsilon fun (t : β) ↦ p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t

Tactic:
rw [index]

Premises used: Besicovitch.TauPackage.index