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
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
⊢ 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)

Tactic:
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
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
⊢ 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:
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
-- 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
this : 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
rw :
  p.c (Classical.epsilon fun (t : β) ↦ 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 : β) ↦ 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]
-- 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
this : 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
⊢ ∃ (t : β), p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t

Tactic:
simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using
        notMem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)

Premises used: not_le, not_exists, Set.mem_setOf_eq, exists_prop, not_lt, notMem_of_lt_csInf, OrderBot.bddBelow, not_and, Classical.not_forall
-- 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
this : 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
⊢ ∃ (t : β), 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
        notMem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)
-- 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
this_1 : 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
this : ∃ (t : β), p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (Classical.epsilon fun (t : β) ↦ 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 : β) ↦ p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)

Tactic:
Classical.epsilon_spec this

Premises used: Set.Mem, Classical.epsilon_spec
-- 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
this_1 : 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
this : ∃ (t : β), p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (Classical.epsilon fun (t : β) ↦ 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 : β) ↦ p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)

Tactic:
exact Classical.epsilon_spec this

Premises used: Classical.epsilon_spec
-- 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, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))

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
        notMem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)
    exact Classical.epsilon_spec 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
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 : ℕ) < G (↑b : ℕ)
⊢ (↑a : ℕ) ≤ N

Tactic:
have ha : (a : ℕ) ≤ N := Nat.lt_succ_iff.1 a.2
-- 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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
⊢ (↑b : ℕ) ≤ N

Tactic:
have hb : (b : ℕ) ≤ N := Nat.lt_succ_iff.1 b.2
-- 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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
hb : (↑b : ℕ) ≤ N
left : p.r (p.index (G (↑a : ℕ))) ≤ dist (p.c (p.index (G (↑a : ℕ)))) (p.c (p.index (G (↑b : ℕ))))
right : p.r (p.index (G (↑b : ℕ))) ≤ p.τ * p.r (p.index (G (↑a : ℕ)))
⊢ p.r (p.index (G (↑a : ℕ))) ≤ dist (p.c (p.index (G (↑a : ℕ)))) (p.c (p.index (G (↑b : ℕ)))) ∧
    p.r (p.index (G (↑b : ℕ))) ≤ p.τ * p.r (p.index (G (↑a : ℕ)))

Tactic:
constructor
-- 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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
hb : (↑b : ℕ) ≤ N
⊢ p.c (p.index (G (↑b : ℕ))) ∉ p.iUnionUpTo (G (↑b : ℕ))

Tactic:
have := (fGn b hb).1
-- 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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
hb : (↑b : ℕ) ≤ N
this : p.c (p.index (G (↑b : ℕ))) ∉ p.iUnionUpTo (G (↑b : ℕ))
⊢ ∀ x < G (↑b : ℕ), p.c (p.index (G (↑b : ℕ))) ∉ ball (p.c (p.index x)) (p.r (p.index x))

Tactic:
simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, not_and,
        Subtype.exists] at this

Premises used: Besicovitch.TauPackage.iUnionUpTo, Set.mem_iUnion, Subtype.exists, not_exists, exists_prop, not_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
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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
hb : (↑b : ℕ) ≤ N
this : ∀ x < G (↑b : ℕ), p.c (p.index (G (↑b : ℕ))) ∉ ball (p.c (p.index x)) (p.r (p.index x))
⊢ p.r (p.index (G (↑a : ℕ))) ≤ dist (p.c (p.index (G (↑a : ℕ)))) (p.c (p.index (G (↑b : ℕ))))

Tactic:
simpa only [dist_comm, mem_ball, not_lt] using this (G a) G_lt

Premises used: dist_comm, Fin.val, Metric.mem_ball, not_lt
-- 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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
hb : (↑b : ℕ) ≤ N
apply : p.r (p.index (G (↑b : ℕ))) ≤ p.R (G (↑a : ℕ))
⊢ p.r (p.index (G (↑b : ℕ))) ≤ p.τ * p.r (p.index (G (↑a : ℕ)))

Tactic:
apply le_trans _ (fGn a ha).2

Premises used: Fin.val, And.right, le_trans
-- 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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
hb : (↑b : ℕ) ≤ N
H : p.c (p.index (G (↑b : ℕ))) ∈ p.iUnionUpTo (G (↑a : ℕ))
⊢ False

Tactic:
(fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)

Premises used: Besicovitch.TauPackage.monotone_iUnionUpTo, LT.lt.le, Fin.val, And.left
-- 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
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 : ℕ) < G (↑b : ℕ)
ha : (↑a : ℕ) ≤ N
hb : (↑b : ℕ) ≤ N
H : p.c (p.index (G (↑b : ℕ))) ∈ p.iUnionUpTo (G (↑a : ℕ))
⊢ False

Tactic:
exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)

Premises used: Besicovitch.TauPackage.monotone_iUnionUpTo, LT.lt.le, Fin.val, And.left