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
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 : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
inl : ∀ (H : G (↑a : ℕ) < G (↑b : ℕ)), G (↑a : ℕ) < G (↑b : ℕ)
inr : ∀ (H : G (↑a : ℕ) = G (↑b : ℕ)), G (↑a : ℕ) < G (↑b : ℕ)
⊢ G (↑a : ℕ) < G (↑b : ℕ)

Tactic:
rcases G_le.lt_or_eq with (H | H)

Premises used: LE.le.lt_or_eq
-- 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
H : G (↑a : ℕ) < G (↑b : ℕ)
⊢ G (↑a : ℕ) < G (↑b : ℕ)

Tactic:
H
-- 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
H : G (↑a : ℕ) < G (↑b : ℕ)
⊢ G (↑a : ℕ) < G (↑b : ℕ)

Tactic:
exact H
-- 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
H : G (↑a : ℕ) = G (↑b : ℕ)
⊢ (↑a : ℕ) ≠ (↑b : ℕ)

Tactic:
have A : (a : ℕ) ≠ b := Fin.val_injective.ne a_ne_b
-- 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_1 : 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_1)
N_mem : N ∈ univ \ A_1
Inf_eq_N : sInf (univ \ A_1) = 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
H : G (↑a : ℕ) = G (↑b : ℕ)
A : (↑a : ℕ) ≠ (↑b : ℕ)
⊢ p.color (G (↑b : ℕ)) ≠ p.color (G (↑b : ℕ))

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

Premises used: Fin.isLt, Fin.val, Nat.lt_succ_iff, Iff.mp
-- 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_1 : 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_1)
N_mem : N ∈ univ \ A_1
Inf_eq_N : sInf (univ \ A_1) = 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
H : G (↑a : ℕ) = G (↑b : ℕ)
A : p.color (G (↑b : ℕ)) ≠ p.color (G (↑b : ℕ))
⊢ G (↑a : ℕ) < G (↑b : ℕ)

Tactic:
(A rfl).elim

Premises used: False.elim, 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_1 : 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_1)
N_mem : N ∈ univ \ A_1
Inf_eq_N : sInf (univ \ A_1) = 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
H : G (↑a : ℕ) = G (↑b : ℕ)
A : p.color (G (↑b : ℕ)) ≠ p.color (G (↑b : ℕ))
⊢ G (↑a : ℕ) < G (↑b : ℕ)

Tactic:
exact (A rfl).elim

Premises used: False.elim, 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
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 : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
⊢ G (↑a : ℕ) < G (↑b : ℕ)

Tactic:
have G_lt : G a < G b := by
          rcases G_le.lt_or_eq with (H | H); · exact H
          have A : (a : ℕ) ≠ b := Fin.val_injective.ne a_ne_b
          rw [← color_G a (Nat.lt_succ_iff.1 a.2), ← color_G b (Nat.lt_succ_iff.1 b.2), H] at A
          exact (A rfl).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
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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
G_lt : G (↑a : ℕ) < G (↑b : 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))) ∨
    p.r (p.index (G (↑b : ℕ))) ≤ dist (p.c (p.index (G (↑b : ℕ)))) (p.c (p.index (G (↑a : ℕ)))) ∧
      p.r (p.index (G (↑a : ℕ))) ≤ p.τ * p.r (p.index (G (↑b : ℕ)))

Tactic:
Or.inl (Gab a b G_lt)

Premises used: Or.inl
-- 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a b : Fin N.succ
a_ne_b : a ≠ b
G_le : G (↑a : ℕ) ≤ G (↑b : ℕ)
G_lt : G (↑a : ℕ) < G (↑b : 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))) ∨
    p.r (p.index (G (↑b : ℕ))) ≤ dist (p.c (p.index (G (↑b : ℕ)))) (p.c (p.index (G (↑a : ℕ)))) ∧
      p.r (p.index (G (↑a : ℕ))) ≤ p.τ * p.r (p.index (G (↑b : ℕ)))

Tactic:
exact Or.inl (Gab a b G_lt)

Premises used: Or.inl
-- 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a : Fin (N + 1)
ha : a < last N
⊢ (↑a : ℕ) < N

Tactic:
have I : (a : ℕ) < N := ha
-- 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a : Fin (N + 1)
ha : a < last N
I : (↑a : ℕ) < N
⊢ G (↑a : ℕ) < G (↑(last N) : ℕ)

Tactic:
simp [G, I.ne, (hg a I).1]

Premises used: Fin.val, And.left, 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
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 : ℕ) < G (↑b : 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)))
a : Fin (N + 1)
ha : a < last N
I : (↑a : ℕ) < N
⊢ G (↑a : ℕ) < G (↑(last N) : ℕ)

Tactic:
have : G a < G (Fin.last N) := by simp [G, I.ne, (hg a I).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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a : Fin (N + 1)
ha : a < last N
I : (↑a : ℕ) < N
this : G (↑a : ℕ) < G (↑(last N) : ℕ)
⊢ p.r (p.index (G (↑a : ℕ))) ≤ dist (p.c (p.index (G (↑a : ℕ)))) (p.c (p.index (G (↑(last N) : ℕ)))) ∧
    p.r (p.index (G (↑(last N) : ℕ))) ≤ p.τ * p.r (p.index (G (↑a : ℕ)))

Tactic:
exact Gab _ _ 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))
Gab :
  ∀ (a b : Fin N.succ),
    G (↑a : ℕ) < G (↑b : 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)))
a : Fin (N + 1)
ha : a < last N
I : (↑a : ℕ) < N
dsimp : G N = i
⊢ G (↑(last N) : ℕ) = i

Tactic:
dsimp