l3lab/ntp-mathlib huggingface dataset:
theorem lastStep_nonempty :
    {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty := by
  by_contra h
  suffices H : Function.Injective p.index; exact not_injective_of_ordinal p.index H
  intro x y hxy
  wlog x_le_y : x ≤ y generalizing x y
  · exact (this hxy.symm (le_of_not_le x_le_y)).symm
  rcases eq_or_lt_of_le x_le_y with (rfl | H); · rfl
  simp only [nonempty_def, not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq,
    not_forall] at h
  specialize h y
  have A : p.c (p.index y) ∉ p.iUnionUpTo y := by
    have :
        p.index y =
          Classical.epsilon fun b : β => p.c b ∉ p.iUnionUpTo y ∧ p.R y ≤ p.τ * p.r b := by
      rw [TauPackage.index]; rfl
    rw [this]
    exact (Classical.epsilon_spec h).1
  simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,
    Subtype.exists, Subtype.coe_mk] at A
  specialize A x H
  simp? [hxy] at A says 

Proof state:
case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x y : Ordinal.{u}
hxy : index p x = index p y
x_le_y : x ≤ y
H : x < y
h : ∃ x, BallPackage.c p.toBallPackage x ∉ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x
A :
  BallPackage.c p.toBallPackage (index p y) ∉
    ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x))
⊢ x = y

Next tactic:
simp only [hxy, mem_ball, dist_self, not_lt] at A
theorem lastStep_nonempty :
    {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty := by
  by_contra h
  suffices H : Function.Injective p.index; exact not_injective_of_ordinal p.index H
  intro x y hxy
  wlog x_le_y : x ≤ y generalizing x y
  · exact (this hxy.symm (le_of_not_le x_le_y)).symm
  rcases eq_or_lt_of_le x_le_y with (rfl | H); · rfl
  simp only [nonempty_def, not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq,
    not_forall] at h
  specialize h y
  have A : p.c (p.index y) ∉ p.iUnionUpTo y := by
    have :
        p.index y =
          Classical.epsilon fun b : β => p.c b ∉ p.iUnionUpTo y ∧ p.R y ≤ p.τ * p.r b := by
      rw [TauPackage.index]; rfl
    rw [this]
    exact (Classical.epsilon_spec h).1
  simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,
    Subtype.exists, Subtype.coe_mk] at A
  specialize A x H
  simp? [hxy] at A says simp only [hxy, mem_ball, dist_self, not_lt] at A
  

Proof state:
case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x y : Ordinal.{u}
hxy : index p x = index p y
x_le_y : x ≤ y
H : x < y
h : ∃ x, BallPackage.c p.toBallPackage x ∉ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x
A : BallPackage.r p.toBallPackage (index p y) ≤ 0
⊢ x = y

Next tactic:
exact (lt_irrefl _ ((p.rpos (p.index y)).trans_le A)).elim
/-- Every point is covered by chosen balls, before `p.lastStep`. -/
theorem mem_iUnionUpTo_lastStep (x : β) : p.c x ∈ p.iUnionUpTo p.lastStep := by
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x : β
⊢ BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p)

Next tactic:
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]
/-- 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
    

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x : β
⊢ ∀ (z : β),
    BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨
      p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p)

Next tactic:
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
/-- 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
    

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
x : β
this :
  lastStep p ∈
    {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
⊢ ∀ (z : β),
    BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨
      p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p)

Next tactic:
simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem]
/-- 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]
  

Proof state:
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)
⊢ BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p)

Next tactic:
by_contra h
/-- 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
  

Proof state:
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)
⊢ False

Next tactic:
rcases A x with (H | H)
/-- 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); · 

Proof state:
case 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 : BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p)
⊢ False

Next tactic:
exact h H
/-- 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
  

Proof state:
case 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)
⊢ False

Next tactic:
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
/-- 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
    

Proof state:
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)
⊢ 0 < R p (lastStep p)

Next tactic:
apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H
/-- 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
  

Proof state:
case 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)
⊢ False

Next tactic:
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
/-- 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
    

Proof state:
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)
⊢ p.τ⁻¹ * R p (lastStep p) < R p (lastStep p)

Next tactic:
conv_rhs => rw [← one_mul (p.R p.lastStep)]
/-- 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 => 

Proof state:
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)
| R p (lastStep p)

Next tactic:
rw [← one_mul (p.R p.lastStep)]
/-- 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)]
    

Proof state:
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)
⊢ p.τ⁻¹ * R p (lastStep p) < 1 * R p (lastStep p)

Next tactic:
exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one
/-- 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
  

Proof state:
case 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)
⊢ False

Next tactic:
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 _⟩
/-- 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
    

Proof state:
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,
    BallPackage.c p.toBallPackage y ∉ iUnionUpTo p (lastStep p) ∧
      p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y

Next tactic:
have := exists_lt_of_lt_csSup ?_ B
/-- 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
    · 

Proof state:
case refine_2
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)
this : ∃ a ∈ range fun b => BallPackage.r p.toBallPackage ↑b, p.τ⁻¹ * R p (lastStep p) < a
⊢ ∃ y,
    BallPackage.c p.toBallPackage y ∉ iUnionUpTo p (lastStep p) ∧
      p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y

Next tactic:
simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists,
      Subtype.coe_mk]
/-- 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]
    

Proof state:
case refine_1
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)
⊢ Set.Nonempty (range fun b => BallPackage.r p.toBallPackage ↑b)

Next tactic:
rw [← image_univ, nonempty_image_iff]
/-- 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]
    

Proof state:
case refine_1
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)
⊢ Set.Nonempty univ

Next tactic:
exact ⟨⟨_, h⟩, mem_univ _⟩
/-- 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 _⟩
  

Proof state:
case inr.intro.intro
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
⊢ False

Next tactic:
rcases A y with (Hy | Hy)