l3lab/ntp-mathlib huggingface dataset:
theorem inter' (i : Fin N.succ) : dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N) := by
  rcases lt_or_le i (last N) with (H | H)
  · exact a.inter i H
  · 

Proof state:
case inr
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
H : last N ≤ i
⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N)

Next tactic:
have I : i = last N := top_le_iff.1 H
theorem inter' (i : Fin N.succ) : dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N) := by
  rcases lt_or_le i (last N) with (H | H)
  · exact a.inter i H
  · have I : i = last N := top_le_iff.1 H
    

Proof state:
case inr
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
H : last N ≤ i
I : i = last N
⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N)

Next tactic:
have := (a.rpos (last N)).le
theorem inter' (i : Fin N.succ) : dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N) := by
  rcases lt_or_le i (last N) with (H | H)
  · exact a.inter i H
  · have I : i = last N := top_le_iff.1 H
    have := (a.rpos (last N)).le
    

Proof state:
case inr
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
H : last N ≤ i
I : i = last N
this : 0 ≤ r a (last N)
⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N)

Next tactic:
simp only [I, add_nonneg this this, dist_self]
theorem hlast' (i : Fin N.succ) (h : 1 ≤ τ) : a.r (last N) ≤ τ * a.r i := by
  

Proof state:
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
h : 1 ≤ τ
⊢ r a (last N) ≤ τ * r a i

Next tactic:
rcases lt_or_le i (last N) with (H | H)
theorem hlast' (i : Fin N.succ) (h : 1 ≤ τ) : a.r (last N) ≤ τ * a.r i := by
  rcases lt_or_le i (last N) with (H | H)
  · 

Proof state:
case inl
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
h : 1 ≤ τ
H : i < last N
⊢ r a (last N) ≤ τ * r a i

Next tactic:
exact (a.hlast i H).2
theorem hlast' (i : Fin N.succ) (h : 1 ≤ τ) : a.r (last N) ≤ τ * a.r i := by
  rcases lt_or_le i (last N) with (H | H)
  · exact (a.hlast i H).2
  · 

Proof state:
case inr
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
h : 1 ≤ τ
H : last N ≤ i
⊢ r a (last N) ≤ τ * r a i

Next tactic:
have : i = last N := top_le_iff.1 H
theorem hlast' (i : Fin N.succ) (h : 1 ≤ τ) : a.r (last N) ≤ τ * a.r i := by
  rcases lt_or_le i (last N) with (H | H)
  · exact (a.hlast i H).2
  · have : i = last N := top_le_iff.1 H
    

Proof state:
case inr
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
h : 1 ≤ τ
H : last N ≤ i
this : i = last N
⊢ r a (last N) ≤ τ * r a i

Next tactic:
rw [this]
theorem hlast' (i : Fin N.succ) (h : 1 ≤ τ) : a.r (last N) ≤ τ * a.r i := by
  rcases lt_or_le i (last N) with (H | H)
  · exact (a.hlast i H).2
  · have : i = last N := top_le_iff.1 H
    rw [this]
    

Proof state:
case inr
α : Type u_1
inst✝ : MetricSpace α
N : ℕ
τ : ℝ
a : SatelliteConfig α N τ
i : Fin (Nat.succ N)
h : 1 ≤ τ
H : last N ≤ i
this : i = last N
⊢ r a (last N) ≤ τ * r a (last N)

Next tactic:
exact le_mul_of_one_le_left (a.rpos _).le h
/-- Choose inductively large balls with centers that are not contained in the union of already
chosen balls. This is a transfinite induction. -/
noncomputable def index : Ordinal.{u} → β
  | i =>
      -- `Z` is the set of points that are covered by already constructed balls
      let Z := ⋃ j : { j // j < i }, ball (p.c (index j)) (p.r (index j))
      -- `R` is the supremum of the radii of balls with centers not in `Z`
      let R := iSup fun b : { b : β // p.c b ∉ Z } => p.r b
      -- return an index `b` for which the center `c b` is not in `Z`, and the radius is at
      -- least `R / τ`, if such an index exists (and garbage otherwise).
      Classical.epsilon fun b : β => p.c b ∉ Z ∧ R ≤ p.τ * p.r b
  termination_by index i => i
  decreasing_by 

Proof state:
a✝ : Ordinal.{u}
i : Ordinal.{u} := a✝
j : { j // j < i }
⊢ (invImage (fun a => a) Ordinal.wellFoundedRelation).1 (↑j) a✝

Next tactic:
exact j.2
theorem monotone_iUnionUpTo : Monotone p.iUnionUpTo := by
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
⊢ Monotone (iUnionUpTo p)

Next tactic:
intro i j hij
theorem monotone_iUnionUpTo : Monotone p.iUnionUpTo := by
  intro i j hij
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i j : Ordinal.{u}
hij : i ≤ j
⊢ iUnionUpTo p i ≤ iUnionUpTo p j

Next tactic:
simp only [iUnionUpTo]
theorem monotone_iUnionUpTo : Monotone p.iUnionUpTo := by
  intro i j hij
  simp only [iUnionUpTo]
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i j : Ordinal.{u}
hij : i ≤ j
⊢ ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ≤
    ⋃ j_1, ball (BallPackage.c p.toBallPackage (index p ↑j_1)) (BallPackage.r p.toBallPackage (index p ↑j_1))

Next tactic:
exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩
/-- Group the balls into disjoint families, by assigning to a ball the smallest color for which
it does not intersect any already chosen ball of this color. -/
noncomputable def color : Ordinal.{u} → ℕ
  | i =>
    let A : Set ℕ :=
      ⋃ (j : { j // j < i })
        (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩
          closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty), {color j}
    sInf (univ \ A)
  termination_by color i => i
  decreasing_by 

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
a✝ : Ordinal.{u}
i : Ordinal.{u} := a✝
j : { j // j < i }
x✝ :
  Set.Nonempty
    (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩
      closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))
⊢ (invImage (fun a => a) Ordinal.wellFoundedRelation).1 (↑j) a✝

Next tactic:
exact j.2
theorem lastStep_nonempty :
    {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty := by
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
⊢ Set.Nonempty
    {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}

Next tactic:
by_contra h
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
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
⊢ False

Next tactic:
suffices H : Function.Injective p.index
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; 

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
H : Function.Injective (index p)
⊢ False
case H
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
⊢ Function.Injective (index p)

Next tactic:
exact not_injective_of_ordinal p.index H
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
  

Proof state:
case H
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
⊢ Function.Injective (index p)

Next tactic:
intro x y hxy
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
  

Proof state:
case H
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
x y : Ordinal.{u}
hxy : index p x = index p y
⊢ x = y

Next tactic:
wlog x_le_y : x ≤ y generalizing x y
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
  · 

Proof state:
case H.inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
x y : Ordinal.{u}
hxy : index p x = index p y
this : ∀ ⦃x y : Ordinal.{u}⦄, index p x = index p y → x ≤ y → x = y
x_le_y : ¬x ≤ y
⊢ x = y

Next tactic:
exact (this hxy.symm (le_of_not_le x_le_y)).symm
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
  

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
x y : Ordinal.{u}
hxy : index p x = index p y
x_le_y : x ≤ y
⊢ x = y

Next tactic:
rcases eq_or_lt_of_le x_le_y with (rfl | H)
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); · 

Proof state:
case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h :
  ¬Set.Nonempty
      {i | ¬∃ b, BallPackage.c p.toBallPackage b ∉ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b}
x : Ordinal.{u}
hxy : index p x = index p x
x_le_y : x ≤ x
⊢ x = x

Next tactic:
rfl
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
  

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

Next tactic:
simp only [nonempty_def, not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq,
    not_forall] at h
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
  

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 : Ordinal.{u}),
    ∃ x_1, BallPackage.c p.toBallPackage x_1 ∉ iUnionUpTo p x ∧ R p x ≤ p.τ * BallPackage.r p.toBallPackage x_1
⊢ x = y

Next tactic:
specialize h y
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
  

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
⊢ x = y

Next tactic:
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
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
    

Proof state:
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
⊢ BallPackage.c p.toBallPackage (index p y) ∉ iUnionUpTo p y

Next tactic:
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
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
      

Proof state:
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
⊢ index p y =
    Classical.epsilon fun b =>
      BallPackage.c p.toBallPackage b ∉ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b

Next tactic:
rw [TauPackage.index]
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]; 

Proof state:
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
⊢ (Classical.epsilon fun b =>
      BallPackage.c p.toBallPackage b ∉
          ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∧
        ⨆ b, BallPackage.r p.toBallPackage ↑b ≤ p.τ * BallPackage.r p.toBallPackage b) =
    Classical.epsilon fun b =>
      BallPackage.c p.toBallPackage b ∉ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b

Next tactic:
rfl
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
    

Proof state:
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
this :
  index p y =
    Classical.epsilon fun b =>
      BallPackage.c p.toBallPackage b ∉ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b
⊢ BallPackage.c p.toBallPackage (index p y) ∉ iUnionUpTo p y

Next tactic:
rw [this]
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]
    

Proof state:
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
this :
  index p y =
    Classical.epsilon fun b =>
      BallPackage.c p.toBallPackage b ∉ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b
⊢ BallPackage.c p.toBallPackage
      (Classical.epsilon fun b =>
        BallPackage.c p.toBallPackage b ∉ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b) ∉
    iUnionUpTo p y

Next tactic:
exact (Classical.epsilon_spec h).1
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
  

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) ∉ iUnionUpTo p y
⊢ x = y

Next tactic:
simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,
    Subtype.exists, Subtype.coe_mk] 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
  

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 :
  ∀ x < y,
    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:
specialize A x H
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
  

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? [hxy] at A says simp only [hxy, mem_ball, dist_self, not_lt] at A