Theorem: Vitali.exists_disjoint_subfamily_covering_enlargment_closedBall
File: Mathlib/MeasureTheory/Covering/Vitali.lean

Proof steps:
state_before: α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case inl
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
x : ι → α
r : ι → ℝ
R τ : ℝ
hτ : 3 < τ
hr : ∀ a ∈ ∅, r a ≤ R
⊢ ∃ u ⊆ ∅,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ ∅, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

case inr
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: rcases eq_empty_or_nonempty t with (rfl | _)
state_before: case inr
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case pos
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∀ a ∈ t, r a < 0
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ¬∀ a ∈ t, r a < 0
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: by_cases ht : ∀ a ∈ t, r a < 0
state_before: case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ¬∀ a ∈ t, r a < 0
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: push_neg at ht
state_before: case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: let t' := { a ∈ t | 0 ≤ r a }
state_before: case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r
      ((τ - 1) / 2) (by linarith) (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
state_before: case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b) := by
    intro a ha
    rcases hu a ha with ⟨b, bu, hb, rb⟩
    refine ⟨b, bu, ?_⟩
    have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb
    apply closedBall_subset_closedBall'
    linarith
state_before: case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
⊢ ∃ u ⊆ t,
    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: refine ⟨u, ut'.trans fun a ha => ha.1, u_disj, fun a ha => ?_⟩
state_before: case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
state_after: case neg.intro.intro.intro.inl
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : 0 ≤ r a
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

case neg.intro.intro.intro.inr
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a < 0
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
tactic: rcases le_or_lt 0 (r a) with (h'a | h'a)