Module: Mathlib.Topology.EMetricSpace.Paracompact

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m
h2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ i, x ∈ s i
w✝ : LinearOrder ι
wf : WellFoundedLT ι
ind : α → ι := fun x ↦ ⋯.min {i | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι}, i < ind x → x ∈ s i → False
D : ℕ → ι → Set α :=
  fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
hD :
  D = fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ x,
        ⋃ (_ : ind x = i), ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (2⁻¹ ^ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔ ∃ x, ind x = i ∧ eball x (3 * 2⁻¹ ^ n) ⊆ s i ∧ (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < 2⁻¹ ^ n
Dcov : ∀ (x : α), ∃ n i, x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
hn : x ∈ D n i
this : D n i ∈ 𝓝 x
k : ℕ
hsub : eball x (2⁻¹ ^ k) ⊆ D n i
B : Set α := eball x (2⁻¹ ^ (n + k + 1))
Hgt : ∀ (m : ℕ), n + k + 1 ≤ m → ∀ (i : ι), Disjoint (D m i) B
m : ℕ
hm : m ≤ n + k
y : α
hyB : y ∈ B
z : α
hzB : z ∈ B
y' : α
hyD : y ∈ D m (ind y')
hsuby : eball y' (3 * 2⁻¹ ^ m) ⊆ s (ind y')
hdisty : edist y y' < 2⁻¹ ^ m
z' : α
hzD : z ∈ D m (ind z')
h' : ind y' ≠ ind z'
h : ind y' < ind z'
hdistz : edist z z' < 2⁻¹ ^ m
⊢ False

Tactic:
suffices edist z' y' < 3 * 2⁻¹ ^ m from notMem_of_lt_ind h (hsuby this)

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m
h2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ i, x ∈ s i
w✝ : LinearOrder ι
wf : WellFoundedLT ι
ind : α → ι := fun x ↦ ⋯.min {i | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι}, i < ind x → x ∈ s i → False
D : ℕ → ι → Set α :=
  fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
hD :
  D = fun t ↦
    Nat.strongRec (motive := fun x ↦ ι → Set α)
      (fun n D' i ↦
        ⋃ x,
          ⋃ (_ : ind x = i),
            ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (2⁻¹ ^ n))
      t
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ x,
        ⋃ (_ : ind x = i), ⋃ (_ : eball x (3 * 2⁻¹ ^ n) ⊆ s i), ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (2⁻¹ ^ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔ ∃ x, ind x = i ∧ eball x (3 * 2⁻¹ ^ n) ⊆ s i ∧ (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < 2⁻¹ ^ n
Dcov : ∀ (x : α), ∃ n i, x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
hn : x ∈ D n i
this : D n i ∈ 𝓝 x
k : ℕ
hsub : eball x (2⁻¹ ^ k) ⊆ D n i
B : Set α := eball x (2⁻¹ ^ (n + k + 1))
Hgt : ∀ (m : ℕ), n + k + 1 ≤ m → ∀ (i : ι), Disjoint (D m i) B
m : ℕ
hm : m ≤ n + k
y : α
hyB : y ∈ B
z : α
hzB : z ∈ B
y' : α
hyD : y ∈ D m (ind y')
hsuby : eball y' (3 * 2⁻¹ ^ m) ⊆ s (ind y')
hdisty : edist y y' < 2⁻¹ ^ m
z' : α
hzD : z ∈ D m (ind z')
h' : ind y' ≠ ind z'
h : ind y' < ind z'
hdistz : edist z z' < 2⁻¹ ^ m
⊢ edist z' y' < 3 * 2⁻¹ ^ m