l3lab/ntp-mathlib huggingface dataset:
theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) := by
  rcases eq_empty_or_nonempty s with (rfl | hne)
  · exact ⟨0, by simp⟩
  have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx
  obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one
  refine' ⟨n, disjoint_left.2 fun y ys hy => ?_⟩
  

Proof state:
case inr.intro
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
hne : Set.Nonempty s
A : 0 < infDist x s
n : ℕ
hn : (1 / 2) ^ n < infDist x s
y : (n : ℕ) → E n
ys : y ∈ s
hy : y ∈ cylinder x n
⊢ False

Next tactic:
apply lt_irrefl (infDist x s)
theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) := by
  rcases eq_empty_or_nonempty s with (rfl | hne)
  · exact ⟨0, by simp⟩
  have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx
  obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one
  refine' ⟨n, disjoint_left.2 fun y ys hy => ?_⟩
  apply lt_irrefl (infDist x s)
  

Proof state:
case inr.intro
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
hne : Set.Nonempty s
A : 0 < infDist x s
n : ℕ
hn : (1 / 2) ^ n < infDist x s
y : (n : ℕ) → E n
ys : y ∈ s
hy : y ∈ cylinder x n
⊢ infDist x s < infDist x s

Next tactic:
calc
    infDist x s ≤ dist x y := infDist_le_dist_of_mem ys
    _ ≤ (1 / 2) ^ n := by
      rw [mem_cylinder_comm] at hy
      exact mem_cylinder_iff_dist_le.1 hy
    _ < infDist x s := hn
theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) := by
  rcases eq_empty_or_nonempty s with (rfl | hne)
  · exact ⟨0, by simp⟩
  have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx
  obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one
  refine' ⟨n, disjoint_left.2 fun y ys hy => ?_⟩
  apply lt_irrefl (infDist x s)
  calc
    infDist x s ≤ dist x y := infDist_le_dist_of_mem ys
    _ ≤ (1 / 2) ^ n := by
      

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
hne : Set.Nonempty s
A : 0 < infDist x s
n : ℕ
hn : (1 / 2) ^ n < infDist x s
y : (n : ℕ) → E n
ys : y ∈ s
hy : y ∈ cylinder x n
⊢ dist x y ≤ (1 / 2) ^ n

Next tactic:
rw [mem_cylinder_comm] at hy
theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) := by
  rcases eq_empty_or_nonempty s with (rfl | hne)
  · exact ⟨0, by simp⟩
  have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx
  obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one
  refine' ⟨n, disjoint_left.2 fun y ys hy => ?_⟩
  apply lt_irrefl (infDist x s)
  calc
    infDist x s ≤ dist x y := infDist_le_dist_of_mem ys
    _ ≤ (1 / 2) ^ n := by
      rw [mem_cylinder_comm] at hy
      

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
hne : Set.Nonempty s
A : 0 < infDist x s
n : ℕ
hn : (1 / 2) ^ n < infDist x s
y : (n : ℕ) → E n
ys : y ∈ s
hy : x ∈ cylinder y n
⊢ dist x y ≤ (1 / 2) ^ n

Next tactic:
exact mem_cylinder_iff_dist_le.1 hy
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
⊢ firstDiff x y < shortestPrefixDiff x s

Next tactic:
have A := exists_disjoint_cylinder hs hx
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  have A := exists_disjoint_cylinder hs hx
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
⊢ firstDiff x y < shortestPrefixDiff x s

Next tactic:
rw [shortestPrefixDiff, dif_pos A]
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  have A := exists_disjoint_cylinder hs hx
  rw [shortestPrefixDiff, dif_pos A]
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
⊢ firstDiff x y < Nat.find A

Next tactic:
have B := Nat.find_spec A
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  have A := exists_disjoint_cylinder hs hx
  rw [shortestPrefixDiff, dif_pos A]
  have B := Nat.find_spec A
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
B : Disjoint s (cylinder x (Nat.find A))
⊢ firstDiff x y < Nat.find A

Next tactic:
contrapose! B
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  have A := exists_disjoint_cylinder hs hx
  rw [shortestPrefixDiff, dif_pos A]
  have B := Nat.find_spec A
  contrapose! B
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A ≤ firstDiff x y
⊢ ¬Disjoint s (cylinder x (Nat.find A))

Next tactic:
rw [not_disjoint_iff_nonempty_inter]
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  have A := exists_disjoint_cylinder hs hx
  rw [shortestPrefixDiff, dif_pos A]
  have B := Nat.find_spec A
  contrapose! B
  rw [not_disjoint_iff_nonempty_inter]
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A ≤ firstDiff x y
⊢ Set.Nonempty (s ∩ cylinder x (Nat.find A))

Next tactic:
refine' ⟨y, hy, _⟩
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  have A := exists_disjoint_cylinder hs hx
  rw [shortestPrefixDiff, dif_pos A]
  have B := Nat.find_spec A
  contrapose! B
  rw [not_disjoint_iff_nonempty_inter]
  refine' ⟨y, hy, _⟩
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A ≤ firstDiff x y
⊢ y ∈ cylinder x (Nat.find A)

Next tactic:
rw [mem_cylinder_comm]
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by
  have A := exists_disjoint_cylinder hs hx
  rw [shortestPrefixDiff, dif_pos A]
  have B := Nat.find_spec A
  contrapose! B
  rw [not_disjoint_iff_nonempty_inter]
  refine' ⟨y, hy, _⟩
  rw [mem_cylinder_comm]
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A ≤ firstDiff x y
⊢ x ∈ cylinder y (Nat.find A)

Next tactic:
exact cylinder_anti y B (mem_cylinder_firstDiff x y)
theorem shortestPrefixDiff_pos {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty)
    {x : ∀ n, E n} (hx : x ∉ s) : 0 < shortestPrefixDiff x s := by
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
⊢ 0 < shortestPrefixDiff x s

Next tactic:
rcases hne with ⟨y, hy⟩
theorem shortestPrefixDiff_pos {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty)
    {x : ∀ n, E n} (hx : x ∉ s) : 0 < shortestPrefixDiff x s := by
  rcases hne with ⟨y, hy⟩
  

Proof state:
case intro
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
y : (n : ℕ) → E n
hy : y ∈ s
⊢ 0 < shortestPrefixDiff x s

Next tactic:
exact (zero_le _).trans_lt (firstDiff_lt_shortestPrefixDiff hs hx hy)
theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y ≤ longestPrefix x s := by
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
⊢ firstDiff x y ≤ longestPrefix x s

Next tactic:
rw [longestPrefix, le_tsub_iff_right]
theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y ≤ longestPrefix x s := by
  rw [longestPrefix, le_tsub_iff_right]
  · 

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
⊢ firstDiff x y + 1 ≤ shortestPrefixDiff x s

Next tactic:
exact firstDiff_lt_shortestPrefixDiff hs hx hy
theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y ≤ longestPrefix x s := by
  rw [longestPrefix, le_tsub_iff_right]
  · exact firstDiff_lt_shortestPrefixDiff hs hx hy
  · 

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
⊢ 1 ≤ shortestPrefixDiff x s

Next tactic:
exact shortestPrefixDiff_pos hs ⟨y, hy⟩ hx
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
⊢ Set.Nonempty (s ∩ cylinder x (longestPrefix x s))

Next tactic:
by_cases hx : x ∈ s
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · 

Proof state:
case pos
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∈ s
⊢ Set.Nonempty (s ∩ cylinder x (longestPrefix x s))

Next tactic:
exact ⟨x, hx, self_mem_cylinder _ _⟩
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  

Proof state:
case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
⊢ Set.Nonempty (s ∩ cylinder x (longestPrefix x s))

Next tactic:
have A := exists_disjoint_cylinder hs hx
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  have A := exists_disjoint_cylinder hs hx
  

Proof state:
case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
A : ∃ n, Disjoint s (cylinder x n)
⊢ Set.Nonempty (s ∩ cylinder x (longestPrefix x s))

Next tactic:
have B : longestPrefix x s < shortestPrefixDiff x s :=
    Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  have A := exists_disjoint_cylinder hs hx
  have B : longestPrefix x s < shortestPrefixDiff x s :=
    Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'
  

Proof state:
case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
A : ∃ n, Disjoint s (cylinder x n)
B : longestPrefix x s < shortestPrefixDiff x s
⊢ Set.Nonempty (s ∩ cylinder x (longestPrefix x s))

Next tactic:
rw [longestPrefix, shortestPrefixDiff, dif_pos A] at B ⊢
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  have A := exists_disjoint_cylinder hs hx
  have B : longestPrefix x s < shortestPrefixDiff x s :=
    Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'
  rw [longestPrefix, shortestPrefixDiff, dif_pos A] at B ⊢
  

Proof state:
case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A - 1 < Nat.find A
⊢ Set.Nonempty (s ∩ cylinder x (Nat.find A - 1))

Next tactic:
obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) := by
    simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  have A := exists_disjoint_cylinder hs hx
  have B : longestPrefix x s < shortestPrefixDiff x s :=
    Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'
  rw [longestPrefix, shortestPrefixDiff, dif_pos A] at B ⊢
  obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) := by
    

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A - 1 < Nat.find A
⊢ ∃ y ∈ s, x ∈ cylinder y (Nat.find A - 1)

Next tactic:
simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  have A := exists_disjoint_cylinder hs hx
  have B : longestPrefix x s < shortestPrefixDiff x s :=
    Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'
  rw [longestPrefix, shortestPrefixDiff, dif_pos A] at B ⊢
  obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) := by
    simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B
  

Proof state:
case neg.intro.intro
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A - 1 < Nat.find A
y : (n : ℕ) → E n
ys : y ∈ s
hy : x ∈ cylinder y (Nat.find A - 1)
⊢ Set.Nonempty (s ∩ cylinder x (Nat.find A - 1))

Next tactic:
refine' ⟨y, ys, _⟩
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  have A := exists_disjoint_cylinder hs hx
  have B : longestPrefix x s < shortestPrefixDiff x s :=
    Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'
  rw [longestPrefix, shortestPrefixDiff, dif_pos A] at B ⊢
  obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) := by
    simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B
  refine' ⟨y, ys, _⟩
  

Proof state:
case neg.intro.intro
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A - 1 < Nat.find A
y : (n : ℕ) → E n
ys : y ∈ s
hy : x ∈ cylinder y (Nat.find A - 1)
⊢ y ∈ cylinder x (Nat.find A - 1)

Next tactic:
rw [mem_cylinder_iff_eq] at hy ⊢
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty := by
  by_cases hx : x ∈ s
  · exact ⟨x, hx, self_mem_cylinder _ _⟩
  have A := exists_disjoint_cylinder hs hx
  have B : longestPrefix x s < shortestPrefixDiff x s :=
    Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'
  rw [longestPrefix, shortestPrefixDiff, dif_pos A] at B ⊢
  obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) := by
    simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B
  refine' ⟨y, ys, _⟩
  rw [mem_cylinder_iff_eq] at hy ⊢
  

Proof state:
case neg.intro.intro
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
x : (n : ℕ) → E n
hx : x ∉ s
A : ∃ n, Disjoint s (cylinder x n)
B : Nat.find A - 1 < Nat.find A
y : (n : ℕ) → E n
ys : y ∈ s
hy : cylinder x (Nat.find A - 1) = cylinder y (Nat.find A - 1)
⊢ cylinder y (Nat.find A - 1) = cylinder x (Nat.find A - 1)

Next tactic:
rw [hy]
theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) := by
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
n : ℕ
hn : longestPrefix x s < n
⊢ Disjoint s (cylinder x n)

Next tactic:
contrapose! hn
theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) := by
  contrapose! hn
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
n : ℕ
hn : ¬Disjoint s (cylinder x n)
⊢ n ≤ longestPrefix x s

Next tactic:
rcases not_disjoint_iff_nonempty_inter.1 hn with ⟨y, ys, hy⟩
theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) := by
  contrapose! hn
  rcases not_disjoint_iff_nonempty_inter.1 hn with ⟨y, ys, hy⟩
  

Proof state:
case intro.intro
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
n : ℕ
hn : ¬Disjoint s (cylinder x n)
y : (n : ℕ) → E n
ys : y ∈ s
hy : y ∈ cylinder x n
⊢ n ≤ longestPrefix x s

Next tactic:
apply le_trans _ (firstDiff_le_longestPrefix hs hx ys)
theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) := by
  contrapose! hn
  rcases not_disjoint_iff_nonempty_inter.1 hn with ⟨y, ys, hy⟩
  apply le_trans _ (firstDiff_le_longestPrefix hs hx ys)
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
n : ℕ
hn : ¬Disjoint s (cylinder x n)
y : (n : ℕ) → E n
ys : y ∈ s
hy : y ∈ cylinder x n
⊢ n ≤ firstDiff x y

Next tactic:
apply (mem_cylinder_iff_le_firstDiff (ne_of_mem_of_not_mem ys hx).symm _).1
theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) := by
  contrapose! hn
  rcases not_disjoint_iff_nonempty_inter.1 hn with ⟨y, ys, hy⟩
  apply le_trans _ (firstDiff_le_longestPrefix hs hx ys)
  apply (mem_cylinder_iff_le_firstDiff (ne_of_mem_of_not_mem ys hx).symm _).1
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
x : (n : ℕ) → E n
hx : x ∉ s
n : ℕ
hn : ¬Disjoint s (cylinder x n)
y : (n : ℕ) → E n
ys : y ∈ s
hy : y ∈ cylinder x n
⊢ x ∈ cylinder y n

Next tactic:
rwa [mem_cylinder_comm]
/-- If two points `x, y` coincide up to length `n`, and the longest common prefix of `x` with `s`
is strictly shorter than `n`, then the longest common prefix of `y` with `s` is the same, and both
cylinders of this length based at `x` and `y` coincide. -/
theorem cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff {x y : ∀ n, E n}
    {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty)
    (H : longestPrefix x s < firstDiff x y) (xs : x ∉ s) (ys : y ∉ s) :
    cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s) := by
  

Proof state:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x y : (n : ℕ) → E n
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : Set.Nonempty s
H : longestPrefix x s < firstDiff x y
xs : x ∉ s
ys : y ∉ s
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)

Next tactic:
have l_eq : longestPrefix y s = longestPrefix x s := by
    rcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)
    · have Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne x
      have Z := disjoint_cylinder_of_longestPrefix_lt hs ys L
      rw [firstDiff_comm] at H
      rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H.le] at Z
      exact (Ax.not_disjoint Z).elim
    · exact L
    · have Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne y
      have A'y : (s ∩ cylinder y (longestPrefix x s).succ).Nonempty :=
        Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))
      have Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)
      rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H] at Z
      exact (A'y.not_disjoint Z).elim