l3lab/ntp-mathlib huggingface dataset:
/-- 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
  have l_eq : longestPrefix y s = longestPrefix x 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
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
rcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)
/-- 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
  have l_eq : longestPrefix y s = longestPrefix x s := by
    rcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)
    · 

Proof state:
case inl
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
L : longestPrefix y s < longestPrefix x s
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
have Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne x
/-- 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
  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
      

Proof state:
case inl
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
L : longestPrefix y s < longestPrefix x s
Ax : Set.Nonempty (s ∩ cylinder x (longestPrefix x s))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
have Z := disjoint_cylinder_of_longestPrefix_lt hs ys L
/-- 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
  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
      

Proof state:
case inl
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
L : longestPrefix y s < longestPrefix x s
Ax : Set.Nonempty (s ∩ cylinder x (longestPrefix x s))
Z : Disjoint s (cylinder y (longestPrefix x s))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
rw [firstDiff_comm] at H
/-- 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
  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
      

Proof state:
case inl
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 y x
xs : x ∉ s
ys : y ∉ s
L : longestPrefix y s < longestPrefix x s
Ax : Set.Nonempty (s ∩ cylinder x (longestPrefix x s))
Z : Disjoint s (cylinder y (longestPrefix x s))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H.le] at Z
/-- 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
  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
      

Proof state:
case inl
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 y x
xs : x ∉ s
ys : y ∉ s
L : longestPrefix y s < longestPrefix x s
Ax : Set.Nonempty (s ∩ cylinder x (longestPrefix x s))
Z : Disjoint s (cylinder x (longestPrefix x s))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
exact (Ax.not_disjoint Z).elim
/-- 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
  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
    · 

Proof state:
case inr.inl
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
L : longestPrefix y s = longestPrefix x s
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
exact L
/-- 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
  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
    · 

Proof state:
case inr.inr
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
L : longestPrefix x s < longestPrefix y s
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
have Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne y
/-- 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
  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
      

Proof state:
case inr.inr
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
L : longestPrefix x s < longestPrefix y s
Ay : Set.Nonempty (s ∩ cylinder y (longestPrefix y s))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
have A'y : (s ∩ cylinder y (longestPrefix x s).succ).Nonempty :=
        Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))
/-- 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
  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))
      

Proof state:
case inr.inr
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
L : longestPrefix x s < longestPrefix y s
Ay : Set.Nonempty (s ∩ cylinder y (longestPrefix y s))
A'y : Set.Nonempty (s ∩ cylinder y (Nat.succ (longestPrefix x s)))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
have Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)
/-- 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
  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 _)
      

Proof state:
case inr.inr
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
L : longestPrefix x s < longestPrefix y s
Ay : Set.Nonempty (s ∩ cylinder y (longestPrefix y s))
A'y : Set.Nonempty (s ∩ cylinder y (Nat.succ (longestPrefix x s)))
Z : Disjoint s (cylinder x (Nat.succ (longestPrefix x s)))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H] at Z
/-- 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
  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
      

Proof state:
case inr.inr
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
L : longestPrefix x s < longestPrefix y s
Ay : Set.Nonempty (s ∩ cylinder y (longestPrefix y s))
A'y : Set.Nonempty (s ∩ cylinder y (Nat.succ (longestPrefix x s)))
Z : Disjoint s (cylinder y (Nat.succ (longestPrefix x s)))
⊢ longestPrefix y s = longestPrefix x s

Next tactic:
exact (A'y.not_disjoint Z).elim
/-- 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
  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
  

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
l_eq : longestPrefix y s = longestPrefix x s
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)

Next tactic:
rw [l_eq, ← mem_cylinder_iff_eq]
/-- 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
  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
  rw [l_eq, ← mem_cylinder_iff_eq]
  

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
l_eq : longestPrefix y s = longestPrefix x s
⊢ x ∈ cylinder y (longestPrefix x s)

Next tactic:
exact cylinder_anti y H.le (mem_cylinder_firstDiff x y)
/-- Given a closed nonempty subset `s` of `Π (n : ℕ), E n`, there exists a Lipschitz retraction
onto this set, i.e., a Lipschitz map with range equal to `s`, equal to the identity on `s`. -/
theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) :
    ∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f := by
  /- The map `f` is defined as follows. For `x ∈ s`, let `f x = x`. Otherwise, consider the longest
    prefix `w` that `x` shares with an element of `s`, and let `f x = z_w` where `z_w` is an element
    of `s` starting with `w`. All the desired properties are clear, except the fact that `f` is
    `1`-Lipschitz: if two points `x, y` belong to a common cylinder of length `n`, one should show
    that their images also belong to a common cylinder of length `n`. This is a case analysis:
    * if both `x, y ∈ s`, then this is clear.
    * if `x ∈ s` but `y ∉ s`, then the longest prefix `w` of `y` shared by an element of `s` is of
    length at least `n` (because of `x`), and then `f y` starts with `w` and therefore stays in the
    same length `n` cylinder.
    * if `x ∉ s`, `y ∉ s`, let `w` be the longest prefix of `x` shared by an element of `s`. If its
    length is `< n`, then it is also the longest prefix of `y`, and we get `f x = f y = z_w`.
    Otherwise, `f x` remains in the same `n`-cylinder as `x`. Similarly for `y`. Finally, `f x` and
    `f y` are again in the same `n`-cylinder, as desired. -/
  

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
⊢ ∃ f, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f

Next tactic:
set f := fun x => if x ∈ s then x else (inter_cylinder_longestPrefix_nonempty hs hne x).some