awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y
⊢ cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)

Tactic:
cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H xs ys

Premises used: PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y
⊢ cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)

Tactic:
exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H xs ys

Premises used: PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) y s < firstDiff (E := E) x y
inr : cylinder (E := E) y (longestPrefix (E := E) y s) = cylinder (E := E) x (longestPrefix (E := E) x s)
⊢ cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)

Tactic:
symm
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) y s < firstDiff (E := E) x y
⊢ longestPrefix (E := E) y s < firstDiff (E := E) y x

Tactic:
rw [firstDiff_comm] at H

Premises used: PiNat.firstDiff_comm
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) y s < firstDiff (E := E) y x
⊢ cylinder (E := E) y (longestPrefix (E := E) y s) = cylinder (E := E) x (longestPrefix (E := E) x s)

Tactic:
cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs

Premises used: PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) y s < firstDiff (E := E) y x
⊢ cylinder (E := E) y (longestPrefix (E := E) y s) = cylinder (E := E) x (longestPrefix (E := E) x s)

Tactic:
exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs

Premises used: PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
⊢ cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)

Tactic:
have : cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s) := by
            rcases H with H | H
            · exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H xs ys
            · symm
              rw [firstDiff_comm] at H
              exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
⊢ f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))

Tactic:
rw [fx, fy] at hfxfy
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
fx₁ : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
⊢ (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty

Tactic:
rw [fx, fy] at hfxfy
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
fx₁ : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay₁ : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
⊢ Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ≠
    Ay₁.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))

Tactic:
rw [fx, fy] at hfxfy
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
fx₁ : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay₁ : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
hfxfy₁ :
  Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ≠
    Ay₁.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
⊢ f y = Ay₁.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))

Tactic:
rw [fx, fy] at hfxfy
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
fx₁ : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay₁ : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
hfxfy₁ :
  Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ≠
    Ay₁.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
fy₁ : f y = Ay₁.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
⊢ longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y

Tactic:
rw [fx, fy] at hfxfy
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
fx₁ : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay₁ : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
hfxfy₁ :
  Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ≠
    Ay₁.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
fy₁ : f y = Ay₁.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H₁ : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
⊢ cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)

Tactic:
rw [fx, fy] at hfxfy
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
hfxfy :
  Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ≠
    Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
apply :
  Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) =
    Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
⊢ firstDiff (E := E) x y ≤ firstDiff (E := E) (f x) (f y)

Tactic:
apply (hfxfy _).elim

Premises used: False.elim
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
hfxfy :
  Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ≠
    Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : longestPrefix (E := E) x s < firstDiff (E := E) x y ∨ longestPrefix (E := E) y s < firstDiff (E := E) x y
this : cylinder (E := E) x (longestPrefix (E := E) x s) = cylinder (E := E) y (longestPrefix (E := E) y s)
⊢ Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) =
    Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))

Tactic:
congr
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : firstDiff (E := E) x y ≤ longestPrefix (E := E) x s ∧ firstDiff (E := E) x y ≤ longestPrefix (E := E) y s
rw : Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ∈ cylinder (E := E) x (firstDiff (E := E) x y)
⊢ cylinder (E := E) (Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))) (firstDiff (E := E) x y) =
    cylinder (E := E) x (firstDiff (E := E) x y)

Tactic:
rw [← mem_cylinder_iff_eq]

Premises used: PiNat.mem_cylinder_iff_eq
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : firstDiff (E := E) x y ≤ longestPrefix (E := E) x s ∧ firstDiff (E := E) x y ≤ longestPrefix (E := E) y s
⊢ Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ∈ cylinder (E := E) x (firstDiff (E := E) x y)

Tactic:
cylinder_anti x H.1 Ax.some_mem.2

Premises used: Set.Nonempty.some_mem, And.right, And.left, PiNat.cylinder_anti
-- Module: Mathlib.Topology.MetricSpace.PiNat
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : (x : (n : ℕ) → E n) → (n : ℕ) → E n :=
  fun (x : (n : ℕ) → E n) ↦ if x ∈ s then x else ⋯.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder (E := E) x (firstDiff (E := E) x y) = cylinder (E := E) y (firstDiff (E := E) x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)).Nonempty
fx : f x = Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))
Ay : (s ∩ cylinder (E := E) y (longestPrefix (E := E) y s)).Nonempty
fy : f y = Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))
H : firstDiff (E := E) x y ≤ longestPrefix (E := E) x s ∧ firstDiff (E := E) x y ≤ longestPrefix (E := E) y s
⊢ Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s)) ∈ cylinder (E := E) x (firstDiff (E := E) x y)

Tactic:
exact cylinder_anti x H.1 Ax.some_mem.2

Premises used: Set.Nonempty.some_mem, And.right, And.left, PiNat.cylinder_anti