-- Mathlib/Topology/MetricSpace/PiNat.lean
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

Proof state:
case refine_2
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
⊢ dist (f x) (f y) ≤ dist x y

Tactic:
rcases eq_or_ne x y with (rfl | hxy)

Result:
case refine_2.inl
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x : (n : ℕ) → E n
⊢ dist (f x) (f x) ≤ dist x x

case refine_2.inr
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
⊢ dist (f x) (f y) ≤ dist x y

Proof state:
case refine_2.inl
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x : (n : ℕ) → E n
⊢ dist (f x) (f x) ≤ dist x x

Tactic:
simp

Result:
no goals

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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : firstDiff x y ≤ longestPrefix x s ∧ firstDiff x y ≤ longestPrefix y s
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
have I1 : cylinder Ax.some (firstDiff x y) = cylinder x (firstDiff x y) := by
  rw [← mem_cylinder_iff_eq]
  exact cylinder_anti x H.1 Ax.some_mem.2

Result:
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : firstDiff x y ≤ longestPrefix x s ∧ firstDiff x y ≤ longestPrefix y s
I1 : cylinder Ax.some (firstDiff x y) = cylinder x (firstDiff x y)
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
x : (n : ℕ) → E n
xs : x ∈ s
⊢ f x = x

Tactic:
simp [f, xs]

Result:
no goals

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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
have Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty :=
  inter_cylinder_longestPrefix_nonempty hs hne y

Result:
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
⊢ cylinder x (firstDiff x y) = cylinder y (firstDiff x y)

Tactic:
rw [← mem_cylinder_iff_eq]

Result:
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
⊢ x ∈ cylinder y (firstDiff x y)

Proof state:
case inr
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : longestPrefix y s < firstDiff y x
⊢ cylinder y (longestPrefix y s) = cylinder x (longestPrefix x s)

Tactic:
exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs

Result:
no goals

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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∈ s
ys : y ∉ s
A : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = A.some
⊢ A.some ∈ cylinder y (firstDiff y x)

Tactic:
apply cylinder_anti y _ A.some_mem.2

Result:
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 : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∈ s
ys : y ∉ s
A : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = A.some
⊢ firstDiff y x ≤ longestPrefix y s