-- 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 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)
I3 : cylinder y (firstDiff x y) = cylinder Ay.some (firstDiff x y)
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
have : cylinder Ax.some (firstDiff x y) = cylinder Ay.some (firstDiff x y) := by
  rw [I1, I2, I3]

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)
I3 : cylinder y (firstDiff x y) = cylinder Ay.some (firstDiff x y)
this : cylinder Ax.some (firstDiff x y) = cylinder Ay.some (firstDiff x y)
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

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 : 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 x y ≤ firstDiff x (f y)

Tactic:
have I : cylinder A.some (firstDiff x y) = cylinder y (firstDiff x y) := by
  rw [← mem_cylinder_iff_eq, firstDiff_comm]
  apply cylinder_anti y _ A.some_mem.2
  exact firstDiff_le_longestPrefix hs ys xs

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 : 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
I : cylinder A.some (firstDiff x y) = cylinder y (firstDiff x y)
⊢ firstDiff x y ≤ firstDiff 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 : 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

Tactic:
exact firstDiff_le_longestPrefix hs ys xs

Result:
no goals

Proof state:
case refine_1.h₂
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
⊢ s ⊆ range f

Tactic:
intro x hx

Result:
case refine_1.h₂
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
hx : x ∈ s
⊢ x ∈ range f

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

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

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

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
⊢ cylinder A.some (firstDiff x y) = cylinder y (firstDiff x y)

Tactic:
rw [← mem_cylinder_iff_eq, firstDiff_comm]

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
⊢ A.some ∈ cylinder y (firstDiff y x)

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
⊢ f y = A.some

Tactic:
simp_rw [f, if_neg ys]

Result:
no goals

Proof state:
case refine_2.inr.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 y : (n : ℕ) → E n
hxy : x ≠ y
h' : f x = f y
⊢ dist (f x) (f y) ≤ dist x y

Tactic:
simp [h', dist_nonneg]

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
y : (n : ℕ) → E n
hy : y ∉ s
⊢ f y ∈ s

Tactic:
simpa [f, if_neg hy] using (inter_cylinder_longestPrefix_nonempty hs hne y).choose_spec.1

Result:
no goals