l3lab/ntp-mathlib huggingface dataset:
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    

Proof state:
case inr
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
⊢ dist (g x) (g y) ≤ 4 * dist x y

Next tactic:
have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [le_zero_iff.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      rw [pow_zero]
      exact hxy
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      

Proof state:
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
⊢ 0 < firstDiff ↑x ↑y

Next tactic:
by_contra! h
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      

Proof state:
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
h : firstDiff ↑x ↑y ≤ 0
⊢ False

Next tactic:
apply apply_firstDiff_ne hne'
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      

Proof state:
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
h : firstDiff ↑x ↑y ≤ 0
⊢ ↑x (firstDiff ↑x ↑y) = ↑y (firstDiff ↑x ↑y)

Next tactic:
rw [le_zero_iff.1 h]
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [le_zero_iff.1 h]
      

Proof state:
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
h : firstDiff ↑x ↑y ≤ 0
⊢ ↑x 0 = ↑y 0

Next tactic:
apply apply_eq_of_dist_lt _ le_rfl
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [le_zero_iff.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      

Proof state:
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
h : firstDiff ↑x ↑y ≤ 0
⊢ dist ↑x ↑y < (1 / 2) ^ 0

Next tactic:
rw [pow_zero]
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [le_zero_iff.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      rw [pow_zero]
      

Proof state:
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
h : firstDiff ↑x ↑y ≤ 0
⊢ dist ↑x ↑y < 1

Next tactic:
exact hxy
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [le_zero_iff.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      rw [pow_zero]
      exact hxy
    

Proof state:
case inr
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
diff_pos : 0 < firstDiff ↑x ↑y
⊢ dist (g x) (g y) ≤ 4 * dist x y

Next tactic:
have hn : firstDiff x.1 y.1 = n + 1 := (Nat.succ_pred_eq_of_pos diff_pos).symm
/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
    (mem_iInter.1 x.2.some_mem n : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [le_zero_iff.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      rw [pow_zero]
      exact hxy
    have hn : firstDiff x.1 y.1 = n + 1 := (Nat.succ_pred_eq_of_pos diff_pos).symm
    

Proof state:
case inr
E : ℕ → Type u_1
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
diff_pos : 0 < firstDiff ↑x ↑y
hn : firstDiff ↑x ↑y = n + 1
⊢ dist (g x) (g y) ≤ 4 * dist x y

Next tactic:
rw [dist', dist_eq_of_ne hne', hn]