awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
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
I1 :
  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)
I3 :
  cylinder (E := E) y (firstDiff (E := E) x y) =
    cylinder (E := E) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (firstDiff (E := E) x y)
rw :
  cylinder (E := E) x (firstDiff (E := E) x y) =
    cylinder (E := E) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (firstDiff (E := E) x y)
rw₁ :
  cylinder (E := E) y (firstDiff (E := E) x y) =
    cylinder (E := E) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (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) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (firstDiff (E := E) x y)
have rw₂ : cylinder (E := E) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (firstDiff (E := E) x y) =
  cylinder (E := E) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (firstDiff (E := E) x y)
Tactic: rw [I1, I2, I3]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
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
I1 :
  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)
I3 :
  cylinder (E := E) y (firstDiff (E := E) x y) =
    cylinder (E := E) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (firstDiff (E := E) x y)
⊢ firstDiff (E := E) x y ≤ firstDiff (E := E) (f x) (f y)
have this : cylinder (E := E) (Ax.some (s := s ∩ cylinder (E := E) x (longestPrefix (E := E) x s))) (firstDiff (E := E) x y) =
  cylinder (E := E) (Ay.some (s := s ∩ cylinder (E := E) y (longestPrefix (E := E) y s))) (firstDiff (E := E) x y)
Tactic: have : cylinder Ax.some (firstDiff x y) = cylinder Ay.some (firstDiff x y) := by
            rw [I1, I2, I3]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
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 : ((n : ℕ) → E n) → (n : ℕ) → E n),
    (∀ x ∈ s, f x = x) ∧ range f = s ∧ Continuous[Pi.topologicalSpace, Pi.topologicalSpace] f
have rcases : ∀ (f : ((n : ℕ) → E n) → (n : ℕ) → E n) (fs : ∀ x ∈ s, f x = x) (frange : range f = s) (hf : LipschitzWith 1 f),
  ∃ (f : ((n : ℕ) → E n) → (n : ℕ) → E n),
    (∀ x ∈ s, f x = x) ∧ range f = s ∧ Continuous[Pi.topologicalSpace, Pi.topologicalSpace] f
Tactic: rcases exists_lipschitz_retraction_of_isClosed hs hne with ⟨f, fs, frange, hf⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
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 : ((n : ℕ) → E n) → (↑s : Type u_1)),
    (∀ (x : (↑s : Type u_1)), f (↑x : (n : ℕ) → E n) = x) ∧
      Surjective f ∧ Continuous[Pi.topologicalSpace, instTopologicalSpaceSubtype] f
have obtain : ∀ (f : ((n : ℕ) → E n) → (n : ℕ) → E n) (f_cont : Continuous[Pi.topologicalSpace, Pi.topologicalSpace] f)
  (hs : IsClosed[Pi.topologicalSpace] (range f)) (hne : (range f).Nonempty) (fs : ∀ x ∈ range f, f x = x),
  ∃ (f_1 : ((n : ℕ) → E n) → (↑(range f) : Type u_1)),
    (∀ (x : (↑(range f) : Type u_1)), f_1 (↑x : (n : ℕ) → E n) = x) ∧
      Surjective f_1 ∧ Continuous[Pi.topologicalSpace, instTopologicalSpaceSubtype] f_1
Tactic: obtain ⟨f, fs, rfl, f_cont⟩ :
    ∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ Continuous f :=
    exists_retraction_of_isClosed hs hne
Cleared hypotheses: s, hs, hne
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
E : ℕ → Type u_1
inst : (n : ℕ) → TopologicalSpace (E n)
inst_1 : ∀ (n : ℕ), DiscreteTopology.{u_1} (E n)
f : ((n : ℕ) → E n) → (n : ℕ) → E n
f_cont : Continuous[Pi.topologicalSpace, Pi.topologicalSpace] f
hs : IsClosed[Pi.topologicalSpace] (range f)
hne : (range f).Nonempty
fs : ∀ x ∈ range f, f x = x
⊢ ∃ (f_1 : ((n : ℕ) → E n) → (↑(range f) : Type u_1)),
    (∀ (x : (↑(range f) : Type u_1)), f_1 (↑x : (n : ℕ) → E n) = x) ∧
      Surjective f_1 ∧ Continuous[Pi.topologicalSpace, instTopologicalSpaceSubtype] f_1
have A : ∀ (x : (↑(range f) : Type u_1)), rangeFactorization f (↑x : (n : ℕ) → E n) = x
Tactic: have A : ∀ x : range f, rangeFactorization f x = x := fun x ↦ Subtype.ext <| fs x x.2
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have I0 : 0 < 1 / 2
Tactic: have I0 : (0 : ℝ) < 1 / 2 := by simp
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have I1 : 1 / 2 < 1
Tactic: have I1 : (1 / 2 : ℝ) < 1 := by norm_num
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have rcases : ∀ (u : ℕ → α) (hu : DenseRange u),
  ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
Tactic: rcases exists_dense_seq α with ⟨u, hu⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
⊢ ∃ (f : (ℕ → ℕ) → α), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f ∧ Surjective f
have A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
Tactic: 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 :)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
have refine : ∀ (y : (↑s : Type)), ContinuousAt g y
Tactic: refine continuous_iff_continuousAt.2 fun y => ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y : (↑s : Type)
⊢ ContinuousAt g y
have refine : ∀ (x : (↑s : Type)) (hxy : dist x y < 1), dist (g x) (g y) ≤ 4 * dist x y
Tactic: refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
⊢ dist (g x) (g y) ≤ 4 * dist x y
have inl : ∀ (hxy : dist x x < 1), dist (g x) (g x) ≤ 4 * dist x x
Tactic: rcases eq_or_ne x y with (rfl | hne)
Cleared hypotheses: y, hxy
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
inl : ∀ (hxy : dist x x < 1), dist (g x) (g x) ≤ 4 * dist x x
⊢ dist (g x) (g y) ≤ 4 * dist x y
have inr : ∀ (hne : x ≠ y), dist (g x) (g y) ≤ 4 * dist x y
Tactic: rcases eq_or_ne x y with (rfl | hne)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
⊢ dist (g x) (g y) ≤ 4 * dist x y
have hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
Tactic: have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
⊢ dist (g x) (g y) ≤ 4 * dist x y
have dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
Tactic: have dist' : dist x y = dist x.1 y.1 := rfl
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
⊢ 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
have by_contra! : ∀ (h : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) ≤ 0), False
Tactic: by_contra! h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
h : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) ≤ 0
⊢ False
have apply : (↑x : ℕ → ℕ) (firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)) =
  (↑y : ℕ → ℕ) (firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ))
Tactic: apply apply_firstDiff_ne hne'
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
h : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) ≤ 0
⊢ (↑x : ℕ → ℕ) (firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)) =
    (↑y : ℕ → ℕ) (firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ))
have rw : (↑x : ℕ → ℕ) 0 = (↑y : ℕ → ℕ) 0
Tactic: rw [Nat.le_zero.1 h]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
h : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) ≤ 0
⊢ (↑x : ℕ → ℕ) 0 = (↑y : ℕ → ℕ) 0
have apply : dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) < HPow.hPow (β := ℕ) (1 / 2) 0
Tactic: apply apply_eq_of_dist_lt _ le_rfl
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
⊢ dist (g x) (g y) ≤ 4 * dist x y
have diff_pos : 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
Tactic: have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [Nat.le_zero.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      rw [pow_zero]
      exact hxy
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
diff_pos : 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
⊢ dist (g x) (g y) ≤ 4 * dist x y
have hn : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) = n + 1
Tactic: have hn : firstDiff x.1 y.1 = n + 1 := (Nat.succ_pred_eq_of_pos diff_pos).symm
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
diff_pos : 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
hn : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) = n + 1
⊢ dist (g x) (g y) ≤ 4 * dist x y
have inr : dist (g x) (g y) ≤ 4 * (1 / 2) ^ (n + 1)
Tactic: rw [dist', dist_eq_of_ne hne', hn]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
diff_pos : 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
hn : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) = n + 1
⊢ dist (g x) (g y) ≤ 4 * (1 / 2) ^ (n + 1)
have B : (↑x : ℕ → ℕ) n = (↑y : ℕ → ℕ) n
Tactic: have B : x.1 n = y.1 n := mem_cylinder_firstDiff x.1 y.1 n (Nat.pred_lt diff_pos.ne')
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
α : Type u_2
inst : MetricSpace α
inst_1 : CompleteSpace α
inst_2 : SecondCountableTopology α
inst_3 : 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 : ℕ → ℕ | Set.Nonempty.{u_2} (α := α) (⋂ (n : ℕ), closedBall (u (x n)) ((1 / 2) ^ n))}
g : (↑s : Type) → α :=
  fun (x : (↑s : Type)) ↦ Set.Nonempty.some (s := ⋂ (n : ℕ), closedBall (u ((↑x : ℕ → ℕ) n)) ((1 / 2) ^ n)) ⋯
A : ∀ (x : (↑s : Type)) (n : ℕ), dist (g x) (u ((↑x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n
y x : (↑s : Type)
hxy : dist x y < 1
hne : x ≠ y
hne' : (↑x : ℕ → ℕ) ≠ (↑y : ℕ → ℕ)
dist' : dist x y = dist (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
n : ℕ := firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) - 1
diff_pos : 0 < firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ)
hn : firstDiff (E := fun (n : ℕ) ↦ ℕ) (↑x : ℕ → ℕ) (↑y : ℕ → ℕ) = n + 1
B : (↑x : ℕ → ℕ) n = (↑y : ℕ → ℕ) n
⊢ dist (g x) (u ((↑x : ℕ → ℕ) n)) + dist (g y) (u ((↑x : ℕ → ℕ) n)) =
    dist (g x) (u ((↑x : ℕ → ℕ) n)) + dist (g y) (u ((↑y : ℕ → ℕ) n))
have rw : dist (g x) (u ((↑x : ℕ → ℕ) n)) + dist (g y) (u ((↑x : ℕ → ℕ) n)) =
  dist (g x) (u ((↑x : ℕ → ℕ) n)) + dist (g y) (u ((↑x : ℕ → ℕ) n))
Tactic: rw [← B]