awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
i : ι
hi : i ∈ (↑K : Set ι)
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i_1 : ℝ≥0∞),
      ⨅ (_ : 0 < i_1),
        𝓟
          {a : ((i : ι) → F i) × ((i : ι) → F i) |
            edist (a.1 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι))
                (a.2 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι)) <
              i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) |
      edist (p.1 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι))
          (p.2 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι)) <
        δ}
have left : Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
  (𝓟
    {a : ((i : ι) → F i) × ((i : ι) → F i) |
      edist (a.1 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι))
          (a.2 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι)) <
        δ})
  {p : ((i : ι) → F i) × ((i : ι) → F i) |
    edist (p.1 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι))
        (p.2 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ (↑K : Set ι)) i hi) : ι)) <
      δ}
Tactic: refine mem_iInf_of_mem δ (mem_iInf_of_mem δpos ?_)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy :
  Membership.mem (γ := Set (((i : ι) → F i) × ((i : ι) → F i)))
    (⋂ (i : (↑(↑K : Set ι) : Type u_2)),
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist (p.1 (↑i : ι)) (p.2 (↑i : ι)) < δ})
    (x, y)
⊢ Membership.mem (γ := Set (((i : ι) → F i) × ((i : ι) → F i)))
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε} (x, y)
have hxy₁ : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
Tactic: simp only [mem_iInter, mem_setOf_eq, SetCoe.forall, Finset.mem_coe] at hxy
Cleared hypotheses: hxy
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
⊢ HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)),
        min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) (edist (α := F (↑i : ι)) (x (↑i : ι)) (y (↑i : ι)))) ≤
    HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, edist (x i) (y i))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)))
have h₁ : ∀ i ∈ K, min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)) ≤ edist (x i) (y i)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
h₁ : ∀ i ∈ K, min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)) ≤ edist (x i) (y i)
⊢ HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)),
        min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) (edist (α := F (↑i : ι)) (x (↑i : ι)) (y (↑i : ι)))) ≤
    HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, edist (x i) (y i))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)))
have h₂ : ∀ (i : (↑(↑K : Set ι)ᶜ : Type u_2)),
  min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) (edist (α := F (↑i : ι)) (x (↑i : ι)) (y (↑i : ι))) ≤
    HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
⊢ HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, edist (x i) (y i))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) <
    ∑ _i ∈ K, δ + ε / 2
have refine : LE.le (α := ℝ≥0∞) (∑ i ∈ K, edist (x i) (y i)) (∑ _i ∈ K, δ)
Tactic: refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ K, edist (x i) (y i)) (∑ _i ∈ K, δ)
have h : ∀ (i : ι) (hi : i ∈ K), edist (x i) (y i) ≤ δ
Tactic: gcongr with i hi
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
⊢ ∑ _i ∈ K, δ + ε / 2 ≤ ε / 2 + ε / 2
have bc : ∑ _i ∈ K, δ ≤ ε / 2
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
⊢ Membership.mem (γ := Set (((i : ι) → F i) × ((i : ι) → F i)))
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε} (x, y)
have edist : HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)))
    (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)),
      min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) (edist (α := F (↑i : ι)) (x (↑i : ι)) (y (↑i : ι)))) ≤
  HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, edist (x i) (y i))
    (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)))
Tactic: calc
          edist x y = ∑' i : ι, min (2⁻¹ ^ encode i) (edist (x i) (y i)) := rfl
          _ = ∑ i ∈ K, min (2⁻¹ ^ encode i) (edist (x i) (y i)) +
                ∑' i : ↑(K : Set ι)ᶜ, min (2⁻¹ ^ encode (i : ι)) (edist (x i) (y i)) :=
            (ENNReal.sum_add_tsum_compl ..).symm
          _ ≤ ∑ i ∈ K, edist (x i) (y i) + ∑' i : ↑(K : Set ι)ᶜ, 2⁻¹ ^ encode (i : ι) := by
            gcongr
            · apply min_le_right
            · apply min_le_left
          _ < ∑ _i ∈ K, δ + ε / 2 := by
            refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
            gcongr with i hi
            exact (hxy i hi).le
          _ ≤ ε / 2 + ε / 2 := by gcongr; simpa [mul_comm] using hδ.le
          _ = ε := ENNReal.add_halves _
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), EDist.edist (x x_1) (y x_1) < δ
edist :
  HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (EDist.edist (x i) (y i)))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)),
        min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) (EDist.edist (α := F (↑i : ι)) (x (↑i : ι)) (y (↑i : ι)))) ≤
    HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, EDist.edist (x i) (y i))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)))
⊢ Membership.mem (γ := Set (((i : ι) → F i) × ((i : ι) → F i)))
    {p : ((i : ι) → F i) × ((i : ι) → F i) | EDist.edist p.1 p.2 < ε} (x, y)
have edist₁ : HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, EDist.edist (x i) (y i))
    (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) <
  ∑ _i ∈ K, δ + ε / 2
Tactic: calc
          edist x y = ∑' i : ι, min (2⁻¹ ^ encode i) (edist (x i) (y i)) := rfl
          _ = ∑ i ∈ K, min (2⁻¹ ^ encode i) (edist (x i) (y i)) +
                ∑' i : ↑(K : Set ι)ᶜ, min (2⁻¹ ^ encode (i : ι)) (edist (x i) (y i)) :=
            (ENNReal.sum_add_tsum_compl ..).symm
          _ ≤ ∑ i ∈ K, edist (x i) (y i) + ∑' i : ↑(K : Set ι)ᶜ, 2⁻¹ ^ encode (i : ι) := by
            gcongr
            · apply min_le_right
            · apply min_le_left
          _ < ∑ _i ∈ K, δ + ε / 2 := by
            refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
            gcongr with i hi
            exact (hxy i hi).le
          _ ≤ ε / 2 + ε / 2 := by gcongr; simpa [mul_comm] using hδ.le
          _ = ε := ENNReal.add_halves _
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), EDist.edist (x x_1) (y x_1) < δ
edist :
  HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (EDist.edist (x i) (y i)))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)),
        min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) (EDist.edist (α := F (↑i : ι)) (x (↑i : ι)) (y (↑i : ι)))) ≤
    HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, EDist.edist (x i) (y i))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)))
edist₁ :
  HAdd.hAdd (α := ℝ≥0∞) (∑ i ∈ K, EDist.edist (x i) (y i))
      (∑' (i : (↑(↑K : Set ι)ᶜ : Type u_2)), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι))) <
    ∑ _i ∈ K, δ + ε / 2
⊢ Membership.mem (γ := Set (((i : ι) → F i) × ((i : ι) → F i)))
    {p : ((i : ι) → F i) × ((i : ι) → F i) | EDist.edist p.1 p.2 < ε} (x, y)
have edist₂ : ∑ _i ∈ K, δ + ε / 2 ≤ ε / 2 + ε / 2
Tactic: calc
          edist x y = ∑' i : ι, min (2⁻¹ ^ encode i) (edist (x i) (y i)) := rfl
          _ = ∑ i ∈ K, min (2⁻¹ ^ encode i) (edist (x i) (y i)) +
                ∑' i : ↑(K : Set ι)ᶜ, min (2⁻¹ ^ encode (i : ι)) (edist (x i) (y i)) :=
            (ENNReal.sum_add_tsum_compl ..).symm
          _ ≤ ∑ i ∈ K, edist (x i) (y i) + ∑' i : ↑(K : Set ι)ᶜ, 2⁻¹ ^ encode (i : ι) := by
            gcongr
            · apply min_le_right
            · apply min_le_left
          _ < ∑ _i ∈ K, δ + ε / 2 := by
            refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
            gcongr with i hi
            exact (hxy i hi).le
          _ ≤ ε / 2 + ε / 2 := by gcongr; simpa [mul_comm] using hδ.le
          _ = ε := ENNReal.add_halves _
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
have left : ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2),
  Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
Tactic: classical
      obtain ⟨K, hK⟩ : ∃ K : Finset ι, ∑' i : {j // j ∉ K}, 2⁻¹ ^ encode (i : ι) < ε / 2 :=
        ((tendsto_order.1 <| ENNReal.tendsto_tsum_compl_atTop_zero
          (tsum_geometric_encode_lt_top ENNReal.one_half_lt_one).ne).2 _
            <| by simpa using hε.ne').exists
      obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ δ * K.card < ε / 2 :=
        ENNReal.exists_pos_mul_lt (by simp) (by simpa using hε.ne')
      apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
          {p : (∀ i : ι, F i) × ∀ i : ι, F i | edist (p.fst i) (p.snd i) < δ}
      · rintro ⟨i, hi⟩
        refine mem_iInf_of_mem δ (mem_iInf_of_mem δpos ?_)
        simp only [mem_principal, Subset.rfl]
      · rintro ⟨x, y⟩ hxy
        simp only [mem_iInter, mem_setOf_eq, SetCoe.forall, Finset.mem_coe] at hxy
        calc
          edist x y = ∑' i : ι, min (2⁻¹ ^ encode i) (edist (x i) (y i)) := rfl
          _ = ∑ i ∈ K, min (2⁻¹ ^ encode i) (edist (x i) (y i)) +
                ∑' i : ↑(K : Set ι)ᶜ, min (2⁻¹ ^ encode (i : ι)) (edist (x i) (y i)) :=
            (ENNReal.sum_add_tsum_compl ..).symm
          _ ≤ ∑ i ∈ K, edist (x i) (y i) + ∑' i : ↑(K : Set ι)ᶜ, 2⁻¹ ^ encode (i : ι) := by
            gcongr
            · apply min_le_right
            · apply min_le_left
          _ < ∑ _i ∈ K, δ + ε / 2 := by
            refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
            gcongr with i hi
            exact (hxy i hi).le
          _ ≤ ε / 2 + ε / 2 := by gcongr; simpa [mul_comm] using hδ.le
          _ = ε := ENNReal.add_halves _
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
left :
  ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2),
    Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
      (⨅ (i : ι),
        ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
      {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
have left₁ : ∀ (K : Finset ι) (hK : ∑' (i : { j : ι // j ∉ K }), HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode (↑i : ι)) < ε / 2) (δ : ℝ≥0∞)
  (δpos : 0 < δ) (hδ : δ * (↑K.card : ℝ≥0∞) < ε / 2),
  Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (i : ι),
      ⨅ (i_1 : ℝ≥0∞), ⨅ (_ : 0 < i_1), 𝓟 {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < i_1})
    {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε}
Tactic: classical
      obtain ⟨K, hK⟩ : ∃ K : Finset ι, ∑' i : {j // j ∉ K}, 2⁻¹ ^ encode (i : ι) < ε / 2 :=
        ((tendsto_order.1 <| ENNReal.tendsto_tsum_compl_atTop_zero
          (tsum_geometric_encode_lt_top ENNReal.one_half_lt_one).ne).2 _
            <| by simpa using hε.ne').exists
      obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ δ * K.card < ε / 2 :=
        ENNReal.exists_pos_mul_lt (by simp) (by simpa using hε.ne')
      apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
          {p : (∀ i : ι, F i) × ∀ i : ι, F i | edist (p.fst i) (p.snd i) < δ}
      · rintro ⟨i, hi⟩
        refine mem_iInf_of_mem δ (mem_iInf_of_mem δpos ?_)
        simp only [mem_principal, Subset.rfl]
      · rintro ⟨x, y⟩ hxy
        simp only [mem_iInter, mem_setOf_eq, SetCoe.forall, Finset.mem_coe] at hxy
        calc
          edist x y = ∑' i : ι, min (2⁻¹ ^ encode i) (edist (x i) (y i)) := rfl
          _ = ∑ i ∈ K, min (2⁻¹ ^ encode i) (edist (x i) (y i)) +
                ∑' i : ↑(K : Set ι)ᶜ, min (2⁻¹ ^ encode (i : ι)) (edist (x i) (y i)) :=
            (ENNReal.sum_add_tsum_compl ..).symm
          _ ≤ ∑ i ∈ K, edist (x i) (y i) + ∑' i : ↑(K : Set ι)ᶜ, 2⁻¹ ^ encode (i : ι) := by
            gcongr
            · apply min_le_right
            · apply min_le_left
          _ < ∑ _i ∈ K, δ + ε / 2 := by
            refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
            gcongr with i hi
            exact (hxy i hi).le
          _ ≤ ε / 2 + ε / 2 := by gcongr; simpa [mul_comm] using hδ.le
          _ = ε := ENNReal.add_halves _
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (hypothesis)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
i : ι
ε : ℝ≥0∞
hε₀ : 0 < ε
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (ε : ℝ≥0∞), ⨅ (_ : 0 < ε), 𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε})
    {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < ε}
have this : 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)
Tactic: have : (0 : ℝ≥0∞) < 2⁻¹ ^ encode i := ENNReal.pow_pos (by norm_num) _
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
i : ι
ε : ℝ≥0∞
hε₀ : 0 < ε
this : 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (⨅ (ε : ℝ≥0∞), ⨅ (_ : 0 < ε), 𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < ε})
    {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < ε}
have right : Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
  (𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) ε})
  {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < ε}
Tactic: refine mem_iInf_of_mem (min (2⁻¹ ^ encode i) ε) <| mem_iInf_of_mem (by positivity) ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoEMetricSpace (F i)
i : ι
ε : ℝ≥0∞
hε₀ : 0 < ε
this : 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)
⊢ Membership.mem (γ := Filter (((i : ι) → F i) × ((i : ι) → F i)))
    (𝓟 {p : ((i : ι) → F i) × ((i : ι) → F i) | edist p.1 p.2 < min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) ε})
    {a : ((i : ι) → F i) × ((i : ι) → F i) | edist (a.1 i) (a.2 i) < ε}
have right : ∀ (a b : (i : ι) → F i), edist a b < HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i) → edist a b < ε → edist (a i) (b i) < ε
Tactic: simp only [and_imp, Prod.forall, setOf_subset_setOf, lt_min_iff, mem_principal]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoMetricSpace (F i)
x y : (i : ι) → F i
⊢ Summable (α := ℝ) fun (i : ι) ↦ min (HPow.hPow (α := ℝ) 2⁻¹ (encode i)) (dist (x i) (y i))
have refine : ∀ (i : ι), 0 ≤ min (HPow.hPow (α := ℝ) 2⁻¹ (encode i)) (dist (x i) (y i))
Tactic: refine .of_nonneg_of_le (fun i => ?_) (fun i => min_le_left _ _) <| by
    simpa [one_div] using summable_geometric_two_encode
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoMetricSpace (F i)
x_1 y_1 : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ 0 ≤ dist x y
have simp : LE.le (α := ℝ) 0 (∑' (i : ι), min (HPow.hPow (α := ℝ) 2 (encode i))⁻¹ (dist (x i) (y i)))
Tactic: simp [dist_eq_tsum]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoMetricSpace (F i)
x_1 y_1 : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ edist x y = ENNReal.ofReal (dist x y)
have rw : Eq (α := ℝ≥0∞) (∑' (i : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)))
  (∑' (n : ι), ENNReal.ofReal (min (HPow.hPow (α := ℝ) 2⁻¹ (encode n)) (dist (x n) (y n))))
Tactic: rw [edist_eq_tsum, dist_eq_tsum,
        ENNReal.ofReal_tsum_of_nonneg (fun _ ↦ by positivity) (dist_summable ..)]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoMetricSpace (F i)
x_1 y_1 : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ Eq (α := ℝ≥0∞) (∑' (i : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (edist (x i) (y i)))
    (∑' (n : ι), ENNReal.ofReal (min (HPow.hPow (α := ℝ) 2⁻¹ (encode n)) (dist (x n) (y n))))
have simp : Eq (α := ℝ≥0∞) (∑' (i : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (PseudoMetricSpace.edist (x i) (y i)))
  (∑' (n : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (ENNReal.ofReal (dist (x n) (y n))))
Tactic: simp [edist, ENNReal.inv_pow]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
E : ℕ → Type u_1
ι : Type u_2
inst : Encodable ι
F : ι → Type u_3
inst_1 : (i : ι) → PseudoMetricSpace (F i)
x_1 y_1 : (i : ι) → F i
i : ι
x y : (i : ι) → F i
⊢ Eq (α := ℝ≥0∞) (∑' (i : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode i)) (PseudoMetricSpace.edist (x i) (y i)))
    (∑' (n : ι), min (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (encode n)) (ENNReal.ofReal (dist (x n) (y n))))
have h : ∀ (a : ι), PseudoMetricSpace.edist (x a) (y a) = ENNReal.ofReal (dist (x a) (y a))
Tactic: congr! with a
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
x y : PiNatEmbed X Y f
hxy : x.ofPiNat (Y := Y) = y.ofPiNat (Y := Y)
⊢ x = y
have toPiNat : ∀ (ofPiNat : X) (hxy : PiNatEmbed.ofPiNat (Y := Y) (f := f) (toPiNat (Y := Y) ofPiNat) = y.ofPiNat (Y := Y)),
  toPiNat (Y := Y) ofPiNat = y
Tactic: cases x
Cleared hypotheses: x, hxy
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst : Encodable ι
inst_1 : (i : ι) → PseudoMetricSpace (Y i)
inst_2 : TopologicalSpace X
continuous_f : ∀ (i : ι), Continuous[inst_2, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
⊢ Continuous[inst_2, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (toPiNat (Y := Y))
have rw : Continuous[instTopologicalSpaceProd, _] fun (x : X × X) ↦
  dist (α := PiNatEmbed X Y f) (toPiNat (Y := Y) x.1) (toPiNat (Y := Y) x.2)
Tactic: rw [continuous_iff_continuous_dist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PiNat (goal)
Goal:
ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst : Encodable ι
inst_1 : (i : ι) → PseudoMetricSpace (Y i)
inst_2 : TopologicalSpace X
continuous_f : ∀ (i : ι), Continuous[inst_2, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
⊢ Continuous[instTopologicalSpaceProd, _] fun (x : X × X) ↦
    dist (α := PiNatEmbed X Y f) (toPiNat (Y := Y) x.1) (toPiNat (Y := Y) x.2)
have simp : Continuous[instTopologicalSpaceProd, _] fun (x : X × X) ↦
  ∑' (i : ι), min (HPow.hPow (α := ℝ) 2⁻¹ (encode i)) (dist (f i x.1) (f i x.2))
Tactic: simp only [dist_def]