awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CauSeqFilter (goal)
Goal:
β : Type v
inst : NormedField β
inst_1 : CauSeq.IsComplete β norm
⊢ CompleteSpace β
have a : ∀ (u : ℕ → β), CauchySeq u → ∃ (a : β), Tendsto u atTop (𝓝 a)
Tactic: apply complete_of_cauchySeq_tendsto
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CauSeqFilter (hypothesis)
Goal:
β : Type v
inst : NormedField β
inst_1 : CauSeq.IsComplete β norm
u : ℕ → β
hu : CauchySeq u
⊢ ∃ (a : β), Tendsto u atTop (𝓝 a)
have C : IsCauSeq norm u
Tactic: have C : IsCauSeq norm u := isCauSeq_iff_cauchySeq.2 hu
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CauSeqFilter (goal)
Goal:
β : Type v
inst : NormedField β
inst_1 : CauSeq.IsComplete β norm
u : ℕ → β
hu : CauchySeq u
C : IsCauSeq norm u
⊢ ∃ (a : β), Tendsto u atTop (𝓝 a)
have a : Tendsto u atTop (𝓝 (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C)))
Tactic: exists CauSeq.lim ⟨u, C⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CauSeqFilter (goal)
Goal:
β : Type v
inst : NormedField β
inst_1 : CauSeq.IsComplete β norm
u : ℕ → β
hu : CauchySeq u
C : IsCauSeq norm u
⊢ Tendsto u atTop (𝓝 (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C)))
have a : ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C)) < ε
Tactic: rw [Metric.tendsto_atTop]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CauSeqFilter (goal)
Goal:
β : Type v
inst : NormedField β
inst_1 : CauSeq.IsComplete β norm
u : ℕ → β
hu : CauchySeq u
C : IsCauSeq norm u
ε : ℝ
εpos : ε > 0
⊢ ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C)) < ε
have a : ∀ (N : ℕ)
  (hN :
    ∀ j ≥ N,
      ‖(↑(Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C -
                  CauSeq.const norm (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C))) :
              ℕ → β)
            j‖ <
        ε),
  ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C)) < ε
Tactic: obtain ⟨N, hN⟩ := (CauSeq.equiv_lim ⟨u, C⟩) _ εpos
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CauSeqFilter (goal)
Goal:
β : Type v
inst : NormedField β
inst_1 : CauSeq.IsComplete β norm
u : ℕ → β
hu : CauchySeq u
C : IsCauSeq norm u
ε : ℝ
εpos : ε > 0
N : ℕ
hN :
  ∀ j ≥ N,
    ‖(↑(Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C -
                CauSeq.const norm (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C))) :
            ℕ → β)
          j‖ <
      ε
⊢ ∃ (N : ℕ), ∀ n ≥ N, dist (u n) (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C)) < ε
have a : ∀ n ≥ N, dist (u n) (CauSeq.lim (Subtype.mk (p := fun (f : ℕ → β) ↦ IsCauSeq norm f) u C)) < ε
Tactic: exists N
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
⊢ UniformCauchySeqOn F atTop s ↔ ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
have mp : UniformCauchySeqOn F atTop s → ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
mp : UniformCauchySeqOn F atTop s → ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
⊢ UniformCauchySeqOn F atTop s ↔ ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
have mpr : (∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε) → UniformCauchySeqOn F atTop s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : UniformCauchySeqOn F atTop s
ε : ℝ
hε : ε > 0
u : Set (α × α) := {a : α × α | dist a.1 a.2 < ε}
⊢ ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
have hu : u ∈ 𝓤 α
Tactic: have hu : u ∈ 𝓤 α := Metric.mem_uniformity_dist.mpr ⟨ε, hε, by simp [u]⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : UniformCauchySeqOn F atTop s
ε : ℝ
hε : ε > 0
u : Set (α × α) := {a : α × α | dist a.1 a.2 < ε}
hu : u ∈ 𝓤 α
⊢ ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
have mp : ∀ᶠ (x : β × β) in atTop, ∀ x_1 ∈ s, dist (F x.1 x_1) (F x.2 x_1) < ε
Tactic: rw [← Filter.eventually_atTop_prod_self' (p := fun m =>
      ∀ x ∈ s, dist (F m.fst x) (F m.snd x) < ε)]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : UniformCauchySeqOn F atTop s
ε : ℝ
hε : ε > 0
u : Set (α × α) := {a : α × α | dist a.1 a.2 < ε}
hu : u ∈ 𝓤 α
⊢ ∀ᶠ (x : β × β) in atTop, ∀ x_1 ∈ s, dist (F x.1 x_1) (F x.2 x_1) < ε
have h₁ : ∀ᶠ (m : β × β) in SProd.sprod (α := Filter β) (β := Filter β) atTop atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
Tactic: specialize h u hu
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
ε : ℝ
hε : ε > 0
u : Set (α × α) := {a : α × α | dist a.1 a.2 < ε}
hu : u ∈ 𝓤 α
h : ∀ᶠ (m : β × β) in SProd.sprod (α := Filter β) (β := Filter β) atTop atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
⊢ ∀ᶠ (x : β × β) in atTop, ∀ x_1 ∈ s, dist (F x.1 x_1) (F x.2 x_1) < ε
have h₁ : ∀ᶠ (m : β × β) in atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
Tactic: rw [prod_atTop_atTop_eq] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
⊢ ∀ᶠ (m : β × β) in SProd.sprod (α := Filter β) (β := Filter β) atTop atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
have mpr : ∀ (ε : ℝ) (hε : ε > 0) (hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u),
  ∀ᶠ (m : β × β) in SProd.sprod (α := Filter β) (β := Filter β) atTop atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
Tactic: rcases Metric.mem_uniformity_dist.mp hu with ⟨ε, hε, hab⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u
⊢ ∀ᶠ (m : β × β) in SProd.sprod (α := Filter β) (β := Filter β) atTop atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
have mpr : ∀ (N : β) (hN : ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε),
  ∀ᶠ (m : β × β) in SProd.sprod (α := Filter β) (β := Filter β) atTop atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
Tactic: rcases h ε hε with ⟨N, hN⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u
N : β
hN : ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
⊢ ∀ᶠ (m : β × β) in SProd.sprod (α := Filter β) (β := Filter β) atTop atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u
have mpr : ∃ (a : β × β), ∀ b ≥ a, ∀ x ∈ s, (F b.1 x, F b.2 x) ∈ u
Tactic: rw [prod_atTop_atTop_eq, eventually_atTop]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u
N : β
hN : ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
⊢ ∃ (a : β × β), ∀ b ≥ a, ∀ x ∈ s, (F b.1 x, F b.2 x) ∈ u
have h₁ : ∀ b ≥ (N, N), ∀ x ∈ s, (F b.1 x, F b.2 x) ∈ u
Tactic: use (N, N)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u
N : β
hN : ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
b : β × β
hb : b ≥ (N, N)
x : γ
hx : x ∈ s
⊢ (F b.1 x, F b.2 x) ∈ u
have hbl : (N, N).1 ≤ b.1
Tactic: rcases hb with ⟨hbl, hbr⟩
Cleared hypotheses: hb
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
β : Type v
inst : PseudoMetricSpace α
inst_1 : Nonempty β
inst_2 : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : ∀ ε > 0, ∃ (N : β), ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u
N : β
hN : ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
b : β × β
hb : b ≥ (N, N)
x : γ
hx : x ∈ s
hbl : (N, N).1 ≤ b.1
⊢ (F b.1 x, F b.2 x) ∈ u
have hbr : (N, N).2 ≤ b.2
Tactic: rcases hb with ⟨hbl, hbr⟩
Cleared hypotheses: hb
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
have rcases : ∀ (N : ℕ) (hN : ∀ n ≥ N, dist (u n) (u N) < 1), ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
Tactic: rcases Metric.cauchySeq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
have rsuffices : ∀ (R : ℝ) (R0 : R > 0) (H : ∀ (n : ℕ), dist (u n) (u N) < R), ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
Tactic: rsuffices ⟨R, R0, H⟩ : ∃ R > 0, ∀ n, dist (u n) (u N) < R
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
rsuffices : ∀ (R : ℝ) (R0 : R > 0) (H : ∀ (n : ℕ), dist (u n) (u N) < R), ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
have rsuffices₁ : ∃ R > 0, ∀ (n : ℕ), dist (u n) (u N) < R
Tactic: rsuffices ⟨R, R0, H⟩ : ∃ R > 0, ∀ n, dist (u n) (u N) < R
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun (n : ℕ) ↦ nndist (u n) (u N)
⊢ ∃ R > 0, ∀ (n : ℕ), dist (u n) (u N) < R
have refine : ∀ (n : ℕ), dist (u n) (u N) < (↑R : ℝ) + 1
Tactic: refine ⟨↑R + 1, add_pos_of_nonneg_of_pos R.2 zero_lt_one, fun n => ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun (n : ℕ) ↦ nndist (u n) (u N)
n : ℕ
⊢ dist (u n) (u N) < (↑R : ℝ) + 1
have inl : ∀ (h : N ≤ n), dist (u n) (u N) < (↑R : ℝ) + 1
Tactic: rcases le_or_gt N n with h | h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun (n : ℕ) ↦ nndist (u n) (u N)
n : ℕ
inl : ∀ (h : N ≤ n), dist (u n) (u N) < (↑R : ℝ) + 1
⊢ dist (u n) (u N) < (↑R : ℝ) + 1
have inr : ∀ (h : n < N), dist (u n) (u N) < (↑R : ℝ) + 1
Tactic: rcases le_or_gt N n with h | h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun (n : ℕ) ↦ nndist (u n) (u N)
n : ℕ
h : n < N
⊢ dist (u n) (u N) < (↑R : ℝ) + 1
have this : nndist (u n) (u N) ≤ R
Tactic: have : _ ≤ R := Finset.le_sup (Finset.mem_range.2 h)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
⊢ ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
have rcases : ∀ (R : ℝ) (hR : ∀ (m n : ℕ), dist (s m) (s n) < R) (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
Tactic: rcases cauchySeq_bdd hs with ⟨R, -, hR⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
R : ℝ
hR : ∀ (m n : ℕ), dist (s m) (s n) < R
⊢ ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
have refine : ∀ (N : ℕ), ∀ y ∈ S N, y ≤ R
Tactic: refine fun N => ⟨R, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
⊢ ∃ (b : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ b n) ∧ (∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)
have hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
Tactic: have hS : ∀ N, ∃ x, ∀ y ∈ S N, y ≤ x := by
      rcases cauchySeq_bdd hs with ⟨R, -, hR⟩
      refine fun N => ⟨R, ?_⟩
      rintro _ ⟨⟨m, n⟩, _, rfl⟩
      exact le_of_lt (hR m n)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
⊢ ∃ (b : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ b n) ∧ (∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)
have ub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)
Tactic: have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N) := fun m n N hm hn =>
      le_csSup (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
ub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)
⊢ ∃ (b : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ b n) ∧ (∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)
have S0m : ∀ (n : ℕ), 0 ∈ S n
Tactic: have S0m : ∀ n, (0 : ℝ) ∈ S n := fun n => ⟨⟨n, n⟩, ⟨le_rfl, le_rfl⟩, dist_self _⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
ub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)
S0m : ∀ (n : ℕ), 0 ∈ S n
⊢ ∃ (b : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ b n) ∧ (∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)
have S0 : ∀ (n : ℕ), 0 ≤ sSup (S n)
Tactic: have S0 := fun n => le_csSup (hS n) (S0m n)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
ub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)
S0m : ∀ (n : ℕ), 0 ∈ S n
S0 : ∀ (n : ℕ), 0 ≤ sSup (S n)
⊢ ∃ (b : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ b n) ∧ (∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0)
have refine : ∀ (ε : ℝ) (ε0 : ε > 0), ∃ (N : ℕ), ∀ n ≥ N, dist (sSup (S n)) 0 < ε
Tactic: refine ⟨fun N => sSup (S N), S0, ub, Metric.tendsto_atTop.2 fun ε ε0 => ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
ub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)
S0m : ∀ (n : ℕ), 0 ∈ S n
S0 : ∀ (n : ℕ), 0 ≤ sSup (S n)
ε : ℝ
ε0 : ε > 0
⊢ ∃ (N : ℕ), ∀ n ≥ N, dist (sSup (S n)) 0 < ε
have refine : ∀ (N : ℕ) (hN : ∀ m ≥ N, ∀ n ≥ N, dist (s m) (s n) < ε / 2) (n : ℕ) (hn : n ≥ N), dist (sSup (S n)) 0 < ε
Tactic: refine (Metric.cauchySeq_iff.1 hs (ε / 2) (half_pos ε0)).imp fun N hN n hn => ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
ub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)
S0m : ∀ (n : ℕ), 0 ∈ S n
S0 : ∀ (n : ℕ), 0 ≤ sSup (S n)
ε : ℝ
ε0 : ε > 0
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, dist (s m) (s n) < ε / 2
n : ℕ
hn : n ≥ N
⊢ dist (sSup (S n)) 0 < ε
have rw : sSup (S n) < ε
Tactic: rw [Real.dist_0_eq_abs, abs_of_nonneg (S0 n)]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : (N : ℕ) → Set ℝ := fun (N : ℕ) ↦ (fun (p : ℕ × ℕ) ↦ dist (s p.1) (s p.2)) '' {p : ℕ × ℕ | p.1 ≥ N ∧ p.2 ≥ N}
hS : ∀ (N : ℕ), ∃ (x : ℝ), ∀ y ∈ S N, y ≤ x
ub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)
S0m : ∀ (n : ℕ), 0 ∈ S n
S0 : ∀ (n : ℕ), 0 ≤ sSup (S n)
ε : ℝ
ε0 : ε > 0
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, dist (s m) (s n) < ε / 2
n : ℕ
hn : n ≥ N
⊢ sSup (S n) < ε
have refine : ∀ b ∈ S n, b ≤ ε / 2
Tactic: refine lt_of_le_of_lt (csSup_le ⟨_, S0m _⟩ ?_) (half_lt_self ε0)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
⊢ ∃ (f : ℕ → ℕ), StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n
have rw : ∀ (hu : ∀ ε > 0, ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε) (b : ℕ → ℝ) (hb : ∀ (n : ℕ), 0 < b n),
  ∃ (f : ℕ → ℕ), StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n
Tactic: rw [cauchySeq_iff] at hu
Cleared hypotheses: hu, b, hb
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k : ℕ
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k
have rw : ∃ (a : ℕ), ∀ b_1 ≥ a, ∀ m ≥ b_1, dist (u m) (u b_1) < b k
Tactic: rw [eventually_atTop]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (goal)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k : ℕ
⊢ ∃ (a : ℕ), ∀ b_1 ≥ a, ∀ m ≥ b_1, dist (u m) (u b_1) < b k
have obtain : ∀ (N : ℕ) (hN : ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < b k), ∃ (a : ℕ), ∀ b_1 ≥ a, ∀ m ≥ b_1, dist (u m) (u b_1) < b k
Tactic: obtain ⟨N, hN⟩ := hu (b k) (hb k)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Cauchy (hypothesis)
Goal:
α : Type u
inst : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ (N : ℕ), ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
⊢ ∃ (f : ℕ → ℕ), StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n
have hu' : ∀ (k : ℕ), ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k
Tactic: have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
    intro k
    rw [eventually_atTop]
    obtain ⟨N, hN⟩ := hu (b k) (hb k)
    exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : hausdorffEDist s t < δ
⊢ Membership.mem (γ := SetRel (Set α) (Set α)) (hausdorffEntourage {p : α × α | edist p.1 p.2 < δ}) (s, t)
have h₁ : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
Tactic: rw [hausdorffEDist, max_lt_iff] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
| (s, t).1 ⊆ SetRel.preimage {p : α × α | edist p.1 p.2 < δ} (s, t).2 ∧
    (s, t).2 ⊆ SetRel.image {p : α × α | edist p.1 p.2 < δ} (s, t).1
have h₁ : ∀ (x : α × α), edist x.1 x.2 < δ
Tactic: enter [2, 2, 1, 1, _]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
x : α × α
| edist x.1 x.2 < δ
have h₁ : edist x.2 x.1 < δ
Tactic: rw [edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
| (s, t).1 ⊆ SetRel.preimage {p : α × α | edist p.1 p.2 < δ} (s, t).2 ∧
    (s, t).2 ⊆ SetRel.image {p : α × α | edist p.1 p.2 < δ} (s, t).1
have h₁ : ∀ (x : α × α), edist x.2 x.1 < δ
Tactic: enter [2, 2, 1, 1, _]; rw [edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
⊢ (s, t).1 ⊆ SetRel.preimage {p : α × α | edist p.1 p.2 < δ} (s, t).2 ∧
    (s, t).2 ⊆ SetRel.image {p : α × α | edist p.1 p.2 < δ} (s, t).1
have conv : (s, t).1 ⊆ SetRel.preimage {p : α × α | edist p.1 p.2 < δ} (s, t).2 ∧
  (s, t).2 ⊆ SetRel.image {x : α × α | edist x.2 x.1 < δ} (s, t).1
Tactic: conv => enter [2, 2, 1, 1, _]; rw [edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
⊢ (s, t).1 ⊆ SetRel.preimage {p : α × α | edist p.1 p.2 < δ} (s, t).2 ∧
    (s, t).2 ⊆ SetRel.image {x : α × α | edist x.2 x.1 < δ} (s, t).1
have this : ∀ {s t : Set α} (h : ⨆ x ∈ s, infEDist x t < δ), s ⊆ SetRel.preimage {p : α × α | edist p.1 p.2 < δ} t
Tactic: have {s t : Set α} (h : ⨆ x ∈ s, infEDist x t < δ) :
      s ⊆ SetRel.preimage {p | edist p.1 p.2 < δ} t := by
    intro x hx
    simpa only [infEDist, iInf_lt_iff, exists_prop] using (le_iSup₂ x hx).trans_lt h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (goal)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : Membership.mem (γ := SetRel (Set α) (Set α)) (hausdorffEntourage {p : α × α | edist p.1 p.2 ≤ δ}) (s, t)
⊢ hausdorffEDist s t ≤ δ
have rw : ⨆ x ∈ s, infEDist x t ≤ δ ∧ ⨆ y ∈ t, infEDist y s ≤ δ
Tactic: rw [hausdorffEDist, max_le_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Closeds (hypothesis)
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : Membership.mem (γ := SetRel (Set α) (Set α)) (hausdorffEntourage {p : α × α | edist p.1 p.2 ≤ δ}) (s, t)
⊢ ⨆ x ∈ s, infEDist x t ≤ δ ∧ ⨆ y ∈ t, infEDist y s ≤ δ
have h₁ : (s, t).1 ⊆ SetRel.preimage {p : α × α | edist p.1 p.2 ≤ δ} (s, t).2 ∧
  (s, t).2 ⊆ SetRel.image {p : α × α | edist p.1 p.2 ≤ δ} (s, t).1
Tactic: rw [hausdorffEntourage, Set.mem_setOf] at h
Cleared hypotheses: h