awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
L : SummationFilter β
inst : CommGroup α
inst_1 : TopologicalSpace α
inst_2 : IsTopologicalGroup α
f : β → α
a : α
inst_3 : L.LeAtTop
inst_4 : DecidableEq β
hf : HasProd f a L
b n : β
⊢ (if n = b then 1 else f n) = update (β := fun (a : β) ↦ α) f b 1 n
have h : (if n = b then 1 else f n) = if n = b then 1 else f n
Tactic: rw [Function.update_apply]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
L : SummationFilter β
inst : CommGroup α
inst_1 : TopologicalSpace α
inst_2 : IsTopologicalGroup α
f : β → α
a : α
inst_3 : L.LeAtTop
inst_4 : DecidableEq β
hf : HasProd f a L
b : β
⊢ a / f b = 1 / f b * a
have h : a / f b = 1 * a / f b
Tactic: rw [div_mul_eq_mul_div, one_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
L : SummationFilter β
inst : CommGroup α
inst_1 : TopologicalSpace α
inst_2 : IsTopologicalGroup α
f : β → α
a : α
inst_3 : L.LeAtTop
inst_4 : DecidableEq β
hf : HasProd f a L
b : β
h : a / f b = 1 * a / f b
⊢ a / f b = 1 / f b * a
have h₁ : a / f b = a / f b
Tactic: rw [div_mul_eq_mul_div, one_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : CommGroup α
inst_1 : TopologicalSpace α
inst_2 : IsTopologicalGroup α
f : β → α
inst_3 : T2Space α
inst_4 : DecidableEq β
hf : Multipliable f
b : β
⊢ ∏' (n : β), f n = f b * ∏' (n : β), if n = b then 1 else f n
have rw : ∏' (n : β), f n = f b * ((∏' (b : β), f b) / f b)
Tactic: rw [(hasProd_ite_div_hasProd hf.hasProd b).tprod_eq]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
⊢ (CauchySeq.{u_1, u_2} (α := α) fun (s : Finset β) ↦ ∏ b ∈ s, f b) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have simp : Tendsto.{u_2, u_1} (β := α) (fun (x : Finset β × Finset β) ↦ HDiv.hDiv (α := α) (∏ b ∈ x.2, f b) (∏ b ∈ x.1, f b)) atTop
    (𝓝 1) ↔
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
Tactic: simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
    uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
⊢ Tendsto.{u_2, u_1} (β := α) (fun (x : Finset β × Finset β) ↦ HDiv.hDiv (α := α) (∏ b ∈ x.2, f b) (∏ b ∈ x.1, f b))
      atTop (𝓝 1) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have rw : (∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) ↔
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
Tactic: rw [tendsto_atTop']
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
⊢ (∀ (s : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) s →
        ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have mp : (∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) →
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
mp :
  (∀ (s : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) s →
        ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) →
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
⊢ (∀ (s : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) s →
        ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have mpr : (∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e) →
  ∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h :
  ∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
⊢ ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have mp : ∀ (s₁ s₂ : Finset β) (h : ∀ b ≥ (s₁, s₂), HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e),
  ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
Tactic: obtain ⟨⟨s₁, s₂⟩, h⟩ := h e he
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
s₁ s₂ : Finset β
h : ∀ b ≥ (s₁, s₂), HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
⊢ ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have h₁ : ∀ (t : Finset β), Disjoint t (s₁ ∪ s₂) → ∏ b ∈ t, f b ∈ e
Tactic: use s₁ ∪ s₂
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
s₁ s₂ : Finset β
h : ∀ b ≥ (s₁, s₂), HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
t : Finset β
ht : Disjoint t (s₁ ∪ s₂)
⊢ ∏ b ∈ t, f b ∈ e
have h₁ : HDiv.hDiv (α := α) (∏ b ∈ (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t).2, f b) (∏ b ∈ (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t).1, f b) ∈ e
Tactic: specialize h (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t) ⟨le_sup_left, le_sup_of_le_left le_sup_right⟩
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
⊢ ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
have mpr : ∀ (d : Set α) (hd : Membership.mem (γ := Filter α) (𝓝 1) d) (hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e),
  ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
Tactic: rcases exists_nhds_split_inv he with ⟨d, hd, hde⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
d : Set α
hd : Membership.mem (γ := Filter α) (𝓝 1) d
hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e
⊢ ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
have mpr : ∀ (s : Finset β) (h : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ d),
  ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
Tactic: rcases h d hd with ⟨s, h⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
d : Set α
hd : Membership.mem (γ := Filter α) (𝓝 1) d
hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e
s : Finset β
h : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ d
⊢ ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
have h₁ : ∀ b ≥ (s, s), HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ e
Tactic: use (s, s)
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
d : Set α
hd : Membership.mem (γ := Filter α) (𝓝 1) d
hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e
s : Finset β
h : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ d
t₁ t₂ : Finset β
ht₁ : (s, s).1 ≤ (t₁, t₂).1
ht₂ : (s, s).2 ≤ (t₁, t₂).2
⊢ HDiv.hDiv (α := α) (∏ b ∈ t₂, f b) (∏ b ∈ t₁, f b) = HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
have rw : (∏ b ∈ t₂, f b) / HMul.hMul (α := α) (∏ x ∈ (t₁, t₂).1 \ (s, s).1, f x) (∏ x ∈ (s, s).1, f x) =
  HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
Tactic: rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
d : Set α
hd : Membership.mem (γ := Filter α) (𝓝 1) d
hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e
s : Finset β
h : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ d
t₁ t₂ : Finset β
ht₁ : (s, s).1 ≤ (t₁, t₂).1
ht₂ : (s, s).2 ≤ (t₁, t₂).2
rw :
  (∏ b ∈ t₂, f b) / HMul.hMul (α := α) (∏ x ∈ (t₁, t₂).1 \ (s, s).1, f x) (∏ x ∈ (s, s).1, f x) =
    HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
⊢ HDiv.hDiv (α := α) (∏ b ∈ t₂, f b) (∏ b ∈ t₁, f b) = HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
have rw₁ : HMul.hMul (α := α) (∏ x ∈ (t₁, t₂).2 \ (s, s).2, f x) (∏ x ∈ (s, s).2, f x) /
    HMul.hMul (α := α) (∏ x ∈ (t₁, t₂).1 \ (s, s).1, f x) (∏ x ∈ (s, s).1, f x) =
  HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
Tactic: rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
d : Set α
hd : Membership.mem (γ := Filter α) (𝓝 1) d
hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e
s : Finset β
h : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ d
t₁ t₂ : Finset β
ht₁ : (s, s).1 ≤ (t₁, t₂).1
ht₂ : (s, s).2 ≤ (t₁, t₂).2
rw :
  (∏ b ∈ t₂, f b) / HMul.hMul (α := α) (∏ x ∈ (t₁, t₂).1 \ (s, s).1, f x) (∏ x ∈ (s, s).1, f x) =
    HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
rw₁ :
  HMul.hMul (α := α) (∏ x ∈ (t₁, t₂).2 \ (s, s).2, f x) (∏ x ∈ (s, s).2, f x) /
      HMul.hMul (α := α) (∏ x ∈ (t₁, t₂).1 \ (s, s).1, f x) (∏ x ∈ (s, s).1, f x) =
    HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
⊢ HDiv.hDiv (α := α) (∏ b ∈ t₂, f b) (∏ b ∈ t₁, f b) = HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
have rw₂ : HDiv.hDiv (α := α) (∏ x ∈ (t₁, t₂).2 \ (s, s).2, f x) (∏ x ∈ (t₁, t₂).1 \ (s, s).1, f x) =
  HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
Tactic: rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (hypothesis)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
d : Set α
hd : Membership.mem (γ := Filter α) (𝓝 1) d
hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e
s : Finset β
h : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ d
t₁ t₂ : Finset β
ht₁ : (s, s).1 ≤ (t₁, t₂).1
ht₂ : (s, s).2 ≤ (t₁, t₂).2
⊢ HDiv.hDiv (α := α) (∏ b ∈ (t₁, t₂).2, f b) (∏ b ∈ (t₁, t₂).1, f b) ∈ e
have this : HDiv.hDiv (α := α) (∏ b ∈ t₂, f b) (∏ b ∈ t₁, f b) = HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
Tactic: have : ((∏ b ∈ t₂, f b) / ∏ b ∈ t₁, f b) = (∏ b ∈ t₂ \ s, f b) / ∏ b ∈ t₁ \ s, f b := by
      rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
h_1 :
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
e : Set α
he : Membership.mem (γ := Filter α) (𝓝 1) e
d : Set α
hd : Membership.mem (γ := Filter α) (𝓝 1) d
hde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e
s : Finset β
h : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ d
t₁ t₂ : Finset β
ht₁ : (s, s).1 ≤ (t₁, t₂).1
ht₂ : (s, s).2 ≤ (t₁, t₂).2
this : HDiv.hDiv (α := α) (∏ b ∈ t₂, f b) (∏ b ∈ t₁, f b) = HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b)
⊢ HDiv.hDiv (α := α) (∏ b ∈ (t₁, t₂).2, f b) (∏ b ∈ (t₁, t₂).1, f b) ∈ e
have h₁ : HDiv.hDiv (α := α) (∏ b ∈ t₂ \ s, f b) (∏ b ∈ t₁ \ s, f b) ∈ e
Tactic: simp only [this]
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
⊢ (CauchySeq.{u_1, u_2} (α := α) fun (s : Finset β) ↦ ∏ b ∈ s, f b) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have simp : Tendsto.{u_2, u_1} (β := α) (fun (x : Finset β × Finset β) ↦ HDiv.hDiv (α := α) (∏ b ∈ x.2, f b) (∏ b ∈ x.1, f b)) atTop
    (𝓝 1) ↔
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
Tactic: classical
  simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
    uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and]
  rw [tendsto_atTop']
  constructor
  · intro h e he
    obtain ⟨⟨s₁, s₂⟩, h⟩ := h e he
    use s₁ ∪ s₂
    intro t ht
    specialize h (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t) ⟨le_sup_left, le_sup_of_le_left le_sup_right⟩
    simpa only [Finset.prod_union ht.symm, mul_div_cancel_left] using h
  · rintro h e he
    rcases exists_nhds_split_inv he with ⟨d, hd, hde⟩
    rcases h d hd with ⟨s, h⟩
    use (s, s)
    rintro ⟨t₁, t₂⟩ ⟨ht₁, ht₂⟩
    have : ((∏ b ∈ t₂, f b) / ∏ b ∈ t₁, f b) = (∏ b ∈ t₂ \ s, f b) / ∏ b ∈ t₁ \ s, f b := by
      rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
    simp only [this]
    exact hde _ (h _ Finset.sdiff_disjoint) _ (h _ Finset.sdiff_disjoint)
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
simp :
  Tendsto.{u_2, u_1} (β := α) (fun (x : Finset β × Finset β) ↦ HDiv.hDiv (α := α) (∏ b ∈ x.2, f b) (∏ b ∈ x.1, f b))
      atTop (𝓝 1) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
⊢ (CauchySeq.{u_1, u_2} (α := α) fun (s : Finset β) ↦ ∏ b ∈ s, f b) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have simp₁ : (∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) ↔
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
Tactic: classical
  simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
    uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and]
  rw [tendsto_atTop']
  constructor
  · intro h e he
    obtain ⟨⟨s₁, s₂⟩, h⟩ := h e he
    use s₁ ∪ s₂
    intro t ht
    specialize h (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t) ⟨le_sup_left, le_sup_of_le_left le_sup_right⟩
    simpa only [Finset.prod_union ht.symm, mul_div_cancel_left] using h
  · rintro h e he
    rcases exists_nhds_split_inv he with ⟨d, hd, hde⟩
    rcases h d hd with ⟨s, h⟩
    use (s, s)
    rintro ⟨t₁, t₂⟩ ⟨ht₁, ht₂⟩
    have : ((∏ b ∈ t₂, f b) / ∏ b ∈ t₁, f b) = (∏ b ∈ t₂ \ s, f b) / ∏ b ∈ t₁ \ s, f b := by
      rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
    simp only [this]
    exact hde _ (h _ Finset.sdiff_disjoint) _ (h _ Finset.sdiff_disjoint)
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
simp :
  Tendsto.{u_2, u_1} (β := α) (fun (x : Finset β × Finset β) ↦ HDiv.hDiv (α := α) (∏ b ∈ x.2, f b) (∏ b ∈ x.1, f b))
      atTop (𝓝 1) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
simp₁ :
  (∀ (s : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) s →
        ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
⊢ (CauchySeq.{u_1, u_2} (α := α) fun (s : Finset β) ↦ ∏ b ∈ s, f b) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have mp : (∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) →
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
Tactic: classical
  simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
    uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and]
  rw [tendsto_atTop']
  constructor
  · intro h e he
    obtain ⟨⟨s₁, s₂⟩, h⟩ := h e he
    use s₁ ∪ s₂
    intro t ht
    specialize h (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t) ⟨le_sup_left, le_sup_of_le_left le_sup_right⟩
    simpa only [Finset.prod_union ht.symm, mul_div_cancel_left] using h
  · rintro h e he
    rcases exists_nhds_split_inv he with ⟨d, hd, hde⟩
    rcases h d hd with ⟨s, h⟩
    use (s, s)
    rintro ⟨t₁, t₂⟩ ⟨ht₁, ht₂⟩
    have : ((∏ b ∈ t₂, f b) / ∏ b ∈ t₁, f b) = (∏ b ∈ t₂ \ s, f b) / ∏ b ∈ t₁ \ s, f b := by
      rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
    simp only [this]
    exact hde _ (h _ Finset.sdiff_disjoint) _ (h _ Finset.sdiff_disjoint)
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
simp :
  Tendsto.{u_2, u_1} (β := α) (fun (x : Finset β × Finset β) ↦ HDiv.hDiv (α := α) (∏ b ∈ x.2, f b) (∏ b ∈ x.1, f b))
      atTop (𝓝 1) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
simp₁ :
  (∀ (s : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) s →
        ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
mp :
  (∀ (s : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) s →
        ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s) →
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
⊢ (CauchySeq.{u_1, u_2} (α := α) fun (s : Finset β) ↦ ∏ b ∈ s, f b) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
have mpr : (∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e → ∃ (s : Finset β), ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e) →
  ∀ (s : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) s →
      ∃ (a : Finset β × Finset β), ∀ b ≥ a, HDiv.hDiv (α := α) (∏ b ∈ b.2, f b) (∏ b ∈ b.1, f b) ∈ s
Tactic: classical
  simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
    uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and]
  rw [tendsto_atTop']
  constructor
  · intro h e he
    obtain ⟨⟨s₁, s₂⟩, h⟩ := h e he
    use s₁ ∪ s₂
    intro t ht
    specialize h (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t) ⟨le_sup_left, le_sup_of_le_left le_sup_right⟩
    simpa only [Finset.prod_union ht.symm, mul_div_cancel_left] using h
  · rintro h e he
    rcases exists_nhds_split_inv he with ⟨d, hd, hde⟩
    rcases h d hd with ⟨s, h⟩
    use (s, s)
    rintro ⟨t₁, t₂⟩ ⟨ht₁, ht₂⟩
    have : ((∏ b ∈ t₂, f b) / ∏ b ∈ t₁, f b) = (∏ b ∈ t₂ \ s, f b) / ∏ b ∈ t₁ \ s, f b := by
      rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
    simp only [this]
    exact hde _ (h _ Finset.sdiff_disjoint) _ (h _ Finset.sdiff_disjoint)
-- Lean 4 proof step from Mathlib.Topology.Algebra.InfiniteSum.Group (goal)
Goal:
α : Type u_1
β : Type u_2
inst : UniformSpace α
inst_1 : CommGroup α
inst_2 : IsUniformGroup α
f : β → α
⊢ (CauchySeq.{u_1, u_2} (α := α) fun (s : Finset β) ↦ ∏ b ∈ s, f b) ↔
    ∀ (e : Set α),
      Membership.mem (γ := Filter α) (𝓝 1) e →
        ∃ (s : Finset β), ∀ (t : Set β), Disjoint t (↑s : Set β) → ∏' (b : (↑t : Type u_2)), f (↑b : β) ∈ e
have simp_rw : (∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e →
      ∃ (s : Finset β), ∀ (t : Finset β), (∀ ⦃a : β⦄, a ∈ t → a ∉ s) → ∏ b ∈ t, f b ∈ e) ↔
  ∀ (e : Set α),
    Membership.mem (γ := Filter α) (𝓝 1) e →
      ∃ (s : Finset β), ∀ (t : Set β), (∀ ⦃a : β⦄, a ∈ t → a ∉ (↑s : Set β)) → ∏' (b : (↑t : Type u_2)), f (↑b : β) ∈ e
Tactic: simp_rw [cauchySeq_finset_iff_prod_vanishing, Set.disjoint_left, disjoint_left]