ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasProd.hasProd_compl_iff
@[to_additive]
theorem HasProd.hasProd_compl_iff {s : Set β} (hf : HasProd (f ∘ (↑) : s → α) a₁) :
    HasProd (f ∘ (↑) : ↑sᶜ → α) a₂ ↔ HasProd f (a₁ * a₂) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} {a₁ a₂ : α} {s : Set β} (hf : HasProd (f ∘ Subtype.val) a₁)
Goal: HasProd (f ∘ Subtype.val) a₂ ↔ HasProd f (a₁ * a₂)
-- HasSum.hasSum_iff_compl
@[to_additive]
theorem HasProd.hasProd_iff_compl {s : Set β} (hf : HasProd (f ∘ (↑) : s → α) a₁) :
    HasProd f a₂ ↔ HasProd (f ∘ (↑) : ↑sᶜ → α) (a₂ / a₁) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} {a₁ a₂ : α} {s : Set β} (hf : HasSum (f ∘ Subtype.val) a₁)
Goal: HasSum f a₂ ↔ HasSum (f ∘ Subtype.val) (a₂ - a₁)
-- HasProd.hasProd_iff_compl
@[to_additive]
theorem HasProd.hasProd_iff_compl {s : Set β} (hf : HasProd (f ∘ (↑) : s → α) a₁) :
    HasProd f a₂ ↔ HasProd (f ∘ (↑) : ↑sᶜ → α) (a₂ / a₁) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} {a₁ a₂ : α} {s : Set β} (hf : HasProd (f ∘ Subtype.val) a₁)
Goal: HasProd f a₂ ↔ HasProd (f ∘ Subtype.val) (a₂ / a₁)
-- Multipliable.multipliable_compl_iff
@[to_additive]
theorem Multipliable.multipliable_compl_iff {s : Set β} (hf : Multipliable (f ∘ (↑) : s → α)) :
    Multipliable (f ∘ (↑) : ↑sᶜ → α) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} {s : Set β} (hf : Multipliable (f ∘ Subtype.val))
Goal: Multipliable (f ∘ Subtype.val) ↔ Multipliable f
-- Summable.summable_compl_iff
@[to_additive]
theorem Multipliable.multipliable_compl_iff {s : Set β} (hf : Multipliable (f ∘ (↑) : s → α)) :
    Multipliable (f ∘ (↑) : ↑sᶜ → α) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} {s : Set β} (hf : Summable (f ∘ Subtype.val))
Goal: Summable (f ∘ Subtype.val) ↔ Summable f
-- Finset.hasProd_compl_iff
@[to_additive]
protected theorem Finset.hasProd_compl_iff (s : Finset β) :
    HasProd (fun x : { x // x ∉ s } ↦ f x) a ↔ HasProd f (a * ∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} {a : α} (s : Finset β)
Goal: HasProd (fun x => f ↑x) a ↔ HasProd f (a * ∏ i ∈ s, f i)
-- Finset.hasSum_compl_iff
@[to_additive]
protected theorem Finset.hasProd_compl_iff (s : Finset β) :
    HasProd (fun x : { x // x ∉ s } ↦ f x) a ↔ HasProd f (a * ∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} {a : α} (s : Finset β)
Goal: HasSum (fun x => f ↑x) a ↔ HasSum f (a + ∑ i ∈ s, f i)
-- Finset.hasProd_iff_compl
@[to_additive]
protected theorem Finset.hasProd_iff_compl (s : Finset β) :
    HasProd f a ↔ HasProd (fun x : { x // x ∉ s } ↦ f x) (a / ∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} {a : α} (s : Finset β)
Goal: HasProd f a ↔ HasProd (fun x => f ↑x) (a / ∏ i ∈ s, f i)
-- Finset.hasSum_iff_compl
@[to_additive]
protected theorem Finset.hasProd_iff_compl (s : Finset β) :
    HasProd f a ↔ HasProd (fun x : { x // x ∉ s } ↦ f x) (a / ∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} {a : α} (s : Finset β)
Goal: HasSum f a ↔ HasSum (fun x => f ↑x) (a - ∑ i ∈ s, f i)
-- Finset.summable_compl_iff
@[to_additive]
protected theorem Finset.multipliable_compl_iff (s : Finset β) :
    (Multipliable fun x : { x // x ∉ s } ↦ f x) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} (s : Finset β)
Goal: (Summable fun x => f ↑x) ↔ Summable f
-- Finset.multipliable_compl_iff
@[to_additive]
protected theorem Finset.multipliable_compl_iff (s : Finset β) :
    (Multipliable fun x : { x // x ∉ s } ↦ f x) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} (s : Finset β)
Goal: (Multipliable fun x => f ↑x) ↔ Multipliable f
-- Set.Finite.multipliable_compl_iff
@[to_additive]
theorem Set.Finite.multipliable_compl_iff {s : Set β} (hs : s.Finite) :
    Multipliable (f ∘ (↑) : ↑sᶜ → α) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} {s : Set β} (hs : s.Finite)
Goal: Multipliable (f ∘ Subtype.val) ↔ Multipliable f
-- Set.Finite.summable_compl_iff
@[to_additive]
theorem Set.Finite.multipliable_compl_iff {s : Set β} (hs : s.Finite) :
    Multipliable (f ∘ (↑) : ↑sᶜ → α) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} {s : Set β} (hs : s.Finite)
Goal: Summable (f ∘ Subtype.val) ↔ Summable f
-- hasProd_ite_div_hasProd
@[to_additive]
theorem hasProd_ite_div_hasProd [DecidableEq β] (hf : HasProd f a) (b : β) :
    HasProd (fun n ↦ ite (n = b) 1 (f n)) (a / f b) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} {a : α} [DecidableEq β] (hf : HasProd f a) (b : β)
Goal: HasProd (fun n => if n = b then 1 else f n) (a / f b)
-- hasSum_ite_sub_hasSum
@[to_additive]
theorem hasProd_ite_div_hasProd [DecidableEq β] (hf : HasProd f a) (b : β) :
    HasProd (fun n ↦ ite (n = b) 1 (f n)) (a / f b) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} {a : α} [DecidableEq β] (hf : HasSum f a) (b : β)
Goal: HasSum (fun n => if n = b then 0 else f n) (a - f b)
-- tsum_neg
@[to_additive]
theorem tprod_inv : ∏' b, (f b)⁻¹ = (∏' b, f b)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} [T2Space α]
Goal: ∑' (b : β), -f b = -∑' (b : β), f b
-- tprod_inv
@[to_additive]
theorem tprod_inv : ∏' b, (f b)⁻¹ = (∏' b, f b)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} [T2Space α]
Goal: ∏' (b : β), (f b)⁻¹ = (∏' (b : β), f b)⁻¹
-- tprod_div
@[to_additive]
theorem tprod_div (hf : Multipliable f) (hg : Multipliable g) :
    ∏' b, (f b / g b) = (∏' b, f b) / ∏' b, g b 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f g : β → α} [T2Space α] (hf : Multipliable f) (hg : Multipliable g)
Goal: ∏' (b : β), f b / g b = (∏' (b : β), f b) / ∏' (b : β), g b
-- tsum_sub
@[to_additive]
theorem tprod_div (hf : Multipliable f) (hg : Multipliable g) :
    ∏' b, (f b / g b) = (∏' b, f b) / ∏' b, g b 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f g : β → α} [T2Space α] (hf : Summable f) (hg : Summable g)
Goal: ∑' (b : β), (f b - g b) = ∑' (b : β), f b - ∑' (b : β), g b
-- sum_add_tsum_compl
@[to_additive]
theorem prod_mul_tprod_compl {s : Finset β} (hf : Multipliable f) :
    (∏ x ∈ s, f x) * ∏' x : ↑(s : Set β)ᶜ, f x = ∏' x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} [T2Space α] {s : Finset β} (hf : Summable f)
Goal: ∑ x ∈ s, f x + ∑' (x : ↑(↑s)ᶜ), f ↑x = ∑' (x : β), f x
-- prod_mul_tprod_compl
@[to_additive]
theorem prod_mul_tprod_compl {s : Finset β} (hf : Multipliable f) :
    (∏ x ∈ s, f x) * ∏' x : ↑(s : Set β)ᶜ, f x = ∏' x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} [T2Space α] {s : Finset β} (hf : Multipliable f)
Goal: (∏ x ∈ s, f x) * ∏' (x : ↑(↑s)ᶜ), f ↑x = ∏' (x : β), f x
-- tsum_eq_add_tsum_ite
@[to_additive "Let `f : β → α` be a summable function and let `b ∈ β` be an index.
Lemma `tsum_eq_add_tsum_ite` writes `Σ' n, f n` as `f b` plus the sum of the
remaining terms."]
theorem tprod_eq_mul_tprod_ite [DecidableEq β] (hf : Multipliable f) (b : β) :
    ∏' n, f n = f b * ∏' n, ite (n = b) 1 (f n) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] {f : β → α} [T2Space α] [DecidableEq β] (hf : Summable f) (b : β)
Goal: ∑' (n : β), f n = f b + ∑' (n : β), if n = b then 0 else f n
-- tprod_eq_mul_tprod_ite
@[to_additive "Let `f : β → α` be a summable function and let `b ∈ β` be an index.
Lemma `tsum_eq_add_tsum_ite` writes `Σ' n, f n` as `f b` plus the sum of the
remaining terms."]
theorem tprod_eq_mul_tprod_ite [DecidableEq β] (hf : Multipliable f) (b : β) :
    ∏' n, f n = f b * ∏' n, ite (n = b) 1 (f n) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [TopologicalSpace α] [TopologicalGroup α] {f : β → α} [T2Space α] [DecidableEq β] (hf : Multipliable f) (b : β)
Goal: ∏' (n : β), f n = f b * ∏' (n : β), if n = b then 1 else f n
-- multipliable_iff_cauchySeq_finset
@[to_additive "The **Cauchy criterion** for infinite sums, also known as the
**Cauchy convergence test**"]
theorem multipliable_iff_cauchySeq_finset [CompleteSpace α] {f : β → α} :
    Multipliable f ↔ CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [CompleteSpace α] {f : β → α}
Goal: Multipliable f ↔ CauchySeq fun s => ∏ b ∈ s, f b
-- summable_iff_cauchySeq_finset
@[to_additive "The **Cauchy criterion** for infinite sums, also known as the
**Cauchy convergence test**"]
theorem multipliable_iff_cauchySeq_finset [CompleteSpace α] {f : β → α} :
    Multipliable f ↔ CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [CompleteSpace α] {f : β → α}
Goal: Summable f ↔ CauchySeq fun s => ∑ b ∈ s, f b
-- cauchySeq_finset_iff_sum_vanishing
@[to_additive]
theorem cauchySeq_finset_iff_prod_vanishing :
    (CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b) ↔
      ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α}
Goal: (CauchySeq fun s => ∑ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 0, ∃ s, ∀ (t : Finset β), Disjoint t s → ∑ b ∈ t, f b ∈ e
-- cauchySeq_finset_iff_prod_vanishing
@[to_additive]
theorem cauchySeq_finset_iff_prod_vanishing :
    (CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b) ↔
      ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α}
Goal: (CauchySeq fun s => ∏ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
-- cauchySeq_finset_iff_tsum_vanishing
@[to_additive]
theorem cauchySeq_finset_iff_tprod_vanishing :
    (CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b) ↔
      ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t : Set β, Disjoint t s → (∏' b : t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α}
Goal: (CauchySeq fun s => ∑ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 0, ∃ s, ∀ (t : Set β), Disjoint t ↑s → ∑' (b : ↑t), f ↑b ∈ e
-- cauchySeq_finset_iff_tprod_vanishing
@[to_additive]
theorem cauchySeq_finset_iff_tprod_vanishing :
    (CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b) ↔
      ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t : Set β, Disjoint t s → (∏' b : t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α}
Goal: (CauchySeq fun s => ∏ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), Disjoint t ↑s → ∏' (b : ↑t), f ↑b ∈ e
-- multipliable_iff_vanishing
@[to_additive]
theorem multipliable_iff_vanishing :
    Multipliable f ↔
    ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α} [CompleteSpace α]
Goal: Multipliable f ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
-- summable_iff_vanishing
@[to_additive]
theorem multipliable_iff_vanishing :
    Multipliable f ↔
    ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α} [CompleteSpace α]
Goal: Summable f ↔ ∀ e ∈ 𝓝 0, ∃ s, ∀ (t : Finset β), Disjoint t s → ∑ b ∈ t, f b ∈ e
-- multipliable_iff_tprod_vanishing
@[to_additive]
theorem multipliable_iff_tprod_vanishing : Multipliable f ↔
    ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t : Set β, Disjoint t s → (∏' b : t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α} [CompleteSpace α]
Goal: Multipliable f ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), Disjoint t ↑s → ∏' (b : ↑t), f ↑b ∈ e
-- summable_iff_tsum_vanishing
@[to_additive]
theorem multipliable_iff_tprod_vanishing : Multipliable f ↔
    ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t : Set β, Disjoint t s → (∏' b : t, f b) ∈ e 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α} [CompleteSpace α]
Goal: Summable f ↔ ∀ e ∈ 𝓝 0, ∃ s, ∀ (t : Set β), Disjoint t ↑s → ∑' (b : ↑t), f ↑b ∈ e
-- Multipliable.multipliable_of_eq_one_or_self
@[to_additive]
theorem Multipliable.multipliable_of_eq_one_or_self (hf : Multipliable f)
    (h : ∀ b, g b = 1 ∨ g b = f b) : Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f g : β → α} [CompleteSpace α] (hf : Multipliable f) (h : ∀ (b : β), g b = 1 ∨ g b = f b)
Goal: Multipliable g
-- Summable.summable_of_eq_zero_or_self
@[to_additive]
theorem Multipliable.multipliable_of_eq_one_or_self (hf : Multipliable f)
    (h : ∀ b, g b = 1 ∨ g b = f b) : Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f g : β → α} [CompleteSpace α] (hf : Summable f) (h : ∀ (b : β), g b = 0 ∨ g b = f b)
Goal: Summable g
-- Summable.indicator
@[to_additive]
protected theorem Multipliable.mulIndicator (hf : Multipliable f) (s : Set β) :
    Multipliable (s.mulIndicator f) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α} [CompleteSpace α] (hf : Summable f) (s : Set β)
Goal: Summable (s.indicator f)
-- Multipliable.mulIndicator
@[to_additive]
protected theorem Multipliable.mulIndicator (hf : Multipliable f) (s : Set β) :
    Multipliable (s.mulIndicator f) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α} [CompleteSpace α] (hf : Multipliable f) (s : Set β)
Goal: Multipliable (s.mulIndicator f)
-- Summable.comp_injective
@[to_additive]
theorem Multipliable.comp_injective {i : γ → β} (hf : Multipliable f) (hi : Injective i) :
    Multipliable (f ∘ i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α} [CompleteSpace α] {i : γ → β} (hf : Summable f) (hi : Function.Injective i)
Goal: Summable (f ∘ i)
-- Multipliable.comp_injective
@[to_additive]
theorem Multipliable.comp_injective {i : γ → β} (hf : Multipliable f) (hi : Injective i) :
    Multipliable (f ∘ i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α} [CompleteSpace α] {i : γ → β} (hf : Multipliable f) (hi : Function.Injective i)
Goal: Multipliable (f ∘ i)
-- Multipliable.subtype
@[to_additive]
theorem Multipliable.subtype (hf : Multipliable f) (s : Set β) : Multipliable (f ∘ (↑) : s → α) 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α} [CompleteSpace α] (hf : Multipliable f) (s : Set β)
Goal: Multipliable (f ∘ Subtype.val)
-- Summable.subtype
@[to_additive]
theorem Multipliable.subtype (hf : Multipliable f) (s : Set β) : Multipliable (f ∘ (↑) : s → α) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α} [CompleteSpace α] (hf : Summable f) (s : Set β)
Goal: Summable (f ∘ Subtype.val)
-- summable_subtype_and_compl
@[to_additive]
theorem multipliable_subtype_and_compl {s : Set β} :
    ((Multipliable fun x : s ↦ f x) ∧ Multipliable fun x : ↑sᶜ ↦ f x) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α} [CompleteSpace α] {s : Set β}
Goal: ((Summable fun x => f ↑x) ∧ Summable fun x => f ↑x) ↔ Summable f
-- multipliable_subtype_and_compl
@[to_additive]
theorem multipliable_subtype_and_compl {s : Set β} :
    ((Multipliable fun x : s ↦ f x) ∧ Multipliable fun x : ↑sᶜ ↦ f x) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] {f : β → α} [CompleteSpace α] {s : Set β}
Goal: ((Multipliable fun x => f ↑x) ∧ Multipliable fun x => f ↑x) ↔ Multipliable f
-- tsum_subtype_add_tsum_subtype_compl
@[to_additive]
theorem tprod_subtype_mul_tprod_subtype_compl [T2Space α] {f : β → α} (hf : Multipliable f)
    (s : Set β) : (∏' x : s, f x) * ∏' x : ↑sᶜ, f x = ∏' x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] [CompleteSpace α] [T2Space α] {f : β → α} (hf : Summable f) (s : Set β)
Goal: ∑' (x : ↑s), f ↑x + ∑' (x : ↑sᶜ), f ↑x = ∑' (x : β), f x
-- tprod_subtype_mul_tprod_subtype_compl
@[to_additive]
theorem tprod_subtype_mul_tprod_subtype_compl [T2Space α] {f : β → α} (hf : Multipliable f)
    (s : Set β) : (∏' x : s, f x) * ∏' x : ↑sᶜ, f x = ∏' x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] [T2Space α] {f : β → α} (hf : Multipliable f) (s : Set β)
Goal: (∏' (x : ↑s), f ↑x) * ∏' (x : ↑sᶜ), f ↑x = ∏' (x : β), f x
-- sum_add_tsum_subtype_compl
@[to_additive]
theorem prod_mul_tprod_subtype_compl [T2Space α] {f : β → α} (hf : Multipliable f) (s : Finset β) :
    (∏ x ∈ s, f x) * ∏' x : { x // x ∉ s }, f x = ∏' x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] [CompleteSpace α] [T2Space α] {f : β → α} (hf : Summable f) (s : Finset β)
Goal: ∑ x ∈ s, f x + ∑' (x : { x // x ∉ s }), f ↑x = ∑' (x : β), f x
-- prod_mul_tprod_subtype_compl
@[to_additive]
theorem prod_mul_tprod_subtype_compl [T2Space α] {f : β → α} (hf : Multipliable f) (s : Finset β) :
    (∏ x ∈ s, f x) * ∏' x : { x // x ∉ s }, f x = ∏' x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] [T2Space α] {f : β → α} (hf : Multipliable f) (s : Finset β)
Goal: (∏ x ∈ s, f x) * ∏' (x : { x // x ∉ s }), f ↑x = ∏' (x : β), f x
-- Summable.vanishing
@[to_additive]
theorem Multipliable.vanishing (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 (1 : G)) :
    ∃ s : Finset α, ∀ t, Disjoint t s → (∏ k ∈ t, f k) ∈ e 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [AddCommGroup G] [TopologicalAddGroup G] {f : α → G} (hf : Summable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 0)
Goal: ∃ s, ∀ (t : Finset α), Disjoint t s → ∑ k ∈ t, f k ∈ e
-- Multipliable.vanishing
@[to_additive]
theorem Multipliable.vanishing (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 (1 : G)) :
    ∃ s : Finset α, ∀ t, Disjoint t s → (∏ k ∈ t, f k) ∈ e 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [CommGroup G] [TopologicalGroup G] {f : α → G} (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 1)
Goal: ∃ s, ∀ (t : Finset α), Disjoint t s → ∏ k ∈ t, f k ∈ e
-- Multipliable.tprod_vanishing
@[to_additive]
theorem Multipliable.tprod_vanishing (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 1) :
    ∃ s : Finset α, ∀ t : Set α, Disjoint t s → (∏' b : t, f b) ∈ e 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [CommGroup G] [TopologicalGroup G] {f : α → G} (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 1)
Goal: ∃ s, ∀ (t : Set α), Disjoint t ↑s → ∏' (b : ↑t), f ↑b ∈ e
-- Summable.tsum_vanishing
@[to_additive]
theorem Multipliable.tprod_vanishing (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 1) :
    ∃ s : Finset α, ∀ t : Set α, Disjoint t s → (∏' b : t, f b) ∈ e 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [AddCommGroup G] [TopologicalAddGroup G] {f : α → G} (hf : Summable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 0)
Goal: ∃ s, ∀ (t : Set α), Disjoint t ↑s → ∑' (b : ↑t), f ↑b ∈ e
-- tendsto_tprod_compl_atTop_one
@[to_additive "The sum over the complement of a finset tends to `0` when the finset grows to cover
the whole space. This does not need a summability assumption, as otherwise all such sums are zero."]
theorem tendsto_tprod_compl_atTop_one (f : α → G) :
    Tendsto (fun s : Finset α ↦ ∏' a : { x // x ∉ s }, f a) atTop (𝓝 1) 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [CommGroup G] [TopologicalGroup G] (f : α → G)
Goal: Filter.Tendsto (fun s => ∏' (a : { x // x ∉ s }), f ↑a) Filter.atTop (𝓝 1)
-- tendsto_tsum_compl_atTop_zero
@[to_additive "The sum over the complement of a finset tends to `0` when the finset grows to cover
the whole space. This does not need a summability assumption, as otherwise all such sums are zero."]
theorem tendsto_tprod_compl_atTop_one (f : α → G) :
    Tendsto (fun s : Finset α ↦ ∏' a : { x // x ∉ s }, f a) atTop (𝓝 1) 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [AddCommGroup G] [TopologicalAddGroup G] (f : α → G)
Goal: Filter.Tendsto (fun s => ∑' (a : { x // x ∉ s }), f ↑a) Filter.atTop (𝓝 0)
-- Summable.tendsto_cofinite_zero
@[to_additive "Series divergence test: if `f` is unconditionally summable, then `f x` tends to zero
along `cofinite`."]
theorem Multipliable.tendsto_cofinite_one (hf : Multipliable f) : Tendsto f cofinite (𝓝 1) 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [AddCommGroup G] [TopologicalAddGroup G] {f : α → G} (hf : Summable f)
Goal: Filter.Tendsto f Filter.cofinite (𝓝 0)
-- Multipliable.tendsto_cofinite_one
@[to_additive "Series divergence test: if `f` is unconditionally summable, then `f x` tends to zero
along `cofinite`."]
theorem Multipliable.tendsto_cofinite_one (hf : Multipliable f) : Tendsto f cofinite (𝓝 1) 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [CommGroup G] [TopologicalGroup G] {f : α → G} (hf : Multipliable f)
Goal: Filter.Tendsto f Filter.cofinite (𝓝 1)
-- Multipliable.countable_mulSupport
@[to_additive]
theorem Multipliable.countable_mulSupport [FirstCountableTopology G] [T1Space G]
    (hf : Multipliable f) : f.mulSupport.Countable 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [CommGroup G] [TopologicalGroup G] {f : α → G} [FirstCountableTopology G] [T1Space G] (hf : Multipliable f)
Goal: (Function.mulSupport f).Countable
-- Summable.countable_support
@[to_additive]
theorem Multipliable.countable_mulSupport [FirstCountableTopology G] [T1Space G]
    (hf : Multipliable f) : f.mulSupport.Countable 
Arguments: {α : Type u_1} {G : Type u_5} [TopologicalSpace G] [AddCommGroup G] [TopologicalAddGroup G] {f : α → G} [FirstCountableTopology G] [T1Space G] (hf : Summable f)
Goal: (Function.support f).Countable
-- multipliable_const_iff
@[to_additive]
theorem multipliable_const_iff [Infinite β] [T2Space G] (a : G) :
    Multipliable (fun _ : β ↦ a) ↔ a = 1 
Arguments: {β : Type u_2} {G : Type u_5} [TopologicalSpace G] [CommGroup G] [TopologicalGroup G] [Infinite β] [T2Space G] (a : G)
Goal: (Multipliable fun x => a) ↔ a = 1
-- summable_const_iff
@[to_additive]
theorem multipliable_const_iff [Infinite β] [T2Space G] (a : G) :
    Multipliable (fun _ : β ↦ a) ↔ a = 1 
Arguments: {β : Type u_2} {G : Type u_5} [TopologicalSpace G] [AddCommGroup G] [TopologicalAddGroup G] [Infinite β] [T2Space G] (a : G)
Goal: (Summable fun x => a) ↔ a = 0
-- tsum_const
@[to_additive (attr := simp)]
theorem tprod_const [T2Space G] (a : G) : ∏' _ : β, a = a ^ (Nat.card β) 
Arguments: {β : Type u_2} {G : Type u_5} [TopologicalSpace G] [AddCommGroup G] [TopologicalAddGroup G] [T2Space G] (a : G)
Goal: ∑' (x : β), a = Nat.card β • a
-- tprod_const
@[to_additive (attr := simp)]
theorem tprod_const [T2Space G] (a : G) : ∏' _ : β, a = a ^ (Nat.card β) 
Arguments: {β : Type u_2} {G : Type u_5} [TopologicalSpace G] [CommGroup G] [TopologicalGroup G] [T2Space G] (a : G)
Goal: ∏' (x : β), a = a ^ Nat.card β