ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Inducing.hasSum_iff
@[to_additive]
protected theorem Inducing.hasProd_iff [CommMonoid γ] [TopologicalSpace γ] {G}
    [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : Inducing g) (f : β → α) (a : α) :
    HasProd (g ∘ f) (g a) ↔ HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [AddCommMonoid γ] [TopologicalSpace γ] {G : Type u_5} [FunLike G α γ] [AddMonoidHomClass G α γ] {g : G} (hg : Inducing ⇑g) (f : β → α) (a : α)
Goal: HasSum (⇑g ∘ f) (g a) ↔ HasSum f a
-- Multipliable.map
@[to_additive]
protected theorem Multipliable.map [CommMonoid γ] [TopologicalSpace γ] (hf : Multipliable f) {G}
    [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) : Multipliable (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} [CommMonoid γ] [TopologicalSpace γ] (hf : Multipliable f) {G : Type u_5} [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous ⇑g)
Goal: Multipliable (⇑g ∘ f)
-- Summable.map
@[to_additive]
protected theorem Multipliable.map [CommMonoid γ] [TopologicalSpace γ] (hf : Multipliable f) {G}
    [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) : Multipliable (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [AddCommMonoid γ] [TopologicalSpace γ] (hf : Summable f) {G : Type u_5} [FunLike G α γ] [AddMonoidHomClass G α γ] (g : G) (hg : Continuous ⇑g)
Goal: Summable (⇑g ∘ f)
-- Summable.map_iff_of_leftInverse
@[to_additive]
protected theorem Multipliable.map_iff_of_leftInverse [CommMonoid γ] [TopologicalSpace γ] {G G'}
    [FunLike G α γ] [MonoidHomClass G α γ] [FunLike G' γ α] [MonoidHomClass G' γ α]
    (g : G) (g' : G') (hg : Continuous g) (hg' : Continuous g') (hinv : Function.LeftInverse g' g) :
    Multipliable (g ∘ f) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [AddCommMonoid γ] [TopologicalSpace γ] {G : Type u_5} {G' : Type u_6} [FunLike G α γ] [AddMonoidHomClass G α γ] [FunLike G' γ α] [AddMonoidHomClass G' γ α] (g : G) (g' : G') (hg : Continuous ⇑g) (hg' : Continuous ⇑g') (hinv : Function.LeftInverse ⇑g' ⇑g)
Goal: Summable (⇑g ∘ f) ↔ Summable f
-- Multipliable.map_iff_of_leftInverse
@[to_additive]
protected theorem Multipliable.map_iff_of_leftInverse [CommMonoid γ] [TopologicalSpace γ] {G G'}
    [FunLike G α γ] [MonoidHomClass G α γ] [FunLike G' γ α] [MonoidHomClass G' γ α]
    (g : G) (g' : G') (hg : Continuous g) (hg' : Continuous g') (hinv : Function.LeftInverse g' g) :
    Multipliable (g ∘ f) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} [CommMonoid γ] [TopologicalSpace γ] {G : Type u_5} {G' : Type u_6} [FunLike G α γ] [MonoidHomClass G α γ] [FunLike G' γ α] [MonoidHomClass G' γ α] (g : G) (g' : G') (hg : Continuous ⇑g) (hg' : Continuous ⇑g') (hinv : Function.LeftInverse ⇑g' ⇑g)
Goal: Multipliable (⇑g ∘ f) ↔ Multipliable f
-- Summable.map_tsum
@[to_additive]
theorem Multipliable.map_tprod [CommMonoid γ] [TopologicalSpace γ] [T2Space γ] (hf : Multipliable f)
    {G} [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) :
    g (∏' i, f i) = ∏' i, g (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [AddCommMonoid γ] [TopologicalSpace γ] [T2Space γ] (hf : Summable f) {G : Type u_5} [FunLike G α γ] [AddMonoidHomClass G α γ] (g : G) (hg : Continuous ⇑g)
Goal: g (∑' (i : β), f i) = ∑' (i : β), g (f i)
-- Multipliable.map_tprod
@[to_additive]
theorem Multipliable.map_tprod [CommMonoid γ] [TopologicalSpace γ] [T2Space γ] (hf : Multipliable f)
    {G} [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) :
    g (∏' i, f i) = ∏' i, g (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} [CommMonoid γ] [TopologicalSpace γ] [T2Space γ] (hf : Multipliable f) {G : Type u_5} [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous ⇑g)
Goal: g (∏' (i : β), f i) = ∏' (i : β), g (f i)
-- Inducing.summable_iff_tsum_comp_mem_range
@[to_additive]
theorem Inducing.multipliable_iff_tprod_comp_mem_range [CommMonoid γ] [TopologicalSpace γ]
    [T2Space γ] {G} [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : Inducing g) (f : β → α) :
    Multipliable f ↔ Multipliable (g ∘ f) ∧ ∏' i, g (f i) ∈ Set.range g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [AddCommMonoid γ] [TopologicalSpace γ] [T2Space γ] {G : Type u_5} [FunLike G α γ] [AddMonoidHomClass G α γ] {g : G} (hg : Inducing ⇑g) (f : β → α)
Goal: Summable f ↔ Summable (⇑g ∘ f) ∧ ∑' (i : β), g (f i) ∈ Set.range ⇑g
-- Inducing.multipliable_iff_tprod_comp_mem_range
@[to_additive]
theorem Inducing.multipliable_iff_tprod_comp_mem_range [CommMonoid γ] [TopologicalSpace γ]
    [T2Space γ] {G} [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : Inducing g) (f : β → α) :
    Multipliable f ↔ Multipliable (g ∘ f) ∧ ∏' i, g (f i) ∈ Set.range g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] [CommMonoid γ] [TopologicalSpace γ] [T2Space γ] {G : Type u_5} [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : Inducing ⇑g) (f : β → α)
Goal: Multipliable f ↔ Multipliable (⇑g ∘ f) ∧ ∏' (i : β), g (f i) ∈ Set.range ⇑g
-- Multipliable.map_iff_of_equiv
@[to_additive "A special case of `Summable.map_iff_of_leftInverse` for convenience"]
protected theorem Multipliable.map_iff_of_equiv [CommMonoid γ] [TopologicalSpace γ] {G}
    [EquivLike G α γ] [MulEquivClass G α γ] (g : G) (hg : Continuous g)
    (hg' : Continuous (EquivLike.inv g : γ → α)) : Multipliable (g ∘ f) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} [CommMonoid γ] [TopologicalSpace γ] {G : Type u_5} [EquivLike G α γ] [MulEquivClass G α γ] (g : G) (hg : Continuous ⇑g) (hg' : Continuous (EquivLike.inv g))
Goal: Multipliable (⇑g ∘ f) ↔ Multipliable f
-- Summable.map_iff_of_equiv
@[to_additive "A special case of `Summable.map_iff_of_leftInverse` for convenience"]
protected theorem Multipliable.map_iff_of_equiv [CommMonoid γ] [TopologicalSpace γ] {G}
    [EquivLike G α γ] [MulEquivClass G α γ] (g : G) (hg : Continuous g)
    (hg' : Continuous (EquivLike.inv g : γ → α)) : Multipliable (g ∘ f) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [AddCommMonoid γ] [TopologicalSpace γ] {G : Type u_5} [EquivLike G α γ] [AddEquivClass G α γ] (g : G) (hg : Continuous ⇑g) (hg' : Continuous (EquivLike.inv g))
Goal: Summable (⇑g ∘ f) ↔ Summable f
-- Function.Surjective.multipliable_iff_of_hasProd_iff
@[to_additive]
theorem Function.Surjective.multipliable_iff_of_hasProd_iff {α' : Type*} [CommMonoid α']
    [TopologicalSpace α'] {e : α' → α} (hes : Function.Surjective e) {f : β → α} {g : γ → α'}
    (he : ∀ {a}, HasProd f (e a) ↔ HasProd g a) : Multipliable f ↔ Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {α' : Type u_5} [CommMonoid α'] [TopologicalSpace α'] {e : α' → α} (hes : Function.Surjective e) {f : β → α} {g : γ → α'} (he : ∀ {a : α'}, HasProd f (e a) ↔ HasProd g a)
Goal: Multipliable f ↔ Multipliable g
-- Function.Surjective.summable_iff_of_hasSum_iff
@[to_additive]
theorem Function.Surjective.multipliable_iff_of_hasProd_iff {α' : Type*} [CommMonoid α']
    [TopologicalSpace α'] {e : α' → α} (hes : Function.Surjective e) {f : β → α} {g : γ → α'}
    (he : ∀ {a}, HasProd f (e a) ↔ HasProd g a) : Multipliable f ↔ Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {α' : Type u_5} [AddCommMonoid α'] [TopologicalSpace α'] {e : α' → α} (hes : Function.Surjective e) {f : β → α} {g : γ → α'} (he : ∀ {a : α'}, HasSum f (e a) ↔ HasSum g a)
Goal: Summable f ↔ Summable g
-- HasSum.add
@[to_additive]
theorem HasProd.mul (hf : HasProd f a) (hg : HasProd g b) :
    HasProd (fun b ↦ f b * g b) (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f g : β → α} {a b : α} [ContinuousAdd α] (hf : HasSum f a) (hg : HasSum g b)
Goal: HasSum (fun b => f b + g b) (a + b)
-- HasProd.mul
@[to_additive]
theorem HasProd.mul (hf : HasProd f a) (hg : HasProd g b) :
    HasProd (fun b ↦ f b * g b) (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f g : β → α} {a b : α} [ContinuousMul α] (hf : HasProd f a) (hg : HasProd g b)
Goal: HasProd (fun b => f b * g b) (a * b)
-- Multipliable.mul
@[to_additive]
theorem Multipliable.mul (hf : Multipliable f) (hg : Multipliable g) :
    Multipliable fun b ↦ f b * g b 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f g : β → α} [ContinuousMul α] (hf : Multipliable f) (hg : Multipliable g)
Goal: Multipliable fun b => f b * g b
-- Summable.add
@[to_additive]
theorem Multipliable.mul (hf : Multipliable f) (hg : Multipliable g) :
    Multipliable fun b ↦ f b * g b 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f g : β → α} [ContinuousAdd α] (hf : Summable f) (hg : Summable g)
Goal: Summable fun b => f b + g b
-- hasProd_prod
@[to_additive]
theorem hasProd_prod {f : γ → β → α} {a : γ → α} {s : Finset γ} :
    (∀ i ∈ s, HasProd (f i) (a i)) → HasProd (fun b ↦ ∏ i ∈ s, f i b) (∏ i ∈ s, a i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] {f : γ → β → α} {a : γ → α} {s : Finset γ}
Goal: (∀ i ∈ s, HasProd (f i) (a i)) → HasProd (fun b => ∏ i ∈ s, f i b) (∏ i ∈ s, a i)
-- hasSum_sum
@[to_additive]
theorem hasProd_prod {f : γ → β → α} {a : γ → α} {s : Finset γ} :
    (∀ i ∈ s, HasProd (f i) (a i)) → HasProd (fun b ↦ ∏ i ∈ s, f i b) (∏ i ∈ s, a i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] {f : γ → β → α} {a : γ → α} {s : Finset γ}
Goal: (∀ i ∈ s, HasSum (f i) (a i)) → HasSum (fun b => ∑ i ∈ s, f i b) (∑ i ∈ s, a i)
-- summable_sum
@[to_additive]
theorem multipliable_prod {f : γ → β → α} {s : Finset γ} (hf : ∀ i ∈ s, Multipliable (f i)) :
    Multipliable fun b ↦ ∏ i ∈ s, f i b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] {f : γ → β → α} {s : Finset γ} (hf : ∀ i ∈ s, Summable (f i))
Goal: Summable fun b => ∑ i ∈ s, f i b
-- multipliable_prod
@[to_additive]
theorem multipliable_prod {f : γ → β → α} {s : Finset γ} (hf : ∀ i ∈ s, Multipliable (f i)) :
    Multipliable fun b ↦ ∏ i ∈ s, f i b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] {f : γ → β → α} {s : Finset γ} (hf : ∀ i ∈ s, Multipliable (f i))
Goal: Multipliable fun b => ∏ i ∈ s, f i b
-- HasSum.add_disjoint
@[to_additive]
theorem HasProd.mul_disjoint {s t : Set β} (hs : Disjoint s t) (ha : HasProd (f ∘ (↑) : s → α) a)
    (hb : HasProd (f ∘ (↑) : t → α) b) : HasProd (f ∘ (↑) : (s ∪ t : Set β) → α) (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousAdd α] {s t : Set β} (hs : Disjoint s t) (ha : HasSum (f ∘ Subtype.val) a) (hb : HasSum (f ∘ Subtype.val) b)
Goal: HasSum (f ∘ Subtype.val) (a + b)
-- HasProd.mul_disjoint
@[to_additive]
theorem HasProd.mul_disjoint {s t : Set β} (hs : Disjoint s t) (ha : HasProd (f ∘ (↑) : s → α) a)
    (hb : HasProd (f ∘ (↑) : t → α) b) : HasProd (f ∘ (↑) : (s ∪ t : Set β) → α) (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousMul α] {s t : Set β} (hs : Disjoint s t) (ha : HasProd (f ∘ Subtype.val) a) (hb : HasProd (f ∘ Subtype.val) b)
Goal: HasProd (f ∘ Subtype.val) (a * b)
-- hasSum_sum_disjoint
@[to_additive]
theorem hasProd_prod_disjoint {ι} (s : Finset ι) {t : ι → Set β} {a : ι → α}
    (hs : (s : Set ι).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, HasProd (f ∘ (↑) : t i → α) (a i)) :
    HasProd (f ∘ (↑) : (⋃ i ∈ s, t i) → α) (∏ i ∈ s, a i) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [ContinuousAdd α] {ι : Type u_5} (s : Finset ι) {t : ι → Set β} {a : ι → α} (hs : (↑s).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, HasSum (f ∘ Subtype.val) (a i))
Goal: HasSum (f ∘ Subtype.val) (∑ i ∈ s, a i)
-- hasProd_prod_disjoint
@[to_additive]
theorem hasProd_prod_disjoint {ι} (s : Finset ι) {t : ι → Set β} {a : ι → α}
    (hs : (s : Set ι).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, HasProd (f ∘ (↑) : t i → α) (a i)) :
    HasProd (f ∘ (↑) : (⋃ i ∈ s, t i) → α) (∏ i ∈ s, a i) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} [ContinuousMul α] {ι : Type u_5} (s : Finset ι) {t : ι → Set β} {a : ι → α} (hs : (↑s).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, HasProd (f ∘ Subtype.val) (a i))
Goal: HasProd (f ∘ Subtype.val) (∏ i ∈ s, a i)
-- HasProd.mul_isCompl
@[to_additive]
theorem HasProd.mul_isCompl {s t : Set β} (hs : IsCompl s t) (ha : HasProd (f ∘ (↑) : s → α) a)
    (hb : HasProd (f ∘ (↑) : t → α) b) : HasProd f (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousMul α] {s t : Set β} (hs : IsCompl s t) (ha : HasProd (f ∘ Subtype.val) a) (hb : HasProd (f ∘ Subtype.val) b)
Goal: HasProd f (a * b)
-- HasSum.add_isCompl
@[to_additive]
theorem HasProd.mul_isCompl {s t : Set β} (hs : IsCompl s t) (ha : HasProd (f ∘ (↑) : s → α) a)
    (hb : HasProd (f ∘ (↑) : t → α) b) : HasProd f (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousAdd α] {s t : Set β} (hs : IsCompl s t) (ha : HasSum (f ∘ Subtype.val) a) (hb : HasSum (f ∘ Subtype.val) b)
Goal: HasSum f (a + b)
-- HasProd.mul_compl
@[to_additive]
theorem HasProd.mul_compl {s : Set β} (ha : HasProd (f ∘ (↑) : s → α) a)
    (hb : HasProd (f ∘ (↑) : (sᶜ : Set β) → α) b) : HasProd f (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousMul α] {s : Set β} (ha : HasProd (f ∘ Subtype.val) a) (hb : HasProd (f ∘ Subtype.val) b)
Goal: HasProd f (a * b)
-- HasSum.add_compl
@[to_additive]
theorem HasProd.mul_compl {s : Set β} (ha : HasProd (f ∘ (↑) : s → α) a)
    (hb : HasProd (f ∘ (↑) : (sᶜ : Set β) → α) b) : HasProd f (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousAdd α] {s : Set β} (ha : HasSum (f ∘ Subtype.val) a) (hb : HasSum (f ∘ Subtype.val) b)
Goal: HasSum f (a + b)
-- Summable.add_compl
@[to_additive]
theorem Multipliable.mul_compl {s : Set β} (hs : Multipliable (f ∘ (↑) : s → α))
    (hsc : Multipliable (f ∘ (↑) : (sᶜ : Set β) → α)) : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [ContinuousAdd α] {s : Set β} (hs : Summable (f ∘ Subtype.val)) (hsc : Summable (f ∘ Subtype.val))
Goal: Summable f
-- Multipliable.mul_compl
@[to_additive]
theorem Multipliable.mul_compl {s : Set β} (hs : Multipliable (f ∘ (↑) : s → α))
    (hsc : Multipliable (f ∘ (↑) : (sᶜ : Set β) → α)) : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} [ContinuousMul α] {s : Set β} (hs : Multipliable (f ∘ Subtype.val)) (hsc : Multipliable (f ∘ Subtype.val))
Goal: Multipliable f
-- HasSum.compl_add
@[to_additive]
theorem HasProd.compl_mul {s : Set β} (ha : HasProd (f ∘ (↑) : (sᶜ : Set β) → α) a)
    (hb : HasProd (f ∘ (↑) : s → α) b) : HasProd f (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousAdd α] {s : Set β} (ha : HasSum (f ∘ Subtype.val) a) (hb : HasSum (f ∘ Subtype.val) b)
Goal: HasSum f (a + b)
-- HasProd.compl_mul
@[to_additive]
theorem HasProd.compl_mul {s : Set β} (ha : HasProd (f ∘ (↑) : (sᶜ : Set β) → α) a)
    (hb : HasProd (f ∘ (↑) : s → α) b) : HasProd f (a * b) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a b : α} [ContinuousMul α] {s : Set β} (ha : HasProd (f ∘ Subtype.val) a) (hb : HasProd (f ∘ Subtype.val) b)
Goal: HasProd f (a * b)
-- Multipliable.compl_add
@[to_additive]
theorem Multipliable.compl_add {s : Set β} (hs : Multipliable (f ∘ (↑) : (sᶜ : Set β) → α))
    (hsc : Multipliable (f ∘ (↑) : s → α)) : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} [ContinuousMul α] {s : Set β} (hs : Multipliable (f ∘ Subtype.val)) (hsc : Multipliable (f ∘ Subtype.val))
Goal: Multipliable f
-- Summable.compl_add
@[to_additive]
theorem Multipliable.compl_add {s : Set β} (hs : Multipliable (f ∘ (↑) : (sᶜ : Set β) → α))
    (hsc : Multipliable (f ∘ (↑) : s → α)) : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [ContinuousAdd α] {s : Set β} (hs : Summable (f ∘ Subtype.val)) (hsc : Summable (f ∘ Subtype.val))
Goal: Summable f
-- HasProd.update'
@[to_additive "Version of `HasSum.update` for `AddCommMonoid` rather than `AddCommGroup`.
Rather than showing that `f.update` has a specific sum in terms of `HasSum`,
it gives a relationship between the sums of `f` and `f.update` given that both exist."]
theorem HasProd.update' {α β : Type*} [TopologicalSpace α] [CommMonoid α] [T2Space α]
    [ContinuousMul α] [DecidableEq β] {f : β → α} {a a' : α} (hf : HasProd f a) (b : β) (x : α)
    (hf' : HasProd (update f b x) a') : a * x = a' * f b 
Arguments: {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [CommMonoid α] [T2Space α] [ContinuousMul α] [DecidableEq β] {f : β → α} {a a' : α} (hf : HasProd f a) (b : β) (x : α) (hf' : HasProd (Function.update f b x) a')
Goal: a * x = a' * f b
-- HasSum.update'
@[to_additive "Version of `HasSum.update` for `AddCommMonoid` rather than `AddCommGroup`.
Rather than showing that `f.update` has a specific sum in terms of `HasSum`,
it gives a relationship between the sums of `f` and `f.update` given that both exist."]
theorem HasProd.update' {α β : Type*} [TopologicalSpace α] [CommMonoid α] [T2Space α]
    [ContinuousMul α] [DecidableEq β] {f : β → α} {a a' : α} (hf : HasProd f a) (b : β) (x : α)
    (hf' : HasProd (update f b x) a') : a * x = a' * f b 
Arguments: {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [AddCommMonoid α] [T2Space α] [ContinuousAdd α] [DecidableEq β] {f : β → α} {a a' : α} (hf : HasSum f a) (b : β) (x : α) (hf' : HasSum (Function.update f b x) a')
Goal: a + x = a' + f b
-- eq_add_of_hasSum_ite
@[to_additive "Version of `hasSum_ite_sub_hasSum` for `AddCommMonoid` rather than `AddCommGroup`.
Rather than showing that the `ite` expression has a specific sum in terms of `HasSum`,
it gives a relationship between the sums of `f` and `ite (n = b) 0 (f n)` given that both exist."]
theorem eq_mul_of_hasProd_ite {α β : Type*} [TopologicalSpace α] [CommMonoid α] [T2Space α]
    [ContinuousMul α] [DecidableEq β] {f : β → α} {a : α} (hf : HasProd f a) (b : β) (a' : α)
    (hf' : HasProd (fun n ↦ ite (n = b) 1 (f n)) a') : a = a' * f b 
Arguments: {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [AddCommMonoid α] [T2Space α] [ContinuousAdd α] [DecidableEq β] {f : β → α} {a : α} (hf : HasSum f a) (b : β) (a' : α) (hf' : HasSum (fun n => if n = b then 0 else f n) a')
Goal: a = a' + f b
-- eq_mul_of_hasProd_ite
@[to_additive "Version of `hasSum_ite_sub_hasSum` for `AddCommMonoid` rather than `AddCommGroup`.
Rather than showing that the `ite` expression has a specific sum in terms of `HasSum`,
it gives a relationship between the sums of `f` and `ite (n = b) 0 (f n)` given that both exist."]
theorem eq_mul_of_hasProd_ite {α β : Type*} [TopologicalSpace α] [CommMonoid α] [T2Space α]
    [ContinuousMul α] [DecidableEq β] {f : β → α} {a : α} (hf : HasProd f a) (b : β) (a' : α)
    (hf' : HasProd (fun n ↦ ite (n = b) 1 (f n)) a') : a = a' * f b 
Arguments: {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [CommMonoid α] [T2Space α] [ContinuousMul α] [DecidableEq β] {f : β → α} {a : α} (hf : HasProd f a) (b : β) (a' : α) (hf' : HasProd (fun n => if n = b then 1 else f n) a')
Goal: a = a' * f b
-- tsum_congr_set_coe
@[to_additive]
theorem tprod_congr_set_coe (f : β → α) {s t : Set β} (h : s = t) :
    ∏' x : s, f x = ∏' x : t, f x 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] (f : β → α) {s t : Set β} (h : s = t)
Goal: ∑' (x : ↑s), f ↑x = ∑' (x : ↑t), f ↑x
-- tprod_congr_set_coe
@[to_additive]
theorem tprod_congr_set_coe (f : β → α) {s t : Set β} (h : s = t) :
    ∏' x : s, f x = ∏' x : t, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] (f : β → α) {s t : Set β} (h : s = t)
Goal: ∏' (x : ↑s), f ↑x = ∏' (x : ↑t), f ↑x
-- tprod_congr_subtype
@[to_additive]
theorem tprod_congr_subtype (f : β → α) {P Q : β → Prop} (h : ∀ x, P x ↔ Q x) :
    ∏' x : {x // P x}, f x = ∏' x : {x // Q x}, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] (f : β → α) {P Q : β → Prop} (h : ∀ (x : β), P x ↔ Q x)
Goal: ∏' (x : { x // P x }), f ↑x = ∏' (x : { x // Q x }), f ↑x
-- tsum_congr_subtype
@[to_additive]
theorem tprod_congr_subtype (f : β → α) {P Q : β → Prop} (h : ∀ x, P x ↔ Q x) :
    ∏' x : {x // P x}, f x = ∏' x : {x // Q x}, f x 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] (f : β → α) {P Q : β → Prop} (h : ∀ (x : β), P x ↔ Q x)
Goal: ∑' (x : { x // P x }), f ↑x = ∑' (x : { x // Q x }), f ↑x
-- tsum_eq_finsum
@[to_additive]
theorem tprod_eq_finprod (hf : (mulSupport f).Finite) :
    ∏' b, f b = ∏ᶠ b, f b 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} (hf : (Function.support f).Finite)
Goal: ∑' (b : β), f b = ∑ᶠ (b : β), f b
-- tprod_eq_finprod
@[to_additive]
theorem tprod_eq_finprod (hf : (mulSupport f).Finite) :
    ∏' b, f b = ∏ᶠ b, f b 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} (hf : (Function.mulSupport f).Finite)
Goal: ∏' (b : β), f b = ∏ᶠ (b : β), f b
-- tprod_eq_prod'
@[to_additive]
theorem tprod_eq_prod' {s : Finset β} (hf : mulSupport f ⊆ s) :
    ∏' b, f b = ∏ b ∈ s, f b 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {s : Finset β} (hf : Function.mulSupport f ⊆ ↑s)
Goal: ∏' (b : β), f b = ∏ b ∈ s, f b
-- tsum_eq_sum'
@[to_additive]
theorem tprod_eq_prod' {s : Finset β} (hf : mulSupport f ⊆ s) :
    ∏' b, f b = ∏ b ∈ s, f b 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {s : Finset β} (hf : Function.support f ⊆ ↑s)
Goal: ∑' (b : β), f b = ∑ b ∈ s, f b
-- tprod_eq_prod
@[to_additive]
theorem tprod_eq_prod {s : Finset β} (hf : ∀ b ∉ s, f b = 1) :
    ∏' b, f b = ∏ b ∈ s, f b 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {s : Finset β} (hf : ∀ b ∉ s, f b = 1)
Goal: ∏' (b : β), f b = ∏ b ∈ s, f b
-- tsum_eq_sum
@[to_additive]
theorem tprod_eq_prod {s : Finset β} (hf : ∀ b ∉ s, f b = 1) :
    ∏' b, f b = ∏ b ∈ s, f b 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {s : Finset β} (hf : ∀ b ∉ s, f b = 0)
Goal: ∑' (b : β), f b = ∑ b ∈ s, f b
-- tprod_one
@[to_additive (attr := simp)]
theorem tprod_one : ∏' _ : β, (1 : α) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α]
Goal: ∏' (x : β), 1 = 1
-- tsum_zero
@[to_additive (attr := simp)]
theorem tprod_one : ∏' _ : β, (1 : α) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α]
Goal: ∑' (x : β), 0 = 0
-- tsum_empty
@[to_additive (attr := simp)]
theorem tprod_empty [IsEmpty β] : ∏' b, f b = 1 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [IsEmpty β]
Goal: ∑' (b : β), f b = 0
-- tprod_empty
@[to_additive (attr := simp)]
theorem tprod_empty [IsEmpty β] : ∏' b, f b = 1 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} [IsEmpty β]
Goal: ∏' (b : β), f b = 1