ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PontryaginDual.map_apply
@[simp]
theorem map_apply (f : ContinuousMonoidHom A B) (x : PontryaginDual B) (y : A) :
    map f x y = x (f y) 
Arguments: {A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousMonoidHom A B) (x : PontryaginDual B) (y : A)
Goal: ((PontryaginDual.map f) x) y = x (f y)
-- PontryaginDual.map_one
@[simp]
theorem map_one : map (one A B) = one (PontryaginDual B) (PontryaginDual A) 
Arguments: {A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: PontryaginDual.map (ContinuousMonoidHom.one A B) = ContinuousMonoidHom.one (PontryaginDual B) (PontryaginDual A)
-- PontryaginDual.map_comp
@[simp]
theorem map_comp (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) :
    map (comp g f) = ContinuousMonoidHom.comp (map f) (map g) 
Arguments: {A : Type u_1} {B : Type u_2} {C : Type u_3} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B)
Goal: PontryaginDual.map (g.comp f) = (PontryaginDual.map f).comp (PontryaginDual.map g)
-- PontryaginDual.map_mul
@[simp]
nonrec theorem map_mul (f g : ContinuousMonoidHom A E) : map (f * g) = map f * map g 
Arguments: {A : Type u_1} {E : Type u_5} [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalGroup E] (f g : ContinuousMonoidHom A E)
Goal: PontryaginDual.map (f * g) = PontryaginDual.map f * PontryaginDual.map g
-- Valuation.inversion_estimate
theorem Valuation.inversion_estimate {x y : K} {γ : Γ₀ˣ} (y_ne : y ≠ 0)
    (h : v (x - y) < min (γ * (v y * v y)) (v y)) : v (x⁻¹ - y⁻¹) < γ 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) {x y : K} {γ : Γ₀ˣ} (y_ne : y ≠ 0) (h : v (x - y) < min (↑γ * (v y * v y)) (v y))
Goal: v (x⁻¹ - y⁻¹) < ↑γ
-- Valued.continuous_valuation
theorem Valued.continuous_valuation [Valued K Γ₀] : Continuous (v : K → Γ₀) 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] [Valued K Γ₀]
Goal: Continuous ⇑Valued.v
-- Valued.continuous_extension
theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) 
Arguments: {K : Type u_1} [Field K] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] [hv : Valued K Γ₀]
Goal: Continuous Valued.extension
-- Valued.extension_extends
@[simp, norm_cast]
theorem extension_extends (x : K) : extension (x : hat K) = v x 
Arguments: {K : Type u_1} [Field K] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] [hv : Valued K Γ₀] (x : K)
Goal: Valued.extension (↑K x) = Valued.v x
-- Valued.closure_coe_completion_v_lt
theorem closure_coe_completion_v_lt {γ : Γ₀ˣ} :
    closure ((↑) '' { x : K | v x < (γ : Γ₀) }) =
    { x : hat K | extensionValuation x < (γ : Γ₀) } 
Arguments: {K : Type u_1} [Field K] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] [hv : Valued K Γ₀] {γ : Γ₀ˣ}
Goal: closure (↑K '' {x | Valued.v x < ↑γ}) = {x | Valued.extensionValuation x < ↑γ}
-- Valued.valuedCompletion_apply
@[simp]
theorem valuedCompletion_apply (x : K) : Valued.v (x : hat K) = v x 
Arguments: {K : Type u_1} [Field K] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] [hv : Valued K Γ₀] (x : K)
Goal: Valued.v (↑K x) = Valued.v x
-- hasProd_one
@[to_additive "Constant zero function has sum `0`"]
theorem hasProd_one : HasProd (fun _ ↦ 1 : β → α) 1 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α]
Goal: HasProd (fun x => 1) 1
-- hasSum_zero
@[to_additive "Constant zero function has sum `0`"]
theorem hasProd_one : HasProd (fun _ ↦ 1 : β → α) 1 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α]
Goal: HasSum (fun x => 0) 0
-- hasSum_empty
@[to_additive]
theorem hasProd_empty [IsEmpty β] : HasProd f 1 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [IsEmpty β]
Goal: HasSum f 0
-- hasProd_empty
@[to_additive]
theorem hasProd_empty [IsEmpty β] : HasProd f 1 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} [IsEmpty β]
Goal: HasProd f 1
-- summable_zero
@[to_additive]
theorem multipliable_one : Multipliable (fun _ ↦ 1 : β → α) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α]
Goal: Summable fun x => 0
-- multipliable_one
@[to_additive]
theorem multipliable_one : Multipliable (fun _ ↦ 1 : β → α) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α]
Goal: Multipliable fun x => 1
-- multipliable_empty
@[to_additive]
theorem multipliable_empty [IsEmpty β] : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} [IsEmpty β]
Goal: Multipliable f
-- summable_empty
@[to_additive]
theorem multipliable_empty [IsEmpty β] : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} [IsEmpty β]
Goal: Summable f
-- multipliable_congr
@[to_additive]
theorem multipliable_congr (hfg : ∀ b, f b = g b) : Multipliable f ↔ Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f g : β → α} (hfg : ∀ (b : β), f b = g b)
Goal: Multipliable f ↔ Multipliable g
-- summable_congr
@[to_additive]
theorem multipliable_congr (hfg : ∀ b, f b = g b) : Multipliable f ↔ Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f g : β → α} (hfg : ∀ (b : β), f b = g b)
Goal: Summable f ↔ Summable g
-- Summable.congr
@[to_additive]
theorem Multipliable.congr (hf : Multipliable f) (hfg : ∀ b, f b = g b) : Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f g : β → α} (hf : Summable f) (hfg : ∀ (b : β), f b = g b)
Goal: Summable g
-- Multipliable.congr
@[to_additive]
theorem Multipliable.congr (hf : Multipliable f) (hfg : ∀ b, f b = g b) : Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f g : β → α} (hf : Multipliable f) (hfg : ∀ (b : β), f b = g b)
Goal: Multipliable g
-- HasSum.congr_fun
@[to_additive]
lemma HasProd.congr_fun (hf : HasProd f a) (h : ∀ x : β, g x = f x) : HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f g : β → α} {a : α} (hf : HasSum f a) (h : ∀ (x : β), g x = f x)
Goal: HasSum g a
-- HasProd.congr_fun
@[to_additive]
lemma HasProd.congr_fun (hf : HasProd f a) (h : ∀ x : β, g x = f x) : HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f g : β → α} {a : α} (hf : HasProd f a) (h : ∀ (x : β), g x = f x)
Goal: HasProd g a
-- HasProd.hasProd_of_prod_eq
@[to_additive]
theorem HasProd.hasProd_of_prod_eq {g : γ → α}
    (h_eq : ∀ u : Finset γ, ∃ v : Finset β, ∀ v', v ⊆ v' →
      ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b)
    (hf : HasProd g a) : HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (h_eq : ∀ (u : Finset γ), ∃ v, ∀ (v' : Finset β), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b) (hf : HasProd g a)
Goal: HasProd f a
-- HasSum.hasSum_of_sum_eq
@[to_additive]
theorem HasProd.hasProd_of_prod_eq {g : γ → α}
    (h_eq : ∀ u : Finset γ, ∃ v : Finset β, ∀ v', v ⊆ v' →
      ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b)
    (hf : HasProd g a) : HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (h_eq : ∀ (u : Finset γ), ∃ v, ∀ (v' : Finset β), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∑ x ∈ u', g x = ∑ b ∈ v', f b) (hf : HasSum g a)
Goal: HasSum f a
-- hasProd_iff_hasProd
@[to_additive]
theorem hasProd_iff_hasProd {g : γ → α}
    (h₁ : ∀ u : Finset γ, ∃ v : Finset β, ∀ v', v ⊆ v' →
      ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b)
    (h₂ : ∀ v : Finset β, ∃ u : Finset γ, ∀ u', u ⊆ u' →
      ∃ v', v ⊆ v' ∧ ∏ b ∈ v', f b = ∏ x ∈ u', g x) :
    HasProd f a ↔ HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (h₁ : ∀ (u : Finset γ), ∃ v, ∀ (v' : Finset β), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b) (h₂ : ∀ (v : Finset β), ∃ u, ∀ (u' : Finset γ), u ⊆ u' → ∃ v', v ⊆ v' ∧ ∏ b ∈ v', f b = ∏ x ∈ u', g x)
Goal: HasProd f a ↔ HasProd g a
-- hasSum_iff_hasSum
@[to_additive]
theorem hasProd_iff_hasProd {g : γ → α}
    (h₁ : ∀ u : Finset γ, ∃ v : Finset β, ∀ v', v ⊆ v' →
      ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b)
    (h₂ : ∀ v : Finset β, ∃ u : Finset γ, ∀ u', u ⊆ u' →
      ∃ v', v ⊆ v' ∧ ∏ b ∈ v', f b = ∏ x ∈ u', g x) :
    HasProd f a ↔ HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (h₁ : ∀ (u : Finset γ), ∃ v, ∀ (v' : Finset β), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∑ x ∈ u', g x = ∑ b ∈ v', f b) (h₂ : ∀ (v : Finset β), ∃ u, ∀ (u' : Finset γ), u ⊆ u' → ∃ v', v ⊆ v' ∧ ∑ b ∈ v', f b = ∑ x ∈ u', g x)
Goal: HasSum f a ↔ HasSum g a
-- Function.Injective.summable_iff
@[to_additive]
theorem Function.Injective.multipliable_iff {g : γ → β} (hg : Injective g)
    (hf : ∀ x ∉ Set.range g, f x = 1) : Multipliable (f ∘ g) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {g : γ → β} (hg : Function.Injective g) (hf : ∀ x ∉ Set.range g, f x = 0)
Goal: Summable (f ∘ g) ↔ Summable f
-- Function.Injective.multipliable_iff
@[to_additive]
theorem Function.Injective.multipliable_iff {g : γ → β} (hg : Injective g)
    (hf : ∀ x ∉ Set.range g, f x = 1) : Multipliable (f ∘ g) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {g : γ → β} (hg : Function.Injective g) (hf : ∀ x ∉ Set.range g, f x = 1)
Goal: Multipliable (f ∘ g) ↔ Multipliable f
-- hasProd_extend_one
@[to_additive (attr := simp)] theorem hasProd_extend_one {g : β → γ} (hg : Injective g) :
    HasProd (extend g f 1) a ↔ HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : β → γ} (hg : Function.Injective g)
Goal: HasProd (Function.extend g f 1) a ↔ HasProd f a
-- hasSum_extend_zero
@[to_additive (attr := simp)] theorem hasProd_extend_one {g : β → γ} (hg : Injective g) :
    HasProd (extend g f 1) a ↔ HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : β → γ} (hg : Function.Injective g)
Goal: HasSum (Function.extend g f 0) a ↔ HasSum f a
-- summable_extend_zero
@[to_additive (attr := simp)] theorem multipliable_extend_one {g : β → γ} (hg : Injective g) :
    Multipliable (extend g f 1) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {g : β → γ} (hg : Function.Injective g)
Goal: Summable (Function.extend g f 0) ↔ Summable f
-- multipliable_extend_one
@[to_additive (attr := simp)] theorem multipliable_extend_one {g : β → γ} (hg : Injective g) :
    Multipliable (extend g f 1) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {g : β → γ} (hg : Function.Injective g)
Goal: Multipliable (Function.extend g f 1) ↔ Multipliable f
-- hasProd_subtype_iff_mulIndicator
@[to_additive]
theorem hasProd_subtype_iff_mulIndicator {s : Set β} :
    HasProd (f ∘ (↑) : s → α) a ↔ HasProd (s.mulIndicator f) a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {s : Set β}
Goal: HasProd (f ∘ Subtype.val) a ↔ HasProd (s.mulIndicator f) a
-- hasSum_subtype_iff_indicator
@[to_additive]
theorem hasProd_subtype_iff_mulIndicator {s : Set β} :
    HasProd (f ∘ (↑) : s → α) a ↔ HasProd (s.mulIndicator f) a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {s : Set β}
Goal: HasSum (f ∘ Subtype.val) a ↔ HasSum (s.indicator f) a
-- multipliable_subtype_iff_mulIndicator
@[to_additive]
theorem multipliable_subtype_iff_mulIndicator {s : Set β} :
    Multipliable (f ∘ (↑) : s → α) ↔ Multipliable (s.mulIndicator f) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {s : Set β}
Goal: Multipliable (f ∘ Subtype.val) ↔ Multipliable (s.mulIndicator f)
-- summable_subtype_iff_indicator
@[to_additive]
theorem multipliable_subtype_iff_mulIndicator {s : Set β} :
    Multipliable (f ∘ (↑) : s → α) ↔ Multipliable (s.mulIndicator f) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {s : Set β}
Goal: Summable (f ∘ Subtype.val) ↔ Summable (s.indicator f)
-- hasSum_subtype_support
@[to_additive (attr := simp)]
theorem hasProd_subtype_mulSupport : HasProd (f ∘ (↑) : mulSupport f → α) a ↔ HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α}
Goal: HasSum (f ∘ Subtype.val) a ↔ HasSum f a
-- hasProd_subtype_mulSupport
@[to_additive (attr := simp)]
theorem hasProd_subtype_mulSupport : HasProd (f ∘ (↑) : mulSupport f → α) a ↔ HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α}
Goal: HasProd (f ∘ Subtype.val) a ↔ HasProd f a
-- Finset.summable
@[to_additive]
protected theorem Finset.multipliable (s : Finset β) (f : β → α) :
    Multipliable (f ∘ (↑) : (↑s : Set β) → α) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] (s : Finset β) (f : β → α)
Goal: Summable (f ∘ Subtype.val)
-- Finset.multipliable
@[to_additive]
protected theorem Finset.multipliable (s : Finset β) (f : β → α) :
    Multipliable (f ∘ (↑) : (↑s : Set β) → α) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] (s : Finset β) (f : β → α)
Goal: Multipliable (f ∘ Subtype.val)
-- Set.Finite.multipliable
@[to_additive]
protected theorem Set.Finite.multipliable {s : Set β} (hs : s.Finite) (f : β → α) :
    Multipliable (f ∘ (↑) : s → α) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {s : Set β} (hs : s.Finite) (f : β → α)
Goal: Multipliable (f ∘ Subtype.val)
-- Set.Finite.summable
@[to_additive]
protected theorem Set.Finite.multipliable {s : Set β} (hs : s.Finite) (f : β → α) :
    Multipliable (f ∘ (↑) : s → α) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {s : Set β} (hs : s.Finite) (f : β → α)
Goal: Summable (f ∘ Subtype.val)
-- summable_of_finite_support
@[to_additive]
theorem multipliable_of_finite_mulSupport (h : (mulSupport f).Finite) : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} (h : (Function.support f).Finite)
Goal: Summable f
-- multipliable_of_finite_mulSupport
@[to_additive]
theorem multipliable_of_finite_mulSupport (h : (mulSupport f).Finite) : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} (h : (Function.mulSupport f).Finite)
Goal: Multipliable f
-- Multipliable.of_finite
@[to_additive]
lemma Multipliable.of_finite [Finite β] {f : β → α} : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [Finite β] {f : β → α}
Goal: Multipliable f
-- Summable.of_finite
@[to_additive]
lemma Multipliable.of_finite [Finite β] {f : β → α} : Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [Finite β] {f : β → α}
Goal: Summable f
-- hasSum_single
@[to_additive]
theorem hasProd_single {f : β → α} (b : β) (hf : ∀ (b') (_ : b' ≠ b), f b' = 1) : HasProd f (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} (b : β) (hf : ∀ (b' : β), b' ≠ b → f b' = 0)
Goal: HasSum f (f b)
-- hasProd_single
@[to_additive]
theorem hasProd_single {f : β → α} (b : β) (hf : ∀ (b') (_ : b' ≠ b), f b' = 1) : HasProd f (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] {f : β → α} (b : β) (hf : ∀ (b' : β), b' ≠ b → f b' = 1)
Goal: HasProd f (f b)
-- hasSum_unique
@[to_additive (attr := simp)] lemma hasProd_unique [Unique β] (f : β → α) : HasProd f (f default) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [Unique β] (f : β → α)
Goal: HasSum f (f default)
-- hasProd_unique
@[to_additive (attr := simp)] lemma hasProd_unique [Unique β] (f : β → α) : HasProd f (f default) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [Unique β] (f : β → α)
Goal: HasProd f (f default)
-- hasSum_singleton
@[to_additive (attr := simp)]
lemma hasProd_singleton (m : β) (f : β → α) : HasProd (({m} : Set β).restrict f) (f m) 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] (m : β) (f : β → α)
Goal: HasSum ({m}.restrict f) (f m)
-- hasProd_singleton
@[to_additive (attr := simp)]
lemma hasProd_singleton (m : β) (f : β → α) : HasProd (({m} : Set β).restrict f) (f m) 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] (m : β) (f : β → α)
Goal: HasProd ({m}.restrict f) (f m)
-- hasSum_ite_eq
@[to_additive]
theorem hasProd_ite_eq (b : β) [DecidablePred (· = b)] (a : α) :
    HasProd (fun b' ↦ if b' = b then a else 1) a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] (b : β) [DecidablePred fun x => x = b] (a : α)
Goal: HasSum (fun b' => if b' = b then a else 0) a
-- hasProd_ite_eq
@[to_additive]
theorem hasProd_ite_eq (b : β) [DecidablePred (· = b)] (a : α) :
    HasProd (fun b' ↦ if b' = b then a else 1) a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] (b : β) [DecidablePred fun x => x = b] (a : α)
Goal: HasProd (fun b' => if b' = b then a else 1) a
-- Equiv.hasSum_iff
@[to_additive]
theorem Equiv.hasProd_iff (e : γ ≃ β) : HasProd (f ∘ e) a ↔ HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} (e : γ ≃ β)
Goal: HasSum (f ∘ ⇑e) a ↔ HasSum f a
-- Equiv.hasProd_iff
@[to_additive]
theorem Equiv.hasProd_iff (e : γ ≃ β) : HasProd (f ∘ e) a ↔ HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} (e : γ ≃ β)
Goal: HasProd (f ∘ ⇑e) a ↔ HasProd f a
-- Function.Injective.hasProd_range_iff
@[to_additive]
theorem Function.Injective.hasProd_range_iff {g : γ → β} (hg : Injective g) :
    HasProd (fun x : Set.range g ↦ f x) a ↔ HasProd (f ∘ g) a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → β} (hg : Function.Injective g)
Goal: HasProd (fun x => f ↑x) a ↔ HasProd (f ∘ g) a
-- Function.Injective.hasSum_range_iff
@[to_additive]
theorem Function.Injective.hasProd_range_iff {g : γ → β} (hg : Injective g) :
    HasProd (fun x : Set.range g ↦ f x) a ↔ HasProd (f ∘ g) a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → β} (hg : Function.Injective g)
Goal: HasSum (fun x => f ↑x) a ↔ HasSum (f ∘ g) a
-- Equiv.summable_iff
@[to_additive]
theorem Equiv.multipliable_iff (e : γ ≃ β) : Multipliable (f ∘ e) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} (e : γ ≃ β)
Goal: Summable (f ∘ ⇑e) ↔ Summable f
-- Equiv.multipliable_iff
@[to_additive]
theorem Equiv.multipliable_iff (e : γ ≃ β) : Multipliable (f ∘ e) ↔ Multipliable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} (e : γ ≃ β)
Goal: Multipliable (f ∘ ⇑e) ↔ Multipliable f
-- Equiv.hasProd_iff_of_mulSupport
@[to_additive]
theorem Equiv.hasProd_iff_of_mulSupport {g : γ → α} (e : mulSupport f ≃ mulSupport g)
    (he : ∀ x : mulSupport f, g (e x) = f x) : HasProd f a ↔ HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (e : ↑(Function.mulSupport f) ≃ ↑(Function.mulSupport g)) (he : ∀ (x : ↑(Function.mulSupport f)), g ↑(e x) = f ↑x)
Goal: HasProd f a ↔ HasProd g a
-- Equiv.hasSum_iff_of_support
@[to_additive]
theorem Equiv.hasProd_iff_of_mulSupport {g : γ → α} (e : mulSupport f ≃ mulSupport g)
    (he : ∀ x : mulSupport f, g (e x) = f x) : HasProd f a ↔ HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (e : ↑(Function.support f) ≃ ↑(Function.support g)) (he : ∀ (x : ↑(Function.support f)), g ↑(e x) = f ↑x)
Goal: HasSum f a ↔ HasSum g a
-- hasProd_iff_hasProd_of_ne_one_bij
@[to_additive]
theorem hasProd_iff_hasProd_of_ne_one_bij {g : γ → α} (i : mulSupport g → β)
    (hi : Injective i) (hf : mulSupport f ⊆ Set.range i)
    (hfg : ∀ x, f (i x) = g x) : HasProd f a ↔ HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (i : ↑(Function.mulSupport g) → β) (hi : Function.Injective i) (hf : Function.mulSupport f ⊆ Set.range i) (hfg : ∀ (x : ↑(Function.mulSupport g)), f (i x) = g ↑x)
Goal: HasProd f a ↔ HasProd g a
-- hasSum_iff_hasSum_of_ne_zero_bij
@[to_additive]
theorem hasProd_iff_hasProd_of_ne_one_bij {g : γ → α} (i : mulSupport g → β)
    (hi : Injective i) (hf : mulSupport f ⊆ Set.range i)
    (hfg : ∀ x, f (i x) = g x) : HasProd f a ↔ HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} {g : γ → α} (i : ↑(Function.support g) → β) (hi : Function.Injective i) (hf : Function.support f ⊆ Set.range i) (hfg : ∀ (x : ↑(Function.support g)), f (i x) = g ↑x)
Goal: HasSum f a ↔ HasSum g a
-- Equiv.multipliable_iff_of_mulSupport
@[to_additive]
theorem Equiv.multipliable_iff_of_mulSupport {g : γ → α} (e : mulSupport f ≃ mulSupport g)
    (he : ∀ x : mulSupport f, g (e x) = f x) : Multipliable f ↔ Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {g : γ → α} (e : ↑(Function.mulSupport f) ≃ ↑(Function.mulSupport g)) (he : ∀ (x : ↑(Function.mulSupport f)), g ↑(e x) = f ↑x)
Goal: Multipliable f ↔ Multipliable g
-- Equiv.summable_iff_of_support
@[to_additive]
theorem Equiv.multipliable_iff_of_mulSupport {g : γ → α} (e : mulSupport f ≃ mulSupport g)
    (he : ∀ x : mulSupport f, g (e x) = f x) : Multipliable f ↔ Multipliable g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {g : γ → α} (e : ↑(Function.support f) ≃ ↑(Function.support g)) (he : ∀ (x : ↑(Function.support f)), g ↑(e x) = f ↑x)
Goal: Summable f ↔ Summable g
-- HasSum.map
@[to_additive]
protected theorem HasProd.map [CommMonoid γ] [TopologicalSpace γ] (hf : HasProd f a) {G}
    [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) :
    HasProd (g ∘ f) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} [AddCommMonoid γ] [TopologicalSpace γ] (hf : HasSum f a) {G : Type u_5} [FunLike G α γ] [AddMonoidHomClass G α γ] (g : G) (hg : Continuous ⇑g)
Goal: HasSum (⇑g ∘ f) (g a)
-- HasProd.map
@[to_additive]
protected theorem HasProd.map [CommMonoid γ] [TopologicalSpace γ] (hf : HasProd f a) {G}
    [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) :
    HasProd (g ∘ f) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β → α} {a : α} [CommMonoid γ] [TopologicalSpace γ] (hf : HasProd f a) {G : Type u_5} [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous ⇑g)
Goal: HasProd (⇑g ∘ f) (g a)
-- Inducing.hasProd_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} [CommMonoid α] [TopologicalSpace α] [CommMonoid γ] [TopologicalSpace γ] {G : Type u_5} [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : Inducing ⇑g) (f : β → α) (a : α)
Goal: HasProd (⇑g ∘ f) (g a) ↔ HasProd f a