ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- exists_open_nhds_one_mul_subset
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
  such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
    ∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ V * V ⊆ U 
Arguments: {M : Type u_3} [TopologicalSpace M] [MulOneClass M] [ContinuousMul M] {U : Set M} (hU : U ∈ 𝓝 1)
Goal: ∃ V, IsOpen V ∧ 1 ∈ V ∧ V * V ⊆ U
-- Submonoid.top_closure_mul_self_subset
@[to_additive]
theorem Submonoid.top_closure_mul_self_subset (s : Submonoid M) :
    _root_.closure (s : Set M) * _root_.closure s ⊆ _root_.closure s 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (s : Submonoid M)
Goal: closure ↑s * closure ↑s ⊆ closure ↑s
-- AddSubmonoid.top_closure_add_self_subset
@[to_additive]
theorem Submonoid.top_closure_mul_self_subset (s : Submonoid M) :
    _root_.closure (s : Set M) * _root_.closure s ⊆ _root_.closure s 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (s : AddSubmonoid M)
Goal: closure ↑s + closure ↑s ⊆ closure ↑s
-- AddSubmonoid.top_closure_add_self_eq
@[to_additive]
theorem Submonoid.top_closure_mul_self_eq (s : Submonoid M) :
    _root_.closure (s : Set M) * _root_.closure s = _root_.closure s 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (s : AddSubmonoid M)
Goal: closure ↑s + closure ↑s = closure ↑s
-- Submonoid.top_closure_mul_self_eq
@[to_additive]
theorem Submonoid.top_closure_mul_self_eq (s : Submonoid M) :
    _root_.closure (s : Set M) * _root_.closure s = _root_.closure s 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (s : Submonoid M)
Goal: closure ↑s * closure ↑s = closure ↑s
-- Submonoid.coe_topologicalClosure
@[to_additive]
theorem Submonoid.coe_topologicalClosure (s : Submonoid M) :
    (s.topologicalClosure : Set M) = _root_.closure (s : Set M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (s : Submonoid M)
Goal: ↑s.topologicalClosure = closure ↑s
-- AddSubmonoid.coe_topologicalClosure
@[to_additive]
theorem Submonoid.coe_topologicalClosure (s : Submonoid M) :
    (s.topologicalClosure : Set M) = _root_.closure (s : Set M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (s : AddSubmonoid M)
Goal: ↑s.topologicalClosure = closure ↑s
-- AddSubmonoid.le_topologicalClosure
@[to_additive]
theorem Submonoid.le_topologicalClosure (s : Submonoid M) : s ≤ s.topologicalClosure 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (s : AddSubmonoid M)
Goal: s ≤ s.topologicalClosure
-- Submonoid.le_topologicalClosure
@[to_additive]
theorem Submonoid.le_topologicalClosure (s : Submonoid M) : s ≤ s.topologicalClosure 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (s : Submonoid M)
Goal: s ≤ s.topologicalClosure
-- AddSubmonoid.isClosed_topologicalClosure
@[to_additive]
theorem Submonoid.isClosed_topologicalClosure (s : Submonoid M) :
    IsClosed (s.topologicalClosure : Set M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (s : AddSubmonoid M)
Goal: IsClosed ↑s.topologicalClosure
-- Submonoid.isClosed_topologicalClosure
@[to_additive]
theorem Submonoid.isClosed_topologicalClosure (s : Submonoid M) :
    IsClosed (s.topologicalClosure : Set M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (s : Submonoid M)
Goal: IsClosed ↑s.topologicalClosure
-- Submonoid.topologicalClosure_minimal
@[to_additive]
theorem Submonoid.topologicalClosure_minimal (s : Submonoid M) {t : Submonoid M} (h : s ≤ t)
    (ht : IsClosed (t : Set M)) : s.topologicalClosure ≤ t 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (s : Submonoid M) {t : Submonoid M} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- AddSubmonoid.topologicalClosure_minimal
@[to_additive]
theorem Submonoid.topologicalClosure_minimal (s : Submonoid M) {t : Submonoid M} (h : s ≤ t)
    (ht : IsClosed (t : Set M)) : s.topologicalClosure ≤ t 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (s : AddSubmonoid M) {t : AddSubmonoid M} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- exists_nhds_zero_quarter
@[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
    ∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {u : Set M} (hu : u ∈ 𝓝 0)
Goal: ∃ V ∈ 𝓝 0, ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v + w + s + t ∈ u
-- exists_nhds_one_split4
@[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
    ∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] {u : Set M} (hu : u ∈ 𝓝 1)
Goal: ∃ V ∈ 𝓝 1, ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u
-- IsCompact.mul
@[to_additive]
theorem IsCompact.mul {s t : Set M} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s * t) 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] {s t : Set M} (hs : IsCompact s) (ht : IsCompact t)
Goal: IsCompact (s * t)
-- IsCompact.add
@[to_additive]
theorem IsCompact.mul {s t : Set M} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s * t) 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {s t : Set M} (hs : IsCompact s) (ht : IsCompact t)
Goal: IsCompact (s + t)
-- tendsto_list_sum
@[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
    ∀ l : List ι,
      (∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
        Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
  
Arguments: {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : ι → α → M} {x : Filter α} {a : ι → M} (l : List ι)
Goal: (∀ i ∈ l, Filter.Tendsto (f i) x (𝓝 (a i))) →
  Filter.Tendsto (fun b => (List.map (fun c => f c b) l).sum) x (𝓝 (List.map a l).sum)
-- tendsto_list_prod
@[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
    ∀ l : List ι,
      (∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
        Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
  
Arguments: {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : ι → α → M} {x : Filter α} {a : ι → M} (l : List ι)
Goal: (∀ i ∈ l, Filter.Tendsto (f i) x (𝓝 (a i))) →
  Filter.Tendsto (fun b => (List.map (fun c => f c b) l).prod) x (𝓝 (List.map a l).prod)
-- continuous_list_sum
@[to_additive (attr := continuity)]
theorem continuous_list_prod {f : ι → X → M} (l : List ι) (h : ∀ i ∈ l, Continuous (f i)) :
    Continuous fun a => (l.map fun i => f i a).prod 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : ι → X → M} (l : List ι) (h : ∀ i ∈ l, Continuous (f i))
Goal: Continuous fun a => (List.map (fun i => f i a) l).sum
-- continuous_list_prod
@[to_additive (attr := continuity)]
theorem continuous_list_prod {f : ι → X → M} (l : List ι) (h : ∀ i ∈ l, Continuous (f i)) :
    Continuous fun a => (l.map fun i => f i a).prod 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : ι → X → M} (l : List ι) (h : ∀ i ∈ l, Continuous (f i))
Goal: Continuous fun a => (List.map (fun i => f i a) l).prod
-- continuousOn_list_prod
@[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
    (h : ∀ i ∈ l, ContinuousOn (f i) t) :
    ContinuousOn (fun a => (l.map fun i => f i a).prod) t 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : ι → X → M} (l : List ι) {t : Set X} (h : ∀ i ∈ l, ContinuousOn (f i) t)
Goal: ContinuousOn (fun a => (List.map (fun i => f i a) l).prod) t
-- continuousOn_list_sum
@[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
    (h : ∀ i ∈ l, ContinuousOn (f i) t) :
    ContinuousOn (fun a => (l.map fun i => f i a).prod) t 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : ι → X → M} (l : List ι) {t : Set X} (h : ∀ i ∈ l, ContinuousOn (f i) t)
Goal: ContinuousOn (fun a => (List.map (fun i => f i a) l).sum) t
-- continuous_nsmul
@[to_additive (attr := continuity)]
theorem continuous_pow : ∀ n : ℕ, Continuous fun a : M => a ^ n
  
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (n : ℕ)
Goal: Continuous fun a => n • a
-- continuous_pow
@[to_additive (attr := continuity)]
theorem continuous_pow : ∀ n : ℕ, Continuous fun a : M => a ^ n
  
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (n : ℕ)
Goal: Continuous fun a => a ^ n
-- Continuous.nsmul
@[to_additive (attr := aesop safe -100 (rule_sets := [Continuous]), fun_prop)]
theorem Continuous.pow {f : X → M} (h : Continuous f) (n : ℕ) : Continuous fun b => f b ^ n 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} (h : Continuous f) (n : ℕ)
Goal: Continuous fun b => n • f b
-- Continuous.pow
@[to_additive (attr := aesop safe -100 (rule_sets := [Continuous]), fun_prop)]
theorem Continuous.pow {f : X → M} (h : Continuous f) (n : ℕ) : Continuous fun b => f b ^ n 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : X → M} (h : Continuous f) (n : ℕ)
Goal: Continuous fun b => f b ^ n
-- continuousOn_nsmul
@[to_additive]
theorem continuousOn_pow {s : Set M} (n : ℕ) : ContinuousOn (fun (x : M) => x ^ n) s 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {s : Set M} (n : ℕ)
Goal: ContinuousOn (fun x => n • x) s
-- continuousOn_pow
@[to_additive]
theorem continuousOn_pow {s : Set M} (n : ℕ) : ContinuousOn (fun (x : M) => x ^ n) s 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] {s : Set M} (n : ℕ)
Goal: ContinuousOn (fun x => x ^ n) s
-- continuousAt_nsmul
@[to_additive]
theorem continuousAt_pow (x : M) (n : ℕ) : ContinuousAt (fun (x : M) => x ^ n) x 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (x : M) (n : ℕ)
Goal: ContinuousAt (fun x => n • x) x
-- continuousAt_pow
@[to_additive]
theorem continuousAt_pow (x : M) (n : ℕ) : ContinuousAt (fun (x : M) => x ^ n) x 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (x : M) (n : ℕ)
Goal: ContinuousAt (fun x => x ^ n) x
-- Filter.Tendsto.pow
@[to_additive]
theorem Filter.Tendsto.pow {l : Filter α} {f : α → M} {x : M} (hf : Tendsto f l (𝓝 x)) (n : ℕ) :
    Tendsto (fun x => f x ^ n) l (𝓝 (x ^ n)) 
Arguments: {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] {l : Filter α} {f : α → M} {x : M} (hf : Filter.Tendsto f l (𝓝 x)) (n : ℕ)
Goal: Filter.Tendsto (fun x => f x ^ n) l (𝓝 (x ^ n))
-- Filter.Tendsto.nsmul
@[to_additive]
theorem Filter.Tendsto.pow {l : Filter α} {f : α → M} {x : M} (hf : Tendsto f l (𝓝 x)) (n : ℕ) :
    Tendsto (fun x => f x ^ n) l (𝓝 (x ^ n)) 
Arguments: {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {l : Filter α} {f : α → M} {x : M} (hf : Filter.Tendsto f l (𝓝 x)) (n : ℕ)
Goal: Filter.Tendsto (fun x => n • f x) l (𝓝 (n • x))
-- ContinuousWithinAt.nsmul
@[to_additive]
theorem ContinuousWithinAt.pow {f : X → M} {x : X} {s : Set X} (hf : ContinuousWithinAt f s x)
    (n : ℕ) : ContinuousWithinAt (fun x => f x ^ n) s x 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} {s : Set X} (hf : ContinuousWithinAt f s x) (n : ℕ)
Goal: ContinuousWithinAt (fun x => n • f x) s x
-- ContinuousWithinAt.pow
@[to_additive]
theorem ContinuousWithinAt.pow {f : X → M} {x : X} {s : Set X} (hf : ContinuousWithinAt f s x)
    (n : ℕ) : ContinuousWithinAt (fun x => f x ^ n) s x 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : X → M} {x : X} {s : Set X} (hf : ContinuousWithinAt f s x) (n : ℕ)
Goal: ContinuousWithinAt (fun x => f x ^ n) s x
-- ContinuousAt.nsmul
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.pow {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) :
    ContinuousAt (fun x => f x ^ n) x 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ)
Goal: ContinuousAt (fun x => n • f x) x
-- ContinuousAt.pow
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.pow {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) :
    ContinuousAt (fun x => f x ^ n) x 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ)
Goal: ContinuousAt (fun x => f x ^ n) x
-- ContinuousOn.pow
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.pow {f : X → M} {s : Set X} (hf : ContinuousOn f s) (n : ℕ) :
    ContinuousOn (fun x => f x ^ n) s 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : X → M} {s : Set X} (hf : ContinuousOn f s) (n : ℕ)
Goal: ContinuousOn (fun x => f x ^ n) s
-- ContinuousOn.nsmul
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.pow {f : X → M} {s : Set X} (hf : ContinuousOn f s) (n : ℕ) :
    ContinuousOn (fun x => f x ^ n) s 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {s : Set X} (hf : ContinuousOn f s) (n : ℕ)
Goal: ContinuousOn (fun x => n • f x) s
-- Filter.tendsto_cocompact_mul_left
theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
    Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] {a b : M} (ha : b * a = 1)
Goal: Filter.Tendsto (fun x => a * x) (Filter.cocompact M) (Filter.cocompact M)
-- Filter.tendsto_cocompact_mul_right
theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
    Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] {a b : M} (ha : a * b = 1)
Goal: Filter.Tendsto (fun x => x * a) (Filter.cocompact M) (Filter.cocompact M)
-- Continuous.addUnits_map
@[to_additive]
theorem Continuous.units_map [Monoid M] [Monoid N] [TopologicalSpace M] [TopologicalSpace N]
    (f : M →* N) (hf : Continuous f) : Continuous (Units.map f) 
Arguments: {M : Type u_3} {N : Type u_4} [AddMonoid M] [AddMonoid N] [TopologicalSpace M] [TopologicalSpace N] (f : M →+ N) (hf : Continuous ⇑f)
Goal: Continuous ⇑(AddUnits.map f)
-- Continuous.units_map
@[to_additive]
theorem Continuous.units_map [Monoid M] [Monoid N] [TopologicalSpace M] [TopologicalSpace N]
    (f : M →* N) (hf : Continuous f) : Continuous (Units.map f) 
Arguments: {M : Type u_3} {N : Type u_4} [Monoid M] [Monoid N] [TopologicalSpace M] [TopologicalSpace N] (f : M →* N) (hf : Continuous ⇑f)
Goal: Continuous ⇑(Units.map f)
-- Submonoid.mem_nhds_one
@[to_additive]
theorem Submonoid.mem_nhds_one (S : Submonoid M) (oS : IsOpen (S : Set M)) :
    (S : Set M) ∈ 𝓝 (1 : M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [CommMonoid M] (S : Submonoid M) (oS : IsOpen ↑S)
Goal: ↑S ∈ 𝓝 1
-- AddSubmonoid.mem_nhds_zero
@[to_additive]
theorem Submonoid.mem_nhds_one (S : Submonoid M) (oS : IsOpen (S : Set M)) :
    (S : Set M) ∈ 𝓝 (1 : M) 
Arguments: {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] (S : AddSubmonoid M) (oS : IsOpen ↑S)
Goal: ↑S ∈ 𝓝 0
-- tendsto_multiset_prod
@[to_additive]
theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
    (∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
      Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) 
Arguments: {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι)
Goal: (∀ i ∈ s, Filter.Tendsto (f i) x (𝓝 (a i))) →
  Filter.Tendsto (fun b => (Multiset.map (fun c => f c b) s).prod) x (𝓝 (Multiset.map a s).prod)
-- tendsto_multiset_sum
@[to_additive]
theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
    (∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
      Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) 
Arguments: {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι)
Goal: (∀ i ∈ s, Filter.Tendsto (f i) x (𝓝 (a i))) →
  Filter.Tendsto (fun b => (Multiset.map (fun c => f c b) s).sum) x (𝓝 (Multiset.map a s).sum)
-- tendsto_finset_sum
@[to_additive]
theorem tendsto_finset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Finset ι) :
    (∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
      Tendsto (fun b => ∏ c ∈ s, f c b) x (𝓝 (∏ c ∈ s, a c)) 
Arguments: {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Finset ι)
Goal: (∀ i ∈ s, Filter.Tendsto (f i) x (𝓝 (a i))) → Filter.Tendsto (fun b => ∑ c ∈ s, f c b) x (𝓝 (∑ c ∈ s, a c))
-- tendsto_finset_prod
@[to_additive]
theorem tendsto_finset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Finset ι) :
    (∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
      Tendsto (fun b => ∏ c ∈ s, f c b) x (𝓝 (∏ c ∈ s, a c)) 
Arguments: {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Finset ι)
Goal: (∀ i ∈ s, Filter.Tendsto (f i) x (𝓝 (a i))) → Filter.Tendsto (fun b => ∏ c ∈ s, f c b) x (𝓝 (∏ c ∈ s, a c))
-- continuous_multiset_prod
@[to_additive (attr := continuity)]
theorem continuous_multiset_prod {f : ι → X → M} (s : Multiset ι) :
    (∀ i ∈ s, Continuous (f i)) → Continuous fun a => (s.map fun i => f i a).prod 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} (s : Multiset ι)
Goal: (∀ i ∈ s, Continuous (f i)) → Continuous fun a => (Multiset.map (fun i => f i a) s).prod
-- continuous_multiset_sum
@[to_additive (attr := continuity)]
theorem continuous_multiset_prod {f : ι → X → M} (s : Multiset ι) :
    (∀ i ∈ s, Continuous (f i)) → Continuous fun a => (s.map fun i => f i a).prod 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Multiset ι)
Goal: (∀ i ∈ s, Continuous (f i)) → Continuous fun a => (Multiset.map (fun i => f i a) s).sum
-- continuousOn_multiset_sum
@[to_additive]
theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
    (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Multiset ι) {t : Set X}
Goal: (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (Multiset.map (fun i => f i a) s).sum) t
-- continuousOn_multiset_prod
@[to_additive]
theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
    (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} (s : Multiset ι) {t : Set X}
Goal: (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (Multiset.map (fun i => f i a) s).prod) t
-- continuous_finset_prod
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_finset_prod {f : ι → X → M} (s : Finset ι) :
    (∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∏ i ∈ s, f i a 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} (s : Finset ι)
Goal: (∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∏ i ∈ s, f i a
-- continuous_finset_sum
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_finset_prod {f : ι → X → M} (s : Finset ι) :
    (∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∏ i ∈ s, f i a 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Finset ι)
Goal: (∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∑ i ∈ s, f i a
-- continuousOn_finset_prod
@[to_additive]
theorem continuousOn_finset_prod {f : ι → X → M} (s : Finset ι) {t : Set X} :
    (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => ∏ i ∈ s, f i a) t 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} (s : Finset ι) {t : Set X}
Goal: (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => ∏ i ∈ s, f i a) t
-- continuousOn_finset_sum
@[to_additive]
theorem continuousOn_finset_prod {f : ι → X → M} (s : Finset ι) {t : Set X} :
    (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => ∏ i ∈ s, f i a) t 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Finset ι) {t : Set X}
Goal: (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => ∑ i ∈ s, f i a) t
-- eventuallyEq_sum
@[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
    {f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i 
Arguments: {ι : Type u_1} {X : Type u_6} {M : Type u_7} [AddCommMonoid M] {s : Finset ι} {l : Filter X} {f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i)
Goal: ∑ i ∈ s, f i =ᶠ[l] ∑ i ∈ s, g i
-- eventuallyEq_prod
@[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
    {f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i 
Arguments: {ι : Type u_1} {X : Type u_6} {M : Type u_7} [CommMonoid M] {s : Finset ι} {l : Filter X} {f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i)
Goal: ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i
-- LocallyFinite.exists_finset_support
@[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
    (hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
    ∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I 
Arguments: {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [AddCommMonoid M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.support (f i)) (x₀ : X)
Goal: ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (Function.support fun i => f i x) ⊆ ↑I
-- LocallyFinite.exists_finset_mulSupport
@[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
    (hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
    ∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I 
Arguments: {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [CommMonoid M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.mulSupport (f i)) (x₀ : X)
Goal: ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (Function.mulSupport fun i => f i x) ⊆ ↑I
-- finprod_eventually_eq_prod
@[to_additive]
theorem finprod_eventually_eq_prod {M : Type*} [CommMonoid M] {f : ι → X → M}
    (hf : LocallyFinite fun i => mulSupport (f i)) (x : X) :
    ∃ s : Finset ι, ∀ᶠ y in 𝓝 x, ∏ᶠ i, f i y = ∏ i ∈ s, f i y 
Arguments: {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [CommMonoid M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.mulSupport (f i)) (x : X)
Goal: ∃ s, ∀ᶠ (y : X) in 𝓝 x, ∏ᶠ (i : ι), f i y = ∏ i ∈ s, f i y