ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- iInf_eq_of_forall_le_of_tendsto
theorem iInf_eq_of_forall_le_of_tendsto {x : R} {as : ι → R} (x_le : ∀ i, x ≤ as i) {F : Filter ι}
    [Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) : ⨅ i, as i = x 
Arguments: {ι : Type u_1} {R : Type u_4} [CompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] {x : R} {as : ι → R} (x_le : ∀ (i : ι), x ≤ as i) {F : Filter ι} [F.NeBot] (as_lim : Filter.Tendsto as F (𝓝 x))
Goal: ⨅ i, as i = x
-- iSup_eq_of_forall_le_of_tendsto
theorem iSup_eq_of_forall_le_of_tendsto {x : R} {as : ι → R} (le_x : ∀ i, as i ≤ x) {F : Filter ι}
    [Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) : ⨆ i, as i = x 
Arguments: {ι : Type u_1} {R : Type u_4} [CompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] {x : R} {as : ι → R} (le_x : ∀ (i : ι), as i ≤ x) {F : Filter ι} [F.NeBot] (as_lim : Filter.Tendsto as F (𝓝 x))
Goal: ⨆ i, as i = x
-- iUnion_Ici_eq_Ioi_of_lt_of_tendsto
theorem iUnion_Ici_eq_Ioi_of_lt_of_tendsto (x : R) {as : ι → R} (x_lt : ∀ i, x < as i)
    {F : Filter ι} [Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) :
    ⋃ i : ι, Ici (as i) = Ioi x 
Arguments: {ι : Type u_1} {R : Type u_4} [CompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (x : R) {as : ι → R} (x_lt : ∀ (i : ι), x < as i) {F : Filter ι} [F.NeBot] (as_lim : Filter.Tendsto as F (𝓝 x))
Goal: ⋃ i, Set.Ici (as i) = Set.Ioi x
-- iUnion_Iic_eq_Iio_of_lt_of_tendsto
theorem iUnion_Iic_eq_Iio_of_lt_of_tendsto (x : R) {as : ι → R} (lt_x : ∀ i, as i < x)
    {F : Filter ι} [Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) :
    ⋃ i : ι, Iic (as i) = Iio x 
Arguments: {ι : Type u_1} {R : Type u_4} [CompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (x : R) {as : ι → R} (lt_x : ∀ (i : ι), as i < x) {F : Filter ι} [F.NeBot] (as_lim : Filter.Tendsto as F (𝓝 x))
Goal: ⋃ i, Set.Iic (as i) = Set.Iio x
-- limsup_eq_tendsto_sum_indicator_nat_atTop
theorem limsup_eq_tendsto_sum_indicator_nat_atTop (s : ℕ → Set α) :
    limsup s atTop = { ω | Tendsto
      (fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω) atTop atTop } 
Arguments: {α : Type u_2} (s : ℕ → Set α)
Goal: Filter.limsup s Filter.atTop =
  {ω | Filter.Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) Filter.atTop Filter.atTop}
-- limsup_eq_tendsto_sum_indicator_atTop
theorem limsup_eq_tendsto_sum_indicator_atTop (R : Type*) [StrictOrderedSemiring R] [Archimedean R]
    (s : ℕ → Set α) : limsup s atTop = { ω | Tendsto
      (fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → R) ω) atTop atTop } 
Arguments: {α : Type u_2} (R : Type u_7) [StrictOrderedSemiring R] [Archimedean R] (s : ℕ → Set α)
Goal: Filter.limsup s Filter.atTop =
  {ω | Filter.Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) Filter.atTop Filter.atTop}
-- limsup_const_add
lemma limsup_const_add (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
    [CovariantClass R R (fun x y ↦ x + y) fun x y ↦ x ≤ y] (f : ι → R) (c : R)
    (bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
    Filter.limsup (fun i ↦ c + f i) F = c + Filter.limsup f F 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [Add R] [ContinuousAdd R] [CovariantClass R R (fun x y => x + y) fun x y => x ≤ y] (f : ι → R) (c : R) (bdd_above : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) F f) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F f)
Goal: Filter.limsup (fun i => c + f i) F = c + Filter.limsup f F
-- limsup_add_const
lemma limsup_add_const (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
    [CovariantClass R R (Function.swap fun x y ↦ x + y) fun x y ↦ x ≤ y] (f : ι → R) (c : R)
    (bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
    Filter.limsup (fun i ↦ f i + c) F = Filter.limsup f F + c 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [Add R] [ContinuousAdd R] [CovariantClass R R (Function.swap fun x y => x + y) fun x y => x ≤ y] (f : ι → R) (c : R) (bdd_above : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) F f) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F f)
Goal: Filter.limsup (fun i => f i + c) F = Filter.limsup f F + c
-- liminf_const_add
lemma liminf_const_add (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
    [CovariantClass R R (fun x y ↦ x + y) fun x y ↦ x ≤ y]  (f : ι → R) (c : R)
    (cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
    Filter.liminf (fun i ↦ c + f i) F = c + Filter.liminf f F 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [Add R] [ContinuousAdd R] [CovariantClass R R (fun x y => x + y) fun x y => x ≤ y] (f : ι → R) (c : R) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F f) (bdd_below : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) F f)
Goal: Filter.liminf (fun i => c + f i) F = c + Filter.liminf f F
-- liminf_add_const
lemma liminf_add_const (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
    [CovariantClass R R (Function.swap fun x y ↦ x + y) fun x y ↦ x ≤ y] (f : ι → R) (c : R)
    (cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
    Filter.liminf (fun i ↦ f i + c) F = Filter.liminf f F + c 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [Add R] [ContinuousAdd R] [CovariantClass R R (Function.swap fun x y => x + y) fun x y => x ≤ y] (f : ι → R) (c : R) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F f) (bdd_below : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) F f)
Goal: Filter.liminf (fun i => f i + c) F = Filter.liminf f F + c
-- limsup_const_sub
lemma limsup_const_sub (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
    [OrderedSub R] [CovariantClass R R (fun x y ↦ x + y) fun x y ↦ x ≤ y] (f : ι → R) (c : R)
    (cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
    Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R] [CovariantClass R R (fun x y => x + y) fun x y => x ≤ y] (f : ι → R) (c : R) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F f) (bdd_below : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) F f)
Goal: Filter.limsup (fun i => c - f i) F = c - Filter.liminf f F
-- limsup_sub_const
lemma limsup_sub_const (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
    [OrderedSub R] (f : ι → R) (c : R)
    (bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
    Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R] (f : ι → R) (c : R) (bdd_above : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) F f) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F f)
Goal: Filter.limsup (fun i => f i - c) F = Filter.limsup f F - c
-- liminf_const_sub
lemma liminf_const_sub (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
    [OrderedSub R] [CovariantClass R R (fun x y ↦ x + y) fun x y ↦ x ≤ y] (f : ι → R) (c : R)
    (bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
    Filter.liminf (fun i ↦ c - f i) F = c - Filter.limsup f F 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R] [CovariantClass R R (fun x y => x + y) fun x y => x ≤ y] (f : ι → R) (c : R) (bdd_above : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) F f) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≤ x_1) F f)
Goal: Filter.liminf (fun i => c - f i) F = c - Filter.limsup f F
-- liminf_sub_const
lemma liminf_sub_const (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
    [OrderedSub R] (f : ι → R) (c : R)
    (cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
    Filter.liminf (fun i ↦ f i - c) F = Filter.liminf f F - c 
Arguments: {ι : Type u_1} {R : Type u_4} [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] (F : Filter ι) [F.NeBot] [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R] (f : ι → R) (c : R) (cobdd : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) F f) (bdd_below : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) F f)
Goal: Filter.liminf (fun i => f i - c) F = Filter.liminf f F - c
-- LinearMap.isClosed_or_dense_ker
theorem LinearMap.isClosed_or_dense_ker (l : M →ₗ[R] N) :
    IsClosed (LinearMap.ker l : Set M) ∨ Dense (LinearMap.ker l : Set M) 
Arguments: {R : Type u} {M : Type v} {N : Type w} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [AddCommGroup N] [Module R M] [ContinuousSMul R M] [Module R N] [ContinuousAdd M] [IsSimpleModule R N] (l : M →ₗ[R] N)
Goal: IsClosed ↑(LinearMap.ker l) ∨ Dense ↑(LinearMap.ker l)
-- hasProd_pi_single
@[to_additive]
theorem hasProd_pi_single [DecidableEq β] (b : β) (a : α) : HasProd (Pi.mulSingle b a) a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [DecidableEq β] (b : β) (a : α)
Goal: HasProd (Pi.mulSingle b a) a
-- hasSum_pi_single
@[to_additive]
theorem hasProd_pi_single [DecidableEq β] (b : β) (a : α) : HasProd (Pi.mulSingle b a) a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [DecidableEq β] (b : β) (a : α)
Goal: HasSum (Pi.single b a) a
-- tsum_pi_single
@[to_additive (attr := simp)]
theorem tprod_pi_single [DecidableEq β] (b : β) (a : α) : ∏' b', Pi.mulSingle b a b' = a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [DecidableEq β] (b : β) (a : α)
Goal: ∑' (b' : β), Pi.single b a b' = a
-- tprod_pi_single
@[to_additive (attr := simp)]
theorem tprod_pi_single [DecidableEq β] (b : β) (a : α) : ∏' b', Pi.mulSingle b a b' = a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [DecidableEq β] (b : β) (a : α)
Goal: ∏' (b' : β), Pi.mulSingle b a b' = a
-- tprod_setProd_singleton_left
@[to_additive tsum_setProd_singleton_left]
lemma tprod_setProd_singleton_left (b : β) (t : Set γ) (f : β × γ → α) :
    (∏' x : {b} ×ˢ t, f x) = ∏' c : t, f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] (b : β) (t : Set γ) (f : β × γ → α)
Goal: ∏' (x : ↑({b} ×ˢ t)), f ↑x = ∏' (c : ↑t), f (b, ↑c)
-- tsum_setProd_singleton_left
@[to_additive tsum_setProd_singleton_left]
lemma tprod_setProd_singleton_left (b : β) (t : Set γ) (f : β × γ → α) :
    (∏' x : {b} ×ˢ t, f x) = ∏' c : t, f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] (b : β) (t : Set γ) (f : β × γ → α)
Goal: ∑' (x : ↑({b} ×ˢ t)), f ↑x = ∑' (c : ↑t), f (b, ↑c)
-- tsum_setProd_singleton_right
@[to_additive tsum_setProd_singleton_right]
lemma tprod_setProd_singleton_right (s : Set β) (c : γ) (f : β × γ → α) :
    (∏' x : s ×ˢ {c}, f x) = ∏' b : s, f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] (s : Set β) (c : γ) (f : β × γ → α)
Goal: ∑' (x : ↑(s ×ˢ {c})), f ↑x = ∑' (b : ↑s), f (↑b, c)
-- tprod_setProd_singleton_right
@[to_additive tsum_setProd_singleton_right]
lemma tprod_setProd_singleton_right (s : Set β) (c : γ) (f : β × γ → α) :
    (∏' x : s ×ˢ {c}, f x) = ∏' b : s, f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] (s : Set β) (c : γ) (f : β × γ → α)
Goal: ∏' (x : ↑(s ×ˢ {c})), f ↑x = ∏' (b : ↑s), f (↑b, c)
-- Multipliable.prod_symm
@[to_additive Summable.prod_symm]
theorem Multipliable.prod_symm {f : β × γ → α} (hf : Multipliable f) :
    Multipliable fun p : γ × β ↦ f p.swap 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] {f : β × γ → α} (hf : Multipliable f)
Goal: Multipliable fun p => f p.swap
-- Summable.prod_symm
@[to_additive Summable.prod_symm]
theorem Multipliable.prod_symm {f : β × γ → α} (hf : Multipliable f) :
    Multipliable fun p : γ × β ↦ f p.swap 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] {f : β × γ → α} (hf : Summable f)
Goal: Summable fun p => f p.swap
-- HasProd.prod_mk
@[to_additive HasSum.prod_mk]
theorem HasProd.prod_mk {f : β → α} {g : β → γ} {a : α} {b : γ}
    (hf : HasProd f a) (hg : HasProd g b) : HasProd (fun x ↦ (⟨f x, g x⟩ : α × γ)) ⟨a, b⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] [CommMonoid γ] [TopologicalSpace γ] {f : β → α} {g : β → γ} {a : α} {b : γ} (hf : HasProd f a) (hg : HasProd g b)
Goal: HasProd (fun x => (f x, g x)) (a, b)
-- HasSum.prod_mk
@[to_additive HasSum.prod_mk]
theorem HasProd.prod_mk {f : β → α} {g : β → γ} {a : α} {b : γ}
    (hf : HasProd f a) (hg : HasProd g b) : HasProd (fun x ↦ (⟨f x, g x⟩ : α × γ)) ⟨a, b⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [AddCommMonoid γ] [TopologicalSpace γ] {f : β → α} {g : β → γ} {a : α} {b : γ} (hf : HasSum f a) (hg : HasSum g b)
Goal: HasSum (fun x => (f x, g x)) (a, b)
-- HasProd.sigma
@[to_additive]
theorem HasProd.sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α} {a : α}
    (ha : HasProd f a) (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) : HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] [RegularSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} {g : β → α} {a : α} (ha : HasProd f a) (hf : ∀ (b : β), HasProd (fun c => f ⟨b, c⟩) (g b))
Goal: HasProd g a
-- HasSum.sigma
@[to_additive]
theorem HasProd.sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α} {a : α}
    (ha : HasProd f a) (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) : HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] [RegularSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} {g : β → α} {a : α} (ha : HasSum f a) (hf : ∀ (b : β), HasSum (fun c => f ⟨b, c⟩) (g b))
Goal: HasSum g a
-- HasProd.prod_fiberwise
@[to_additive HasSum.prod_fiberwise "If a series `f` on `β × γ` has sum `a` and for each `b` the
restriction of `f` to `{b} × γ` has sum `g b`, then the series `g` has sum `a`."]
theorem HasProd.prod_fiberwise {f : β × γ → α} {g : β → α} {a : α} (ha : HasProd f a)
    (hf : ∀ b, HasProd (fun c ↦ f (b, c)) (g b)) : HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] [RegularSpace α] {f : β × γ → α} {g : β → α} {a : α} (ha : HasProd f a) (hf : ∀ (b : β), HasProd (fun c => f (b, c)) (g b))
Goal: HasProd g a
-- HasSum.prod_fiberwise
@[to_additive HasSum.prod_fiberwise "If a series `f` on `β × γ` has sum `a` and for each `b` the
restriction of `f` to `{b} × γ` has sum `g b`, then the series `g` has sum `a`."]
theorem HasProd.prod_fiberwise {f : β × γ → α} {g : β → α} {a : α} (ha : HasProd f a)
    (hf : ∀ b, HasProd (fun c ↦ f (b, c)) (g b)) : HasProd g a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] [RegularSpace α] {f : β × γ → α} {g : β → α} {a : α} (ha : HasSum f a) (hf : ∀ (b : β), HasSum (fun c => f (b, c)) (g b))
Goal: HasSum g a
-- Summable.sigma'
@[to_additive]
theorem Multipliable.sigma' {γ : β → Type*} {f : (Σb : β, γ b) → α} (ha : Multipliable f)
    (hf : ∀ b, Multipliable fun c ↦ f ⟨b, c⟩) : Multipliable fun b ↦ ∏' c, f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] [RegularSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (ha : Summable f) (hf : ∀ (b : β), Summable fun c => f ⟨b, c⟩)
Goal: Summable fun b => ∑' (c : γ b), f ⟨b, c⟩
-- Multipliable.sigma'
@[to_additive]
theorem Multipliable.sigma' {γ : β → Type*} {f : (Σb : β, γ b) → α} (ha : Multipliable f)
    (hf : ∀ b, Multipliable fun c ↦ f ⟨b, c⟩) : Multipliable fun b ↦ ∏' c, f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] [RegularSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (ha : Multipliable f) (hf : ∀ (b : β), Multipliable fun c => f ⟨b, c⟩)
Goal: Multipliable fun b => ∏' (c : γ b), f ⟨b, c⟩
-- HasProd.sigma_of_hasProd
@[to_additive]
theorem HasProd.sigma_of_hasProd {γ : β → Type*} {f : (Σb : β, γ b) → α} {g : β → α}
    {a : α} (ha : HasProd g a) (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) (hf' : Multipliable f) :
    HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] [T3Space α] {γ : β → Type u_5} {f : (b : β) × γ b → α} {g : β → α} {a : α} (ha : HasProd g a) (hf : ∀ (b : β), HasProd (fun c => f ⟨b, c⟩) (g b)) (hf' : Multipliable f)
Goal: HasProd f a
-- HasSum.sigma_of_hasSum
@[to_additive]
theorem HasProd.sigma_of_hasProd {γ : β → Type*} {f : (Σb : β, γ b) → α} {g : β → α}
    {a : α} (ha : HasProd g a) (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) (hf' : Multipliable f) :
    HasProd f a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] [T3Space α] {γ : β → Type u_5} {f : (b : β) × γ b → α} {g : β → α} {a : α} (ha : HasSum g a) (hf : ∀ (b : β), HasSum (fun c => f ⟨b, c⟩) (g b)) (hf' : Summable f)
Goal: HasSum f a
-- tsum_sigma'
@[to_additive]
theorem tprod_sigma' {γ : β → Type*} {f : (Σb : β, γ b) → α}
    (h₁ : ∀ b, Multipliable fun c ↦ f ⟨b, c⟩) (h₂ : Multipliable f) :
    ∏' p, f p = ∏' (b) (c), f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] [T3Space α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (h₁ : ∀ (b : β), Summable fun c => f ⟨b, c⟩) (h₂ : Summable f)
Goal: ∑' (p : (b : β) × γ b), f p = ∑' (b : β) (c : γ b), f ⟨b, c⟩
-- tprod_sigma'
@[to_additive]
theorem tprod_sigma' {γ : β → Type*} {f : (Σb : β, γ b) → α}
    (h₁ : ∀ b, Multipliable fun c ↦ f ⟨b, c⟩) (h₂ : Multipliable f) :
    ∏' p, f p = ∏' (b) (c), f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] [T3Space α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (h₁ : ∀ (b : β), Multipliable fun c => f ⟨b, c⟩) (h₂ : Multipliable f)
Goal: ∏' (p : (b : β) × γ b), f p = ∏' (b : β) (c : γ b), f ⟨b, c⟩
-- tprod_prod'
@[to_additive tsum_prod']
theorem tprod_prod' {f : β × γ → α} (h : Multipliable f)
    (h₁ : ∀ b, Multipliable fun c ↦ f (b, c)) :
    ∏' p, f p = ∏' (b) (c), f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] [T3Space α] {f : β × γ → α} (h : Multipliable f) (h₁ : ∀ (b : β), Multipliable fun c => f (b, c))
Goal: ∏' (p : β × γ), f p = ∏' (b : β) (c : γ), f (b, c)
-- tsum_prod'
@[to_additive tsum_prod']
theorem tprod_prod' {f : β × γ → α} (h : Multipliable f)
    (h₁ : ∀ b, Multipliable fun c ↦ f (b, c)) :
    ∏' p, f p = ∏' (b) (c), f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] [T3Space α] {f : β × γ → α} (h : Summable f) (h₁ : ∀ (b : β), Summable fun c => f (b, c))
Goal: ∑' (p : β × γ), f p = ∑' (b : β) (c : γ), f (b, c)
-- tprod_comm'
@[to_additive]
theorem tprod_comm' {f : β → γ → α} (h : Multipliable (Function.uncurry f))
    (h₁ : ∀ b, Multipliable (f b)) (h₂ : ∀ c, Multipliable fun b ↦ f b c) :
    ∏' (c) (b), f b c = ∏' (b) (c), f b c 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [TopologicalSpace α] [ContinuousMul α] [T3Space α] {f : β → γ → α} (h : Multipliable (Function.uncurry f)) (h₁ : ∀ (b : β), Multipliable (f b)) (h₂ : ∀ (c : γ), Multipliable fun b => f b c)
Goal: ∏' (c : γ) (b : β), f b c = ∏' (b : β) (c : γ), f b c
-- tsum_comm'
@[to_additive]
theorem tprod_comm' {f : β → γ → α} (h : Multipliable (Function.uncurry f))
    (h₁ : ∀ b, Multipliable (f b)) (h₂ : ∀ c, Multipliable fun b ↦ f b c) :
    ∏' (c) (b), f b c = ∏' (b) (c), f b c 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [TopologicalSpace α] [ContinuousAdd α] [T3Space α] {f : β → γ → α} (h : Summable (Function.uncurry f)) (h₁ : ∀ (b : β), Summable (f b)) (h₂ : ∀ (c : γ), Summable fun b => f b c)
Goal: ∑' (c : γ) (b : β), f b c = ∑' (b : β) (c : γ), f b c
-- Summable.sigma_factor
@[to_additive]
theorem Multipliable.sigma_factor {γ : β → Type*} {f : (Σb : β, γ b) → α}
    (ha : Multipliable f) (b : β) :
    Multipliable fun c ↦ f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] [CompleteSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (ha : Summable f) (b : β)
Goal: Summable fun c => f ⟨b, c⟩
-- Multipliable.sigma_factor
@[to_additive]
theorem Multipliable.sigma_factor {γ : β → Type*} {f : (Σb : β, γ b) → α}
    (ha : Multipliable f) (b : β) :
    Multipliable fun c ↦ f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (ha : Multipliable f) (b : β)
Goal: Multipliable fun c => f ⟨b, c⟩
-- Multipliable.sigma
@[to_additive]
theorem Multipliable.sigma {γ : β → Type*} {f : (Σb : β, γ b) → α} (ha : Multipliable f) :
    Multipliable fun b ↦ ∏' c, f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (ha : Multipliable f)
Goal: Multipliable fun b => ∏' (c : γ b), f ⟨b, c⟩
-- Summable.sigma
@[to_additive]
theorem Multipliable.sigma {γ : β → Type*} {f : (Σb : β, γ b) → α} (ha : Multipliable f) :
    Multipliable fun b ↦ ∏' c, f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] [CompleteSpace α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (ha : Summable f)
Goal: Summable fun b => ∑' (c : γ b), f ⟨b, c⟩
-- Multipliable.prod_factor
@[to_additive Summable.prod_factor]
theorem Multipliable.prod_factor {f : β × γ → α} (h : Multipliable f) (b : β) :
    Multipliable fun c ↦ f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] {f : β × γ → α} (h : Multipliable f) (b : β)
Goal: Multipliable fun c => f (b, c)
-- Summable.prod_factor
@[to_additive Summable.prod_factor]
theorem Multipliable.prod_factor {f : β × γ → α} (h : Multipliable f) (b : β) :
    Multipliable fun c ↦ f (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] [CompleteSpace α] {f : β × γ → α} (h : Summable f) (b : β)
Goal: Summable fun c => f (b, c)
-- HasProd.tprod_fiberwise
@[to_additive]
lemma HasProd.tprod_fiberwise [T2Space α] {f : β → α} {a : α} (hf : HasProd f a) (g : β → γ) :
    HasProd (fun c : γ ↦ ∏' b : g ⁻¹' {c}, f b) a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] [T2Space α] {f : β → α} {a : α} (hf : HasProd f a) (g : β → γ)
Goal: HasProd (fun c => ∏' (b : ↑(g ⁻¹' {c})), f ↑b) a
-- HasSum.tsum_fiberwise
@[to_additive]
lemma HasProd.tprod_fiberwise [T2Space α] {f : β → α} {a : α} (hf : HasProd f a) (g : β → γ) :
    HasProd (fun c : γ ↦ ∏' b : g ⁻¹' {c}, f b) a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] [CompleteSpace α] [T2Space α] {f : β → α} {a : α} (hf : HasSum f a) (g : β → γ)
Goal: HasSum (fun c => ∑' (b : ↑(g ⁻¹' {c})), f ↑b) a
-- tprod_sigma
@[to_additive]
theorem tprod_sigma {γ : β → Type*} {f : (Σb : β, γ b) → α} (ha : Multipliable f) :
    ∏' p, f p = ∏' (b) (c), f ⟨b, c⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α] [T0Space α] {γ : β → Type u_5} {f : (b : β) × γ b → α} (ha : Multipliable f)
Goal: ∏' (p : (b : β) × γ b), f p = ∏' (b : β) (c : γ b), f ⟨b, c⟩