ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMap.compStarAlgHom_apply
@[simps]
def compStarAlgHom (φ : A →⋆ₐ[𝕜] B) (hφ : Continuous φ) :
    C(X, A) →⋆ₐ[𝕜] C(X, B) 
Arguments: (X : Type u_1) {𝕜 : Type u_2} {A : Type u_3} {B : Type u_4} [TopologicalSpace X] [CommSemiring 𝕜] [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A] [TopologicalSpace B] [Semiring B] [TopologicalSemiring B] [Star B] [ContinuousStar B] [Algebra 𝕜 B] (φ : A →⋆ₐ[𝕜] B) (hφ : Continuous ⇑φ) (f : C(X, A))
Goal: (ContinuousMap.compStarAlgHom X φ hφ) f = { toFun := ⇑φ, continuous_toFun := hφ }.comp f
-- ContinuousMap.compStarAlgHom_id
lemma compStarAlgHom_id : compStarAlgHom X (.id 𝕜 A) continuous_id = .id 𝕜 C(X, A) 
Arguments: (X : Type u_1) {𝕜 : Type u_2} {A : Type u_3} [TopologicalSpace X] [CommSemiring 𝕜] [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A]
Goal: ContinuousMap.compStarAlgHom X (StarAlgHom.id 𝕜 A) ⋯ = StarAlgHom.id 𝕜 C(X, A)
-- ContinuousMap.compStarAlgHom_comp
lemma compStarAlgHom_comp (φ : A →⋆ₐ[𝕜] B) (ψ : B →⋆ₐ[𝕜] C) (hφ : Continuous φ)
    (hψ : Continuous ψ) : compStarAlgHom X (ψ.comp φ) (hψ.comp hφ) =
      (compStarAlgHom X ψ hψ).comp (compStarAlgHom X φ hφ) 
Arguments: (X : Type u_1) {𝕜 : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [TopologicalSpace X] [CommSemiring 𝕜] [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A] [TopologicalSpace B] [Semiring B] [TopologicalSemiring B] [Star B] [ContinuousStar B] [Algebra 𝕜 B] [TopologicalSpace C] [Semiring C] [TopologicalSemiring C] [Star C] [ContinuousStar C] [Algebra 𝕜 C] (φ : A →⋆ₐ[𝕜] B) (ψ : B →⋆ₐ[𝕜] C) (hφ : Continuous ⇑φ) (hψ : Continuous ⇑ψ)
Goal: ContinuousMap.compStarAlgHom X (ψ.comp φ) ⋯ =
  (ContinuousMap.compStarAlgHom X ψ hψ).comp (ContinuousMap.compStarAlgHom X φ hφ)
-- ContinuousMap.periodic_tsum_comp_add_zsmul
theorem periodic_tsum_comp_add_zsmul [AddCommGroup X] [TopologicalAddGroup X] [AddCommMonoid Y]
    [ContinuousAdd Y] [T2Space Y] (f : C(X, Y)) (p : X) :
    Function.Periodic (⇑(∑' n : ℤ, f.comp (ContinuousMap.addRight (n • p)))) p 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [AddCommGroup X] [TopologicalAddGroup X] [AddCommMonoid Y] [ContinuousAdd Y] [T2Space Y] (f : C(X, Y)) (p : X)
Goal: Function.Periodic (⇑(∑' (n : ℤ), f.comp (ContinuousMap.addRight (n • p)))) p
-- Homeomorph.compStarAlgEquiv'_symm_apply
@[simps]
def compStarAlgEquiv' (f : X ≃ₜ Y) : C(Y, A) ≃⋆ₐ[𝕜] C(X, A) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (𝕜 : Type u_3) [CommSemiring 𝕜] (A : Type u_4) [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [StarRing A] [ContinuousStar A] [Algebra 𝕜 A] (f : X ≃ₜ Y) (a : C(X, A))
Goal: (Homeomorph.compStarAlgEquiv' 𝕜 A f).symm a = (ContinuousMap.compStarAlgHom' 𝕜 A f.symm.toContinuousMap) a
-- Homeomorph.compStarAlgEquiv'_apply
@[simps]
def compStarAlgEquiv' (f : X ≃ₜ Y) : C(Y, A) ≃⋆ₐ[𝕜] C(X, A) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (𝕜 : Type u_3) [CommSemiring 𝕜] (A : Type u_4) [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [StarRing A] [ContinuousStar A] [Algebra 𝕜 A] (f : X ≃ₜ Y) (a : C(Y, A))
Goal: (Homeomorph.compStarAlgEquiv' 𝕜 A f) a = (ContinuousMap.compStarAlgHom' 𝕜 A f.toContinuousMap) a
-- ContinuousMap.evalAlgHom_apply
@[simps]
def ContinuousMap.evalAlgHom (x : X) : C(X, R) →ₐ[S] R 
Arguments: {X : Type u_1} (S : Type u_2) (R : Type u_3) [TopologicalSpace X] [CommSemiring S] [CommSemiring R] [Algebra S R] [TopologicalSpace R] [TopologicalSemiring R] (x : X) (f : C(X, R))
Goal: (ContinuousMap.evalAlgHom S R x) f = f x
-- ContinuousMap.evalStarAlgHom_apply
@[simps!]
def ContinuousMap.evalStarAlgHom [StarRing R] [ContinuousStar R] (x : X) :
    C(X, R) →⋆ₐ[S] R 
Arguments: {X : Type u_1} (S : Type u_2) (R : Type u_3) [TopologicalSpace X] [CommSemiring S] [CommSemiring R] [Algebra S R] [TopologicalSpace R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] (x : X) (f : C(X, R))
Goal: (ContinuousMap.evalStarAlgHom S R x) f = f x
-- tsum_le_of_sum_range_le
@[to_additive]
theorem tprod_le_of_prod_range_le (hf : Multipliable f) (h : ∀ n, ∏ i ∈ range n, f i ≤ c) :
    ∏' n, f n ≤ c 
Arguments: {α : Type u_3} [Preorder α] [AddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] [T2Space α] {f : ℕ → α} {c : α} (hf : Summable f) (h : ∀ (n : ℕ), ∑ i ∈ Finset.range n, f i ≤ c)
Goal: ∑' (n : ℕ), f n ≤ c
-- tprod_le_of_prod_range_le
@[to_additive]
theorem tprod_le_of_prod_range_le (hf : Multipliable f) (h : ∀ n, ∏ i ∈ range n, f i ≤ c) :
    ∏' n, f n ≤ c 
Arguments: {α : Type u_3} [Preorder α] [CommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] [T2Space α] {f : ℕ → α} {c : α} (hf : Multipliable f) (h : ∀ (n : ℕ), ∏ i ∈ Finset.range n, f i ≤ c)
Goal: ∏' (n : ℕ), f n ≤ c
-- hasSum_le
@[to_additive]
theorem hasProd_le (h : ∀ i, f i ≤ g i) (hf : HasProd f a₁) (hg : HasProd g a₂) : a₁ ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} (h : ∀ (i : ι), f i ≤ g i) (hf : HasSum f a₁) (hg : HasSum g a₂)
Goal: a₁ ≤ a₂
-- hasProd_le
@[to_additive]
theorem hasProd_le (h : ∀ i, f i ≤ g i) (hf : HasProd f a₁) (hg : HasProd g a₂) : a₁ ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} (h : ∀ (i : ι), f i ≤ g i) (hf : HasProd f a₁) (hg : HasProd g a₂)
Goal: a₁ ≤ a₂
-- hasSum_mono
@[to_additive (attr := mono)]
theorem hasProd_mono (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f ≤ g) : a₁ ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} (hf : HasSum f a₁) (hg : HasSum g a₂) (h : f ≤ g)
Goal: a₁ ≤ a₂
-- hasProd_mono
@[to_additive (attr := mono)]
theorem hasProd_mono (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f ≤ g) : a₁ ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f ≤ g)
Goal: a₁ ≤ a₂
-- hasSum_le_of_sum_le
@[to_additive]
theorem hasProd_le_of_prod_le (hf : HasProd f a) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : a ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a a₂ : α} (hf : HasSum f a) (h : ∀ (s : Finset ι), ∑ i ∈ s, f i ≤ a₂)
Goal: a ≤ a₂
-- hasProd_le_of_prod_le
@[to_additive]
theorem hasProd_le_of_prod_le (hf : HasProd f a) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : a ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a a₂ : α} (hf : HasProd f a) (h : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂)
Goal: a ≤ a₂
-- le_hasProd_of_le_prod
@[to_additive]
theorem le_hasProd_of_le_prod (hf : HasProd f a) (h : ∀ s, a₂ ≤ ∏ i ∈ s, f i) : a₂ ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a a₂ : α} (hf : HasProd f a) (h : ∀ (s : Finset ι), a₂ ≤ ∏ i ∈ s, f i)
Goal: a₂ ≤ a
-- le_hasSum_of_le_sum
@[to_additive]
theorem le_hasProd_of_le_prod (hf : HasProd f a) (h : ∀ s, a₂ ≤ ∏ i ∈ s, f i) : a₂ ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a a₂ : α} (hf : HasSum f a) (h : ∀ (s : Finset ι), a₂ ≤ ∑ i ∈ s, f i)
Goal: a₂ ≤ a
-- hasProd_le_inj
@[to_additive]
theorem hasProd_le_inj {g : κ → α} (e : ι → κ) (he : Injective e)
    (hs : ∀ c, c ∉ Set.range e → 1 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : HasProd f a₁)
    (hg : HasProd g a₂) : a₁ ≤ a₂ 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a₁ a₂ : α} {g : κ → α} (e : ι → κ) (he : Function.Injective e) (hs : ∀ c ∉ Set.range e, 1 ≤ g c) (h : ∀ (i : ι), f i ≤ g (e i)) (hf : HasProd f a₁) (hg : HasProd g a₂)
Goal: a₁ ≤ a₂
-- hasSum_le_inj
@[to_additive]
theorem hasProd_le_inj {g : κ → α} (e : ι → κ) (he : Injective e)
    (hs : ∀ c, c ∉ Set.range e → 1 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : HasProd f a₁)
    (hg : HasProd g a₂) : a₁ ≤ a₂ 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a₁ a₂ : α} {g : κ → α} (e : ι → κ) (he : Function.Injective e) (hs : ∀ c ∉ Set.range e, 0 ≤ g c) (h : ∀ (i : ι), f i ≤ g (e i)) (hf : HasSum f a₁) (hg : HasSum g a₂)
Goal: a₁ ≤ a₂
-- tprod_le_tprod_of_inj
@[to_additive]
theorem tprod_le_tprod_of_inj {g : κ → α} (e : ι → κ) (he : Injective e)
    (hs : ∀ c, c ∉ Set.range e → 1 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : Multipliable f)
    (hg : Multipliable g) : tprod f ≤ tprod g 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {g : κ → α} (e : ι → κ) (he : Function.Injective e) (hs : ∀ c ∉ Set.range e, 1 ≤ g c) (h : ∀ (i : ι), f i ≤ g (e i)) (hf : Multipliable f) (hg : Multipliable g)
Goal: tprod f ≤ tprod g
-- tsum_le_tsum_of_inj
@[to_additive]
theorem tprod_le_tprod_of_inj {g : κ → α} (e : ι → κ) (he : Injective e)
    (hs : ∀ c, c ∉ Set.range e → 1 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : Multipliable f)
    (hg : Multipliable g) : tprod f ≤ tprod g 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {g : κ → α} (e : ι → κ) (he : Function.Injective e) (hs : ∀ c ∉ Set.range e, 0 ≤ g c) (h : ∀ (i : ι), f i ≤ g (e i)) (hf : Summable f) (hg : Summable g)
Goal: tsum f ≤ tsum g
-- tprod_subtype_le
@[to_additive]
lemma tprod_subtype_le {κ γ : Type*} [OrderedCommGroup γ] [UniformSpace γ] [UniformGroup γ]
    [OrderClosedTopology γ] [CompleteSpace γ] (f : κ → γ) (β : Set κ) (h : ∀ a : κ, 1 ≤ f a)
    (hf : Multipliable f) : (∏' (b : β), f b) ≤ (∏' (a : κ), f a) 
Arguments: {κ : Type u_4} {γ : Type u_5} [OrderedCommGroup γ] [UniformSpace γ] [UniformGroup γ] [OrderClosedTopology γ] [CompleteSpace γ] (f : κ → γ) (β : Set κ) (h : ∀ (a : κ), 1 ≤ f a) (hf : Multipliable f)
Goal: ∏' (b : ↑β), f ↑b ≤ ∏' (a : κ), f a
-- tsum_subtype_le
@[to_additive]
lemma tprod_subtype_le {κ γ : Type*} [OrderedCommGroup γ] [UniformSpace γ] [UniformGroup γ]
    [OrderClosedTopology γ] [CompleteSpace γ] (f : κ → γ) (β : Set κ) (h : ∀ a : κ, 1 ≤ f a)
    (hf : Multipliable f) : (∏' (b : β), f b) ≤ (∏' (a : κ), f a) 
Arguments: {κ : Type u_4} {γ : Type u_5} [OrderedAddCommGroup γ] [UniformSpace γ] [UniformAddGroup γ] [OrderClosedTopology γ] [CompleteSpace γ] (f : κ → γ) (β : Set κ) (h : ∀ (a : κ), 0 ≤ f a) (hf : Summable f)
Goal: ∑' (b : ↑β), f ↑b ≤ ∑' (a : κ), f a
-- sum_le_hasSum
@[to_additive]
theorem prod_le_hasProd (s : Finset ι) (hs : ∀ i, i ∉ s → 1 ≤ f i) (hf : HasProd f a) :
    ∏ i ∈ s, f i ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (s : Finset ι) (hs : ∀ i ∉ s, 0 ≤ f i) (hf : HasSum f a)
Goal: ∑ i ∈ s, f i ≤ a
-- prod_le_hasProd
@[to_additive]
theorem prod_le_hasProd (s : Finset ι) (hs : ∀ i, i ∉ s → 1 ≤ f i) (hf : HasProd f a) :
    ∏ i ∈ s, f i ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (s : Finset ι) (hs : ∀ i ∉ s, 1 ≤ f i) (hf : HasProd f a)
Goal: ∏ i ∈ s, f i ≤ a
-- isLUB_hasSum
@[to_additive]
theorem isLUB_hasProd (h : ∀ i, 1 ≤ f i) (hf : HasProd f a) :
    IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (h : ∀ (i : ι), 0 ≤ f i) (hf : HasSum f a)
Goal: IsLUB (Set.range fun s => ∑ i ∈ s, f i) a
-- isLUB_hasProd
@[to_additive]
theorem isLUB_hasProd (h : ∀ i, 1 ≤ f i) (hf : HasProd f a) :
    IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (h : ∀ (i : ι), 1 ≤ f i) (hf : HasProd f a)
Goal: IsLUB (Set.range fun s => ∏ i ∈ s, f i) a
-- le_hasSum
@[to_additive]
theorem le_hasProd (hf : HasProd f a) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) : f i ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (hf : HasSum f a) (i : ι) (hb : ∀ (j : ι), j ≠ i → 0 ≤ f j)
Goal: f i ≤ a
-- le_hasProd
@[to_additive]
theorem le_hasProd (hf : HasProd f a) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) : f i ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (hf : HasProd f a) (i : ι) (hb : ∀ (j : ι), j ≠ i → 1 ≤ f j)
Goal: f i ≤ a
-- sum_le_tsum
@[to_additive]
theorem prod_le_tprod {f : ι → α} (s : Finset ι) (hs : ∀ i, i ∉ s → 1 ≤ f i) (hf : Multipliable f) :
    ∏ i ∈ s, f i ≤ ∏' i, f i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (s : Finset ι) (hs : ∀ i ∉ s, 0 ≤ f i) (hf : Summable f)
Goal: ∑ i ∈ s, f i ≤ ∑' (i : ι), f i
-- prod_le_tprod
@[to_additive]
theorem prod_le_tprod {f : ι → α} (s : Finset ι) (hs : ∀ i, i ∉ s → 1 ≤ f i) (hf : Multipliable f) :
    ∏ i ∈ s, f i ≤ ∏' i, f i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (s : Finset ι) (hs : ∀ i ∉ s, 1 ≤ f i) (hf : Multipliable f)
Goal: ∏ i ∈ s, f i ≤ ∏' (i : ι), f i
-- le_tsum
@[to_additive]
theorem le_tprod (hf : Multipliable f) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) : f i ≤ ∏' i, f i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Summable f) (i : ι) (hb : ∀ (j : ι), j ≠ i → 0 ≤ f j)
Goal: f i ≤ ∑' (i : ι), f i
-- le_tprod
@[to_additive]
theorem le_tprod (hf : Multipliable f) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) : f i ≤ ∏' i, f i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : Multipliable f) (i : ι) (hb : ∀ (j : ι), j ≠ i → 1 ≤ f j)
Goal: f i ≤ ∏' (i : ι), f i
-- tprod_le_tprod
@[to_additive]
theorem tprod_le_tprod (h : ∀ i, f i ≤ g i) (hf : Multipliable f) (hg : Multipliable g) :
    ∏' i, f i ≤ ∏' i, g i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} (h : ∀ (i : ι), f i ≤ g i) (hf : Multipliable f) (hg : Multipliable g)
Goal: ∏' (i : ι), f i ≤ ∏' (i : ι), g i
-- tsum_le_tsum
@[to_additive]
theorem tprod_le_tprod (h : ∀ i, f i ≤ g i) (hf : Multipliable f) (hg : Multipliable g) :
    ∏' i, f i ≤ ∏' i, g i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} (h : ∀ (i : ι), f i ≤ g i) (hf : Summable f) (hg : Summable g)
Goal: ∑' (i : ι), f i ≤ ∑' (i : ι), g i
-- tsum_mono
@[to_additive (attr := mono)]
theorem tprod_mono (hf : Multipliable f) (hg : Multipliable g) (h : f ≤ g) :
    ∏' n, f n ≤ ∏' n, g n 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} (hf : Summable f) (hg : Summable g) (h : f ≤ g)
Goal: ∑' (n : ι), f n ≤ ∑' (n : ι), g n
-- tprod_mono
@[to_additive (attr := mono)]
theorem tprod_mono (hf : Multipliable f) (hg : Multipliable g) (h : f ≤ g) :
    ∏' n, f n ≤ ∏' n, g n 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α} (hf : Multipliable f) (hg : Multipliable g) (h : f ≤ g)
Goal: ∏' (n : ι), f n ≤ ∏' (n : ι), g n
-- tprod_le_of_prod_le
@[to_additive]
theorem tprod_le_of_prod_le (hf : Multipliable f) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : ∏' i, f i ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a₂ : α} (hf : Multipliable f) (h : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂)
Goal: ∏' (i : ι), f i ≤ a₂
-- tsum_le_of_sum_le
@[to_additive]
theorem tprod_le_of_prod_le (hf : Multipliable f) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : ∏' i, f i ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a₂ : α} (hf : Summable f) (h : ∀ (s : Finset ι), ∑ i ∈ s, f i ≤ a₂)
Goal: ∑' (i : ι), f i ≤ a₂
-- tprod_le_of_prod_le'
@[to_additive]
theorem tprod_le_of_prod_le' (ha₂ : 1 ≤ a₂) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : ∏' i, f i ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a₂ : α} (ha₂ : 1 ≤ a₂) (h : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂)
Goal: ∏' (i : ι), f i ≤ a₂
-- tsum_le_of_sum_le'
@[to_additive]
theorem tprod_le_of_prod_le' (ha₂ : 1 ≤ a₂) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : ∏' i, f i ≤ a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a₂ : α} (ha₂ : 0 ≤ a₂) (h : ∀ (s : Finset ι), ∑ i ∈ s, f i ≤ a₂)
Goal: ∑' (i : ι), f i ≤ a₂
-- HasProd.one_le
@[to_additive]
theorem HasProd.one_le (h : ∀ i, 1 ≤ g i) (ha : HasProd g a) : 1 ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {g : ι → α} {a : α} (h : ∀ (i : ι), 1 ≤ g i) (ha : HasProd g a)
Goal: 1 ≤ a
-- HasSum.nonneg
@[to_additive]
theorem HasProd.one_le (h : ∀ i, 1 ≤ g i) (ha : HasProd g a) : 1 ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {g : ι → α} {a : α} (h : ∀ (i : ι), 0 ≤ g i) (ha : HasSum g a)
Goal: 0 ≤ a
-- HasSum.nonpos
@[to_additive]
theorem HasProd.le_one (h : ∀ i, g i ≤ 1) (ha : HasProd g a) : a ≤ 1 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {g : ι → α} {a : α} (h : ∀ (i : ι), g i ≤ 0) (ha : HasSum g a)
Goal: a ≤ 0
-- HasProd.le_one
@[to_additive]
theorem HasProd.le_one (h : ∀ i, g i ≤ 1) (ha : HasProd g a) : a ≤ 1 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {g : ι → α} {a : α} (h : ∀ (i : ι), g i ≤ 1) (ha : HasProd g a)
Goal: a ≤ 1
-- tsum_nonneg
@[to_additive tsum_nonneg]
theorem one_le_tprod (h : ∀ i, 1 ≤ g i) : 1 ≤ ∏' i, g i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {g : ι → α} (h : ∀ (i : ι), 0 ≤ g i)
Goal: 0 ≤ ∑' (i : ι), g i
-- one_le_tprod
@[to_additive tsum_nonneg]
theorem one_le_tprod (h : ∀ i, 1 ≤ g i) : 1 ≤ ∏' i, g i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {g : ι → α} (h : ∀ (i : ι), 1 ≤ g i)
Goal: 1 ≤ ∏' (i : ι), g i
-- tprod_le_one
@[to_additive]
theorem tprod_le_one (h : ∀ i, f i ≤ 1) : ∏' i, f i ≤ 1 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (h : ∀ (i : ι), f i ≤ 1)
Goal: ∏' (i : ι), f i ≤ 1
-- tsum_nonpos
@[to_additive]
theorem tprod_le_one (h : ∀ i, f i ≤ 1) : ∏' i, f i ≤ 1 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (h : ∀ (i : ι), f i ≤ 0)
Goal: ∑' (i : ι), f i ≤ 0
-- hasSum_zero_iff_of_nonneg
@[to_additive]
theorem hasProd_one_iff_of_one_le (hf : ∀ i, 1 ≤ f i) : HasProd f 1 ↔ f = 1 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : ∀ (i : ι), 0 ≤ f i)
Goal: HasSum f 0 ↔ f = 0
-- hasProd_one_iff_of_one_le
@[to_additive]
theorem hasProd_one_iff_of_one_le (hf : ∀ i, 1 ≤ f i) : HasProd f 1 ↔ f = 1 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} (hf : ∀ (i : ι), 1 ≤ f i)
Goal: HasProd f 1 ↔ f = 1
-- hasSum_lt
@[to_additive]
theorem hasProd_lt (h : f ≤ g) (hi : f i < g i) (hf : HasProd f a₁) (hg : HasProd g a₂) :
    a₁ < a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} {i : ι} (h : f ≤ g) (hi : f i < g i) (hf : HasSum f a₁) (hg : HasSum g a₂)
Goal: a₁ < a₂
-- hasProd_lt
@[to_additive]
theorem hasProd_lt (h : f ≤ g) (hi : f i < g i) (hf : HasProd f a₁) (hg : HasProd g a₂) :
    a₁ < a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommGroup α] [TopologicalSpace α] [TopologicalGroup α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} {i : ι} (h : f ≤ g) (hi : f i < g i) (hf : HasProd f a₁) (hg : HasProd g a₂)
Goal: a₁ < a₂
-- hasProd_strict_mono
@[to_additive (attr := mono)]
theorem hasProd_strict_mono (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f < g) : a₁ < a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommGroup α] [TopologicalSpace α] [TopologicalGroup α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f < g)
Goal: a₁ < a₂
-- hasSum_strict_mono
@[to_additive (attr := mono)]
theorem hasProd_strict_mono (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f < g) : a₁ < a₂ 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} (hf : HasSum f a₁) (hg : HasSum g a₂) (h : f < g)
Goal: a₁ < a₂
-- tsum_lt_tsum
@[to_additive]
theorem tprod_lt_tprod (h : f ≤ g) (hi : f i < g i) (hf : Multipliable f) (hg : Multipliable g) :
    ∏' n, f n < ∏' n, g n 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] [OrderClosedTopology α] {f g : ι → α} {i : ι} (h : f ≤ g) (hi : f i < g i) (hf : Summable f) (hg : Summable g)
Goal: ∑' (n : ι), f n < ∑' (n : ι), g n
-- tprod_lt_tprod
@[to_additive]
theorem tprod_lt_tprod (h : f ≤ g) (hi : f i < g i) (hf : Multipliable f) (hg : Multipliable g) :
    ∏' n, f n < ∏' n, g n 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommGroup α] [TopologicalSpace α] [TopologicalGroup α] [OrderClosedTopology α] {f g : ι → α} {i : ι} (h : f ≤ g) (hi : f i < g i) (hf : Multipliable f) (hg : Multipliable g)
Goal: ∏' (n : ι), f n < ∏' (n : ι), g n
-- tprod_strict_mono
@[to_additive (attr := mono)]
theorem tprod_strict_mono (hf : Multipliable f) (hg : Multipliable g) (h : f < g) :
    ∏' n, f n < ∏' n, g n 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommGroup α] [TopologicalSpace α] [TopologicalGroup α] [OrderClosedTopology α] {f g : ι → α} (hf : Multipliable f) (hg : Multipliable g) (h : f < g)
Goal: ∏' (n : ι), f n < ∏' (n : ι), g n
-- tsum_strict_mono
@[to_additive (attr := mono)]
theorem tprod_strict_mono (hf : Multipliable f) (hg : Multipliable g) (h : f < g) :
    ∏' n, f n < ∏' n, g n 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] [OrderClosedTopology α] {f g : ι → α} (hf : Summable f) (hg : Summable g) (h : f < g)
Goal: ∑' (n : ι), f n < ∑' (n : ι), g n
-- tsum_pos
@[to_additive tsum_pos]
theorem one_lt_tprod (hsum : Multipliable g) (hg : ∀ i, 1 ≤ g i) (i : ι) (hi : 1 < g i) :
    1 < ∏' i, g i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedAddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α] [OrderClosedTopology α] {g : ι → α} (hsum : Summable g) (hg : ∀ (i : ι), 0 ≤ g i) (i : ι) (hi : 0 < g i)
Goal: 0 < ∑' (i : ι), g i
-- one_lt_tprod
@[to_additive tsum_pos]
theorem one_lt_tprod (hsum : Multipliable g) (hg : ∀ i, 1 ≤ g i) (i : ι) (hi : 1 < g i) :
    1 < ∏' i, g i 
Arguments: {ι : Type u_1} {α : Type u_3} [OrderedCommGroup α] [TopologicalSpace α] [TopologicalGroup α] [OrderClosedTopology α] {g : ι → α} (hsum : Multipliable g) (hg : ∀ (i : ι), 1 ≤ g i) (i : ι) (hi : 1 < g i)
Goal: 1 < ∏' (i : ι), g i
-- le_hasSum'
@[to_additive]
theorem le_hasProd' (hf : HasProd f a) (i : ι) : f i ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (hf : HasSum f a) (i : ι)
Goal: f i ≤ a
-- le_hasProd'
@[to_additive]
theorem le_hasProd' (hf : HasProd f a) (i : ι) : f i ≤ a 
Arguments: {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α} {a : α} (hf : HasProd f a) (i : ι)
Goal: f i ≤ a