ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ENNReal.iInf_mul_left
theorem iInf_mul_left {ι} [Nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
    (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) : ⨅ i, a * f i = a * ⨅ i, f i 
Arguments: {ι : Sort u_4} [Nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞} (h : a = ⊤ → ⨅ i, f i = 0 → ∃ i, f i = 0)
Goal: ⨅ i, a * f i = a * ⨅ i, f i
-- ENNReal.iInf_mul_right'
theorem iInf_mul_right' {ι} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0)
    (h0 : a = 0 → Nonempty ι) : ⨅ i, f i * a = (⨅ i, f i) * a 
Arguments: {ι : Sort u_4} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} (h : a = ⊤ → ⨅ i, f i = 0 → ∃ i, f i = 0) (h0 : a = 0 → Nonempty ι)
Goal: ⨅ i, f i * a = (⨅ i, f i) * a
-- ENNReal.iInf_mul_right
theorem iInf_mul_right {ι} [Nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
    (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) : ⨅ i, f i * a = (⨅ i, f i) * a 
Arguments: {ι : Sort u_4} [Nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞} (h : a = ⊤ → ⨅ i, f i = 0 → ∃ i, f i = 0)
Goal: ⨅ i, f i * a = (⨅ i, f i) * a
-- ENNReal.inv_map_iInf
theorem inv_map_iInf {ι : Sort*} {x : ι → ℝ≥0∞} : (iInf x)⁻¹ = ⨆ i, (x i)⁻¹ 
Arguments: {ι : Sort u_4} {x : ι → ℝ≥0∞}
Goal: (iInf x)⁻¹ = ⨆ i, (x i)⁻¹
-- ENNReal.inv_map_iSup
theorem inv_map_iSup {ι : Sort*} {x : ι → ℝ≥0∞} : (iSup x)⁻¹ = ⨅ i, (x i)⁻¹ 
Arguments: {ι : Sort u_4} {x : ι → ℝ≥0∞}
Goal: (iSup x)⁻¹ = ⨅ i, (x i)⁻¹
-- ENNReal.inv_limsup
theorem inv_limsup {ι : Sort _} {x : ι → ℝ≥0∞} {l : Filter ι} :
    (limsup x l)⁻¹ = liminf (fun i => (x i)⁻¹) l 
Arguments: {ι : Type u_4} {x : ι → ℝ≥0∞} {l : Filter ι}
Goal: (Filter.limsup x l)⁻¹ = Filter.liminf (fun i => (x i)⁻¹) l
-- ENNReal.inv_liminf
theorem inv_liminf {ι : Sort _} {x : ι → ℝ≥0∞} {l : Filter ι} :
    (liminf x l)⁻¹ = limsup (fun i => (x i)⁻¹) l 
Arguments: {ι : Type u_4} {x : ι → ℝ≥0∞} {l : Filter ι}
Goal: (Filter.liminf x l)⁻¹ = Filter.limsup (fun i => (x i)⁻¹) l
-- ENNReal.tendsto_inv_iff
@[simp] -- Porting note (#11215): TODO: generalize to `[InvolutiveInv _] [ContinuousInv _]`
protected theorem tendsto_inv_iff {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} :
    Tendsto (fun x => (m x)⁻¹) f (𝓝 a⁻¹) ↔ Tendsto m f (𝓝 a) 
Arguments: {α : Type u_1} {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞}
Goal: Filter.Tendsto (fun x => (m x)⁻¹) f (𝓝 a⁻¹) ↔ Filter.Tendsto m f (𝓝 a)
-- ENNReal.Tendsto.div
protected theorem Tendsto.div {f : Filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
    (hma : Tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) (hmb : Tendsto mb f (𝓝 b))
    (hb : b ≠ ∞ ∨ a ≠ ∞) : Tendsto (fun a => ma a / mb a) f (𝓝 (a / b)) 
Arguments: {α : Type u_1} {f : Filter α} {ma mb : α → ℝ≥0∞} {a b : ℝ≥0∞} (hma : Filter.Tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) (hmb : Filter.Tendsto mb f (𝓝 b)) (hb : b ≠ ⊤ ∨ a ≠ ⊤)
Goal: Filter.Tendsto (fun a => ma a / mb a) f (𝓝 (a / b))
-- ENNReal.Tendsto.const_div
protected theorem Tendsto.const_div {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
    (hm : Tendsto m f (𝓝 b)) (hb : b ≠ ∞ ∨ a ≠ ∞) : Tendsto (fun b => a / m b) f (𝓝 (a / b)) 
Arguments: {α : Type u_1} {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞} (hm : Filter.Tendsto m f (𝓝 b)) (hb : b ≠ ⊤ ∨ a ≠ ⊤)
Goal: Filter.Tendsto (fun b => a / m b) f (𝓝 (a / b))
-- ENNReal.Tendsto.div_const
protected theorem Tendsto.div_const {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
    (hm : Tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) : Tendsto (fun x => m x / b) f (𝓝 (a / b)) 
Arguments: {α : Type u_1} {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞} (hm : Filter.Tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0)
Goal: Filter.Tendsto (fun x => m x / b) f (𝓝 (a / b))
-- ENNReal.tendsto_inv_nat_nhds_zero
protected theorem tendsto_inv_nat_nhds_zero : Tendsto (fun n : ℕ => (n : ℝ≥0∞)⁻¹) atTop (𝓝 0) 
Goal: Filter.Tendsto (fun n => (↑n)⁻¹) Filter.atTop (𝓝 0)
-- ENNReal.iSup_add
theorem iSup_add {ι : Sort*} {s : ι → ℝ≥0∞} [Nonempty ι] : iSup s + a = ⨆ b, s b + a 
Arguments: {a : ℝ≥0∞} {ι : Sort u_4} {s : ι → ℝ≥0∞} [Nonempty ι]
Goal: iSup s + a = ⨆ b, s b + a
-- ENNReal.biSup_add'
theorem biSup_add' {ι : Sort*} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞} :
    (⨆ (i) (_ : p i), f i) + a = ⨆ (i) (_ : p i), f i + a 
Arguments: {a : ℝ≥0∞} {ι : Sort u_4} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞}
Goal: (⨆ i, ⨆ (_ : p i), f i) + a = ⨆ i, ⨆ (_ : p i), f i + a
-- ENNReal.add_biSup'
theorem add_biSup' {ι : Sort*} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞} :
    (a + ⨆ (i) (_ : p i), f i) = ⨆ (i) (_ : p i), a + f i 
Arguments: {a : ℝ≥0∞} {ι : Sort u_4} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞}
Goal: a + ⨆ i, ⨆ (_ : p i), f i = ⨆ i, ⨆ (_ : p i), a + f i
-- ENNReal.biSup_add
theorem biSup_add {ι} {s : Set ι} (hs : s.Nonempty) {f : ι → ℝ≥0∞} :
    (⨆ i ∈ s, f i) + a = ⨆ i ∈ s, f i + a 
Arguments: {a : ℝ≥0∞} {ι : Type u_4} {s : Set ι} (hs : s.Nonempty) {f : ι → ℝ≥0∞}
Goal: (⨆ i ∈ s, f i) + a = ⨆ i ∈ s, f i + a
-- ENNReal.add_biSup
theorem add_biSup {ι} {s : Set ι} (hs : s.Nonempty) {f : ι → ℝ≥0∞} :
    (a + ⨆ i ∈ s, f i) = ⨆ i ∈ s, a + f i 
Arguments: {a : ℝ≥0∞} {ι : Type u_4} {s : Set ι} (hs : s.Nonempty) {f : ι → ℝ≥0∞}
Goal: a + ⨆ i ∈ s, f i = ⨆ i ∈ s, a + f i
-- ENNReal.sSup_add
theorem sSup_add {s : Set ℝ≥0∞} (hs : s.Nonempty) : sSup s + a = ⨆ b ∈ s, b + a 
Arguments: {a : ℝ≥0∞} {s : Set ℝ≥0∞} (hs : s.Nonempty)
Goal: sSup s + a = ⨆ b ∈ s, b + a
-- ENNReal.add_iSup
theorem add_iSup {ι : Sort*} {s : ι → ℝ≥0∞} [Nonempty ι] : a + iSup s = ⨆ b, a + s b 
Arguments: {a : ℝ≥0∞} {ι : Sort u_4} {s : ι → ℝ≥0∞} [Nonempty ι]
Goal: a + iSup s = ⨆ b, a + s b
-- ENNReal.iSup_add_iSup_le
theorem iSup_add_iSup_le {ι ι' : Sort*} [Nonempty ι] [Nonempty ι'] {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞}
    {a : ℝ≥0∞} (h : ∀ i j, f i + g j ≤ a) : iSup f + iSup g ≤ a 
Arguments: {ι : Sort u_4} {ι' : Sort u_5} [Nonempty ι] [Nonempty ι'] {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ (i : ι) (j : ι'), f i + g j ≤ a)
Goal: iSup f + iSup g ≤ a
-- ENNReal.biSup_add_biSup_le'
theorem biSup_add_biSup_le' {ι ι'} {p : ι → Prop} {q : ι' → Prop} (hp : ∃ i, p i) (hq : ∃ j, q j)
    {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i, p i → ∀ j, q j → f i + g j ≤ a) :
    ((⨆ (i) (_ : p i), f i) + ⨆ (j) (_ : q j), g j) ≤ a 
Arguments: {ι : Sort u_4} {ι' : Sort u_5} {p : ι → Prop} {q : ι' → Prop} (hp : ∃ i, p i) (hq : ∃ j, q j) {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ (i : ι), p i → ∀ (j : ι'), q j → f i + g j ≤ a)
Goal: (⨆ i, ⨆ (_ : p i), f i) + ⨆ j, ⨆ (_ : q j), g j ≤ a
-- ENNReal.biSup_add_biSup_le
theorem biSup_add_biSup_le {ι ι'} {s : Set ι} {t : Set ι'} (hs : s.Nonempty) (ht : t.Nonempty)
    {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i ∈ s, ∀ j ∈ t, f i + g j ≤ a) :
    ((⨆ i ∈ s, f i) + ⨆ j ∈ t, g j) ≤ a 
Arguments: {ι : Type u_4} {ι' : Type u_5} {s : Set ι} {t : Set ι'} (hs : s.Nonempty) (ht : t.Nonempty) {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i ∈ s, ∀ j ∈ t, f i + g j ≤ a)
Goal: (⨆ i ∈ s, f i) + ⨆ j ∈ t, g j ≤ a
-- ENNReal.iSup_add_iSup
theorem iSup_add_iSup {ι : Sort*} {f g : ι → ℝ≥0∞} (h : ∀ i j, ∃ k, f i + g j ≤ f k + g k) :
    iSup f + iSup g = ⨆ a, f a + g a 
Arguments: {ι : Sort u_4} {f g : ι → ℝ≥0∞} (h : ∀ (i j : ι), ∃ k, f i + g j ≤ f k + g k)
Goal: iSup f + iSup g = ⨆ a, f a + g a
-- ENNReal.iSup_add_iSup_of_monotone
theorem iSup_add_iSup_of_monotone {ι : Type*} [Preorder ι] [IsDirected ι (· ≤ ·)]
    {f g : ι → ℝ≥0∞} (hf : Monotone f) (hg : Monotone g) : iSup f + iSup g = ⨆ a, f a + g a 
Arguments: {ι : Type u_4} [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {f g : ι → ℝ≥0∞} (hf : Monotone f) (hg : Monotone g)
Goal: iSup f + iSup g = ⨆ a, f a + g a
-- ENNReal.finsetSum_iSup
theorem finsetSum_iSup {α ι : Type*} {s : Finset α} {f : α → ι → ℝ≥0∞}
    (hf : ∀ i j, ∃ k, ∀ a, f a i ≤ f a k ∧ f a j ≤ f a k) :
    ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i 
Arguments: {α : Type u_4} {ι : Type u_5} {s : Finset α} {f : α → ι → ℝ≥0∞} (hf : ∀ (i j : ι), ∃ k, ∀ (a : α), f a i ≤ f a k ∧ f a j ≤ f a k)
Goal: ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i
-- ENNReal.finsetSum_iSup_of_monotone
theorem finsetSum_iSup_of_monotone {α} {ι} [Preorder ι] [IsDirected ι (· ≤ ·)]
    {s : Finset α} {f : α → ι → ℝ≥0∞} (hf : ∀ a, Monotone (f a)) :
    (∑ a ∈ s, iSup (f a)) = ⨆ n, ∑ a ∈ s, f a n 
Arguments: {α : Type u_4} {ι : Type u_5} [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {s : Finset α} {f : α → ι → ℝ≥0∞} (hf : ∀ (a : α), Monotone (f a))
Goal: ∑ a ∈ s, iSup (f a) = ⨆ n, ∑ a ∈ s, f a n
-- ENNReal.mul_iSup
theorem mul_iSup {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : a * iSup f = ⨆ i, a * f i 
Arguments: {ι : Sort u_4} {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
Goal: a * iSup f = ⨆ i, a * f i
-- ENNReal.mul_sSup
theorem mul_sSup {s : Set ℝ≥0∞} {a : ℝ≥0∞} : a * sSup s = ⨆ i ∈ s, a * i 
Arguments: {s : Set ℝ≥0∞} {a : ℝ≥0∞}
Goal: a * sSup s = ⨆ i ∈ s, a * i
-- ENNReal.iSup_mul
theorem iSup_mul {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : iSup f * a = ⨆ i, f i * a 
Arguments: {ι : Sort u_4} {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
Goal: iSup f * a = ⨆ i, f i * a
-- ENNReal.smul_iSup
theorem smul_iSup {ι : Sort*} {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (f : ι → ℝ≥0∞)
    (c : R) : (c • ⨆ i, f i) = ⨆ i, c • f i 
Arguments: {ι : Sort u_4} {R : Type u_5} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (f : ι → ℝ≥0∞) (c : R)
Goal: c • ⨆ i, f i = ⨆ i, c • f i
-- ENNReal.smul_sSup
theorem smul_sSup {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (s : Set ℝ≥0∞) (c : R) :
    c • sSup s = ⨆ i ∈ s, c • i 
Arguments: {R : Type u_4} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (s : Set ℝ≥0∞) (c : R)
Goal: c • sSup s = ⨆ i ∈ s, c • i
-- ENNReal.iSup_div
theorem iSup_div {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : iSup f / a = ⨆ i, f i / a 
Arguments: {ι : Sort u_4} {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
Goal: iSup f / a = ⨆ i, f i / a
-- ENNReal.tendsto_coe_sub
protected theorem tendsto_coe_sub {b : ℝ≥0∞} :
    Tendsto (fun b : ℝ≥0∞ => ↑r - b) (𝓝 b) (𝓝 (↑r - b)) 
Arguments: {r : ℝ≥0} {b : ℝ≥0∞}
Goal: Filter.Tendsto (fun b => ↑r - b) (𝓝 b) (𝓝 (↑r - b))
-- ENNReal.sub_iSup
theorem sub_iSup {ι : Sort*} [Nonempty ι] {b : ι → ℝ≥0∞} (hr : a < ∞) :
    (a - ⨆ i, b i) = ⨅ i, a - b i 
Arguments: {a : ℝ≥0∞} {ι : Sort u_4} [Nonempty ι] {b : ι → ℝ≥0∞} (hr : a < ⊤)
Goal: a - ⨆ i, b i = ⨅ i, a - b i
-- ENNReal.exists_countable_dense_no_zero_top
theorem exists_countable_dense_no_zero_top :
    ∃ s : Set ℝ≥0∞, s.Countable ∧ Dense s ∧ 0 ∉ s ∧ ∞ ∉ s 
Goal: ∃ s, s.Countable ∧ Dense s ∧ 0 ∉ s ∧ ⊤ ∉ s
-- ENNReal.exists_lt_add_of_lt_add
theorem exists_lt_add_of_lt_add {x y z : ℝ≥0∞} (h : x < y + z) (hy : y ≠ 0) (hz : z ≠ 0) :
    ∃ y' z', y' < y ∧ z' < z ∧ x < y' + z' 
Arguments: {x y z : ℝ≥0∞} (h : x < y + z) (hy : y ≠ 0) (hz : z ≠ 0)
Goal: ∃ y' z', y' < y ∧ z' < z ∧ x < y' + z'
-- ENNReal.ofReal_cinfi
theorem ofReal_cinfi (f : α → ℝ) [Nonempty α] :
    ENNReal.ofReal (⨅ i, f i) = ⨅ i, ENNReal.ofReal (f i) 
Arguments: {α : Type u_1} (f : α → ℝ) [Nonempty α]
Goal: ENNReal.ofReal (⨅ i, f i) = ⨅ i, ENNReal.ofReal (f i)
-- ENNReal.exists_frequently_lt_of_liminf_ne_top
theorem exists_frequently_lt_of_liminf_ne_top {ι : Type*} {l : Filter ι} {x : ι → ℝ}
    (hx : liminf (fun n => (Real.nnabs (x n) : ℝ≥0∞)) l ≠ ∞) : ∃ R, ∃ᶠ n in l, x n < R 
Arguments: {ι : Type u_4} {l : Filter ι} {x : ι → ℝ} (hx : Filter.liminf (fun n => ↑(Real.nnabs (x n))) l ≠ ⊤)
Goal: ∃ R, ∃ᶠ (n : ι) in l, x n < R
-- ENNReal.exists_frequently_lt_of_liminf_ne_top'
theorem exists_frequently_lt_of_liminf_ne_top' {ι : Type*} {l : Filter ι} {x : ι → ℝ}
    (hx : liminf (fun n => (Real.nnabs (x n) : ℝ≥0∞)) l ≠ ∞) : ∃ R, ∃ᶠ n in l, R < x n 
Arguments: {ι : Type u_4} {l : Filter ι} {x : ι → ℝ} (hx : Filter.liminf (fun n => ↑(Real.nnabs (x n))) l ≠ ⊤)
Goal: ∃ R, ∃ᶠ (n : ι) in l, R < x n
-- ENNReal.exists_upcrossings_of_not_bounded_under
theorem exists_upcrossings_of_not_bounded_under {ι : Type*} {l : Filter ι} {x : ι → ℝ}
    (hf : liminf (fun i => (Real.nnabs (x i) : ℝ≥0∞)) l ≠ ∞)
    (hbdd : ¬IsBoundedUnder (· ≤ ·) l fun i => |x i|) :
    ∃ a b : ℚ, a < b ∧ (∃ᶠ i in l, x i < a) ∧ ∃ᶠ i in l, ↑b < x i 
Arguments: {ι : Type u_4} {l : Filter ι} {x : ι → ℝ} (hf : Filter.liminf (fun i => ↑(Real.nnabs (x i))) l ≠ ⊤) (hbdd : ¬Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun i => |x i|)
Goal: ∃ a b, a < b ∧ (∃ᶠ (i : ι) in l, x i < ↑a) ∧ ∃ᶠ (i : ι) in l, ↑b < x i
-- ENNReal.hasSum_coe
@[norm_cast]
protected theorem hasSum_coe {f : α → ℝ≥0} {r : ℝ≥0} :
    HasSum (fun a => (f a : ℝ≥0∞)) ↑r ↔ HasSum f r 
Arguments: {α : Type u_1} {f : α → ℝ≥0} {r : ℝ≥0}
Goal: HasSum (fun a => ↑(f a)) ↑r ↔ HasSum f r
-- ENNReal.tsum_coe_eq
protected theorem tsum_coe_eq {f : α → ℝ≥0} (h : HasSum f r) : (∑' a, (f a : ℝ≥0∞)) = r 
Arguments: {α : Type u_1} {r : ℝ≥0} {f : α → ℝ≥0} (h : HasSum f r)
Goal: ∑' (a : α), ↑(f a) = ↑r
-- ENNReal.coe_tsum
protected theorem coe_tsum {f : α → ℝ≥0} : Summable f → ↑(tsum f) = ∑' a, (f a : ℝ≥0∞)
  
Arguments: {α : Type u_1} {f : α → ℝ≥0}
Goal: Summable f → ↑(tsum f) = ∑' (a : α), ↑(f a)
-- ENNReal.hasSum
protected theorem hasSum : HasSum f (⨆ s : Finset α, ∑ a ∈ s, f a) 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞}
Goal: HasSum f (⨆ s, ∑ a ∈ s, f a)
-- ENNReal.summable
@[simp]
protected theorem summable : Summable f 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞}
Goal: Summable f
-- ENNReal.tsum_coe_ne_top_iff_summable
theorem tsum_coe_ne_top_iff_summable {f : β → ℝ≥0} : (∑' b, (f b : ℝ≥0∞)) ≠ ∞ ↔ Summable f 
Arguments: {β : Type u_2} {f : β → ℝ≥0}
Goal: ∑' (b : β), ↑(f b) ≠ ⊤ ↔ Summable f
-- ENNReal.tsum_eq_iSup_sum
protected theorem tsum_eq_iSup_sum : ∑' a, f a = ⨆ s : Finset α, ∑ a ∈ s, f a 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞}
Goal: ∑' (a : α), f a = ⨆ s, ∑ a ∈ s, f a
-- ENNReal.tsum_eq_iSup_sum'
protected theorem tsum_eq_iSup_sum' {ι : Type*} (s : ι → Finset α) (hs : ∀ t, ∃ i, t ⊆ s i) :
    ∑' a, f a = ⨆ i, ∑ a ∈ s i, f a 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} {ι : Type u_4} (s : ι → Finset α) (hs : ∀ (t : Finset α), ∃ i, t ⊆ s i)
Goal: ∑' (a : α), f a = ⨆ i, ∑ a ∈ s i, f a
-- ENNReal.tsum_sigma
protected theorem tsum_sigma {β : α → Type*} (f : ∀ a, β a → ℝ≥0∞) :
    ∑' p : Σa, β a, f p.1 p.2 = ∑' (a) (b), f a b 
Arguments: {α : Type u_1} {β : α → Type u_4} (f : (a : α) → β a → ℝ≥0∞)
Goal: ∑' (p : (a : α) × β a), f p.fst p.snd = ∑' (a : α) (b : β a), f a b
-- ENNReal.tsum_sigma'
protected theorem tsum_sigma' {β : α → Type*} (f : (Σa, β a) → ℝ≥0∞) :
    ∑' p : Σa, β a, f p = ∑' (a) (b), f ⟨a, b⟩ 
Arguments: {α : Type u_1} {β : α → Type u_4} (f : (a : α) × β a → ℝ≥0∞)
Goal: ∑' (p : (a : α) × β a), f p = ∑' (a : α) (b : β a), f ⟨a, b⟩
-- ENNReal.tsum_prod
protected theorem tsum_prod {f : α → β → ℝ≥0∞} : ∑' p : α × β, f p.1 p.2 = ∑' (a) (b), f a b 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β → ℝ≥0∞}
Goal: ∑' (p : α × β), f p.1 p.2 = ∑' (a : α) (b : β), f a b
-- ENNReal.tsum_prod'
protected theorem tsum_prod' {f : α × β → ℝ≥0∞} : ∑' p : α × β, f p = ∑' (a) (b), f (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α × β → ℝ≥0∞}
Goal: ∑' (p : α × β), f p = ∑' (a : α) (b : β), f (a, b)
-- ENNReal.tsum_comm
protected theorem tsum_comm {f : α → β → ℝ≥0∞} : ∑' a, ∑' b, f a b = ∑' b, ∑' a, f a b 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β → ℝ≥0∞}
Goal: ∑' (a : α) (b : β), f a b = ∑' (b : β) (a : α), f a b
-- ENNReal.tsum_add
protected theorem tsum_add : ∑' a, (f a + g a) = ∑' a, f a + ∑' a, g a 
Arguments: {α : Type u_1} {f g : α → ℝ≥0∞}
Goal: ∑' (a : α), (f a + g a) = ∑' (a : α), f a + ∑' (a : α), g a
-- ENNReal.tsum_le_tsum
protected theorem tsum_le_tsum (h : ∀ a, f a ≤ g a) : ∑' a, f a ≤ ∑' a, g a 
Arguments: {α : Type u_1} {f g : α → ℝ≥0∞} (h : ∀ (a : α), f a ≤ g a)
Goal: ∑' (a : α), f a ≤ ∑' (a : α), g a
-- GCongr.ennreal_tsum_le_tsum
@[gcongr]
protected theorem _root_.GCongr.ennreal_tsum_le_tsum (h : ∀ a, f a ≤ g a) : tsum f ≤ tsum g 
Arguments: {α : Type u_1} {f g : α → ℝ≥0∞} (h : ∀ (a : α), f a ≤ g a)
Goal: tsum f ≤ tsum g
-- ENNReal.sum_le_tsum
protected theorem sum_le_tsum {f : α → ℝ≥0∞} (s : Finset α) : ∑ x ∈ s, f x ≤ ∑' x, f x 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (s : Finset α)
Goal: ∑ x ∈ s, f x ≤ ∑' (x : α), f x
-- ENNReal.tsum_eq_iSup_nat'
protected theorem tsum_eq_iSup_nat' {f : ℕ → ℝ≥0∞} {N : ℕ → ℕ} (hN : Tendsto N atTop atTop) :
    ∑' i : ℕ, f i = ⨆ i : ℕ, ∑ a ∈ Finset.range (N i), f a 
Arguments: {f : ℕ → ℝ≥0∞} {N : ℕ → ℕ} (hN : Filter.Tendsto N Filter.atTop Filter.atTop)
Goal: ∑' (i : ℕ), f i = ⨆ i, ∑ a ∈ Finset.range (N i), f a
-- ENNReal.tsum_eq_iSup_nat
protected theorem tsum_eq_iSup_nat {f : ℕ → ℝ≥0∞} :
    ∑' i : ℕ, f i = ⨆ i : ℕ, ∑ a ∈ Finset.range i, f a 
Arguments: {f : ℕ → ℝ≥0∞}
Goal: ∑' (i : ℕ), f i = ⨆ i, ∑ a ∈ Finset.range i, f a
-- ENNReal.tsum_eq_liminf_sum_nat
protected theorem tsum_eq_liminf_sum_nat {f : ℕ → ℝ≥0∞} :
    ∑' i, f i = liminf (fun n => ∑ i ∈ Finset.range n, f i) atTop 
Arguments: {f : ℕ → ℝ≥0∞}
Goal: ∑' (i : ℕ), f i = Filter.liminf (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop
-- ENNReal.tsum_eq_limsup_sum_nat
protected theorem tsum_eq_limsup_sum_nat {f : ℕ → ℝ≥0∞} :
    ∑' i, f i = limsup (fun n => ∑ i ∈ Finset.range n, f i) atTop 
Arguments: {f : ℕ → ℝ≥0∞}
Goal: ∑' (i : ℕ), f i = Filter.limsup (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop
-- ENNReal.le_tsum
protected theorem le_tsum (a : α) : f a ≤ ∑' a, f a 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (a : α)
Goal: f a ≤ ∑' (a : α), f a
-- ENNReal.tsum_eq_zero
@[simp]
protected theorem tsum_eq_zero : ∑' i, f i = 0 ↔ ∀ i, f i = 0 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞}
Goal: ∑' (i : α), f i = 0 ↔ ∀ (i : α), f i = 0
-- ENNReal.tsum_eq_top_of_eq_top
protected theorem tsum_eq_top_of_eq_top : (∃ a, f a = ∞) → ∑' a, f a = ∞
  
Arguments: {α : Type u_1} {f : α → ℝ≥0∞}
Goal: (∃ a, f a = ⊤) → ∑' (a : α), f a = ⊤
-- ENNReal.lt_top_of_tsum_ne_top
protected theorem lt_top_of_tsum_ne_top {a : α → ℝ≥0∞} (tsum_ne_top : ∑' i, a i ≠ ∞) (j : α) :
    a j < ∞ 
Arguments: {α : Type u_1} {a : α → ℝ≥0∞} (tsum_ne_top : ∑' (i : α), a i ≠ ⊤) (j : α)
Goal: a j < ⊤
-- ENNReal.tsum_top
@[simp]
protected theorem tsum_top [Nonempty α] : ∑' _ : α, ∞ = ∞ 
Arguments: {α : Type u_1} [Nonempty α]
Goal: ∑' (x : α), ⊤ = ⊤
-- ENNReal.tsum_const_eq_top_of_ne_zero
theorem tsum_const_eq_top_of_ne_zero {α : Type*} [Infinite α] {c : ℝ≥0∞} (hc : c ≠ 0) :
    ∑' _ : α, c = ∞ 
Arguments: {α : Type u_4} [Infinite α] {c : ℝ≥0∞} (hc : c ≠ 0)
Goal: ∑' (x : α), c = ⊤
-- ENNReal.ne_top_of_tsum_ne_top
protected theorem ne_top_of_tsum_ne_top (h : ∑' a, f a ≠ ∞) (a : α) : f a ≠ ∞ 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (h : ∑' (a : α), f a ≠ ⊤) (a : α)
Goal: f a ≠ ⊤
-- ENNReal.tsum_mul_left
protected theorem tsum_mul_left : ∑' i, a * f i = a * ∑' i, f i 
Arguments: {α : Type u_1} {a : ℝ≥0∞} {f : α → ℝ≥0∞}
Goal: ∑' (i : α), a * f i = a * ∑' (i : α), f i
-- ENNReal.tsum_mul_right
protected theorem tsum_mul_right : ∑' i, f i * a = (∑' i, f i) * a 
Arguments: {α : Type u_1} {a : ℝ≥0∞} {f : α → ℝ≥0∞}
Goal: ∑' (i : α), f i * a = (∑' (i : α), f i) * a
-- ENNReal.tsum_const_smul
protected theorem tsum_const_smul {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (a : R) :
    ∑' i, a • f i = a • ∑' i, f i 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} {R : Type u_4} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (a : R)
Goal: ∑' (i : α), a • f i = a • ∑' (i : α), f i
-- ENNReal.tsum_iSup_eq
@[simp]
theorem tsum_iSup_eq {α : Type*} (a : α) {f : α → ℝ≥0∞} : (∑' b : α, ⨆ _ : a = b, f b) = f a 
Arguments: {α : Type u_4} (a : α) {f : α → ℝ≥0∞}
Goal: ∑' (b : α), ⨆ (_ : a = b), f b = f a
-- ENNReal.hasSum_iff_tendsto_nat
theorem hasSum_iff_tendsto_nat {f : ℕ → ℝ≥0∞} (r : ℝ≥0∞) :
    HasSum f r ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 r) 
Arguments: {f : ℕ → ℝ≥0∞} (r : ℝ≥0∞)
Goal: HasSum f r ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 r)
-- ENNReal.tendsto_nat_tsum
theorem tendsto_nat_tsum (f : ℕ → ℝ≥0∞) :
    Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 (∑' n, f n)) 
Arguments: (f : ℕ → ℝ≥0∞)
Goal: Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 (∑' (n : ℕ), f n))
-- ENNReal.toNNReal_apply_of_tsum_ne_top
theorem toNNReal_apply_of_tsum_ne_top {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' i, f i ≠ ∞) (x : α) :
    (((ENNReal.toNNReal ∘ f) x : ℝ≥0) : ℝ≥0∞) = f x 
Arguments: {α : Type u_4} {f : α → ℝ≥0∞} (hf : ∑' (i : α), f i ≠ ⊤) (x : α)
Goal: ↑((ENNReal.toNNReal ∘ f) x) = f x
-- ENNReal.summable_toNNReal_of_tsum_ne_top
theorem summable_toNNReal_of_tsum_ne_top {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' i, f i ≠ ∞) :
    Summable (ENNReal.toNNReal ∘ f) 
Arguments: {α : Type u_4} {f : α → ℝ≥0∞} (hf : ∑' (i : α), f i ≠ ⊤)
Goal: Summable (ENNReal.toNNReal ∘ f)
-- ENNReal.tendsto_cofinite_zero_of_tsum_ne_top
theorem tendsto_cofinite_zero_of_tsum_ne_top {α} {f : α → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
    Tendsto f cofinite (𝓝 0) 
Arguments: {α : Type u_4} {f : α → ℝ≥0∞} (hf : ∑' (x : α), f x ≠ ⊤)
Goal: Filter.Tendsto f Filter.cofinite (𝓝 0)
-- ENNReal.tendsto_atTop_zero_of_tsum_ne_top
theorem tendsto_atTop_zero_of_tsum_ne_top {f : ℕ → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
    Tendsto f atTop (𝓝 0) 
Arguments: {f : ℕ → ℝ≥0∞} (hf : ∑' (x : ℕ), f x ≠ ⊤)
Goal: Filter.Tendsto f Filter.atTop (𝓝 0)
-- ENNReal.tendsto_tsum_compl_atTop_zero
theorem tendsto_tsum_compl_atTop_zero {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
    Tendsto (fun s : Finset α => ∑' b : { x // x ∉ s }, f b) atTop (𝓝 0) 
Arguments: {α : Type u_4} {f : α → ℝ≥0∞} (hf : ∑' (x : α), f x ≠ ⊤)
Goal: Filter.Tendsto (fun s => ∑' (b : { x // x ∉ s }), f ↑b) Filter.atTop (𝓝 0)
-- ENNReal.tsum_apply
protected theorem tsum_apply {ι α : Type*} {f : ι → α → ℝ≥0∞} {x : α} :
    (∑' i, f i) x = ∑' i, f i x 
Arguments: {ι : Type u_4} {α : Type u_5} {f : ι → α → ℝ≥0∞} {x : α}
Goal: tsum (fun i => f i) x = ∑' (i : ι), f i x
-- ENNReal.tsum_sub
theorem tsum_sub {f : ℕ → ℝ≥0∞} {g : ℕ → ℝ≥0∞} (h₁ : ∑' i, g i ≠ ∞) (h₂ : g ≤ f) :
    ∑' i, (f i - g i) = ∑' i, f i - ∑' i, g i 
Arguments: {f g : ℕ → ℝ≥0∞} (h₁ : ∑' (i : ℕ), g i ≠ ⊤) (h₂ : g ≤ f)
Goal: ∑' (i : ℕ), (f i - g i) = ∑' (i : ℕ), f i - ∑' (i : ℕ), g i
-- ENNReal.tsum_comp_le_tsum_of_injective
theorem tsum_comp_le_tsum_of_injective {f : α → β} (hf : Injective f) (g : β → ℝ≥0∞) :
    ∑' x, g (f x) ≤ ∑' y, g y 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) (g : β → ℝ≥0∞)
Goal: ∑' (x : α), g (f x) ≤ ∑' (y : β), g y
-- ENNReal.tsum_le_tsum_comp_of_surjective
theorem tsum_le_tsum_comp_of_surjective {f : α → β} (hf : Surjective f) (g : β → ℝ≥0∞) :
    ∑' y, g y ≤ ∑' x, g (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Surjective f) (g : β → ℝ≥0∞)
Goal: ∑' (y : β), g y ≤ ∑' (x : α), g (f x)
-- ENNReal.tsum_mono_subtype
theorem tsum_mono_subtype (f : α → ℝ≥0∞) {s t : Set α} (h : s ⊆ t) :
    ∑' x : s, f x ≤ ∑' x : t, f x 
Arguments: {α : Type u_1} (f : α → ℝ≥0∞) {s t : Set α} (h : s ⊆ t)
Goal: ∑' (x : ↑s), f ↑x ≤ ∑' (x : ↑t), f ↑x
-- ENNReal.tsum_iUnion_le_tsum
theorem tsum_iUnion_le_tsum {ι : Type*} (f : α → ℝ≥0∞) (t : ι → Set α) :
    ∑' x : ⋃ i, t i, f x ≤ ∑' i, ∑' x : t i, f x 
Arguments: {α : Type u_1} {ι : Type u_4} (f : α → ℝ≥0∞) (t : ι → Set α)
Goal: ∑' (x : ↑(⋃ i, t i)), f ↑x ≤ ∑' (i : ι) (x : ↑(t i)), f ↑x
-- ENNReal.tsum_biUnion_le_tsum
theorem tsum_biUnion_le_tsum {ι : Type*} (f : α → ℝ≥0∞) (s : Set ι) (t : ι → Set α) :
    ∑' x : ⋃ i ∈ s , t i, f x ≤ ∑' i : s, ∑' x : t i, f x 
Arguments: {α : Type u_1} {ι : Type u_4} (f : α → ℝ≥0∞) (s : Set ι) (t : ι → Set α)
Goal: ∑' (x : ↑(⋃ i ∈ s, t i)), f ↑x ≤ ∑' (i : ↑s) (x : ↑(t ↑i)), f ↑x
-- ENNReal.tsum_biUnion_le
theorem tsum_biUnion_le {ι : Type*} (f : α → ℝ≥0∞) (s : Finset ι) (t : ι → Set α) :
    ∑' x : ⋃ i ∈ s, t i, f x ≤ ∑ i ∈ s, ∑' x : t i, f x 
Arguments: {α : Type u_1} {ι : Type u_4} (f : α → ℝ≥0∞) (s : Finset ι) (t : ι → Set α)
Goal: ∑' (x : ↑(⋃ i ∈ s, t i)), f ↑x ≤ ∑ i ∈ s, ∑' (x : ↑(t i)), f ↑x
-- ENNReal.tsum_iUnion_le
theorem tsum_iUnion_le {ι : Type*} [Fintype ι] (f : α → ℝ≥0∞) (t : ι → Set α) :
    ∑' x : ⋃ i, t i, f x ≤ ∑ i, ∑' x : t i, f x 
Arguments: {α : Type u_1} {ι : Type u_4} [Fintype ι] (f : α → ℝ≥0∞) (t : ι → Set α)
Goal: ∑' (x : ↑(⋃ i, t i)), f ↑x ≤ ∑ i : ι, ∑' (x : ↑(t i)), f ↑x
-- ENNReal.tsum_union_le
theorem tsum_union_le (f : α → ℝ≥0∞) (s t : Set α) :
    ∑' x : ↑(s ∪ t), f x ≤ ∑' x : s, f x + ∑' x : t, f x 
Arguments: {α : Type u_1} (f : α → ℝ≥0∞) (s t : Set α)
Goal: ∑' (x : ↑(s ∪ t)), f ↑x ≤ ∑' (x : ↑s), f ↑x + ∑' (x : ↑t), f ↑x
-- ENNReal.tsum_eq_add_tsum_ite
theorem tsum_eq_add_tsum_ite {f : β → ℝ≥0∞} (b : β) :
    ∑' x, f x = f b + ∑' x, ite (x = b) 0 (f x) 
Arguments: {β : Type u_2} {f : β → ℝ≥0∞} (b : β)
Goal: ∑' (x : β), f x = f b + ∑' (x : β), if x = b then 0 else f x