ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsClosed.upperSemicontinuousWithinAt_indicator
theorem IsClosed.upperSemicontinuousWithinAt_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
    UpperSemicontinuousWithinAt (indicator s fun _x => y) t x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s t : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : 0 ≤ y)
Goal: UpperSemicontinuousWithinAt (s.indicator fun _x => y) t x
-- upperSemicontinuous_iff_isOpen_preimage
theorem upperSemicontinuous_iff_isOpen_preimage :
    UpperSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Iio y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β}
Goal: UpperSemicontinuous f ↔ ∀ (y : β), IsOpen (f ⁻¹' Set.Iio y)
-- UpperSemicontinuous.isOpen_preimage
theorem UpperSemicontinuous.isOpen_preimage (hf : UpperSemicontinuous f) (y : β) :
    IsOpen (f ⁻¹' Iio y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (hf : UpperSemicontinuous f) (y : β)
Goal: IsOpen (f ⁻¹' Set.Iio y)
-- upperSemicontinuous_iff_isClosed_preimage
theorem upperSemicontinuous_iff_isClosed_preimage {f : α → γ} :
    UpperSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Ici y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] {f : α → γ}
Goal: UpperSemicontinuous f ↔ ∀ (y : γ), IsClosed (f ⁻¹' Set.Ici y)
-- UpperSemicontinuous.isClosed_preimage
theorem UpperSemicontinuous.isClosed_preimage {f : α → γ} (hf : UpperSemicontinuous f) (y : γ) :
    IsClosed (f ⁻¹' Ici y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] {f : α → γ} (hf : UpperSemicontinuous f) (y : γ)
Goal: IsClosed (f ⁻¹' Set.Ici y)
-- ContinuousWithinAt.upperSemicontinuousWithinAt
theorem ContinuousWithinAt.upperSemicontinuousWithinAt {f : α → γ} (h : ContinuousWithinAt f s x) :
    UpperSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : ContinuousWithinAt f s x)
Goal: UpperSemicontinuousWithinAt f s x
-- ContinuousAt.upperSemicontinuousAt
theorem ContinuousAt.upperSemicontinuousAt {f : α → γ} (h : ContinuousAt f x) :
    UpperSemicontinuousAt f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : ContinuousAt f x)
Goal: UpperSemicontinuousAt f x
-- ContinuousOn.upperSemicontinuousOn
theorem ContinuousOn.upperSemicontinuousOn {f : α → γ} (h : ContinuousOn f s) :
    UpperSemicontinuousOn f s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : ContinuousOn f s)
Goal: UpperSemicontinuousOn f s
-- Continuous.upperSemicontinuous
theorem Continuous.upperSemicontinuous {f : α → γ} (h : Continuous f) : UpperSemicontinuous f 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : Continuous f)
Goal: UpperSemicontinuous f
-- upperSemicontinuousWithinAt_iff_limsup_le
theorem upperSemicontinuousWithinAt_iff_limsup_le {f : α → γ} :
    UpperSemicontinuousWithinAt f s x ↔ limsup f (𝓝[s] x) ≤ f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: UpperSemicontinuousWithinAt f s x ↔ Filter.limsup f (𝓝[s] x) ≤ f x
-- upperSemicontinuousAt_iff_limsup_le
theorem upperSemicontinuousAt_iff_limsup_le {f : α → γ} :
    UpperSemicontinuousAt f x ↔ limsup f (𝓝 x) ≤ f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: UpperSemicontinuousAt f x ↔ Filter.limsup f (𝓝 x) ≤ f x
-- upperSemicontinuous_iff_limsup_le
theorem upperSemicontinuous_iff_limsup_le {f : α → γ} :
    UpperSemicontinuous f ↔ ∀ x, limsup f (𝓝 x) ≤ f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: UpperSemicontinuous f ↔ ∀ (x : α), Filter.limsup f (𝓝 x) ≤ f x
-- upperSemicontinuousOn_iff_limsup_le
theorem upperSemicontinuousOn_iff_limsup_le {f : α → γ} :
    UpperSemicontinuousOn f s ↔ ∀ x ∈ s, limsup f (𝓝[s] x) ≤ f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: UpperSemicontinuousOn f s ↔ ∀ x ∈ s, Filter.limsup f (𝓝[s] x) ≤ f x
-- upperSemicontinuous_iff_IsClosed_hypograph
theorem upperSemicontinuous_iff_IsClosed_hypograph {f : α → γ} :
    UpperSemicontinuous f ↔ IsClosed {p : α × γ | p.2 ≤ f p.1} 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ}
Goal: UpperSemicontinuous f ↔ IsClosed {p | p.2 ≤ f p.1}
-- ContinuousAt.comp_upperSemicontinuousWithinAt
theorem ContinuousAt.comp_upperSemicontinuousWithinAt {g : γ → δ} {f : α → γ}
    (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Monotone g) :
    UpperSemicontinuousWithinAt (g ∘ f) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Monotone g)
Goal: UpperSemicontinuousWithinAt (g ∘ f) s x
-- ContinuousAt.comp_upperSemicontinuousAt
theorem ContinuousAt.comp_upperSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x))
    (hf : UpperSemicontinuousAt f x) (gmon : Monotone g) : UpperSemicontinuousAt (g ∘ f) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousAt f x) (gmon : Monotone g)
Goal: UpperSemicontinuousAt (g ∘ f) x
-- Continuous.comp_upperSemicontinuousOn
theorem Continuous.comp_upperSemicontinuousOn {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : UpperSemicontinuousOn f s) (gmon : Monotone g) : UpperSemicontinuousOn (g ∘ f) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuousOn f s) (gmon : Monotone g)
Goal: UpperSemicontinuousOn (g ∘ f) s
-- Continuous.comp_upperSemicontinuous
theorem Continuous.comp_upperSemicontinuous {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : UpperSemicontinuous f) (gmon : Monotone g) : UpperSemicontinuous (g ∘ f) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuous f) (gmon : Monotone g)
Goal: UpperSemicontinuous (g ∘ f)
-- ContinuousAt.comp_upperSemicontinuousWithinAt_antitone
theorem ContinuousAt.comp_upperSemicontinuousWithinAt_antitone {g : γ → δ} {f : α → γ}
    (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Antitone g) :
    LowerSemicontinuousWithinAt (g ∘ f) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Antitone g)
Goal: LowerSemicontinuousWithinAt (g ∘ f) s x
-- ContinuousAt.comp_upperSemicontinuousAt_antitone
theorem ContinuousAt.comp_upperSemicontinuousAt_antitone {g : γ → δ} {f : α → γ}
    (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousAt f x) (gmon : Antitone g) :
    LowerSemicontinuousAt (g ∘ f) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousAt f x) (gmon : Antitone g)
Goal: LowerSemicontinuousAt (g ∘ f) x
-- Continuous.comp_upperSemicontinuousOn_antitone
theorem Continuous.comp_upperSemicontinuousOn_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : UpperSemicontinuousOn f s) (gmon : Antitone g) : LowerSemicontinuousOn (g ∘ f) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuousOn f s) (gmon : Antitone g)
Goal: LowerSemicontinuousOn (g ∘ f) s
-- Continuous.comp_upperSemicontinuous_antitone
theorem Continuous.comp_upperSemicontinuous_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : UpperSemicontinuous f) (gmon : Antitone g) : LowerSemicontinuous (g ∘ f) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuous f) (gmon : Antitone g)
Goal: LowerSemicontinuous (g ∘ f)
-- UpperSemicontinuousAt.comp_continuousAt
theorem UpperSemicontinuousAt.comp_continuousAt {f : α → β} {g : ι → α} {x : ι}
    (hf : UpperSemicontinuousAt f (g x)) (hg : ContinuousAt g x) :
    UpperSemicontinuousAt (fun x ↦ f (g x)) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {ι : Type u_5} [TopologicalSpace ι] {f : α → β} {g : ι → α} {x : ι} (hf : UpperSemicontinuousAt f (g x)) (hg : ContinuousAt g x)
Goal: UpperSemicontinuousAt (fun x => f (g x)) x
-- UpperSemicontinuousAt.comp_continuousAt_of_eq
theorem UpperSemicontinuousAt.comp_continuousAt_of_eq {f : α → β} {g : ι → α} {y : α} {x : ι}
    (hf : UpperSemicontinuousAt f y) (hg : ContinuousAt g x) (hy : g x = y) :
    UpperSemicontinuousAt (fun x ↦ f (g x)) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {ι : Type u_5} [TopologicalSpace ι] {f : α → β} {g : ι → α} {y : α} {x : ι} (hf : UpperSemicontinuousAt f y) (hg : ContinuousAt g x) (hy : g x = y)
Goal: UpperSemicontinuousAt (fun x => f (g x)) x
-- UpperSemicontinuous.comp_continuous
theorem UpperSemicontinuous.comp_continuous {f : α → β} {g : ι → α}
    (hf : UpperSemicontinuous f) (hg : Continuous g) : UpperSemicontinuous fun x ↦ f (g x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {ι : Type u_5} [TopologicalSpace ι] {f : α → β} {g : ι → α} (hf : UpperSemicontinuous f) (hg : Continuous g)
Goal: UpperSemicontinuous fun x => f (g x)
-- UpperSemicontinuousWithinAt.add'
theorem UpperSemicontinuousWithinAt.add' {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x)
    (hg : UpperSemicontinuousWithinAt g s x)
    (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    UpperSemicontinuousWithinAt (fun z => f z + g z) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x) (hg : UpperSemicontinuousWithinAt g s x) (hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: UpperSemicontinuousWithinAt (fun z => f z + g z) s x
-- UpperSemicontinuousAt.add'
theorem UpperSemicontinuousAt.add' {f g : α → γ} (hf : UpperSemicontinuousAt f x)
    (hg : UpperSemicontinuousAt g x)
    (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    UpperSemicontinuousAt (fun z => f z + g z) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] {f g : α → γ} (hf : UpperSemicontinuousAt f x) (hg : UpperSemicontinuousAt g x) (hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: UpperSemicontinuousAt (fun z => f z + g z) x
-- UpperSemicontinuousOn.add'
theorem UpperSemicontinuousOn.add' {f g : α → γ} (hf : UpperSemicontinuousOn f s)
    (hg : UpperSemicontinuousOn g s)
    (hcont : ∀ x ∈ s, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    UpperSemicontinuousOn (fun z => f z + g z) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] {f g : α → γ} (hf : UpperSemicontinuousOn f s) (hg : UpperSemicontinuousOn g s) (hcont : ∀ x ∈ s, ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: UpperSemicontinuousOn (fun z => f z + g z) s
-- UpperSemicontinuous.add'
theorem UpperSemicontinuous.add' {f g : α → γ} (hf : UpperSemicontinuous f)
    (hg : UpperSemicontinuous g)
    (hcont : ∀ x, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    UpperSemicontinuous fun z => f z + g z 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] {f g : α → γ} (hf : UpperSemicontinuous f) (hg : UpperSemicontinuous g) (hcont : ∀ (x : α), ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: UpperSemicontinuous fun z => f z + g z
-- UpperSemicontinuousWithinAt.add
theorem UpperSemicontinuousWithinAt.add {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x)
    (hg : UpperSemicontinuousWithinAt g s x) :
    UpperSemicontinuousWithinAt (fun z => f z + g z) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x) (hg : UpperSemicontinuousWithinAt g s x)
Goal: UpperSemicontinuousWithinAt (fun z => f z + g z) s x
-- UpperSemicontinuousAt.add
theorem UpperSemicontinuousAt.add {f g : α → γ} (hf : UpperSemicontinuousAt f x)
    (hg : UpperSemicontinuousAt g x) : UpperSemicontinuousAt (fun z => f z + g z) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f g : α → γ} (hf : UpperSemicontinuousAt f x) (hg : UpperSemicontinuousAt g x)
Goal: UpperSemicontinuousAt (fun z => f z + g z) x
-- UpperSemicontinuousOn.add
theorem UpperSemicontinuousOn.add {f g : α → γ} (hf : UpperSemicontinuousOn f s)
    (hg : UpperSemicontinuousOn g s) : UpperSemicontinuousOn (fun z => f z + g z) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f g : α → γ} (hf : UpperSemicontinuousOn f s) (hg : UpperSemicontinuousOn g s)
Goal: UpperSemicontinuousOn (fun z => f z + g z) s
-- UpperSemicontinuous.add
theorem UpperSemicontinuous.add {f g : α → γ} (hf : UpperSemicontinuous f)
    (hg : UpperSemicontinuous g) : UpperSemicontinuous fun z => f z + g z 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f g : α → γ} (hf : UpperSemicontinuous f) (hg : UpperSemicontinuous g)
Goal: UpperSemicontinuous fun z => f z + g z
-- upperSemicontinuousWithinAt_sum
theorem upperSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, UpperSemicontinuousWithinAt (f i) s x) :
    UpperSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {ι : Type u_3} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, UpperSemicontinuousWithinAt (f i) s x)
Goal: UpperSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x
-- upperSemicontinuousAt_sum
theorem upperSemicontinuousAt_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, UpperSemicontinuousAt (f i) x) :
    UpperSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {ι : Type u_3} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, UpperSemicontinuousAt (f i) x)
Goal: UpperSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x
-- upperSemicontinuousOn_sum
theorem upperSemicontinuousOn_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, UpperSemicontinuousOn (f i) s) :
    UpperSemicontinuousOn (fun z => ∑ i ∈ a, f i z) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {ι : Type u_3} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, UpperSemicontinuousOn (f i) s)
Goal: UpperSemicontinuousOn (fun z => ∑ i ∈ a, f i z) s
-- upperSemicontinuous_sum
theorem upperSemicontinuous_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, UpperSemicontinuous (f i)) : UpperSemicontinuous fun z => ∑ i ∈ a, f i z 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_3} {γ : Type u_4} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ] {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, UpperSemicontinuous (f i))
Goal: UpperSemicontinuous fun z => ∑ i ∈ a, f i z
-- upperSemicontinuousWithinAt_ciInf
theorem upperSemicontinuousWithinAt_ciInf {f : ι → α → δ'}
    (bdd : ∀ᶠ y in 𝓝[s] x, BddBelow (range fun i => f i y))
    (h : ∀ i, UpperSemicontinuousWithinAt (f i) s x) :
    UpperSemicontinuousWithinAt (fun x' => ⨅ i, f i x') s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {ι : Sort u_3} {δ' : Type u_5} [ConditionallyCompleteLinearOrder δ'] {f : ι → α → δ'} (bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddBelow (Set.range fun i => f i y)) (h : ∀ (i : ι), UpperSemicontinuousWithinAt (f i) s x)
Goal: UpperSemicontinuousWithinAt (fun x' => ⨅ i, f i x') s x
-- upperSemicontinuousWithinAt_iInf
theorem upperSemicontinuousWithinAt_iInf {f : ι → α → δ}
    (h : ∀ i, UpperSemicontinuousWithinAt (f i) s x) :
    UpperSemicontinuousWithinAt (fun x' => ⨅ i, f i x') s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {f : ι → α → δ} (h : ∀ (i : ι), UpperSemicontinuousWithinAt (f i) s x)
Goal: UpperSemicontinuousWithinAt (fun x' => ⨅ i, f i x') s x
-- upperSemicontinuousWithinAt_biInf
theorem upperSemicontinuousWithinAt_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, UpperSemicontinuousWithinAt (f i hi) s x) :
    UpperSemicontinuousWithinAt (fun x' => ⨅ (i) (hi), f i hi x') s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {p : ι → Prop} {f : (i : ι) → p i → α → δ} (h : ∀ (i : ι) (hi : p i), UpperSemicontinuousWithinAt (f i hi) s x)
Goal: UpperSemicontinuousWithinAt (fun x' => ⨅ i, ⨅ (hi : p i), f i hi x') s x
-- upperSemicontinuousAt_ciInf
theorem upperSemicontinuousAt_ciInf {f : ι → α → δ'}
    (bdd : ∀ᶠ y in 𝓝 x, BddBelow (range fun i => f i y)) (h : ∀ i, UpperSemicontinuousAt (f i) x) :
    UpperSemicontinuousAt (fun x' => ⨅ i, f i x') x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {ι : Sort u_3} {δ' : Type u_5} [ConditionallyCompleteLinearOrder δ'] {f : ι → α → δ'} (bdd : ∀ᶠ (y : α) in 𝓝 x, BddBelow (Set.range fun i => f i y)) (h : ∀ (i : ι), UpperSemicontinuousAt (f i) x)
Goal: UpperSemicontinuousAt (fun x' => ⨅ i, f i x') x
-- upperSemicontinuousAt_iInf
theorem upperSemicontinuousAt_iInf {f : ι → α → δ} (h : ∀ i, UpperSemicontinuousAt (f i) x) :
    UpperSemicontinuousAt (fun x' => ⨅ i, f i x') x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {f : ι → α → δ} (h : ∀ (i : ι), UpperSemicontinuousAt (f i) x)
Goal: UpperSemicontinuousAt (fun x' => ⨅ i, f i x') x
-- upperSemicontinuousAt_biInf
theorem upperSemicontinuousAt_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, UpperSemicontinuousAt (f i hi) x) :
    UpperSemicontinuousAt (fun x' => ⨅ (i) (hi), f i hi x') x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {p : ι → Prop} {f : (i : ι) → p i → α → δ} (h : ∀ (i : ι) (hi : p i), UpperSemicontinuousAt (f i hi) x)
Goal: UpperSemicontinuousAt (fun x' => ⨅ i, ⨅ (hi : p i), f i hi x') x
-- upperSemicontinuousOn_ciInf
theorem upperSemicontinuousOn_ciInf {f : ι → α → δ'}
    (bdd : ∀ x ∈ s, BddBelow (range fun i => f i x)) (h : ∀ i, UpperSemicontinuousOn (f i) s) :
    UpperSemicontinuousOn (fun x' => ⨅ i, f i x') s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {ι : Sort u_3} {δ' : Type u_5} [ConditionallyCompleteLinearOrder δ'] {f : ι → α → δ'} (bdd : ∀ x ∈ s, BddBelow (Set.range fun i => f i x)) (h : ∀ (i : ι), UpperSemicontinuousOn (f i) s)
Goal: UpperSemicontinuousOn (fun x' => ⨅ i, f i x') s
-- upperSemicontinuousOn_iInf
theorem upperSemicontinuousOn_iInf {f : ι → α → δ} (h : ∀ i, UpperSemicontinuousOn (f i) s) :
    UpperSemicontinuousOn (fun x' => ⨅ i, f i x') s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {f : ι → α → δ} (h : ∀ (i : ι), UpperSemicontinuousOn (f i) s)
Goal: UpperSemicontinuousOn (fun x' => ⨅ i, f i x') s
-- upperSemicontinuousOn_biInf
theorem upperSemicontinuousOn_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, UpperSemicontinuousOn (f i hi) s) :
    UpperSemicontinuousOn (fun x' => ⨅ (i) (hi), f i hi x') s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {p : ι → Prop} {f : (i : ι) → p i → α → δ} (h : ∀ (i : ι) (hi : p i), UpperSemicontinuousOn (f i hi) s)
Goal: UpperSemicontinuousOn (fun x' => ⨅ i, ⨅ (hi : p i), f i hi x') s
-- upperSemicontinuous_ciInf
theorem upperSemicontinuous_ciInf {f : ι → α → δ'} (bdd : ∀ x, BddBelow (range fun i => f i x))
    (h : ∀ i, UpperSemicontinuous (f i)) : UpperSemicontinuous fun x' => ⨅ i, f i x' 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Sort u_3} {δ' : Type u_5} [ConditionallyCompleteLinearOrder δ'] {f : ι → α → δ'} (bdd : ∀ (x : α), BddBelow (Set.range fun i => f i x)) (h : ∀ (i : ι), UpperSemicontinuous (f i))
Goal: UpperSemicontinuous fun x' => ⨅ i, f i x'
-- upperSemicontinuous_iInf
theorem upperSemicontinuous_iInf {f : ι → α → δ} (h : ∀ i, UpperSemicontinuous (f i)) :
    UpperSemicontinuous fun x' => ⨅ i, f i x' 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {f : ι → α → δ} (h : ∀ (i : ι), UpperSemicontinuous (f i))
Goal: UpperSemicontinuous fun x' => ⨅ i, f i x'
-- upperSemicontinuous_biInf
theorem upperSemicontinuous_biInf {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, UpperSemicontinuous (f i hi)) :
    UpperSemicontinuous fun x' => ⨅ (i) (hi), f i hi x' 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Sort u_3} {δ : Type u_4} [CompleteLinearOrder δ] {p : ι → Prop} {f : (i : ι) → p i → α → δ} (h : ∀ (i : ι) (hi : p i), UpperSemicontinuous (f i hi))
Goal: UpperSemicontinuous fun x' => ⨅ i, ⨅ (hi : p i), f i hi x'
-- continuousWithinAt_iff_lower_upperSemicontinuousWithinAt
theorem continuousWithinAt_iff_lower_upperSemicontinuousWithinAt {f : α → γ} :
    ContinuousWithinAt f s x ↔
      LowerSemicontinuousWithinAt f s x ∧ UpperSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ}
Goal: ContinuousWithinAt f s x ↔ LowerSemicontinuousWithinAt f s x ∧ UpperSemicontinuousWithinAt f s x
-- continuousAt_iff_lower_upperSemicontinuousAt
theorem continuousAt_iff_lower_upperSemicontinuousAt {f : α → γ} :
    ContinuousAt f x ↔ LowerSemicontinuousAt f x ∧ UpperSemicontinuousAt f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ}
Goal: ContinuousAt f x ↔ LowerSemicontinuousAt f x ∧ UpperSemicontinuousAt f x