ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousAt.comp_lowerSemicontinuousWithinAt_antitone
theorem ContinuousAt.comp_lowerSemicontinuousWithinAt_antitone {g : γ → δ} {f : α → γ}
    (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Antitone 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 : LowerSemicontinuousWithinAt f s x) (gmon : Antitone g)
Goal: UpperSemicontinuousWithinAt (g ∘ f) s x
-- ContinuousAt.comp_lowerSemicontinuousAt_antitone
theorem ContinuousAt.comp_lowerSemicontinuousAt_antitone {g : γ → δ} {f : α → γ}
    (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousAt f x) (gmon : Antitone 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 : LowerSemicontinuousAt f x) (gmon : Antitone g)
Goal: UpperSemicontinuousAt (g ∘ f) x
-- Continuous.comp_lowerSemicontinuousOn_antitone
theorem Continuous.comp_lowerSemicontinuousOn_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : LowerSemicontinuousOn f s) (gmon : Antitone 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 : LowerSemicontinuousOn f s) (gmon : Antitone g)
Goal: UpperSemicontinuousOn (g ∘ f) s
-- Continuous.comp_lowerSemicontinuous_antitone
theorem Continuous.comp_lowerSemicontinuous_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : LowerSemicontinuous f) (gmon : Antitone 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 : LowerSemicontinuous f) (gmon : Antitone g)
Goal: UpperSemicontinuous (g ∘ f)
-- LowerSemicontinuousAt.comp_continuousAt
theorem LowerSemicontinuousAt.comp_continuousAt {f : α → β} {g : ι → α} {x : ι}
    (hf : LowerSemicontinuousAt f (g x)) (hg : ContinuousAt g x) :
    LowerSemicontinuousAt (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 : LowerSemicontinuousAt f (g x)) (hg : ContinuousAt g x)
Goal: LowerSemicontinuousAt (fun x => f (g x)) x
-- LowerSemicontinuousAt.comp_continuousAt_of_eq
theorem LowerSemicontinuousAt.comp_continuousAt_of_eq {f : α → β} {g : ι → α} {y : α} {x : ι}
    (hf : LowerSemicontinuousAt f y) (hg : ContinuousAt g x) (hy : g x = y) :
    LowerSemicontinuousAt (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 : LowerSemicontinuousAt f y) (hg : ContinuousAt g x) (hy : g x = y)
Goal: LowerSemicontinuousAt (fun x => f (g x)) x
-- LowerSemicontinuous.comp_continuous
theorem LowerSemicontinuous.comp_continuous {f : α → β} {g : ι → α}
    (hf : LowerSemicontinuous f) (hg : Continuous g) : LowerSemicontinuous 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 : LowerSemicontinuous f) (hg : Continuous g)
Goal: LowerSemicontinuous fun x => f (g x)
-- LowerSemicontinuousWithinAt.add'
theorem LowerSemicontinuousWithinAt.add' {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x)
    (hg : LowerSemicontinuousWithinAt g s x)
    (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    LowerSemicontinuousWithinAt (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 : LowerSemicontinuousWithinAt f s x) (hg : LowerSemicontinuousWithinAt g s x) (hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: LowerSemicontinuousWithinAt (fun z => f z + g z) s x
-- LowerSemicontinuousAt.add'
theorem LowerSemicontinuousAt.add' {f g : α → γ} (hf : LowerSemicontinuousAt f x)
    (hg : LowerSemicontinuousAt g x)
    (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    LowerSemicontinuousAt (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 : LowerSemicontinuousAt f x) (hg : LowerSemicontinuousAt g x) (hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: LowerSemicontinuousAt (fun z => f z + g z) x
-- LowerSemicontinuousOn.add'
theorem LowerSemicontinuousOn.add' {f g : α → γ} (hf : LowerSemicontinuousOn f s)
    (hg : LowerSemicontinuousOn g s)
    (hcont : ∀ x ∈ s, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    LowerSemicontinuousOn (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 : LowerSemicontinuousOn f s) (hg : LowerSemicontinuousOn g s) (hcont : ∀ x ∈ s, ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: LowerSemicontinuousOn (fun z => f z + g z) s
-- LowerSemicontinuous.add'
theorem LowerSemicontinuous.add' {f g : α → γ} (hf : LowerSemicontinuous f)
    (hg : LowerSemicontinuous g)
    (hcont : ∀ x, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    LowerSemicontinuous 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 : LowerSemicontinuous f) (hg : LowerSemicontinuous g) (hcont : ∀ (x : α), ContinuousAt (fun p => p.1 + p.2) (f x, g x))
Goal: LowerSemicontinuous fun z => f z + g z
-- LowerSemicontinuousWithinAt.add
theorem LowerSemicontinuousWithinAt.add {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x)
    (hg : LowerSemicontinuousWithinAt g s x) :
    LowerSemicontinuousWithinAt (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 : LowerSemicontinuousWithinAt f s x) (hg : LowerSemicontinuousWithinAt g s x)
Goal: LowerSemicontinuousWithinAt (fun z => f z + g z) s x
-- LowerSemicontinuousAt.add
theorem LowerSemicontinuousAt.add {f g : α → γ} (hf : LowerSemicontinuousAt f x)
    (hg : LowerSemicontinuousAt g x) : LowerSemicontinuousAt (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 : LowerSemicontinuousAt f x) (hg : LowerSemicontinuousAt g x)
Goal: LowerSemicontinuousAt (fun z => f z + g z) x
-- LowerSemicontinuousOn.add
theorem LowerSemicontinuousOn.add {f g : α → γ} (hf : LowerSemicontinuousOn f s)
    (hg : LowerSemicontinuousOn g s) : LowerSemicontinuousOn (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 : LowerSemicontinuousOn f s) (hg : LowerSemicontinuousOn g s)
Goal: LowerSemicontinuousOn (fun z => f z + g z) s
-- LowerSemicontinuous.add
theorem LowerSemicontinuous.add {f g : α → γ} (hf : LowerSemicontinuous f)
    (hg : LowerSemicontinuous g) : LowerSemicontinuous 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 : LowerSemicontinuous f) (hg : LowerSemicontinuous g)
Goal: LowerSemicontinuous fun z => f z + g z
-- lowerSemicontinuousWithinAt_sum
theorem lowerSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, LowerSemicontinuousWithinAt (f i) s x) :
    LowerSemicontinuousWithinAt (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, LowerSemicontinuousWithinAt (f i) s x)
Goal: LowerSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x
-- lowerSemicontinuousAt_sum
theorem lowerSemicontinuousAt_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, LowerSemicontinuousAt (f i) x) :
    LowerSemicontinuousAt (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, LowerSemicontinuousAt (f i) x)
Goal: LowerSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x
-- lowerSemicontinuousOn_sum
theorem lowerSemicontinuousOn_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, LowerSemicontinuousOn (f i) s) :
    LowerSemicontinuousOn (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, LowerSemicontinuousOn (f i) s)
Goal: LowerSemicontinuousOn (fun z => ∑ i ∈ a, f i z) s
-- lowerSemicontinuous_sum
theorem lowerSemicontinuous_sum {f : ι → α → γ} {a : Finset ι}
    (ha : ∀ i ∈ a, LowerSemicontinuous (f i)) : LowerSemicontinuous 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, LowerSemicontinuous (f i))
Goal: LowerSemicontinuous fun z => ∑ i ∈ a, f i z
-- lowerSemicontinuousWithinAt_ciSup
theorem lowerSemicontinuousWithinAt_ciSup {f : ι → α → δ'}
    (bdd : ∀ᶠ y in 𝓝[s] x, BddAbove (range fun i => f i y))
    (h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
    LowerSemicontinuousWithinAt (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, BddAbove (Set.range fun i => f i y)) (h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x)
Goal: LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x
-- lowerSemicontinuousWithinAt_iSup
theorem lowerSemicontinuousWithinAt_iSup {f : ι → α → δ}
    (h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
    LowerSemicontinuousWithinAt (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), LowerSemicontinuousWithinAt (f i) s x)
Goal: LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x
-- lowerSemicontinuousWithinAt_biSup
theorem lowerSemicontinuousWithinAt_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, LowerSemicontinuousWithinAt (f i hi) s x) :
    LowerSemicontinuousWithinAt (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), LowerSemicontinuousWithinAt (f i hi) s x)
Goal: LowerSemicontinuousWithinAt (fun x' => ⨆ i, ⨆ (hi : p i), f i hi x') s x
-- lowerSemicontinuousAt_ciSup
theorem lowerSemicontinuousAt_ciSup {f : ι → α → δ'}
    (bdd : ∀ᶠ y in 𝓝 x, BddAbove (range fun i => f i y)) (h : ∀ i, LowerSemicontinuousAt (f i) x) :
    LowerSemicontinuousAt (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, BddAbove (Set.range fun i => f i y)) (h : ∀ (i : ι), LowerSemicontinuousAt (f i) x)
Goal: LowerSemicontinuousAt (fun x' => ⨆ i, f i x') x
-- lowerSemicontinuousAt_iSup
theorem lowerSemicontinuousAt_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousAt (f i) x) :
    LowerSemicontinuousAt (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), LowerSemicontinuousAt (f i) x)
Goal: LowerSemicontinuousAt (fun x' => ⨆ i, f i x') x
-- lowerSemicontinuousAt_biSup
theorem lowerSemicontinuousAt_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, LowerSemicontinuousAt (f i hi) x) :
    LowerSemicontinuousAt (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), LowerSemicontinuousAt (f i hi) x)
Goal: LowerSemicontinuousAt (fun x' => ⨆ i, ⨆ (hi : p i), f i hi x') x
-- lowerSemicontinuousOn_ciSup
theorem lowerSemicontinuousOn_ciSup {f : ι → α → δ'}
    (bdd : ∀ x ∈ s, BddAbove (range fun i => f i x)) (h : ∀ i, LowerSemicontinuousOn (f i) s) :
    LowerSemicontinuousOn (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, BddAbove (Set.range fun i => f i x)) (h : ∀ (i : ι), LowerSemicontinuousOn (f i) s)
Goal: LowerSemicontinuousOn (fun x' => ⨆ i, f i x') s
-- lowerSemicontinuousOn_iSup
theorem lowerSemicontinuousOn_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousOn (f i) s) :
    LowerSemicontinuousOn (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), LowerSemicontinuousOn (f i) s)
Goal: LowerSemicontinuousOn (fun x' => ⨆ i, f i x') s
-- lowerSemicontinuousOn_biSup
theorem lowerSemicontinuousOn_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, LowerSemicontinuousOn (f i hi) s) :
    LowerSemicontinuousOn (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), LowerSemicontinuousOn (f i hi) s)
Goal: LowerSemicontinuousOn (fun x' => ⨆ i, ⨆ (hi : p i), f i hi x') s
-- lowerSemicontinuous_ciSup
theorem lowerSemicontinuous_ciSup {f : ι → α → δ'} (bdd : ∀ x, BddAbove (range fun i => f i x))
    (h : ∀ i, LowerSemicontinuous (f i)) : LowerSemicontinuous 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), BddAbove (Set.range fun i => f i x)) (h : ∀ (i : ι), LowerSemicontinuous (f i))
Goal: LowerSemicontinuous fun x' => ⨆ i, f i x'
-- lowerSemicontinuous_iSup
theorem lowerSemicontinuous_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuous (f i)) :
    LowerSemicontinuous 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), LowerSemicontinuous (f i))
Goal: LowerSemicontinuous fun x' => ⨆ i, f i x'
-- lowerSemicontinuous_biSup
theorem lowerSemicontinuous_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
    (h : ∀ i hi, LowerSemicontinuous (f i hi)) :
    LowerSemicontinuous 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), LowerSemicontinuous (f i hi))
Goal: LowerSemicontinuous fun x' => ⨆ i, ⨆ (hi : p i), f i hi x'
-- lowerSemicontinuousWithinAt_tsum
theorem lowerSemicontinuousWithinAt_tsum {f : ι → α → ℝ≥0∞}
    (h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
    LowerSemicontinuousWithinAt (fun x' => ∑' i, f i x') s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {ι : Type u_3} {f : ι → α → ℝ≥0∞} (h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x)
Goal: LowerSemicontinuousWithinAt (fun x' => ∑' (i : ι), f i x') s x
-- lowerSemicontinuousAt_tsum
theorem lowerSemicontinuousAt_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuousAt (f i) x) :
    LowerSemicontinuousAt (fun x' => ∑' i, f i x') x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {ι : Type u_3} {f : ι → α → ℝ≥0∞} (h : ∀ (i : ι), LowerSemicontinuousAt (f i) x)
Goal: LowerSemicontinuousAt (fun x' => ∑' (i : ι), f i x') x
-- lowerSemicontinuousOn_tsum
theorem lowerSemicontinuousOn_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuousOn (f i) s) :
    LowerSemicontinuousOn (fun x' => ∑' i, f i x') s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {ι : Type u_3} {f : ι → α → ℝ≥0∞} (h : ∀ (i : ι), LowerSemicontinuousOn (f i) s)
Goal: LowerSemicontinuousOn (fun x' => ∑' (i : ι), f i x') s
-- lowerSemicontinuous_tsum
theorem lowerSemicontinuous_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuous (f i)) :
    LowerSemicontinuous fun x' => ∑' i, f i x' 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_3} {f : ι → α → ℝ≥0∞} (h : ∀ (i : ι), LowerSemicontinuous (f i))
Goal: LowerSemicontinuous fun x' => ∑' (i : ι), f i x'
-- UpperSemicontinuousWithinAt.mono
theorem UpperSemicontinuousWithinAt.mono (h : UpperSemicontinuousWithinAt f s x) (hst : t ⊆ s) :
    UpperSemicontinuousWithinAt f t x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α} {s t : Set α} (h : UpperSemicontinuousWithinAt f s x) (hst : t ⊆ s)
Goal: UpperSemicontinuousWithinAt f t x
-- upperSemicontinuousWithinAt_univ_iff
theorem upperSemicontinuousWithinAt_univ_iff :
    UpperSemicontinuousWithinAt f univ x ↔ UpperSemicontinuousAt f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α}
Goal: UpperSemicontinuousWithinAt f Set.univ x ↔ UpperSemicontinuousAt f x
-- UpperSemicontinuousAt.upperSemicontinuousWithinAt
theorem UpperSemicontinuousAt.upperSemicontinuousWithinAt (s : Set α)
    (h : UpperSemicontinuousAt f x) : UpperSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α} (s : Set α) (h : UpperSemicontinuousAt f x)
Goal: UpperSemicontinuousWithinAt f s x
-- UpperSemicontinuousOn.upperSemicontinuousWithinAt
theorem UpperSemicontinuousOn.upperSemicontinuousWithinAt (h : UpperSemicontinuousOn f s)
    (hx : x ∈ s) : UpperSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α} {s : Set α} (h : UpperSemicontinuousOn f s) (hx : x ∈ s)
Goal: UpperSemicontinuousWithinAt f s x
-- UpperSemicontinuousOn.mono
theorem UpperSemicontinuousOn.mono (h : UpperSemicontinuousOn f s) (hst : t ⊆ s) :
    UpperSemicontinuousOn f t 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {s t : Set α} (h : UpperSemicontinuousOn f s) (hst : t ⊆ s)
Goal: UpperSemicontinuousOn f t
-- upperSemicontinuousOn_univ_iff
theorem upperSemicontinuousOn_univ_iff : UpperSemicontinuousOn f univ ↔ UpperSemicontinuous f 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β}
Goal: UpperSemicontinuousOn f Set.univ ↔ UpperSemicontinuous f
-- UpperSemicontinuous.upperSemicontinuousAt
theorem UpperSemicontinuous.upperSemicontinuousAt (h : UpperSemicontinuous f) (x : α) :
    UpperSemicontinuousAt f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (h : UpperSemicontinuous f) (x : α)
Goal: UpperSemicontinuousAt f x
-- UpperSemicontinuous.upperSemicontinuousWithinAt
theorem UpperSemicontinuous.upperSemicontinuousWithinAt (h : UpperSemicontinuous f) (s : Set α)
    (x : α) : UpperSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (h : UpperSemicontinuous f) (s : Set α) (x : α)
Goal: UpperSemicontinuousWithinAt f s x
-- UpperSemicontinuous.upperSemicontinuousOn
theorem UpperSemicontinuous.upperSemicontinuousOn (h : UpperSemicontinuous f) (s : Set α) :
    UpperSemicontinuousOn f s 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (h : UpperSemicontinuous f) (s : Set α)
Goal: UpperSemicontinuousOn f s
-- upperSemicontinuousWithinAt_const
theorem upperSemicontinuousWithinAt_const : UpperSemicontinuousWithinAt (fun _x => z) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s : Set α} {z : β}
Goal: UpperSemicontinuousWithinAt (fun _x => z) s x
-- upperSemicontinuousAt_const
theorem upperSemicontinuousAt_const : UpperSemicontinuousAt (fun _x => z) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {z : β}
Goal: UpperSemicontinuousAt (fun _x => z) x
-- upperSemicontinuousOn_const
theorem upperSemicontinuousOn_const : UpperSemicontinuousOn (fun _x => z) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s : Set α} {z : β}
Goal: UpperSemicontinuousOn (fun _x => z) s
-- upperSemicontinuous_const
theorem upperSemicontinuous_const : UpperSemicontinuous fun _x : α => z 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {z : β}
Goal: UpperSemicontinuous fun _x => z
-- IsOpen.upperSemicontinuous_indicator
theorem IsOpen.upperSemicontinuous_indicator (hs : IsOpen s) (hy : y ≤ 0) :
    UpperSemicontinuous (indicator s fun _x => y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s : Set α} {y : β} [Zero β] (hs : IsOpen s) (hy : y ≤ 0)
Goal: UpperSemicontinuous (s.indicator fun _x => y)
-- IsOpen.upperSemicontinuousOn_indicator
theorem IsOpen.upperSemicontinuousOn_indicator (hs : IsOpen s) (hy : y ≤ 0) :
    UpperSemicontinuousOn (indicator s fun _x => y) t 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s t : Set α} {y : β} [Zero β] (hs : IsOpen s) (hy : y ≤ 0)
Goal: UpperSemicontinuousOn (s.indicator fun _x => y) t
-- IsOpen.upperSemicontinuousAt_indicator
theorem IsOpen.upperSemicontinuousAt_indicator (hs : IsOpen s) (hy : y ≤ 0) :
    UpperSemicontinuousAt (indicator s fun _x => y) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s : Set α} {y : β} [Zero β] (hs : IsOpen s) (hy : y ≤ 0)
Goal: UpperSemicontinuousAt (s.indicator fun _x => y) x
-- IsOpen.upperSemicontinuousWithinAt_indicator
theorem IsOpen.upperSemicontinuousWithinAt_indicator (hs : IsOpen s) (hy : y ≤ 0) :
    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 : IsOpen s) (hy : y ≤ 0)
Goal: UpperSemicontinuousWithinAt (s.indicator fun _x => y) t x
-- IsClosed.upperSemicontinuous_indicator
theorem IsClosed.upperSemicontinuous_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
    UpperSemicontinuous (indicator s fun _x => y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : 0 ≤ y)
Goal: UpperSemicontinuous (s.indicator fun _x => y)
-- IsClosed.upperSemicontinuousOn_indicator
theorem IsClosed.upperSemicontinuousOn_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
    UpperSemicontinuousOn (indicator s fun _x => y) t 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s t : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : 0 ≤ y)
Goal: UpperSemicontinuousOn (s.indicator fun _x => y) t
-- IsClosed.upperSemicontinuousAt_indicator
theorem IsClosed.upperSemicontinuousAt_indicator (hs : IsClosed s) (hy : 0 ≤ y) :
    UpperSemicontinuousAt (indicator s fun _x => y) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : 0 ≤ y)
Goal: UpperSemicontinuousAt (s.indicator fun _x => y) x