ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.Tendsto.inf_nhds
lemma inf_nhds [Inf L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (fun i => f i ⊓ g i) l (𝓝 (x ⊓ y)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {α : Type u_3} {l : Filter α} {f g : α → L} {x y : L} [Inf L] [ContinuousInf L] (hf : Filter.Tendsto f l (𝓝 x)) (hg : Filter.Tendsto g l (𝓝 y))
Goal: Filter.Tendsto (fun i => f i ⊓ g i) l (𝓝 (x ⊓ y))
-- Filter.Tendsto.finset_sup'_nhds
lemma finset_sup'_nhds [SemilatticeSup L] [ContinuousSup L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (s.sup' hne f) l (𝓝 (s.sup' hne g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeSup L] [ContinuousSup L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (s.sup' hne f) l (𝓝 (s.sup' hne g))
-- Filter.Tendsto.finset_sup'_nhds_apply
lemma finset_sup'_nhds_apply [SemilatticeSup L] [ContinuousSup L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.sup' hne (f · a)) l (𝓝 (s.sup' hne g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeSup L] [ContinuousSup L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (fun a => s.sup' hne fun x => f x a) l (𝓝 (s.sup' hne g))
-- Filter.Tendsto.finset_inf'_nhds
lemma finset_inf'_nhds [SemilatticeInf L] [ContinuousInf L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (s.inf' hne f) l (𝓝 (s.inf' hne g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeInf L] [ContinuousInf L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (s.inf' hne f) l (𝓝 (s.inf' hne g))
-- Filter.Tendsto.finset_inf'_nhds_apply
lemma finset_inf'_nhds_apply [SemilatticeInf L] [ContinuousInf L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.inf' hne (f · a)) l (𝓝 (s.inf' hne g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeInf L] [ContinuousInf L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (fun a => s.inf' hne fun x => f x a) l (𝓝 (s.inf' hne g))
-- Filter.Tendsto.finset_sup_nhds
lemma finset_sup_nhds [SemilatticeSup L] [OrderBot L] [ContinuousSup L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.sup f) l (𝓝 (s.sup g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (s.sup f) l (𝓝 (s.sup g))
-- Filter.Tendsto.finset_sup_nhds_apply
lemma finset_sup_nhds_apply [SemilatticeSup L] [OrderBot L] [ContinuousSup L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.sup (f · a)) l (𝓝 (s.sup g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (fun a => s.sup fun x => f x a) l (𝓝 (s.sup g))
-- Filter.Tendsto.finset_inf_nhds
lemma finset_inf_nhds [SemilatticeInf L] [OrderTop L] [ContinuousInf L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.inf f) l (𝓝 (s.inf g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (s.inf f) l (𝓝 (s.inf g))
-- Filter.Tendsto.finset_inf_nhds_apply
lemma finset_inf_nhds_apply [SemilatticeInf L] [OrderTop L] [ContinuousInf L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.inf (f · a)) l (𝓝 (s.inf g)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {ι : Type u_3} {α : Type u_4} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] (hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i)))
Goal: Filter.Tendsto (fun a => s.inf fun x => f x a) l (𝓝 (s.inf g))
-- ContinuousAt.sup'
lemma ContinuousAt.sup' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (f ⊔ g) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (f ⊔ g) x
-- ContinuousAt.sup
@[fun_prop]
lemma ContinuousAt.sup (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (fun a ↦ f a ⊔ g a) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (fun a => f a ⊔ g a) x
-- ContinuousWithinAt.sup'
lemma ContinuousWithinAt.sup' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (f ⊔ g) s x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} {s : Set X} {x : X} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (f ⊔ g) s x
-- ContinuousWithinAt.sup
lemma ContinuousWithinAt.sup (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun a ↦ f a ⊔ g a) s x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} {s : Set X} {x : X} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (fun a => f a ⊔ g a) s x
-- ContinuousOn.sup'
lemma ContinuousOn.sup' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (f ⊔ g) s 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (f ⊔ g) s
-- ContinuousOn.sup
@[fun_prop]
lemma ContinuousOn.sup (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun a ↦ f a ⊔ g a) s 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun a => f a ⊔ g a) s
-- Continuous.sup'
lemma Continuous.sup' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊔ g) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous (f ⊔ g)
-- ContinuousAt.inf'
lemma ContinuousAt.inf' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (f ⊓ g) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (f ⊓ g) x
-- ContinuousAt.inf
@[fun_prop]
lemma ContinuousAt.inf (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (fun a ↦ f a ⊓ g a) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (fun a => f a ⊓ g a) x
-- ContinuousWithinAt.inf'
lemma ContinuousWithinAt.inf' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (f ⊓ g) s x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} {s : Set X} {x : X} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (f ⊓ g) s x
-- ContinuousWithinAt.inf
lemma ContinuousWithinAt.inf (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun a ↦ f a ⊓ g a) s x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} {s : Set X} {x : X} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (fun a => f a ⊓ g a) s x
-- ContinuousOn.inf'
lemma ContinuousOn.inf' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (f ⊓ g) s 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (f ⊓ g) s
-- ContinuousOn.inf
@[fun_prop]
lemma ContinuousOn.inf (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun a ↦ f a ⊓ g a) s 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun a => f a ⊓ g a) s
-- Continuous.inf'
lemma Continuous.inf' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊓ g) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous (f ⊓ g)
-- ContinuousAt.finset_sup'_apply
lemma ContinuousAt.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.sup' hne (f · a)) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (fun a => s.sup' hne fun x => f x a) x
-- ContinuousAt.finset_sup'
lemma ContinuousAt.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.sup' hne f) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (s.sup' hne f) x
-- ContinuousWithinAt.finset_sup'_apply
lemma ContinuousWithinAt.finset_sup'_apply (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.sup' hne (f · a)) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (fun a => s.sup' hne fun x => f x a) t x
-- ContinuousWithinAt.finset_sup'
lemma ContinuousWithinAt.finset_sup' (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup' hne f) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (s.sup' hne f) t x
-- ContinuousOn.finset_sup'_apply
lemma ContinuousOn.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.sup' hne (f · a)) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (fun a => s.sup' hne fun x => f x a) t
-- ContinuousOn.finset_sup'
lemma ContinuousOn.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.sup' hne f) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (s.sup' hne f) t
-- Continuous.finset_sup'_apply
lemma Continuous.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.sup' hne (f · a)) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous fun a => s.sup' hne fun x => f x a
-- Continuous.finset_sup'
lemma Continuous.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (s.sup' hne f) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous (s.sup' hne f)
-- ContinuousAt.finset_sup_apply
lemma ContinuousAt.finset_sup_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.sup (f · a)) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {x : X} (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (fun a => s.sup fun x => f x a) x
-- ContinuousAt.finset_sup
lemma ContinuousAt.finset_sup (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.sup f) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {x : X} (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (s.sup f) x
-- ContinuousWithinAt.finset_sup_apply
lemma ContinuousWithinAt.finset_sup_apply
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.sup (f · a)) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (fun a => s.sup fun x => f x a) t x
-- ContinuousWithinAt.finset_sup
lemma ContinuousWithinAt.finset_sup
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup f) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (s.sup f) t x
-- ContinuousOn.finset_sup_apply
lemma ContinuousOn.finset_sup_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.sup (f · a)) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (fun a => s.sup fun x => f x a) t
-- ContinuousOn.finset_sup
lemma ContinuousOn.finset_sup (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.sup f) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (s.sup f) t
-- Continuous.finset_sup_apply
lemma Continuous.finset_sup_apply (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.sup (f · a)) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous fun a => s.sup fun x => f x a
-- Continuous.finset_sup
lemma Continuous.finset_sup (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.sup f) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous (s.sup f)
-- ContinuousAt.finset_inf'_apply
lemma ContinuousAt.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.inf' hne (f · a)) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (fun a => s.inf' hne fun x => f x a) x
-- ContinuousAt.finset_inf'
lemma ContinuousAt.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.inf' hne f) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (s.inf' hne f) x
-- ContinuousWithinAt.finset_inf'_apply
lemma ContinuousWithinAt.finset_inf'_apply (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.inf' hne (f · a)) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (fun a => s.inf' hne fun x => f x a) t x
-- ContinuousWithinAt.finset_inf'
lemma ContinuousWithinAt.finset_inf' (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf' hne f) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (s.inf' hne f) t x
-- ContinuousOn.finset_inf'_apply
lemma ContinuousOn.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.inf' hne (f · a)) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (fun a => s.inf' hne fun x => f x a) t
-- ContinuousOn.finset_inf'
lemma ContinuousOn.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.inf' hne f) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (s.inf' hne f) t
-- Continuous.finset_inf'_apply
lemma Continuous.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.inf' hne (f · a)) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous fun a => s.inf' hne fun x => f x a
-- Continuous.finset_inf'
lemma Continuous.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (s.inf' hne f) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous (s.inf' hne f)
-- ContinuousAt.finset_inf_apply
lemma ContinuousAt.finset_inf_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.inf (f · a)) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {x : X} (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (fun a => s.inf fun x => f x a) x
-- ContinuousAt.finset_inf
lemma ContinuousAt.finset_inf (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.inf f) x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {x : X} (hs : ∀ i ∈ s, ContinuousAt (f i) x)
Goal: ContinuousAt (s.inf f) x
-- ContinuousWithinAt.finset_inf_apply
lemma ContinuousWithinAt.finset_inf_apply
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.inf (f · a)) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (fun a => s.inf fun x => f x a) t x
-- ContinuousWithinAt.finset_inf
lemma ContinuousWithinAt.finset_inf
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf f) t x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x)
Goal: ContinuousWithinAt (s.inf f) t x
-- ContinuousOn.finset_inf_apply
lemma ContinuousOn.finset_inf_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.inf (f · a)) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (fun a => s.inf fun x => f x a) t
-- ContinuousOn.finset_inf
lemma ContinuousOn.finset_inf (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.inf f) t 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} (hs : ∀ i ∈ s, ContinuousOn (f i) t)
Goal: ContinuousOn (s.inf f) t
-- Continuous.finset_inf_apply
lemma Continuous.finset_inf_apply (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.inf (f · a)) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous fun a => s.inf fun x => f x a
-- Continuous.finset_inf
lemma Continuous.finset_inf (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.inf f) 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] {ι : Type u_3} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} (hs : ∀ i ∈ s, Continuous (f i))
Goal: Continuous (s.inf f)
-- ShrinkingLemma.PartialRefinement.isOpen
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
@[ext] structure PartialRefinement (u : ι → Set X) (s : Set X) where
    toFun : ι → Set X
    carrier : Set ι
    protected isOpen : ∀ i, IsOpen (toFun i)
    subset_iUnion : s ⊆ ⋃ i, toFun i
    closure_subset : ∀ {i}, i ∈ carrier → closure (toFun i) ⊆ u i
    apply_eq : ∀ {i}, i ∉ carrier → toFun i = u i
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} (self : ShrinkingLemma.PartialRefinement u s) (i : ι)
Goal: IsOpen (self.toFun i)