ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ici_mem_nhdsSet_Ico
theorem Ici_mem_nhdsSet_Ico (h : a < b) : Ici a ∈ 𝓝ˢ (Ico b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : a < b)
Goal: Set.Ici a ∈ 𝓝ˢ (Set.Ico b c)
-- Ioi_mem_nhdsSet_Ico
theorem Ioi_mem_nhdsSet_Ico (h : a < b) : Ioi a ∈ 𝓝ˢ (Ico b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : a < b)
Goal: Set.Ioi a ∈ 𝓝ˢ (Set.Ico b c)
-- Iio_mem_nhdsSet_Ico
theorem Iio_mem_nhdsSet_Ico (h : b ≤ c) : Iio c ∈ 𝓝ˢ (Ico a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : b ≤ c)
Goal: Set.Iio c ∈ 𝓝ˢ (Set.Ico a b)
-- Iic_mem_nhdsSet_Ico
theorem Iic_mem_nhdsSet_Ico (h : b ≤ c) : Iic c ∈ 𝓝ˢ (Ico a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : b ≤ c)
Goal: Set.Iic c ∈ 𝓝ˢ (Set.Ico a b)
-- Ioo_mem_nhdsSet_Ico
theorem Ioo_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioo a d ∈ 𝓝ˢ (Ico b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c ≤ d)
Goal: Set.Ioo a d ∈ 𝓝ˢ (Set.Ico b c)
-- Icc_mem_nhdsSet_Ico
theorem Icc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Icc a d ∈ 𝓝ˢ (Ico b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c ≤ d)
Goal: Set.Icc a d ∈ 𝓝ˢ (Set.Ico b c)
-- Ioc_mem_nhdsSet_Ico
theorem Ioc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioc a d ∈ 𝓝ˢ (Ico b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c ≤ d)
Goal: Set.Ioc a d ∈ 𝓝ˢ (Set.Ico b c)
-- Ico_mem_nhdsSet_Ico
theorem Ico_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ico a d ∈ 𝓝ˢ (Ico b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a < b) (h' : c ≤ d)
Goal: Set.Ico a d ∈ 𝓝ˢ (Set.Ico b c)
-- Ioi_mem_nhdsSet_Ioc
theorem Ioi_mem_nhdsSet_Ioc (h : a ≤ b) : Ioi a ∈ 𝓝ˢ (Ioc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : a ≤ b)
Goal: Set.Ioi a ∈ 𝓝ˢ (Set.Ioc b c)
-- Iio_mem_nhdsSet_Ioc
theorem Iio_mem_nhdsSet_Ioc (h : b < c) : Iio c ∈ 𝓝ˢ (Ioc a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : b < c)
Goal: Set.Iio c ∈ 𝓝ˢ (Set.Ioc a b)
-- Ici_mem_nhdsSet_Ioc
theorem Ici_mem_nhdsSet_Ioc (h : a ≤ b) : Ici a ∈ 𝓝ˢ (Ioc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : a ≤ b)
Goal: Set.Ici a ∈ 𝓝ˢ (Set.Ioc b c)
-- Iic_mem_nhdsSet_Ioc
theorem Iic_mem_nhdsSet_Ioc (h : b < c) : Iic c ∈ 𝓝ˢ (Ioc a b) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c : α} (h : b < c)
Goal: Set.Iic c ∈ 𝓝ˢ (Set.Ioc a b)
-- Ioo_mem_nhdsSet_Ioc
theorem Ioo_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Ioc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a ≤ b) (h' : c < d)
Goal: Set.Ioo a d ∈ 𝓝ˢ (Set.Ioc b c)
-- Icc_mem_nhdsSet_Ioc
theorem Icc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Ioc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a ≤ b) (h' : c < d)
Goal: Set.Icc a d ∈ 𝓝ˢ (Set.Ioc b c)
-- Ioc_mem_nhdsSet_Ioc
theorem Ioc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Ioc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a ≤ b) (h' : c < d)
Goal: Set.Ioc a d ∈ 𝓝ˢ (Set.Ioc b c)
-- Ico_mem_nhdsSet_Ioc
theorem Ico_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Ioc b c) 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} (h : a ≤ b) (h' : c < d)
Goal: Set.Ico a d ∈ 𝓝ˢ (Set.Ioc b c)
-- hasBasis_nhdsSet_Iic_Iio
theorem hasBasis_nhdsSet_Iic_Iio (a : α) [h : Nonempty (Ioi a)] :
    HasBasis (𝓝ˢ (Iic a)) (a < ·) Iio 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] (a : α) [h : Nonempty ↑(Set.Ioi a)]
Goal: (𝓝ˢ (Set.Iic a)).HasBasis (fun x => a < x) Set.Iio
-- hasBasis_nhdsSet_Iic_Iic
theorem hasBasis_nhdsSet_Iic_Iic (a : α) [NeBot (𝓝[>] a)] :
    HasBasis (𝓝ˢ (Iic a)) (a < ·) Iic 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] (a : α) [(𝓝[>] a).NeBot]
Goal: (𝓝ˢ (Set.Iic a)).HasBasis (fun x => a < x) Set.Iic
-- Iic_mem_nhdsSet_Iic_iff
@[simp]
theorem Iic_mem_nhdsSet_Iic_iff {a b : α} [NeBot (𝓝[>] b)] : Iic a ∈ 𝓝ˢ (Iic b) ↔ b < a 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] {a b : α} [(𝓝[>] b).NeBot]
Goal: Set.Iic a ∈ 𝓝ˢ (Set.Iic b) ↔ b < a
-- hasBasis_nhdsSet_Ici_Ioi
theorem hasBasis_nhdsSet_Ici_Ioi (a : α) [Nonempty (Iio a)] :
    HasBasis (𝓝ˢ (Ici a)) (· < a) Ioi 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] (a : α) [Nonempty ↑(Set.Iio a)]
Goal: (𝓝ˢ (Set.Ici a)).HasBasis (fun x => x < a) Set.Ioi
-- hasBasis_nhdsSet_Ici_Ici
theorem hasBasis_nhdsSet_Ici_Ici (a : α) [NeBot (𝓝[<] a)] :
    HasBasis (𝓝ˢ (Ici a)) (· < a) Ici 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] (a : α) [(𝓝[<] a).NeBot]
Goal: (𝓝ˢ (Set.Ici a)).HasBasis (fun x => x < a) Set.Ici
-- Ici_mem_nhdsSet_Ici_iff
@[simp]
theorem Ici_mem_nhdsSet_Ici_iff {a b : α} [NeBot (𝓝[<] b)] : Ici a ∈ 𝓝ˢ (Ici b) ↔ a < b 
Arguments: {α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] {a b : α} [(𝓝[<] b).NeBot]
Goal: Set.Ici a ∈ 𝓝ˢ (Set.Ici b) ↔ a < b
-- StrictMonoOn.continuousWithinAt_right_of_exists_between
theorem StrictMonoOn.continuousWithinAt_right_of_exists_between {f : α → β} {s : Set α} {a : α}
    (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
    ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Set.Ioc (f a) b)
Goal: ContinuousWithinAt f (Set.Ici a) a
-- continuousWithinAt_right_of_monotoneOn_of_exists_between
theorem continuousWithinAt_right_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
    (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
    ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Set.Ioo (f a) b)
Goal: ContinuousWithinAt f (Set.Ici a) a
-- continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin
theorem continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
    {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a)
    (hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : closure (f '' s) ∈ 𝓝[≥] f a)
Goal: ContinuousWithinAt f (Set.Ici a) a
-- continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin
theorem continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β]
    {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a)
    (hfs : f '' s ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : f '' s ∈ 𝓝[≥] f a)
Goal: ContinuousWithinAt f (Set.Ici a) a
-- StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin
theorem StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
    {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a)
    (hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : closure (f '' s) ∈ 𝓝[≥] f a)
Goal: ContinuousWithinAt f (Set.Ici a) a
-- StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin
theorem StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
    {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : f '' s ∈ 𝓝[≥] f a) :
    ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : f '' s ∈ 𝓝[≥] f a)
Goal: ContinuousWithinAt f (Set.Ici a) a
-- StrictMonoOn.continuousWithinAt_right_of_surjOn
theorem StrictMonoOn.continuousWithinAt_right_of_surjOn {f : α → β} {s : Set α} {a : α}
    (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : SurjOn f s (Ioi (f a))) :
    ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : Set.SurjOn f s (Set.Ioi (f a)))
Goal: ContinuousWithinAt f (Set.Ici a) a
-- StrictMonoOn.continuousWithinAt_left_of_exists_between
theorem StrictMonoOn.continuousWithinAt_left_of_exists_between {f : α → β} {s : Set α} {a : α}
    (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) :
    ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Set.Ico b (f a))
Goal: ContinuousWithinAt f (Set.Iic a) a
-- continuousWithinAt_left_of_monotoneOn_of_exists_between
theorem continuousWithinAt_left_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
    (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) :
    ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Set.Ioo b (f a))
Goal: ContinuousWithinAt f (Set.Iic a) a
-- continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin
theorem continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
    {f : α → β} {s : Set α} {a : α} (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a)
    (hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : closure (f '' s) ∈ 𝓝[≤] f a)
Goal: ContinuousWithinAt f (Set.Iic a) a
-- continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin
theorem continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
    {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) :
    ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a)
Goal: ContinuousWithinAt f (Set.Iic a) a
-- StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin
theorem StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
    {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a)
    (hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : closure (f '' s) ∈ 𝓝[≤] f a)
Goal: ContinuousWithinAt f (Set.Iic a) a
-- StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin
theorem StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
    {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) :
    ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a)
Goal: ContinuousWithinAt f (Set.Iic a) a
-- StrictMonoOn.continuousWithinAt_left_of_surjOn
theorem StrictMonoOn.continuousWithinAt_left_of_surjOn {f : α → β} {s : Set α} {a : α}
    (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : SurjOn f s (Iio (f a))) :
    ContinuousWithinAt f (Iic a) a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : Set.SurjOn f s (Set.Iio (f a)))
Goal: ContinuousWithinAt f (Set.Iic a) a
-- StrictMonoOn.continuousAt_of_exists_between
theorem StrictMonoOn.continuousAt_of_exists_between {f : α → β} {s : Set α} {a : α}
    (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a))
    (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) : ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Set.Ico b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Set.Ioc (f a) b)
Goal: ContinuousAt f a
-- StrictMonoOn.continuousAt_of_closure_image_mem_nhds
theorem StrictMonoOn.continuousAt_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β}
    {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a)
    (hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs : closure (f '' s) ∈ 𝓝 (f a))
Goal: ContinuousAt f a
-- StrictMonoOn.continuousAt_of_image_mem_nhds
theorem StrictMonoOn.continuousAt_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α}
    {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) :
    ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a))
Goal: ContinuousAt f a
-- continuousAt_of_monotoneOn_of_exists_between
theorem continuousAt_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
    (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a))
    (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) : ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Set.Ioo b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Set.Ioo (f a) b)
Goal: ContinuousAt f a
-- continuousAt_of_monotoneOn_of_closure_image_mem_nhds
theorem continuousAt_of_monotoneOn_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β}
    {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a)
    (hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs : closure (f '' s) ∈ 𝓝 (f a))
Goal: ContinuousAt f a
-- continuousAt_of_monotoneOn_of_image_mem_nhds
theorem continuousAt_of_monotoneOn_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α}
    {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) : ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a))
Goal: ContinuousAt f a
-- Monotone.continuous_of_denseRange
theorem Monotone.continuous_of_denseRange [DenselyOrdered β] {f : α → β} (h_mono : Monotone f)
    (h_dense : DenseRange f) : Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} (h_mono : Monotone f) (h_dense : DenseRange f)
Goal: Continuous f
-- Monotone.continuous_of_surjective
theorem Monotone.continuous_of_surjective [DenselyOrdered β] {f : α → β} (h_mono : Monotone f)
    (h_surj : Function.Surjective f) : Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f)
Goal: Continuous f
-- OrderIso.continuous
protected theorem continuous (e : α ≃o β) : Continuous e 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [TopologicalSpace α] [TopologicalSpace β] [OrderTopology α] [OrderTopology β] (e : α ≃o β)
Goal: Continuous ⇑e
-- OrderIso.coe_toHomeomorph
@[simp]
theorem coe_toHomeomorph (e : α ≃o β) : ⇑e.toHomeomorph = e 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [TopologicalSpace α] [TopologicalSpace β] [OrderTopology α] [OrderTopology β] (e : α ≃o β)
Goal: ⇑e.toHomeomorph = ⇑e
-- OrderIso.coe_toHomeomorph_symm
@[simp]
theorem coe_toHomeomorph_symm (e : α ≃o β) : ⇑e.toHomeomorph.symm = e.symm 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [TopologicalSpace α] [TopologicalSpace β] [OrderTopology α] [OrderTopology β] (e : α ≃o β)
Goal: ⇑e.toHomeomorph.symm = ⇑e.symm
-- Filter.Tendsto.IccExtend
protected theorem Filter.Tendsto.IccExtend (f : γ → Icc a b → β) {la : Filter α} {lb : Filter β}
    {lc : Filter γ} (hf : Tendsto (↿f) (lc ×ˢ la.map (projIcc a b h)) lb) :
    Tendsto (↿(IccExtend h ∘ f)) (lc ×ˢ la) lb 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LinearOrder α] {a b : α} {h : a ≤ b} (f : γ → ↑(Set.Icc a b) → β) {la : Filter α} {lb : Filter β} {lc : Filter γ} (hf : Filter.Tendsto (↿f) (lc ×ˢ Filter.map (Set.projIcc a b h) la) lb)
Goal: Filter.Tendsto (↿(Set.IccExtend h ∘ f)) (lc ×ˢ la) lb
-- continuous_projIcc
@[continuity]
theorem continuous_projIcc : Continuous (projIcc a b h) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} {h : a ≤ b} [TopologicalSpace α] [OrderTopology α]
Goal: Continuous (Set.projIcc a b h)
-- quotientMap_projIcc
theorem quotientMap_projIcc : QuotientMap (projIcc a b h) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} {h : a ≤ b} [TopologicalSpace α] [OrderTopology α]
Goal: QuotientMap (Set.projIcc a b h)
-- continuous_IccExtend_iff
@[simp]
theorem continuous_IccExtend_iff {f : Icc a b → β} : Continuous (IccExtend h f) ↔ Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} {h : a ≤ b} [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : ↑(Set.Icc a b) → β}
Goal: Continuous (Set.IccExtend h f) ↔ Continuous f
-- Continuous.IccExtend
protected theorem Continuous.IccExtend {f : γ → Icc a b → β} {g : γ → α} (hf : Continuous ↿f)
    (hg : Continuous g) : Continuous fun a => IccExtend h (f a) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LinearOrder α] [TopologicalSpace γ] {a b : α} {h : a ≤ b} [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : γ → ↑(Set.Icc a b) → β} {g : γ → α} (hf : Continuous ↿f) (hg : Continuous g)
Goal: Continuous fun a_1 => Set.IccExtend h (f a_1) (g a_1)
-- Continuous.Icc_extend'
@[continuity]
protected theorem Continuous.Icc_extend' {f : Icc a b → β} (hf : Continuous f) :
    Continuous (IccExtend h f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} {h : a ≤ b} [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : ↑(Set.Icc a b) → β} (hf : Continuous f)
Goal: Continuous (Set.IccExtend h f)
-- ContinuousAt.IccExtend
theorem ContinuousAt.IccExtend {x : γ} (f : γ → Icc a b → β) {g : γ → α}
    (hf : ContinuousAt (↿f) (x, projIcc a b h (g x))) (hg : ContinuousAt g x) :
    ContinuousAt (fun a => IccExtend h (f a) (g a)) x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LinearOrder α] [TopologicalSpace γ] {a b : α} {h : a ≤ b} [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {x : γ} (f : γ → ↑(Set.Icc a b) → β) {g : γ → α} (hf : ContinuousAt (↿f) (x, Set.projIcc a b h (g x))) (hg : ContinuousAt g x)
Goal: ContinuousAt (fun a_1 => Set.IccExtend h (f a_1) (g a_1)) x
-- PriestleySpace.priestley
class PriestleySpace (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
  priestley {x y : α} : ¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U
Arguments: {α : Type u_2} [Preorder α] [TopologicalSpace α] [self : PriestleySpace α] {x y : α}
Goal: ¬x ≤ y → ∃ U, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U
-- exists_isClopen_upper_of_not_le
theorem exists_isClopen_upper_of_not_le :
    ¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [PriestleySpace α] {x y : α}
Goal: ¬x ≤ y → ∃ U, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U
-- exists_isClopen_lower_of_not_le
theorem exists_isClopen_lower_of_not_le (h : ¬x ≤ y) :
    ∃ U : Set α, IsClopen U ∧ IsLowerSet U ∧ x ∉ U ∧ y ∈ U 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [PriestleySpace α] {x y : α} (h : ¬x ≤ y)
Goal: ∃ U, IsClopen U ∧ IsLowerSet U ∧ x ∉ U ∧ y ∈ U
-- exists_isClopen_upper_or_lower_of_ne
theorem exists_isClopen_upper_or_lower_of_ne (h : x ≠ y) :
    ∃ U : Set α, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ y ∉ U 
Arguments: {α : Type u_1} [TopologicalSpace α] [PartialOrder α] [PriestleySpace α] {x y : α} (h : x ≠ y)
Goal: ∃ U, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ y ∉ U
-- LocallyBoundedMap.comap_cobounded_le'
structure LocallyBoundedMap (α β : Type*) [Bornology α] [Bornology β] where
    toFun : α → β
    comap_cobounded_le' : (cobounded β).comap toFun ≤ cobounded α
Arguments: {α : Type u_6} {β : Type u_7} [Bornology α] [Bornology β] (self : LocallyBoundedMap α β)
Goal: Filter.comap self.toFun (Bornology.cobounded β) ≤ Bornology.cobounded α