ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- absorbs_biUnion_finset
@[simp]
lemma _root_.absorbs_biUnion_finset {ι : Type*} {t : ι → Set α} {I : Finset ι} :
    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α} {ι : Type u_3} {t : ι → Set α} {I : Finset ι}
Goal: Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i)
-- Absorbs.add
protected lemma add [AddZeroClass E] [DistribSMul M E]
    (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) : Absorbs M (s₁ + s₂) (t₁ + t₂) 
Arguments: {M : Type u_1} {E : Type u_2} [Bornology M] {s₁ s₂ t₁ t₂ : Set E} [AddZeroClass E] [DistribSMul M E] (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂)
Goal: Absorbs M (s₁ + s₂) (t₁ + t₂)
-- Absorbs.zero
protected lemma zero [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 ∈ s) : Absorbs M s 0 
Arguments: {M : Type u_1} {E : Type u_2} [Bornology M] [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 ∈ s)
Goal: Absorbs M s 0
-- Absorbs.univ
@[simp]
protected lemma univ : Absorbs G₀ univ s 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {s : Set α}
Goal: Absorbs G₀ Set.univ s
-- absorbs_iff_eventually_cobounded_mapsTo
lemma _root_.absorbs_iff_eventually_cobounded_mapsTo :
    Absorbs G₀ s t ↔ ∀ᶠ c in cobounded G₀, MapsTo (c⁻¹ • ·) t s 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {s t : Set α}
Goal: Absorbs G₀ s t ↔ ∀ᶠ (c : G₀) in Bornology.cobounded G₀, Set.MapsTo (fun x => c⁻¹ • x) t s
-- Set.Finite.absorbs_sInter
lemma _root_.Set.Finite.absorbs_sInter (hS : S.Finite) :
    Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {t : Set α} {S : Set (Set α)} (hS : S.Finite)
Goal: Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t
-- absorbs_inter
@[simp]
lemma _root_.absorbs_inter : Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {s t u : Set α}
Goal: Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u
-- Absorbs.inter
protected lemma inter (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u) : Absorbs G₀ (s ∩ t) u 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {s t u : Set α} (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u)
Goal: Absorbs G₀ (s ∩ t) u
-- absorbs_iInter
@[simp]
lemma _root_.absorbs_iInter {ι : Sort*} [Finite ι] {s : ι → Set α} :
    Absorbs G₀ (⋂ i, s i) t ↔ ∀ i, Absorbs G₀ (s i) t 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {t : Set α} {ι : Sort u_3} [Finite ι] {s : ι → Set α}
Goal: Absorbs G₀ (⋂ i, s i) t ↔ ∀ (i : ι), Absorbs G₀ (s i) t
-- Set.Finite.absorbs_biInter
lemma _root_.Set.Finite.absorbs_biInter {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
    Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {t : Set α} {ι : Type u_3} {I : Set ι} (hI : I.Finite) {s : ι → Set α}
Goal: Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t
-- absorbs_zero_iff
@[simp]
lemma _root_.absorbs_zero_iff [NeBot (cobounded G₀)] {E : Type*} [AddMonoid E]
    [DistribMulAction G₀ E] {s : Set E} : Absorbs G₀ s 0 ↔ 0 ∈ s 
Arguments: {G₀ : Type u_1} [GroupWithZero G₀] [Bornology G₀] [(Bornology.cobounded G₀).NeBot] {E : Type u_3} [AddMonoid E] [DistribMulAction G₀ E] {s : Set E}
Goal: Absorbs G₀ s 0 ↔ 0 ∈ s
-- absorbs_neg_neg
@[simp]
lemma absorbs_neg_neg {s t : Set E} : Absorbs M (-s) (-t) ↔ Absorbs M s t 
Arguments: {M : Type u_1} {E : Type u_2} [Monoid M] [AddGroup E] [DistribMulAction M E] [Bornology M] {s t : Set E}
Goal: Absorbs M (-s) (-t) ↔ Absorbs M s t
-- Absorbs.sub
lemma Absorbs.sub {s₁ s₂ t₁ t₂ : Set E} (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) :
    Absorbs M (s₁ - s₂) (t₁ - t₂) 
Arguments: {M : Type u_1} {E : Type u_2} [Monoid M] [AddGroup E] [DistribMulAction M E] [Bornology M] {s₁ s₂ t₁ t₂ : Set E} (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂)
Goal: Absorbs M (s₁ - s₂) (t₁ - t₂)
-- Absorbent.mono
protected theorem mono (ht : Absorbent M s) (hsub : s ⊆ t) : Absorbent M t 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s t : Set α} (ht : Absorbent M s) (hsub : s ⊆ t)
Goal: Absorbent M t
-- absorbent_iff_forall_absorbs_singleton
theorem _root_.absorbent_iff_forall_absorbs_singleton : Absorbent M s ↔ ∀ x, Absorbs M s {x} 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α}
Goal: Absorbent M s ↔ ∀ (x : α), Absorbs M s {x}
-- Absorbent.absorbs
protected theorem absorbs (hs : Absorbent M s) {x : α} : Absorbs M s {x} 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s : Set α} (hs : Absorbent M s) {x : α}
Goal: Absorbs M s {x}
-- Absorbent.absorbs_finite
theorem absorbs_finite (hs : Absorbent M s) (ht : t.Finite) : Absorbs M s t 
Arguments: {M : Type u_1} {α : Type u_2} [Bornology M] [SMul M α] {s t : Set α} (hs : Absorbent M s) (ht : t.Finite)
Goal: Absorbs M s t
-- Absorbent.vadd_absorbs
theorem vadd_absorbs {M E : Type*} [Bornology M] [AddZeroClass E] [DistribSMul M E]
    {s₁ s₂ t : Set E} {x : E} (h₁ : Absorbent M s₁) (h₂ : Absorbs M s₂ t) :
    Absorbs M (s₁ + s₂) (x +ᵥ t) 
Arguments: {M : Type u_1} {E : Type u_2} [Bornology M] [AddZeroClass E] [DistribSMul M E] {s₁ s₂ t : Set E} {x : E} (h₁ : Absorbent M s₁) (h₂ : Absorbs M s₂ t)
Goal: Absorbs M (s₁ + s₂) (x +ᵥ t)
-- absorbent_univ
lemma absorbent_univ : Absorbent G₀ (univ : Set α) 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α]
Goal: Absorbent G₀ Set.univ
-- absorbent_iff_inv_smul
lemma absorbent_iff_inv_smul {s : Set α} :
    Absorbent G₀ s ↔ ∀ x, ∀ᶠ c in cobounded G₀, c⁻¹ • x ∈ s 
Arguments: {G₀ : Type u_1} {α : Type u_2} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α] {s : Set α}
Goal: Absorbent G₀ s ↔ ∀ (x : α), ∀ᶠ (c : G₀) in Bornology.cobounded G₀, c⁻¹ • x ∈ s
-- Absorbent.zero_mem
lemma Absorbent.zero_mem [NeBot (cobounded G₀)] [AddMonoid E] [DistribMulAction G₀ E]
    {s : Set E} (hs : Absorbent G₀ s) : (0 : E) ∈ s 
Arguments: {G₀ : Type u_1} {E : Type u_3} [GroupWithZero G₀] [Bornology G₀] [(Bornology.cobounded G₀).NeBot] [AddMonoid E] [DistribMulAction G₀ E] {s : Set E} (hs : Absorbent G₀ s)
Goal: 0 ∈ s
-- leftLim_eq_of_tendsto
theorem leftLim_eq_of_tendsto [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β]
    {f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : Tendsto f (𝓝[<] a) (𝓝 y)) :
    leftLim f a = y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace β] [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β] {f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : Filter.Tendsto f (𝓝[<] a) (𝓝 y))
Goal: Function.leftLim f a = y
-- leftLim_eq_of_eq_bot
theorem leftLim_eq_of_eq_bot [hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α}
    (h : 𝓝[<] a = ⊥) : leftLim f a = f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace β] [hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α} (h : 𝓝[<] a = ⊥)
Goal: Function.leftLim f a = f a
-- rightLim_eq_of_tendsto
theorem rightLim_eq_of_tendsto [TopologicalSpace α] [OrderTopology α] [T2Space β]
    {f : α → β} {a : α} {y : β} (h : 𝓝[>] a ≠ ⊥) (h' : Tendsto f (𝓝[>] a) (𝓝 y)) :
    Function.rightLim f a = y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace β] [TopologicalSpace α] [OrderTopology α] [T2Space β] {f : α → β} {a : α} {y : β} (h : 𝓝[>] a ≠ ⊥) (h' : Filter.Tendsto f (𝓝[>] a) (𝓝 y))
Goal: Function.rightLim f a = y
-- rightLim_eq_of_eq_bot
theorem rightLim_eq_of_eq_bot [TopologicalSpace α] [OrderTopology α] (f : α → β) {a : α}
    (h : 𝓝[>] a = ⊥) : rightLim f a = f a 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [TopologicalSpace β] [TopologicalSpace α] [OrderTopology α] (f : α → β) {a : α} (h : 𝓝[>] a = ⊥)
Goal: Function.rightLim f a = f a
-- Monotone.leftLim_eq_sSup
theorem leftLim_eq_sSup [TopologicalSpace α] [OrderTopology α] (h : 𝓝[<] x ≠ ⊥) :
    leftLim f x = sSup (f '' Iio x) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x : α} [TopologicalSpace α] [OrderTopology α] (h : 𝓝[<] x ≠ ⊥)
Goal: Function.leftLim f x = sSup (f '' Set.Iio x)
-- Monotone.rightLim_eq_sInf
theorem rightLim_eq_sInf [TopologicalSpace α] [OrderTopology α] (h : 𝓝[>] x ≠ ⊥) :
    rightLim f x = sInf (f '' Ioi x) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x : α} [TopologicalSpace α] [OrderTopology α] (h : 𝓝[>] x ≠ ⊥)
Goal: Function.rightLim f x = sInf (f '' Set.Ioi x)
-- Monotone.leftLim_le
theorem leftLim_le (h : x ≤ y) : leftLim f x ≤ f y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x y : α} (h : x ≤ y)
Goal: Function.leftLim f x ≤ f y
-- Monotone.le_leftLim
theorem le_leftLim (h : x < y) : f x ≤ leftLim f y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x y : α} (h : x < y)
Goal: f x ≤ Function.leftLim f y
-- Monotone.leftLim
@[mono]
protected theorem leftLim : Monotone (leftLim f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f)
Goal: Monotone (Function.leftLim f)
-- Monotone.le_rightLim
theorem le_rightLim (h : x ≤ y) : f x ≤ rightLim f y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x y : α} (h : x ≤ y)
Goal: f x ≤ Function.rightLim f y
-- Monotone.rightLim_le
theorem rightLim_le (h : x < y) : rightLim f x ≤ f y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x y : α} (h : x < y)
Goal: Function.rightLim f x ≤ f y
-- Monotone.rightLim
@[mono]
protected theorem rightLim : Monotone (rightLim f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f)
Goal: Monotone (Function.rightLim f)
-- Monotone.leftLim_le_rightLim
theorem leftLim_le_rightLim (h : x ≤ y) : leftLim f x ≤ rightLim f y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x y : α} (h : x ≤ y)
Goal: Function.leftLim f x ≤ Function.rightLim f y
-- Monotone.rightLim_le_leftLim
theorem rightLim_le_leftLim (h : x < y) : rightLim f x ≤ leftLim f y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x y : α} (h : x < y)
Goal: Function.rightLim f x ≤ Function.leftLim f y
-- Monotone.tendsto_leftLim
theorem tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[<] x) (𝓝 (Function.leftLim f x))
-- Monotone.tendsto_leftLim_within
theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[<] x) (𝓝[≤] Function.leftLim f x)
-- Monotone.tendsto_rightLim
theorem tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[>] x) (𝓝 (Function.rightLim f x))
-- Monotone.tendsto_rightLim_within
theorem tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≥] rightLim f x) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[>] x) (𝓝[≥] Function.rightLim f x)
-- Monotone.continuousWithinAt_Iio_iff_leftLim_eq
theorem continuousWithinAt_Iio_iff_leftLim_eq :
    ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x : α} [TopologicalSpace α] [OrderTopology α]
Goal: ContinuousWithinAt f (Set.Iio x) x ↔ Function.leftLim f x = f x
-- Monotone.continuousWithinAt_Ioi_iff_rightLim_eq
theorem continuousWithinAt_Ioi_iff_rightLim_eq :
    ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x : α} [TopologicalSpace α] [OrderTopology α]
Goal: ContinuousWithinAt f (Set.Ioi x) x ↔ Function.rightLim f x = f x
-- Monotone.continuousAt_iff_leftLim_eq_rightLim
theorem continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x : α} [TopologicalSpace α] [OrderTopology α]
Goal: ContinuousAt f x ↔ Function.leftLim f x = Function.rightLim f x
-- Monotone.countable_not_continuousWithinAt_Ioi
theorem countable_not_continuousWithinAt_Ioi [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousWithinAt f (Ioi x) x } 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) [TopologicalSpace α] [OrderTopology α] [SecondCountableTopology β]
Goal: {x | ¬ContinuousWithinAt f (Set.Ioi x) x}.Countable
-- Monotone.countable_not_continuousWithinAt_Iio
theorem countable_not_continuousWithinAt_Iio [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousWithinAt f (Iio x) x } 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) [TopologicalSpace α] [OrderTopology α] [SecondCountableTopology β]
Goal: {x | ¬ContinuousWithinAt f (Set.Iio x) x}.Countable
-- Monotone.countable_not_continuousAt
theorem countable_not_continuousAt [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousAt f x } 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) [TopologicalSpace α] [OrderTopology α] [SecondCountableTopology β]
Goal: {x | ¬ContinuousAt f x}.Countable
-- Antitone.le_leftLim
theorem le_leftLim (h : x ≤ y) : f y ≤ leftLim f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x y : α} (h : x ≤ y)
Goal: f y ≤ Function.leftLim f x
-- Antitone.leftLim_le
theorem leftLim_le (h : x < y) : leftLim f y ≤ f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x y : α} (h : x < y)
Goal: Function.leftLim f y ≤ f x
-- Antitone.leftLim
@[mono]
protected theorem leftLim : Antitone (leftLim f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f)
Goal: Antitone (Function.leftLim f)
-- Antitone.rightLim_le
theorem rightLim_le (h : x ≤ y) : rightLim f y ≤ f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x y : α} (h : x ≤ y)
Goal: Function.rightLim f y ≤ f x
-- Antitone.le_rightLim
theorem le_rightLim (h : x < y) : f y ≤ rightLim f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x y : α} (h : x < y)
Goal: f y ≤ Function.rightLim f x
-- Antitone.rightLim
@[mono]
protected theorem rightLim : Antitone (rightLim f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f)
Goal: Antitone (Function.rightLim f)
-- Antitone.rightLim_le_leftLim
theorem rightLim_le_leftLim (h : x ≤ y) : rightLim f y ≤ leftLim f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x y : α} (h : x ≤ y)
Goal: Function.rightLim f y ≤ Function.leftLim f x
-- Antitone.leftLim_le_rightLim
theorem leftLim_le_rightLim (h : x < y) : leftLim f y ≤ rightLim f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x y : α} (h : x < y)
Goal: Function.leftLim f y ≤ Function.rightLim f x
-- Antitone.tendsto_leftLim
theorem tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[<] x) (𝓝 (Function.leftLim f x))
-- Antitone.tendsto_leftLim_within
theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≥] leftLim f x) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[<] x) (𝓝[≥] Function.leftLim f x)
-- Antitone.tendsto_rightLim
theorem tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[>] x) (𝓝 (Function.rightLim f x))
-- Antitone.tendsto_rightLim_within
theorem tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≤] rightLim f x) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) [TopologicalSpace α] [OrderTopology α] (x : α)
Goal: Filter.Tendsto f (𝓝[>] x) (𝓝[≤] Function.rightLim f x)
-- Antitone.continuousWithinAt_Iio_iff_leftLim_eq
theorem continuousWithinAt_Iio_iff_leftLim_eq :
    ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x : α} [TopologicalSpace α] [OrderTopology α]
Goal: ContinuousWithinAt f (Set.Iio x) x ↔ Function.leftLim f x = f x
-- Antitone.continuousWithinAt_Ioi_iff_rightLim_eq
theorem continuousWithinAt_Ioi_iff_rightLim_eq :
    ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x : α} [TopologicalSpace α] [OrderTopology α]
Goal: ContinuousWithinAt f (Set.Ioi x) x ↔ Function.rightLim f x = f x
-- Antitone.continuousAt_iff_leftLim_eq_rightLim
theorem continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x : α} [TopologicalSpace α] [OrderTopology α]
Goal: ContinuousAt f x ↔ Function.leftLim f x = Function.rightLim f x
-- Antitone.countable_not_continuousAt
theorem countable_not_continuousAt [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousAt f x } 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) [TopologicalSpace α] [OrderTopology α] [SecondCountableTopology β]
Goal: {x | ¬ContinuousAt f x}.Countable
-- TopologicalSpace.Closeds.closed'
structure Closeds (α : Type*) [TopologicalSpace α] where
    carrier : Set α
  closed' : IsClosed carrier
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.Closeds α)
Goal: IsClosed self.carrier
-- TopologicalSpace.Closeds.closed
theorem closed (s : Closeds α) : IsClosed (s : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α)
Goal: IsClosed ↑s
-- TopologicalSpace.Closeds.ext
@[ext]
protected theorem ext {s t : Closeds α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_2} [TopologicalSpace α] {s t : TopologicalSpace.Closeds α} (h : ↑s = ↑t)
Goal: s = t
-- TopologicalSpace.Closeds.coe_mk
@[simp]
theorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : Set α) (h : IsClosed s)
Goal: ↑{ carrier := s, closed' := h } = s
-- TopologicalSpace.Closeds.coe_closure
@[simps]
protected def closure (s : Set α) : Closeds α 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : Set α)
Goal: ↑(TopologicalSpace.Closeds.closure s) = closure s
-- TopologicalSpace.Closeds.gc
theorem gc : GaloisConnection Closeds.closure ((↑) : Closeds α → Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: GaloisConnection TopologicalSpace.Closeds.closure SetLike.coe
-- TopologicalSpace.Closeds.coe_sup
@[simp, norm_cast]
theorem coe_sup (s t : Closeds α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Closeds α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- TopologicalSpace.Closeds.coe_inf
@[simp, norm_cast]
theorem coe_inf (s t : Closeds α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Closeds α)
Goal: ↑(s ⊓ t) = ↑s ∩ ↑t
-- TopologicalSpace.Closeds.coe_top
@[simp, norm_cast]
theorem coe_top : (↑(⊤ : Closeds α) : Set α) = univ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: ↑⊤ = Set.univ
-- TopologicalSpace.Closeds.coe_eq_univ
@[simp, norm_cast]
theorem coe_eq_univ {s : Closeds α} : (s : Set α) = univ ↔ s = ⊤ 
Arguments: {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Closeds α}
Goal: ↑s = Set.univ ↔ s = ⊤
-- TopologicalSpace.Closeds.coe_bot
@[simp, norm_cast]
theorem coe_bot : (↑(⊥ : Closeds α) : Set α) = ∅ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: ↑⊥ = ∅
-- TopologicalSpace.Closeds.coe_eq_empty
@[simp, norm_cast]
theorem coe_eq_empty {s : Closeds α} : (s : Set α) = ∅ ↔ s = ⊥ 
Arguments: {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Closeds α}
Goal: ↑s = ∅ ↔ s = ⊥
-- TopologicalSpace.Closeds.coe_nonempty
theorem coe_nonempty {s : Closeds α} : (s : Set α).Nonempty ↔ s ≠ ⊥ 
Arguments: {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Closeds α}
Goal: (↑s).Nonempty ↔ s ≠ ⊥
-- TopologicalSpace.Closeds.coe_sInf
@[simp, norm_cast]
theorem coe_sInf {S : Set (Closeds α)} : (↑(sInf S) : Set α) = ⋂ i ∈ S, ↑i 
Arguments: {α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Closeds α)}
Goal: ↑(sInf S) = ⋂ i ∈ S, ↑i
-- TopologicalSpace.Closeds.coe_sSup
@[simp]
lemma coe_sSup {S : Set (Closeds α)} : ((sSup S : Closeds α) : Set α) =
    closure (⋃₀ ((↑) '' S)) 
Arguments: {α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Closeds α)}
Goal: ↑(sSup S) = closure (⋃₀ (SetLike.coe '' S))