ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- eventually_lt_nhds
theorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (hab : a < b)
Goal: ∀ᶠ (x : α) in 𝓝 a, x < b
-- Iic_mem_nhds
theorem Iic_mem_nhds (h : a < b) : Iic b ∈ 𝓝 a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (h : a < b)
Goal: Set.Iic b ∈ 𝓝 a
-- eventually_le_nhds
theorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (hab : a < b)
Goal: ∀ᶠ (x : α) in 𝓝 a, x ≤ b
-- eventually_lt_of_tendsto_lt
theorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
    (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u 
Arguments: {α : Type u} {γ : Type w} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u) (h : Filter.Tendsto f l (𝓝 v))
Goal: ∀ᶠ (a : γ) in l, f a < u
-- eventually_le_of_tendsto_lt
theorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
    (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u 
Arguments: {α : Type u} {γ : Type w} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u) (h : Filter.Tendsto f l (𝓝 v))
Goal: ∀ᶠ (a : γ) in l, f a ≤ u
-- Dense.exists_lt
protected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :
    ∃ y ∈ s, y < x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α)
Goal: ∃ y ∈ s, y < x
-- Dense.exists_le
protected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :
    ∃ y ∈ s, y ≤ x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α)
Goal: ∃ y ∈ s, y ≤ x
-- Dense.exists_le'
theorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :
    ∃ y ∈ s, y ≤ x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {s : Set α} (hs : Dense s) (hbot : ∀ (x : α), IsBot x → x ∈ s) (x : α)
Goal: ∃ y ∈ s, y ≤ x
-- Ioo_mem_nhdsWithin_Ioi
theorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ico a c)
Goal: Set.Ioo a c ∈ 𝓝[>] b
-- Ioo_mem_nhdsWithin_Ioi'
theorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (H : a < b)
Goal: Set.Ioo a b ∈ 𝓝[>] a
-- CovBy.nhdsWithin_Ioi
theorem CovBy.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (h : a ⋖ b)
Goal: 𝓝[>] a = ⊥
-- Ioc_mem_nhdsWithin_Ioi
theorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ico a c)
Goal: Set.Ioc a c ∈ 𝓝[>] b
-- Ioc_mem_nhdsWithin_Ioi'
theorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (H : a < b)
Goal: Set.Ioc a b ∈ 𝓝[>] a
-- Ico_mem_nhdsWithin_Ioi
theorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ico a c)
Goal: Set.Ico a c ∈ 𝓝[>] b
-- Ico_mem_nhdsWithin_Ioi'
theorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (H : a < b)
Goal: Set.Ico a b ∈ 𝓝[>] a
-- Icc_mem_nhdsWithin_Ioi
theorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ico a c)
Goal: Set.Icc a c ∈ 𝓝[>] b
-- Icc_mem_nhdsWithin_Ioi'
theorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (H : a < b)
Goal: Set.Icc a b ∈ 𝓝[>] a
-- nhdsWithin_Ioc_eq_nhdsWithin_Ioi
@[simp]
theorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Ioc a b] a = 𝓝[>] a
-- nhdsWithin_Ioo_eq_nhdsWithin_Ioi
@[simp]
theorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Ioo a b] a = 𝓝[>] a
-- continuousWithinAt_Ioc_iff_Ioi
@[simp]
theorem continuousWithinAt_Ioc_iff_Ioi (h : a < b) :
    ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Ioc a b) a ↔ ContinuousWithinAt f (Set.Ioi a) a
-- continuousWithinAt_Ioo_iff_Ioi
@[simp]
theorem continuousWithinAt_Ioo_iff_Ioi (h : a < b) :
    ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Ioo a b) a ↔ ContinuousWithinAt f (Set.Ioi a) a
-- Ico_mem_nhdsWithin_Ici'
theorem Ico_mem_nhdsWithin_Ici' (H : a < b) : Ico a b ∈ 𝓝[≥] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (H : a < b)
Goal: Set.Ico a b ∈ 𝓝[≥] a
-- Ico_mem_nhdsWithin_Ici
theorem Ico_mem_nhdsWithin_Ici (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ico a c)
Goal: Set.Ico a c ∈ 𝓝[≥] b
-- Ioo_mem_nhdsWithin_Ici
theorem Ioo_mem_nhdsWithin_Ici (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ioo a c)
Goal: Set.Ioo a c ∈ 𝓝[≥] b
-- Ioc_mem_nhdsWithin_Ici
theorem Ioc_mem_nhdsWithin_Ici (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ioo a c)
Goal: Set.Ioc a c ∈ 𝓝[≥] b
-- Icc_mem_nhdsWithin_Ici
theorem Icc_mem_nhdsWithin_Ici (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b c : α} (H : b ∈ Set.Ico a c)
Goal: Set.Icc a c ∈ 𝓝[≥] b
-- Icc_mem_nhdsWithin_Ici'
theorem Icc_mem_nhdsWithin_Ici' (H : a < b) : Icc a b ∈ 𝓝[≥] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (H : a < b)
Goal: Set.Icc a b ∈ 𝓝[≥] a
-- nhdsWithin_Icc_eq_nhdsWithin_Ici
@[simp]
theorem nhdsWithin_Icc_eq_nhdsWithin_Ici (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Icc a b] a = 𝓝[≥] a
-- nhdsWithin_Ico_eq_nhdsWithin_Ici
@[simp]
theorem nhdsWithin_Ico_eq_nhdsWithin_Ici (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {a b : α} (h : a < b)
Goal: 𝓝[Set.Ico a b] a = 𝓝[≥] a
-- continuousWithinAt_Icc_iff_Ici
@[simp]
theorem continuousWithinAt_Icc_iff_Ici (h : a < b) :
    ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Icc a b) a ↔ ContinuousWithinAt f (Set.Ici a) a
-- continuousWithinAt_Ico_iff_Ici
@[simp]
theorem continuousWithinAt_Ico_iff_Ici (h : a < b) :
    ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b)
Goal: ContinuousWithinAt f (Set.Ico a b) a ↔ ContinuousWithinAt f (Set.Ici a) a
-- isClosed_le_prod
theorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]
Goal: IsClosed {p | p.1 ≤ p.2}
-- isClosed_le
theorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
    IsClosed { b | f b ≤ g b } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g)
Goal: IsClosed {b | f b ≤ g b}
-- isClosed_Icc
theorem isClosed_Icc {a b : α} : IsClosed (Icc a b) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] {a b : α}
Goal: IsClosed (Set.Icc a b)
-- closure_Icc
@[simp]
theorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] (a b : α)
Goal: closure (Set.Icc a b) = Set.Icc a b
-- le_of_tendsto_of_tendsto
theorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]
    (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] {f g : β → α} {b : Filter β} {a₁ a₂ : α} [b.NeBot] (hf : Filter.Tendsto f b (𝓝 a₁)) (hg : Filter.Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g)
Goal: a₁ ≤ a₂
-- le_of_tendsto_of_tendsto'
theorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]
    (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] {f g : β → α} {b : Filter β} {a₁ a₂ : α} [b.NeBot] (hf : Filter.Tendsto f b (𝓝 a₁)) (hg : Filter.Tendsto g b (𝓝 a₂)) (h : ∀ (x : β), f x ≤ g x)
Goal: a₁ ≤ a₂
-- closure_le_eq
@[simp]
theorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
    closure { b | f b ≤ g b } = { b | f b ≤ g b } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g)
Goal: closure {b | f b ≤ g b} = {b | f b ≤ g b}
-- closure_lt_subset_le
theorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
    (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g)
Goal: closure {b | f b < g b} ⊆ {b | f b ≤ g b}
-- ContinuousWithinAt.closure_le
theorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}
    (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
    (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β} (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) (h : ∀ y ∈ s, f y ≤ g y)
Goal: f x ≤ g x
-- IsClosed.isClosed_le
theorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)
    (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s) (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: IsClosed {x | x ∈ s ∧ f x ≤ g x}
-- le_on_closure
theorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)
    (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :
    f x ≤ g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x) (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x : β⦄ (hx : x ∈ closure s)
Goal: f x ≤ g x
-- IsClosed.epigraph
theorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)
    (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s) (hf : ContinuousOn f s)
Goal: IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ p.2}
-- IsClosed.hypograph
theorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)
    (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s) (hf : ContinuousOn f s)
Goal: IsClosed {p | p.1 ∈ s ∧ p.2 ≤ f p.1}
-- isOpen_lt
theorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
    IsOpen { b | f b < g b } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g)
Goal: IsOpen {b | f b < g b}
-- isOpen_lt_prod
theorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]
Goal: IsOpen {p | p.1 < p.2}
-- isOpen_Ioo
theorem isOpen_Ioo : IsOpen (Ioo a b) 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: IsOpen (Set.Ioo a b)
-- interior_Ioo
@[simp]
theorem interior_Ioo : interior (Ioo a b) = Ioo a b 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: interior (Set.Ioo a b) = Set.Ioo a b
-- Ioo_subset_closure_interior
theorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: Set.Ioo a b ⊆ closure (interior (Set.Ioo a b))
-- Ioo_mem_nhds
theorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {a b x : α} (ha : a < x) (hb : x < b)
Goal: Set.Ioo a b ∈ 𝓝 x
-- Ioc_mem_nhds
theorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {a b x : α} (ha : a < x) (hb : x < b)
Goal: Set.Ioc a b ∈ 𝓝 x
-- Ico_mem_nhds
theorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {a b x : α} (ha : a < x) (hb : x < b)
Goal: Set.Ico a b ∈ 𝓝 x
-- Icc_mem_nhds
theorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {a b x : α} (ha : a < x) (hb : x < b)
Goal: Set.Icc a b ∈ 𝓝 x
-- lt_subset_interior_le
theorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :
    { b | f b < g b } ⊆ interior { b | f b ≤ g b } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] (hf : Continuous f) (hg : Continuous g)
Goal: {b | f b < g b} ⊆ interior {b | f b ≤ g b}
-- frontier_le_subset_eq
theorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :
    frontier { b | f b ≤ g b } ⊆ { b | f b = g b } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] (hf : Continuous f) (hg : Continuous g)
Goal: frontier {b | f b ≤ g b} ⊆ {b | f b = g b}
-- frontier_Iic_subset
theorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] (a : α)
Goal: frontier (Set.Iic a) ⊆ {a}
-- frontier_Ici_subset
theorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] (a : α)
Goal: frontier (Set.Ici a) ⊆ {a}
-- frontier_lt_subset_eq
theorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :
    frontier { b | f b < g b } ⊆ { b | f b = g b } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] (hf : Continuous f) (hg : Continuous g)
Goal: frontier {b | f b < g b} ⊆ {b | f b = g b}
-- continuous_if_le
theorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}
    (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })
    (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :
    Continuous fun x => if f x ≤ g x then f' x else g' x 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] [TopologicalSpace γ] [(x : β) → Decidable (f x ≤ g x)] {f' g' : β → γ} (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' {x | f x ≤ g x}) (hg' : ContinuousOn g' {x | g x ≤ f x}) (hfg : ∀ (x : β), f x = g x → f' x = g' x)
Goal: Continuous fun x => if f x ≤ g x then f' x else g' x
-- Continuous.if_le
theorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}
    (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)
    (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] [TopologicalSpace γ] [(x : β) → Decidable (f x ≤ g x)] {f' g' : β → γ} (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g) (hfg : ∀ (x : β), f x = g x → f' x = g' x)
Goal: Continuous fun x => if f x ≤ g x then f' x else g' x
-- Filter.Tendsto.eventually_lt
theorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))
    (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x 
Arguments: {α : Type u} {γ : Type w} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {l : Filter γ} {f g : γ → α} {y z : α} (hf : Filter.Tendsto f l (𝓝 y)) (hg : Filter.Tendsto g l (𝓝 z)) (hyz : y < z)
Goal: ∀ᶠ (x : γ) in l, f x < g x
-- ContinuousAt.eventually_lt
nonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)
    (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀) (hfg : f x₀ < g x₀)
Goal: ∀ᶠ (x : β) in 𝓝 x₀, f x < g x
-- Continuous.min
@[continuity, fun_prop]
protected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :
    Continuous fun b => min (f b) (g b) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun b => min (f b) (g b)
-- Continuous.max
@[continuity, fun_prop]
protected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :
    Continuous fun b => max (f b) (g b) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun b => max (f b) (g b)
-- continuous_min
theorem continuous_min : Continuous fun p : α × α => min p.1 p.2 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]
Goal: Continuous fun p => min p.1 p.2
-- continuous_max
theorem continuous_max : Continuous fun p : α × α => max p.1 p.2 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]
Goal: Continuous fun p => max p.1 p.2
-- Filter.Tendsto.max
protected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))
    (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} {b : Filter β} {a₁ a₂ : α} (hf : Filter.Tendsto f b (𝓝 a₁)) (hg : Filter.Tendsto g b (𝓝 a₂))
Goal: Filter.Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂))
-- Filter.Tendsto.min
protected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))
    (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} {b : Filter β} {a₁ a₂ : α} (hf : Filter.Tendsto f b (𝓝 a₁)) (hg : Filter.Tendsto g b (𝓝 a₂))
Goal: Filter.Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂))
-- Filter.Tendsto.max_right
protected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
    Tendsto (fun i => max a (f i)) l (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun i => max a (f i)) l (𝓝 a)
-- Filter.Tendsto.max_left
protected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
    Tendsto (fun i => max (f i) a) l (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun i => max (f i) a) l (𝓝 a)
-- Filter.tendsto_nhds_max_right
theorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :
    Tendsto (fun i => max a (f i)) l (𝓝[>] a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝[>] a))
Goal: Filter.Tendsto (fun i => max a (f i)) l (𝓝[>] a)
-- Filter.tendsto_nhds_max_left
theorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :
    Tendsto (fun i => max (f i) a) l (𝓝[>] a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝[>] a))
Goal: Filter.Tendsto (fun i => max (f i) a) l (𝓝[>] a)
-- Filter.Tendsto.min_right
theorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
    Tendsto (fun i => min a (f i)) l (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun i => min a (f i)) l (𝓝 a)
-- Filter.Tendsto.min_left
theorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
    Tendsto (fun i => min (f i) a) l (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun i => min (f i) a) l (𝓝 a)
-- Filter.tendsto_nhds_min_right
theorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :
    Tendsto (fun i => min a (f i)) l (𝓝[<] a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f : β → α} {l : Filter β} {a : α} (h : Filter.Tendsto f l (𝓝[<] a))
Goal: Filter.Tendsto (fun i => min a (f i)) l (𝓝[<] a)