ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.iUnion_Icc_left
@[simp]
theorem iUnion_Icc_left (b : α) : ⋃ a, Icc a b = Iic b 
Arguments: {α : Type v} [Preorder α] (b : α)
Goal: ⋃ a, Set.Icc a b = Set.Iic b
-- Set.iUnion_Ico_left
@[simp]
theorem iUnion_Ico_left (b : α) : ⋃ a, Ico a b = Iio b 
Arguments: {α : Type v} [Preorder α] (b : α)
Goal: ⋃ a, Set.Ico a b = Set.Iio b
-- Set.iUnion_Iio
@[simp]
theorem iUnion_Iio [NoMaxOrder α] : ⋃ a : α, Iio a = univ 
Arguments: {α : Type v} [Preorder α] [NoMaxOrder α]
Goal: ⋃ a, Set.Iio a = Set.univ
-- Set.iUnion_Ioi
@[simp]
theorem iUnion_Ioi [NoMinOrder α] : ⋃ a : α, Ioi a = univ 
Arguments: {α : Type v} [Preorder α] [NoMinOrder α]
Goal: ⋃ a, Set.Ioi a = Set.univ
-- Set.iUnion_Ico_right
@[simp]
theorem iUnion_Ico_right [NoMaxOrder α] (a : α) : ⋃ b, Ico a b = Ici a 
Arguments: {α : Type v} [Preorder α] [NoMaxOrder α] (a : α)
Goal: ⋃ b, Set.Ico a b = Set.Ici a
-- Set.iUnion_Ioo_right
@[simp]
theorem iUnion_Ioo_right [NoMaxOrder α] (a : α) : ⋃ b, Ioo a b = Ioi a 
Arguments: {α : Type v} [Preorder α] [NoMaxOrder α] (a : α)
Goal: ⋃ b, Set.Ioo a b = Set.Ioi a
-- Set.iUnion_Ioc_left
@[simp]
theorem iUnion_Ioc_left [NoMinOrder α] (b : α) : ⋃ a, Ioc a b = Iic b 
Arguments: {α : Type v} [Preorder α] [NoMinOrder α] (b : α)
Goal: ⋃ a, Set.Ioc a b = Set.Iic b
-- Set.iUnion_Ioo_left
@[simp]
theorem iUnion_Ioo_left [NoMinOrder α] (b : α) : ⋃ a, Ioo a b = Iio b 
Arguments: {α : Type v} [Preorder α] [NoMinOrder α] (b : α)
Goal: ⋃ a, Set.Ioo a b = Set.Iio b
-- Set.Ico_disjoint_Ico
@[simp]
theorem Ico_disjoint_Ico : Disjoint (Ico a₁ a₂) (Ico b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ 
Arguments: {α : Type v} [LinearOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: Disjoint (Set.Ico a₁ a₂) (Set.Ico b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁
-- Set.Ioc_disjoint_Ioc
@[simp]
theorem Ioc_disjoint_Ioc : Disjoint (Ioc a₁ a₂) (Ioc b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ 
Arguments: {α : Type v} [LinearOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: Disjoint (Set.Ioc a₁ a₂) (Set.Ioc b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁
-- Set.Ioo_disjoint_Ioo
@[simp]
theorem Ioo_disjoint_Ioo [DenselyOrdered α] :
    Disjoint (Set.Ioo a₁ a₂) (Set.Ioo b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ 
Arguments: {α : Type v} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} [DenselyOrdered α]
Goal: Disjoint (Set.Ioo a₁ a₂) (Set.Ioo b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁
-- Set.eq_of_Ico_disjoint
theorem eq_of_Ico_disjoint {x₁ x₂ y₁ y₂ : α} (h : Disjoint (Ico x₁ x₂) (Ico y₁ y₂)) (hx : x₁ < x₂)
    (h2 : x₂ ∈ Ico y₁ y₂) : y₁ = x₂ 
Arguments: {α : Type v} [LinearOrder α] {x₁ x₂ y₁ y₂ : α} (h : Disjoint (Set.Ico x₁ x₂) (Set.Ico y₁ y₂)) (hx : x₁ < x₂) (h2 : x₂ ∈ Set.Ico y₁ y₂)
Goal: y₁ = x₂
-- Set.iUnion_Ico_eq_Iio_self_iff
@[simp]
theorem iUnion_Ico_eq_Iio_self_iff {f : ι → α} {a : α} :
    ⋃ i, Ico (f i) a = Iio a ↔ ∀ x < a, ∃ i, f i ≤ x 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {f : ι → α} {a : α}
Goal: ⋃ i, Set.Ico (f i) a = Set.Iio a ↔ ∀ x < a, ∃ i, f i ≤ x
-- Set.iUnion_Ioc_eq_Ioi_self_iff
@[simp]
theorem iUnion_Ioc_eq_Ioi_self_iff {f : ι → α} {a : α} :
    ⋃ i, Ioc a (f i) = Ioi a ↔ ∀ x, a < x → ∃ i, x ≤ f i 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {f : ι → α} {a : α}
Goal: ⋃ i, Set.Ioc a (f i) = Set.Ioi a ↔ ∀ (x : α), a < x → ∃ i, x ≤ f i
-- Set.biUnion_Ico_eq_Iio_self_iff
@[simp]
theorem biUnion_Ico_eq_Iio_self_iff {p : ι → Prop} {f : ∀ i, p i → α} {a : α} :
    ⋃ (i) (hi : p i), Ico (f i hi) a = Iio a ↔ ∀ x < a, ∃ i hi, f i hi ≤ x 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {p : ι → Prop} {f : (i : ι) → p i → α} {a : α}
Goal: ⋃ i, ⋃ (hi : p i), Set.Ico (f i hi) a = Set.Iio a ↔ ∀ x < a, ∃ i, ∃ (hi : p i), f i hi ≤ x
-- Set.biUnion_Ioc_eq_Ioi_self_iff
@[simp]
theorem biUnion_Ioc_eq_Ioi_self_iff {p : ι → Prop} {f : ∀ i, p i → α} {a : α} :
    ⋃ (i) (hi : p i), Ioc a (f i hi) = Ioi a ↔ ∀ x, a < x → ∃ i hi, x ≤ f i hi 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {p : ι → Prop} {f : (i : ι) → p i → α} {a : α}
Goal: ⋃ i, ⋃ (hi : p i), Set.Ioc a (f i hi) = Set.Ioi a ↔ ∀ (x : α), a < x → ∃ i, ∃ (hi : p i), x ≤ f i hi
-- IsGLB.biUnion_Ioi_eq
theorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : ⋃ x ∈ s, Ioi x = Ioi a 
Arguments: {α : Type v} [LinearOrder α] {s : Set α} {a : α} (h : IsGLB s a)
Goal: ⋃ x ∈ s, Set.Ioi x = Set.Ioi a
-- IsGLB.iUnion_Ioi_eq
theorem IsGLB.iUnion_Ioi_eq (h : IsGLB (range f) a) : ⋃ x, Ioi (f x) = Ioi a 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {a : α} {f : ι → α} (h : IsGLB (Set.range f) a)
Goal: ⋃ x, Set.Ioi (f x) = Set.Ioi a
-- IsLUB.biUnion_Iio_eq
theorem IsLUB.biUnion_Iio_eq (h : IsLUB s a) : ⋃ x ∈ s, Iio x = Iio a 
Arguments: {α : Type v} [LinearOrder α] {s : Set α} {a : α} (h : IsLUB s a)
Goal: ⋃ x ∈ s, Set.Iio x = Set.Iio a
-- IsLUB.iUnion_Iio_eq
theorem IsLUB.iUnion_Iio_eq (h : IsLUB (range f) a) : ⋃ x, Iio (f x) = Iio a 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {a : α} {f : ι → α} (h : IsLUB (Set.range f) a)
Goal: ⋃ x, Set.Iio (f x) = Set.Iio a
-- IsGLB.biUnion_Ici_eq_Ioi
theorem IsGLB.biUnion_Ici_eq_Ioi (a_glb : IsGLB s a) (a_not_mem : a ∉ s) :
    ⋃ x ∈ s, Ici x = Ioi a 
Arguments: {α : Type v} [LinearOrder α] {s : Set α} {a : α} (a_glb : IsGLB s a) (a_not_mem : a ∉ s)
Goal: ⋃ x ∈ s, Set.Ici x = Set.Ioi a
-- IsGLB.biUnion_Ici_eq_Ici
theorem IsGLB.biUnion_Ici_eq_Ici (a_glb : IsGLB s a) (a_mem : a ∈ s) :
    ⋃ x ∈ s, Ici x = Ici a 
Arguments: {α : Type v} [LinearOrder α] {s : Set α} {a : α} (a_glb : IsGLB s a) (a_mem : a ∈ s)
Goal: ⋃ x ∈ s, Set.Ici x = Set.Ici a
-- IsLUB.biUnion_Iic_eq_Iio
theorem IsLUB.biUnion_Iic_eq_Iio (a_lub : IsLUB s a) (a_not_mem : a ∉ s) :
    ⋃ x ∈ s, Iic x = Iio a 
Arguments: {α : Type v} [LinearOrder α] {s : Set α} {a : α} (a_lub : IsLUB s a) (a_not_mem : a ∉ s)
Goal: ⋃ x ∈ s, Set.Iic x = Set.Iio a
-- IsLUB.biUnion_Iic_eq_Iic
theorem IsLUB.biUnion_Iic_eq_Iic (a_lub : IsLUB s a) (a_mem : a ∈ s) : ⋃ x ∈ s, Iic x = Iic a 
Arguments: {α : Type v} [LinearOrder α] {s : Set α} {a : α} (a_lub : IsLUB s a) (a_mem : a ∈ s)
Goal: ⋃ x ∈ s, Set.Iic x = Set.Iic a
-- iUnion_Ici_eq_Ioi_iInf
theorem iUnion_Ici_eq_Ioi_iInf {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (no_least_elem : ⨅ i, f i ∉ range f) : ⋃ i : ι, Ici (f i) = Ioi (⨅ i, f i) 
Arguments: {ι : Sort u} {R : Type u_1} [CompleteLinearOrder R] {f : ι → R} (no_least_elem : ⨅ i, f i ∉ Set.range f)
Goal: ⋃ i, Set.Ici (f i) = Set.Ioi (⨅ i, f i)
-- iUnion_Iic_eq_Iio_iSup
theorem iUnion_Iic_eq_Iio_iSup {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (no_greatest_elem : (⨆ i, f i) ∉ range f) : ⋃ i : ι, Iic (f i) = Iio (⨆ i, f i) 
Arguments: {ι : Sort u} {R : Type u_1} [CompleteLinearOrder R] {f : ι → R} (no_greatest_elem : ⨆ i, f i ∉ Set.range f)
Goal: ⋃ i, Set.Iic (f i) = Set.Iio (⨆ i, f i)
-- iUnion_Ici_eq_Ici_iInf
theorem iUnion_Ici_eq_Ici_iInf {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (has_least_elem : (⨅ i, f i) ∈ range f) : ⋃ i : ι, Ici (f i) = Ici (⨅ i, f i) 
Arguments: {ι : Sort u} {R : Type u_1} [CompleteLinearOrder R] {f : ι → R} (has_least_elem : ⨅ i, f i ∈ Set.range f)
Goal: ⋃ i, Set.Ici (f i) = Set.Ici (⨅ i, f i)
-- iUnion_Iic_eq_Iic_iSup
theorem iUnion_Iic_eq_Iic_iSup {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (has_greatest_elem : (⨆ i, f i) ∈ range f) : ⋃ i : ι, Iic (f i) = Iic (⨆ i, f i) 
Arguments: {ι : Sort u} {R : Type u_1} [CompleteLinearOrder R] {f : ι → R} (has_greatest_elem : ⨆ i, f i ∈ Set.range f)
Goal: ⋃ i, Set.Iic (f i) = Set.Iic (⨆ i, f i)
-- iUnion_Iio_eq_univ_iff
theorem iUnion_Iio_eq_univ_iff : ⋃ i, Iio (f i) = univ ↔ (¬ BddAbove (range f)) 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {f : ι → α}
Goal: ⋃ i, Set.Iio (f i) = Set.univ ↔ ¬BddAbove (Set.range f)
-- iUnion_Iic_of_not_bddAbove_range
theorem iUnion_Iic_of_not_bddAbove_range (hf : ¬ BddAbove (range f)) : ⋃ i, Iic (f i) = univ 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {f : ι → α} (hf : ¬BddAbove (Set.range f))
Goal: ⋃ i, Set.Iic (f i) = Set.univ
-- iInter_Iic_eq_empty_iff
theorem iInter_Iic_eq_empty_iff : ⋂ i, Iic (f i) = ∅ ↔ ¬ BddBelow (range f) 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {f : ι → α}
Goal: ⋂ i, Set.Iic (f i) = ∅ ↔ ¬BddBelow (Set.range f)
-- iInter_Iio_of_not_bddBelow_range
theorem iInter_Iio_of_not_bddBelow_range (hf : ¬ BddBelow (range f)) : ⋂ i, Iio (f i) = ∅ 
Arguments: {ι : Sort u} {α : Type v} [LinearOrder α] {f : ι → α} (hf : ¬BddBelow (Set.range f))
Goal: ⋂ i, Set.Iio (f i) = ∅
-- WithTop.preimage_coe_top
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) 
Arguments: {α : Type u_1}
Goal: WithTop.some ⁻¹' {⊤} = ∅
-- WithTop.range_coe
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ 
Arguments: {α : Type u_1} [Preorder α]
Goal: Set.range WithTop.some = Set.Iio ⊤
-- WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some ⁻¹' Set.Ioi ↑a = Set.Ioi a
-- WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some ⁻¹' Set.Ici ↑a = Set.Ici a
-- WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some ⁻¹' Set.Iio ↑a = Set.Iio a
-- WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some ⁻¹' Set.Iic ↑a = Set.Iic a
-- WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some ⁻¹' Set.Icc ↑a ↑b = Set.Icc a b
-- WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some ⁻¹' Set.Ico ↑a ↑b = Set.Ico a b
-- WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some ⁻¹' Set.Ioc ↑a ↑b = Set.Ioc a b
-- WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some ⁻¹' Set.Ioo ↑a ↑b = Set.Ioo a b
-- WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ 
Arguments: {α : Type u_1} [Preorder α]
Goal: WithTop.some ⁻¹' Set.Iio ⊤ = Set.univ
-- WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some ⁻¹' Set.Ico ↑a ⊤ = Set.Ici a
-- WithTop.preimage_coe_Ioo_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some ⁻¹' Set.Ioo ↑a ⊤ = Set.Ioi a
-- WithTop.image_coe_Ioi
theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some '' Set.Ioi a = Set.Ioo ↑a ⊤
-- WithTop.image_coe_Ici
theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some '' Set.Ici a = Set.Ico ↑a ⊤
-- WithTop.image_coe_Iio
theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some '' Set.Iio a = Set.Iio ↑a
-- WithTop.image_coe_Iic
theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithTop.some '' Set.Iic a = Set.Iic ↑a
-- WithTop.image_coe_Icc
theorem image_coe_Icc : (some : α → WithTop α) '' Icc a b = Icc (a : WithTop α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some '' Set.Icc a b = Set.Icc ↑a ↑b
-- WithTop.image_coe_Ico
theorem image_coe_Ico : (some : α → WithTop α) '' Ico a b = Ico (a : WithTop α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some '' Set.Ico a b = Set.Ico ↑a ↑b
-- WithTop.image_coe_Ioc
theorem image_coe_Ioc : (some : α → WithTop α) '' Ioc a b = Ioc (a : WithTop α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some '' Set.Ioc a b = Set.Ioc ↑a ↑b
-- WithTop.image_coe_Ioo
theorem image_coe_Ioo : (some : α → WithTop α) '' Ioo a b = Ioo (a : WithTop α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithTop.some '' Set.Ioo a b = Set.Ioo ↑a ↑b
-- WithBot.preimage_coe_bot
@[simp]
theorem preimage_coe_bot : (some : α → WithBot α) ⁻¹' {⊥} = (∅ : Set α) 
Arguments: {α : Type u_1}
Goal: WithBot.some ⁻¹' {⊥} = ∅
-- WithBot.range_coe
theorem range_coe : range (some : α → WithBot α) = Ioi ⊥ 
Arguments: {α : Type u_1} [Preorder α]
Goal: Set.range WithBot.some = Set.Ioi ⊥
-- WithBot.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ioi : (some : α → WithBot α) ⁻¹' Ioi a = Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some ⁻¹' Set.Ioi ↑a = Set.Ioi a
-- WithBot.preimage_coe_Ici
@[simp]
theorem preimage_coe_Ici : (some : α → WithBot α) ⁻¹' Ici a = Ici a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some ⁻¹' Set.Ici ↑a = Set.Ici a
-- WithBot.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iio : (some : α → WithBot α) ⁻¹' Iio a = Iio a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some ⁻¹' Set.Iio ↑a = Set.Iio a
-- WithBot.preimage_coe_Iic
@[simp]
theorem preimage_coe_Iic : (some : α → WithBot α) ⁻¹' Iic a = Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some ⁻¹' Set.Iic ↑a = Set.Iic a
-- WithBot.preimage_coe_Icc
@[simp]
theorem preimage_coe_Icc : (some : α → WithBot α) ⁻¹' Icc a b = Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some ⁻¹' Set.Icc ↑a ↑b = Set.Icc a b
-- WithBot.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ico : (some : α → WithBot α) ⁻¹' Ico a b = Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some ⁻¹' Set.Ico ↑a ↑b = Set.Ico a b
-- WithBot.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioc : (some : α → WithBot α) ⁻¹' Ioc a b = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some ⁻¹' Set.Ioc ↑a ↑b = Set.Ioc a b
-- WithBot.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Ioo : (some : α → WithBot α) ⁻¹' Ioo a b = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some ⁻¹' Set.Ioo ↑a ↑b = Set.Ioo a b
-- WithBot.preimage_coe_Ioi_bot
@[simp]
theorem preimage_coe_Ioi_bot : (some : α → WithBot α) ⁻¹' Ioi ⊥ = univ 
Arguments: {α : Type u_1} [Preorder α]
Goal: WithBot.some ⁻¹' Set.Ioi ⊥ = Set.univ
-- WithBot.preimage_coe_Ioc_bot
@[simp]
theorem preimage_coe_Ioc_bot : (some : α → WithBot α) ⁻¹' Ioc ⊥ a = Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some ⁻¹' Set.Ioc ⊥ ↑a = Set.Iic a
-- WithBot.preimage_coe_Ioo_bot
@[simp]
theorem preimage_coe_Ioo_bot : (some : α → WithBot α) ⁻¹' Ioo ⊥ a = Iio a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some ⁻¹' Set.Ioo ⊥ ↑a = Set.Iio a
-- WithBot.image_coe_Iio
theorem image_coe_Iio : (some : α → WithBot α) '' Iio a = Ioo (⊥ : WithBot α) a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some '' Set.Iio a = Set.Ioo ⊥ ↑a
-- WithBot.image_coe_Iic
theorem image_coe_Iic : (some : α → WithBot α) '' Iic a = Ioc (⊥ : WithBot α) a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some '' Set.Iic a = Set.Ioc ⊥ ↑a
-- WithBot.image_coe_Ioi
theorem image_coe_Ioi : (some : α → WithBot α) '' Ioi a = Ioi (a : WithBot α) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some '' Set.Ioi a = Set.Ioi ↑a
-- WithBot.image_coe_Ici
theorem image_coe_Ici : (some : α → WithBot α) '' Ici a = Ici (a : WithBot α) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: WithBot.some '' Set.Ici a = Set.Ici ↑a
-- WithBot.image_coe_Icc
theorem image_coe_Icc : (some : α → WithBot α) '' Icc a b = Icc (a : WithBot α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some '' Set.Icc a b = Set.Icc ↑a ↑b
-- WithBot.image_coe_Ioc
theorem image_coe_Ioc : (some : α → WithBot α) '' Ioc a b = Ioc (a : WithBot α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some '' Set.Ioc a b = Set.Ioc ↑a ↑b
-- WithBot.image_coe_Ico
theorem image_coe_Ico : (some : α → WithBot α) '' Ico a b = Ico (a : WithBot α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some '' Set.Ico a b = Set.Ico ↑a ↑b
-- WithBot.image_coe_Ioo
theorem image_coe_Ioo : (some : α → WithBot α) '' Ioo a b = Ioo (a : WithBot α) b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: WithBot.some '' Set.Ioo a b = Set.Ioo ↑a ↑b
-- OrderIso.preimage_Iic
@[simp]
theorem preimage_Iic (e : α ≃o β) (b : β) : e ⁻¹' Iic b = Iic (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β)
Goal: ⇑e ⁻¹' Set.Iic b = Set.Iic (e.symm b)
-- OrderIso.preimage_Ici
@[simp]
theorem preimage_Ici (e : α ≃o β) (b : β) : e ⁻¹' Ici b = Ici (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β)
Goal: ⇑e ⁻¹' Set.Ici b = Set.Ici (e.symm b)
-- OrderIso.preimage_Iio
@[simp]
theorem preimage_Iio (e : α ≃o β) (b : β) : e ⁻¹' Iio b = Iio (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β)
Goal: ⇑e ⁻¹' Set.Iio b = Set.Iio (e.symm b)
-- OrderIso.preimage_Ioi
@[simp]
theorem preimage_Ioi (e : α ≃o β) (b : β) : e ⁻¹' Ioi b = Ioi (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β)
Goal: ⇑e ⁻¹' Set.Ioi b = Set.Ioi (e.symm b)
-- OrderIso.preimage_Icc
@[simp]
theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : β)
Goal: ⇑e ⁻¹' Set.Icc a b = Set.Icc (e.symm a) (e.symm b)
-- OrderIso.preimage_Ico
@[simp]
theorem preimage_Ico (e : α ≃o β) (a b : β) : e ⁻¹' Ico a b = Ico (e.symm a) (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : β)
Goal: ⇑e ⁻¹' Set.Ico a b = Set.Ico (e.symm a) (e.symm b)
-- OrderIso.preimage_Ioc
@[simp]
theorem preimage_Ioc (e : α ≃o β) (a b : β) : e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : β)
Goal: ⇑e ⁻¹' Set.Ioc a b = Set.Ioc (e.symm a) (e.symm b)
-- OrderIso.preimage_Ioo
@[simp]
theorem preimage_Ioo (e : α ≃o β) (a b : β) : e ⁻¹' Ioo a b = Ioo (e.symm a) (e.symm b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : β)
Goal: ⇑e ⁻¹' Set.Ioo a b = Set.Ioo (e.symm a) (e.symm b)
-- OrderIso.image_Iic
@[simp]
theorem image_Iic (e : α ≃o β) (a : α) : e '' Iic a = Iic (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α)
Goal: ⇑e '' Set.Iic a = Set.Iic (e a)
-- OrderIso.image_Ici
@[simp]
theorem image_Ici (e : α ≃o β) (a : α) : e '' Ici a = Ici (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α)
Goal: ⇑e '' Set.Ici a = Set.Ici (e a)
-- OrderIso.image_Iio
@[simp]
theorem image_Iio (e : α ≃o β) (a : α) : e '' Iio a = Iio (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α)
Goal: ⇑e '' Set.Iio a = Set.Iio (e a)
-- OrderIso.image_Ioi
@[simp]
theorem image_Ioi (e : α ≃o β) (a : α) : e '' Ioi a = Ioi (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α)
Goal: ⇑e '' Set.Ioi a = Set.Ioi (e a)
-- OrderIso.image_Ioo
@[simp]
theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : α)
Goal: ⇑e '' Set.Ioo a b = Set.Ioo (e a) (e b)
-- OrderIso.image_Ioc
@[simp]
theorem image_Ioc (e : α ≃o β) (a b : α) : e '' Ioc a b = Ioc (e a) (e b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : α)
Goal: ⇑e '' Set.Ioc a b = Set.Ioc (e a) (e b)
-- OrderIso.image_Ico
@[simp]
theorem image_Ico (e : α ≃o β) (a b : α) : e '' Ico a b = Ico (e a) (e b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : α)
Goal: ⇑e '' Set.Ico a b = Set.Ico (e a) (e b)
-- OrderIso.image_Icc
@[simp]
theorem image_Icc (e : α ≃o β) (a b : α) : e '' Icc a b = Icc (e a) (e b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a b : α)
Goal: ⇑e '' Set.Icc a b = Set.Icc (e a) (e b)
-- mem_upperBounds
theorem mem_upperBounds : a ∈ upperBounds s ↔ ∀ x ∈ s, x ≤ a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α}
Goal: a ∈ upperBounds s ↔ ∀ x ∈ s, x ≤ a
-- mem_lowerBounds
theorem mem_lowerBounds : a ∈ lowerBounds s ↔ ∀ x ∈ s, a ≤ x 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α}
Goal: a ∈ lowerBounds s ↔ ∀ x ∈ s, a ≤ x
-- mem_upperBounds_iff_subset_Iic
lemma mem_upperBounds_iff_subset_Iic : a ∈ upperBounds s ↔ s ⊆ Iic a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α}
Goal: a ∈ upperBounds s ↔ s ⊆ Set.Iic a
-- mem_lowerBounds_iff_subset_Ici
lemma mem_lowerBounds_iff_subset_Ici : a ∈ lowerBounds s ↔ s ⊆ Ici a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α}
Goal: a ∈ lowerBounds s ↔ s ⊆ Set.Ici a
-- bddAbove_def
theorem bddAbove_def : BddAbove s ↔ ∃ x, ∀ y ∈ s, y ≤ x 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: BddAbove s ↔ ∃ x, ∀ y ∈ s, y ≤ x
-- bddBelow_def
theorem bddBelow_def : BddBelow s ↔ ∃ x, ∀ y ∈ s, x ≤ y 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: BddBelow s ↔ ∃ x, ∀ y ∈ s, x ≤ y
-- bot_mem_lowerBounds
theorem bot_mem_lowerBounds [OrderBot α] (s : Set α) : ⊥ ∈ lowerBounds s 
Arguments: {α : Type u} [Preorder α] [OrderBot α] (s : Set α)
Goal: ⊥ ∈ lowerBounds s
-- top_mem_upperBounds
theorem top_mem_upperBounds [OrderTop α] (s : Set α) : ⊤ ∈ upperBounds s 
Arguments: {α : Type u} [Preorder α] [OrderTop α] (s : Set α)
Goal: ⊤ ∈ upperBounds s
-- isLeast_bot_iff
@[simp]
theorem isLeast_bot_iff [OrderBot α] : IsLeast s ⊥ ↔ ⊥ ∈ s 
Arguments: {α : Type u} [Preorder α] {s : Set α} [OrderBot α]
Goal: IsLeast s ⊥ ↔ ⊥ ∈ s
-- isGreatest_top_iff
@[simp]
theorem isGreatest_top_iff [OrderTop α] : IsGreatest s ⊤ ↔ ⊤ ∈ s 
Arguments: {α : Type u} [Preorder α] {s : Set α} [OrderTop α]
Goal: IsGreatest s ⊤ ↔ ⊤ ∈ s
-- not_bddAbove_iff'
theorem not_bddAbove_iff' : ¬BddAbove s ↔ ∀ x, ∃ y ∈ s, ¬y ≤ x 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: ¬BddAbove s ↔ ∀ (x : α), ∃ y ∈ s, ¬y ≤ x
-- not_bddBelow_iff'
theorem not_bddBelow_iff' : ¬BddBelow s ↔ ∀ x, ∃ y ∈ s, ¬x ≤ y 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: ¬BddBelow s ↔ ∀ (x : α), ∃ y ∈ s, ¬x ≤ y
-- not_bddAbove_iff
theorem not_bddAbove_iff {α : Type*} [LinearOrder α] {s : Set α} :
    ¬BddAbove s ↔ ∀ x, ∃ y ∈ s, x < y 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α}
Goal: ¬BddAbove s ↔ ∀ (x : α), ∃ y ∈ s, x < y
-- not_bddBelow_iff
theorem not_bddBelow_iff {α : Type*} [LinearOrder α] {s : Set α} :
    ¬BddBelow s ↔ ∀ x, ∃ y ∈ s, y < x 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α}
Goal: ¬BddBelow s ↔ ∀ (x : α), ∃ y ∈ s, y < x
-- bddBelow_preimage_ofDual
@[simp] lemma bddBelow_preimage_ofDual {s : Set α} : BddBelow (ofDual ⁻¹' s) ↔ BddAbove s 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: BddBelow (⇑OrderDual.ofDual ⁻¹' s) ↔ BddAbove s
-- bddAbove_preimage_ofDual
@[simp] lemma bddAbove_preimage_ofDual {s : Set α} : BddAbove (ofDual ⁻¹' s) ↔ BddBelow s 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: BddAbove (⇑OrderDual.ofDual ⁻¹' s) ↔ BddBelow s
-- bddBelow_preimage_toDual
@[simp] lemma bddBelow_preimage_toDual {s : Set αᵒᵈ} :
    BddBelow (toDual ⁻¹' s) ↔ BddAbove s 
Arguments: {α : Type u} [Preorder α] {s : Set αᵒᵈ}
Goal: BddBelow (⇑OrderDual.toDual ⁻¹' s) ↔ BddAbove s
-- bddAbove_preimage_toDual
@[simp] lemma bddAbove_preimage_toDual {s : Set αᵒᵈ} :
    BddAbove (toDual ⁻¹' s) ↔ BddBelow s 
Arguments: {α : Type u} [Preorder α] {s : Set αᵒᵈ}
Goal: BddAbove (⇑OrderDual.toDual ⁻¹' s) ↔ BddBelow s
-- BddAbove.dual
theorem BddAbove.dual (h : BddAbove s) : BddBelow (ofDual ⁻¹' s) 
Arguments: {α : Type u} [Preorder α] {s : Set α} (h : BddAbove s)
Goal: BddBelow (⇑OrderDual.ofDual ⁻¹' s)
-- BddBelow.dual
theorem BddBelow.dual (h : BddBelow s) : BddAbove (ofDual ⁻¹' s) 
Arguments: {α : Type u} [Preorder α] {s : Set α} (h : BddBelow s)
Goal: BddAbove (⇑OrderDual.ofDual ⁻¹' s)