ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- StrictMono.apply_eq_bot_iff
theorem StrictMono.apply_eq_bot_iff (hf : StrictMono f) : f a = f ⊥ ↔ a = ⊥ 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [OrderBot α] [Preorder β] {f : α → β} {a : α} (hf : StrictMono f)
Goal: f a = f ⊥ ↔ a = ⊥
-- StrictAnti.apply_eq_bot_iff
theorem StrictAnti.apply_eq_bot_iff (hf : StrictAnti f) : f a = f ⊥ ↔ a = ⊥ 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [OrderBot α] [Preorder β] {f : α → β} {a : α} (hf : StrictAnti f)
Goal: f a = f ⊥ ↔ a = ⊥
-- not_isMax_bot
theorem not_isMax_bot : ¬IsMax (⊥ : α) 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] [Nontrivial α]
Goal: ¬IsMax ⊥
-- StrictMono.minimal_preimage_bot
theorem StrictMono.minimal_preimage_bot [LinearOrder α] [PartialOrder β] [OrderBot β] {f : α → β}
    (H : StrictMono f) {a} (h_bot : f a = ⊥) (x : α) : a ≤ x 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [PartialOrder β] [OrderBot β] {f : α → β} (H : StrictMono f) {a : α} (h_bot : f a = ⊥) (x : α)
Goal: a ≤ x
-- OrderBot.ext_bot
theorem OrderBot.ext_bot {α} {hA : PartialOrder α} (A : OrderBot α) {hB : PartialOrder α}
    (B : OrderBot α) (H : ∀ x y : α, (haveI := hA; x ≤ y) ↔ x ≤ y) :
    (@Bot.bot α (@OrderBot.toBot α hA.toLE A)) = (@Bot.bot α (@OrderBot.toBot α hB.toLE B)) 
Arguments: {α : Type u_3} {hA : PartialOrder α} (A : OrderBot α) {hB : PartialOrder α} (B : OrderBot α) (H : ∀ (x y : α), x ≤ y ↔ x ≤ y)
Goal: ⊥ = ⊥
-- top_sup_eq
theorem top_sup_eq (a : α) : ⊤ ⊔ a = ⊤ 
Arguments: {α : Type u} [SemilatticeSup α] [OrderTop α] (a : α)
Goal: ⊤ ⊔ a = ⊤
-- sup_top_eq
theorem sup_top_eq (a : α) : a ⊔ ⊤ = ⊤ 
Arguments: {α : Type u} [SemilatticeSup α] [OrderTop α] (a : α)
Goal: a ⊔ ⊤ = ⊤
-- bot_sup_eq
theorem bot_sup_eq (a : α) : ⊥ ⊔ a = a 
Arguments: {α : Type u} [SemilatticeSup α] [OrderBot α] (a : α)
Goal: ⊥ ⊔ a = a
-- sup_bot_eq
theorem sup_bot_eq (a : α) : a ⊔ ⊥ = a 
Arguments: {α : Type u} [SemilatticeSup α] [OrderBot α] (a : α)
Goal: a ⊔ ⊥ = a
-- sup_eq_bot_iff
@[simp]
theorem sup_eq_bot_iff : a ⊔ b = ⊥ ↔ a = ⊥ ∧ b = ⊥ 
Arguments: {α : Type u} [SemilatticeSup α] [OrderBot α] {a b : α}
Goal: a ⊔ b = ⊥ ↔ a = ⊥ ∧ b = ⊥
-- top_inf_eq
lemma top_inf_eq (a : α) : ⊤ ⊓ a = a 
Arguments: {α : Type u} [SemilatticeInf α] [OrderTop α] (a : α)
Goal: ⊤ ⊓ a = a
-- inf_top_eq
lemma inf_top_eq (a : α) : a ⊓ ⊤ = a 
Arguments: {α : Type u} [SemilatticeInf α] [OrderTop α] (a : α)
Goal: a ⊓ ⊤ = a
-- inf_eq_top_iff
@[simp]
theorem inf_eq_top_iff : a ⊓ b = ⊤ ↔ a = ⊤ ∧ b = ⊤ 
Arguments: {α : Type u} [SemilatticeInf α] [OrderTop α] {a b : α}
Goal: a ⊓ b = ⊤ ↔ a = ⊤ ∧ b = ⊤
-- bot_inf_eq
lemma bot_inf_eq (a : α) : ⊥ ⊓ a = ⊥ 
Arguments: {α : Type u} [SemilatticeInf α] [OrderBot α] (a : α)
Goal: ⊥ ⊓ a = ⊥
-- inf_bot_eq
lemma inf_bot_eq (a : α) : a ⊓ ⊥ = ⊥ 
Arguments: {α : Type u} [SemilatticeInf α] [OrderBot α] (a : α)
Goal: a ⊓ ⊥ = ⊥
-- monotone_and
theorem monotone_and {p q : α → Prop} (m_p : Monotone p) (m_q : Monotone q) :
    Monotone fun x => p x ∧ q x 
Arguments: {α : Type u} [Preorder α] {p q : α → Prop} (m_p : Monotone p) (m_q : Monotone q)
Goal: Monotone fun x => p x ∧ q x
-- monotone_or
theorem monotone_or {p q : α → Prop} (m_p : Monotone p) (m_q : Monotone q) :
    Monotone fun x => p x ∨ q x 
Arguments: {α : Type u} [Preorder α] {p q : α → Prop} (m_p : Monotone p) (m_q : Monotone q)
Goal: Monotone fun x => p x ∨ q x
-- monotone_le
theorem monotone_le {x : α} : Monotone (x ≤ ·) 
Arguments: {α : Type u} [Preorder α] {x : α}
Goal: Monotone fun x_1 => x ≤ x_1
-- monotone_lt
theorem monotone_lt {x : α} : Monotone (x < ·) 
Arguments: {α : Type u} [Preorder α] {x : α}
Goal: Monotone fun x_1 => x < x_1
-- antitone_le
theorem antitone_le {x : α} : Antitone (· ≤ x) 
Arguments: {α : Type u} [Preorder α] {x : α}
Goal: Antitone fun x_1 => x_1 ≤ x
-- antitone_lt
theorem antitone_lt {x : α} : Antitone (· < x) 
Arguments: {α : Type u} [Preorder α] {x : α}
Goal: Antitone fun x_1 => x_1 < x
-- Monotone.forall
theorem Monotone.forall {P : β → α → Prop} (hP : ∀ x, Monotone (P x)) :
    Monotone fun y => ∀ x, P x y 
Arguments: {α : Type u} {β : Type v} [Preorder α] {P : β → α → Prop} (hP : ∀ (x : β), Monotone (P x))
Goal: Monotone fun y => ∀ (x : β), P x y
-- Antitone.forall
theorem Antitone.forall {P : β → α → Prop} (hP : ∀ x, Antitone (P x)) :
    Antitone fun y => ∀ x, P x y 
Arguments: {α : Type u} {β : Type v} [Preorder α] {P : β → α → Prop} (hP : ∀ (x : β), Antitone (P x))
Goal: Antitone fun y => ∀ (x : β), P x y
-- Monotone.ball
theorem Monotone.ball {P : β → α → Prop} {s : Set β} (hP : ∀ x ∈ s, Monotone (P x)) :
    Monotone fun y => ∀ x ∈ s, P x y 
Arguments: {α : Type u} {β : Type v} [Preorder α] {P : β → α → Prop} {s : Set β} (hP : ∀ (x : β), x ∈ s → Monotone (P x))
Goal: Monotone fun y => ∀ (x : β), x ∈ s → P x y
-- Antitone.ball
theorem Antitone.ball {P : β → α → Prop} {s : Set β} (hP : ∀ x ∈ s, Antitone (P x)) :
    Antitone fun y => ∀ x ∈ s, P x y 
Arguments: {α : Type u} {β : Type v} [Preorder α] {P : β → α → Prop} {s : Set β} (hP : ∀ (x : β), x ∈ s → Antitone (P x))
Goal: Antitone fun y => ∀ (x : β), x ∈ s → P x y
-- Monotone.exists
theorem Monotone.exists {P : β → α → Prop} (hP : ∀ x, Monotone (P x)) :
    Monotone fun y => ∃ x, P x y 
Arguments: {α : Type u} {β : Type v} [Preorder α] {P : β → α → Prop} (hP : ∀ (x : β), Monotone (P x))
Goal: Monotone fun y => ∃ x, P x y
-- Antitone.exists
theorem Antitone.exists {P : β → α → Prop} (hP : ∀ x, Antitone (P x)) :
    Antitone fun y => ∃ x, P x y 
Arguments: {α : Type u} {β : Type v} [Preorder α] {P : β → α → Prop} (hP : ∀ (x : β), Antitone (P x))
Goal: Antitone fun y => ∃ x, P x y
-- forall_ge_iff
theorem forall_ge_iff {P : α → Prop} {x₀ : α} (hP : Monotone P) :
    (∀ x ≥ x₀, P x) ↔ P x₀ 
Arguments: {α : Type u} [Preorder α] {P : α → Prop} {x₀ : α} (hP : Monotone P)
Goal: (∀ (x : α), x ≥ x₀ → P x) ↔ P x₀
-- forall_le_iff
theorem forall_le_iff {P : α → Prop} {x₀ : α} (hP : Antitone P) :
    (∀ x ≤ x₀, P x) ↔ P x₀ 
Arguments: {α : Type u} [Preorder α] {P : α → Prop} {x₀ : α} (hP : Antitone P)
Goal: (∀ (x : α), x ≤ x₀ → P x) ↔ P x₀
-- exists_ge_and_iff_exists
theorem exists_ge_and_iff_exists {P : α → Prop} {x₀ : α} (hP : Monotone P) :
    (∃ x, x₀ ≤ x ∧ P x) ↔ ∃ x, P x 
Arguments: {α : Type u} [SemilatticeSup α] {P : α → Prop} {x₀ : α} (hP : Monotone P)
Goal: (∃ x, x₀ ≤ x ∧ P x) ↔ ∃ x, P x
-- exists_le_and_iff_exists
theorem exists_le_and_iff_exists {P : α → Prop} {x₀ : α} (hP : Antitone P) :
    (∃ x, x ≤ x₀ ∧ P x) ↔ ∃ x, P x 
Arguments: {α : Type u} [SemilatticeInf α] {P : α → Prop} {x₀ : α} (hP : Antitone P)
Goal: (∃ x, x ≤ x₀ ∧ P x) ↔ ∃ x, P x
-- Pi.bot_apply
@[simp]
theorem bot_apply [∀ i, Bot (α' i)] (i : ι) : (⊥ : ∀ i, α' i) i = ⊥ 
Arguments: {ι : Type u_3} {α' : ι → Type u_4} [(i : ι) → Bot (α' i)] (i : ι)
Goal: ⊥ i = ⊥
-- Pi.bot_def
theorem bot_def [∀ i, Bot (α' i)] : (⊥ : ∀ i, α' i) = fun _ => ⊥ 
Arguments: {ι : Type u_3} {α' : ι → Type u_4} [(i : ι) → Bot (α' i)]
Goal: ⊥ = fun x => ⊥
-- Pi.top_apply
@[simp]
theorem top_apply [∀ i, Top (α' i)] (i : ι) : (⊤ : ∀ i, α' i) i = ⊤ 
Arguments: {ι : Type u_3} {α' : ι → Type u_4} [(i : ι) → Top (α' i)] (i : ι)
Goal: ⊤ i = ⊤
-- Pi.top_def
theorem top_def [∀ i, Top (α' i)] : (⊤ : ∀ i, α' i) = fun _ => ⊤ 
Arguments: {ι : Type u_3} {α' : ι → Type u_4} [(i : ι) → Top (α' i)]
Goal: ⊤ = fun x => ⊤
-- eq_bot_of_bot_eq_top
theorem eq_bot_of_bot_eq_top (hα : (⊥ : α) = ⊤) (x : α) : x = (⊥ : α) 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α] (hα : ⊥ = ⊤) (x : α)
Goal: x = ⊥
-- eq_top_of_bot_eq_top
theorem eq_top_of_bot_eq_top (hα : (⊥ : α) = ⊤) (x : α) : x = (⊤ : α) 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α] (hα : ⊥ = ⊤) (x : α)
Goal: x = ⊤
-- subsingleton_of_top_le_bot
theorem subsingleton_of_top_le_bot (h : (⊤ : α) ≤ (⊥ : α)) : Subsingleton α 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α] (h : ⊤ ≤ ⊥)
Goal: Subsingleton α
-- subsingleton_of_bot_eq_top
theorem subsingleton_of_bot_eq_top (hα : (⊥ : α) = (⊤ : α)) : Subsingleton α 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α] (hα : ⊥ = ⊤)
Goal: Subsingleton α
-- subsingleton_iff_bot_eq_top
theorem subsingleton_iff_bot_eq_top : (⊥ : α) = (⊤ : α) ↔ Subsingleton α 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α]
Goal: ⊥ = ⊤ ↔ Subsingleton α
-- Subtype.mk_bot
@[simp]
theorem mk_bot [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥) : mk ⊥ hbot = ⊥ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥)
Goal: ⟨⊥, hbot⟩ = ⊥
-- Subtype.mk_top
@[simp]
theorem mk_top [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤) : mk ⊤ htop = ⊤ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤)
Goal: ⟨⊤, htop⟩ = ⊤
-- Subtype.coe_bot
theorem coe_bot [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥) : ((⊥ : Subtype p) : α) = ⊥ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥)
Goal: ↑⊥ = ⊥
-- Subtype.coe_top
theorem coe_top [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤) : ((⊤ : Subtype p) : α) = ⊤ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤)
Goal: ↑⊤ = ⊤
-- Subtype.coe_eq_bot_iff
@[simp]
theorem coe_eq_bot_iff [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥) {x : { x // p x }} :
    (x : α) = ⊥ ↔ x = ⊥ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥) {x : { x // p x }}
Goal: ↑x = ⊥ ↔ x = ⊥
-- Subtype.coe_eq_top_iff
@[simp]
theorem coe_eq_top_iff [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤) {x : { x // p x }} :
    (x : α) = ⊤ ↔ x = ⊤ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤) {x : { x // p x }}
Goal: ↑x = ⊤ ↔ x = ⊤
-- Subtype.mk_eq_bot_iff
@[simp]
theorem mk_eq_bot_iff [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥) {x : α} (hx : p x) :
    (⟨x, hx⟩ : Subtype p) = ⊥ ↔ x = ⊥ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderBot α] [OrderBot (Subtype p)] (hbot : p ⊥) {x : α} (hx : p x)
Goal: ⟨x, hx⟩ = ⊥ ↔ x = ⊥
-- Subtype.mk_eq_top_iff
@[simp]
theorem mk_eq_top_iff [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤) {x : α} (hx : p x) :
    (⟨x, hx⟩ : Subtype p) = ⊤ ↔ x = ⊤ 
Arguments: {α : Type u} {p : α → Prop} [PartialOrder α] [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤) {x : α} (hx : p x)
Goal: ⟨x, hx⟩ = ⊤ ↔ x = ⊤
-- Prod.fst_top
theorem fst_top [Top α] [Top β] : (⊤ : α × β).fst = ⊤ 
Arguments: (α : Type u) (β : Type v) [Top α] [Top β]
Goal: ⊤.1 = ⊤
-- Prod.snd_top
theorem snd_top [Top α] [Top β] : (⊤ : α × β).snd = ⊤ 
Arguments: (α : Type u) (β : Type v) [Top α] [Top β]
Goal: ⊤.2 = ⊤
-- Prod.fst_bot
theorem fst_bot [Bot α] [Bot β] : (⊥ : α × β).fst = ⊥ 
Arguments: (α : Type u) (β : Type v) [Bot α] [Bot β]
Goal: ⊥.1 = ⊥
-- Prod.snd_bot
theorem snd_bot [Bot α] [Bot β] : (⊥ : α × β).snd = ⊥ 
Arguments: (α : Type u) (β : Type v) [Bot α] [Bot β]
Goal: ⊥.2 = ⊥
-- ULift.up_top
@[simp] theorem up_top [Top α] : up (⊤ : α) = ⊤ 
Arguments: {α : Type u} [Top α]
Goal: { down := ⊤ } = ⊤
-- ULift.down_top
@[simp] theorem down_top [Top α] : down (⊤ : ULift α) = ⊤ 
Arguments: {α : Type u} [Top α]
Goal: ⊤.down = ⊤
-- ULift.up_bot
@[simp] theorem up_bot [Bot α] : up (⊥ : α) = ⊥ 
Arguments: {α : Type u} [Bot α]
Goal: { down := ⊥ } = ⊥
-- ULift.down_bot
@[simp] theorem down_bot [Bot α] : down (⊥ : ULift α) = ⊥ 
Arguments: {α : Type u} [Bot α]
Goal: ⊥.down = ⊥
-- min_bot_left
theorem min_bot_left [OrderBot α] (a : α) : min ⊥ a = ⊥ 
Arguments: {α : Type u} [LinearOrder α] [OrderBot α] (a : α)
Goal: min ⊥ a = ⊥
-- max_top_left
theorem max_top_left [OrderTop α] (a : α) : max ⊤ a = ⊤ 
Arguments: {α : Type u} [LinearOrder α] [OrderTop α] (a : α)
Goal: max ⊤ a = ⊤
-- min_top_left
theorem min_top_left [OrderTop α] (a : α) : min ⊤ a = a 
Arguments: {α : Type u} [LinearOrder α] [OrderTop α] (a : α)
Goal: min ⊤ a = a
-- max_bot_left
theorem max_bot_left [OrderBot α] (a : α) : max ⊥ a = a 
Arguments: {α : Type u} [LinearOrder α] [OrderBot α] (a : α)
Goal: max ⊥ a = a
-- min_top_right
theorem min_top_right [OrderTop α] (a : α) : min a ⊤ = a 
Arguments: {α : Type u} [LinearOrder α] [OrderTop α] (a : α)
Goal: min a ⊤ = a
-- max_bot_right
theorem max_bot_right [OrderBot α] (a : α) : max a ⊥ = a 
Arguments: {α : Type u} [LinearOrder α] [OrderBot α] (a : α)
Goal: max a ⊥ = a
-- min_bot_right
theorem min_bot_right [OrderBot α] (a : α) : min a ⊥ = ⊥ 
Arguments: {α : Type u} [LinearOrder α] [OrderBot α] (a : α)
Goal: min a ⊥ = ⊥
-- max_top_right
theorem max_top_right [OrderTop α] (a : α) : max a ⊤ = ⊤ 
Arguments: {α : Type u} [LinearOrder α] [OrderTop α] (a : α)
Goal: max a ⊤ = ⊤
-- min_eq_bot
@[simp]
theorem min_eq_bot [OrderBot α] {a b : α} : min a b = ⊥ ↔ a = ⊥ ∨ b = ⊥ 
Arguments: {α : Type u} [LinearOrder α] [OrderBot α] {a b : α}
Goal: min a b = ⊥ ↔ a = ⊥ ∨ b = ⊥
-- max_eq_top
@[simp]
theorem max_eq_top [OrderTop α] {a b : α} : max a b = ⊤ ↔ a = ⊤ ∨ b = ⊤ 
Arguments: {α : Type u} [LinearOrder α] [OrderTop α] {a b : α}
Goal: max a b = ⊤ ↔ a = ⊤ ∨ b = ⊤
-- max_eq_bot
@[simp]
theorem max_eq_bot [OrderBot α] {a b : α} : max a b = ⊥ ↔ a = ⊥ ∧ b = ⊥ 
Arguments: {α : Type u} [LinearOrder α] [OrderBot α] {a b : α}
Goal: max a b = ⊥ ↔ a = ⊥ ∧ b = ⊥
-- min_eq_top
@[simp]
theorem min_eq_top [OrderTop α] {a b : α} : min a b = ⊤ ↔ a = ⊤ ∧ b = ⊤ 
Arguments: {α : Type u} [LinearOrder α] [OrderTop α] {a b : α}
Goal: min a b = ⊤ ↔ a = ⊤ ∧ b = ⊤
-- bot_ne_top
@[simp]
theorem bot_ne_top : (⊥ : α) ≠ ⊤ 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α] [Nontrivial α]
Goal: ⊥ ≠ ⊤
-- top_ne_bot
@[simp]
theorem top_ne_bot : (⊤ : α) ≠ ⊥ 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α] [Nontrivial α]
Goal: ⊤ ≠ ⊥
-- bot_lt_top
@[simp]
theorem bot_lt_top : (⊥ : α) < ⊤ 
Arguments: {α : Type u} [PartialOrder α] [BoundedOrder α] [Nontrivial α]
Goal: ⊥ < ⊤
-- top_eq_true
@[simp]
theorem top_eq_true : ⊤ = true 
Goal: ⊤ = true
-- bot_eq_false
@[simp]
theorem bot_eq_false : ⊥ = false 
Goal: ⊥ = false
-- StrictMonoOn.union
protected theorem StrictMonoOn.union {s t : Set α} {c : α} (h₁ : StrictMonoOn f s)
    (h₂ : StrictMonoOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : StrictMonoOn f (s ∪ t) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {f : α → β} {s t : Set α} {c : α} (h₁ : StrictMonoOn f s) (h₂ : StrictMonoOn f t) (hs : IsGreatest s c) (ht : IsLeast t c)
Goal: StrictMonoOn f (s ∪ t)
-- StrictMonoOn.Iic_union_Ici
protected theorem StrictMonoOn.Iic_union_Ici (h₁ : StrictMonoOn f (Iic a))
    (h₂ : StrictMonoOn f (Ici a)) : StrictMono f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a : α} {f : α → β} (h₁ : StrictMonoOn f (Set.Iic a)) (h₂ : StrictMonoOn f (Set.Ici a))
Goal: StrictMono f
-- StrictAntiOn.union
protected theorem StrictAntiOn.union {s t : Set α} {c : α} (h₁ : StrictAntiOn f s)
    (h₂ : StrictAntiOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : StrictAntiOn f (s ∪ t) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {f : α → β} {s t : Set α} {c : α} (h₁ : StrictAntiOn f s) (h₂ : StrictAntiOn f t) (hs : IsGreatest s c) (ht : IsLeast t c)
Goal: StrictAntiOn f (s ∪ t)
-- StrictAntiOn.Iic_union_Ici
protected theorem StrictAntiOn.Iic_union_Ici (h₁ : StrictAntiOn f (Iic a))
    (h₂ : StrictAntiOn f (Ici a)) : StrictAnti f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a : α} {f : α → β} (h₁ : StrictAntiOn f (Set.Iic a)) (h₂ : StrictAntiOn f (Set.Ici a))
Goal: StrictAnti f
-- MonotoneOn.union_right
protected theorem MonotoneOn.union_right {s t : Set α} {c : α} (h₁ : MonotoneOn f s)
    (h₂ : MonotoneOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : MonotoneOn f (s ∪ t) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {f : α → β} {s t : Set α} {c : α} (h₁ : MonotoneOn f s) (h₂ : MonotoneOn f t) (hs : IsGreatest s c) (ht : IsLeast t c)
Goal: MonotoneOn f (s ∪ t)
-- MonotoneOn.Iic_union_Ici
protected theorem MonotoneOn.Iic_union_Ici (h₁ : MonotoneOn f (Iic a)) (h₂ : MonotoneOn f (Ici a)) :
    Monotone f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a : α} {f : α → β} (h₁ : MonotoneOn f (Set.Iic a)) (h₂ : MonotoneOn f (Set.Ici a))
Goal: Monotone f
-- AntitoneOn.union_right
protected theorem AntitoneOn.union_right {s t : Set α} {c : α} (h₁ : AntitoneOn f s)
    (h₂ : AntitoneOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : AntitoneOn f (s ∪ t) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {f : α → β} {s t : Set α} {c : α} (h₁ : AntitoneOn f s) (h₂ : AntitoneOn f t) (hs : IsGreatest s c) (ht : IsLeast t c)
Goal: AntitoneOn f (s ∪ t)
-- AntitoneOn.Iic_union_Ici
protected theorem AntitoneOn.Iic_union_Ici (h₁ : AntitoneOn f (Iic a)) (h₂ : AntitoneOn f (Ici a)) :
    Antitone f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a : α} {f : α → β} (h₁ : AntitoneOn f (Set.Iic a)) (h₂ : AntitoneOn f (Set.Ici a))
Goal: Antitone f
-- IsAtom.Iic
theorem IsAtom.Iic (ha : IsAtom a) (hax : a ≤ x) : IsAtom (⟨a, hax⟩ : Set.Iic x) 
Arguments: {α : Type u_2} [Preorder α] [OrderBot α] {a x : α} (ha : IsAtom a) (hax : a ≤ x)
Goal: IsAtom ⟨a, hax⟩
-- IsAtom.of_isAtom_coe_Iic
theorem IsAtom.of_isAtom_coe_Iic {a : Set.Iic x} (ha : IsAtom a) : IsAtom (a : α) 
Arguments: {α : Type u_2} [Preorder α] [OrderBot α] {x : α} {a : ↑(Set.Iic x)} (ha : IsAtom a)
Goal: IsAtom ↑a
-- isAtom_iff_le_of_ge
theorem isAtom_iff_le_of_ge : IsAtom a ↔ a ≠ ⊥ ∧ ∀ b ≠ ⊥, b ≤ a → a ≤ b 
Arguments: {α : Type u_2} [Preorder α] [OrderBot α] {a : α}
Goal: IsAtom a ↔ a ≠ ⊥ ∧ ∀ (b : α), b ≠ ⊥ → b ≤ a → a ≤ b
-- IsAtom.lt_iff
theorem IsAtom.lt_iff (h : IsAtom a) : x < a ↔ x = ⊥ 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] {a x : α} (h : IsAtom a)
Goal: x < a ↔ x = ⊥
-- IsAtom.le_iff
theorem IsAtom.le_iff (h : IsAtom a) : x ≤ a ↔ x = ⊥ ∨ x = a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] {a x : α} (h : IsAtom a)
Goal: x ≤ a ↔ x = ⊥ ∨ x = a
-- IsAtom.le_iff_eq
lemma IsAtom.le_iff_eq (ha : IsAtom a) (hb : b ≠ ⊥) : b ≤ a ↔ b = a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] {a b : α} (ha : IsAtom a) (hb : b ≠ ⊥)
Goal: b ≤ a ↔ b = a
-- IsAtom.Iic_eq
theorem IsAtom.Iic_eq (h : IsAtom a) : Set.Iic a = {⊥, a} 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] {a : α} (h : IsAtom a)
Goal: Set.Iic a = {⊥, a}
-- bot_covBy_iff
@[simp]
theorem bot_covBy_iff : ⊥ ⋖ a ↔ IsAtom a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] {a : α}
Goal: ⊥ ⋖ a ↔ IsAtom a
-- atom_le_iSup
theorem atom_le_iSup [Order.Frame α] {a : α} (ha : IsAtom a) {f : ι → α} :
    a ≤ iSup f ↔ ∃ i, a ≤ f i 
Arguments: {ι : Sort u_1} {α : Type u_2} [Order.Frame α] {a : α} (ha : IsAtom a) {f : ι → α}
Goal: a ≤ iSup f ↔ ∃ i, a ≤ f i
-- isCoatom_dual_iff_isAtom
@[simp]
theorem isCoatom_dual_iff_isAtom [OrderBot α] {a : α} :
    IsCoatom (OrderDual.toDual a) ↔ IsAtom a 
Arguments: {α : Type u_2} [Preorder α] [OrderBot α] {a : α}
Goal: IsCoatom (OrderDual.toDual a) ↔ IsAtom a
-- isAtom_dual_iff_isCoatom
@[simp]
theorem isAtom_dual_iff_isCoatom [OrderTop α] {a : α} :
    IsAtom (OrderDual.toDual a) ↔ IsCoatom a 
Arguments: {α : Type u_2} [Preorder α] [OrderTop α] {a : α}
Goal: IsAtom (OrderDual.toDual a) ↔ IsCoatom a
-- IsCoatom.Ici
theorem IsCoatom.Ici (ha : IsCoatom a) (hax : x ≤ a) : IsCoatom (⟨a, hax⟩ : Set.Ici x) 
Arguments: {α : Type u_2} [Preorder α] [OrderTop α] {a x : α} (ha : IsCoatom a) (hax : x ≤ a)
Goal: IsCoatom ⟨a, hax⟩
-- IsCoatom.of_isCoatom_coe_Ici
theorem IsCoatom.of_isCoatom_coe_Ici {a : Set.Ici x} (ha : IsCoatom a) : IsCoatom (a : α) 
Arguments: {α : Type u_2} [Preorder α] [OrderTop α] {x : α} {a : ↑(Set.Ici x)} (ha : IsCoatom a)
Goal: IsCoatom ↑a
-- isCoatom_iff_ge_of_le
theorem isCoatom_iff_ge_of_le : IsCoatom a ↔ a ≠ ⊤ ∧ ∀ b ≠ ⊤, a ≤ b → b ≤ a 
Arguments: {α : Type u_2} [Preorder α] [OrderTop α] {a : α}
Goal: IsCoatom a ↔ a ≠ ⊤ ∧ ∀ (b : α), b ≠ ⊤ → a ≤ b → b ≤ a
-- IsCoatom.lt_iff
theorem IsCoatom.lt_iff (h : IsCoatom a) : a < x ↔ x = ⊤ 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] {a x : α} (h : IsCoatom a)
Goal: a < x ↔ x = ⊤
-- IsCoatom.le_iff
theorem IsCoatom.le_iff (h : IsCoatom a) : a ≤ x ↔ x = ⊤ ∨ x = a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] {a x : α} (h : IsCoatom a)
Goal: a ≤ x ↔ x = ⊤ ∨ x = a
-- IsCoatom.le_iff_eq
lemma IsCoatom.le_iff_eq (ha : IsCoatom a) (hb : b ≠ ⊤) : a ≤ b ↔ b = a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] {a b : α} (ha : IsCoatom a) (hb : b ≠ ⊤)
Goal: a ≤ b ↔ b = a
-- IsCoatom.Ici_eq
theorem IsCoatom.Ici_eq (h : IsCoatom a) : Set.Ici a = {⊤, a} 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] {a : α} (h : IsCoatom a)
Goal: Set.Ici a = {⊤, a}
-- covBy_top_iff
@[simp]
theorem covBy_top_iff : a ⋖ ⊤ ↔ IsCoatom a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] {a : α}
Goal: a ⋖ ⊤ ↔ IsCoatom a
-- iInf_le_coatom
theorem iInf_le_coatom [Order.Coframe α] {a : α} (ha : IsCoatom a) {f : ι → α} :
    iInf f ≤ a ↔ ∃ i, f i ≤ a 
Arguments: {ι : Sort u_1} {α : Type u_2} [Order.Coframe α] {a : α} (ha : IsCoatom a) {f : ι → α}
Goal: iInf f ≤ a ↔ ∃ i, f i ≤ a
-- Set.Ici.isAtom_iff
@[simp]
theorem Set.Ici.isAtom_iff {b : Set.Ici a} : IsAtom b ↔ a ⋖ b 
Arguments: {α : Type u_2} [PartialOrder α] {a : α} {b : ↑(Set.Ici a)}
Goal: IsAtom b ↔ a ⋖ ↑b
-- Set.Iic.isCoatom_iff
@[simp]
theorem Set.Iic.isCoatom_iff {a : Set.Iic b} : IsCoatom a ↔ ↑a ⋖ b 
Arguments: {α : Type u_2} [PartialOrder α] {b : α} {a : ↑(Set.Iic b)}
Goal: IsCoatom a ↔ ↑a ⋖ b
-- covBy_iff_atom_Ici
theorem covBy_iff_atom_Ici (h : a ≤ b) : a ⋖ b ↔ IsAtom (⟨b, h⟩ : Set.Ici a) 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: a ⋖ b ↔ IsAtom ⟨b, h⟩
-- covBy_iff_coatom_Iic
theorem covBy_iff_coatom_Iic (h : a ≤ b) : a ⋖ b ↔ IsCoatom (⟨a, h⟩ : Set.Iic b) 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: a ⋖ b ↔ IsCoatom ⟨a, h⟩
-- IsAtom.inf_eq_bot_of_ne
theorem IsAtom.inf_eq_bot_of_ne [SemilatticeInf α] [OrderBot α] {a b : α} (ha : IsAtom a)
    (hb : IsAtom b) (hab : a ≠ b) : a ⊓ b = ⊥ 
Arguments: {α : Type u_2} [SemilatticeInf α] [OrderBot α] {a b : α} (ha : IsAtom a) (hb : IsAtom b) (hab : a ≠ b)
Goal: a ⊓ b = ⊥
-- IsAtom.disjoint_of_ne
theorem IsAtom.disjoint_of_ne [SemilatticeInf α] [OrderBot α] {a b : α} (ha : IsAtom a)
    (hb : IsAtom b) (hab : a ≠ b) : Disjoint a b 
Arguments: {α : Type u_2} [SemilatticeInf α] [OrderBot α] {a b : α} (ha : IsAtom a) (hb : IsAtom b) (hab : a ≠ b)
Goal: Disjoint a b
-- IsCoatom.sup_eq_top_of_ne
theorem IsCoatom.sup_eq_top_of_ne [SemilatticeSup α] [OrderTop α] {a b : α} (ha : IsCoatom a)
    (hb : IsCoatom b) (hab : a ≠ b) : a ⊔ b = ⊤ 
Arguments: {α : Type u_2} [SemilatticeSup α] [OrderTop α] {a b : α} (ha : IsCoatom a) (hb : IsCoatom b) (hab : a ≠ b)
Goal: a ⊔ b = ⊤
-- IsAtomic.eq_bot_or_exists_atom_le
@[mk_iff]
class IsAtomic [OrderBot α] : Prop where
    eq_bot_or_exists_atom_le : ∀ b : α, b = ⊥ ∨ ∃ a : α, IsAtom a ∧ a ≤ b
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] [self : IsAtomic α] (b : α)
Goal: b = ⊥ ∨ ∃ a, IsAtom a ∧ a ≤ b
-- isAtomic_iff
@[mk_iff]
class IsAtomic [OrderBot α] : Prop where
    eq_bot_or_exists_atom_le : ∀ b : α, b = ⊥ ∨ ∃ a : α, IsAtom a ∧ a ≤ b
Arguments: (α : Type u_2) [PartialOrder α] [OrderBot α]
Goal: IsAtomic α ↔ ∀ (b : α), b = ⊥ ∨ ∃ a, IsAtom a ∧ a ≤ b
-- IsCoatomic.eq_top_or_exists_le_coatom
@[mk_iff]
class IsCoatomic [OrderTop α] : Prop where
    eq_top_or_exists_le_coatom : ∀ b : α, b = ⊤ ∨ ∃ a : α, IsCoatom a ∧ b ≤ a
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] [self : IsCoatomic α] (b : α)
Goal: b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a
-- isCoatomic_iff
@[mk_iff]
class IsCoatomic [OrderTop α] : Prop where
    eq_top_or_exists_le_coatom : ∀ b : α, b = ⊤ ∨ ∃ a : α, IsCoatom a ∧ b ≤ a
Arguments: (α : Type u_2) [PartialOrder α] [OrderTop α]
Goal: IsCoatomic α ↔ ∀ (b : α), b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a
-- IsAtomic.exists_atom
lemma IsAtomic.exists_atom [OrderBot α] [Nontrivial α] [IsAtomic α] : ∃ a : α, IsAtom a 
Arguments: (α : Type u_2) [PartialOrder α] [OrderBot α] [Nontrivial α] [IsAtomic α]
Goal: ∃ a, IsAtom a
-- IsCoatomic.exists_coatom
lemma IsCoatomic.exists_coatom [OrderTop α] [Nontrivial α] [IsCoatomic α] : ∃ a : α, IsCoatom a 
Arguments: (α : Type u_2) [PartialOrder α] [OrderTop α] [Nontrivial α] [IsCoatomic α]
Goal: ∃ a, IsCoatom a