ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Codisjoint.mono_right
theorem Codisjoint.mono_right : b ≤ c → Codisjoint a b → Codisjoint a c 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b c : α}
Goal: b ≤ c → Codisjoint a b → Codisjoint a c
-- codisjoint_self
@[simp]
theorem codisjoint_self : Codisjoint a a ↔ a = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a : α}
Goal: Codisjoint a a ↔ a = ⊤
-- Codisjoint.ne
theorem Codisjoint.ne (ha : a ≠ ⊤) (hab : Codisjoint a b) : a ≠ b 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (ha : a ≠ ⊤) (hab : Codisjoint a b)
Goal: a ≠ b
-- Codisjoint.eq_top_of_le
theorem Codisjoint.eq_top_of_le (hab : Codisjoint a b) (h : b ≤ a) : a = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b) (h : b ≤ a)
Goal: a = ⊤
-- Codisjoint.eq_top_of_ge
theorem Codisjoint.eq_top_of_ge (hab : Codisjoint a b) : a ≤ b → b = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b)
Goal: a ≤ b → b = ⊤
-- Codisjoint.eq_iff
lemma Codisjoint.eq_iff (hab : Codisjoint a b) : a = b ↔ a = ⊤ ∧ b = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b)
Goal: a = b ↔ a = ⊤ ∧ b = ⊤
-- Codisjoint.ne_iff
lemma Codisjoint.ne_iff (hab : Codisjoint a b) : a ≠ b ↔ a ≠ ⊤ ∨ b ≠ ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b)
Goal: a ≠ b ↔ a ≠ ⊤ ∨ b ≠ ⊤
-- codisjoint_bot
@[simp]
theorem codisjoint_bot : Codisjoint a ⊥ ↔ a = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α}
Goal: Codisjoint a ⊥ ↔ a = ⊤
-- bot_codisjoint
@[simp]
theorem bot_codisjoint : Codisjoint ⊥ a ↔ a = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α}
Goal: Codisjoint ⊥ a ↔ a = ⊤
-- Codisjoint.ne_bot_of_ne_top
lemma Codisjoint.ne_bot_of_ne_top (h : Codisjoint a b) (ha : a ≠ ⊤) : b ≠ ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : α} (h : Codisjoint a b) (ha : a ≠ ⊤)
Goal: b ≠ ⊥
-- Codisjoint.ne_bot_of_ne_top'
lemma Codisjoint.ne_bot_of_ne_top' (h : Codisjoint a b) (hb : b ≠ ⊤) : a ≠ ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : α} (h : Codisjoint a b) (hb : b ≠ ⊤)
Goal: a ≠ ⊥
-- codisjoint_iff_le_sup
theorem codisjoint_iff_le_sup : Codisjoint a b ↔ ⊤ ≤ a ⊔ b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α}
Goal: Codisjoint a b ↔ ⊤ ≤ a ⊔ b
-- codisjoint_iff
theorem codisjoint_iff : Codisjoint a b ↔ a ⊔ b = ⊤ 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α}
Goal: Codisjoint a b ↔ a ⊔ b = ⊤
-- Codisjoint.top_le
theorem Codisjoint.top_le : Codisjoint a b → ⊤ ≤ a ⊔ b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α}
Goal: Codisjoint a b → ⊤ ≤ a ⊔ b
-- Codisjoint.eq_top
theorem Codisjoint.eq_top : Codisjoint a b → a ⊔ b = ⊤ 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α}
Goal: Codisjoint a b → a ⊔ b = ⊤
-- codisjoint_assoc
theorem codisjoint_assoc : Codisjoint (a ⊔ b) c ↔ Codisjoint a (b ⊔ c) 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α}
Goal: Codisjoint (a ⊔ b) c ↔ Codisjoint a (b ⊔ c)
-- codisjoint_left_comm
theorem codisjoint_left_comm : Codisjoint a (b ⊔ c) ↔ Codisjoint b (a ⊔ c) 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α}
Goal: Codisjoint a (b ⊔ c) ↔ Codisjoint b (a ⊔ c)
-- codisjoint_right_comm
theorem codisjoint_right_comm : Codisjoint (a ⊔ b) c ↔ Codisjoint (a ⊔ c) b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α}
Goal: Codisjoint (a ⊔ b) c ↔ Codisjoint (a ⊔ c) b
-- Codisjoint.sup_left
theorem Codisjoint.sup_left (h : Codisjoint a b) : Codisjoint (a ⊔ c) b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b)
Goal: Codisjoint (a ⊔ c) b
-- Codisjoint.sup_left'
theorem Codisjoint.sup_left' (h : Codisjoint a b) : Codisjoint (c ⊔ a) b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b)
Goal: Codisjoint (c ⊔ a) b
-- Codisjoint.sup_right
theorem Codisjoint.sup_right (h : Codisjoint a b) : Codisjoint a (b ⊔ c) 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b)
Goal: Codisjoint a (b ⊔ c)
-- Codisjoint.sup_right'
theorem Codisjoint.sup_right' (h : Codisjoint a b) : Codisjoint a (c ⊔ b) 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b)
Goal: Codisjoint a (c ⊔ b)
-- Codisjoint.of_codisjoint_sup_of_le
theorem Codisjoint.of_codisjoint_sup_of_le (h : Codisjoint (a ⊔ b) c) (hle : c ≤ a) :
    Codisjoint a b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α} (h : Codisjoint (a ⊔ b) c) (hle : c ≤ a)
Goal: Codisjoint a b
-- Codisjoint.of_codisjoint_sup_of_le'
theorem Codisjoint.of_codisjoint_sup_of_le' (h : Codisjoint (a ⊔ b) c) (hle : c ≤ b) :
    Codisjoint a b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α} (h : Codisjoint (a ⊔ b) c) (hle : c ≤ b)
Goal: Codisjoint a b
-- codisjoint_inf_left
@[simp]
theorem codisjoint_inf_left : Codisjoint (a ⊓ b) c ↔ Codisjoint a c ∧ Codisjoint b c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α}
Goal: Codisjoint (a ⊓ b) c ↔ Codisjoint a c ∧ Codisjoint b c
-- codisjoint_inf_right
@[simp]
theorem codisjoint_inf_right : Codisjoint a (b ⊓ c) ↔ Codisjoint a b ∧ Codisjoint a c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α}
Goal: Codisjoint a (b ⊓ c) ↔ Codisjoint a b ∧ Codisjoint a c
-- Codisjoint.inf_left
theorem Codisjoint.inf_left (ha : Codisjoint a c) (hb : Codisjoint b c) : Codisjoint (a ⊓ b) c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (ha : Codisjoint a c) (hb : Codisjoint b c)
Goal: Codisjoint (a ⊓ b) c
-- Codisjoint.inf_right
theorem Codisjoint.inf_right (hb : Codisjoint a b) (hc : Codisjoint a c) : Codisjoint a (b ⊓ c) 
Arguments: {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (hb : Codisjoint a b) (hc : Codisjoint a c)
Goal: Codisjoint a (b ⊓ c)
-- Codisjoint.left_le_of_le_inf_right
theorem Codisjoint.left_le_of_le_inf_right (h : a ⊓ b ≤ c) (hd : Codisjoint b c) : a ≤ c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (h : a ⊓ b ≤ c) (hd : Codisjoint b c)
Goal: a ≤ c
-- Codisjoint.left_le_of_le_inf_left
theorem Codisjoint.left_le_of_le_inf_left (h : b ⊓ a ≤ c) (hd : Codisjoint b c) : a ≤ c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (h : b ⊓ a ≤ c) (hd : Codisjoint b c)
Goal: a ≤ c
-- Disjoint.dual
theorem Disjoint.dual [SemilatticeInf α] [OrderBot α] {a b : α} :
    Disjoint a b → Codisjoint (toDual a) (toDual b) 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α}
Goal: Disjoint a b → Codisjoint (OrderDual.toDual a) (OrderDual.toDual b)
-- Codisjoint.dual
theorem Codisjoint.dual [SemilatticeSup α] [OrderTop α] {a b : α} :
    Codisjoint a b → Disjoint (toDual a) (toDual b) 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α}
Goal: Codisjoint a b → Disjoint (OrderDual.toDual a) (OrderDual.toDual b)
-- disjoint_toDual_iff
@[simp]
theorem disjoint_toDual_iff [SemilatticeSup α] [OrderTop α] {a b : α} :
    Disjoint (toDual a) (toDual b) ↔ Codisjoint a b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α}
Goal: Disjoint (OrderDual.toDual a) (OrderDual.toDual b) ↔ Codisjoint a b
-- disjoint_ofDual_iff
@[simp]
theorem disjoint_ofDual_iff [SemilatticeInf α] [OrderBot α] {a b : αᵒᵈ} :
    Disjoint (ofDual a) (ofDual b) ↔ Codisjoint a b 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : αᵒᵈ}
Goal: Disjoint (OrderDual.ofDual a) (OrderDual.ofDual b) ↔ Codisjoint a b
-- codisjoint_toDual_iff
@[simp]
theorem codisjoint_toDual_iff [SemilatticeInf α] [OrderBot α] {a b : α} :
    Codisjoint (toDual a) (toDual b) ↔ Disjoint a b 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α}
Goal: Codisjoint (OrderDual.toDual a) (OrderDual.toDual b) ↔ Disjoint a b
-- codisjoint_ofDual_iff
@[simp]
theorem codisjoint_ofDual_iff [SemilatticeSup α] [OrderTop α] {a b : αᵒᵈ} :
    Codisjoint (ofDual a) (ofDual b) ↔ Disjoint a b 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : αᵒᵈ}
Goal: Codisjoint (OrderDual.ofDual a) (OrderDual.ofDual b) ↔ Disjoint a b
-- Disjoint.le_of_codisjoint
theorem Disjoint.le_of_codisjoint (hab : Disjoint a b) (hbc : Codisjoint b c) : a ≤ c 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b c : α} (hab : Disjoint a b) (hbc : Codisjoint b c)
Goal: a ≤ c
-- IsCompl.codisjoint
structure IsCompl [PartialOrder α] [BoundedOrder α] (x y : α) : Prop where
    protected disjoint : Disjoint x y
    protected codisjoint : Codisjoint x y
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α} (self : IsCompl x y)
Goal: Codisjoint x y
-- IsCompl.disjoint
structure IsCompl [PartialOrder α] [BoundedOrder α] (x y : α) : Prop where
    protected disjoint : Disjoint x y
    protected codisjoint : Codisjoint x y
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α} (self : IsCompl x y)
Goal: Disjoint x y
-- isCompl_iff
theorem isCompl_iff [PartialOrder α] [BoundedOrder α] {a b : α} :
    IsCompl a b ↔ Disjoint a b ∧ Codisjoint a b 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : α}
Goal: IsCompl a b ↔ Disjoint a b ∧ Codisjoint a b
-- IsCompl.symm
@[symm]
protected theorem symm (h : IsCompl x y) : IsCompl y x 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α} (h : IsCompl x y)
Goal: IsCompl y x
-- isCompl_comm
lemma _root_.isCompl_comm : IsCompl x y ↔ IsCompl y x 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α}
Goal: IsCompl x y ↔ IsCompl y x
-- IsCompl.dual
theorem dual (h : IsCompl x y) : IsCompl (toDual x) (toDual y) 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α} (h : IsCompl x y)
Goal: IsCompl (OrderDual.toDual x) (OrderDual.toDual y)
-- IsCompl.ofDual
theorem ofDual {a b : αᵒᵈ} (h : IsCompl a b) : IsCompl (ofDual a) (ofDual b) 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : αᵒᵈ} (h : IsCompl a b)
Goal: IsCompl (OrderDual.ofDual a) (OrderDual.ofDual b)
-- IsCompl.of_le
theorem of_le (h₁ : x ⊓ y ≤ ⊥) (h₂ : ⊤ ≤ x ⊔ y) : IsCompl x y 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h₁ : x ⊓ y ≤ ⊥) (h₂ : ⊤ ≤ x ⊔ y)
Goal: IsCompl x y
-- IsCompl.of_eq
theorem of_eq (h₁ : x ⊓ y = ⊥) (h₂ : x ⊔ y = ⊤) : IsCompl x y 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h₁ : x ⊓ y = ⊥) (h₂ : x ⊔ y = ⊤)
Goal: IsCompl x y
-- IsCompl.inf_eq_bot
theorem inf_eq_bot (h : IsCompl x y) : x ⊓ y = ⊥ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h : IsCompl x y)
Goal: x ⊓ y = ⊥
-- IsCompl.sup_eq_top
theorem sup_eq_top (h : IsCompl x y) : x ⊔ y = ⊤ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h : IsCompl x y)
Goal: x ⊔ y = ⊤
-- IsCompl.inf_left_le_of_le_sup_right
theorem inf_left_le_of_le_sup_right (h : IsCompl x y) (hle : a ≤ b ⊔ y) : a ⊓ x ≤ b 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b x y : α} (h : IsCompl x y) (hle : a ≤ b ⊔ y)
Goal: a ⊓ x ≤ b
-- IsCompl.le_sup_right_iff_inf_left_le
theorem le_sup_right_iff_inf_left_le {a b} (h : IsCompl x y) : a ≤ b ⊔ y ↔ a ⊓ x ≤ b 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y a b : α} (h : IsCompl x y)
Goal: a ≤ b ⊔ y ↔ a ⊓ x ≤ b
-- IsCompl.inf_left_eq_bot_iff
theorem inf_left_eq_bot_iff (h : IsCompl y z) : x ⊓ y = ⊥ ↔ x ≤ z 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z)
Goal: x ⊓ y = ⊥ ↔ x ≤ z
-- IsCompl.inf_right_eq_bot_iff
theorem inf_right_eq_bot_iff (h : IsCompl y z) : x ⊓ z = ⊥ ↔ x ≤ y 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z)
Goal: x ⊓ z = ⊥ ↔ x ≤ y
-- IsCompl.disjoint_left_iff
theorem disjoint_left_iff (h : IsCompl y z) : Disjoint x y ↔ x ≤ z 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z)
Goal: Disjoint x y ↔ x ≤ z
-- IsCompl.disjoint_right_iff
theorem disjoint_right_iff (h : IsCompl y z) : Disjoint x z ↔ x ≤ y 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z)
Goal: Disjoint x z ↔ x ≤ y
-- IsCompl.le_left_iff
theorem le_left_iff (h : IsCompl x y) : z ≤ x ↔ Disjoint z y 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y)
Goal: z ≤ x ↔ Disjoint z y
-- IsCompl.le_right_iff
theorem le_right_iff (h : IsCompl x y) : z ≤ y ↔ Disjoint z x 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y)
Goal: z ≤ y ↔ Disjoint z x
-- IsCompl.left_le_iff
theorem left_le_iff (h : IsCompl x y) : x ≤ z ↔ Codisjoint z y 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y)
Goal: x ≤ z ↔ Codisjoint z y
-- IsCompl.right_le_iff
theorem right_le_iff (h : IsCompl x y) : y ≤ z ↔ Codisjoint z x 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y)
Goal: y ≤ z ↔ Codisjoint z x
-- IsCompl.Antitone
protected theorem Antitone {x' y'} (h : IsCompl x y) (h' : IsCompl x' y') (hx : x ≤ x') : y' ≤ y 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y x' y' : α} (h : IsCompl x y) (h' : IsCompl x' y') (hx : x ≤ x')
Goal: y' ≤ y
-- IsCompl.right_unique
theorem right_unique (hxy : IsCompl x y) (hxz : IsCompl x z) : y = z 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (hxy : IsCompl x y) (hxz : IsCompl x z)
Goal: y = z
-- IsCompl.left_unique
theorem left_unique (hxz : IsCompl x z) (hyz : IsCompl y z) : x = y 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (hxz : IsCompl x z) (hyz : IsCompl y z)
Goal: x = y
-- IsCompl.sup_inf
theorem sup_inf {x' y'} (h : IsCompl x y) (h' : IsCompl x' y') : IsCompl (x ⊔ x') (y ⊓ y') 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y x' y' : α} (h : IsCompl x y) (h' : IsCompl x' y')
Goal: IsCompl (x ⊔ x') (y ⊓ y')
-- IsCompl.inf_sup
theorem inf_sup {x' y'} (h : IsCompl x y) (h' : IsCompl x' y') : IsCompl (x ⊓ x') (y ⊔ y') 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y x' y' : α} (h : IsCompl x y) (h' : IsCompl x' y')
Goal: IsCompl (x ⊓ x') (y ⊔ y')
-- Prod.disjoint_iff
protected theorem disjoint_iff [OrderBot α] [OrderBot β] {x y : α × β} :
    Disjoint x y ↔ Disjoint x.1 y.1 ∧ Disjoint x.2 y.2 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] {x y : α × β}
Goal: Disjoint x y ↔ Disjoint x.1 y.1 ∧ Disjoint x.2 y.2
-- Prod.codisjoint_iff
protected theorem codisjoint_iff [OrderTop α] [OrderTop β] {x y : α × β} :
    Codisjoint x y ↔ Codisjoint x.1 y.1 ∧ Codisjoint x.2 y.2 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] {x y : α × β}
Goal: Codisjoint x y ↔ Codisjoint x.1 y.1 ∧ Codisjoint x.2 y.2
-- Prod.isCompl_iff
protected theorem isCompl_iff [BoundedOrder α] [BoundedOrder β] {x y : α × β} :
    IsCompl x y ↔ IsCompl x.1 y.1 ∧ IsCompl x.2 y.2 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [BoundedOrder α] [BoundedOrder β] {x y : α × β}
Goal: IsCompl x y ↔ IsCompl x.1 y.1 ∧ IsCompl x.2 y.2
-- isCompl_toDual_iff
@[simp]
theorem isCompl_toDual_iff : IsCompl (toDual a) (toDual b) ↔ IsCompl a b 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : α}
Goal: IsCompl (OrderDual.toDual a) (OrderDual.toDual b) ↔ IsCompl a b
-- isCompl_ofDual_iff
@[simp]
theorem isCompl_ofDual_iff {a b : αᵒᵈ} : IsCompl (ofDual a) (ofDual b) ↔ IsCompl a b 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : αᵒᵈ}
Goal: IsCompl (OrderDual.ofDual a) (OrderDual.ofDual b) ↔ IsCompl a b
-- isCompl_bot_top
theorem isCompl_bot_top : IsCompl (⊥ : α) ⊤ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: IsCompl ⊥ ⊤
-- isCompl_top_bot
theorem isCompl_top_bot : IsCompl (⊤ : α) ⊥ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: IsCompl ⊤ ⊥
-- eq_top_of_isCompl_bot
theorem eq_top_of_isCompl_bot (h : IsCompl x ⊥) : x = ⊤ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl x ⊥)
Goal: x = ⊤
-- eq_top_of_bot_isCompl
theorem eq_top_of_bot_isCompl (h : IsCompl ⊥ x) : x = ⊤ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl ⊥ x)
Goal: x = ⊤
-- eq_bot_of_isCompl_top
theorem eq_bot_of_isCompl_top (h : IsCompl x ⊤) : x = ⊥ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl x ⊤)
Goal: x = ⊥
-- eq_bot_of_top_isCompl
theorem eq_bot_of_top_isCompl (h : IsCompl ⊤ x) : x = ⊥ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl ⊤ x)
Goal: x = ⊥
-- isComplemented_bot
theorem isComplemented_bot : IsComplemented (⊥ : α) 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: IsComplemented ⊥
-- isComplemented_top
theorem isComplemented_top : IsComplemented (⊤ : α) 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: IsComplemented ⊤
-- IsComplemented.sup
theorem IsComplemented.sup : IsComplemented a → IsComplemented b → IsComplemented (a ⊔ b) 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α}
Goal: IsComplemented a → IsComplemented b → IsComplemented (a ⊔ b)
-- IsComplemented.inf
theorem IsComplemented.inf : IsComplemented a → IsComplemented b → IsComplemented (a ⊓ b) 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α}
Goal: IsComplemented a → IsComplemented b → IsComplemented (a ⊓ b)
-- ComplementedLattice.exists_isCompl
class ComplementedLattice (α) [Lattice α] [BoundedOrder α] : Prop where
    exists_isCompl : ∀ a : α, ∃ b : α, IsCompl a b
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [self : ComplementedLattice α] (a : α)
Goal: ∃ b, IsCompl a b
-- Complementeds.coe_injective
theorem coe_injective : Injective ((↑) : Complementeds α → α) 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: Function.Injective Subtype.val
-- Complementeds.coe_inj
@[simp, norm_cast]
theorem coe_inj : (a : α) = b ↔ a = b 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : Complementeds α}
Goal: ↑a = ↑b ↔ a = b
-- Complementeds.coe_le_coe
@[norm_cast]
theorem coe_le_coe : (a : α) ≤ b ↔ a ≤ b 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : Complementeds α}
Goal: ↑a ≤ ↑b ↔ a ≤ b
-- Complementeds.coe_lt_coe
@[norm_cast]
theorem coe_lt_coe : (a : α) < b ↔ a < b 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : Complementeds α}
Goal: ↑a < ↑b ↔ a < b
-- Complementeds.coe_bot
@[simp, norm_cast]
theorem coe_bot : ((⊥ : Complementeds α) : α) = ⊥ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: ↑⊥ = ⊥
-- Complementeds.coe_top
@[simp, norm_cast]
theorem coe_top : ((⊤ : Complementeds α) : α) = ⊤ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: ↑⊤ = ⊤
-- Complementeds.mk_bot
theorem mk_bot : (⟨⊥, isComplemented_bot⟩ : Complementeds α) = ⊥ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: ⟨⊥, ⋯⟩ = ⊥
-- Complementeds.mk_top
theorem mk_top : (⟨⊤, isComplemented_top⟩ : Complementeds α) = ⊤ 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α]
Goal: ⟨⊤, ⋯⟩ = ⊤
-- Complementeds.coe_sup
@[simp, norm_cast]
theorem coe_sup (a b : Complementeds α) : ↑(a ⊔ b) = (a : α) ⊔ b 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] (a b : Complementeds α)
Goal: ↑(a ⊔ b) = ↑a ⊔ ↑b
-- Complementeds.coe_inf
@[simp, norm_cast]
theorem coe_inf (a b : Complementeds α) : ↑(a ⊓ b) = (a : α) ⊓ b 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] (a b : Complementeds α)
Goal: ↑(a ⊓ b) = ↑a ⊓ ↑b
-- Complementeds.mk_sup_mk
@[simp]
theorem mk_sup_mk {a b : α} (ha : IsComplemented a) (hb : IsComplemented b) :
    (⟨a, ha⟩ ⊔ ⟨b, hb⟩ : Complementeds α) = ⟨a ⊔ b, ha.sup hb⟩ 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α} (ha : IsComplemented a) (hb : IsComplemented b)
Goal: ⟨a, ha⟩ ⊔ ⟨b, hb⟩ = ⟨a ⊔ b, ⋯⟩
-- Complementeds.mk_inf_mk
@[simp]
theorem mk_inf_mk {a b : α} (ha : IsComplemented a) (hb : IsComplemented b) :
    (⟨a, ha⟩ ⊓ ⟨b, hb⟩ : Complementeds α) = ⟨a ⊓ b, ha.inf hb⟩ 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α} (ha : IsComplemented a) (hb : IsComplemented b)
Goal: ⟨a, ha⟩ ⊓ ⟨b, hb⟩ = ⟨a ⊓ b, ⋯⟩
-- Complementeds.disjoint_coe
@[simp, norm_cast]
theorem disjoint_coe : Disjoint (a : α) b ↔ Disjoint a b 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : Complementeds α}
Goal: Disjoint ↑a ↑b ↔ Disjoint a b
-- Complementeds.codisjoint_coe
@[simp, norm_cast]
theorem codisjoint_coe : Codisjoint (a : α) b ↔ Codisjoint a b 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : Complementeds α}
Goal: Codisjoint ↑a ↑b ↔ Codisjoint a b
-- Complementeds.isCompl_coe
@[simp, norm_cast]
theorem isCompl_coe : IsCompl (a : α) b ↔ IsCompl a b 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : Complementeds α}
Goal: IsCompl ↑a ↑b ↔ IsCompl a b
-- Filter.mem_zero
@[to_additive (attr := simp)]
theorem mem_one : s ∈ (1 : Filter α) ↔ (1 : α) ∈ s 
Arguments: {α : Type u_2} [Zero α] {s : Set α}
Goal: s ∈ 0 ↔ 0 ∈ s
-- Filter.mem_one
@[to_additive (attr := simp)]
theorem mem_one : s ∈ (1 : Filter α) ↔ (1 : α) ∈ s 
Arguments: {α : Type u_2} [One α] {s : Set α}
Goal: s ∈ 1 ↔ 1 ∈ s
-- Filter.one_mem_one
@[to_additive]
theorem one_mem_one : (1 : Set α) ∈ (1 : Filter α) 
Arguments: {α : Type u_2} [One α]
Goal: 1 ∈ 1
-- Filter.zero_mem_zero
@[to_additive]
theorem one_mem_one : (1 : Set α) ∈ (1 : Filter α) 
Arguments: {α : Type u_2} [Zero α]
Goal: 0 ∈ 0
-- Filter.pure_zero
@[to_additive (attr := simp)]
theorem pure_one : pure 1 = (1 : Filter α) 
Arguments: {α : Type u_2} [Zero α]
Goal: pure 0 = 0
-- Filter.pure_one
@[to_additive (attr := simp)]
theorem pure_one : pure 1 = (1 : Filter α) 
Arguments: {α : Type u_2} [One α]
Goal: pure 1 = 1
-- Filter.principal_zero
@[to_additive (attr := simp)]
theorem principal_one : 𝓟 1 = (1 : Filter α) 
Arguments: {α : Type u_2} [Zero α]
Goal: 𝓟 0 = 0
-- Filter.principal_one
@[to_additive (attr := simp)]
theorem principal_one : 𝓟 1 = (1 : Filter α) 
Arguments: {α : Type u_2} [One α]
Goal: 𝓟 1 = 1
-- Filter.one_neBot
@[to_additive]
theorem one_neBot : (1 : Filter α).NeBot 
Arguments: {α : Type u_2} [One α]
Goal: Filter.NeBot 1
-- Filter.zero_neBot
@[to_additive]
theorem one_neBot : (1 : Filter α).NeBot 
Arguments: {α : Type u_2} [Zero α]
Goal: Filter.NeBot 0
-- Filter.map_zero'
@[to_additive (attr := simp)]
protected theorem map_one' (f : α → β) : (1 : Filter α).map f = pure (f 1) 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] (f : α → β)
Goal: Filter.map f 0 = pure (f 0)
-- Filter.map_one'
@[to_additive (attr := simp)]
protected theorem map_one' (f : α → β) : (1 : Filter α).map f = pure (f 1) 
Arguments: {α : Type u_2} {β : Type u_3} [One α] (f : α → β)
Goal: Filter.map f 1 = pure (f 1)
-- Filter.le_one_iff
@[to_additive (attr := simp)]
theorem le_one_iff : f ≤ 1 ↔ (1 : Set α) ∈ f 
Arguments: {α : Type u_2} [One α] {f : Filter α}
Goal: f ≤ 1 ↔ 1 ∈ f
-- Filter.nonpos_iff
@[to_additive (attr := simp)]
theorem le_one_iff : f ≤ 1 ↔ (1 : Set α) ∈ f 
Arguments: {α : Type u_2} [Zero α] {f : Filter α}
Goal: f ≤ 0 ↔ 0 ∈ f
-- Filter.NeBot.le_one_iff
@[to_additive]
protected theorem NeBot.le_one_iff (h : f.NeBot) : f ≤ 1 ↔ f = 1 
Arguments: {α : Type u_2} [One α] {f : Filter α} (h : f.NeBot)
Goal: f ≤ 1 ↔ f = 1
-- Filter.NeBot.nonpos_iff
@[to_additive]
protected theorem NeBot.le_one_iff (h : f.NeBot) : f ≤ 1 ↔ f = 1 
Arguments: {α : Type u_2} [Zero α] {f : Filter α} (h : f.NeBot)
Goal: f ≤ 0 ↔ f = 0
-- Filter.eventually_one
@[to_additive (attr := simp)]
theorem eventually_one {p : α → Prop} : (∀ᶠ x in 1, p x) ↔ p 1 
Arguments: {α : Type u_2} [One α] {p : α → Prop}
Goal: (∀ᶠ (x : α) in 1, p x) ↔ p 1
-- Filter.eventually_zero
@[to_additive (attr := simp)]
theorem eventually_one {p : α → Prop} : (∀ᶠ x in 1, p x) ↔ p 1 
Arguments: {α : Type u_2} [Zero α] {p : α → Prop}
Goal: (∀ᶠ (x : α) in 0, p x) ↔ p 0
-- Filter.tendsto_one
@[to_additive (attr := simp)]
theorem tendsto_one {a : Filter β} {f : β → α} : Tendsto f a 1 ↔ ∀ᶠ x in a, f x = 1 
Arguments: {α : Type u_2} {β : Type u_3} [One α] {a : Filter β} {f : β → α}
Goal: Filter.Tendsto f a 1 ↔ ∀ᶠ (x : β) in a, f x = 1