File: Mathlib/Order/LatticeIntervals.lean

full_name: Set.Ico.semilatticeInf
code: instance semilatticeInf [SemilatticeInf α] {a b : α} : SemilatticeInf (Ico a b) 
goal: SemilatticeInf ↑(Set.Ico a b)

full_name: Set.Ico.orderBot
code: /-- `Ico a b` has a bottom element whenever `a < b`. -/
protected abbrev orderBot [PartialOrder α] {a b : α} (h : a < b) : OrderBot (Ico a b) 
goal: OrderBot ↑(Set.Ico a b)

full_name: Set.Iio.semilatticeInf
code: instance semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Iio a) 
goal: SemilatticeInf ↑(Set.Iio a)

full_name: Set.Ioc.semilatticeSup
code: instance semilatticeSup [SemilatticeSup α] {a b : α} : SemilatticeSup (Ioc a b) 
goal: SemilatticeSup ↑(Set.Ioc a b)

full_name: Set.Ioc.orderTop
code: /-- `Ioc a b` has a top element whenever `a < b`. -/
protected abbrev orderTop [PartialOrder α] {a b : α} (h : a < b) : OrderTop (Ioc a b) 
goal: OrderTop ↑(Set.Ioc a b)

full_name: Set.Ioi.semilatticeSup
code: instance semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Ioi a) 
goal: SemilatticeSup ↑(Set.Ioi a)

full_name: Set.Iic.semilatticeInf
code: instance semilatticeInf [SemilatticeInf α] : SemilatticeInf (Iic a) 
goal: SemilatticeInf ↑(Set.Iic a)

full_name: Set.Iic.coe_inf
code: @[simp, norm_cast]
protected lemma coe_inf [SemilatticeInf α] {x y : Iic a} :
    (↑(x ⊓ y) : α) = (x : α) ⊓ (y : α) 
goal: ↑(x ⊓ y) = ↑x ⊓ ↑y

full_name: Set.Iic.semilatticeSup
code: instance semilatticeSup [SemilatticeSup α] : SemilatticeSup (Iic a) 
goal: SemilatticeSup ↑(Set.Iic a)

full_name: Set.Iic.coe_sup
code: @[simp, norm_cast]
protected lemma coe_sup [SemilatticeSup α] {x y : Iic a} :
    (↑(x ⊔ y) : α) = (x : α) ⊔ (y : α) 
goal: ↑(x ⊔ y) = ↑x ⊔ ↑y

full_name: Set.Iic.instLatticeElem
code: instance [Lattice α] : Lattice (Iic a) 
goal: Lattice ↑(Set.Iic a)

full_name: Set.Iic.orderTop
code: instance orderTop [Preorder α] :
    OrderTop (Iic a) 
goal: OrderTop ↑(Set.Iic a)

full_name: Set.Iic.coe_top
code: @[simp]
theorem coe_top [Preorder α] : (⊤ : Iic a) = a 
goal: ↑⊤ = a

full_name: Set.Iic.eq_top_iff
code: protected lemma eq_top_iff [Preorder α] {x : Iic a} :
    x = ⊤ ↔ (x : α) = a 
goal: x = ⊤ ↔ ↑x = a

full_name: Set.Iic.orderBot
code: instance orderBot [Preorder α] [OrderBot α] :
    OrderBot (Iic a) 
goal: OrderBot ↑(Set.Iic a)

full_name: Set.Iic.coe_bot
code: @[simp]
theorem coe_bot [Preorder α] [OrderBot α] : (⊥ : Iic a) = (⊥ : α) 
goal: ↑⊥ = ⊥

full_name: Set.Iic.instBoundedOrderElemOfOrderBot
code: instance [Preorder α] [OrderBot α] : BoundedOrder (Iic a) 
goal: BoundedOrder ↑(Set.Iic a)

full_name: Set.Iic.disjoint_iff
code: protected lemma disjoint_iff [SemilatticeInf α] [OrderBot α] {x y : Iic a} :
    Disjoint x y ↔ Disjoint (x : α) (y : α) 
goal: Disjoint x y ↔ Disjoint ↑x ↑y

full_name: Set.Iic.codisjoint_iff
code: protected lemma codisjoint_iff [SemilatticeSup α] {x y : Iic a} :
    Codisjoint x y ↔ (x : α) ⊔ (y : α) = a 
goal: Codisjoint x y ↔ ↑x ⊔ ↑y = a

full_name: Set.Iic.isCompl_iff
code: protected lemma isCompl_iff [Lattice α] [BoundedOrder α] {x y : Iic a} :
    IsCompl x y ↔ Disjoint (x : α) (y : α) ∧ (x : α) ⊔ (y : α) = a 
goal: IsCompl x y ↔ Disjoint ↑x ↑y ∧ ↑x ⊔ ↑y = a

full_name: Set.Ici.semilatticeInf
code: instance semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Ici a) 
goal: SemilatticeInf ↑(Set.Ici a)

full_name: Set.Ici.semilatticeSup
code: instance semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Ici a) 
goal: SemilatticeSup ↑(Set.Ici a)

full_name: Set.Ici.lattice
code: instance lattice [Lattice α] {a : α} : Lattice (Ici a) 
goal: Lattice ↑(Set.Ici a)

full_name: Set.Ici.distribLattice
code: instance distribLattice [DistribLattice α] {a : α} : DistribLattice (Ici a) 
goal: DistribLattice ↑(Set.Ici a)

full_name: Set.Ici.orderBot
code: instance orderBot [Preorder α] {a : α} :
    OrderBot (Ici a) 
goal: OrderBot ↑(Set.Ici a)

full_name: Set.Ici.coe_bot
code: @[simp]
theorem coe_bot [Preorder α] {a : α} : ↑(⊥ : Ici a) = a 
goal: ↑⊥ = a

full_name: Set.Ici.orderTop
code: instance orderTop [Preorder α] [OrderTop α] {a : α} :
    OrderTop (Ici a) 
goal: OrderTop ↑(Set.Ici a)

full_name: Set.Ici.coe_top
code: @[simp]
theorem coe_top [Preorder α] [OrderTop α] {a : α} : ↑(⊤ : Ici a) = (⊤ : α) 
goal: ↑⊤ = ⊤

full_name: Set.Ici.boundedOrder
code: instance boundedOrder [Preorder α] [OrderTop α] {a : α} : BoundedOrder (Ici a) 
goal: BoundedOrder ↑(Set.Ici a)

full_name: Set.Icc.semilatticeInf
code: instance semilatticeInf [SemilatticeInf α] {a b : α} : SemilatticeInf (Icc a b) 
goal: SemilatticeInf ↑(Set.Icc a b)

full_name: Set.Icc.semilatticeSup
code: instance semilatticeSup [SemilatticeSup α] {a b : α} : SemilatticeSup (Icc a b) 
goal: SemilatticeSup ↑(Set.Icc a b)

full_name: Set.Icc.lattice
code: instance lattice [Lattice α] {a b : α} : Lattice (Icc a b) 
goal: Lattice ↑(Set.Icc a b)

full_name: Set.Icc.orderBot
code: /-- `Icc a b` has a bottom element whenever `a ≤ b`. -/
protected abbrev orderBot [Preorder α] {a b : α} (h : a ≤ b) : OrderBot (Icc a b) 
goal: OrderBot ↑(Set.Icc a b)

full_name: Set.Icc.orderTop
code: /-- `Icc a b` has a top element whenever `a ≤ b`. -/
protected abbrev orderTop [Preorder α] {a b : α} (h : a ≤ b) : OrderTop (Icc a b) 
goal: OrderTop ↑(Set.Icc a b)

full_name: Set.Icc.boundedOrder
code: /-- `Icc a b` is a `BoundedOrder` whenever `a ≤ b`. -/
protected abbrev boundedOrder [Preorder α] {a b : α} (h : a ≤ b) : BoundedOrder (Icc a b) 
goal: BoundedOrder ↑(Set.Icc a b)