File: Mathlib/Order/Hom/Order.lean

full_name: OrderHom.instSup
code: instance [SemilatticeSup β] : Sup (α →o β) 
goal: Sup (α →o β)

full_name: OrderHom.coe_sup
code: @[simp] lemma coe_sup [SemilatticeSup β] (f g : α →o β) :
  ((f ⊔ g : α →o β) : α → β) = (f : α → β) ⊔ g 
goal: ⇑(f ⊔ g) = ⇑f ⊔ ⇑g

full_name: OrderHom.instSemilatticeSup
code: instance [SemilatticeSup β] : SemilatticeSup (α →o β) 
goal: SemilatticeSup (α →o β)

full_name: OrderHom.instInf
code: instance [SemilatticeInf β] : Inf (α →o β) 
goal: Inf (α →o β)

full_name: OrderHom.coe_inf
code: @[simp] lemma coe_inf [SemilatticeInf β] (f g : α →o β) :
  ((f ⊓ g : α →o β) : α → β) = (f : α → β) ⊓ g 
goal: ⇑(f ⊓ g) = ⇑f ⊓ ⇑g

full_name: OrderHom.instSemilatticeInf
code: instance [SemilatticeInf β] : SemilatticeInf (α →o β) 
goal: SemilatticeInf (α →o β)

full_name: OrderHom.lattice
code: instance lattice [Lattice β] : Lattice (α →o β) 
goal: Lattice (α →o β)

full_name: OrderHom.instBotOfOrderBot
code: @[simps]
instance [Preorder β] [OrderBot β] : Bot (α →o β) 
goal: Bot (α →o β)

full_name: OrderHom.instBotOfOrderBot_bot
code: @[simps]
instance [Preorder β] [OrderBot β] : Bot (α →o β) 
goal: ⊥ = (OrderHom.const α) ⊥

full_name: OrderHom.orderBot
code: instance orderBot [Preorder β] [OrderBot β] : OrderBot (α →o β) 
goal: OrderBot (α →o β)

full_name: OrderHom.instTopOrderHom
code: @[simps]
instance instTopOrderHom [Preorder β] [OrderTop β] : Top (α →o β) 
goal: Top (α →o β)

full_name: OrderHom.instTopOrderHom_top
code: @[simps]
instance instTopOrderHom [Preorder β] [OrderTop β] : Top (α →o β) 
goal: ⊤ = (OrderHom.const α) ⊤

full_name: OrderHom.orderTop
code: instance orderTop [Preorder β] [OrderTop β] : OrderTop (α →o β) 
goal: OrderTop (α →o β)

full_name: OrderHom.instInfSet
code: instance [CompleteLattice β] : InfSet (α →o β) 
goal: InfSet (α →o β)

full_name: OrderHom.sInf_apply
code: @[simp]
theorem sInf_apply [CompleteLattice β] (s : Set (α →o β)) (x : α) :
    sInf s x = ⨅ f ∈ s, (f : _) x 
goal: (sInf s) x = ⨅ f ∈ s, f x

full_name: OrderHom.iInf_apply
code: theorem iInf_apply {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) (x : α) :
    (⨅ i, f i) x = ⨅ i, f i x 
goal: (⨅ i, f i) x = ⨅ i, (f i) x

full_name: OrderHom.coe_iInf
code: @[simp, norm_cast]
theorem coe_iInf {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) :
    ((⨅ i, f i : α →o β) : α → β) = ⨅ i, (f i : α → β) 
goal: ⇑(⨅ i, f i) = ⨅ i, ⇑(f i)

full_name: OrderHom.instSupSet
code: instance [CompleteLattice β] : SupSet (α →o β) 
goal: SupSet (α →o β)

full_name: OrderHom.sSup_apply
code: @[simp]
theorem sSup_apply [CompleteLattice β] (s : Set (α →o β)) (x : α) :
    sSup s x = ⨆ f ∈ s, (f : _) x 
goal: (sSup s) x = ⨆ f ∈ s, f x

full_name: OrderHom.iSup_apply
code: theorem iSup_apply {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) (x : α) :
    (⨆ i, f i) x = ⨆ i, f i x 
goal: (⨆ i, f i) x = ⨆ i, (f i) x

full_name: OrderHom.coe_iSup
code: @[simp, norm_cast]
theorem coe_iSup {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) :
    ((⨆ i, f i : α →o β) : α → β) = ⨆ i, (f i : α → β) 
goal: ⇑(⨆ i, f i) = ⨆ i, ⇑(f i)

full_name: OrderHom.instCompleteLattice
code: instance [CompleteLattice β] : CompleteLattice (α →o β) 
goal: CompleteLattice (α →o β)

full_name: OrderHom.iterate_sup_le_sup_iff
code: theorem iterate_sup_le_sup_iff {α : Type*} [SemilatticeSup α] (f : α →o α) :
    (∀ n₁ n₂ a₁ a₂, f^[n₁ + n₂] (a₁ ⊔ a₂) ≤ f^[n₁] a₁ ⊔ f^[n₂] a₂) ↔
      ∀ a₁ a₂, f (a₁ ⊔ a₂) ≤ f a₁ ⊔ a₂ 
goal: (∀ (n₁ n₂ : ℕ) (a₁ a₂ : α), (⇑f)^[n₁ + n₂] (a₁ ⊔ a₂) ≤ (⇑f)^[n₁] a₁ ⊔ (⇑f)^[n₂] a₂) ↔
  ∀ (a₁ a₂ : α), f (a₁ ⊔ a₂) ≤ f a₁ ⊔ a₂