ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- sSupHom.setImage_toFun
@[simps]
def sSupHom.setImage (f : α → β) : sSupHom (Set α) (Set β) 
Arguments: {α : Type u_2} {β : Type u_3} (f : α → β) (s : Set α)
Goal: (sSupHom.setImage f) s = f '' s
-- Equiv.toOrderIsoSet_apply
@[simps]
def Equiv.toOrderIsoSet (e : α ≃ β) : Set α ≃o Set β 
Arguments: {α : Type u_2} {β : Type u_3} (e : α ≃ β) (s : Set α)
Goal: e.toOrderIsoSet s = ⇑e '' s
-- Equiv.toOrderIsoSet_symm_apply
@[simps]
def Equiv.toOrderIsoSet (e : α ≃ β) : Set α ≃o Set β 
Arguments: {α : Type u_2} {β : Type u_3} (e : α ≃ β) (s : Set β)
Goal: (RelIso.symm e.toOrderIsoSet) s = ⇑e.symm '' s
-- supsSupHom_apply
@[simp, norm_cast]
theorem supsSupHom_apply : supsSupHom x = x.1 ⊔ x.2 
Arguments: {α : Type u_2} [CompleteLattice α] (x : α × α)
Goal: supsSupHom x = x.1 ⊔ x.2
-- infsInfHom_apply
@[simp, norm_cast]
theorem infsInfHom_apply : infsInfHom x = x.1 ⊓ x.2 
Arguments: {α : Type u_2} [CompleteLattice α] (x : α × α)
Goal: infsInfHom x = x.1 ⊓ x.2
-- OmegaCompletePartialOrder.Chain.isChain_range
lemma isChain_range : IsChain (· ≤ ·) (Set.range c) 
Arguments: {α : Type u} [Preorder α] (c : OmegaCompletePartialOrder.Chain α)
Goal: IsChain (fun x x_1 => x ≤ x_1) (Set.range ⇑c)
-- OmegaCompletePartialOrder.Chain.directed
lemma directed : Directed (· ≤ ·) c 
Arguments: {α : Type u} [Preorder α] (c : OmegaCompletePartialOrder.Chain α)
Goal: Directed (fun x x_1 => x ≤ x_1) ⇑c
-- OmegaCompletePartialOrder.Chain.map_coe
@[simp] theorem map_coe : ⇑(map c f) = f ∘ c 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] (c : OmegaCompletePartialOrder.Chain α) (f : α →o β)
Goal: ⇑(c.map f) = ⇑f ∘ ⇑c
-- OmegaCompletePartialOrder.Chain.mem_map
theorem mem_map (x : α) : x ∈ c → f x ∈ Chain.map c f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] (c : OmegaCompletePartialOrder.Chain α) {f : α →o β} (x : α)
Goal: x ∈ c → f x ∈ c.map f
-- OmegaCompletePartialOrder.Chain.exists_of_mem_map
theorem exists_of_mem_map {b : β} : b ∈ c.map f → ∃ a, a ∈ c ∧ f a = b 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] (c : OmegaCompletePartialOrder.Chain α) {f : α →o β} {b : β}
Goal: b ∈ c.map f → ∃ a ∈ c, f a = b
-- OmegaCompletePartialOrder.Chain.mem_map_iff
@[simp]
theorem mem_map_iff {b : β} : b ∈ c.map f ↔ ∃ a, a ∈ c ∧ f a = b 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] (c : OmegaCompletePartialOrder.Chain α) {f : α →o β} {b : β}
Goal: b ∈ c.map f ↔ ∃ a ∈ c, f a = b
-- OmegaCompletePartialOrder.Chain.map_id
@[simp]
theorem map_id : c.map OrderHom.id = c 
Arguments: {α : Type u} [Preorder α] (c : OmegaCompletePartialOrder.Chain α)
Goal: c.map OrderHom.id = c
-- OmegaCompletePartialOrder.Chain.map_comp
theorem map_comp : (c.map f).map g = c.map (g.comp f) 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} [Preorder α] [Preorder β] [Preorder γ] (c : OmegaCompletePartialOrder.Chain α) {f : α →o β} (g : β →o γ)
Goal: (c.map f).map g = c.map (g.comp f)
-- OmegaCompletePartialOrder.Chain.map_le_map
@[mono]
theorem map_le_map {g : α →o β} (h : f ≤ g) : c.map f ≤ c.map g 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] (c : OmegaCompletePartialOrder.Chain α) {f g : α →o β} (h : f ≤ g)
Goal: c.map f ≤ c.map g
-- OmegaCompletePartialOrder.Chain.zip_coe
@[simp] theorem zip_coe (c₀ : Chain α) (c₁ : Chain β) (n : ℕ) : c₀.zip c₁ n = (c₀ n, c₁ n) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] (c₀ : OmegaCompletePartialOrder.Chain α) (c₁ : OmegaCompletePartialOrder.Chain β) (n : ℕ)
Goal: (c₀.zip c₁) n = (c₀ n, c₁ n)
-- OmegaCompletePartialOrder.le_ωSup
class OmegaCompletePartialOrder (α : Type*) extends PartialOrder α where
    ωSup : Chain α → α
    le_ωSup : ∀ c : Chain α, ∀ i, c i ≤ ωSup c
    ωSup_le : ∀ (c : Chain α) (x), (∀ i, c i ≤ x) → ωSup c ≤ x
Arguments: {α : Type u_1} [self : OmegaCompletePartialOrder α] (c : OmegaCompletePartialOrder.Chain α) (i : ℕ)
Goal: c i ≤ OmegaCompletePartialOrder.ωSup c
-- OmegaCompletePartialOrder.ωSup_le
class OmegaCompletePartialOrder (α : Type*) extends PartialOrder α where
    ωSup : Chain α → α
    le_ωSup : ∀ c : Chain α, ∀ i, c i ≤ ωSup c
    ωSup_le : ∀ (c : Chain α) (x), (∀ i, c i ≤ x) → ωSup c ≤ x
Arguments: {α : Type u_1} [self : OmegaCompletePartialOrder α] (c : OmegaCompletePartialOrder.Chain α) (x : α)
Goal: (∀ (i : ℕ), c i ≤ x) → OmegaCompletePartialOrder.ωSup c ≤ x
-- OmegaCompletePartialOrder.le_ωSup_of_le
theorem le_ωSup_of_le {c : Chain α} {x : α} (i : ℕ) (h : x ≤ c i) : x ≤ ωSup c 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {c : OmegaCompletePartialOrder.Chain α} {x : α} (i : ℕ) (h : x ≤ c i)
Goal: x ≤ OmegaCompletePartialOrder.ωSup c
-- OmegaCompletePartialOrder.ωSup_total
theorem ωSup_total {c : Chain α} {x : α} (h : ∀ i, c i ≤ x ∨ x ≤ c i) : ωSup c ≤ x ∨ x ≤ ωSup c 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {c : OmegaCompletePartialOrder.Chain α} {x : α} (h : ∀ (i : ℕ), c i ≤ x ∨ x ≤ c i)
Goal: OmegaCompletePartialOrder.ωSup c ≤ x ∨ x ≤ OmegaCompletePartialOrder.ωSup c
-- OmegaCompletePartialOrder.ωSup_le_ωSup_of_le
@[mono]
theorem ωSup_le_ωSup_of_le {c₀ c₁ : Chain α} (h : c₀ ≤ c₁) : ωSup c₀ ≤ ωSup c₁ 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {c₀ c₁ : OmegaCompletePartialOrder.Chain α} (h : c₀ ≤ c₁)
Goal: OmegaCompletePartialOrder.ωSup c₀ ≤ OmegaCompletePartialOrder.ωSup c₁
-- OmegaCompletePartialOrder.ωSup_le_iff
theorem ωSup_le_iff (c : Chain α) (x : α) : ωSup c ≤ x ↔ ∀ i, c i ≤ x 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] (c : OmegaCompletePartialOrder.Chain α) (x : α)
Goal: OmegaCompletePartialOrder.ωSup c ≤ x ↔ ∀ (i : ℕ), c i ≤ x
-- OmegaCompletePartialOrder.isLUB_range_ωSup
lemma isLUB_range_ωSup (c : Chain α) : IsLUB (Set.range c) (ωSup c) 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] (c : OmegaCompletePartialOrder.Chain α)
Goal: IsLUB (Set.range ⇑c) (OmegaCompletePartialOrder.ωSup c)
-- OmegaCompletePartialOrder.ωSup_eq_of_isLUB
lemma ωSup_eq_of_isLUB {c : Chain α} {a : α} (h : IsLUB (Set.range c) a) : a = ωSup c 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {c : OmegaCompletePartialOrder.Chain α} {a : α} (h : IsLUB (Set.range ⇑c) a)
Goal: a = OmegaCompletePartialOrder.ωSup c
-- OmegaCompletePartialOrder.isLUB_of_scottContinuous
lemma isLUB_of_scottContinuous {c : Chain α} {f : α → β} (hf : ScottContinuous f) :
    IsLUB (Set.range (Chain.map c ⟨f, (ScottContinuous.monotone hf)⟩)) (f (ωSup c)) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] {c : OmegaCompletePartialOrder.Chain α} {f : α → β} (hf : ScottContinuous f)
Goal: IsLUB (Set.range ⇑(c.map { toFun := f, monotone' := ⋯ })) (f (OmegaCompletePartialOrder.ωSup c))
-- OmegaCompletePartialOrder.ScottContinuous.continuous'
lemma ScottContinuous.continuous' {f : α → β} (hf : ScottContinuous f) : Continuous' f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] {f : α → β} (hf : ScottContinuous f)
Goal: OmegaCompletePartialOrder.Continuous' f
-- OmegaCompletePartialOrder.Continuous'.to_monotone
theorem Continuous'.to_monotone {f : α → β} (hf : Continuous' f) : Monotone f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] {f : α → β} (hf : OmegaCompletePartialOrder.Continuous' f)
Goal: Monotone f
-- OmegaCompletePartialOrder.Continuous.of_bundled
theorem Continuous.of_bundled (f : α → β) (hf : Monotone f) (hf' : Continuous ⟨f, hf⟩) :
    Continuous' f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α → β) (hf : Monotone f) (hf' : OmegaCompletePartialOrder.Continuous { toFun := f, monotone' := hf })
Goal: OmegaCompletePartialOrder.Continuous' f
-- OmegaCompletePartialOrder.Continuous.of_bundled'
theorem Continuous.of_bundled' (f : α →o β) (hf' : Continuous f) : Continuous' f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α →o β) (hf' : OmegaCompletePartialOrder.Continuous f)
Goal: OmegaCompletePartialOrder.Continuous' ⇑f
-- OmegaCompletePartialOrder.Continuous'.to_bundled
theorem Continuous'.to_bundled (f : α → β) (hf : Continuous' f) : Continuous ⟨f, hf.to_monotone⟩ 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α → β) (hf : OmegaCompletePartialOrder.Continuous' f)
Goal: OmegaCompletePartialOrder.Continuous { toFun := f, monotone' := ⋯ }
-- OmegaCompletePartialOrder.continuous'_coe
@[simp, norm_cast]
theorem continuous'_coe : ∀ {f : α →o β}, Continuous' f ↔ Continuous f
  
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] {f : α →o β}
Goal: OmegaCompletePartialOrder.Continuous' ⇑f ↔ OmegaCompletePartialOrder.Continuous f
-- OmegaCompletePartialOrder.continuous_id
theorem continuous_id : Continuous (@OrderHom.id α _) 
Arguments: {α : Type u} [OmegaCompletePartialOrder α]
Goal: OmegaCompletePartialOrder.Continuous OrderHom.id
-- OmegaCompletePartialOrder.continuous_comp
theorem continuous_comp (hfc : Continuous f) (hgc : Continuous g) : Continuous (g.comp f) 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] [OmegaCompletePartialOrder γ] (f : α →o β) (g : β →o γ) (hfc : OmegaCompletePartialOrder.Continuous f) (hgc : OmegaCompletePartialOrder.Continuous g)
Goal: OmegaCompletePartialOrder.Continuous (g.comp f)
-- OmegaCompletePartialOrder.id_continuous'
theorem id_continuous' : Continuous' (@id α) 
Arguments: {α : Type u} [OmegaCompletePartialOrder α]
Goal: OmegaCompletePartialOrder.Continuous' id
-- OmegaCompletePartialOrder.continuous_const
theorem continuous_const (x : β) : Continuous (OrderHom.const α x) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (x : β)
Goal: OmegaCompletePartialOrder.Continuous ((OrderHom.const α) x)
-- OmegaCompletePartialOrder.const_continuous'
theorem const_continuous' (x : β) : Continuous' (Function.const α x) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (x : β)
Goal: OmegaCompletePartialOrder.Continuous' (Function.const α x)
-- Part.eq_of_chain
theorem eq_of_chain {c : Chain (Part α)} {a b : α} (ha : some a ∈ c) (hb : some b ∈ c) : a = b 
Arguments: {α : Type u} {c : OmegaCompletePartialOrder.Chain (Part α)} {a b : α} (ha : Part.some a ∈ c) (hb : Part.some b ∈ c)
Goal: a = b
-- Part.ωSup_eq_some
theorem ωSup_eq_some {c : Chain (Part α)} {a : α} (h : some a ∈ c) : Part.ωSup c = some a 
Arguments: {α : Type u} {c : OmegaCompletePartialOrder.Chain (Part α)} {a : α} (h : Part.some a ∈ c)
Goal: Part.ωSup c = Part.some a
-- Part.ωSup_eq_none
theorem ωSup_eq_none {c : Chain (Part α)} (h : ¬∃ a, some a ∈ c) : Part.ωSup c = none 
Arguments: {α : Type u} {c : OmegaCompletePartialOrder.Chain (Part α)} (h : ¬∃ a, Part.some a ∈ c)
Goal: Part.ωSup c = Part.none
-- Part.mem_chain_of_mem_ωSup
theorem mem_chain_of_mem_ωSup {c : Chain (Part α)} {a : α} (h : a ∈ Part.ωSup c) : some a ∈ c 
Arguments: {α : Type u} {c : OmegaCompletePartialOrder.Chain (Part α)} {a : α} (h : a ∈ Part.ωSup c)
Goal: Part.some a ∈ c
-- Part.mem_ωSup
theorem mem_ωSup (x : α) (c : Chain (Part α)) : x ∈ ωSup c ↔ some x ∈ c 
Arguments: {α : Type u} (x : α) (c : OmegaCompletePartialOrder.Chain (Part α))
Goal: x ∈ OmegaCompletePartialOrder.ωSup c ↔ Part.some x ∈ c
-- Pi.OmegaCompletePartialOrder.flip₁_continuous'
theorem flip₁_continuous' (f : ∀ x : α, γ → β x) (a : α) (hf : Continuous' fun x y => f y x) :
    Continuous' (f a) 
Arguments: {α : Type u_1} {β : α → Type u_2} {γ : Type u_3} [(x : α) → OmegaCompletePartialOrder (β x)] [OmegaCompletePartialOrder γ] (f : (x : α) → γ → β x) (a : α) (hf : OmegaCompletePartialOrder.Continuous' fun x y => f y x)
Goal: OmegaCompletePartialOrder.Continuous' (f a)
-- Pi.OmegaCompletePartialOrder.flip₂_continuous'
theorem flip₂_continuous' (f : γ → ∀ x, β x) (hf : ∀ x, Continuous' fun g => f g x) :
    Continuous' f 
Arguments: {α : Type u_1} {β : α → Type u_2} {γ : Type u_3} [(x : α) → OmegaCompletePartialOrder (β x)] [OmegaCompletePartialOrder γ] (f : γ → (x : α) → β x) (hf : ∀ (x : α), OmegaCompletePartialOrder.Continuous' fun g => f g x)
Goal: OmegaCompletePartialOrder.Continuous' f
-- Prod.ωSup_snd
@[simps]
protected def ωSup (c : Chain (α × β)) : α × β 
Arguments: {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (c : OmegaCompletePartialOrder.Chain (α × β))
Goal: (Prod.ωSup c).2 = OmegaCompletePartialOrder.ωSup (c.map OrderHom.snd)
-- Prod.ωSup_fst
@[simps]
protected def ωSup (c : Chain (α × β)) : α × β 
Arguments: {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (c : OmegaCompletePartialOrder.Chain (α × β))
Goal: (Prod.ωSup c).1 = OmegaCompletePartialOrder.ωSup (c.map OrderHom.fst)
-- Prod.instOmegaCompletePartialOrder_ωSup_fst
@[simps! ωSup_fst ωSup_snd]
instance : OmegaCompletePartialOrder (α × β) 
Arguments: {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (c : OmegaCompletePartialOrder.Chain (α × β))
Goal: (OmegaCompletePartialOrder.ωSup c).1 = OmegaCompletePartialOrder.ωSup (c.map OrderHom.fst)
-- Prod.instOmegaCompletePartialOrder_ωSup_snd
@[simps! ωSup_fst ωSup_snd]
instance : OmegaCompletePartialOrder (α × β) 
Arguments: {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (c : OmegaCompletePartialOrder.Chain (α × β))
Goal: (OmegaCompletePartialOrder.ωSup c).2 = OmegaCompletePartialOrder.ωSup (c.map OrderHom.snd)
-- Prod.ωSup_zip
theorem ωSup_zip (c₀ : Chain α) (c₁ : Chain β) : ωSup (c₀.zip c₁) = (ωSup c₀, ωSup c₁) 
Arguments: {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (c₀ : OmegaCompletePartialOrder.Chain α) (c₁ : OmegaCompletePartialOrder.Chain β)
Goal: OmegaCompletePartialOrder.ωSup (c₀.zip c₁) = (OmegaCompletePartialOrder.ωSup c₀, OmegaCompletePartialOrder.ωSup c₁)
-- CompleteLattice.sSup_continuous
theorem sSup_continuous (s : Set <| α →o β) (hs : ∀ f ∈ s, Continuous f) : Continuous (sSup s) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [CompleteLattice β] (s : Set (α →o β)) (hs : ∀ f ∈ s, OmegaCompletePartialOrder.Continuous f)
Goal: OmegaCompletePartialOrder.Continuous (sSup s)
-- CompleteLattice.iSup_continuous
theorem iSup_continuous {ι : Sort*} {f : ι → α →o β} (h : ∀ i, Continuous (f i)) :
    Continuous (⨆ i, f i) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [CompleteLattice β] {ι : Sort u_1} {f : ι → α →o β} (h : ∀ (i : ι), OmegaCompletePartialOrder.Continuous (f i))
Goal: OmegaCompletePartialOrder.Continuous (⨆ i, f i)
-- CompleteLattice.sSup_continuous'
theorem sSup_continuous' (s : Set (α → β)) (hc : ∀ f ∈ s, Continuous' f) :
    Continuous' (sSup s) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [CompleteLattice β] (s : Set (α → β)) (hc : ∀ f ∈ s, OmegaCompletePartialOrder.Continuous' f)
Goal: OmegaCompletePartialOrder.Continuous' (sSup s)
-- CompleteLattice.sup_continuous
theorem sup_continuous {f g : α →o β} (hf : Continuous f) (hg : Continuous g) :
    Continuous (f ⊔ g) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [CompleteLattice β] {f g : α →o β} (hf : OmegaCompletePartialOrder.Continuous f) (hg : OmegaCompletePartialOrder.Continuous g)
Goal: OmegaCompletePartialOrder.Continuous (f ⊔ g)
-- CompleteLattice.top_continuous
theorem top_continuous : Continuous (⊤ : α →o β) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [CompleteLattice β]
Goal: OmegaCompletePartialOrder.Continuous ⊤
-- CompleteLattice.bot_continuous
theorem bot_continuous : Continuous (⊥ : α →o β) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [CompleteLattice β]
Goal: OmegaCompletePartialOrder.Continuous ⊥
-- CompleteLattice.inf_continuous
theorem inf_continuous (f g : α →o β) (hf : Continuous f) (hg : Continuous g) :
    Continuous (f ⊓ g) 
Arguments: {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [CompleteLinearOrder β] (f g : α →o β) (hf : OmegaCompletePartialOrder.Continuous f) (hg : OmegaCompletePartialOrder.Continuous g)
Goal: OmegaCompletePartialOrder.Continuous (f ⊓ g)
-- CompleteLattice.inf_continuous'
theorem inf_continuous' {f g : α → β} (hf : Continuous' f) (hg : Continuous' g) :
    Continuous' (f ⊓ g) 
Arguments: {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [CompleteLinearOrder β] {f g : α → β} (hf : OmegaCompletePartialOrder.Continuous' f) (hg : OmegaCompletePartialOrder.Continuous' g)
Goal: OmegaCompletePartialOrder.Continuous' (f ⊓ g)
-- OmegaCompletePartialOrder.OrderHom.ωSup_coe
@[simps]
protected def ωSup (c : Chain (α →o β)) : α →o β 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (c : OmegaCompletePartialOrder.Chain (α →o β)) (a : α)
Goal: (OmegaCompletePartialOrder.OrderHom.ωSup c) a = OmegaCompletePartialOrder.ωSup (c.map (OrderHom.apply a))
-- OmegaCompletePartialOrder.OrderHom.omegaCompletePartialOrder_ωSup_coe
@[simps! ωSup_coe]
instance omegaCompletePartialOrder : OmegaCompletePartialOrder (α →o β) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (c : OmegaCompletePartialOrder.Chain (α →o β)) (a : α)
Goal: (OmegaCompletePartialOrder.ωSup c) a = OmegaCompletePartialOrder.ωSup (c.map (OrderHom.apply a))
-- OmegaCompletePartialOrder.ContinuousHom.cont
structure ContinuousHom extends OrderHom α β where
    cont : Continuous toOrderHom
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (self : α →𝒄 β)
Goal: OmegaCompletePartialOrder.Continuous self.toOrderHom
-- OmegaCompletePartialOrder.ContinuousHom.toOrderHom_eq_coe
theorem toOrderHom_eq_coe (f : α →𝒄 β) : f.1 = f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α →𝒄 β)
Goal: f.toOrderHom = ↑f
-- OmegaCompletePartialOrder.ContinuousHom.coe_mk
@[simp] theorem coe_mk (f : α →o β) (hf : Continuous f) : ⇑(mk f hf) = f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α →o β) (hf : OmegaCompletePartialOrder.Continuous f)
Goal: ⇑{ toOrderHom := f, cont := hf } = ⇑f
-- OmegaCompletePartialOrder.ContinuousHom.coe_toOrderHom
@[simp] theorem coe_toOrderHom (f : α →𝒄 β) : ⇑f.1 = f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α →𝒄 β)
Goal: ⇑f.toOrderHom = ⇑f
-- OmegaCompletePartialOrder.ContinuousHom.congr_fun
protected theorem congr_fun {f g : α →𝒄 β} (h : f = g) (x : α) : f x = g x 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] {f g : α →𝒄 β} (h : f = g) (x : α)
Goal: f x = g x
-- OmegaCompletePartialOrder.ContinuousHom.congr_arg
protected theorem congr_arg (f : α →𝒄 β) {x y : α} (h : x = y) : f x = f y 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α →𝒄 β) {x y : α} (h : x = y)
Goal: f x = f y
-- OmegaCompletePartialOrder.ContinuousHom.monotone
protected theorem monotone (f : α →𝒄 β) : Monotone f 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α →𝒄 β)
Goal: Monotone ⇑f
-- OmegaCompletePartialOrder.ContinuousHom.apply_mono
@[mono]
theorem apply_mono {f g : α →𝒄 β} {x y : α} (h₁ : f ≤ g) (h₂ : x ≤ y) : f x ≤ g y 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] {f g : α →𝒄 β} {x y : α} (h₁ : f ≤ g) (h₂ : x ≤ y)
Goal: f x ≤ g y
-- OmegaCompletePartialOrder.ContinuousHom.ite_continuous'
theorem ite_continuous' {p : Prop} [hp : Decidable p] (f g : α → β) (hf : Continuous' f)
    (hg : Continuous' g) : Continuous' fun x => if p then f x else g x 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] {p : Prop} [hp : Decidable p] (f g : α → β) (hf : OmegaCompletePartialOrder.Continuous' f) (hg : OmegaCompletePartialOrder.Continuous' g)
Goal: OmegaCompletePartialOrder.Continuous' fun x => if p then f x else g x
-- OmegaCompletePartialOrder.ContinuousHom.ωSup_bind
theorem ωSup_bind {β γ : Type v} (c : Chain α) (f : α →o Part β) (g : α →o β → Part γ) :
    ωSup (c.map (f.partBind g)) = ωSup (c.map f) >>= ωSup (c.map g) 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {β γ : Type v} (c : OmegaCompletePartialOrder.Chain α) (f : α →o Part β) (g : α →o β → Part γ)
Goal: OmegaCompletePartialOrder.ωSup (c.map (f.partBind g)) =
  OmegaCompletePartialOrder.ωSup (c.map f) >>= OmegaCompletePartialOrder.ωSup (c.map g)
-- OmegaCompletePartialOrder.ContinuousHom.bind_continuous'
theorem bind_continuous' {β γ : Type v} (f : α → Part β) (g : α → β → Part γ) :
    Continuous' f → Continuous' g → Continuous' fun x => f x >>= g x
  
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {β γ : Type v} (f : α → Part β) (g : α → β → Part γ)
Goal: OmegaCompletePartialOrder.Continuous' f →
  OmegaCompletePartialOrder.Continuous' g → OmegaCompletePartialOrder.Continuous' fun x => f x >>= g x
-- OmegaCompletePartialOrder.ContinuousHom.map_continuous'
theorem map_continuous' {β γ : Type v} (f : β → γ) (g : α → Part β) (hg : Continuous' g) :
    Continuous' fun x => f <$> g x 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {β γ : Type v} (f : β → γ) (g : α → Part β) (hg : OmegaCompletePartialOrder.Continuous' g)
Goal: OmegaCompletePartialOrder.Continuous' fun x => f <$> g x
-- OmegaCompletePartialOrder.ContinuousHom.seq_continuous'
theorem seq_continuous' {β γ : Type v} (f : α → Part (β → γ)) (g : α → Part β) (hf : Continuous' f)
    (hg : Continuous' g) : Continuous' fun x => f x <*> g x 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] {β γ : Type v} (f : α → Part (β → γ)) (g : α → Part β) (hf : OmegaCompletePartialOrder.Continuous' f) (hg : OmegaCompletePartialOrder.Continuous' g)
Goal: OmegaCompletePartialOrder.Continuous' fun x => Seq.seq (f x) fun x_1 => g x
-- OmegaCompletePartialOrder.ContinuousHom.continuous
theorem continuous (F : α →𝒄 β) (C : Chain α) : F (ωSup C) = ωSup (C.map F) 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (F : α →𝒄 β) (C : OmegaCompletePartialOrder.Chain α)
Goal: F (OmegaCompletePartialOrder.ωSup C) = OmegaCompletePartialOrder.ωSup (C.map ↑F)
-- OmegaCompletePartialOrder.ContinuousHom.copy_apply
-- Porting note: removed `@[reducible]`
@[simps!]
def copy (f : α → β) (g : α →𝒄 β) (h : f = g) : α →𝒄 β 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : α → β) (g : α →𝒄 β) (h : f = ⇑g)
Goal: ∀ (a : α), (OmegaCompletePartialOrder.ContinuousHom.copy f g h) a = f a
-- OmegaCompletePartialOrder.ContinuousHom.id_apply
@[simps!]
def id : α →𝒄 α 
Arguments: {α : Type u} [OmegaCompletePartialOrder α] (a : α)
Goal: OmegaCompletePartialOrder.ContinuousHom.id a = a
-- OmegaCompletePartialOrder.ContinuousHom.comp_apply
@[simps!]
def comp (f : β →𝒄 γ) (g : α →𝒄 β) : α →𝒄 γ 
Arguments: {α : Type u} {β : Type v} {γ : Type u_3} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] [OmegaCompletePartialOrder γ] (f : β →𝒄 γ) (g : α →𝒄 β)
Goal: ∀ (a : α), (f.comp g) a = f (g a)
-- OmegaCompletePartialOrder.ContinuousHom.ext
@[ext]
protected theorem ext (f g : α →𝒄 β) (h : ∀ x, f x = g x) : f = g 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f g : α →𝒄 β) (h : ∀ (x : α), f x = g x)
Goal: f = g
-- OmegaCompletePartialOrder.ContinuousHom.coe_inj
protected theorem coe_inj (f g : α →𝒄 β) (h : (f : α → β) = g) : f = g 
Arguments: {α : Type u} {β : Type v} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f g : α →𝒄 β) (h : ⇑f = ⇑g)
Goal: f = g
-- OmegaCompletePartialOrder.ContinuousHom.comp_id
@[simp]
theorem comp_id (f : β →𝒄 γ) : f.comp id = f 
Arguments: {β : Type v} {γ : Type u_3} [OmegaCompletePartialOrder β] [OmegaCompletePartialOrder γ] (f : β →𝒄 γ)
Goal: f.comp OmegaCompletePartialOrder.ContinuousHom.id = f
-- OmegaCompletePartialOrder.ContinuousHom.id_comp
@[simp]
theorem id_comp (f : β →𝒄 γ) : id.comp f = f 
Arguments: {β : Type v} {γ : Type u_3} [OmegaCompletePartialOrder β] [OmegaCompletePartialOrder γ] (f : β →𝒄 γ)
Goal: OmegaCompletePartialOrder.ContinuousHom.id.comp f = f