ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrderHom.sSup_apply
@[simp]
theorem sSup_apply [CompleteLattice β] (s : Set (α →o β)) (x : α) :
    sSup s x = ⨆ f ∈ s, (f : _) x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [CompleteLattice β] (s : Set (α →o β)) (x : α)
Goal: (sSup s) x = ⨆ f ∈ s, f x
-- OrderHom.iSup_apply
theorem iSup_apply {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) (x : α) :
    (⨆ i, f i) x = ⨆ i, f i x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] {ι : Sort u_3} [CompleteLattice β] (f : ι → α →o β) (x : α)
Goal: (⨆ i, f i) x = ⨆ i, (f i) x
-- OrderHom.coe_iSup
@[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) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] {ι : Sort u_3} [CompleteLattice β] (f : ι → α →o β)
Goal: ⇑(⨆ i, f i) = ⨆ i, ⇑(f i)
-- OrderHom.iterate_sup_le_sup_iff
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 
Arguments: {α : Type u_3} [SemilatticeSup α] (f : α →o α)
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
-- self_le_wellOrderSucc
lemma self_le_wellOrderSucc (a : α) : a ≤ wellOrderSucc a 
Arguments: {α : Type u_1} [LinearOrder α] [IsWellOrder α fun x x_1 => x < x_1] (a : α)
Goal: a ≤ wellOrderSucc a
-- wellOrderSucc_le
lemma wellOrderSucc_le {a b : α} (ha : a < b) : wellOrderSucc a ≤ b 
Arguments: {α : Type u_1} [LinearOrder α] [IsWellOrder α fun x x_1 => x < x_1] {a b : α} (ha : a < b)
Goal: wellOrderSucc a ≤ b
-- self_lt_wellOrderSucc
lemma self_lt_wellOrderSucc {a b : α} (h : a < b) : a < wellOrderSucc a 
Arguments: {α : Type u_1} [LinearOrder α] [IsWellOrder α fun x x_1 => x < x_1] {a b : α} (h : a < b)
Goal: a < wellOrderSucc a
-- le_of_lt_wellOrderSucc
lemma le_of_lt_wellOrderSucc {a b : α} (h : a < wellOrderSucc b) : a ≤ b 
Arguments: {α : Type u_1} [LinearOrder α] [IsWellOrder α fun x x_1 => x < x_1] {a b : α} (h : a < wellOrderSucc b)
Goal: a ≤ b
-- IsWellOrderLimitElement.not_succ
class IsWellOrderLimitElement (a : α) : Prop where
  not_bot : ∃ (b : α), b < a
  not_succ (b : α) (hb : b < a) : ∃ (c : α), b < c ∧ c < a
Arguments: {α : Type u_1} [LinearOrder α] {a : α} [self : IsWellOrderLimitElement a] (b : α) (hb : b < a)
Goal: ∃ c, b < c ∧ c < a
-- IsWellOrderLimitElement.not_bot
class IsWellOrderLimitElement (a : α) : Prop where
  not_bot : ∃ (b : α), b < a
  not_succ (b : α) (hb : b < a) : ∃ (c : α), b < c ∧ c < a
Arguments: {α : Type u_1} [LinearOrder α] {a : α} [self : IsWellOrderLimitElement a]
Goal: ∃ b, b < a
-- IsWellOrderLimitElement.neq_bot
lemma IsWellOrderLimitElement.neq_bot [OrderBot α] : a ≠ ⊥ 
Arguments: {α : Type u_1} [LinearOrder α] (a : α) [ha : IsWellOrderLimitElement a] [OrderBot α]
Goal: a ≠ ⊥
-- IsWellOrderLimitElement.bot_lt
lemma IsWellOrderLimitElement.bot_lt [OrderBot α] : ⊥ < a 
Arguments: {α : Type u_1} [LinearOrder α] (a : α) [ha : IsWellOrderLimitElement a] [OrderBot α]
Goal: ⊥ < a
-- IsWellOrderLimitElement.wellOrderSucc_lt
lemma IsWellOrderLimitElement.wellOrderSucc_lt {b : α} (hb : b < a) :
    wellOrderSucc b < a 
Arguments: {α : Type u_1} [LinearOrder α] {a : α} [ha : IsWellOrderLimitElement a] [IsWellOrder α fun x x_1 => x < x_1] {b : α} (hb : b < a)
Goal: wellOrderSucc b < a
-- eq_bot_or_eq_succ_or_isWellOrderLimitElement
lemma eq_bot_or_eq_succ_or_isWellOrderLimitElement [OrderBot α] (a : α) :
    a = ⊥ ∨ (∃ b, a = wellOrderSucc b ∧ b < a) ∨ IsWellOrderLimitElement a 
Arguments: {α : Type u_1} [LinearOrder α] [IsWellOrder α fun x x_1 => x < x_1] [OrderBot α] (a : α)
Goal: a = ⊥ ∨ (∃ b, a = wellOrderSucc b ∧ b < a) ∨ IsWellOrderLimitElement a
-- IsWellOrderLimitElement.neq_succ
lemma IsWellOrderLimitElement.neq_succ (a : α) (ha : a < wellOrderSucc a)
    [IsWellOrderLimitElement (wellOrderSucc a)] : False 
Arguments: {α : Type u_1} [LinearOrder α] [IsWellOrder α fun x x_1 => x < x_1] (a : α) (ha : a < wellOrderSucc a) [IsWellOrderLimitElement (wellOrderSucc a)]
Goal: False
-- Nat.wellOrderSucc_eq
@[simp]
lemma Nat.wellOrderSucc_eq (a : ℕ) : wellOrderSucc a = succ a 
Arguments: (a : ℕ)
Goal: wellOrderSucc a = a.succ
-- Nat.not_isWellOrderLimitElement
lemma Nat.not_isWellOrderLimitElement (a : ℕ) [IsWellOrderLimitElement a] : False 
Arguments: (a : ℕ) [IsWellOrderLimitElement a]
Goal: False
-- TopHom.map_top'
structure TopHom (α β : Type*) [Top α] [Top β] where
    toFun : α → β
    map_top' : toFun ⊤ = ⊤
Arguments: {α : Type u_6} {β : Type u_7} [Top α] [Top β] (self : TopHom α β)
Goal: self.toFun ⊤ = ⊤
-- BotHom.map_bot'
structure BotHom (α β : Type*) [Bot α] [Bot β] where
    toFun : α → β
    map_bot' : toFun ⊥ = ⊥
Arguments: {α : Type u_6} {β : Type u_7} [Bot α] [Bot β] (self : BotHom α β)
Goal: self.toFun ⊥ = ⊥
-- BoundedOrderHom.map_bot'
structure BoundedOrderHom (α β : Type*) [Preorder α] [Preorder β] [BoundedOrder α]
  [BoundedOrder β] extends OrderHom α β where
    map_top' : toFun ⊤ = ⊤
    map_bot' : toFun ⊥ = ⊥
Arguments: {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (self : BoundedOrderHom α β)
Goal: self.toFun ⊥ = ⊥
-- BoundedOrderHom.map_top'
structure BoundedOrderHom (α β : Type*) [Preorder α] [Preorder β] [BoundedOrder α]
  [BoundedOrder β] extends OrderHom α β where
    map_top' : toFun ⊤ = ⊤
    map_bot' : toFun ⊥ = ⊥
Arguments: {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (self : BoundedOrderHom α β)
Goal: self.toFun ⊤ = ⊤
-- TopHomClass.map_top
class TopHomClass (F α β : Type*) [Top α] [Top β] [FunLike F α β] : Prop where
    map_top (f : F) : f ⊤ = ⊤
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [Top α] [Top β] [FunLike F α β] [self : TopHomClass F α β] (f : F)
Goal: f ⊤ = ⊤
-- BotHomClass.map_bot
class BotHomClass (F α β : Type*) [Bot α] [Bot β] [FunLike F α β] : Prop where
    map_bot (f : F) : f ⊥ = ⊥
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [Bot α] [Bot β] [FunLike F α β] [self : BotHomClass F α β] (f : F)
Goal: f ⊥ = ⊥
-- BoundedOrderHomClass.map_top
class BoundedOrderHomClass (F α β : Type*) [LE α] [LE β]
  [BoundedOrder α] [BoundedOrder β] [FunLike F α β]
  extends RelHomClass F ((· ≤ ·) : α → α → Prop) ((· ≤ ·) : β → β → Prop) : Prop where
    map_top (f : F) : f ⊤ = ⊤
    map_bot (f : F) : f ⊥ = ⊥
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [LE α] [LE β] [BoundedOrder α] [BoundedOrder β] [FunLike F α β] [self : BoundedOrderHomClass F α β] (f : F)
Goal: f ⊤ = ⊤
-- BoundedOrderHomClass.map_bot
class BoundedOrderHomClass (F α β : Type*) [LE α] [LE β]
  [BoundedOrder α] [BoundedOrder β] [FunLike F α β]
  extends RelHomClass F ((· ≤ ·) : α → α → Prop) ((· ≤ ·) : β → β → Prop) : Prop where
    map_top (f : F) : f ⊤ = ⊤
    map_bot (f : F) : f ⊥ = ⊥
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [LE α] [LE β] [BoundedOrder α] [BoundedOrder β] [FunLike F α β] [self : BoundedOrderHomClass F α β] (f : F)
Goal: f ⊥ = ⊥
-- map_eq_top_iff
@[simp]
theorem map_eq_top_iff [LE α] [OrderTop α] [PartialOrder β] [OrderTop β] [OrderIsoClass F α β]
    (f : F) {a : α} : f a = ⊤ ↔ a = ⊤ 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [LE α] [OrderTop α] [PartialOrder β] [OrderTop β] [OrderIsoClass F α β] (f : F) {a : α}
Goal: f a = ⊤ ↔ a = ⊤
-- map_eq_bot_iff
@[simp]
theorem map_eq_bot_iff [LE α] [OrderBot α] [PartialOrder β] [OrderBot β] [OrderIsoClass F α β]
    (f : F) {a : α} : f a = ⊥ ↔ a = ⊥ 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [LE α] [OrderBot α] [PartialOrder β] [OrderBot β] [OrderIsoClass F α β] (f : F) {a : α}
Goal: f a = ⊥ ↔ a = ⊥
-- TopHom.ext
@[ext]
theorem ext {f g : TopHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [Top β] {f g : TopHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- TopHom.coe_copy
@[simp]
theorem coe_copy (f : TopHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- TopHom.copy_eq
theorem copy_eq (f : TopHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- TopHom.coe_id
@[simp]
theorem coe_id : ⇑(TopHom.id α) = id 
Arguments: (α : Type u_2) [Top α]
Goal: ⇑(TopHom.id α) = id
-- TopHom.id_apply
@[simp]
theorem id_apply (a : α) : TopHom.id α a = a 
Arguments: {α : Type u_2} [Top α] (a : α)
Goal: (TopHom.id α) a = a
-- TopHom.coe_comp
@[simp]
theorem coe_comp (f : TopHom β γ) (g : TopHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] (f : TopHom β γ) (g : TopHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- TopHom.comp_apply
@[simp]
theorem comp_apply (f : TopHom β γ) (g : TopHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] (f : TopHom β γ) (g : TopHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- TopHom.comp_assoc
@[simp]
theorem comp_assoc (f : TopHom γ δ) (g : TopHom β γ) (h : TopHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Top α] [Top β] [Top γ] [Top δ] (f : TopHom γ δ) (g : TopHom β γ) (h : TopHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- TopHom.comp_id
@[simp]
theorem comp_id (f : TopHom α β) : f.comp (TopHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β)
Goal: f.comp (TopHom.id α) = f
-- TopHom.id_comp
@[simp]
theorem id_comp (f : TopHom α β) : (TopHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β)
Goal: (TopHom.id β).comp f = f
-- TopHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : TopHom β γ} {f : TopHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] {g₁ g₂ : TopHom β γ} {f : TopHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- TopHom.cancel_left
@[simp]
theorem cancel_left {g : TopHom β γ} {f₁ f₂ : TopHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] {g : TopHom β γ} {f₁ f₂ : TopHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- TopHom.coe_top
@[simp]
theorem coe_top : ⇑(⊤ : TopHom α β) = ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [LE β] [OrderTop β]
Goal: ⇑⊤ = ⊤
-- TopHom.top_apply
@[simp]
theorem top_apply (a : α) : (⊤ : TopHom α β) a = ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [LE β] [OrderTop β] (a : α)
Goal: ⊤ a = ⊤
-- TopHom.coe_inf
@[simp]
theorem coe_inf : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeInf β] [OrderTop β] (f g : TopHom α β)
Goal: ⇑(f ⊓ g) = ⇑f ⊓ ⇑g
-- TopHom.inf_apply
@[simp]
theorem inf_apply (a : α) : (f ⊓ g) a = f a ⊓ g a 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeInf β] [OrderTop β] (f g : TopHom α β) (a : α)
Goal: (f ⊓ g) a = f a ⊓ g a
-- TopHom.coe_sup
@[simp]
theorem coe_sup : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeSup β] [OrderTop β] (f g : TopHom α β)
Goal: ⇑(f ⊔ g) = ⇑f ⊔ ⇑g
-- TopHom.sup_apply
@[simp]
theorem sup_apply (a : α) : (f ⊔ g) a = f a ⊔ g a 
Arguments: {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeSup β] [OrderTop β] (f g : TopHom α β) (a : α)
Goal: (f ⊔ g) a = f a ⊔ g a
-- BotHom.ext
@[ext]
theorem ext {f g : BotHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] {f g : BotHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- BotHom.coe_copy
@[simp]
theorem coe_copy (f : BotHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- BotHom.copy_eq
theorem copy_eq (f : BotHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- BotHom.coe_id
@[simp]
theorem coe_id : ⇑(BotHom.id α) = id 
Arguments: (α : Type u_2) [Bot α]
Goal: ⇑(BotHom.id α) = id
-- BotHom.id_apply
@[simp]
theorem id_apply (a : α) : BotHom.id α a = a 
Arguments: {α : Type u_2} [Bot α] (a : α)
Goal: (BotHom.id α) a = a
-- BotHom.coe_comp
@[simp]
theorem coe_comp (f : BotHom β γ) (g : BotHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] (f : BotHom β γ) (g : BotHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- BotHom.comp_apply
@[simp]
theorem comp_apply (f : BotHom β γ) (g : BotHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] (f : BotHom β γ) (g : BotHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- BotHom.comp_assoc
@[simp]
theorem comp_assoc (f : BotHom γ δ) (g : BotHom β γ) (h : BotHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Bot α] [Bot β] [Bot γ] [Bot δ] (f : BotHom γ δ) (g : BotHom β γ) (h : BotHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- BotHom.comp_id
@[simp]
theorem comp_id (f : BotHom α β) : f.comp (BotHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β)
Goal: f.comp (BotHom.id α) = f
-- BotHom.id_comp
@[simp]
theorem id_comp (f : BotHom α β) : (BotHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β)
Goal: (BotHom.id β).comp f = f
-- BotHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : BotHom β γ} {f : BotHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] {g₁ g₂ : BotHom β γ} {f : BotHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- BotHom.cancel_left
@[simp]
theorem cancel_left {g : BotHom β γ} {f₁ f₂ : BotHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] {g : BotHom β γ} {f₁ f₂ : BotHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- BotHom.coe_bot
@[simp]
theorem coe_bot : ⇑(⊥ : BotHom α β) = ⊥ 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [LE β] [OrderBot β]
Goal: ⇑⊥ = ⊥
-- BotHom.bot_apply
@[simp]
theorem bot_apply (a : α) : (⊥ : BotHom α β) a = ⊥ 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [LE β] [OrderBot β] (a : α)
Goal: ⊥ a = ⊥
-- BotHom.coe_inf
@[simp]
theorem coe_inf : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeInf β] [OrderBot β] (f g : BotHom α β)
Goal: ⇑(f ⊓ g) = ⇑f ⊓ ⇑g
-- BotHom.inf_apply
@[simp]
theorem inf_apply (a : α) : (f ⊓ g) a = f a ⊓ g a 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeInf β] [OrderBot β] (f g : BotHom α β) (a : α)
Goal: (f ⊓ g) a = f a ⊓ g a
-- BotHom.coe_sup
@[simp]
theorem coe_sup : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeSup β] [OrderBot β] (f g : BotHom α β)
Goal: ⇑(f ⊔ g) = ⇑f ⊔ ⇑g
-- BotHom.sup_apply
@[simp]
theorem sup_apply (a : α) : (f ⊔ g) a = f a ⊔ g a 
Arguments: {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeSup β] [OrderBot β] (f g : BotHom α β) (a : α)
Goal: (f ⊔ g) a = f a ⊔ g a
-- BoundedOrderHom.ext
@[ext]
theorem ext {f g : BoundedOrderHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] {f g : BoundedOrderHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- BoundedOrderHom.coe_copy
@[simp]
theorem coe_copy (f : BoundedOrderHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- BoundedOrderHom.copy_eq
theorem copy_eq (f : BoundedOrderHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- BoundedOrderHom.coe_id
@[simp]
theorem coe_id : ⇑(BoundedOrderHom.id α) = id 
Arguments: (α : Type u_2) [Preorder α] [BoundedOrder α]
Goal: ⇑(BoundedOrderHom.id α) = id
-- BoundedOrderHom.id_apply
@[simp]
theorem id_apply (a : α) : BoundedOrderHom.id α a = a 
Arguments: {α : Type u_2} [Preorder α] [BoundedOrder α] (a : α)
Goal: (BoundedOrderHom.id α) a = a
-- BoundedOrderHom.coe_comp
@[simp]
theorem coe_comp (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- BoundedOrderHom.comp_apply
@[simp]
theorem comp_apply (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) (a : α) :
    (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- BoundedOrderHom.coe_comp_orderHom
@[simp]
theorem coe_comp_orderHom (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) :
    (f.comp g : OrderHom α γ) = (f : OrderHom β γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- BoundedOrderHom.coe_comp_topHom
@[simp]
theorem coe_comp_topHom (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) :
    (f.comp g : TopHom α γ) = (f : TopHom β γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- BoundedOrderHom.coe_comp_botHom
@[simp]
theorem coe_comp_botHom (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) :
    (f.comp g : BotHom α γ) = (f : BotHom β γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- BoundedOrderHom.comp_assoc
@[simp]
theorem comp_assoc (f : BoundedOrderHom γ δ) (g : BoundedOrderHom β γ) (h : BoundedOrderHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] [BoundedOrder δ] (f : BoundedOrderHom γ δ) (g : BoundedOrderHom β γ) (h : BoundedOrderHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- BoundedOrderHom.comp_id
@[simp]
theorem comp_id (f : BoundedOrderHom α β) : f.comp (BoundedOrderHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β)
Goal: f.comp (BoundedOrderHom.id α) = f
-- BoundedOrderHom.id_comp
@[simp]
theorem id_comp (f : BoundedOrderHom α β) : (BoundedOrderHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β)
Goal: (BoundedOrderHom.id β).comp f = f
-- BoundedOrderHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : BoundedOrderHom β γ} {f : BoundedOrderHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g₁ g₂ : BoundedOrderHom β γ} {f : BoundedOrderHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- BoundedOrderHom.cancel_left
@[simp]
theorem cancel_left {g : BoundedOrderHom β γ} {f₁ f₂ : BoundedOrderHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g : BoundedOrderHom β γ} {f₁ f₂ : BoundedOrderHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- TopHom.dual_apply_apply
@[simps]
protected def dual :
    TopHom α β ≃ BotHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [OrderTop α] [LE β] [OrderTop β] (f : TopHom α β) (a : α)
Goal: (TopHom.dual f) a = f a
-- TopHom.dual_symm_apply_apply
@[simps]
protected def dual :
    TopHom α β ≃ BotHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [OrderTop α] [LE β] [OrderTop β] (f : BotHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ)
Goal: (TopHom.dual.symm f) a = f a
-- TopHom.dual_id
@[simp]
theorem dual_id : TopHom.dual (TopHom.id α) = BotHom.id _ 
Arguments: {α : Type u_2} [LE α] [OrderTop α]
Goal: TopHom.dual (TopHom.id α) = BotHom.id αᵒᵈ
-- TopHom.dual_comp
@[simp]
theorem dual_comp (g : TopHom β γ) (f : TopHom α β) :
    TopHom.dual (g.comp f) = g.dual.comp (TopHom.dual f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderTop α] [LE β] [OrderTop β] [LE γ] [OrderTop γ] (g : TopHom β γ) (f : TopHom α β)
Goal: TopHom.dual (g.comp f) = (TopHom.dual g).comp (TopHom.dual f)
-- TopHom.symm_dual_id
@[simp]
theorem symm_dual_id : TopHom.dual.symm (BotHom.id _) = TopHom.id α 
Arguments: {α : Type u_2} [LE α] [OrderTop α]
Goal: TopHom.dual.symm (BotHom.id αᵒᵈ) = TopHom.id α
-- TopHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : BotHom βᵒᵈ γᵒᵈ) (f : BotHom αᵒᵈ βᵒᵈ) :
    TopHom.dual.symm (g.comp f) = (TopHom.dual.symm g).comp (TopHom.dual.symm f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderTop α] [LE β] [OrderTop β] [LE γ] [OrderTop γ] (g : BotHom βᵒᵈ γᵒᵈ) (f : BotHom αᵒᵈ βᵒᵈ)
Goal: TopHom.dual.symm (g.comp f) = (TopHom.dual.symm g).comp (TopHom.dual.symm f)
-- BotHom.dual_apply_apply
@[simps]
protected def dual :
    BotHom α β ≃ TopHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [OrderBot α] [LE β] [OrderBot β] (f : BotHom α β) (a : α)
Goal: (BotHom.dual f) a = f a
-- BotHom.dual_symm_apply_apply
@[simps]
protected def dual :
    BotHom α β ≃ TopHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [OrderBot α] [LE β] [OrderBot β] (f : TopHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ)
Goal: (BotHom.dual.symm f) a = f a
-- BotHom.dual_id
@[simp]
theorem dual_id : BotHom.dual (BotHom.id α) = TopHom.id _ 
Arguments: {α : Type u_2} [LE α] [OrderBot α]
Goal: BotHom.dual (BotHom.id α) = TopHom.id αᵒᵈ
-- BotHom.dual_comp
@[simp]
theorem dual_comp (g : BotHom β γ) (f : BotHom α β) :
    BotHom.dual (g.comp f) = g.dual.comp (BotHom.dual f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderBot α] [LE β] [OrderBot β] [LE γ] [OrderBot γ] (g : BotHom β γ) (f : BotHom α β)
Goal: BotHom.dual (g.comp f) = (BotHom.dual g).comp (BotHom.dual f)
-- BotHom.symm_dual_id
@[simp]
theorem symm_dual_id : BotHom.dual.symm (TopHom.id _) = BotHom.id α 
Arguments: {α : Type u_2} [LE α] [OrderBot α]
Goal: BotHom.dual.symm (TopHom.id αᵒᵈ) = BotHom.id α
-- BotHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : TopHom βᵒᵈ γᵒᵈ) (f : TopHom αᵒᵈ βᵒᵈ) :
    BotHom.dual.symm (g.comp f) = (BotHom.dual.symm g).comp (BotHom.dual.symm f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderBot α] [LE β] [OrderBot β] [LE γ] [OrderBot γ] (g : TopHom βᵒᵈ γᵒᵈ) (f : TopHom αᵒᵈ βᵒᵈ)
Goal: BotHom.dual.symm (g.comp f) = (BotHom.dual.symm g).comp (BotHom.dual.symm f)
-- BoundedOrderHom.dual_apply_toOrderHom
@[simps]
protected def dual :
    BoundedOrderHom α β ≃
      BoundedOrderHom αᵒᵈ
        βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [BoundedOrder α] [Preorder β] [BoundedOrder β] (f : BoundedOrderHom α β)
Goal: (BoundedOrderHom.dual f).toOrderHom = OrderHom.dual f.toOrderHom