ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsCompl.map
theorem IsCompl.map (h : IsCompl a b) : IsCompl (f a) (f b) 
Arguments: {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [FunLike F α β] [BoundedLatticeHomClass F α β] (f : F) {a b : α} (h : IsCompl a b)
Goal: IsCompl (f a) (f b)
-- map_compl'
theorem map_compl' (a : α) : f aᶜ = (f a)ᶜ 
Arguments: {F : Type u_1} {α : Type u_3} {β : Type u_4} [BooleanAlgebra α] [BooleanAlgebra β] [FunLike F α β] [BoundedLatticeHomClass F α β] (f : F) (a : α)
Goal: f aᶜ = (f a)ᶜ
-- map_sdiff'
theorem map_sdiff' (a b : α) : f (a \ b) = f a \ f b 
Arguments: {F : Type u_1} {α : Type u_3} {β : Type u_4} [BooleanAlgebra α] [BooleanAlgebra β] [FunLike F α β] [BoundedLatticeHomClass F α β] (f : F) (a b : α)
Goal: f (a \ b) = f a \ f b
-- map_symmDiff'
theorem map_symmDiff' (a b : α) : f (a ∆ b) = f a ∆ f b 
Arguments: {F : Type u_1} {α : Type u_3} {β : Type u_4} [BooleanAlgebra α] [BooleanAlgebra β] [FunLike F α β] [BoundedLatticeHomClass F α β] (f : F) (a b : α)
Goal: f (symmDiff a b) = symmDiff (f a) (f b)
-- SupHom.toFun_eq_coe
@[simp] lemma toFun_eq_coe (f : SupHom α β) : f.toFun = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β)
Goal: f.toFun = ⇑f
-- SupHom.coe_mk
@[simp, norm_cast] lemma coe_mk (f : α → β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : α → β) (hf : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b)
Goal: ⇑{ toFun := f, map_sup' := hf } = f
-- SupHom.ext
@[ext]
theorem ext {f g : SupHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] {f g : SupHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- SupHom.coe_copy
@[simp]
theorem coe_copy (f : SupHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- SupHom.copy_eq
theorem copy_eq (f : SupHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- SupHom.coe_id
@[simp]
theorem coe_id : ⇑(SupHom.id α) = id 
Arguments: (α : Type u_3) [Sup α]
Goal: ⇑(SupHom.id α) = id
-- SupHom.id_apply
@[simp]
theorem id_apply (a : α) : SupHom.id α a = a 
Arguments: {α : Type u_3} [Sup α] (a : α)
Goal: (SupHom.id α) a = a
-- SupHom.coe_comp
@[simp]
theorem coe_comp (f : SupHom β γ) (g : SupHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (f : SupHom β γ) (g : SupHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- SupHom.comp_apply
@[simp]
theorem comp_apply (f : SupHom β γ) (g : SupHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (f : SupHom β γ) (g : SupHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- SupHom.comp_assoc
@[simp]
theorem comp_assoc (f : SupHom γ δ) (g : SupHom β γ) (h : SupHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Sup α] [Sup β] [Sup γ] [Sup δ] (f : SupHom γ δ) (g : SupHom β γ) (h : SupHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- SupHom.comp_id
@[simp] theorem comp_id (f : SupHom α β) : f.comp (SupHom.id α) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β)
Goal: f.comp (SupHom.id α) = f
-- SupHom.id_comp
@[simp] theorem id_comp (f : SupHom α β) : (SupHom.id β).comp f = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β)
Goal: (SupHom.id β).comp f = f
-- SupHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : SupHom β γ} {f : SupHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] {g₁ g₂ : SupHom β γ} {f : SupHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- SupHom.cancel_left
@[simp]
theorem cancel_left {g : SupHom β γ} {f₁ f₂ : SupHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] {g : SupHom β γ} {f₁ f₂ : SupHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- SupHom.coe_const
@[simp]
theorem coe_const (b : β) : ⇑(const α b) = Function.const α b 
Arguments: (α : Type u_3) {β : Type u_4} [Sup α] [SemilatticeSup β] (b : β)
Goal: ⇑(SupHom.const α b) = Function.const α b
-- SupHom.const_apply
@[simp]
theorem const_apply (b : β) (a : α) : const α b a = b 
Arguments: (α : Type u_3) {β : Type u_4} [Sup α] [SemilatticeSup β] (b : β) (a : α)
Goal: (SupHom.const α b) a = b
-- SupHom.coe_sup
@[simp]
theorem coe_sup (f g : SupHom α β) : DFunLike.coe (f ⊔ g) = f ⊔ g 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] (f g : SupHom α β)
Goal: ⇑(f ⊔ g) = ⇑(f ⊔ g)
-- SupHom.coe_bot
@[simp]
theorem coe_bot [Bot β] : ⇑(⊥ : SupHom α β) = ⊥ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Bot β]
Goal: ⇑⊥ = ⊥
-- SupHom.coe_top
@[simp]
theorem coe_top [Top β] : ⇑(⊤ : SupHom α β) = ⊤ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Top β]
Goal: ⇑⊤ = ⊤
-- SupHom.sup_apply
@[simp]
theorem sup_apply (f g : SupHom α β) (a : α) : (f ⊔ g) a = f a ⊔ g a 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] (f g : SupHom α β) (a : α)
Goal: (f ⊔ g) a = f a ⊔ g a
-- SupHom.bot_apply
@[simp]
theorem bot_apply [Bot β] (a : α) : (⊥ : SupHom α β) a = ⊥ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Bot β] (a : α)
Goal: ⊥ a = ⊥
-- SupHom.top_apply
@[simp]
theorem top_apply [Top β] (a : α) : (⊤ : SupHom α β) a = ⊤ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Top β] (a : α)
Goal: ⊤ a = ⊤
-- InfHom.toFun_eq_coe
@[simp] lemma toFun_eq_coe (f : InfHom α β) : f.toFun = (f : α → β) 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β)
Goal: f.toFun = ⇑f
-- InfHom.coe_mk
@[simp, norm_cast] lemma coe_mk (f : α → β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : α → β) (hf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b)
Goal: ⇑{ toFun := f, map_inf' := hf } = f
-- InfHom.ext
@[ext]
theorem ext {f g : InfHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] {f g : InfHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- InfHom.coe_copy
@[simp]
theorem coe_copy (f : InfHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- InfHom.copy_eq
theorem copy_eq (f : InfHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- InfHom.coe_id
@[simp]
theorem coe_id : ⇑(InfHom.id α) = id 
Arguments: (α : Type u_3) [Inf α]
Goal: ⇑(InfHom.id α) = id
-- InfHom.id_apply
@[simp]
theorem id_apply (a : α) : InfHom.id α a = a 
Arguments: {α : Type u_3} [Inf α] (a : α)
Goal: (InfHom.id α) a = a
-- InfHom.coe_comp
@[simp]
theorem coe_comp (f : InfHom β γ) (g : InfHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (f : InfHom β γ) (g : InfHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- InfHom.comp_apply
@[simp]
theorem comp_apply (f : InfHom β γ) (g : InfHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (f : InfHom β γ) (g : InfHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- InfHom.comp_assoc
@[simp]
theorem comp_assoc (f : InfHom γ δ) (g : InfHom β γ) (h : InfHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Inf α] [Inf β] [Inf γ] [Inf δ] (f : InfHom γ δ) (g : InfHom β γ) (h : InfHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- InfHom.comp_id
@[simp] theorem comp_id (f : InfHom α β) : f.comp (InfHom.id α) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β)
Goal: f.comp (InfHom.id α) = f
-- InfHom.id_comp
@[simp] theorem id_comp (f : InfHom α β) : (InfHom.id β).comp f = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β)
Goal: (InfHom.id β).comp f = f
-- InfHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : InfHom β γ} {f : InfHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] {g₁ g₂ : InfHom β γ} {f : InfHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- InfHom.cancel_left
@[simp]
theorem cancel_left {g : InfHom β γ} {f₁ f₂ : InfHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] {g : InfHom β γ} {f₁ f₂ : InfHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- InfHom.coe_const
@[simp]
theorem coe_const (b : β) : ⇑(const α b) = Function.const α b 
Arguments: (α : Type u_3) {β : Type u_4} [Inf α] [SemilatticeInf β] (b : β)
Goal: ⇑(InfHom.const α b) = Function.const α b
-- InfHom.const_apply
@[simp]
theorem const_apply (b : β) (a : α) : const α b a = b 
Arguments: (α : Type u_3) {β : Type u_4} [Inf α] [SemilatticeInf β] (b : β) (a : α)
Goal: (InfHom.const α b) a = b
-- InfHom.coe_inf
@[simp]
theorem coe_inf (f g : InfHom α β) : DFunLike.coe (f ⊓ g) = f ⊓ g 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] (f g : InfHom α β)
Goal: ⇑(f ⊓ g) = ⇑(f ⊓ g)
-- InfHom.coe_bot
@[simp]
theorem coe_bot [Bot β] : ⇑(⊥ : InfHom α β) = ⊥ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Bot β]
Goal: ⇑⊥ = ⊥
-- InfHom.coe_top
@[simp]
theorem coe_top [Top β] : ⇑(⊤ : InfHom α β) = ⊤ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Top β]
Goal: ⇑⊤ = ⊤
-- InfHom.inf_apply
@[simp]
theorem inf_apply (f g : InfHom α β) (a : α) : (f ⊓ g) a = f a ⊓ g a 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] (f g : InfHom α β) (a : α)
Goal: (f ⊓ g) a = f a ⊓ g a
-- InfHom.bot_apply
@[simp]
theorem bot_apply [Bot β] (a : α) : (⊥ : InfHom α β) a = ⊥ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Bot β] (a : α)
Goal: ⊥ a = ⊥
-- InfHom.top_apply
@[simp]
theorem top_apply [Top β] (a : α) : (⊤ : InfHom α β) a = ⊤ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Top β] (a : α)
Goal: ⊤ a = ⊤
-- SupBotHom.toFun_eq_coe
lemma toFun_eq_coe (f : SupBotHom α β) : f.toFun = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β)
Goal: f.toFun = ⇑f
-- SupBotHom.coe_toSupHom
@[simp] lemma coe_toSupHom (f : SupBotHom α β) : ⇑f.toSupHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β)
Goal: ⇑f.toSupHom = ⇑f
-- SupBotHom.coe_toBotHom
@[simp] lemma coe_toBotHom (f : SupBotHom α β) : ⇑f.toBotHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β)
Goal: ⇑f.toBotHom = ⇑f
-- SupBotHom.coe_mk
@[simp] lemma coe_mk (f : SupHom α β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupHom α β) (hf : f.toFun ⊥ = ⊥)
Goal: ⇑{ toSupHom := f, map_bot' := hf } = ⇑f
-- SupBotHom.ext
@[ext]
theorem ext {f g : SupBotHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] {f g : SupBotHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- SupBotHom.coe_copy
@[simp]
theorem coe_copy (f : SupBotHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- SupBotHom.copy_eq
theorem copy_eq (f : SupBotHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- SupBotHom.id_toSupHom
@[simps]
protected def id : SupBotHom α α 
Arguments: (α : Type u_3) [Sup α] [Bot α]
Goal: (SupBotHom.id α).toSupHom = SupHom.id α
-- SupBotHom.coe_id
@[simp]
theorem coe_id : ⇑(SupBotHom.id α) = id 
Arguments: (α : Type u_3) [Sup α] [Bot α]
Goal: ⇑(SupBotHom.id α) = id
-- SupBotHom.id_apply
@[simp]
theorem id_apply (a : α) : SupBotHom.id α a = a 
Arguments: {α : Type u_3} [Sup α] [Bot α] (a : α)
Goal: (SupBotHom.id α) a = a
-- SupBotHom.coe_comp
@[simp]
theorem coe_comp (f : SupBotHom β γ) (g : SupBotHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] (f : SupBotHom β γ) (g : SupBotHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- SupBotHom.comp_apply
@[simp]
theorem comp_apply (f : SupBotHom β γ) (g : SupBotHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] (f : SupBotHom β γ) (g : SupBotHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- SupBotHom.comp_assoc
@[simp]
theorem comp_assoc (f : SupBotHom γ δ) (g : SupBotHom β γ) (h : SupBotHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] [Sup δ] [Bot δ] (f : SupBotHom γ δ) (g : SupBotHom β γ) (h : SupBotHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- SupBotHom.comp_id
@[simp] theorem comp_id (f : SupBotHom α β) : f.comp (SupBotHom.id α) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β)
Goal: f.comp (SupBotHom.id α) = f
-- SupBotHom.id_comp
@[simp] theorem id_comp (f : SupBotHom α β) : (SupBotHom.id β).comp f = f 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β)
Goal: (SupBotHom.id β).comp f = f
-- SupBotHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : SupBotHom β γ} {f : SupBotHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] {g₁ g₂ : SupBotHom β γ} {f : SupBotHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- SupBotHom.cancel_left
@[simp]
theorem cancel_left {g : SupBotHom β γ} {f₁ f₂ : SupBotHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] {g : SupBotHom β γ} {f₁ f₂ : SupBotHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- SupBotHom.coe_sup
@[simp]
theorem coe_sup (f g : SupBotHom α β) : DFunLike.coe (f ⊔ g) = f ⊔ g 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] (f g : SupBotHom α β)
Goal: ⇑(f ⊔ g) = ⇑(f ⊔ g)
-- SupBotHom.coe_bot
@[simp]
theorem coe_bot : ⇑(⊥ : SupBotHom α β) = ⊥ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β]
Goal: ⇑⊥ = ⊥
-- SupBotHom.sup_apply
@[simp]
theorem sup_apply (f g : SupBotHom α β) (a : α) : (f ⊔ g) a = f a ⊔ g a 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] (f g : SupBotHom α β) (a : α)
Goal: (f ⊔ g) a = f a ⊔ g a
-- SupBotHom.bot_apply
@[simp]
theorem bot_apply (a : α) : (⊥ : SupBotHom α β) a = ⊥ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] (a : α)
Goal: ⊥ a = ⊥
-- InfTopHom.toFun_eq_coe
theorem toFun_eq_coe (f : InfTopHom α β) : f.toFun = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β)
Goal: f.toFun = ⇑f
-- InfTopHom.coe_toInfHom
@[simp] lemma coe_toInfHom (f : InfTopHom α β) : ⇑f.toInfHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β)
Goal: ⇑f.toInfHom = ⇑f
-- InfTopHom.coe_toTopHom
@[simp] lemma coe_toTopHom (f : InfTopHom α β) : ⇑f.toTopHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β)
Goal: ⇑f.toTopHom = ⇑f
-- InfTopHom.coe_mk
@[simp] lemma coe_mk (f : InfHom α β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfHom α β) (hf : f.toFun ⊤ = ⊤)
Goal: ⇑{ toInfHom := f, map_top' := hf } = ⇑f
-- InfTopHom.ext
@[ext]
theorem ext {f g : InfTopHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] {f g : InfTopHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- InfTopHom.coe_copy
@[simp]
theorem coe_copy (f : InfTopHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- InfTopHom.copy_eq
theorem copy_eq (f : InfTopHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- InfTopHom.id_toInfHom
@[simps]
protected def id : InfTopHom α α 
Arguments: (α : Type u_3) [Inf α] [Top α]
Goal: (InfTopHom.id α).toInfHom = InfHom.id α
-- InfTopHom.coe_id
@[simp]
theorem coe_id : ⇑(InfTopHom.id α) = id 
Arguments: (α : Type u_3) [Inf α] [Top α]
Goal: ⇑(InfTopHom.id α) = id
-- InfTopHom.id_apply
@[simp]
theorem id_apply (a : α) : InfTopHom.id α a = a 
Arguments: {α : Type u_3} [Inf α] [Top α] (a : α)
Goal: (InfTopHom.id α) a = a
-- InfTopHom.coe_comp
@[simp]
theorem coe_comp (f : InfTopHom β γ) (g : InfTopHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] (f : InfTopHom β γ) (g : InfTopHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- InfTopHom.comp_apply
@[simp]
theorem comp_apply (f : InfTopHom β γ) (g : InfTopHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] (f : InfTopHom β γ) (g : InfTopHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- InfTopHom.comp_assoc
@[simp]
theorem comp_assoc (f : InfTopHom γ δ) (g : InfTopHom β γ) (h : InfTopHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] [Inf δ] [Top δ] (f : InfTopHom γ δ) (g : InfTopHom β γ) (h : InfTopHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- InfTopHom.comp_id
@[simp] theorem comp_id (f : InfTopHom α β) : f.comp (InfTopHom.id α) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β)
Goal: f.comp (InfTopHom.id α) = f
-- InfTopHom.id_comp
@[simp] theorem id_comp (f : InfTopHom α β) : (InfTopHom.id β).comp f = f 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β)
Goal: (InfTopHom.id β).comp f = f
-- InfTopHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : InfTopHom β γ} {f : InfTopHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] {g₁ g₂ : InfTopHom β γ} {f : InfTopHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- InfTopHom.cancel_left
@[simp]
theorem cancel_left {g : InfTopHom β γ} {f₁ f₂ : InfTopHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] {g : InfTopHom β γ} {f₁ f₂ : InfTopHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- InfTopHom.coe_inf
@[simp]
theorem coe_inf (f g : InfTopHom α β) : DFunLike.coe (f ⊓ g) = f ⊓ g 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] (f g : InfTopHom α β)
Goal: ⇑(f ⊓ g) = ⇑(f ⊓ g)
-- InfTopHom.coe_top
@[simp]
theorem coe_top : ⇑(⊤ : InfTopHom α β) = ⊤ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β]
Goal: ⇑⊤ = ⊤
-- InfTopHom.inf_apply
@[simp]
theorem inf_apply (f g : InfTopHom α β) (a : α) : (f ⊓ g) a = f a ⊓ g a 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] (f g : InfTopHom α β) (a : α)
Goal: (f ⊓ g) a = f a ⊓ g a
-- InfTopHom.top_apply
@[simp]
theorem top_apply (a : α) : (⊤ : InfTopHom α β) a = ⊤ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] (a : α)
Goal: ⊤ a = ⊤
-- LatticeHom.toFun_eq_coe
lemma toFun_eq_coe (f : LatticeHom α β) : f.toFun = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: f.toFun = ⇑f
-- LatticeHom.coe_toSupHom
@[simp] lemma coe_toSupHom (f : LatticeHom α β) : ⇑f.toSupHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: ⇑f.toSupHom = ⇑f
-- LatticeHom.coe_toInfHom
@[simp] lemma coe_toInfHom (f : LatticeHom α β) : ⇑f.toInfHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: ⇑f.toInfHom = ⇑f
-- LatticeHom.coe_mk
@[simp] lemma coe_mk (f : SupHom α β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : SupHom α β) (hf : ∀ (a b : α), f.toFun (a ⊓ b) = f.toFun a ⊓ f.toFun b)
Goal: ⇑{ toSupHom := f, map_inf' := hf } = ⇑f
-- LatticeHom.ext
@[ext]
theorem ext {f g : LatticeHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] {f g : LatticeHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- LatticeHom.coe_copy
@[simp]
theorem coe_copy (f : LatticeHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- LatticeHom.copy_eq
theorem copy_eq (f : LatticeHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- LatticeHom.coe_id
@[simp]
theorem coe_id : ⇑(LatticeHom.id α) = id 
Arguments: (α : Type u_3) [Lattice α]
Goal: ⇑(LatticeHom.id α) = id
-- LatticeHom.id_apply
@[simp]
theorem id_apply (a : α) : LatticeHom.id α a = a 
Arguments: {α : Type u_3} [Lattice α] (a : α)
Goal: (LatticeHom.id α) a = a
-- LatticeHom.coe_comp
@[simp]
theorem coe_comp (f : LatticeHom β γ) (g : LatticeHom α β) : (f.comp g : α → γ) = f ∘ g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- LatticeHom.comp_apply
@[simp]
theorem comp_apply (f : LatticeHom β γ) (g : LatticeHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- LatticeHom.coe_comp_sup_hom'
@[simp]
-- Porting note: `simp`-normal form of `coe_comp_sup_hom`
theorem coe_comp_sup_hom' (f : LatticeHom β γ) (g : LatticeHom α β) :
    ⟨f ∘ g, map_sup (f.comp g)⟩ = (f : SupHom β γ).comp g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β)
Goal: { toFun := ⇑f ∘ ⇑g, map_sup' := ⋯ } = { toFun := ⇑f, map_sup' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯ }
-- LatticeHom.coe_comp_sup_hom
theorem coe_comp_sup_hom (f : LatticeHom β γ) (g : LatticeHom α β) :
    (f.comp g : SupHom α γ) = (f : SupHom β γ).comp g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β)
Goal: { toFun := ⇑(f.comp g), map_sup' := ⋯ } = { toFun := ⇑f, map_sup' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯ }