ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LatticeHom.coe_comp_inf_hom'
@[simp]
-- Porting note: `simp`-normal form of `coe_comp_inf_hom`
theorem coe_comp_inf_hom' (f : LatticeHom β γ) (g : LatticeHom α β) :
    ⟨f ∘ g, map_inf (f.comp g)⟩ = (f : InfHom 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_inf' := ⋯ } = { toFun := ⇑f, map_inf' := ⋯ }.comp { toFun := ⇑g, map_inf' := ⋯ }
-- LatticeHom.coe_comp_inf_hom
theorem coe_comp_inf_hom (f : LatticeHom β γ) (g : LatticeHom α β) :
    (f.comp g : InfHom α γ) = (f : InfHom 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_inf' := ⋯ } = { toFun := ⇑f, map_inf' := ⋯ }.comp { toFun := ⇑g, map_inf' := ⋯ }
-- LatticeHom.comp_assoc
@[simp]
theorem comp_assoc (f : LatticeHom γ δ) (g : LatticeHom β γ) (h : LatticeHom 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} [Lattice α] [Lattice β] [Lattice γ] [Lattice δ] (f : LatticeHom γ δ) (g : LatticeHom β γ) (h : LatticeHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- LatticeHom.comp_id
@[simp]
theorem comp_id (f : LatticeHom α β) : f.comp (LatticeHom.id α) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: f.comp (LatticeHom.id α) = f
-- LatticeHom.id_comp
@[simp]
theorem id_comp (f : LatticeHom α β) : (LatticeHom.id β).comp f = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: (LatticeHom.id β).comp f = f
-- LatticeHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : LatticeHom β γ} {f : LatticeHom 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} [Lattice α] [Lattice β] [Lattice γ] {g₁ g₂ : LatticeHom β γ} {f : LatticeHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- LatticeHom.cancel_left
@[simp]
theorem cancel_left {g : LatticeHom β γ} {f₁ f₂ : LatticeHom 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} [Lattice α] [Lattice β] [Lattice γ] {g : LatticeHom β γ} {f₁ f₂ : LatticeHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- OrderHomClass.coe_to_lattice_hom
@[simp]
theorem coe_to_lattice_hom (f : F) : ⇑(toLatticeHom α β f) = f 
Arguments: {F : Type u_1} (α : Type u_3) (β : Type u_4) [LinearOrder α] [Lattice β] [FunLike F α β] [OrderHomClass F α β] (f : F)
Goal: ⇑(OrderHomClass.toLatticeHom α β f) = ⇑f
-- OrderHomClass.to_lattice_hom_apply
@[simp]
theorem to_lattice_hom_apply (f : F) (a : α) : toLatticeHom α β f a = f a 
Arguments: {F : Type u_1} (α : Type u_3) (β : Type u_4) [LinearOrder α] [Lattice β] [FunLike F α β] [OrderHomClass F α β] (f : F) (a : α)
Goal: (OrderHomClass.toLatticeHom α β f) a = f a
-- BoundedLatticeHom.toFun_eq_coe
@[simp] lemma toFun_eq_coe (f : BoundedLatticeHom α β) : f.toFun = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: f.toFun = ⇑f
-- BoundedLatticeHom.coe_toLatticeHom
@[simp] lemma coe_toLatticeHom (f : BoundedLatticeHom α β) : ⇑f.toLatticeHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: ⇑f.toLatticeHom = ⇑f
-- BoundedLatticeHom.coe_toSupBotHom
@[simp] lemma coe_toSupBotHom (f : BoundedLatticeHom α β) : ⇑f.toSupBotHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: ⇑f.toSupBotHom = ⇑f
-- BoundedLatticeHom.coe_toInfTopHom
@[simp] lemma coe_toInfTopHom (f : BoundedLatticeHom α β) : ⇑f.toInfTopHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: ⇑f.toInfTopHom = ⇑f
-- BoundedLatticeHom.coe_toBoundedOrderHom
@[simp] lemma coe_toBoundedOrderHom (f : BoundedLatticeHom α β) : ⇑f.toBoundedOrderHom = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: ⇑f.toBoundedOrderHom = ⇑f
-- BoundedLatticeHom.coe_mk
@[simp] lemma coe_mk (f : LatticeHom α β) (hf hf') : ⇑(mk f hf hf') = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : LatticeHom α β) (hf : f.toFun ⊤ = ⊤) (hf' : f.toFun ⊥ = ⊥)
Goal: ⇑{ toLatticeHom := f, map_top' := hf, map_bot' := hf' } = ⇑f
-- BoundedLatticeHom.ext
@[ext]
theorem ext {f g : BoundedLatticeHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] {f g : BoundedLatticeHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- BoundedLatticeHom.coe_copy
@[simp]
theorem coe_copy (f : BoundedLatticeHom 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] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- BoundedLatticeHom.copy_eq
theorem copy_eq (f : BoundedLatticeHom 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] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- BoundedLatticeHom.coe_id
@[simp]
theorem coe_id : ⇑(BoundedLatticeHom.id α) = id 
Arguments: (α : Type u_3) [Lattice α] [BoundedOrder α]
Goal: ⇑(BoundedLatticeHom.id α) = id
-- BoundedLatticeHom.id_apply
@[simp]
theorem id_apply (a : α) : BoundedLatticeHom.id α a = a 
Arguments: {α : Type u_3} [Lattice α] [BoundedOrder α] (a : α)
Goal: (BoundedLatticeHom.id α) a = a
-- BoundedLatticeHom.coe_comp
@[simp]
theorem coe_comp (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom 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] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- BoundedLatticeHom.comp_apply
@[simp]
theorem comp_apply (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom 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] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- BoundedLatticeHom.coe_comp_lattice_hom'
@[simp]
-- Porting note: `simp`-normal form of `coe_comp_lattice_hom`
theorem coe_comp_lattice_hom' (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) :
    (⟨(f : SupHom β γ).comp g, map_inf (f.comp g)⟩ : LatticeHom α γ) =
      (f : LatticeHom β γ).comp g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β)
Goal: { toSupHom := { toFun := ⇑f, map_sup' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯ }, map_inf' := ⋯ } =
  { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯, map_inf' := ⋯ }
-- BoundedLatticeHom.coe_comp_lattice_hom
theorem coe_comp_lattice_hom (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) :
    (f.comp g : LatticeHom α γ) = (f : LatticeHom β γ).comp g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β)
Goal: { toFun := ⇑(f.comp g), map_sup' := ⋯, map_inf' := ⋯ } =
  { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯, map_inf' := ⋯ }
-- BoundedLatticeHom.coe_comp_sup_hom'
@[simp]
-- Porting note: `simp`-normal form of `coe_comp_sup_hom`
theorem coe_comp_sup_hom' (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom 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] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β)
Goal: { toFun := ⇑f ∘ ⇑g, map_sup' := ⋯ } = { toFun := ⇑f, map_sup' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯ }
-- BoundedLatticeHom.coe_comp_sup_hom
theorem coe_comp_sup_hom (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom 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] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β)
Goal: { toFun := ⇑(f.comp g), map_sup' := ⋯ } = { toFun := ⇑f, map_sup' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯ }
-- BoundedLatticeHom.coe_comp_inf_hom'
@[simp]
-- Porting note: `simp`-normal form of `coe_comp_inf_hom`
theorem coe_comp_inf_hom' (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) :
    ⟨f ∘ g, map_inf (f.comp g)⟩ = (f : InfHom β γ).comp g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β)
Goal: { toFun := ⇑f ∘ ⇑g, map_inf' := ⋯ } = { toFun := ⇑f, map_inf' := ⋯ }.comp { toFun := ⇑g, map_inf' := ⋯ }
-- BoundedLatticeHom.coe_comp_inf_hom
theorem coe_comp_inf_hom (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) :
    (f.comp g : InfHom α γ) = (f : InfHom β γ).comp g 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β)
Goal: { toFun := ⇑(f.comp g), map_inf' := ⋯ } = { toFun := ⇑f, map_inf' := ⋯ }.comp { toFun := ⇑g, map_inf' := ⋯ }
-- BoundedLatticeHom.comp_assoc
@[simp]
theorem comp_assoc (f : BoundedLatticeHom γ δ) (g : BoundedLatticeHom β γ)
    (h : BoundedLatticeHom 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} [Lattice α] [Lattice β] [Lattice γ] [Lattice δ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] [BoundedOrder δ] (f : BoundedLatticeHom γ δ) (g : BoundedLatticeHom β γ) (h : BoundedLatticeHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- BoundedLatticeHom.comp_id
@[simp] theorem comp_id (f : BoundedLatticeHom α β) : f.comp (BoundedLatticeHom.id α) = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: f.comp (BoundedLatticeHom.id α) = f
-- BoundedLatticeHom.id_comp
@[simp] theorem id_comp (f : BoundedLatticeHom α β) : (BoundedLatticeHom.id β).comp f = f 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: (BoundedLatticeHom.id β).comp f = f
-- BoundedLatticeHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : BoundedLatticeHom β γ} {f : BoundedLatticeHom 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} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g₁ g₂ : BoundedLatticeHom β γ} {f : BoundedLatticeHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- BoundedLatticeHom.cancel_left
@[simp]
theorem cancel_left {g : BoundedLatticeHom β γ} {f₁ f₂ : BoundedLatticeHom 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} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g : BoundedLatticeHom β γ} {f₁ f₂ : BoundedLatticeHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- SupHom.dual_apply_toFun
@[simps]
protected def dual : SupHom α β ≃ InfHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) (a : α)
Goal: (SupHom.dual f) a = f a
-- SupHom.dual_symm_apply_toFun
@[simps]
protected def dual : SupHom α β ≃ InfHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : InfHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ)
Goal: (SupHom.dual.symm f) a = f a
-- SupHom.dual_id
@[simp]
theorem dual_id : SupHom.dual (SupHom.id α) = InfHom.id _ 
Arguments: {α : Type u_3} [Sup α]
Goal: SupHom.dual (SupHom.id α) = InfHom.id αᵒᵈ
-- SupHom.dual_comp
@[simp]
theorem dual_comp (g : SupHom β γ) (f : SupHom α β) :
    SupHom.dual (g.comp f) = (SupHom.dual g).comp (SupHom.dual f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (g : SupHom β γ) (f : SupHom α β)
Goal: SupHom.dual (g.comp f) = (SupHom.dual g).comp (SupHom.dual f)
-- SupHom.symm_dual_id
@[simp]
theorem symm_dual_id : SupHom.dual.symm (InfHom.id _) = SupHom.id α 
Arguments: {α : Type u_3} [Sup α]
Goal: SupHom.dual.symm (InfHom.id αᵒᵈ) = SupHom.id α
-- SupHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : InfHom βᵒᵈ γᵒᵈ) (f : InfHom αᵒᵈ βᵒᵈ) :
    SupHom.dual.symm (g.comp f) =
      (SupHom.dual.symm g).comp (SupHom.dual.symm f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (g : InfHom βᵒᵈ γᵒᵈ) (f : InfHom αᵒᵈ βᵒᵈ)
Goal: SupHom.dual.symm (g.comp f) = (SupHom.dual.symm g).comp (SupHom.dual.symm f)
-- InfHom.dual_symm_apply_toFun
@[simps]
protected def dual : InfHom α β ≃ SupHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : SupHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ)
Goal: (InfHom.dual.symm f) a = f a
-- InfHom.dual_apply_toFun
@[simps]
protected def dual : InfHom α β ≃ SupHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) (a : α)
Goal: (InfHom.dual f) a = f a
-- InfHom.dual_id
@[simp]
theorem dual_id : InfHom.dual (InfHom.id α) = SupHom.id _ 
Arguments: {α : Type u_3} [Inf α]
Goal: InfHom.dual (InfHom.id α) = SupHom.id αᵒᵈ
-- InfHom.dual_comp
@[simp]
theorem dual_comp (g : InfHom β γ) (f : InfHom α β) :
    InfHom.dual (g.comp f) = (InfHom.dual g).comp (InfHom.dual f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (g : InfHom β γ) (f : InfHom α β)
Goal: InfHom.dual (g.comp f) = (InfHom.dual g).comp (InfHom.dual f)
-- InfHom.symm_dual_id
@[simp]
theorem symm_dual_id : InfHom.dual.symm (SupHom.id _) = InfHom.id α 
Arguments: {α : Type u_3} [Inf α]
Goal: InfHom.dual.symm (SupHom.id αᵒᵈ) = InfHom.id α
-- InfHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : SupHom βᵒᵈ γᵒᵈ) (f : SupHom αᵒᵈ βᵒᵈ) :
    InfHom.dual.symm (g.comp f) =
      (InfHom.dual.symm g).comp (InfHom.dual.symm f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (g : SupHom βᵒᵈ γᵒᵈ) (f : SupHom αᵒᵈ βᵒᵈ)
Goal: InfHom.dual.symm (g.comp f) = (InfHom.dual.symm g).comp (InfHom.dual.symm f)
-- SupBotHom.dual_id
@[simp] theorem dual_id : SupBotHom.dual (SupBotHom.id α) = InfTopHom.id _ 
Arguments: {α : Type u_3} [Sup α] [Bot α]
Goal: SupBotHom.dual (SupBotHom.id α) = InfTopHom.id αᵒᵈ
-- SupBotHom.dual_comp
@[simp]
theorem dual_comp (g : SupBotHom β γ) (f : SupBotHom α β) :
    SupBotHom.dual (g.comp f) = (SupBotHom.dual g).comp (SupBotHom.dual f) 
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) (f : SupBotHom α β)
Goal: SupBotHom.dual (g.comp f) = (SupBotHom.dual g).comp (SupBotHom.dual f)
-- SupBotHom.symm_dual_id
@[simp]
theorem symm_dual_id : SupBotHom.dual.symm (InfTopHom.id _) = SupBotHom.id α 
Arguments: {α : Type u_3} [Sup α] [Bot α]
Goal: SupBotHom.dual.symm (InfTopHom.id αᵒᵈ) = SupBotHom.id α
-- SupBotHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : InfTopHom βᵒᵈ γᵒᵈ) (f : InfTopHom αᵒᵈ βᵒᵈ) :
    SupBotHom.dual.symm (g.comp f) =
      (SupBotHom.dual.symm g).comp (SupBotHom.dual.symm f) 
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 : InfTopHom βᵒᵈ γᵒᵈ) (f : InfTopHom αᵒᵈ βᵒᵈ)
Goal: SupBotHom.dual.symm (g.comp f) = (SupBotHom.dual.symm g).comp (SupBotHom.dual.symm f)
-- InfTopHom.dual_symm_apply_toInfHom
@[simps]
protected def dual : InfTopHom α β ≃ SupBotHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : SupBotHom αᵒᵈ βᵒᵈ)
Goal: (InfTopHom.dual.symm f).toInfHom = InfHom.dual.symm f.toSupHom
-- InfTopHom.dual_apply_toSupHom
@[simps]
protected def dual : InfTopHom α β ≃ SupBotHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β)
Goal: (InfTopHom.dual f).toSupHom = InfHom.dual f.toInfHom
-- InfTopHom.dual_id
@[simp]
theorem dual_id : InfTopHom.dual (InfTopHom.id α) = SupBotHom.id _ 
Arguments: {α : Type u_3} [Inf α] [Top α]
Goal: InfTopHom.dual (InfTopHom.id α) = SupBotHom.id αᵒᵈ
-- InfTopHom.dual_comp
@[simp]
theorem dual_comp (g : InfTopHom β γ) (f : InfTopHom α β) :
    InfTopHom.dual (g.comp f) = (InfTopHom.dual g).comp (InfTopHom.dual f) 
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) (f : InfTopHom α β)
Goal: InfTopHom.dual (g.comp f) = (InfTopHom.dual g).comp (InfTopHom.dual f)
-- InfTopHom.symm_dual_id
@[simp]
theorem symm_dual_id : InfTopHom.dual.symm (SupBotHom.id _) = InfTopHom.id α 
Arguments: {α : Type u_3} [Inf α] [Top α]
Goal: InfTopHom.dual.symm (SupBotHom.id αᵒᵈ) = InfTopHom.id α
-- InfTopHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : SupBotHom βᵒᵈ γᵒᵈ) (f : SupBotHom αᵒᵈ βᵒᵈ) :
    InfTopHom.dual.symm (g.comp f) =
      (InfTopHom.dual.symm g).comp (InfTopHom.dual.symm f) 
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 : SupBotHom βᵒᵈ γᵒᵈ) (f : SupBotHom αᵒᵈ βᵒᵈ)
Goal: InfTopHom.dual.symm (g.comp f) = (InfTopHom.dual.symm g).comp (InfTopHom.dual.symm f)
-- LatticeHom.dual_apply_toSupHom
@[simps]
protected def dual : LatticeHom α β ≃ LatticeHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: (LatticeHom.dual f).toSupHom = InfHom.dual f.toInfHom
-- LatticeHom.dual_symm_apply_toSupHom
@[simps]
protected def dual : LatticeHom α β ≃ LatticeHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom αᵒᵈ βᵒᵈ)
Goal: (LatticeHom.dual.symm f).toSupHom = SupHom.dual.symm f.toInfHom
-- LatticeHom.dual_id
@[simp] theorem dual_id : LatticeHom.dual (LatticeHom.id α) = LatticeHom.id _ 
Arguments: {α : Type u_3} [Lattice α]
Goal: LatticeHom.dual (LatticeHom.id α) = LatticeHom.id αᵒᵈ
-- LatticeHom.dual_comp
@[simp]
theorem dual_comp (g : LatticeHom β γ) (f : LatticeHom α β) :
    LatticeHom.dual (g.comp f) = (LatticeHom.dual g).comp (LatticeHom.dual f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (g : LatticeHom β γ) (f : LatticeHom α β)
Goal: LatticeHom.dual (g.comp f) = (LatticeHom.dual g).comp (LatticeHom.dual f)
-- LatticeHom.symm_dual_id
@[simp]
theorem symm_dual_id : LatticeHom.dual.symm (LatticeHom.id _) = LatticeHom.id α 
Arguments: {α : Type u_3} [Lattice α]
Goal: LatticeHom.dual.symm (LatticeHom.id αᵒᵈ) = LatticeHom.id α
-- LatticeHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : LatticeHom βᵒᵈ γᵒᵈ) (f : LatticeHom αᵒᵈ βᵒᵈ) :
    LatticeHom.dual.symm (g.comp f) =
      (LatticeHom.dual.symm g).comp (LatticeHom.dual.symm f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (g : LatticeHom βᵒᵈ γᵒᵈ) (f : LatticeHom αᵒᵈ βᵒᵈ)
Goal: LatticeHom.dual.symm (g.comp f) = (LatticeHom.dual.symm g).comp (LatticeHom.dual.symm f)
-- BoundedLatticeHom.dual_apply_toLatticeHom
@[simps]
protected def dual : BoundedLatticeHom α β ≃ BoundedLatticeHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] (f : BoundedLatticeHom α β)
Goal: (BoundedLatticeHom.dual f).toLatticeHom = LatticeHom.dual f.toLatticeHom
-- BoundedLatticeHom.dual_symm_apply_toLatticeHom
@[simps]
protected def dual : BoundedLatticeHom α β ≃ BoundedLatticeHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] (f : BoundedLatticeHom αᵒᵈ βᵒᵈ)
Goal: (BoundedLatticeHom.dual.symm f).toLatticeHom = LatticeHom.dual.symm f.toLatticeHom
-- BoundedLatticeHom.dual_id
@[simp]
theorem dual_id : BoundedLatticeHom.dual (BoundedLatticeHom.id α) = BoundedLatticeHom.id _ 
Arguments: {α : Type u_3} [Lattice α] [BoundedOrder α]
Goal: BoundedLatticeHom.dual (BoundedLatticeHom.id α) = BoundedLatticeHom.id αᵒᵈ
-- BoundedLatticeHom.dual_comp
@[simp]
theorem dual_comp (g : BoundedLatticeHom β γ) (f : BoundedLatticeHom α β) :
    BoundedLatticeHom.dual (g.comp f) =
      (BoundedLatticeHom.dual g).comp (BoundedLatticeHom.dual f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [Lattice γ] [BoundedOrder γ] (g : BoundedLatticeHom β γ) (f : BoundedLatticeHom α β)
Goal: BoundedLatticeHom.dual (g.comp f) = (BoundedLatticeHom.dual g).comp (BoundedLatticeHom.dual f)
-- BoundedLatticeHom.symm_dual_id
@[simp]
theorem symm_dual_id :
    BoundedLatticeHom.dual.symm (BoundedLatticeHom.id _) = BoundedLatticeHom.id α 
Arguments: {α : Type u_3} [Lattice α] [BoundedOrder α]
Goal: BoundedLatticeHom.dual.symm (BoundedLatticeHom.id αᵒᵈ) = BoundedLatticeHom.id α
-- BoundedLatticeHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : BoundedLatticeHom βᵒᵈ γᵒᵈ) (f : BoundedLatticeHom αᵒᵈ βᵒᵈ) :
    BoundedLatticeHom.dual.symm (g.comp f) =
      (BoundedLatticeHom.dual.symm g).comp (BoundedLatticeHom.dual.symm f) 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [Lattice γ] [BoundedOrder γ] (g : BoundedLatticeHom βᵒᵈ γᵒᵈ) (f : BoundedLatticeHom αᵒᵈ βᵒᵈ)
Goal: BoundedLatticeHom.dual.symm (g.comp f) = (BoundedLatticeHom.dual.symm g).comp (BoundedLatticeHom.dual.symm f)
-- LatticeHom.coe_fst
@[simp, norm_cast] lemma coe_fst : ⇑(fst (α := α) (β := β)) = Prod.fst 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β]
Goal: ⇑LatticeHom.fst = Prod.fst
-- LatticeHom.coe_snd
@[simp, norm_cast] lemma coe_snd : ⇑(snd (α := α) (β := β)) = Prod.snd 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β]
Goal: ⇑LatticeHom.snd = Prod.snd
-- LatticeHom.fst_apply
lemma fst_apply (x : α × β) : fst x = x.fst 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (x : α × β)
Goal: LatticeHom.fst x = x.1
-- LatticeHom.snd_apply
lemma snd_apply (x : α × β) : snd x = x.snd 
Arguments: {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (x : α × β)
Goal: LatticeHom.snd x = x.2
-- Pi.coe_evalLatticeHom
@[simp, norm_cast]
lemma coe_evalLatticeHom (i : ι) : ⇑(evalLatticeHom (α := α) i) = Function.eval i 
Arguments: {ι : Type u_7} {α : ι → Type u_8} [(i : ι) → Lattice (α i)] (i : ι)
Goal: ⇑(Pi.evalLatticeHom i) = Function.eval i
-- Pi.evalLatticeHom_apply
lemma evalLatticeHom_apply (i : ι) (f : ∀ i, α i) : evalLatticeHom i f = f i 
Arguments: {ι : Type u_7} {α : ι → Type u_8} [(i : ι) → Lattice (α i)] (i : ι) (f : (i : ι) → α i)
Goal: (Pi.evalLatticeHom i) f = f i
-- SupHom.withTop_toFun
@[simps]
protected def withTop (f : SupHom α β) : SupHom (WithTop α) (WithTop β) 
Arguments: {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] (f : SupHom α β)
Goal: ∀ (a : WithTop α), f.withTop a = WithTop.map (⇑f) a
-- SupHom.withTop_id
@[simp]
theorem withTop_id : (SupHom.id α).withTop = SupHom.id _ 
Arguments: {α : Type u_3} [SemilatticeSup α]
Goal: (SupHom.id α).withTop = SupHom.id (WithTop α)
-- SupHom.withTop_comp
@[simp]
theorem withTop_comp (f : SupHom β γ) (g : SupHom α β) :
    (f.comp g).withTop = f.withTop.comp g.withTop 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [SemilatticeSup α] [SemilatticeSup β] [SemilatticeSup γ] (f : SupHom β γ) (g : SupHom α β)
Goal: (f.comp g).withTop = f.withTop.comp g.withTop
-- SupHom.withBot_toSupHom_toFun
@[simps]
protected def withBot (f : SupHom α β) : SupBotHom (WithBot α) (WithBot β) 
Arguments: {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] (f : SupHom α β)
Goal: ∀ (a : Option α), f.withBot.toSupHom a = Option.map (⇑f) a