ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrderAddMonoidHom.ext
@[to_additive (attr := ext)]
theorem ext (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] {f g : α →+o β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- OrderMonoidHom.ext
@[to_additive (attr := ext)]
theorem ext (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] {f g : α →*o β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- OrderAddMonoidHom.toFun_eq_coe
@[to_additive]
theorem toFun_eq_coe (f : α →*o β) : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (f : α →+o β)
Goal: (↑f.toAddMonoidHom).toFun = ⇑f
-- OrderMonoidHom.toFun_eq_coe
@[to_additive]
theorem toFun_eq_coe (f : α →*o β) : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (f : α →*o β)
Goal: (↑f.toMonoidHom).toFun = ⇑f
-- OrderAddMonoidHom.coe_mk
@[to_additive (attr := simp)]
theorem coe_mk (f : α →* β) (h) : (OrderMonoidHom.mk f h : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (f : α →+ β) (h : Monotone (↑f).toFun)
Goal: ⇑{ toAddMonoidHom := f, monotone' := h } = ⇑f
-- OrderMonoidHom.coe_mk
@[to_additive (attr := simp)]
theorem coe_mk (f : α →* β) (h) : (OrderMonoidHom.mk f h : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (f : α →* β) (h : Monotone (↑f).toFun)
Goal: ⇑{ toMonoidHom := f, monotone' := h } = ⇑f
-- OrderAddMonoidHom.mk_coe
@[to_additive (attr := simp)]
theorem mk_coe (f : α →*o β) (h) : OrderMonoidHom.mk (f : α →* β) h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (f : α →+o β) (h : Monotone (↑↑f).toFun)
Goal: { toAddMonoidHom := ↑f, monotone' := h } = f
-- OrderMonoidHom.mk_coe
@[to_additive (attr := simp)]
theorem mk_coe (f : α →*o β) (h) : OrderMonoidHom.mk (f : α →* β) h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (f : α →*o β) (h : Monotone (↑↑f).toFun)
Goal: { toMonoidHom := ↑f, monotone' := h } = f
-- OrderAddMonoidHom.coe_addMonoidHom
@[to_additive (attr := simp)]
theorem coe_monoidHom (f : α →*o β) : ((f : α →* β) : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (f : α →+o β)
Goal: ⇑↑f = ⇑f
-- OrderMonoidHom.coe_monoidHom
@[to_additive (attr := simp)]
theorem coe_monoidHom (f : α →*o β) : ((f : α →* β) : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (f : α →*o β)
Goal: ⇑↑f = ⇑f
-- OrderMonoidHom.coe_orderHom
@[to_additive (attr := simp)]
theorem coe_orderHom (f : α →*o β) : ((f : α →o β) : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (f : α →*o β)
Goal: ⇑↑f = ⇑f
-- OrderAddMonoidHom.coe_orderHom
@[to_additive (attr := simp)]
theorem coe_orderHom (f : α →*o β) : ((f : α →o β) : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (f : α →+o β)
Goal: ⇑↑f = ⇑f
-- OrderMonoidHom.toMonoidHom_injective
@[to_additive]
theorem toMonoidHom_injective : Injective (toMonoidHom : _ → α →* β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β]
Goal: Function.Injective OrderMonoidHom.toMonoidHom
-- OrderAddMonoidHom.toAddMonoidHom_injective
@[to_additive]
theorem toMonoidHom_injective : Injective (toMonoidHom : _ → α →* β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β]
Goal: Function.Injective OrderAddMonoidHom.toAddMonoidHom
-- OrderMonoidHom.toOrderHom_injective
@[to_additive]
theorem toOrderHom_injective : Injective (toOrderHom : _ → α →o β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β]
Goal: Function.Injective OrderMonoidHom.toOrderHom
-- OrderAddMonoidHom.toOrderHom_injective
@[to_additive]
theorem toOrderHom_injective : Injective (toOrderHom : _ → α →o β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β]
Goal: Function.Injective OrderAddMonoidHom.toOrderHom
-- OrderMonoidHom.coe_copy
@[to_additive (attr := simp)]
theorem coe_copy (f : α →*o 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] [MulOneClass α] [MulOneClass β] (f : α →*o β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- OrderAddMonoidHom.coe_copy
@[to_additive (attr := simp)]
theorem coe_copy (f : α →*o 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] [AddZeroClass α] [AddZeroClass β] (f : α →+o β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- OrderMonoidHom.copy_eq
@[to_additive]
theorem copy_eq (f : α →*o 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] [MulOneClass α] [MulOneClass β] (f : α →*o β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- OrderAddMonoidHom.copy_eq
@[to_additive]
theorem copy_eq (f : α →*o 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] [AddZeroClass α] [AddZeroClass β] (f : α →+o β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- OrderAddMonoidHom.coe_id
@[to_additive (attr := simp)]
theorem coe_id : ⇑(OrderMonoidHom.id α) = id 
Arguments: (α : Type u_2) [Preorder α] [AddZeroClass α]
Goal: ⇑(OrderAddMonoidHom.id α) = id
-- OrderMonoidHom.coe_id
@[to_additive (attr := simp)]
theorem coe_id : ⇑(OrderMonoidHom.id α) = id 
Arguments: (α : Type u_2) [Preorder α] [MulOneClass α]
Goal: ⇑(OrderMonoidHom.id α) = id
-- OrderMonoidHom.coe_comp
@[to_additive (attr := simp)]
theorem coe_comp (f : β →*o γ) (g : α →*o 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] [MulOneClass α] [MulOneClass β] [MulOneClass γ] (f : β →*o γ) (g : α →*o β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- OrderAddMonoidHom.coe_comp
@[to_additive (attr := simp)]
theorem coe_comp (f : β →*o γ) (g : α →*o 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] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] (f : β →+o γ) (g : α →+o β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- OrderAddMonoidHom.comp_apply
@[to_additive (attr := simp)]
theorem comp_apply (f : β →*o γ) (g : α →*o 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] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] (f : β →+o γ) (g : α →+o β) (a : α)
Goal: (f.comp g) a = f (g a)
-- OrderMonoidHom.comp_apply
@[to_additive (attr := simp)]
theorem comp_apply (f : β →*o γ) (g : α →*o 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] [MulOneClass α] [MulOneClass β] [MulOneClass γ] (f : β →*o γ) (g : α →*o β) (a : α)
Goal: (f.comp g) a = f (g a)
-- OrderAddMonoidHom.coe_comp_addMonoidHom
@[to_additive]
theorem coe_comp_monoidHom (f : β →*o γ) (g : α →*o β) :
    (f.comp g : α →* γ) = (f : β →* γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] (f : β →+o γ) (g : α →+o β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- OrderMonoidHom.coe_comp_monoidHom
@[to_additive]
theorem coe_comp_monoidHom (f : β →*o γ) (g : α →*o β) :
    (f.comp g : α →* γ) = (f : β →* γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [MulOneClass α] [MulOneClass β] [MulOneClass γ] (f : β →*o γ) (g : α →*o β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- OrderAddMonoidHom.coe_comp_orderHom
@[to_additive]
theorem coe_comp_orderHom (f : β →*o γ) (g : α →*o β) :
    (f.comp g : α →o γ) = (f : β →o γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] (f : β →+o γ) (g : α →+o β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- OrderMonoidHom.coe_comp_orderHom
@[to_additive]
theorem coe_comp_orderHom (f : β →*o γ) (g : α →*o β) :
    (f.comp g : α →o γ) = (f : β →o γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [MulOneClass α] [MulOneClass β] [MulOneClass γ] (f : β →*o γ) (g : α →*o β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- OrderMonoidHom.comp_assoc
@[to_additive (attr := simp)]
theorem comp_assoc (f : γ →*o δ) (g : β →*o γ) (h : α →*o 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] [MulOneClass α] [MulOneClass β] [MulOneClass γ] [MulOneClass δ] (f : γ →*o δ) (g : β →*o γ) (h : α →*o β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- OrderAddMonoidHom.comp_assoc
@[to_additive (attr := simp)]
theorem comp_assoc (f : γ →*o δ) (g : β →*o γ) (h : α →*o 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] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] [AddZeroClass δ] (f : γ →+o δ) (g : β →+o γ) (h : α →+o β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- OrderAddMonoidHom.comp_id
@[to_additive (attr := simp)]
theorem comp_id (f : α →*o β) : f.comp (OrderMonoidHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (f : α →+o β)
Goal: f.comp (OrderAddMonoidHom.id α) = f
-- OrderMonoidHom.comp_id
@[to_additive (attr := simp)]
theorem comp_id (f : α →*o β) : f.comp (OrderMonoidHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (f : α →*o β)
Goal: f.comp (OrderMonoidHom.id α) = f
-- OrderAddMonoidHom.id_comp
@[to_additive (attr := simp)]
theorem id_comp (f : α →*o β) : (OrderMonoidHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (f : α →+o β)
Goal: (OrderAddMonoidHom.id β).comp f = f
-- OrderMonoidHom.id_comp
@[to_additive (attr := simp)]
theorem id_comp (f : α →*o β) : (OrderMonoidHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (f : α →*o β)
Goal: (OrderMonoidHom.id β).comp f = f
-- OrderMonoidHom.cancel_right
@[to_additive (attr := simp)]
theorem cancel_right {g₁ g₂ : β →*o γ} {f : α →*o β} (hf : Function.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] [MulOneClass α] [MulOneClass β] [MulOneClass γ] {g₁ g₂ : β →*o γ} {f : α →*o β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- OrderAddMonoidHom.cancel_right
@[to_additive (attr := simp)]
theorem cancel_right {g₁ g₂ : β →*o γ} {f : α →*o β} (hf : Function.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] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] {g₁ g₂ : β →+o γ} {f : α →+o β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- OrderMonoidHom.cancel_left
@[to_additive (attr := simp)]
theorem cancel_left {g : β →*o γ} {f₁ f₂ : α →*o β} (hg : Function.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] [MulOneClass α] [MulOneClass β] [MulOneClass γ] {g : β →*o γ} {f₁ f₂ : α →*o β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- OrderAddMonoidHom.cancel_left
@[to_additive (attr := simp)]
theorem cancel_left {g : β →*o γ} {f₁ f₂ : α →*o β} (hg : Function.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] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] {g : β →+o γ} {f₁ f₂ : α →+o β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- OrderMonoidHom.coe_one
@[to_additive (attr := simp)]
theorem coe_one : ⇑(1 : α →*o β) = 1 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β]
Goal: ⇑1 = 1
-- OrderAddMonoidHom.coe_zero
@[to_additive (attr := simp)]
theorem coe_one : ⇑(1 : α →*o β) = 1 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β]
Goal: ⇑0 = 0
-- OrderMonoidHom.one_apply
@[to_additive (attr := simp)]
theorem one_apply (a : α) : (1 : α →*o β) a = 1 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] (a : α)
Goal: 1 a = 1
-- OrderAddMonoidHom.zero_apply
@[to_additive (attr := simp)]
theorem one_apply (a : α) : (1 : α →*o β) a = 1 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [AddZeroClass α] [AddZeroClass β] (a : α)
Goal: 0 a = 0
-- OrderMonoidHom.one_comp
@[to_additive (attr := simp)]
theorem one_comp (f : α →*o β) : (1 : β →*o γ).comp f = 1 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [MulOneClass α] [MulOneClass β] [MulOneClass γ] (f : α →*o β)
Goal: OrderMonoidHom.comp 1 f = 1
-- OrderAddMonoidHom.zero_comp
@[to_additive (attr := simp)]
theorem one_comp (f : α →*o β) : (1 : β →*o γ).comp f = 1 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] (f : α →+o β)
Goal: OrderAddMonoidHom.comp 0 f = 0
-- OrderAddMonoidHom.comp_zero
@[to_additive (attr := simp)]
theorem comp_one (f : β →*o γ) : f.comp (1 : α →*o β) = 1 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [AddZeroClass α] [AddZeroClass β] [AddZeroClass γ] (f : β →+o γ)
Goal: f.comp 0 = 0
-- OrderMonoidHom.comp_one
@[to_additive (attr := simp)]
theorem comp_one (f : β →*o γ) : f.comp (1 : α →*o β) = 1 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [MulOneClass α] [MulOneClass β] [MulOneClass γ] (f : β →*o γ)
Goal: f.comp 1 = 1
-- OrderMonoidHom.coe_mul
@[to_additive (attr := simp)]
theorem coe_mul (f g : α →*o β) : ⇑(f * g) = f * g 
Arguments: {α : Type u_2} {β : Type u_3} [OrderedCommMonoid α] [OrderedCommMonoid β] (f g : α →*o β)
Goal: ⇑(f * g) = ⇑f * ⇑g
-- OrderAddMonoidHom.coe_add
@[to_additive (attr := simp)]
theorem coe_mul (f g : α →*o β) : ⇑(f * g) = f * g 
Arguments: {α : Type u_2} {β : Type u_3} [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] (f g : α →+o β)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- OrderMonoidHom.mul_apply
@[to_additive (attr := simp)]
theorem mul_apply (f g : α →*o β) (a : α) : (f * g) a = f a * g a 
Arguments: {α : Type u_2} {β : Type u_3} [OrderedCommMonoid α] [OrderedCommMonoid β] (f g : α →*o β) (a : α)
Goal: (f * g) a = f a * g a
-- OrderAddMonoidHom.add_apply
@[to_additive (attr := simp)]
theorem mul_apply (f g : α →*o β) (a : α) : (f * g) a = f a * g a 
Arguments: {α : Type u_2} {β : Type u_3} [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] (f g : α →+o β) (a : α)
Goal: (f + g) a = f a + g a
-- OrderAddMonoidHom.add_comp
@[to_additive]
theorem mul_comp (g₁ g₂ : β →*o γ) (f : α →*o β) : (g₁ * g₂).comp f = g₁.comp f * g₂.comp f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [OrderedAddCommMonoid γ] (g₁ g₂ : β →+o γ) (f : α →+o β)
Goal: (g₁ + g₂).comp f = g₁.comp f + g₂.comp f
-- OrderMonoidHom.mul_comp
@[to_additive]
theorem mul_comp (g₁ g₂ : β →*o γ) (f : α →*o β) : (g₁ * g₂).comp f = g₁.comp f * g₂.comp f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [OrderedCommMonoid α] [OrderedCommMonoid β] [OrderedCommMonoid γ] (g₁ g₂ : β →*o γ) (f : α →*o β)
Goal: (g₁ * g₂).comp f = g₁.comp f * g₂.comp f
-- OrderAddMonoidHom.comp_add
@[to_additive]
theorem comp_mul (g : β →*o γ) (f₁ f₂ : α →*o β) : g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [OrderedAddCommMonoid γ] (g : β →+o γ) (f₁ f₂ : α →+o β)
Goal: g.comp (f₁ + f₂) = g.comp f₁ + g.comp f₂
-- OrderMonoidHom.comp_mul
@[to_additive]
theorem comp_mul (g : β →*o γ) (f₁ f₂ : α →*o β) : g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [OrderedCommMonoid α] [OrderedCommMonoid β] [OrderedCommMonoid γ] (g : β →*o γ) (f₁ f₂ : α →*o β)
Goal: g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂
-- OrderMonoidHom.toMonoidHom_eq_coe
@[to_additive (attr := simp)]
theorem toMonoidHom_eq_coe (f : α →*o β) : f.toMonoidHom = f 
Arguments: {α : Type u_2} {β : Type u_3} {hα : OrderedCommMonoid α} {hβ : OrderedCommMonoid β} (f : α →*o β)
Goal: f.toMonoidHom = ↑f
-- OrderAddMonoidHom.toAddMonoidHom_eq_coe
@[to_additive (attr := simp)]
theorem toMonoidHom_eq_coe (f : α →*o β) : f.toMonoidHom = f 
Arguments: {α : Type u_2} {β : Type u_3} {hα : OrderedAddCommMonoid α} {hβ : OrderedAddCommMonoid β} (f : α →+o β)
Goal: f.toAddMonoidHom = ↑f
-- OrderAddMonoidHom.toOrderHom_eq_coe
@[to_additive (attr := simp)]
theorem toOrderHom_eq_coe (f : α →*o β) : f.toOrderHom = f 
Arguments: {α : Type u_2} {β : Type u_3} {hα : OrderedAddCommMonoid α} {hβ : OrderedAddCommMonoid β} (f : α →+o β)
Goal: f.toOrderHom = ↑f
-- OrderMonoidHom.toOrderHom_eq_coe
@[to_additive (attr := simp)]
theorem toOrderHom_eq_coe (f : α →*o β) : f.toOrderHom = f 
Arguments: {α : Type u_2} {β : Type u_3} {hα : OrderedCommMonoid α} {hβ : OrderedCommMonoid β} (f : α →*o β)
Goal: f.toOrderHom = ↑f
-- OrderMonoidWithZeroHom.ext
@[ext]
theorem ext (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] {f g : α →*₀o β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- OrderMonoidWithZeroHom.toFun_eq_coe
theorem toFun_eq_coe (f : α →*₀o β) : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β)
Goal: (↑f.toMonoidWithZeroHom).toFun = ⇑f
-- OrderMonoidWithZeroHom.coe_mk
@[simp]
theorem coe_mk (f : α →*₀ β) (h) : (OrderMonoidWithZeroHom.mk f h : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀ β) (h : Monotone (↑f).toFun)
Goal: ⇑{ toMonoidWithZeroHom := f, monotone' := h } = ⇑f
-- OrderMonoidWithZeroHom.mk_coe
@[simp]
theorem mk_coe (f : α →*₀o β) (h) : OrderMonoidWithZeroHom.mk (f : α →*₀ β) h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β) (h : Monotone (↑↑f).toFun)
Goal: { toMonoidWithZeroHom := ↑f, monotone' := h } = f
-- OrderMonoidWithZeroHom.coe_monoidWithZeroHom
@[simp]
theorem coe_monoidWithZeroHom (f : α →*₀o β) : ⇑(f : α →*₀ β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β)
Goal: ⇑↑f = ⇑f
-- OrderMonoidWithZeroHom.coe_orderMonoidHom
@[simp]
theorem coe_orderMonoidHom (f : α →*₀o β) : ⇑(f : α →*o β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β)
Goal: ⇑↑f = ⇑f
-- OrderMonoidWithZeroHom.toOrderMonoidHom_injective
theorem toOrderMonoidHom_injective : Injective (toOrderMonoidHom : _ → α →*o β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β]
Goal: Function.Injective OrderMonoidWithZeroHom.toOrderMonoidHom
-- OrderMonoidWithZeroHom.toMonoidWithZeroHom_injective
theorem toMonoidWithZeroHom_injective : Injective (toMonoidWithZeroHom : _ → α →*₀ β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β]
Goal: Function.Injective OrderMonoidWithZeroHom.toMonoidWithZeroHom
-- OrderMonoidWithZeroHom.coe_copy
@[simp]
theorem coe_copy (f : α →*₀o 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] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- OrderMonoidWithZeroHom.copy_eq
theorem copy_eq (f : α →*₀o 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] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = ↑f
-- OrderMonoidWithZeroHom.coe_id
@[simp]
theorem coe_id : ⇑(OrderMonoidWithZeroHom.id α) = id 
Arguments: (α : Type u_2) [Preorder α] [MulZeroOneClass α]
Goal: ⇑(OrderMonoidWithZeroHom.id α) = id
-- OrderMonoidWithZeroHom.coe_comp
@[simp]
theorem coe_comp (f : β →*₀o γ) (g : α →*₀o 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] [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ] (f : β →*₀o γ) (g : α →*₀o β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- OrderMonoidWithZeroHom.comp_apply
@[simp]
theorem comp_apply (f : β →*₀o γ) (g : α →*₀o 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] [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ] (f : β →*₀o γ) (g : α →*₀o β) (a : α)
Goal: (f.comp g) a = f (g a)
-- OrderMonoidWithZeroHom.coe_comp_monoidWithZeroHom
theorem coe_comp_monoidWithZeroHom (f : β →*₀o γ) (g : α →*₀o β) :
    (f.comp g : α →*₀ γ) = (f : β →*₀ γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ] (f : β →*₀o γ) (g : α →*₀o β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- OrderMonoidWithZeroHom.coe_comp_orderMonoidHom
theorem coe_comp_orderMonoidHom (f : β →*₀o γ) (g : α →*₀o β) :
    (f.comp g : α →*o γ) = (f : β →*o γ).comp g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ] (f : β →*₀o γ) (g : α →*₀o β)
Goal: ↑(f.comp g) = (↑f).comp ↑g
-- OrderMonoidWithZeroHom.comp_assoc
@[simp]
theorem comp_assoc (f : γ →*₀o δ) (g : β →*₀o γ) (h : α →*₀o 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] [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ] [MulZeroOneClass δ] (f : γ →*₀o δ) (g : β →*₀o γ) (h : α →*₀o β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- OrderMonoidWithZeroHom.comp_id
@[simp]
theorem comp_id (f : α →*₀o β) : f.comp (OrderMonoidWithZeroHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β)
Goal: f.comp (OrderMonoidWithZeroHom.id α) = f
-- OrderMonoidWithZeroHom.id_comp
@[simp]
theorem id_comp (f : α →*₀o β) : (OrderMonoidWithZeroHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [MulZeroOneClass α] [MulZeroOneClass β] (f : α →*₀o β)
Goal: (OrderMonoidWithZeroHom.id β).comp f = f
-- OrderMonoidWithZeroHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : β →*₀o γ} {f : α →*₀o β} (hf : Function.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] [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ] {g₁ g₂ : β →*₀o γ} {f : α →*₀o β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- OrderMonoidWithZeroHom.cancel_left
@[simp]
theorem cancel_left {g : β →*₀o γ} {f₁ f₂ : α →*₀o β} (hg : Function.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] [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ] {g : β →*₀o γ} {f₁ f₂ : α →*₀o β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂