Ordered ring homomorphisms

Homomorphisms between ordered (semi)rings that respect the ordering.

Main definitions

• OrderRingHom : Monotone semiring homomorphisms.
• OrderRingIso : Monotone semiring isomorphisms.

Notation

• →+*o: Ordered ring homomorphisms.
• ≃+*o: Ordered ring isomorphisms.

Implementation notes

This file used to define typeclasses for order-preserving ring homomorphisms and isomorphisms. In https://github.com/leanprover-community/mathlib4/pull/10544, we migrated from assumptions like [FunLike F R S] [OrderRingHomClass F R S] to assumptions like [FunLike F R S] [OrderHomClass F R S] [RingHomClass F R S], making some typeclasses and instances irrelevant.

Tags

ordered ring homomorphism, order homomorphism

structure OrderRingHom (α : Type u_6) (β : Type u_7) [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] extends α →+* β : Type (max u_6 u_7)

OrderRingHom α β, denoted α →+*o β, is the type of monotone semiring homomorphisms from α to β.

When possible, instead of parametrizing results over (f : OrderRingHom α β), you should parametrize over (F : Type*) [OrderRingHomClass F α β] (f : F).

When you extend this structure, make sure to extend OrderRingHomClass.

• toFun : α → β
• map_one' : (↑↑self.toRingHom).toFun 1 = 1
• map_mul' (x y : α) : (↑↑self.toRingHom).toFun (x * y) = (↑↑self.toRingHom).toFun x * (↑↑self.toRingHom).toFun y
• map_zero' : (↑↑self.toRingHom).toFun 0 = 0
• map_add' (x y : α) : (↑↑self.toRingHom).toFun (x + y) = (↑↑self.toRingHom).toFun x + (↑↑self.toRingHom).toFun y
• monotone' : Monotone (↑↑self.toRingHom).toFun

  The proposition that the function preserves the order.

def «term_→+*o_» : Lean.TrailingParserDescr

OrderRingHom α β, denoted α →+*o β, is the type of monotone semiring homomorphisms from α to β.

When possible, instead of parametrizing results over (f : OrderRingHom α β), you should parametrize over (F : Type*) [OrderRingHomClass F α β] (f : F).

When you extend this structure, make sure to extend OrderRingHomClass.

structure OrderRingIso (α : Type u_6) (β : Type u_7) [Mul α] [Mul β] [Add α] [Add β] [LE α] [LE β] extends α ≃+* β : Type (max u_6 u_7)

OrderRingIso α β, denoted as α ≃+*o β, is the type of order-preserving semiring isomorphisms between α and β.

When possible, instead of parametrizing results over (f : OrderRingIso α β), you should parametrize over (F : Type*) [OrderRingIsoClass F α β] (f : F).

When you extend this structure, make sure to extend OrderRingIsoClass.

• toFun : α → β
• invFun : β → α
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• map_mul' (x y : α) : self.toFun (x * y) = self.toFun x * self.toFun y
• map_add' (x y : α) : self.toFun (x + y) = self.toFun x + self.toFun y
• map_le_map_iff' {a b : α} : self.toFun a ≤ self.toFun b ↔ a ≤ b

  The proposition that the function preserves the order bijectively.

def «term_≃+*o_» : Lean.TrailingParserDescr

OrderRingIso α β, denoted as α ≃+*o β, is the type of order-preserving semiring isomorphisms between α and β.

When possible, instead of parametrizing results over (f : OrderRingIso α β), you should parametrize over (F : Type*) [OrderRingIsoClass F α β] (f : F).

When you extend this structure, make sure to extend OrderRingIsoClass.

def OrderRingHomClass.toOrderRingHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [OrderHomClass F α β] [RingHomClass F α β] (f : F) : α →+*o β

Turn an element of a type F satisfying OrderHomClass F α β and RingHomClass F α β into an actual OrderRingHom. This is declared as the default coercion from F to α →+*o β.

instance instCoeTCOrderRingHomOfOrderHomClassOfRingHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [OrderHomClass F α β] [RingHomClass F α β] : CoeTC F (α →+*o β)

Any type satisfying OrderRingHomClass can be cast into OrderRingHom via OrderRingHomClass.toOrderRingHom.

def OrderRingIsoClass.toOrderRingIso {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [OrderIsoClass F α β] [RingEquivClass F α β] (f : F) : α ≃+*o β

Turn an element of a type F satisfying OrderIsoClass F α β and RingEquivClass F α β into an actual OrderRingIso. This is declared as the default coercion from F to α ≃+*o β.

instance instCoeTCOrderRingIsoOfOrderIsoClassOfRingEquivClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [OrderIsoClass F α β] [RingEquivClass F α β] : CoeTC F (α ≃+*o β)

Any type satisfying OrderRingIsoClass can be cast into OrderRingIso via OrderRingIsoClass.toOrderRingIso.

Ordered ring homomorphisms

def OrderRingHom.toOrderAddMonoidHom {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : α →+o β

Reinterpret an ordered ring homomorphism as an ordered additive monoid homomorphism.

def OrderRingHom.toOrderMonoidWithZeroHom {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : α →*₀o β

Reinterpret an ordered ring homomorphism as an order homomorphism.

instance OrderRingHom.instFunLike {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] : FunLike (α →+*o β) α β

instance OrderRingHom.instOrderHomClass {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] : OrderHomClass (α →+*o β) α β

instance OrderRingHom.instRingHomClass {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] : RingHomClass (α →+*o β) α β

theorem OrderRingHom.toFun_eq_coe {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : (↑↑f.toRingHom).toFun = ⇑f

theorem OrderRingHom.ext {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] {f g : α →+*o β} (h : ∀ (a : α), f a = g a) : f = g

theorem OrderRingHom.ext_iff {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] {f g : α →+*o β} : f = g ↔ ∀ (a : α), f a = g a

@[simp]

theorem OrderRingHom.toRingHom_eq_coe {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : f.toRingHom = ↑f

@[simp]

theorem OrderRingHom.toOrderAddMonoidHom_eq_coe {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : f.toOrderAddMonoidHom = ↑f

@[simp]

theorem OrderRingHom.toOrderMonoidWithZeroHom_eq_coe {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : f.toOrderMonoidWithZeroHom = ↑f

@[simp]

theorem OrderRingHom.coe_coe_ringHom {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : ⇑↑f = ⇑f

@[simp]

theorem OrderRingHom.coe_coe_orderAddMonoidHom {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : ⇑↑f = ⇑f

@[simp]

theorem OrderRingHom.coe_coe_orderMonoidWithZeroHom {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : ⇑↑f = ⇑f

theorem OrderRingHom.coe_ringHom_apply {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) (a : α) : ↑f a = f a

theorem OrderRingHom.coe_orderAddMonoidHom_apply {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) (a : α) : ↑f a = f a

theorem OrderRingHom.coe_orderMonoidWithZeroHom_apply {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) (a : α) : ↑f a = f a

def OrderRingHom.copy {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) (f' : α → β) (h : f' = ⇑f) : α →+*o β

Copy of an OrderRingHom with a new toFun equal to the old one. Useful to fix definitional equalities.

@[simp]

theorem OrderRingHom.coe_copy {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) (f' : α → β) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem OrderRingHom.copy_eq {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) (f' : α → β) (h : f' = ⇑f) : f.copy f' h = f

def OrderRingHom.id (α : Type u_2) [NonAssocSemiring α] [Preorder α] : α →+*o α

The identity as an ordered ring homomorphism.

instance OrderRingHom.instInhabited (α : Type u_2) [NonAssocSemiring α] [Preorder α] : Inhabited (α →+*o α)

@[simp]

theorem OrderRingHom.coe_id (α : Type u_2) [NonAssocSemiring α] [Preorder α] : ⇑(OrderRingHom.id α) = id

@[simp]

theorem OrderRingHom.id_apply {α : Type u_2} [NonAssocSemiring α] [Preorder α] (a : α) : (OrderRingHom.id α) a = a

@[simp]

theorem OrderRingHom.coe_ringHom_id {α : Type u_2} [NonAssocSemiring α] [Preorder α] : ↑(OrderRingHom.id α) = RingHom.id α

@[simp]

theorem OrderRingHom.coe_orderAddMonoidHom_id {α : Type u_2} [NonAssocSemiring α] [Preorder α] : ↑(OrderRingHom.id α) = OrderAddMonoidHom.id α

@[simp]

theorem OrderRingHom.coe_orderMonoidWithZeroHom_id {α : Type u_2} [NonAssocSemiring α] [Preorder α] : ↑(OrderRingHom.id α) = OrderMonoidWithZeroHom.id α

def OrderRingHom.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [NonAssocSemiring γ] [Preorder γ] (f : β →+*o γ) (g : α →+*o β) : α →+*o γ

Composition of two OrderRingHoms as an OrderRingHom.

@[simp]

theorem OrderRingHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [NonAssocSemiring γ] [Preorder γ] (f : β →+*o γ) (g : α →+*o β) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem OrderRingHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [NonAssocSemiring γ] [Preorder γ] (f : β →+*o γ) (g : α →+*o β) (a : α) : (f.comp g) a = f (g a)

theorem OrderRingHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [NonAssocSemiring γ] [Preorder γ] [NonAssocSemiring δ] [Preorder δ] (f : γ →+*o δ) (g : β →+*o γ) (h : α →+*o β) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem OrderRingHom.comp_id {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : f.comp (OrderRingHom.id α) = f

@[simp]

theorem OrderRingHom.id_comp {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α →+*o β) : (OrderRingHom.id β).comp f = f

@[simp]

theorem OrderRingHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [NonAssocSemiring γ] [Preorder γ] {f₁ f₂ : β →+*o γ} {g : α →+*o β} (hg : Function.Surjective ⇑g) : f₁.comp g = f₂.comp g ↔ f₁ = f₂

@[simp]

theorem OrderRingHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] [NonAssocSemiring γ] [Preorder γ] {f : β →+*o γ} {g₁ g₂ : α →+*o β} (hf : Function.Injective ⇑f) : f.comp g₁ = f.comp g₂ ↔ g₁ = g₂

instance OrderRingHom.instPreorder {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] : Preorder (α →+*o β)

instance OrderRingHom.instPartialOrder {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [PartialOrder β] : PartialOrder (α →+*o β)

Ordered ring isomorphisms

def OrderRingIso.toOrderIso {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (f : α ≃+*o β) : α ≃o β

Reinterpret an ordered ring isomorphism as an order isomorphism.

instance OrderRingIso.instEquivLike {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] : EquivLike (α ≃+*o β) α β

instance OrderRingIso.instOrderIsoClass {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] : OrderIsoClass (α ≃+*o β) α β

instance OrderRingIso.instRingEquivClass {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] : RingEquivClass (α ≃+*o β) α β

theorem OrderRingIso.toFun_eq_coe {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (f : α ≃+*o β) : f.toFun = ⇑f

theorem OrderRingIso.ext {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] {f g : α ≃+*o β} (h : ∀ (a : α), f a = g a) : f = g

theorem OrderRingIso.ext_iff {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] {f g : α ≃+*o β} : f = g ↔ ∀ (a : α), f a = g a

@[simp]

theorem OrderRingIso.coe_mk {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+* β) (h : ∀ {a b : α}, e.toFun a ≤ e.toFun b ↔ a ≤ b) : ⇑{ toRingEquiv := e, map_le_map_iff' := h } = ⇑e

@[simp]

theorem OrderRingIso.mk_coe {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+*o β) (h : ∀ {a b : α}, (↑e).toFun a ≤ (↑e).toFun b ↔ a ≤ b) : { toRingEquiv := ↑e, map_le_map_iff' := h } = e

@[simp]

theorem OrderRingIso.toRingEquiv_eq_coe {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (f : α ≃+*o β) : f.toRingEquiv = ↑f

@[simp]

theorem OrderRingIso.toOrderIso_eq_coe {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (f : α ≃+*o β) : ↑f = ↑f

@[simp]

theorem OrderRingIso.coe_toRingEquiv {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (f : α ≃+*o β) : ⇑↑f = ⇑f

@[simp]

theorem OrderRingIso.coe_toOrderIso {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (f : α ≃+*o β) : ⇑↑f = ⇑f

def OrderRingIso.refl (α : Type u_2) [Mul α] [Add α] [LE α] : α ≃+*o α

The identity map as an ordered ring isomorphism.

instance OrderRingIso.instInhabited (α : Type u_2) [Mul α] [Add α] [LE α] : Inhabited (α ≃+*o α)

@[simp]

theorem OrderRingIso.refl_apply (α : Type u_2) [Mul α] [Add α] [LE α] (x : α) : (OrderRingIso.refl α) x = x

@[simp]

theorem OrderRingIso.coe_ringEquiv_refl (α : Type u_2) [Mul α] [Add α] [LE α] : ↑(OrderRingIso.refl α) = RingEquiv.refl α

@[simp]

theorem OrderRingIso.coe_orderIso_refl (α : Type u_2) [Mul α] [Add α] [LE α] : ↑(OrderRingIso.refl α) = OrderIso.refl α

def OrderRingIso.symm {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+*o β) : β ≃+*o α

The inverse of an ordered ring isomorphism as an ordered ring isomorphism.

def OrderRingIso.Simps.symm_apply {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+*o β) : β → α

See Note [custom simps projection]

@[simp]

theorem OrderRingIso.symm_symm {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+*o β) : e.symm.symm = e

theorem OrderRingIso.symm_bijective {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] : Function.Bijective OrderRingIso.symm

def OrderRingIso.trans {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [Mul γ] [Add γ] [LE γ] (f : α ≃+*o β) (g : β ≃+*o γ) : α ≃+*o γ

Composition of OrderRingIsos as an OrderRingIso.

theorem OrderRingIso.trans_toRingEquiv {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [Mul γ] [Add γ] [LE γ] (f : α ≃+*o β) (g : β ≃+*o γ) : (f.trans g).toRingEquiv = f.trans g.toRingEquiv

This lemma used to be generated by [simps] on trans, but the lhs of this simplifies under simp. Removed [simps] attribute and added aux version below.

@[simp]

theorem OrderRingIso.trans_toRingEquiv_aux {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [Mul γ] [Add γ] [LE γ] (f : α ≃+*o β) (g : β ≃+*o γ) : ↑(f.trans g) = f.trans g.toRingEquiv

simp-normal form of trans_toRingEquiv.

@[simp]

theorem OrderRingIso.trans_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] [Mul γ] [Add γ] [LE γ] (f : α ≃+*o β) (g : β ≃+*o γ) (a : α) : (f.trans g) a = g (f a)

@[simp]

theorem OrderRingIso.self_trans_symm {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+*o β) : e.trans e.symm = OrderRingIso.refl α

@[simp]

theorem OrderRingIso.symm_trans_self {α : Type u_2} {β : Type u_3} [Mul α] [Add α] [LE α] [Mul β] [Add β] [LE β] (e : α ≃+*o β) : e.symm.trans e = OrderRingIso.refl β

def OrderRingIso.toOrderRingHom {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α ≃+*o β) : α →+*o β

Reinterpret an ordered ring isomorphism as an ordered ring homomorphism.

@[simp]

theorem OrderRingIso.toOrderRingHom_eq_coe {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α ≃+*o β) : f.toOrderRingHom = ↑f

@[simp]

theorem OrderRingIso.coe_toOrderRingHom {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] (f : α ≃+*o β) : ⇑↑f = ⇑f

@[simp]

theorem OrderRingIso.coe_toOrderRingHom_refl {α : Type u_2} [NonAssocSemiring α] [Preorder α] : ↑(OrderRingIso.refl α) = OrderRingHom.id α

theorem OrderRingIso.toOrderRingHom_injective {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [Preorder α] [NonAssocSemiring β] [Preorder β] : Function.Injective toOrderRingHom