Linearly ordered commutative groups and monoids with a zero element adjoined

This file sets up a special class of linearly ordered commutative monoids that show up as the target of so-called “valuations” in algebraic number theory.

Usually, in the informal literature, these objects are constructed by taking a linearly ordered commutative group Γ and formally adjoining a zero element: Γ ∪ {0}.

The disadvantage is that a type such as NNReal is not of that form, whereas it is a very common target for valuations. The solutions is to use a typeclass, and that is exactly what we do in this file.

class LinearOrderedCommMonoidWithZero (α : Type u_2) extends LinearOrderedCommMonoid α, CommMonoidWithZero α : Type u_2

A linearly ordered commutative monoid with a zero element.

• mul : α → α → α
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• npow : ℕ → α → α
• npow_zero : ∀ (x : α), Monoid.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : α), Monoid.npow (n + 1) x = Monoid.npow n x * x
• mul_comm : ∀ (a b : α), a * b = b * a
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
• mul_le_mul_left : ∀ (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b
• min : α → α → α
• max : α → α → α
• compare : α → α → Ordering
• le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
• decidableLE : DecidableRel fun (x1 x2 : α) => x1 ≤ x2
• decidableEq : DecidableEq α
• decidableLT : DecidableRel fun (x1 x2 : α) => x1 < x2
• min_def : ∀ (a b : α), a ⊓ b = if a ≤ b then a else b
• max_def : ∀ (a b : α), a ⊔ b = if a ≤ b then b else a
• compare_eq_compareOfLessAndEq : ∀ (a b : α), compare a b = compareOfLessAndEq a b
• zero : α
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• zero_le_one : 0 ≤ 1

  0 ≤ 1 in any linearly ordered commutative monoid.

class LinearOrderedCommGroupWithZero (α : Type u_2) extends LinearOrderedCommMonoidWithZero α, CommGroupWithZero α : Type u_2

A linearly ordered commutative group with a zero element.

• mul : α → α → α
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• npow : ℕ → α → α
• npow_zero : ∀ (x : α), Monoid.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : α), Monoid.npow (n + 1) x = Monoid.npow n x * x
• mul_comm : ∀ (a b : α), a * b = b * a
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
• mul_le_mul_left : ∀ (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b
• min : α → α → α
• max : α → α → α
• compare : α → α → Ordering
• le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a
• decidableLE : DecidableRel fun (x1 x2 : α) => x1 ≤ x2
• decidableEq : DecidableEq α
• decidableLT : DecidableRel fun (x1 x2 : α) => x1 < x2
• min_def : ∀ (a b : α), a ⊓ b = if a ≤ b then a else b
• max_def : ∀ (a b : α), a ⊔ b = if a ≤ b then b else a
• compare_eq_compareOfLessAndEq : ∀ (a b : α), compare a b = compareOfLessAndEq a b
• zero : α
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• zero_le_one : 0 ≤ 1
• inv : α → α
• div : α → α → α
• div_eq_mul_inv : ∀ (a b : α), a / b = a * b⁻¹
• zpow : ℤ → α → α
• zpow_zero' : ∀ (a : α), LinearOrderedCommGroupWithZero.zpow 0 a = 1
• zpow_succ' : ∀ (n : ℕ) (a : α), LinearOrderedCommGroupWithZero.zpow (↑n.succ) a = LinearOrderedCommGroupWithZero.zpow (↑n) a * a
• zpow_neg' : ∀ (n : ℕ) (a : α), LinearOrderedCommGroupWithZero.zpow (Int.negSucc n) a = (LinearOrderedCommGroupWithZero.zpow (↑n.succ) a)⁻¹
• exists_pair_ne : ∃ (x : α), ∃ (y : α), x ≠ y
• inv_zero : 0⁻¹ = 0
• mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1

@[instance 100]

instance LinearOrderedCommMonoidWithZero.toZeroLeOneClass {α : Type u_1} [LinearOrderedCommMonoidWithZero α] : ZeroLEOneClass α

Equations

• ⋯ = ⋯

@[instance 100]

instance canonicallyOrderedAddCommMonoid.toZeroLeOneClass {α : Type u_1} [CanonicallyOrderedAddCommMonoid α] [One α] : ZeroLEOneClass α

Equations

• ⋯ = ⋯

@[reducible, inline]

abbrev Function.Injective.linearOrderedCommMonoidWithZero {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {β : Type u_2} [Zero β] [One β] [Mul β] [Pow β ℕ] [Max β] [Min β] (f : β → α) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) (hsup : ∀ (x y : β), f (x ⊔ y) = f x ⊔ f y) (hinf : ∀ (x y : β), f (x ⊓ y) = f x ⊓ f y) : LinearOrderedCommMonoidWithZero β

Pullback a LinearOrderedCommMonoidWithZero under an injective map. See note [reducible non-instances].

Equations

• Function.Injective.linearOrderedCommMonoidWithZero f hf zero one mul npow hsup hinf = LinearOrderedCommMonoidWithZero.mk ⋯ ⋯ ⋯

@[simp]

theorem zero_le' {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α} : 0 ≤ a

@[simp]

theorem not_lt_zero' {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α} : ¬a < 0

@[simp]

theorem le_zero_iff {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α} : a ≤ 0 ↔ a = 0

theorem zero_lt_iff {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α} : 0 < a ↔ a ≠ 0

theorem ne_zero_of_lt {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a b : α} (h : b < a) : a ≠ 0

instance instLinearOrderedAddCommMonoidWithTopAdditiveOrderDual {α : Type u_1} [LinearOrderedCommMonoidWithZero α] : LinearOrderedAddCommMonoidWithTop (Additive αᵒᵈ)

Equations

• instLinearOrderedAddCommMonoidWithTopAdditiveOrderDual = LinearOrderedAddCommMonoidWithTop.mk ⋯

theorem pow_pos_iff {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α} {n : ℕ} [NoZeroDivisors α] (hn : n ≠ 0) : 0 < a ^ n ↔ 0 < a

@[instance 100]

instance LinearOrderedCommGroupWithZero.toMulPosMono {α : Type u_1} [LinearOrderedCommGroupWithZero α] : MulPosMono α

Equations

• ⋯ = ⋯

@[instance 100]

instance LinearOrderedCommGroupWithZero.toPosMulMono {α : Type u_1} [LinearOrderedCommGroupWithZero α] : PosMulMono α

Equations

• ⋯ = ⋯

@[instance 100]

instance LinearOrderedCommGroupWithZero.toPosMulReflectLE {α : Type u_1} [LinearOrderedCommGroupWithZero α] : PosMulReflectLE α

Equations

• ⋯ = ⋯

@[instance 100]

instance LinearOrderedCommGroupWithZero.toMulPosReflectLE {α : Type u_1} [LinearOrderedCommGroupWithZero α] : MulPosReflectLE α

Equations

• ⋯ = ⋯

@[instance 100]

instance LinearOrderedCommGroupWithZero.toPosMulReflectLT {α : Type u_1} [LinearOrderedCommGroupWithZero α] : PosMulReflectLT α

Equations

• ⋯ = ⋯

@[instance 100]

instance LinearOrderedCommGroupWithZero.toPosMulStrictMono {α : Type u_1} [LinearOrderedCommGroupWithZero α] : PosMulStrictMono α

Equations

• ⋯ = ⋯

@[instance 100]

instance LinearOrderedCommGroupWithZero.toMulPosStrictMono {α : Type u_1} [LinearOrderedCommGroupWithZero α] : MulPosStrictMono α

Equations

• ⋯ = ⋯

@[deprecated one_le_mul]

theorem one_le_mul₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) : 1 ≤ a * b

Alias of one_le_mul' for unification.

@[deprecated mul_le_mul_right]

theorem le_of_le_mul_right {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b

@[deprecated le_mul_inv_iff₀]

theorem le_mul_inv_of_mul_le {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : c ≠ 0) (hab : a * c ≤ b) : a ≤ b * c⁻¹

theorem mul_inv_le_of_le_mul {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hab : a ≤ b * c) : a * c⁻¹ ≤ b

@[simp]

theorem Units.zero_lt {α : Type u_1} [LinearOrderedCommGroupWithZero α] (u : αˣ) : 0 < ↑u

theorem mul_lt_mul_of_lt_of_le₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c d : α} (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d

theorem mul_lt_mul₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c d : α} (hab : a < b) (hcd : c < d) : a * c < b * d

theorem mul_inv_lt_of_lt_mul₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : a < b * c) : a * c⁻¹ < b

theorem inv_mul_lt_of_lt_mul₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : a < b * c) : b⁻¹ * a < c

@[deprecated mul_lt_mul_of_pos_right]

theorem mul_lt_right₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b : α} (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c

theorem lt_of_mul_lt_mul_of_le₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c d : α} (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d

@[deprecated mul_le_mul_right]

theorem mul_le_mul_right₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0) : a * c ≤ b * c ↔ a ≤ b

@[deprecated mul_le_mul_left]

theorem mul_le_mul_left₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (ha : a ≠ 0) : a * b ≤ a * c ↔ b ≤ c

theorem div_le_div_right₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0) : a / c ≤ b / c ↔ a ≤ b

theorem div_le_div_left₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : a / b ≤ a / c ↔ c ≤ b

def OrderIso.mulLeft₀' {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) : α ≃o α

Equiv.mulLeft₀ as an OrderIso on a LinearOrderedCommGroupWithZero..

Note that OrderIso.mulLeft₀ refers to the LinearOrderedField version.

Equations

• OrderIso.mulLeft₀' ha = { toEquiv := Equiv.mulLeft₀ a ha, map_rel_iff' := ⋯ }

@[simp]

theorem OrderIso.mulLeft₀'_toEquiv {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) : (OrderIso.mulLeft₀' ha).toEquiv = Equiv.mulLeft₀ a ha

@[simp]

theorem OrderIso.mulLeft₀'_apply {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) (x : α) : (OrderIso.mulLeft₀' ha) x = a * x

theorem OrderIso.mulLeft₀'_symm {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) : (OrderIso.mulLeft₀' ha).symm = OrderIso.mulLeft₀' ⋯

def OrderIso.mulRight₀' {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) : α ≃o α

Equiv.mulRight₀ as an OrderIso on a LinearOrderedCommGroupWithZero..

Note that OrderIso.mulRight₀ refers to the LinearOrderedField version.

Equations

• OrderIso.mulRight₀' ha = { toEquiv := Equiv.mulRight₀ a ha, map_rel_iff' := ⋯ }

@[simp]

theorem OrderIso.mulRight₀'_apply {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) (x : α) : (OrderIso.mulRight₀' ha) x = x * a

@[simp]

theorem OrderIso.mulRight₀'_toEquiv {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) : (OrderIso.mulRight₀' ha).toEquiv = Equiv.mulRight₀ a ha

theorem OrderIso.mulRight₀'_symm {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) : (OrderIso.mulRight₀' ha).symm = OrderIso.mulRight₀' ⋯

instance instLinearOrderedAddCommGroupWithTopAdditiveOrderDual {α : Type u_1} [LinearOrderedCommGroupWithZero α] : LinearOrderedAddCommGroupWithTop (Additive αᵒᵈ)

Equations

• instLinearOrderedAddCommGroupWithTopAdditiveOrderDual = LinearOrderedAddCommGroupWithTop.mk ⋯ SubNegMonoid.zsmul ⋯ ⋯ ⋯ ⋯ ⋯

theorem pow_lt_pow_succ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} {n : ℕ} (ha : 1 < a) : a ^ n < a ^ n.succ

theorem pow_lt_pow_right₀ {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} {m n : ℕ} (ha : 1 < a) (hmn : m < n) : a ^ m < a ^ n

instance instLinearOrderedCommMonoidWithZeroMultiplicativeOrderDual {α : Type u_1} [LinearOrderedAddCommMonoidWithTop α] : LinearOrderedCommMonoidWithZero (Multiplicative αᵒᵈ)

Equations

• instLinearOrderedCommMonoidWithZeroMultiplicativeOrderDual = LinearOrderedCommMonoidWithZero.mk ⋯ ⋯ ⋯

instance instLinearOrderedCommGroupWithZeroMultiplicativeOrderDualOfLinearOrderedAddCommGroupWithTop {α : Type u_1} [LinearOrderedAddCommGroupWithTop α] : LinearOrderedCommGroupWithZero (Multiplicative αᵒᵈ)

Equations

• instLinearOrderedCommGroupWithZeroMultiplicativeOrderDualOfLinearOrderedAddCommGroupWithTop = LinearOrderedCommGroupWithZero.mk ⋯ DivInvMonoid.zpow ⋯ ⋯ ⋯ ⋯ ⋯

instance WithZero.preorder {α : Type u_1} [Preorder α] : Preorder (WithZero α)

Equations

• WithZero.preorder = WithBot.preorder

instance WithZero.orderBot {α : Type u_1} [Preorder α] : OrderBot (WithZero α)

Equations

• WithZero.orderBot = WithBot.orderBot

theorem WithZero.zero_le {α : Type u_1} [Preorder α] (a : WithZero α) : 0 ≤ a

theorem WithZero.zero_lt_coe {α : Type u_1} [Preorder α] (a : α) : 0 < ↑a

theorem WithZero.zero_eq_bot {α : Type u_1} [Preorder α] : 0 = ⊥

@[simp]

theorem WithZero.coe_lt_coe {α : Type u_1} [Preorder α] {a b : α} : ↑a < ↑b ↔ a < b

@[simp]

theorem WithZero.coe_le_coe {α : Type u_1} [Preorder α] {a b : α} : ↑a ≤ ↑b ↔ a ≤ b

@[simp]

theorem WithZero.one_lt_coe {α : Type u_1} [Preorder α] {a : α} [One α] : 1 < ↑a ↔ 1 < a

@[simp]

theorem WithZero.one_le_coe {α : Type u_1} [Preorder α] {a : α} [One α] : 1 ≤ ↑a ↔ 1 ≤ a

@[simp]

theorem WithZero.coe_lt_one {α : Type u_1} [Preorder α] {a : α} [One α] : ↑a < 1 ↔ a < 1

@[simp]

theorem WithZero.coe_le_one {α : Type u_1} [Preorder α] {a : α} [One α] : ↑a ≤ 1 ↔ a ≤ 1

theorem WithZero.coe_le_iff {α : Type u_1} [Preorder α] {a : α} {x : WithZero α} : ↑a ≤ x ↔ ∃ (b : α), x = ↑b ∧ a ≤ b

instance WithZero.mulLeftMono {α : Type u_1} [Preorder α] [Mul α] [MulLeftMono α] : MulLeftMono (WithZero α)

Equations

• ⋯ = ⋯

theorem WithZero.addLeftMono {α : Type u_1} [Preorder α] [AddZeroClass α] [AddLeftMono α] (h : ∀ (a : α), 0 ≤ a) : AddLeftMono (WithZero α)

instance WithZero.existsAddOfLE {α : Type u_1} [Preorder α] [Add α] [ExistsAddOfLE α] : ExistsAddOfLE (WithZero α)

Equations

• ⋯ = ⋯

instance WithZero.partialOrder {α : Type u_1} [PartialOrder α] : PartialOrder (WithZero α)

Equations

• WithZero.partialOrder = WithBot.partialOrder

instance WithZero.mulLeftReflectLT {α : Type u_1} [PartialOrder α] [Mul α] [MulLeftReflectLT α] : MulLeftReflectLT (WithZero α)

Equations

• ⋯ = ⋯

instance WithZero.lattice {α : Type u_1} [Lattice α] : Lattice (WithZero α)

Equations

• WithZero.lattice = WithBot.lattice

instance WithZero.linearOrder {α : Type u_1} [LinearOrder α] : LinearOrder (WithZero α)

Equations

• WithZero.linearOrder = WithBot.linearOrder

theorem WithZero.le_max_iff {α : Type u_1} [LinearOrder α] {a b c : α} : ↑a ≤ ↑b ⊔ ↑c ↔ a ≤ b ⊔ c

theorem WithZero.min_le_iff {α : Type u_1} [LinearOrder α] {a b c : α} : ↑a ⊓ ↑b ≤ ↑c ↔ a ⊓ b ≤ c

instance WithZero.orderedCommMonoid {α : Type u_1} [OrderedCommMonoid α] : OrderedCommMonoid (WithZero α)

Equations

• WithZero.orderedCommMonoid = OrderedCommMonoid.mk ⋯

@[reducible, inline]

abbrev WithZero.orderedAddCommMonoid {α : Type u_1} [OrderedAddCommMonoid α] (zero_le : ∀ (a : α), 0 ≤ a) : OrderedAddCommMonoid (WithZero α)

If 0 is the least element in α, then WithZero α is an OrderedAddCommMonoid.

Equations

• WithZero.orderedAddCommMonoid zero_le = OrderedAddCommMonoid.mk ⋯

instance WithZero.canonicallyOrderedAddCommMonoid {α : Type u_1} [CanonicallyOrderedAddCommMonoid α] : CanonicallyOrderedAddCommMonoid (WithZero α)

Adding a new zero to a canonically ordered additive monoid produces another one.

Equations

• WithZero.canonicallyOrderedAddCommMonoid = CanonicallyOrderedAddCommMonoid.mk ⋯ ⋯

instance WithZero.canonicallyLinearOrderedAddCommMonoid {α : Type u_1} [CanonicallyLinearOrderedAddCommMonoid α] : CanonicallyLinearOrderedAddCommMonoid (WithZero α)

Equations

• WithZero.canonicallyLinearOrderedAddCommMonoid = CanonicallyLinearOrderedAddCommMonoid.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯

instance WithZero.instLinearOrderedCommMonoidWithZero {α : Type u_1} [LinearOrderedCommMonoid α] : LinearOrderedCommMonoidWithZero (WithZero α)

Equations

• WithZero.instLinearOrderedCommMonoidWithZero = LinearOrderedCommMonoidWithZero.mk ⋯ ⋯ ⋯

instance WithZero.instLinearOrderedCommGroupWithZero {α : Type u_1} [LinearOrderedCommGroup α] : LinearOrderedCommGroupWithZero (WithZero α)

Equations

• WithZero.instLinearOrderedCommGroupWithZero = LinearOrderedCommGroupWithZero.mk ⋯ CommGroupWithZero.zpow ⋯ ⋯ ⋯ ⋯ ⋯

def Multiplicative.termℕₘ₀ : Lean.ParserDescr

Notation for WithZero (Multiplicative ℕ)

Equations

• Multiplicative.termℕₘ₀ = Lean.ParserDescr.node `Multiplicative.termℕₘ₀ 1024 (Lean.ParserDescr.symbol "ℕₘ₀")

def Multiplicative.termℤₘ₀ : Lean.ParserDescr

Notation for WithZero (Multiplicative ℤ)

Equations

• Multiplicative.termℤₘ₀ = Lean.ParserDescr.node `Multiplicative.termℤₘ₀ 1024 (Lean.ParserDescr.symbol "ℤₘ₀")