# Documentation

Mathlib.Algebra.Order.AbsoluteValue

# Absolute values #

This file defines a bundled type of absolute values AbsoluteValue R S.

## Main definitions #

• AbsoluteValue R S is the type of absolute values on R mapping to S.
• AbsoluteValue.abs is the "standard" absolute value on S, mapping negative x to -x.
• AbsoluteValue.toMonoidWithZeroHom: absolute values mapping to a linear ordered field preserve 0, * and 1
• IsAbsoluteValue: a type class stating that f : β → α satisfies the axioms of an absolute value
structure AbsoluteValue (R : Type u_1) (S : Type u_2) [] [] extends :
Type (max u_1 u_2)

AbsoluteValue R S is the type of absolute values on R mapping to S: the maps that preserve *, are nonnegative, positive definite and satisfy the triangle equality.

Instances For
instance AbsoluteValue.zeroHomClass {R : Type u_1} {S : Type u_2} [] [] :
instance AbsoluteValue.mulHomClass {R : Type u_1} {S : Type u_2} [] [] :
instance AbsoluteValue.nonnegHomClass {R : Type u_1} {S : Type u_2} [] [] :
instance AbsoluteValue.subadditiveHomClass {R : Type u_1} {S : Type u_2} [] [] :
@[simp]
theorem AbsoluteValue.coe_mk {R : Type u_1} {S : Type u_2} [] [] (f : R →ₙ* S) {h₁ : ∀ (x : R), 0 } {h₂ : ∀ (x : R), = 0 x = 0} {h₃ : ∀ (x y : R), MulHom.toFun f (x + y) + } :
{ toMulHom := f, nonneg' := h₁, eq_zero' := h₂, add_le' := h₃ } = f
theorem AbsoluteValue.ext {R : Type u_1} {S : Type u_2} [] [] ⦃f : ⦃g : :
(∀ (x : R), f x = g x) → f = g
def AbsoluteValue.Simps.apply {R : Type u_1} {S : Type u_2} [] [] (f : ) :
RS

See Note [custom simps projection].

Instances For
instance AbsoluteValue.instCoeFunAbsoluteValueForAll {R : Type u_1} {S : Type u_2} [] [] :
CoeFun () fun x => RS

Helper instance for when there's too many metavariables to apply FunLike.has_coe_to_fun directly.

@[simp]
theorem AbsoluteValue.coe_toMulHom {R : Type u_1} {S : Type u_2} [] [] (abv : ) :
abv.toMulHom = abv
theorem AbsoluteValue.nonneg {R : Type u_1} {S : Type u_2} [] [] (abv : ) (x : R) :
0 abv x
@[simp]
theorem AbsoluteValue.eq_zero {R : Type u_1} {S : Type u_2} [] [] (abv : ) {x : R} :
abv x = 0 x = 0
theorem AbsoluteValue.add_le {R : Type u_1} {S : Type u_2} [] [] (abv : ) (x : R) (y : R) :
abv (x + y) abv x + abv y
theorem AbsoluteValue.map_mul {R : Type u_1} {S : Type u_2} [] [] (abv : ) (x : R) (y : R) :
abv (x * y) = abv x * abv y
theorem AbsoluteValue.ne_zero_iff {R : Type u_1} {S : Type u_2} [] [] (abv : ) {x : R} :
abv x 0 x 0
theorem AbsoluteValue.pos {R : Type u_1} {S : Type u_2} [] [] (abv : ) {x : R} (hx : x 0) :
0 < abv x
@[simp]
theorem AbsoluteValue.pos_iff {R : Type u_1} {S : Type u_2} [] [] (abv : ) {x : R} :
0 < abv x x 0
theorem AbsoluteValue.ne_zero {R : Type u_1} {S : Type u_2} [] [] (abv : ) {x : R} (hx : x 0) :
abv x 0
theorem AbsoluteValue.map_one_of_isLeftRegular {R : Type u_1} {S : Type u_2} [] [] (abv : ) (h : IsLeftRegular (abv 1)) :
abv 1 = 1
theorem AbsoluteValue.map_zero {R : Type u_1} {S : Type u_2} [] [] (abv : ) :
abv 0 = 0
theorem AbsoluteValue.sub_le {R : Type u_1} {S : Type u_2} [Ring R] [] (abv : ) (a : R) (b : R) (c : R) :
abv (a - c) abv (a - b) + abv (b - c)
@[simp]
theorem AbsoluteValue.map_sub_eq_zero_iff {R : Type u_1} {S : Type u_2} [Ring R] [] (abv : ) (a : R) (b : R) :
abv (a - b) = 0 a = b
theorem AbsoluteValue.map_one {R : Type u_1} {S : Type u_2} [] [] (abv : ) [] [] :
abv 1 = 1
instance AbsoluteValue.monoidWithZeroHomClass {R : Type u_1} {S : Type u_2} [] [] [] [] :
def AbsoluteValue.toMonoidWithZeroHom {R : Type u_1} {S : Type u_2} [] [] (abv : ) [] [] :

Absolute values from a nontrivial R to a linear ordered ring preserve *, 0 and 1.

Instances For
@[simp]
theorem AbsoluteValue.coe_toMonoidWithZeroHom {R : Type u_1} {S : Type u_2} [] [] (abv : ) [] [] :
= abv
def AbsoluteValue.toMonoidHom {R : Type u_1} {S : Type u_2} [] [] (abv : ) [] [] :
R →* S

Absolute values from a nontrivial R to a linear ordered ring preserve * and 1.

Instances For
@[simp]
theorem AbsoluteValue.coe_toMonoidHom {R : Type u_1} {S : Type u_2} [] [] (abv : ) [] [] :
↑() = abv
theorem AbsoluteValue.map_pow {R : Type u_1} {S : Type u_2} [] [] (abv : ) [] [] (a : R) (n : ) :
abv (a ^ n) = abv a ^ n
theorem AbsoluteValue.le_sub {R : Type u_1} {S : Type u_2} [Ring R] [] (abv : ) (a : R) (b : R) :
abv a - abv b abv (a - b)
@[simp]
theorem AbsoluteValue.map_neg {R : Type u_1} {S : Type u_2} [Ring R] [] (abv : ) [] (a : R) :
abv (-a) = abv a
theorem AbsoluteValue.map_sub {R : Type u_1} {S : Type u_2} [Ring R] [] (abv : ) [] (a : R) (b : R) :
abv (a - b) = abv (b - a)
@[simp]
theorem AbsoluteValue.abs_toFun {S : Type u_2} :
∀ (a : S), AbsoluteValue.abs a = |a|
@[simp]
theorem AbsoluteValue.abs_apply {S : Type u_2} :
∀ (a : S), AbsoluteValue.abs a = |a|
def AbsoluteValue.abs {S : Type u_2} :

AbsoluteValue.abs is abs as a bundled AbsoluteValue.

Instances For
theorem AbsoluteValue.abs_abv_sub_le_abv_sub {R : Type u_1} {S : Type u_2} [Ring R] (abv : ) (a : R) (b : R) :
|abv a - abv b| abv (a - b)
class IsAbsoluteValue {S : Type u_1} [] {R : Type u_2} [] (f : RS) :
• abv_nonneg' : ∀ (x : R), 0 f x

The absolute value is nonnegative

• abv_eq_zero' : ∀ {x : R}, f x = 0 x = 0

The absolute value is positive definitive

• abv_add' : ∀ (x y : R), f (x + y) f x + f y

The absolute value satisfies the triangle inequality

• abv_mul' : ∀ (x y : R), f (x * y) = f x * f y

The absolute value is multiplicative

A function f is an absolute value if it is nonnegative, zero only at 0, additive, and multiplicative.

See also the type AbsoluteValue which represents a bundled version of absolute values.

Instances
theorem IsAbsoluteValue.abv_nonneg {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] (x : R) :
0 abv x

The positivity extension which identifies expressions of the form abv a.

Instances For
theorem IsAbsoluteValue.abv_eq_zero {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] {x : R} :
abv x = 0 x = 0
theorem IsAbsoluteValue.abv_add {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] (x : R) (y : R) :
abv (x + y) abv x + abv y
theorem IsAbsoluteValue.abv_mul {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] (x : R) (y : R) :
abv (x * y) = abv x * abv y
instance AbsoluteValue.isAbsoluteValue {S : Type u_1} [] {R : Type u_2} [] (abv : ) :

A bundled absolute value is an absolute value.

@[simp]
theorem IsAbsoluteValue.toAbsoluteValue_apply {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] :
∀ (a : R), ↑() a = abv a
@[simp]
theorem IsAbsoluteValue.toAbsoluteValue_toFun {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] :
∀ (a : R), ↑() a = abv a
def IsAbsoluteValue.toAbsoluteValue {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] :

Convert an unbundled IsAbsoluteValue to a bundled AbsoluteValue.

Instances For
theorem IsAbsoluteValue.abv_zero {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] :
abv 0 = 0
theorem IsAbsoluteValue.abv_pos {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] {a : R} :
0 < abv a a 0
theorem IsAbsoluteValue.abv_one {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] [] [] :
abv 1 = 1
def IsAbsoluteValue.abvHom {S : Type u_1} [] {R : Type u_2} [] (abv : RS) [] [] [] :

abv as a MonoidWithZeroHom.

Instances For
theorem IsAbsoluteValue.abv_pow {S : Type u_1} [] {R : Type u_2} [] [] [] (abv : RS) [] (a : R) (n : ) :
abv (a ^ n) = abv a ^ n
theorem IsAbsoluteValue.abv_sub_le {S : Type u_1} [] {R : Type u_2} [Ring R] (abv : RS) [] (a : R) (b : R) (c : R) :
abv (a - c) abv (a - b) + abv (b - c)
theorem IsAbsoluteValue.sub_abv_le_abv_sub {S : Type u_1} [] {R : Type u_2} [Ring R] (abv : RS) [] (a : R) (b : R) :
abv a - abv b abv (a - b)
theorem IsAbsoluteValue.abv_neg {S : Type u_1} [] {R : Type u_2} [Ring R] (abv : RS) [] [] (a : R) :
abv (-a) = abv a
theorem IsAbsoluteValue.abv_sub {S : Type u_1} [] {R : Type u_2} [Ring R] (abv : RS) [] [] (a : R) (b : R) :
abv (a - b) = abv (b - a)
theorem IsAbsoluteValue.abs_abv_sub_le_abv_sub {S : Type u_1} {R : Type u_2} [Ring R] (abv : RS) [] (a : R) (b : R) :
|abv a - abv b| abv (a - b)
theorem IsAbsoluteValue.abv_one' {S : Type u_1} {R : Type u_2} [] [] (abv : RS) [] :
abv 1 = 1
def IsAbsoluteValue.abvHom' {S : Type u_1} {R : Type u_2} [] [] (abv : RS) [] :

An absolute value as a monoid with zero homomorphism, assuming the target is a semifield.

Instances For
theorem IsAbsoluteValue.abv_inv {S : Type u_1} {R : Type u_2} [] (abv : RS) [] (a : R) :
abv a⁻¹ = (abv a)⁻¹
theorem IsAbsoluteValue.abv_div {S : Type u_1} {R : Type u_2} [] (abv : RS) [] (a : R) (b : R) :
abv (a / b) = abv a / abv b