mathlib documentation

algebra.order.absolute_value

Absolute values #

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

Main definitions #

structure absolute_value (R : Type u_1) (S : Type u_2) [semiring R] [ordered_semiring S] :
Type (max u_1 u_2)

absolute_value 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.

@[instance]
def absolute_value.has_coe_to_fun {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] :
has_coe_to_fun (absolute_value R S) (λ (f : absolute_value R S), R → S)
Equations
@[simp]
theorem absolute_value.coe_to_mul_hom {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) :
theorem absolute_value.nonneg {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) (x : R) :
0 abv x
@[simp]
theorem absolute_value.eq_zero {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) {x : R} :
abv x = 0 x = 0
theorem absolute_value.add_le {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) (x y : R) :
abv (x + y) abv x + abv y
@[simp]
theorem absolute_value.map_mul {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) (x y : R) :
abv (x * y) = (abv x) * abv y
theorem absolute_value.pos {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) {x : R} (hx : x 0) :
0 < abv x
@[simp]
theorem absolute_value.pos_iff {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) {x : R} :
0 < abv x x 0
theorem absolute_value.ne_zero {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) {x : R} (hx : x 0) :
abv x 0
@[simp]
theorem absolute_value.map_zero {R : Type u_1} {S : Type u_2} [semiring R] [ordered_semiring S] (abv : absolute_value R S) :
abv 0 = 0
theorem absolute_value.sub_le {R : Type u_1} {S : Type u_2} [ring R] [ordered_ring S] (abv : absolute_value R S) (a b c : R) :
abv (a - c) abv (a - b) + abv (b - c)
theorem absolute_value.le_sub {R : Type u_1} {S : Type u_2} [ring R] [ordered_ring S] (abv : absolute_value R S) (a b : R) :
abv a - abv b abv (a - b)
@[simp]
theorem absolute_value.map_sub_eq_zero_iff {R : Type u_1} {S : Type u_2} [ring R] [ordered_ring S] (abv : absolute_value R S) (a b : R) :
abv (a - b) = 0 a = b
@[simp]
theorem absolute_value.abs_apply {S : Type u_2} [linear_ordered_ring S] (ᾰ : S) :
@[simp]
@[simp]
theorem absolute_value.map_one {R : Type u_1} {S : Type u_2} [semiring R] [linear_ordered_ring S] (abv : absolute_value R S) [nontrivial R] :
abv 1 = 1

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

Equations
@[simp]
theorem absolute_value.coe_to_monoid_with_zero_hom {R : Type u_1} {S : Type u_2} [semiring R] [linear_ordered_ring S] (abv : absolute_value R S) [nontrivial R] :
def absolute_value.to_monoid_hom {R : Type u_1} {S : Type u_2} [semiring R] [linear_ordered_ring S] (abv : absolute_value R S) [nontrivial R] :
R →* S

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

Equations
@[simp]
theorem absolute_value.coe_to_monoid_hom {R : Type u_1} {S : Type u_2} [semiring R] [linear_ordered_ring S] (abv : absolute_value R S) [nontrivial R] :
@[simp]
theorem absolute_value.map_pow {R : Type u_1} {S : Type u_2} [semiring R] [linear_ordered_ring S] (abv : absolute_value R S) [nontrivial R] (a : R) (n : ) :
abv (a ^ n) = abv a ^ n
@[simp]
theorem absolute_value.map_neg {R : Type u_1} {S : Type u_2} [ring R] [linear_ordered_comm_ring S] (abv : absolute_value R S) (a : R) :
abv (-a) = abv a
theorem absolute_value.map_sub {R : Type u_1} {S : Type u_2} [ring R] [linear_ordered_comm_ring S] (abv : absolute_value R S) (a b : R) :
abv (a - b) = abv (b - a)
theorem absolute_value.abs_abv_sub_le_abv_sub {R : Type u_1} {S : Type u_2} [ring R] [linear_ordered_comm_ring S] (abv : absolute_value R S) (a b : R) :
|abv a - abv b| abv (a - b)
@[simp]
theorem absolute_value.map_inv {R : Type u_1} {S : Type u_2} [field R] [linear_ordered_field S] (abv : absolute_value R S) (a : R) :
abv a⁻¹ = (abv a)⁻¹
@[simp]
theorem absolute_value.map_div {R : Type u_1} {S : Type u_2} [field R] [linear_ordered_field S] (abv : absolute_value R S) (a b : R) :
abv (a / b) = abv a / abv b
@[class]
structure is_absolute_value {S : Type u_1} [ordered_semiring S] {R : Type u_2} [semiring R] (f : R → S) :
Prop
  • abv_nonneg : ∀ (x : R), 0 f x
  • abv_eq_zero : ∀ {x : R}, f x = 0 x = 0
  • abv_add : ∀ (x y : R), f (x + y) f x + f y
  • abv_mul : ∀ (x y : R), f (x * y) = (f x) * f y

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

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

Instances
@[instance]

A bundled absolute value is an absolute value.

def is_absolute_value.to_absolute_value {S : Type u_1} [ordered_semiring S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] :

Convert an unbundled is_absolute_value to a bundled absolute_value.

Equations
@[simp]
theorem is_absolute_value.to_absolute_value_to_mul_hom_apply {S : Type u_1} [ordered_semiring S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] (ᾰ : R) :
@[simp]
theorem is_absolute_value.to_absolute_value_apply {S : Type u_1} [ordered_semiring S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] (ᾰ : R) :
theorem is_absolute_value.abv_zero {S : Type u_1} [ordered_semiring S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] :
abv 0 = 0
theorem is_absolute_value.abv_pos {S : Type u_1} [ordered_semiring S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] {a : R} :
0 < abv a a 0
theorem is_absolute_value.abv_one {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] [nontrivial R] :
abv 1 = 1
def is_absolute_value.abv_hom {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] [nontrivial R] :

abv as a monoid_with_zero_hom.

Equations
theorem is_absolute_value.abv_pow {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [semiring R] [nontrivial R] (abv : R → S) [is_absolute_value abv] (a : R) (n : ) :
abv (a ^ n) = abv a ^ n
theorem is_absolute_value.abv_neg {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [ring R] (abv : R → S) [is_absolute_value abv] (a : R) :
abv (-a) = abv a
theorem is_absolute_value.abv_sub {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [ring R] (abv : R → S) [is_absolute_value abv] (a b : R) :
abv (a - b) = abv (b - a)
theorem is_absolute_value.abv_sub_le {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [ring R] (abv : R → S) [is_absolute_value abv] (a b c : R) :
abv (a - c) abv (a - b) + abv (b - c)
theorem is_absolute_value.sub_abv_le_abv_sub {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [ring R] (abv : R → S) [is_absolute_value abv] (a b : R) :
abv a - abv b abv (a - b)
theorem is_absolute_value.abs_abv_sub_le_abv_sub {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [ring R] (abv : R → S) [is_absolute_value abv] (a b : R) :
|abv a - abv b| abv (a - b)
theorem is_absolute_value.abv_inv {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [field R] (abv : R → S) [is_absolute_value abv] (a : R) :
abv a⁻¹ = (abv a)⁻¹
theorem is_absolute_value.abv_div {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [field R] (abv : R → S) [is_absolute_value abv] (a b : R) :
abv (a / b) = abv a / abv b