algebra.order.absolute_value

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

Type (max u_1 u_2)

to_mul_hom : R →ₙ* S
nonneg' : ∀ (x : R), 0 ≤ self.to_mul_hom.to_fun x
eq_zero' : ∀ (x : R), self.to_mul_hom.to_fun x = 0 ↔ x = 0
add_le' : ∀ (x y : R), self.to_mul_hom.to_fun (x + y) ≤ self.to_mul_hom.to_fun x + self.to_mul_hom.to_fun y

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.

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@[protected, 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

absolute_value.has_coe_to_fun = {coe := λ (f : absolute_value R S), f.to_mul_hom.to_fun}

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@[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) :

⇑(abv.to_mul_hom) = ⇑abv

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@[protected]

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

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@[protected, 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

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@[protected]

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

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@[protected, 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

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@[protected]

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

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@[protected, 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

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@[protected]

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

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@[protected, 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

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@[protected]

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)

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@[protected]

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)

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@[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

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@[protected]

def absolute_value.abs {S : Type u_2} [linear_ordered_ring S] :

absolute_value S S

absolute_value.abs is abs as a bundled absolute_value.

Equations

absolute_value.abs = {to_mul_hom := {to_fun := abs has_neg.to_has_abs, map_mul' := _}, nonneg' := _, eq_zero' := _, add_le' := _}

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@[simp]

theorem absolute_value.abs_apply {S : Type u_2} [linear_ordered_ring S] (ᾰ : S) :

absolute_value.abs.to_mul_hom.to_fun ᾰ = |ᾰ|

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@[simp]

theorem absolute_value.abs_to_mul_hom_apply {S : Type u_2} [linear_ordered_ring S] (ᾰ : S) :

⇑(absolute_value.abs.to_mul_hom) ᾰ = |ᾰ|

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@[protected, instance]

def absolute_value.inhabited {S : Type u_2} [linear_ordered_ring S] :

inhabited (absolute_value S S)

Equations

absolute_value.inhabited = {default := absolute_value.abs _inst_2}

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@[protected, 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

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def absolute_value.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] :

R →*₀ S

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

Equations

abv.to_monoid_with_zero_hom = {to_fun := ⇑abv, map_zero' := _, map_one' := _, map_mul' := _}

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@[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] :

⇑(abv.to_monoid_with_zero_hom) = ⇑abv

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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

abv.to_monoid_hom = {to_fun := ⇑abv, map_one' := _, map_mul' := _}

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@[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] :

⇑(abv.to_monoid_hom) = ⇑abv

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@[protected, 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 : ℕ) :

⇑abv (a ^ n) = ⇑abv a ^ n

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@[protected, 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

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@[protected]

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)

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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)

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@[protected, simp]

theorem absolute_value.map_inv {R : Type u_1} {S : Type u_2} [division_ring R] [linear_ordered_field S] (abv : absolute_value R S) (a : R) :

⇑abv a⁻¹ = (⇑abv a)⁻¹

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@[protected, simp]

theorem absolute_value.map_div {R : Type u_1} {S : Type u_2} [division_ring R] [linear_ordered_field S] (abv : absolute_value R S) (a b : R) :

⇑abv (a / b) = ⇑abv a / ⇑abv b

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@[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

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@[protected, instance]

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

is_absolute_value ⇑abv

A bundled absolute value is an absolute value.

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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] :

absolute_value R S

Convert an unbundled is_absolute_value to a bundled absolute_value.

Equations

is_absolute_value.to_absolute_value abv = {to_mul_hom := {to_fun := abv, map_mul' := _}, nonneg' := _, eq_zero' := _, add_le' := _}

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@[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) :

⇑((is_absolute_value.to_absolute_value abv).to_mul_hom) ᾰ = abv ᾰ

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@[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) :

(is_absolute_value.to_absolute_value abv).to_mul_hom.to_fun ᾰ = abv ᾰ

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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

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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

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@[protected, instance]

def is_absolute_value.abs_is_absolute_value {S : Type u_1} [linear_ordered_ring S] :

is_absolute_value abs

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theorem is_absolute_value.abv_one {S : Type u_1} [linear_ordered_comm_ring S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] [nontrivial R] :

abv 1 = 1

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def is_absolute_value.abv_hom {S : Type u_1} [linear_ordered_comm_ring S] {R : Type u_2} [semiring R] (abv : R → S) [is_absolute_value abv] [nontrivial R] :

R →*₀ S

abv as a monoid_with_zero_hom.

Equations

is_absolute_value.abv_hom abv = {to_fun := abv, map_zero' := _, map_one' := _, map_mul' := _}

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theorem is_absolute_value.abv_pow {S : Type u_1} [linear_ordered_comm_ring S] {R : Type u_2} [semiring R] [nontrivial R] (abv : R → S) [is_absolute_value abv] (a : R) (n : ℕ) :

abv (a ^ n) = abv a ^ n

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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

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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)

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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)

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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)

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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)

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theorem is_absolute_value.abv_inv {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [division_ring R] (abv : R → S) [is_absolute_value abv] (a : R) :

abv a⁻¹ = (abv a)⁻¹

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theorem is_absolute_value.abv_div {S : Type u_1} [linear_ordered_field S] {R : Type u_2} [division_ring R] (abv : R → S) [is_absolute_value abv] (a b : R) :

abv (a / b) = abv a / abv b

mathlib documentation

Absolute values #

Main definitions #