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_5) (S : Type u_6) [Semiring R] [OrderedSemiring S] extends MulHom :

Type (max u_5 u_6)

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.

toFun : R → S
map_mul' : ∀ (x y : R), self.toFun (x * y) = self.toFun x * self.toFun y
nonneg' : ∀ (x : R), 0 ≤ self.toFun x
The absolute value is nonnegative
eq_zero' : ∀ (x : R), self.toFun x = 0 ↔ x = 0
The absolute value is positive definitive
add_le' : ∀ (x y : R), self.toFun (x + y) ≤ self.toFun x + self.toFun y
The absolute value satisfies the triangle inequality

Instances For

source

theorem AbsoluteValue.nonneg' {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (self : AbsoluteValue R S) (x : R) :

0 ≤ self.toFun x

The absolute value is nonnegative

source

theorem AbsoluteValue.eq_zero' {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (self : AbsoluteValue R S) (x : R) :

self.toFun x = 0 ↔ x = 0

The absolute value is positive definitive

source

theorem AbsoluteValue.add_le' {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (self : AbsoluteValue R S) (x : R) (y : R) :

self.toFun (x + y) ≤ self.toFun x + self.toFun y

The absolute value satisfies the triangle inequality

source

instance AbsoluteValue.funLike {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] :

FunLike (AbsoluteValue R S) R S

Equations

AbsoluteValue.funLike = { coe := fun (f : AbsoluteValue R S) => f.toFun, coe_injective' := ⋯ }

source

instance AbsoluteValue.zeroHomClass {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] :

ZeroHomClass (AbsoluteValue R S) R S

Equations

⋯ = ⋯

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instance AbsoluteValue.mulHomClass {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] :

MulHomClass (AbsoluteValue R S) R S

Equations

⋯ = ⋯

source

instance AbsoluteValue.nonnegHomClass {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] :

NonnegHomClass (AbsoluteValue R S) R S

Equations

⋯ = ⋯

source

instance AbsoluteValue.subadditiveHomClass {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] :

SubadditiveHomClass (AbsoluteValue R S) R S

Equations

⋯ = ⋯

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@[simp]

theorem AbsoluteValue.coe_mk {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (f : R →ₙ* S) {h₁ : ∀ (x : R), 0 ≤ f.toFun x} {h₂ : ∀ (x : R), f.toFun x = 0 ↔ x = 0} {h₃ : ∀ (x y : R), f.toFun (x + y) ≤ f.toFun x + f.toFun y} :

⇑{ toMulHom := f, nonneg' := h₁, eq_zero' := h₂, add_le' := h₃ } = ⇑f

source

theorem AbsoluteValue.ext {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] ⦃f : AbsoluteValue R S⦄ ⦃g : AbsoluteValue R S⦄ :

(∀ (x : R), f x = g x) → f = g

source

def AbsoluteValue.Simps.apply {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (f : AbsoluteValue R S) :

R → S

See Note [custom simps projection].

Equations

AbsoluteValue.Simps.apply f = ⇑f

source

instance AbsoluteValue.instCoeFunForall {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] :

CoeFun (AbsoluteValue R S) fun (x : AbsoluteValue R S) => R → S

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

Equations

AbsoluteValue.instCoeFunForall = DFunLike.hasCoeToFun

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@[simp]

theorem AbsoluteValue.coe_toMulHom {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) :

⇑abv.toMulHom = ⇑abv

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theorem AbsoluteValue.nonneg {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (x : R) :

0 ≤ abv x

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@[simp]

theorem AbsoluteValue.eq_zero {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) {x : R} :

abv x = 0 ↔ x = 0

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theorem AbsoluteValue.add_le {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (x : R) (y : R) :

abv (x + y) ≤ abv x + abv y

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theorem AbsoluteValue.map_mul {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (x : R) (y : R) :

abv (x * y) = abv x * abv y

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theorem AbsoluteValue.ne_zero_iff {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) {x : R} :

abv x ≠ 0 ↔ x ≠ 0

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theorem AbsoluteValue.pos {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) {x : R} (hx : x ≠ 0) :

0 < abv x

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@[simp]

theorem AbsoluteValue.pos_iff {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) {x : R} :

0 < abv x ↔ x ≠ 0

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theorem AbsoluteValue.ne_zero {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) {x : R} (hx : x ≠ 0) :

abv x ≠ 0

source

theorem AbsoluteValue.map_one_of_isLeftRegular {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (h : IsLeftRegular (abv 1)) :

abv 1 = 1

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theorem AbsoluteValue.map_zero {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) :

abv 0 = 0

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theorem AbsoluteValue.sub_le {R : Type u_5} {S : Type u_6} [Ring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (a : R) (b : R) (c : R) :

abv (a - c) ≤ abv (a - b) + abv (b - c)

source

@[simp]

theorem AbsoluteValue.map_sub_eq_zero_iff {R : Type u_5} {S : Type u_6} [Ring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (a : R) (b : R) :

abv (a - b) = 0 ↔ a = b

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theorem AbsoluteValue.map_one {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedRing S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] :

abv 1 = 1

source

instance AbsoluteValue.monoidWithZeroHomClass {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedRing S] [IsDomain S] [Nontrivial R] :

MonoidWithZeroHomClass (AbsoluteValue R S) R S

Equations

⋯ = ⋯

source

def AbsoluteValue.toMonoidWithZeroHom {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedRing S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] :

R →*₀ S

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

Equations

abv.toMonoidWithZeroHom = ↑abv

source

@[simp]

theorem AbsoluteValue.coe_toMonoidWithZeroHom {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedRing S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] :

⇑abv.toMonoidWithZeroHom = ⇑abv

source

def AbsoluteValue.toMonoidHom {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedRing S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] :

R →* S

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

Equations

abv.toMonoidHom = ↑abv

source

@[simp]

theorem AbsoluteValue.coe_toMonoidHom {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedRing S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] :

⇑abv.toMonoidHom = ⇑abv

source

theorem AbsoluteValue.map_pow {R : Type u_5} {S : Type u_6} [Semiring R] [OrderedRing S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] (a : R) (n : ℕ) :

abv (a ^ n) = abv a ^ n

source

theorem AbsoluteValue.le_sub {R : Type u_5} {S : Type u_6} [Ring R] [OrderedRing S] (abv : AbsoluteValue R S) (a : R) (b : R) :

abv a - abv b ≤ abv (a - b)

source

@[simp]

theorem AbsoluteValue.map_neg {R : Type u_3} {S : Type u_4} [OrderedCommRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a : R) :

abv (-a) = abv a

source

theorem AbsoluteValue.map_sub {R : Type u_3} {S : Type u_4} [OrderedCommRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a : R) (b : R) :

abv (a - b) = abv (b - a)

source

theorem AbsoluteValue.le_add {R : Type u_3} {S : Type u_4} [OrderedCommRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a : R) (b : R) :

abv a - abv b ≤ abv (a + b)

Bound abv (a + b) from below

source

theorem AbsoluteValue.sub_le_add {R : Type u_3} {S : Type u_4} [OrderedCommRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a : R) (b : R) :

abv (a - b) ≤ abv a + abv b

Bound abv (a - b) from above

source

instance AbsoluteValue.instMulRingNormClassOfNontrivialOfIsDomain {R : Type u_3} {S : Type u_4} [OrderedCommRing S] [Ring R] [NoZeroDivisors S] [Nontrivial R] [IsDomain S] :

MulRingNormClass (AbsoluteValue R S) R S

Equations

⋯ = ⋯

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@[simp]

theorem AbsoluteValue.abs_toFun {S : Type u_6} [LinearOrderedRing S] (a : S) :

AbsoluteValue.abs a = |a|

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@[simp]

theorem AbsoluteValue.abs_apply {S : Type u_6} [LinearOrderedRing S] (a : S) :

AbsoluteValue.abs a = |a|

source

def AbsoluteValue.abs {S : Type u_6} [LinearOrderedRing S] :

AbsoluteValue S S

AbsoluteValue.abs is abs as a bundled AbsoluteValue.

Equations

AbsoluteValue.abs = { toFun := abs, map_mul' := ⋯, nonneg' := ⋯, eq_zero' := ⋯, add_le' := ⋯ }

source

instance AbsoluteValue.instInhabited {S : Type u_6} [LinearOrderedRing S] :

Inhabited (AbsoluteValue S S)

Equations

AbsoluteValue.instInhabited = { default := AbsoluteValue.abs }

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theorem AbsoluteValue.abs_abv_sub_le_abv_sub {R : Type u_5} {S : Type u_6} [Ring R] [LinearOrderedCommRing S] (abv : AbsoluteValue R S) (a : R) (b : R) :

|abv a - abv b| ≤ abv (a - b)

source

class IsAbsoluteValue {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (f : R → S) :

Prop

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.

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

Instances

source

theorem IsAbsoluteValue.abv_nonneg' {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] {f : R → S} [self : IsAbsoluteValue f] (x : R) :

0 ≤ f x

The absolute value is nonnegative

source

theorem IsAbsoluteValue.abv_eq_zero' {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] {f : R → S} [self : IsAbsoluteValue f] {x : R} :

f x = 0 ↔ x = 0

The absolute value is positive definitive

source

theorem IsAbsoluteValue.abv_add' {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] {f : R → S} [self : IsAbsoluteValue f] (x : R) (y : R) :

f (x + y) ≤ f x + f y

The absolute value satisfies the triangle inequality

source

theorem IsAbsoluteValue.abv_mul' {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] {f : R → S} [self : IsAbsoluteValue f] (x : R) (y : R) :

f (x * y) = f x * f y

The absolute value is multiplicative

source

theorem IsAbsoluteValue.abv_nonneg {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] (x : R) :

0 ≤ abv x

source

def IsAbsoluteValue.Mathlib.Meta.Positivity.evalAbv :

Mathlib.Meta.Positivity.PositivityExt

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

Equations

One or more equations did not get rendered due to their size.

source

theorem IsAbsoluteValue.abv_eq_zero {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] {x : R} :

abv x = 0 ↔ x = 0

source

theorem IsAbsoluteValue.abv_add {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] (x : R) (y : R) :

abv (x + y) ≤ abv x + abv y

source

theorem IsAbsoluteValue.abv_mul {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] (x : R) (y : R) :

abv (x * y) = abv x * abv y

source

instance AbsoluteValue.isAbsoluteValue {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : AbsoluteValue R S) :

IsAbsoluteValue ⇑abv

A bundled absolute value is an absolute value.

Equations

⋯ = ⋯

source

@[simp]

theorem IsAbsoluteValue.toAbsoluteValue_apply {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] :

∀ (a : R), (IsAbsoluteValue.toAbsoluteValue abv) a = abv a

source

@[simp]

theorem IsAbsoluteValue.toAbsoluteValue_toFun {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] :

∀ (a : R), (IsAbsoluteValue.toAbsoluteValue abv) a = abv a

source

def IsAbsoluteValue.toAbsoluteValue {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] :

AbsoluteValue R S

Convert an unbundled IsAbsoluteValue to a bundled AbsoluteValue.

Equations

IsAbsoluteValue.toAbsoluteValue abv = { toFun := abv, map_mul' := ⋯, nonneg' := ⋯, eq_zero' := ⋯, add_le' := ⋯ }

source

theorem IsAbsoluteValue.abv_zero {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] :

abv 0 = 0

source

theorem IsAbsoluteValue.abv_pos {S : Type u_5} [OrderedSemiring S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] {a : R} :

0 < abv a ↔ a ≠ 0

source

instance IsAbsoluteValue.abs_isAbsoluteValue {S : Type u_5} [LinearOrderedRing S] :

IsAbsoluteValue abs

Equations

⋯ = ⋯

source

theorem IsAbsoluteValue.abv_one {S : Type u_5} [OrderedRing S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] [IsDomain S] [Nontrivial R] :

abv 1 = 1

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def IsAbsoluteValue.abvHom {S : Type u_5} [OrderedRing S] {R : Type u_6} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] [IsDomain S] [Nontrivial R] :

R →*₀ S

abv as a MonoidWithZeroHom.

Equations

IsAbsoluteValue.abvHom abv = (IsAbsoluteValue.toAbsoluteValue abv).toMonoidWithZeroHom

source

theorem IsAbsoluteValue.abv_pow {S : Type u_5} [OrderedRing S] {R : Type u_6} [Semiring R] [IsDomain S] [Nontrivial R] (abv : R → S) [IsAbsoluteValue abv] (a : R) (n : ℕ) :

abv (a ^ n) = abv a ^ n

source

theorem IsAbsoluteValue.abv_sub_le {S : Type u_5} [OrderedRing S] {R : Type u_6} [Ring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) (b : R) (c : R) :

abv (a - c) ≤ abv (a - b) + abv (b - c)

source

theorem IsAbsoluteValue.sub_abv_le_abv_sub {S : Type u_5} [OrderedRing S] {R : Type u_6} [Ring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) (b : R) :

abv a - abv b ≤ abv (a - b)

source

theorem IsAbsoluteValue.abv_neg {R : Type u_3} {S : Type u_4} [OrderedCommRing S] [NoZeroDivisors S] [Ring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) :

abv (-a) = abv a

source

theorem IsAbsoluteValue.abv_sub {R : Type u_3} {S : Type u_4} [OrderedCommRing S] [NoZeroDivisors S] [Ring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) (b : R) :

abv (a - b) = abv (b - a)

source

theorem IsAbsoluteValue.abs_abv_sub_le_abv_sub {S : Type u_5} [LinearOrderedCommRing S] {R : Type u_6} [Ring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) (b : R) :

|abv a - abv b| ≤ abv (a - b)

source

theorem IsAbsoluteValue.abv_one' {S : Type u_5} [LinearOrderedSemifield S] {R : Type u_6} [Semiring R] [Nontrivial R] (abv : R → S) [IsAbsoluteValue abv] :

abv 1 = 1

source

def IsAbsoluteValue.abvHom' {S : Type u_5} [LinearOrderedSemifield S] {R : Type u_6} [Semiring R] [Nontrivial R] (abv : R → S) [IsAbsoluteValue abv] :

R →*₀ S

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

Equations

IsAbsoluteValue.abvHom' abv = { toFun := abv, map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯ }

source

theorem IsAbsoluteValue.abv_inv {S : Type u_5} [LinearOrderedSemifield S] {R : Type u_6} [DivisionSemiring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) :

abv a⁻¹ = (abv a)⁻¹

source

theorem IsAbsoluteValue.abv_div {S : Type u_5} [LinearOrderedSemifield S] {R : Type u_6} [DivisionSemiring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) (b : R) :

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

Documentation

Mathlib.Algebra.Order.AbsoluteValue

Absolute values #

Main definitions #