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] [Semiring S] [PartialOrder S] extends R →ₙ* S :

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 inequality.

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

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

FunLike (AbsoluteValue R S) R S

Equations

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

ZeroHomClass (AbsoluteValue R S) R S

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

MulHomClass (AbsoluteValue R S) R S

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

NonnegHomClass (AbsoluteValue R S) R S

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

SubadditiveHomClass (AbsoluteValue R S) R S

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@[simp]

theorem AbsoluteValue.coe_mk {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [PartialOrder 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

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

(∀ (x : R), f x = g x) → f = g

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theorem AbsoluteValue.ext_iff {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [PartialOrder S] {f g : AbsoluteValue R S} :

f = g ↔ ∀ (x : R), f x = g x

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

R → S

See Note [custom simps projection].

Equations

Instances For

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@[simp]

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

abv (x + y) ≤ abv x + abv y

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theorem AbsoluteValue.listSum_le {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [AddLeftMono S] (l : List R) :

abv l.sum ≤ (List.map (⇑abv) l).sum

The triangle inequality for an AbsoluteValue applied to a list.

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@[simp]

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

abv x ≠ 0 ↔ x ≠ 0

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

x ≠ 0 → abv x ≠ 0

Alias of the reverse direction of AbsoluteValue.ne_zero_iff.

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@[simp]

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

0 < abv x ↔ x ≠ 0

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

x ≠ 0 → 0 < abv x

Alias of the reverse direction of AbsoluteValue.pos_iff.

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

abv 1 = 1

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@[simp]

theorem AbsoluteValue.map_zero {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [PartialOrder 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] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) (a b c : R) :

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

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@[simp]

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

abv (a - b) = 0 ↔ a = b

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@[simp]

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

abv 1 = 1

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

MonoidWithZeroHomClass (AbsoluteValue R S) R S

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def AbsoluteValue.toMonoidWithZeroHom {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [PartialOrder 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.

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@[simp]

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

⇑abv.toMonoidWithZeroHom = ⇑abv

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def AbsoluteValue.toMonoidHom {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [PartialOrder 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.

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@[simp]

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

⇑abv.toMonoidHom = ⇑abv

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@[simp]

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

abv (a ^ n) = abv a ^ n

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theorem AbsoluteValue.apply_nat_le_self {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [IsOrderedRing S] (n : ℕ) :

abv ↑n ≤ ↑n

An absolute value satisfies f (n : R) ≤ n for every n : ℕ.

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

abv a - abv b ≤ abv (a - b)

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@[simp]

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

abv (-a) = abv a

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

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

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

abv a - abv b ≤ abv (a + b)

Bound abv (a + b) from below

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

abv (a - b) ≤ abv a + abv b

Bound abv (a - b) from above

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

MulRingNormClass (AbsoluteValue R S) R S

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theorem AbsoluteValue.apply_natAbs_eq {R : Type u_3} {S : Type u_4} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (x : ℤ) :

abv ↑x.natAbs = abv ↑x

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theorem AbsoluteValue.eq_on_nat_iff_eq_on_int {R : Type u_3} {S : Type u_4} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] [NoZeroDivisors S] {f g : AbsoluteValue R S} :

(∀ (n : ℕ), f ↑n = g ↑n) ↔ ∀ (n : ℤ), f ↑n = g ↑n

Values of an absolute value coincide on the image of ℕ in R if and only if they coincide on the image of ℤ in R.

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def AbsoluteValue.abs {S : Type u_6} [Ring S] [LinearOrder S] [IsStrictOrderedRing S] :

AbsoluteValue S S

AbsoluteValue.abs is abs as a bundled AbsoluteValue.

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@[simp]

theorem AbsoluteValue.abs_apply {S : Type u_6} [Ring S] [LinearOrder S] [IsStrictOrderedRing S] (a : S) :

AbsoluteValue.abs a = |a|

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instance AbsoluteValue.instInhabited {S : Type u_6} [Ring S] [LinearOrder S] [IsStrictOrderedRing S] :

Inhabited (AbsoluteValue S S)

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

|abv a - abv b| ≤ abv (a - b)

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def AbsoluteValue.trivial {R : Type u_5} [Semiring R] [DecidablePred fun (x : R) => x = 0] [NoZeroDivisors R] {S : Type u_6} [Semiring S] [PartialOrder S] [IsOrderedRing S] [Nontrivial S] :

AbsoluteValue R S

The trivial absolute value takes the value 1 on all nonzero elements.

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@[simp]

theorem AbsoluteValue.trivial_apply {R : Type u_5} [Semiring R] [DecidablePred fun (x : R) => x = 0] [NoZeroDivisors R] {S : Type u_6} [Semiring S] [PartialOrder S] [IsOrderedRing S] [Nontrivial S] {x : R} (hx : x ≠ 0) :

AbsoluteValue.trivial x = 1

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def AbsoluteValue.IsNontrivial {R : Type u_5} [Semiring R] {S : Type u_6} [Semiring S] [PartialOrder S] (v : AbsoluteValue R S) :

Prop

An absolute value on a semiring R without zero divisors is nontrivial if it takes a value ≠ 1 on a nonzero element.

This has the advantage over v ≠ .trivial that it does not require decidability of · = 0 in R.

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Instances For

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theorem AbsoluteValue.isNontrivial_iff_ne_trivial {R : Type u_5} [Semiring R] {S : Type u_6} [Semiring S] [PartialOrder S] [IsOrderedRing S] [DecidablePred fun (x : R) => x = 0] [NoZeroDivisors R] [Nontrivial S] (v : AbsoluteValue R S) :

v.IsNontrivial ↔ v ≠ AbsoluteValue.trivial

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theorem AbsoluteValue.not_isNontrivial_iff {R : Type u_5} [Semiring R] {S : Type u_6} [Semiring S] [PartialOrder S] (v : AbsoluteValue R S) :

¬v.IsNontrivial ↔ ∀ (x : R), x ≠ 0 → v x = 1

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@[simp]

theorem AbsoluteValue.not_isNontrivial_apply {R : Type u_5} [Semiring R] {S : Type u_6} [Semiring S] [PartialOrder S] {v : AbsoluteValue R S} (hv : ¬v.IsNontrivial) {x : R} (hx : x ≠ 0) :

v x = 1

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theorem AbsoluteValue.IsNontrivial.exists_abv_gt_one {R : Type u_3} {S : Type u_4} [Field R] [Semifield S] [LinearOrder S] [IsStrictOrderedRing S] [ExistsAddOfLE S] {v : AbsoluteValue R S} (h : v.IsNontrivial) :

∃ (x : R), 1 < v x

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theorem AbsoluteValue.IsNontrivial.exists_abv_lt_one {R : Type u_3} {S : Type u_4} [Field R] [Semifield S] [LinearOrder S] [IsStrictOrderedRing S] [ExistsAddOfLE S] {v : AbsoluteValue R S} (h : v.IsNontrivial) :

∃ (x : R), x ≠ 0 ∧ v x < 1

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class IsAbsoluteValue {S : Type u_5} [Semiring S] [PartialOrder 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

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

0 ≤ abv x

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def IsAbsoluteValue.Mathlib.Meta.Positivity.evalAbv :

Mathlib.Meta.Positivity.PositivityExt

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

Equations

Instances For

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

abv x = 0 ↔ x = 0

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

abv (x + y) ≤ abv x + abv y

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

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

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

IsAbsoluteValue ⇑abv

A bundled absolute value is an absolute value.

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

AbsoluteValue R S

Convert an unbundled IsAbsoluteValue to a bundled AbsoluteValue.

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@[simp]

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

(toAbsoluteValue abv) a✝ = abv a✝

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

abv 0 = 0

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

0 < abv a ↔ a ≠ 0

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instance IsAbsoluteValue.abs_isAbsoluteValue {S : Type u_5} [Ring S] [LinearOrder S] [IsStrictOrderedRing S] :

IsAbsoluteValue abs

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

R →*₀ S

abv as a MonoidWithZeroHom.

Equations

Instances For

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theorem IsAbsoluteValue.abv_pow {S : Type u_5} [Ring S] [PartialOrder 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

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

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

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

abv a - abv b ≤ abv (a - b)

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

abv (-a) = abv a

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

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

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

|abv a - abv b| ≤ abv (a - b)

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

abv 1 = 1

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def IsAbsoluteValue.abvHom' {S : Type u_5} [Semiring S] [PartialOrder S] [IsCancelMulZero 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

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

abv a⁻¹ = (abv a)⁻¹

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

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

Documentation

Mathlib.Algebra.Order.AbsoluteValue.Basic

Absolute values #

Main definitions #