WithAbs

WithAbs v is a type synonym for a semiring R which depends on an absolute value. The point of this is to allow the type class inference system to handle multiple sources of instances that arise from absolute values. See NumberTheory.NumberField.Completion for an example of this being used to define Archimedean completions of a number field.

Main definitions

• WithAbs : type synonym for a semiring which depends on an absolute value. This is a function that takes an absolute value on a semiring and returns the semiring. This can be used to assign and infer instances on a semiring that depend on absolute values.
• WithAbs.equiv v : the canonical (type) equivalence between WithAbs v and R.
• WithAbs.ringEquiv v : The canonical ring equivalence between WithAbs v and R.
• AbsoluteValue.completion : the uniform space completion of a field K according to the uniform structure defined by the specified real absolute value.

def WithAbs {R : Type u_1} {S : Type u_2} [Semiring R] [OrderedSemiring S] : AbsoluteValue R S → Type u_1

Type synonym for a semiring which depends on an absolute value. This is a function that takes an absolute value on a semiring and returns the semiring. We use this to assign and infer instances on a semiring that depend on absolute values.

Equations

• WithAbs x = R

def WithAbs.equiv {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) : WithAbs v ≃ R

Canonical equivalence between WithAbs v and R.

Equations

• WithAbs.equiv v = Equiv.refl (WithAbs v)

instance WithAbs.instNonTrivial {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) [Nontrivial R] : Nontrivial (WithAbs v)

Equations

• ⋯ = ⋯

instance WithAbs.instUnique {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) [Unique R] : Unique (WithAbs v)

Equations

• WithAbs.instUnique v = inferInstanceAs (Unique R)

instance WithAbs.instSemiring {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) : Semiring (WithAbs v)

Equations

• WithAbs.instSemiring v = inferInstanceAs (Semiring R)

instance WithAbs.instRing {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) [Ring R] : Ring (WithAbs v)

Equations

• WithAbs.instRing v = inferInstanceAs (Ring R)

instance WithAbs.instInhabited {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) : Inhabited (WithAbs v)

Equations

• WithAbs.instInhabited v = { default := 0 }

instance WithAbs.normedRing {R : Type u_4} [Ring R] (v : AbsoluteValue R ℝ) : NormedRing (WithAbs v)

Equations

• WithAbs.normedRing v = v.toNormedRing

instance WithAbs.normedField {K : Type u_3} [Field K] (v : AbsoluteValue K ℝ) : NormedField (WithAbs v)

Equations

• WithAbs.normedField v = v.toNormedField

WithAbs.equiv preserves the ring structure.

@[simp]

theorem WithAbs.equiv_zero {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) : (WithAbs.equiv v) 0 = 0

@[simp]

theorem WithAbs.equiv_symm_zero {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) : (WithAbs.equiv v).symm 0 = 0

@[simp]

theorem WithAbs.equiv_add {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) (y : WithAbs v) : (WithAbs.equiv v) (x + y) = (WithAbs.equiv v) x + (WithAbs.equiv v) y

@[simp]

theorem WithAbs.equiv_symm_add {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (r : R) (s : R) : (WithAbs.equiv v).symm (r + s) = (WithAbs.equiv v).symm r + (WithAbs.equiv v).symm s

@[simp]

theorem WithAbs.equiv_sub {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) (y : WithAbs v) [Ring R] : (WithAbs.equiv v) (x - y) = (WithAbs.equiv v) x - (WithAbs.equiv v) y

@[simp]

theorem WithAbs.equiv_symm_sub {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (r : R) (s : R) [Ring R] : (WithAbs.equiv v).symm (r - s) = (WithAbs.equiv v).symm r - (WithAbs.equiv v).symm s

@[simp]

theorem WithAbs.equiv_neg {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) [Ring R] : (WithAbs.equiv v) (-x) = -(WithAbs.equiv v) x

@[simp]

theorem WithAbs.equiv_symm_neg {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (r : R) [Ring R] : (WithAbs.equiv v).symm (-r) = -(WithAbs.equiv v).symm r

@[simp]

theorem WithAbs.equiv_mul {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) (y : WithAbs v) : (WithAbs.equiv v) (x * y) = (WithAbs.equiv v) x * (WithAbs.equiv v) y

@[simp]

theorem WithAbs.equiv_symm_mul {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) (y : WithAbs v) : (WithAbs.equiv v).symm (x * y) = (WithAbs.equiv v).symm x * (WithAbs.equiv v).symm y

def WithAbs.ringEquiv {R : Type u_1} [Semiring R] (v : AbsoluteValue R ℝ) : WithAbs v ≃+* R

WithAbs.equiv as a ring equivalence.

Equations

• WithAbs.ringEquiv v = RingEquiv.refl (WithAbs v)

The completion of a field at an absolute value.

theorem WithAbs.isometry_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : Isometry ⇑f

If the absolute value v factors through an embedding f into a normed field, then f is an isometry.

theorem WithAbs.pseudoMetricSpace_induced_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : PseudoMetricSpace.induced (⇑f) inferInstance = MetricSpace.toPseudoMetricSpace

If the absolute value v factors through an embedding f into a normed field, then the pseudo metric space associated to the absolute value is the same as the pseudo metric space induced by f.

theorem WithAbs.uniformSpace_comap_eq_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : UniformSpace.comap (⇑f) inferInstance = PseudoMetricSpace.toUniformSpace

If the absolute value v factors through an embedding f into a normed field, then the uniform structure associated to the absolute value is the same as the uniform structure induced by f.

theorem WithAbs.isUniformInducing_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : IsUniformInducing ⇑f

If the absolute value v factors through an embedding f into a normed field, then f is uniform inducing.

@[reducible, inline]

abbrev AbsoluteValue.completion {K : Type u_4} [Field K] (v : AbsoluteValue K ℝ) : Type u_4

The completion of a field with respect to a real absolute value.

Equations

• v.completion = UniformSpace.Completion (WithAbs v)

instance AbsoluteValue.Completion.instCoeCompletion {K : Type u_4} [Field K] (v : AbsoluteValue K ℝ) : Coe K v.completion

Equations

• AbsoluteValue.Completion.instCoeCompletion v = inferInstanceAs (Coe (WithAbs v) (UniformSpace.Completion (WithAbs v)))

@[reducible, inline]

abbrev AbsoluteValue.Completion.extensionEmbedding_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] [CompleteSpace L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : v.completion →+* L

If the absolute value of a normed field factors through an embedding into another normed field L, then we can extend that embedding to an embedding on the completion v.completion →+* L.

Equations

• AbsoluteValue.Completion.extensionEmbedding_of_comp h = UniformSpace.Completion.extensionHom f ⋯

theorem AbsoluteValue.Completion.extensionEmbedding_of_comp_coe {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] [CompleteSpace L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) (x : K) : (AbsoluteValue.Completion.extensionEmbedding_of_comp h) (↑(WithAbs v) x) = f x

theorem AbsoluteValue.Completion.extensionEmbedding_dist_eq_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] [CompleteSpace L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) (x : v.completion) (y : v.completion) : dist ((AbsoluteValue.Completion.extensionEmbedding_of_comp h) x) ((AbsoluteValue.Completion.extensionEmbedding_of_comp h) y) = dist x y

If the absolute value of a normed field factors through an embedding into another normed field, then the extended embedding v.completion →+* L preserves distances.

theorem AbsoluteValue.Completion.isometry_extensionEmbedding_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] [CompleteSpace L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : Isometry ⇑(AbsoluteValue.Completion.extensionEmbedding_of_comp h)

If the absolute value of a normed field factors through an embedding into another normed field, then the extended embedding v.completion →+* L is an isometry.

theorem AbsoluteValue.Completion.isClosedEmbedding_extensionEmbedding_of_comp {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] [CompleteSpace L] {f : WithAbs v →+* L} (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : IsClosedEmbedding ⇑(AbsoluteValue.Completion.extensionEmbedding_of_comp h)

If the absolute value of a normed field factors through an embedding into another normed field, then the extended embedding v.completion →+* L is a closed embedding.

theorem AbsoluteValue.Completion.locallyCompactSpace {K : Type u_4} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_5} [NormedField L] [CompleteSpace L] {f : WithAbs v →+* L} [LocallyCompactSpace L] (h : ∀ (x : WithAbs v), ‖f x‖ = v x) : LocallyCompactSpace v.completion

If the absolute value of a normed field factors through an embedding into another normed field that is locally compact, then the completion of the first normed field is also locally compact.