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.

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.

def WithAbs {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] : 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.

This is also helpful when dealing with several absolute values on the same semiring.

Equations

• WithAbs x✝ = R

instance WithAbs.instNontrivial {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) [Nontrivial R] : Nontrivial (WithAbs v)

instance WithAbs.instUnique {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) [Unique R] : Unique (WithAbs v)

Equations

• WithAbs.instUnique v = inferInstanceAs (Unique R)

instance WithAbs.instSemiring {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Semiring (WithAbs v)

Equations

• WithAbs.instSemiring v = inferInstanceAs (Semiring R)

instance WithAbs.instInhabited {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Inhabited (WithAbs v)

Equations

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

def WithAbs.equiv {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : WithAbs v ≃+* R

The canonical (semiring) equivalence between WithAbs v and R.

Equations

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

def WithAbs.equivWithAbs {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v w : AbsoluteValue R S) : WithAbs v ≃+* WithAbs w

The canonical (semiring) equivalence between WithAbs v and WithAbs w, for any two absolute values v and w on R.

Equations

• WithAbs.equivWithAbs v w = (WithAbs.equiv v).trans (WithAbs.equiv w).symm

theorem WithAbs.equivWithAbs_symm {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v w : AbsoluteValue R S) : (equivWithAbs v w).symm = equivWithAbs w v

@[simp]

theorem WithAbs.equiv_equivWithAbs_symm_apply {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] {v w : AbsoluteValue R S} {x : WithAbs w} : (equiv v) ((equivWithAbs v w).symm x) = (equiv w) x

@[simp]

theorem WithAbs.equivWithAbs_equiv_symm_apply {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] {v w : AbsoluteValue R S} {x : R} : (equivWithAbs v w) ((equiv v).symm x) = (equiv w).symm x

@[simp]

theorem WithAbs.equivWithAbs_symm_equiv_symm_apply {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] {v w : AbsoluteValue R S} {x : R} : (equivWithAbs v w).symm ((equiv w).symm x) = (equiv v).symm x

instance WithAbs.instCommSemiring {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [CommSemiring R] (v : AbsoluteValue R S) : CommSemiring (WithAbs v)

Equations

• WithAbs.instCommSemiring v = inferInstanceAs (CommSemiring R)

instance WithAbs.instRing {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Ring R] (v : AbsoluteValue R S) : Ring (WithAbs v)

Equations

• WithAbs.instRing v = inferInstanceAs (Ring R)

instance WithAbs.instCommRing {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [CommRing R] (v : AbsoluteValue R S) : CommRing (WithAbs v)

Equations

• WithAbs.instCommRing v = inferInstanceAs (CommRing R)

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

Equations

• WithAbs.normedRing v = v.toNormedRing

theorem WithAbs.norm_eq_abv {R : Type u_1} [Ring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) : ‖x‖ = v ((equiv v) x)

instance WithAbs.instModule_left {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [Semiring R] [AddCommGroup R'] [Module R R'] (v : AbsoluteValue R S) : Module (WithAbs v) R'

Equations

• WithAbs.instModule_left v = inferInstanceAs (Module R R')

instance WithAbs.instModule_right {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [Semiring R] [Semiring R'] [Module R R'] (v : AbsoluteValue R' S) : Module R (WithAbs v)

Equations

• WithAbs.instModule_right v = inferInstanceAs (Module R R')

instance WithAbs.instAlgebra_left {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [CommSemiring R] [Semiring R'] [Algebra R R'] (v : AbsoluteValue R S) : Algebra (WithAbs v) R'

Equations

• WithAbs.instAlgebra_left v = inferInstanceAs (Algebra R R')

theorem WithAbs.algebraMap_left_apply {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [CommSemiring R] [Semiring R'] [Algebra R R'] (v : AbsoluteValue R S) (x : WithAbs v) : (algebraMap (WithAbs v) R') x = (algebraMap R R') ((equiv v) x)

instance WithAbs.instAlgebra_right {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [CommSemiring R] [Semiring R'] [Algebra R R'] (v : AbsoluteValue R' S) : Algebra R (WithAbs v)

Equations

• WithAbs.instAlgebra_right v = inferInstanceAs (Algebra R R')

theorem WithAbs.algebraMap_right_apply {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [CommSemiring R] [Semiring R'] [Algebra R R'] (v : AbsoluteValue R' S) (x : R) : (algebraMap R (WithAbs v)) x = (equiv v) ((algebraMap R R') x)

theorem WithAbs.equiv_algebraMap_apply {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [CommSemiring R] [Semiring R'] [Algebra R R'] (v : AbsoluteValue R S) (w : AbsoluteValue R' S) (x : WithAbs v) : (equiv w) ((algebraMap (WithAbs v) (WithAbs w)) x) = (algebraMap R R') ((equiv v) x)

def WithAbs.algEquiv {R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] {R' : Type u_3} [CommSemiring R] [Semiring R'] [Algebra R R'] (v : AbsoluteValue R' S) : WithAbs v ≃ₐ[R] R'

The canonical algebra isomorphism from an R-algebra R' with an absolute value v to R'.

Equations

• WithAbs.algEquiv v = AlgEquiv.refl

class AbsoluteValue.LiesOver {K : Type u_3} {L : Type u_4} {S : Type u_5} [CommRing K] [IsSimpleRing K] [CommRing L] [Algebra K L] [PartialOrder S] [Nontrivial L] [Semiring S] (w : AbsoluteValue L S) (v : AbsoluteValue K S) : Prop

An absolute value w of L / K lies over the absolute value v of K if v is the restriction of w to K.

• comp_eq' : w.comp ⋯ = v

theorem AbsoluteValue.LiesOver.comp_eq {K : Type u_3} {L : Type u_4} {S : Type u_5} [CommRing K] [IsSimpleRing K] [CommRing L] [Algebra K L] [PartialOrder S] [Nontrivial L] [Semiring S] (w : AbsoluteValue L S) (v : AbsoluteValue K S) [w.LiesOver v] : w.comp ⋯ = v