Valuative Relations as Valued

In this temporary file, we provide a helper instance for Valued R Γ derived from a ValuativeRel R, so that downstream files can refer to ValuativeRel R, to facilitate a refactor.

Alternate constructors

theorem IsValuativeTopology.of_zero {R : Type u_1} [Ring R] [ValuativeRel R] [TopologicalSpace R] [ContinuousConstVAdd R R] (h₀ : ∀ (s : Set R), s ∈ nhds 0 ↔ ∃ (γ : (ValuativeRel.ValueGroupWithZero R)ˣ), {z : R | (ValuativeRel.valuation R) z < ↑γ} ⊆ s) : IsValuativeTopology R

Assuming ContinuousConstVAdd R R, we only need to check the neighbourhood of 0 in order to prove IsValuativeTopology R.

@[instance_reducible, instance 100]

instance IsValuativeTopology.instValuedValueGroupWithZeroOfIsUniformAddGroup {R : Type u_2} [Ring R] [ValuativeRel R] [UniformSpace R] [IsUniformAddGroup R] [IsValuativeTopology R] : Valued R (ValuativeRel.ValueGroupWithZero R)

Helper Valued instance when ValuativeTopology R over a UniformSpace R, for use in porting files from Valued to ValuativeRel.

Equations

• IsValuativeTopology.instValuedValueGroupWithZeroOfIsUniformAddGroup = { toUniformSpace := inst✝², toIsUniformAddGroup := inst✝¹, v := ValuativeRel.valuation R, is_topological_valuation := ⋯ }

theorem IsValuativeTopology.v_eq_valuation {R : Type u_2} [Ring R] [ValuativeRel R] [UniformSpace R] [IsUniformAddGroup R] [IsValuativeTopology R] : Valued.v = ValuativeRel.valuation R

theorem IsValuativeTopology.continuous_valuation {R : Type u_1} [Ring R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] : Continuous ⇑(ValuativeRel.valuation R)

def ValuativeRel.«term𝒪[_]» : Lean.ParserDescr

The ring of integers under a given valuation is the subring of elements with valuation ≤ 1.

Equations

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

def ValuativeRel.«term𝓂[_]» : Lean.ParserDescr

The ideal of elements that are not units.

Equations

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

def ValuativeRel.«term𝓀[_]» : Lean.ParserDescr

The residue field of a local ring is the quotient of the ring by its maximal ideal.

Equations

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