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} [CommRing 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.

theorem IsValuativeTopology.mem_nhds_iff' {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {s : Set R} {x : R} : s ∈ nhds x ↔ ∃ (γ : (ValuativeRel.ValueGroupWithZero R)ˣ), {z : R | (ValuativeRel.valuation R) (z - x) < ↑γ} ⊆ s

A version mentioning subtraction.

@[deprecated IsValuativeTopology.mem_nhds_iff' (since := "2025-08-01")]

theorem ValuativeTopology.mem_nhds {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {s : Set R} {x : R} : s ∈ nhds x ↔ ∃ (γ : (ValuativeRel.ValueGroupWithZero R)ˣ), {z : R | (ValuativeRel.valuation R) (z - x) < ↑γ} ⊆ s

Alias of IsValuativeTopology.mem_nhds_iff'.

---

A version mentioning subtraction.

theorem IsValuativeTopology.mem_nhds_zero_iff {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (s : Set R) : s ∈ nhds 0 ↔ ∃ (γ : (ValuativeRel.ValueGroupWithZero R)ˣ), {x : R | (ValuativeRel.valuation R) x < ↑γ} ⊆ s

@[deprecated IsValuativeTopology.mem_nhds_zero_iff (since := "2025-08-04")]

theorem ValuativeTopology.mem_nhds_iff {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (s : Set R) : s ∈ nhds 0 ↔ ∃ (γ : (ValuativeRel.ValueGroupWithZero R)ˣ), {x : R | (ValuativeRel.valuation R) x < ↑γ} ⊆ s

Alias of IsValuativeTopology.mem_nhds_zero_iff.

@[instance 100]

instance IsValuativeTopology.instValuedValueGroupWithZeroOfIsUniformAddGroup {R : Type u_2} [CommRing 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.hasBasis_nhds {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (x : R) : (nhds x).HasBasis (fun (x : (ValuativeRel.ValueGroupWithZero R)ˣ) => True) fun (γ : (ValuativeRel.ValueGroupWithZero R)ˣ) => {z : R | (ValuativeRel.valuation R) (z - x) < ↑γ}

theorem IsValuativeTopology.hasBasis_nhds_zero (R : Type u_1) [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] : (nhds 0).HasBasis (fun (x : (ValuativeRel.ValueGroupWithZero R)ˣ) => True) fun (γ : (ValuativeRel.ValueGroupWithZero R)ˣ) => {x : R | (ValuativeRel.valuation R) x < ↑γ}

@[deprecated IsValuativeTopology.hasBasis_nhds_zero (since := "2025-08-01")]

theorem ValuativeTopology.hasBasis_nhds_zero (R : Type u_1) [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] : (nhds 0).HasBasis (fun (x : (ValuativeRel.ValueGroupWithZero R)ˣ) => True) fun (γ : (ValuativeRel.ValueGroupWithZero R)ˣ) => {x : R | (ValuativeRel.valuation R) x < ↑γ}

Alias of IsValuativeTopology.hasBasis_nhds_zero.

@[instance 100]

instance IsValuativeTopology.isTopologicalAddGroup (R : Type u_1) [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] : IsTopologicalAddGroup R

@[instance 100]

instance IsValuativeTopology.instIsTopologicalRing {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] : IsTopologicalRing R

theorem IsValuativeTopology.isOpen_ball {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsOpen {x : R | (ValuativeRel.valuation R) x < r}

@[deprecated IsValuativeTopology.isOpen_ball (since := "2025-08-01")]

theorem ValuativeTopology.isOpen_ball {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsOpen {x : R | (ValuativeRel.valuation R) x < r}

Alias of IsValuativeTopology.isOpen_ball.

theorem IsValuativeTopology.isClosed_ball {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsClosed {x : R | (ValuativeRel.valuation R) x < r}

@[deprecated IsValuativeTopology.isClosed_ball (since := "2025-08-01")]

theorem ValuativeTopology.isClosed_ball {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsClosed {x : R | (ValuativeRel.valuation R) x < r}

Alias of IsValuativeTopology.isClosed_ball.

theorem IsValuativeTopology.isClopen_ball {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsClopen {x : R | (ValuativeRel.valuation R) x < r}

@[deprecated IsValuativeTopology.isClopen_ball (since := "2025-08-01")]

theorem ValuativeTopology.isClopen_ball {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsClopen {x : R | (ValuativeRel.valuation R) x < r}

Alias of IsValuativeTopology.isClopen_ball.

theorem IsValuativeTopology.isOpen_closedBall {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsOpen {x : R | (ValuativeRel.valuation R) x ≤ r}

@[deprecated IsValuativeTopology.isOpen_closedBall (since := "2025-08-01")]

theorem ValuativeTopology.isOpen_closedBall {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsOpen {x : R | (ValuativeRel.valuation R) x ≤ r}

Alias of IsValuativeTopology.isOpen_closedBall.

theorem IsValuativeTopology.isClosed_closedBall {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsClosed {x : R | (ValuativeRel.valuation R) x ≤ r}

@[deprecated IsValuativeTopology.isClosed_closedBall (since := "2025-08-01")]

theorem ValuativeTopology.isClosed_closedBall {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] (r : ValuativeRel.ValueGroupWithZero R) : IsClosed {x : R | (ValuativeRel.valuation R) x ≤ r}

Alias of IsValuativeTopology.isClosed_closedBall.

theorem IsValuativeTopology.isClopen_closedBall {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsClopen {x : R | (ValuativeRel.valuation R) x ≤ r}

@[deprecated IsValuativeTopology.isClopen_closedBall (since := "2025-08-01")]

theorem ValuativeTopology.isClopen_closedBall {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsClopen {x : R | (ValuativeRel.valuation R) x ≤ r}

Alias of IsValuativeTopology.isClopen_closedBall.

theorem IsValuativeTopology.isClopen_sphere {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsClopen {x : R | (ValuativeRel.valuation R) x = r}

@[deprecated IsValuativeTopology.isClopen_sphere (since := "2025-08-01")]

theorem ValuativeTopology.isClopen_sphere {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsClopen {x : R | (ValuativeRel.valuation R) x = r}

Alias of IsValuativeTopology.isClopen_sphere.

theorem IsValuativeTopology.isOpen_sphere {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsOpen {x : R | (ValuativeRel.valuation R) x = r}

@[deprecated IsValuativeTopology.isOpen_sphere (since := "2025-08-01")]

theorem ValuativeTopology.isOpen_sphere {R : Type u_1} [CommRing R] [ValuativeRel R] [TopologicalSpace R] [IsValuativeTopology R] {r : ValuativeRel.ValueGroupWithZero R} (hr : r ≠ 0) : IsOpen {x : R | (ValuativeRel.valuation R) x = r}

Alias of IsValuativeTopology.isOpen_sphere.

theorem IsValuativeTopology.continuous_valuation {R : Type u_1} [CommRing 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.