The basics of valuation theory.

The basic theory of valuations (non-archimedean norms) on a commutative ring, following T. Wedhorn's unpublished notes “Adic Spaces” ([wedhorn_adic]).

The definition of a valuation we use here is Definition 1.22 of [wedhorn_adic]. A valuation on a ring R is a monoid homomorphism v to a linearly ordered commutative monoid with zero, that in addition satisfies the following two axioms:

• v 0 = 0
• ∀ x y, v (x + y) ≤ max (v x) (v y)

Valuation R Γ₀is the type of valuations R → Γ₀, with a coercion to the underlying function. If v is a valuation from R to Γ₀ then the induced group homomorphism units(R) → Γ₀ is called unit_map v.

The equivalence "relation" IsEquiv v₁ v₂ : Prop defined in 1.27 of [wedhorn_adic] is not strictly speaking a relation, because v₁ : Valuation R Γ₁ and v₂ : Valuation R Γ₂ might not have the same type. This corresponds in ZFC to the set-theoretic difficulty that the class of all valuations (as Γ₀ varies) on a ring R is not a set. The "relation" is however reflexive, symmetric and transitive in the obvious sense. Note that we use 1.27(iii) of [wedhorn_adic] as the definition of equivalence.

Main definitions

• Valuation R Γ₀, the type of valuations on R with values in Γ₀

• Valuation.IsEquiv, the heterogeneous equivalence relation on valuations

• Valuation.supp, the support of a valuation

• AddValuation R Γ₀, the type of additive valuations on R with values in a linearly ordered additive commutative group with a top element, Γ₀.

Implementation Details

AddValuation R Γ₀ is implemented as Valuation R (Multiplicative Γ₀)ᵒᵈ.

Notation

In the DiscreteValuation locale:

• ℕₘ₀ is a shorthand for WithZero (Multiplicative ℕ)
• ℤₘ₀ is a shorthand for WithZero (Multiplicative ℤ)

TODO

If ever someone extends Valuation, we should fully comply to the DFunLike by migrating the boilerplate lemmas to ValuationClass.

structure Valuation (R : Type u_3) (Γ₀ : Type u_4) [LinearOrderedCommMonoidWithZero Γ₀] [Ring R] extends MonoidWithZeroHom : Type (max u_3 u_4)

The type of Γ₀-valued valuations on R.

When you extend this structure, make sure to extend ValuationClass.

• toFun : R → Γ₀
• map_zero' : (↑self.toMonoidWithZeroHom).toFun 0 = 0
• map_one' : (↑self.toMonoidWithZeroHom).toFun 1 = 1
• map_mul' : ∀ (x y : R), (↑self.toMonoidWithZeroHom).toFun (x * y) = (↑self.toMonoidWithZeroHom).toFun x * (↑self.toMonoidWithZeroHom).toFun y
• map_add_le_max' : ∀ (x y : R), (↑self.toMonoidWithZeroHom).toFun (x + y) ≤ max ((↑self.toMonoidWithZeroHom).toFun x) ((↑self.toMonoidWithZeroHom).toFun y)

  The valuation of a a sum is less that the sum of the valuations

theorem Valuation.map_add_le_max' {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedCommMonoidWithZero Γ₀] [Ring R] (self : Valuation R Γ₀) (x : R) (y : R) : (↑self.toMonoidWithZeroHom).toFun (x + y) ≤ max ((↑self.toMonoidWithZeroHom).toFun x) ((↑self.toMonoidWithZeroHom).toFun y)

The valuation of a a sum is less that the sum of the valuations

class ValuationClass (F : Type u_7) (R : outParam (Type u_5)) (Γ₀ : outParam (Type u_6)) [LinearOrderedCommMonoidWithZero Γ₀] [Ring R] [FunLike F R Γ₀] extends MonoidWithZeroHomClass : Prop

ValuationClass F α β states that F is a type of valuations.

You should also extend this typeclass when you extend Valuation.

• map_mul : ∀ (f : F) (x y : R), f (x * y) = f x * f y
• map_one : ∀ (f : F), f 1 = 1
• map_zero : ∀ (f : F), f 0 = 0
• map_add_le_max : ∀ (f : F) (x y : R), f (x + y) ≤ max (f x) (f y)

  The valuation of a a sum is less that the sum of the valuations

theorem ValuationClass.map_add_le_max {F : Type u_7} {R : outParam (Type u_5)} {Γ₀ : outParam (Type u_6)} [LinearOrderedCommMonoidWithZero Γ₀] [Ring R] [FunLike F R Γ₀] [self : ValuationClass F R Γ₀] (f : F) (x : R) (y : R) : f (x + y) ≤ max (f x) (f y)

The valuation of a a sum is less that the sum of the valuations

instance instCoeTCValuationOfValuationClass (F : Type u_2) (R : Type u_3) (Γ₀ : Type u_4) [LinearOrderedCommMonoidWithZero Γ₀] [Ring R] [FunLike F R Γ₀] [ValuationClass F R Γ₀] : CoeTC F (Valuation R Γ₀)

Equations

• instCoeTCValuationOfValuationClass F R Γ₀ = { coe := fun (f : F) => { toFun := ⇑f, map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯, map_add_le_max' := ⋯ } }

instance Valuation.instFunLike {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] : FunLike (Valuation R Γ₀) R Γ₀

Equations

• Valuation.instFunLike = { coe := fun (f : Valuation R Γ₀) => (↑f.toMonoidWithZeroHom).toFun, coe_injective' := ⋯ }

instance Valuation.instValuationClass {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] : ValuationClass (Valuation R Γ₀) R Γ₀

Equations

• ⋯ = ⋯

@[simp]

theorem Valuation.coe_mk {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (f : R →*₀ Γ₀) (h : ∀ (x y : R), (↑f).toFun (x + y) ≤ max ((↑f).toFun x) ((↑f).toFun y)) : ⇑{ toMonoidWithZeroHom := f, map_add_le_max' := h } = ⇑f

theorem Valuation.toFun_eq_coe {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : (↑v.toMonoidWithZeroHom).toFun = ⇑v

@[simp]

theorem Valuation.toMonoidWithZeroHom_coe_eq_coe {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : ⇑v.toMonoidWithZeroHom = ⇑v

theorem Valuation.ext {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ₀} (h : ∀ (r : R), v₁ r = v₂ r) : v₁ = v₂

@[simp]

theorem Valuation.coe_coe {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : ⇑↑v = ⇑v

theorem Valuation.map_zero {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : v 0 = 0

theorem Valuation.map_one {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : v 1 = 1

theorem Valuation.map_mul {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x : R) (y : R) : v (x * y) = v x * v y

theorem Valuation.map_add {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x : R) (y : R) : v (x + y) ≤ max (v x) (v y)

@[simp]

theorem Valuation.map_add' {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x : R) (y : R) : v (x + y) ≤ v x ∨ v (x + y) ≤ v y

theorem Valuation.map_add_le {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {x : R} {y : R} {g : Γ₀} (hx : v x ≤ g) (hy : v y ≤ g) : v (x + y) ≤ g

theorem Valuation.map_add_lt {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {x : R} {y : R} {g : Γ₀} (hx : v x < g) (hy : v y < g) : v (x + y) < g

theorem Valuation.map_sum_le {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {ι : Type u_7} {s : Finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, v (f i) ≤ g) : v (∑ i ∈ s, f i) ≤ g

theorem Valuation.map_sum_lt {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {ι : Type u_7} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ 0) (hf : ∀ i ∈ s, v (f i) < g) : v (∑ i ∈ s, f i) < g

theorem Valuation.map_sum_lt' {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {ι : Type u_7} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : 0 < g) (hf : ∀ i ∈ s, v (f i) < g) : v (∑ i ∈ s, f i) < g

theorem Valuation.map_pow {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x : R) (n : ℕ) : v (x ^ n) = v x ^ n

theorem Valuation.ext_iff {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ₀} : v₁ = v₂ ↔ ∀ (r : R), v₁ r = v₂ r

Deprecated. Use DFunLike.ext_iff.

def Valuation.toPreorder {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : Preorder R

A valuation gives a preorder on the underlying ring.

Equations

• v.toPreorder = Preorder.lift ⇑v

theorem Valuation.zero_iff {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommMonoidWithZero Γ₀] [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : v x = 0 ↔ x = 0

If v is a valuation on a division ring then v(x) = 0 iff x = 0.

theorem Valuation.ne_zero_iff {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommMonoidWithZero Γ₀] [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : v x ≠ 0 ↔ x ≠ 0

theorem Valuation.unit_map_eq {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (u : Rˣ) : ↑((Units.map ↑v) u) = v ↑u

def Valuation.comap {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {S : Type u_7} [Ring S] (f : S →+* R) (v : Valuation R Γ₀) : Valuation S Γ₀

A ring homomorphism S → R induces a map Valuation R Γ₀ → Valuation S Γ₀.

Equations

• Valuation.comap f v = let __src := v.comp f.toMonoidWithZeroHom; { toFun := ⇑v ∘ ⇑f, map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯, map_add_le_max' := ⋯ }

@[simp]

theorem Valuation.comap_apply {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {S : Type u_7} [Ring S] (f : S →+* R) (v : Valuation R Γ₀) (s : S) : (Valuation.comap f v) s = v (f s)

@[simp]

theorem Valuation.comap_id {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : Valuation.comap (RingHom.id R) v = v

theorem Valuation.comap_comp {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {S₁ : Type u_7} {S₂ : Type u_8} [Ring S₁] [Ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R) : Valuation.comap (g.comp f) v = Valuation.comap f (Valuation.comap g v)

def Valuation.map {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] (f : Γ₀ →*₀ Γ'₀) (hf : Monotone ⇑f) (v : Valuation R Γ₀) : Valuation R Γ'₀

A ≤-preserving group homomorphism Γ₀ → Γ'₀ induces a map Valuation R Γ₀ → Valuation R Γ'₀.

Equations

• Valuation.map f hf v = let __src := f.comp v.toMonoidWithZeroHom; { toFun := ⇑f ∘ ⇑v, map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯, map_add_le_max' := ⋯ }

def Valuation.IsEquiv {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] (v₁ : Valuation R Γ₀) (v₂ : Valuation R Γ'₀) : Prop

Two valuations on R are defined to be equivalent if they induce the same preorder on R.

Equations

• v₁.IsEquiv v₂ = ∀ (r s : R), v₁ r ≤ v₁ s ↔ v₂ r ≤ v₂ s

@[simp]

theorem Valuation.map_neg {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (x : R) : v (-x) = v x

theorem Valuation.map_sub_swap {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (x : R) (y : R) : v (x - y) = v (y - x)

theorem Valuation.map_sub {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (x : R) (y : R) : v (x - y) ≤ max (v x) (v y)

theorem Valuation.map_sub_le {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} {y : R} {g : Γ₀} (hx : v x ≤ g) (hy : v y ≤ g) : v (x - y) ≤ g

theorem Valuation.map_add_of_distinct_val {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} {y : R} (h : v x ≠ v y) : v (x + y) = max (v x) (v y)

theorem Valuation.map_add_eq_of_lt_right {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} {y : R} (h : v x < v y) : v (x + y) = v y

theorem Valuation.map_add_eq_of_lt_left {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} {y : R} (h : v y < v x) : v (x + y) = v x

theorem Valuation.map_eq_of_sub_lt {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} {y : R} (h : v (y - x) < v x) : v y = v x

theorem Valuation.map_one_add_of_lt {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} (h : v x < 1) : v (1 + x) = 1

theorem Valuation.map_one_sub_of_lt {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} (h : v x < 1) : v (1 - x) = 1

theorem Valuation.one_lt_val_iff {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : 1 < v x ↔ v x⁻¹ < 1

def Valuation.ltAddSubgroup {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (γ : Γ₀ˣ) : AddSubgroup R

The subgroup of elements whose valuation is less than a certain unit.

Equations

• v.ltAddSubgroup γ = { carrier := {x : R | v x < ↑γ}, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }

theorem Valuation.IsEquiv.refl {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v : Valuation R Γ₀} : v.IsEquiv v

theorem Valuation.IsEquiv.symm {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} (h : v₁.IsEquiv v₂) : v₂.IsEquiv v₁

theorem Valuation.IsEquiv.trans {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} {Γ''₀ : Type u_6} [LinearOrderedCommMonoidWithZero Γ''₀] [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} {v₃ : Valuation R Γ''₀} (h₁₂ : v₁.IsEquiv v₂) (h₂₃ : v₂.IsEquiv v₃) : v₁.IsEquiv v₃

theorem Valuation.IsEquiv.of_eq {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v : Valuation R Γ₀} {v' : Valuation R Γ₀} (h : v = v') : v.IsEquiv v'

theorem Valuation.IsEquiv.map {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v : Valuation R Γ₀} {v' : Valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀) (hf : Monotone ⇑f) (inf : Function.Injective ⇑f) (h : v.IsEquiv v') : (Valuation.map f hf v).IsEquiv (Valuation.map f hf v')

theorem Valuation.IsEquiv.comap {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} {S : Type u_7} [Ring S] (f : S →+* R) (h : v₁.IsEquiv v₂) : (Valuation.comap f v₁).IsEquiv (Valuation.comap f v₂)

comap preserves equivalence.

theorem Valuation.IsEquiv.val_eq {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} (h : v₁.IsEquiv v₂) {r : R} {s : R} : v₁ r = v₁ s ↔ v₂ r = v₂ s

theorem Valuation.IsEquiv.ne_zero {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} (h : v₁.IsEquiv v₂) {r : R} : v₁ r ≠ 0 ↔ v₂ r ≠ 0

theorem Valuation.isEquiv_of_map_strictMono {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] [Ring R] {v : Valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀) (H : StrictMono ⇑f) : (Valuation.map f ⋯ v).IsEquiv v

theorem Valuation.isEquiv_of_val_le_one {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) (h : ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1) : v.IsEquiv v'

theorem Valuation.isEquiv_iff_val_le_one {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) : v.IsEquiv v' ↔ ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1

theorem Valuation.isEquiv_iff_val_eq_one {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) : v.IsEquiv v' ↔ ∀ {x : K}, v x = 1 ↔ v' x = 1

theorem Valuation.isEquiv_iff_val_lt_one {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) : v.IsEquiv v' ↔ ∀ {x : K}, v x < 1 ↔ v' x < 1

theorem Valuation.isEquiv_iff_val_sub_one_lt_one {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) : v.IsEquiv v' ↔ ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1

theorem Valuation.isEquiv_tfae {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) : [v.IsEquiv v', ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1, ∀ {x : K}, v x = 1 ↔ v' x = 1, ∀ {x : K}, v x < 1 ↔ v' x < 1, ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1].TFAE

def Valuation.supp {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) : Ideal R

The support of a valuation v : R → Γ₀ is the ideal of R where v vanishes.

Equations

• v.supp = { carrier := {x : R | v x = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

@[simp]

theorem Valuation.mem_supp_iff {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x : R) : x ∈ v.supp ↔ v x = 0

instance Valuation.instIsPrimeSuppOfNontrivialOfNoZeroDivisors {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) [Nontrivial Γ₀] [NoZeroDivisors Γ₀] : v.supp.IsPrime

The support of a valuation is a prime ideal.

Equations

• ⋯ = ⋯

theorem Valuation.map_add_supp {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (a : R) {s : R} (h : s ∈ v.supp) : v (a + s) = v a

theorem Valuation.comap_supp {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {S : Type u_7} [CommRing S] (f : S →+* R) : (Valuation.comap f v).supp = Ideal.comap f v.supp

def AddValuation (R : Type u_3) [Ring R] (Γ₀ : Type u_4) [LinearOrderedAddCommMonoidWithTop Γ₀] : Type (max u_3 u_4)

The type of Γ₀-valued additive valuations on R.

Equations

• AddValuation R Γ₀ = Valuation R (Multiplicative Γ₀ᵒᵈ)

instance AddValuation.instFunLike (R : Type u_6) (Γ₀ : Type u_7) [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] : FunLike (AddValuation R Γ₀) R Γ₀

A valuation is coerced to the underlying function R → Γ₀.

Equations

• AddValuation.instFunLike R Γ₀ = { coe := fun (v : AddValuation R Γ₀) => (↑v.toMonoidWithZeroHom).toFun, coe_injective' := ⋯ }

def AddValuation.of {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (f : R → Γ₀) (h0 : f 0 = ⊤) (h1 : f 1 = 0) (hadd : ∀ (x y : R), min (f x) (f y) ≤ f (x + y)) (hmul : ∀ (x y : R), f (x * y) = f x + f y) : AddValuation R Γ₀

An alternate constructor of AddValuation, that doesn't reference Multiplicative Γ₀ᵒᵈ

Equations

• AddValuation.of f h0 h1 hadd hmul = { toFun := f, map_zero' := h0, map_one' := h1, map_mul' := hmul, map_add_le_max' := hadd }

@[simp]

theorem AddValuation.of_apply {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (f : R → Γ₀) {h0 : f 0 = ⊤} {h1 : f 1 = 0} {hadd : ∀ (x y : R), min (f x) (f y) ≤ f (x + y)} {hmul : ∀ (x y : R), f (x * y) = f x + f y} {r : R} : (AddValuation.of f h0 h1 hadd hmul) r = f r

def AddValuation.valuation {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) : Valuation R (Multiplicative Γ₀ᵒᵈ)

The Valuation associated to an AddValuation (useful if the latter is constructed using AddValuation.of).

Equations

• v.valuation = v

@[simp]

theorem AddValuation.valuation_apply {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (r : R) : v.valuation r = Multiplicative.ofAdd (OrderDual.toDual (v r))

@[simp]

theorem AddValuation.map_zero {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) : v 0 = ⊤

@[simp]

theorem AddValuation.map_one {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) : v 1 = 0

def AddValuation.asFun {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) : R → Γ₀

A helper function for Lean to inferring types correctly

Equations

• v.asFun = ⇑v

@[simp]

theorem AddValuation.map_mul {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x : R) (y : R) : v (x * y) = v x + v y

theorem AddValuation.map_add {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x : R) (y : R) : min (v x) (v y) ≤ v (x + y)

@[simp]

theorem AddValuation.map_add' {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x : R) (y : R) : v x ≤ v (x + y) ∨ v y ≤ v (x + y)

theorem AddValuation.map_le_add {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {x : R} {y : R} {g : Γ₀} (hx : g ≤ v x) (hy : g ≤ v y) : g ≤ v (x + y)

theorem AddValuation.map_lt_add {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {x : R} {y : R} {g : Γ₀} (hx : g < v x) (hy : g < v y) : g < v (x + y)

theorem AddValuation.map_le_sum {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {ι : Type u_6} {s : Finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, g ≤ v (f i)) : g ≤ v (∑ i ∈ s, f i)

theorem AddValuation.map_lt_sum {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {ι : Type u_6} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ ⊤) (hf : ∀ i ∈ s, g < v (f i)) : g < v (∑ i ∈ s, f i)

theorem AddValuation.map_lt_sum' {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {ι : Type u_6} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g < ⊤) (hf : ∀ i ∈ s, g < v (f i)) : g < v (∑ i ∈ s, f i)

@[simp]

theorem AddValuation.map_pow {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x : R) (n : ℕ) : v (x ^ n) = n • v x

theorem AddValuation.ext {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ₀} (h : ∀ (r : R), v₁ r = v₂ r) : v₁ = v₂

theorem AddValuation.ext_iff {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ₀} : v₁ = v₂ ↔ ∀ (r : R), v₁ r = v₂ r

def AddValuation.toPreorder {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) : Preorder R

A valuation gives a preorder on the underlying ring.

Equations

• v.toPreorder = Preorder.lift ⇑v

@[simp]

theorem AddValuation.top_iff {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Nontrivial Γ₀] (v : AddValuation K Γ₀) {x : K} : v x = ⊤ ↔ x = 0

If v is an additive valuation on a division ring then v(x) = ⊤ iff x = 0.

theorem AddValuation.ne_top_iff {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Nontrivial Γ₀] (v : AddValuation K Γ₀) {x : K} : v x ≠ ⊤ ↔ x ≠ 0

def AddValuation.comap {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] {S : Type u_6} [Ring S] (f : S →+* R) (v : AddValuation R Γ₀) : AddValuation S Γ₀

A ring homomorphism S → R induces a map AddValuation R Γ₀ → AddValuation S Γ₀.

Equations

• AddValuation.comap f v = Valuation.comap f v

@[simp]

theorem AddValuation.comap_id {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) : AddValuation.comap (RingHom.id R) v = v

theorem AddValuation.comap_comp {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {S₁ : Type u_6} {S₂ : Type u_7} [Ring S₁] [Ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R) : AddValuation.comap (g.comp f) v = AddValuation.comap f (AddValuation.comap g v)

def AddValuation.map {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] (f : Γ₀ →+ Γ'₀) (ht : f ⊤ = ⊤) (hf : Monotone ⇑f) (v : AddValuation R Γ₀) : AddValuation R Γ'₀

A ≤-preserving, ⊤-preserving group homomorphism Γ₀ → Γ'₀ induces a map AddValuation R Γ₀ → AddValuation R Γ'₀.

Equations

• AddValuation.map f ht hf v = Valuation.map { toFun := ⇑f, map_zero' := ht, map_one' := ⋯, map_mul' := ⋯ } ⋯ v

def AddValuation.IsEquiv {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] (v₁ : AddValuation R Γ₀) (v₂ : AddValuation R Γ'₀) : Prop

Two additive valuations on R are defined to be equivalent if they induce the same preorder on R.

Equations

• v₁.IsEquiv v₂ = Valuation.IsEquiv v₁ v₂

@[simp]

theorem AddValuation.map_inv {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] (v : AddValuation K Γ₀) {x : K} : v x⁻¹ = -v x

@[simp]

theorem AddValuation.map_neg {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) (x : R) : v (-x) = v x

theorem AddValuation.map_sub_swap {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) (x : R) (y : R) : v (x - y) = v (y - x)

theorem AddValuation.map_sub {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) (x : R) (y : R) : min (v x) (v y) ≤ v (x - y)

theorem AddValuation.map_le_sub {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x : R} {y : R} {g : Γ₀} (hx : g ≤ v x) (hy : g ≤ v y) : g ≤ v (x - y)

theorem AddValuation.map_add_of_distinct_val {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x : R} {y : R} (h : v x ≠ v y) : v (x + y) = min (v x) (v y)

theorem AddValuation.map_eq_of_lt_sub {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x : R} {y : R} (h : v x < v (y - x)) : v y = v x

theorem AddValuation.IsEquiv.refl {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Ring R] {v : AddValuation R Γ₀} : v.IsEquiv v

theorem AddValuation.IsEquiv.symm {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} (h : v₁.IsEquiv v₂) : v₂.IsEquiv v₁

theorem AddValuation.IsEquiv.trans {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {Γ''₀ : Type u_6} [LinearOrderedAddCommMonoidWithTop Γ''₀] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} {v₃ : AddValuation R Γ''₀} (h₁₂ : v₁.IsEquiv v₂) (h₂₃ : v₂.IsEquiv v₃) : v₁.IsEquiv v₃

theorem AddValuation.IsEquiv.of_eq {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Ring R] {v : AddValuation R Γ₀} {v' : AddValuation R Γ₀} (h : v = v') : v.IsEquiv v'

theorem AddValuation.IsEquiv.map {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v : AddValuation R Γ₀} {v' : AddValuation R Γ₀} (f : Γ₀ →+ Γ'₀) (ht : f ⊤ = ⊤) (hf : Monotone ⇑f) (inf : Function.Injective ⇑f) (h : v.IsEquiv v') : (AddValuation.map f ht hf v).IsEquiv (AddValuation.map f ht hf v')

theorem AddValuation.IsEquiv.comap {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} {S : Type u_7} [Ring S] (f : S →+* R) (h : v₁.IsEquiv v₂) : (AddValuation.comap f v₁).IsEquiv (AddValuation.comap f v₂)

comap preserves equivalence.

theorem AddValuation.IsEquiv.val_eq {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} (h : v₁.IsEquiv v₂) {r : R} {s : R} : v₁ r = v₁ s ↔ v₂ r = v₂ s

theorem AddValuation.IsEquiv.ne_top {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} (h : v₁.IsEquiv v₂) {r : R} : v₁ r ≠ ⊤ ↔ v₂ r ≠ ⊤

def AddValuation.supp {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [CommRing R] (v : AddValuation R Γ₀) : Ideal R

The support of an additive valuation v : R → Γ₀ is the ideal of R where v x = ⊤

Equations

• v.supp = Valuation.supp v

@[simp]

theorem AddValuation.mem_supp_iff {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [CommRing R] (v : AddValuation R Γ₀) (x : R) : x ∈ v.supp ↔ v x = ⊤

theorem AddValuation.map_add_supp {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [CommRing R] (v : AddValuation R Γ₀) (a : R) {s : R} (h : s ∈ v.supp) : v (a + s) = v a

def DiscreteValuation.termℕₘ₀ : Lean.ParserDescr

Notation for WithZero (Multiplicative ℕ)

Equations

• DiscreteValuation.termℕₘ₀ = Lean.ParserDescr.node `DiscreteValuation.termℕₘ₀ 1024 (Lean.ParserDescr.symbol "ℕₘ₀")

def DiscreteValuation.termℤₘ₀ : Lean.ParserDescr

Notation for WithZero (Multiplicative ℤ)

Equations

• DiscreteValuation.termℤₘ₀ = Lean.ParserDescr.node `DiscreteValuation.termℤₘ₀ 1024 (Lean.ParserDescr.symbol "ℤₘ₀")