mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
s : Set R
x : R
⊢ s ∈ 𝓝 x ↔ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
s : Set R
⊢ s ∈ 𝓝 0 ↔ ∃ γ, {x | v.restrict x < ↑γ} ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
x : R
⊢ (𝓝 x).HasBasis (fun x ↦ True) fun γ ↦ {z | v.restrict (z - x) < ↑γ}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
⊢ (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | v.restrict x < ↑γ}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
⊢ ∀ (x y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (if h : x ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑s ⟨x, h⟩) =
          if h : y ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑t ⟨y, h⟩

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
x : R
h : v x ≠ 0
⊢ v.restrict x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ a : R
x : ↥(posSubmonoid R)
ha : 0 < ValueGroupWithZero.mk a x
b : R
y : ↥(posSubmonoid R)
c : R
z : ↥(posSubmonoid R)
hbc : ValueGroupWithZero.mk b y < ValueGroupWithZero.mk c z
⊢ ValueGroupWithZero.mk a x * ValueGroupWithZero.mk b y < ValueGroupWithZero.mk a x * ValueGroupWithZero.mk c z

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
⊢ ∃ x y, x ≠ y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
⊢ Cauchy F ↔ F.NeBot ∧ ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
x : R
h : v x ≠ 0
⊢ {y | v y = v x} ∈ 𝓝 x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z x : R
y : ↥(posSubmonoid R)
h : ValueGroupWithZero.mk x y ≠ 0
⊢ ValueGroupWithZero.mk x y * (ValueGroupWithZero.mk x y)⁻¹ = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝³ : Semiring R✝
inst✝² : ValuativeRel R✝
x✝² x' y y' z : R✝
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x x✝¹ x✝ : R
⊢ ValueGroupWithZero.mk (x✝¹ * x✝) 1 = ValueGroupWithZero.mk x✝¹ 1 * ValueGroupWithZero.mk x✝ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
x z : R
γ : (ofClass (valuation R)).ValueGroup₀ˣ
x✝ : True
y : R
hy : y ∈ {z_1 | (valuation R).restrict (z_1 - z) < ↑γ}
⊢ x +ᵥ y ∈ {z_1 | (valuation R).restrict (z_1 - (x + z)) < ↑γ}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝³ : Semiring R✝
inst✝² : ValuativeRel R✝
x✝ x' y y' z : R✝
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x : R
⊢ ∀ (x y : R), ValueGroupWithZero.mk (x + y) 1 ≤ max (ValueGroupWithZero.mk x 1) (ValueGroupWithZero.mk y 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝³ : Semiring R✝
inst✝² : ValuativeRel R✝
x✝² x' y y' z : R✝
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x x✝¹ x✝ : R
⊢ x✝¹ ≤ᵥ x✝ ↔ (valuation R) x✝¹ ≤ (valuation R) x✝

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x : ↥(posSubmonoid R)
⊢ (valuation R) ↑x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
r : R
s : ↥(posSubmonoid R)
⊢ ValueGroupWithZero.mk r s = (valuation R) r / (valuation R) ↑s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
x✝² x' y y' z : R✝
R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
x : R
S : Type u_3
Γ : Type u_4
inst✝¹ : Ring S
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation S Γ
x✝¹ y✝ : S
x✝ : v x✝¹ ≤ v y✝
h : S
⊢ v (x✝¹ * h) ≤ v (y✝ * h)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ IsOpen[_i.toTopologicalSpace] {x | v.restrict x < r}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
x✝¹ x' y y' z : R✝
R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
x : R
S : Type u_3
Γ : Type u_4
inst✝¹ : Ring S
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation S Γ
x✝ y✝ z✝ : S
h0 : ¬v z✝ ≤ v 0
h : v (x✝ * z✝) ≤ v (y✝ * z✝)
⊢ v x✝ ≤ v y✝

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
x✝ x' y y' z : R✝
R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
x : R
S : Type u_3
Γ : Type u_4
inst✝¹ : Ring S
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation S Γ
⊢ ¬v 1 ≤ v 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
x✝ x' y y' z : R✝
R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
x : R
S : Type u_3
Γ : Type u_4
inst✝¹ : Ring S
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation S Γ
⊢ ∀ {x y : S}, v (x * y) ≤ v (y * x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₁ : Type u_3
Γ₂ : Type u_4
inst✝³ : LinearOrderedCommMonoidWithZero Γ₁
inst✝² : LinearOrderedCommMonoidWithZero Γ₂
v₁ : Valuation R Γ₁
v₂ : Valuation R Γ₂
inst✝¹ : v₁.Compatible
inst✝ : v₂.Compatible
⊢ v₁.IsEquiv v₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ IsClosed[_i.toTopologicalSpace] {x | v.restrict x < r}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x y : R
⊢ x <ᵥ y ↔ v x < v y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x y : R
⊢ v x ≤ v y ∧ v y ≤ v x ↔ v x = v y

Tactic:
antisymm_iff

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x y : R
⊢ x =ᵥ y ↔ v x = v y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x : R
⊢ x ≤ᵥ 1 ↔ v x ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x : R
⊢ x <ᵥ 1 ↔ v x < 1

Tactic:
simp [v.vlt_iff_lt]

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x : R
⊢ x <ᵥ 1 ↔ v x < 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x : R
⊢ 1 ≤ᵥ x ↔ 1 ≤ v x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x : R
⊢ 1 <ᵥ x ↔ 1 < v x

Tactic:
simp [v.vlt_iff_lt]

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x : R
⊢ 1 <ᵥ x ↔ 1 < v x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x : ↥(posSubmonoid R)
⊢ v ↑x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y : R
⊢ x <ᵥ y ↔ x ≤ᵥ y ∧ ¬y ≤ᵥ x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x✝ : R
h : x✝ ∈ {x | x ≤ᵥ 0}
⊢ x • x✝ ∈ {x | x ≤ᵥ 0}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
a✝ x✝ : R
h : a✝ ∈ supp R
⊢ a✝ * x✝ ∈ supp R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
inst✝ : ValuativeRel R
⊢ supp R = (valuation R).supp

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
⊢ supp R ≠ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ y✝ : R
h : x✝ * y✝ ∈ supp R
⊢ x✝ ∈ supp R ∨ y✝ ∈ supp R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a b : R
hab : a ≤ᵥ b
n : ℕ
⊢ a ^ n ≤ᵥ b ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ IsOpen[_i.toTopologicalSpace] {x | v.restrict x ≤ r}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a b : R
hab : a <ᵥ b
n : ℕ
hn : n ≠ 0
⊢ a ^ n <ᵥ b ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a : R
ha : a ≤ᵥ 1
n m : ℕ
hnm : n ≤ m
⊢ a ^ m ≤ᵥ a ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a : R
ha : 1 ≤ᵥ a
n m : ℕ
hnm : n ≤ m
⊢ a ^ n ≤ᵥ a ^ m

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a : K
⊢ a ≤ᵥ 0 ↔ a = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a : K
⊢ 0 <ᵥ a ↔ a ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a : K
⊢ a = 0 → a =ᵥ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a : K
⊢ 0 =ᵥ a ↔ 0 = a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a b c : K
hc : c ≠ 0
⊢ 0 <ᵥ ?m.26

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a b c : K
hc : c ≠ 0
⊢ c ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a b c : K
hc : c ≠ 0
⊢ a ≤ᵥ b / c ↔ a * c ≤ᵥ b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
γ : (ofClass (valuation R)).ValueGroup₀ˣ
x✝ : True
y : R
hy : y ∈ {x | (valuation R).restrict x < ↑γ}
⊢ -y ∈ {x | (valuation R).restrict x < ↑γ}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a b c : K
hc : c ≠ 0
⊢ a / c ≤ᵥ b ↔ a ≤ᵥ b * c

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a b : K
hb : b ≠ 0
⊢ 1 ≤ᵥ a / b ↔ b ≤ᵥ a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a b : K
hb : b ≠ 0
⊢ a / b ≤ᵥ 1 ↔ a ≤ᵥ b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
x : K
hx : x ≠ 0
⊢ 1 ≤ᵥ x⁻¹ ↔ x ≤ᵥ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
x : K
hx : x ≠ 0
⊢ x⁻¹ ≤ᵥ 1 ↔ 1 ≤ᵥ x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
inst✝¹ : ValuativeRel R
inst✝ : IsNontrivial R
γ : ValueGroupWithZero R
h0 : γ ≠ 0
h1✝ : γ ≠ 1
h1 : 1 < γ
⊢ 0 < γ⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
inst✝¹ : ValuativeRel R
inst✝ : IsNontrivial R
γ : ValueGroupWithZero R
h0 : γ ≠ 0
h1✝ : γ ≠ 1
h1 : 1 < γ
⊢ γ⁻¹ < 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
inst✝¹ : ValuativeRel R
inst✝ : IsNontrivial R
⊢ ∃ γ, 0 < γ ∧ γ < 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
s : (ValueGroupWithZero R)ˣ
h : 1 ≠ s
⊢ ↑s ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
s : (ValueGroupWithZero R)ˣ
h : 1 ≠ s
⊢ ↑s ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ IsClosed[_i.toTopologicalSpace] {x | v.restrict x ≤ r}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
r s : (ValueGroupWithZero R)ˣ
h : r ≠ s
hr : r ≠ 1
⊢ ↑r ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ {x | v.restrict x = r} = {x | v.restrict x ≤ r} \ {x | v.restrict x < r}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
r s : (ValueGroupWithZero R)ˣ
h : r ≠ s
hr : r ≠ 1
⊢ ↑r ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
⊢ IsNontrivial R ↔ Nontrivial (ValueGroupWithZero R)ˣ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ IsClopen {x | v.restrict x = r}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
⊢ IsTopologicalAddGroup R

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ IsClosed[_i.toTopologicalSpace] {x | v.restrict x = r}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
⊢ IsOpen[_i.toTopologicalSpace] ↑v.integer

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
⊢ IsClosed[_i.toTopologicalSpace] ↑v.integer

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ (𝓤 R).HasBasis (fun x ↦ True) fun γ ↦ {p | v.restrict (p.2 - p.1) < ↑γ}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_3
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
r : R
s : ↥(posSubmonoid R)
hr : ValueGroupWithZero.mk r s ≠ 0
hr' : ValueGroupWithZero.mk r s ≠ 1
⊢ v r ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ _u = IsTopologicalAddGroup.rightUniformSpace R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_3
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
r : R
s : ↥(posSubmonoid R)
hr : ValueGroupWithZero.mk r s ≠ 0
hr' : ValueGroupWithZero.mk r s ≠ 1
hγ : v r ≠ 0
⊢ v r ≤ v ↑s → v r < v ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁴ : UniformSpace α
inst✝³ : MonoidWithZero M
inst✝² : Zero α
inst✝¹ : MulActionWithZero M α
inst✝ : UniformContinuousConstSMul M α
r : M
⊢ r • 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁴ : UniformSpace α
inst✝³ : MonoidWithZero M
inst✝² : Zero α
inst✝¹ : MulActionWithZero M α
inst✝ : UniformContinuousConstSMul M α
a : α
⊢ 0 • ↑a = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ x.2.1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ x.2.2

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_3
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
r : R
s : ↥(posSubmonoid R)
hr✝ : ValueGroupWithZero.mk r s ≠ 0
hr' : ValueGroupWithZero.mk r s ≠ 1
hγ : v r ≠ 0
hγ' : v r ≤ v ↑s → v r < v ↑s
hr : v r = 1
⊢ v ↑s ≠ 0

Tactic:
by

Goal state after: