mathlib-initiative/mathlib-tactics huggingface dataset:
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
h : v ↑s = 1
⊢ False

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 r ≠ 0

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
hr : v r ≠ 0
hr' : v r ≠ 1
⊢ (valuation R) r ≠ 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✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
M : Type ?u.13
inst✝² : Monoid M
inst✝¹ : DistribMulAction 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✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
M : Type ?u.13
inst✝² : Monoid M
inst✝¹ : DistribMulAction M α
inst✝ : UniformContinuousConstSMul M α
r : M
x y : Completion α
a b : α
⊢ r • (↑a + ↑b) = r • ↑a + r • ↑b

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
⊢ IsNontrivial R ↔ v.IsNontrivial

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 : AddCommGroup α
inst✝ : IsUniformAddGroup α
a b : Completion α
⊢ Continuous fun x ↦ x.1 + x.2

Tactic:
by

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

Goal state before:
pp: R✝ : Type u_1
inst✝⁶ : Semiring R✝
inst✝⁵ : ValuativeRel R✝
R : Type u_2
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommMonoidWithZero Γ₀
inst✝¹ : IsNontrivial R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ v.IsNontrivial

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 : AddCommGroup α
inst✝ : IsUniformAddGroup α
a b : Completion α
⊢ Continuous fun x ↦ x.2 + x.1

Tactic:
by

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

Goal state before:
pp: K : Type u_3
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
x : K
y : ↥(posSubmonoid K)
⊢ ValueGroupWithZero.mk x y = (valuation K) (x / ↑y)

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 : AddCommGroup α
inst✝ : IsUniformAddGroup α
a b : Completion α
x y : α
⊢ ↑x + ↑y = ↑y + ↑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
γ : ValueGroupWithZero R
⊢ ∃ a b, (valuation R) a / (valuation R) ↑b = γ

Tactic:
induction γ using ValueGroupWithZero.ind with | mk a b
  use a, b
  simp [valuation, div_eq_mul_inv, ValueGroupWithZero.inv_mk (b : R) 1 b.prop]

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

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
γ : ValueGroupWithZero R
⊢ ∃ a b, (valuation R) a / (valuation R) ↑b = γ

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✝⁵ : UniformSpace α
inst✝⁴ : AddCommGroup α
inst✝³ : IsUniformAddGroup α
inst✝² : Semiring R
inst✝¹ : Module R α
inst✝ : UniformContinuousConstSMul R α
a b : R
x : α
⊢ (a + b) • ↑x = a • ↑x + b • ↑x

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
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✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ _t = ⋯.topology

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
γ : (ValueGroupWithZero R)ˣ
a : R
b : ↥(posSubmonoid R)
hab : a ∉ posSubmonoid R
⊢ (valuation R) a / (valuation R) ↑b ≠ ↑γ

Tactic:
posSubmonoid_def,

Goal state after:
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
γ : (ValueGroupWithZero R)ˣ
a : R
b : ↥(posSubmonoid R)
hab : ¬0 <ᵥ a
⊢ (valuation R) a / (valuation R) ↑b ≠ ↑γ
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁸ : Ring R
inst✝⁷ : ValuativeRel R
K : Type u_2
inst✝⁶ : DivisionRing K
inst✝⁵ : ValuativeRel K
Γ₀ : Type u_3
inst✝⁴ : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝³ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝² : v.Compatible
inst✝¹ : TopologicalSpace K
inst✝ : IsValuativeTopology K
⊢ IsTopologicalRing R

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✝⁷ : UniformSpace α
inst✝⁶ : AddGroup α
inst✝⁵ : IsUniformAddGroup α
inst✝⁴ : UniformSpace β
inst✝³ : AddGroup β
inst✝² : IsUniformAddGroup β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α →+ β
hf✝ : Continuous ⇑f
hf : UniformContinuous ⇑f
⊢ Completion.extension (⇑f) 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✝⁷ : UniformSpace α
inst✝⁶ : AddGroup α
inst✝⁵ : IsUniformAddGroup α
inst✝⁴ : UniformSpace β
inst✝³ : AddGroup β
inst✝² : IsUniformAddGroup β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α →+ β
hf✝ : Continuous ⇑f
hf : UniformContinuous ⇑f
a b : Completion α
⊢ Continuous fun x ↦ Completion.extension (⇑f) (x.1 + x.2)

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
γ : (ValueGroupWithZero R)ˣ
a : R
b : ↥(posSubmonoid R)
hab : (valuation R) a / (valuation R) ↑b = ↑γ
⊢ a ∈ posSubmonoid R

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✝⁷ : UniformSpace α
inst✝⁶ : AddGroup α
inst✝⁵ : IsUniformAddGroup α
inst✝⁴ : UniformSpace β
inst✝³ : AddGroup β
inst✝² : IsUniformAddGroup β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α →+ β
hf✝ : Continuous ⇑f
hf : UniformContinuous ⇑f
a b : Completion α
⊢ Continuous fun x ↦ Completion.extension (⇑f) x.1 + Completion.extension (⇑f) x.2

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
⊢ W'.negAddY P Q * (P z * Q z) ^ 3 =
    (P y * Q z ^ 3 - Q y * P z ^ 3) * (W'.addX P Q * (P z * Q z) ^ 2 - P x * Q z ^ 2 * addZ P Q ^ 2) +
      P y * Q z ^ 3 * addZ P Q ^ 3

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
h : ∀ (x : R), x ≠ 0 → v x = 1
⊢ DiscreteTopology 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
γ : (ValueGroupWithZero R)ˣ
⊢ ∃ a b, (valuation R) ↑a / (valuation R) ↑b = ↑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
inst✝¹ : IsDiscrete R
inst✝ : IsNontrivial R
⊢ 0 < uniformizer R

Tactic:
by

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

Goal state before:
pp: K : Type u_2
inst✝⁶ : DivisionRing K
inst✝⁵ : ValuativeRel K
Γ₀ : Type u_3
inst✝⁴ : LinearOrderedCommGroupWithZero Γ₀
inst✝³ : TopologicalSpace K
inst✝² : IsValuativeTopology K
inst✝¹ : MulArchimedean Γ₀
v : Valuation K Γ₀
inst✝ : v.Compatible
r : Γ₀
hr : r ≠ 0
h : ∀ (x : K), v x ≠ 0 → r < v x
⊢ ∀ (x : K), x ≠ 0 → v x = 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
inst✝¹ : IsDiscrete R
inst✝ : IsNontrivial R
a : ValueGroupWithZero R
⊢ (uniformizer R)⁻¹ ≤ a ↔ 1 < a

Tactic:
by

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

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
inst✝² : UniformSpace β
inst✝¹ : AddGroup β
inst✝ : IsUniformAddGroup β
⊢ completion 0 ⋯ = 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hPz : P z ≠ 0
hQz : Q z ≠ 0
⊢ W.negAddY P Q =
    ((P y * Q z ^ 3 - Q y * P z ^ 3) * (W.addX P Q * (P z * Q z) ^ 2 - P x * Q z ^ 2 * addZ P Q ^ 2) +
        P y * Q z ^ 3 * addZ P Q ^ 3) /
      (P z * Q z) ^ 3

Tactic:
by

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

Goal state before:
pp: α : Type u_3
inst✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
γ : Type u_5
inst✝² : AddCommGroup γ
inst✝¹ : UniformSpace γ
inst✝ : IsUniformAddGroup γ
f g : α →+ γ
hf : Continuous ⇑f
hg : Continuous ⇑g
⊢ (f + g).completion ⋯ = f.completion hf + g.completion hg

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
⊢ Continuous fun p ↦ p.1 * p.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ x.1 * x.2.1 * x.2.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ x.1 * (x.2.1 * x.2.2)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ b✝ c✝ : Completion α
a b c : α
⊢ ↑a * ↑b * ↑c = ↑a * (↑b * ↑c)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a : Completion α
⊢ Continuous (HMul.hMul 1)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ 1 * ↑a = ↑a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a : Completion α
⊢ Continuous fun a ↦ a * 1

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
⊢ IsOpen[_t] {x | v.restrict x < r}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑a * 1 = ↑a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a : Completion α
⊢ Continuous (HMul.hMul 0)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ 0 * ↑a = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a : Completion α
⊢ Continuous fun a ↦ a * 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑a * 0 = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
⊢ IsClosed[_t] {x | v.restrict x < r}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ x.1 * (x.2.1 + x.2.2)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ x.1 * x.2.1 + x.1 * x.2.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ b✝ c✝ : Completion α
a b c : α
⊢ ↑a * (↑b + ↑c) = ↑a * ↑b + ↑a * ↑c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ (x.1 + x.2.1) * x.2.2

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ IsOpen[_t] {x | v.restrict x ≤ r}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a b c : Completion α
⊢ Continuous fun x ↦ x.1 * x.2.2 + x.2.1 * x.2.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ b✝ c✝ : Completion α
a b c : α
⊢ (↑a + ↑b) * ↑c = ↑a * ↑c + ↑b * ↑c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
⊢ Completion.extension (⇑f) 1 = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a b : Completion α
⊢ Continuous fun x ↦ Completion.extension (⇑f) (x.1 * x.2)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
⊢ IsClosed[_t] {x | v.restrict x ≤ r}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a b : Completion α
⊢ Continuous fun x ↦ Completion.extension (⇑f) x.1 * Completion.extension (⇑f) x.2

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
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✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
⊢ ∀ (x y : R) (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (restrict₀ (ofClass v)) x / (restrict₀ (ofClass v)) ↑s = (restrict₀ (ofClass v)) y / (restrict₀ (ofClass v)) ↑t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a✝ b✝ : Completion α
a b : α
⊢ Completion.extension (⇑f) (↑a * ↑b) = Completion.extension ⇑f ↑a * Completion.extension ⇑f ↑b

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✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
⊢ ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v)) ↑s) ⋯ 0 = 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✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
⊢ ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v)) ↑s) ⋯ 1 = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
⊢ Completion.extension (⇑f) 0 = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a b : Completion α
⊢ Continuous fun x ↦ Completion.extension (⇑f) (x.1 + x.2)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ IsClopen {x | v.restrict x = r}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a b : Completion α
⊢ Continuous fun x ↦ Completion.extension (⇑f) x.1 + Completion.extension (⇑f) x.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a✝ b✝ : Completion α
a b : α
⊢ Completion.extension (⇑f) (↑a + ↑b) = Completion.extension ⇑f ↑a + Completion.extension ⇑f ↑b

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
⊢ IsClosed[_t] {x | v.restrict x = r}

Tactic:
by

Goal state after: