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
x : R
h : v x ≠ 0
⊢ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ {y | v y = v x}

Tactic:
]

Goal state after:
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
⊢ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ {y | v y = v x}
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
⊢ a * ↑x * (b * ↑z) <ᵥ a * ↑x * (c * ↑y)

Tactic:
]

Goal state after:
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
⊢ a * ↑x * (b * ↑z) <ᵥ a * ↑x * (c * ↑y)
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: case h
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
h' : v.restrict x ≠ 0
⊢ {z | v.restrict (z - x) < v.restrict x} ⊆ {y | v y = v x}

Tactic:
]

Goal state after:
case h
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
h' : v.restrict x ≠ 0
⊢ {z | v.restrict (z - x) < v.restrict x} ⊆ {y | v y = v x}
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
h : 1 * ↑1 ≤ᵥ 0 * ↑1
⊢ False

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
h : 1 * ↑1 ≤ᵥ 0 * ↑1
⊢ False
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 : ¬(x * ↑1 ≤ᵥ 0 * ↑y ∧ 0 * ↑y ≤ᵥ x * ↑1)
⊢ ValueGroupWithZero.mk x y * (ValueGroupWithZero.mk x y)⁻¹ = 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z x : R
y : ↥(posSubmonoid R)
h : ¬(x * ↑1 ≤ᵥ 0 * ↑y ∧ 0 * ↑y ≤ᵥ x * ↑1)
⊢ ValueGroupWithZero.mk x y * (ValueGroupWithZero.mk x y)⁻¹ = 1
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 : ¬x ≤ᵥ 0
⊢ ↑y * x * ↑1 ≤ᵥ 1 * ↑(y * ⟨x, h⟩) ∧ 1 * ↑(y * ⟨x, h⟩) ≤ᵥ ↑y * x * ↑1

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z x : R
y : ↥(posSubmonoid R)
h : ¬x ≤ᵥ 0
⊢ ↑y * x * ↑1 ≤ᵥ 1 * ↑(y * ⟨x, h⟩) ∧ 1 * ↑(y * ⟨x, h⟩) ≤ᵥ ↑y * x * ↑1
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
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

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

Tactic:
]

Goal state after:
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x : ↥(posSubmonoid R)
⊢ ¬↑x ≤ᵥ 0
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) ↑s = (valuation R) r

Tactic:
]

Goal state after:
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
r : R
s : ↥(posSubmonoid R)
⊢ ValueGroupWithZero.mk r s * (valuation R) ↑s = (valuation R) r
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case inr
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 : R
hx : x ∈ {x | v.restrict x < r}
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ {x | v.restrict x < r}

Tactic:
]

Goal state after:
case inr
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 : R
hx : x ∈ {x | v.restrict x < r}
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ {x | v.restrict x < r}
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 ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 x

Tactic:
]

Goal state after:
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 ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 x
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 : R
hx : x ∈ {x | v.restrict x ≤ r}
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ {x | v.restrict x ≤ r}

Tactic:
]

Goal state after:
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 : R
hx : x ∈ {x | v.restrict x ≤ r}
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ {x | v.restrict x ≤ r}
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

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

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
⊢ 1 ∉ supp R
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 : R
hx : x ∈ {x | v.restrict x ≤ r}
⊢ ∃ γ, ∀ (a : R), v.restrict (a - x) < ↑γ → v.restrict a ≤ r

Tactic:
exact ⟨Units.mk0 _ hr, fun y hy ↦
    (sub_add_cancel y x).symm ▸ le_trans (v.restrict.map_add _ _) (max_le (le_of_lt hy) hx)⟩

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 = 0

Tactic:
]

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
⊢ ∀ x ∈ {x | v.restrict x ≤ r}ᶜ, {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ ∀ x ∈ {x | v.restrict x ≤ r}ᶜ, {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x
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 * c ≤ᵥ b

Tactic:
]

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 ≤ᵥ b * c

Tactic:
]

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
x : R
hx : r < v.restrict x
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ {x | v.restrict x ≤ r}ᶜ

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
x : R
hx : r < v.restrict x
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ {x | v.restrict x ≤ r}ᶜ
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
h : {x | v.restrict x = r} = {x | v.restrict x ≤ r} \ {x | v.restrict x < r}
⊢ IsClopen ({x | v.restrict x ≤ r} \ {x | v.restrict x < r})

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
h : {x | v.restrict x = r} = {x | v.restrict x ≤ r} \ {x | v.restrict x < r}
⊢ IsClopen ({x | v.restrict x ≤ r} \ {x | v.restrict x < r})
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:
rw [negAddY, addX, addZ]

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
⊢ (-P y * Q x ^ 3 * P z ^ 3 + 2 * P y * Q y ^ 2 * P z ^ 3 - 3 * P x ^ 2 * Q x * Q y * P z ^ 2 * Q z +
                                        3 * P x * P y * Q x ^ 2 * P z * Q z ^ 2 +
                                      P x ^ 3 * Q y * Q z ^ 3 -
                                    2 * P y ^ 2 * Q y * Q z ^ 3 +
                                  W'.a₁ * P x * Q y ^ 2 * P z ^ 4 +
                                W'.a₁ * P y * Q x * Q y * P z ^ 3 * Q z -
                              W'.a₁ * P x * P y * Q y * P z * Q z ^ 3 -
                            W'.a₁ * P y ^ 2 * Q x * Q z ^ 4 -
                          2 * W'.a₂ * P x * Q x * Q y * P z ^ 4 * Q z +
                        2 * W'.a₂ * P x * P y * Q x * P z * Q z ^ 4 +
                      W'.a₃ * Q y ^ 2 * P z ^ 6 -
                    W'.a₃ * P y ^ 2 * Q z ^ 6 -
                  W'.a₄ * Q x * Q y * P z ^ 6 * Q z -
                W'.a₄ * P x * Q y * P z ^ 4 * Q z ^ 3 +
              W'.a₄ * P y * Q x * P z ^ 3 * Q z ^ 4 +
            W'.a₄ * P x * P y * P z * Q z ^ 6 -
          2 * W'.a₆ * Q y * P z ^ 6 * Q z ^ 3 +
        2 * W'.a₆ * P y * P z ^ 3 * Q z ^ 6) *
      (P z * Q z) ^ 3 =
    (P y * Q z ^ 3 - Q y * P z ^ 3) *
        ((P x * Q x ^ 2 * P z ^ 2 - 2 * P y * Q y * P z * Q z + P x ^ 2 * Q x * Q z ^ 2 -
                            W'.a₁ * P x * Q y * P z ^ 2 * Q z -
                          W'.a₁ * P y * Q x * P z * Q z ^ 2 +
                        2 * W'.a₂ * P x * Q x * P z ^ 2 * Q z ^ 2 -
                      W'.a₃ * Q y * P z ^ 4 * Q z -
                    W'.a₃ * P y * P z * Q z ^ 4 +
                  W'.a₄ * Q x * P z ^ 4 * Q z ^ 2 +
                W'.a₄ * P x * P z ^ 2 * Q z ^ 4 +
              2 * W'.a₆ * P z ^ 4 * Q z ^ 4) *
            (P z * Q z) ^ 2 -
          P x * Q z ^ 2 * (P x * Q z ^ 2 - Q x * P z ^ 2) ^ 2) +
      P y * Q z ^ 3 * (P x * Q z ^ 2 - Q x * P z ^ 2) ^ 3
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
⊢ (Filter.comap (fun x ↦ x.2 - x.1) (𝓝 0)).HasBasis (fun x ↦ True) fun γ ↦ {p | v.restrict (p.2 - p.1) < ↑γ}

Tactic:
]

Goal state after:
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
⊢ (Filter.comap (fun x ↦ x.2 - x.1) (𝓝 0)).HasBasis (fun x ↦ True) fun γ ↦ {p | v.restrict (p.2 - p.1) < ↑γ}
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
⊢ (F.NeBot ∧ ∀ U ∈ ⋯.toRingFilterBasis.toAddGroupFilterBasis, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U) ↔
    F.NeBot ∧ ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
]

Goal state after:
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
⊢ (F.NeBot ∧ ∀ U ∈ ⋯.toRingFilterBasis.toAddGroupFilterBasis, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U) ↔
    F.NeBot ∧ ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
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:
← coe_smul,

Goal state after:
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
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
⊢ ↑0 = ↑0

Tactic:
]

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:
← coe_smul,

Goal state after:
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
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 = 0

Tactic:
]

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✝ : Completion α
a : α
⊢ ↑a = ↑a

Tactic:
]

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✝ : Completion α
a : α
⊢ 0 • ↑a = 0

Tactic:
← coe_smul,

Goal state after:
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✝ : Completion α
a : α
⊢ ↑(0 • a) = 0
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✝ : Completion α
a : α
⊢ ↑0 = ↑0

Tactic:
]

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
n : ℕ
a✝ : Completion α
a : α
⊢ (n + 1) • ↑a = n • ↑a + ↑a

Tactic:
← coe_smul,

Goal state after:
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
n : ℕ
a✝ : Completion α
a : α
⊢ ↑((n + 1) • a) = n • ↑a + ↑a
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
n : ℕ
a✝ : Completion α
a : α
⊢ n • ↑a + ↑a = n • ↑a + ↑a

Tactic:
]

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
n : ℕ
a✝ : Completion α
a : α
⊢ ↑n.succ • ↑a = ↑n • ↑a + ↑a

Tactic:
← coe_smul,

Goal state after:
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
n : ℕ
a✝ : Completion α
a : α
⊢ ↑(↑n.succ • a) = ↑n • ↑a + ↑a
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
n : ℕ
a✝ : Completion α
a : α
⊢ ↑n • ↑a + ↑a = ↑n • ↑a + ↑a

Tactic:
]

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
n : ℕ
a✝ : Completion α
a : α
⊢ Int.negSucc n • ↑a = -(↑n.succ • ↑a)

Tactic:
← coe_smul,

Goal state after:
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
n : ℕ
a✝ : Completion α
a : α
⊢ ↑(Int.negSucc n • a) = -(↑n.succ • ↑a)
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
n : ℕ
a✝ : Completion α
a : α
⊢ -(↑n.succ • ↑a) = -(↑n.succ • ↑a)

Tactic:
]

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:
← coe_smul,

Goal state after:
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
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
⊢ ↑0 = ↑0

Tactic:
]

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
⊢ IsNontrivial R

Tactic:
]

Goal state after:
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
⊢ IsNontrivial R
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)
⊢ (valuation K) x / (valuation K) ↑y = (valuation K) x / (valuation K) ↑y

Tactic:
]

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
⊢ ↑(y + x) = ↑(y + x)

Tactic:
]

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:
← coe_smul,

Goal state after:
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
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 • ↑x + b • ↑x = a • ↑x + b • ↑x

Tactic:
]

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
⊢ 0 = 0

Tactic:
]

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 = 0
⊢ (valuation R) a / (valuation R) ↑b ≠ ↑γ

Tactic:
]

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

Goal state before:
pp: case h
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
x : R
hx : v x < 1
hx' : x ≠ 0
⊢ ∃ x, x ≠ 0 ∧ v x ≠ 1

Tactic:
]

Goal state after:
case h
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
x : R
hx : v x < 1
hx' : x ≠ 0
⊢ ∃ x, x ≠ 0 ∧ v x ≠ 1
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
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x

Tactic:
]

Goal state after:
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
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝³ : Semiring R
inst✝² : ValuativeRel R
inst✝¹ : IsDiscrete R
inst✝ : IsNontrivial R
a : ValueGroupWithZero R
ha : 0 < a
⊢ 1 < a ↔ 1 < a

Tactic:
]

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

Goal state before:
pp: case refine_2
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
x : Completion α
⊢ ∀ (a : α), (completion 0 ⋯) ↑a = 0 ↑a

Tactic:
· simp [(0 : α →+ β).completion_coe continuous_const, coe_zero]

Goal state after: