mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
x_ne : x ≠ 0
s : Set K
γ : (ofClass v).ValueGroup₀ˣ
hs : {y | v.restrict (y - x⁻¹) < ↑γ} ⊆ s
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ Inv.inv ⁻¹' s

Tactic:
]

Goal state after:
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
x_ne : x ≠ 0
s : Set K
γ : (ofClass v).ValueGroup₀ˣ
hs : {y | v.restrict (y - x⁻¹) < ↑γ} ⊆ s
⊢ ∃ γ, {y | v.restrict (y - x) < ↑γ} ⊆ Inv.inv ⁻¹' s
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case h.e_a.e_a
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
H : (maximalIdeal A).intValuation π ^ n / (maximalIdeal A).intValuation π ^ m ≤ 1
h_mn : m ≤ n
⊢ (algebraMap A K) π ^ n * ((algebraMap A K) π ^ m)⁻¹ = (algebraMap A K) π ^ n * ((algebraMap A K) π ^ m)⁻¹

pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
H : (maximalIdeal A).intValuation π ^ n / (maximalIdeal A).intValuation π ^ m ≤ 1
h_mn : m ≤ n
⊢ (algebraMap A K) π ≠ 0

Tactic:
]

Goal state after:
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
H : (maximalIdeal A).intValuation π ^ n / (maximalIdeal A).intValuation π ^ m ≤ 1
h_mn : m ≤ n
⊢ (algebraMap A K) π ≠ 0
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
H : DenselyOrdered X
⊢ ¬∃ x y, x < y

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
H : DenselyOrdered X
⊢ ¬∃ x y, x < y
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
⊢ ¬∃ x y, x < y

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
⊢ ¬∃ x y, x < y
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H✝ : DenselyOrdered ↑s
this : (some ⁻¹' s).Subsingleton
x y : ↑s
H : x < y
⊢ False

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H✝ : DenselyOrdered ↑s
this : (some ⁻¹' s).Subsingleton
x y : ↑s
H : x < y
⊢ False
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H✝ : DenselyOrdered ↑s
this : (some ⁻¹' s).Subsingleton
x y : ↑s
H : x < y
z : ↑s
hz : x < z
hz' : z < y
hz0 : ⊥ < ↑z
⊢ ↑z < ↑y

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H✝ : DenselyOrdered ↑s
this : (some ⁻¹' s).Subsingleton
x y : ↑s
H : x < y
z : ↑s
hz : x < z
hz' : z < y
hz0 : ⊥ < ↑z
⊢ ↑z < ↑y
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
H : (maximalIdeal A).intValuation π ^ n / (maximalIdeal A).intValuation π ^ m ≤ 1
h_mn : m ≤ n
⊢ π ≠ 0

Tactic:
]

Goal state after:
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
H : (maximalIdeal A).intValuation π ^ n / (maximalIdeal A).intValuation π ^ m ≤ 1
h_mn : m ≤ n
⊢ π ≠ 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
x_ne : x ≠ 0
⊢ ∃ γ, {y | v.restrict (y - 0) < ↑γ} ⊆ {k | v k < v x}

Tactic:
]

Goal state after:
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
x_ne : x ≠ 0
⊢ ∃ γ, {y | v.restrict (y - 0) < ↑γ} ⊆ {k | v k < v x}
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case refine_1
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
b : A
h : b ∉ (maximalIdeal A).asIdeal
⊢ ∃ x ∈ IsUnit.submonoid A, (algebraMap A K) x = (algebraMap A K) b

Tactic:
]

Goal state after:
case refine_1
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
b : A
h : b ∉ (maximalIdeal A).asIdeal
⊢ ∃ x ∈ IsUnit.submonoid A, (algebraMap A K) x = (algebraMap A K) b
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: case refine_2
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
x : X
hx : ↑x ∈ s
y : X
hy : ↑y ∈ s
hxy : x < y
this : ⟨↑x, hx⟩ < ⟨↑y, hy⟩
z : { x // x ∈ s }
hz : ↑x < ↑z
hz' : ↑z < ↑y
⊢ ↑x < ↑((↑z).unbot ⋯)

Tactic:
]

Goal state after:
case refine_2
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
x : X
hx : ↑x ∈ s
y : X
hy : ↑y ∈ s
hxy : x < y
this : ⟨↑x, hx⟩ < ⟨↑y, hy⟩
z : { x // x ∈ s }
hz : ↑x < ↑z
hz' : ↑z < ↑y
⊢ ↑x < ↑((↑z).unbot ⋯)
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ ∀ (x : K), ContinuousAt (⇑v.restrict) x

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ ∀ (x : K), ContinuousAt (⇑v.restrict) x
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: case refine_3
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
x : X
hx : ↑x ∈ s
y : X
hy : ↑y ∈ s
hxy : x < y
this : ⟨↑x, hx⟩ < ⟨↑y, hy⟩
z : { x // x ∈ s }
hz : ↑x < ↑z
hz' : ↑z < ↑y
⊢ ↑((↑z).unbot ⋯) < ↑y

Tactic:
]

Goal state after:
case refine_3
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
x : X
hx : ↑x ∈ s
y : X
hy : ↑y ∈ s
hxy : x < y
this : ⟨↑x, hx⟩ < ⟨↑y, hy⟩
z : { x // x ∈ s }
hz : ↑x < ↑z
hz' : ↑z < ↑y
⊢ ↑((↑z).unbot ⋯) < ↑y
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case pos
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x y : K
h : (valuation K (maximalIdeal A)) y = (valuation K (maximalIdeal A)) x
hx : x = 0
⊢ ∃ u, u • x = y

Tactic:
]

Goal state after:
case pos
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x y : K
h : (valuation K (maximalIdeal A)) y = (valuation K (maximalIdeal A)) x
hx : x = 0
⊢ ∃ u, u • x = y
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ ∀ (γ₀ : (ofClass v).ValueGroup₀), γ₀ ≠ 0 → ∀ᶠ (x : K) in 𝓝 0, v.restrict x < γ₀

Tactic:
]

Goal state after:
case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ ∀ (γ₀ : (ofClass v).ValueGroup₀), γ₀ ≠ 0 → ∀ᶠ (x : K) in 𝓝 0, v.restrict x < γ₀
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x y : K
h : (valuation K (maximalIdeal A)) x = (valuation K (maximalIdeal A)) y
hx : ¬x = 0
hy : ¬y = 0
this : y / x ∈ Submonoid.map (algebraMap A K) (IsUnit.submonoid A)
⊢ ∃ u, u • x = y

Tactic:
]

Goal state after:
case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x y : K
h : (valuation K (maximalIdeal A)) x = (valuation K (maximalIdeal A)) y
hx : ¬x = 0
hy : ¬y = 0
this : y / x ∈ Submonoid.map (algebraMap A K) (IsUnit.submonoid A)
⊢ ∃ u, u • x = y
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
γ : (ofClass v).ValueGroup₀
hγ : γ ≠ 0
⊢ ∃ γ_1, {x | v.restrict x < ↑γ_1} ⊆ {x | v.restrict x < γ}

Tactic:
]

Goal state after:
case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
γ : (ofClass v).ValueGroup₀
hγ : γ ≠ 0
⊢ ∃ γ_1, {x | v.restrict x < ↑γ_1} ⊆ {x | v.restrict x < γ}
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithTop X)
he : StrictAnti ⇑(WithTop.toDual.image s)
⊢ s.Subsingleton ↔ s.Subsingleton

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case refine_2
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
y : A
h : (algebraMap A K) y ∈ (valuation K (maximalIdeal A)).valuationSubring.toSubring
⊢ ∃ x ∈ ⊤, (algebraMap A K) x = (algebraMap A K) y

Tactic:
]

Goal state after:
case refine_2
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
y : A
h : (algebraMap A K) y ∈ (valuation K (maximalIdeal A)).valuationSubring.toSubring
⊢ ∃ x ∈ ⊤, (algebraMap A K) x = (algebraMap A K) y
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inr
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
h : x ≠ 0
v_ne : v.restrict x ≠ 0
⊢ ∀ᶠ (x_1 : K) in 𝓝 x, v.restrict x_1 = v.restrict x

Tactic:
]

Goal state after:
case inr
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
h : x ≠ 0
v_ne : v.restrict x ≠ 0
⊢ ∀ᶠ (x_1 : K) in 𝓝 x, v.restrict x_1 = v.restrict x
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
⊢ ∀ (x : K), ContinuousAt (⇑v) x

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
⊢ ∀ (x : K), ContinuousAt (⇑v) x
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
⊢ ∀ (γ₀ : Γ₀), γ₀ ≠ 0 → ∀ᶠ (x : K) in 𝓝 0, v x < γ₀

Tactic:
]

Goal state after:
case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
⊢ ∀ (γ₀ : Γ₀), γ₀ ≠ 0 → ∀ᶠ (x : K) in 𝓝 0, v x < γ₀
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
γ : Γ₀
hγ : γ ≠ 0
⊢ ∃ γ_1, {x | v.restrict x < ↑γ_1} ⊆ {x | v x < γ}

Tactic:
]

Goal state after:
case inl
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
γ : Γ₀
hγ : γ ≠ 0
⊢ ∃ γ_1, {x | v.restrict x < ↑γ_1} ⊆ {x | v x < γ}
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inr
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
x : K
h : x ≠ 0
h0 : v x ≠ 0
⊢ ∀ᶠ (x_1 : K) in 𝓝 x, v x_1 = v x

Tactic:
]

Goal state after:
case inr
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
x : K
h : x ≠ 0
h0 : v x ≠ 0
⊢ ∀ᶠ (x_1 : K) in 𝓝 x, v x_1 = v x
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : Cauchy F
h0 : 𝓝 0 ⊓ F = ⊥
U : Set K
U_in : U ∈ 𝓝 0
M : Set K
M_in : M ∈ F
H : U ∩ M = ∅
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
hU : {x | v.restrict x < ↑γ₀} ⊆ U
x : K
xM : x ∈ M
hyp : v.restrict x < ↑γ₀
this : x ∈ ∅
⊢ False

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : Cauchy F
h0 : 𝓝 0 ⊓ F = ⊥
U : Set K
U_in : U ∈ 𝓝 0
M : Set K
M_in : M ∈ F
H : U ∩ M = ∅
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
hU : {x | v.restrict x < ↑γ₀} ⊆ U
x : K
xM : x ∈ M
hyp : v.restrict x < ↑γ₀
this : x ∈ ∅
⊢ False
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝² : CommGroup G
inst✝¹ : LinearOrder G
inst✝ : IsOrderedMonoid G
a b : G
ha : a ≠ 1
⊢ (∃ n, a ^ n = b) ↔ ∃! k, a ^ k = b

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : CommGroup G
inst✝¹ : LinearOrder G
inst✝ : IsOrderedMonoid G
a b : G
ha : a ≠ 1
⊢ (∃ n, a ^ n = b) ↔ ∃! k, a ^ k = b
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : F.NeBot ∧ ∀ (γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v.restrict x
⊢ (map (fun x ↦ x⁻¹) F).NeBot ∧
    ∀ (γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ),
      ∃ M ∈ map (fun x ↦ x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : F.NeBot ∧ ∀ (γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v.restrict x
⊢ (map (fun x ↦ x⁻¹) F).NeBot ∧
    ∀ (γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ),
      ∃ M ∈ map (fun x ↦ x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: e : ℤ ≃+ ℤ
he : ¬IsOfFinAddOrder (e 1)
⊢ AddSubgroup.zmultiples (e 1) = AddSubgroup.zmultiples 1

Tactic:
]

Goal state after:
e : ℤ ≃+ ℤ
he : ¬IsOfFinAddOrder (e 1)
⊢ AddSubgroup.zmultiples (e 1) = AddSubgroup.zmultiples 1
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: e : ℤ ≃+o ℤ
H : e 1 = -1
⊢ 0 < 1

Tactic:
]

Goal state after:
e : ℤ ≃+o ℤ
H : e 1 = -1
⊢ 0 < 1
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: e : ℤ ≃+o ℤᵒᵈ
H : e 1 = 1
⊢ 0 < 1

Tactic:
]

Goal state after:
e : ℤ ≃+o ℤᵒᵈ
H : e 1 = 1
⊢ 0 < 1
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case h.left
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v.restrict x
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ (fun x ↦ x⁻¹) ⁻¹' (fun x ↦ x⁻¹) '' (M₀ ∩ M₁) ∈ F

Tactic:
]

Goal state after:
case h.left
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v.restrict x
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ (fun x ↦ x⁻¹) ⁻¹' (fun x ↦ x⁻¹) '' (M₀ ∩ M₁) ∈ F
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
H₁ : v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
x_in₀ : ↑γ₀ ≤ 0
h : v.restrict x = 0
⊢ False

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
H₁ : v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
x_in₀ : ↑γ₀ ≤ 0
h : v.restrict x = 0
⊢ False
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_1
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
Hy : ∀ (z : ↥(closure {y})), z ∈ zpowers ⟨y', ⋯⟩
⊢ orderOf x' = orderOf y'

Tactic:
]

Goal state after:
case refine_1
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
Hy : ∀ (z : ↥(closure {y})), z ∈ zpowers ⟨y', ⋯⟩
⊢ orderOf x' = orderOf y'
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case h
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
H₁ : v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
x_in₀ : ↑γ₀ ≤ v.restrict x
⊢ min (↑γ * (v.restrict x * v.restrict x)) (v.restrict x) ≤ min (↑γ * (v.restrict x * v.restrict x)) (v.restrict x)

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
Hy : ∀ (z : ↥(closure {y})), z ∈ zpowers ⟨y', ⋯⟩
a b : ↥(closure {x})
⊢ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun a ≤ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun b ↔ a ≤ b

Tactic:
obtain ⟨m, rfl⟩ := Hx a

Goal state after:
case refine_2
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
Hy : ∀ (z : ↥(closure {y})), z ∈ zpowers ⟨y', ⋯⟩
b : ↥(closure {x})
m : ℤ
⊢ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun ((fun x_1 ↦ ⟨x', ⋯⟩ ^ x_1) m) ≤ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun b ↔
    (fun x_1 ↦ ⟨x', ⋯⟩ ^ x_1) m ≤ b
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Tendsto (⇑v.restrict) (𝓝 0) (𝓝 0)

Tactic:
]

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Tendsto (⇑v.restrict) (𝓝 0) (𝓝 0)
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Tendsto (⇑v.restrict) (Filter.comap Completion.coe' (𝓝 0)) (𝓝 0)

Tactic:
rw [← Completion.coe_zero, ← Completion.isDenseInducing_coe.isInducing.nhds_eq_comap]

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Tendsto (⇑v.restrict) (𝓝 0) (𝓝 0)
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
⊢ 1 ≠ 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
⊢ 1 ≠ 0
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case mp
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
a b : G
h : IsLeast {y | y ∈ closure {a} ∧ 1 < y} b
this : closure {a} = closure {b}
ha : ∃ n, b ^ n = a
⊢ b = |a|ₘ ∧ 1 < b

Tactic:
]

Goal state after:
case mp
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
a b : G
h : IsLeast {y | y ∈ closure {a} ∧ 1 < y} b
this : closure {a} = closure {b}
ha : ∃ n, b ^ n = a
⊢ b = |a|ₘ ∧ 1 < b
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
this : v 1 ≠ 0
x : K
⊢ v x = 1 ↔ v x = 1

Tactic:
]

Goal state after: