mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
⊢ (b ^ n) ^ m = b ^ 1

Tactic:
hm,

Goal state after:
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
⊢ b = b ^ 1
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
⊢ b = b

Tactic:
]

Goal state after:
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
preimage_one : ∃ t ∈ 𝓝 ↑1, Completion.coe' ⁻¹' t ⊆ ⇑v ⁻¹' {1}
⊢ ∃ V ∈ 𝓝 1, ∀ (x : K), ↑x ∈ V → v x = 1

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
preimage_one : ∃ t ∈ 𝓝 ↑1, Completion.coe' ⁻¹' t ⊆ ⇑v ⁻¹' {1}
⊢ ∃ V ∈ 𝓝 1, ∀ (x : K), ↑x ∈ V → v x = 1
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
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
key : m = 1 ∧ n = 1 ∨ m = -1 ∧ n = -1
⊢ b = |b ^ n|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
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
key : m = 1 ∧ n = 1 ∨ m = -1 ∧ n = -1
⊢ b = |b ^ n|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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
⊢ Tendsto (fun p ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 1)

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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
⊢ Tendsto (fun p ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 1)
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
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
key : m = 1 ∧ n = 1 ∨ m = -1 ∧ n = -1
⊢ |b ^ n|ₘ = b ∧ 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
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
key : m = 1 ∧ n = 1 ∨ m = -1 ∧ n = -1
⊢ |b ^ n|ₘ = b ∧ 1 < b
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case mpr.inr
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
a b : G
h : b = |a|ₘ ∧ 1 < b
this : ∀ {a : G}, b = |a|ₘ ∧ 1 < b → 1 ≤ a → IsLeast {y | y ∈ closure {a} ∧ 1 < y} b
ha : ¬1 ≤ a
⊢ b = |a|ₘ ∧ 1 < b

Tactic:
]

Goal state after:
case mpr.inr
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
a b : G
h : b = |a|ₘ ∧ 1 < b
this : ∀ {a : G}, b = |a|ₘ ∧ 1 < b → 1 ≤ a → IsLeast {y | y ∈ closure {a} ∧ 1 < y} b
ha : ¬1 ≤ a
⊢ b = |a|ₘ ∧ 1 < b
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b a : G
h : b = a ∧ 1 < b
ha : 1 ≤ a
⊢ IsLeast {y | y ∈ closure {a} ∧ 1 < y} b

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b a : G
h : b = a ∧ 1 < b
ha : 1 ≤ a
⊢ IsLeast {y | y ∈ closure {a} ∧ 1 < y} 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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
r : Function.RightInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
c : Continuous fun x ↦ x * x₀⁻¹
⊢ (fun x ↦ x * x₀⁻¹) ⁻¹' V' ∈ 𝓝 x₀

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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
r : Function.RightInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
c : Continuous fun x ↦ x * x₀⁻¹
⊢ (fun x ↦ x * x₀⁻¹) ⁻¹' V' ∈ 𝓝 x₀
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : 1 < b
ha : 1 ≤ b
k : ℤ
hk : 1 < b ^ k
⊢ 0 < k

Tactic:
]

Goal state after:
case refine_2
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : 1 < b
ha : 1 ≤ b
k : ℤ
hk : 1 < b ^ k
⊢ 0 < k
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : 1 < b
ha : 1 ≤ b
k : ℤ
hk : k ≤ 0
⊢ 1 ≤ b ^ (-k)

Tactic:
]

Goal state after:
case refine_2
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : 1 < b
ha : 1 ≤ b
k : ℤ
hk : k ≤ 0
⊢ 1 ≤ b ^ (-k)
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 (x₀ * x₀⁻¹)
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
r : Function.RightInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
c : Continuous fun x ↦ x * x₀⁻¹
⊢ (fun x ↦ x * x₀⁻¹) ⁻¹' V' ∈ 𝓝 x₀

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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 (x₀ * x₀⁻¹)
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
r : Function.RightInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
c : Continuous fun x ↦ x * x₀⁻¹
⊢ (fun x ↦ x * x₀⁻¹) ⁻¹' V' ∈ 𝓝 x₀
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
⊢ z₀ ≠ 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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
⊢ z₀ ≠ 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inr
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
⊢ ∃ c, Tendsto (⇑v.restrict) (Filter.comap Completion.coe' (𝓝 x₀)) (𝓝 c)

Tactic:
have vz₀_ne : v.restrict z₀ ≠ 0 := by rwa [Valuation.ne_zero_iff]

Goal state after:
case inr
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
⊢ ∃ c, Tendsto (⇑v.restrict) (Filter.comap Completion.coe' (𝓝 x₀)) (𝓝 c)
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inr
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
⊢ ∀ᶠ (b : hat K) in 𝓝 x₀, ∀ (a : K), ↑a = b → v.restrict a = v.restrict z₀

Tactic:
]

Goal state after:
case inr
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
⊢ ∀ᶠ (b : hat K) in 𝓝 x₀, ∀ (a : K), ↑a = b → v.restrict a = v.restrict z₀
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G

Tactic:
by_cases! H : ∃ x, IsLeast {y : G | 0 < y} x

Goal state after:
case pos
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
H : ∃ x, IsLeast {y | 0 < y} x
⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G

case neg
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
H : ∀ (x : G), ¬IsLeast {y | 0 < y} x
⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
⊢ (ofClass v) (a * z₀⁻¹) = 1

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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
⊢ (ofClass v) (a * z₀⁻¹) = 1
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h
R : Type u_1
inst✝³ : CommRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
W : WeierstrassCurve K
inst✝ : IsIntegral R W
W_int : WeierstrassCurve R
hW_int : W = W_int⁄K
⊢ (algebraMap R K) W_int.Δ = (algebraMap R K) W_int.Δ

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
e : G ≃+o ℤ
H : DenselyOrdered ℤ
⊢ False

Tactic:
]

Goal state after:
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
e : G ≃+o ℤ
H : DenselyOrdered ℤ
⊢ False
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: A : Type u_2
inst✝⁴ : AddCommGroup A
inst✝³ : LinearOrder A
inst✝² : IsOrderedAddMonoid A
inst✝¹ : Archimedean A
inst✝ : Nontrivial A
⊢ Nonempty (A ≃+o ℤ) ↔ Nonempty (A ≃+o ℤ)

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
⊢ Nonempty (Additive G ≃+o ℤ) ∨ DenselyOrdered G

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
⊢ Nonempty (Additive G ≃+o ℤ) ∨ DenselyOrdered G
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
e : G ≃o Additive G := OrderIso.refl G
⊢ ¬DenselyOrdered (Additive G) ↔ ¬DenselyOrdered (Additive G)

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝⁴ : CommGroup G
inst✝³ : LinearOrder G
inst✝² : IsOrderedMonoid G
inst✝¹ : MulArchimedean G
inst✝ : Nontrivial G
⊢ ¬DenselyOrdered (Additive G) ↔ ¬DenselyOrdered G

Tactic:
]

Goal state after:
G : Type u_1
inst✝⁴ : CommGroup G
inst✝³ : LinearOrder G
inst✝² : IsOrderedMonoid G
inst✝¹ : MulArchimedean G
inst✝ : Nontrivial G
⊢ ¬DenselyOrdered (Additive G) ↔ ¬DenselyOrdered G
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
⊢ Nonempty (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered Gˣ

Tactic:
]

Goal state after:
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
⊢ Nonempty (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered Gˣ
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
⊢ Nonempty (G ≃*o WithZero (Multiplicative ℤ)) ↔ Nonempty (Gˣ ≃*o Multiplicative ℤ)

Tactic:
]

Goal state after:
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
⊢ Nonempty (G ≃*o WithZero (Multiplicative ℤ)) ↔ Nonempty (Gˣ ≃*o Multiplicative ℤ)
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_1
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
a✝ b✝ : Gˣ
⊢ f ↑a✝ ≤ f ↑b✝ ↔ f ↑a✝ ≤ f ↑b✝

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
this : ({x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃+o ℤ)
⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ {x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2

Tactic:
]

Goal state after:
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
this : ({x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃+o ℤ)
⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ {x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
this : ↑z₀⁻¹ = (↑z₀)⁻¹
⊢ y * y₀⁻¹ ∈ V

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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
this : ↑z₀⁻¹ = (↑z₀)⁻¹
⊢ y * y₀⁻¹ ∈ V
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
this : v.restrict (a * z₀⁻¹) = 1
⊢ v.restrict a = v.restrict a

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case mp
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
⊢ Nonempty (G ≃+o ℤ)

Tactic:
]

Goal state after:
case mp
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
⊢ Nonempty (G ≃+o ℤ)
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case pos
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : Archimedean G
⊢ ¬DenselyOrdered G

Tactic:
]

Goal state after:
case pos
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : Archimedean G
⊢ ¬DenselyOrdered G
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
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
this : v.restrict (a * z₀⁻¹) = 1
⊢ v.restrict z₀ = v.restrict z₀

Tactic:
]

Goal state after:
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
x : K
⊢ Tendsto (⇑v.restrict) (𝓝 x) (𝓝 (v.restrict x))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
⊢ Tendsto (⇑v.restrict) (𝓝 x) (𝓝 (v.restrict 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
⊢ extension 0 = extension 0

Tactic:
]

Goal state after:
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
⊢ 1 = 1

Tactic:
]

Goal state after:
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
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ extension (x * y) = extension x * extension y

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ extension (x * y) = extension x * extension y
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
⊢ ({x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2) ↔ {x | -g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2

Tactic:
]

Goal state after:
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
⊢ ({x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2) ↔ {x | -g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
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
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ extension (x + y) ≤ extension x ∨ extension (x + y) ≤ extension y

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ extension (x + y) ≤ extension x ∨ extension (x + y) ≤ extension y
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_1
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : {x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2
x✝ : G
⊢ x✝ ∈ {x | -g ≤ x} → (fun x ↦ -x) x✝ ∈ {x | x ≤ g}

Tactic:
simp [Function.onFun, neg_le]

Goal state after: