mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ Continuous extension

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : lmax ≤ 1
⊢ (valuation R K) W.a₄ ∈ l

Tactic:
by

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
⊢ extension ↑x = v.restrict x

Tactic:
by

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
⊢ (⇑embedding ∘ extension) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : lmax ≤ 1
⊢ (valuation R K) W.a₆ ∈ l

Tactic:
by

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
⊢ (⇑embedding ∘ extension) 1 = 1

Tactic:
by

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
f : G ≃+o ℤ
this : LocallyFiniteOrder G
⊢ 0 ∈ lowerBounds {x | 0 ≤ x}

Tactic:
by

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
⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃+o ℤ)

Tactic:
by

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)
⊢ (⇑embedding ∘ extension) (x * y) = (⇑embedding ∘ extension) x * (⇑embedding ∘ extension) y

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
i : ℕ := List.idxOf lmax l
hi : i < l.length
⊢ (valuation R K) l₀[i] = l[i]

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
⊢ ∃ a, (valuation R K) a = lmax

Tactic:
by

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)
cont : Continuous extension
⊢ Continuous fun x ↦ extension (x.1 + x.2)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
⊢ a ≠ 0

Tactic:
by

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
⊢ ({x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2) ↔ Nonempty (G ≃+o ℤ)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
⊢ ∃ C, IsIntegral R (C • W)

Tactic:
have ha₀ : a ≠ 0 := by
      by_contra ha₀; simp only [ha₀, map_zero] at ha
      exact (ha ▸ hlmax_le_1) zero_le_one

Goal state after:
case neg
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
ha₀ : a ≠ 0
⊢ ∃ C, IsIntegral R (C • W)
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)
⊢ (⇑embedding ∘ extension) (x + y) ≤ max ((⇑embedding ∘ extension) x) ((⇑embedding ∘ extension) y)

Tactic:
by

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
⊢ extensionValuation ↑x = v x

Tactic:
by

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 : hat K
this : extensionValuation x = 0 ↔ x = 0
⊢ extension x = 0 ↔ x = 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : Nontrivial G
g : G
e : G ≃o Additive G := OrderIso.refl G
this : ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ {x | e g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃*o Multiplicative ℤ)

Tactic:
by

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 : hat K
⊢ extension x = 0 ↔ x = 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

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

Tactic:
by

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 : hat K
h : x ≠ 0
a : K
⊢ ∃ r, extensionValuation ↑a = v r

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
ha₀ : a ≠ 0
⊢ (valuation R K) W.a₁ ∈ l

Tactic:
by

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 : hat K
h : x ≠ 0
r : K
⊢ embedding ((restrict₀ (ofClass v)) r) = v r

Tactic:
by

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 : hat K
h : x ≠ 0
hr : ∀ (r : K), embedding ((restrict₀ (ofClass v)) r) = v r
a b : (ofClass v).ValueGroup₀
⊢ embedding a = embedding b ↔ a = b

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
ha₀ : a ≠ 0
⊢ (valuation R K) W.a₂ ∈ l

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : Nontrivial G
g : G
⊢ ({x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2) ↔ Nonempty (G ≃*o Multiplicative ℤ)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
ha₀ : a ≠ 0
⊢ (valuation R K) W.a₃ ∈ l

Tactic:
by

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 : hat K
⊢ ∃ r, extensionValuation x = v r

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
ha₀ : a ≠ 0
⊢ (valuation R K) W.a₄ ∈ l

Tactic:
by

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
γ : Γ₀ˣ
x : hat K
γ₀' : (ofClass v).ValueGroup₀ := extension x
hγ₀'_def : γ₀' = extension x
γ₀ : Γ₀ := extensionValuation x
hγ₀_def : γ₀ = extensionValuation x
heq : γ₀ = embedding γ₀'
this : γ₀ ≠ 0 → (x ∈ closure (Completion.coe' '' {x | v x < ↑γ}) ↔ γ₀ < ↑γ)
h : γ₀ = 0
⊢ 0 ∈ {x | v x < ↑γ} ∧ ↑0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
ha₀ : a ≠ 0
⊢ (valuation R K) W.a₆ ∈ l

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
this : ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ {x | Units.mk0 g hg ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G₀ ≃*o WithZero (Multiplicative ℤ))

Tactic:
by

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
γ : Γ₀ˣ
x : hat K
γ₀' : (ofClass v).ValueGroup₀ := extension x
hγ₀'_def : γ₀' = extension x
γ₀ : Γ₀ := extensionValuation x
hγ₀_def : γ₀ = extensionValuation x
heq : γ₀ = embedding γ₀'
this : γ₀ ≠ 0 → (x ∈ closure (Completion.coe' '' {x | v x < ↑γ}) ↔ γ₀ < ↑γ)
⊢ x ∈ closure (Completion.coe' '' {x | v x < ↑γ}) ↔ x ∈ {x | extensionValuation x < ↑γ}

Tactic:
by

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
γ : Γ₀ˣ
x : hat K
γ₀' : (ofClass v).ValueGroup₀ := extension x
hγ₀'_def : γ₀' = extension x
γ₀ : Γ₀ := extensionValuation x
hγ₀_def : γ₀ = extensionValuation x
heq : γ₀ = embedding γ₀'
h : γ₀ ≠ 0
⊢ γ₀' ≠ 0

Tactic:
by

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
γ : Γ₀ˣ
x : hat K
γ₀' : (ofClass v).ValueGroup₀ := extension x
hγ₀'_def : γ₀' = extension x
γ₀ : Γ₀ := extensionValuation x
hγ₀_def : γ₀ = extensionValuation x
heq : γ₀ = embedding γ₀'
h : γ₀ ≠ 0
h' : γ₀' ≠ 0
hγ₀ : extension ⁻¹' {γ₀'} ∈ 𝓝 x
hx : ∀ t ∈ 𝓝 x, ∃ y, ↑y ∈ t
y : K
hy₁ : v y < ↑γ
hy₂ : ↑⟨↑y, ⋯⟩ ∈ extension ⁻¹' {γ₀'}
⊢ v y = γ₀

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G₀ ≃*o WithZero (Multiplicative ℤ))

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
⊢ g⁻¹ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
⊢ ∃ C, IsIntegral R (C • W)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDiscreteValuationRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
h : IsIntegral R W
⊢ (valuation K (maximalIdeal R)) W.Δ ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDiscreteValuationRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
hW : IsIntegral R W
⊢ ↑(valuation_Δ_aux R W) = (valuation K (maximalIdeal R)) W.Δ

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsDiscreteValuationRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
W : WeierstrassCurve K
inst✝ : IsMinimal R W
⊢ IsIntegral R W

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDiscreteValuationRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
C : VariableChange K
hC : MaximalFor (fun C ↦ IsIntegral R (C • W)) (fun C ↦ valuation_Δ_aux R (C • W)) C
⊢ (fun C_1 ↦ IsIntegral R (C_1 • C • W)) 1

Tactic:
by

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
γ : Γ₀ˣ
x : hat K
γ₀' : (ofClass v).ValueGroup₀ := extension x
hγ₀'_def : γ₀' = extension x
γ₀ : Γ₀ := extensionValuation x
hγ₀_def : γ₀ = extensionValuation x
heq : γ₀ = embedding γ₀'
h : γ₀ ≠ 0
h' : γ₀' ≠ 0
hγ₀ : extension ⁻¹' {γ₀'} ∈ 𝓝 x
hx : γ₀ < ↑γ
s : Set (hat K)
hs : s ∈ 𝓝 x
y : K
hy₁ : ↑y ∈ extension ⁻¹' {γ₀'}
hy₂ : ↑y ∈ s
⊢ v y = γ₀

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDiscreteValuationRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
⊢ ∃ C, IsMinimal R (C • W)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
hg' : g⁻¹ ≠ 0
h : ({x | x ≤ g} \ {0}).WellFoundedOn fun x1 x2 ↦ x1 > x2
a b : G₀
x✝ : g⁻¹ ≤ b
ha : a = 0
⊢ a < g⁻¹

Tactic:
simp [zero_lt_iff, ha, hg]

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
γ : Γ₀ˣ
⊢ closure (Completion.coe' '' {x | v x < ↑γ}) = {x | extensionValuation x < ↑γ}

Tactic:
by

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
r s : K
hr : r ≠ 0
hs : s ≠ 0
⊢ v s / v r ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDiscreteValuationRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
hW : IsMinimal R W
⊢ HasGoodReduction R W ↔ (reduction R W).IsElliptic

Tactic:
by

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
r s : K
hr : r ≠ 0
hs : s ≠ 0
⊢ closure (Completion.coe' '' {x | v x * v r < v s}) = {x | extensionValuation x * v r < v s}

Tactic:
by

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
⊢ extensionValuation.restrict ↑⋯.choose = 0

Tactic:
by

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
⊢ extensionValuation.restrict ↑⋯.choose = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
⊢ ({x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2) ↔ Nonempty (G₀ ≃*o WithZero (Multiplicative ℤ))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsDiscreteValuationRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
W : WeierstrassCurve K
inst✝ : IsMinimal R W
⊢ HasGoodReduction R W ∨ HasMultiplicativeReduction R W ∨ HasAdditiveReduction R W

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : CommMonoid α
inst✝³ : PartialOrder α
inst✝² : CommMonoid β
inst✝¹ : PartialOrder β
e : α ≃*o β
inst✝ : MulArchimedean α
x y : β
hxy : 1 < y
⊢ 1 < e.symm y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : CommMonoid α
inst✝³ : PartialOrder α
inst✝² : CommMonoid β
inst✝¹ : PartialOrder β
e : α ≃*o β
inst✝ : MulArchimedean α
⊢ MulArchimedean β

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: α : Type u_2
inst✝¹ : CommGroup α
inst✝ : PartialOrder α
⊢ MulArchimedean (WithZero α) ↔ MulArchimedean α

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero G₀
⊢ MulArchimedean G₀ˣ ↔ MulArchimedean G₀

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : MulArchimedean G
X : Type u_2
inst✝¹ : Preorder X
inst✝ : LocallyFiniteOrder X
⊢ ∀ {a b : Multiplicative X}, Multiplicative.toAdd a ≤ Multiplicative.toAdd b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : MulArchimedean G
X : Type u_2
inst✝¹ : Preorder X
inst✝ : LocallyFiniteOrder X
⊢ ∀ {a b : Additive X}, Additive.toMul a ≤ Additive.toMul b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
inst✝⁶ : CommGroup G
inst✝⁵ : LinearOrder G
inst✝⁴ : IsOrderedMonoid G
inst✝³ : MulArchimedean G
X : Type u_2
inst✝² : Preorder X
inst✝¹ : LocallyFiniteOrder X
inst✝ : Monoid X
⊢ ∀ {a b : Xˣ}, { toFun := Units.val, inj' := ⋯ } a ≤ { toFun := Units.val, inj' := ⋯ } b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Northcott

Goal state before:
pp: α : Type u_1
β : Type u_2
h : α → β
inst✝¹ : LinearOrder β
inst✝ : NoMaxOrder β
⊢ Northcott h ↔ Tendsto h cofinite atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Northcott

Goal state before:
pp: α : Type u_1
β : Type u_2
h : α → β
inst✝¹ : LinearOrder β
inst✝ : Northcott h
s : Set α
hs : s.Nonempty
⊢ ∃ a ∈ s, ∀ a' ∈ s, h a ≤ h a'

Tactic:
by

Goal state after: