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

Goal state before:
pp: case mpr
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
⊢ IsLeast {y | y ∈ closure {a} ∧ 1 < y} b

Tactic:
·

Goal state after:
case mpr
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
⊢ IsLeast {y | y ∈ closure {a} ∧ 1 < y} 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
⊢ IsLeast {y | y ∈ closure {a} ∧ 1 < y} b

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|ₘ ∧ 1 < b
ha : 1 ≤ a
⊢ IsLeast {y | y ∈ closure {a} ∧ 1 < y} 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
⊢ IsLeast {y | y ∈ closure {a} ∧ 1 < y} 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
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
x✝ : G
⊢ x✝ ∈ closure {a} ↔ x✝ ∈ closure {a⁻¹}

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⁻¹|ₘ ∧ 1 < b

Tactic:
·

Goal state after:
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
x✝ : G
⊢ x✝ ∈ closure {a} ↔ x✝ ∈ closure {a⁻¹}
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⁻¹|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⁻¹|ₘ ∧ 1 < b
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_1
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
⊢ b ∈ {y | y ∈ closure {b} ∧ 1 < y}

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
⊢ b ∈ lowerBounds {y | y ∈ closure {b} ∧ 1 < y}

Tactic:
·

Goal state after:
case refine_1
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
⊢ b ∈ {y | y ∈ closure {b} ∧ 1 < y}
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
⊢ b ∈ lowerBounds {y | y ∈ closure {b} ∧ 1 < y}

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
⊢ b ∈ lowerBounds {y | y ∈ closure {b} ∧ 1 < y}
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.a₁
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₁ = (algebraMap R K) h₁.choose

pp: case h.a₂
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₂ = (algebraMap R K) h₂.choose

pp: case h.a₃
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₃ = (algebraMap R K) h₃.choose

pp: case h.a₄
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₄ = (algebraMap R K) h₄.choose

pp: case h.a₆
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₆ = (algebraMap R K) h₆.choose

Tactic:
·

Goal state after:
case h.a₁
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₁ = (algebraMap R K) h₁.choose
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.a₂
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₂ = (algebraMap R K) h₂.choose

pp: case h.a₃
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₃ = (algebraMap R K) h₃.choose

pp: case h.a₄
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₄ = (algebraMap R K) h₄.choose

pp: case h.a₆
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₆ = (algebraMap R K) h₆.choose

Tactic:
·

Goal state after:
case h.a₂
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₂ = (algebraMap R K) h₂.choose
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.a₃
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₃ = (algebraMap R K) h₃.choose

pp: case h.a₄
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₄ = (algebraMap R K) h₄.choose

pp: case h.a₆
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₆ = (algebraMap R K) h₆.choose

Tactic:
·

Goal state after:
case h.a₃
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₃ = (algebraMap R K) h₃.choose
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.a₄
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₄ = (algebraMap R K) h₄.choose

pp: case h.a₆
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₆ = (algebraMap R K) h₆.choose

Tactic:
·

Goal state after:
case h.a₄
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₄ = (algebraMap R K) h₄.choose
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✝ : Archimedean G
H : ∃ x, IsLeast {y | 0 < y} x
⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G

pp: 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

Tactic:
·

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
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.a₆
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₆ = (algebraMap R K) h₆.choose

Tactic:
·

Goal state after:
case h.a₆
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₆ = (algebraMap R K) h₆.choose
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: 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

Tactic:
·

Goal state after:
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.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
x y : G
H : ¬IsLeast {y | 0 < y} (y - x)
hxy : x < y
⊢ ∃ z, 0 < z ∧ z < y - x

Tactic:
by
      contrapose! H
      refine ⟨by simp [hxy], fun _ ↦ H _⟩

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case neg.refine_1
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
x y : G
H : ¬IsLeast {y | 0 < y} (y - x)
hxy : x < y
z : G
hz : 0 < z ∧ z < y - x
⊢ x < x + z

pp: case neg.refine_2
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
x y : G
H : ¬IsLeast {y | 0 < y} (y - x)
hxy : x < y
z : G
hz : 0 < z ∧ z < y - x
⊢ x + z < y

Tactic:
·

Goal state after:
case neg.refine_1
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
x y : G
H : ¬IsLeast {y | 0 < y} (y - x)
hxy : x < y
z : G
hz : 0 < z ∧ z < y - x
⊢ x < x + z
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case neg.refine_2
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
x y : G
H : ¬IsLeast {y | 0 < y} (y - x)
hxy : x < y
z : G
hz : 0 < z ∧ z < y - x
⊢ x + z < y

Tactic:
·

Goal state after:
case neg.refine_2
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
x y : G
H : ¬IsLeast {y | 0 < y} (y - x)
hxy : x < y
z : G
hz : 0 < z ∧ z < y - x
⊢ x + z < y
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case inl
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
this : ∀ (x : G ≃+o ℤ), ¬DenselyOrdered G
h : G ≃+o ℤ
⊢ Nonempty (G ≃+o ℤ) ↔ ¬DenselyOrdered G

pp: case inr
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
this : ∀ (x : G ≃+o ℤ), ¬DenselyOrdered G
h : DenselyOrdered G
⊢ Nonempty (G ≃+o ℤ) ↔ ¬DenselyOrdered G

Tactic:
·

Goal state after:
case inl
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
this : ∀ (x : G ≃+o ℤ), ¬DenselyOrdered G
h : G ≃+o ℤ
⊢ Nonempty (G ≃+o ℤ) ↔ ¬DenselyOrdered G
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case inr
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
this : ∀ (x : G ≃+o ℤ), ¬DenselyOrdered G
h : DenselyOrdered G
⊢ Nonempty (G ≃+o ℤ) ↔ ¬DenselyOrdered G

Tactic:
·

Goal state after:
case inr
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
this : ∀ (x : G ≃+o ℤ), ¬DenselyOrdered G
h : DenselyOrdered G
⊢ Nonempty (G ≃+o ℤ) ↔ ¬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 G

Tactic:
← denselyOrdered_units_iff,

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: case refine_1
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
⊢ Gˣ ≃*o Multiplicative ℤ

pp: case refine_2
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ G ≃*o WithZero (Multiplicative ℤ)

Tactic:
·

Goal state after:
case refine_1
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
⊢ Gˣ ≃*o Multiplicative ℤ
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ G ≃*o WithZero (Multiplicative ℤ)

Tactic:
·

Goal state after:
case refine_2
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ G ≃*o WithZero (Multiplicative ℤ)
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
this : ({x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃+o ℤ)
h : {x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ MapsTo (fun x ↦ x + g) {x | 0 ≤ x} {x | g ≤ x}

Tactic:
·

Goal state after:
case refine_1
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)
h : {x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ MapsTo (fun x ↦ x + g) {x | 0 ≤ x} {x | g ≤ x}
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
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)
h : {x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ ∀ a ∈ {x | 0 ≤ x}, ∀ b ∈ {x | 0 ≤ x}, a < b → Function.onFun (fun x1 x2 ↦ x1 < x2) (fun x ↦ x + g) a b

Tactic:
·

Goal state after:
case refine_2
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)
h : {x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ ∀ a ∈ {x | 0 ≤ x}, ∀ b ∈ {x | 0 ≤ x}, a < b → Function.onFun (fun x1 x2 ↦ x1 < x2) (fun x ↦ x + g) a b
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_3
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)
h : {x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ MapsTo (fun x ↦ x - g) {x | g ≤ x} {x | 0 ≤ x}

Tactic:
·

Goal state after:
case refine_3
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)
h : {x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ MapsTo (fun x ↦ x - g) {x | g ≤ x} {x | 0 ≤ x}
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_4
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)
h : {x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ ∀ a ∈ {x | g ≤ x}, ∀ b ∈ {x | g ≤ x}, a < b → Function.onFun (fun x1 x2 ↦ x1 < x2) (fun x ↦ x - g) a b

Tactic:
·

Goal state after:
case refine_4
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)
h : {x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ ∀ a ∈ {x | g ≤ x}, ∀ b ∈ {x | g ≤ x}, a < b → Function.onFun (fun x1 x2 ↦ x1 < x2) (fun x ↦ x - g) a b
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
⊢ ({x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) → Nonempty (G ≃+o ℤ)

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

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
⊢ ({x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) → Nonempty (G ≃+o ℤ)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case pos
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
⊢ ∃ C, IsIntegral R (C • W)

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
⊢ ∃ C, IsIntegral R (C • W)

Tactic:
·

Goal state after:
case pos
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
⊢ ∃ C, IsIntegral R (C • W)
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 : ∀ (x : G) {y : G}, 0 < y → ∃ n, x ≤ n • y
⊢ Nonempty (G ≃+o ℤ)

pp: case neg
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 : ∃ x y, 0 < y ∧ ∀ (n : ℕ), n • y < x
⊢ Nonempty (G ≃+o ℤ)

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 : ∀ (x : G) {y : G}, 0 < y → ∃ n, x ≤ n • y
⊢ Nonempty (G ≃+o ℤ)
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case pos.inr
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
hd : DenselyOrdered G
y : G
hy : y ≠ 0
this : ∀ (y : G), y ≠ 0 → 0 < y → False
hy' : y ≤ 0
⊢ False

pp: 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
hd : DenselyOrdered G
y : G
hy : y ≠ 0
hy' : 0 < y
⊢ False

Tactic:
·

Goal state after:
case pos.inr
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
hd : DenselyOrdered G
y : G
hy : y ≠ 0
this : ∀ (y : G), y ≠ 0 → 0 < y → False
hy' : y ≤ 0
⊢ False