awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
simp_rw :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a b)
simp_rw₁ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y'}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (zpowers y') a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
⊢ ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
have simp_rw₃ : ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a),
  ∃ (k : ℤ),
    Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' sorry ^ k =
      Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a sorry
Tactic: simp_rw [Subtype.forall, hyc, ← zpowers_eq_closure, mem_zpowers_iff, Subtype.ext_iff,
        SubgroupClass.coe_zpow, imp_self, implies_true]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
simp_rw :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a b)
simp_rw₁ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y'}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (zpowers y') a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
simp_rw₃ :
  ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a),
    ∃ (k : ℤ),
      Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯ ^ k =
        Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯
⊢ ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
have simp_rw₄ : ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a),
  ∃ (k : ℤ), (↑(Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' sorry ^ k) : G') = a
Tactic: simp_rw [Subtype.forall, hyc, ← zpowers_eq_closure, mem_zpowers_iff, Subtype.ext_iff,
        SubgroupClass.coe_zpow, imp_self, implies_true]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
simp_rw :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a b)
simp_rw₁ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y'}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (zpowers y') a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
simp_rw₃ :
  ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a),
    ∃ (k : ℤ),
      Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯ ^ k =
        Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯
simp_rw₄ :
  ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a),
    ∃ (k : ℤ), (↑(Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯ ^ k) : G') = a
⊢ ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
have simp_rw₅ : ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a), ∃ (k : ℤ), y' ^ k = a
Tactic: simp_rw [Subtype.forall, hyc, ← zpowers_eq_closure, mem_zpowers_iff, Subtype.ext_iff,
        SubgroupClass.coe_zpow, imp_self, implies_true]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
simp_rw :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a b)
simp_rw₁ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (closure {y'}) a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G') (b : Membership.mem (γ := Subgroup G') (zpowers y') a),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯))
      (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯)
simp_rw₃ :
  ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a),
    ∃ (k : ℤ),
      Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯ ^ k =
        Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a ⋯
simp_rw₄ :
  ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a),
    ∃ (k : ℤ), (↑(Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯ ^ k) : G') = a
simp_rw₅ : ∀ (a : G') (b : ∃ (k : ℤ), y' ^ k = a), ∃ (k : ℤ), y' ^ k = a
⊢ ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
have simp_rw₆ : ∀ (a : G'), True
Tactic: simp_rw [Subtype.forall, hyc, ← zpowers_eq_closure, mem_zpowers_iff, Subtype.ext_iff,
        SubgroupClass.coe_zpow, imp_self, implies_true]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have Hy : ∀ (z : ↥(closure {y})),
  Membership.mem (γ := Subgroup ↥(closure {y}))
    (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' sorry)) z
Tactic: have Hy : ∀ (z : closure {y}), z ∈ zpowers ⟨y', by simp [hyc]⟩ := by
      simp_rw [Subtype.forall, hyc, ← zpowers_eq_closure, mem_zpowers_iff, Subtype.ext_iff,
        SubgroupClass.coe_zpow, imp_self, implies_true]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
Hy :
  ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have refine_1 : orderOf (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry) =
  orderOf (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' sorry)
Tactic: refine OrderMonoidIso.mk (mulEquivOfOrderOfEq Hx Hy ?_) ?_
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
Hy :
  ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
refine_1 :
  orderOf (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯) =
    orderOf (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have refine_2 : ∀ {a b : ↥(closure {x})}, (mulEquivOfOrderOfEq Hx Hy sorry).toFun a ≤ (mulEquivOfOrderOfEq Hx Hy sorry).toFun b ↔ a ≤ b
Tactic: refine OrderMonoidIso.mk (mulEquivOfOrderOfEq Hx Hy ?_) ?_
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
Hy :
  ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
⊢ orderOf (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯) =
    orderOf (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)
have refine_1 : orderOf x' = orderOf y'
Tactic: rw [orderOf_mk, orderOf_mk]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
Hy :
  ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
a b : ↥(closure {x})
⊢ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun a ≤ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun b ↔ a ≤ b
have refine_2 : ∀ (m : ℤ),
  (mulEquivOfOrderOfEq Hx Hy sorry).toFun
        ((fun (x_1 : ℤ) ↦
            Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' sorry ^ x_1)
          m) ≤
      (mulEquivOfOrderOfEq Hx Hy sorry).toFun b ↔
    (fun (x_1 : ℤ) ↦ Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' sorry ^ x_1)
        m ≤
      b
Tactic: obtain ⟨m, rfl⟩ := Hx a
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
Hy :
  ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
b : ↥(closure {x})
m : ℤ
⊢ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun
        ((fun (x_1 : ℤ) ↦
            Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' ⋯ ^ x_1)
          m) ≤
      (mulEquivOfOrderOfEq Hx Hy ⋯).toFun b ↔
    (fun (x_1 : ℤ) ↦ Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' ⋯ ^ x_1)
        m ≤
      b
have refine_2 : ∀ (n : ℤ),
  (mulEquivOfOrderOfEq Hx Hy sorry).toFun
        ((fun (x_1 : ℤ) ↦
            Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' sorry ^ x_1)
          m) ≤
      (mulEquivOfOrderOfEq Hx Hy sorry).toFun
        ((fun (x_1 : ℤ) ↦
            Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' sorry ^ x_1)
          n) ↔
    LE.le.{u_1} (α := ↥(closure {x}))
      ((fun (x_1 : ℤ) ↦ Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' sorry ^ x_1)
        m)
      ((fun (x_1 : ℤ) ↦ Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' sorry ^ x_1)
        n)
Tactic: obtain ⟨n, rfl⟩ := Hx b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
G' : Type u_2
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : CommGroup G'
inst_4 : LinearOrder G'
inst_5 : 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 : Eq (α := Subgroup G) (closure {x}) (closure {x'})
hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Hx :
  ∀ (z : ↥(closure {x})),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯)) z
Hy :
  ∀ (z : ↥(closure {y})),
    Membership.mem (γ := Subgroup ↥(closure {y}))
      (zpowers (Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' ⋯)) z
m n : ℤ
⊢ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun
        ((fun (x_1 : ℤ) ↦
            Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' ⋯ ^ x_1)
          m) ≤
      (mulEquivOfOrderOfEq Hx Hy ⋯).toFun
        ((fun (x_1 : ℤ) ↦
            Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' ⋯ ^ x_1)
          n) ↔
    LE.le.{u_1} (α := ↥(closure {x}))
      ((fun (x_1 : ℤ) ↦ Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' ⋯ ^ x_1)
        m)
      ((fun (x_1 : ℤ) ↦ Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) x' ⋯ ^ x_1)
        n)
have refine_2 : Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' sorry ^ m ≤
    Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) y' sorry ^ n ↔
  Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry ^ m ≤
    Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry ^ n
Tactic: simp only [MulEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe, map_zpow,
        mulEquivOfOrderOfEq_apply_gen]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
a b : G
⊢ IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b ↔ b = |a|ₘ ∧ 1 < b
have mp : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b → b = |a|ₘ ∧ 1 < b
Tactic: constructor
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
a b : G
mp : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b → b = |a|ₘ ∧ 1 < b
⊢ IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b ↔ b = |a|ₘ ∧ 1 < b
have mpr : b = |a|ₘ ∧ 1 < b → IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b
Tactic: constructor
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
a b : G
h : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b
⊢ b = |a|ₘ ∧ 1 < b
have this : Eq (α := Subgroup G) (closure {a}) (closure {b})
Tactic: have := Subgroup.cyclic_of_min h
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
a b : G
h : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b
this : Eq (α := Subgroup G) (closure {a}) (closure {b})
⊢ b = |a|ₘ ∧ 1 < b
have ha : Membership.mem (γ := Subgroup G) (closure {b}) a
Tactic: have ha : a ∈ closure ({b} : Set G) := by
      simp [← this]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
a b : G
h : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b
this : Eq (α := Subgroup G) (closure {a}) (closure {b})
ha : Membership.mem (γ := Subgroup G) (closure {b}) a
⊢ b = |a|ₘ ∧ 1 < b
have ha₁ : ∃ (n : ℤ), b ^ n = a
Tactic: rw [mem_closure_singleton] at ha
Cleared hypotheses: ha
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
a b : G
h : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {a}) y ∧ 1 < y} b
this : Eq (α := Subgroup G) (closure {a}) (closure {b})
ha : ∃ (n : ℤ), b ^ n = a
⊢ b = |a|ₘ ∧ 1 < b
have mp : ∀ (n : ℤ) (h : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {b ^ n}) y ∧ 1 < y} b)
  (this : Eq (α := Subgroup G) (closure {b ^ n}) (closure {b})), b = |b ^ n|ₘ ∧ 1 < b
Tactic: obtain ⟨n, rfl⟩ := ha
Cleared hypotheses: a, h, this, ha
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
b : G
n : ℤ
h : IsLeast {y : G | Membership.mem (γ := Subgroup G) (closure {b ^ n}) y ∧ 1 < y} b
this : Eq (α := Subgroup G) (closure {b ^ n}) (closure {b})
⊢ b = |b ^ n|ₘ ∧ 1 < b
have this₁ : Membership.mem (γ := Set G) {y : G | Membership.mem (γ := Subgroup G) (closure {b ^ n}) y ∧ 1 < y} b
Tactic: have := h.left