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'
inl : max x x⁻¹ = x ∧ x⁻¹ ≤ x → Eq (α := Subgroup G) (closure {x}) (closure {x'})
⊢ Eq (α := Subgroup G) (closure {x}) (closure {x'})
have inr : max x x⁻¹ = x⁻¹ ∧ x < x⁻¹ → Eq (α := Subgroup G) (closure {x}) (closure {x'})
Tactic: rcases max_cases x x⁻¹
-- 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'
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have hxc : Eq (α := Subgroup G) (closure {x}) (closure {x'})
Tactic: have hxc : closure {x} = closure {x'} := by rcases max_cases x x⁻¹ <;> simp_all
-- 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'})
⊢ Eq (α := Subgroup G') (closure {y}) (closure {y'})
have inl : max y y⁻¹ = y ∧ y⁻¹ ≤ y → Eq (α := Subgroup G') (closure {y}) (closure {y'})
Tactic: rcases max_cases y y⁻¹
-- 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'})
inl : max y y⁻¹ = y ∧ y⁻¹ ≤ y → Eq (α := Subgroup G') (closure {y}) (closure {y'})
⊢ Eq (α := Subgroup G') (closure {y}) (closure {y'})
have inr : max y y⁻¹ = y⁻¹ ∧ y < y⁻¹ → Eq (α := Subgroup G') (closure {y}) (closure {y'})
Tactic: rcases max_cases y y⁻¹
-- 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'})
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have hyc : Eq (α := Subgroup G') (closure {y}) (closure {y'})
Tactic: have hyc : closure {y} = closure {y'} := by rcases max_cases y y⁻¹ <;> simp_all
-- 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'})
⊢ ∀ (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
have simp_rw : ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x}) a),
  Membership.mem (γ := Subgroup ↥(closure {x}))
    (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry))
    (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a b)
Tactic: simp_rw [Subtype.forall, hxc, ← 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'})
simp_rw :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a b)
⊢ ∀ (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
have simp_rw₁ : ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x'}) a),
  Membership.mem (γ := Subgroup ↥(closure {x}))
    (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry))
    (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a sorry)
Tactic: simp_rw [Subtype.forall, hxc, ← 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'})
simp_rw :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a b)
simp_rw₁ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x'}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
⊢ ∀ (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
have simp_rw₂ : ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (zpowers x') a),
  Membership.mem (γ := Subgroup ↥(closure {x}))
    (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry))
    (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a sorry)
Tactic: simp_rw [Subtype.forall, hxc, ← 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'})
simp_rw :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a b)
simp_rw₁ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x'}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (zpowers x') a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
⊢ ∀ (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
have simp_rw₃ : ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a),
  ∃ (k : ℤ),
    Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry ^ k =
      Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a sorry
Tactic: simp_rw [Subtype.forall, hxc, ← 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'})
simp_rw :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a b)
simp_rw₁ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x'}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (zpowers x') a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
simp_rw₃ :
  ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a),
    ∃ (k : ℤ),
      Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯ ^ k =
        Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯
⊢ ∀ (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
have simp_rw₄ : ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a),
  ∃ (k : ℤ), (↑(Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' sorry ^ k) : G) = a
Tactic: simp_rw [Subtype.forall, hxc, ← 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'})
simp_rw :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a b)
simp_rw₁ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x'}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (zpowers x') a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
simp_rw₃ :
  ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a),
    ∃ (k : ℤ),
      Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯ ^ k =
        Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯
simp_rw₄ :
  ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a),
    ∃ (k : ℤ),
      (↑(Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯ ^ k) : G) = a
⊢ ∀ (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
have simp_rw₅ : ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a), ∃ (k : ℤ), x' ^ k = a
Tactic: simp_rw [Subtype.forall, hxc, ← 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'})
simp_rw :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a b)
simp_rw₁ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (closure {x'}) a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
simp_rw₂ :
  ∀ (a : G) (b : Membership.mem (γ := Subgroup G) (zpowers x') a),
    Membership.mem (γ := Subgroup ↥(closure {x}))
      (zpowers (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯))
      (Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯)
simp_rw₃ :
  ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a),
    ∃ (k : ℤ),
      Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯ ^ k =
        Subtype.mk (p := fun (a : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) a) a ⋯
simp_rw₄ :
  ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a),
    ∃ (k : ℤ),
      (↑(Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) x' ⋯ ^ k) : G) = a
simp_rw₅ : ∀ (a : G) (b : ∃ (k : ℤ), x' ^ k = a), ∃ (k : ℤ), x' ^ k = a
⊢ ∀ (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
have simp_rw₆ : ∀ (a : G), True
Tactic: simp_rw [Subtype.forall, hxc, ← 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'})
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have 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' sorry)) z
Tactic: have Hx : ∀ (z : closure {x}), z ∈ zpowers ⟨x', by simp [hxc]⟩ := by
      simp_rw [Subtype.forall, hxc, ← 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
⊢ ∀ (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_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' sorry))
    (Subtype.mk (p := fun (a : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) a) a b)
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)
⊢ ∀ (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_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' sorry))
    (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 ⋯)
⊢ ∀ (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_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' sorry))
    (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]