awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
H : Subgroup G
hbot : H ≠ ⊥
a : G
h₀ : 1 < a
hd : Disjoint (↑H : Set G) (Ioo 1 a)
hex : ∀ g > 1, ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
this : ∃ (n : ℕ), ((↑H : Set G) ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
n : ℕ
hmin : ∀ m < n, ¬((↑H : Set G) ∩ Ioc (a ^ m) (a ^ (m + 1))).Nonempty
x : G
hxH : x ∈ (↑H : Set G)
hnx : a ^ n < x
hxn : x ≤ a ^ (n + 1)
hxmin : ∀ (x : G), x ∈ H ∧ 1 < x → ∃ (x_1 : G) (_ : x_1 ∈ H ∧ 1 < x_1), x_1 < x
y : G
hxy : y < x
hyH : y ∈ H
hy₀ : 1 < y
m : ℕ
hm : a ^ m < y
hya : y ≤ a ^ (m + 1)
hnm : n ≤ m
⊢ x < y * a
have calc : a ^ (n + 1) ≤ a ^ (m + 1)
Tactic: calc x ≤ a ^ (n + 1) := hxn
    _ ≤ a ^ (m + 1) := by grw [hnm]; exact h₀.le
    _ = a ^ m * a := pow_succ _ _
    _ < y * a := by gcongr
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
H : Subgroup G
hbot : H ≠ ⊥
a : G
h₀ : 1 < a
hd : Disjoint (↑H : Set G) (Ioo 1 a)
hex : ∀ g > 1, ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
this : ∃ (n : ℕ), ((↑H : Set G) ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
n : ℕ
hmin : ∀ m < n, ¬((↑H : Set G) ∩ Ioc (a ^ m) (a ^ (m + 1))).Nonempty
x : G
hxH : x ∈ (↑H : Set G)
hnx : a ^ n < x
hxn : x ≤ a ^ (n + 1)
hxmin : ∀ (x : G), x ∈ H ∧ 1 < x → ∃ (x_1 : G) (_ : x_1 ∈ H ∧ 1 < x_1), x_1 < x
y : G
hxy : y < x
hyH : y ∈ H
hy₀ : 1 < y
m : ℕ
hm : a ^ m < y
hya : y ≤ a ^ (m + 1)
hnm : n ≤ m
calc : a ^ (n + 1) ≤ a ^ (m + 1)
⊢ x < y * a
have calc₁ : a ^ m * a < y * a
Tactic: calc x ≤ a ^ (n + 1) := hxn
    _ ≤ a ^ (m + 1) := by grw [hnm]; exact h₀.le
    _ = a ^ m * a := pow_succ _ _
    _ < y * a := by gcongr
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
H : Subgroup G
a : G
h₀ : 1 < a
hd : Disjoint (↑H : Set G) (Ioo 1 a)
⊢ ∃ (b : G), H = closure {b}
have inl : ∀ (hd : Disjoint (↑⊥ : Set G) (Ioo 1 a)), ∃ (b : G), Eq (α := Subgroup G) ⊥ (closure {b})
Tactic: rcases eq_or_ne H ⊥ with rfl | hbot
Cleared hypotheses: H, hd
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
H : Subgroup G
a : G
h₀ : 1 < a
hd : Disjoint (↑H : Set G) (Ioo 1 a)
inl : ∀ (hd : Disjoint (↑⊥ : Set G) (Ioo 1 a)), ∃ (b : G), Eq (α := Subgroup G) ⊥ (closure {b})
⊢ ∃ (b : G), H = closure {b}
have inr : ∀ (hbot : H ≠ ⊥), ∃ (b : G), H = closure {b}
Tactic: rcases eq_or_ne H ⊥ with rfl | hbot
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : LinearOrderedCommGroupWithZero G
inst_1 : MulArchimedean G
a : G
ha : a < 1
b : Gˣ
⊢ ∃ (n : ℕ), a ^ n < (↑b : G)
have inl : ∀ (ha : LT.lt.{u_1} (α := G) 0 1), ∃ (n : ℕ), HPow.hPow (α := G) 0 n < (↑b : G)
Tactic: rcases eq_or_ne a 0 with rfl | ha'
Cleared hypotheses: a, ha
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : LinearOrderedCommGroupWithZero G
inst_1 : MulArchimedean G
a : G
ha : a < 1
b : Gˣ
inl : ∀ (ha : LT.lt.{u_1} (α := G) 0 1), ∃ (n : ℕ), HPow.hPow (α := G) 0 n < (↑b : G)
⊢ ∃ (n : ℕ), a ^ n < (↑b : G)
have inr : ∀ (ha' : a ≠ 0), ∃ (n : ℕ), a ^ n < (↑b : G)
Tactic: rcases eq_or_ne a 0 with rfl | ha'
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : LinearOrderedCommGroupWithZero G
inst_1 : MulArchimedean G
b : Gˣ
ha : LT.lt.{u_1} (α := G) 0 1
⊢ ∃ (n : ℕ), HPow.hPow (α := G) 0 n < (↑b : G)
have h : HPow.hPow (α := G) (β := ℕ) 0 1 < (↑b : G)
Tactic: use 1
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : LinearOrderedCommGroupWithZero G
inst_1 : MulArchimedean G
a : G
ha : a < 1
b : Gˣ
ha' : a ≠ 0
⊢ ∃ (n : ℕ), a ^ n < (↑b : G)
have inr : ∀ (a : Gˣ) (ha : (↑a : G) < 1) (ha' : (↑a : G) ≠ 0), ∃ (n : ℕ), (↑a : G) ^ n < (↑b : G)
Tactic: lift a to Gˣ using IsUnit.mk0 _ ha'
Cleared hypotheses: a, ha, 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
a b : G
ha : a ≠ 1
⊢ Membership.mem (γ := Subgroup G) (closure {a}) b ↔ ∃! k : ℤ, a ^ k = b
have rw : (∃ (n : ℤ), a ^ n = b) ↔ ∃! k : ℤ, a ^ k = b
Tactic: rw [mem_closure_singleton]
-- 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
a b : G
ha : a ≠ 1
⊢ (∃ (n : ℤ), a ^ n = b) ↔ ∃! k : ℤ, a ^ k = b
have mp : (∃ (n : ℤ), a ^ n = b) → ∃! k : ℤ, a ^ k = 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
a b : G
ha : a ≠ 1
mp : (∃ (n : ℤ), a ^ n = b) → ∃! k : ℤ, a ^ k = b
⊢ (∃ (n : ℤ), a ^ n = b) ↔ ∃! k : ℤ, a ^ k = b
have mpr : (∃! k : ℤ, a ^ k = b) → ∃ (n : ℤ), a ^ n = 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
a b : G
ha : a ≠ 1
⊢ (∃ (n : ℤ), a ^ n = b) → ∃! k : ℤ, a ^ k = b
have mp : Function.Injective.{1, u_1 + 1} (β := G) fun (x : ℤ) ↦ a ^ x
Tactic: suffices Function.Injective (a ^ · : ℤ → G) by
      rintro ⟨m, rfl⟩
      exact ⟨m, rfl, fun k hk ↦ this hk⟩
-- 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
a b : G
ha : a ≠ 1
⊢ Function.Injective.{1, u_1 + 1} (β := G) fun (x : ℤ) ↦ a ^ x
have mp : ∀ (ha : a < 1), Function.Injective.{1, u_1 + 1} (β := G) fun (x : ℤ) ↦ a ^ x
Tactic: rcases ha.lt_or_gt with ha | 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
a b : G
ha : a ≠ 1
mp : ∀ (ha : a < 1), Function.Injective.{1, u_1 + 1} (β := G) fun (x : ℤ) ↦ a ^ x
⊢ Function.Injective.{1, u_1 + 1} (β := G) fun (x : ℤ) ↦ a ^ x
have mp₁ : ∀ (ha : 1 < a), Function.Injective.{1, u_1 + 1} (β := G) fun (x : ℤ) ↦ a ^ x
Tactic: rcases ha.lt_or_gt with ha | ha
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
e : ℤ ≃+ ℤ
⊢ e = AddEquiv.refl ℤ ∨ e = AddEquiv.neg ℤ
have suffices : Eq (α := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ) (1 : ℤ) ∨ Eq (α := ℤ) (-(e : (a : ℤ) → ℤ) 1 : ℤ) (1 : ℤ)
Tactic: suffices e 1 = 1 ∨ -e 1 = 1 by simpa [AddEquiv.ext_int_iff, neg_eq_iff_eq_neg]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
e : ℤ ≃+ ℤ
⊢ Eq (α := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ) (1 : ℤ) ∨ Eq (α := ℤ) (-(e : (a : ℤ) → ℤ) 1 : ℤ) (1 : ℤ)
have he : ¬IsOfFinAddOrder (G := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ)
Tactic: have he : ¬IsOfFinAddOrder (e 1) :=
    not_isOfFinAddOrder_of_isAddTorsionFree ((AddEquiv.map_ne_zero_iff e).mpr Int.one_ne_zero)
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
e : ℤ ≃+ ℤ
he : ¬IsOfFinAddOrder (G := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ)
⊢ Eq (α := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ) (1 : ℤ) ∨ Eq (α := ℤ) (-(e : (a : ℤ) → ℤ) 1 : ℤ) (1 : ℤ)
have rw : Eq (α := AddSubgroup ℤ) (AddSubgroup.zmultiples ((e : (a : ℤ) → ℤ) 1) : AddSubgroup ℤ)
  (AddSubgroup.zmultiples 1 : AddSubgroup ℤ)
Tactic: rw [← AddSubgroup.zmultiples_eq_zmultiples_iff he]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
x : ℤ ≃+ ℤ
⊢ x ∈ Finset.cons (AddEquiv.neg ℤ) {AddEquiv.refl ℤ} instFintypeAddEquivInt._proof_1
have inl : AddEquiv.refl ℤ ∈ Finset.cons (AddEquiv.neg ℤ) {AddEquiv.refl ℤ} instFintypeAddEquivInt._proof_1
Tactic: obtain rfl | rfl := Int.addEquiv_eq_refl_or_neg x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
x : ℤ ≃+ ℤ
inl : AddEquiv.refl ℤ ∈ Finset.cons (AddEquiv.neg ℤ) {AddEquiv.refl ℤ} instFintypeAddEquivInt._proof_1
⊢ x ∈ Finset.cons (AddEquiv.neg ℤ) {AddEquiv.refl ℤ} instFintypeAddEquivInt._proof_1
have inr : AddEquiv.neg ℤ ∈ Finset.cons (AddEquiv.neg ℤ) {AddEquiv.refl ℤ} instFintypeAddEquivInt._proof_1
Tactic: obtain rfl | rfl := Int.addEquiv_eq_refl_or_neg x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
e : ℤ ≃+o ℤ
H : Eq (α := ℤ ≃+ ℤ) (↑e : ℤ ≃+ ℤ) (AddEquiv.neg ℤ)
⊢ False
have H₁ : Eq (α := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ) (-1 : ℤ)
Tactic: replace H : e 1 = -1 := congr($H 1)
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
e : ℤ ≃+o ℤ
H : Eq (α := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ) (-1 : ℤ)
⊢ LT.lt (α := ℤ) (0 : ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ)
have rw : LT.lt (α := ℤ) 0 1
Tactic: rw [← map_zero e, map_lt_map_iff]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
e : ℤ ≃+o ℤ
H : Eq (α := ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ) (-1 : ℤ)
⊢ False
have h1 : LT.lt (α := ℤ) (0 : ℤ) ((e : (a : ℤ) → ℤ) 1 : ℤ)
Tactic: have h1 : 0 < e 1 := by
        rw [← map_zero e, map_lt_map_iff]
        simp
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
e : ℤ ≃+o ℤᵒᵈ
⊢ e.toAddEquiv =
    { toAddEquiv := AddEquiv.trans (AddEquiv.neg ℤ) { toEquiv := toDual, map_add' := ⋯ : ℤ ≃+ ℤᵒᵈ },
        map_le_map_iff' := @instUniqueOrderAddMonoidIsoIntOrderDual._proof_1 : ℤ ≃+o ℤᵒᵈ }.toAddEquiv
have simp : Eq (α := ℤ ≃+ ℤᵒᵈ) (↑e : ℤ ≃+ ℤᵒᵈ) (AddEquiv.trans (AddEquiv.neg ℤ) { toEquiv := toDual, map_add' := sorry })
Tactic: simp only [OrderAddMonoidIso.toAddEquiv_eq_coe]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
e : ℤ ≃+o ℤᵒᵈ
H : AddEquiv.trans (↑e : ℤ ≃+ ℤᵒᵈ) { toEquiv := toDual, map_add' := ⋯ : ℤᵒᵈ ≃+ ℤ } = AddEquiv.refl ℤ
⊢ False
have H₁ : Eq (α := ℤᵒᵈ) ((e : (a : ℤ) → ℤᵒᵈ) 1 : ℤᵒᵈ) (1 : ℤᵒᵈ)
Tactic: replace H : e 1 = 1 := congr($H 1)
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
e : ℤ ≃+o ℤᵒᵈ
H : Eq (α := ℤᵒᵈ) ((e : (a : ℤ) → ℤᵒᵈ) 1 : ℤᵒᵈ) (1 : ℤᵒᵈ)
⊢ LT.lt (α := ℤᵒᵈ) (0 : ℤᵒᵈ) ((e : (a : ℤ) → ℤᵒᵈ) 1 : ℤᵒᵈ)
have rw : LT.lt (α := ℤ) 0 1
Tactic: rw [← map_zero e, map_lt_map_iff]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (hypothesis)
Goal:
e : ℤ ≃+o ℤᵒᵈ
H : Eq (α := ℤᵒᵈ) ((e : (a : ℤ) → ℤᵒᵈ) 1 : ℤᵒᵈ) (1 : ℤᵒᵈ)
⊢ False
have h1 : LT.lt (α := ℤᵒᵈ) (0 : ℤᵒᵈ) ((e : (a : ℤ) → ℤᵒᵈ) 1 : ℤᵒᵈ)
Tactic: have h1 : 0 < e 1 := by
        rw [← map_zero e, map_lt_map_iff]
        simp
-- 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
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have refine_1 : Function.LeftInverse
  (fun (x_1 : ↥(closure {y})) ↦
    Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 sorry)
  fun (x_1 : ↥(closure {x})) ↦ Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 sorry
Tactic: refine ⟨⟨⟨fun _ ↦ ⟨1, by simp [hxy.mp hx]⟩, fun _ ↦ ⟨1, by simp [hx]⟩, ?_, ?_⟩, ?_⟩, ?_⟩
-- 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
refine_1 :
  Function.LeftInverse
    (fun (x_1 : ↥(closure {y})) ↦
      Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯)
    fun (x_1 : ↥(closure {x})) ↦ Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have refine_2 : Function.RightInverse
  (fun (x_1 : ↥(closure {y})) ↦
    Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 sorry)
  fun (x_1 : ↥(closure {x})) ↦ Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 sorry
Tactic: refine ⟨⟨⟨fun _ ↦ ⟨1, by simp [hxy.mp hx]⟩, fun _ ↦ ⟨1, by simp [hx]⟩, ?_, ?_⟩, ?_⟩, ?_⟩
-- 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
refine_1 :
  Function.LeftInverse
    (fun (x_1 : ↥(closure {y})) ↦
      Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯)
    fun (x_1 : ↥(closure {x})) ↦ Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯
refine_2 :
  Function.RightInverse
    (fun (x_1 : ↥(closure {y})) ↦
      Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯)
    fun (x_1 : ↥(closure {x})) ↦ Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have refine_3 : ∀ (x_1 y_1 : ↥(closure {x})),
  Equiv.toFun
      {
        toFun := fun (x_2 : ↥(closure {x})) ↦
          Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 sorry,
        invFun := fun (x_2 : ↥(closure {y})) ↦
          Subtype.mk (p := fun (x_3 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_3) 1 sorry,
        left_inv := sorry, right_inv := sorry }
      (x_1 * y_1) =
    HMul.hMul (α := ↥(closure {y}))
      (Equiv.toFun
        {
          toFun := fun (x_2 : ↥(closure {x})) ↦
            Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 sorry,
          invFun := fun (x_2 : ↥(closure {y})) ↦
            Subtype.mk (p := fun (x_3 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_3) 1 sorry,
          left_inv := sorry, right_inv := sorry : Equiv.{u_1 + 1, u_2 + 1} ↥(closure {x}) ↥(closure {y}) }
        x_1)
      (Equiv.toFun
        {
          toFun := fun (x_2 : ↥(closure {x})) ↦
            Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 sorry,
          invFun := fun (x_2 : ↥(closure {y})) ↦
            Subtype.mk (p := fun (x_3 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_3) 1 sorry,
          left_inv := sorry, right_inv := sorry : Equiv.{u_1 + 1, u_2 + 1} ↥(closure {x}) ↥(closure {y}) }
        y_1)
Tactic: refine ⟨⟨⟨fun _ ↦ ⟨1, by simp [hxy.mp hx]⟩, fun _ ↦ ⟨1, by simp [hx]⟩, ?_, ?_⟩, ?_⟩, ?_⟩
-- 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
a : G
ha : Membership.mem (γ := Subgroup G) (closure {x}) a
⊢ ∀ {b : ↥(closure {x})},
    LE.le.{u_2} (α := ↥(closure {y}))
        (Equiv.toFun
          (MulEquiv.toEquiv
            {
              toFun := fun (x_1 : ↥(closure {x})) ↦
                Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯,
              invFun := fun (x_1 : ↥(closure {y})) ↦
                Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯,
              left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ })
          (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha))
        (Equiv.toFun
          (MulEquiv.toEquiv
            {
              toFun := fun (x_1 : ↥(closure {x})) ↦
                Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯,
              invFun := fun (x_1 : ↥(closure {y})) ↦
                Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯,
              left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ })
          b) ↔
      Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha ≤ b
have refine_4 : ∀ (b : G) (hb : Membership.mem (γ := Subgroup G) (closure {x}) b),
  LE.le.{u_2} (α := ↥(closure {y}))
      (Equiv.toFun
        (MulEquiv.toEquiv
          {
            toFun := fun (x_1 : ↥(closure {x})) ↦
              Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 sorry,
            invFun := fun (x_1 : ↥(closure {y})) ↦
              Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 sorry,
            left_inv := sorry, right_inv := sorry, map_mul' := sorry })
        (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha))
      (Equiv.toFun
        (MulEquiv.toEquiv
          {
            toFun := fun (x_1 : ↥(closure {x})) ↦
              Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 sorry,
            invFun := fun (x_1 : ↥(closure {y})) ↦
              Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 sorry,
            left_inv := sorry, right_inv := sorry, map_mul' := sorry })
        (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) b hb)) ↔
    Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha ≤
      Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) b hb
Tactic: ⟨b, hb⟩
-- 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
a : G
ha : Membership.mem (γ := Subgroup G) (closure {x}) a
b : G
hb : Membership.mem (γ := Subgroup G) (closure {x}) b
⊢ LE.le.{u_2} (α := ↥(closure {y}))
      (Equiv.toFun
        (MulEquiv.toEquiv
          {
            toFun := fun (x_1 : ↥(closure {x})) ↦
              Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯,
            invFun := fun (x_1 : ↥(closure {y})) ↦
              Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯,
            left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ })
        (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha))
      (Equiv.toFun
        (MulEquiv.toEquiv
          {
            toFun := fun (x_1 : ↥(closure {x})) ↦
              Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯,
            invFun := fun (x_1 : ↥(closure {y})) ↦
              Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯,
            left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ })
        (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) b hb)) ↔
    Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha ≤
      Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) b hb
have ha₁ : a = 1
Tactic: simp only [hx, closure_singleton_one, mem_bot] at ha hb
-- 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
a : G
ha : Membership.mem (γ := Subgroup G) (closure {x}) a
b : G
hb : Membership.mem (γ := Subgroup G) (closure {x}) b
ha₁ : a = 1
⊢ LE.le.{u_2} (α := ↥(closure {y}))
      (Equiv.toFun
        (MulEquiv.toEquiv
          {
            toFun := fun (x_1 : ↥(closure {x})) ↦
              Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯,
            invFun := fun (x_1 : ↥(closure {y})) ↦
              Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯,
            left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ })
        (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha))
      (Equiv.toFun
        (MulEquiv.toEquiv
          {
            toFun := fun (x_1 : ↥(closure {x})) ↦
              Subtype.mk (p := fun (x : G') ↦ Membership.mem (γ := Subgroup G') (closure {y}) x) 1 ⋯,
            invFun := fun (x_1 : ↥(closure {y})) ↦
              Subtype.mk (p := fun (x_2 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_2) 1 ⋯,
            left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ })
        (Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) b hb)) ↔
    Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) a ha ≤
      Subtype.mk (p := fun (x_1 : G) ↦ Membership.mem (γ := Subgroup G) (closure {x}) x_1) b hb
have hb₁ : b = 1
Tactic: simp only [hx, closure_singleton_one, mem_bot] at ha hb
-- 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
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have hx' : let x' := max x x⁻¹;
x' = max x x⁻¹
Tactic: set x' := max x x⁻¹ with hx'
-- 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
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have xpos : 1 < x'
Tactic: have xpos : 1 < x' := by
      simp [hx', eq_comm, hx]
-- 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'
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have hy' : let y' := max y y⁻¹;
y' = max y y⁻¹
Tactic: set y' := max y y⁻¹ with hy'
-- 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
⊢ OrderMonoidIso.{u_1, u_2} ↥(closure {x}) ↥(closure {y})
have ypos : 1 < y'
Tactic: have ypos : 1 < y' := by
      simp [hy', eq_comm, ← hxy, hx]
-- 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'
⊢ Eq (α := Subgroup G) (closure {x}) (closure {x'})
have inl : max x x⁻¹ = x ∧ x⁻¹ ≤ x → Eq (α := Subgroup G) (closure {x}) (closure {x'})
Tactic: rcases max_cases x x⁻¹