awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : PresheafedSpace.{v, u, v} C
α : X ⟶ Y
inst_2 : IsIso α
x : (↑(↑X : TopCat) : Type v)
β : Y ⟶ X := CategoryTheory.inv α
h_eq :
  (ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (α ≫ β).base :
        (a : (↑(↑X : TopCat) : Type v)) → (↑(↑X : TopCat) : Type v))
      x =
    x
refine_2 :
  eqToHom ⋯ ≫
      Hom.stalkMap β
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                α.base :
              (a : (↑(↑X : TopCat) : Type v)) → (↑(↑Y : TopCat) : Type v))
            x) ≫
        Hom.stalkMap α x =
    𝟙 (X.presheaf.stalk x)
refine_2₁ : eqToHom ⋯ ≫ Hom.stalkMap (α ≫ β) x = 𝟙 (X.presheaf.stalk x)
refine_2₂ :
  CategoryStruct.comp (Y :=
      X.presheaf.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (α ≫ β).base :
            (a : (↑(↑X : TopCat) : Type v)) → (↑(↑X : TopCat) : Type v))
          x))
      (eqToHom ⋯) (eqToHom ⋯ ≫ Hom.stalkMap (𝟙 X) x) =
    𝟙 (X.presheaf.stalk x)
refine_2₃ :
  Eq.{v + 1} (α := X.presheaf.stalk x ⟶ X.presheaf.stalk x)
    (CategoryStruct.comp (Y :=
      X.presheaf.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (α ≫ β).base :
            (a : (↑(↑X : TopCat) : Type v)) → (↑(↑X : TopCat) : Type v))
          x))
      (eqToHom ⋯) (eqToHom ⋯ ≫ 𝟙 (X.presheaf.stalk x)))
    (𝟙 (X.presheaf.stalk x))
refine_2₄ :
  Eq.{v + 1} (α := X.presheaf.stalk x ⟶ X.presheaf.stalk x) (eqToHom ⋯ ≫ 𝟙 (X.presheaf.stalk x))
    (𝟙 (X.presheaf.stalk x))
refine_2₅ :
  Eq.{v + 1} (α := X.presheaf.stalk x ⟶ X.presheaf.stalk x) (𝟙 (X.presheaf.stalk x) ≫ 𝟙 (X.presheaf.stalk x))
    (𝟙 (X.presheaf.stalk x))
⊢ (eqToHom ⋯ ≫
        Hom.stalkMap β
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                α.base :
              (a : (↑(↑X : TopCat) : Type v)) → (↑(↑Y : TopCat) : Type v))
            x)) ≫
      Hom.stalkMap α x =
    𝟙 (X.presheaf.stalk x)
have refine_2₆ : Eq.{v + 1} (α := X.presheaf.stalk x ⟶ X.presheaf.stalk x) (𝟙 (X.presheaf.stalk x)) (𝟙 (X.presheaf.stalk x))
Tactic: rw [Category.assoc, ← stalkMap.comp, congr_hom _ _ (IsIso.hom_inv_id α), stalkMap.id,
        eqToHom_trans_assoc, eqToHom_refl, Category.id_comp]
-- Lean 4 proof step from Mathlib.GroupTheory.Abelianization.Defs (goal)
Goal:
G : Type u
inst : Group G
A : Type v
inst_1 : CommGroup A
f : G →* A
⊢ commutator G ≤ f.ker
have rw : commutatorSet G ⊆ (↑f.ker : Set G)
Tactic: rw [commutator_eq_closure, Subgroup.closure_le]
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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
ha : IsLeast {g : G | g ∈ H ∧ 1 < g} a
⊢ H = closure {a}
have a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
Tactic: obtain ⟨⟨a_in, a_pos⟩, a_min⟩ := ha
Cleared hypotheses: ha
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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
ha : IsLeast {g : G | g ∈ H ∧ 1 < g} a
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
⊢ H = closure {a}
have a_in : a ∈ H
Tactic: obtain ⟨⟨a_in, a_pos⟩, a_min⟩ := ha
Cleared hypotheses: ha
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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
ha : IsLeast {g : G | g ∈ H ∧ 1 < g} a
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
⊢ H = closure {a}
have a_pos : 1 < a
Tactic: obtain ⟨⟨a_in, a_pos⟩, a_min⟩ := ha
Cleared hypotheses: ha
-- 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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
⊢ H = closure {a}
have refine : H ≤ closure {a}
Tactic: refine le_antisymm ?_ (H.closure_le.mpr <| by simp [a_in])
-- 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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
⊢ Membership.mem (γ := Subgroup G) (closure {a}) g
have obtain : ∀ (k : ℤ),
  (∀ (y : ℤ), (fun (k : ℤ) ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k) →
    ∀ (nonneg : a ^ k ≤ g) (lt : g < a ^ (k + 1)), Membership.mem (γ := Subgroup G) (closure {a}) g
Tactic: obtain ⟨k, ⟨nonneg, lt⟩, _⟩ := existsUnique_zpow_near_of_one_lt a_pos g
-- 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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
k : ℤ
right : ∀ (y : ℤ), (fun (k : ℤ) ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k
nonneg : a ^ k ≤ g
lt : g < a ^ (k + 1)
⊢ g / a ^ k = 1
have by_contra : ∀ (h : ¬g / a ^ k = 1), False
Tactic: by_contra h
-- 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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
k : ℤ
right : ∀ (y : ℤ), (fun (k : ℤ) ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k
nonneg : a ^ k ≤ g
lt : g < a ^ (k + 1)
h : ¬g / a ^ k = 1
⊢ a ≤ g / a ^ k
have refine_1 : g / a ^ k ∈ H
Tactic: refine a_min ⟨?_, ?_⟩
-- 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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
k : ℤ
right : ∀ (y : ℤ), (fun (k : ℤ) ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k
nonneg : a ^ k ≤ g
lt : g < a ^ (k + 1)
h : ¬g / a ^ k = 1
refine_1 : g / a ^ k ∈ H
⊢ a ≤ g / a ^ k
have refine_2 : 1 < g / a ^ k
Tactic: refine a_min ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
k : ℤ
right : ∀ (y : ℤ), (fun (k : ℤ) ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k
nonneg : a ^ k ≤ g
lt : g < a ^ (k + 1)
h : ¬g / a ^ k = 1
⊢ False
have h₁ : a ≤ g / a ^ k
Tactic: have h : a ≤ g / (a ^ k) := by
      refine a_min ⟨?_, ?_⟩
      · exact Subgroup.div_mem H g_in (Subgroup.zpow_mem H a_in k)
      · exact lt_of_le_of_ne (by simpa using nonneg) (Ne.symm h)
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
k : ℤ
right : ∀ (y : ℤ), (fun (k : ℤ) ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k
nonneg : a ^ k ≤ g
lt : g < a ^ (k + 1)
h_1 : ¬g / a ^ k = 1
h : a ≤ g / a ^ k
⊢ False
have h' : ¬a ≤ g / a ^ k
Tactic: have h' : ¬a ≤ g / (a ^ k) := not_le.mpr (by simpa [zpow_add_one, div_lt_iff_lt_mul'] using lt)
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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
a_min : Membership.mem (γ := Set G) (lowerBounds {g : G | g ∈ H ∧ 1 < g}) a
a_in : a ∈ H
a_pos : 1 < a
g : G
g_in : g ∈ H
k : ℤ
right : ∀ (y : ℤ), (fun (k : ℤ) ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k
nonneg : a ^ k ≤ g
lt : g < a ^ (k + 1)
⊢ Membership.mem (γ := Subgroup G) (closure {a}) g
have h_zero : g / a ^ k = 1
Tactic: have h_zero : g / (a ^ k) = 1 := by
    by_contra h
    have h : a ≤ g / (a ^ k) := by
      refine a_min ⟨?_, ?_⟩
      · exact Subgroup.div_mem H g_in (Subgroup.zpow_mem H a_in k)
      · exact lt_of_le_of_ne (by simpa using nonneg) (Ne.symm h)
    have h' : ¬a ≤ g / (a ^ k) := not_le.mpr (by simpa [zpow_add_one, div_lt_iff_lt_mul'] using lt)
    contradiction
-- 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)
g : G
hg : g > 1
⊢ ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
have rcases : ∀ (m : ℤ) (hm : g / a ≤ 1 * a ^ m) (hm' : 1 * a ^ m < g / a * a), ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
Tactic: rcases existsUnique_mul_zpow_mem_Ico h₀ 1 (g / a) with ⟨m, ⟨hm, hm'⟩, -⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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)
g : G
hg : g > 1
m : ℤ
hm : g / a ≤ 1 * a ^ m
hm' : 1 * a ^ m < g / a * a
⊢ ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
have hm₁ : g ≤ a ^ (m + 1)
Tactic: simp only [one_mul, div_le_iff_le_mul, div_mul_cancel, ← zpow_add_one] at hm hm'
Cleared hypotheses: hm, hm'
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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)
g : G
hg : g > 1
m : ℤ
hm : g / a ≤ 1 * a ^ m
hm' : 1 * a ^ m < g / a * a
hm₁ : g ≤ a ^ (m + 1)
⊢ ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
have hm'₁ : a ^ m < g
Tactic: simp only [one_mul, div_le_iff_le_mul, div_mul_cancel, ← zpow_add_one] at hm hm'
Cleared hypotheses: hm, hm'
-- 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)
g : G
hg : g > 1
m : ℤ
hm : g ≤ a ^ (m + 1)
hm' : a ^ m < g
⊢ ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
have lift : 0 ≤ m
Tactic: lift m to ℕ
-- 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)
g : G
hg : g > 1
m : ℤ
hm : g ≤ a ^ (m + 1)
hm' : a ^ m < g
lift : 0 ≤ m
⊢ ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
have lift₁ : ∀ (m : ℕ) (hm : g ≤ a ^ HAdd.hAdd (α := ℤ) (↑m : ℤ) 1) (hm' : HPow.hPow (β := ℤ) a (↑m : ℤ) < g),
  ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
Tactic: lift m to ℕ
Cleared hypotheses: m, hm, hm'
-- 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)
g : G
hg : g > 1
m : ℤ
hm : g ≤ a ^ (m + 1)
hm' : a ^ m < g
⊢ 0 ≤ m
have rw : 1 < a ^ (m + 1)
Tactic: rw [← Int.lt_add_one_iff, ← zpow_lt_zpow_iff_right h₀, zpow_zero]
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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)
g : G
hg : g > 1
m : ℕ
hm : g ≤ a ^ HAdd.hAdd (α := ℤ) (↑m : ℤ) 1
hm' : HPow.hPow (β := ℤ) a (↑m : ℤ) < g
⊢ ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
have hm₁ : g ≤ a ^ m.succ
Tactic: simp only [← Nat.cast_succ, zpow_natCast] at hm hm'
Cleared hypotheses: hm, hm'
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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)
g : G
hg : g > 1
m : ℕ
hm : g ≤ a ^ HAdd.hAdd (α := ℤ) (↑m : ℤ) 1
hm' : HPow.hPow (β := ℤ) a (↑m : ℤ) < g
hm₁ : g ≤ a ^ m.succ
⊢ ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
have hm'₁ : a ^ m < g
Tactic: simp only [← Nat.cast_succ, zpow_natCast] at hm hm'
Cleared hypotheses: hm, hm'
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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)
⊢ ∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
have hex : ∀ g > 1, ∃ (n : ℕ), g ∈ Ioc (a ^ n) (a ^ (n + 1))
Tactic: have hex : ∀ g > 1, ∃ n : ℕ, g ∈ Ioc (a ^ n) (a ^ (n + 1)) := fun g hg => by
    rcases existsUnique_mul_zpow_mem_Ico h₀ 1 (g / a) with ⟨m, ⟨hm, hm'⟩, -⟩
    simp only [one_mul, div_le_iff_le_mul, div_mul_cancel, ← zpow_add_one] at hm hm'
    lift m to ℕ
    · rw [← Int.lt_add_one_iff, ← zpow_lt_zpow_iff_right h₀, zpow_zero]
      exact hg.trans_le hm
    · simp only [← Nat.cast_succ, zpow_natCast] at hm hm'
      exact ⟨m, hm', hm⟩
-- 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))
⊢ ∃ (n : ℕ), ((↑H : Set G) ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
have rcases : ∀ (g : G) (hgH : g ∈ H) (hg₀ : g ≠ 1), ∃ (n : ℕ), ((↑H : Set G) ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
Tactic: rcases (bot_or_exists_ne_one H).resolve_left hbot with ⟨g, hgH, hg₀⟩
-- 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))
g : G
hgH : g ∈ H
hg₀ : g ≠ 1
⊢ ∃ (n : ℕ), ((↑H : Set G) ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
have rcases : ∀ (n : ℕ) (hn : |g|ₘ ∈ Ioc (a ^ n) (a ^ (n + 1))), ∃ (n : ℕ), ((↑H : Set G) ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
Tactic: rcases hex |g|ₘ (one_lt_mabs.2 hg₀) with ⟨n, hn⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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))
⊢ ∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
have this : ∃ (n : ℕ), ((↑H : Set G) ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
Tactic: have : ∃ n : ℕ, Set.Nonempty (H ∩ Ioc (a ^ n) (a ^ (n + 1))) := by
    rcases (bot_or_exists_ne_one H).resolve_left hbot with ⟨g, hgH, hg₀⟩
    rcases hex |g|ₘ (one_lt_mabs.2 hg₀) with ⟨n, hn⟩
    exact ⟨n, _, (@mabs_mem_iff (Subgroup G) G _ _).2 hgH, hn⟩
-- 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
⊢ ∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
have rcases : ∀ (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)), ∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
Tactic: rcases Nat.findX this with ⟨n, ⟨x, hxH, hnx, hxn⟩, hmin⟩
-- 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
⊢ ∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
have classical : ∀ (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)), ∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
Tactic: classical rcases Nat.findX this with ⟨n, ⟨x, hxH, hnx, hxn⟩, hmin⟩
-- 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)
⊢ ∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
have by_contra : ∀ (hxmin : ¬∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b), False
Tactic: by_contra hxmin
-- Lean 4 proof step from Mathlib.GroupTheory.Archimedean (hypothesis)
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 : ¬∃ (b : G), IsLeast {g : G | g ∈ H ∧ 1 < g} b
⊢ False
have hxmin₁ : ∀ (x : G), x ∈ H ∧ 1 < x → ∃ (x_1 : G) (_ : x_1 ∈ H ∧ 1 < x_1), x_1 < x
Tactic: simp only [IsLeast, not_and, mem_setOf_eq, mem_lowerBounds, not_exists, not_forall,
    not_le] at hxmin
Cleared hypotheses: hxmin
-- 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
⊢ False
have rcases : ∀ (y : G) (hxy : y < x) (hyH : y ∈ H) (hy₀ : 1 < y), False
Tactic: rcases hxmin x ⟨hxH, (one_le_pow_of_one_le'  h₀.le _).trans_lt hnx⟩ with ⟨y, ⟨hyH, hy₀⟩, hxy⟩
-- 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
⊢ False
have obtain : ∀ (m : ℕ) (hm : a ^ m < y) (hya : y ≤ a ^ (m + 1)), False
Tactic: obtain ⟨m, hm, hya⟩ := hex y hy₀
-- 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)
⊢ False
have inl : ∀ (hmn : m < n), False
Tactic: rcases lt_or_ge m n with hmn | hnm
-- 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)
inl : ∀ (hmn : m < n), False
⊢ False
have inr : ∀ (hnm : n ≤ m), False
Tactic: rcases lt_or_ge m n with hmn | hnm
-- 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
⊢ False
have inr : x < y * a
Tactic: refine disjoint_left.1 hd (div_mem hxH hyH) ⟨one_lt_div'.2 hxy, div_lt_iff_lt_mul'.2 ?_⟩
-- 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
⊢ a ^ (n + 1) ≤ a ^ (m + 1)
have h₁ : 1 ≤ a
Tactic: grw [hnm]