awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Pointwise (goal)
Goal:
G : Type u_2
inst : Group G
H : Subgroup G
g : G
⊢ g ∈ { carrier := {a : G | ∀ (b : G), b * a * b⁻¹ ∈ H}, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ : Subgroup G } ↔
    ∀ (i : ConjAct G), i⁻¹ • g ∈ H
have h : ∀ (h : ∀ (i : ConjAct G), i⁻¹ • g ∈ H) (x : G), x * g * x⁻¹ ∈ H
Tactic: refine ⟨fun h x ↦ h x⁻¹, fun h x ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Pointwise (goal)
Goal:
G : Type u_2
inst : Group G
H : Subgroup G
g : G
⊢ HSMul.hSMul (α := ConjAct G) ((ConjAct.toConjAct : (a : G) → ConjAct G) g : ConjAct G) H = H ↔
    Membership.mem (γ := Subgroup G) (normalizer (↑H : Set G)) g
have rw : HSMul.hSMul (α := ConjAct G) ((ConjAct.toConjAct : (a : G) → ConjAct G) g : ConjAct G) H = H ↔
  Membership.mem (γ := Subgroup G) (normalizer (↑H : Set G)) g⁻¹
Tactic: rw [← (normalizer H : Subgroup G).inv_mem_iff]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
⊢ Eq (α := Subgroup G) (zpowers g) (closure {g})
have h : ∀ (x : G), Membership.mem (γ := Subgroup G) (zpowers g) x ↔ Membership.mem (γ := Subgroup G) (closure {g}) x
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
N : Type u_3
inst_1 : Group N
f : G →* N
x : G
⊢ Subgroup.map f (Subgroup.zpowers x) = Subgroup.zpowers ((f : (a : G) → N) x)
have rw : Subgroup.map f (Subgroup.closure {x}) = Subgroup.zpowers ((f : (a : G) → N) x)
Tactic: rw [Subgroup.zpowers_eq_closure, Subgroup.zpowers_eq_closure, f.map_closure, Set.image_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
N : Type u_3
inst_1 : Group N
f : G →* N
x : G
rw : Subgroup.map f (Subgroup.closure {x}) = Subgroup.zpowers ((f : (a : G) → N) x)
⊢ Subgroup.map f (Subgroup.zpowers x) = Subgroup.zpowers ((f : (a : G) → N) x)
have rw₁ : Subgroup.map f (Subgroup.closure {x}) = Subgroup.closure (singleton (α := N) ((f : (a : G) → N) x : N))
Tactic: rw [Subgroup.zpowers_eq_closure, Subgroup.zpowers_eq_closure, f.map_closure, Set.image_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
N : Type u_3
inst_1 : Group N
f : G →* N
x : G
rw : Subgroup.map f (Subgroup.closure {x}) = Subgroup.zpowers ((f : (a : G) → N) x)
rw₁ : Subgroup.map f (Subgroup.closure {x}) = Subgroup.closure (singleton (α := N) ((f : (a : G) → N) x : N))
⊢ Subgroup.map f (Subgroup.zpowers x) = Subgroup.zpowers ((f : (a : G) → N) x)
have rw₂ : Eq (α := Subgroup N) (Subgroup.closure (Set.image (α := G) (⇑f : (a : G) → N) {x}))
  (Subgroup.closure (singleton (α := N) ((f : (a : G) → N) x : N)))
Tactic: rw [Subgroup.zpowers_eq_closure, Subgroup.zpowers_eq_closure, f.map_closure, Set.image_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
N : Type u_3
inst_1 : Group N
f : G →* N
x : G
rw : Subgroup.map f (Subgroup.closure {x}) = Subgroup.zpowers ((f : (a : G) → N) x)
rw₁ : Subgroup.map f (Subgroup.closure {x}) = Subgroup.closure (singleton (α := N) ((f : (a : G) → N) x : N))
rw₂ :
  Eq (α := Subgroup N) (Subgroup.closure (Set.image (α := G) (⇑f : (a : G) → N) {x}))
    (Subgroup.closure (singleton (α := N) ((f : (a : G) → N) x : N)))
⊢ Subgroup.map f (Subgroup.zpowers x) = Subgroup.zpowers ((f : (a : G) → N) x)
have rw₃ : Eq (α := Subgroup N) (Subgroup.closure (singleton (α := N) ((f : (a : G) → N) x : N)))
  (Subgroup.closure (singleton (α := N) ((f : (a : G) → N) x : N)))
Tactic: rw [Subgroup.zpowers_eq_closure, Subgroup.zpowers_eq_closure, f.map_closure, Set.image_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
a b k : ℤ
⊢ (fun (x : ℤ) ↦ x • a) k = b ↔ b = a * k
have rw : (fun (x : ℤ) ↦ x • a) k = b ↔ b = k * a
Tactic: rw [mul_comm, eq_comm, ← smul_eq_mul]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
a b k : ℤ
rw : (fun (x : ℤ) ↦ x • a) k = b ↔ b = k * a
⊢ (fun (x : ℤ) ↦ x • a) k = b ↔ b = a * k
have rw₁ : b = (fun (x : ℤ) ↦ x • a) k ↔ b = k * a
Tactic: rw [mul_comm, eq_comm, ← smul_eq_mul]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
a b k : ℤ
rw : (fun (x : ℤ) ↦ x • a) k = b ↔ b = k * a
rw₁ : b = (fun (x : ℤ) ↦ x • a) k ↔ b = k * a
⊢ (fun (x : ℤ) ↦ x • a) k = b ↔ b = a * k
have rw₂ : b = (fun (x : ℤ) ↦ x • a) k ↔ b = k • a
Tactic: rw [mul_comm, eq_comm, ← smul_eq_mul]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
⊢ Eq (α := AddSubgroup ℤ) (AddSubgroup.zmultiples 1) ⊤
have h : ∀ (z : ℤ), Membership.mem (γ := AddSubgroup ℤ) (AddSubgroup.zmultiples 1) z ↔ Membership.mem (γ := AddSubgroup ℤ) ⊤ z
Tactic: ext z
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
x : G
⊢ (⇑Additive.ofMul : (a : G) → Additive G) '' (↑(Subgroup.zpowers x) : Set G) =
    (↑(AddSubgroup.zmultiples ((Additive.ofMul : (a : G) → Additive G) x) : AddSubgroup (Additive G)) :
      Set (Additive G))
have h : ∀ (x_1 : Additive G),
  Membership.mem (γ := Set (Additive G)) ((⇑Additive.ofMul : (a : G) → Additive G) '' (↑(Subgroup.zpowers x) : Set G))
      x_1 ↔
    x_1 ∈
      (↑(AddSubgroup.zmultiples ((Additive.ofMul : (a : G) → Additive G) x) : AddSubgroup (Additive G)) :
        Set (Additive G))
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
A : Type u_2
inst : AddGroup A
x : A
⊢ (⇑Multiplicative.ofAdd : (a : A) → Multiplicative A) '' (↑(AddSubgroup.zmultiples x) : Set A) =
    (↑(Subgroup.zpowers ((Multiplicative.ofAdd : (a : A) → Multiplicative A) x) : Subgroup (Multiplicative A)) :
      Set (Multiplicative A))
have symm : (↑(Subgroup.zpowers ((Multiplicative.ofAdd : (a : A) → Multiplicative A) x) : Subgroup (Multiplicative A)) :
    Set (Multiplicative A)) =
  (⇑Multiplicative.ofAdd : (a : A) → Multiplicative A) '' (↑(AddSubgroup.zmultiples x) : Set A)
Tactic: symm
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
A : Type u_2
inst : AddGroup A
x : A
⊢ (↑(Subgroup.zpowers ((Multiplicative.ofAdd : (a : A) → Multiplicative A) x) : Subgroup (Multiplicative A)) :
      Set (Multiplicative A)) =
    (⇑Multiplicative.ofAdd : (a : A) → Multiplicative A) '' (↑(AddSubgroup.zmultiples x) : Set A)
have rw : (⇑(Equiv.symm Multiplicative.ofAdd) : (a : Multiplicative A) → A) ''
    (↑(Subgroup.zpowers ((Multiplicative.ofAdd : (a : A) → Multiplicative A) x) : Subgroup (Multiplicative A)) :
      Set (Multiplicative A)) =
  (↑(AddSubgroup.zmultiples x) : Set A)
Tactic: rw [Equiv.eq_image_iff_symm_image_eq]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
H : Subgroup G
⊢ zpowers g ≤ H ↔ g ∈ H
have rw : closure {g} ≤ H ↔ g ∈ H
Tactic: rw [zpowers_eq_closure, closure_le, Set.singleton_subset_iff, SetLike.mem_coe]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
H : Subgroup G
rw : closure {g} ≤ H ↔ g ∈ H
⊢ zpowers g ≤ H ↔ g ∈ H
have rw₁ : {g} ⊆ (↑H : Set G) ↔ g ∈ H
Tactic: rw [zpowers_eq_closure, closure_le, Set.singleton_subset_iff, SetLike.mem_coe]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
H : Subgroup G
rw : closure {g} ≤ H ↔ g ∈ H
rw₁ : {g} ⊆ (↑H : Set G) ↔ g ∈ H
⊢ zpowers g ≤ H ↔ g ∈ H
have rw₂ : g ∈ (↑H : Set G) ↔ g ∈ H
Tactic: rw [zpowers_eq_closure, closure_le, Set.singleton_subset_iff, SetLike.mem_coe]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
H : Subgroup G
rw : closure {g} ≤ H ↔ g ∈ H
rw₁ : {g} ⊆ (↑H : Set G) ↔ g ∈ H
rw₂ : g ∈ (↑H : Set G) ↔ g ∈ H
⊢ zpowers g ≤ H ↔ g ∈ H
have rw₃ : g ∈ H ↔ g ∈ H
Tactic: rw [zpowers_eq_closure, closure_le, Set.singleton_subset_iff, SetLike.mem_coe]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
⊢ Eq (α := Subgroup G) (zpowers g) ⊥ ↔ g = 1
have rw : LE.le (α := Subgroup G) (zpowers g) ⊥ ↔ g = 1
Tactic: rw [eq_bot_iff, zpowers_le, mem_bot]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
rw : LE.le (α := Subgroup G) (zpowers g) ⊥ ↔ g = 1
⊢ Eq (α := Subgroup G) (zpowers g) ⊥ ↔ g = 1
have rw₁ : Membership.mem (γ := Subgroup G) ⊥ g ↔ g = 1
Tactic: rw [eq_bot_iff, zpowers_le, mem_bot]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
G : Type u_1
inst : Group G
g : G
rw : LE.le (α := Subgroup G) (zpowers g) ⊥ ↔ g = 1
rw₁ : Membership.mem (γ := Subgroup G) ⊥ g ↔ g = 1
⊢ Eq (α := Subgroup G) (zpowers g) ⊥ ↔ g = 1
have rw₂ : g = 1 ↔ g = 1
Tactic: rw [eq_bot_iff, zpowers_le, mem_bot]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Basic (goal)
Goal:
⊢ Eq (α := AddSubgroup ℤ) (AddSubgroup.closure {1}) ⊤
have h : ∀ (x : ℤ), Membership.mem (γ := AddSubgroup ℤ) (AddSubgroup.closure {1}) x ↔ Membership.mem (γ := AddSubgroup ℤ) ⊤ x
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
A : Type u_4
inst : AddGroupWithOne A
⊢ Eq.{u_4 + 1} (α := AddSubgroup A) (AddMonoidHom.range (castAddHom A)) (zmultiples 1)
have h : ∀ (a : A),
  Membership.mem (γ := AddSubgroup A) (AddMonoidHom.range (castAddHom A)) a ↔
    Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
Tactic: ext a
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
A : Type u_4
inst : AddGroupWithOne A
a : A
⊢ Membership.mem (γ := AddSubgroup A) (AddMonoidHom.range (castAddHom A)) a ↔
    Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
have h : (∃ (x : ℤ), (castAddHom A : (a : ℤ) → A) x = a) ↔ Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
Tactic: simp_rw [AddMonoidHom.mem_range, Int.coe_castAddHom, AddSubgroup.mem_zmultiples_iff, zsmul_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
A : Type u_4
inst : AddGroupWithOne A
a : A
h : (∃ (x : ℤ), (castAddHom A : (a : ℤ) → A) x = a) ↔ Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
⊢ Membership.mem (γ := AddSubgroup A) (AddMonoidHom.range (castAddHom A)) a ↔
    Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
have h₁ : (∃ (x : ℤ), (↑x : A) = a) ↔ Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
Tactic: simp_rw [AddMonoidHom.mem_range, Int.coe_castAddHom, AddSubgroup.mem_zmultiples_iff, zsmul_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
A : Type u_4
inst : AddGroupWithOne A
a : A
h : (∃ (x : ℤ), (castAddHom A : (a : ℤ) → A) x = a) ↔ Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
h₁ : (∃ (x : ℤ), (↑x : A) = a) ↔ Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
⊢ Membership.mem (γ := AddSubgroup A) (AddMonoidHom.range (castAddHom A)) a ↔
    Membership.mem (γ := AddSubgroup A) (zmultiples 1) a
have h₂ : (∃ (x : ℤ), (↑x : A) = a) ↔ ∃ (k : ℤ), HSMul.hSMul (β := A) k 1 = a
Tactic: simp_rw [AddMonoidHom.mem_range, Int.coe_castAddHom, AddSubgroup.mem_zmultiples_iff, zsmul_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
n : ℕ
⊢ Eq (α := AddSubgroup ℤ) (AddMonoidHom.range (nsmulAddMonoidHom n)) (zmultiples (↑n : ℤ))
have h : ∀ (m : ℤ),
  Membership.mem (γ := AddSubgroup ℤ) (AddMonoidHom.range (nsmulAddMonoidHom n)) m ↔
    Membership.mem (γ := AddSubgroup ℤ) (zmultiples (↑n : ℤ)) m
Tactic: ext m : 1
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
n : ℕ
m : ℤ
⊢ Membership.mem (γ := AddSubgroup ℤ) (AddMonoidHom.range (nsmulAddMonoidHom n)) m ↔
    Membership.mem (γ := AddSubgroup ℤ) (zmultiples (↑n : ℤ)) m
have h : (∃ (x : ℤ), (↑n : ℤ) * x = m) ↔ ∃ (c : ℤ), m = (↑n : ℤ) * c
Tactic: simp [mem_zmultiples_iff, dvd_def]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
a b : ℤ
⊢ Eq (α := AddSubgroup ℤ) (closure {a, b}) (zmultiples (↑(a.gcd b) : ℤ))
have a₁ : LE.le (α := AddSubgroup ℤ) (closure {a, b}) (zmultiples (↑(a.gcd b) : ℤ))
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas (goal)
Goal:
a b : ℤ
a₁ : LE.le (α := AddSubgroup ℤ) (closure {a, b}) (zmultiples (↑(a.gcd b) : ℤ))
⊢ Eq (α := AddSubgroup ℤ) (closure {a, b}) (zmultiples (↑(a.gcd b) : ℤ))
have a₂ : LE.le (α := AddSubgroup ℤ) (zmultiples (↑(a.gcd b) : ℤ)) (closure {a, b})
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
motive :
  (x y : M) → Membership.mem (γ := Submonoid M) (closure s) x → Membership.mem (γ := Submonoid M) (closure s) y → Prop
mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), motive x y ⋯ ⋯
one_left : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive 1 x ⋯ hx
one_right : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive x 1 hx ⋯
mul_left :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive x z hx hz → motive y z hy hz → motive (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive z x hz hx → motive z y hz hy → motive z (x * y) hz ⋯
x y : M
hx : Membership.mem (γ := Submonoid M) (closure s) x
z : M
hz : z ∈ s
⊢ motive x z hx ⋯
have mem₁ : ∀ (x : M) (h : x ∈ s), motive x z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem _ _ h hz
    | one => exact one_left _ (subset_closure hz)
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
motive :
  (x y : M) → Membership.mem (γ := Submonoid M) (closure s) x → Membership.mem (γ := Submonoid M) (closure s) y → Prop
mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), motive x y ⋯ ⋯
one_left : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive 1 x ⋯ hx
one_right : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive x 1 hx ⋯
mul_left :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive x z hx hz → motive y z hy hz → motive (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive z x hz hx → motive z y hz hy → motive z (x * y) hz ⋯
x y : M
hx : Membership.mem (γ := Submonoid M) (closure s) x
z : M
hz : z ∈ s
mem₁ : ∀ (x : M) (h : x ∈ s), motive x z ⋯ ⋯
⊢ motive x z hx ⋯
have mem₂ : motive 1 z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem _ _ h hz
    | one => exact one_left _ (subset_closure hz)
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
motive :
  (x y : M) → Membership.mem (γ := Submonoid M) (closure s) x → Membership.mem (γ := Submonoid M) (closure s) y → Prop
mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), motive x y ⋯ ⋯
one_left : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive 1 x ⋯ hx
one_right : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive x 1 hx ⋯
mul_left :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive x z hx hz → motive y z hy hz → motive (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive z x hz hx → motive z y hz hy → motive z (x * y) hz ⋯
x y : M
hx : Membership.mem (γ := Submonoid M) (closure s) x
z : M
hz : z ∈ s
mem₁ : ∀ (x : M) (h : x ∈ s), motive x z ⋯ ⋯
mem₂ : motive 1 z ⋯ ⋯
⊢ motive x z hx ⋯
have mem₃ : ∀ (x y : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
  (hy : Membership.mem (γ := Submonoid M) (closure s) y) (h₁ : motive x z hx sorry) (h₂ : motive y z hy sorry),
  motive (x * y) z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem _ _ h hz
    | one => exact one_left _ (subset_closure hz)
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
motive :
  (x y : M) → Membership.mem (γ := Submonoid M) (closure s) x → Membership.mem (γ := Submonoid M) (closure s) y → Prop
mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), motive x y ⋯ ⋯
one_left : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive 1 x ⋯ hx
one_right : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive x 1 hx ⋯
mul_left :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive x z hx hz → motive y z hy hz → motive (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive z x hz hx → motive z y hz hy → motive z (x * y) hz ⋯
x y : M
hx : Membership.mem (γ := Submonoid M) (closure s) x
hy : Membership.mem (γ := Submonoid M) (closure s) y
⊢ motive x y hx hy
have mem₁ : ∀ (z : M) (hz : z ∈ s), motive x z hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem _ _ h hz
    | one => exact one_left _ (subset_closure hz)
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ hx h₁ h₂
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
motive :
  (x y : M) → Membership.mem (γ := Submonoid M) (closure s) x → Membership.mem (γ := Submonoid M) (closure s) y → Prop
mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), motive x y ⋯ ⋯
one_left : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive 1 x ⋯ hx
one_right : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive x 1 hx ⋯
mul_left :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive x z hx hz → motive y z hy hz → motive (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive z x hz hx → motive z y hz hy → motive z (x * y) hz ⋯
x y : M
hx : Membership.mem (γ := Submonoid M) (closure s) x
hy : Membership.mem (γ := Submonoid M) (closure s) y
mem₁ : ∀ (z : M) (hz : z ∈ s), motive x z hx ⋯
⊢ motive x y hx hy
have one : motive x 1 hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem _ _ h hz
    | one => exact one_left _ (subset_closure hz)
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ hx h₁ h₂
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
motive :
  (x y : M) → Membership.mem (γ := Submonoid M) (closure s) x → Membership.mem (γ := Submonoid M) (closure s) y → Prop
mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), motive x y ⋯ ⋯
one_left : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive 1 x ⋯ hx
one_right : ∀ (x : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x), motive x 1 hx ⋯
mul_left :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive x z hx hz → motive y z hy hz → motive (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : M) (hx : Membership.mem (γ := Submonoid M) (closure s) x)
    (hy : Membership.mem (γ := Submonoid M) (closure s) y) (hz : Membership.mem (γ := Submonoid M) (closure s) z),
    motive z x hz hx → motive z y hz hy → motive z (x * y) hz ⋯
x y : M
hx : Membership.mem (γ := Submonoid M) (closure s) x
hy : Membership.mem (γ := Submonoid M) (closure s) y
mem₁ : ∀ (z : M) (hz : z ∈ s), motive x z hx ⋯
one : motive x 1 hx ⋯
⊢ motive x y hx hy
have mul : ∀ (x_1 y : M) (hx_1 : Membership.mem (γ := Submonoid M) (closure s) x_1)
  (hy : Membership.mem (γ := Submonoid M) (closure s) y) (h₁ : motive x x_1 hx hx_1) (h₂ : motive x y hx hy),
  motive x (x_1 * y) hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem _ _ h hz
    | one => exact one_left _ (subset_closure hz)
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ hx h₁ h₂
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (hypothesis)
Goal:
M : Type u_1
inst : MulOneClass M
motive : M → Prop
s : Set M
closure : Eq (α := Submonoid M) (Submonoid.closure s) ⊤
mem : ∀ x ∈ s, motive x
one : motive 1
mul : ∀ (x y : M), motive x → motive y → motive (x * y)
x : M
⊢ motive x
have x₁ : Membership.mem (γ := Submonoid M) (Submonoid.closure s) x
Tactic: induction closure.symm ▸ mem_top x using closure_induction with
  | mem _ h => exact mem _ h
  | one => exact one
  | mul _ _ _ _ h₁ h₂ => exact mul _ _ h₁ h₂
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
⊢ (↑(closure s) : Set M) = {1} ∪ (↑(Subsemigroup.closure s) : Set M)
have a : (↑(closure s) : Set M) ≤ {1} ∪ (↑(Subsemigroup.closure s) : Set M)
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (goal)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
a : (↑(closure s) : Set M) ≤ {1} ∪ (↑(Subsemigroup.closure s) : Set M)
⊢ (↑(closure s) : Set M) = {1} ∪ (↑(Subsemigroup.closure s) : Set M)
have a₁ : {1} ∪ (↑(Subsemigroup.closure s) : Set M) ≤ (↑(closure s) : Set M)
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Algebra.Group.Submonoid.Basic (hypothesis)
Goal:
M : Type u_1
inst : MulOneClass M
s : Set M
x_1 x hx_1 : M
y : Membership.mem (γ := Submonoid M) (closure s) x
hy_1 : Membership.mem (γ := Submonoid M) (closure s) hx_1
hx : x ∈ {1} ∪ (↑(Subsemigroup.closure s) : Set M)
hy : hx_1 ∈ {1} ∪ (↑(Subsemigroup.closure s) : Set M)
⊢ x * hx_1 ∈ {1} ∪ (↑(Subsemigroup.closure s) : Set M)
have hx₁ : x = 1 ∨ Membership.mem (γ := Subsemigroup M) (Subsemigroup.closure s) x
Tactic: push _ ∈ _ at hx hy
Cleared hypotheses: hx, hy