awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
this : Nontrivial.{u_1} ↥H ↔ H ≠ ⊥
inl : Nontrivial.{u_1} ↥H → H ≠ ⊥ → H = ⊥ ∨ Nontrivial.{u_1} ↥H
inr : ¬Nontrivial.{u_1} ↥H → ¬H ≠ ⊥ → H = ⊥ ∨ Nontrivial.{u_1} ↥H
inl₁ : Nontrivial.{u_1} ↥H → H ≠ ⊥ → H = ⊥ ∨ Nontrivial.{u_1} ↥H
inr₁ : ¬Nontrivial.{u_1} ↥H → ¬H ≠ ⊥ → H = ⊥ ∨ Nontrivial.{u_1} ↥H
inl₂ : Nontrivial.{u_1} ↥H → H ≠ ⊥ → ¬H = ⊥ → Nontrivial.{u_1} ↥H
inr₂ : ¬Nontrivial.{u_1} ↥H → ¬H ≠ ⊥ → ¬H = ⊥ → Nontrivial.{u_1} ↥H
⊢ H = ⊥ ∨ Nontrivial.{u_1} ↥H
have inr₃ : ¬Nontrivial.{u_1} ↥H → H = ⊥ → ¬H = ⊥ → Nontrivial.{u_1} ↥H
Tactic: tauto
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
⊢ H = ⊥ ∨ ∃ x ∈ H, x ≠ 1
have h : (∃ x ∈ H, x ≠ 1) ↔ Nontrivial.{u_1} ↥H
Tactic: convert! H.bot_or_nontrivial
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
⊢ (∃ x ∈ H, x ≠ 1) ↔ Nontrivial.{u_1} ↥H
have h : (∃ x ∈ H, x ≠ 1) ↔ ∃ x ∈ H, x ≠ 1
Tactic: rw [nontrivial_iff_exists_ne_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
⊢ H ≠ ⊥ ↔ ∃ (a : ↥H), a ≠ 1
have rw : (∃ x ∈ H, x ≠ 1) ↔ ∃ (a : ↥H), a ≠ 1
Tactic: rw [← nontrivial_iff_ne_bot, nontrivial_iff_exists_ne_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
H K : Subgroup G
inst_1 : Nontrivial G
⊢ Nontrivial.{u_1} ↥⊤
have rw : Ne.{u_1 + 1} (α := Subgroup G) ⊤ ⊥
Tactic: rw [nontrivial_iff_ne_bot]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
z : G
hz : z ∈ k
⊢ p x z hx ⋯
have mem₁ : ∀ (x : G) (h : x ∈ k), p 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
    | inv _ _ h => exact inv_left _ _ _ _ h
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
z : G
hz : z ∈ k
mem₁ : ∀ (x : G) (h : x ∈ k), p x z ⋯ ⋯
⊢ p x z hx ⋯
have mem₂ : p 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
    | inv _ _ h => exact inv_left _ _ _ _ h
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
z : G
hz : z ∈ k
mem₁ : ∀ (x : G) (h : x ∈ k), p x z ⋯ ⋯
mem₂ : p 1 z ⋯ ⋯
⊢ p x z hx ⋯
have mem₃ : ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
  (h₁ : p x z hx sorry) (h₂ : p y z hy sorry), p (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
    | inv _ _ h => exact inv_left _ _ _ _ h
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
z : G
hz : z ∈ k
mem₁ : ∀ (x : G) (h : x ∈ k), p x z ⋯ ⋯
mem₂ : p 1 z ⋯ ⋯
mem₃ :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x * y) z ⋯ ⋯
⊢ p x z hx ⋯
have mem₄ : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x) (h : p x z hx sorry), p x⁻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
    | inv _ _ h => exact inv_left _ _ _ _ h
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
hy : Membership.mem (γ := Subgroup G) (closure k) y
⊢ p x y hx hy
have mem₁ : ∀ (z : G) (hz : z ∈ k), p 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
    | inv _ _ h => exact inv_left _ _ _ _ h
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ hx h₁ h₂
  | inv _ _ h => exact inv_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
hy : Membership.mem (γ := Subgroup G) (closure k) y
mem₁ : ∀ (z : G) (hz : z ∈ k), p x z hx ⋯
⊢ p x y hx hy
have one : p 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
    | inv _ _ h => exact inv_left _ _ _ _ h
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ hx h₁ h₂
  | inv _ _ h => exact inv_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
hy : Membership.mem (γ := Subgroup G) (closure k) y
mem₁ : ∀ (z : G) (hz : z ∈ k), p x z hx ⋯
one : p x 1 hx ⋯
⊢ p x y hx hy
have mul : ∀ (x_1 y : G) (hx_1 : Membership.mem (γ := Subgroup G) (closure k) x_1)
  (hy : Membership.mem (γ := Subgroup G) (closure k) y) (h₁ : p x x_1 hx hx_1) (h₂ : p x y hx hy), p 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
    | inv _ _ h => exact inv_left _ _ _ _ h
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ hx h₁ h₂
  | inv _ _ h => exact inv_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
k : Set G
p : (x y : G) → Membership.mem (γ := Subgroup G) (closure k) x → Membership.mem (γ := Subgroup G) (closure k) y → Prop
mem : ∀ (x y : G) (hx : x ∈ k) (hy : y ∈ k), p x y ⋯ ⋯
one_left : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p 1 x ⋯ hx
one_right : ∀ (x : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x), p x 1 hx ⋯
mul_left :
  ∀ (x y z : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (hz : Membership.mem (γ := Subgroup G) (closure k) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (y z x : G) (hy : Membership.mem (γ := Subgroup G) (closure k) y)
    (hz : Membership.mem (γ := Subgroup G) (closure k) z) (hx : Membership.mem (γ := Subgroup G) (closure k) x),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
inv_left :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x⁻¹ y ⋯ hy
inv_right :
  ∀ (x y : G) (hx : Membership.mem (γ := Subgroup G) (closure k) x)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y), p x y hx hy → p x y⁻¹ hx ⋯
x y : G
hx : Membership.mem (γ := Subgroup G) (closure k) x
hy : Membership.mem (γ := Subgroup G) (closure k) y
mem₁ : ∀ (z : G) (hz : z ∈ k), p x z hx ⋯
one : p x 1 hx ⋯
mul :
  ∀ (x_1 y : G) (hx_1 : Membership.mem (γ := Subgroup G) (closure k) x_1)
    (hy : Membership.mem (γ := Subgroup G) (closure k) y) (h₁ : p x x_1 hx hx_1) (h₂ : p x y hx hy), p x (x_1 * y) hx ⋯
⊢ p x y hx hy
have inv : ∀ (x_1 : G) (hx_1 : Membership.mem (γ := Subgroup G) (closure k) x_1) (h : p x x_1 hx hx_1), p x x_1⁻¹ 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
    | inv _ _ h => exact inv_left _ _ _ _ h
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ hx h₁ h₂
  | inv _ _ h => exact inv_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
H H' : Subgroup G
⊢ H ⊔ H' = closure ((↑H : Set G) ∪ (↑H' : Set G))
have simp_rw : H ⊔ H' = closure (↑H : Set G) ⊔ closure (↑H' : Set G)
Tactic: simp_rw [closure_union, closure_eq]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
ι : Sort u_2
p : ι → Subgroup G
⊢ Eq.{u_1 + 1} (α := Subgroup G) (⨆ (i : ι), p i) (closure (⋃ (i : ι), (↑(p i) : Set G)))
have simp_rw : Eq.{u_1 + 1} (α := Subgroup G) (⨆ (i : ι), p i) (⨆ (i : ι), closure (↑(p i) : Set G))
Tactic: simp_rw [closure_iUnion, closure_eq]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
x y : G
⊢ Membership.mem (γ := Subgroup G) (closure {x}) y ↔ ∃ (n : ℤ), x ^ n = y
have refine_1 : ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y) (x_1 : G) (hx : Membership.mem (γ := Set G) {x} x_1),
  ∃ (n : ℤ), x ^ n = x_1
Tactic: refine
    ⟨fun hy => closure_induction ?_ ?_ ?_ ?_ hy, fun ⟨n, hn⟩ =>
      hn ▸ zpow_mem (subset_closure <| mem_singleton x) n⟩
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
x y : G
refine_1 :
  ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y) (x_1 : G) (hx : Membership.mem (γ := Set G) {x} x_1),
    ∃ (n : ℤ), x ^ n = x_1
⊢ Membership.mem (γ := Subgroup G) (closure {x}) y ↔ ∃ (n : ℤ), x ^ n = y
have refine_2 : ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y), ∃ (n : ℤ), x ^ n = 1
Tactic: refine
    ⟨fun hy => closure_induction ?_ ?_ ?_ ?_ hy, fun ⟨n, hn⟩ =>
      hn ▸ zpow_mem (subset_closure <| mem_singleton x) n⟩
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
x y : G
refine_1 :
  ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y) (x_1 : G) (hx : Membership.mem (γ := Set G) {x} x_1),
    ∃ (n : ℤ), x ^ n = x_1
refine_2 : ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y), ∃ (n : ℤ), x ^ n = 1
⊢ Membership.mem (γ := Subgroup G) (closure {x}) y ↔ ∃ (n : ℤ), x ^ n = y
have refine_3 : ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y) (x_1 y : G)
  (hx : Membership.mem (γ := Subgroup G) (closure {x}) x_1) (hy : Membership.mem (γ := Subgroup G) (closure {x}) y),
  (∃ (n : ℤ), x ^ n = x_1) → (∃ (n : ℤ), x ^ n = y) → ∃ (n : ℤ), x ^ n = x_1 * y
Tactic: refine
    ⟨fun hy => closure_induction ?_ ?_ ?_ ?_ hy, fun ⟨n, hn⟩ =>
      hn ▸ zpow_mem (subset_closure <| mem_singleton x) n⟩
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
x y : G
refine_1 :
  ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y) (x_1 : G) (hx : Membership.mem (γ := Set G) {x} x_1),
    ∃ (n : ℤ), x ^ n = x_1
refine_2 : ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y), ∃ (n : ℤ), x ^ n = 1
refine_3 :
  ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y) (x_1 y : G)
    (hx : Membership.mem (γ := Subgroup G) (closure {x}) x_1) (hy : Membership.mem (γ := Subgroup G) (closure {x}) y),
    (∃ (n : ℤ), x ^ n = x_1) → (∃ (n : ℤ), x ^ n = y) → ∃ (n : ℤ), x ^ n = x_1 * y
⊢ Membership.mem (γ := Subgroup G) (closure {x}) y ↔ ∃ (n : ℤ), x ^ n = y
have refine_4 : ∀ (hy : Membership.mem (γ := Subgroup G) (closure {x}) y) (x_1 : G)
  (hx : Membership.mem (γ := Subgroup G) (closure {x}) x_1), (∃ (n : ℤ), x ^ n = x_1) → ∃ (n : ℤ), x ^ n = x_1⁻¹
Tactic: refine
    ⟨fun hy => closure_induction ?_ ?_ ?_ ?_ hy, fun ⟨n, hn⟩ =>
      hn ▸ zpow_mem (subset_closure <| mem_singleton x) n⟩
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
x y_1 : G
hy_1 : Membership.mem (γ := Subgroup G) (closure {x}) y_1
y : G
hy : Membership.mem (γ := Set G) {x} y
⊢ ∃ (n : ℤ), x ^ n = y
have refine_1 : ∃ (n : ℤ), x ^ n = x
Tactic: rw [eq_of_mem_singleton hy]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
s : Set G
⊢ Eq (α := Subgroup G) (closure (insert 1 s)) (closure s)
have rw : Eq (α := Subgroup G) (closure {1} ⊔ closure s) (closure s)
Tactic: rw [insert_eq, closure_union]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
s : Set G
⊢ Eq (α := Subgroup G) (closure (s ∪ {1})) (closure s)
have rw : Eq (α := Subgroup G) (closure (insert 1 s)) (closure s)
Tactic: rw [union_singleton, closure_insert_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
s : Set G
rw : Eq (α := Subgroup G) (closure (insert 1 s)) (closure s)
⊢ Eq (α := Subgroup G) (closure (s ∪ {1})) (closure s)
have rw₁ : Eq (α := Subgroup G) (closure s) (closure s)
Tactic: rw [union_singleton, closure_insert_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
s : Set G
⊢ Eq (α := Subgroup G) (closure (s \ {1})) (closure s)
have rw : Eq (α := Subgroup G) (closure (s \ {1} ∪ {1})) (closure s)
Tactic: rw [← closure_union_one (s \ {1}), diff_union_self, closure_union_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
s : Set G
rw : Eq (α := Subgroup G) (closure (s \ {1} ∪ {1})) (closure s)
⊢ Eq (α := Subgroup G) (closure (s \ {1})) (closure s)
have rw₁ : Eq (α := Subgroup G) (closure (s ∪ {1})) (closure s)
Tactic: rw [← closure_union_one (s \ {1}), diff_union_self, closure_union_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
s : Set G
rw : Eq (α := Subgroup G) (closure (s \ {1} ∪ {1})) (closure s)
rw₁ : Eq (α := Subgroup G) (closure (s ∪ {1})) (closure s)
⊢ Eq (α := Subgroup G) (closure (s \ {1})) (closure s)
have rw₂ : Eq (α := Subgroup G) (closure s) (closure s)
Tactic: rw [← closure_union_one (s \ {1}), diff_union_self, closure_union_one]
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
ι : Type u_2
p : ι → Prop
K : ι → Subgroup G
i : ι
hp : p i
hK : DirectedOn ((fun (x1 x2 : Subgroup G) ↦ x1 ≤ x2) on K) {i : ι | p i}
x : G
⊢ Membership.mem (γ := Subgroup G) (⨆ (i : ι), ⨆ (_ : p i), K i) x ↔ ∃ (i : ι), p i ∧ x ∈ K i
have refine_1 : Membership.mem (γ := Subgroup G) (⨆ (i : ι), ⨆ (_ : p i), K i) x → ∃ (i : ι), p i ∧ x ∈ K i
Tactic: refine ⟨?_, fun ⟨i, hi', hi⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Algebra.Group.Subgroup.Lattice (goal)
Goal:
G : Type u_1
inst : Group G
ι : Type u_2
p : ι → Prop
K : ι → Subgroup G
i : ι
hp : p i
hK : DirectedOn ((fun (x1 x2 : Subgroup G) ↦ x1 ≤ x2) on K) {i : ι | p i}
x : G
refine_1 : Membership.mem (γ := Subgroup G) (⨆ (i : ι), ⨆ (_ : p i), K i) x → ∃ (i : ι), p i ∧ x ∈ K i
⊢ Membership.mem (γ := Subgroup G) (⨆ (i : ι), ⨆ (_ : p i), K i) x ↔ ∃ (i : ι), p i ∧ x ∈ K i
have refine_2 : (∃ (i : ι), p i ∧ x ∈ K i) →
  ∀ (i : ι) (hi' : p i) (hi : x ∈ K i), Membership.mem (γ := Subgroup G) (⨆ (i : ι), ⨆ (_ : p i), K i) x
Tactic: refine ⟨?_, fun ⟨i, hi', hi⟩ ↦ ?_⟩