awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
p : (x y : R) → Membership.mem (γ := Subring R) (closure s) x → Membership.mem (γ := Subring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 1 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subring R) (closure s) x
hy : Membership.mem (γ := Subring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
⊢ p x y hx hy
have zero : p x 0 hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
  | neg _ _ h => exact neg_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
p : (x y : R) → Membership.mem (γ := Subring R) (closure s) x → Membership.mem (γ := Subring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 1 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subring R) (closure s) x
hy : Membership.mem (γ := Subring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
zero : p x 0 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_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
  | neg _ _ h => exact neg_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
p : (x y : R) → Membership.mem (γ := Subring R) (closure s) x → Membership.mem (γ := Subring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 1 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subring R) (closure s) x
hy : Membership.mem (γ := Subring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
zero : p x 0 hx ⋯
one : p x 1 hx ⋯
⊢ p x y hx hy
have mul : ∀ (x_1 y : R) (hx_1 : Membership.mem (γ := Subring R) (closure s) x_1)
  (hy : Membership.mem (γ := Subring R) (closure s) 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_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
  | neg _ _ h => exact neg_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
p : (x y : R) → Membership.mem (γ := Subring R) (closure s) x → Membership.mem (γ := Subring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x 1 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subring R) (closure s) x)
    (hy : Membership.mem (γ := Subring R) (closure s) y) (hz : Membership.mem (γ := Subring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subring R) (closure s) x
hy : Membership.mem (γ := Subring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
zero : p x 0 hx ⋯
one : p x 1 hx ⋯
mul :
  ∀ (x_1 y : R) (hx_1 : Membership.mem (γ := Subring R) (closure s) x_1)
    (hy : Membership.mem (γ := Subring R) (closure s) 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 add : ∀ (x_1 y : R) (hx_1 : Membership.mem (γ := Subring R) (closure s) x_1)
  (hy : Membership.mem (γ := Subring R) (closure s) 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_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
  | neg _ _ h => exact neg_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x x_1 y : R
hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x_1
hy : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x_1 * y)
have mul : ∀ (x y : R) (hx : x ∈ (↑(Submonoid.closure s) : Set R)) (hy : y ∈ (↑(Submonoid.closure s) : Set R)),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * y)
Tactic: induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | zero_left => simp
      | zero_right => simp
      | add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h
Cleared hypotheses: hx, hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x x_1 y : R
hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x_1
hy : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y
mul :
  ∀ (x y : R) (hx : x ∈ (↑(Submonoid.closure s) : Set R)) (hy : y ∈ (↑(Submonoid.closure s) : Set R)),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * y)
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x_1 * y)
have mul₁ : ∀ (x : R),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (0 * x)
Tactic: induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | zero_left => simp
      | zero_right => simp
      | add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h
Cleared hypotheses: hx, hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x x_1 y : R
hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x_1
hy : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y
mul :
  ∀ (x y : R) (hx : x ∈ (↑(Submonoid.closure s) : Set R)) (hy : y ∈ (↑(Submonoid.closure s) : Set R)),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * y)
mul₁ :
  ∀ (x : R),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (0 * x)
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x_1 * y)
have mul₂ : ∀ (x : R),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * 0)
Tactic: induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | zero_left => simp
      | zero_right => simp
      | add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h
Cleared hypotheses: hx, hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x x_1 y : R
hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x_1
hy : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y
mul :
  ∀ (x y : R) (hx : x ∈ (↑(Submonoid.closure s) : Set R)) (hy : y ∈ (↑(Submonoid.closure s) : Set R)),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * y)
mul₁ :
  ∀ (x : R),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (0 * x)
mul₂ :
  ∀ (x : R),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * 0)
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x_1 * y)
have mul₃ : ∀ (x y z : R),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) z →
        ∀ (h₁ : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * z))
          (h₂ : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (y * z)),
          Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) ((x + y) * z)
Tactic: induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | zero_left => simp
      | zero_right => simp
      | add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h
Cleared hypotheses: hx, hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x x_1 y : R
hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x_1
hy : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y
mul :
  ∀ (x y : R) (hx : x ∈ (↑(Submonoid.closure s) : Set R)) (hy : y ∈ (↑(Submonoid.closure s) : Set R)),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * y)
mul₁ :
  ∀ (x : R),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (0 * x)
mul₂ :
  ∀ (x : R),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * 0)
mul₃ :
  ∀ (x y z : R),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y →
        Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) z →
          ∀ (h₁ : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * z))
            (h₂ : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (y * z)),
            Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) ((x + y) * z)
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x_1 * y)
have mul₄ : ∀ (y z x : R),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) y →
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) z →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x →
        ∀ (h₁ : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * y))
          (h₂ : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * z)),
          Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) (x * (y + z))
Tactic: induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | zero_left => simp
      | zero_right => simp
      | add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h
Cleared hypotheses: hx, hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x : R
h : Membership.mem (γ := Subring R) (closure s) x
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x
have mem : ∀ (x : R) (hx : x ∈ s), Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x
Tactic: induction h using closure_induction with
    | mem _ hx => exact AddSubgroup.subset_closure (Submonoid.subset_closure hx)
    | zero => exact zero_mem _
    | one => exact AddSubgroup.subset_closure (one_mem _)
    | add _ _ _ _ hx hy => exact add_mem hx hy
    | neg _ _ hx => exact neg_mem hx
    | mul _ _ _hx _hy hx hy =>
      clear _hx _hy
      induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | zero_left => simp
      | zero_right => simp
      | add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x : R
h : Membership.mem (γ := Subring R) (closure s) x
mem :
  ∀ (x : R) (hx : x ∈ s), Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) x
have zero : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Submonoid.closure s) : Set R)) 0
Tactic: induction h using closure_induction with
    | mem _ hx => exact AddSubgroup.subset_closure (Submonoid.subset_closure hx)
    | zero => exact zero_mem _
    | one => exact AddSubgroup.subset_closure (one_mem _)
    | add _ _ _ _ hx hy => exact add_mem hx hy
    | neg _ _ hx => exact neg_mem hx
    | mul _ _ _hx _hy hx hy =>
      clear _hx _hy
      induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | zero_left => simp
      | zero_right => simp
      | add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h
Cleared hypotheses: h