awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
M_1 : Type u_1
R : Type u_2
A : Type u_3
inst : NonAssocSemiring R
M : AddSubmonoid R
rw : M * closure 1 = M
rw₁ : HMul.hMul (α := AddSubmonoid R) (closure (↑M : Set R)) (closure 1) = closure (↑M : Set R)
⊢ M * 1 = M
have rw₂ : Eq (α := AddSubmonoid R) (closure ((↑M : Set R) * 1)) (closure (↑M : Set R))
Tactic: rw [one_eq_closure_one_set, ← closure_eq M, closure_mul_closure, mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
M_1 : Type u_1
R : Type u_2
A : Type u_3
inst : NonAssocSemiring R
M : AddSubmonoid R
rw : M * closure 1 = M
rw₁ : HMul.hMul (α := AddSubmonoid R) (closure (↑M : Set R)) (closure 1) = closure (↑M : Set R)
rw₂ : Eq (α := AddSubmonoid R) (closure ((↑M : Set R) * 1)) (closure (↑M : Set R))
⊢ M * 1 = M
have rw₃ : Eq (α := AddSubmonoid R) (closure (↑M : Set R)) (closure (↑M : Set R))
Tactic: rw [one_eq_closure_one_set, ← closure_eq M, closure_mul_closure, mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : Set R
⊢ HPow.hPow (α := AddSubmonoid R) (β := ℕ) (closure s) 0 = closure (HPow.hPow (β := ℕ) s 0)
have rw : Eq (α := AddSubmonoid R) 1 (closure (HPow.hPow (β := ℕ) s 0))
Tactic: rw [pow_zero, pow_zero, one_eq_closure_one_set]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : Set R
rw : Eq (α := AddSubmonoid R) 1 (closure (HPow.hPow (β := ℕ) s 0))
⊢ HPow.hPow (α := AddSubmonoid R) (β := ℕ) (closure s) 0 = closure (HPow.hPow (β := ℕ) s 0)
have rw₁ : Eq.{u_2 + 1} (α := AddSubmonoid R) 1 (closure 1)
Tactic: rw [pow_zero, pow_zero, one_eq_closure_one_set]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : Set R
rw : Eq (α := AddSubmonoid R) 1 (closure (HPow.hPow (β := ℕ) s 0))
rw₁ : Eq.{u_2 + 1} (α := AddSubmonoid R) 1 (closure 1)
⊢ HPow.hPow (α := AddSubmonoid R) (β := ℕ) (closure s) 0 = closure (HPow.hPow (β := ℕ) s 0)
have rw₂ : Eq.{u_2 + 1} (α := AddSubmonoid R) (closure 1) (closure 1)
Tactic: rw [pow_zero, pow_zero, one_eq_closure_one_set]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : Set R
n : ℕ
⊢ HPow.hPow (α := AddSubmonoid R) (closure s) (n + 1) = closure (s ^ (n + 1))
have rw : HPow.hPow (α := AddSubmonoid R) (closure s) n * closure s = closure (s ^ (n + 1))
Tactic: rw [pow_succ, pow_succ, closure_pow s n, closure_mul_closure]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : Set R
n : ℕ
rw : HPow.hPow (α := AddSubmonoid R) (closure s) n * closure s = closure (s ^ (n + 1))
⊢ HPow.hPow (α := AddSubmonoid R) (closure s) (n + 1) = closure (s ^ (n + 1))
have rw₁ : HPow.hPow (α := AddSubmonoid R) (closure s) n * closure s = closure (s ^ n * s)
Tactic: rw [pow_succ, pow_succ, closure_pow s n, closure_mul_closure]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : Set R
n : ℕ
rw : HPow.hPow (α := AddSubmonoid R) (closure s) n * closure s = closure (s ^ (n + 1))
rw₁ : HPow.hPow (α := AddSubmonoid R) (closure s) n * closure s = closure (s ^ n * s)
⊢ HPow.hPow (α := AddSubmonoid R) (closure s) (n + 1) = closure (s ^ (n + 1))
have rw₂ : HMul.hMul (α := AddSubmonoid R) (closure (s ^ n)) (closure s) = closure (s ^ n * s)
Tactic: rw [pow_succ, pow_succ, closure_pow s n, closure_mul_closure]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : Set R
n : ℕ
rw : HPow.hPow (α := AddSubmonoid R) (closure s) n * closure s = closure (s ^ (n + 1))
rw₁ : HPow.hPow (α := AddSubmonoid R) (closure s) n * closure s = closure (s ^ n * s)
rw₂ : HMul.hMul (α := AddSubmonoid R) (closure (s ^ n)) (closure s) = closure (s ^ n * s)
⊢ HPow.hPow (α := AddSubmonoid R) (closure s) (n + 1) = closure (s ^ (n + 1))
have rw₃ : Eq (α := AddSubmonoid R) (closure (s ^ n * s)) (closure (s ^ n * s))
Tactic: rw [pow_succ, pow_succ, closure_pow s n, closure_mul_closure]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : AddSubmonoid R
n : ℕ
⊢ s ^ n = closure ((↑s : Set R) ^ n)
have rw : s ^ n = HPow.hPow (α := AddSubmonoid R) (closure (↑s : Set R)) n
Tactic: rw [← closure_pow, closure_eq]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Submonoid.Pointwise (goal)
Goal:
R : Type u_2
inst : Semiring R
s : AddSubmonoid R
n : ℕ
rw : s ^ n = HPow.hPow (α := AddSubmonoid R) (closure (↑s : Set R)) n
⊢ s ^ n = closure ((↑s : Set R) ^ n)
have rw₁ : s ^ n = s ^ n
Tactic: rw [← closure_pow, closure_eq]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.Basic (goal)
Goal:
R : Type u
S : Type v
inst : NonAssocRing R
inst_1 : NonAssocRing S
f : R →+* S
⊢ f.range = Subring.map f ⊤
have h : ∀ (x : S), x ∈ f.range ↔ x ∈ Subring.map f ⊤
Tactic: ext
-- 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
z : R
hz : z ∈ s
⊢ p x z hx ⋯
have mem : ∀ (x : R) (h : x ∈ s), p x z sorry sorry
Tactic: 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
Cleared hypotheses: hx
-- 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
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
⊢ p x z hx ⋯
have mem₁ : p 0 z sorry sorry
Tactic: 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
Cleared hypotheses: hx
-- 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
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
⊢ p x z hx ⋯
have mem₂ : p 1 z sorry sorry
Tactic: 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
Cleared hypotheses: hx
-- 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
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
mem₂ : p 1 z ⋯ ⋯
⊢ p x z hx ⋯
have mem₃ : ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) 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_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
Cleared hypotheses: hx
-- 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
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
mem₂ : p 1 z ⋯ ⋯
mem₃ :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y)
    (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x * y) z ⋯ ⋯
⊢ p x z hx ⋯
have mem₄ : ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) 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_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
Cleared hypotheses: hx
-- 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
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
mem₂ : p 1 z ⋯ ⋯
mem₃ :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y)
    (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x * y) z ⋯ ⋯
mem₄ :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y)
    (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x + y) z ⋯ ⋯
⊢ p x z hx ⋯
have mem₅ : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (h : p x z hx sorry), p (-x) z sorry sorry
Tactic: 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
Cleared hypotheses: hx
-- 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
⊢ p x y hx hy
have mem : ∀ (z : R) (hz : z ∈ s), 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_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