awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 : R
h : x_1 ∈ s
⊢ p x_1 z ⋯ ⋯

Tactic:
mem_mem _ _ h hz
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 : R
h : x_1 ∈ s
⊢ p x_1 z ⋯ ⋯

Tactic:
exact mem_mem _ _ h hz
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
⊢ p 0 z ⋯ ⋯

Tactic:
zero_left _ _
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
⊢ p 0 z ⋯ ⋯

Tactic:
exact zero_left _ _
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 y_1 : R
hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y_1
h₁ : p x_1 z hx ⋯
h₂ : p y_1 z hy ⋯
⊢ p (x_1 * y_1) z ⋯ ⋯

Tactic:
mul_left _ _ _ _ _ _ h₁ h₂
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 y_1 : R
hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y_1
h₁ : p x_1 z hx ⋯
h₂ : p y_1 z hy ⋯
⊢ p (x_1 * y_1) z ⋯ ⋯

Tactic:
exact mul_left _ _ _ _ _ _ h₁ h₂
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 y_1 : R
hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y_1
h₁ : p x_1 z hx ⋯
h₂ : p y_1 z hy ⋯
⊢ p (x_1 + y_1) z ⋯ ⋯

Tactic:
add_left _ _ _ _ _ _ h₁ h₂
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 y_1 : R
hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y_1
h₁ : p x_1 z hx ⋯
h₂ : p y_1 z hy ⋯
⊢ p (x_1 + y_1) z ⋯ ⋯

Tactic:
exact add_left _ _ _ _ _ _ h₁ h₂
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 : R
hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
h : p x_1 z hx ⋯
⊢ p (-x_1) z ⋯ ⋯

Tactic:
neg_left _ _ _ _ h
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring R) (closure s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : R
hz : z ∈ s
x_1 : R
hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
h : p x_1 z hx ⋯
⊢ p (-x_1) z ⋯ ⋯

Tactic:
exact neg_left _ _ _ _ h
-- Module: Mathlib.RingTheory.NonUnitalSubring.Basic
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x), p x 0 hx ⋯
neg_left :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p (-x) y ⋯ hy
neg_right :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubring 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 := NonUnitalSubring R) (closure s) x
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
mem₂ :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y) (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x * y) z ⋯ ⋯
mem₃ :
  ∀ (x y : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y) (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x + y) z ⋯ ⋯
mem₄ : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x) (h : p x z hx ⋯), p (-x) z ⋯ ⋯
⊢ p x z hx ⋯

Tactic:
induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h