awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
p p' : Subring R
x : R
⊢ x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

Tactic:
Iff.rfl

Premises used: Set.Mem, Iff.rfl, setOf, Set.inter
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
S : Type v
T : Type w
inst : NonAssocRing R
inst_1 : NonAssocRing S
inst_2 : NonAssocRing T
s : Set (Subring R)
⊢ (↑(⨅ t ∈ s, t.toSubmonoid) : Set R) = ⋂ t ∈ s, (↑t : Set R)

Tactic:
simp
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
S : Type v
T : Type w
inst : NonAssocRing R
inst_1 : NonAssocRing S
inst_2 : NonAssocRing T
s : Set (Subring R)
⊢ (↑(⨅ t ∈ s, t.toAddSubgroup) : Set R) = ⋂ t ∈ s, (↑t : Set R)

Tactic:
simp
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
S : Set (Subring R)
⊢ (↑(sInf S) : Set R) = ⋂ s ∈ S, (↑s : Set R)

Tactic:
rfl

Premises used: iInf, Set.iInter, setOf, rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
S : Set (Subring R)
x : R
⊢ x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

Tactic:
Set.mem_iInter₂

Premises used: iInf, Set.Mem, Set.iInter, setOf, Set.mem_iInter₂
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
ι : Sort u_1
S : ι → Subring R
⊢ (↑(⨅ (i : ι), S i) : Set R) = ⋂ (i : ι), (↑(S i) : Set R)

Tactic:
simp only [iInf, coe_sInf, Set.biInter_range]

Premises used: iInf, Set.biInter_range, Subring.coe_sInf
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
ι : Sort u_1
S : ι → Subring R
x : R
⊢ Membership.mem (γ := Subring R) (⨅ (i : ι), S i) x ↔ ∀ (i : ι), x ∈ S i

Tactic:
simp only [iInf, mem_sInf, Set.forall_mem_range]

Premises used: iInf, Set.forall_mem_range, Subring.mem_sInf
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set (Subring R)
⊢ (sInf s).toSubmonoid = ⨅ t ∈ s, t.toSubmonoid

Tactic:
mk'_toSubmonoid _ _

Premises used: Subring.mk'_toSubmonoid
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set (Subring R)
⊢ (sInf s).toAddSubgroup = ⨅ t ∈ s, t.toAddSubgroup

Tactic:
mk'_toAddSubgroup _ _

Premises used: Subring.mk'_toAddSubgroup
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
A : Subring R
⊢ A = ⊤ ↔ ∀ (x : R), x ∈ A

Tactic:
eq_top_iff.trans ⟨fun h m => h <| mem_top m, fun h m _ => h m⟩

Premises used: eq_top_iff, Subring.mem_top, Iff.intro, Iff.trans
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
⊢ (↑(center R) : Set R) = Set.center R

Tactic:
rfl

Premises used: setOf, Set.center, rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
⊢ Eq (α := Subsemiring R) (toSubsemiring (center R)) (Subsemiring.center R)

Tactic:
rfl

Premises used: setOf, Set.center, Subsemiring.center, rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
z : R
⊢ Membership.mem (γ := Subring R) (center R) z ↔ ∀ (g : R), g * z = z * g

Tactic:
Subsemigroup.mem_center_iff

Premises used: Set.Mem, Subsemigroup.mem_center_iff
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : CommRing R
⊢ Eq (α := Subring R) (center R) ⊤

Tactic:
SetLike.coe_injective (Set.center_eq_univ R)

Premises used: Set.center_eq_univ, SetLike.coe_injective
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
K : Type u
inst : DivisionRing K
a : ↥(center K)
⊢ (↑a⁻¹ : K) = (↑a : K)⁻¹

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
K : Type u
inst : DivisionRing K
a b : ↥(center K)
⊢ (↑(a / b) : K) = (↑a : K) / (↑b : K)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s : Set R
⊢ (↑(centralizer s) : Set R) = s.centralizer

Tactic:
rfl

Premises used: setOf, Set.centralizer, rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s : Set R
⊢ Eq (α := Submonoid R) (Subsemiring.toSubmonoid (toSubsemiring (centralizer s))) (Submonoid.centralizer s)

Tactic:
rfl

Premises used: Submonoid.centralizer, rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s : Set R
⊢ Eq (α := Subsemiring R) (toSubsemiring (centralizer s)) (Subsemiring.centralizer s)

Tactic:
rfl

Premises used: Subsemiring.centralizer, rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s : Set R
⊢ Eq (α := NonUnitalSubring R) (toNonUnitalSubring (centralizer s)) (NonUnitalSubring.centralizer s)

Tactic:
rfl

Premises used: setOf, Set.centralizer, NonUnitalSubring.centralizer, rfl
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s : Set R
z : R
⊢ Membership.mem (γ := Subring R) (centralizer s) z ↔ ∀ g ∈ s, g * z = z * g

Tactic:
Iff.rfl

Premises used: Set.Mem, Iff.rfl, setOf, Set.centralizer
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s : Set R
⊢ LE.le (α := Subring R) (center R) (centralizer s)

Tactic:
s.center_subset_centralizer

Premises used: Set.Mem, setOf, Set.center, Set.centralizer, Set.Subset, Set.center_subset_centralizer
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s t : Set R
h : s ⊆ t
⊢ LE.le (α := Subring R) (centralizer t) (centralizer s)

Tactic:
Set.centralizer_subset h

Premises used: Set.Mem, setOf, Set.centralizer, Set.Subset, Set.centralizer_subset
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
s : Set R
⊢ Eq (α := Subring R) (centralizer s) ⊤ ↔ s ⊆ (↑(center R) : Set R)

Tactic:
SetLike.ext'_iff.trans Set.centralizer_eq_top_iff_subset

Premises used: Set.centralizer_eq_top_iff_subset, SetLike.ext'_iff, Iff.trans
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u_1
inst : Ring R
⊢ Eq (α := Subring R) (centralizer Set.univ) (center R)

Tactic:
SetLike.ext' (Set.centralizer_univ R)

Premises used: SetLike.ext', Set.centralizer_univ
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
x : R
s : Set R
⊢ Membership.mem (γ := Subring R) (closure s) x ↔ ∀ (S : Subring R), s ⊆ (↑S : Set R) → x ∈ S

Tactic:
mem_sInf

Premises used: Set.Mem, setOf, Subring.mem_sInf
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
⊢ s ⊆ (↑(closure s) : Set R)

Tactic:
fun _ hx => mem_closure.2 fun _ hS => hS hx

Premises used: Iff.mpr, Subring.mem_closure, Set.Subset
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
x : R
hx : x ∈ s
⊢ Membership.mem (γ := Subring R) (closure s) x

Tactic:
subset_closure hx

Premises used: Subring.subset_closure
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
P : R
hP : P ∉ closure s
⊢ P ∉ s

Tactic:
fun h =>
  hP (subset_closure h)

Premises used: Subring.subset_closure, Not
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
t : Subring R
⊢ closure s ≤ t ↔ s ⊆ (↑t : Set R)

Tactic:
⟨Set.Subset.trans subset_closure, fun h => sInf_le h⟩

Premises used: Set.Subset.trans, Subring.subset_closure, sInf_le, Iff.intro
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s t : Set R
h : s ⊆ t
⊢ LE.le (α := Subring R) (closure s) (closure t)

Tactic:
closure_le.2 <| Set.Subset.trans h subset_closure

Premises used: Iff.mpr, Set.Subset.trans, Subring.subset_closure, Subring.closure_le
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
t : Subring R
h₁ : s ⊆ (↑t : Set R)
h₂ : t ≤ closure s
⊢ closure s = t

Tactic:
le_antisymm (closure_le.2 h₁) h₂

Premises used: Iff.mpr, le_antisymm, Subring.closure_le
-- Module: Mathlib.Algebra.Ring.Subring.Basic
Goal:
R : Type u
inst : NonAssocRing R
s : Set R
p : (x : R) → Membership.mem (γ := Subring R) (closure s) x → Prop
mem : ∀ (x : R) (hx : x ∈ s), p x ⋯
zero : p 0 ⋯
one : p 1 ⋯
add :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x hx → p y hy → p (x + y) ⋯
neg : ∀ (x : R) (hx : Membership.mem (γ := Subring R) (closure s) x), p x hx → p (-x) ⋯
mul :
  ∀ (x y : R) (hx : Membership.mem (γ := Subring R) (closure s) x) (hy : Membership.mem (γ := Subring R) (closure s) y),
    p x hx → p y hy → p (x * y) ⋯
x : R
hx : Membership.mem (γ := Subring R) (closure s) x
⊢ p x hx

Tactic:
let K : Subring R :=
    { carrier := { x | ∃ hx, p x hx }
      mul_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, mul _ _ _ _ hpx hpy⟩
      add_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, add _ _ _ _ hpx hpy⟩
      neg_mem' := fun ⟨_, hpx⟩ ↦ ⟨_, neg _ _ hpx⟩
      zero_mem' := ⟨_, zero⟩
      one_mem' := ⟨_, one⟩ }
  closure_le (t := K) |>.mpr (fun y hy ↦ ⟨subset_closure hy, mem y hy⟩) hx |>.elim fun _ ↦ id

Premises used: Iff.mpr, HMul.hMul, Subsemigroup.mk, setOf, Membership.mem, Exists, id, Subring.subset_closure, HAdd.hAdd, _private.0.Subring.closure_induction.match_1, Subsemiring.mk, Exists.intro, AddMemClass.add_mem, Submonoid.mk, ZeroMemClass.zero_mem, OneMemClass.one_mem, Subring.closure_le, Subring.mk, Neg.neg, Subring.closure, Exists.elim, NegMemClass.neg_mem, MulMemClass.mul_mem
-- Module: Mathlib.Algebra.Ring.Subring.Basic
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 z : R
hz : z ∈ s
x_1 : R
h : x_1 ∈ s
⊢ p x_1 z ⋯ ⋯

Tactic:
mem_mem _ _ h hz
-- Module: Mathlib.Algebra.Ring.Subring.Basic
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 z : R
hz : z ∈ s
x_1 : R
h : x_1 ∈ s
⊢ p x_1 z ⋯ ⋯

Tactic:
exact mem_mem _ _ h hz
-- Module: Mathlib.Algebra.Ring.Subring.Basic
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 z : R
hz : z ∈ s
⊢ p 0 z ⋯ ⋯

Tactic:
zero_left _ _
-- Module: Mathlib.Algebra.Ring.Subring.Basic
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 z : R
hz : z ∈ s
⊢ p 0 z ⋯ ⋯

Tactic:
exact zero_left _ _
-- Module: Mathlib.Algebra.Ring.Subring.Basic
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 z : R
hz : z ∈ s
⊢ p 1 z ⋯ ⋯

Tactic:
one_left _ _
-- Module: Mathlib.Algebra.Ring.Subring.Basic
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 z : R
hz : z ∈ s
⊢ p 1 z ⋯ ⋯

Tactic:
exact one_left _ _
-- Module: Mathlib.Algebra.Ring.Subring.Basic
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 z : R
hz : z ∈ s
x_1 y_1 : R
hx : Membership.mem (γ := Subring R) (closure s) x_1
hy : Membership.mem (γ := Subring 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