awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
x : R
h : Membership.mem (γ := NonUnitalSubring R) (closure s) x
mem :
  ∀ (x : R) (hx : x ∈ s),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x
zero : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) 0
add :
  ∀ (x y : R),
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubring R) (closure s) y →
        ∀ (hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x)
          (hy : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) y),
          Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) (x + y)
neg :
  ∀ (x : R),
    Membership.mem (γ := NonUnitalSubring R) (closure s) x →
      ∀ (hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x),
        Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) (-x)
⊢ Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x
have mul : ∀ (x y : R) (_hx : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
  (_hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y)
  (hx : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x)
  (hy : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) y),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) (x * y)
Tactic: induction h using closure_induction with
    | mem _ hx => exact AddSubgroup.subset_closure (Subsemigroup.subset_closure hx)
    | zero => exact zero_mem _
    | add _ _ _ _ hx hy => exact add_mem hx hy
    | neg x _ 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.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
x x_1 : R
hx : x_1 ∈ (↑(Subsemigroup.closure s) : Set R)
⊢ Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
have mem : ∀ (x : R) (h : x ∈ s), Membership.mem (γ := NonUnitalSubring R) (closure s) x
Tactic: induction hx using Subsemigroup.closure_induction with
      | mem _ h => exact subset_closure h
      | mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
x x_1 : R
hx : x_1 ∈ (↑(Subsemigroup.closure s) : Set R)
mem : ∀ (x : R) (h : x ∈ s), Membership.mem (γ := NonUnitalSubring R) (closure s) x
⊢ Membership.mem (γ := NonUnitalSubring R) (closure s) x_1
have mem₁ : ∀ (x y : R),
  Membership.mem (γ := Subsemigroup R) (Subsemigroup.closure s) x →
    Membership.mem (γ := Subsemigroup R) (Subsemigroup.closure s) y →
      ∀ (h₁ : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
        (h₂ : Membership.mem (γ := NonUnitalSubring R) (closure s) y),
        Membership.mem (γ := NonUnitalSubring R) (closure s) (x * y)
Tactic: induction hx using Subsemigroup.closure_induction with
      | mem _ h => exact subset_closure h
      | mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
x : R
h : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x
⊢ Membership.mem (γ := NonUnitalSubring R) (closure s) x
have mem : ∀ (x : R) (hx : x ∈ (↑(Subsemigroup.closure s) : Set R)), Membership.mem (γ := NonUnitalSubring R) (closure s) x
Tactic: induction h using AddSubgroup.closure_induction with
    | mem _ hx => induction hx using Subsemigroup.closure_induction with
      | mem _ h => exact subset_closure h
      | mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
    | zero => exact zero_mem _
    | add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
    | neg _ _ h => exact neg_mem h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
x : R
h : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x
mem : ∀ (x : R) (hx : x ∈ (↑(Subsemigroup.closure s) : Set R)), Membership.mem (γ := NonUnitalSubring R) (closure s) x
⊢ Membership.mem (γ := NonUnitalSubring R) (closure s) x
have zero : Membership.mem (γ := NonUnitalSubring R) (closure s) 0
Tactic: induction h using AddSubgroup.closure_induction with
    | mem _ hx => induction hx using Subsemigroup.closure_induction with
      | mem _ h => exact subset_closure h
      | mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
    | zero => exact zero_mem _
    | add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
    | neg _ _ h => exact neg_mem h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
x : R
h : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x
mem : ∀ (x : R) (hx : x ∈ (↑(Subsemigroup.closure s) : Set R)), Membership.mem (γ := NonUnitalSubring R) (closure s) x
zero : Membership.mem (γ := NonUnitalSubring R) (closure s) 0
⊢ Membership.mem (γ := NonUnitalSubring R) (closure s) x
have add : ∀ (x y : R),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x →
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) y →
      ∀ (h₁ : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
        (h₂ : Membership.mem (γ := NonUnitalSubring R) (closure s) y),
        Membership.mem (γ := NonUnitalSubring R) (closure s) (x + y)
Tactic: induction h using AddSubgroup.closure_induction with
    | mem _ hx => induction hx using Subsemigroup.closure_induction with
      | mem _ h => exact subset_closure h
      | mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
    | zero => exact zero_mem _
    | add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
    | neg _ _ h => exact neg_mem h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
s : Set R
x : R
h : Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x
mem : ∀ (x : R) (hx : x ∈ (↑(Subsemigroup.closure s) : Set R)), Membership.mem (γ := NonUnitalSubring R) (closure s) x
zero : Membership.mem (γ := NonUnitalSubring R) (closure s) 0
add :
  ∀ (x y : R),
    Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x →
      Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) y →
        ∀ (h₁ : Membership.mem (γ := NonUnitalSubring R) (closure s) x)
          (h₂ : Membership.mem (γ := NonUnitalSubring R) (closure s) y),
          Membership.mem (γ := NonUnitalSubring R) (closure s) (x + y)
⊢ Membership.mem (γ := NonUnitalSubring R) (closure s) x
have neg : ∀ (x : R),
  Membership.mem (γ := AddSubgroup R) (AddSubgroup.closure (↑(Subsemigroup.closure s) : Set R)) x →
    ∀ (h : Membership.mem (γ := NonUnitalSubring R) (closure s) x),
      Membership.mem (γ := NonUnitalSubring R) (closure s) (-x)
Tactic: induction h using AddSubgroup.closure_induction with
    | mem _ hx => induction hx using Subsemigroup.closure_induction with
      | mem _ h => exact subset_closure h
      | mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
    | zero => exact zero_mem _
    | add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
    | neg _ _ h => exact neg_mem h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
F : Type w
R : Type u
S : Type v
inst : NonUnitalNonAssocRing R
inst_1 : NonUnitalNonAssocRing S
inst_2 : FunLike F R S
inst_3 : NonUnitalRingHomClass F R S
ι : Sort u_1
inst_4 : Nonempty ι
f : F
hf : Function.Injective.{u + 1, v + 1} (α := R) (β := S) (⇑f : (a : R) → S)
s : ι → NonUnitalSubring R
⊢ Eq (α := NonUnitalSubring S) (map f (iInf s)) (⨅ (i : ι), map f (s i))
have a : (↑(map f (iInf s)) : Set S) = (↑(⨅ (i : ι), map f (s i)) : Set S)
Tactic: apply SetLike.coe_injective
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
ι : Sort u_1
hι : Nonempty ι
S : ι → NonUnitalSubring R
hS : Directed (fun (x1 x2 : NonUnitalSubring R) ↦ x1 ≤ x2) S
x : R
⊢ Membership.mem (γ := NonUnitalSubring R) (⨆ (i : ι), S i) x ↔ ∃ (i : ι), x ∈ S i
have refine : Membership.mem (γ := NonUnitalSubring R) (⨆ (i : ι), S i) x → ∃ (i : ι), x ∈ S i
Tactic: refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
ι : Sort u_1
hι : Nonempty ι
S : ι → NonUnitalSubring R
hS : Directed (fun (x1 x2 : NonUnitalSubring R) ↦ x1 ≤ x2) S
x : R
U : NonUnitalSubring R :=
  NonUnitalSubring.mk' (⋃ (i : ι), (↑(S i) : Set R)) (⨆ (i : ι), (S i).toSubsemigroup) (⨆ (i : ι), (S i).toAddSubgroup)
    ⋯ ⋯
⊢ Membership.mem (γ := NonUnitalSubring R) (⨆ (i : ι), S i) x → ∃ (i : ι), x ∈ S i
have suffices : ⨆ (i : ι), S i ≤ U
Tactic: suffices ⨆ i, S i ≤ U by simpa [U] using @this x
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (hypothesis)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
S : Set (NonUnitalSubring R)
Sne : S.Nonempty
hS : DirectedOn (fun (x1 x2 : NonUnitalSubring R) ↦ x1 ≤ x2) S
x : R
⊢ x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s
have this : Nonempty.{u + 1} (↑S : Type u)
Tactic: haveI : Nonempty S := Sne.to_subtype
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalNonAssocRing R
inst_1 : NonUnitalNonAssocRing S
f : R →ₙ+* S
⊢ (↑f.range : Set S) = (↑⊤ : Set S) ↔
    Eq.{v + 1} (α := Set S) (Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S)) Set.univ
have rw : Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S) = (↑⊤ : Set S) ↔
  Eq.{v + 1} (α := Set S) (Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S)) Set.univ
Tactic: rw [coe_range, coe_top]
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalNonAssocRing R
inst_1 : NonUnitalNonAssocRing S
f : R →ₙ+* S
rw :
  Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S) = (↑⊤ : Set S) ↔
    Eq.{v + 1} (α := Set S) (Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S)) Set.univ
⊢ (↑f.range : Set S) = (↑⊤ : Set S) ↔
    Eq.{v + 1} (α := Set S) (Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S)) Set.univ
have rw₁ : Eq.{v + 1} (α := Set S) (Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S)) Set.univ ↔
  Eq.{v + 1} (α := Set S) (Set.range.{v, u + 1} (ι := R) (⇑f : (a : R) → S)) Set.univ
Tactic: rw [coe_range, coe_top]
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalRing R
inst_1 : NonUnitalRing S
s t : NonUnitalSubring R
g : S → R
f : R →ₙ+* S
h : Function.LeftInverse g (⇑f : (a : R) → S)
x : ↥f.range
x' : R
hx' : (f : (a : R) → S) x' = (↑x : S)
⊢ (f : (a : R) → S) (g (↑x : S)) = (↑x : S)
have rw : Eq.{v + 1} (α := S) ((f : (a : R) → S) (g ((f : (a : R) → S) x')) : S) ((f : (a : R) → S) x' : S)
Tactic: rw [← hx', h x']
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalRing R
inst_1 : NonUnitalRing S
s t : NonUnitalSubring R
g : S → R
f : R →ₙ+* S
h : Function.LeftInverse g (⇑f : (a : R) → S)
x : ↥f.range
x' : R
hx' : (f : (a : R) → S) x' = (↑x : S)
rw : Eq.{v + 1} (α := S) ((f : (a : R) → S) (g ((f : (a : R) → S) x')) : S) ((f : (a : R) → S) x' : S)
⊢ (f : (a : R) → S) (g (↑x : S)) = (↑x : S)
have rw₁ : Eq.{v + 1} (α := S) ((f : (a : R) → S) x' : S) ((f : (a : R) → S) x' : S)
Tactic: rw [← hx', h x']
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Defs (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalNonAssocRing R
p q : NonUnitalSubring R
h :
  Eq.{u + 1} (α := Set R) ((fun (s : NonUnitalSubring R) ↦ s.carrier) p) ((fun (s : NonUnitalSubring R) ↦ s.carrier) q)
⊢ p = q
have mk : ∀ (toNonUnitalSubsemiring : NonUnitalSubsemiring R)
  (neg_mem' : ∀ {x : R}, x ∈ toNonUnitalSubsemiring.carrier → -x ∈ toNonUnitalSubsemiring.carrier)
  (h :
    Eq.{u + 1} (α := Set R)
      ((fun (s : NonUnitalSubring R) ↦ s.carrier)
        { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' })
      ((fun (s : NonUnitalSubring R) ↦ s.carrier) q)),
  { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' : NonUnitalSubring R } = q
Tactic: cases p
Cleared hypotheses: p, h
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Defs (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalNonAssocRing R
q : NonUnitalSubring R
toNonUnitalSubsemiring : NonUnitalSubsemiring R
neg_mem' : ∀ {x : R}, x ∈ toNonUnitalSubsemiring.carrier → -x ∈ toNonUnitalSubsemiring.carrier
h :
  Eq.{u + 1} (α := Set R)
    ((fun (s : NonUnitalSubring R) ↦ s.carrier)
      { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' })
    ((fun (s : NonUnitalSubring R) ↦ s.carrier) q)
⊢ { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' : NonUnitalSubring R } = q
have mk : ∀ (toNonUnitalSubsemiring_1 : NonUnitalSubsemiring R)
  (neg_mem'_1 : ∀ {x : R}, x ∈ toNonUnitalSubsemiring_1.carrier → -x ∈ toNonUnitalSubsemiring_1.carrier)
  (h :
    Eq.{u + 1} (α := Set R)
      ((fun (s : NonUnitalSubring R) ↦ s.carrier)
        { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' })
      ((fun (s : NonUnitalSubring R) ↦ s.carrier)
        { toNonUnitalSubsemiring := toNonUnitalSubsemiring_1, neg_mem' := @neg_mem'_1 })),
  { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' : NonUnitalSubring R } =
    { toNonUnitalSubsemiring := toNonUnitalSubsemiring_1, neg_mem' := @neg_mem'_1 }
Tactic: cases q
Cleared hypotheses: q, h
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Defs (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalNonAssocRing R
toNonUnitalSubsemiring : NonUnitalSubsemiring R
neg_mem' : ∀ {x : R}, x ∈ toNonUnitalSubsemiring.carrier → -x ∈ toNonUnitalSubsemiring.carrier
toNonUnitalSubsemiring_1 : NonUnitalSubsemiring R
neg_mem'_1 : ∀ {x : R}, x ∈ toNonUnitalSubsemiring_1.carrier → -x ∈ toNonUnitalSubsemiring_1.carrier
h :
  Eq.{u + 1} (α := Set R)
    ((fun (s : NonUnitalSubring R) ↦ s.carrier)
      { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' })
    ((fun (s : NonUnitalSubring R) ↦ s.carrier)
      { toNonUnitalSubsemiring := toNonUnitalSubsemiring_1, neg_mem' := @neg_mem'_1 })
⊢ { toNonUnitalSubsemiring := toNonUnitalSubsemiring, neg_mem' := @neg_mem' : NonUnitalSubring R } =
    { toNonUnitalSubsemiring := toNonUnitalSubsemiring_1, neg_mem' := @neg_mem'_1 }
have mk : toNonUnitalSubsemiring = toNonUnitalSubsemiring_1
Tactic: congr
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
S : Type v
inst : NonUnitalNonAssocSemiring R
inst_1 : NonUnitalNonAssocSemiring S
F : Type u_1
inst_2 : FunLike F R S
inst_3 : NonUnitalRingHomClass F R S
f : F
⊢ Eq (α := NonUnitalSubsemiring S) (srange f) (map f ⊤)
have h : ∀ (x : S),
  Membership.mem (γ := NonUnitalSubsemiring S) (srange f) x ↔ Membership.mem (γ := NonUnitalSubsemiring S) (map f ⊤) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
a : R
⊢ a ∈ Set.center R ↔
    Eq (α := R →+ R) (AddMonoidHom.mulLeft a) (AddMonoidHom.mulRight a) ∧
      Eq (α := R →+ R →+ R) (AddMonoidHom.compr₂ AddMonoidHom.mul (AddMonoidHom.mulLeft a))
          (AddMonoidHom.comp AddMonoidHom.mul (AddMonoidHom.mulLeft a)) ∧
        Eq (α := R →+ R →+ R) (AddMonoidHom.compr₂ AddMonoidHom.mul (AddMonoidHom.mulRight a))
          (AddMonoidHom.compl₂ AddMonoidHom.mul (AddMonoidHom.mulRight a))
have rw : ((∀ (a_1 : R), Commute a a_1) ∧ (∀ (b c : R), a * (b * c) = a * b * c) ∧ ∀ (a_1 b : R), a_1 * b * a = a_1 * (b * a)) ↔
  Eq (α := R →+ R) (AddMonoidHom.mulLeft a) (AddMonoidHom.mulRight a) ∧
    Eq (α := R →+ R →+ R) (AddMonoidHom.compr₂ AddMonoidHom.mul (AddMonoidHom.mulLeft a))
        (AddMonoidHom.comp AddMonoidHom.mul (AddMonoidHom.mulLeft a)) ∧
      Eq (α := R →+ R →+ R) (AddMonoidHom.compr₂ AddMonoidHom.mul (AddMonoidHom.mulRight a))
        (AddMonoidHom.compl₂ AddMonoidHom.mul (AddMonoidHom.mulRight a))
Tactic: rw [Set.mem_center_iff, isMulCentral_iff]
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u_1
inst : NonUnitalSemiring R
z : R
⊢ Membership.mem (γ := NonUnitalSubsemiring R) (center R) z ↔ ∀ (g : R), g * z = z * g
have rw : Membership.mem (γ := NonUnitalSubsemiring R) (center R) z ↔ z ∈ Set.center R
Tactic: rw [← Semigroup.mem_center_iff]
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
M : Subsemigroup R
⊢ M.nonUnitalSubsemiringClosure = NonUnitalSubsemiring.closure (↑M : Set R)
have h : ∀ (x : R),
  x ∈ M.nonUnitalSubsemiringClosure ↔
    Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
M : Subsemigroup R
x : R
⊢ x ∈ M.nonUnitalSubsemiringClosure ↔
    Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
have h : ∀ (hx : x ∈ M.nonUnitalSubsemiringClosure),
  Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
Tactic: refine ⟨fun hx => ?_,
    fun hx => (NonUnitalSubsemiring.mem_closure.mp hx) M.nonUnitalSubsemiringClosure fun s sM => ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
M : Subsemigroup R
x : R
h :
  ∀ (hx : x ∈ M.nonUnitalSubsemiringClosure),
    Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
⊢ x ∈ M.nonUnitalSubsemiringClosure ↔
    Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
have h₁ : ∀ (hx : Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x) (s : R)
  (sM : s ∈ (↑M : Set R)), s ∈ (↑M.nonUnitalSubsemiringClosure : Set R)
Tactic: refine ⟨fun hx => ?_,
    fun hx => (NonUnitalSubsemiring.mem_closure.mp hx) M.nonUnitalSubsemiringClosure fun s sM => ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
s : Set R
⊢ Eq (α := NonUnitalSubsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) (closure s)
have h : ∀ (x : R),
  Membership.mem (γ := NonUnitalSubsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x ↔
    Membership.mem (γ := NonUnitalSubsemiring R) (closure s) x
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
s : Set R
x : R
⊢ Membership.mem (γ := NonUnitalSubsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x ↔
    Membership.mem (γ := NonUnitalSubsemiring R) (closure s) x
have h : ∀ (hx : Membership.mem (γ := NonUnitalSubsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x),
  Membership.mem (γ := NonUnitalSubsemiring R) (closure s) x
Tactic: refine ⟨fun hx => ?_, fun hx => closure_mono AddSubmonoid.subset_closure hx⟩
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
s : Set R
x : R
hx : Membership.mem (γ := NonUnitalSubsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x
H : NonUnitalSubsemiring R
J : Membership.mem (γ := Set (NonUnitalSubsemiring R)) {S : NonUnitalSubsemiring R | s ⊆ (↑S : Set R)} H
⊢ Membership.mem (γ := Set R)
    ((fun (h : Membership.mem (γ := Set (NonUnitalSubsemiring R)) {S : NonUnitalSubsemiring R | s ⊆ (↑S : Set R)} H) ↦
        (↑H : Set R))
      J)
    x
have h : ∀ (y : R) (hy : y ∈ (↑(Subsemigroup.closure (↑(AddSubmonoid.closure s) : Set R)) : Set R)),
  y ∈ (↑H.toAddSubmonoid : Set R)
Tactic: refine (AddSubmonoid.mem_closure.mp (mem_closure_iff.mp hx)) H.toAddSubmonoid fun y hy => ?_
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
s : Set R
x : R
hx : Membership.mem (γ := NonUnitalSubsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x
H : NonUnitalSubsemiring R
J : Membership.mem (γ := Set (NonUnitalSubsemiring R)) {S : NonUnitalSubsemiring R | s ⊆ (↑S : Set R)} H
y : R
hy : y ∈ (↑(Subsemigroup.closure (↑(AddSubmonoid.closure s) : Set R)) : Set R)
⊢ y ∈ (↑H.toAddSubmonoid : Set R)
have h : ∀ (z : R) (hz : z ∈ (↑(AddSubmonoid.closure s) : Set R)), z ∈ (↑H.toSubsemigroup : Set R)
Tactic: refine (Subsemigroup.mem_closure.mp hy) H.toSubsemigroup fun z hz => ?_
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubsemiring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := NonUnitalSubsemiring R) (closure s) x →
      Membership.mem (γ := NonUnitalSubsemiring R) (closure s) y → Prop
mem_mem : ∀ (x : R) (hx : x ∈ s) (y : R) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubsemiring R) (closure s) x), p x 0 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := NonUnitalSubsemiring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubsemiring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubsemiring 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 := NonUnitalSubsemiring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubsemiring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubsemiring 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 := NonUnitalSubsemiring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubsemiring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubsemiring 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 := NonUnitalSubsemiring R) (closure s) x)
    (hy : Membership.mem (γ := NonUnitalSubsemiring R) (closure s) y)
    (hz : Membership.mem (γ := NonUnitalSubsemiring 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 := NonUnitalSubsemiring 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 _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx