awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Noetherian.OfPrime (goal)
Goal:
R : Type u_1
inst : CommRing R
H : ∀ (I : Ideal R), I.IsPrime → I.FG
C : Set (Ideal R)
hC₁ : C ⊆ {I : Ideal R | ¬I.FG}
hC₂ : IsChain (fun (x1 x2 : Ideal R) ↦ x1 ≤ x2) C
I : Ideal R
hI : I ∈ C
G : Finset R
hG : span (↑G : Set R) = sSup C
⊢ sSup C ∈ C
have obtain : ∀ (J : Ideal R) (J_mem_C : J ∈ C) (G_subset_J : (↑G : Set R) ⊆ (↑J : Set R)), sSup C ∈ C
Tactic: obtain ⟨J, J_mem_C, G_subset_J⟩ : ∃ J ∈ C, (G : Set R) ⊆ J := by
    refine hC₂.directedOn.exists_mem_subset_of_finset_subset_biUnion ⟨I, hI⟩ (fun _ hx ↦ ?_)
    simp only [Set.mem_iUnion, SetLike.mem_coe, exists_prop]
    exact (Submodule.mem_sSup_of_directed ⟨I, hI⟩ hC₂.directedOn).1 <| hG ▸ subset_span hx
-- Lean 4 proof step from Mathlib.RingTheory.Noetherian.OfPrime (goal)
Goal:
R : Type u_1
inst : CommRing R
H : ∀ (I : Ideal R), I.IsPrime → I.FG
C : Set (Ideal R)
hC₁ : C ⊆ {I : Ideal R | ¬I.FG}
hC₂ : IsChain (fun (x1 x2 : Ideal R) ↦ x1 ≤ x2) C
I : Ideal R
hI : I ∈ C
G : Finset R
hG : span (↑G : Set R) = sSup C
J : Ideal R
J_mem_C : J ∈ C
G_subset_J : (↑G : Set R) ⊆ (↑J : Set R)
⊢ sSup C ∈ C
have suffices : J = sSup C
Tactic: suffices J_eq_sSup : J = sSup C from J_eq_sSup ▸ J_mem_C
-- Lean 4 proof step from Mathlib.RingTheory.Noetherian.Orzech (hypothesis)
Goal:
R : Type u_1
M : Type u_2
N : Type w
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsNoetherian R M
i f : N →ₗ[R] M
hi : Injective.{w + 1, u_2 + 1} (α := N) (β := M) (⇑i : (a : N) → M)
hf : Surjective.{w + 1, u_2 + 1} (α := N) (β := M) (⇑f : (a : N) → M)
⊢ Injective.{w + 1, u_2 + 1} (α := N) (β := M) (⇑f : (a : N) → M)
have this : IsNoetherian R N
Tactic: haveI := isNoetherian_of_injective i hi
-- Lean 4 proof step from Mathlib.RingTheory.Noetherian.Orzech (goal)
Goal:
R : Type u_1
M : Type u_2
N : Type w
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsNoetherian R M
i f : N →ₗ[R] M
hi : Injective.{w + 1, u_2 + 1} (α := N) (β := M) (⇑i : (a : N) → M)
hf : Surjective.{w + 1, u_2 + 1} (α := N) (β := M) (⇑f : (a : N) → M)
this : IsNoetherian R N
⊢ Injective.{w + 1, u_2 + 1} (α := N) (β := M) (⇑f : (a : N) → M)
have obtain : ∀ (n : ℕ)
  (H :
    ∀ (m : ℕ),
      n ≤ m →
        Eq (α := Submodule R N)
          (({ toFun := fun (n : ℕ) ↦ f.iterateMapComap i n ⊥, monotone' := sorry : ℕ →o Submodule R N } :
                (a : ℕ) → Submodule R N)
              n :
            Submodule R N)
          (({ toFun := fun (n : ℕ) ↦ f.iterateMapComap i n ⊥, monotone' := sorry : ℕ →o Submodule R N } :
                (a : ℕ) → Submodule R N)
              m :
            Submodule R N)),
  Injective.{w + 1, u_2 + 1} (α := N) (β := M) (⇑f : (a : N) → M)
Tactic: obtain ⟨n, H⟩ := monotone_stabilizes_iff_noetherian.2 ‹_›
    ⟨_, monotone_nat_of_le_succ <| f.iterateMapComap_le_succ i ⊥ (by simp)⟩
-- Lean 4 proof step from Mathlib.RingTheory.Noetherian.Orzech (hypothesis)
Goal:
R : Type u_1
M : Type u_2
N : Type w
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsNoetherian R M
f : M × N →ₗ[R] M
i : Injective (α := M × N) (β := M) (⇑f : (a : M × N) → M)
x y : N
⊢ x = y
have h : Injective (α := M × N) (β := M) (⇑(LinearMap.fst R M N : M × N →ₗ[R] M) : (a : M × N) → M)
Tactic: have h := IsNoetherian.injective_of_surjective_of_injective f _ i LinearMap.fst_surjective
-- 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 = NonUnitalSubring.map f ⊤
have h : ∀ (x : S), x ∈ f.range ↔ Membership.mem (γ := NonUnitalSubring S) (NonUnitalSubring.map f ⊤) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
x : R
⊢ x ∈ (↑⊥ : Set R) ↔ x = 0
have rw : Membership.mem (γ := Set R) {0} x ↔ x = 0
Tactic: rw [coe_bot, Set.mem_singleton_iff]
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
Goal:
R : Type u
inst : NonUnitalNonAssocRing R
x : R
rw : Membership.mem (γ := Set R) {0} x ↔ x = 0
⊢ x ∈ (↑⊥ : Set R) ↔ x = 0
have rw₁ : x = 0 ↔ x = 0
Tactic: rw [coe_bot, Set.mem_singleton_iff]
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
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
⊢ 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
    | neg _ _ h => exact neg_left _ _ _ _ h
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
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 ⋯ ⋯
⊢ 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 _ _
    | 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.RingTheory.NonUnitalSubring.Basic (goal)
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 ⋯ ⋯
⊢ p x z hx ⋯
have 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 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 _ _
    | 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.RingTheory.NonUnitalSubring.Basic (goal)
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 ⋯ ⋯
⊢ p x z hx ⋯
have 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 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 _ _
    | 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.RingTheory.NonUnitalSubring.Basic (goal)
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 ⋯ ⋯
⊢ p x z hx ⋯
have mem₄ : ∀ (x : R) (hx : Membership.mem (γ := NonUnitalSubring 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 _ _
    | 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.RingTheory.NonUnitalSubring.Basic (goal)
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
hy : Membership.mem (γ := NonUnitalSubring 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 _ _
    | 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
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
  | neg _ _ h => exact neg_right _ _ _ _ h
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.RingTheory.NonUnitalSubring.Basic (goal)
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
hy : Membership.mem (γ := NonUnitalSubring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
⊢ p x y hx hy
have zero : p x 0 hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | 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
  | 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