awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.MulOpposite (goal)
Goal:
R : Type u_2
inst : NonAssocRing R
s : Set R
⊢ Eq.{u_2 + 1} (α := Subring Rᵐᵒᵖ) (sInf {a : Subring Rᵐᵒᵖ | s ⊆ MulOpposite.op ⁻¹' (↑a : Set Rᵐᵒᵖ)})
    (sInf {S : Subring Rᵐᵒᵖ | MulOpposite.unop ⁻¹' s ⊆ (↑S : Set Rᵐᵒᵖ)})
have e_a : ∀ (a : Subring Rᵐᵒᵖ), s ⊆ MulOpposite.op ⁻¹' (↑a : Set Rᵐᵒᵖ) ↔ MulOpposite.unop ⁻¹' s ⊆ (↑a : Set Rᵐᵒᵖ)
Tactic: congr with a
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.MulOpposite (goal)
Goal:
R : Type u_2
inst : NonAssocRing R
s : Set Rᵐᵒᵖ
⊢ Eq (α := Subring R) (Subring.unop (closure s)) (closure (MulOpposite.op ⁻¹' s))
have rw : Eq (α := Subring Rᵐᵒᵖ) (closure s) (closure (MulOpposite.unop ⁻¹' MulOpposite.op ⁻¹' s))
Tactic: rw [← op_inj, op_unop, op_closure]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.MulOpposite (goal)
Goal:
R : Type u_2
inst : NonAssocRing R
s : Set Rᵐᵒᵖ
⊢ Eq (α := Subring Rᵐᵒᵖ) (closure s) (closure (MulOpposite.unop ⁻¹' MulOpposite.op ⁻¹' s))
have simp_rw : Eq (α := Subring Rᵐᵒᵖ) (closure s) (closure ((fun (x : Rᵐᵒᵖ) ↦ MulOpposite.op (MulOpposite.unop x)) ⁻¹' s))
Tactic: simp_rw [Set.preimage_preimage, MulOpposite.op_unop, Set.preimage_id']
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subring.MulOpposite (goal)
Goal:
R : Type u_2
inst : NonAssocRing R
s : Set Rᵐᵒᵖ
simp_rw : Eq (α := Subring Rᵐᵒᵖ) (closure s) (closure ((fun (x : Rᵐᵒᵖ) ↦ MulOpposite.op (MulOpposite.unop x)) ⁻¹' s))
⊢ Eq (α := Subring Rᵐᵒᵖ) (closure s) (closure (MulOpposite.unop ⁻¹' MulOpposite.op ⁻¹' s))
have simp_rw₁ : Eq (α := Subring Rᵐᵒᵖ) (closure s) (closure ((fun (x : Rᵐᵒᵖ) ↦ x) ⁻¹' s))
Tactic: simp_rw [Set.preimage_preimage, MulOpposite.op_unop, Set.preimage_id']
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
S : Type v
inst : NonAssocSemiring R
inst_1 : NonAssocSemiring S
f : R →+* S
⊢ f.rangeS = Subsemiring.map f ⊤
have h : ∀ (x : S), x ∈ f.rangeS ↔ x ∈ Subsemiring.map f ⊤
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
⊢ Eq (α := Subsemiring R) (centralizer s.centralizer.centralizer) (centralizer s)
have a : (↑(centralizer s.centralizer.centralizer) : Set R) = (↑(centralizer s) : Set R)
Tactic: apply SetLike.coe_injective
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
⊢ M.subsemiringClosure.toNonUnitalSubsemiring = NonUnitalSubsemiring.closure (↑M : Set R)
have refine_1 : (↑M : Set R) ⊆ (↑M.subsemiringClosure.toNonUnitalSubsemiring : Set R)
Tactic: refine Eq.symm (NonUnitalSubsemiring.closure_eq_of_le ?_ fun _ hx ↦ ?_)
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
refine_1 : (↑M : Set R) ⊆ (↑M.subsemiringClosure.toNonUnitalSubsemiring : Set R)
⊢ M.subsemiringClosure.toNonUnitalSubsemiring = NonUnitalSubsemiring.closure (↑M : Set R)
have refine_2 : ∀ (x : R) (hx : x ∈ M.subsemiringClosure.toNonUnitalSubsemiring),
  Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
Tactic: refine Eq.symm (NonUnitalSubsemiring.closure_eq_of_le ?_ fun _ hx ↦ ?_)
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (hypothesis)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
x : R
hx : x ∈ M.subsemiringClosure.toNonUnitalSubsemiring
⊢ Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
have hx₁ : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑M : Set R)) x
Tactic: simp only [Subsemiring.mem_toNonUnitalSubsemiring, subsemiringClosure_mem] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
x : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑M : Set R)) x
⊢ Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
have refine_2 : ∀ x ∈ (↑M : Set R), Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
Tactic: induction hx using AddSubmonoid.closure_induction
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
x : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑M : Set R)) x
refine_2 :
  ∀ x ∈ (↑M : Set R), Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
⊢ Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
have refine_2₁ : Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) 0
Tactic: induction hx using AddSubmonoid.closure_induction
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
x : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑M : Set R)) x
refine_2 :
  ∀ x ∈ (↑M : Set R), Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
refine_2₁ : Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) 0
⊢ Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x
have refine_2₂ : ∀ (x y : R),
  Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑M : Set R)) x →
    Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑M : Set R)) y →
      Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) x →
        Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) y →
          Membership.mem (γ := NonUnitalSubsemiring R) (NonUnitalSubsemiring.closure (↑M : Set R)) (x + y)
Tactic: induction hx using AddSubmonoid.closure_induction
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
⊢ M.subsemiringClosure = Subsemiring.closure (↑M : Set R)
have h : ∀ (x : R), x ∈ M.subsemiringClosure ↔ Membership.mem (γ := Subsemiring R) (Subsemiring.closure (↑M : Set R)) x
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
x : R
⊢ x ∈ M.subsemiringClosure ↔ Membership.mem (γ := Subsemiring R) (Subsemiring.closure (↑M : Set R)) x
have h : ∀ (hx : x ∈ M.subsemiringClosure), Membership.mem (γ := Subsemiring R) (Subsemiring.closure (↑M : Set R)) x
Tactic: refine
    ⟨fun hx => ?_, fun hx =>
      (Subsemiring.mem_closure.mp hx) M.subsemiringClosure fun s sM => ?_⟩
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
M : Submonoid R
x : R
h : ∀ (hx : x ∈ M.subsemiringClosure), Membership.mem (γ := Subsemiring R) (Subsemiring.closure (↑M : Set R)) x
⊢ x ∈ M.subsemiringClosure ↔ Membership.mem (γ := Subsemiring R) (Subsemiring.closure (↑M : Set R)) x
have h₁ : ∀ (hx : Membership.mem (γ := Subsemiring R) (Subsemiring.closure (↑M : Set R)) x) (s : R) (sM : s ∈ (↑M : Set R)),
  s ∈ (↑M.subsemiringClosure : Set R)
Tactic: refine
    ⟨fun hx => ?_, fun hx =>
      (Subsemiring.mem_closure.mp hx) M.subsemiringClosure fun s sM => ?_⟩
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
⊢ Eq (α := Subsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) (closure s)
have h : ∀ (x : R),
  Membership.mem (γ := Subsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x ↔
    Membership.mem (γ := Subsemiring R) (closure s) x
Tactic: ext x
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
x : R
⊢ Membership.mem (γ := Subsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x ↔
    Membership.mem (γ := Subsemiring R) (closure s) x
have h : ∀ (hx : Membership.mem (γ := Subsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x),
  Membership.mem (γ := Subsemiring R) (closure s) x
Tactic: refine ⟨fun hx => ?_, fun hx => closure_mono AddSubmonoid.subset_closure hx⟩
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
x : R
hx : Membership.mem (γ := Subsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x
H : Subsemiring R
J : Membership.mem (γ := Set (Subsemiring R)) {S : Subsemiring R | s ⊆ (↑S : Set R)} H
⊢ Membership.mem (γ := Set R)
    ((fun (h : Membership.mem (γ := Set (Subsemiring R)) {S : Subsemiring R | s ⊆ (↑S : Set R)} H) ↦ (↑H : Set R)) J) x
have h : ∀ (y : R) (hy : y ∈ (↑(Submonoid.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.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
x : R
hx : Membership.mem (γ := Subsemiring R) (closure (↑(AddSubmonoid.closure s) : Set R)) x
H : Subsemiring R
J : Membership.mem (γ := Set (Subsemiring R)) {S : Subsemiring R | s ⊆ (↑S : Set R)} H
y : R
hy : y ∈ (↑(Submonoid.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.toSubmonoid : Set R)
Tactic: refine (Submonoid.mem_closure.mp hy) H.toSubmonoid fun z hz => ?_
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x
z : R
hz : z ∈ s
⊢ p x z hx ⋯
have mem : ∀ (x : R) (h : x ∈ s), p x z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
⊢ p x z hx ⋯
have mem₁ : p 0 z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
⊢ p x z hx ⋯
have mem₂ : p 1 z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
mem₂ : p 1 z ⋯ ⋯
⊢ p x z hx ⋯
have mem₃ : ∀ (x y : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
  (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (h₁ : p x z hx sorry) (h₂ : p y z hy sorry), p (x * y) z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x
z : R
hz : z ∈ s
mem : ∀ (x : R) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : p 0 z ⋯ ⋯
mem₂ : p 1 z ⋯ ⋯
mem₃ :
  ∀ (x y : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x * y) z ⋯ ⋯
⊢ p x z hx ⋯
have mem₄ : ∀ (x y : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
  (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (h₁ : p x z hx sorry) (h₂ : p y z hy sorry), p (x + y) z sorry sorry
Tactic: induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring 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 := Subsemiring R) (closure s) x
hy : Membership.mem (γ := Subsemiring R) (closure s) y
⊢ p x y hx hy
have mem : ∀ (z : R) (hz : z ∈ s), p x z hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hy