awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
toNonUnitalNonAssocRing : NonUnitalNonAssocRing R
toOne : One R
one_mul : ∀ (a : R), 1 * a = a
mul_one : ∀ (a : R), a * 1 = a
toNatCast : NatCast R
natCast_zero : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
toIntCast : IntCast R
intCast_ofNat : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R)
intCast_negSucc : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R))
toNonUnitalNonAssocRing_1 : NonUnitalNonAssocRing R
toOne_1 : One R
one_mul_1 : ∀ (a : R), 1 * a = a
mul_one_1 : ∀ (a : R), a * 1 = a
toNatCast_1 : NatCast R
natCast_zero_1 : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ_1 : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
toIntCast_1 : IntCast R
intCast_ofNat_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R)
intCast_negSucc_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R))
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₁ : Eq.{u + 1} (α := NonUnitalNonAssocRing R) NonAssocRing.toNonUnitalNonAssocRing NonAssocRing.toNonUnitalNonAssocRing
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h₃ : Eq.{u + 1} (α := AddCommGroupWithOne R) toAddCommGroupWithOne toAddCommGroupWithOne
⊢ { toNonUnitalNonAssocRing := toNonUnitalNonAssocRing, toOne := toOne, one_mul := one_mul, mul_one := mul_one,
      toNatCast := toNatCast, natCast_zero := natCast_zero, natCast_succ := natCast_succ, toIntCast := toIntCast,
      intCast_ofNat := intCast_ofNat, intCast_negSucc := intCast_negSucc : NonAssocRing R } =
    { toNonUnitalNonAssocRing := toNonUnitalNonAssocRing_1, toOne := toOne_1, one_mul := one_mul_1,
      mul_one := mul_one_1, toNatCast := toNatCast_1, natCast_zero := natCast_zero_1, natCast_succ := natCast_succ_1,
      toIntCast := toIntCast_1, intCast_ofNat := intCast_ofNat_1, intCast_negSucc := intCast_negSucc_1 }
have mk : toOne = toOne_1
Tactic: congr
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
toNonUnitalNonAssocRing : NonUnitalNonAssocRing R
toOne : One R
one_mul : ∀ (a : R), 1 * a = a
mul_one : ∀ (a : R), a * 1 = a
toNatCast : NatCast R
natCast_zero : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
toIntCast : IntCast R
intCast_ofNat : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R)
intCast_negSucc : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R))
toNonUnitalNonAssocRing_1 : NonUnitalNonAssocRing R
toOne_1 : One R
one_mul_1 : ∀ (a : R), 1 * a = a
mul_one_1 : ∀ (a : R), a * 1 = a
toNatCast_1 : NatCast R
natCast_zero_1 : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ_1 : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
toIntCast_1 : IntCast R
intCast_ofNat_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R)
intCast_negSucc_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R))
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₁ : Eq.{u + 1} (α := NonUnitalNonAssocRing R) NonAssocRing.toNonUnitalNonAssocRing NonAssocRing.toNonUnitalNonAssocRing
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h₃ : Eq.{u + 1} (α := AddCommGroupWithOne R) toAddCommGroupWithOne toAddCommGroupWithOne
mk : toOne = toOne_1
⊢ { toNonUnitalNonAssocRing := toNonUnitalNonAssocRing, toOne := toOne, one_mul := one_mul, mul_one := mul_one,
      toNatCast := toNatCast, natCast_zero := natCast_zero, natCast_succ := natCast_succ, toIntCast := toIntCast,
      intCast_ofNat := intCast_ofNat, intCast_negSucc := intCast_negSucc : NonAssocRing R } =
    { toNonUnitalNonAssocRing := toNonUnitalNonAssocRing_1, toOne := toOne_1, one_mul := one_mul_1,
      mul_one := mul_one_1, toNatCast := toNatCast_1, natCast_zero := natCast_zero_1, natCast_succ := natCast_succ_1,
      toIntCast := toIntCast_1, intCast_ofNat := intCast_ofNat_1, intCast_negSucc := intCast_negSucc_1 }
have mk₁ : toNatCast = toNatCast_1
Tactic: congr
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
toNonUnitalNonAssocRing : NonUnitalNonAssocRing R
toOne : One R
one_mul : ∀ (a : R), 1 * a = a
mul_one : ∀ (a : R), a * 1 = a
toNatCast : NatCast R
natCast_zero : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
toIntCast : IntCast R
intCast_ofNat : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R)
intCast_negSucc : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R))
toNonUnitalNonAssocRing_1 : NonUnitalNonAssocRing R
toOne_1 : One R
one_mul_1 : ∀ (a : R), 1 * a = a
mul_one_1 : ∀ (a : R), a * 1 = a
toNatCast_1 : NatCast R
natCast_zero_1 : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ_1 : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
toIntCast_1 : IntCast R
intCast_ofNat_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R)
intCast_negSucc_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R))
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₁ : Eq.{u + 1} (α := NonUnitalNonAssocRing R) NonAssocRing.toNonUnitalNonAssocRing NonAssocRing.toNonUnitalNonAssocRing
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h₃ : Eq.{u + 1} (α := AddCommGroupWithOne R) toAddCommGroupWithOne toAddCommGroupWithOne
mk : toOne = toOne_1
mk₁ : toNatCast = toNatCast_1
⊢ { toNonUnitalNonAssocRing := toNonUnitalNonAssocRing, toOne := toOne, one_mul := one_mul, mul_one := mul_one,
      toNatCast := toNatCast, natCast_zero := natCast_zero, natCast_succ := natCast_succ, toIntCast := toIntCast,
      intCast_ofNat := intCast_ofNat, intCast_negSucc := intCast_negSucc : NonAssocRing R } =
    { toNonUnitalNonAssocRing := toNonUnitalNonAssocRing_1, toOne := toOne_1, one_mul := one_mul_1,
      mul_one := mul_one_1, toNatCast := toNatCast_1, natCast_zero := natCast_zero_1, natCast_succ := natCast_succ_1,
      toIntCast := toIntCast_1, intCast_ofNat := intCast_ofNat_1, intCast_negSucc := intCast_negSucc_1 }
have mk₂ : toIntCast = toIntCast_1
Tactic: congr
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : NonAssocRing R
h : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
⊢ a₁ = a₂
have h_add : ∀ (x y : R), x + y = x + y
Tactic: ext x y
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : NonAssocRing R
h : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h_add : ∀ (x y : R), x + y = x + y
⊢ a₁ = a₂
have h_mul : ∀ (x y : R), x * y = x * y
Tactic: ext x y
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : NonAssocRing R
h : Eq.{u + 1} (α := NonUnitalNonAssocRing R) toNonUnitalNonAssocRing toNonUnitalNonAssocRing
⊢ a₁ = a₂
have h_add : ∀ (x x_1 : R), x + x_1 = x + x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : NonAssocRing R
h : Eq.{u + 1} (α := NonUnitalNonAssocRing R) toNonUnitalNonAssocRing toNonUnitalNonAssocRing
h_add : ∀ (x x_1 : R), x + x_1 = x + x_1
⊢ a₁ = a₂
have h_mul : ∀ (x x_1 : R), x * x_1 = x * x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : Semiring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
⊢ inst₁ = inst₂
have h₀ : Eq.{u + 1} (α := AddCommMonoid R) toAddCommMonoid toAddCommMonoid
Tactic: have h₀ : inst₁.toAddCommMonoid = inst₂.toAddCommMonoid := by
    ext : 1 <;> assumption
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : Semiring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₀ : Eq.{u + 1} (α := AddCommMonoid R) toAddCommMonoid toAddCommMonoid
⊢ inst₁ = inst₂
have h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
Tactic: have h₁ : inst₁.toNonUnitalSemiring = inst₂.toNonUnitalSemiring := by
    ext : 1 <;> assumption
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : Semiring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₀ : Eq.{u + 1} (α := AddCommMonoid R) toAddCommMonoid toAddCommMonoid
h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
⊢ inst₁ = inst₂
have h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
Tactic: have h₂ : inst₁.toNonAssocSemiring = inst₂.toNonAssocSemiring := by
    ext : 1 <;> assumption
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : Semiring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₀ : Eq.{u + 1} (α := AddCommMonoid R) toAddCommMonoid toAddCommMonoid
h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
⊢ inst₁ = inst₂
have h₃ : Eq.{u + 1} (α := Monoid R) MonoidWithZero.toMonoid MonoidWithZero.toMonoid
Tactic: have h₃ : (inst₁.toMonoidWithZero).toMonoid = (inst₂.toMonoidWithZero).toMonoid := by
    ext : 1; exact h_mul
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : Semiring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₀ : Eq.{u + 1} (α := AddCommMonoid R) toAddCommMonoid toAddCommMonoid
h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h₃ : Eq.{u + 1} (α := Monoid R) MonoidWithZero.toMonoid MonoidWithZero.toMonoid
⊢ inst₁ = inst₂
have mk : ∀ [toAddCommMonoid : AddCommMonoid R] [toMonoid : Monoid R] (zero_mul : ∀ (a : R), 0 * a = 0)
  (mul_zero : ∀ (a : R), a * 0 = 0) (left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c)
  (right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c) [toNatCast : NatCast R]
  (natCast_zero : Eq.{u + 1} (α := R) (↑0) 0) (natCast_succ : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1)
  (h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd) (h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul)
  (h₀ : Eq.{u + 1} (α := AddCommMonoid R) Semiring.toAddCommMonoid Semiring.toAddCommMonoid)
  (h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring)
  (h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring)
  (h₃ : Eq.{u + 1} (α := Monoid R) MonoidWithZero.toMonoid MonoidWithZero.toMonoid),
  { toAddCommMonoid := toAddCommMonoid, toMonoid := toMonoid, zero_mul := zero_mul, mul_zero := mul_zero,
      left_distrib := left_distrib, right_distrib := right_distrib, toNatCast := toNatCast,
      natCast_zero := natCast_zero, natCast_succ := natCast_succ : Semiring R } =
    inst₂
Tactic: cases inst₁
Cleared hypotheses: inst₁, h_add, h_mul, h₀, h₁, h₂, h₃
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₂ : Semiring R
toAddCommMonoid : AddCommMonoid R
toMonoid : Monoid R
zero_mul : ∀ (a : R), 0 * a = 0
mul_zero : ∀ (a : R), a * 0 = 0
left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c
right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c
toNatCast : NatCast R
natCast_zero : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₀ : Eq.{u + 1} (α := AddCommMonoid R) Semiring.toAddCommMonoid Semiring.toAddCommMonoid
h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h₃ : Eq.{u + 1} (α := Monoid R) MonoidWithZero.toMonoid MonoidWithZero.toMonoid
⊢ { toAddCommMonoid := toAddCommMonoid, toMonoid := toMonoid, zero_mul := zero_mul, mul_zero := mul_zero,
      left_distrib := left_distrib, right_distrib := right_distrib, toNatCast := toNatCast,
      natCast_zero := natCast_zero, natCast_succ := natCast_succ : Semiring R } =
    inst₂
have mk : ∀ [toAddCommMonoid_1 : AddCommMonoid R] [toMonoid_1 : Monoid R] (zero_mul_1 : ∀ (a : R), 0 * a = 0)
  (mul_zero_1 : ∀ (a : R), a * 0 = 0) (left_distrib_1 : ∀ (a b c : R), a * (b + c) = a * b + a * c)
  (right_distrib_1 : ∀ (a b c : R), (a + b) * c = a * c + b * c) [toNatCast_1 : NatCast R]
  (natCast_zero_1 : Eq.{u + 1} (α := R) (↑0) 0) (natCast_succ_1 : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1)
  (h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd) (h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul)
  (h₀ : Eq.{u + 1} (α := AddCommMonoid R) Semiring.toAddCommMonoid Semiring.toAddCommMonoid)
  (h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring)
  (h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring)
  (h₃ : Eq.{u + 1} (α := Monoid R) MonoidWithZero.toMonoid MonoidWithZero.toMonoid),
  { toAddCommMonoid := toAddCommMonoid, toMonoid := toMonoid, zero_mul := zero_mul, mul_zero := mul_zero,
      left_distrib := left_distrib, right_distrib := right_distrib, toNatCast := toNatCast,
      natCast_zero := natCast_zero, natCast_succ := natCast_succ : Semiring R } =
    { toAddCommMonoid := toAddCommMonoid_1, toMonoid := toMonoid_1, zero_mul := zero_mul_1, mul_zero := mul_zero_1,
      left_distrib := left_distrib_1, right_distrib := right_distrib_1, toNatCast := toNatCast_1,
      natCast_zero := natCast_zero_1, natCast_succ := natCast_succ_1 }
Tactic: cases inst₂
Cleared hypotheses: inst₂, h_add, h_mul, h₀, h₁, h₂, h₃
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
toAddCommMonoid : AddCommMonoid R
toMonoid : Monoid R
zero_mul : ∀ (a : R), 0 * a = 0
mul_zero : ∀ (a : R), a * 0 = 0
left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c
right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c
toNatCast : NatCast R
natCast_zero : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
toAddCommMonoid_1 : AddCommMonoid R
toMonoid_1 : Monoid R
zero_mul_1 : ∀ (a : R), 0 * a = 0
mul_zero_1 : ∀ (a : R), a * 0 = 0
left_distrib_1 : ∀ (a b c : R), a * (b + c) = a * b + a * c
right_distrib_1 : ∀ (a b c : R), (a + b) * c = a * c + b * c
toNatCast_1 : NatCast R
natCast_zero_1 : Eq.{u + 1} (α := R) (↑0) 0
natCast_succ_1 : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₀ : Eq.{u + 1} (α := AddCommMonoid R) Semiring.toAddCommMonoid Semiring.toAddCommMonoid
h₁ : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h₃ : Eq.{u + 1} (α := Monoid R) MonoidWithZero.toMonoid MonoidWithZero.toMonoid
⊢ { toAddCommMonoid := toAddCommMonoid, toMonoid := toMonoid, zero_mul := zero_mul, mul_zero := mul_zero,
      left_distrib := left_distrib, right_distrib := right_distrib, toNatCast := toNatCast,
      natCast_zero := natCast_zero, natCast_succ := natCast_succ : Semiring R } =
    { toAddCommMonoid := toAddCommMonoid_1, toMonoid := toMonoid_1, zero_mul := zero_mul_1, mul_zero := mul_zero_1,
      left_distrib := left_distrib_1, right_distrib := right_distrib_1, toNatCast := toNatCast_1,
      natCast_zero := natCast_zero_1, natCast_succ := natCast_succ_1 }
have mk : toNatCast = toNatCast_1
Tactic: congr
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : Semiring R
h : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
⊢ a₁ = a₂
have h_add : ∀ (x y : R), x + y = x + y
Tactic: ext x y
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : Semiring R
h : Eq.{u + 1} (α := NonUnitalSemiring R) toNonUnitalSemiring toNonUnitalSemiring
h_add : ∀ (x y : R), x + y = x + y
⊢ a₁ = a₂
have h_mul : ∀ (x y : R), x * y = x * y
Tactic: ext x y
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : Semiring R
h : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
⊢ a₁ = a₂
have h_add : ∀ (x y : R), x + y = x + y
Tactic: ext x y
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
a₁ a₂ : Semiring R
h : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h_add : ∀ (x y : R), x + y = x + y
⊢ a₁ = a₂
have h_mul : ∀ (x y : R), x * y = x * y
Tactic: ext x y
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : Ring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
⊢ inst₁ = inst₂
have h₁ : Eq.{u + 1} (α := Semiring R) toSemiring toSemiring
Tactic: have h₁ : inst₁.toSemiring = inst₂.toSemiring := by
    ext : 1 <;> assumption
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : Ring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₁ : Eq.{u + 1} (α := Semiring R) toSemiring toSemiring
⊢ inst₁ = inst₂
have h₂ : Eq.{u + 1} (α := NonAssocRing R) toNonAssocRing toNonAssocRing
Tactic: have h₂ : inst₁.toNonAssocRing = inst₂.toNonAssocRing := by
    ext : 1 <;> assumption
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : Ring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₁ : Eq.{u + 1} (α := Semiring R) toSemiring toSemiring
h₂ : Eq.{u + 1} (α := NonAssocRing R) toNonAssocRing toNonAssocRing
⊢ inst₁ = inst₂
have h₃ : Eq.{u + 1} (α := SubNegMonoid R) AddGroup.toSubNegMonoid AddGroup.toSubNegMonoid
Tactic: have h₃ : (inst₁.toAddCommGroup).toAddGroup.toSubNegMonoid
            = (inst₂.toAddCommGroup).toAddGroup.toSubNegMonoid :=
    congrArg (@AddGroup.toSubNegMonoid R) <| by ext : 1; exact h_add
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : Ring R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₁ : Eq.{u + 1} (α := Semiring R) toSemiring toSemiring
h₂ : Eq.{u + 1} (α := NonAssocRing R) toNonAssocRing toNonAssocRing
h₃ : Eq.{u + 1} (α := SubNegMonoid R) AddGroup.toSubNegMonoid AddGroup.toSubNegMonoid
⊢ inst₁ = inst₂
have mk : ∀ [toSemiring : Semiring R] [toNeg : Neg R] [toSub : Sub R] (sub_eq_add_neg : ∀ (a b : R), a - b = a + -b)
  (zsmul : ℤ → R → R) (zsmul_zero' : ∀ (a : R), zsmul 0 a = 0)
  (zsmul_succ' : ∀ (n : ℕ) (a : R), zsmul (↑n.succ : ℤ) a = zsmul (↑n : ℤ) a + a)
  (zsmul_neg' : ∀ (n : ℕ) (a : R), zsmul (Int.negSucc n) a = -zsmul (↑n.succ : ℤ) a)
  (neg_add_cancel : ∀ (a : R), -a + a = 0) [toIntCast : IntCast R]
  (intCast_ofNat : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R))
  (intCast_negSucc : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R)))
  (h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd) (h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul)
  (h₁ : Eq.{u + 1} (α := Semiring R) Ring.toSemiring Ring.toSemiring)
  (h₂ : Eq.{u + 1} (α := NonAssocRing R) toNonAssocRing toNonAssocRing)
  (h₃ : Eq.{u + 1} (α := SubNegMonoid R) AddGroup.toSubNegMonoid AddGroup.toSubNegMonoid),
  { toSemiring := toSemiring, toNeg := toNeg, toSub := toSub, sub_eq_add_neg := sub_eq_add_neg, zsmul := zsmul,
      zsmul_zero' := zsmul_zero', zsmul_succ' := zsmul_succ', zsmul_neg' := zsmul_neg',
      neg_add_cancel := neg_add_cancel, toIntCast := toIntCast, intCast_ofNat := intCast_ofNat,
      intCast_negSucc := intCast_negSucc : Ring R } =
    inst₂
Tactic: cases inst₁
Cleared hypotheses: inst₁, h_add, h_mul, h₁, h₂, h₃
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₂ : Ring R
toSemiring : Semiring R
toNeg : Neg R
toSub : Sub R
sub_eq_add_neg : ∀ (a b : R), a - b = a + -b
zsmul : ℤ → R → R
zsmul_zero' : ∀ (a : R), zsmul 0 a = 0
zsmul_succ' : ∀ (n : ℕ) (a : R), zsmul (↑n.succ : ℤ) a = zsmul (↑n : ℤ) a + a
zsmul_neg' : ∀ (n : ℕ) (a : R), zsmul (Int.negSucc n) a = -zsmul (↑n.succ : ℤ) a
neg_add_cancel : ∀ (a : R), -a + a = 0
toIntCast : IntCast R
intCast_ofNat : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R)
intCast_negSucc : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R))
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul
h₁ : Eq.{u + 1} (α := Semiring R) Ring.toSemiring Ring.toSemiring
h₂ : Eq.{u + 1} (α := NonAssocRing R) toNonAssocRing toNonAssocRing
h₃ : Eq.{u + 1} (α := SubNegMonoid R) AddGroup.toSubNegMonoid AddGroup.toSubNegMonoid
⊢ { toSemiring := toSemiring, toNeg := toNeg, toSub := toSub, sub_eq_add_neg := sub_eq_add_neg, zsmul := zsmul,
      zsmul_zero' := zsmul_zero', zsmul_succ' := zsmul_succ', zsmul_neg' := zsmul_neg',
      neg_add_cancel := neg_add_cancel, toIntCast := toIntCast, intCast_ofNat := intCast_ofNat,
      intCast_negSucc := intCast_negSucc : Ring R } =
    inst₂
have mk : ∀ [toSemiring_1 : Semiring R] [toNeg_1 : Neg R] [toSub_1 : Sub R] (sub_eq_add_neg_1 : ∀ (a b : R), a - b = a + -b)
  (zsmul_1 : ℤ → R → R) (zsmul_zero'_1 : ∀ (a : R), zsmul_1 0 a = 0)
  (zsmul_succ'_1 : ∀ (n : ℕ) (a : R), zsmul_1 (↑n.succ : ℤ) a = zsmul_1 (↑n : ℤ) a + a)
  (zsmul_neg'_1 : ∀ (n : ℕ) (a : R), zsmul_1 (Int.negSucc n) a = -zsmul_1 (↑n.succ : ℤ) a)
  (neg_add_cancel_1 : ∀ (a : R), -a + a = 0) [toIntCast_1 : IntCast R]
  (intCast_ofNat_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (↑n : ℤ)) (↑n : R))
  (intCast_negSucc_1 : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (-(↑(n + 1) : R)))
  (h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd) (h_mul : Eq (α := R → R → R) HMul.hMul HMul.hMul)
  (h₁ : Eq.{u + 1} (α := Semiring R) Ring.toSemiring Ring.toSemiring)
  (h₂ : Eq.{u + 1} (α := NonAssocRing R) toNonAssocRing toNonAssocRing)
  (h₃ : Eq.{u + 1} (α := SubNegMonoid R) AddGroup.toSubNegMonoid AddGroup.toSubNegMonoid),
  { toSemiring := toSemiring, toNeg := toNeg, toSub := toSub, sub_eq_add_neg := sub_eq_add_neg, zsmul := zsmul,
      zsmul_zero' := zsmul_zero', zsmul_succ' := zsmul_succ', zsmul_neg' := zsmul_neg',
      neg_add_cancel := neg_add_cancel, toIntCast := toIntCast, intCast_ofNat := intCast_ofNat,
      intCast_negSucc := intCast_negSucc : Ring R } =
    { toSemiring := toSemiring_1, toNeg := toNeg_1, toSub := toSub_1, sub_eq_add_neg := sub_eq_add_neg_1,
      zsmul := zsmul_1, zsmul_zero' := zsmul_zero'_1, zsmul_succ' := zsmul_succ'_1, zsmul_neg' := zsmul_neg'_1,
      neg_add_cancel := neg_add_cancel_1, toIntCast := toIntCast_1, intCast_ofNat := intCast_ofNat_1,
      intCast_negSucc := intCast_negSucc_1 }
Tactic: cases inst₂
Cleared hypotheses: inst₂, h_add, h_mul, h₁, h₂, h₃