awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : AddGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddMonoidWithOne R) toAddMonoidWithOne toAddMonoidWithOne
this : Eq.{u + 1} (α := NatCast R) AddMonoidWithOne.toNatCast AddMonoidWithOne.toNatCast
toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) AddMonoidWithOne.toAddMonoid AddMonoidWithOne.toAddMonoid
toNeg_eq : Eq.{u + 1} (α := Neg R) toNeg toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) toSub toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) AddGroupWithOne.zsmul AddGroupWithOne.zsmul
n : ℕ
⊢ Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (IntCast.intCast (Int.negSucc n))
have h : Eq.{u + 1} (α := R) (-(↑(n + 1) : R)) (-(↑(n + 1) : R))
Tactic: rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : AddGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddMonoidWithOne R) toAddMonoidWithOne toAddMonoidWithOne
this : Eq.{u + 1} (α := NatCast R) AddMonoidWithOne.toNatCast AddMonoidWithOne.toNatCast
toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) AddMonoidWithOne.toAddMonoid AddMonoidWithOne.toAddMonoid
toNeg_eq : Eq.{u + 1} (α := Neg R) toNeg toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) toSub toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) AddGroupWithOne.zsmul AddGroupWithOne.zsmul
n : ℤ
⊢ Eq.{u + 1} (α := R) (IntCast.intCast n) (IntCast.intCast n)
have h : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.ofNat n)) (IntCast.intCast (Int.ofNat n))
Tactic: cases n with
    | ofNat n => rewrite [Int.ofNat_eq_natCast, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
    | negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : AddGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddMonoidWithOne R) toAddMonoidWithOne toAddMonoidWithOne
this : Eq.{u + 1} (α := NatCast R) AddMonoidWithOne.toNatCast AddMonoidWithOne.toNatCast
toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) AddMonoidWithOne.toAddMonoid AddMonoidWithOne.toAddMonoid
toNeg_eq : Eq.{u + 1} (α := Neg R) toNeg toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) toSub toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) AddGroupWithOne.zsmul AddGroupWithOne.zsmul
n : ℤ
h : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.ofNat n)) (IntCast.intCast (Int.ofNat n))
⊢ Eq.{u + 1} (α := R) (IntCast.intCast n) (IntCast.intCast n)
have h₁ : ∀ (n : ℕ), Eq.{u + 1} (α := R) (IntCast.intCast (Int.negSucc n)) (IntCast.intCast (Int.negSucc n))
Tactic: cases n with
    | ofNat n => rewrite [Int.ofNat_eq_natCast, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
    | negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddMonoidWithOne R) toAddMonoidWithOne toAddMonoidWithOne
this : Eq.{u + 1} (α := NatCast R) AddMonoidWithOne.toNatCast AddMonoidWithOne.toNatCast
toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) AddMonoidWithOne.toAddMonoid AddMonoidWithOne.toAddMonoid
toNeg_eq : Eq.{u + 1} (α := Neg R) toNeg toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) toSub toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) AddGroupWithOne.zsmul AddGroupWithOne.zsmul
⊢ inst₁ = inst₂
have this₁ : Eq.{u + 1} (α := ℤ → R) IntCast.intCast IntCast.intCast
Tactic: have : inst₁.toIntCast.intCast = inst₂.toIntCast.intCast := by
    funext n; cases n with
    | ofNat n => rewrite [Int.ofNat_eq_natCast, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
    | negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : AddGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddMonoidWithOne R) toAddMonoidWithOne toAddMonoidWithOne
this_2 : Eq.{u + 1} (α := NatCast R) AddMonoidWithOne.toNatCast AddMonoidWithOne.toNatCast
toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) AddMonoidWithOne.toAddMonoid AddMonoidWithOne.toAddMonoid
toNeg_eq : Eq.{u + 1} (α := Neg R) toNeg toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) toSub toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) AddGroupWithOne.zsmul AddGroupWithOne.zsmul
this : Eq.{u + 1} (α := ℤ → R) IntCast.intCast IntCast.intCast
⊢ inst₁ = inst₂
have rcases : ∀ [toAddMonoidWithOne : AddMonoidWithOne 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) (intCast : ℤ → 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_one : Eq.{u + 1} (α := R) One.one One.one)
  (this : Eq.{u + 1} (α := AddMonoidWithOne R) AddGroupWithOne.toAddMonoidWithOne AddGroupWithOne.toAddMonoidWithOne)
  (this : Eq.{u + 1} (α := NatCast R) AddMonoidWithOne.toNatCast AddMonoidWithOne.toNatCast),
  Eq.{u + 1} (α := AddMonoid R) AddMonoidWithOne.toAddMonoid AddMonoidWithOne.toAddMonoid →
    Eq.{u + 1} (α := Neg R) AddGroupWithOne.toNeg AddGroupWithOne.toNeg →
      Eq.{u + 1} (α := Sub R) AddGroupWithOne.toSub AddGroupWithOne.toSub →
        Eq.{u + 1} (α := ℤ → R → R) AddGroupWithOne.zsmul AddGroupWithOne.zsmul →
          ∀ (this : Eq.{u + 1} (α := ℤ → R) IntCast.intCast IntCast.intCast),
            { intCast := intCast, toAddMonoidWithOne := toAddMonoidWithOne, 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,
                intCast_ofNat := intCast_ofNat, intCast_negSucc := intCast_negSucc : AddGroupWithOne R } =
              inst₂
Tactic: rcases inst₁ with @⟨⟨⟩⟩
Cleared hypotheses: inst₁, h_add, h_one, this_1, this_2, toAddMonoid_eq, toNeg_eq, toSub_eq, zsmul_eq, this
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
⊢ inst₁ = inst₂
have this : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
Tactic: have : inst₁.toAddCommGroup = inst₂.toAddCommGroup :=
    AddCommGroup.ext h_add
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
⊢ inst₁ = inst₂
have this₁ : Eq.{u + 1} (α := AddGroupWithOne R) toAddGroupWithOne toAddGroupWithOne
Tactic: have : inst₁.toAddGroupWithOne = inst₂.toAddGroupWithOne :=
    AddGroupWithOne.ext h_add h_one
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
this : Eq.{u + 1} (α := AddGroupWithOne R) toAddGroupWithOne toAddGroupWithOne
⊢ inst₁ = inst₂
have toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
Tactic: injection this with _ h_addMonoidWithOne
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
this : Eq.{u + 1} (α := AddGroupWithOne R) toAddGroupWithOne toAddGroupWithOne
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
⊢ inst₁ = inst₂
have h_addMonoidWithOne : Eq.{u + 1} (α := AddMonoidWithOne R)
  { toNatCast := inst₁.toNatCast, toAddMonoid := inst₁.toAddMonoid, toOne := inst₁.toOne, natCast_zero := sorry,
    natCast_succ := sorry }
  { toNatCast := inst₂.toNatCast, toAddMonoid := inst₂.toAddMonoid, toOne := inst₂.toOne, natCast_zero := sorry,
    natCast_succ := sorry }
Tactic: injection this with _ h_addMonoidWithOne
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
this : Eq.{u + 1} (α := AddGroupWithOne R) toAddGroupWithOne toAddGroupWithOne
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
h_addMonoidWithOne :
  Eq.{u + 1} (α := AddMonoidWithOne R)
    { toNatCast := inst₁.toNatCast, toAddMonoid := inst₁.toAddMonoid, toOne := inst₁.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
    { toNatCast := inst₂.toNatCast, toAddMonoid := inst₂.toAddMonoid, toOne := inst₂.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
⊢ inst₁ = inst₂
have toNeg_eq : Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg
Tactic: injection this with _ h_addMonoidWithOne
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
this : Eq.{u + 1} (α := AddGroupWithOne R) toAddGroupWithOne toAddGroupWithOne
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
h_addMonoidWithOne :
  Eq.{u + 1} (α := AddMonoidWithOne R)
    { toNatCast := inst₁.toNatCast, toAddMonoid := inst₁.toAddMonoid, toOne := inst₁.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
    { toNatCast := inst₂.toNatCast, toAddMonoid := inst₂.toAddMonoid, toOne := inst₂.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
toNeg_eq : Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg
⊢ inst₁ = inst₂
have toSub_eq : Eq.{u + 1} (α := Sub R) SubNegMonoid.toSub SubNegMonoid.toSub
Tactic: injection this with _ h_addMonoidWithOne
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this_1 : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
this : Eq.{u + 1} (α := AddGroupWithOne R) toAddGroupWithOne toAddGroupWithOne
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
h_addMonoidWithOne :
  Eq.{u + 1} (α := AddMonoidWithOne R)
    { toNatCast := inst₁.toNatCast, toAddMonoid := inst₁.toAddMonoid, toOne := inst₁.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
    { toNatCast := inst₂.toNatCast, toAddMonoid := inst₂.toAddMonoid, toOne := inst₂.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
toNeg_eq : Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) SubNegMonoid.toSub SubNegMonoid.toSub
⊢ inst₁ = inst₂
have zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) SubNegMonoid.zsmul SubNegMonoid.zsmul
Tactic: injection this with _ h_addMonoidWithOne
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
h_addMonoidWithOne :
  Eq.{u + 1} (α := AddMonoidWithOne R)
    { toNatCast := inst₁.toNatCast, toAddMonoid := inst₁.toAddMonoid, toOne := inst₁.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
    { toNatCast := inst₂.toNatCast, toAddMonoid := inst₂.toAddMonoid, toOne := inst₂.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
toNeg_eq : Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) SubNegMonoid.toSub SubNegMonoid.toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) SubNegMonoid.zsmul SubNegMonoid.zsmul
⊢ inst₁ = inst₂
have toNatCast_eq : Eq.{u + 1} (α := NatCast R) toNatCast toNatCast
Tactic: injection h_addMonoidWithOne
Cleared hypotheses: h_addMonoidWithOne
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
h_addMonoidWithOne :
  Eq.{u + 1} (α := AddMonoidWithOne R)
    { toNatCast := inst₁.toNatCast, toAddMonoid := inst₁.toAddMonoid, toOne := inst₁.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
    { toNatCast := inst₂.toNatCast, toAddMonoid := inst₂.toAddMonoid, toOne := inst₂.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
toNeg_eq : Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) SubNegMonoid.toSub SubNegMonoid.toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) SubNegMonoid.zsmul SubNegMonoid.zsmul
toNatCast_eq : Eq.{u + 1} (α := NatCast R) toNatCast toNatCast
⊢ inst₁ = inst₂
have toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) SubNegMonoid.toAddMonoid SubNegMonoid.toAddMonoid
Tactic: injection h_addMonoidWithOne
Cleared hypotheses: h_addMonoidWithOne
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
h_addMonoidWithOne :
  Eq.{u + 1} (α := AddMonoidWithOne R)
    { toNatCast := inst₁.toNatCast, toAddMonoid := inst₁.toAddMonoid, toOne := inst₁.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
    { toNatCast := inst₂.toNatCast, toAddMonoid := inst₂.toAddMonoid, toOne := inst₂.toOne, natCast_zero := ⋯,
      natCast_succ := ⋯ }
toNeg_eq : Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) SubNegMonoid.toSub SubNegMonoid.toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) SubNegMonoid.zsmul SubNegMonoid.zsmul
toNatCast_eq : Eq.{u + 1} (α := NatCast R) toNatCast toNatCast
toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) SubNegMonoid.toAddMonoid SubNegMonoid.toAddMonoid
⊢ inst₁ = inst₂
have toOne_eq : Eq.{u + 1} (α := One R) toOne toOne
Tactic: injection h_addMonoidWithOne
Cleared hypotheses: h_addMonoidWithOne
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : AddCommGroupWithOne R
h_add : Eq (α := R → R → R) HAdd.hAdd HAdd.hAdd
h_one : Eq.{u + 1} (α := R) One.one One.one
this : Eq.{u + 1} (α := AddCommGroup R) toAddCommGroup toAddCommGroup
toIntCast_eq : Eq.{u + 1} (α := IntCast R) toIntCast toIntCast
toNeg_eq : Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg
toSub_eq : Eq.{u + 1} (α := Sub R) SubNegMonoid.toSub SubNegMonoid.toSub
zsmul_eq : Eq.{u + 1} (α := ℤ → R → R) SubNegMonoid.zsmul SubNegMonoid.zsmul
toNatCast_eq : Eq.{u + 1} (α := NatCast R) toNatCast toNatCast
toAddMonoid_eq : Eq.{u + 1} (α := AddMonoid R) SubNegMonoid.toAddMonoid SubNegMonoid.toAddMonoid
toOne_eq : Eq.{u + 1} (α := One R) toOne toOne
⊢ inst₁ = inst₂
have mk : ∀ [toAddCommGroup : AddCommGroup R] [toIntCast : IntCast R] [toNatCast : NatCast R] [toOne : One R]
  (natCast_zero : Eq.{u + 1} (α := R) (↑0) 0) (natCast_succ : ∀ (n : ℕ), ↑(n + 1) = HAdd.hAdd (α := R) (↑n) 1)
  (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_one : Eq.{u + 1} (α := R) One.one One.one)
  (this : Eq.{u + 1} (α := AddCommGroup R) AddCommGroupWithOne.toAddCommGroup AddCommGroupWithOne.toAddCommGroup),
  Eq.{u + 1} (α := IntCast R) AddCommGroupWithOne.toIntCast AddCommGroupWithOne.toIntCast →
    Eq.{u + 1} (α := Neg R) SubNegMonoid.toNeg SubNegMonoid.toNeg →
      Eq.{u + 1} (α := Sub R) SubNegMonoid.toSub SubNegMonoid.toSub →
        Eq.{u + 1} (α := ℤ → R → R) SubNegMonoid.zsmul SubNegMonoid.zsmul →
          Eq.{u + 1} (α := NatCast R) AddCommGroupWithOne.toNatCast AddCommGroupWithOne.toNatCast →
            Eq.{u + 1} (α := AddMonoid R) SubNegMonoid.toAddMonoid SubNegMonoid.toAddMonoid →
              Eq.{u + 1} (α := One R) AddCommGroupWithOne.toOne AddCommGroupWithOne.toOne →
                { toAddCommGroup := toAddCommGroup, toIntCast := toIntCast, toNatCast := toNatCast, toOne := toOne,
                    natCast_zero := natCast_zero, natCast_succ := natCast_succ, intCast_ofNat := intCast_ofNat,
                    intCast_negSucc := intCast_negSucc : AddCommGroupWithOne R } =
                  inst₂
Tactic: cases inst₁
Cleared hypotheses: inst₁, h_add, h_one, this, toIntCast_eq, toNeg_eq, toSub_eq, zsmul_eq, toNatCast_eq, toAddMonoid_eq, toOne_eq
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : NonAssocRing 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 := NonUnitalNonAssocRing R) toNonUnitalNonAssocRing toNonUnitalNonAssocRing
Tactic: have h₁ : inst₁.toNonUnitalNonAssocRing = inst₂.toNonUnitalNonAssocRing := by
    ext : 1 <;> assumption
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (hypothesis)
Goal:
R : Type u
inst₁ inst₂ : NonAssocRing 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) toNonUnitalNonAssocRing toNonUnitalNonAssocRing
⊢ 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 : NonAssocRing 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) toNonUnitalNonAssocRing toNonUnitalNonAssocRing
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
⊢ inst₁ = inst₂
have h₃ : Eq.{u + 1} (α := AddCommGroupWithOne R) toAddCommGroupWithOne toAddCommGroupWithOne
Tactic: have h₃ : inst₁.toAddCommGroupWithOne = inst₂.toAddCommGroupWithOne :=
    AddCommGroupWithOne.ext h_add (congrArg (·.toOne.one) h₂)
-- Lean 4 proof step from Mathlib.Algebra.Ring.Ext (goal)
Goal:
R : Type u
inst₁ inst₂ : NonAssocRing 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) toNonUnitalNonAssocRing toNonUnitalNonAssocRing
h₂ : Eq.{u + 1} (α := NonAssocSemiring R) toNonAssocSemiring toNonAssocSemiring
h₃ : Eq.{u + 1} (α := AddCommGroupWithOne R) toAddCommGroupWithOne toAddCommGroupWithOne
⊢ inst₁ = inst₂
have mk : ∀ [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)))
  (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 } =
    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 : NonAssocRing R
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))
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 } =
    inst₂
have mk : ∀ [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 }
Tactic: cases inst₂
Cleared hypotheses: inst₂, h_add, h_mul, h₁, h₂, h₃