l3lab/ntp-mathlib huggingface dataset:
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | rfl
    | 

Proof state:
case refine'_15
x y : ℝ
n✝ : ℕ
⊢ NatCast.natCast (n✝ + 1) = NatCast.natCast n✝ + 1

Next tactic:
apply ext_cauchy
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | rfl
    | apply ext_cauchy
      

Proof state:
case refine'_15.a
x y : ℝ
n✝ : ℕ
⊢ (NatCast.natCast (n✝ + 1)).cauchy = (NatCast.natCast n✝ + 1).cauchy

Next tactic:
simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_16
x y : ℝ
⊢ ∀ (x : ℝ), npowRec 0 x = 1

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_16
x y x✝ : ℝ
⊢ npowRec 0 x✝ = 1

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_16
x y x✝ : ℝ
⊢ npowRec 0 x✝ = 1

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_17
x y : ℝ
⊢ ∀ (n : ℕ) (x : ℝ), npowRec (n + 1) x = x * npowRec n x

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_17
x y : ℝ
n✝ : ℕ
x✝ : ℝ
⊢ npowRec (n✝ + 1) x✝ = x✝ * npowRec n✝ x✝

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_17
x y : ℝ
n✝ : ℕ
x✝ : ℝ
⊢ npowRec (n✝ + 1) x✝ = x✝ * npowRec n✝ x✝

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_18
x y : ℝ
⊢ ∀ (a b : ℝ), a - b = a + -b

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_18
x y a✝ b✝ : ℝ
⊢ a✝ - b✝ = a✝ + -b✝

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_18
x y a✝ b✝ : ℝ
⊢ a✝ - b✝ = a✝ + -b✝

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_19
x y : ℝ
⊢ ∀ (a : ℝ), zsmulRec 0 a = 0

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_19
x y a✝ : ℝ
⊢ zsmulRec 0 a✝ = 0

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_19
x y a✝ : ℝ
⊢ zsmulRec 0 a✝ = 0

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_20
x y : ℝ
⊢ ∀ (n : ℕ) (a : ℝ), zsmulRec (Int.ofNat (Nat.succ n)) a = a + zsmulRec (Int.ofNat n) a

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_20
x y : ℝ
n✝ : ℕ
a✝ : ℝ
⊢ zsmulRec (Int.ofNat (Nat.succ n✝)) a✝ = a✝ + zsmulRec (Int.ofNat n✝) a✝

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_20
x y : ℝ
n✝ : ℕ
a✝ : ℝ
⊢ zsmulRec (Int.ofNat (Nat.succ n✝)) a✝ = a✝ + zsmulRec (Int.ofNat n✝) a✝

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_21
x y : ℝ
⊢ ∀ (n : ℕ) (a : ℝ), zsmulRec (Int.negSucc n) a = -zsmulRec (↑(Nat.succ n)) a

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_21
x y : ℝ
n✝ : ℕ
a✝ : ℝ
⊢ zsmulRec (Int.negSucc n✝) a✝ = -zsmulRec (↑(Nat.succ n✝)) a✝

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_21
x y : ℝ
n✝ : ℕ
a✝ : ℝ
⊢ zsmulRec (Int.negSucc n✝) a✝ = -zsmulRec (↑(Nat.succ n✝)) a✝

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_22
x y : ℝ
⊢ ∀ (a : ℝ), -a + a = 0

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_22
x y a✝ : ℝ
⊢ -a✝ + a✝ = 0

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_22
x y a✝ : ℝ
⊢ -a✝ + a✝ = 0

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | rfl
    | 

Proof state:
case refine'_22
x y a✝ : ℝ
⊢ -a✝ + a✝ = 0

Next tactic:
apply ext_cauchy
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | rfl
    | apply ext_cauchy
      

Proof state:
case refine'_22.a
x y a✝ : ℝ
⊢ (-a✝ + a✝).cauchy = 0.cauchy

Next tactic:
simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_23
x y : ℝ
⊢ ∀ (n : ℕ), IntCast.intCast ↑n = ↑n

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_23
x y : ℝ
n✝ : ℕ
⊢ IntCast.intCast ↑n✝ = ↑n✝

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_23
x y : ℝ
n✝ : ℕ
⊢ IntCast.intCast ↑n✝ = ↑n✝

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_24
x y : ℝ
⊢ ∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_24
x y : ℝ
n✝ : ℕ
⊢ IntCast.intCast (Int.negSucc n✝) = -↑(n✝ + 1)

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | 

Proof state:
case refine'_24
x y : ℝ
n✝ : ℕ
⊢ IntCast.intCast (Int.negSucc n✝) = -↑(n✝ + 1)

Next tactic:
rfl
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | rfl
    | 

Proof state:
case refine'_24
x y : ℝ
n✝ : ℕ
⊢ IntCast.intCast (Int.negSucc n✝) = -↑(n✝ + 1)

Next tactic:
apply ext_cauchy
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    first
    | rfl
    | apply ext_cauchy
      

Proof state:
case refine'_24.a
x y : ℝ
n✝ : ℕ
⊢ (IntCast.intCast (Int.negSucc n✝)).cauchy = (-↑(n✝ + 1)).cauchy

Next tactic:
simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    

Proof state:
case refine'_25
x y : ℝ
⊢ ∀ (a b : ℝ), a * b = b * a

Next tactic:
intros
instance commRing : CommRing ℝ := by
  refine' { natCast := fun n => ⟨n⟩
            intCast := fun z => ⟨z⟩
            zero := (0 : ℝ)
            one := (1 : ℝ)
            mul := (· * ·)
            add := (· + ·)
            neg := @Neg.neg ℝ _
            sub := @Sub.sub ℝ _
            npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩
            nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩
            zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩,
            .. }
  all_goals
    intros
    

Proof state:
case refine'_25
x y a✝ b✝ : ℝ
⊢ a✝ * b✝ = b✝ * a✝

Next tactic:
first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | apply add_comm
        | apply left_distrib
        | apply right_distrib
        | apply mul_assoc
        | apply mul_comm