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
    

Proof state:
case refine'_10
x y a✝ : ℝ
⊢ a✝ * 0 = 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'_10
x y a✝ : ℝ
⊢ a✝ * 0 = 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'_10
x y a✝ : ℝ
⊢ a✝ * 0 = 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'_10.a
x y a✝ : ℝ
⊢ (a✝ * 0).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'_11
x y : ℝ
⊢ ∀ (a b c : ℝ), a * b * c = a * (b * c)

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'_11
x y a✝ b✝ c✝ : ℝ
⊢ a✝ * b✝ * c✝ = a✝ * (b✝ * c✝)

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'_11
x y a✝ b✝ c✝ : ℝ
⊢ a✝ * b✝ * c✝ = a✝ * (b✝ * c✝)

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'_11
x y a✝ b✝ c✝ : ℝ
⊢ a✝ * b✝ * c✝ = a✝ * (b✝ * c✝)

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'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ (a✝ * b✝ * c✝).cauchy = (a✝ * (b✝ * c✝)).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
    intros
    first
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      

Proof state:
case refine'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy * c✝.cauchy = a✝.cauchy * (b✝.cauchy * c✝.cauchy)

Next tactic:
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
    | rfl
    | apply ext_cauchy
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | 

Proof state:
case refine'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy * c✝.cauchy = a✝.cauchy * (b✝.cauchy * c✝.cauchy)

Next tactic:
done
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
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | 

Proof state:
case refine'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy * c✝.cauchy = a✝.cauchy * (b✝.cauchy * c✝.cauchy)

Next tactic:
apply add_assoc
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
      simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul,
        cauchy_natCast, cauchy_intCast]
      first
        | done
        | apply add_assoc
        | 

Proof state:
case refine'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy * c✝.cauchy = a✝.cauchy * (b✝.cauchy * c✝.cauchy)

Next tactic:
apply add_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
    | 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
        | 

Proof state:
case refine'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy * c✝.cauchy = a✝.cauchy * (b✝.cauchy * c✝.cauchy)

Next tactic:
apply left_distrib
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
      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
        | 

Proof state:
case refine'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy * c✝.cauchy = a✝.cauchy * (b✝.cauchy * c✝.cauchy)

Next tactic:
apply right_distrib
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
      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
        | 

Proof state:
case refine'_11.a
x y a✝ b✝ c✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy * c✝.cauchy = a✝.cauchy * (b✝.cauchy * c✝.cauchy)

Next tactic:
apply mul_assoc
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'_12
x y : ℝ
⊢ ∀ (a : ℝ), 1 * a = 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'_12
x y a✝ : ℝ
⊢ 1 * a✝ = 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'_12
x y a✝ : ℝ
⊢ 1 * a✝ = 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
    intros
    first
    | rfl
    | 

Proof state:
case refine'_12
x y a✝ : ℝ
⊢ 1 * a✝ = a✝

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'_12.a
x y a✝ : ℝ
⊢ (1 * a✝).cauchy = a✝.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'_13
x y : ℝ
⊢ ∀ (a : ℝ), a * 1 = 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'_13
x y a✝ : ℝ
⊢ a✝ * 1 = 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'_13
x y a✝ : ℝ
⊢ a✝ * 1 = 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
    intros
    first
    | rfl
    | 

Proof state:
case refine'_13
x y a✝ : ℝ
⊢ a✝ * 1 = a✝

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'_13.a
x y a✝ : ℝ
⊢ (a✝ * 1).cauchy = a✝.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'_14
x y : ℝ
⊢ NatCast.natCast 0 = 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'_14
x y : ℝ
⊢ NatCast.natCast 0 = 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'_14
x y : ℝ
⊢ NatCast.natCast 0 = 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'_14
x y : ℝ
⊢ NatCast.natCast 0 = 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'_14.a
x y : ℝ
⊢ (NatCast.natCast 0).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'_15
x y : ℝ
⊢ ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast 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'_15
x y : ℝ
n✝ : ℕ
⊢ NatCast.natCast (n✝ + 1) = NatCast.natCast 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'_15
x y : ℝ
n✝ : ℕ
⊢ NatCast.natCast (n✝ + 1) = NatCast.natCast n✝ + 1

Next tactic:
rfl