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
    | 

Proof state:
case refine'_25
x y a✝ b✝ : ℝ
⊢ a✝ * b✝ = b✝ * 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'_25
x y a✝ b✝ : ℝ
⊢ a✝ * b✝ = b✝ * 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'_25.a
x y a✝ b✝ : ℝ
⊢ (a✝ * b✝).cauchy = (b✝ * 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
    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'_25.a
x y a✝ b✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy = b✝.cauchy * a✝.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'_25.a
x y a✝ b✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy = b✝.cauchy * a✝.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'_25.a
x y a✝ b✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy = b✝.cauchy * a✝.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'_25.a
x y a✝ b✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy = b✝.cauchy * a✝.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'_25.a
x y a✝ b✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy = b✝.cauchy * a✝.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'_25.a
x y a✝ b✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy = b✝.cauchy * a✝.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'_25.a
x y a✝ b✝ : ℝ
⊢ a✝.cauchy * b✝.cauchy = b✝.cauchy * a✝.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
    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
        | apply mul_assoc
        | 

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

Next tactic:
apply mul_comm
instance : Ring ℝ := by 

Proof state:
x y : ℝ
⊢ Ring ℝ

Next tactic:
infer_instance
instance : CommSemiring ℝ := by 

Proof state:
x y : ℝ
⊢ CommSemiring ℝ

Next tactic:
infer_instance
instance semiring : Semiring ℝ := by 

Proof state:
x y : ℝ
⊢ Semiring ℝ

Next tactic:
infer_instance
instance : CommMonoidWithZero ℝ := by 

Proof state:
x y : ℝ
⊢ CommMonoidWithZero ℝ

Next tactic:
infer_instance
instance : MonoidWithZero ℝ := by 

Proof state:
x y : ℝ
⊢ MonoidWithZero ℝ

Next tactic:
infer_instance
instance : AddCommGroup ℝ := by 

Proof state:
x y : ℝ
⊢ AddCommGroup ℝ

Next tactic:
infer_instance
instance : AddGroup ℝ := by 

Proof state:
x y : ℝ
⊢ AddGroup ℝ

Next tactic:
infer_instance
instance : AddCommMonoid ℝ := by 

Proof state:
x y : ℝ
⊢ AddCommMonoid ℝ

Next tactic:
infer_instance
instance : AddMonoid ℝ := by 

Proof state:
x y : ℝ
⊢ AddMonoid ℝ

Next tactic:
infer_instance
instance : AddLeftCancelSemigroup ℝ := by 

Proof state:
x y : ℝ
⊢ AddLeftCancelSemigroup ℝ

Next tactic:
infer_instance
instance : AddRightCancelSemigroup ℝ := by 

Proof state:
x y : ℝ
⊢ AddRightCancelSemigroup ℝ

Next tactic:
infer_instance
instance : AddCommSemigroup ℝ := by 

Proof state:
x y : ℝ
⊢ AddCommSemigroup ℝ

Next tactic:
infer_instance
instance : AddSemigroup ℝ := by 

Proof state:
x y : ℝ
⊢ AddSemigroup ℝ

Next tactic:
infer_instance
instance : CommMonoid ℝ := by 

Proof state:
x y : ℝ
⊢ CommMonoid ℝ

Next tactic:
infer_instance
instance : Monoid ℝ := by 

Proof state:
x y : ℝ
⊢ Monoid ℝ

Next tactic:
infer_instance
instance : CommSemigroup ℝ := by 

Proof state:
x y : ℝ
⊢ CommSemigroup ℝ

Next tactic:
infer_instance
instance : Semigroup ℝ := by 

Proof state:
x y : ℝ
⊢ Semigroup ℝ

Next tactic:
infer_instance
theorem lt_cauchy {f g} : (⟨⟦f⟧⟩ : ℝ) < ⟨⟦g⟧⟩ ↔ f < g :=
  show lt _ _ ↔ _ by 

Proof state:
x y : ℝ
f g : CauSeq ℚ abs
⊢ Real.lt { cauchy := ⟦f⟧ } { cauchy := ⟦g⟧ } ↔ f < g

Next tactic:
rw [lt_def]
theorem lt_cauchy {f g} : (⟨⟦f⟧⟩ : ℝ) < ⟨⟦g⟧⟩ ↔ f < g :=
  show lt _ _ ↔ _ by rw [lt_def]; 

Proof state:
x y : ℝ
f g : CauSeq ℚ abs
⊢ (match { cauchy := ⟦f⟧ }, { cauchy := ⟦g⟧ } with
    | { cauchy := x }, { cauchy := y } => Quotient.liftOn₂ x y (fun x x_1 => x < x_1) Real.definition.proof_1✝) ↔
    f < g

Next tactic:
rfl
theorem mk_zero : mk 0 = 0 := by 

Proof state:
x y : ℝ
⊢ mk 0 = 0

Next tactic:
rw [← ofCauchy_zero]
theorem mk_zero : mk 0 = 0 := by rw [← ofCauchy_zero]; 

Proof state:
x y : ℝ
⊢ mk 0 = { cauchy := 0 }

Next tactic:
rfl
theorem mk_one : mk 1 = 1 := by 

Proof state:
x y : ℝ
⊢ mk 1 = 1

Next tactic:
rw [← ofCauchy_one]
theorem mk_one : mk 1 = 1 := by rw [← ofCauchy_one]; 

Proof state:
x y : ℝ
⊢ mk 1 = { cauchy := 1 }

Next tactic:
rfl
theorem mk_add {f g : CauSeq ℚ abs} : mk (f + g) = mk f + mk g := by 

Proof state:
x y : ℝ
f g : CauSeq ℚ abs
⊢ mk (f + g) = mk f + mk g

Next tactic:
simp [mk, ← ofCauchy_add]
theorem mk_mul {f g : CauSeq ℚ abs} : mk (f * g) = mk f * mk g := by 

Proof state:
x y : ℝ
f g : CauSeq ℚ abs
⊢ mk (f * g) = mk f * mk g

Next tactic:
simp [mk, ← ofCauchy_mul]
theorem mk_neg {f : CauSeq ℚ abs} : mk (-f) = -mk f := by 

Proof state:
x y : ℝ
f : CauSeq ℚ abs
⊢ mk (-f) = -mk f

Next tactic:
simp [mk, ← ofCauchy_neg]
@[simp]
theorem mk_pos {f : CauSeq ℚ abs} : 0 < mk f ↔ Pos f := by
  

Proof state:
x y : ℝ
f : CauSeq ℚ abs
⊢ 0 < mk f ↔ Pos f

Next tactic:
rw [← mk_zero, mk_lt]
@[simp]
theorem mk_pos {f : CauSeq ℚ abs} : 0 < mk f ↔ Pos f := by
  rw [← mk_zero, mk_lt]
  

Proof state:
x y : ℝ
f : CauSeq ℚ abs
⊢ 0 < f ↔ Pos f

Next tactic:
exact iff_of_eq (congr_arg Pos (sub_zero f))
private theorem le_def' {x y : ℝ} : x ≤ y ↔ x < y ∨ x = y :=
  show le _ _ ↔ _ by 

Proof state:
x✝ y✝ x y : ℝ
⊢ Real.le x y ↔ x < y ∨ x = y

Next tactic:
rw [le_def]
@[simp]
theorem mk_le {f g : CauSeq ℚ abs} : mk f ≤ mk g ↔ f ≤ g := by 

Proof state:
x y : ℝ
f g : CauSeq ℚ abs
⊢ mk f ≤ mk g ↔ f ≤ g

Next tactic:
simp [le_def', mk_eq]
@[simp]
theorem mk_le {f g : CauSeq ℚ abs} : mk f ≤ mk g ↔ f ≤ g := by simp [le_def', mk_eq]; 

Proof state:
x y : ℝ
f g : CauSeq ℚ abs
⊢ f < g ∨ f ≈ g ↔ f ≤ g

Next tactic:
rfl
@[elab_as_elim]
protected theorem ind_mk {C : Real → Prop} (x : Real) (h : ∀ y, C (mk y)) : C x := by
  

Proof state:
x✝ y : ℝ
C : ℝ → Prop
x : ℝ
h : ∀ (y : CauSeq ℚ abs), C (mk y)
⊢ C x

Next tactic:
cases' x with x
@[elab_as_elim]
protected theorem ind_mk {C : Real → Prop} (x : Real) (h : ∀ y, C (mk y)) : C x := by
  cases' x with x
  

Proof state:
case ofCauchy
x✝ y : ℝ
C : ℝ → Prop
h : ∀ (y : CauSeq ℚ abs), C (mk y)
x : Cauchy abs
⊢ C { cauchy := x }

Next tactic:
induction' x using Quot.induction_on with x
@[elab_as_elim]
protected theorem ind_mk {C : Real → Prop} (x : Real) (h : ∀ y, C (mk y)) : C x := by
  cases' x with x
  induction' x using Quot.induction_on with x
  

Proof state:
case ofCauchy.h
x✝ y : ℝ
C : ℝ → Prop
h : ∀ (y : CauSeq ℚ abs), C (mk y)
x : CauSeq ℚ abs
⊢ C { cauchy := Quot.mk Setoid.r x }

Next tactic:
exact h x
theorem add_lt_add_iff_left {a b : ℝ} (c : ℝ) : c + a < c + b ↔ a < b := by
  

Proof state:
x y a b c : ℝ
⊢ c + a < c + b ↔ a < b

Next tactic:
induction a using Real.ind_mk
theorem add_lt_add_iff_left {a b : ℝ} (c : ℝ) : c + a < c + b ↔ a < b := by
  induction a using Real.ind_mk
  

Proof state:
case h
x y b c : ℝ
y✝ : CauSeq ℚ abs
⊢ c + mk y✝ < c + b ↔ mk y✝ < b

Next tactic:
induction b using Real.ind_mk
theorem add_lt_add_iff_left {a b : ℝ} (c : ℝ) : c + a < c + b ↔ a < b := by
  induction a using Real.ind_mk
  induction b using Real.ind_mk
  

Proof state:
case h.h
x y c : ℝ
y✝¹ y✝ : CauSeq ℚ abs
⊢ c + mk y✝¹ < c + mk y✝ ↔ mk y✝¹ < mk y✝

Next tactic:
induction c using Real.ind_mk
theorem add_lt_add_iff_left {a b : ℝ} (c : ℝ) : c + a < c + b ↔ a < b := by
  induction a using Real.ind_mk
  induction b using Real.ind_mk
  induction c using Real.ind_mk
  

Proof state:
case h.h.h
x y : ℝ
y✝² y✝¹ y✝ : CauSeq ℚ abs
⊢ mk y✝ + mk y✝² < mk y✝ + mk y✝¹ ↔ mk y✝² < mk y✝¹

Next tactic:
simp only [mk_lt, ← mk_add]
theorem add_lt_add_iff_left {a b : ℝ} (c : ℝ) : c + a < c + b ↔ a < b := by
  induction a using Real.ind_mk
  induction b using Real.ind_mk
  induction c using Real.ind_mk
  simp only [mk_lt, ← mk_add]
  

Proof state:
case h.h.h
x y : ℝ
y✝² y✝¹ y✝ : CauSeq ℚ abs
⊢ y✝ + y✝² < y✝ + y✝¹ ↔ y✝² < y✝¹

Next tactic:
show Pos _ ↔ Pos _
theorem add_lt_add_iff_left {a b : ℝ} (c : ℝ) : c + a < c + b ↔ a < b := by
  induction a using Real.ind_mk
  induction b using Real.ind_mk
  induction c using Real.ind_mk
  simp only [mk_lt, ← mk_add]
  show Pos _ ↔ Pos _; 

Proof state:
case h.h.h
x y : ℝ
y✝² y✝¹ y✝ : CauSeq ℚ abs
⊢ Pos (y✝ + y✝¹ - (y✝ + y✝²)) ↔ Pos (y✝¹ - y✝²)

Next tactic:
rw [add_sub_add_left_eq_sub]
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    

Proof state:
x y a : ℝ
⊢ a ≤ a

Next tactic:
induction' a using Real.ind_mk with a
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    

Proof state:
case h
x y : ℝ
a : CauSeq ℚ abs
⊢ mk a ≤ mk a

Next tactic:
rw [mk_le]
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    rw [mk_le]
  le_trans a b c := by
    

Proof state:
x y a b c : ℝ
⊢ a ≤ b → b ≤ c → a ≤ c

Next tactic:
induction' a using Real.ind_mk with a
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    rw [mk_le]
  le_trans a b c := by
    induction' a using Real.ind_mk with a
    

Proof state:
case h
x y b c : ℝ
a : CauSeq ℚ abs
⊢ mk a ≤ b → b ≤ c → mk a ≤ c

Next tactic:
induction' b using Real.ind_mk with b
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    rw [mk_le]
  le_trans a b c := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    

Proof state:
case h.h
x y c : ℝ
a b : CauSeq ℚ abs
⊢ mk a ≤ mk b → mk b ≤ c → mk a ≤ c

Next tactic:
induction' c using Real.ind_mk with c
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    rw [mk_le]
  le_trans a b c := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    induction' c using Real.ind_mk with c
    

Proof state:
case h.h.h
x y : ℝ
a b c : CauSeq ℚ abs
⊢ mk a ≤ mk b → mk b ≤ mk c → mk a ≤ mk c

Next tactic:
simpa using le_trans
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    

Proof state:
x y a b : ℝ
⊢ a < b ↔ a ≤ b ∧ ¬b ≤ a

Next tactic:
induction' a using Real.ind_mk with a
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    

Proof state:
case h
x y b : ℝ
a : CauSeq ℚ abs
⊢ mk a < b ↔ mk a ≤ b ∧ ¬b ≤ mk a

Next tactic:
induction' b using Real.ind_mk with b
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    

Proof state:
case h.h
x y : ℝ
a b : CauSeq ℚ abs
⊢ mk a < mk b ↔ mk a ≤ mk b ∧ ¬mk b ≤ mk a

Next tactic:
simpa using lt_iff_le_not_le
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    rw [mk_le]
  le_trans a b c := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    induction' c using Real.ind_mk with c
    simpa using le_trans
  le_antisymm a b := by
    

Proof state:
x y a b : ℝ
⊢ a ≤ b → b ≤ a → a = b

Next tactic:
induction' a using Real.ind_mk with a
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    rw [mk_le]
  le_trans a b c := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    induction' c using Real.ind_mk with c
    simpa using le_trans
  le_antisymm a b := by
    induction' a using Real.ind_mk with a
    

Proof state:
case h
x y b : ℝ
a : CauSeq ℚ abs
⊢ mk a ≤ b → b ≤ mk a → mk a = b

Next tactic:
induction' b using Real.ind_mk with b
instance partialOrder : PartialOrder ℝ where
  le := (· ≤ ·)
  lt := (· < ·)
  lt_iff_le_not_le a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    simpa using lt_iff_le_not_le
  le_refl a := by
    induction' a using Real.ind_mk with a
    rw [mk_le]
  le_trans a b c := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    induction' c using Real.ind_mk with c
    simpa using le_trans
  le_antisymm a b := by
    induction' a using Real.ind_mk with a
    induction' b using Real.ind_mk with b
    

Proof state:
case h.h
x y : ℝ
a b : CauSeq ℚ abs
⊢ mk a ≤ mk b → mk b ≤ mk a → mk a = mk b

Next tactic:
simpa [mk_eq] using @CauSeq.le_antisymm _ _ a b
instance : Preorder ℝ := by 

Proof state:
x y : ℝ
⊢ Preorder ℝ

Next tactic:
infer_instance
theorem ratCast_lt {x y : ℚ} : (x : ℝ) < (y : ℝ) ↔ x < y := by
  

Proof state:
x✝ y✝ : ℝ
x y : ℚ
⊢ ↑x < ↑y ↔ x < y

Next tactic:
erw [mk_lt]
theorem ratCast_lt {x y : ℚ} : (x : ℝ) < (y : ℝ) ↔ x < y := by
  erw [mk_lt]
  

Proof state:
x✝ y✝ : ℝ
x y : ℚ
⊢ const abs ↑x < const abs ↑y ↔ x < y

Next tactic:
exact const_lt
protected theorem zero_lt_one : (0 : ℝ) < 1 := by
  

Proof state:
x y : ℝ
⊢ 0 < 1

Next tactic:
convert ratCast_lt.2 zero_lt_one
protected theorem zero_lt_one : (0 : ℝ) < 1 := by
  convert ratCast_lt.2 zero_lt_one <;> 

Proof state:
case h.e'_3
x y : ℝ
⊢ 0 = ↑0

Next tactic:
simp [← ofCauchy_ratCast, ofCauchy_one, ofCauchy_zero]
protected theorem zero_lt_one : (0 : ℝ) < 1 := by
  convert ratCast_lt.2 zero_lt_one <;> 

Proof state:
case h.e'_4
x y : ℝ
⊢ 1 = ↑1

Next tactic:
simp [← ofCauchy_ratCast, ofCauchy_one, ofCauchy_zero]
protected theorem mul_pos {a b : ℝ} : 0 < a → 0 < b → 0 < a * b := by
  

Proof state:
x y a b : ℝ
⊢ 0 < a → 0 < b → 0 < a * b

Next tactic:
induction' a using Real.ind_mk with a
protected theorem mul_pos {a b : ℝ} : 0 < a → 0 < b → 0 < a * b := by
  induction' a using Real.ind_mk with a
  

Proof state:
case h
x y b : ℝ
a : CauSeq ℚ abs
⊢ 0 < mk a → 0 < b → 0 < mk a * b

Next tactic:
induction' b using Real.ind_mk with b
protected theorem mul_pos {a b : ℝ} : 0 < a → 0 < b → 0 < a * b := by
  induction' a using Real.ind_mk with a
  induction' b using Real.ind_mk with b
  

Proof state:
case h.h
x y : ℝ
a b : CauSeq ℚ abs
⊢ 0 < mk a → 0 < mk b → 0 < mk a * mk b

Next tactic:
simpa only [mk_lt, mk_pos, ← mk_mul] using CauSeq.mul_pos
instance : StrictOrderedCommRing ℝ :=
  { Real.commRing, Real.partialOrder,
    Real.semiring with
    exists_pair_ne := ⟨0, 1, Real.zero_lt_one.ne⟩
    add_le_add_left := by
      

Proof state:
x y : ℝ
src✝² : CommRing ℝ := commRing
src✝¹ : PartialOrder ℝ := partialOrder
src✝ : Semiring ℝ := semiring
⊢ ∀ (a b : ℝ), a ≤ b → ∀ (c : ℝ), c + a ≤ c + b

Next tactic:
simp only [le_iff_eq_or_lt]
instance : StrictOrderedCommRing ℝ :=
  { Real.commRing, Real.partialOrder,
    Real.semiring with
    exists_pair_ne := ⟨0, 1, Real.zero_lt_one.ne⟩
    add_le_add_left := by
      simp only [le_iff_eq_or_lt]
      

Proof state:
x y : ℝ
src✝² : CommRing ℝ := commRing
src✝¹ : PartialOrder ℝ := partialOrder
src✝ : Semiring ℝ := semiring
⊢ ∀ (a b : ℝ), a = b ∨ a < b → ∀ (c : ℝ), c + a = c + b ∨ c + a < c + b

Next tactic:
rintro a b ⟨rfl, h⟩
instance : StrictOrderedCommRing ℝ :=
  { Real.commRing, Real.partialOrder,
    Real.semiring with
    exists_pair_ne := ⟨0, 1, Real.zero_lt_one.ne⟩
    add_le_add_left := by
      simp only [le_iff_eq_or_lt]
      rintro a b ⟨rfl, h⟩
      · 

Proof state:
case inl.refl
x y : ℝ
src✝² : CommRing ℝ := commRing
src✝¹ : PartialOrder ℝ := partialOrder
src✝ : Semiring ℝ := semiring
a : ℝ
⊢ ∀ (c : ℝ), c + a = c + a ∨ c + a < c + a

Next tactic:
simp only [lt_self_iff_false, or_false, forall_const]