l3lab/ntp-mathlib huggingface dataset:
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      

Proof state:
case h
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
b c : ℝ
a : CauSeq ℚ abs
⊢ mk a ≤ b → mk a ≤ c → mk a ≤ b ⊓ c

Next tactic:
induction' b using Real.ind_mk with b
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      

Proof state:
case h.h
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
c : ℝ
a b : CauSeq ℚ abs
⊢ mk a ≤ mk b → mk a ≤ c → mk a ≤ mk b ⊓ c

Next tactic:
induction' c using Real.ind_mk with c
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      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
src✝ : PartialOrder ℝ := partialOrder
a b c : CauSeq ℚ abs
⊢ mk a ≤ mk b → mk a ≤ mk c → mk a ≤ mk b ⊓ mk c

Next tactic:
simp_rw [← mk_inf, mk_le]
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      

Proof state:
case h.h.h
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
a b c : CauSeq ℚ abs
⊢ a ≤ b → a ≤ c → a ≤ b ⊓ c

Next tactic:
exact CauSeq.le_inf
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      exact CauSeq.le_inf
    le_sup_inf := by
      

Proof state:
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
⊢ ∀ (x y z : ℝ), (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z

Next tactic:
intros a b c
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      exact CauSeq.le_inf
    le_sup_inf := by
      intros a b c
      

Proof state:
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
a b c : ℝ
⊢ (a ⊔ b) ⊓ (a ⊔ c) ≤ a ⊔ b ⊓ c

Next tactic:
induction' a using Real.ind_mk with a
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      exact CauSeq.le_inf
    le_sup_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      

Proof state:
case h
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
b c : ℝ
a : CauSeq ℚ abs
⊢ (mk a ⊔ b) ⊓ (mk a ⊔ c) ≤ mk a ⊔ b ⊓ c

Next tactic:
induction' b using Real.ind_mk with b
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      exact CauSeq.le_inf
    le_sup_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      

Proof state:
case h.h
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
c : ℝ
a b : CauSeq ℚ abs
⊢ (mk a ⊔ mk b) ⊓ (mk a ⊔ c) ≤ mk a ⊔ mk b ⊓ c

Next tactic:
induction' c using Real.ind_mk with c
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      exact CauSeq.le_inf
    le_sup_inf := by
      intros a b c
      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
src✝ : PartialOrder ℝ := partialOrder
a b c : CauSeq ℚ abs
⊢ (mk a ⊔ mk b) ⊓ (mk a ⊔ mk c) ≤ mk a ⊔ mk b ⊓ mk c

Next tactic:
apply Eq.le
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      exact CauSeq.le_inf
    le_sup_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      apply Eq.le
      

Proof state:
case h.h.h.a
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
a b c : CauSeq ℚ abs
⊢ (mk a ⊔ mk b) ⊓ (mk a ⊔ mk c) = mk a ⊔ mk b ⊓ mk c

Next tactic:
simp only [← mk_sup, ← mk_inf]
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_left
    le_sup_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_sup, mk_le]
      exact CauSeq.le_sup_right
    sup_le := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_sup, mk_le]
      exact CauSeq.sup_le
    inf := (· ⊓ ·)
    inf_le_left := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_left
    inf_le_right := by
      intros a b
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      rw [← mk_inf, mk_le]
      exact CauSeq.inf_le_right
    le_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      simp_rw [← mk_inf, mk_le]
      exact CauSeq.le_inf
    le_sup_inf := by
      intros a b c
      induction' a using Real.ind_mk with a
      induction' b using Real.ind_mk with b
      induction' c using Real.ind_mk with c
      apply Eq.le
      simp only [← mk_sup, ← mk_inf]
      

Proof state:
case h.h.h.a
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
a b c : CauSeq ℚ abs
⊢ mk ((a ⊔ b) ⊓ (a ⊔ c)) = mk (a ⊔ b ⊓ c)

Next tactic:
exact congr_arg mk (CauSeq.sup_inf_distrib_left _ _ _).symm
instance : IsTotal ℝ (· ≤ ·) :=
  ⟨by
    

Proof state:
x y : ℝ
⊢ ∀ (a b : ℝ), a ≤ b ∨ b ≤ a

Next tactic:
intros a b
instance : IsTotal ℝ (· ≤ ·) :=
  ⟨by
    intros a b
    

Proof state:
x y a b : ℝ
⊢ a ≤ b ∨ b ≤ a

Next tactic:
induction' a using Real.ind_mk with a
instance : IsTotal ℝ (· ≤ ·) :=
  ⟨by
    intros a b
    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

Next tactic:
induction' b using Real.ind_mk with b
instance : IsTotal ℝ (· ≤ ·) :=
  ⟨by
    intros a b
    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

Next tactic:
simpa using le_total a b
noncomputable instance : LinearOrderedRing ℝ := by 

Proof state:
x y : ℝ
⊢ LinearOrderedRing ℝ

Next tactic:
infer_instance
noncomputable instance : LinearOrderedSemiring ℝ := by 

Proof state:
x y : ℝ
⊢ LinearOrderedSemiring ℝ

Next tactic:
infer_instance
noncomputable instance : LinearOrderedField ℝ :=
  { Real.linearOrderedCommRing with
    inv := Inv.inv
    mul_inv_cancel := by
      

Proof state:
x y : ℝ
src✝ : LinearOrderedCommRing ℝ := linearOrderedCommRing
⊢ ∀ (a : ℝ), a ≠ 0 → a * a⁻¹ = 1

Next tactic:
rintro ⟨a⟩ h
noncomputable instance : LinearOrderedField ℝ :=
  { Real.linearOrderedCommRing with
    inv := Inv.inv
    mul_inv_cancel := by
      rintro ⟨a⟩ h
      

Proof state:
case ofCauchy
x y : ℝ
src✝ : LinearOrderedCommRing ℝ := linearOrderedCommRing
a : Cauchy abs
h : { cauchy := a } ≠ 0
⊢ { cauchy := a } * { cauchy := a }⁻¹ = 1

Next tactic:
rw [mul_comm]
noncomputable instance : LinearOrderedField ℝ :=
  { Real.linearOrderedCommRing with
    inv := Inv.inv
    mul_inv_cancel := by
      rintro ⟨a⟩ h
      rw [mul_comm]
      

Proof state:
case ofCauchy
x y : ℝ
src✝ : LinearOrderedCommRing ℝ := linearOrderedCommRing
a : Cauchy abs
h : { cauchy := a } ≠ 0
⊢ { cauchy := a }⁻¹ * { cauchy := a } = 1

Next tactic:
simp only [← ofCauchy_inv, ← ofCauchy_mul, ← ofCauchy_one, ← ofCauchy_zero,
        Ne.def, ofCauchy.injEq] at *
noncomputable instance : LinearOrderedField ℝ :=
  { Real.linearOrderedCommRing with
    inv := Inv.inv
    mul_inv_cancel := by
      rintro ⟨a⟩ h
      rw [mul_comm]
      simp only [← ofCauchy_inv, ← ofCauchy_mul, ← ofCauchy_one, ← ofCauchy_zero,
        Ne.def, ofCauchy.injEq] at *
      

Proof state:
case ofCauchy
x y : ℝ
src✝ : LinearOrderedCommRing ℝ := linearOrderedCommRing
a : Cauchy abs
h : ¬a = 0
⊢ a⁻¹ * a = 1

Next tactic:
exact CauSeq.Completion.inv_mul_cancel h
noncomputable instance : LinearOrderedField ℝ :=
  { Real.linearOrderedCommRing with
    inv := Inv.inv
    mul_inv_cancel := by
      rintro ⟨a⟩ h
      rw [mul_comm]
      simp only [← ofCauchy_inv, ← ofCauchy_mul, ← ofCauchy_one, ← ofCauchy_zero,
        Ne.def, ofCauchy.injEq] at *
      exact CauSeq.Completion.inv_mul_cancel h
    inv_zero := by 

Proof state:
x y : ℝ
src✝ : LinearOrderedCommRing ℝ := linearOrderedCommRing
⊢ 0⁻¹ = 0

Next tactic:
simp [← ofCauchy_zero, ← ofCauchy_inv]
noncomputable instance : LinearOrderedField ℝ :=
  { Real.linearOrderedCommRing with
    inv := Inv.inv
    mul_inv_cancel := by
      rintro ⟨a⟩ h
      rw [mul_comm]
      simp only [← ofCauchy_inv, ← ofCauchy_mul, ← ofCauchy_one, ← ofCauchy_zero,
        Ne.def, ofCauchy.injEq] at *
      exact CauSeq.Completion.inv_mul_cancel h
    inv_zero := by simp [← ofCauchy_zero, ← ofCauchy_inv]
    ratCast := (↑)
    ratCast_mk := fun n d hd h2 => by
      

Proof state:
x y : ℝ
src✝ : LinearOrderedCommRing ℝ := linearOrderedCommRing
n : ℤ
d : ℕ
hd : d ≠ 0
h2 : Nat.Coprime (Int.natAbs n) d
⊢ ↑(Rat.mk' n d) = ↑n * (↑d)⁻¹

Next tactic:
rw [← ofCauchy_ratCast, Rat.cast_mk', ofCauchy_mul, ofCauchy_inv, ofCauchy_natCast,
        ofCauchy_intCast]
noncomputable instance : LinearOrderedAddCommGroup ℝ := by 

Proof state:
x y : ℝ
⊢ LinearOrderedAddCommGroup ℝ

Next tactic:
infer_instance
noncomputable instance field : Field ℝ := by 

Proof state:
x y : ℝ
⊢ Field ℝ

Next tactic:
infer_instance
noncomputable instance : DivisionRing ℝ := by 

Proof state:
x y : ℝ
⊢ DivisionRing ℝ

Next tactic:
infer_instance
noncomputable instance decidableLT (a b : ℝ) : Decidable (a < b) := by 

Proof state:
x y a b : ℝ
⊢ Decidable (a < b)

Next tactic:
infer_instance
noncomputable instance decidableLE (a b : ℝ) : Decidable (a ≤ b) := by 

Proof state:
x y a b : ℝ
⊢ Decidable (a ≤ b)

Next tactic:
infer_instance
noncomputable instance decidableEq (a b : ℝ) : Decidable (a = b) := by 

Proof state:
x y a b : ℝ
⊢ Decidable (a = b)

Next tactic:
infer_instance
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f := by
  

Proof state:
x y : ℝ
f : CauSeq ℚ abs
⊢ (∃ i, ∀ j ≥ i, x ≤ ↑(↑f j)) → x ≤ mk f

Next tactic:
intro h
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f := by
  intro h
  

Proof state:
x y : ℝ
f : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, x ≤ ↑(↑f j)
⊢ x ≤ mk f

Next tactic:
induction' x using Real.ind_mk with x
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f := by
  intro h
  induction' x using Real.ind_mk with x
  

Proof state:
case h
x✝ y : ℝ
f x : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, mk x ≤ ↑(↑f j)
⊢ mk x ≤ mk f

Next tactic:
apply le_of_not_lt
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f := by
  intro h
  induction' x using Real.ind_mk with x
  apply le_of_not_lt
  

Proof state:
case h.h
x✝ y : ℝ
f x : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, mk x ≤ ↑(↑f j)
⊢ ¬mk f < mk x

Next tactic:
rw [mk_lt]
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f := by
  intro h
  induction' x using Real.ind_mk with x
  apply le_of_not_lt
  rw [mk_lt]
  

Proof state:
case h.h
x✝ y : ℝ
f x : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, mk x ≤ ↑(↑f j)
⊢ ¬f < x

Next tactic:
rintro ⟨K, K0, hK⟩
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f := by
  intro h
  induction' x using Real.ind_mk with x
  apply le_of_not_lt
  rw [mk_lt]
  rintro ⟨K, K0, hK⟩
  

Proof state:
case h.h.intro.intro
x✝ y : ℝ
f x : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, mk x ≤ ↑(↑f j)
K : ℚ
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ ↑(x - f) j
⊢ False

Next tactic:
obtain ⟨i, H⟩ := exists_forall_ge_and h (exists_forall_ge_and hK (f.cauchy₃ <| half_pos K0))
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f := by
  intro h
  induction' x using Real.ind_mk with x
  apply le_of_not_lt
  rw [mk_lt]
  rintro ⟨K, K0, hK⟩
  obtain ⟨i, H⟩ := exists_forall_ge_and h (exists_forall_ge_and hK (f.cauchy₃ <| half_pos K0))
  

Proof state:
case h.h.intro.intro.intro
x✝ y : ℝ
f x : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, mk x ≤ ↑(↑f j)
K : ℚ
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ ↑(x - f) j
i : ℕ
H : ∀ j ≥ i, mk x ≤ ↑(↑f j) ∧ K ≤ ↑(x - f) j ∧ ∀ k ≥ j, |↑f k - ↑f j| < K / 2
⊢ False

Next tactic:
apply not_lt_of_le (H _ le_rfl).1