l3lab/ntp-mathlib huggingface dataset:
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⟩
      · simp only [lt_self_iff_false, or_false, forall_const]
      · 

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

Next tactic:
exact fun c => Or.inr ((add_lt_add_iff_left c).2 ‹_›)
theorem ofCauchy_sup (a b) : (⟨⟦a ⊔ b⟧⟩ : ℝ) = ⟨⟦a⟧⟩ ⊔ ⟨⟦b⟧⟩ :=
  show _ = sup _ _ by
    

Proof state:
x y : ℝ
a b : CauSeq ℚ abs
⊢ { cauchy := ⟦a ⊔ b⟧ } = Real.sup { cauchy := ⟦a⟧ } { cauchy := ⟦b⟧ }

Next tactic:
rw [sup_def]
theorem ofCauchy_sup (a b) : (⟨⟦a ⊔ b⟧⟩ : ℝ) = ⟨⟦a⟧⟩ ⊔ ⟨⟦b⟧⟩ :=
  show _ = sup _ _ by
    rw [sup_def]
    

Proof state:
x y : ℝ
a b : CauSeq ℚ abs
⊢ { cauchy := ⟦a ⊔ b⟧ } =
    match { cauchy := ⟦a⟧ }, { cauchy := ⟦b⟧ } with
    | { cauchy := x }, { cauchy := y } =>
      {
        cauchy :=
          Quotient.map₂ (fun x x_1 => x ⊔ x_1)
            (_ : ∀ (x x_1 : CauSeq ℚ abs), x ≈ x_1 → ∀ (x_2 x_3 : CauSeq ℚ abs), x_2 ≈ x_3 → x ⊔ x_2 ≈ x_1 ⊔ x_3) x y }

Next tactic:
rfl
theorem ofCauchy_inf (a b) : (⟨⟦a ⊓ b⟧⟩ : ℝ) = ⟨⟦a⟧⟩ ⊓ ⟨⟦b⟧⟩ :=
  show _ = inf _ _ by
    

Proof state:
x y : ℝ
a b : CauSeq ℚ abs
⊢ { cauchy := ⟦a ⊓ b⟧ } = Real.inf { cauchy := ⟦a⟧ } { cauchy := ⟦b⟧ }

Next tactic:
rw [inf_def]
theorem ofCauchy_inf (a b) : (⟨⟦a ⊓ b⟧⟩ : ℝ) = ⟨⟦a⟧⟩ ⊓ ⟨⟦b⟧⟩ :=
  show _ = inf _ _ by
    rw [inf_def]
    

Proof state:
x y : ℝ
a b : CauSeq ℚ abs
⊢ { cauchy := ⟦a ⊓ b⟧ } =
    match { cauchy := ⟦a⟧ }, { cauchy := ⟦b⟧ } with
    | { cauchy := x }, { cauchy := y } =>
      {
        cauchy :=
          Quotient.map₂ (fun x x_1 => x ⊓ x_1)
            (_ : ∀ (x x_1 : CauSeq ℚ abs), x ≈ x_1 → ∀ (x_2 x_3 : CauSeq ℚ abs), x_2 ≈ x_3 → x ⊓ x_2 ≈ x_1 ⊓ x_3) x y }

Next tactic:
rfl
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      

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

Next tactic:
intros a b
instance : DistribLattice ℝ :=
  { Real.partialOrder with
    sup := (· ⊔ ·)
    le := (· ≤ ·)
    le_sup_left := by
      intros a b
      

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

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
      

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

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
      

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

Next tactic:
rw [← mk_sup, 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]
      

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

Next tactic:
exact CauSeq.le_sup_left
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
      

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

Next tactic:
intros a 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
      

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

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
      

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

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
      

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

Next tactic:
rw [← mk_sup, 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]
      

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

Next tactic:
exact CauSeq.le_sup_right
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
      

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

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
      

Proof state:
x y : ℝ
src✝ : PartialOrder ℝ := partialOrder
a b c : ℝ
⊢ a ≤ c → b ≤ 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
      

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

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

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

Next tactic:
simp_rw [← mk_sup, 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]
      

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

Next tactic:
exact CauSeq.sup_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
      

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

Next tactic:
intros a 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
      

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

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
      

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

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
      

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

Next tactic:
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]
      

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

Next tactic:
exact CauSeq.inf_le_left
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
      

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

Next tactic:
intros a 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
      

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

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
      

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

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
      

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

Next tactic:
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]
      

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

Next tactic:
exact CauSeq.inf_le_right
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
      

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

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
      

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