l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
    (single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by
  ext n
  rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
  by_cases h : x.order ≤ n
  · rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
      powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
      add_sub_cancel'_right]
  · rw [ofPowerSeries_apply, embDomain_notin_range]
    · 

Proof state:
case neg
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
n : ℤ
h : ¬order x ≤ n
⊢ 0 = HahnSeries.coeff x n

Next tactic:
contrapose! h
@[simp]
theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
    (single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by
  ext n
  rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
  by_cases h : x.order ≤ n
  · rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
      powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
      add_sub_cancel'_right]
  · rw [ofPowerSeries_apply, embDomain_notin_range]
    · contrapose! h
      

Proof state:
case neg
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
n : ℤ
h : 0 ≠ HahnSeries.coeff x n
⊢ order x ≤ n

Next tactic:
exact order_le_of_coeff_ne_zero h.symm
@[simp]
theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
    (single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by
  ext n
  rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
  by_cases h : x.order ≤ n
  · rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
      powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
      add_sub_cancel'_right]
  · rw [ofPowerSeries_apply, embDomain_notin_range]
    · contrapose! h
      exact order_le_of_coeff_ne_zero h.symm
    · 

Proof state:
case neg
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
n : ℤ
h : ¬order x ≤ n
⊢ n - order x ∉
    Set.range
      ⇑{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) },
          map_rel_iff' :=
            (_ :
              ∀ {a b : ℕ},
                { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } a ≤
                    { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } b ↔
                  a ≤ b) }

Next tactic:
contrapose! h
@[simp]
theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
    (single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by
  ext n
  rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
  by_cases h : x.order ≤ n
  · rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
      powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
      add_sub_cancel'_right]
  · rw [ofPowerSeries_apply, embDomain_notin_range]
    · contrapose! h
      exact order_le_of_coeff_ne_zero h.symm
    · contrapose! h
      

Proof state:
case neg
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
n : ℤ
h :
  n - order x ∈
    Set.range
      ⇑{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) },
          map_rel_iff' :=
            (_ :
              ∀ {a b : ℕ},
                { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } a ≤
                    { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } b ↔
                  a ≤ b) }
⊢ order x ≤ n

Next tactic:
simp only [Set.mem_range, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk] at h
@[simp]
theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
    (single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by
  ext n
  rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
  by_cases h : x.order ≤ n
  · rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
      powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
      add_sub_cancel'_right]
  · rw [ofPowerSeries_apply, embDomain_notin_range]
    · contrapose! h
      exact order_le_of_coeff_ne_zero h.symm
    · contrapose! h
      simp only [Set.mem_range, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk] at h
      

Proof state:
case neg
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
n : ℤ
h : ∃ y, ↑y = n - order x
⊢ order x ≤ n

Next tactic:
obtain ⟨m, hm⟩ := h
@[simp]
theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
    (single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by
  ext n
  rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
  by_cases h : x.order ≤ n
  · rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
      powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
      add_sub_cancel'_right]
  · rw [ofPowerSeries_apply, embDomain_notin_range]
    · contrapose! h
      exact order_le_of_coeff_ne_zero h.symm
    · contrapose! h
      simp only [Set.mem_range, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk] at h
      obtain ⟨m, hm⟩ := h
      

Proof state:
case neg.intro
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
n : ℤ
m : ℕ
hm : ↑m = n - order x
⊢ order x ≤ n

Next tactic:
rw [← sub_nonneg, ← hm]
@[simp]
theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
    (single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by
  ext n
  rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
  by_cases h : x.order ≤ n
  · rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
      powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
      add_sub_cancel'_right]
  · rw [ofPowerSeries_apply, embDomain_notin_range]
    · contrapose! h
      exact order_le_of_coeff_ne_zero h.symm
    · contrapose! h
      simp only [Set.mem_range, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk] at h
      obtain ⟨m, hm⟩ := h
      rw [← sub_nonneg, ← hm]
      

Proof state:
case neg.intro
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
n : ℤ
m : ℕ
hm : ↑m = n - order x
⊢ 0 ≤ ↑m

Next tactic:
simp only [Nat.cast_nonneg]
theorem ofPowerSeries_powerSeriesPart (x : LaurentSeries R) :
    ofPowerSeries ℤ R x.powerSeriesPart = single (-x.order) 1 * x := by
  

Proof state:
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
⊢ (ofPowerSeries ℤ R) (powerSeriesPart x) = (single (-order x)) 1 * x

Next tactic:
refine' Eq.trans _ (congr rfl x.single_order_mul_powerSeriesPart)
theorem ofPowerSeries_powerSeriesPart (x : LaurentSeries R) :
    ofPowerSeries ℤ R x.powerSeriesPart = single (-x.order) 1 * x := by
  refine' Eq.trans _ (congr rfl x.single_order_mul_powerSeriesPart)
  

Proof state:
R : Type u
inst✝ : Semiring R
x : LaurentSeries R
⊢ (ofPowerSeries ℤ R) (powerSeriesPart x) =
    (single (-order x)) 1 * ((single (order x)) 1 * (ofPowerSeries ℤ R) (powerSeriesPart x))

Next tactic:
rw [← mul_assoc, single_mul_single, neg_add_self, mul_one, ← C_apply, C_one, one_mul]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    

Proof state:
R : Type u
inst✝ : CommRing R
⊢ ∀ (y : ↥(Submonoid.powers PowerSeries.X)), IsUnit ((algebraMap (PowerSeries R) (LaurentSeries R)) ↑y)

Next tactic:
rintro ⟨_, n, rfl⟩
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    

Proof state:
case mk.intro
R : Type u
inst✝ : CommRing R
n : ℕ
⊢ IsUnit
    ((algebraMap (PowerSeries R) (LaurentSeries R))
      ↑{ val := (fun x => PowerSeries.X ^ x) n,
          property := (_ : ∃ y, (fun x => PowerSeries.X ^ x) y = (fun x => PowerSeries.X ^ x) n) })

Next tactic:
refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · 

Proof state:
case mk.intro.refine'_1
R : Type u
inst✝ : CommRing R
n : ℕ
⊢ (single ↑n) 1 * (single (-↑n)) 1 = 1

Next tactic:
simp only [single_mul_single, mul_one, add_right_neg]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      

Proof state:
case mk.intro.refine'_1
R : Type u
inst✝ : CommRing R
n : ℕ
⊢ (single 0) 1 = 1

Next tactic:
rfl
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · 

Proof state:
case mk.intro.refine'_2
R : Type u
inst✝ : CommRing R
n : ℕ
⊢ (single (-↑n)) 1 * (single ↑n) 1 = 1

Next tactic:
simp only [single_mul_single, mul_one, add_left_neg]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      

Proof state:
case mk.intro.refine'_2
R : Type u
inst✝ : CommRing R
n : ℕ
⊢ (single 0) 1 = 1

Next tactic:
rfl
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · 

Proof state:
case mk.intro.refine'_3
R : Type u
inst✝ : CommRing R
n : ℕ
⊢ ↑{ val := (single ↑n) 1, inv := (single (-↑n)) 1, val_inv := (_ : (single ↑n) 1 * (single (-↑n)) 1 = 1),
        inv_val := (_ : (single (-↑n)) 1 * (single ↑n) 1 = 1) } =
    (algebraMap (PowerSeries R) (LaurentSeries R))
      ↑{ val := (fun x => PowerSeries.X ^ x) n,
          property := (_ : ∃ y, (fun x => PowerSeries.X ^ x) y = (fun x => PowerSeries.X ^ x) n) }

Next tactic:
dsimp
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; 

Proof state:
case mk.intro.refine'_3
R : Type u
inst✝ : CommRing R
n : ℕ
⊢ (single ↑n) 1 = (ofPowerSeries ℤ R) (PowerSeries.X ^ n)

Next tactic:
rw [ofPowerSeries_X_pow]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    

Proof state:
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
⊢ ∃ x, z * (algebraMap (PowerSeries R) (LaurentSeries R)) ↑x.2 = (algebraMap (PowerSeries R) (LaurentSeries R)) x.1

Next tactic:
by_cases h : 0 ≤ z.order
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · 

Proof state:
case pos
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : 0 ≤ order z
⊢ ∃ x, z * (algebraMap (PowerSeries R) (LaurentSeries R)) ↑x.2 = (algebraMap (PowerSeries R) (LaurentSeries R)) x.1

Next tactic:
refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      

Proof state:
case pos
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : 0 ≤ order z
⊢ z * (algebraMap (PowerSeries R) (LaurentSeries R)) ↑(PowerSeries.X ^ Int.natAbs (order z) * powerSeriesPart z, 1).2 =
    (algebraMap (PowerSeries R) (LaurentSeries R)) (PowerSeries.X ^ Int.natAbs (order z) * powerSeriesPart z, 1).1

Next tactic:
simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
      

Proof state:
case pos
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : 0 ≤ order z
⊢ z = (single ↑(Int.natAbs (order z))) 1 * (ofPowerSeries ℤ R) (powerSeriesPart z)

Next tactic:
rw [Int.natAbs_of_nonneg h, single_order_mul_powerSeriesPart]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
      rw [Int.natAbs_of_nonneg h, single_order_mul_powerSeriesPart]
    · 

Proof state:
case neg
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : ¬0 ≤ order z
⊢ ∃ x, z * (algebraMap (PowerSeries R) (LaurentSeries R)) ↑x.2 = (algebraMap (PowerSeries R) (LaurentSeries R)) x.1

Next tactic:
refine' ⟨⟨powerSeriesPart z, PowerSeries.X ^ Int.natAbs z.order, ⟨_, rfl⟩⟩, _⟩
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
      rw [Int.natAbs_of_nonneg h, single_order_mul_powerSeriesPart]
    · refine' ⟨⟨powerSeriesPart z, PowerSeries.X ^ Int.natAbs z.order, ⟨_, rfl⟩⟩, _⟩
      

Proof state:
case neg
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : ¬0 ≤ order z
⊢ z *
      (algebraMap (PowerSeries R) (LaurentSeries R))
        ↑(powerSeriesPart z,
              { val := PowerSeries.X ^ Int.natAbs (order z),
                property := (_ : ∃ y, (fun x => PowerSeries.X ^ x) y = PowerSeries.X ^ Int.natAbs (order z)) }).2 =
    (algebraMap (PowerSeries R) (LaurentSeries R))
      (powerSeriesPart z,
          { val := PowerSeries.X ^ Int.natAbs (order z),
            property := (_ : ∃ y, (fun x => PowerSeries.X ^ x) y = PowerSeries.X ^ Int.natAbs (order z)) }).1

Next tactic:
simp only [coe_algebraMap, ofPowerSeries_powerSeriesPart]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
      rw [Int.natAbs_of_nonneg h, single_order_mul_powerSeriesPart]
    · refine' ⟨⟨powerSeriesPart z, PowerSeries.X ^ Int.natAbs z.order, ⟨_, rfl⟩⟩, _⟩
      simp only [coe_algebraMap, ofPowerSeries_powerSeriesPart]
      

Proof state:
case neg
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : ¬0 ≤ order z
⊢ z * (ofPowerSeries ℤ R) (PowerSeries.X ^ Int.natAbs (order z)) = (single (-order z)) 1 * z

Next tactic:
rw [mul_comm _ z]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
      rw [Int.natAbs_of_nonneg h, single_order_mul_powerSeriesPart]
    · refine' ⟨⟨powerSeriesPart z, PowerSeries.X ^ Int.natAbs z.order, ⟨_, rfl⟩⟩, _⟩
      simp only [coe_algebraMap, ofPowerSeries_powerSeriesPart]
      rw [mul_comm _ z]
      

Proof state:
case neg
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : ¬0 ≤ order z
⊢ z * (ofPowerSeries ℤ R) (PowerSeries.X ^ Int.natAbs (order z)) = z * (single (-order z)) 1

Next tactic:
refine' congr rfl _
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
      rw [Int.natAbs_of_nonneg h, single_order_mul_powerSeriesPart]
    · refine' ⟨⟨powerSeriesPart z, PowerSeries.X ^ Int.natAbs z.order, ⟨_, rfl⟩⟩, _⟩
      simp only [coe_algebraMap, ofPowerSeries_powerSeriesPart]
      rw [mul_comm _ z]
      refine' congr rfl _
      

Proof state:
case neg
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : ¬0 ≤ order z
⊢ (ofPowerSeries ℤ R) (PowerSeries.X ^ Int.natAbs (order z)) = (single (-order z)) 1

Next tactic:
rw [ofPowerSeries_X_pow, Int.ofNat_natAbs_of_nonpos]
/-- The localization map from power series to Laurent series. -/
@[simps (config := { rhsMd := .all, simpRhs := true })]
instance of_powerSeries_localization [CommRing R] :
    IsLocalization (Submonoid.powers (PowerSeries.X : PowerSeries R)) (LaurentSeries R) where
  map_units' := by
    rintro ⟨_, n, rfl⟩
    refine' ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩
    · simp only [single_mul_single, mul_one, add_right_neg]
      rfl
    · simp only [single_mul_single, mul_one, add_left_neg]
      rfl
    · dsimp; rw [ofPowerSeries_X_pow]
  surj' z := by
    by_cases h : 0 ≤ z.order
    · refine' ⟨⟨PowerSeries.X ^ Int.natAbs z.order * powerSeriesPart z, 1⟩, _⟩
      simp only [RingHom.map_one, mul_one, RingHom.map_mul, coe_algebraMap, ofPowerSeries_X_pow,
        Submonoid.coe_one]
      rw [Int.natAbs_of_nonneg h, single_order_mul_powerSeriesPart]
    · refine' ⟨⟨powerSeriesPart z, PowerSeries.X ^ Int.natAbs z.order, ⟨_, rfl⟩⟩, _⟩
      simp only [coe_algebraMap, ofPowerSeries_powerSeriesPart]
      rw [mul_comm _ z]
      refine' congr rfl _
      rw [ofPowerSeries_X_pow, Int.ofNat_natAbs_of_nonpos]
      

Proof state:
case neg
R : Type u
inst✝ : CommRing R
z : LaurentSeries R
h : ¬0 ≤ order z
⊢ order z ≤ 0

Next tactic:
exact le_of_not_ge h