l3lab/ntp-mathlib huggingface dataset:
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      

Proof state:
case refine_2.intro
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this : 0 < ↑p ^ (-↑n)
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, ‖↑x' k - ↑(const norm (CauSeq.lim x')) j‖ < ↑p ^ (-↑n)
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
by_cases hin : i ≤ n
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      by_cases hin : i ≤ n
      · 

Proof state:
case pos
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this : 0 < ↑p ^ (-↑n)
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, ‖↑x' k - ↑(const norm (CauSeq.lim x')) j‖ < ↑p ^ (-↑n)
hin : i ≤ n
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
exact (hi i le_rfl n hin).le
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      by_cases hin : i ≤ n
      · exact (hi i le_rfl n hin).le
      · 

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this : 0 < ↑p ^ (-↑n)
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, ‖↑x' k - ↑(const norm (CauSeq.lim x')) j‖ < ↑p ^ (-↑n)
hin : ¬i ≤ n
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
push_neg at hin
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      by_cases hin : i ≤ n
      · exact (hi i le_rfl n hin).le
      · push_neg at hin
        

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this : 0 < ↑p ^ (-↑n)
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, ‖↑x' k - ↑(const norm (CauSeq.lim x')) j‖ < ↑p ^ (-↑n)
hin : n < i
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
specialize hi i le_rfl i le_rfl
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      by_cases hin : i ≤ n
      · exact (hi i le_rfl n hin).le
      · push_neg at hin
        specialize hi i le_rfl i le_rfl
        

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this : 0 < ↑p ^ (-↑n)
i : ℕ
hin : n < i
hi : ‖↑x' i - ↑(const norm (CauSeq.lim x')) i‖ < ↑p ^ (-↑n)
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
specialize hx hin.le
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      by_cases hin : i ≤ n
      · exact (hi i le_rfl n hin).le
      · push_neg at hin
        specialize hi i le_rfl i le_rfl
        specialize hx hin.le
        

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx✝ : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this : 0 < ↑p ^ (-↑n)
i : ℕ
hin : n < i
hi : ‖↑x' i - ↑(const norm (CauSeq.lim x')) i‖ < ↑p ^ (-↑n)
hx : ‖x n - x i‖ ≤ ↑p ^ (-↑n)
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
have := nonarchimedean (x n - x i : ℤ_[p]) (x i - x'.lim)
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      by_cases hin : i ≤ n
      · exact (hi i le_rfl n hin).le
      · push_neg at hin
        specialize hi i le_rfl i le_rfl
        specialize hx hin.le
        have := nonarchimedean (x n - x i : ℤ_[p]) (x i - x'.lim)
        

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx✝ : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this✝ : 0 < ↑p ^ (-↑n)
i : ℕ
hin : n < i
hi : ‖↑x' i - ↑(const norm (CauSeq.lim x')) i‖ < ↑p ^ (-↑n)
hx : ‖x n - x i‖ ≤ ↑p ^ (-↑n)
this : ‖x n - x i + (x i - CauSeq.lim x')‖ ≤ max ‖x n - x i‖ ‖x i - CauSeq.lim x'‖
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
rw [sub_add_sub_cancel] at this
instance : IsAdicComplete (maximalIdeal ℤ_[p]) ℤ_[p] where
  prec' x hx := by
    simp only [← Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p,
      Ideal.span_singleton_pow, ← norm_le_pow_iff_mem_span_pow] at hx ⊢
    let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩; swap
    · intro ε hε
      obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
      refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
      rw [← neg_sub, norm_neg]
      exact hx hn
    · refine ⟨x'.lim, fun n => ?_⟩
      have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := by
        apply zpow_pos_of_pos
        exact mod_cast hp.1.pos
      obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
      by_cases hin : i ≤ n
      · exact (hi i le_rfl n hin).le
      · push_neg at hin
        specialize hi i le_rfl i le_rfl
        specialize hx hin.le
        have := nonarchimedean (x n - x i : ℤ_[p]) (x i - x'.lim)
        rw [sub_add_sub_cancel] at this
        

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℕ → ℤ_[p]
hx✝ : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)
x' : CauSeq ℤ_[p] norm := { val := x, property := (_ : ∀ ε > 0, ∃ i, ∀ j ≥ i, ‖x j - x i‖ < ε) }
n : ℕ
this✝ : 0 < ↑p ^ (-↑n)
i : ℕ
hin : n < i
hi : ‖↑x' i - ↑(const norm (CauSeq.lim x')) i‖ < ↑p ^ (-↑n)
hx : ‖x n - x i‖ ≤ ↑p ^ (-↑n)
this : ‖x n - CauSeq.lim x'‖ ≤ max ‖x n - x i‖ ‖x i - CauSeq.lim x'‖
⊢ ‖x n - CauSeq.lim x'‖ ≤ ↑p ^ (-↑n)

Next tactic:
refine' this.trans (max_le_iff.mpr ⟨hx, hi.le⟩)
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x✝ : ↥(nonZeroDivisors ℤ_[p])
x : ℤ_[p]
hx : x ∈ nonZeroDivisors ℤ_[p]
⊢ IsUnit ((algebraMap ℤ_[p] ℚ_[p]) ↑{ val := x, property := hx })

Next tactic:
rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1

Next tactic:
by_cases hx : ‖x‖ ≤ 1
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · 

Proof state:
case pos
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ‖x‖ ≤ 1
⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1

Next tactic:
use (⟨x, hx⟩, 1)
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      

Proof state:
case h
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ‖x‖ ≤ 1
⊢ x * (algebraMap ℤ_[p] ℚ_[p]) ↑({ val := x, property := hx }, 1).2 =
    (algebraMap ℤ_[p] ℚ_[p]) ({ val := x, property := hx }, 1).1

Next tactic:
rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · 

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1

Next tactic:
set n := Int.toNat (-x.valuation) with hn
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1

Next tactic:
have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
⊢ ↑n = -Padic.valuation x

Next tactic:
rw [hn, Int.toNat_of_nonneg]
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
⊢ 0 ≤ -Padic.valuation x

Next tactic:
rw [Right.nonneg_neg_iff]
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
⊢ Padic.valuation x ≤ 0

Next tactic:
rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : Padic.valuation x < 0
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
⊢ Padic.valuation x ≤ 0

Next tactic:
exact hx.le
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
      

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
hn_coe : ↑n = -Padic.valuation x
⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1

Next tactic:
set a := x * (p : ℚ_[p]) ^ n with ha
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
      set a := x * (p : ℚ_[p]) ^ n with ha
      

Proof state:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
hn_coe : ↑n = -Padic.valuation x
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1

Next tactic:
have ha_norm : ‖a‖ = 1 := by
        have hx : x ≠ 0 := by
          intro h0
          rw [h0, norm_zero] at hx
          exact hx zero_le_one
        rw [ha, padicNormE.mul, padicNormE.norm_p_pow, Padic.norm_eq_pow_val hx, ← zpow_add',
          hn_coe, neg_neg, add_left_neg, zpow_zero]
        exact Or.inl (Nat.cast_ne_zero.mpr (NeZero.ne p))
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
      set a := x * (p : ℚ_[p]) ^ n with ha
      have ha_norm : ‖a‖ = 1 := by
        

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
hn_coe : ↑n = -Padic.valuation x
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
⊢ ‖a‖ = 1

Next tactic:
have hx : x ≠ 0 := by
          intro h0
          rw [h0, norm_zero] at hx
          exact hx zero_le_one
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
      set a := x * (p : ℚ_[p]) ^ n with ha
      have ha_norm : ‖a‖ = 1 := by
        have hx : x ≠ 0 := by
          

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
hn_coe : ↑n = -Padic.valuation x
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
⊢ x ≠ 0

Next tactic:
intro h0
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
      set a := x * (p : ℚ_[p]) ^ n with ha
      have ha_norm : ‖a‖ = 1 := by
        have hx : x ≠ 0 := by
          intro h0
          

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
hn_coe : ↑n = -Padic.valuation x
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
h0 : x = 0
⊢ False

Next tactic:
rw [h0, norm_zero] at hx
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
      set a := x * (p : ℚ_[p]) ^ n with ha
      have ha_norm : ‖a‖ = 1 := by
        have hx : x ≠ 0 := by
          intro h0
          rw [h0, norm_zero] at hx
          

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬0 ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
hn_coe : ↑n = -Padic.valuation x
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
h0 : x = 0
⊢ False

Next tactic:
exact hx zero_le_one
instance isFractionRing : IsFractionRing ℤ_[p] ℚ_[p] where
  map_units' := fun ⟨x, hx⟩ => by
    rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
  surj' x := by
    by_cases hx : ‖x‖ ≤ 1
    · use (⟨x, hx⟩, 1)
      rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
    · set n := Int.toNat (-x.valuation) with hn
      have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
      set a := x * (p : ℚ_[p]) ^ n with ha
      have ha_norm : ‖a‖ = 1 := by
        have hx : x ≠ 0 := by
          intro h0
          rw [h0, norm_zero] at hx
          exact hx zero_le_one
        

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx✝ : ¬‖x‖ ≤ 1
n : ℕ := Int.toNat (-Padic.valuation x)
hn : n = Int.toNat (-Padic.valuation x)
hn_coe : ↑n = -Padic.valuation x
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
hx : x ≠ 0
⊢ ‖a‖ = 1

Next tactic:
rw [ha, padicNormE.mul, padicNormE.norm_p_pow, Padic.norm_eq_pow_val hx, ← zpow_add',
          hn_coe, neg_neg, add_left_neg, zpow_zero]