l3lab/ntp-mathlib huggingface dataset:
theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
    ‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by
  

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
⊢ ‖↑k‖ ≤ ↑p ^ (-↑n) ↔ ↑p ^ n ∣ k

Next tactic:
have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp
theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
    ‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by
  have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by 

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
⊢ ↑p ^ (-↑n) = ↑↑p ^ (-↑n)

Next tactic:
simp
theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
    ‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by
  have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp
  

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
this : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)
⊢ ‖↑k‖ ≤ ↑p ^ (-↑n) ↔ ↑p ^ n ∣ k

Next tactic:
rw [show (k : ℚ_[p]) = ((k : ℚ) : ℚ_[p]) by norm_cast, eq_padicNorm, this]
theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
    ‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by
  have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp
  rw [show (k : ℚ_[p]) = ((k : ℚ) : ℚ_[p]) by 

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
this : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)
⊢ ↑k = ↑↑k

Next tactic:
norm_cast
theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
    ‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by
  have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp
  rw [show (k : ℚ_[p]) = ((k : ℚ) : ℚ_[p]) by norm_cast, eq_padicNorm, this]
  

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
this : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)
⊢ ↑(padicNorm p ↑k) ≤ ↑↑p ^ (-↑n) ↔ ↑p ^ n ∣ k

Next tactic:
norm_cast
theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
    ‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by
  have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp
  rw [show (k : ℚ_[p]) = ((k : ℚ) : ℚ_[p]) by norm_cast, eq_padicNorm, this]
  norm_cast
  

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
n : ℕ
this : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)
⊢ padicNorm p ↑k ≤ ↑p ^ (-↑n) ↔ ↑(p ^ n) ∣ k

Next tactic:
rw [← padicNorm.dvd_iff_norm_le]
theorem eq_of_norm_add_lt_right {z1 z2 : ℚ_[p]} (h : ‖z1 + z2‖ < ‖z2‖) : ‖z1‖ = ‖z2‖ :=
  _root_.by_contradiction fun hne ↦
    not_lt_of_ge (by 

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℚ_[p]
h : ‖z1 + z2‖ < ‖z2‖
hne : ¬‖z1‖ = ‖z2‖
⊢ ‖z1 + z2‖ ≥ ‖z2‖

Next tactic:
rw [padicNormE.add_eq_max_of_ne hne]
theorem eq_of_norm_add_lt_right {z1 z2 : ℚ_[p]} (h : ‖z1 + z2‖ < ‖z2‖) : ‖z1‖ = ‖z2‖ :=
  _root_.by_contradiction fun hne ↦
    not_lt_of_ge (by rw [padicNormE.add_eq_max_of_ne hne]; 

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℚ_[p]
h : ‖z1 + z2‖ < ‖z2‖
hne : ¬‖z1‖ = ‖z2‖
⊢ max ‖z1‖ ‖z2‖ ≥ ‖z2‖

Next tactic:
apply le_max_right
theorem eq_of_norm_add_lt_left {z1 z2 : ℚ_[p]} (h : ‖z1 + z2‖ < ‖z1‖) : ‖z1‖ = ‖z2‖ :=
  _root_.by_contradiction fun hne ↦
    not_lt_of_ge (by 

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℚ_[p]
h : ‖z1 + z2‖ < ‖z1‖
hne : ¬‖z1‖ = ‖z2‖
⊢ ‖z1 + z2‖ ≥ ‖z1‖

Next tactic:
rw [padicNormE.add_eq_max_of_ne hne]
theorem eq_of_norm_add_lt_left {z1 z2 : ℚ_[p]} (h : ‖z1 + z2‖ < ‖z1‖) : ‖z1‖ = ‖z2‖ :=
  _root_.by_contradiction fun hne ↦
    not_lt_of_ge (by rw [padicNormE.add_eq_max_of_ne hne]; 

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℚ_[p]
h : ‖z1 + z2‖ < ‖z1‖
hne : ¬‖z1‖ = ‖z2‖
⊢ max ‖z1‖ ‖z2‖ ≥ ‖z1‖

Next tactic:
apply le_max_left
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
⊢ ∃ b, f ≈ const norm b

Next tactic:
have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
ε : ℚ
hε : ε > 0
⊢ ∃ i, ∀ j ≥ i, padicNormE (↑f j - ↑f i) < ε

Next tactic:
have h := isCauSeq f ε (mod_cast hε)
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
ε : ℚ
hε : ε > 0
h : ∃ i, ∀ j ≥ i, ‖↑f j - ↑f i‖ < ↑ε
⊢ ∃ i, ∀ j ≥ i, padicNormE (↑f j - ↑f i) < ε

Next tactic:
dsimp [norm] at h
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
ε : ℚ
hε : ε > 0
h : ∃ i, ∀ j ≥ i, ↑(padicNormE (↑f j - ↑f i)) < ↑ε
⊢ ∃ i, ∀ j ≥ i, padicNormE (↑f j - ↑f i) < ε

Next tactic:
exact mod_cast h
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
⊢ ∃ b, f ≈ const norm b

Next tactic:
cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    

Proof state:
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
⊢ ∃ b, f ≈ const norm b

Next tactic:
exists q
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    

Proof state:
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
⊢ f ≈ const norm q

Next tactic:
intro ε hε
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    

Proof state:
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
⊢ ∃ i, ∀ j ≥ i, ‖↑(f - const norm q) j‖ < ε

Next tactic:
cases' exists_rat_btwn hε with ε' hε'
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    

Proof state:
case intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ↑ε' ∧ ↑ε' < ε
⊢ ∃ i, ∀ j ≥ i, ‖↑(f - const norm q) j‖ < ε

Next tactic:
norm_cast at hε'
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    

Proof state:
case intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
⊢ ∃ i, ∀ j ≥ i, ‖↑(f - const norm q) j‖ < ε

Next tactic:
cases' hq ε' hε'.1 with N hN
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    cases' hq ε' hε'.1 with N hN
    

Proof state:
case intro.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
⊢ ∃ i, ∀ j ≥ i, ‖↑(f - const norm q) j‖ < ε

Next tactic:
exists N
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    cases' hq ε' hε'.1 with N hN
    exists N
    

Proof state:
case intro.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
⊢ ∀ j ≥ N, ‖↑(f - const norm q) j‖ < ε

Next tactic:
intro i hi
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    cases' hq ε' hε'.1 with N hN
    exists N
    intro i hi
    

Proof state:
case intro.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
i : ℕ
hi : i ≥ N
⊢ ‖↑(f - const norm q) i‖ < ε

Next tactic:
have h := hN i hi
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    cases' hq ε' hε'.1 with N hN
    exists N
    intro i hi
    have h := hN i hi
    

Proof state:
case intro.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
i : ℕ
hi : i ≥ N
h : padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
⊢ ‖↑(f - const norm q) i‖ < ε

Next tactic:
change norm (f i - q) < ε
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    cases' hq ε' hε'.1 with N hN
    exists N
    intro i hi
    have h := hN i hi
    change norm (f i - q) < ε
    

Proof state:
case intro.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
i : ℕ
hi : i ≥ N
h : padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
⊢ ‖↑f i - q‖ < ε

Next tactic:
refine lt_trans ?_ hε'.2
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    cases' hq ε' hε'.1 with N hN
    exists N
    intro i hi
    have h := hN i hi
    change norm (f i - q) < ε
    refine lt_trans ?_ hε'.2
    

Proof state:
case intro.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
i : ℕ
hi : i ≥ N
h : padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
⊢ ‖↑f i - q‖ < ↑ε'

Next tactic:
dsimp [norm]
instance complete : CauSeq.IsComplete ℚ_[p] norm where
  isComplete := fun f => by
    have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h
    -- Porting note: Padic.complete' works with `f i - q`, but the goal needs `q - f i`,
    -- using `rewrite [padicNormE.map_sub]` causes time out, so a separate lemma is created
    cases' Padic.complete'' ⟨f, cau_seq_norm_e⟩ with q hq
    exists q
    intro ε hε
    cases' exists_rat_btwn hε with ε' hε'
    norm_cast at hε'
    cases' hq ε' hε'.1 with N hN
    exists N
    intro i hi
    have h := hN i hi
    change norm (f i - q) < ε
    refine lt_trans ?_ hε'.2
    dsimp [norm]
    

Proof state:
case intro.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
i : ℕ
hi : i ≥ N
h : padicNormE (↑{ val := ↑f, property := cau_seq_norm_e } i - q) < ε'
⊢ ↑(padicNormE (↑f i - q)) < ↑ε'

Next tactic:
exact mod_cast h
theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (ha : 0 < a) (hf : ∀ i, ‖f i‖ ≤ a) :
    ‖f.lim‖ ≤ a := by
  -- Porting note: `Setoid.symm` cannot work out which `Setoid` to use, so instead swap the order
  -- now, I use a rewrite to swap it later
  

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
⊢ ‖CauSeq.lim f‖ ≤ a

Next tactic:
obtain ⟨N, hN⟩ := (CauSeq.equiv_lim f) _ ha
theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (ha : 0 < a) (hf : ∀ i, ‖f i‖ ≤ a) :
    ‖f.lim‖ ≤ a := by
  -- Porting note: `Setoid.symm` cannot work out which `Setoid` to use, so instead swap the order
  -- now, I use a rewrite to swap it later
  obtain ⟨N, hN⟩ := (CauSeq.equiv_lim f) _ ha
  

Proof state:
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
N : ℕ
hN : ∀ j ≥ N, ‖↑(f - const norm (CauSeq.lim f)) j‖ < a
⊢ ‖CauSeq.lim f‖ ≤ a

Next tactic:
rw [← sub_add_cancel f.lim (f N)]
theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (ha : 0 < a) (hf : ∀ i, ‖f i‖ ≤ a) :
    ‖f.lim‖ ≤ a := by
  -- Porting note: `Setoid.symm` cannot work out which `Setoid` to use, so instead swap the order
  -- now, I use a rewrite to swap it later
  obtain ⟨N, hN⟩ := (CauSeq.equiv_lim f) _ ha
  rw [← sub_add_cancel f.lim (f N)]
  

Proof state:
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
N : ℕ
hN : ∀ j ≥ N, ‖↑(f - const norm (CauSeq.lim f)) j‖ < a
⊢ ‖CauSeq.lim f - ↑f N + ↑f N‖ ≤ a

Next tactic:
refine le_trans (padicNormE.nonarchimedean _ _) ?_
theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (ha : 0 < a) (hf : ∀ i, ‖f i‖ ≤ a) :
    ‖f.lim‖ ≤ a := by
  -- Porting note: `Setoid.symm` cannot work out which `Setoid` to use, so instead swap the order
  -- now, I use a rewrite to swap it later
  obtain ⟨N, hN⟩ := (CauSeq.equiv_lim f) _ ha
  rw [← sub_add_cancel f.lim (f N)]
  refine le_trans (padicNormE.nonarchimedean _ _) ?_
  

Proof state:
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
N : ℕ
hN : ∀ j ≥ N, ‖↑(f - const norm (CauSeq.lim f)) j‖ < a
⊢ max ‖CauSeq.lim f - ↑f N‖ ‖↑f N‖ ≤ a

Next tactic:
rw [norm_sub_rev]
theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (ha : 0 < a) (hf : ∀ i, ‖f i‖ ≤ a) :
    ‖f.lim‖ ≤ a := by
  -- Porting note: `Setoid.symm` cannot work out which `Setoid` to use, so instead swap the order
  -- now, I use a rewrite to swap it later
  obtain ⟨N, hN⟩ := (CauSeq.equiv_lim f) _ ha
  rw [← sub_add_cancel f.lim (f N)]
  refine le_trans (padicNormE.nonarchimedean _ _) ?_
  rw [norm_sub_rev]
  

Proof state:
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
N : ℕ
hN : ∀ j ≥ N, ‖↑(f - const norm (CauSeq.lim f)) j‖ < a
⊢ max ‖↑f N - CauSeq.lim f‖ ‖↑f N‖ ≤ a

Next tactic:
exact max_le (le_of_lt (hN _ le_rfl)) (hf _)
instance : CompleteSpace ℚ_[p] := by
  

Proof state:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ CompleteSpace ℚ_[p]

Next tactic:
apply complete_of_cauchySeq_tendsto
instance : CompleteSpace ℚ_[p] := by
  apply complete_of_cauchySeq_tendsto
  

Proof state:
case a
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ∀ (u : ℕ → ℚ_[p]), CauchySeq u → ∃ a, Tendsto u atTop (nhds a)

Next tactic:
intro u hu
instance : CompleteSpace ℚ_[p] := by
  apply complete_of_cauchySeq_tendsto
  intro u hu
  

Proof state:
case a
p : ℕ
hp : Fact (Nat.Prime p)
u : ℕ → ℚ_[p]
hu : CauchySeq u
⊢ ∃ a, Tendsto u atTop (nhds a)

Next tactic:
let c : CauSeq ℚ_[p] norm := ⟨u, Metric.cauchySeq_iff'.mp hu⟩
instance : CompleteSpace ℚ_[p] := by
  apply complete_of_cauchySeq_tendsto
  intro u hu
  let c : CauSeq ℚ_[p] norm := ⟨u, Metric.cauchySeq_iff'.mp hu⟩
  

Proof state:
case a
p : ℕ
hp : Fact (Nat.Prime p)
u : ℕ → ℚ_[p]
hu : CauchySeq u
c : CauSeq ℚ_[p] norm := { val := u, property := (_ : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε) }
⊢ ∃ a, Tendsto u atTop (nhds a)

Next tactic:
refine' ⟨c.lim, fun s h ↦ _⟩
instance : CompleteSpace ℚ_[p] := by
  apply complete_of_cauchySeq_tendsto
  intro u hu
  let c : CauSeq ℚ_[p] norm := ⟨u, Metric.cauchySeq_iff'.mp hu⟩
  refine' ⟨c.lim, fun s h ↦ _⟩
  

Proof state:
case a
p : ℕ
hp : Fact (Nat.Prime p)
u : ℕ → ℚ_[p]
hu : CauchySeq u
c : CauSeq ℚ_[p] norm := { val := u, property := (_ : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε) }
s : Set ℚ_[p]
h : s ∈ nhds (CauSeq.lim c)
⊢ s ∈ map u atTop

Next tactic:
rcases Metric.mem_nhds_iff.1 h with ⟨ε, ε0, hε⟩
instance : CompleteSpace ℚ_[p] := by
  apply complete_of_cauchySeq_tendsto
  intro u hu
  let c : CauSeq ℚ_[p] norm := ⟨u, Metric.cauchySeq_iff'.mp hu⟩
  refine' ⟨c.lim, fun s h ↦ _⟩
  rcases Metric.mem_nhds_iff.1 h with ⟨ε, ε0, hε⟩
  

Proof state:
case a.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
u : ℕ → ℚ_[p]
hu : CauchySeq u
c : CauSeq ℚ_[p] norm := { val := u, property := (_ : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε) }
s : Set ℚ_[p]
h : s ∈ nhds (CauSeq.lim c)
ε : ℝ
ε0 : ε > 0
hε : ball (CauSeq.lim c) ε ⊆ s
⊢ s ∈ map u atTop

Next tactic:
have := c.equiv_lim ε ε0
instance : CompleteSpace ℚ_[p] := by
  apply complete_of_cauchySeq_tendsto
  intro u hu
  let c : CauSeq ℚ_[p] norm := ⟨u, Metric.cauchySeq_iff'.mp hu⟩
  refine' ⟨c.lim, fun s h ↦ _⟩
  rcases Metric.mem_nhds_iff.1 h with ⟨ε, ε0, hε⟩
  have := c.equiv_lim ε ε0
  

Proof state:
case a.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
u : ℕ → ℚ_[p]
hu : CauchySeq u
c : CauSeq ℚ_[p] norm := { val := u, property := (_ : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε) }
s : Set ℚ_[p]
h : s ∈ nhds (CauSeq.lim c)
ε : ℝ
ε0 : ε > 0
hε : ball (CauSeq.lim c) ε ⊆ s
this : ∃ i, ∀ j ≥ i, ‖↑(c - const norm (CauSeq.lim c)) j‖ < ε
⊢ s ∈ map u atTop

Next tactic:
simp only [mem_map, mem_atTop_sets, mem_setOf_eq]
instance : CompleteSpace ℚ_[p] := by
  apply complete_of_cauchySeq_tendsto
  intro u hu
  let c : CauSeq ℚ_[p] norm := ⟨u, Metric.cauchySeq_iff'.mp hu⟩
  refine' ⟨c.lim, fun s h ↦ _⟩
  rcases Metric.mem_nhds_iff.1 h with ⟨ε, ε0, hε⟩
  have := c.equiv_lim ε ε0
  simp only [mem_map, mem_atTop_sets, mem_setOf_eq]
  

Proof state:
case a.intro.intro
p : ℕ
hp : Fact (Nat.Prime p)
u : ℕ → ℚ_[p]
hu : CauchySeq u
c : CauSeq ℚ_[p] norm := { val := u, property := (_ : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε) }
s : Set ℚ_[p]
h : s ∈ nhds (CauSeq.lim c)
ε : ℝ
ε0 : ε > 0
hε : ball (CauSeq.lim c) ε ⊆ s
this : ∃ i, ∀ j ≥ i, ‖↑(c - const norm (CauSeq.lim c)) j‖ < ε
⊢ ∃ a, ∀ b ≥ a, b ∈ u ⁻¹' s

Next tactic:
exact this.imp fun N hN n hn ↦ hε (hN n hn)