l3lab/ntp-mathlib huggingface dataset:
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      

Proof state:
case a
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2)) + padicNormE (↑f (max N N2) - ↑(limSeq f (max N N2))) <
    ε / 3 + ε / 3 + ε / 3

Next tactic:
apply _root_.add_lt_add
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · 

Proof state:
case a.h₁
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2)) < ε / 3 + ε / 3

Next tactic:
suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
this : padicNormE (↑(limSeq f j) - ↑f j + (↑f j - ↑f (max N N2))) < ε / 3 + ε / 3
⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2)) < ε / 3 + ε / 3

Next tactic:
simpa only [sub_add_sub_cancel]
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        

Proof state:
case a.h₁
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f j + (↑f j - ↑f (max N N2))) < ε / 3 + ε / 3

Next tactic:
apply lt_of_le_of_lt
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · 

Proof state:
case a.h₁.a
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f j + (↑f j - ↑f (max N N2))) ≤ ?a.h₁.b✝

Next tactic:
apply padicNormE.add_le
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · apply padicNormE.add_le
        · 

Proof state:
case a.h₁.a
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f j) + padicNormE (↑f j - ↑f (max N N2)) < ε / 3 + ε / 3

Next tactic:
apply _root_.add_lt_add
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · apply padicNormE.add_le
        · apply _root_.add_lt_add
          · 

Proof state:
case a.h₁.a.h₁
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f j) < ε / 3

Next tactic:
rw [padicNormE.map_sub]
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · apply padicNormE.add_le
        · apply _root_.add_lt_add
          · rw [padicNormE.map_sub]
            

Proof state:
case a.h₁.a.h₁
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑f j - ↑(limSeq f j)) < ε / 3

Next tactic:
apply mod_cast hN j
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · apply padicNormE.add_le
        · apply _root_.add_lt_add
          · rw [padicNormE.map_sub]
            apply mod_cast hN j
            

Proof state:
case a.h₁.a.h₁
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ j ≥ N

Next tactic:
exact le_of_max_le_left hj
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · apply padicNormE.add_le
        · apply _root_.add_lt_add
          · rw [padicNormE.map_sub]
            apply mod_cast hN j
            exact le_of_max_le_left hj
          · 

Proof state:
case a.h₁.a.h₂
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑f j - ↑f (max N N2)) < ε / 3

Next tactic:
exact hN2 _ (le_of_max_le_right hj) _ (le_max_right _ _)
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · apply padicNormE.add_le
        · apply _root_.add_lt_add
          · rw [padicNormE.map_sub]
            apply mod_cast hN j
            exact le_of_max_le_left hj
          · exact hN2 _ (le_of_max_le_right hj) _ (le_max_right _ _)
      · 

Proof state:
case a.h₂
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑f (max N N2) - ↑(limSeq f (max N N2))) < ε / 3

Next tactic:
apply mod_cast hN (max N N2)
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
  have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
  let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
  let ⟨N2, hN2⟩ := f.cauchy₂ hε3
  exists max N N2
  intro j hj
  suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
  · apply lt_of_le_of_lt
    · apply padicNormE.add_le
    · rw [← add_thirds ε]
      apply _root_.add_lt_add
      · suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
          simpa only [sub_add_sub_cancel]
        apply lt_of_le_of_lt
        · apply padicNormE.add_le
        · apply _root_.add_lt_add
          · rw [padicNormE.map_sub]
            apply mod_cast hN j
            exact le_of_max_le_left hj
          · exact hN2 _ (le_of_max_le_right hj) _ (le_max_right _ _)
      · apply mod_cast hN (max N N2)
        

Proof state:
case a.h₂
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ max N N2 ≥ N

Next tactic:
apply le_max_left
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
⊢ ∃ N, ∀ i ≥ N, padicNormE (Padic.lim f - ↑f i) < ε

Next tactic:
obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    

Proof state:
case intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
⊢ ∃ N, ∀ i ≥ N, padicNormE (Padic.lim f - ↑f i) < ε

Next tactic:
obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
⊢ ∃ N, ∀ i ≥ N, padicNormE (Padic.lim f - ↑f i) < ε

Next tactic:
refine' ⟨max N N2, fun i hi ↦ _⟩
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    refine' ⟨max N N2, fun i hi ↦ _⟩
    

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
i : ℕ
hi : i ≥ max N N2
⊢ padicNormE (Padic.lim f - ↑f i) < ε

Next tactic:
rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    refine' ⟨max N N2, fun i hi ↦ _⟩
    rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]
    

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
i : ℕ
hi : i ≥ max N N2
⊢ padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i) + (↑(↑(Padic.lim' f) i) - ↑f i)) < ε

Next tactic:
refine' (padicNormE.add_le _ _).trans_lt _
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    refine' ⟨max N N2, fun i hi ↦ _⟩
    rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]
    refine' (padicNormE.add_le _ _).trans_lt _
    

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
i : ℕ
hi : i ≥ max N N2
⊢ padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) + padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε

Next tactic:
rw [← add_halves ε]
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    refine' ⟨max N N2, fun i hi ↦ _⟩
    rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]
    refine' (padicNormE.add_le _ _).trans_lt _
    rw [← add_halves ε]
    

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
i : ℕ
hi : i ≥ max N N2
⊢ padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) + padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε / 2 + ε / 2

Next tactic:
apply _root_.add_lt_add
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    refine' ⟨max N N2, fun i hi ↦ _⟩
    rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]
    refine' (padicNormE.add_le _ _).trans_lt _
    rw [← add_halves ε]
    apply _root_.add_lt_add
    · 

Proof state:
case intro.intro.h₁
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
i : ℕ
hi : i ≥ max N N2
⊢ padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) < ε / 2

Next tactic:
apply hN2 _ (le_of_max_le_right hi)
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    refine' ⟨max N N2, fun i hi ↦ _⟩
    rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]
    refine' (padicNormE.add_le _ _).trans_lt _
    rw [← add_halves ε]
    apply _root_.add_lt_add
    · apply hN2 _ (le_of_max_le_right hi)
    · 

Proof state:
case intro.intro.h₂
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
i : ℕ
hi : i ≥ max N N2
⊢ padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε / 2

Next tactic:
rw [padicNormE.map_sub]
theorem complete' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - f i : ℚ_[p]) < ε :=
  ⟨lim f, fun ε hε ↦ by
    obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)
    obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)
    refine' ⟨max N N2, fun i hi ↦ _⟩
    rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]
    refine' (padicNormE.add_le _ _).trans_lt _
    rw [← add_halves ε]
    apply _root_.add_lt_add
    · apply hN2 _ (le_of_max_le_right hi)
    · rw [padicNormE.map_sub]
      

Proof state:
case intro.intro.h₂
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 2
N2 : ℕ
hN2 : ∀ i ≥ N2, padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2
i : ℕ
hi : i ≥ max N N2
⊢ padicNormE (↑f i - ↑(↑(Padic.lim' f) i)) < ε / 2

Next tactic:
exact hN _ (le_of_max_le_left hi)
theorem complete'' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (f i - q : ℚ_[p]) < ε := by
  

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
⊢ ∃ q, ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑f i - q) < ε

Next tactic:
obtain ⟨x, hx⟩ := complete' f
theorem complete'' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (f i - q : ℚ_[p]) < ε := by
  obtain ⟨x, hx⟩ := complete' f
  

Proof state:
case intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
x : ℚ_[p]
hx : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (x - ↑f i) < ε
⊢ ∃ q, ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑f i - q) < ε

Next tactic:
refine ⟨x, fun ε hε => ?_⟩
theorem complete'' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (f i - q : ℚ_[p]) < ε := by
  obtain ⟨x, hx⟩ := complete' f
  refine ⟨x, fun ε hε => ?_⟩
  

Proof state:
case intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
x : ℚ_[p]
hx : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (x - ↑f i) < ε
ε : ℚ
hε : ε > 0
⊢ ∃ N, ∀ i ≥ N, padicNormE (↑f i - x) < ε

Next tactic:
obtain ⟨N, hN⟩ := hx ε hε
theorem complete'' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (f i - q : ℚ_[p]) < ε := by
  obtain ⟨x, hx⟩ := complete' f
  refine ⟨x, fun ε hε => ?_⟩
  obtain ⟨N, hN⟩ := hx ε hε
  

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
x : ℚ_[p]
hx : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (x - ↑f i) < ε
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (x - ↑f i) < ε
⊢ ∃ N, ∀ i ≥ N, padicNormE (↑f i - x) < ε

Next tactic:
refine ⟨N, fun i hi => ?_⟩
theorem complete'' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (f i - q : ℚ_[p]) < ε := by
  obtain ⟨x, hx⟩ := complete' f
  refine ⟨x, fun ε hε => ?_⟩
  obtain ⟨N, hN⟩ := hx ε hε
  refine ⟨N, fun i hi => ?_⟩
  

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
x : ℚ_[p]
hx : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (x - ↑f i) < ε
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (x - ↑f i) < ε
i : ℕ
hi : i ≥ N
⊢ padicNormE (↑f i - x) < ε

Next tactic:
rw [padicNormE.map_sub]
theorem complete'' : ∃ q : ℚ_[p], ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (f i - q : ℚ_[p]) < ε := by
  obtain ⟨x, hx⟩ := complete' f
  refine ⟨x, fun ε hε => ?_⟩
  obtain ⟨N, hN⟩ := hx ε hε
  refine ⟨N, fun i hi => ?_⟩
  rw [padicNormE.map_sub]
  

Proof state:
case intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
x : ℚ_[p]
hx : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (x - ↑f i) < ε
ε : ℚ
hε : ε > 0
N : ℕ
hN : ∀ i ≥ N, padicNormE (x - ↑f i) < ε
i : ℕ
hi : i ≥ N
⊢ padicNormE (x - ↑f i) < ε

Next tactic:
exact hN i hi
instance metricSpace : MetricSpace ℚ_[p] where
  dist_self := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ∀ (x : ℚ_[p]), dist x x = 0

Next tactic:
simp [dist]
instance metricSpace : MetricSpace ℚ_[p] where
  dist_self := by simp [dist]
  dist := dist
  dist_comm x y := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
x y : ℚ_[p]
⊢ dist x y = dist y x

Next tactic:
simp [dist, ← padicNormE.map_neg (x - y : ℚ_[p])]
instance metricSpace : MetricSpace ℚ_[p] where
  dist_self := by simp [dist]
  dist := dist
  dist_comm x y := by simp [dist, ← padicNormE.map_neg (x - y : ℚ_[p])]
  dist_triangle x y z := by
    

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
x y z : ℚ_[p]
⊢ dist x z ≤ dist x y + dist y z

Next tactic:
dsimp [dist]