-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
theorem defn (f : PadicSeq p) {ε : ℚ} (hε : 0 < ε) :
    ∃ N, ∀ i ≥ N, padicNormE (Padic.mk f - f i : ℚ_[p]) < ε

Proof state:
case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
hge : ε ≤ padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
hngen : stationaryPoint hne ≤ N
⊢ ε > padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))

Tactic:
have := stationaryPoint_spec hne le_rfl hngen

Result:
case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
hge : ε ≤ padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
hngen : stationaryPoint hne ≤ N
this :
  padicNorm p (↑(f - const (padicNorm p) (↑f i)) N) =
    padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
⊢ ε > padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))

Proof state:
case intro.intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hge :
  ε ≤
    if hf : f - const (padicNorm p) (↑f i) ≈ 0 then 0
    else padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hf))
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
⊢ False

Tactic:
split_ifs at hge

Result:
case intro.intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
hge : ε ≤ padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
⊢ False

Proof state:
case intro.intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hge : ε ≤ (f - const (padicNorm p) (↑f i)).norm
⊢ False

Tactic:
have hne : ¬f - const (padicNorm p) (f i) ≈ 0 := fun h ↦ by
  rw [PadicSeq.norm, dif_pos h] at hge
  exact not_lt_of_ge hge hε

Result:
case intro.intro.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hge : ε ≤ (f - const (padicNorm p) (↑f i)).norm
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
⊢ False

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
⊢ ∃ N, ∀ i ≥ N, Quotient.lift PadicSeq.norm ⋯ (Padic.mk f - ↑(↑f i)) < ε

Tactic:
suffices hyp : ∃ N, ∀ i ≥ N, (f - const _ (f i)).norm < ε by peel hyp with N; use N

Result:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
⊢ ∃ N, ∀ i ≥ N, (f - const (padicNorm p) (↑f i)).norm < ε

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
hge : ε ≤ padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
⊢ ε > padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))

Tactic:
cases _root_.le_total N (stationaryPoint hne) with
| inl hgen =>
  exact hN _ hgen _ hi
| inr hngen =>
  have := stationaryPoint_spec hne le_rfl hngen
  rw [← this]
  exact hN _ le_rfl _ hi

Result:
no goals