awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : ℚ
h :
  LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
inr :
  ∀
    (h' :
      Eq (α := WithZero (Multiplicative ℤ))
        ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ))),
    LT.lt (α := WithZero (Multiplicative ℤ))
      ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
      ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
have inr₁ : ∀
  (h' :
    LT.lt (α := WithZero (Multiplicative ℤ))
      ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))),
  LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
Tactic: rcases (Int.padicValuation_le_one p x.num).eq_or_lt with h' | h'
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : ℚ
h :
  LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
h' :
  LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
have h₁ : Dvd.dvd (α := ℤ) (↑p : ℤ) (↑x.den : ℤ)
Tactic: rw [Int.padicValuation_lt_one_iff] at h h'
Cleared hypotheses: h, h'
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : ℚ
h :
  LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
h' :
  LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
h₁ : Dvd.dvd (α := ℤ) (↑p : ℤ) (↑x.den : ℤ)
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
have h'₁ : (↑p : ℤ) ∣ x.num
Tactic: rw [Int.padicValuation_lt_one_iff] at h h'
Cleared hypotheses: h, h'
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : ℚ
h : Dvd.dvd (α := ℤ) (↑p : ℤ) (↑x.den : ℤ)
h' : (↑p : ℤ) ∣ x.num
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) (↑x.den : ℤ) : WithZero (Multiplicative ℤ))
    ((Int.padicValuation p : (a : ℤ) → WithZero (Multiplicative ℤ)) x.num : WithZero (Multiplicative ℤ))
have inr : False
Tactic: exfalso
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : ℚ
h : Dvd.dvd (α := ℤ) (↑p : ℤ) (↑x.den : ℤ)
h' : (↑p : ℤ) ∣ x.num
⊢ False
have h₁ : p ∣ x.den
Tactic: rw [Int.natCast_dvd_natCast] at h
Cleared hypotheses: h, h'
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : ℚ
h : p ∣ x.den
h' : (↑p : ℤ) ∣ x.num
⊢ False
have h'₁ : p ∣ x.num.natAbs
Tactic: rw [Int.natCast_dvd] at h'
Cleared hypotheses: h'
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : Multiplicative ℤ
⊢ ∃ (a : ℚ), (padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a = (↑x : WithZero (Multiplicative ℤ))
have coe : ∀ (x : ℤ),
  ∃ (a : ℚ),
    Eq (α := WithZero (Multiplicative ℤ))
      ((padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) x) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
Tactic: induction x with | ofAdd x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ
⊢ ∃ (a : ℚ),
    Eq (α := WithZero (Multiplicative ℤ))
      (({
              toFun := fun (x : ℚ) ↦
                if x = 0 then 0
                else
                  (↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-padicValRat p x)) :
                    WithZero (Multiplicative ℤ)),
              map_zero' := ⋯ } :
            (a : ℚ) → WithZero (Multiplicative ℤ))
          a :
        WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) x) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
have coe : ∀ (hx : 0 ≤ x),
  ∃ (a : ℚ),
    Eq (α := WithZero (Multiplicative ℤ))
      (({
              toFun := fun (x : ℚ) ↦
                if x = 0 then 0
                else
                  (↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-padicValRat p x)) :
                    WithZero (Multiplicative ℤ)),
              map_zero' := sorry } :
            (a : ℚ) → WithZero (Multiplicative ℤ))
          a :
        WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) x) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
Tactic: rcases le_or_gt 0 x with (hx | hx)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ
coe :
  ∀ (hx : 0 ≤ x),
    ∃ (a : ℚ),
      Eq (α := WithZero (Multiplicative ℤ))
        (({
                toFun := fun (x : ℚ) ↦
                  if x = 0 then 0
                  else
                    (↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-padicValRat p x)) :
                      WithZero (Multiplicative ℤ)),
                map_zero' := ⋯ } :
              (a : ℚ) → WithZero (Multiplicative ℤ))
            a :
          WithZero (Multiplicative ℤ))
        ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) x) : WithZero (Multiplicative ℤ)) :
          WithZero (Multiplicative ℤ))
⊢ ∃ (a : ℚ),
    Eq (α := WithZero (Multiplicative ℤ))
      (({
              toFun := fun (x : ℚ) ↦
                if x = 0 then 0
                else
                  (↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-padicValRat p x)) :
                    WithZero (Multiplicative ℤ)),
              map_zero' := ⋯ } :
            (a : ℚ) → WithZero (Multiplicative ℤ))
          a :
        WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) x) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
have coe₁ : ∀ (hx : x < 0),
  ∃ (a : ℚ),
    Eq (α := WithZero (Multiplicative ℤ))
      (({
              toFun := fun (x : ℚ) ↦
                if x = 0 then 0
                else
                  (↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-padicValRat p x)) :
                    WithZero (Multiplicative ℤ)),
              map_zero' := sorry } :
            (a : ℚ) → WithZero (Multiplicative ℤ))
          a :
        WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) x) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
Tactic: rcases le_or_gt 0 x with (hx | hx)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : WithZero (Multiplicative ℤ)
⊢ ∃ (a : ℚ), (padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a = x
have zero : ∃ (a : ℚ),
  Eq (α := WithZero (Multiplicative ℤ))
    ((padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
Tactic: induction x with
  | zero => simp
  | coe x =>
    induction x with | ofAdd x
    simp_rw [Rat.padicValuation, WithZero.exp, Valuation.coe_mk, MonoidWithZeroHom.coe_mk]
    rcases le_or_gt 0 x with (hx | hx)
    · exact ⟨(p ^ x.natAbs)⁻¹, by simp [hp.out.ne_zero, hx]⟩
    · exact ⟨p ^ x.natAbs, by simp [hp.out.ne_zero, padicValRat.pow, abs_eq_neg_self.2 hx.le]⟩
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : WithZero (Multiplicative ℤ)
zero :
  ∃ (a : ℚ),
    Eq (α := WithZero (Multiplicative ℤ))
      ((padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a : WithZero (Multiplicative ℤ))
      (0 : WithZero (Multiplicative ℤ))
⊢ ∃ (a : ℚ), (padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a = x
have coe : ∀ (x : Multiplicative ℤ),
  ∃ (a : ℚ), (padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a = (↑x : WithZero (Multiplicative ℤ))
Tactic: induction x with
  | zero => simp
  | coe x =>
    induction x with | ofAdd x
    simp_rw [Rat.padicValuation, WithZero.exp, Valuation.coe_mk, MonoidWithZeroHom.coe_mk]
    rcases le_or_gt 0 x with (hx | hx)
    · exact ⟨(p ^ x.natAbs)⁻¹, by simp [hp.out.ne_zero, hx]⟩
    · exact ⟨p ^ x.natAbs, by simp [hp.out.ne_zero, padicValRat.pow, abs_eq_neg_self.2 hx.le]⟩
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
⊢ padicNorm p ((↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
have this₁ : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
Tactic: have : padicNorm p (f n - f m) < ε := hN2 _ (max_le_iff.1 hn).2 _ (max_le_iff.1 hm).2
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this_1 : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
⊢ padicNorm p ((↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
have this₁ : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < padicNorm p ((↑f : ℕ → ℚ) n)
Tactic: have : padicNorm p (f n - f m) < padicNorm p (f n) :=
      lt_of_lt_of_le this <| hN1 _ (max_le_iff.1 hn).1
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this_1 : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this_2 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
this : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < padicNorm p ((↑f : ℕ → ℚ) n)
⊢ padicNorm p ((↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
have this₁ : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
Tactic: have : padicNorm p (f n - f m) < max (padicNorm p (f n)) (padicNorm p (f m)) :=
      lt_max_iff.2 (Or.inl this)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this_1 : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this_2 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
this_3 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < padicNorm p ((↑f : ℕ → ℚ) n)
this : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
⊢ padicNorm p ((↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
have by_contra : ∀ (hne : ¬padicNorm p ((↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)), False
Tactic: by_contra hne
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this_1 : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this_2 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
this_3 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < padicNorm p ((↑f : ℕ → ℚ) n)
this : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
hne : ¬padicNorm p ((↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
⊢ False
have hne₁ : ¬padicNorm p (-(↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
Tactic: rw [← padicNorm.neg (f m)] at hne
Cleared hypotheses: hne
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this_1 : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this_2 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
this_3 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < padicNorm p ((↑f : ℕ → ℚ) n)
this : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
hne : ¬padicNorm p (-(↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
⊢ False
have hnam : padicNorm p (-(↑f : ℕ → ℚ) m + (↑f : ℕ → ℚ) n) = max (padicNorm p (-(↑f : ℕ → ℚ) m)) (padicNorm p ((↑f : ℕ → ℚ) n))
Tactic: have hnam := add_eq_max_of_ne hne
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this_1 : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this_2 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
this_3 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < padicNorm p ((↑f : ℕ → ℚ) n)
this : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
hne : ¬padicNorm p (-(↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
hnam :
  padicNorm p (-(↑f : ℕ → ℚ) m + (↑f : ℕ → ℚ) n) = max (padicNorm p (-(↑f : ℕ → ℚ) m)) (padicNorm p ((↑f : ℕ → ℚ) n))
⊢ False
have hnam₁ : padicNorm p (-(↑f : ℕ → ℚ) m + (↑f : ℕ → ℚ) n) = max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
Tactic: rw [padicNorm.neg, max_comm] at hnam
Cleared hypotheses: hnam
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this_1 : ∃ ε > 0, ∃ (N1 : ℕ), ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p ((↑f : ℕ → ℚ) j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p ((↑f : ℕ → ℚ) j - (↑f : ℕ → ℚ) k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this_2 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < ε
this_3 : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < padicNorm p ((↑f : ℕ → ℚ) n)
this : padicNorm p ((↑f : ℕ → ℚ) n - (↑f : ℕ → ℚ) m) < max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
hne : ¬padicNorm p (-(↑f : ℕ → ℚ) m) = padicNorm p ((↑f : ℕ → ℚ) n)
hnam :
  padicNorm p (-(↑f : ℕ → ℚ) m + (↑f : ℕ → ℚ) n) = max (padicNorm p ((↑f : ℕ → ℚ) n)) (padicNorm p ((↑f : ℕ → ℚ) m))
⊢ False
have this₁ : padicNorm p (-(↑f : ℕ → ℚ) m + (↑f : ℕ → ℚ) n) < padicNorm p (-(↑f : ℕ → ℚ) m + (↑f : ℕ → ℚ) n)
Tactic: rw [← hnam, sub_eq_add_neg, add_comm] at this
Cleared hypotheses: this, hne, hnam
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
⊢ f.norm = 0 ↔ f ≈ 0
have mp : f.norm = 0 → f ≈ 0
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
mp : f.norm = 0 → f ≈ 0
⊢ f.norm = 0 ↔ f ≈ 0
have mpr : f ≈ 0 → f.norm = 0
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
h : f.norm = 0
⊢ f ≈ 0
have mp : ∀ (hf : ¬f ≈ 0), False
Tactic: by_contra hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
h : Eq (α := ℚ) (if hf : f ≈ 0 then 0 else padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf))) 0
hf : ¬f ≈ 0
⊢ False
have h₁ : padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = 0
Tactic: split_ifs at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
h : padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = 0
⊢ False
have mp : f ≈ 0
Tactic: apply hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
h : padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = 0
ε : ℚ
hε : ε > 0
⊢ ∃ (i : ℕ), ∀ j ≥ i, padicNorm p ((↑(f - 0) : ℕ → ℚ) j) < ε
have mp : ∀ j ≥ stationaryPoint hf, padicNorm p ((↑(f - 0) : ℕ → ℚ) j) < ε
Tactic: exists stationaryPoint hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
h : padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = 0
ε : ℚ
hε : ε > 0
j : ℕ
hj : j ≥ stationaryPoint hf
⊢ padicNorm p ((↑(f - 0) : ℕ → ℚ) j) < ε
have heq : padicNorm p ((↑f : ℕ → ℚ) j) = padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf))
Tactic: have heq := stationaryPoint_spec hf le_rfl hj
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v2 v3 : ℕ
⊢ padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = padicNorm p ((↑f : ℕ → ℚ) (max (stationaryPoint hf) (max v2 v3)))
have a : stationaryPoint hf ≤ max (stationaryPoint hf) (max v2 v3)
Tactic: apply stationaryPoint_spec hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v2 v3 : ℕ
a : stationaryPoint hf ≤ max (stationaryPoint hf) (max v2 v3)
⊢ padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = padicNorm p ((↑f : ℕ → ℚ) (max (stationaryPoint hf) (max v2 v3)))
have a₁ : stationaryPoint hf ≤ stationaryPoint hf
Tactic: apply stationaryPoint_spec hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v3 : ℕ
⊢ padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = padicNorm p ((↑f : ℕ → ℚ) (max v1 (max (stationaryPoint hf) v3)))
have a : stationaryPoint hf ≤ max v1 (max (stationaryPoint hf) v3)
Tactic: apply stationaryPoint_spec hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v3 : ℕ
a : stationaryPoint hf ≤ max v1 (max (stationaryPoint hf) v3)
⊢ padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = padicNorm p ((↑f : ℕ → ℚ) (max v1 (max (stationaryPoint hf) v3)))
have a₁ : stationaryPoint hf ≤ stationaryPoint hf
Tactic: apply stationaryPoint_spec hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v3 : ℕ
⊢ stationaryPoint hf ≤ max v1 (max (stationaryPoint hf) v3)
have a : stationaryPoint hf ≤ max (stationaryPoint hf) v3
Tactic: apply le_trans
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v3 : ℕ
a : stationaryPoint hf ≤ max (stationaryPoint hf) v3
⊢ stationaryPoint hf ≤ max v1 (max (stationaryPoint hf) v3)
have a₁ : max (stationaryPoint hf) v3 ≤ max v1 (max (stationaryPoint hf) v3)
Tactic: apply le_trans
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v2 : ℕ
⊢ padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = padicNorm p ((↑f : ℕ → ℚ) (max v1 (max v2 (stationaryPoint hf))))
have a : stationaryPoint hf ≤ max v1 (max v2 (stationaryPoint hf))
Tactic: apply stationaryPoint_spec hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v2 : ℕ
a : stationaryPoint hf ≤ max v1 (max v2 (stationaryPoint hf))
⊢ padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) = padicNorm p ((↑f : ℕ → ℚ) (max v1 (max v2 (stationaryPoint hf))))
have a₁ : stationaryPoint hf ≤ stationaryPoint hf
Tactic: apply stationaryPoint_spec hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v2 : ℕ
⊢ stationaryPoint hf ≤ max v1 (max v2 (stationaryPoint hf))
have a : stationaryPoint hf ≤ max v2 (stationaryPoint hf)
Tactic: apply le_trans
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v2 : ℕ
a : stationaryPoint hf ≤ max v2 (stationaryPoint hf)
⊢ stationaryPoint hf ≤ max v1 (max v2 (stationaryPoint hf))
have a₁ : max v2 (stationaryPoint hf) ≤ max v1 (max v2 (stationaryPoint hf))
Tactic: apply le_trans
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
⊢ f.norm = HPow.hPow (α := ℚ) (↑p : ℚ) (-f.valuation)
have hnc : ¬(↑f : ℕ → ℚ) (stationaryPoint hf) = 0
Tactic: rw [norm, valuation, dif_neg hf, dif_neg hf, padicNorm, if_neg]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
H : (↑f : ℕ → ℚ) (stationaryPoint hf) = 0
⊢ False
have hnc : LimZero f
Tactic: apply CauSeq.not_limZero_of_not_congr_zero hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
H : (↑f : ℕ → ℚ) (stationaryPoint hf) = 0
ε : ℚ
hε : ε > 0
⊢ ∃ (i : ℕ), ∀ j ≥ i, padicNorm p ((↑f : ℕ → ℚ) j) < ε
have h : ∀ j ≥ stationaryPoint hf, padicNorm p ((↑f : ℕ → ℚ) j) < ε
Tactic: use stationaryPoint hf
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNumbers (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
H : (↑f : ℕ → ℚ) (stationaryPoint hf) = 0
ε : ℚ
hε : ε > 0
n : ℕ
hn : n ≥ stationaryPoint hf
⊢ padicNorm p ((↑f : ℕ → ℚ) n) < ε
have h : padicNorm p ((↑f : ℕ → ℚ) (stationaryPoint hf)) < ε
Tactic: rw [stationaryPoint_spec hf le_rfl hn]