awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : norm (E := ℚ_[p]) (↑r : ℚ_[p]) ≤ 1
norm_denom_lt : norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) < 1
hr : norm (E := ℚ_[p]) (↑r : ℚ_[p]) * norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
key : norm (E := ℚ_[p]) (↑r.num : ℚ_[p]) < 1
this : (↑p : ℤ) ∣ r.num ∧ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑r.den : ℤ)
⊢ False
have apply : p ∣ 1
Tactic: apply hp_prime.1.not_dvd_one
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : norm (E := ℚ_[p]) (↑r : ℚ_[p]) ≤ 1
norm_denom_lt : norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) < 1
hr : norm (E := ℚ_[p]) (↑r : ℚ_[p]) * norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
key : norm (E := ℚ_[p]) (↑r.num : ℚ_[p]) < 1
this : (↑p : ℤ) ∣ r.num ∧ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑r.den : ℤ)
⊢ p ∣ 1
have rwa : p ∣ r.num.natAbs.gcd r.den
Tactic: rwa [← r.reduced.gcd_eq_one, Nat.dvd_gcd_iff, ← Int.natCast_dvd, ← Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : norm (E := ℚ_[p]) (↑r : ℚ_[p]) ≤ 1
norm_denom_lt : norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) < 1
hr : norm (E := ℚ_[p]) (↑r : ℚ_[p]) * norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
key : norm (E := ℚ_[p]) (↑r.num : ℚ_[p]) < 1
this : (↑p : ℤ) ∣ r.num ∧ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑r.den : ℤ)
rwa : p ∣ r.num.natAbs.gcd r.den
⊢ p ∣ 1
have rwa₁ : p ∣ r.num.natAbs ∧ p ∣ r.den
Tactic: rwa [← r.reduced.gcd_eq_one, Nat.dvd_gcd_iff, ← Int.natCast_dvd, ← Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : norm (E := ℚ_[p]) (↑r : ℚ_[p]) ≤ 1
norm_denom_lt : norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) < 1
hr : norm (E := ℚ_[p]) (↑r : ℚ_[p]) * norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
key : norm (E := ℚ_[p]) (↑r.num : ℚ_[p]) < 1
this : (↑p : ℤ) ∣ r.num ∧ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑r.den : ℤ)
rwa : p ∣ r.num.natAbs.gcd r.den
rwa₁ : p ∣ r.num.natAbs ∧ p ∣ r.den
⊢ p ∣ 1
have rwa₂ : (↑p : ℤ) ∣ r.num ∧ p ∣ r.den
Tactic: rwa [← r.reduced.gcd_eq_one, Nat.dvd_gcd_iff, ← Int.natCast_dvd, ← Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
⊢ ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔ n ≤ x.valuation
have rw : LT.lt (α := ℝ) 1 (↑p : ℝ)
Tactic: rw [norm_eq_zpow_neg_valuation hx, zpow_le_zpow_iff_right₀, neg_le_neg_iff, Nat.cast_le]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
⊢ Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x ↔ n ≤ x.valuation
have rw : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ x ↔ n ≤ x.valuation
Tactic: rw [Ideal.mem_span_singleton]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ x ↔ n ≤ x.valuation
have mp : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ x → n ≤ x.valuation
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
mp : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ x → n ≤ x.valuation
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ x ↔ n ≤ x.valuation
have mpr : n ≤ x.valuation → HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ x
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hx : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c ≠ 0
this : c ≠ 0
⊢ n ≤ (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c).valuation
have rw : n ≤ n + c.valuation
Tactic: rw [valuation_p_pow_mul _ _ this]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hx : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c ≠ 0
⊢ n ≤ (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c).valuation
have mp : c ≠ 0
Tactic: suffices c ≠ 0 by
      rw [valuation_p_pow_mul _ _ this]
      exact le_self_add
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hx : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c ≠ 0
⊢ c ≠ 0
have mp : ∀ (hx : c = 0), HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c = 0
Tactic: contrapose hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hx : c = 0
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c = 0
have mp : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * 0 = 0
Tactic: rw [hx, mul_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hx : c = 0
mp : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * 0 = 0
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c = 0
have mp₁ : Eq (α := ℤ_[p]) 0 0
Tactic: rw [hx, mul_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
⊢ n ≤ x.valuation → HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ x
have mpr : n ≤ x.valuation →
  HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑(unitCoeff hx) : ℤ_[p]) * HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) x.valuation
Tactic: nth_rewrite 2 [unitCoeff_spec hx]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℕ
⊢ ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
have pos : ∀ (hx : x = 0),
  ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
Tactic: by_cases hx : x = 0
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℕ
pos :
  ∀ (hx : x = 0),
    ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
      Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
⊢ ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
have neg : ∀ (hx : ¬x = 0),
  ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
Tactic: by_cases hx : x = 0
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℕ
hx : x = 0
⊢ ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
have pos : norm (E := ℤ_[p]) 0 ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
  Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) 0
Tactic: subst hx
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ norm (E := ℤ_[p]) 0 ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) 0
have pos : 0 ≤ HPow.hPow (α := ℝ) (↑p : ℝ) n
Tactic: simp only [norm_zero, zpow_neg, zpow_natCast, inv_nonneg, iff_true, Submodule.zero_mem]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℕ
hx : ¬x = 0
⊢ ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
have neg : n ≤ x.valuation ↔ Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
Tactic: rw [norm_le_pow_iff_le_valuation x hx, mem_span_pow_iff_le_valuation x hx]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℕ
hx : ¬x = 0
neg : n ≤ x.valuation ↔ Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
⊢ ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) x
have neg₁ : n ≤ x.valuation ↔ n ≤ x.valuation
Tactic: rw [norm_le_pow_iff_le_valuation x hx, mem_span_pow_iff_le_valuation x hx]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℤ
⊢ ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) n ↔ ‖x‖ ≤ HPow.hPow (α := ℝ) (↑p : ℝ) (n - 1)
have rw : ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) n ↔ ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) (n - 1 + 1)
Tactic: rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℤ
rw : ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) n ↔ ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) (n - 1 + 1)
⊢ ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) n ↔ ‖x‖ ≤ HPow.hPow (α := ℝ) (↑p : ℝ) (n - 1)
have rw₁ : ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) n ↔ ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) n
Tactic: rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
⊢ ‖x‖ < 1 ↔ (↑p : ℤ_[p]) ∣ x
have this : ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔
  Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (β := ℕ) (↑p : ℤ_[p]) 1}) x
Tactic: have := norm_le_pow_iff_mem_span_pow x 1
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
this :
  ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (β := ℕ) (↑p : ℤ_[p]) 1}) x
⊢ ‖x‖ < 1 ↔ (↑p : ℤ_[p]) ∣ x
have this₁ : ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔ (↑p : ℤ_[p]) ∣ x
Tactic: rw [Ideal.mem_span_singleton, pow_one] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
this : ‖x‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔ (↑p : ℤ_[p]) ∣ x
⊢ ‖x‖ < 1 ↔ (↑p : ℤ_[p]) ∣ x
have rw : ‖x‖ < 1 ↔ ‖x‖ < HPow.hPow (α := ℝ) (↑p : ℝ) (HAdd.hAdd (α := ℤ) (-(↑1 : ℤ)) 1)
Tactic: rw [← this, norm_le_pow_iff_norm_lt_pow_add_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
a : ℤ
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
have rw : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
Tactic: rw [← Nat.cast_pow, ← norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow,
    Ideal.mem_span_singleton, Nat.cast_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
a : ℤ
rw : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
have rw₁ : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔
  norm (E := ℤ_[p]) (↑a : ℤ_[p]) ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: rw [← Nat.cast_pow, ← norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow,
    Ideal.mem_span_singleton, Nat.cast_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
a : ℤ
rw : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
rw₁ :
  Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔
    norm (E := ℤ_[p]) (↑a : ℤ_[p]) ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
have rw₂ : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔
  Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) (↑a : ℤ_[p])
Tactic: rw [← Nat.cast_pow, ← norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow,
    Ideal.mem_span_singleton, Nat.cast_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
a : ℤ
rw : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
rw₁ :
  Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔
    norm (E := ℤ_[p]) (↑a : ℤ_[p]) ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
rw₂ :
  Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) (↑a : ℤ_[p])
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
have rw₃ : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p])
Tactic: rw [← Nat.cast_pow, ← norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow,
    Ideal.mem_span_singleton, Nat.cast_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
a : ℤ
rw : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
rw₁ :
  Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔
    norm (E := ℤ_[p]) (↑a : ℤ_[p]) ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
rw₂ :
  Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n}) (↑a : ℤ_[p])
rw₃ : Dvd.dvd (α := ℤ_[p]) (↑(p ^ n) : ℤ_[p]) (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p])
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ) (↑p : ℤ) n ∣ a
have rw₄ : HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p]) ↔ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n ∣ (↑a : ℤ_[p])
Tactic: rw [← Nat.cast_pow, ← norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow,
    Ideal.mem_span_singleton, Nat.cast_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ ∀ (a b : ℤ_[p]),
    Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) a →
      Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) b → Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (a + b)
have simp : ∀ (a b : ℤ_[p]), ‖a‖ < 1 → ‖b‖ < 1 → ‖a + b‖ < 1
Tactic: simp only [mem_nonunits]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (↑p : ℤ_[p])
have this : LT.lt (α := ℝ) (↑p : ℝ)⁻¹ 1
Tactic: have : (p : ℝ)⁻¹ < 1 := inv_lt_one_of_one_lt₀ <| mod_cast hp.out.one_lt
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
this : LT.lt (α := ℝ) (↑p : ℝ)⁻¹ 1
⊢ Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (↑p : ℤ_[p])
have rwa : ∀ (this : norm (E := ℤ_[p]) (↑p : ℤ_[p]) < 1), Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (↑p : ℤ_[p])
Tactic: rwa [← norm_p, ← mem_nonunits] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
this : LT.lt (α := ℝ) (↑p : ℝ)⁻¹ 1
rwa : ∀ (this : norm (E := ℤ_[p]) (↑p : ℤ_[p]) < 1), Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (↑p : ℤ_[p])
⊢ Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (↑p : ℤ_[p])
have rwa₁ : ∀ (this : Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (↑p : ℤ_[p])),
  Membership.mem (γ := Set ℤ_[p]) (nonunits ℤ_[p]) (↑p : ℤ_[p])
Tactic: rwa [← norm_p, ← mem_nonunits] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ Eq (α := Ideal ℤ_[p]) (maximalIdeal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
have a : LE.le (α := Ideal ℤ_[p]) (maximalIdeal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
a : LE.le (α := Ideal ℤ_[p]) (maximalIdeal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
⊢ Eq (α := Ideal ℤ_[p]) (maximalIdeal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
have a₁ : LE.le (α := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) (maximalIdeal ℤ_[p])
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : Membership.mem (γ := Submodule ℤ_[p] ℤ_[p]) (maximalIdeal ℤ_[p]) x
⊢ Membership.mem (γ := Submodule ℤ_[p] ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
have hx₁ : ‖x‖ < 1
Tactic: simp only [IsLocalRing.mem_maximalIdeal, mem_nonunits] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : ‖x‖ < 1
⊢ Membership.mem (γ := Submodule ℤ_[p] ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
have a : (↑p : ℤ_[p]) ∣ x
Tactic: rwa [Ideal.mem_span_singleton, ← norm_lt_one_iff_dvd]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ LE.le (α := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) (maximalIdeal ℤ_[p])
have a : (↑p : ℤ_[p]) ∈ (↑(maximalIdeal ℤ_[p]) : Set ℤ_[p])
Tactic: rw [Ideal.span_le, Set.singleton_subset_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ Prime (M := ℤ_[p]) (↑p : ℤ_[p])
have rw : Ideal.IsPrime (α := ℤ_[p]) (maximalIdeal ℤ_[p])
Tactic: rw [← Ideal.span_singleton_prime, ← maximalIdeal_eq_span_p]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
rw : Ideal.IsPrime (α := ℤ_[p]) (maximalIdeal ℤ_[p])
⊢ Prime (M := ℤ_[p]) (↑p : ℤ_[p])
have rw₁ : Ne (α := ℤ_[p]) (↑p : ℤ_[p]) 0
Tactic: rw [← Ideal.span_singleton_prime, ← maximalIdeal_eq_span_p]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y x : ℤ_[p]
hx : x ≠ 0
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) x.valuation * (↑(unitCoeff hx) : ℤ_[p]) = x
have rw : (↑(unitCoeff hx) : ℤ_[p]) * HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) x.valuation = x
Tactic: rw [mul_comm, ← unitCoeff_spec hx]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y x : ℤ_[p]
hx : x ≠ 0
rw : (↑(unitCoeff hx) : ℤ_[p]) * HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) x.valuation = x
⊢ HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) x.valuation * (↑(unitCoeff hx) : ℤ_[p]) = x
have rw₁ : x = x
Tactic: rw [mul_comm, ← unitCoeff_spec hx]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx :
  ∀ {m n : ℕ},
    m ≤ n →
      x m ≡ x n [SMOD HSMul.hSMul (β := Submodule ℤ_[p] ℤ_[p]) (HPow.hPow (α := Ideal ℤ_[p]) (maximalIdeal ℤ_[p]) m) ⊤]
⊢ ∃ (L : ℤ_[p]),
    ∀ (n : ℕ),
      x n ≡ L [SMOD HSMul.hSMul (β := Submodule ℤ_[p] ℤ_[p]) (HPow.hPow (α := Ideal ℤ_[p]) (maximalIdeal ℤ_[p]) n) ⊤]
have simp : ∀ (hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))),
  ∃ (L : ℤ_[p]), ∀ (n : ℕ), ‖x n - L‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: 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 ⊢
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
refine_2 : ∃ (L : ℤ_[p]), ∀ (n : ℕ), ‖x n - L‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
⊢ ∃ (L : ℤ_[p]), ∀ (n : ℕ), ‖x n - L‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have refine_1 : IsCauSeq norm x
Tactic: let x' : CauSeq ℤ_[p] norm := ⟨x, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
ε : ℝ
hε : ε > 0
⊢ ∃ (i : ℕ), ∀ j ≥ i, ‖x j - x i‖ < ε
have refine_1 : ∀ (m : ℕ) (hm : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ)) < ε), ∃ (i : ℕ), ∀ j ≥ i, ‖x j - x i‖ < ε
Tactic: obtain ⟨m, hm⟩ := exists_pow_neg_lt p hε
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
ε : ℝ
hε : ε > 0
m : ℕ
hm : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ)) < ε
⊢ ∃ (i : ℕ), ∀ j ≥ i, ‖x j - x i‖ < ε
have refine_1 : ∀ (n : ℕ) (hn : n ≥ m), ‖x n - x m‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
Tactic: refine ⟨m, fun n hn => lt_of_le_of_lt ?_ hm⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
ε : ℝ
hε : ε > 0
m : ℕ
hm : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ)) < ε
n : ℕ
hn : n ≥ m
⊢ ‖x n - x m‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
have refine_1 : ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
Tactic: rw [← neg_sub, norm_neg]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
⊢ ∃ (L : ℤ_[p]), ∀ (n : ℕ), ‖x n - L‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have refine_2 : ∀ (n : ℕ), ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: refine ⟨x'.lim, fun n => ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have this : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: have : (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) := zpow_pos (mod_cast hp.out.pos) _
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
this : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have refine_2 : ∀ (i : ℕ)
  (hi :
    ∀ j ≥ i,
      ∀ k ≥ j,
        ‖(↑x' : ℕ → ℤ_[p]) k - (↑(const norm x'.lim) : ℕ → ℤ_[p]) j‖ <
          HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))),
  ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: obtain ⟨i, hi⟩ := equiv_def₃ (equiv_lim x') this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
this : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
i : ℕ
hi :
  ∀ j ≥ i,
    ∀ k ≥ j,
      ‖(↑x' : ℕ → ℤ_[p]) k - (↑(const norm x'.lim) : ℕ → ℤ_[p]) j‖ < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have pos : ∀ (hin : i ≤ n), ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: by_cases! hin : i ≤ n