awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
seq : ℕ → ℤ
h : IsCauSeq (padicNorm p) fun (n : ℕ) ↦ (↑(seq n) : ℚ)
hne : Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h ≈ 0
⊢ LE.le (α := ℝ) (↑0 : ℝ) 1
have pos : LE.le (α := ℕ) 0 1
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
seq : ℕ → ℤ
h : IsCauSeq (padicNorm p) fun (n : ℕ) ↦ (↑(seq n) : ℚ)
hne : ¬Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h ≈ 0
⊢ LE.le (α := ℝ)
    (↑(padicNorm p
          ((↑(Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h) : ℕ → ℚ)
            (PadicSeq.stationaryPoint hne))) :
      ℝ)
    1
have neg : padicNorm p
    ((↑(Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h) : ℕ → ℚ)
      (PadicSeq.stationaryPoint hne)) ≤
  1
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℤ_[p]
h : ‖z1 + z2‖ < ‖z2‖
hne : ¬‖z1‖ = ‖z2‖
⊢ ‖z2‖ ≤ ‖z1 + z2‖
have rw : ‖z2‖ ≤ max ‖z1‖ ‖z2‖
Tactic: rw [norm_add_eq_max_of_ne hne]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℤ_[p]
h : ‖z1 + z2‖ < ‖z1‖
hne : ¬‖z1‖ = ‖z2‖
⊢ ‖z1‖ ≤ ‖z1 + z2‖
have rw : ‖z1‖ ≤ max ‖z1‖ ‖z2‖
Tactic: rw [norm_add_eq_max_of_ne hne]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
⊢ ∃ (k : ℕ), HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < ε
have obtain : ∀ (k : ℕ) (hk : ε⁻¹ < (↑k : ℝ)), ∃ (k : ℕ), HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < ε
Tactic: obtain ⟨k, hk⟩ := exists_nat_gt ε⁻¹
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
⊢ ∃ (k : ℕ), HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < ε
have h : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < ε
Tactic: use k
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
⊢ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < ε
have h : ε⁻¹ < (HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)))⁻¹
Tactic: rw [← inv_lt_inv₀ hε (zpow_pos _ _)]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
h : ε⁻¹ < (HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)))⁻¹
⊢ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < ε
have rw : LT.lt (α := ℝ) 0 (↑p : ℝ)
Tactic: rw [← inv_lt_inv₀ hε (zpow_pos _ _)]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
⊢ ε⁻¹ < (HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)))⁻¹
have h : ε⁻¹ < HPow.hPow (α := ℝ) (↑p : ℝ) k
Tactic: rw [zpow_neg, inv_inv, zpow_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
⊢ ε⁻¹ < HPow.hPow (α := ℝ) (↑p : ℝ) k
have h : (↑k : ℝ) ≤ HPow.hPow (α := ℝ) (↑p : ℝ) k
Tactic: apply lt_of_lt_of_le hk
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
⊢ (↑k : ℝ) ≤ HPow.hPow (α := ℝ) (↑p : ℝ) k
have h : k ≤ p ^ k
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
⊢ k ≤ p ^ k
have h : k < p ^ k
Tactic: apply le_of_lt
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < (↑k : ℝ)
⊢ k < p ^ k
have h : 1 < p
Tactic: convert! Nat.lt_pow_self _ using 1
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℚ
hε : 0 < ε
⊢ ∃ (k : ℕ), HPow.hPow (α := ℚ) (β := ℤ) (↑p : ℚ) (-(↑k : ℤ)) < ε
have obtain : ∀ (k : ℕ) (hk : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < (↑ε : ℝ)),
  ∃ (k : ℕ), HPow.hPow (α := ℚ) (β := ℤ) (↑p : ℚ) (-(↑k : ℤ)) < ε
Tactic: obtain ⟨k, hk⟩ := @exists_pow_neg_lt p _ ε (mod_cast hε)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℚ
hε : 0 < ε
k : ℕ
hk : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < (↑ε : ℝ)
⊢ ∃ (k : ℕ), HPow.hPow (α := ℚ) (β := ℤ) (↑p : ℚ) (-(↑k : ℤ)) < ε
have h : HPow.hPow (α := ℚ) (β := ℤ) (↑p : ℚ) (-(↑k : ℤ)) < ε
Tactic: use k
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℚ
hε : 0 < ε
k : ℕ
hk : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑k : ℤ)) < (↑ε : ℝ)
⊢ HPow.hPow (α := ℚ) (β := ℤ) (↑p : ℚ) (-(↑k : ℤ)) < ε
have hk₁ : HPow.hPow (α := ℝ) (β := ℤ) (↑(↑p : ℚ) : ℝ) (-(↑k : ℤ)) < (↑ε : ℝ)
Tactic: rw [show (p : ℝ) = (p : ℚ) by simp] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ norm (E := ℤ_[p]) (↑n : ℤ_[p]) = 1 ↔ p.Coprime n
have rw : norm (E := ℚ_[p]) (↑(↑n : ℤ_[p]) : ℚ_[p]) = 1 ↔ p.Coprime n
Tactic: rw [norm_def, coe_natCast, Padic.norm_natCast_eq_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
rw : norm (E := ℚ_[p]) (↑(↑n : ℤ_[p]) : ℚ_[p]) = 1 ↔ p.Coprime n
⊢ norm (E := ℤ_[p]) (↑n : ℤ_[p]) = 1 ↔ p.Coprime n
have rw₁ : norm (E := ℚ_[p]) (↑n : ℚ_[p]) = 1 ↔ p.Coprime n
Tactic: rw [norm_def, coe_natCast, Padic.norm_natCast_eq_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
rw : norm (E := ℚ_[p]) (↑(↑n : ℤ_[p]) : ℚ_[p]) = 1 ↔ p.Coprime n
rw₁ : norm (E := ℚ_[p]) (↑n : ℚ_[p]) = 1 ↔ p.Coprime n
⊢ norm (E := ℤ_[p]) (↑n : ℤ_[p]) = 1 ↔ p.Coprime n
have rw₂ : p.Coprime n ↔ p.Coprime n
Tactic: rw [norm_def, coe_natCast, Padic.norm_natCast_eq_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ norm (E := ℤ_[p]) (↑n : ℤ_[p]) < 1 ↔ p ∣ n
have rw : norm (E := ℚ_[p]) (↑(↑n : ℤ_[p]) : ℚ_[p]) < 1 ↔ p ∣ n
Tactic: rw [norm_def, coe_natCast, Padic.norm_natCast_lt_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
rw : norm (E := ℚ_[p]) (↑(↑n : ℤ_[p]) : ℚ_[p]) < 1 ↔ p ∣ n
⊢ norm (E := ℤ_[p]) (↑n : ℤ_[p]) < 1 ↔ p ∣ n
have rw₁ : norm (E := ℚ_[p]) (↑n : ℚ_[p]) < 1 ↔ p ∣ n
Tactic: rw [norm_def, coe_natCast, Padic.norm_natCast_lt_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
rw : norm (E := ℚ_[p]) (↑(↑n : ℤ_[p]) : ℚ_[p]) < 1 ↔ p ∣ n
rw₁ : norm (E := ℚ_[p]) (↑n : ℚ_[p]) < 1 ↔ p ∣ n
⊢ norm (E := ℤ_[p]) (↑n : ℤ_[p]) < 1 ↔ p ∣ n
have rw₂ : p ∣ n ↔ p ∣ n
Tactic: rw [norm_def, coe_natCast, Padic.norm_natCast_lt_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
⊢ norm (E := ℤ_[p]) (↑z : ℤ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
have rw : norm (E := ℚ_[p]) (↑(↑z : ℤ_[p]) : ℚ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
Tactic: rw [norm_def, coe_intCast, Padic.norm_intCast_eq_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
rw : norm (E := ℚ_[p]) (↑(↑z : ℤ_[p]) : ℚ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
⊢ norm (E := ℤ_[p]) (↑z : ℤ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
have rw₁ : norm (E := ℚ_[p]) (↑z : ℚ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
Tactic: rw [norm_def, coe_intCast, Padic.norm_intCast_eq_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
rw : norm (E := ℚ_[p]) (↑(↑z : ℤ_[p]) : ℚ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
rw₁ : norm (E := ℚ_[p]) (↑z : ℚ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
⊢ norm (E := ℤ_[p]) (↑z : ℤ_[p]) = 1 ↔ IsCoprime z (↑p : ℤ)
have rw₂ : IsCoprime z (↑p : ℤ) ↔ IsCoprime z (↑p : ℤ)
Tactic: rw [norm_def, coe_intCast, Padic.norm_intCast_eq_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
⊢ norm (E := ℤ_[p]) (↑z : ℤ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
have rw : norm (E := ℚ_[p]) (↑(↑z : ℤ_[p]) : ℚ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
Tactic: rw [norm_def, coe_intCast, Padic.norm_intCast_lt_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
rw : norm (E := ℚ_[p]) (↑(↑z : ℤ_[p]) : ℚ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
⊢ norm (E := ℤ_[p]) (↑z : ℤ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
have rw₁ : norm (E := ℚ_[p]) (↑z : ℚ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
Tactic: rw [norm_def, coe_intCast, Padic.norm_intCast_lt_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
rw : norm (E := ℚ_[p]) (↑(↑z : ℤ_[p]) : ℚ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
rw₁ : norm (E := ℚ_[p]) (↑z : ℚ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
⊢ norm (E := ℤ_[p]) (↑z : ℤ_[p]) < 1 ↔ (↑p : ℤ) ∣ z
have rw₂ : (↑p : ℤ) ∣ z ↔ (↑p : ℤ) ∣ z
Tactic: rw [norm_def, coe_intCast, Padic.norm_intCast_lt_one_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
⊢ 0 ≤ (↑x : ℚ_[p]).valuation
have inl : 0 ≤ valuation (p := p) (↑0 : ℚ_[p])
Tactic: obtain rfl | hx := eq_or_ne x 0
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
inl : 0 ≤ valuation (p := p) (↑0 : ℚ_[p])
⊢ 0 ≤ (↑x : ℚ_[p]).valuation
have inr : ∀ (hx : x ≠ 0), 0 ≤ (↑x : ℚ_[p]).valuation
Tactic: obtain rfl | hx := eq_or_ne x 0
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
⊢ 0 ≤ (↑x : ℚ_[p]).valuation
have this : ‖(↑x : ℚ_[p])‖ ≤ 1
Tactic: have := x.2
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
this : ‖(↑x : ℚ_[p])‖ ≤ 1
⊢ 0 ≤ (↑x : ℚ_[p]).valuation
have inr : ∀ (this : HPow.hPow (α := ℝ) (↑p : ℝ) (-(↑x : ℚ_[p]).valuation) ≤ 1), LT.lt (α := ℝ) 1 (↑p : ℝ)
Tactic: rwa [Padic.norm_eq_zpow_neg_valuation <| coe_ne_zero.2 hx, zpow_le_one_iff_right₀, neg_nonpos]
    at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
hxy : x + y ≠ 0
⊢ min x.valuation y.valuation ≤ (x + y).valuation
have zify : LE.le (α := ℤ) (min (↑x.valuation : ℤ) (↑y.valuation : ℤ)) (↑(x + y).valuation : ℤ)
Tactic: zify
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
hx : x ≠ 0
hy : y ≠ 0
⊢ (x * y).valuation = x.valuation + y.valuation
have zify : (↑(x * y).valuation : ℤ) = HAdd.hAdd (α := ℤ) (↑x.valuation : ℤ) (↑y.valuation : ℤ)
Tactic: zify
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℕ
⊢ (x ^ n).valuation = n * x.valuation
have zify : (↑(x ^ n).valuation : ℤ) = HMul.hMul (α := ℤ) (↑n : ℤ) (↑x.valuation : ℤ)
Tactic: zify
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hc : c ≠ 0
⊢ (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c).valuation = n + c.valuation
have rw : (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n).valuation + c.valuation = n + c.valuation
Tactic: rw [valuation_mul (NeZero.ne _) hc, valuation_pow, valuation_p, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hc : c ≠ 0
rw : (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n).valuation + c.valuation = n + c.valuation
⊢ (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c).valuation = n + c.valuation
have rw₁ : n * valuation (p := p) (↑p : ℤ_[p]) + c.valuation = n + c.valuation
Tactic: rw [valuation_mul (NeZero.ne _) hc, valuation_pow, valuation_p, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hc : c ≠ 0
rw : (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n).valuation + c.valuation = n + c.valuation
rw₁ : n * valuation (p := p) (↑p : ℤ_[p]) + c.valuation = n + c.valuation
⊢ (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c).valuation = n + c.valuation
have rw₂ : n * 1 + c.valuation = n + c.valuation
Tactic: rw [valuation_mul (NeZero.ne _) hc, valuation_pow, valuation_p, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hc : c ≠ 0
rw : (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n).valuation + c.valuation = n + c.valuation
rw₁ : n * valuation (p := p) (↑p : ℤ_[p]) + c.valuation = n + c.valuation
rw₂ : n * 1 + c.valuation = n + c.valuation
⊢ (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n * c).valuation = n + c.valuation
have rw₃ : n + c.valuation = n + c.valuation
Tactic: rw [valuation_mul (NeZero.ne _) hc, valuation_pow, valuation_p, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℚ_[p]
property : ‖k‖ ≤ 1
h : norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property) = 1
⊢ Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property *
      inv (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property) =
    1
have hk : k ≠ 0
Tactic: have hk : k ≠ 0 := fun h' => zero_ne_one' ℚ_[p] (by simp [h'] at h)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℚ_[p]
property : ‖k‖ ≤ 1
h : norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property) = 1
hk : k ≠ 0
⊢ HMul.hMul (α := ℤ_[p]) (β := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property)
      (match (motive := ℤ_[p] → ℤ_[p]) Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property with
        | ⟨k, property⟩ => if h : ‖k‖ = 1 then Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k⁻¹ ⋯ else 0 :
        ℤ_[p]) =
    1
have h₁ : ‖k‖ = 1
Tactic: rw [norm_eq_padic_norm] at h
Cleared hypotheses: h, hk
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℚ_[p]
property : ‖k‖ ≤ 1
h : ‖k‖ = 1
hk : k ≠ 0
⊢ HMul.hMul (α := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property)
      (if h : ‖k‖ = 1 then Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k⁻¹ ⋯ else 0) =
    1
have rw : HMul.hMul (α := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property)
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k⁻¹ sorry) =
  1
Tactic: rw [dif_pos h]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℚ_[p]
property : ‖k‖ ≤ 1
h : ‖k‖ = 1
hk : k ≠ 0
⊢ HMul.hMul (α := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property)
      (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k⁻¹ ⋯) =
    1
have apply : (↑(HMul.hMul (α := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k property)
        (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) k⁻¹ sorry)) :
    ℚ_[p]) =
  (↑1 : ℚ_[p])
Tactic: apply Subtype.ext_iff.2
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
hz : ‖z‖ = 1
⊢ z.inv * z = 1
have rw : z * z.inv = 1
Tactic: rw [mul_comm, mul_inv hz]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
hz : ‖z‖ = 1
rw : z * z.inv = 1
⊢ z.inv * z = 1
have rw₁ : Eq (α := ℤ_[p]) 1 1
Tactic: rw [mul_comm, mul_inv hz]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
⊢ ‖z‖ = 1
have rcases : ∀ (w : ℤ_[p]) (eq : 1 = z * w), ‖z‖ = 1
Tactic: rcases isUnit_iff_dvd_one.1 h with ⟨w, eq⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
⊢ ‖z‖ = 1
have refine : 1 ≤ ‖z‖
Tactic: refine le_antisymm (norm_le_one _) ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
⊢ 1 ≤ ‖z‖
have this : ‖z‖ * ‖w‖ ≤ ‖z‖ * 1
Tactic: have := mul_le_mul_of_nonneg_left (norm_le_one w) (norm_nonneg z)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
this : ‖z‖ * ‖w‖ ≤ ‖z‖ * 1
⊢ 1 ≤ ‖z‖
have rwa : ∀ (this : ‖z‖ * ‖w‖ ≤ ‖z‖), 1 ≤ ‖z‖
Tactic: rwa [mul_one, ← norm_mul, ← eq, norm_one] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
this : ‖z‖ * ‖w‖ ≤ ‖z‖ * 1
rwa : ∀ (this : ‖z‖ * ‖w‖ ≤ ‖z‖), 1 ≤ ‖z‖
⊢ 1 ≤ ‖z‖
have rwa₁ : ∀ (this : ‖z * w‖ ≤ ‖z‖), 1 ≤ ‖z‖
Tactic: rwa [mul_one, ← norm_mul, ← eq, norm_one] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
this : ‖z‖ * ‖w‖ ≤ ‖z‖ * 1
rwa : ∀ (this : ‖z‖ * ‖w‖ ≤ ‖z‖), 1 ≤ ‖z‖
rwa₁ : ∀ (this : ‖z * w‖ ≤ ‖z‖), 1 ≤ ‖z‖
⊢ 1 ≤ ‖z‖
have rwa₂ : ∀ (this : norm (E := ℤ_[p]) 1 ≤ ‖z‖), 1 ≤ ‖z‖
Tactic: rwa [mul_one, ← norm_mul, ← eq, norm_one] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
this : ‖z‖ * ‖w‖ ≤ ‖z‖ * 1
rwa : ∀ (this : ‖z‖ * ‖w‖ ≤ ‖z‖), 1 ≤ ‖z‖
rwa₁ : ∀ (this : ‖z * w‖ ≤ ‖z‖), 1 ≤ ‖z‖
rwa₂ : ∀ (this : norm (E := ℤ_[p]) 1 ≤ ‖z‖), 1 ≤ ‖z‖
⊢ 1 ≤ ‖z‖
have rwa₃ : ∀ (this : 1 ≤ ‖z‖), 1 ≤ ‖z‖
Tactic: rwa [mul_one, ← norm_mul, ← eq, norm_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]
hx : x ≠ 0
⊢ x = (↑(unitCoeff hx) : ℤ_[p]) * HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) x.valuation
have a : Eq (α := ℚ_[p]) ((fun (a : { x : ℚ_[p] // ‖x‖ ≤ 1 }) ↦ (↑a : ℚ_[p])) x)
  ((fun (a : { x : ℚ_[p] // ‖x‖ ≤ 1 }) ↦ (↑a : ℚ_[p]))
    ((↑(unitCoeff hx) : ℤ_[p]) * HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) x.valuation))
Tactic: apply Subtype.coe_injective
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
⊢ (↑x : ℚ_[p]) = (↑(↑(unitCoeff hx) : ℤ_[p]) : ℚ_[p]) * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) x.valuation
have a : (↑x : ℚ_[p]) =
  (↑x : ℚ_[p]) * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) (HAdd.hAdd (α := ℤ) (-(↑x.valuation : ℤ)) (↑x.valuation : ℤ))
Tactic: rw [unitCoeff_coe, mul_assoc, ← zpow_natCast, ← zpow_add₀]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
a :
  (↑x : ℚ_[p]) =
    (↑x : ℚ_[p]) * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) (HAdd.hAdd (α := ℤ) (-(↑x.valuation : ℤ)) (↑x.valuation : ℤ))
⊢ (↑x : ℚ_[p]) = (↑(↑(unitCoeff hx) : ℤ_[p]) : ℚ_[p]) * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) x.valuation
have a₁ : Ne (α := ℚ_[p]) (↑p : ℚ_[p]) 0
Tactic: rw [unitCoeff_coe, mul_assoc, ← zpow_natCast, ← zpow_add₀]
-- 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
⊢ IsUnit (M := ℤ_[p]) (↑r.den : ℤ_[p])
have rw : norm (E := ℤ_[p]) (↑r.den : ℤ_[p]) = 1
Tactic: rw [isUnit_iff]
-- 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 (E := ℤ_[p]) (↑r.den : ℤ_[p]) = 1
have apply : 1 ≤ norm (E := ℚ_[p]) (↑(↑r.den : ℤ_[p]) : ℚ_[p])
Tactic: apply le_antisymm (r.den : ℤ_[p]).2
-- 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
⊢ 1 ≤ norm (E := ℚ_[p]) (↑(↑r.den : ℤ_[p]) : ℚ_[p])
have rw : ¬norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) < 1
Tactic: rw [← not_lt, coe_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
⊢ ‖HMul.hMul (α := ℚ_[p]) (↑r : ℚ_[p]) (↑r.den : ℚ_[p])‖ = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
have e_a : HMul.hMul (α := ℚ_[p]) (↑r : ℚ_[p]) (↑r.den : ℚ_[p]) = (↑r.num : ℚ_[p])
Tactic: congr
-- 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
⊢ HMul.hMul (α := ℚ_[p]) (↑r : ℚ_[p]) (↑r.den : ℚ_[p]) = (↑r.num : ℚ_[p])
have e_a : ∀ (h : norm (E := ℚ_[p]) (↑r : ℚ_[p]) ≤ 1) (norm_denom_lt : norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) < 1),
  r * (↑r.den : ℚ) = (↑r.num : ℚ)
Tactic: rw_mod_cast [@Rat.mul_den_eq_num r]
Cleared hypotheses: h, norm_denom_lt
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
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
⊢ False
have hr : ‖HMul.hMul (α := ℚ_[p]) (↑r : ℚ_[p]) (↑r.den : ℚ_[p])‖ = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
Tactic: have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
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 : ‖HMul.hMul (α := ℚ_[p]) (↑r : ℚ_[p]) (↑r.den : ℚ_[p])‖ = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
⊢ False
have hr₁ : norm (E := ℚ_[p]) (↑r : ℚ_[p]) * norm (E := ℚ_[p]) (↑r.den : ℚ_[p]) = norm (E := ℚ_[p]) (↑r.num : ℚ_[p])
Tactic: rw [padicNormE.mul] at hr
Cleared hypotheses: hr
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
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])
⊢ False
have key : norm (E := ℚ_[p]) (↑r.num : ℚ_[p]) < 1
Tactic: have key : ‖(r.num : ℚ_[p])‖ < 1 := by
    calc
      _ = _ := hr.symm
      _ < 1 * 1 := mul_lt_mul' h norm_denom_lt (norm_nonneg _) zero_lt_one
      _ = 1 := mul_one 1
-- 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
⊢ (↑p : ℤ) ∣ r.num ∧ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑r.den : ℤ)
have simp : norm (E := ℚ_[p]) (↑(↑r.num : ℤ_[p]) : ℚ_[p]) < 1 ∧ norm (E := ℚ_[p]) (↑(↑(↑r.den : ℤ) : ℤ_[p]) : ℚ_[p]) < 1
Tactic: simp only [← norm_int_lt_one_iff_dvd, ← padic_norm_e_of_padicInt]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
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
⊢ False
have this : (↑p : ℤ) ∣ r.num ∧ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑r.den : ℤ)
Tactic: have : ↑p ∣ r.num ∧ (p : ℤ) ∣ r.den := by
    simp only [← norm_int_lt_one_iff_dvd, ← padic_norm_e_of_padicInt]
    exact ⟨key, norm_denom_lt⟩