l3lab/ntp-mathlib huggingface dataset:
/-- **Liouville's Theorem** -/
protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x := by
  -- Proceed by contradiction: if `x` is algebraic, then `x` is the root (`ef0`) of a
  -- non-zero (`f0`) polynomial `f`
  rintro ⟨f : ℤ[X], f0, ef0⟩
  -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.
  replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0;
  · rwa [aeval_def, ← eval_map] at ef0
  -- There is a "large" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a / (b + 1))| * A`
  -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.
  obtain ⟨A, hA, h⟩ : ∃ A : ℝ, 0 < A ∧ ∀ (a : ℤ) (b : ℕ),
      (1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|x - a / (b + 1)| * A) :=
    exists_pos_real_of_irrational_root lx.irrational f0 ef0
  -- Since the real numbers are Archimedean, a power of `2` exceeds `A`: `hn : A < 2 ^ r`.
  rcases pow_unbounded_of_one_lt A (lt_add_one 1) with ⟨r, hn⟩
  -- Use the Liouville property, with exponent `r + deg f`.
  obtain ⟨a, b, b1, -, a1⟩ : ∃ a b : ℤ, 1 < b ∧ x ≠ a / b ∧
      |x - a / b| < 1 / (b : ℝ) ^ (r + f.natDegree) :=
    lx (r + f.natDegree)
  have b0 : (0 : ℝ) < b := zero_lt_one.trans (by rw [← Int.cast_one]; exact Int.cast_lt.mpr b1)
  -- Prove that `b ^ f.nat_degree * abs (x - a / b)` is strictly smaller than itself
  -- recall, this is a proof by contradiction!
  refine' lt_irrefl ((b : ℝ) ^ f.natDegree * |x - ↑a / ↑b|) _
  -- clear denominators at `a1`
  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1
  -- split the inequality via `1 / A`.
  refine' (_ : (b : ℝ) ^ f.natDegree * |x - a / b| < 1 / A).trans_le _
  -- This branch of the proof uses the Liouville condition and the Archimedean property
  · refine' (lt_div_iff' hA).mpr _
    refine' lt_of_le_of_lt _ a1
    gcongr
    refine' hn.le.trans _
    rw [one_add_one_eq_two]
    gcongr
    exact Int.cast_two.symm.le.trans (Int.cast_le.mpr (Int.add_one_le_iff.mpr b1))
  -- this branch of the proof exploits the "integrality" of evaluations of polynomials
  -- at ratios of integers.
  · lift b to ℕ using zero_le_one.trans b1.le
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.intro
x : ℝ
lx : Liouville x
f : ℤ[X]
f0 : f ≠ 0
ef0 : eval x (map (algebraMap ℤ ℝ) f) = 0
A : ℝ
hA : 0 < A
h : ∀ (a : ℤ) (b : ℕ), 1 ≤ (↑b + 1) ^ natDegree f * (|x - ↑a / (↑b + 1)| * A)
r : ℕ
hn : A < (1 + 1) ^ r
a : ℤ
b : ℕ
b1 : 1 < ↑b
a1 : ↑↑b ^ r * (↑↑b ^ natDegree f * |x - ↑a / ↑↑b|) < 1
b0 : 0 < ↑↑b
⊢ 1 / A ≤ ↑↑b ^ natDegree f * |x - ↑a / ↑↑b|

Next tactic:
specialize h a b.pred
/-- **Liouville's Theorem** -/
protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x := by
  -- Proceed by contradiction: if `x` is algebraic, then `x` is the root (`ef0`) of a
  -- non-zero (`f0`) polynomial `f`
  rintro ⟨f : ℤ[X], f0, ef0⟩
  -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.
  replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0;
  · rwa [aeval_def, ← eval_map] at ef0
  -- There is a "large" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a / (b + 1))| * A`
  -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.
  obtain ⟨A, hA, h⟩ : ∃ A : ℝ, 0 < A ∧ ∀ (a : ℤ) (b : ℕ),
      (1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|x - a / (b + 1)| * A) :=
    exists_pos_real_of_irrational_root lx.irrational f0 ef0
  -- Since the real numbers are Archimedean, a power of `2` exceeds `A`: `hn : A < 2 ^ r`.
  rcases pow_unbounded_of_one_lt A (lt_add_one 1) with ⟨r, hn⟩
  -- Use the Liouville property, with exponent `r + deg f`.
  obtain ⟨a, b, b1, -, a1⟩ : ∃ a b : ℤ, 1 < b ∧ x ≠ a / b ∧
      |x - a / b| < 1 / (b : ℝ) ^ (r + f.natDegree) :=
    lx (r + f.natDegree)
  have b0 : (0 : ℝ) < b := zero_lt_one.trans (by rw [← Int.cast_one]; exact Int.cast_lt.mpr b1)
  -- Prove that `b ^ f.nat_degree * abs (x - a / b)` is strictly smaller than itself
  -- recall, this is a proof by contradiction!
  refine' lt_irrefl ((b : ℝ) ^ f.natDegree * |x - ↑a / ↑b|) _
  -- clear denominators at `a1`
  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1
  -- split the inequality via `1 / A`.
  refine' (_ : (b : ℝ) ^ f.natDegree * |x - a / b| < 1 / A).trans_le _
  -- This branch of the proof uses the Liouville condition and the Archimedean property
  · refine' (lt_div_iff' hA).mpr _
    refine' lt_of_le_of_lt _ a1
    gcongr
    refine' hn.le.trans _
    rw [one_add_one_eq_two]
    gcongr
    exact Int.cast_two.symm.le.trans (Int.cast_le.mpr (Int.add_one_le_iff.mpr b1))
  -- this branch of the proof exploits the "integrality" of evaluations of polynomials
  -- at ratios of integers.
  · lift b to ℕ using zero_le_one.trans b1.le
    specialize h a b.pred
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.intro
x : ℝ
lx : Liouville x
f : ℤ[X]
f0 : f ≠ 0
ef0 : eval x (map (algebraMap ℤ ℝ) f) = 0
A : ℝ
hA : 0 < A
r : ℕ
hn : A < (1 + 1) ^ r
a : ℤ
b : ℕ
b1 : 1 < ↑b
a1 : ↑↑b ^ r * (↑↑b ^ natDegree f * |x - ↑a / ↑↑b|) < 1
b0 : 0 < ↑↑b
h : 1 ≤ (↑(Nat.pred b) + 1) ^ natDegree f * (|x - ↑a / (↑(Nat.pred b) + 1)| * A)
⊢ 1 / A ≤ ↑↑b ^ natDegree f * |x - ↑a / ↑↑b|

Next tactic:
rwa [← Nat.cast_succ, Nat.succ_pred_eq_of_pos (zero_lt_one.trans _), ← mul_assoc, ←
      div_le_iff hA] at h
/-- **Liouville's Theorem** -/
protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x := by
  -- Proceed by contradiction: if `x` is algebraic, then `x` is the root (`ef0`) of a
  -- non-zero (`f0`) polynomial `f`
  rintro ⟨f : ℤ[X], f0, ef0⟩
  -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.
  replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0;
  · rwa [aeval_def, ← eval_map] at ef0
  -- There is a "large" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a / (b + 1))| * A`
  -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.
  obtain ⟨A, hA, h⟩ : ∃ A : ℝ, 0 < A ∧ ∀ (a : ℤ) (b : ℕ),
      (1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|x - a / (b + 1)| * A) :=
    exists_pos_real_of_irrational_root lx.irrational f0 ef0
  -- Since the real numbers are Archimedean, a power of `2` exceeds `A`: `hn : A < 2 ^ r`.
  rcases pow_unbounded_of_one_lt A (lt_add_one 1) with ⟨r, hn⟩
  -- Use the Liouville property, with exponent `r + deg f`.
  obtain ⟨a, b, b1, -, a1⟩ : ∃ a b : ℤ, 1 < b ∧ x ≠ a / b ∧
      |x - a / b| < 1 / (b : ℝ) ^ (r + f.natDegree) :=
    lx (r + f.natDegree)
  have b0 : (0 : ℝ) < b := zero_lt_one.trans (by rw [← Int.cast_one]; exact Int.cast_lt.mpr b1)
  -- Prove that `b ^ f.nat_degree * abs (x - a / b)` is strictly smaller than itself
  -- recall, this is a proof by contradiction!
  refine' lt_irrefl ((b : ℝ) ^ f.natDegree * |x - ↑a / ↑b|) _
  -- clear denominators at `a1`
  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1
  -- split the inequality via `1 / A`.
  refine' (_ : (b : ℝ) ^ f.natDegree * |x - a / b| < 1 / A).trans_le _
  -- This branch of the proof uses the Liouville condition and the Archimedean property
  · refine' (lt_div_iff' hA).mpr _
    refine' lt_of_le_of_lt _ a1
    gcongr
    refine' hn.le.trans _
    rw [one_add_one_eq_two]
    gcongr
    exact Int.cast_two.symm.le.trans (Int.cast_le.mpr (Int.add_one_le_iff.mpr b1))
  -- this branch of the proof exploits the "integrality" of evaluations of polynomials
  -- at ratios of integers.
  · lift b to ℕ using zero_le_one.trans b1.le
    specialize h a b.pred
    rwa [← Nat.cast_succ, Nat.succ_pred_eq_of_pos (zero_lt_one.trans _), ← mul_assoc, ←
      div_le_iff hA] at h
    

Proof state:
x : ℝ
lx : Liouville x
f : ℤ[X]
f0 : f ≠ 0
ef0 : eval x (map (algebraMap ℤ ℝ) f) = 0
A : ℝ
hA : 0 < A
r : ℕ
hn : A < (1 + 1) ^ r
a : ℤ
b : ℕ
b1 : 1 < ↑b
a1 : ↑↑b ^ r * (↑↑b ^ natDegree f * |x - ↑a / ↑↑b|) < 1
b0 : 0 < ↑↑b
h : 1 ≤ ↑(Nat.succ (Nat.pred b)) ^ natDegree f * (|x - ↑a / ↑(Nat.succ (Nat.pred b))| * A)
⊢ 1 < b

Next tactic:
exact Int.ofNat_lt.mp b1
theorem modPart_lt_p : modPart p r < p := by
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
⊢ modPart p r < ↑p

Next tactic:
convert Int.emod_lt _ _
theorem modPart_lt_p : modPart p r < p := by
  convert Int.emod_lt _ _
  · 

Proof state:
case h.e'_4
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
⊢ ↑p = |↑p|

Next tactic:
simp
theorem modPart_lt_p : modPart p r < p := by
  convert Int.emod_lt _ _
  · simp
  · 

Proof state:
case convert_3
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
⊢ ↑p ≠ 0

Next tactic:
exact mod_cast hp_prime.1.ne_zero
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
⊢ IsUnit ↑r.den

Next tactic:
rw [isUnit_iff]
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
⊢ ‖↑r.den‖ = 1

Next tactic:
apply le_antisymm (r.den : ℤ_[p]).2
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
⊢ 1 ≤ ‖↑↑r.den‖

Next tactic:
rw [← not_lt, coe_nat_cast]
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
⊢ ¬‖↑r.den‖ < 1

Next tactic:
intro norm_denom_lt
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
⊢ False

Next tactic:
have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
⊢ ‖↑r * ↑r.den‖ = ‖↑r.num‖

Next tactic:
congr
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    

Proof state:
case e_a
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
⊢ ↑r * ↑r.den = ↑r.num

Next tactic:
rw_mod_cast [@Rat.mul_den_eq_num r]
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r * ↑r.den‖ = ‖↑r.num‖
⊢ False

Next tactic:
rw [padicNormE.mul] at hr
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  rw [padicNormE.mul] at hr
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
⊢ False

Next 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
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  rw [padicNormE.mul] at hr
  have key : ‖(r.num : ℚ_[p])‖ < 1 := by
    

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
⊢ ‖↑r.num‖ < 1

Next tactic:
calc
      _ = _ := hr.symm
      _ < 1 * 1 := (mul_lt_mul' h norm_denom_lt (norm_nonneg _) zero_lt_one)
      _ = 1 := mul_one 1
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  rw [padicNormE.mul] at hr
  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

  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
key : ‖↑r.num‖ < 1
⊢ False

Next 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⟩
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  rw [padicNormE.mul] at hr
  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

  have : ↑p ∣ r.num ∧ (p : ℤ) ∣ r.den := by
    

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
key : ‖↑r.num‖ < 1
⊢ ↑p ∣ r.num ∧ ↑p ∣ ↑r.den

Next tactic:
simp only [← norm_int_lt_one_iff_dvd, ← padic_norm_e_of_padicInt]
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  rw [padicNormE.mul] at hr
  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

  have : ↑p ∣ r.num ∧ (p : ℤ) ∣ r.den := by
    simp only [← norm_int_lt_one_iff_dvd, ← padic_norm_e_of_padicInt]
    

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
key : ‖↑r.num‖ < 1
⊢ ‖↑↑r.num‖ < 1 ∧ ‖↑↑↑r.den‖ < 1

Next tactic:
exact ⟨key, norm_denom_lt⟩
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  rw [padicNormE.mul] at hr
  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

  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⟩
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
key : ‖↑r.num‖ < 1
this : ↑p ∣ r.num ∧ ↑p ∣ ↑r.den
⊢ False

Next tactic:
apply hp_prime.1.not_dvd_one
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by
  rw [isUnit_iff]
  apply le_antisymm (r.den : ℤ_[p]).2
  rw [← not_lt, coe_nat_cast]
  intro norm_denom_lt
  have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
    congr
    rw_mod_cast [@Rat.mul_den_eq_num r]
  rw [padicNormE.mul] at hr
  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

  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⟩
  apply hp_prime.1.not_dvd_one
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
key : ‖↑r.num‖ < 1
this : ↑p ∣ r.num ∧ ↑p ∣ ↑r.den
⊢ p ∣ 1

Next tactic:
rwa [← r.reduced.gcd_eq_one, Nat.dvd_gcd_iff, ← Int.coe_nat_dvd_left, ← Int.coe_nat_dvd]
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
⊢ ↑p ∣ r.num - r.num * gcdA r.den p % ↑p * ↑r.den

Next tactic:
rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
⊢ ↑(r.num - r.num * gcdA r.den p % ↑p * ↑r.den) = 0

Next tactic:
simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
⊢ ↑r.num - ↑(r.num * gcdA r.den p % ↑p) * ↑r.den = 0

Next tactic:
have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : (fun x => ↑x % ↑p) ↑(Nat.gcd r.den p) = (fun x => ↑x % ↑p) (↑r.den * gcdA r.den p + ↑p * gcdB r.den p)
⊢ ↑r.num - ↑(r.num * gcdA r.den p % ↑p) * ↑r.den = 0

Next tactic:
simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : ↑(Nat.gcd r.den p) = ↑r.den * ↑(gcdA r.den p)
⊢ ↑r.num - ↑(r.num * gcdA r.den p % ↑p) * ↑r.den = 0

Next tactic:
push_cast
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
  push_cast
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : ↑(Nat.gcd r.den p) = ↑r.den * ↑(gcdA r.den p)
⊢ ↑r.num - ↑r.num * ↑(gcdA r.den p) * ↑r.den = 0

Next tactic:
rw [mul_right_comm, mul_assoc, ← this]
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
  push_cast
  rw [mul_right_comm, mul_assoc, ← this]
  

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : ↑(Nat.gcd r.den p) = ↑r.den * ↑(gcdA r.den p)
⊢ ↑r.num - ↑r.num * ↑(Nat.gcd r.den p) = 0

Next tactic:
suffices rdcp : r.den.Coprime p
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
  push_cast
  rw [mul_right_comm, mul_assoc, ← this]
  suffices rdcp : r.den.Coprime p
  · 

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : ↑(Nat.gcd r.den p) = ↑r.den * ↑(gcdA r.den p)
rdcp : Coprime r.den p
⊢ ↑r.num - ↑r.num * ↑(Nat.gcd r.den p) = 0

Next tactic:
rw [rdcp.gcd_eq_one]
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
  push_cast
  rw [mul_right_comm, mul_assoc, ← this]
  suffices rdcp : r.den.Coprime p
  · rw [rdcp.gcd_eq_one]
    

Proof state:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : ↑(Nat.gcd r.den p) = ↑r.den * ↑(gcdA r.den p)
rdcp : Coprime r.den p
⊢ ↑r.num - ↑r.num * ↑1 = 0

Next tactic:
simp only [mul_one, cast_one, sub_self]
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
  push_cast
  rw [mul_right_comm, mul_assoc, ← this]
  suffices rdcp : r.den.Coprime p
  · rw [rdcp.gcd_eq_one]
    simp only [mul_one, cast_one, sub_self]
  

Proof state:
case rdcp
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : ↑(Nat.gcd r.den p) = ↑r.den * ↑(gcdA r.den p)
⊢ Coprime r.den p

Next tactic:
apply Coprime.symm
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by
  rw [← ZMod.int_cast_zmod_eq_zero_iff_dvd]
  simp only [Int.cast_ofNat, ZMod.nat_cast_mod, Int.cast_mul, Int.cast_sub]
  have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
  simp only [Int.cast_ofNat, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
    Int.cast_mul, zero_mul, add_zero] at this
  push_cast
  rw [mul_right_comm, mul_assoc, ← this]
  suffices rdcp : r.den.Coprime p
  · rw [rdcp.gcd_eq_one]
    simp only [mul_one, cast_one, sub_self]
  apply Coprime.symm
  

Proof state:
case rdcp.a
p : ℕ
hp_prime : Fact (Nat.Prime p)
r✝ r : ℚ
h : ‖↑r‖ ≤ 1
this : ↑(Nat.gcd r.den p) = ↑r.den * ↑(gcdA r.den p)
⊢ Coprime p r.den

Next tactic:
apply (coprime_or_dvd_of_prime hp_prime.1 _).resolve_right