l3lab/ntp-mathlib huggingface dataset:
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
| -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i

Next tactic:
congr
    congr
    · skip
    ext i
    rw [coeff_map, hφ i.1 (lt_of_lt_of_le i.2 (natDegree_map_le _ _)),
      RingHom.map_mul, mul_assoc]
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
| -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i

Next tactic:
congr
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    

Proof state:
case a
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
| ∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i

Next tactic:
congr
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    congr
    

Proof state:
case a.s
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
| univ
case a.f
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
| fun i => coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i

Next tactic:
· skip
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    congr
    · 

Proof state:
case a.s
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
| univ

Next tactic:
skip
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    congr
    · skip
    

Proof state:
case a.f
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
| fun i => coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i

Next tactic:
ext i
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    congr
    · skip
    ext i
    

Proof state:
case a.f.h
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i
φ : ℕ → R
hφ : ∀ n < natDegree f, coeff f n = p * φ n
i : Fin (natDegree (Polynomial.map (algebraMap R S) f))
| coeff (Polynomial.map (algebraMap R S) f) ↑i * x ^ ↑i

Next tactic:
rw [coeff_map, hφ i.1 (lt_of_lt_of_le i.2 (natDegree_map_le _ _)),
      RingHom.map_mul, mul_assoc]
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    congr
    · skip
    ext i
    rw [coeff_map, hφ i.1 (lt_of_lt_of_le i.2 (natDegree_map_le _ _)),
      RingHom.map_mul, mul_assoc]
  

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
φ : ℕ → R
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), (algebraMap R S) p * ((algebraMap R S) (φ ↑i) * x ^ ↑i)
hφ : ∀ n < natDegree f, coeff f n = p * φ n
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)

Next tactic:
rw [hx, ← mul_sum, neg_eq_neg_one_mul, ← mul_assoc (-1 : S), mul_comm (-1 : S), mul_assoc]
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    congr
    · skip
    ext i
    rw [coeff_map, hφ i.1 (lt_of_lt_of_le i.2 (natDegree_map_le _ _)),
      RingHom.map_mul, mul_assoc]
  rw [hx, ← mul_sum, neg_eq_neg_one_mul, ← mul_assoc (-1 : S), mul_comm (-1 : S), mul_assoc]
  

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
φ : ℕ → R
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), (algebraMap R S) p * ((algebraMap R S) (φ ↑i) * x ^ ↑i)
hφ : ∀ n < natDegree f, coeff f n = p * φ n
⊢ ∃ y ∈ adjoin R {x},
    (algebraMap R S) p * y =
      (algebraMap R S) p *
        (-1 * ∑ x_1 : Fin (natDegree (Polynomial.map (algebraMap R S) f)), (algebraMap R S) (φ ↑x_1) * x ^ ↑x_1)

Next tactic:
refine'
    ⟨-1 * ∑ i : Fin (f.map (algebraMap R S)).natDegree, (algebraMap R S) (φ i.1) * x ^ i.1, _, rfl⟩
theorem exists_mem_adjoin_mul_eq_pow_natDegree {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) : ∃ y ∈ adjoin R ({x} : Set S),
    (algebraMap R S) p * y = x ^ (f.map (algebraMap R S)).natDegree := by
  rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
    sum_insert not_mem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
    one_mul] at hx
  replace hx := eq_neg_of_add_eq_zero_left hx
  have : ∀ n < f.natDegree, p ∣ f.coeff n := by
    intro n hn
    refine' mem_span_singleton.1 (by simpa using hf.mem hn)
  choose! φ hφ using this
  conv_rhs at hx =>
    congr
    congr
    · skip
    ext i
    rw [coeff_map, hφ i.1 (lt_of_lt_of_le i.2 (natDegree_map_le _ _)),
      RingHom.map_mul, mul_assoc]
  rw [hx, ← mul_sum, neg_eq_neg_one_mul, ← mul_assoc (-1 : S), mul_comm (-1 : S), mul_assoc]
  refine'
    ⟨-1 * ∑ i : Fin (f.map (algebraMap R S)).natDegree, (algebraMap R S) (φ i.1) * x ^ i.1, _, rfl⟩
  

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
φ : ℕ → R
hx :
  x ^ natDegree (Polynomial.map (algebraMap R S) f) =
    -∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), (algebraMap R S) p * ((algebraMap R S) (φ ↑i) * x ^ ↑i)
hφ : ∀ n < natDegree f, coeff f n = p * φ n
⊢ -1 * ∑ i : Fin (natDegree (Polynomial.map (algebraMap R S) f)), (algebraMap R S) (φ ↑i) * x ^ ↑i ∈ adjoin R {x}

Next tactic:
exact
    Subalgebra.mul_mem _ (Subalgebra.neg_mem _ (Subalgebra.one_mem _))
      (Subalgebra.sum_mem _ fun i _ =>
        Subalgebra.mul_mem _ (Subalgebra.algebraMap_mem _ _)
          (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton x)) _))
theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
        ∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by
  

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
⊢ ∀ (i : ℕ), natDegree (Polynomial.map (algebraMap R S) f) ≤ i → ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i

Next tactic:
intro i hi
theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
        ∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by
  intro i hi
  

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
i : ℕ
hi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i

Next tactic:
obtain ⟨k, hk⟩ := exists_add_of_le hi
theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
        ∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  

Proof state:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
i : ℕ
hi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i
k : ℕ
hk : i = natDegree (Polynomial.map (algebraMap R S) f) + k
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i

Next tactic:
rw [hk, pow_add]
theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
        ∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  

Proof state:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
i : ℕ
hi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i
k : ℕ
hk : i = natDegree (Polynomial.map (algebraMap R S) f) + k
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k

Next tactic:
obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_natDegree hx hmo hf
theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
        ∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_natDegree hx hmo hf
  

Proof state:
case intro.intro.intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
i : ℕ
hi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i
k : ℕ
hk : i = natDegree (Polynomial.map (algebraMap R S) f) + k
y : S
hy : y ∈ adjoin R {x}
H : (algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k

Next tactic:
refine' ⟨y * x ^ k, _, _⟩
theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
        ∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_natDegree hx hmo hf
  refine' ⟨y * x ^ k, _, _⟩
  · 

Proof state:
case intro.intro.intro.refine'_1
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
i : ℕ
hi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i
k : ℕ
hk : i = natDegree (Polynomial.map (algebraMap R S) f) + k
y : S
hy : y ∈ adjoin R {x}
H : (algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)
⊢ y * x ^ k ∈ adjoin R {x}

Next tactic:
exact Subalgebra.mul_mem _ hy (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton x)) _)
theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
    (hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
        ∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_natDegree hx hmo hf
  refine' ⟨y * x ^ k, _, _⟩
  · exact Subalgebra.mul_mem _ hy (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton x)) _)
  · 

Proof state:
case intro.intro.intro.refine'_2
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : Monic f
hf : IsWeaklyEisensteinAt f (Submodule.span R {p})
i : ℕ
hi : natDegree (Polynomial.map (algebraMap R S) f) ≤ i
k : ℕ
hk : i = natDegree (Polynomial.map (algebraMap R S) f) + k
y : S
hy : y ∈ adjoin R {x}
H : (algebraMap R S) p * y = x ^ natDegree (Polynomial.map (algebraMap R S) f)
⊢ (algebraMap R S) p * (y * x ^ k) = x ^ natDegree (Polynomial.map (algebraMap R S) f) * x ^ k

Next tactic:
rw [← mul_assoc _ y, H]
theorem pow_natDegree_le_of_root_of_monic_mem {x : R} (hroot : IsRoot f x) (hmo : f.Monic) :
    ∀ i, f.natDegree ≤ i → x ^ i ∈ 𝓟 := by
  

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : R
hroot : IsRoot f x
hmo : Monic f
⊢ ∀ (i : ℕ), natDegree f ≤ i → x ^ i ∈ 𝓟

Next tactic:
intro i hi
theorem pow_natDegree_le_of_root_of_monic_mem {x : R} (hroot : IsRoot f x) (hmo : f.Monic) :
    ∀ i, f.natDegree ≤ i → x ^ i ∈ 𝓟 := by
  intro i hi
  

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : R
hroot : IsRoot f x
hmo : Monic f
i : ℕ
hi : natDegree f ≤ i
⊢ x ^ i ∈ 𝓟

Next tactic:
obtain ⟨k, hk⟩ := exists_add_of_le hi
theorem pow_natDegree_le_of_root_of_monic_mem {x : R} (hroot : IsRoot f x) (hmo : f.Monic) :
    ∀ i, f.natDegree ≤ i → x ^ i ∈ 𝓟 := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  

Proof state:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : R
hroot : IsRoot f x
hmo : Monic f
i : ℕ
hi : natDegree f ≤ i
k : ℕ
hk : i = natDegree f + k
⊢ x ^ i ∈ 𝓟

Next tactic:
rw [hk, pow_add]
theorem pow_natDegree_le_of_root_of_monic_mem {x : R} (hroot : IsRoot f x) (hmo : f.Monic) :
    ∀ i, f.natDegree ≤ i → x ^ i ∈ 𝓟 := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  

Proof state:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : R
hroot : IsRoot f x
hmo : Monic f
i : ℕ
hi : natDegree f ≤ i
k : ℕ
hk : i = natDegree f + k
⊢ x ^ natDegree f * x ^ k ∈ 𝓟

Next tactic:
suffices x ^ f.natDegree ∈ 𝓟 by exact mul_mem_right (x ^ k) 𝓟 this
theorem pow_natDegree_le_of_root_of_monic_mem {x : R} (hroot : IsRoot f x) (hmo : f.Monic) :
    ∀ i, f.natDegree ≤ i → x ^ i ∈ 𝓟 := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  suffices x ^ f.natDegree ∈ 𝓟 by 

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : R
hroot : IsRoot f x
hmo : Monic f
i : ℕ
hi : natDegree f ≤ i
k : ℕ
hk : i = natDegree f + k
this : x ^ natDegree f ∈ 𝓟
⊢ x ^ natDegree f * x ^ k ∈ 𝓟

Next tactic:
exact mul_mem_right (x ^ k) 𝓟 this
theorem pow_natDegree_le_of_root_of_monic_mem {x : R} (hroot : IsRoot f x) (hmo : f.Monic) :
    ∀ i, f.natDegree ≤ i → x ^ i ∈ 𝓟 := by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  suffices x ^ f.natDegree ∈ 𝓟 by exact mul_mem_right (x ^ k) 𝓟 this
  

Proof state:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf : IsWeaklyEisensteinAt f 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : R
hroot : IsRoot f x
hmo : Monic f
i : ℕ
hi : natDegree f ≤ i
k : ℕ
hk : i = natDegree f + k
⊢ x ^ natDegree f ∈ 𝓟

Next tactic:
rw [IsRoot.def, eval_eq_sum_range, Finset.range_add_one,
    Finset.sum_insert Finset.not_mem_range_self, Finset.sum_range, hmo.coeff_natDegree, one_mul] at
    *