/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
      exfalso
      exact hp.ne_zero ((injective_iff_map_eq_zero _).1 this _ H)
    · rw [H₁]
      exact Subalgebra.zero_mem _
  -- It is enough to prove that all coefficients of `Q` are divisible by `p`, by induction.
  -- The base case is `dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt`.
  refine mem_adjoin_of_dvd_coeff_of_dvd_aeval hp.ne_zero (fun i => ?_) hQ
  induction' i using Nat.case_strong_induction_on with j hind
  · intro _
    exact dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt hp hBint hQ hzint hei
  · intro hj
    convert hp.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd _ hndiv
    exact n
    -- Two technical results we will need about `P.natDegree` and `Q.natDegree`.
    have H := degree_modByMonic_lt Q₁ (minpoly.monic hBint)
    rw [← hQ₁, ← hP] at H
    replace H := Nat.lt_iff_add_one_le.1
      (lt_of_lt_of_le
        (lt_of_le_of_lt (Nat.lt_iff_add_one_le.1 (Nat.lt_of_succ_lt_succ (mem_range.1 hj)))
          (lt_succ_self _)) (Nat.lt_iff_add_one_le.1 ((natDegree_lt_natDegree_iff hQzero).2 H)))
    rw [add_assoc] at H
    have Hj : Q.natDegree + 1 = j + 1 + (Q.natDegree - j) := by
      rw [← add_comm 1, ← add_comm 1, add_assoc, add_right_inj,
        ← Nat.add_sub_assoc (Nat.lt_of_succ_lt_succ (mem_range.1 hj)).le, add_comm,
        Nat.add_sub_cancel]
    -- By induction hypothesis we can find `g : ℕ → R` such that
    -- `k ∈ range (j + 1) → Q.coeff k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k`-
    choose! g hg using hind
    replace hg : ∀ k ∈ range (j + 1), Q.coeff k • B.gen ^ k = algebraMap R L p * g k • B.gen ^ k
    · intro k hk
      rw [hg k (mem_range_succ_iff.1 hk)
        (mem_range_succ_iff.2
          (le_trans (mem_range_succ_iff.1 hk) (succ_le_iff.1 (mem_range_succ_iff.1 hj)).le)),
        Algebra.smul_def, Algebra.smul_def, RingHom.map_mul, mul_assoc]
    -- Since `minpoly R B.gen` is Eiseinstein, we can find `f : ℕ → L` such that
    -- `(map (algebraMap R L) (minpoly R B.gen)).nat_degree ≤ i` implies `f i ∈ adjoin R {B.gen}`
    -- and `(algebraMap R L) p * f i = B.gen ^ i`. We will also need `hf₁`, a reformulation of this
    -- property.
    choose! f hf using
      IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree_le (minpoly.aeval R B.gen)
        (minpoly.monic hBint) hei.isWeaklyEisensteinAt
    have hf₁ : ∀ k ∈ (range (Q.natDegree - j)).erase 0,
        Q.coeff (j + 1 + k) • B.gen ^ (j + 1 + k) * B.gen ^ (P.natDegree - (j + 2)) =
        (algebraMap R L) p * Q.coeff (j + 1 + k) • f (k + P.natDegree - 1) := by
      intro k hk
      rw [smul_mul_assoc, ← pow_add, ← Nat.add_sub_assoc H, ← add_assoc j 1 1, add_comm (j + 1) 1,
        add_assoc (j + 1), add_comm _ (k + P.natDegree), Nat.add_sub_add_right,
        ← (hf (k + P.natDegree - 1) _).2, mul_smul_comm]
      rw [(minpoly.monic hBint).natDegree_map, add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
      · exact Nat.zero_le _
      · refine one_le_iff_ne_zero.2 fun h => ?_
        rw [h] at hk
        simp at hk

    -- The Eisenstein condition shows that `p` divides `Q.coeff j`
    -- if `p^n.succ` divides the following multiple of `Q.coeff (succ j)^n.succ`:
    suffices
        p ^ n.succ ∣ Q.coeff (succ j) ^ n.succ *
          (minpoly R B.gen).coeff 0 ^ (succ j + (P.natDegree - (j + 2))) by
      convert this
      rw [Nat.succ_eq_add_one, add_assoc, ← Nat.add_sub_assoc H, ← add_assoc, add_comm (j + 1),
        Nat.add_sub_add_left, ← Nat.add_sub_assoc, Nat.add_sub_add_left, hP, ←
        (minpoly.monic hBint).natDegree_map (algebraMap R K), ←
        minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, natDegree_minpoly, hn, Nat.sub_one,
        Nat.pred_succ]
      linarith

    -- Using `hQ : aeval B.gen Q = p • z`, we write `p • z` as a sum of terms of degree less than
    -- `j+1`, that are multiples of `p` by induction, and terms of degree at least `j+1`.
    rw [aeval_eq_sum_range, Hj, range_add, sum_union (disjoint_range_addLeftEmbedding _ _),
      sum_congr rfl hg, add_comm] at hQ
    -- We multiply this equality by `B.gen ^ (P.natDegree-(j+2))`, so we can use `hf₁` on the terms
    -- we didn't know were multiples of `p`, and we take the norm on both sides.
    replace hQ := congr_arg (fun x => x * B.gen ^ (P.natDegree - (j + 2))) hQ
    simp_rw [sum_map, addLeftEmbedding_apply, add_mul, sum_mul, mul_assoc] at hQ
    rw [← insert_erase
      (mem_range.2 (tsub_pos_iff_lt.2 <| Nat.lt_of_succ_lt_succ <| mem_range.1 hj)),
      sum_insert (not_mem_erase 0 _), add_zero, sum_congr rfl hf₁, ← mul_sum, ← mul_sum, add_assoc,
      ← mul_add, smul_mul_assoc, ← pow_add, Algebra.smul_def] at hQ
    replace hQ := congr_arg (norm K) (eq_sub_of_add_eq hQ)

    -- We obtain an equality of elements of `K`, but everything is integral, so we can move to `R`
    -- and simplify `hQ`.
    have hintsum : IsIntegral R (z * B.gen ^ (P.natDegree - (j + 2)) -
        (∑ x : ℕ in (range (Q.natDegree - j)).erase 0,
          Q.coeff (j + 1 + x) • f (x + P.natDegree - 1) +
            ∑ x : ℕ in range (j + 1), g x • B.gen ^ x * B.gen ^ (P.natDegree - (j + 2)))) := by
      refine (hzint.mul (hBint.pow _)).sub
        (.add (.sum _ fun k hk => .smul _ ?_)
          (.sum _ fun k _ => .mul (.smul _ (.pow hBint _)) (hBint.pow _)))
      refine adjoin_le_integralClosure hBint (hf _ ?_).1
      rw [(minpoly.monic hBint).natDegree_map (algebraMap R L)]
      rw [add_comm, Nat.add_sub_assoc, le_add_iff_nonneg_right]
      · exact _root_.zero_le _
      · refine one_le_iff_ne_zero.2 fun h => ?_
        rw [h] at hk
        simp at hk
    obtain ⟨r, hr⟩ := isIntegral_iff.1 (isIntegral_norm K hintsum)
    

Proof state:
case neg.hi.convert_3.intro
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : ¬Q = 0
j : ℕ
hj : succ j ∈ range (natDegree Q + 1)
H : j + (1 + 1) ≤ natDegree P
Hj : natDegree Q + 1 = j + 1 + (natDegree Q - j)
g : ℕ → R
hg : ∀ k ∈ range (j + 1), coeff Q k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k
f : ℕ → L
hf :
  ∀ (i : ℕ),
    natDegree (Polynomial.map (algebraMap R L) (minpoly R B.gen)) ≤ i →
      f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
hf₁ :
  ∀ k ∈ Finset.erase (range (natDegree Q - j)) 0,
    coeff Q (j + 1 + k) • B.gen ^ (j + 1 + k) * B.gen ^ (natDegree P - (j + 2)) =
      (algebraMap R L) p * coeff Q (j + 1 + k) • f (k + natDegree P - 1)
hQ :
  (Algebra.norm K)
      ((algebraMap R L) (coeff (Q₁ %ₘ minpoly R B.gen) (j + 1)) *
        B.gen ^ (j + 1 + (natDegree (minpoly R B.gen) - (j + 2)))) =
    (Algebra.norm K)
      (p • z * B.gen ^ (natDegree (minpoly R B.gen) - (j + 2)) -
        (algebraMap R L) p *
          (∑ x in Finset.erase (range (natDegree Q - j)) 0, coeff Q (j + 1 + x) • f (x + natDegree P - 1) +
            ∑ x in range (j + 1), g x • B.gen ^ x * B.gen ^ (natDegree (minpoly R B.gen) - (j + 2))))
hintsum :
  IsIntegral R
    (z * B.gen ^ (natDegree P - (j + 2)) -
      (∑ x in Finset.erase (range (natDegree Q - j)) 0, coeff Q (j + 1 + x) • f (x + natDegree P - 1) +
        ∑ x in range (j + 1), g x • B.gen ^ x * B.gen ^ (natDegree P - (j + 2))))
r : R
hr :
  (algebraMap R K) r =
    (Algebra.norm K)
      (z * B.gen ^ (natDegree P - (j + 2)) -
        (∑ x in Finset.erase (range (natDegree Q - j)) 0, coeff Q (j + 1 + x) • f (x + natDegree P - 1) +
          ∑ x in range (j + 1), g x • B.gen ^ x * B.gen ^ (natDegree P - (j + 2))))
⊢ p ^ succ n ∣ coeff Q (succ j) ^ succ n * coeff (minpoly R B.gen) 0 ^ (succ j + (natDegree P - (j + 2)))

Next tactic:
rw [Algebra.smul_def, mul_assoc, ← mul_sub, _root_.map_mul, algebraMap_apply R K L, map_pow,
      Algebra.norm_algebraMap, _root_.map_mul, algebraMap_apply R K L, Algebra.norm_algebraMap,
      finrank B, ← hr, PowerBasis.norm_gen_eq_coeff_zero_minpoly,
      minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map,
      show (-1 : K) = algebraMap R K (-1) by simp, ← map_pow, ← map_pow, ← _root_.map_mul, ←
      map_pow, ← _root_.map_mul, ← map_pow, ← _root_.map_mul] at hQ