/-- Formula for the discriminant of a power basis using the norm of the field extension. -/
-- Porting note: `(minpoly K pb.gen).derivative` does not work anymore.
theorem discr_powerBasis_eq_norm [IsSeparable K L] :
    discr K pb.basis =
      (-1) ^ (n * (n - 1) / 2) *
      norm K (aeval pb.gen (derivative (R := K) (minpoly K pb.gen))) := by
  let E := AlgebraicClosure L
  letI := fun a b : E => Classical.propDecidable (Eq a b)
  have e : Fin pb.dim ≃ (L →ₐ[K] E) := by
    refine' equivOfCardEq _
    rw [Fintype.card_fin, AlgHom.card]
    exact (PowerBasis.finrank pb).symm
  have hnodup : ((minpoly K pb.gen).aroots E).Nodup :=
    nodup_roots (Separable.map (IsSeparable.separable K pb.gen))
  have hroots : ∀ σ : L →ₐ[K] E, σ pb.gen ∈ (minpoly K pb.gen).aroots E := by
    intro σ
    rw [mem_roots, IsRoot.def, eval_map, ← aeval_def, aeval_algHom_apply]
    repeat' simp [minpoly.ne_zero (IsSeparable.isIntegral K pb.gen)]
  apply (algebraMap K E).injective
  rw [RingHom.map_mul, RingHom.map_pow, RingHom.map_neg, RingHom.map_one,
    discr_powerBasis_eq_prod'' _ _ _ e]
  congr
  rw [norm_eq_prod_embeddings, prod_prod_Ioi_mul_eq_prod_prod_off_diag]
  conv_rhs =>
    congr
    rfl
    ext σ
    rw [← aeval_algHom_apply,
      aeval_root_derivative_of_splits (minpoly.monic (IsSeparable.isIntegral K pb.gen))
        (IsAlgClosed.splits_codomain _) (hroots σ),
      ← Finset.prod_mk _ (hnodup.erase _)]
  rw [prod_sigma', prod_sigma']
  refine'
    prod_bij (fun i _ => ⟨e i.2, e i.1 pb.gen⟩) (fun i hi => _) (fun i _ => by simp)
      (fun i j hi hj hij => _) fun σ hσ => _
  · simp only [true_and_iff, Finset.mem_mk, mem_univ, mem_sigma]
    rw [Multiset.mem_erase_of_ne fun h => ?_]
    · exact hroots _
-- Porting note: `@mem_compl` was not necessary.
    · simp only [true_and_iff, mem_univ, Ne.def, mem_sigma, @mem_compl _ _ _ (_),
        mem_singleton] at hi
      rw [← PowerBasis.liftEquiv_apply_coe, ← PowerBasis.liftEquiv_apply_coe] at h
      exact hi (e.injective <| pb.liftEquiv.injective <| Subtype.eq h.symm)
  · simp only [Sigma.mk.inj_iff, EmbeddingLike.apply_eq_iff_eq, heq_eq_eq] at hij
    have h := hij.2
    rw [← PowerBasis.liftEquiv_apply_coe, ← PowerBasis.liftEquiv_apply_coe] at h
    refine' Sigma.eq (Equiv.injective e (Equiv.injective _ (Subtype.eq h))) (by simp [hij.1])
  · simp only [true_and_iff, Finset.mem_mk, mem_univ, mem_sigma] at hσ ⊢
    simp only [Sigma.exists, exists_prop, mem_compl, mem_singleton, Ne.def]
    refine' ⟨e.symm (PowerBasis.lift pb σ.2 _), e.symm σ.1, ⟨_, Sigma.eq _ _⟩⟩
    · rw [aeval_def, eval₂_eq_eval_map, ← IsRoot.def, ← mem_roots]
      · exact Multiset.erase_subset _ _ hσ
      · simp [minpoly.ne_zero (IsSeparable.isIntegral K pb.gen)]
-- Porting note: the `simp only` was not needed.
    · simp only [@mem_compl _ _ _ (_), mem_singleton]
      intro h
      replace h := AlgHom.congr_fun (Equiv.injective _ h) pb.gen
      rw [PowerBasis.lift_gen] at h
      rw [← h] at hσ
      exact hnodup.not_mem_erase hσ
    

Proof state:
case a.e_a.refine'_3.refine'_3
A : Type u
B : Type v
C : Type z
ι : Type w
inst✝¹⁶ : DecidableEq ι
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
ι' : Type u_1
inst✝¹⁰ : Fintype ι'
inst✝⁹ : Fintype ι
inst✝⁸ : DecidableEq ι'
K : Type u
L : Type v
E✝ : Type z
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field E✝
inst✝⁴ : Algebra K L
inst✝³ : Algebra K E✝
inst✝² : Module.Finite K L
inst✝¹ : IsAlgClosed E✝
b : ι → L
pb : PowerBasis K L
inst✝ : IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b => Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : Multiset.Nodup (aroots (minpoly K pb.gen) E)
hroots : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ aroots (minpoly K pb.gen) E
σ : (_ : L →ₐ[K] E) × E
hσ : σ.snd ∈ Multiset.erase (aroots (minpoly K pb.gen) E) (σ.fst pb.gen)
⊢ σ.fst =
    { fst := e (e.symm σ.fst),
        snd := (e (e.symm (PowerBasis.lift pb σ.snd (_ : (aeval σ.snd) (minpoly K pb.gen) = 0)))) pb.gen }.fst
case a.e_a.refine'_3.refine'_4
A : Type u
B : Type v
C : Type z
ι : Type w
inst✝¹⁶ : DecidableEq ι
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
ι' : Type u_1
inst✝¹⁰ : Fintype ι'
inst✝⁹ : Fintype ι
inst✝⁸ : DecidableEq ι'
K : Type u
L : Type v
E✝ : Type z
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field E✝
inst✝⁴ : Algebra K L
inst✝³ : Algebra K E✝
inst✝² : Module.Finite K L
inst✝¹ : IsAlgClosed E✝
b : ι → L
pb : PowerBasis K L
inst✝ : IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b => Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : Multiset.Nodup (aroots (minpoly K pb.gen) E)
hroots : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ aroots (minpoly K pb.gen) E
σ : (_ : L →ₐ[K] E) × E
hσ : σ.snd ∈ Multiset.erase (aroots (minpoly K pb.gen) E) (σ.fst pb.gen)
⊢ Eq.recOn ?a.e_a.refine'_3.refine'_3✝ σ.snd =
    { fst := e (e.symm σ.fst),
        snd := (e (e.symm (PowerBasis.lift pb σ.snd (_ : (aeval σ.snd) (minpoly K pb.gen) = 0)))) pb.gen }.snd

Next tactic:
all_goals simp