Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: case inr.refine_1
R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Module.Finite R S
Q : Ideal S
inst✝¹ : Q.IsPrime
inst✝ : IsUnramifiedAt R Q
h✝ : Nontrivial S
this✝ : Nontrivial R
P : Ideal R := Ideal.under R Q
this : Algebra (Localization.AtPrime P) (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver P Q
x : S
hx : R[x] = ⊤
hRx : IsIntegral R x
I : Ideal R[X] := RingHom.ker (aeval x).toRingHom
hx' : Function.Surjective ⇑(aeval x)
p : R[X]
hpI : p ∈ I
hp :
  Ideal.span {Polynomial.map (algebraMap R P.ResidueField) p} = Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I
hI' : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = RingHom.ker (aeval (1 ⊗ₜ[R] x)).toRingHom
m : P.ResidueField[X] := minpoly P.ResidueField ((algebraMap S Q.ResidueField) x)
hm : Prime m
hmp₁ : m ∣ Polynomial.map (algebraMap R P.ResidueField) p
hmp₂ : ¬m ^ 2 ∣ Polynomial.map (algebraMap R P.ResidueField) p
q : R[X] := minpoly R x ^ (p.natDegree + 2) + p
⊢ q.Monic

pp: case inr.refine_2
R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Module.Finite R S
Q : Ideal S
inst✝¹ : Q.IsPrime
inst✝ : IsUnramifiedAt R Q
h✝ : Nontrivial S
this✝ : Nontrivial R
P : Ideal R := Ideal.under R Q
this : Algebra (Localization.AtPrime P) (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver P Q
x : S
hx : R[x] = ⊤
hRx : IsIntegral R x
I : Ideal R[X] := RingHom.ker (aeval x).toRingHom
hx' : Function.Surjective ⇑(aeval x)
p : R[X]
hpI : p ∈ I
hp :
  Ideal.span {Polynomial.map (algebraMap R P.ResidueField) p} = Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I
hI' : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = RingHom.ker (aeval (1 ⊗ₜ[R] x)).toRingHom
m : P.ResidueField[X] := minpoly P.ResidueField ((algebraMap S Q.ResidueField) x)
hm : Prime m
hmp₁ : m ∣ Polynomial.map (algebraMap R P.ResidueField) p
hmp₂ : ¬m ^ 2 ∣ Polynomial.map (algebraMap R P.ResidueField) p
q : R[X] := minpoly R x ^ (p.natDegree + 2) + p
⊢ (aeval x) (derivative q) ∉ Q

Tactic:
· refine ((minpoly.monic hRx).pow _).add_of_left (degree_lt_degree ?_)
    grw [natDegree_pow' (by simp [minpoly.monic hRx]),
      ← Nat.le_mul_of_pos_right _ (minpoly.natDegree_pos hRx)]; lia

Goal state after:
case inr.refine_2
R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Module.Finite R S
Q : Ideal S
inst✝¹ : Q.IsPrime
inst✝ : IsUnramifiedAt R Q
h✝ : Nontrivial S
this✝ : Nontrivial R
P : Ideal R := Ideal.under R Q
this : Algebra (Localization.AtPrime P) (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver P Q
x : S
hx : R[x] = ⊤
hRx : IsIntegral R x
I : Ideal R[X] := RingHom.ker (aeval x).toRingHom
hx' : Function.Surjective ⇑(aeval x)
p : R[X]
hpI : p ∈ I
hp :
  Ideal.span {Polynomial.map (algebraMap R P.ResidueField) p} = Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I
hI' : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = RingHom.ker (aeval (1 ⊗ₜ[R] x)).toRingHom
m : P.ResidueField[X] := minpoly P.ResidueField ((algebraMap S Q.ResidueField) x)
hm : Prime m
hmp₁ : m ∣ Polynomial.map (algebraMap R P.ResidueField) p
hmp₂ : ¬m ^ 2 ∣ Polynomial.map (algebraMap R P.ResidueField) p
q : R[X] := minpoly R x ^ (p.natDegree + 2) + p
⊢ (aeval x) (derivative q) ∉ Q