Module: Mathlib.NumberTheory.NumberField.ClassNumber

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
hlies : P.LiesOver (span {p})
hspan : span {↑p.natAbs} = span {p}
hple :
  p.natAbs ^ P.inertiaDeg ℤ ≤
    ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊
hpabsprime : Nat.Prime p.natAbs
⊢ Submodule.IsPrincipal ↑⟨P, HP⟩

Tactic:
refine h _ ?_ hpabsprime _ ⟨hP, ?_⟩ hple

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
hlies : P.LiesOver (span {p})
hspan : span {↑p.natAbs} = span {p}
hple :
  p.natAbs ^ P.inertiaDeg ℤ ≤
    ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊
hpabsprime : Nat.Prime p.natAbs
⊢ p.natAbs ∈
    Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊

case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h :
  ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,
    Nat.Prime p →
      ∀ P ∈ (span {↑p}).primesOver (𝓞 K),
        p ^ P.inertiaDeg ℤ ≤
            ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊ →
          Submodule.IsPrincipal P
x✝ : ↥(Ideal (𝓞 K))⁰
P : Ideal (𝓞 K)
HP : P ∈ (Ideal (𝓞 K))⁰
hP : (↑⟨P, HP⟩).IsPrime
hPN :
  ↑(absNorm ↑⟨P, HP⟩) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)
p : ℤ
hp : under ℤ P = ℤ ∙ p
hp0 : p ≠ 0
hpprime : (under ℤ P).IsPrime → _root_.Prime p
hlies : P.LiesOver (span {p})
hspan : span {↑p.natAbs} = span {p}
hple :
  p.natAbs ^ P.inertiaDeg ℤ ≤
    ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊
hpabsprime : Nat.Prime p.natAbs
⊢ (↑⟨P, HP⟩).LiesOver (span {↑p.natAbs})