Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case neg
n : ℕ
P : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop
hP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }
hP₂ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),
    (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e
hP₃ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i j : Fin n),
    (e.val i).Monic →
      (e.val i).degree ≤ (e.val j).degree → i ≠ j → P R { val := Function.update e.val j (e.val j %ₘ e.val i) } → P R e
hP₄ :
  ∀ (R : Type u) [inst : CommRing R] (c : R) (i : Fin n) (e : InductionObj R n),
    c = (e.val i).leadingCoeff →
      c ≠ 0 →
        P (Away c)
            { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom (algebraMap R (Away c))) ∘ e.val } →
          P (R ⧸ Ideal.span {c}) { val := ⇑(mapRingHom (algebraMap R (R ⧸ Ideal.span {c}))) ∘ e.val } → P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n), y = e.degree → P R e
R : Type u
inst✝ : CommRing R
e : InductionObj R n
hv : v = e.degree
h✝ : Nontrivial R
he0 : ∃ x, ¬e.val x = 0
H : ¬∃ i, (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
i : Fin n
hi : e.val i ≠ 0
i_min : ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
⊢ P R e

Tactic:
apply hP₄ _ _ i e rfl (by simpa using hi) (H_IH _ ?_ _ rfl) (H_IH _ ?_ _ rfl)

Goal state after:
n : ℕ
P : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop
hP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }
hP₂ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),
    (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e
hP₃ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i j : Fin n),
    (e.val i).Monic →
      (e.val i).degree ≤ (e.val j).degree → i ≠ j → P R { val := Function.update e.val j (e.val j %ₘ e.val i) } → P R e
hP₄ :
  ∀ (R : Type u) [inst : CommRing R] (c : R) (i : Fin n) (e : InductionObj R n),
    c = (e.val i).leadingCoeff →
      c ≠ 0 →
        P (Away c)
            { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom (algebraMap R (Away c))) ∘ e.val } →
          P (R ⧸ Ideal.span {c}) { val := ⇑(mapRingHom (algebraMap R (R ⧸ Ideal.span {c}))) ∘ e.val } → P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n), y = e.degree → P R e
R : Type u
inst✝ : CommRing R
e : InductionObj R n
hv : v = e.degree
h✝ : Nontrivial R
he0 : ∃ x, ¬e.val x = 0
H : ¬∃ i, (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
i : Fin n
hi : e.val i ≠ 0
i_min : ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
⊢ {
        val :=
          Polynomial.C (IsLocalization.Away.invSelf (e.val i).leadingCoeff) •
            ⇑(mapRingHom (algebraMap R (Away (e.val i).leadingCoeff))) ∘ e.val }.degree <
    v

n : ℕ
P : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop
hP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }
hP₂ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),
    (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e
hP₃ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i j : Fin n),
    (e.val i).Monic →
      (e.val i).degree ≤ (e.val j).degree → i ≠ j → P R { val := Function.update e.val j (e.val j %ₘ e.val i) } → P R e
hP₄ :
  ∀ (R : Type u) [inst : CommRing R] (c : R) (i : Fin n) (e : InductionObj R n),
    c = (e.val i).leadingCoeff →
      c ≠ 0 →
        P (Away c)
            { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom (algebraMap R (Away c))) ∘ e.val } →
          P (R ⧸ Ideal.span {c}) { val := ⇑(mapRingHom (algebraMap R (R ⧸ Ideal.span {c}))) ∘ e.val } → P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n), y = e.degree → P R e
R : Type u
inst✝ : CommRing R
e : InductionObj R n
hv : v = e.degree
h✝ : Nontrivial R
he0 : ∃ x, ¬e.val x = 0
H : ¬∃ i, (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
i : Fin n
hi : e.val i ≠ 0
i_min : ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
⊢ { val := ⇑(mapRingHom (algebraMap R (R ⧸ Ideal.span {(e.val i).leadingCoeff}))) ∘ e.val }.degree < v