awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
this : ∃ (n_1 : ℕ) (i : Fin n), (e.val i).degree = (↑n_1 : WithBot ℕ) ∧ e.val i ≠ 0
m : ℕ := Nat.find (p := fun (n_1 : ℕ) ↦ ∃ (i : Fin n), (e.val i).degree = (↑n_1 : WithBot ℕ) ∧ e.val i ≠ 0) this
i : Fin n
hi : (e.val i).degree = (↑m : WithBot ℕ)
hi' : e.val i ≠ 0
j : Fin n
hj : e.val j ≠ 0
⊢ m ≤ (e.val j).natDegree

Tactic:
Nat.find_min' _ ⟨j, degree_eq_natDegree hj, hj⟩

Premises used: Polynomial.degree_eq_natDegree, Nat.find_min', And.intro, Exists.intro
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
this : ∃ (n_1 : ℕ) (i : Fin n), (e.val i).degree = (↑n_1 : WithBot ℕ) ∧ e.val i ≠ 0
m : ℕ := Nat.find (p := fun (n_1 : ℕ) ↦ ∃ (i : Fin n), (e.val i).degree = (↑n_1 : WithBot ℕ) ∧ e.val i ≠ 0) this
i : Fin n
hi : (e.val i).degree = (↑m : WithBot ℕ)
hi' : e.val i ≠ 0
j : Fin n
hj : e.val j ≠ 0
⊢ m ≤ (e.val j).natDegree

Tactic:
exact Nat.find_min' _ ⟨j, degree_eq_natDegree hj, hj⟩

Premises used: Nat.find_min'
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
neg : ∀ (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
⊢ P R e

Tactic:
obtain ⟨i, hi, i_min⟩ : ∃ i, e.1 i ≠ 0 ∧ ∀ j, e.1 j ≠ 0 → (e.1 i).degree ≤ (e.1 j).degree := by
      have : ∃ n : ℕ, ∃ i, (e.1 i).degree = n ∧ (e.1 i) ≠ 0 := by
        obtain ⟨i, hi⟩ := he0; exact ⟨(e.1 i).natDegree, i, degree_eq_natDegree hi, hi⟩
      let m := Nat.find this
      obtain ⟨i, hi, hi'⟩ : ∃ i, (e.1 i).degree = m ∧ (e.1 i) ≠ 0 := Nat.find_spec this
      refine ⟨i, hi', fun j hj ↦ ?_⟩
      refine hi.le.trans ?_
      rw [degree_eq_natDegree hj, Nat.cast_le]
      exact Nat.find_min' _ ⟨j, degree_eq_natDegree hj, hj⟩
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (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
⊢ (e.val i).leadingCoeff ≠ 0

Tactic:
simpa using hi
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (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
apply :
  {
        val :=
          HSMul.hSMul (α := (Away (e.val i).leadingCoeff)[X]) (β := Fin n → (Away (e.val i).leadingCoeff)[X])
            ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
              (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
            ((⇑(mapRingHom (algebraMap R (Away (e.val i).leadingCoeff))) :
                (a : R[X]) → (Away (e.val i).leadingCoeff)[X]) ∘
              e.val) :
        InductionObj (Away (e.val i).leadingCoeff) n }.degree <
    v
apply₁ :
  InductionObj.degree (R := HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff}))
      {
        val :=
          (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})))) :
              (a : R[X]) → (R ⧸ Ideal.span {(e.val i).leadingCoeff})[X]) ∘
            e.val :
        InductionObj (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})) n } <
    v
⊢ P R e

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

Premises used: rfl
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (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
j : Fin n
⊢ (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
        ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
          (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
        (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val j))).degree ≤
    (e.val j).degree

Tactic:
((degree_mul_le _ _).trans (add_le_add degree_C_le degree_map_le)).trans ?_

Premises used: Polynomial.degree_map_le, Polynomial.degree_mul_le, add_le_add, LE.le.trans, Polynomial.degree_C_le
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (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
j : Fin n
left : 0 + (e.val j).degree ≤ (e.val j).degree
⊢ (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
        ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
          (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
        (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val j))).degree ≤
    (e.val j).degree

Tactic:
refine ((degree_mul_le _ _).trans (add_le_add degree_C_le degree_map_le)).trans ?_

Premises used: Polynomial.degree_map_le, Polynomial.degree_mul_le, add_le_add, LE.le.trans, Polynomial.degree_C_le
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (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
j : Fin n
⊢ 0 + (e.val j).degree ≤ (e.val j).degree

Tactic:
simp
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H_1 : ¬∃ (i : Fin n), (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
h_eq :
  ∀ (x : Fin n),
    (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
          ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
            (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
          (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val x))).degree =
      (e.val x).degree
j : Fin n
hj :
  ¬HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
        ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
          (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
        (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val j)) =
      0
H : e.val j = 0
⊢ False

Tactic:
simp [H] at hj
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
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 (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            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 :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : 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 : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (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
h_eq :
  ∀ (x : Fin n),
    (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
          ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
            (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
          (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val x))).degree =
      (e.val x).degree
j : Fin n
hj :
  ¬HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
        ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
          (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
        (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val j)) =
      0
⊢ (e.val i).degree ≤ (e.val j).degree

Tactic:
exact i_min j fun H ↦ (by simp [H] at hj)