awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R₀ : Type u_1
R : Type u_2
inst : CommRing R₀
inst_1 : CommRing R
inst_2 : Algebra R₀ R
n : ℕ
e : InductionObj R n
i : Fin n
d : ℕ
⊢ Membership.mem (γ := Submodule R₀ R) (coeffSubmodule R₀ e) ((e.val i).coeff d)

Tactic:
Submodule.subset_span <| .inr <| Set.mem_iUnion.2 ⟨i, Set.mem_range_self _⟩

Premises used: Set.mem_range_self, Iff.mpr, Set.Mem, Set.mem_iUnion, Exists.intro, Submodule.subset_span, Or.inr
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R₀ : Type u_1
R : Type u_2
inst : CommRing R₀
inst_1 : CommRing R
inst_2 : Algebra R₀ R
n : ℕ
e : InductionObj R n
⊢ Membership.mem (γ := Submodule R₀ R) (coeffSubmodule R₀ e) 1

Tactic:
Submodule.subset_span (.inl rfl)

Premises used: Set.Mem, Or.inl, Submodule.subset_span, rfl
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R₀ : Type u_1
R : Type u_2
inst : CommRing R₀
inst_1 : CommRing R
inst_2 : Algebra R₀ R
n : ℕ
e : InductionObj R n
rw : 1 ∈ (↑(coeffSubmodule R₀ e) : Set R)
⊢ LE.le (α := Submodule R₀ R) 1 (coeffSubmodule R₀ e)

Tactic:
rw [Submodule.one_eq_span, Submodule.span_le, Set.singleton_subset_iff]

Premises used: Set.singleton_subset_iff, Submodule.span_le, Submodule.one_eq_span
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R₀ : Type u_1
R : Type u_2
inst : CommRing R₀
inst_1 : CommRing R
inst_2 : Algebra R₀ R
n : ℕ
e : InductionObj R n
⊢ 1 ∈ (↑(coeffSubmodule R₀ e) : Set R)

Tactic:
one_mem_coeffSubmodule

Premises used: Set.Mem, _private.0.ChevalleyThm.PolynomialC.InductionObj.one_mem_coeffSubmodule
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R₀ : Type u_1
R : Type u_2
inst : CommRing R₀
inst_1 : CommRing R
inst_2 : Algebra R₀ R
n : ℕ
e : InductionObj R n
⊢ 1 ∈ (↑(coeffSubmodule R₀ e) : Set R)

Tactic:
exact one_mem_coeffSubmodule

Premises used: _private.0.ChevalleyThm.PolynomialC.InductionObj.one_mem_coeffSubmodule
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R : Type u_2
inst : CommRing R
n : ℕ
e : InductionObj R n
i : Fin n
⊢ ((ofLex : (a : Lex ((Fin n → WithBot ℕ) × Prop)) → (Fin n → WithBot ℕ) × Prop) e.degree).1 i = (e.val i).degree

Tactic:
rfl

Premises used: Multiset.foldr, Finset.fold, Polynomial.degree, Finset.max, Finset.sup, rfl, Multiset.fold
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R : Type u_2
inst : CommRing R
n : ℕ
e : InductionObj R n
⊢ ((ofLex : (a : Lex ((Fin n → WithBot ℕ) × Prop)) → (Fin n → WithBot ℕ) × Prop) e.degree).2 ↔
    ¬∃ (i : Fin n), (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree

Tactic:
.rfl

Premises used: Iff.rfl, Not
-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R : Type u_2
inst : CommRing R
n : ℕ
e : InductionObj R n
⊢ e.powBound ≠ 0

Tactic:
Nat.pow_self_pos.ne'

Premises used: _private.0.ChevalleyThm.PolynomialC.InductionObj.powBound, LT.lt.ne', Nat.pow_self_pos
-- 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
R : Type u
inst : CommRing R
e : InductionObj R n
set :
  let v := e.degree;
  ∀ (hv : v = e.degree), P R e
⊢ P R e

Tactic:
set v := e.degree with hv

Premises used: _private.0.ChevalleyThm.PolynomialC.InductionObj.degree
-- 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
R : Type u
inst : CommRing R
e : InductionObj R n
v : DegreeType n := e.degree
hv : v = e.degree
clear_value : ∀ (v : DegreeType n) (hv : v = e.degree), P R e
⊢ P R e

Tactic:
clear_value v
-- 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
pos : ∀ (he0 : e = { val := 0 }), P R e
neg : ∀ (he0 : ¬e = { val := 0 }), P R e
⊢ P R e

Tactic:
by_cases he0 : e = ⟨0⟩
-- 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
he0 : e = { val := 0 }
⊢ P R e

Tactic:
he0 ▸ hP₁ R

Premises used: Eq.rec, Eq.symm
-- 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
he0 : e = { val := 0 }
⊢ P R e

Tactic:
exact he0 ▸ hP₁ R
-- 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
he0 : ¬e = { val := 0 }
neg : Subsingleton R → P R e
neg₁ : Nontrivial R → P R e
⊢ P R e

Tactic:
cases subsingleton_or_nontrivial R

Premises used: subsingleton_or_nontrivial
-- 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
he0 : ¬e = { val := 0 }
h : Subsingleton R
⊢ P R { val := 0 }

Tactic:
hP₁ R
-- 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
he0 : ¬e = { val := 0 }
h_1 : Subsingleton R
h : e.val = 0
⊢ P R e

Tactic:
convert! hP₁ R
-- 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
he0 : ¬e = { val := 0 }
h_1 : Subsingleton R
h : ∀ (x : Fin n) (n_1 : ℕ), (e.val x).coeff n_1 = Polynomial.coeff (0 x) n_1
⊢ e.val = 0

Tactic:
ext
-- 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
he0 : ¬e = { val := 0 }
h : Subsingleton R
x : Fin n
n_1 : ℕ
⊢ (e.val x).coeff n_1 = Polynomial.coeff (0 x) n_1

Tactic:
Subsingleton.elim _ _

Premises used: Subsingleton.elim
-- 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
he0 : ¬e = { val := 0 }
h : Subsingleton R
x : Fin n
n_1 : ℕ
⊢ (e.val x).coeff n_1 = Polynomial.coeff (0 x) n_1

Tactic:
exact Subsingleton.elim _ _

Premises used: Subsingleton.elim
-- 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
he0 : ¬e = { val := 0 }
h : Nontrivial R
⊢ ∃ (x : Fin n), ¬e.val x = 0

Tactic:
simp only [InductionObj.ext_iff, funext_iff, Pi.zero_apply, not_forall] at he0

Premises used: funext_iff, Pi.zero_apply, _private.0.ChevalleyThm.PolynomialC.InductionObj.ext_iff, Classical.not_forall