awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
hP₃ :
  comap Polynomial.C '' (zeroLocus (Set.range c.val) \ zeroLocus {f}) =
    comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f})
⊢ ∃ (T : ConstructibleSetData R),
    comap Polynomial.C '' (zeroLocus (Set.range c.val) \ zeroLocus {f}) = T.toSet ∧
      ∀ C ∈ T,
        C.n ≤ c.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ HPow.hPow (α := Submodule R₀ R) (coeffSubmodule R₀ c) c.powBound
have hP₄ : ∀ C ∈ S, C.n ≤ c.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ HPow.hPow (α := Submodule R₀ R) (coeffSubmodule R₀ c) c.powBound
Tactic: refine ⟨S, Eq.trans ?_ hS, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
⊢ comap Polynomial.C '' (zeroLocus (Set.range c.val) \ zeroLocus {f}) =
    comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f})
have hP₃ : comap Polynomial.C '' (zeroLocus (↑(Ideal.span (Set.range c.val)) : Set R[X]) \ zeroLocus {f}) =
  comap Polynomial.C ''
    (zeroLocus
        (Set.range
          { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
              InductionObj R n }.val) \
      zeroLocus {f})
Tactic: rw [← zeroLocus_span (Set.range _), ← zeroLocus_span (Set.range _),
        idealSpan_range_update_divByMonic hne]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
hP₃ :
  comap Polynomial.C '' (zeroLocus (↑(Ideal.span (Set.range c.val)) : Set R[X]) \ zeroLocus {f}) =
    comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f})
⊢ comap Polynomial.C '' (zeroLocus (Set.range c.val) \ zeroLocus {f}) =
    comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f})
have hP₄ : comap Polynomial.C '' (zeroLocus (↑(Ideal.span (Set.range c.val)) : Set R[X]) \ zeroLocus {f}) =
  comap Polynomial.C ''
    (zeroLocus
        (↑(Ideal.span
              (Set.range
                { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                    InductionObj R n }.val)) :
          Set R[X]) \
      zeroLocus {f})
Tactic: rw [← zeroLocus_span (Set.range _), ← zeroLocus_span (Set.range _),
        idealSpan_range_update_divByMonic hne]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
hP₃ :
  comap Polynomial.C '' (zeroLocus (↑(Ideal.span (Set.range c.val)) : Set R[X]) \ zeroLocus {f}) =
    comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f})
hP₄ :
  comap Polynomial.C '' (zeroLocus (↑(Ideal.span (Set.range c.val)) : Set R[X]) \ zeroLocus {f}) =
    comap Polynomial.C ''
      (zeroLocus
          (↑(Ideal.span
                (Set.range
                  { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                      InductionObj R n }.val)) :
            Set R[X]) \
        zeroLocus {f})
⊢ comap Polynomial.C '' (zeroLocus (Set.range c.val) \ zeroLocus {f}) =
    comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f})
have hP₅ : comap Polynomial.C '' (zeroLocus (↑(Ideal.span (Set.range c.val)) : Set R[X]) \ zeroLocus {f}) =
  comap Polynomial.C '' (zeroLocus (↑(Ideal.span (Set.range c.val)) : Set R[X]) \ zeroLocus {f})
Tactic: rw [← zeroLocus_span (Set.range _), ← zeroLocus_span (Set.range _),
        idealSpan_range_update_divByMonic hne]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
⊢ c'.degBound ≤ c.degBound
have dsimp : LE.le (α := ℕ)
  (∑ i_1 : Fin n, (Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) i_1).degree.succ)
  (∑ i : Fin n, (c.val i).degree.succ)
Tactic: dsimp [InductionObj.degBound, c']
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
⊢ LE.le (α := ℕ)
    (∑ i_1 : Fin n, (Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) i_1).degree.succ)
    (∑ i : Fin n, (c.val i).degree.succ)
have h : ∀ (k : Fin n),
  Membership.mem (γ := Finset (Fin n)) Finset.univ k →
    (Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) k).degree ≤ (c.val k).degree
Tactic: gcongr with k
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
k : Fin n
h_1 : Membership.mem (γ := Finset (Fin n)) Finset.univ k
⊢ (Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) k).degree ≤ (c.val k).degree
have h : (if k = j then c.val j %ₘ c.val i else c.val k).degree ≤ (c.val k).degree
Tactic: rw [update_apply]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
k : Fin n
h_1 : Membership.mem (γ := Finset (Fin n)) Finset.univ k
⊢ (if k = j then c.val j %ₘ c.val i else c.val k).degree ≤ (c.val k).degree
have pos : ∀ (hkj : k = j), (c.val j %ₘ c.val i).degree ≤ (c.val k).degree
Tactic: split_ifs with hkj
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
k : Fin n
h_1 : Membership.mem (γ := Finset (Fin n)) Finset.univ k
pos : ∀ (hkj : k = j), (c.val j %ₘ c.val i).degree ≤ (c.val k).degree
⊢ (if k = j then c.val j %ₘ c.val i else c.val k).degree ≤ (c.val k).degree
have neg : ∀ (hkj : ¬k = j), (c.val k).degree ≤ (c.val k).degree
Tactic: split_ifs with hkj
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (goal)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
k : Fin n
h_1 : Membership.mem (γ := Finset (Fin n)) Finset.univ k
hkj : k = j
⊢ (c.val j %ₘ c.val i).degree ≤ (c.val k).degree
have pos : ∀ (hle : (c.val i).degree ≤ (c.val k).degree) (hne : i ≠ k)
  (H :
    ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
      Statement R₀ R n
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) : InductionObj R n })
  (hS :
    comap Polynomial.C ''
        (zeroLocus
            (Set.range
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
                  InductionObj R n }.val) \
          zeroLocus {f}) =
      S.toSet)
  (hS' :
    ∀ C ∈ S,
      C.n ≤
          { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
              InductionObj R n }.degBound ∧
        ∀ (i_1 : Fin C.n),
          C.g i_1 ∈
            HPow.hPow (α := Submodule R₀ R)
              (coeffSubmodule R₀
                { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
                  InductionObj R n })
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
                  InductionObj R n }.powBound),
  (c.val k %ₘ c.val i).degree ≤ (c.val k).degree
Tactic: subst hkj
Cleared hypotheses: j, hle, hne, H, hS, hS', c', hkj
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity (hypothesis)
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
⊢ C.n ≤ c.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ HPow.hPow (α := Submodule R₀ R) (coeffSubmodule R₀ c) c.powBound
have deg_bound₁ : c'.degBound ≤ c.degBound
Tactic: have deg_bound₁ : c'.degBound ≤ c.degBound := by
        dsimp [InductionObj.degBound, c']
        gcongr with k
        · rw [update_apply]
          split_ifs with hkj
          · subst hkj; exact (degree_modByMonic_le _ hi).trans hle
          · rfl