awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
x y : R
inst : Semiring R
h_comm : Commute x y
hx : IsNilpotent x
hy : IsNilpotent y
⊢ IsNilpotent (x + y)
have obtain : ∀ (n : ℕ) (hn : x ^ n = 0), IsNilpotent (x + y)
Tactic: obtain ⟨n, hn⟩ := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
x y : R
inst : Semiring R
h_comm : Commute x y
hy : IsNilpotent y
n : ℕ
hn : x ^ n = 0
⊢ IsNilpotent (x + y)
have obtain : ∀ (m : ℕ) (hm : y ^ m = 0), IsNilpotent (x + y)
Tactic: obtain ⟨m, hm⟩ := hy
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
s : Finset ι
f : ι → R
hnp : ∀ i ∈ s, IsNilpotent (f i)
h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
have insert : ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
  (ih :
    ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
      IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
  (hnp : ∀ i ∈ insert j s, IsNilpotent (f i))
  (h_comm : ∀ (i j_1 : ι), i ∈ insert j s → j_1 ∈ insert j s → Commute (f i) (f j_1)),
  IsNilpotent.{u_1} (R := R) (∑ i ∈ insert j s, f i)
Tactic: induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
Cleared hypotheses: s, hnp, h_comm
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
j : ι
s : Finset ι
hj : j ∉ s
ih :
  ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
hnp : ∀ i ∈ insert j s, IsNilpotent (f i)
h_comm : ∀ (i j_1 : ι), i ∈ insert j s → j_1 ∈ insert j s → Commute (f i) (f j_1)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ insert j s, f i)
have insert : IsNilpotent (f j + ∑ x ∈ s, f x)
Tactic: rw [Finset.sum_insert hj]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
j : ι
s : Finset ι
hj : j ∉ s
ih :
  ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
hnp : ∀ i ∈ insert j s, IsNilpotent (f i)
h_comm : ∀ (i j_1 : ι), i ∈ insert j s → j_1 ∈ insert j s → Commute (f i) (f j_1)
⊢ IsNilpotent (f j + ∑ x ∈ s, f x)
have insert : Commute (f j) (∑ x ∈ s, f x)
Tactic: apply Commute.isNilpotent_add
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
j : ι
s : Finset ι
hj : j ∉ s
ih :
  ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i)
h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)
insert : Commute (f j) (∑ x ∈ s, f x)
⊢ IsNilpotent (f j + ∑ x ∈ s, f x)
have insert₁ : IsNilpotent (f j)
Tactic: apply Commute.isNilpotent_add
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
j : ι
s : Finset ι
hj : j ∉ s
ih :
  ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i)
h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)
insert : Commute (f j) (∑ x ∈ s, f x)
insert₁ : IsNilpotent (f j)
⊢ IsNilpotent (f j + ∑ x ∈ s, f x)
have insert₂ : IsNilpotent.{u_1} (R := R) (∑ x ∈ s, f x)
Tactic: apply Commute.isNilpotent_add
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
j : ι
s : Finset ι
hj : j ∉ s
ih :
  ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
hnp : ∀ i ∈ insert j s, IsNilpotent (f i)
h_comm : ∀ (i j_1 : ι), i ∈ insert j s → j_1 ∈ insert j s → Commute (f i) (f j_1)
⊢ IsNilpotent (f j)
have insert : j ∈ insert j s
Tactic: apply hnp
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
s : Finset ι
f : ι → R
hnp : ∀ i ∈ s, IsNilpotent (f i)
h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
have empty : ∀ (hnp : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → IsNilpotent (f i))
  (h_comm : ∀ (i j : ι), Membership.mem (γ := Finset ι) ∅ i → Membership.mem (γ := Finset ι) ∅ j → Commute (f i) (f j)),
  IsNilpotent.{u_1} (R := R) (∑ i ∈ ∅, f i)
Tactic: classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
Cleared hypotheses: s, hnp, h_comm
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
s : Finset ι
f : ι → R
hnp : ∀ i ∈ s, IsNilpotent (f i)
h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)
empty :
  ∀ (hnp : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → IsNilpotent (f i))
    (h_comm :
      ∀ (i j : ι), Membership.mem (γ := Finset ι) ∅ i → Membership.mem (γ := Finset ι) ∅ j → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ ∅, f i)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
have insert : ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
  (ih :
    ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
      IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
  (hnp : ∀ i ∈ insert j s, IsNilpotent (f i))
  (h_comm : ∀ (i j_1 : ι), i ∈ insert j s → j_1 ∈ insert j s → Commute (f i) (f j_1)),
  IsNilpotent.{u_1} (R := R) (∑ i ∈ insert j s, f i)
Tactic: classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
Cleared hypotheses: s, hnp, h_comm
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
s : Finset ι
f : ι → R
hnp : ∀ i ∈ s, IsNilpotent (f i)
h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)
empty :
  ∀ (hnp : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → IsNilpotent (f i))
    (h_comm :
      ∀ (i j : ι), Membership.mem (γ := Finset ι) ∅ i → Membership.mem (γ := Finset ι) ∅ j → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ ∅, f i)
insert insert₁ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ Insert.insert j s, f i)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
have insert₂ : ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
  (ih :
    ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
      IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
  (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
  (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
  IsNilpotent (f j + ∑ x ∈ s, f x)
Tactic: classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
Cleared hypotheses: s, hnp, h_comm
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
s : Finset ι
f : ι → R
hnp : ∀ i ∈ s, IsNilpotent (f i)
h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)
empty :
  ∀ (hnp : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → IsNilpotent (f i))
    (h_comm :
      ∀ (i j : ι), Membership.mem (γ := Finset ι) ∅ i → Membership.mem (γ := Finset ι) ∅ j → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ ∅, f i)
insert insert₁ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ Insert.insert j s, f i)
insert₂ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    IsNilpotent (f j + ∑ x ∈ s, f x)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
have insert₃ : ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
  (ih :
    ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
      IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
  (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
  (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
  Commute (f j) (∑ x ∈ s, f x)
Tactic: classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
Cleared hypotheses: s, hnp, h_comm
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
s : Finset ι
f : ι → R
hnp : ∀ i ∈ s, IsNilpotent (f i)
h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)
empty :
  ∀ (hnp : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → IsNilpotent (f i))
    (h_comm :
      ∀ (i j : ι), Membership.mem (γ := Finset ι) ∅ i → Membership.mem (γ := Finset ι) ∅ j → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ ∅, f i)
insert insert₁ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ Insert.insert j s, f i)
insert₂ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    IsNilpotent (f j + ∑ x ∈ s, f x)
insert₃ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    Commute (f j) (∑ x ∈ s, f x)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
have insert₄ : ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
  (ih :
    ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
      IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
  (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
  (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)), IsNilpotent (f j)
Tactic: classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
Cleared hypotheses: s, hnp, h_comm
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
s : Finset ι
f : ι → R
hnp : ∀ i ∈ s, IsNilpotent (f i)
h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)
empty :
  ∀ (hnp : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → IsNilpotent (f i))
    (h_comm :
      ∀ (i j : ι), Membership.mem (γ := Finset ι) ∅ i → Membership.mem (γ := Finset ι) ∅ j → Commute (f i) (f j)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ ∅, f i)
insert insert₁ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    IsNilpotent.{u_1} (R := R) (∑ i ∈ Insert.insert j s, f i)
insert₂ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    IsNilpotent (f j + ∑ x ∈ s, f x)
insert₃ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
    Commute (f j) (∑ x ∈ s, f x)
insert₄ :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (ih :
      ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
        IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
    (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
    (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)), IsNilpotent (f j)
⊢ IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i)
have insert₅ : ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
  (ih :
    ∀ (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)),
      IsNilpotent.{u_1} (R := R) (∑ i ∈ s, f i))
  (hnp : ∀ i ∈ Insert.insert j s, IsNilpotent (f i))
  (h_comm : ∀ (i j_1 : ι), i ∈ Insert.insert j s → j_1 ∈ Insert.insert j s → Commute (f i) (f j_1)),
  IsNilpotent.{u_1} (R := R) (∑ x ∈ s, f x)
Tactic: classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
Cleared hypotheses: s, hnp, h_comm
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
hf : ∀ (b : ι), IsNilpotent (f b)
h_comm : ∀ (i j : ι), Commute (f i) (f j)
⊢ IsNilpotent (finsum f)
have pos : ∀ (h : HasFiniteSupport f), IsNilpotent (finsum f)
Tactic: by_cases h : HasFiniteSupport f
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
hf : ∀ (b : ι), IsNilpotent (f b)
h_comm : ∀ (i j : ι), Commute (f i) (f j)
pos : ∀ (h : HasFiniteSupport f), IsNilpotent (finsum f)
⊢ IsNilpotent (finsum f)
have neg : ∀ (h : ¬HasFiniteSupport f), IsNilpotent (finsum f)
Tactic: by_cases h : HasFiniteSupport f
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
hf : ∀ (b : ι), IsNilpotent (f b)
h_comm : ∀ (i j : ι), Commute (f i) (f j)
h : HasFiniteSupport f
⊢ IsNilpotent (finsum f)
have pos : IsNilpotent.{u_1} (R := R) (∑ i ∈ Finite.toFinset (s := support f) h, f i)
Tactic: rw [finsum_def, dif_pos h]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
hf : ∀ (b : ι), IsNilpotent (f b)
h_comm : ∀ (i j : ι), Commute (f i) (f j)
⊢ IsNilpotent (finsum f)
have pos : ∀ (h : HasFiniteSupport f), IsNilpotent (finsum f)
Tactic: classical
  by_cases h : HasFiniteSupport f
  · rw [finsum_def, dif_pos h]
    exact Commute.isNilpotent_sum (fun b _ ↦ hf b) (fun _ _ _ _ ↦ h_comm _ _)
  · simp only [finsum_def, dif_neg h, IsNilpotent.zero]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
ι : Type u_3
f : ι → R
hf : ∀ (b : ι), IsNilpotent (f b)
h_comm : ∀ (i j : ι), Commute (f i) (f j)
pos : ∀ (h : HasFiniteSupport f), IsNilpotent (finsum f)
⊢ IsNilpotent (finsum f)
have neg : ∀ (h : ¬HasFiniteSupport f), IsNilpotent (finsum f)
Tactic: classical
  by_cases h : HasFiniteSupport f
  · rw [finsum_def, dif_pos h]
    exact Commute.isNilpotent_sum (fun b _ ↦ hf b) (fun _ _ _ _ ↦ h_comm _ _)
  · simp only [finsum_def, dif_neg h, IsNilpotent.zero]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
x y : R
inst : Semiring R
h_comm : Commute x y
hy : Membership.mem (γ := Submonoid R) (nonZeroDivisorsRight R) y
⊢ IsNilpotent (x * y) ↔ IsNilpotent x
have refine : IsNilpotent (x * y) → IsNilpotent x
Tactic: refine ⟨?_, h_comm.isNilpotent_mul_right⟩
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (hypothesis)
Goal:
R : Type u_1
x y : R
inst : Semiring R
h_comm : Commute x y
hy : Membership.mem (γ := Submonoid R) (nonZeroDivisorsRight R) y
k : ℕ
hk : (x * y) ^ k = 0
⊢ IsNilpotent x
have hk₁ : x ^ k * y ^ k = 0
Tactic: rw [mul_pow h_comm] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
x y : R
inst : Semiring R
h_comm : Commute x y
hx : Membership.mem (γ := Submonoid R) (nonZeroDivisorsLeft R) x
⊢ IsNilpotent (x * y) ↔ IsNilpotent y
have refine : IsNilpotent (x * y) → IsNilpotent y
Tactic: refine ⟨?_, h_comm.isNilpotent_mul_left⟩
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (hypothesis)
Goal:
R : Type u_1
x y : R
inst : Semiring R
h_comm : Commute x y
hx : Membership.mem (γ := Submonoid R) (nonZeroDivisorsLeft R) x
k : ℕ
hk : (x * y) ^ k = 0
⊢ IsNilpotent y
have hk₁ : x ^ k * y ^ k = 0
Tactic: rw [mul_pow h_comm] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (hypothesis)
Goal:
R : Type u_1
x y : R
inst : Ring R
h_comm : Commute x y
hx : IsNilpotent x
hy : IsNilpotent y
⊢ IsNilpotent (x - y)
have h_comm₁ : Commute x (-y)
Tactic: rw [← neg_right_iff] at h_comm
Cleared hypotheses: h_comm, hx, hy
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (hypothesis)
Goal:
R : Type u_1
x y : R
inst : Ring R
h_comm : Commute x (-y)
hx : IsNilpotent x
hy : IsNilpotent y
⊢ IsNilpotent (x - y)
have hy₁ : IsNilpotent (-y)
Tactic: rw [← isNilpotent_neg_iff] at hy
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Basic (goal)
Goal:
R : Type u_1
x y : R
inst : Ring R
h_comm : Commute x (-y)
hx : IsNilpotent x
hy : IsNilpotent (-y)
⊢ IsNilpotent (x - y)
have rw : IsNilpotent (x + -y)
Tactic: rw [sub_eq_add_neg]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
S : Type u_2
inst : MonoidWithZero R
inst_1 : MonoidWithZero S
r : R
F : Type u_3
inst_2 : FunLike F R S
inst_3 : MonoidWithZeroHomClass F R S
hr : IsNilpotent r
f : F
⊢ IsNilpotent.{u_2} (R := S) ((f : (a : R) → S) r : S)
have h : HPow.hPow (α := S) ((f : (a : R) → S) r : S) (Exists.choose (p := fun (n : ℕ) ↦ r ^ n = 0) hr) = 0
Tactic: use hr.choose
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
S : Type u_2
inst : MonoidWithZero R
inst_1 : MonoidWithZero S
r : R
F : Type u_3
inst_2 : FunLike F R S
inst_3 : MonoidWithZeroHomClass F R S
hr : IsNilpotent r
f : F
⊢ HPow.hPow (α := S) ((f : (a : R) → S) r : S) (Exists.choose (p := fun (n : ℕ) ↦ r ^ n = 0) hr) = 0
have h : Eq.{u_2 + 1} (α := S) ((f : (a : R) → S) (r ^ Exists.choose (p := fun (n : ℕ) ↦ r ^ n = 0) hr) : S) (0 : S)
Tactic: rw [← map_pow, hr.choose_spec, map_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
S : Type u_2
inst : MonoidWithZero R
inst_1 : MonoidWithZero S
r : R
F : Type u_3
inst_2 : FunLike F R S
inst_3 : MonoidWithZeroHomClass F R S
hr : IsNilpotent r
f : F
h : Eq.{u_2 + 1} (α := S) ((f : (a : R) → S) (r ^ Exists.choose (p := fun (n : ℕ) ↦ r ^ n = 0) hr) : S) (0 : S)
⊢ HPow.hPow (α := S) ((f : (a : R) → S) r : S) (Exists.choose (p := fun (n : ℕ) ↦ r ^ n = 0) hr) = 0
have h₁ : Eq.{u_2 + 1} (α := S) ((f : (a : R) → S) 0 : S) (0 : S)
Tactic: rw [← map_pow, hr.choose_spec, map_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
S : Type u_2
inst : MonoidWithZero R
inst_1 : MonoidWithZero S
r : R
F : Type u_3
inst_2 : FunLike F R S
inst_3 : MonoidWithZeroHomClass F R S
hr : IsNilpotent r
f : F
h : Eq.{u_2 + 1} (α := S) ((f : (a : R) → S) (r ^ Exists.choose (p := fun (n : ℕ) ↦ r ^ n = 0) hr) : S) (0 : S)
h₁ : Eq.{u_2 + 1} (α := S) ((f : (a : R) → S) 0 : S) (0 : S)
⊢ HPow.hPow (α := S) ((f : (a : R) → S) r : S) (Exists.choose (p := fun (n : ℕ) ↦ r ^ n = 0) hr) = 0
have h₂ : Eq.{u_2 + 1} (α := S) 0 0
Tactic: rw [← map_pow, hr.choose_spec, map_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
inst : MonoidWithZero R
r u : R
hu : IsUnit u
h_comm : Commute r u
n : ℕ
⊢ (r * u) ^ n = 0 ↔ r ^ n = 0
have rw : r ^ n * u ^ n = 0 ↔ r ^ n = 0
Tactic: rw [h_comm.mul_pow, (hu.pow n).mul_left_eq_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
inst : MonoidWithZero R
r u : R
hu : IsUnit u
h_comm : Commute r u
n : ℕ
rw : r ^ n * u ^ n = 0 ↔ r ^ n = 0
⊢ (r * u) ^ n = 0 ↔ r ^ n = 0
have rw₁ : r ^ n = 0 ↔ r ^ n = 0
Tactic: rw [h_comm.mul_pow, (hu.pow n).mul_left_eq_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
x : R
inst : Zero R
inst_1 : Pow R ℕ
inst_2 : Subsingleton R
s : Set ℕ := {k : ℕ | x ^ k = 0}
⊢ nilpotencyClass x = 0
have suffices : s = univ
Tactic: suffices s = univ by change sInf _ = 0; simp [s] at this; simp [this]
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
x : R
inst : Zero R
inst_1 : Pow R ℕ
hx : 0 < nilpotencyClass x
s : Set ℕ := {k : ℕ | x ^ k = 0}
⊢ IsNilpotent x
have change : s.Nonempty
Tactic: change s.Nonempty
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (hypothesis)
Goal:
R : Type u_1
x : R
inst : Zero R
inst_1 : Pow R ℕ
hx : 0 < nilpotencyClass x
s : Set ℕ := {k : ℕ | x ^ k = 0}
⊢ s.Nonempty
have hx₁ : 0 < sInf s
Tactic: change 0 < sInf s at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.Nilpotent.Defs (goal)
Goal:
R : Type u_1
x : R
inst : Zero R
inst_1 : Pow R ℕ
s : Set ℕ := {k : ℕ | x ^ k = 0}
hx : 0 < sInf s
⊢ s.Nonempty
have by_contra : ∀ (contra : ¬s.Nonempty), False
Tactic: by_contra contra