-- Mathlib/RingTheory/Nilpotent/Exp.lean
theorem exp_add_of_commute {a b : A} (h₁ : Commute a b) (h₂ : IsNilpotent a) (h₃ : IsNilpotent b) :
    exp (a + b) = exp a * exp b

Proof state:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ filter (fun ij => ¬ij.1 + ij.2 ≤ 2 * N) (R2N ×ˢ R2N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ filter (fun x => x.1 + x.2 ≤ 2 * N) (R2N ×ˢ R2N), ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
i : ℕ × ℕ
hi : i ∈ R2N ×ˢ R2N ∧ (i.1 ≤ N → N < i.2)
⊢ ((↑i.1!)⁻¹ * (↑i.2!)⁻¹) • (a ^ i.1 * b ^ i.2) = 0

Tactic:
cases le_or_lt (N + 1) i.1 with
  | inl h => rw [pow_eq_zero_of_le h h₄, zero_mul, smul_zero]
  | inr h => rw [pow_eq_zero_of_le (hi.2 (Nat.le_of_lt_succ h)) h₅, mul_zero, smul_zero]

Result:
no goals

Proof state:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ≤ i
⊢ ↑(j ! * (i - j)!) ≠ 0

Tactic:
field_simp

Result:
no goals

Proof state:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
⊢ n₁ ≤ N + 1

Tactic:
omega

Result:
no goals

Proof state:
case calc_3.i_surj
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
⊢ ∀ (a b : ℕ),
    a < 2 * N + 1 →
      b < 2 * N + 1 → a + b ≤ 2 * N → ∃ a_4 b_1, (a_4 < 2 * N + 1 ∧ b_1 < a_4 + 1) ∧ b_1 = a ∧ a_4 - b_1 = b

Tactic:
exact fun x y _ _ _ ↦ ⟨x + y, x, by omega⟩

Result:
no goals

Proof state:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
⊢ N + 1 + (N + 1) ≤ 2 * N + 1 + 1

Tactic:
omega

Result:
no goals

Proof state:
case intro.intro
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ :
  ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i =
    ∑ ij ∈ filter (fun ij => ij.1 + ij.2 ≤ 2 * N) (R2N ×ˢ R2N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
z₁ : ∑ ij ∈ filter (fun ij => ¬ij.1 + ij.2 ≤ 2 * N) (R2N ×ˢ R2N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ filter (fun x => x.1 + x.2 ≤ 2 * N) (R2N ×ˢ R2N), ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i

Tactic:
rw [split₁] at s₁

Result:
case intro.intro
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ filter (fun ij => ¬ij.1 + ij.2 ≤ 2 * N) (R2N ×ˢ R2N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ filter (fun x => x.1 + x.2 ≤ 2 * N) (R2N ×ˢ R2N), ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i

Proof state:
case calc_2
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ∈ range (i + 1)
⊢ (↑i !)⁻¹ • (a ^ j * b ^ (i - j) * ↑(i.choose j)) = ((↑j !)⁻¹ * (↑(i - j)!)⁻¹) • (a ^ j * b ^ (i - j))

Tactic:
simp only [mem_range] at hi hj

Result:
case calc_2
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j < i + 1
⊢ (↑i !)⁻¹ • (a ^ j * b ^ (i - j) * ↑(i.choose j)) = ((↑j !)⁻¹ * (↑(i - j)!)⁻¹) • (a ^ j * b ^ (i - j))