English
The set of derivations Derivation R A A forms a Lie ring with the bracket given by the commutator, compatible with addition and scalar multiplication.
Русский
Множество производных Derivation R A A образует брейк Ли с скобкой, заданной коммутатором, совместимой с сложением и скалярным умножением.
LaTeX
$$$\text{Derivation}(R,A,A) \text{ is a Lie ring with bracket } [D_1,D_2](a) = D_1(D_2(a)) - D_2(D_1(a)).$$$
Lean4
instance : LieRing (Derivation R A A)
where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply]; ring_nf; simp
leibniz_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_sub]; ring
