induction as
α : Type u
inst✝ : AddMonoid α
as bs : List α
⊢ sum (as ++ bs) = sum as + sum bs
case nil
α : Type u
inst✝ : AddMonoid α
bs : List α
⊢ sum ([] ++ bs) = sum [] + sum bs

case cons
α : Type u
inst✝ : AddMonoid α
bs : List α
head✝ : α
tail✝ : List α
tail_ih✝ : sum (tail✝ ++ bs) = sum tail✝ + sum bs
⊢ sum (head✝ :: tail✝ ++ bs) = sum (head✝ :: tail✝) + sum bs