def prod_go

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/lists.bend:30

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S

30def prod_go(+xs: List<&2, Nat>, +acc: Nat) -> {M.prod_go(xs, acc) == Nat.mul(acc, S.lprod(xs)) : Nat}:
31  match xs:
32    case Nil{}:
33      Equal.sym(Nat, Nat.mul(acc, 1n), acc, A.mul_one(acc))
34    case Con{+x, +t}:
35      Equal.trans(Nat, M.prod_go(t, Nat.mul(acc, x)), Nat.mul(Nat.mul(acc, x), S.lprod(t)), Nat.mul(acc, Nat.mul(x, S.lprod(t))), prod_go(t, Nat.mul(acc, x)), A.mul_assoc(acc, x, S.lprod(t)))