Lists.prod_go: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend as Lists
Lines 30-35 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend&start_line=30&end_line=35

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

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

Uses (9):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=lprod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=prod_go