BendMlAutograd.ndist_l: def of bend-ml-autograd@0.1.1.0
Import: import bend-ml-autograd@0.1.1.0/main.bend as BendMlAutograd
Lines 67-75 of bend-ml-autograd@0.1.1.0/main.bend: https://jend.leeeet.dev/read.txt?path=bend-ml-autograd@0.1.1.0/main.bend&start_line=67&end_line=75

import Base
import bend-ml-nat-lemmas@0.1.0.0/main.bend as NL

# a * (b + c) == a*b + a*c   (left distributivity; nat-lemmas only has the right one)
def ndist_l(+a: Nat, +b: Nat, +c: Nat) -> {Nat.mul(a, Nat.add(b, c)) == Nat.add(Nat.mul(a, b), Nat.mul(a, c)) : Nat}:
  Equal.trans(Nat, Nat.mul(a, Nat.add(b, c)), Nat.mul(Nat.add(b, c), a), Nat.add(Nat.mul(a, b), Nat.mul(a, c)),
    NL.mul_comm(a, Nat.add(b, c)),
    Equal.trans(Nat, Nat.mul(Nat.add(b, c), a), Nat.add(Nat.mul(b, a), Nat.mul(c, a)), Nat.add(Nat.mul(a, b), Nat.mul(a, c)),
      Equal.sym(Nat, Nat.add(Nat.mul(b, a), Nat.mul(c, a)), Nat.mul(Nat.add(b, c), a), NL.mul_dist(b, c, a)),
      Equal.trans(Nat, Nat.add(Nat.mul(b, a), Nat.mul(c, a)), Nat.add(Nat.mul(a, b), Nat.mul(c, a)), Nat.add(Nat.mul(a, b), Nat.mul(a, c)),
        Equal.cong(Nat, Nat, k => Nat.add(k, Nat.mul(c, a)), Nat.mul(b, a), Nat.mul(a, b), NL.mul_comm(b, a)),
        Equal.cong(Nat, Nat, k => Nat.add(Nat.mul(a, b), k), Nat.mul(c, a), Nat.mul(a, c), NL.mul_comm(c, a)))))

Uses (8):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-ml-nat-lemmas@0.1.2.0/main.bend&name=mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-ml-nat-lemmas@0.1.2.0/main.bend&name=mul_dist