Roots.two_mul: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/roots.bend as Roots
Lines 68-70 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/roots.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/roots.bend&start_line=68&end_line=70

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A

# x 2 == x + x
def two_mul(+x: Nat) -> {Nat.mul(x, 2n) == Nat.add(x, x) : Nat}:
  Equal.trans(Nat, Nat.mul(x, 2n), Nat.mul(2n, x), Nat.add(x, x), A.mul_comm(x, 2n), Equal.cong(Nat, Nat, z => Nat.add(x, z), Nat.add(x, 0n), x, A.add_zero(x)))

Uses (7):
  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.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-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_comm