PROOF.ur.r16: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 1395-1400 of bend-osdev-u32@0.1.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-u32@0.1.0.0/PROOF.bend&start_line=1395&end_line=1400

import Base
import bend-osdev-u32@0.1.0.0/LAWS.bend as Laws

def ur.r16(+x0: Nat, +x1: Nat) -> {Nat.mul(x0, Nat.double(x1)) == Nat.mul(Nat.double(x0), x1) : Nat}:
  %Equal.sym(Nat, Nat.double(x1), Nat.add(x1, x1), Laws.nat_double(x1)) : {Nat.mul(x0, _) == Nat.mul(Nat.double(x0), x1) : Nat}
  %Equal.sym(Nat, Nat.mul(x0, Nat.add(x1, x1)), Nat.add(Nat.mul(x0, x1), Nat.mul(x0, x1)), Laws.nat_mul_add(x0, x1, x1)) : {_ == Nat.mul(Nat.double(x0), x1) : Nat}
  %Equal.sym(Nat, Nat.add(Nat.mul(x0, x1), Nat.mul(x0, x1)), Nat.mul(Nat.add(x0, x0), x1), Equal.sym(Nat, Nat.mul(Nat.add(x0, x0), x1), Nat.add(Nat.mul(x0, x1), Nat.mul(x0, x1)), Laws.nat_add_mul(x0, x0, x1))) : {_ == Nat.mul(Nat.double(x0), x1) : Nat}
  %Equal.sym(Nat, Nat.add(x0, x0), Nat.double(x0), Equal.sym(Nat, Nat.double(x0), Nat.add(x0, x0), Laws.nat_double(x0))) : {Nat.mul(_, x1) == Nat.mul(Nat.double(x0), x1) : Nat}
  {==}

Uses (8):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  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.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_add_mul
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_double
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_mul_add