PROOF.ur.r8: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 1229-1234 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=1229&end_line=1234

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

def ur.r8(+x0: Nat, +x1: Nat, +x2: Nat) -> {Nat.add(Nat.mul(x0, U.U32Nat.dbl(x1, 1n)), x2) == Nat.add(x2, U.U32Nat.dbl(x1, x0)) : Nat}:
  %Equal.sym(Nat, Nat.mul(x0, U.U32Nat.dbl(x1, 1n)), U.U32Nat.dbl(x1, Nat.mul(x0, 1n)), Laws.dbl_mul_r(x1, x0, 1n)) : {Nat.add(_, x2) == Nat.add(x2, U.U32Nat.dbl(x1, x0)) : Nat}
  %Equal.sym(Nat, Nat.mul(x0, 1n), Nat.mul(1n, x0), Laws.nat_mul_comm(x0, 1n)) : {Nat.add(U.U32Nat.dbl(x1, _), x2) == Nat.add(x2, U.U32Nat.dbl(x1, x0)) : Nat}
  %Equal.sym(Nat, Nat.mul(1n, x0), x0, Laws.nat_one_mul(x0)) : {Nat.add(U.U32Nat.dbl(x1, _), x2) == Nat.add(x2, U.U32Nat.dbl(x1, x0)) : Nat}
  %Equal.sym(Nat, Nat.add(U.U32Nat.dbl(x1, x0), x2), Nat.add(x2, U.U32Nat.dbl(x1, x0)), Laws.nat_add_comm(U.U32Nat.dbl(x1, x0), x2)) : {_ == Nat.add(x2, U.U32Nat.dbl(x1, x0)) : Nat}
  {==}

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=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-osdev-u32@0.1.0.0/LAWS.bend&name=dbl_mul_r
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_add_comm
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_one_mul
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/u32.bend&name=U32Nat.dbl