PROOF.ur.r34: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 1708-1712 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=1708&end_line=1712

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.r34(+x0: Nat, +x1: Nat, +x2: Nat) -> {Nat.add(x0, Nat.mul(0n, x1)) == Nat.add(x0, U.U32Nat.dbl(32n, Nat.mul(0n, x2))) : Nat}:
  %Equal.sym(Nat, Nat.add(x0, 0n), x0, Laws.nat_add_zero(x0)) : {_ == Nat.add(x0, U.U32Nat.dbl(32n, Nat.mul(0n, x2))) : Nat}
  %Equal.sym(Nat, x0, Nat.add(x0, 0n), Equal.sym(Nat, Nat.add(x0, 0n), x0, Laws.nat_add_zero(x0))) : {_ == Nat.add(x0, U.U32Nat.dbl(32n, Nat.mul(0n, x2))) : Nat}
  %Equal.sym(Nat, 0n, U.U32Nat.dbl(32n, 0n), Equal.sym(Nat, U.U32Nat.dbl(32n, 0n), 0n, Laws.dbl_zero(32n))) : {Nat.add(x0, _) == Nat.add(x0, U.U32Nat.dbl(32n, Nat.mul(0n, x2))) : Nat}
  {==}

Uses (7):
  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_zero
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/u32.bend&name=U32Nat.dbl