PROOF.ur.r1: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 1055-1061 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=1055&end_line=1061

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

def ur.r1(+x0: Nat, +x1: Nat, +x2: Nat) -> {Nat.add(x0, Nat.add(Nat.mul(x1, x0), x2)) == Nat.add(Nat.mul(1n+x1, x0), x2) : Nat}:
  %Equal.sym(Nat, Nat.mul(x1, x0), Nat.mul(x0, x1), Laws.nat_mul_comm(x1, x0)) : {Nat.add(x0, Nat.add(_, x2)) == Nat.add(Nat.mul(1n+x1, x0), x2) : Nat}
  %Equal.sym(Nat, Nat.add(x0, Nat.add(Nat.mul(x0, x1), x2)), Nat.add(Nat.add(x0, Nat.mul(x0, x1)), x2), Equal.sym(Nat, Nat.add(Nat.add(x0, Nat.mul(x0, x1)), x2), Nat.add(x0, Nat.add(Nat.mul(x0, x1), x2)), Laws.nat_add_assoc(x0, Nat.mul(x0, x1), x2))) : {_ == Nat.add(Nat.mul(1n+x1, x0), x2) : Nat}
  %Equal.sym(Nat, Nat.mul(x0, x1), Nat.mul(x1, x0), Equal.sym(Nat, Nat.mul(x1, x0), Nat.mul(x0, x1), Laws.nat_mul_comm(x1, x0))) : {Nat.add(Nat.add(x0, _), x2) == Nat.add(Nat.mul(1n+x1, x0), x2) : Nat}
  %Equal.sym(Nat, x0, Nat.mul(1n, x0), Equal.sym(Nat, Nat.mul(1n, x0), x0, Laws.nat_one_mul(x0))) : {Nat.add(Nat.add(_, Nat.mul(x1, x0)), x2) == Nat.add(Nat.mul(1n+x1, x0), x2) : Nat}
  %Equal.sym(Nat, Nat.add(Nat.mul(1n, x0), Nat.mul(x1, x0)), Nat.mul(Nat.add(1n, x1), x0), Equal.sym(Nat, Nat.mul(Nat.add(1n, x1), x0), Nat.add(Nat.mul(1n, x0), Nat.mul(x1, x0)), Laws.nat_add_mul(1n, x1, x0))) : {Nat.add(_, x2) == Nat.add(Nat.mul(1n+x1, x0), x2) : 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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_add_assoc
  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_mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/LAWS.bend&name=nat_one_mul