PROOF.un.double_mul: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF.bend as PROOF
Lines 979-980 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=979&end_line=980

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

def un.double_mul(+x: Nat, +y: Nat) -> {Nat.mul(Nat.double(x), y) == Nat.double(Nat.mul(x, y)) : Nat}:
  Equal.trans(Nat, Nat.mul(Nat.double(x), y), Nat.add(Nat.mul(x, y), Nat.mul(x, y)), Nat.double(Nat.mul(x, y)), Equal.trans(Nat, Nat.mul(Nat.double(x), y), Nat.mul(Nat.add(x, x), y), Nat.add(Nat.mul(x, y), Nat.mul(x, y)), Equal.cong(Nat, Nat, zz => Nat.mul(zz, y), Nat.double(x), Nat.add(x, x), Laws.nat_double(x)), Laws.nat_add_mul(x, x, y)), Equal.sym(Nat, Nat.double(Nat.mul(x, y)), Nat.add(Nat.mul(x, y), Nat.mul(x, y)), Laws.nat_double(Nat.mul(x, y))))

Uses (9):
  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.sym
  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.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