LAWS.to_nat_or_split: law of lawful-words@0.1.0.0
Import: import lawful-words@0.1.0.0/LAWS.bend as LAWS
Lines 266-274 of lawful-words@0.1.0.0/LAWS.bend: https://jend.leeeet.dev/read.txt?path=lawful-words@0.1.0.0/LAWS.bend&start_line=266&end_line=274

import Base

# Or of a word with only bits from k up and a word below 2^k is their sum
# (no bit in common). Xor likewise.
law to_nat_or_split:
  for a: U32
  for b: U32
  for k: Nat
  for ha: {Nat.mod(U32.to_nat(a), Nat.pow(2n, k)) == 0n : Nat}
  for hb: {Nat.is_lt(U32.to_nat(b), Nat.pow(2n, k)) == True{} : Bool}
  {U32.to_nat(U32.or(a, b)) == Nat.add(U32.to_nat(a), U32.to_nat(b)) : Nat}

Uses (10):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mod
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.pow
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.or
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.to_nat
  https://jend.leeeet.dev/definition.txt?path=lawful-words@0.1.0.0/proof_u32.bend&name=to_nat_or_split