Lemire.pair_eq: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend as Lemire
Lines 384-398 of bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend&start_line=384&end_line=398

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C

# [2a == k] + [2a + 1 == k] == [a == k / 2]
def pair_eq(+a: Nat, +k: Nat) -> {Nat.add(b2n(Nat.is_eq(Nat.double(a), k)), b2n(Nat.is_eq(1n+Nat.double(a), k))) == b2n(Nat.is_eq(a, C.half(k))) : Nat}:
  match a k:
    case 0n 0n:
      {==}
    case 0n 1n:
      {==}
    case 0n 2n+q:
      {==}
    case 1n+p 0n:
      {==}
    case 1n+p 1n:
      {==}
    case 1n+p 2n+q:
      pair_eq(p, q)

Uses (6):
  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.is_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend&name=b2n
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=half