Shuffle.lt_succ_split: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/shuffle.bend as Shuffle
Lines 118-129 of bend-collections-laws-math@1.0.0.0/proofs/math/random/shuffle.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/shuffle.bend&start_line=118&end_line=129

import Base
import bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend as SR
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N

# [p == v] + [v < p] == [v < p + 1]
def lt_succ_split(+v: Nat, +p: Nat) -> {Nat.add(SR.b2n(Nat.is_eq(p, v)), SR.b2n(Nat.is_lt(v, p))) == SR.b2n(Nat.is_lt(v, 1n+p)) : Nat}:
  match v p:
    case 0n 0n:
      {==}
    case 0n 1n+q:
      {==}
    case 1n+u 0n:
      %Equal.sym(Bool, Nat.is_lt(u, 0n), False{}, N.not_lt_zero(u)) : {0n == SR.b2n(_) : Nat}
      {==}
    case 1n+u 1n+q:
      lt_succ_split(u, q)

Uses (8):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.is_eq
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=not_lt_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend&name=b2n