Bytes.prefix: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/random/bytes.bend as Bytes
Lines 72-89 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/random/bytes.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/random/bytes.bend&start_line=72&end_line=89

import Base
import bend-collections-laws-crypto@1.0.0.0/src/math/random/chacha8.bend as C8
import bend-collections-laws-crypto@1.0.0.0/spec/math/random/source.bend as SRC
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/random.bend as SCR
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N

# taking j bytes never looks past the first j outputs
def prefix(+j: Nat, +r: List<&2, U32>, +c: C8.ChaCha8, +d: Nat) -> {SCR.take(j, SCR.append(r, SCR.flat(outs(Nat.add(j, d), c)))) == SCR.take(j, SCR.append(r, SCR.flat(outs(j, c)))) : List<&2, U32>}:
  match j r:
    case 0n _:
      {==}
    case 1n+i Con{+b, +rest}:
      Equal.cong(List<&2, U32>, List<&2, U32>, l => Con{b, l}, SCR.take(i, SCR.append(rest, SCR.flat(outs(1n+Nat.add(i, d), c)))), SCR.take(i, SCR.append(rest, SCR.flat(outs(1n+i, c)))),
        Equal.trans(List<&2, U32>, SCR.take(i, SCR.append(rest, SCR.flat(outs(1n+Nat.add(i, d), c)))), SCR.take(i, SCR.append(rest, SCR.flat(outs(i, c)))), SCR.take(i, SCR.append(rest, SCR.flat(outs(1n+i, c)))),
          %N.add_succ(i, d) : {SCR.take(i, SCR.append(rest, SCR.flat(outs(_, c)))) == SCR.take(i, SCR.append(rest, SCR.flat(outs(i, c)))) : List<&2, U32>}
          prefix(i, rest, c, 1n+d),
          Equal.sym(List<&2, U32>, SCR.take(i, SCR.append(rest, SCR.flat(outs(1n+i, c)))), SCR.take(i, SCR.append(rest, SCR.flat(outs(i, c)))),
            %Equal.trans(Nat, Nat.add(i, 1n), 1n+Nat.add(i, 0n), 1n+i, N.add_succ(i, 0n), Equal.cong(Nat, Nat, z => 1n+z, Nat.add(i, 0n), i, N.add_zero(i))) : {SCR.take(i, SCR.append(rest, SCR.flat(outs(_, c)))) == SCR.take(i, SCR.append(rest, SCR.flat(outs(i, c)))) : List<&2, U32>}
            prefix(i, rest, c, 1n))))
    case 1n+i Nil{}:
      +x = SRC.fst64(C8.ChaCha8, C8.next(c))
      +c1 = SRC.snd64(C8.ChaCha8, C8.next(c))
      %Equal.sym(List<&2, U32>, SCR.le_bytes(x), Con{b0(x), b7(x)}, le_split(x)) : {SCR.take(1n+i, SCR.append(_, SCR.flat(outs(Nat.add(i, d), c1)))) == SCR.take(1n+i, SCR.append(_, SCR.flat(outs(i, c1)))) : List<&2, U32>}
      Equal.cong(List<&2, U32>, List<&2, U32>, l => Con{b0(x), l}, SCR.take(i, SCR.append(b7(x), SCR.flat(outs(Nat.add(i, d), c1)))), SCR.take(i, SCR.append(b7(x), SCR.flat(outs(i, c1)))), prefix(i, b7(x), c1, d))

Uses (21):
  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=List
  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=U32
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/random/bytes.bend&name=b0
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/random/bytes.bend&name=b7
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/random/bytes.bend&name=le_split
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/random/bytes.bend&name=outs
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=add_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/random.bend&name=append
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/random.bend&name=flat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/random.bend&name=le_bytes
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/random.bend&name=take
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/source.bend&name=fst64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/source.bend&name=snd64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/random/chacha8.bend&name=ChaCha8
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/random/chacha8.bend&name=next