Memory.flat_zero: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend as Memory
Lines 580-589 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&start_line=580&end_line=589

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as SC
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/list.bend as LL
import bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/types.bend as A
import bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/sub.bend as SB
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/argon2/blamka.bend as SG

def flat_zero(+n: Nat) -> {flat(SC.replicate(A.Block, n, SG.zero())) == SC.replicate(U32, Nat.mul(n, 256n), 0) : List<&2, U32>}:
  match n:
    case 0n:
      {==}
    case 1n+ +k:
      Equal.trans(List<&2, U32>, SC.append(U32, SB.words(SG.zero()), flat(SC.replicate(A.Block, k, SG.zero()))), SC.append(U32, SC.replicate(U32, 256n, 0), SC.replicate(U32, Nat.mul(k, 256n), 0)), SC.replicate(U32, Nat.add(256n, Nat.mul(k, 256n)), 0),
        Equal.trans(List<&2, U32>, SC.append(U32, SB.words(SG.zero()), flat(SC.replicate(A.Block, k, SG.zero()))), SC.append(U32, SC.replicate(U32, 256n, 0), flat(SC.replicate(A.Block, k, SG.zero()))), SC.append(U32, SC.replicate(U32, 256n, 0), SC.replicate(U32, Nat.mul(k, 256n), 0)),
          Equal.cong(List<&2, U32>, List<&2, U32>, z => SC.append(U32, z, flat(SC.replicate(A.Block, k, SG.zero()))), SB.words(SG.zero()), SC.replicate(U32, 256n, 0), zero_words()),
          Equal.cong(List<&2, U32>, List<&2, U32>, z => SC.append(U32, SC.replicate(U32, 256n, 0), z), flat(SC.replicate(A.Block, k, SG.zero())), SC.replicate(U32, Nat.mul(k, 256n), 0), flat_zero(k))),
        LL.replicate_add(U32, 256n, Nat.mul(k, 256n), 0))

Uses (15):
  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.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=Nat.mul
  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/argon2/memory.bend&name=flat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&name=zero_words
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/list.bend&name=replicate_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/argon2/blamka.bend&name=zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=append
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=replicate
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/sub.bend&name=words
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/types.bend&name=Block