Memory.W_splice: 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 556-566 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=556&end_line=566

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/proofs/lib/logic.bend as L
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/argon2.bend as SA

def W_splice(+d: Nat, +mem: List<&2, A.Block>, +b: Nat, +v: A.Block, +hb: {Nat.is_lt(b, SC.length(A.Block, mem)) == True{} : Bool}) -> {splice(SB.words(v), Nat.mul(b, 256n), W(d, mem)) == W(d, SA.set(mem, b, v)) : List<&2, U32>}:
  +F = flat(mem)
  +hn = mul_lt(b, SC.length(A.Block, mem), hb)
  +hfl = Equal.sym(Nat, SC.length(U32, F), Nat.mul(SC.length(A.Block, mem), 256n), flat_len(mem))
  +h = L.subst(Nat, z => {Nat.is_le(Nat.add(Nat.mul(b, 256n), z), SC.length(U32, F)) == True{} : Bool}, 256n, SC.length(U32, SB.words(v)), Equal.sym(Nat, SC.length(U32, SB.words(v)), 256n, words_len(v)), L.subst(Nat, z => {Nat.is_le(Nat.add(Nat.mul(b, 256n), 256n), z) == True{} : Bool}, Nat.mul(SC.length(A.Block, mem), 256n), SC.length(U32, F), hfl, hn))
  +P = pad(d, SC.length(A.Block, mem))
  Equal.trans(List<&2, U32>, splice(SB.words(v), Nat.mul(b, 256n), SC.append(U32, F, P)), SC.append(U32, splice(SB.words(v), Nat.mul(b, 256n), F), P), W(d, SA.set(mem, b, v)),
    splice_app_left(SB.words(v), Nat.mul(b, 256n), F, P, h),
    Equal.trans(List<&2, U32>, SC.append(U32, splice(SB.words(v), Nat.mul(b, 256n), F), P), SC.append(U32, flat(SA.set(mem, b, v)), P), W(d, SA.set(mem, b, v)),
      Equal.cong(List<&2, U32>, List<&2, U32>, z => SC.append(U32, z, P), splice(SB.words(v), Nat.mul(b, 256n), F), flat(SA.set(mem, b, v)), flat_splice(mem, b, v, hb)),
      Equal.cong(Nat, List<&2, U32>, z => SC.append(U32, flat(SA.set(mem, b, v)), pad(d, z)), SC.length(A.Block, mem), SC.length(A.Block, SA.set(mem, b, v)), Equal.sym(Nat, SC.length(A.Block, SA.set(mem, b, v)), SC.length(A.Block, mem), LL.length_update(A.Block, mem, b, v)))))

Uses (27):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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=Nat.is_le
  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.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=W
  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=flat_len
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&name=flat_splice
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&name=mul_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&name=pad
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&name=splice
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&name=splice_app_left
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/memory.bend&name=words_len
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/list.bend&name=length_update
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/argon2/argon2.bend&name=set
  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=length
  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