Phc.byte_rt: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend as Phc
Lines 337-346 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend&start_line=337&end_line=346

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/words.bend as LW
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as NR
import bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/phc.bend as PH

# A byte is the byte of its crumbs.
def byte_rt(+x: U32, +h: {Nat.is_lt(U32.to_nat(x), 256n) == True{} : Bool}) -> {PH.byte_of(PH.c3(x), PH.c2(x), PH.c1(x), PH.c0(x)) == x : U32}:
  %Equal.sym(Nat, U32.to_nat(U32.from_nat(Nat.div(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n))), Nat.div(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n), tn(Nat.div(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n), t3_lt(x, h))) : {U32.from_nat(Nat.add(Nat.mul(Nat.add(Nat.mul(Nat.add(Nat.mul(_, 4n), U32.to_nat(PH.c2(x))), 4n), U32.to_nat(PH.c1(x))), 4n), U32.to_nat(PH.c0(x)))) == x : U32}
  %Equal.sym(Nat, U32.to_nat(U32.from_nat(Nat.mod(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n))), Nat.mod(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n), tn(Nat.mod(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n), NR.dm_lt(3n, Nat.div(Nat.div(U32.to_nat(x), 4n), 4n)))) : {U32.from_nat(Nat.add(Nat.mul(Nat.add(Nat.mul(Nat.add(Nat.mul(Nat.div(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n), 4n), _), 4n), U32.to_nat(PH.c1(x))), 4n), U32.to_nat(PH.c0(x)))) == x : U32}
  %Equal.sym(Nat, U32.to_nat(U32.from_nat(Nat.mod(Nat.div(U32.to_nat(x), 4n), 4n))), Nat.mod(Nat.div(U32.to_nat(x), 4n), 4n), tn(Nat.mod(Nat.div(U32.to_nat(x), 4n), 4n), NR.dm_lt(3n, Nat.div(U32.to_nat(x), 4n)))) : {U32.from_nat(Nat.add(Nat.mul(Nat.add(Nat.mul(Nat.add(Nat.mul(Nat.div(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n), 4n), Nat.mod(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n)), 4n), _), 4n), U32.to_nat(PH.c0(x)))) == x : U32}
  %Equal.sym(Nat, U32.to_nat(U32.from_nat(Nat.mod(U32.to_nat(x), 4n))), Nat.mod(U32.to_nat(x), 4n), tn(Nat.mod(U32.to_nat(x), 4n), NR.dm_lt(3n, U32.to_nat(x)))) : {U32.from_nat(Nat.add(Nat.mul(Nat.add(Nat.mul(Nat.add(Nat.mul(Nat.div(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n), 4n), Nat.mod(Nat.div(Nat.div(U32.to_nat(x), 4n), 4n), 4n)), 4n), Nat.mod(Nat.div(U32.to_nat(x), 4n), 4n)), 4n), _)) == x : U32}
  %NR.dm_eq(3n, Nat.div(Nat.div(U32.to_nat(x), 4n), 4n)) : {U32.from_nat(Nat.add(Nat.mul(Nat.add(Nat.mul(_, 4n), Nat.mod(Nat.div(U32.to_nat(x), 4n), 4n)), 4n), Nat.mod(U32.to_nat(x), 4n))) == x : U32}
  %NR.dm_eq(3n, Nat.div(U32.to_nat(x), 4n)) : {U32.from_nat(Nat.add(Nat.mul(_, 4n), Nat.mod(U32.to_nat(x), 4n))) == x : U32}
  %NR.dm_eq(3n, U32.to_nat(x)) : {U32.from_nat(_) == x : U32}
  LW.rt(x)

Uses (21):
  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.div
  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.mod
  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=Base/base.bend&name=U32.from_nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.to_nat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend&name=t3_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend&name=tn
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/words.bend&name=rt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=dm_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=dm_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/phc.bend&name=byte_of
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/phc.bend&name=c0
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/phc.bend&name=c1
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/phc.bend&name=c2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/argon2/phc.bend&name=c3