Memory.mul_lt: 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 393-401 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=393&end_line=401

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

def mul_lt(+b: Nat, +n: Nat, +hb: {Nat.is_lt(b, n) == True{} : Bool}) -> {Nat.is_le(Nat.add(Nat.mul(b, 256n), 256n), Nat.mul(n, 256n)) == True{} : Bool}:
  match b n:
    case _ 0n:
      Empty.absurd({Nat.is_le(Nat.add(Nat.mul(b, 256n), 256n), 0n) == True{} : Bool}, N.lt_zero_absurd(b, hb))
    case 0n 1n+m:
      N.le_add_right(256n, Nat.mul(m, 256n))
    case 1n+ +c 1n+ +m:
      %Equal.sym(Nat, Nat.add(Nat.add(256n, Nat.mul(c, 256n)), 256n), Nat.add(256n, Nat.add(Nat.mul(c, 256n), 256n)), NA.add_assoc(256n, Nat.mul(c, 256n), 256n)) : {Nat.is_le(_, Nat.add(256n, Nat.mul(m, 256n))) == True{} : Bool}
      N.le_add_left(Nat.add(Nat.mul(c, 256n), 256n), Nat.mul(m, 256n), 256n, mul_lt(c, m, hb))

Uses (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  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_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=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=le_add_left
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_zero_absurd