Fill.lt_mul_add: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/fill.bend as Fill
Lines 34-39 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/fill.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/fill.bend&start_line=34&end_line=39

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/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend as AR2
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

# a < p and b < q give a * q + b < p * q.
def lt_mul_add(+a: Nat, +p: Nat, +b: Nat, +q: Nat, +ha: {Nat.is_lt(a, p) == True{} : Bool}, +hb: {Nat.is_lt(b, q) == True{} : Bool}) -> {Nat.is_lt(Nat.add(Nat.mul(a, q), b), Nat.mul(p, q)) == True{} : Bool}:
  +h1 = N.lt_add_left(b, q, Nat.mul(a, q), hb)
  +h2 = AR2.mul_le(1n+a, p, q, N.lt_succ_le_succ(a, p, ha))
  N.lt_le_trans(Nat.add(Nat.mul(a, q), b), Nat.add(Nat.mul(a, q), q), Nat.mul(p, q), h1,
    L.subst(Nat, z => {Nat.is_le(z, Nat.mul(p, q)) == True{} : Bool}, Nat.mul(1n+a, q), Nat.add(Nat.mul(a, q), q), NA.add_comm(q, Nat.mul(a, q)), h2))

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=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/arith.bend&name=mul_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_comm
  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/proofs/lib/nat.bend&name=lt_add_left
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_succ_le_succ