def lt_mul_add

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/argon2/fill.bend:34

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

34# a < p and b < q give a * q + b < p * q.
35def 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}:
36  +h1 = N.lt_add_left(b, q, Nat.mul(a, q), hb)
37  +h2 = AR2.mul_le(1n+a, p, q, N.lt_succ_le_succ(a, p, ha))
38  N.lt_le_trans(Nat.add(Nat.mul(a, q), b), Nat.add(Nat.mul(a, q), q), Nat.mul(p, q), h1,
39    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))