def p38

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/freeze.bend:55

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/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
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/proofs/crypto/curve25519/num.bend as M

55# 38 <= p
56def p38(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}) -> {Nat.is_le(38n, 1n+pp) == True{} : Bool}:
57  +P = {1n+pp : Nat}
58  +a = L.subst(Nat, o => {Nat.is_le(64n, WD.sc(6n, o)) == True{} : Bool}, 1n, one, Equal.sym(Nat, one, 1n, h1), {==})
59  +b = N.le_trans(64n, WD.sc(6n, one), WD.sc(255n, one), a, M.sc_mono(249n, 6n, one))
60  +c = L.subst(Nat, z => {Nat.is_le(64n, z) == True{} : Bool}, WD.sc(255n, one), Nat.add(P, 19n), Equal.sym(Nat, Nat.add(P, 19n), WD.sc(255n, one), hP), b)
61  +c2 = L.subst(Nat, z => {Nat.is_le(64n, z) == True{} : Bool}, Nat.add(P, 19n), Nat.add(19n, P), NA.add_comm(P, 19n), c)
62  +d = L.subst(Nat, z => {Nat.is_le(z, Nat.add(19n, P)) == True{} : Bool}, 64n, Nat.add(19n, 45n), {==}, c2)
63  +e = L.subst(Bool, z => {z == True{} : Bool}, Nat.is_le(Nat.add(19n, 45n), Nat.add(19n, P)), Nat.is_le(45n, P), NR.le_add_cancel(19n, 45n, P), d)
64  N.le_trans(38n, 45n, P, {==}, e)