def mz

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/ed25519/scalar.bend:51

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend as F
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/u32.bend as U
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/crypto/curve25519/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/fold.bend as FD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/reduce.bend as RD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/consts.bend as K

51# case c < 1: x < M
52def mz(+one: Nat, +h1: {one == 1n : Nat}, +mm: Nat, +x: List<&2, U32>, +cp: List<&2, U32>, +hx: {LM.okb(32n, x, 255n) == True{} : Bool}, +hk: {LM.okb(32n, cp, 255n) == True{} : Bool}, +hv: {Nat.add(K.valo(one, cp), 1n+mm) == T(one) : Nat}, +hc: {Nat.is_lt(v(LM.cc(F.carry(F.addl(x, cp), 0))), one) == True{} : Bool}) -> {Nat.is_lt(FS.value(x), 1n+mm) == True{} : Bool}:
53  +r = F.carry(F.addl(x, cp), 0)
54  +c = v(LM.cc(r))
55  +vs = FS.value(LM.cl(r))
56  +VC = FS.value(cp)
57  +c0 = U.lt_one_zero(c, L.subst(Nat, o => {Nat.is_lt(c, o) == True{} : Bool}, one, 1n, h1, hc))
58  +ev = L.subst(Nat, z => {Nat.add(vs, LM.shn(32n, z)) == Nat.add(FS.value(x), VC) : Nat}, c, 0n, c0, cvx(one, h1, mm, x, cp, hx, hk))
59  +ev1 = Equal.trans(Nat, vs, Nat.add(vs, 0n), Nat.add(FS.value(x), VC), Equal.sym(Nat, Nat.add(vs, 0n), vs, N.add_zero(vs)), ev)
60  +lt = RD.val_lt(one, h1, 32n, LM.cl(r), FD.cr_okb(one, h1, F.addl(x, cp), ad_okb(one, h1, x, cp, hx, hk)))
61  +lt2 = L.subst(Nat, z => {Nat.is_lt(vs, z) == True{} : Bool}, T(one), Nat.add(VC, 1n+mm), Equal.sym(Nat, Nat.add(VC, 1n+mm), T(one), cp_val(one, h1, mm, cp, hv)), lt)
62  +lt3 = L.subst(Nat, z => {Nat.is_lt(z, Nat.add(VC, 1n+mm)) == True{} : Bool}, vs, Nat.add(FS.value(x), VC), ev1, lt2)
63  +lt4 = L.subst(Nat, z => {Nat.is_lt(Nat.add(FS.value(x), VC), z) == True{} : Bool}, Nat.add(VC, 1n+mm), Nat.add(1n+mm, VC), NA.add_comm(VC, 1n+mm), lt3)
64  RD.lt_cancel_r(FS.value(x), 1n+mm, VC, lt4)