def two_lt

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/reduce.bend:104

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

104# after two passes the value is below 2^256
105def two_lt(+one: Nat, +h1: {one == 1n : Nat}, +xs: List<&2, U32>, +hx: {LM.okb(32n, xs, WD.sc(28n, one)) == True{} : Bool}) -> {Nat.is_lt(FS.value(F.pass(F.pass(xs))), LM.shn(32n, one)) == True{} : Bool}:
106  +T = LM.shn(32n, one)
107  +K = WD.sc(27n, one)
108  +r1 = F.carry(xs, 0)
109  +a1 = FS.value(LM.cl(r1))
110  +c1 = v(LM.cc(r1))
111  +y1 = F.pass(xs)
112  +hy1 = FD.pass_okb(one, h1, xs, hx)
113  +la = val_lt(one, h1, 32n, LM.cl(r1), FD.cr_okb(one, h1, xs, hx))
114  +lc = N.le_trans(Nat.mul(38n, c1), Nat.mul(38n, WD.sc(21n, one)), WD.sc(6n, WD.sc(21n, one)), M.le_mul2(38n, c1, 38n, WD.sc(21n, one), N.le_refl(38n), FD.cr_cb(one, h1, xs, hx)), FD.m38(WD.sc(21n, one)))
115  +l1 = N.lt_le_trans(Nat.add(a1, Nat.mul(38n, c1)), Nat.add(T, Nat.mul(38n, c1)), Nat.add(T, K), N.lt_add_r2(a1, T, Nat.mul(38n, c1), la), N.le_add_left(Nat.mul(38n, c1), K, T, lc))
116  +hV = L.subst(Nat, z => {Nat.is_lt(z, Nat.add(T, K)) == True{} : Bool}, Nat.add(a1, Nat.mul(38n, c1)), FS.value(y1), Equal.sym(Nat, FS.value(y1), Nat.add(a1, Nat.mul(38n, c1)), FD.pass_val(one, h1, xs, hx)), l1)
117  +r2 = F.carry(y1, 0)
118  +a2 = FS.value(LM.cl(r2))
119  +c2 = v(LM.cc(r2))
120  +la2 = val_lt(one, h1, 32n, LM.cl(r2), FD.cr_okb(one, h1, y1, hy1))
121  +lt2 = small_by(one, h1, a2, c2, FS.value(y1), Nat.is_lt(c2, one), {==}, la2, FD.cr_val(one, h1, y1, hy1), hV)
122  L.subst(Nat, z => {Nat.is_lt(z, T) == True{} : Bool}, Nat.add(a2, Nat.mul(38n, c2)), FS.value(F.pass(y1)), Equal.sym(Nat, FS.value(F.pass(y1)), Nat.add(a2, Nat.mul(38n, c2)), FD.pass_val(one, h1, y1, hy1)), lt2)