def small_r

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/fieldops.bend:188

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/secp256k1/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/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 Lg
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend as LV
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/ineq.bend as I
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/consts.bend as K
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/listops.bend as LO

188def small_r(+one: Nat, +h1: {one == 1n : Nat}, +k: Nat, +hk: {Nat.is_lt(k, FS.prime(one)) == True{} : Bool}) -> {FS.reduced(one, FS.prime(one), F.small(k)) == True{} : Bool}:
189  +hk2 = I.lt_subst_r(k, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1), hk)
190  +hkt = N.lt_le_trans(k, 1n+K.pp(one), C.shift(256n, one), hk2, I.le_subst_r(1n+K.pp(one), Nat.add(1n+K.pp(one), FS.cp(one)), C.shift(256n, one), K.hm_p(one, h1), N.le_add_right(1n+K.pp(one), FS.cp(one))))
191  +ev = LO.small_v(one, h1, k, hkt)
192  +lt = I.lt_subst_l(k, FS.value(one, F.small(k)), FS.prime(one), Equal.sym(Nat, FS.value(one, F.small(k)), k, Equal.trans(Nat, FS.value(one, F.small(k)), LV.ev(F.small(k)), k, LV.ev_value(one, h1, F.small(k)), ev)), hk)
193  Lg.and_intro(FS.limbs16(one, 16n, F.small(k)), Nat.is_lt(FS.value(one, F.small(k)), FS.prime(one)), LO.small_l(one, h1, k), lt)