def small_lt_p

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/fieldops.bend:213

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/proofs/lib/nat.bend as N
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

213def small_lt_p(+one: Nat, +h1: {one == 1n : Nat}, +k: Nat, +hk: {Nat.is_lt(k, C.shift(10n, 1n)) == True{} : Bool}) -> {Nat.is_lt(k, FS.prime(one)) == True{} : Bool}:
214  +hk1 = I.lt_subst_r(k, C.shift(10n, 1n), C.shift(10n, one), Equal.cong(Nat, Nat, z => C.shift(10n, z), 1n, one, Equal.sym(Nat, one, 1n, h1)), hk)
215  +s1 = N.lt_le_trans(k, C.shift(10n, one), C.shift(255n, one), hk1, K.shift_le_k(10n, 255n, one, 245n, {==}))
216  +pp = K.pp(one)
217  # 2^255 <= p: p + c = 2^256 = 2^255 + 2^255 and c <= 2^33 <= 2^255
218  +c = FS.cp(one)
219  +e1 = Equal.trans(Nat, Nat.add(1n+pp, c), C.shift(256n, one), Nat.add(C.shift(255n, one), C.shift(255n, one)), K.hm_p(one, h1), Equal.sym(Nat, Nat.add(C.shift(255n, one), C.shift(255n, one)), C.shift(256n, one), K.shift_twice(255n, one)))
220  +ch = N.le_trans(c, C.shift(33n, one), C.shift(255n, one), K.cp_le(one, h1), K.shift_le_k(33n, 255n, one, 222n, {==}))
221  +s2 = lt_p_half(C.shift(255n, one), 1n+pp, c, e1, ch)
222  I.lt_subst_r(k, 1n+pp, FS.prime(one), Equal.sym(Nat, FS.prime(one), 1n+pp, K.prime_eq(one, h1)), N.lt_le_trans(k, C.shift(255n, one), 1n+pp, s1, s2))