def cn_lt

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/consts.bend:171

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/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/bounds.bend as B

171def cn_lt(+one: Nat, +h1: {one == 1n : Nat}) -> {Nat.is_lt(FS.cn(one), C.shift(129n, one)) == True{} : Bool}:
172  %Equal.sym(Nat, FS.cn(one), Nat.add(FS.digits(one, cn8()), LV.pos(8n, one)), cn_split(one, h1)) : {Nat.is_lt(_, C.shift(129n, one)) == True{} : Bool}
173  %Equal.sym(Nat, LV.pos(8n, one), C.shift(128n, one), B.pos_shift(8n, one)) : {Nat.is_lt(Nat.add(FS.digits(one, cn8()), _), C.shift(129n, one)) == True{} : Bool}
174  %Equal.sym(Nat, C.shift(129n, one), Nat.add(C.shift(128n, one), C.shift(128n, one)), Equal.sym(Nat, Nat.add(C.shift(128n, one), C.shift(128n, one)), C.shift(129n, one), shift_twice(128n, one))) : {Nat.is_lt(Nat.add(FS.digits(one, cn8()), C.shift(128n, one)), _) == True{} : Bool}
175  I.lt_le_add(FS.digits(one, cn8()), C.shift(128n, one), C.shift(128n, one), C.shift(128n, one), I.lt_subst_r(FS.digits(one, cn8()), LV.pos(8n, one), C.shift(128n, one), B.pos_shift(8n, one), digits_lt(one, h1, 8n, cn8(), cn8_limbs(one, h1))), N.le_refl(C.shift(128n, one)))