def by_lt

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/canon.bend:54

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/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/reduce.bend as RD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/consts.bend as K
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/freeze.bend as FZ

54def by_lt(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}, +x: List<&2, U32>, +cp: List<&2, U32>, +hx: {LM.okb(32n, x, 255n) == True{} : Bool}, +hk: {LM.okb(32n, cp, 128n) == True{} : Bool}, +hv: {K.valo(one, cp) == Nat.add(1n+pp, 38n) : Nat}, +bB: Nat, +q: Nat, +hxB: {Nat.is_lt(FS.value(x), Nat.add(1n+pp, bB)) == True{} : Bool}, +hPQ: {Nat.is_le(1n+pp, q) == True{} : Bool}, +hBQ: {Nat.is_le(bB, q) == True{} : Bool}, +b: Bool, +hb: {Nat.is_lt(v(LM.cc(F.carry(F.addl(x, cp), 0))), one) == b : Bool}) -> {Nat.is_lt(FS.value(FZ.csg(x, cp)), q) == True{} : Bool}:
55  match b:
56    case True{}:
57      %Equal.sym(List<&2, U32>, FZ.csg(x, cp), x, FZ.cs_zero(one, h1, x, cp, hx, hk, hb)) : {Nat.is_lt(FS.value(_), q) == True{} : Bool}
58      N.lt_le_trans(FS.value(x), 1n+pp, q, FZ.z_case(one, h1, pp, hP, x, cp, hx, hk, hv, hb), hPQ)
59    case False{}:
60      +s = LM.cl(F.carry(F.addl(x, cp), 0))
61      %Equal.sym(List<&2, U32>, FZ.csg(x, cp), s, FZ.cs_one(one, h1, pp, hP, x, cp, hx, hk, hv, hb)) : {Nat.is_lt(FS.value(_), q) == True{} : Bool}
62      +e1 = FZ.o_case(one, h1, pp, hP, x, cp, hx, hk, hv, hb)
63      +l1 = L.subst(Nat, z => {Nat.is_lt(z, Nat.add(1n+pp, bB)) == True{} : Bool}, FS.value(x), Nat.add(FS.value(s), 1n+pp), Equal.sym(Nat, Nat.add(FS.value(s), 1n+pp), FS.value(x), e1), hxB)
64      +l2 = L.subst(Nat, z => {Nat.is_lt(Nat.add(FS.value(s), 1n+pp), z) == True{} : Bool}, Nat.add(1n+pp, bB), Nat.add(bB, 1n+pp), NA.add_comm(1n+pp, bB), l1)
65      N.lt_le_trans(FS.value(s), bB, q, RD.lt_cancel_r(FS.value(s), bB, 1n+pp, l2), hBQ)