def rc_sub

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/rel.bend:32

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/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as NR
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/fieldops.bend as FO
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend as G

32def rc_sub(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}, +x: List<&2, U32>, +y: List<&2, U32>, +a: Nat, +b: Nat, +ox: {LM.okb(32n, x, 255n) == True{} : Bool}, +cx: G.ceq(pp, FS.value(x), a), +oy: {LM.okb(32n, y, 255n) == True{} : Bool}, +cy: G.ceq(pp, FS.value(y), b)) -> G.ceq(pp, FS.value(F.sub(x, y)), FS.fsub(1n+pp, a, b)):
33  +P = {1n+pp : Nat}
34  +mb = Nat.mod(b, P)
35  +e1 = FO.sub_mod(one, h1, pp, hP, x, y, ox, oy)
36  +e2 = G.c_eq(pp, Nat.sub(Nat.add(FS.value(x), P), Nat.mod(FS.value(y), P)), Nat.sub(Nat.add(FS.value(x), P), mb), Equal.cong(Nat, Nat, z => Nat.sub(Nat.add(FS.value(x), P), z), Nat.mod(FS.value(y), P), mb, cy))
37  +e3 = G.c_subm(pp, FS.value(x), a, mb, cx, N.lt_le(mb, P, NR.dm_lt(pp, b)))
38  +e4 = G.c_sym(pp, Nat.mod(Nat.sub(Nat.add(a, P), mb), P), Nat.sub(Nat.add(a, P), mb), G.c_mod(pp, Nat.sub(Nat.add(a, P), mb)))
39  +e = G.c_trans(pp, FS.value(F.sub(x, y)), Nat.sub(Nat.add(FS.value(x), P), Nat.mod(FS.value(y), P)), FS.fsub(P, a, b), e1, G.c_trans(pp, Nat.sub(Nat.add(FS.value(x), P), Nat.mod(FS.value(y), P)), Nat.sub(Nat.add(FS.value(x), P), mb), FS.fsub(P, a, b), e2, G.c_trans(pp, Nat.sub(Nat.add(FS.value(x), P), mb), Nat.sub(Nat.add(a, P), mb), FS.fsub(P, a, b), e3, e4)))
40  e