def sum_val

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/ed25519/scalar.bend:157

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/crypto/curve25519/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend as PL
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/fold.bend as FD
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/fieldops.bend as FO

157# a + b < 2 M <= 2^256: no carry out, the limbs hold a + b
158def sum_val(+one: Nat, +h1: {one == 1n : Nat}, +mm: Nat, +h2: {Nat.is_le(Nat.add(1n+mm, 1n+mm), T(one)) == True{} : Bool}, +a: List<&2, U32>, +b: List<&2, U32>, +ha: {LM.okb(32n, a, 255n) == True{} : Bool}, +hb: {LM.okb(32n, b, 255n) == True{} : Bool}, +la: {Nat.is_lt(FS.value(a), 1n+mm) == True{} : Bool}, +lb: {Nat.is_lt(FS.value(b), 1n+mm) == True{} : Bool}) -> {FS.value(F.cr_limbs(F.carry(F.addl(a, b), 0))) == Nat.add(FS.value(a), FS.value(b)) : Nat}:
159  +ys = F.addl(a, b)
160  +r = F.carry(ys, 0)
161  +V = Nat.add(FS.value(a), FS.value(b))
162  +hyo = FO.addl_okb(one, h1, a, b, ha, hb)
163  +ev = Equal.trans(Nat, Nat.add(FS.value(LM.cl(r)), LM.shn(32n, v(LM.cc(r)))), FS.value(ys), V, FD.cr_val(one, h1, ys, hyo), PL.addl_val(one, h1, a, b, 255n, 255n, LM.okb_lea(32n, a, 255n, ha), LM.okb_lea(32n, b, 255n, hb), FO.lt32_small(one, h1, Nat.add(255n, 255n), {==})))
164  +l2 = N.lt_le_trans(V, Nat.add(1n+mm, 1n+mm), T(one), N.lt_le_trans(V, Nat.add(1n+mm, FS.value(b)), Nat.add(1n+mm, 1n+mm), N.lt_add_r2(FS.value(a), 1n+mm, FS.value(b), la), N.lt_le(Nat.add(1n+mm, FS.value(b)), Nat.add(1n+mm, 1n+mm), N.lt_add_left(FS.value(b), 1n+mm, 1n+mm, lb))), h2)
165  +z = RD.last_zero(one, h1, FS.value(LM.cl(r)), v(LM.cc(r)), V, ev, l2, Nat.is_lt(v(LM.cc(r)), one), {==})
166  +ev0 = L.subst(Nat, w => {Nat.add(FS.value(LM.cl(r)), LM.shn(32n, w)) == V : Nat}, v(LM.cc(r)), 0n, z, ev)
167  Equal.trans(Nat, FS.value(LM.cl(r)), Nat.add(FS.value(LM.cl(r)), 0n), V, Equal.sym(Nat, Nat.add(FS.value(LM.cl(r)), 0n), FS.value(LM.cl(r)), N.add_zero(FS.value(LM.cl(r)))), ev0)