def sc_add

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/num.bend:37

import Base
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

37# 2^k x + 2^k y == 2^k (x + y)
38def sc_add(+k: Nat, +x: Nat, +y: Nat) -> {Nat.add(WD.sc(k, x), WD.sc(k, y)) == WD.sc(k, Nat.add(x, y)) : Nat}:
39  match k:
40    case 0n:
41      {==}
42    case 1n+q:
43      +e = sc_add(q, x, y)
44      +d = Equal.sym(Nat, Nat.double(Nat.add(WD.sc(q, x), WD.sc(q, y))), Nat.add(Nat.double(WD.sc(q, x)), Nat.double(WD.sc(q, y))), NA.double_add(WD.sc(q, x), WD.sc(q, y)))
45      Equal.trans(Nat, Nat.add(Nat.double(WD.sc(q, x)), Nat.double(WD.sc(q, y))), Nat.double(Nat.add(WD.sc(q, x), WD.sc(q, y))), Nat.double(WD.sc(q, Nat.add(x, y))), d, Equal.cong(Nat, Nat, t => Nat.double(t), Nat.add(WD.sc(q, x), WD.sc(q, y)), WD.sc(q, Nat.add(x, y)), e))