def prod_limbs

bend-collections-laws-crypto@1.0.0.0 · proofs/math/typed/u32laws.bend:239

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/proofs/lib/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/w64mul.bend as W64M
import bend-collections-laws-crypto@1.0.0.0/src/math/w64.bend as X

239# the product value as the two limbs of the full product
240def prod_limbs(+a: U32, +b: U32) -> {Nat.add(v(X.lo(X.mul32(a, b))), WD.sc(32n, v(X.hi(X.mul32(a, b))))) == Nat.mul(v(a), v(b)) : Nat}:
241  e = W64M.mul32_value(a, b)
242  +lo = X.lo(X.mul32(a, b))
243  +hi = X.hi(X.mul32(a, b))
244  Equal.trans(Nat, Nat.add(v(lo), WD.sc(32n, v(hi))), Nat.add(v(lo), C.shift(32n, v(hi))), Nat.mul(v(a), v(b)), W64M.cong_r(v(lo), WD.sc(32n, v(hi)), C.shift(32n, v(hi)), Equal.sym(Nat, C.shift(32n, v(hi)), WD.sc(32n, v(hi)), W64M.shift_sc(32n, v(hi)))), e)