def add

bend-collections-laws-math@1.0.0.0 · proofs/math/u64/u64.bend:222

import Base
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as U
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/modular_addition.bend as MA
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/word_addition.bend as WA
import bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend as WD

222# THEOREM: add is 64-bit wrapping addition.
223def add(+a: U.U64, +b: U.U64) -> {bits(U.add(a, b)) == Word.add(64n, bits(a), bits(b)) : Word(64n)}:
224  Equal.trans(Word(64n), bits(U.add(a, b)), S.from_nat(64n, Nat.add(val(a), val(b))), Word.add(64n, bits(a), bits(b)),
225    Equal.trans(Word(64n), bits(U.add(a, b)), S.from_nat(64n, Nat.add(val(U.add(a, b)), WD.sc(64n, add_over(a, b)))), S.from_nat(64n, Nat.add(val(a), val(b))),
226      Equal.sym(Word(64n), S.from_nat(64n, Nat.add(val(U.add(a, b)), WD.sc(64n, add_over(a, b)))), bits(U.add(a, b)), MA.reconstruct(64n, bits(U.add(a, b)), add_over(a, b))),
227      Equal.cong(Nat, Word(64n), z => S.from_nat(64n, z), Nat.add(val(U.add(a, b)), WD.sc(64n, add_over(a, b))), Nat.add(val(a), val(b)), add_value(a, b))),
228    Equal.sym(Word(64n), Word.add(64n, bits(a), bits(b)), S.from_nat(64n, Nat.add(val(a), val(b))), WA.refines(64n, bits(a), bits(b))))