def andw

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/u32and.bend:26

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/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend as S
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32.bend as U
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/width.bend as WW

26# the low k bits of a word
27def andw(+n: Nat, +k: Nat, +w: Word(n)) -> {S.unsigned(n, Word.and(n, w, WD.mask(n, k))) == C.low(k, S.unsigned(n, w)) : Nat}:
28  +a = S.unsigned(n, Word.and(n, w, WD.mask(n, k)))
29  +hp = WD.hi_part(n, k, w)
30  +e1 = WD.mask_split(n, k, w)
31  +e2 = Equal.trans(Nat, S.unsigned(n, w), Nat.add(a, S.scale_binary(k, hp)), Nat.add(a, C.shift(k, hp)), e1, Equal.cong(Nat, Nat, z => Nat.add(a, z), S.scale_binary(k, hp), C.shift(k, hp), sb(k, hp)))
32  +hl = WD.mask_lt(n, k, 1n, {==}, w)
33  +hl2 = L.subst(Nat, z => {Nat.is_lt(a, z) == True{} : Bool}, S.scale_binary(k, 1n), C.pow2(k), Equal.sym(Nat, C.pow2(k), S.scale_binary(k, 1n), U.pow2_scale(k)), hl)
34  Equal.sym(Nat, C.low(k, S.unsigned(n, w)), a, Equal.trans(Nat, C.low(k, S.unsigned(n, w)), C.low(k, Nat.add(a, C.shift(k, hp))), a, Equal.cong(Nat, Nat, z => C.low(k, z), S.unsigned(n, w), Nat.add(a, C.shift(k, hp)), e2), WW.low_uniq(k, a, hp, hl2)))