def xs

bend-collections-laws-math@1.0.0.0 · proofs/math/random/proof_pcg.bend:42

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/spec/math/random/pcg.bend as SP
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/src/math/random/pcg.bend as P
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64sh.bend as SH
import bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/xor.bend as XO

42# x ^ (x >> k)
43def xs(+a: WU.U64, +k: Nat) -> {SW.value(P.xor64(a, X.shr(a, k))) == SP.xor_bits(64n, SW.value(a), C.high(k, SW.value(a))) : Nat}:
44  Equal.trans(Nat, SW.value(P.xor64(a, X.shr(a, k))), SP.xor_bits(64n, SW.value(a), SW.value(X.shr(a, k))), SP.xor_bits(64n, SW.value(a), C.high(k, SW.value(a))),
45    XO.xor64_value(a, X.shr(a, k)),
46    Equal.cong(Nat, Nat, z => SP.xor_bits(64n, SW.value(a), z), SW.value(X.shr(a, k)), C.high(k, SW.value(a)), SH.shr_value(a, k)))