def xor_word

bend-collections-laws-math@1.0.0.0 · proofs/math/random/pcg/xor.bend:51

import Base
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/proofs/lib/lemmas/spec/numeric.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend as WD

51# the value of a word xor is xor_bits of the values
52def xor_word(+n: Nat, +x: Word(n), +y: Word(n)) -> {WD.uw(n, Word.xor(n, x, y)) == SP.xor_bits(n, WD.uw(n, x), WD.uw(n, y)) : Nat}:
53  match n x y:
54    case 0n WNil{} WNil{}:
55      {==}
56    case 1n+p WCon{+a, +xt} WCon{+b, +yt}:
57      +M = Nat.add(SP.b2n(Bool.not(Nat.is_eq(S.bit_value(a), S.bit_value(b)))), Nat.double(SP.xor_bits(p, WD.uw(p, xt), WD.uw(p, yt))))
58      +e1 = Equal.trans(Nat, Nat.add(S.bit_value(Bool.xor(a, b)), Nat.double(WD.uw(p, Word.xor(p, xt, yt)))), Nat.add(SP.b2n(Bool.not(Nat.is_eq(S.bit_value(a), S.bit_value(b)))), Nat.double(WD.uw(p, Word.xor(p, xt, yt)))), M,
59        Equal.cong(Nat, Nat, z => Nat.add(z, Nat.double(WD.uw(p, Word.xor(p, xt, yt)))), S.bit_value(Bool.xor(a, b)), SP.b2n(Bool.not(Nat.is_eq(S.bit_value(a), S.bit_value(b)))), xbit(a, b)),
60        Equal.cong(Nat, Nat, z => Nat.add(SP.b2n(Bool.not(Nat.is_eq(S.bit_value(a), S.bit_value(b)))), Nat.double(z)), WD.uw(p, Word.xor(p, xt, yt)), SP.xor_bits(p, WD.uw(p, xt), WD.uw(p, yt)), xor_word(p, xt, yt)))
61      Equal.trans(Nat, Nat.add(S.bit_value(Bool.xor(a, b)), Nat.double(WD.uw(p, Word.xor(p, xt, yt)))), M, SP.xor_bits(1n+p, Nat.add(S.bit_value(a), Nat.double(WD.uw(p, xt))), Nat.add(S.bit_value(b), Nat.double(WD.uw(p, yt)))),
62        e1, Equal.sym(Nat, SP.xor_bits(1n+p, Nat.add(S.bit_value(a), Nat.double(WD.uw(p, xt))), Nat.add(S.bit_value(b), Nat.double(WD.uw(p, yt)))), M, step_eq(p, a, WD.uw(p, xt), b, WD.uw(p, yt))))