Xor.xor_word: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/xor.bend as Xor
Lines 51-62 of bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/xor.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/xor.bend&start_line=51&end_line=62

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

# the value of a word xor is xor_bits of the values
def 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}:
  match n x y:
    case 0n WNil{} WNil{}:
      {==}
    case 1n+p WCon{+a, +xt} WCon{+b, +yt}:
      +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))))
      +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,
        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)),
        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)))
      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)))),
        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))))

Uses (19):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.not
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.xor
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_eq
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.Con
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.Nil
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.xor
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=bit_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend&name=uw
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/xor.bend&name=step_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/xor.bend&name=xbit
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/pcg.bend&name=b2n
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/pcg.bend&name=xor_bits