Definition eqb (a b : byte) : bool
  := let '(a0, (a1, (a2, (a3, (a4, (a5, (a6, a7))))))) := to_bits a in
     let '(b0, (b1, (b2, (b3, (b4, (b5, (b6, b7))))))) := to_bits b in
     (Bool.eqb a0 b0 && Bool.eqb a1 b1 && Bool.eqb a2 b2 && Bool.eqb a3 b3 &&
          Bool.eqb a4 b4 && Bool.eqb a5 b5 && Bool.eqb a6 b6 && Bool.eqb a7 b7)%bool.Infix "=?" := eqb (at level 70) : byte_scope.

Lemma byte_dec_lb x y : x = y -> eqb x y = true.