Confusions about the technique for verifying implementations of linearizable objects in [Herlihy and Wing, 1990]
Herlihy and Wing write on p. 477: 


  In conclusion, the rep invariant $\mathbf{I}$ must be continually satisfied and the abstraction function continually defined, not only between abstract operations, but also between rep operations implementing abstract operations. The abstraction function maps each rep value to a nonempty set of abstract values:
  $$
\mathbf{A}: \mathrm{REP} \to 2^{\mathrm{ABS}}
$$


I am tempted to read into this that even when an abstract operation is implemented by a sequence of multiple rep operations, the rep invariant has to hold after each of those rep operations, and when it holds, then the rep value must have at least one abstract counterpart.

Many people have addressed some of the challenges posed in this paper. One of the more interesting and recent developments is  R Jagadeesan and J Riely's Quantitative Quiescent Consistency [http://arxiv.org/abs/1402.4043].