A specification $s$ that is field by field a special case of $t$ embeds its instances into those of $t$ (the lift): same kernel instance, data read with the more general values, and schedules unchanged. A reading lifts along $s \le t$ if every word of an instance of $s$ is also a word of its lift, and every word that writes a schedule of the lift writes the same schedule of the instance.

Theorem. If the reading lifts along $s \le t$, every algorithm for $t$ is one for $s$ — the same program: deciding within $O(T)$ becomes deciding within $O(T \circ \mathrm{lift})$, and likewise for approximations within a running time, randomized ones and approximation schemes, with the ratio read on the lift. The same holds for approximations and schemes of the kept weight, if the lift keeps the total weight, and for deciding with wide words, also on the instances satisfying a predicate: on those of $s$ whose lift satisfies it.

Theorem. The standard reading of the operations family lifts along every $s \le t$ with the same value of the encoding field, and the standard reading of the batching family lifts along every $s \le t$.

The second theorem holds because the standard words write every field by its view, and reading data with a more general value does not change its view.

A reading determines the answer for $t$ if two instances of $t$ with thresholds written by one word have the same threshold and the same answer.

Theorem. The standard reading of the operations family determines the answer for every specification with the standard encoding, and the standard reading of the batching family for every specification: the words parse uniquely, give the same kernel instance and the same views, and constraints and costs read only the views.

Theorem. Let the reading lift along $s \le t$ and determine the answer for $t$. Then a word of an instance of $s$ has the same answer for $s$ and for $t$, and:

1. hardness goes up, by the identity on words: NP-hardness and strong NP-hardness of $s$ give those of $t$; W[t]-hardness for $\kappa_s$ gives W[t]-hardness for $\kappa_t$ if, on every word, $\kappa_t$ of an instance of $t$ is at most a computable function of $\kappa_s$ of an instance of $s$, and para-NP-hardness likewise for any function;
2. the coarse classes go down: in P, pseudo-polynomial, in P for every fixed value (with $\kappa_t$ bounded by a function of $\kappa_s$ on every word), and FPT (with a computable bound, for a parameter of $s$ computable in polynomial time).