A statement about an algorithm has the form "there is an algorithm A such that for all parameters there is a constant c such that for all inputs a, A computes b from a in time O(T)". The program and the exponent e come first, so neither depends on the parameters or on the word length. The constant c is O-notation with explicit constants. It bounds both the running time and the additive term of the word length.

The word length bound includes the numbers of the input, since the machine reduces every number modulo 2^w, so a number that does not fit into a word cannot be read. The sum of the numbers dominates each of them and also covers the empty input.

The first clause is soundness at every word length of the hypothesis. An algorithm that calls another one runs it at word lengths fixed by its own memory, where the running time of the callee need not fit into a word. It can trust any output it receives. The second clause claims halting only once the time bound fits into a word, the usual word RAM convention that a word holds the running time. This matters when the output is large. Its size can be polynomial of a degree that grows with parameters fixed after e, and then at some word lengths of the hypothesis the output cannot be written. Since the machine is deterministic, the two clauses together say that the run is correct whenever it halts, and that it halts in time whenever the time fits.

The time bound is a real number, so that bounds like n^(3+ε) can be written directly. The additive c lets T vanish on small inputs, as O-notation does. Several outputs are computed as one pair.