[see also: decrease, diminish, lower, cut down]

We claim that, by setting $w$ to zero on this interval, the value of $F(w)$ is reduced.

Kim announces that (by a tedious proof) the upper bound can be reduced to 10.

The length of $F$ is thus reduced by half.

By induction, we are reduced to proving the following lemma.

The problem now reduces to establishing that ......

We first prove a reduced form of the theorem.

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