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An equation expressing the optimum or count for a dynamic programming state in terms of smaller or otherwise previously computable states. A valid dependency order, base cases, and a proof that every candidate solution is represented are required. The minimum coin change recurrence follows by classifying a nonempty solution by its last coin.

After fixing an initial decision in an optimal solution, the remaining decisions must solve the resulting residual problem optimally, with the relevant state and constraints retained. Otherwise substituting a better continuation would improve the original solution. This replacement argument justifies dynamic programming recurrences when the chosen dynamic programming state contains all information needed by future decisions.

For a directed graph with adjacency matrix $A$, let $D[S,v]$ record whether a path visits precisely the vertices of $S$ once and ends at $v$. The singleton states are true. A larger state is true exactly when some predecessor $u\in S\setminus \{v\}$ has a true state for $S\setminus \{v\}$ and an edge to $v$. Removing or adjoining the last edge proves this recurrence by mathematical induction on $|S|$. There are $n{2}^{n-1}$ states, each requiring at most $n-1$ predecessor checks, giving $O(n^2{2}^n)$ time and $O(n{2}^n)$ storage. A true state for $S=V$ certifies a Hamiltonian path.

For nonnegative maximum jump lengths in an array, let $D[0]=0$ and let unreachable positions have value $\infty$. Classifying a route to $j$ by its penultimate position proves this dynamic programming recurrence. Increasing $j$ is a valid dependency order because every jump advances. Scanning all previous positions uses $\Theta (n^2)$ time complexity and $\Theta (n)$ space complexity. Store a minimizing predecessor to reconstruct a shortest route.

Double dispatch selects behavior using the runtime types of two participating software objects, often by arranging two ordinary dynamic dispatch steps. The visitor pattern supplies a useful example.

Suppose Shape has an overridable accept(Renderer) operation, and Renderer has overridable overloads visit(Circle) and visit(Rectangle). A Circle implementation of accept calls renderer.visit(this). Calling shape.accept(renderer) first selects Circle.accept using the shape's runtime type. Inside that method, this has the static type Circle because the method is defined in the Circle software class. Method overloading therefore chooses the visit(Circle) signature. The renderer's runtime type then selects the implementation of that signature.

The result depends on both the concrete shape and concrete renderer. Ordinary method overloading alone would not inspect an argument's runtime type to choose the shape overload. The static type established in the first dispatched method is the bridge to the second dispatch.^[ref]

Retain the first occurrence of each distinct value in a list. The right fold foldr (\x ys -> x : filter (/= x) ys) [] preserves this order under lazy evaluation. With terminating equality tests, an infinite input containing infinitely many distinct values produces every finite output prefix. If only finitely many distinct values occur, those values appear but the remaining infinite input cannot generally be certified empty: the result has an undefined tail rather than a terminating empty list.

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