in practice, theorem provers can solve many hard problems, even in models that are not fully described by any first order theory (such as the integers). Related problems A simpler, but related, problem is proof verification, where an existing proof for a theorem is certified valid. For this, it is generally required that each individual proof step can be verified by a primitive recursive function or program, and hence the problem is always decidable. Since the proofs generated by automated theorem provers are typically very large, the problem of proof compression is crucial and various techniques aiming at making the prover's output smaller, and consequently more easily understandable and checkable, have been developed. Proof assistants require a human user to give hints to the system. Depending on the degree of automation, the prover can essentially be reduced to a proof checker, with the user providing the proof in a formal way, or significant proof tasks can be performed automatically. Interactive provers are used for a variety of tasks, but even fully automatic systems have proved a number of interesting and hard theorems, including at least one that has eluded human mathematicians for a long time, namely the Robbins conjecture. However, these successes are sporadic, and work on hard problems usually requires a proficient user. Another distinction is sometimes drawn between theorem proving and other techniques, where a process is considered to be theorem proving if it consists of a traditional proof, starting with axioms and producing new inference steps using rules of inference. Other techniques would include model checking, which, in the simplest case, involves brute-force enumeration of many possible states (although the actual implementation of model checkers requires much cleverness, and does not simply reduce to brute force). There are hybrid theorem proving systems which use model checking as an inference rule. There are also programs which were written to prove a particular theorem, with a (usually informal) proof that if the program finishes with a certain result, then the theorem is true. A good example of this was the machine-aided proof of the four color theorem, which was very controversial as the first claimed mathematical proof which was essentially impossible to verify by humans due to the enormous size of the program's calculation (such proofs are called non-surveyable proofs). Another example of a program-assisted proof is the one that shows that the game of Connect Four can always