# Communications of the ACM

Review Articles

## Mechanical Mathematicians

In the 1960s, researchers dreamed of automatic theorem provers powerful enough to develop lengthy proofs of conjectures that no human was able to prove before. Some even thought these "mechanical mathematicians" would eventually replace flesh-and-blood mathematicians. However, for many decades, automatic provers remained too weak to help, let alone replace, mathematicians. Instead, mathematicians turned to computer algebra systems, which help not so much with proofs as with computations, and automatic provers turned to other application areas. These areas include hardware and software verification, either directly or via an interactive verification platform (for example, Atelier B, Dafny, F*, Frama-C, SPARK 2014, Spec#, VCC, Why3). In addition, automatic provers are used as back ends to general-purpose proof assistants (for example, ACL2, Coq, HOL, Isabelle, Lean, Mizar, PVS). As mathematicians are slowly embracing proof assistants, automatic provers are finally becoming useful to them, for discharging straight-forward but tedious proof tasks.

### Key Insights

As a simple example from elementary mathematics, consider the formula

The logical symbols ∧ and ⇒ mean "and" and "implies," gcd is the greatest common divisor, and | is the "divides" relation between two numbers (for example, d | ab means ab is divisible by d). Let us check a special case: If a = 12, b = 35, and d = 5, we have a' = 1, b' = 5, and indeed a'b' = 5 divides d = 5 and vice versa. Mathematicians can easily see that the formula holds in general, but to write a formal proof with an acceptable level of detail for a proof assistant can easily take 15 or 30 minutes. In contrast, automatic theorem provers can prove such formulas within seconds.

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