Computer-assisted proof

A mathematical proof in which computation performs part of the reasoning or verifies cases too numerous for practical hand checking.

Formal verification: The use of formal methods to check whether a system or argument satisfies specified properties. Proof assistants apply formal verification to mathematical derivations.
Four color theorem: The theorem that every planar map can be colored with at most four colors so adjacent regions differ. Its first proof relied on computer-checked configurations and reducibility.
Constructive proof: A proof that explicitly provides an object or method establishing an existence claim. A computer-assisted existence proof may verify cases without yielding a simple construction.
Four Color Problem: The historical problem of deciding whether every planar map can be colored with four colors. Its long history culminated in the first widely debated computer-assisted proof.
Formal proof: A proof written in a precise formal language whose steps follow explicit inference rules. Formalization offers a route from complex machine-assisted arguments to mechanically checked derivations.
Proof assistant: Software that helps construct and mechanically check formal proofs. It can verify each inference in a computer-assisted proof.
Kepler conjecture: The claim that no arrangement of equal spheres packs space more densely than the face-centered cubic packing. Thomas Hales combined geometric arguments with extensive computer calculations.
Probabilistic proof: A proof that uses probability to establish the existence of an object or the truth of a claim. Randomized reasoning differs from deterministic exhaustive verification.
Kenneth Appel: An American mathematician who, with Wolfgang Haken, proved the four color theorem. Appel helped lead the computation that checked thousands of configurations.
Mathematical rigor: The standard of precision and justification required for mathematical arguments. Computer-assisted results test how rigor applies when no person checks every calculation.

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