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Automated deduction uses an algorithm to construct or check a mathematical proof from explicit assumptions and inference rules. The representation might use propositional logic, first-order logic or a specialized theory such as linear arithmetic. Soundness means the rules preserve truth; finding a proof additionally requires a search strategy.

For a small resolution example, the clauses $P\vee Q$, $\neg P$ and $\neg Q$ yield $Q$ and then the empty clause. This is a contradiction, so the original clauses cannot all hold. To prove a conclusion from assumptions, a refutation method adds its negation and searches for such a contradiction. In first-order logic, unification determines which terms can instantiate matching predicates. A procedure that eventually finds every valid proof need not terminate on every invalid claim: this distinction explains why proof search and proof checking have different computational costs.

Linear arithmetic is the fragment of arithmetic built from equalities and inequalities between linear expressions $a_1{x}_1+\cdots +a_n{x}_n+b$, combined by logical operations and, in some versions, quantifiers. Coefficients are fixed numbers; multiplying two unknowns, as in $xy$, leaves this fragment. One must specify whether the unknowns range over the integers, rational numbers or real numbers.^[ref]

For example, $2x+3y\le 7$ and $x-y=1$ are linear constraints. Substitution gives $5y+2\le 7$, hence $y\le 1$. But $2x=1$ has a real number solution and no integer solution, so a procedure for one number system cannot simply be reused for the other. This distinction makes linear arithmetic useful in automated reasoning about counts, schedules and bounds.