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An abstraction function maps a valid concrete representation to its abstract value. Write $R_{\mathrm{valid}}=\{r:I(r)\}$ and $\alpha :R_{\mathrm{valid}}\to Q$. Different storage states can denote the same abstract value. Correct implementation requires both preservation of the representation invariant and agreement with the specified operation after applying the abstraction function; preserving a length bound alone would not prove first-in, first-out behavior.

The representation invariant answers "is this storage state allowed?" The abstraction function answers "which value does it represent?" Thus $I(r)$ is a truth value, whereas $\alpha (r)$ is a value of the abstract data type. Treating $\alpha$ as a partial function on all storage states, or as a function defined on $R_{\mathrm{valid}}$ only, expresses the same restriction. The abstraction function may forget storage details, and need not be executable code: its purpose is to connect an implementation to its formal specification.^[ref]

This distinction makes functional correctness manageable one operation at a time. The unreduced rational representation below separates valid storage from mathematical meaning; correct operations through an abstraction function then checks whether an implementation performs the promised addition. For an example with hidden mutable storage and failure cases, the bounded queue with encapsulation supplies a different abstraction function, reading an occupied prefix of an array as a sequence.

For a deterministic binary operation on an abstract data type, let $f$ be its specified operation on abstract values and $F$ the implementation on concrete states. For every permitted valid input pair $(r,s)$, require bothThe first condition preserves the representation invariant. The second makes a commutative diagram: either execute $F$ and then apply the abstraction function, or apply the abstraction function to both inputs and then execute the abstract operation $f$; both paths reach the same value. These conditions state functional correctness of a completed operation; termination is an additional obligation for algorithm correctness.^[ref]

For addition in the unreduced rational representation, chooseFor example, $F((1,2),(1,3))=(5,6)$ represents $5/6$, precisely the sum of $1/2$ and $1/3$. The short correctness argument uses nonzero denominators: $b\ne 0$ and $d\ne 0$ imply $bd\ne 0$, preserving the representation invariant, and

Now consider an incorrect implementation that uses addition on the numerators and multiplication on the denominators:It preserves the representation invariant for every valid input pair because its denominator is still $bd\ne 0$. Yet $B((1,2),(1,3))=(2,6)$ represents $1/3$, not the specified sum $5/6$. Even perfect checks of the representation invariant before and after this call would miss the error: the result is a legal representation of the wrong rational number.

The incorrect operation also exposes an accidental dependence on representation. Replace $(1,2)$ by the equivalent state $(2,4)$ while keeping $(1,3)$. The correct operation gives $(10,12)$, still representing $5/6$; the incorrect operation gives $(3,12)$, representing $1/4$ instead of its previous $1/3$. A correct addition must yield the same abstract result for equivalent inputs, although its concrete outputs may differ. This is why checks against the formal specification should compare abstract values, including unreduced representations and representations with a negative denominator, rather than demand identical stored pairs.

The calculation assumes exact integers. An implementation with bounded storage must additionally specify how it handles products and sums outside its supported range; the mathematical argument alone does not establish its functional correctness. The bounded queue with encapsulation illustrates the corresponding need to account for returned status values and failed operations, alongside the abstract state transition.