QUANTIFIERS

```Major Section:  DEFUN-SK
```

ACL2 supports first-order quantifiers `exists` and `forall` by way of the `defun-sk` event. However, proof support for quantification is quite limited. Therefore, you may prefer using recursion in place of `defun-sk` when possible (following common ACL2 practice).

Some Related Topics

• QUANTIFIERS-USING-RECURSION -- recursion for implementing quantification

For example, the notion ``every member of `x` has property `p`'' can be defined either with recursion or explicit quantification, but proofs may be simpler when recursion is used. We illustrate this point with two proofs of the same informal claim, one of which uses recursion which the other uses explicit quantification. Notice that with recursion, the proof goes through fully automatically; but this is far from true with explicit quantification (especially notable is the ugly hint).

The informal claim for our examples is: If every member `a` of each of two lists satisfies the predicate `(p a)`, then this holds of their `append`; and, conversely.

See quantifiers-using-recursion for a solution to this example using recursion.

See quantifiers-using-defun-sk for a solution to this example using `defun-sk`. Also See quantifiers-using-defun-sk-extended for an elaboration on that solution.

But perhaps first, see defun-sk for an ACL2 utility to introduce first-order quantification in a direct way. Examples of the use of `defun-sk` are also available: see defun-sk-example and see Tutorial4-Defun-Sk-Example for basic examples, and see quantifier-tutorial for a more complete, careful beginner's introduction that takes you through typical kinds of quantifier-based reasoning in ACL2.