Example of quantified notions

This example illustrates the use of `defun-sk` and `defthm` events to reason about quantifiers. See defun-sk. For
a more through, systematic beginner's introduction to quantification in ACL2,
see quantifier-tutorial.

Many users prefer to avoid the use of quantifiers, since ACL2 provides only very limited support for reasoning about quantifiers.

Here is a list of events that proves that if there are arbitrarily
large numbers satisfying the disjunction

; Introduce undefined predicates p and r. (defstub p (x) t) (defstub r (x) t) ; Define the notion that something bigger than x satisfies p. (defun-sk some-bigger-p (x) (exists y (and (< x y) (p y)))) ; Define the notion that something bigger than x satisfies r. (defun-sk some-bigger-r (x) (exists y (and (< x y) (r y)))) ; Define the notion that arbitrarily large x satisfy p. (defun-sk arb-lg-p () (forall x (some-bigger-p x))) ; Define the notion that arbitrarily large x satisfy r. (defun-sk arb-lg-r () (forall x (some-bigger-r x))) ; Define the notion that something bigger than x satisfies p or r. (defun-sk some-bigger-p-or-r (x) (exists y (and (< x y) (or (p y) (r y))))) ; Define the notion that arbitrarily large x satisfy p or r. (defun-sk arb-lg-p-or-r () (forall x (some-bigger-p-or-r x))) ; Prove the theorem promised above. Notice that the functions open ; automatically, but that we have to provide help for some rewrite ; rules because they have free variables in the hypotheses. The ; ``witness functions'' mentioned below were introduced by DEFUN-SK. (thm (implies (arb-lg-p-or-r) (or (arb-lg-p) (arb-lg-r))) :hints (("Goal" :use ((:instance some-bigger-p-suff (x (arb-lg-p-witness)) (y (some-bigger-p-or-r-witness (max (arb-lg-p-witness) (arb-lg-r-witness))))) (:instance some-bigger-r-suff (x (arb-lg-r-witness)) (y (some-bigger-p-or-r-witness (max (arb-lg-p-witness) (arb-lg-r-witness))))) (:instance arb-lg-p-or-r-necc (x (max (arb-lg-p-witness) (arb-lg-r-witness)))))))) ; And finally, here's a cute little example. We have already ; defined above the notion (some-bigger-p x), which says that ; something bigger than x satisfies p. Let us introduce a notion ; that asserts that there exists both y and z bigger than x which ; satisfy p. On first glance this new notion may appear to be ; stronger than the old one, but careful inspection shows that y and ; z do not have to be distinct. In fact ACL2 realizes this, and ; proves the theorem below automatically. (defun-sk two-bigger-p (x) (exists (y z) (and (< x y) (p y) (< x z) (p z)))) (thm (implies (some-bigger-p x) (two-bigger-p x))) ; A technical point: ACL2 fails to prove the theorem above ; automatically if we take its contrapositive, unless we disable ; two-bigger-p as shown below. That is because ACL2 needs to expand ; some-bigger-p before applying the rewrite rule introduced for ; two-bigger-p, which contains free variables. The moral of the ; story is: Don't expect too much automatic support from ACL2 for ; reasoning about quantified notions. (thm (implies (not (two-bigger-p x)) (not (some-bigger-p x))) :hints (("Goal" :in-theory (disable two-bigger-p))))