Here is the theorem prover's output when it processes the defthm command
for the associativity of
app. We have highlighted text for which we
offer some explanation, and broken the presentation into several pages. (The
most recent version of ACL2 may print slightly different output but the
basics are the same.) Just follow the Walking Tour after exploring the
However, before exploring this output you should understand that ACL2 users rarely read successful proofs! Instead, they look at certain subgoals printed in failed proofs, figure whether and how those subgoals can be proved, and give ACL2 directions for proving them, usually by simply proving other lemmas. Furthermore, to be a good user of ACL2 you do not have to understand how the theorem prover works. You just have to understand how to interact with it. We explain this in great detail later. But basically all new users are curious to know how ACL2 works and this little tour attempts to give some answers, just to satisfy your curiosity.
ACL2!>(defthm associativity-of-app (equal (app (app a b) c) (app a (app b c))))
Name the formula above *1.
Perhaps we can prove *1 by induction. Three induction schemes are suggested by this conjecture. Subsumption reduces that number to two. However, one of these is flawed and so we are left with one viable candidate.
We will induct according to a scheme suggested by (APP A B). If we let (:P A B C) denote *1 above then the induction scheme we'll use is (AND (IMPLIES (AND (NOT (ENDP A)) (:P (CDR A) B C)) (:P A B C)) (IMPLIES (ENDP A) (:P A B C))). This induction is justified by the same argument used to admit APP, namely, the measure (ACL2-COUNT A) is decreasing according to the relation O< (which is known to be well-founded on the domain recognized by O-P). When applied to the goal at hand the above induction scheme produces the following two nontautological subgoals.