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. Just follow the Walking Tour after exploring the
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.