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The Proof of the Associativity of App
Here is the theorem prover's output when it processes the defthm
command for the associativity of
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. The first command below,
ACL2!>:set-gag-mode nil
<state>
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.