Major Section: EVENTS
Encapsulate provides a way to execute a sequence of events and then
hide some of the resulting effects. There are two kinds of encapsulations:
``trivial'' and ``non-trivial''. We discuss these briefly before providing
detailed documentation.
A trivial encapsulation is an event of the following form.
(encapsulate () ; nil here indicates "trivial" <event-1> ... <event-k>)We use the term ``sub-events'' to refer to
<event-1> through
<event-k>. Each sub-event <event-i> may be ``local'', that is,
of the form (local <event-i'>); the other sub-events are called
``non-local''. When this encapsulate form is submitted to ACL2, it is
processed in two passes. On the first pass, each sub-event is processed in
sequence; admission of the encapsulate fails if any <event-i> fails
to be admitted. Then a second pass is made after rolling back the logical
world to what it was just before executing the encapsulate form. In
the second pass, only the non-local forms <event-i> are evaluated,
again in order, and proofs are skipped.For example, the following trivial encapsulation exports a single event,
member-equal-reverse. The lemma member-revappend is used (as a
rewrite rule) to prove member-equal-reverse on the first pass, but
since member-revappend is local, it is ignored on the second (final)
pass.
(encapsulate
()
(local
(defthm member-revappend
(iff (member-equal a (revappend x y))
(or (member-equal a x)
(member-equal a y)))
:hints (("Goal" :induct (revappend x y)))))
(defthm member-equal-reverse
(iff (member-equal a (reverse x))
(member-equal a x))))
Of course, one might prefer to prove these events at the top level,
rather than within an encapsulation; but the point here is to illustrate that
you can have local events that do not become part of the logical
world. (Such a capability is also provided at the level of books;
in particular, see include-book.)On the other hand, non-trivial encapsulations provide a way to introduce axioms about new function symbols, without introducing inconsistency and without introducing complete definitions. The following example illustrates how that works.
(encapsulate
; The following list has a single signature, introducing a function foo of
; one argument that returns one value. (The list is non-empty, so we call
; this a "non-trivial" encapsulation.)
( ((foo *) => *) )
; Introduce a ``witness'' (example) for foo, marked as local so that
; it is not exported:
(local (defun foo (x) x))
; Introduce a non-local property to be exported:
(defthm foo-preserves-consp
(implies (consp x)
(consp (foo x))))
)
The form above introduces a new function symbol, foo, with the indicated
property and no definition. In fact, the output from ACL2 concludes as
follows.
The following constraint is associated with the function FOO: (IMPLIES (CONSP X) (CONSP (FOO X)))To understand this example, we consider how non-trivial encapsulations are processed. The same two passes are made as for trivial encapsulations, and the (local) definition of
foo is ignored on the second pass, and
hence does not appear in the resulting ACL2 logical world. But before
the second pass, each signature is stored in the world. Thus, when
the theorem foo-preserves-consp is encountered in the second pass,
foo is a known function symbol with the indicated signature.We turn now to more complete documentation. But discussion of redundancy for
encapsulate events may be found elsewhere; see redundant-encapsulate.
Other Examples:
(encapsulate (((an-element *) => *))
; The list of signatures above could also be written
; ((an-element (lst) t))
(local (defun an-element (lst)
(if (consp lst) (car lst) nil)))
(local (defthm member-equal-car
(implies (and lst (true-listp lst))
(member-equal (car lst) lst))))
(defthm thm1
(implies (null lst) (null (an-element lst))))
(defthm thm2
(implies (and (true-listp lst)
(not (null lst)))
(member-equal (an-element lst) lst))))
(encapsulate
() ; empty signature: no constrained functions indicated
(local (defthm hack
(implies (and (syntaxp (quotep x))
(syntaxp (quotep y)))
(equal (+ x y z)
(+ (+ x y) z)))))
(defthm nthcdr-add1-conditional
(implies (not (zp (1+ n)))
(equal (nthcdr (1+ n) x)
(nthcdr n (cdr x))))))
Some Related Topics
encapsulate events
signature is a well-formed signature, each signature
describes a different function symbol, and each evi is an embedded event
form (See embedded-event-form). Also see signature, in particular for a
discussion of how a signature can assign a guard to a function symbol.
There must be at least one evi. The evi inside local special
forms are called ``local'' events below. Events that are not
local are sometimes said to be ``exported'' by the encapsulation. We
make the further restriction that no defaxiom event may be introduced
in the scope of an encapsulate (not even by encapsulate or
include-book events that are among the evi). Furthermore, no
non-local include-book event is permitted in the scope of any
encapsulate with a non-empty list of signatures.To be well-formed, an encapsulate event must have the properties that
each event in the body (including the local ones) can be successfully
executed in sequence and that in the resulting theory, each function
mentioned among the signatures was introduced via a local event
and has the signature listed. (A utility is provided to assist in
debugging failures of such execution; see redo-flat.) In addition, the body
may contain no ``local incompatibilities'' which, roughly stated, means that
the events that are not local must not syntactically require
symbols defined by local events, except for the functions listed
in the signatures. See local-incompatibility. Finally, no
non-local recursive definition in the body may involve in its suggested
induction scheme any function symbol listed among the signatures.
See subversive-recursions.
Observe that if the signatures list is empty, the resulting ``trivial''
encapsulate may still be useful for deriving theorems to be exported
whose proofs require lemmas you prefer to hide (i.e., made local).
Whether trivial or not (i.e., whether the signature is empty or not),
encapsulate exports the results of evaluating its non-local
events, but its local events are ignored for the resulting
logical world.
The result of a non-trivial encapsulate event is an extension of the
logic in which, roughly speaking, the functions listed in the signatures
are constrained to have the signatures listed and to satisfy the
non-local theorems proved about them. In fact, other functions
introduced in the encapsulate event may be considered to have
``constraints'' as well. (See constraint for details, which are only
relevant to functional instantiation.) Since the constraints were all
theorems in the ``ephemeral'' or ``local'' theory, we are assured that the
extension produced by encapsulate is sound. In essence, the local
definitions of the constrained functions are just ``witness functions'' that
establish the consistency of the constraints. Because those definitions
are local, they are not present in the theory produced by
encapsulation. After a non-trivial encapsulate event is admitted,
theorems about the constrained function symbols may then be proved --
theorems whose proofs necessarily employ only the constraints. Thus,
those theorems may be later functionally instantiated, as with the
:functional-instance lemma instance (see lemma-instance), to derive
analogous theorems about different functions, provided the
constraints (see constraint) can be proved about the new functions.
The default-defun-mode for the first event in an encapsulation is
the default defun-mode ``outside'' the encapsulation. But since
events changing the defun-mode are permitted within the body of an
encapsulate, the default defun-mode may be changed. However,
defun-mode changes occurring within the body of the encapsulate
are not exported. In particular, the acl2-defaults-table after
an encapsulate is always the same as it was before the
encapsulate, even though the encapsulate body might contain
defun-mode changing events, :program and :logic.
See defun-mode. More generally, after execution of an
encapsulate event, the value of acl2-defaults-table is
restored to what it was immediately before that event was executed.
See acl2-defaults-table.
We make some remarks on guards and evaluation. Calls of functions
introduced in the signatures list cannot be evaluated in the ACL2
read-eval-print loop. See defattach for a way to overcome this limitation.
Moreover, any :guard supplied in the signature is automatically
associated in the world with its corresponding function symbol, with no
requirement other than that the guard is a legal term all of whose function
symbols are in :logic mode with their guards verified. In
particular, there need not be any relationship between a guard in a signature
and the guard in a local witness function. Finally, note that for
functions introduced non-locally inside a non-trivial encapsulate
event, guard verification is illegal unless ACL2 determines that the
proof obligations hold outside the encapsulate event as well.
(encapsulate ((f (x) t)) (local (defun f (x) (declare (xargs :guard t)) (consp x))) ;; ERROR! (defun g (x) (declare (xargs :guard (f x))) (car x)))
The order of the events in the vicinity of an encapsulate is
confusing. We discuss it in some detail here because when logical names are
being used with theory functions to compute sets of rules, it is sometimes
important to know the order in which events were executed.
(See logical-name and see theory-functions.) What, for example, is the set
of function names extant in the middle of an encapsulation?
If the most recent event is previous and then you execute an
encapsulate constraining an-element with two non-local
events in its body, thm1 and thm2, then the order of the
events after the encapsulation is (reading chronologically forward):
previous, thm1, thm2, an-element (the encapsulate
itself). Actually, between previous and thm1 certain extensions were
made to the world by the superior encapsulate, to permit
an-element to be used as a function symbol in thm1.
Remark for ACL2(r) (see real). For ACL2(r), encapsulate can be used
to introduce classical and non-classical functions, as determined by the
signatures; see signature. Those marked as classical (respectively
non-classical) must have classical (respectively, non-classical) local
witness functions. A related requirement applies to functional
instantiation; see lemma-instance.