Major Section: RULE-CLASSES
See rule-classes for a general discussion of rule classes and
how they are used to build rules from formulas. An example
corollary formula from which a
:equivalence rule might be built is
as follows. (We assume that
r-equal has been defined.)
Example: (and (booleanp (r-equal x y)) (r-equal x x) (implies (r-equal x y) (r-equal y x)) (implies (and (r-equal x y) (r-equal y z)) (r-equal x z))).Also see defequiv.
General Form: (and (booleanp (equiv x y)) (equiv x x) (implies (equiv x y) (equiv y x)) (implies (and (equiv x y) (equiv y z)) (equiv x z)))except that the order of the conjuncts and terms and the choice of variable symbols is unimportant. The effect of such a rule is to identify
equivas an equivalence relation. Note that only Boolean 2-place function symbols can be treated as equivalence relations. See congruence and see refinement for closely related concepts.
The macro form
(defequiv equiv) is an abbreviation for a
:equivalence that establishes that
equiv is an
equivalence relation. It generates the formula shown above.
equiv is marked as an equivalence relation, its reflexivity,
symmetry, and transitivity are built into the system in a deeper way
rewrite rules. More importantly, after
equiv has been
shown to be an equivalence relation, lemmas about
(implies hyps (equiv lhs rhs)),when stored as
rewriterules, cause the system to rewrite certain occurrences of (instances of)
lhsto (instances of)
rhs. Roughly speaking, an occurrence of
kthargument of some
(fn ... lhs' ...), can be rewritten to produce
(fn ... rhs' ...), provided the system ``knows'' that the value of
fnis unaffected by
equiv-substitution in the
kthargument. Such knowledge is communicated to the system via ``congruence lemmas.''
For example, suppose that
r-equal is known to be an equivalence
(implies (r-equal s1 s2) (equal (fn s1 n) (fn s2 n)))informs the rewriter that, while rewriting the first argument of
fn-expressions, it is permitted to use
r-equalrewrite-rules. See congruence for details about
congruencelemmas. Interestingly, congruence lemmas are automatically created when an equivalence relation is stored, saying that either of the equivalence relation's arguments may be replaced by an equivalent argument. That is, if the equivalence relation is
fn, we store congruence rules that state the following fact:
(implies (and (fn x1 y1) (fn x2 y2)) (iff (fn x1 x2) (fn y1 y2)))Another aspect of equivalence relations is that of ``refinement.'' We say
(equiv1 x y)implies
(equiv2 x y).
refinementrules permit you to establish such connections between your equivalence relations. The value of refinements is that if the system is trying to rewrite something while maintaining
equiv2it is permitted to use as a
rewriterule any refinement of
equiv2. Thus, if
equiv1is a refinement of
equiv2and there are
equiv1rewrite-rules available, they can be brought to bear while maintaining
equiv2. See refinement.
The system initially has knowledge of two equivalence relations,
equality, denoted by the symbol
equal, and propositional
equivalence, denoted by
Equal is known to be a refinement of
all equivalence relations and to preserve equality across all
arguments of all functions.
Typically there are five steps involved in introducing and using a new equivalence relation, equiv.
(2) prove the
(3) prove the
congruencelemmas that show where
equivcan be used to maintain known relations,
(4) prove the
refinementlemmas that relate
equivto known relations other than equal, and
(5) develop the theory of conditional
rewriterules that drive equiv rewriting.
More will be written about this as we develop the techniques. For now, here is an example that shows how to make use of equivalence relations in rewriting.
Among the theorems proved below is
(defthm insert-sort-is-id (perm (insert-sort x) x))Here
permis defined as usual with
deleteand is proved to be an equivalence relation and to be a congruence relation for
Then we prove the lemma
(defthm insert-is-cons (perm (insert a x) (cons a x)))which you must think of as you would
(insert a x) = (cons a x).
(perm (insert-sort x) x). The base case is trivial. The
induction step is
(consp x) & (perm (insert-sort (cdr x)) (cdr x)) -> (perm (insert-sort x) x).Opening
insert-sortmakes the conclusion be
(perm (insert (car x) (insert-sort (cdr x))) x).Then apply the induction hypothesis (rewriting
(insert-sort (cdr x))to
(cdr x)), to make the conclusion be
(perm (insert (car x) (cdr x)) x)Then apply
(perm (cons (car x) (cdr x)) x). But we know that
(cons (car x) (cdr x))is
x, so we get
(perm x x)which is trivial, since
permis an equivalence relation.
Here are the events.
(encapsulate (((lt * *) => *)) (local (defun lt (x y) (declare (ignore x y)) nil)) (defthm lt-non-symmetric (implies (lt x y) (not (lt y x))))) (defun insert (x lst) (cond ((atom lst) (list x)) ((lt x (car lst)) (cons x lst)) (t (cons (car lst) (insert x (cdr lst)))))) (defun insert-sort (lst) (cond ((atom lst) nil) (t (insert (car lst) (insert-sort (cdr lst)))))) (defun del (x lst) (cond ((atom lst) nil) ((equal x (car lst)) (cdr lst)) (t (cons (car lst) (del x (cdr lst)))))) (defun mem (x lst) (cond ((atom lst) nil) ((equal x (car lst)) t) (t (mem x (cdr lst))))) (defun perm (lst1 lst2) (cond ((atom lst1) (atom lst2)) ((mem (car lst1) lst2) (perm (cdr lst1) (del (car lst1) lst2))) (t nil))) (defthm perm-reflexive (perm x x)) (defthm perm-cons (implies (mem a x) (equal (perm x (cons a y)) (perm (del a x) y))) :hints (("Goal" :induct (perm x y)))) (defthm perm-symmetric (implies (perm x y) (perm y x))) (defthm mem-del (implies (mem a (del b x)) (mem a x))) (defthm perm-mem (implies (and (perm x y) (mem a x)) (mem a y))) (defthm mem-del2 (implies (and (mem a x) (not (equal a b))) (mem a (del b x)))) (defthm comm-del (equal (del a (del b x)) (del b (del a x)))) (defthm perm-del (implies (perm x y) (perm (del a x) (del a y)))) (defthm perm-transitive (implies (and (perm x y) (perm y z)) (perm x z))) (defequiv perm) (in-theory (disable perm perm-reflexive perm-symmetric perm-transitive)) (defcong perm perm (cons x y) 2) (defcong perm iff (mem x y) 2) (defthm atom-perm (implies (not (consp x)) (perm x nil)) :rule-classes :forward-chaining :hints (("Goal" :in-theory (enable perm)))) (defthm insert-is-cons (perm (insert a x) (cons a x))) (defthm insert-sort-is-id (perm (insert-sort x) x)) (defun app (x y) (if (consp x) (cons (car x) (app (cdr x) y)) y)) (defun rev (x) (if (consp x) (app (rev (cdr x)) (list (car x))) nil)) (defcong perm perm (app x y) 2) (defthm app-cons (perm (app a (cons b c)) (cons b (app a c)))) (defthm app-commutes (perm (app a b) (app b a))) (defcong perm perm (app x y) 1 :hints (("Goal" :induct (app y x)))) (defthm rev-is-id (perm (rev x) x)) (defun == (x y) (if (consp x) (if (consp y) (and (equal (car x) (car y)) (== (cdr x) (cdr y))) nil) (not (consp y)))) (defthm ==-reflexive (== x x)) (defthm ==-symmetric (implies (== x y) (== y x))) (defequiv ==) (in-theory (disable ==-symmetric ==-reflexive)) (defcong == == (cons x y) 2) (defcong == iff (consp x) 1) (defcong == == (app x y) 2) (defcong == == (app x y) 1) (defthm rev-rev (== (rev (rev x)) x))