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 `:definition`

rule might be built is:

Example: (implies (true-listp x) (equal (len x) (if (null x) 0 (if (null (cdr x)) 1 (+ 2 (len (cddr x))))))) General Form: (implies hyp (equiv (fn a1 ... an) body))where

`equiv`

is an equivalence relation and `fn`

is a function symbol
other than `if`

, `hide`

, `force`

or `case-split`

. Such rules
allow ``alternative'' definitions of `fn`

to be proved as theorems but used
as definitions. These rules are not true ``definitions'' in the sense that
they (a) cannot introduce new function symbols and (b) do not have to be
terminating recursion schemes. They are just conditional rewrite rules that
are controlled the same way we control recursive definitions. We call these
``definition rules'' or ``generalized definitions''.Consider the general form above. Generalized definitions are stored among
the `:`

`rewrite`

rules for the function ``defined,'' `fn`

above, but
the procedure for applying them is a little different. During rewriting,
instances of `(fn a1 ... an)`

are replaced by corresponding instances of
`body`

provided the `hyp`

s can be established as for a `:`

`rewrite`

rule and the result of rewriting `body`

satisfies the criteria for function
expansion. There are two primary criteria, either of which permits
expansion. The first is that the ``recursive'' calls of `fn`

in the
rewritten body have arguments that already occur in the goal conjecture. The
second is that the ``controlling'' arguments to `fn`

are simpler in the
rewritten body.

The notions of ``recursive call'' and ``controllers'' are complicated by the provisions for mutually recursive definitions. Consider a ``clique'' of mutually recursive definitions. Then a ``recursive call'' is a call to any function defined in the clique and an argument is a ``controller'' if it is involved in the measure that decreases in all recursive calls. These notions are precisely defined by the definitional principle and do not necessarily make sense in the context of generalized definitional equations as implemented here.

But because the heuristics governing the use of generalized definitions
require these notions, it is generally up to the user to specify which calls
in body are to be considered recursive and what the controlling arguments
are. This information is specified in the `:clique`

and
`:controller-alist`

fields of the `:definition`

rule class.

The `:clique`

field is the list of function symbols to be considered
recursive calls of `fn`

. In the case of a non-recursive definition, the
`:clique`

field is empty; in a singly recursive definition, it should
consist of the singleton list containing `fn`

; otherwise it should be a
list of all of the functions in the mutually recursive clique with this
definition of `fn`

.

If the `:clique`

field is not provided it defaults to `nil`

if `fn`

does not occur as a function symbol in `body`

and it defaults to the
singleton list containing `fn`

otherwise. Thus, `:clique`

must be
supplied by the user only when the generalized definition rule is to be
treated as one of several in a mutually recursive clique.

The `:controller-alist`

is an alist that maps each function symbol in the
`:clique`

to a mask specifying which arguments are considered controllers.
The mask for a given member of the clique, `fn`

, must be a list of `t`

's
and `nil`

's of length equal to the arity of `fn`

. A `t`

should be in
each argument position that is considered a ``controller'' of the recursion.
For a function admitted under the principle of definition, an argument
controls the recursion if it is one of the arguments measured in the
termination argument for the function. But in generalized definition rules,
the user is free to designate any subset of the arguments as controllers.
Failure to choose wisely may result in the ``infinite expansion'' of
definitional rules but cannot render ACL2 unsound since the rule being
misused is a theorem.

If the `:controller-alist`

is omitted it can sometimes be defaulted
automatically by the system. If the `:clique`

is `nil`

, the
`:controller-alist`

defaults to `nil`

. If the `:clique`

is a singleton
containing `fn`

, the `:controller-alist`

defaults to the controller alist
computed by `(defun fn args body)`

. (The user can obtain some control over
this analysis by setting the default ruler-extenders;
see ruler-extenders.) If the `:clique`

contains more than one function,
the user must supply the `:controller-alist`

specifying the controllers for
each function in the clique. This is necessary since the system cannot
determine and thus cannot analyze the other definitional equations to be
included in the clique.

For example, suppose `fn1`

and `fn2`

have been defined one way and it is
desired to make ``alternative'' mutually recursive definitions available to
the rewriter. Then one would prove two theorems and store each as a
`:definition`

rule. These two theorems would exhibit equations
``defining'' `fn1`

and `fn2`

in terms of each other. No provision is
here made for exhibiting these two equations as a system of equations. One
is proved and then the other. It just so happens that the user intends them
to be treated as mutually recursive definitions. To achieve this end, both
`:definition`

rules should specify the `:clique`

`(fn1 fn2)`

and should
specify a suitable `:controller-alist`

. If, for example, the new
definition of `fn1`

is controlled by its first argument and the new
definition of `fn2`

is controlled by its second and third (and they each
take three arguments) then a suitable `:controller-alist`

would be
`((fn1 t nil nil) (fn2 nil t t))`

. The order of the pairs in the alist is
unimportant, but there must be a pair for each function in the clique.

Inappropriate heuristic advice via `:clique`

and `:controller-alist`

can
cause ``infinite expansion'' of generalized definitions, but cannot render
ACL2 unsound.

Note that the actual definition of `fn1`

has the runic name
`(:definition fn1)`

. The runic name of the alternative definition is
`(:definition lemma)`

, where `lemma`

is the name given to the event that
created the generalized `:definition`

rule. This allows theories to switch
between various ``definitions'' of the functions.

By default, a `:definition`

rule establishes the so-called ``body'' of a
function. The body is used by `:expand`

hints, and it is also used
heuristically by the theorem prover's preprocessing (the initial
simplification using ``simple'' rules that is controlled by the
`preprocess`

symbol in `:do-not`

hints), induction analysis, and the
determination for when to warn about non-recursive functions in rules. The
body is also used by some heuristics involving whether a function is
recursively defined, and by the `expand`

, `x`

, and `x-dumb`

commands of
the proof-checker.

See rule-classes for a discussion of the optional field `:install-body`

of
`:definition`

rules, which controls whether a `:definition`

rule is used
as described in the paragraph above. Note that even if `:install-body nil`

is supplied, the rewriter will still rewrite with the `:definition`

rule;
in that case, ACL2 just won't install a new body for the top function symbol
of the left-hand side of the rule, which for example affects the application
of `:expand`

hints as described in the preceding paragraph. Also
see set-body and see show-bodies for how to change the body of a function
symbol.

Note only that if you prove a definition rule for function `foo`

, say,
`foo-new-def`

, you will need to refer to that definition as `foo-new-def`

or as `(:DEFINITION foo-new-def)`

. That is because a `:definition`

rule
does not change the meaning of the symbol `foo`

for `:use`

hints,
nor does it change the meaning of the symbol `foo`

in theory expressions;
see theories, in particular the discussion there of runic designators.
Similarly `:`

`pe`

`foo`

and `:`

`pf`

`foo`

will still show the
original definition of `foo`

.

The definitional principle, `defun`

, actually adds `:definition`

rules.
Thus the handling of generalized definitions is exactly the same as for
``real'' definitions because no distinction is made in the implementation.
Suppose `(fn x y)`

is `defun`

'd to be `body`

. Note that `defun`

(or `defuns`

or `mutual-recursion`

) can compute the clique for `fn`

from the syntactic presentation and it can compute the controllers from the
termination analysis. Provided the definition is admissible, `defun`

adds the `:definition`

rule `(equal (fn x y) body)`

.