Major Section: ACL2 Documentation

Example: '((:definition app) (:executable-counterpart app) rv (rv) assoc-of-app)See:

### CURRENT-THEORY -- currently enabled rules as of logical name

### DEFTHEORY -- define a theory (to enable or disable a set of rules)

### DISABLE -- deletes names from current theory

### E/D -- enable/disable rules

### ENABLE -- adds names to current theory

### EXECUTABLE-COUNTERPART-THEORY -- executable counterpart rules as of logical name

### FUNCTION-THEORY -- function symbol rules as of logical name

### GROUND-ZERO -- enabled rules in the startup theory

### INCOMPATIBLE -- declaring that two rules should not both be enabled

### INTERSECTION-THEORIES -- intersect two theories

### MINIMAL-THEORY -- a minimal theory to enable

### RULE-NAMES -- How rules are named.

### RUNE -- a rule name

### SET-DIFFERENCE-THEORIES -- difference of two theories

### THEORY -- retrieve named theory

### THEORY-FUNCTIONS -- functions for obtaining or producing theories

### UNION-THEORIES -- union two theories

### UNIVERSAL-THEORY -- all rules as of logical name

`in-theory`

event, extended with the rule names introduced
since then. Inside the theorem prover, the `:`

`in-theory`

hint
(see hints) can be used to select a particular theory as
current during the proof attempt for a particular goal.
Theories are generally constructed by ``theory expressions.''
Formally, a theory expression is any term, containing at most the
single free variable `world`

, that when evaluated with `world`

bound to
the current ACL2 world (see world) produces a theory. ACL2
provides various functions for the convenient construction and
manipulation of theories. These are called ``theory functions''
(see theory-functions). For example, the theory function
`union-theories`

takes two theories and produces their union. The
theory function `universal-theory`

returns the theory containing all
known rule names as of the introduction of a given logical name.
But a theory expression can contain constants, e.g.,

'(assoc (assoc) (:rewrite car-cons) car-cdr-elim)and user-defined functions. The only important criterion is that a theory expression mention no variable freely except

`world`

and
evaluate to a theory.
More often than not, theory expressions typed by the user do not
mention the variable `world`

. This is because user-typed theory
expressions are generally composed of applications of ACL2's theory
functions. These ``functions'' are actually macros that expand into
terms in which `world`

is used freely and appropriately. Thus, the
technical definition of ``theory expression'' should not mislead you
into thinking that interestng theory expressions must mention `world`

;
they probably do and you just didn't know it!

One aspect of this arrangement is that theory expressions cannot
generally be evaluated at the top-level of ACL2, because `world`

is
not bound. To see the value of a theory expression, `expr`

, at the
top-level, type

ACL2 !>(LET ((WORLD (W STATE))) expr).However, because the built-in theories are quite long, you may be sorry you printed the value of a theory expression!

A theory is a true list of runic designators and to each theory there corresponds a set of runes, obtained by unioning together the sets of runes denoted by each runic designator. For example, the theory constant

'(assoc (assoc) (:rewrite car-cons) car-cdr-elim)corresponds to the set of runes

{(:definition assoc) (:executable-counterpart assoc) (:rewrite car-cons) (:rewrite car-cdr-elim) (:elim car-cdr-elim)} .Observe that the theory contains four elements but its runic correspondent contains five. That is because some designators denote sets of several runes. If the above theory were selected as current then the five rules named in its runic counterpart would be enabled and all other rules would be disabled.

We now precisely define the runic designators and the set of runes denoted by each.

These conventions attempt to implement the Nqthm-1992 treatment of theories. For example, including a function name, e.g.,o A rune is a runic designator and denotes the singleton set containing that rune.

o If

`symb`

is a function symbol introduced with a`defun`

(or`defuns`

) event, then`symb`

is a runic designator and denotes the set containing the runes`(:definition symb)`

and`(:induction symb)`

.o If

`symb`

is a function symbol introduced with a`defun`

(or`defuns`

) event, then`(symb)`

is a runic designator and denotes the singleton set containing the rune`(:executable-counterpart symb)`

.o If

`symb`

is the name of a`defthm`

(or`defaxiom`

) event that introduced at least one rule, then`symb`

is a runic designator and denotes the set of the names of all rules introduced by the named event.o If

`str`

is the string naming some`defpkg`

event and`symb`

is the symbol returned by`(intern str "ACL2")`

, then`symb`

is a runic designator and denotes the singleton set containing`(:rewrite symb)`

, which is the name of the rule stating the conditions under which the`symbol-package-name`

of`(intern x str)`

is`str`

.o If

`symb`

is the name of a`deftheory`

event, then`symb`

is a runic designator and denotes the runic theory corresponding to`symb`

.

`assoc`

, in
the current theory enables that function but does not enable the
executable counterpart. Similarly, including `(assoc)`

enables the
executable counterpart (Nqthm's `*1*assoc`

) but not the symbolic
definition. And including the name of a proved lemma enables all of
the rules added by the event. These conventions are entirely
consistent with Nqthm usage. Of course, in ACL2 one can include
explicitly the runes naming the rules in question and so can avoid
entirely the use of non-runic elements in theories.
Because a rune is a runic designator denoting the set containing
that rune, a list of runes is a theory and denotes itself. We call
such theories ``runic theories.'' To every theory there corresponds
a runic theory obtained by unioning together the sets denoted by
each designator in the theory. When a theory is selected as
``current'' it is actually its runic correspondent that is
effectively used. That is, a rune is enabled iff it is a member of
the runic correspondent of the current theory. The value of a
theory defined with `deftheory`

is the runic correspondent of the
theory computed by the defining theory expression. The theory
manipulation functions, e.g., `union-theories`

, actually convert their
theory arguments to their runic correspondents before performing the
required set operation. The manipulation functions always return
runic theories. Thus, it is sometimes convenient to think of
(non-runic) theories as merely abbreviations for their runic
correspondents, abbreviations which are ``expanded'' at the first
opportunity by theory manipulation functions and the ``theory
consumer'' functions such as `in-theory`

and `deftheory`

.