the three senses of well-formed ACL2 expressions or formulas

Examples of Terms:
(cond ((caar x) (cons t x)) (t 0))   ; an untranslated term

(if (car (car x)) (cons 't x) '0) ; a translated term

(car (cons x y) 'nil v) ; a pseudo-term

In traditional first-order predicate calculus a ``term'' is a syntactic entity denoting some object in the universe of individuals. Often, for example, the syntactic characterization of a term is that it is either a variable symbol or the application of a function symbol to the appropriate number of argument terms. Traditionally, ``atomic formulas'' are built from terms with predicate symbols such as ``equal'' and ``member;'' ``formulas'' are then built from atomic formulas with propositional ``operators'' like ``not,'' ``and,'' and ``implies.'' Theorems are formulas. Theorems are ``valid'' in the sense that the value of a theorem is true, in any model of the axioms and under all possible assignments of individuals to variables.

However, in ACL2, terms are used in place of both atomic formulas and formulas. ACL2 does not have predicate symbols or propositional operators as distinguished syntactic entities. The ACL2 universe of individuals includes a ``true'' object (denoted by t) and a ``false'' object (denoted by nil), predicates and propositional operators are functions that return these objects. Theorems in ACL2 are terms and the ``validity'' of a term means that, under no assignment to the variables does the term evaluate to nil.

We use the word ``term'' in ACL2 in three distinct senses. We will speak of ``translated'' terms, ``untranslated'' terms, and ``pseudo-'' terms.

Translated Terms: The Strict Sense and Internal Form

In its most strict sense, a ``term'' is either a legal variable symbol, a quoted constant, or the application of an n-ary function symbol or closed lambda expression to a true list of n terms.

The legal variable symbols are symbols other than t or nil which are not in the keyword package, do not start with ampersand, do not start and end with asterisks, and if in the main Lisp package, do not violate an appropriate restriction (see name).

Quoted constants are expressions of the form (quote x), where x is any ACL2 object. Such expressions may also be written 'x.

Closed lambda expressions are expressions of the form (lambda (v1 ... vn) body) where the vi are distinct legal variable symbols, body is a term, and the only free variables in body are among the vi.

The function termp, which takes two arguments, an alleged term x and a logical world w (see world), recognizes terms of a given extension of the logic. Termp is defined in :program mode. Its definition may be inspected with :pe termp for a complete specification of what we mean by ``term'' in the most strict sense. Most ACL2 term-processing functions deal with terms in this strict sense and use termp as a guard. That is, the ``internal form'' of a term satisfies termp, the strict sense of the word ``term.''

Untranslated Terms: What the User Types

While terms in the strict sense are easy to explore (because their structure is so regular and simple) they can be cumbersome to type. Thus, ACL2 supports a more sugary syntax that includes uses of macros and constant symbols. Very roughly speaking, macros are functions that produce terms as their results. Constants are symbols that are associated with quoted objects. Terms in this sugary syntax are ``translated'' to terms in the strict sense; the sugary syntax is more often called ``untranslated.'' Roughly speaking, translation just implements macroexpansion, the replacement of constant symbols by their quoted values, and the checking of all the rules governing the strict sense of ``term.''

More precisely, macro symbols are as described in the documentation for defmacro. A macro, mac, can be thought of as a function, mac-fn, from ACL2 objects to an ACL2 object to be treated as an untranslated term. For example, caar is defined as a macro symbol; the associated macro function maps the object x into the object (car (car x)). A macro form is a ``call'' of a macro symbol, i.e., a list whose car is the macro symbol and whose cdr is an arbitrary true list of objects, used as a term. Macroexpansion is the process of replacing in an untranslated term every occurrence of a macro form by the result of applying the macro function to the appropriate arguments. The ``appropriate'' arguments are determined by the exact form of the definition of the macro; macros support positional, keyword, optional and other kinds of arguments. See defmacro.

In addition to macroexpansion and constant symbol dereferencing, translation implements the mapping of let and let* forms into applications of lambda expressions and closes lambda expressions containing free variables. Thus, the translation of

(let ((x (1+ i))) (cons x k))
can be seen as a two-step process that first produces
((lambda (x) (cons x k)) (1+ i))
and then
((lambda (x k) (cons x k)) (1+ i) k) .
Observe that the body of the let and of the first lambda expression contains a free k which is finally bound and passed into the second lambda expression.

When we say, of an event-level function such as defun or defthm, that some argument ``must be a term'' we mean an untranslated term. The event functions translate their term-like arguments.

To better understand the mapping between untranslated terms and translated terms it is convenient to use the keyword command :trans to see examples of translations. See trans and also see trans1.

Finally, we note that the theorem prover prints terms in untranslated form. But there can be multiple correct untranslated terms corresponding to a given translated term. For example, the translated term (if x y 'nil) can be untranslated as (if x y nil) and can also be untranslated as (and x y). The theorem prover attempts to print an untranslated term that is as helpful to the user as possible. In particular, consider a term of the form (nth k st) where st is a single-threaded object (see stobj) and the kth accessor of st is, say, kn. The theorem prover typically would expand (kn st) to (nth k st). If k is large then it could be difficult for the user to make sense out of a proof transcript that mentions the expanded term. Fortunately, the untranslation of (nth k st) would be (nth *kn* st); here *kn* would be a constant (see defconst) added by the defstobj event introducing st, defined to have value k. The user can extend this user-friendly style of printing applications of nth to stobjs; see add-nth-alias. These remarks about printing applications of function nth extend naturally to function update-nth. Moreover, the prover will attempt to treat terms as stobjs for the above purpose when appropriate. For example, if function foo has signature ((foo * st) => (mv * * * st)), where st is introduced with (defstobj st f0 f1), then the term (nth '1 (mv-nth '3 (foo x st0))) will be printed as (nth *f1* (mv-nth 3 (foo x st0))).

Pseudo-Terms: A Common Guard for Metafunctions

Because termp is defined in :program mode, it cannot be used effectively in conjectures to be proved. Furthermore, from the perspective of merely guarding a term processing function, termp often checks more than is required. Finally, because termp requires the logical world as one of its arguments it is impossible to use termp as a guard in places where the logical world is not itself one of the arguments.

For these reasons we support the idea of ``pseudo-terms.'' A pseudo-term is either a symbol (but not necessarily one having the syntax of a legal variable symbol), a true list beginning with quote (but not necessarily well-formed), or the ``application of'' a symbol or pseudo lambda expression to a true list of pseudo-terms. A pseudo lambda expression is an expression of the form (lambda (v1 ... vn) body) where the vi are all symbols and body is a pseudo-term.

Pseudo-terms are recognized by the unary function pseudo-termp. If (termp x w) is true, then (pseudo-termp x) is true. However, if x fails to be a (strict) term it may nevertheless still be a pseudo-term. For example, (car a b) is not a term, because car is applied to the wrong number of arguments, but it is a pseudo-term.

The structures recognized by pseudo-termp can be recursively explored with the same simplicity that terms can be. In particular, if x is not a variablep or an fquotep, then (ffn-symb x) is the function (symbol or lambda expression) and (fargs x) is the list of argument pseudo-terms. A metafunction may use pseudo-termp as the guard.