TYPE-SPEC

type specifiers in declarations
```Major Section:  PROGRAMMING
```

```Examples:
The symbol INTEGER in (declare (type INTEGER i j k)) is a type-spec.  Other
type-specs supported by ACL2 include RATIONAL, COMPLEX, (INTEGER 0 127),
(RATIONAL 1 *), CHARACTER, and ATOM.
```

The type-specs and their meanings (when applied to the variable `x` as in `(declare (type type-spec x))` are given below.

```type-spec              meaning
(AND type1 ... typek)  (AND (p1 X) ... (pk X))
where (pj x) is the meaning for type-spec typej
ATOM                   (ATOM X)
BIT                    (OR (EQUAL X 1) (EQUAL X 0))
CHARACTER              (CHARACTERP X)
COMPLEX,               (AND (COMPLEX-RATIONALP X)
(COMPLEX RATIONAL)          (RATIONALP (REALPART X))
(RATIONALP (IMAGPART X)))
(COMPLEX type)         (AND (COMPLEX-RATIONALP X)
(p (REALPART X))
(p (IMAGPART X)))
where (p x) is the meaning for type-spec type
CONS                   (CONSP X)
INTEGER                (INTEGERP X)
(INTEGER i j)          (AND (INTEGERP X)   ; See notes below
(<= i X)
(<= X j))
(MEMBER x1 ... xn)     (MEMBER X '(x1 ... xn))
(MOD i)                same as (INTEGER 0 i-1)
NIL                    NIL
(NOT type)             (NOT (p X))
where (p x) is the meaning for type-spec type
NULL                   (EQ X NIL)
(OR type1 ... typek)   (OR (p1 X) ... (pk X))
where (pj x) is the meaning for type-spec typej
RATIO                  (AND (RATIONALP X) (NOT (INTEGERP X)))
RATIONAL               (RATIONALP X)
(RATIONAL i j)         (AND (RATIONALP X)  ; See notes below
(<= i X)
(<= X j))
REAL                   (RATIONALP X)       ; (REALP X) in ACL2(r)
(REAL i j)             (AND (RATIONALP X)  ; See notes below
(<= i X)
(<= X j))
(SATISFIES pred)       (pred X)
SIGNED-BYTE            (INTEGERP X)
(SIGNED-BYTE i)        same as (INTEGER -2**i-1 (2**i-1)-1)
STANDARD-CHAR          (STANDARD-CHARP X)
STRING                 (STRINGP X)
(STRING max)           (AND (STRINGP X) (EQUAL (LENGTH X) max))
SYMBOL                 (SYMBOLP X)
T                      T
UNSIGNED-BYTE          same as (INTEGER 0 *)
(UNSIGNED-BYTE i)      same as (INTEGER 0 (2**i)-1)
```
Notes:

In general, `(integer i j)` means

```     (AND (INTEGERP X)
(<= i X)
(<= X j)).
```
But if `i` is the symbol `*`, the first inequality is omitted. If `j` is the symbol `*`, the second inequality is omitted. If instead of being an integer, the second element of the type specification is a list containing an integer, `(i)`, then the first inequality is made strict. An analogous remark holds for the `(j)` case. The `RATIONAL` and `REAL` type specifiers are similarly generalized.  