sxbuiltins - a collections of predefined functions to work with symbolic expressions

This package provides a collection of functions that can be used when symbolic expressions are evaluated (as defined in package sxeval).

Callable

Callable functions allow to compute with objects. All arguments are evaluated before the function is called with these evaluated values as arguments.

For example, (+ 2 5 7) calculates the sum of these three numbers, resulting in the number value 14. Similar, * is the symbol that evaluates to the multiplication function, while the symbol / evaluates to the division function. (= a b) returns a boolean "true" value, if the symbols a and b evaluate to equal values.

Syntax

Syntax functions provide a way to interpret some symbolic expression with a special evaluation scheme. In most cases, it is not desired to evaluate all arguments of a function.

For example, the expression (if COND TRUE FALSE) should evaluate the value of COND. Depending on its value, either TRUE or FALSE must be evaluated, but not true. Otherwise, the expression (if (= a 0) 'error (/ 10 a)) will not work if a is equal to the number zero, when all arguments to a hypothetical if function are evaluated before calling the function. An error would be raised.

An interesting syntax function is (lambda PARAMS OBJ1 OBJ2 ...). When evaluated, it creates a user-defined callable function. ARGS is typically a list that specifies the parameters of the function, while OBJ1, OBJ2, ... are objects that are evaluated sequentially in the freshly created new environment of that function. The result of the last object is the result of the function call. The environment is first the mapping of the parameter names to the argument values, and second the pre-existing environment where lambda was executed.

A simple user-defined function is a function that adds the number seven to its single argument: (lambda (x) (+ c 7)). Since the result of the lambda function is a callable, you are allowed to put it as a first object into a list: ((lambda (x) (+ x 7)) 10). This binds the number 10 to the symbol x and calls the function, resulting in a value of 17.

Other syntax functions provide a way to update the current evaluation environment. (defvar SYMBOL OBJ) binds the value of OBJ to the symbol SYMBOL. Later, the binding may be changed.

Since (defvar add7 (lambda (x) (+ x 7))) is a little verbose, there is a simpler form: (defun add7 (x) (+x 7)). Then you can evaluate (add7 10).

While a lambda creates an user-defined callable function, the is a form to create an user-defined syntax function: (defmacro NAME PARAMS OBJ1 ...). Such a syntax function typically creates a list that will be evaluated separately. For example, there is an user-defined syntax function that sequentially evaluates its arguments and stops if a boolean "false" value is found: (and OBJ1 ...). If all arguments evaluate to a boolean "true" value, and returns a "true" value, of course. Here is its definition (T is bound to the symbol T):

(defmacro and args
    (cond ((null? args)       T)
          ((null? (cdr args)) (car args))
          (T                  `(if ,(car args) (and ,@(cdr args))))))

cond and if are predefined syntax functions, null?, cdr, and car are predefined callable functions.