20 February 1998

Revised(5) Report on the Algorithmic Language Scheme
****************************************************


     RICHARD KELSEY, WILLIAM CLINGER, AND JONATHAN REES (Editors)

     H. ABELSON         R. K. DYBVIG       C. T. HAYNES       G. J. ROZAS
     N. I. ADAMS IV     D. P. FRIEDMAN     E. KOHLBECKER      G. L. STEELE JR.
     D. H. BARTLEY      R. HALSTEAD        D. OXLEY           G. J. SUSSMAN
     G. BROOKS          C. HANSON          K. M. PITMAN       M. WAND



Dedicated to the Memory of Robert Hieb




Summary
*******

The report gives a defining description of the programming language
Scheme.  Scheme is a statically scoped and properly tail-recursive
dialect of the Lisp programming language invented by Guy Lewis Steele
Jr. and Gerald Jay Sussman.  It was designed to have an exceptionally
clear and simple semantics and few different ways to form expressions.
A wide variety of programming paradigms, including imperative,
functional, and message passing styles, find convenient expression in
Scheme.

The introduction offers a brief history of the language and of the
report.

The first three chapters present the fundamental ideas of the language
and describe the notational conventions used for describing the language
and for writing programs in the language.

Chapters *note Expressions:: and *note Program structure:: describe the
syntax and semantics of expressions, programs, and definitions.

Chapter *note Standard procedures:: describes Scheme's built-in
procedures, which include all of the language's data manipulation and
input/output primitives.

Chapter *note Formal syntax and semantics:: provides a formal syntax for
Scheme written in extended BNF, along with a formal denotational
semantics.  An example of the use of the language follows the formal
syntax and semantics.

The report concludes with a list of references and an alphabetic index.

Contents
********

Introduction
************

Programming languages should be designed not by piling feature on top of
feature, but by removing the weaknesses and restrictions that make
additional features appear necessary.  Scheme demonstrates that a very
small number of rules for forming expressions, with no restrictions on
how they are composed, suffice to form a practical and efficient
programming language that is flexible enough to support most of the
major programming paradigms in use today.

Scheme was one of the first programming languages to incorporate first
class procedures as in the lambda calculus, thereby proving the
usefulness of static scope rules and block structure in a dynamically
typed language.  Scheme was the first major dialect of Lisp to
distinguish procedures from lambda expressions and symbols, to use a
single lexical environment for all variables, and to evaluate the
operator position of a procedure call in the same way as an operand
position.  By relying entirely on procedure calls to express iteration,
Scheme emphasized the fact that tail-recursive procedure calls are
essentially goto's that pass arguments.  Scheme was the first widely
used programming language to embrace first class escape procedures, from
which all previously known sequential control structures can be
synthesized.  A subsequent version of Scheme introduced the concept of
exact and inexact numbers, an extension of Common Lisp's generic
arithmetic.  More recently, Scheme became the first programming language
to support hygienic macros, which permit the syntax of a
block-structured language to be extended in a consistent and reliable
manner.

Background
==========

The first description of Scheme was written in 1975 [Scheme75].  A
revised report [Scheme78] appeared in 1978, which described the
evolution of the language as its MIT implementation was upgraded to
support an innovative compiler [Rabbit].  Three distinct projects began
in 1981 and 1982 to use variants of Scheme for courses at MIT, Yale, and
Indiana University [Rees82], [MITScheme], [Scheme311].  An introductory
computer science textbook using Scheme was published in 1984 [SICP].

As Scheme became more widespread, local dialects began to diverge until
students and researchers occasionally found it difficult to understand
code written at other sites.  Fifteen representatives of the major
implementations of Scheme therefore met in October 1984 to work toward a
better and more widely accepted standard for Scheme.

Their report [RRRS] was published at MIT and Indiana University in the
summer of 1985.  Further revision took place in the spring of 1986
[R3RS], and in the spring of 1988 [R4RS]. The present report reflects
further revisions agreed upon in a meeting at Xerox PARC in June 1992.




We intend this report to belong to the entire Scheme community, and so
we grant permission to copy it in whole or in part without fee.  In
particular, we encourage implementors of Scheme to use this report as a
starting point for manuals and other documentation, modifying it as
necessary.

Acknowledgements
================

We would like to thank the following people for their help: Alan Bawden,
Michael Blair, George Carrette, Andy Cromarty, Pavel Curtis, Jeff
Dalton, Olivier Danvy, Ken Dickey, Bruce Duba, Marc Feeley, Andy
Freeman, Richard Gabriel, Yekta G"ursel, Ken Haase, Robert Hieb, Paul
Hudak, Morry Katz, Chris Lindblad, Mark Meyer, Jim Miller, Jim Philbin,
John Ramsdell, Mike Shaff, Jonathan Shapiro, Julie Sussman, Perry Wagle,
Daniel Weise, Henry Wu, and Ozan Yigit.  We thank Carol Fessenden,
Daniel Friedman, and Christopher Haynes for permission to use text from
the Scheme 311 version 4 reference manual.  We thank Texas Instruments,
Inc.  for permission to use text from the _TI Scheme Language Reference
Manual_[TImanual85].  We gladly acknowledge the influence of manuals for
MIT Scheme[MITScheme], T[Rees84], Scheme 84[Scheme84],Common Lisp[CLtL],
and Algol 60[Naur63].

We also thank Betty Dexter for the extreme effort she put into setting
this report in TeX, and Donald Knuth for designing the program that
caused her troubles.

The Artificial Intelligence Laboratory of the Massachusetts Institute of
Technology, the Computer Science Department of Indiana University, the
Computer and Information Sciences Department of the University of
Oregon, and the NEC Research Institute supported the preparation of this
report.  Support for the MIT work was provided in part by the Advanced
Research Projects Agency of the Department of Defense under Office of
Naval Research contract N00014-80-C-0505.  Support for the Indiana
University work was provided by NSF grants NCS 83-04567 and NCS
83-03325.



1 Overview of Scheme
********************

1.1 Semantics
=============

This section gives an overview of Scheme's semantics.  A detailed
informal semantics is the subject of chapters *note Basic concepts::
through *note Standard procedures::.  For reference purposes, section
*note Formal semantics:: provides a formal semantics of Scheme.

Following Algol, Scheme is a statically scoped programming language.
Each use of a variable is associated with a lexically apparent binding
of that variable.

Scheme has latent as opposed to manifest types.  Types are associated
with values (also called objects) rather than with variables.  (Some
authors refer to languages with latent types as weakly typed or
dynamically typed languages.)  Other languages with latent types are
APL, Snobol, and other dialects of Lisp.  Languages with manifest types
(sometimes referred to as strongly typed or statically typed languages)
include Algol 60, Pascal, and C.

All objects created in the course of a Scheme computation, including
procedures and continuations, have unlimited extent.  No Scheme object
is ever destroyed.  The reason that implementations of Scheme do not
(usually!)  run out of storage is that they are permitted to reclaim the
storage occupied by an object if they can prove that the object cannot
possibly matter to any future computation.  Other languages in which
most objects have unlimited extent include APL and other Lisp dialects.

Implementations of Scheme are required to be properly tail-recursive.
This allows the execution of an iterative computation in constant space,
even if the iterative computation is described by a syntactically
recursive procedure.  Thus with a properly tail-recursive
implementation, iteration can be expressed using the ordinary
procedure-call mechanics, so that special iteration constructs are
useful only as syntactic sugar.  See section *note Proper tail
recursion::.

Scheme procedures are objects in their own right.  Procedures can be
created dynamically, stored in data structures, returned as results of
procedures, and so on.  Other languages with these properties include
Common Lisp and ML.

One distinguishing feature of Scheme is that continuations, which in
most other languages only operate behind the scenes, also have
"first-class" status.  Continuations are useful for implementing a wide
variety of advanced control constructs, including non-local exits,
backtracking, and coroutines.  See section *note Control features::.

Arguments to Scheme procedures are always passed by value, which means
that the actual argument expressions are evaluated before the procedure
gains control, whether the procedure needs the result of the evaluation
or not.  ML, C, and APL are three other languages that always pass
arguments by value.  This is distinct from the lazy-evaluation semantics
of Haskell, or the call-by-name semantics of Algol 60, where an argument
expression is not evaluated unless its value is needed by the procedure.

Scheme's model of arithmetic is designed to remain as independent as
possible of the particular ways in which numbers are represented within
a computer.  In Scheme, every integer is a rational number, every
rational is a real, and every real is a complex number.  Thus the
distinction between integer and real arithmetic, so important to many
programming languages, does not appear in Scheme.  In its place is a
distinction between exact arithmetic, which corresponds to the
mathematical ideal, and inexact arithmetic on approximations.  As in
Common Lisp, exact arithmetic is not limited to integers.

1.2 Syntax
==========

Scheme, like most dialects of Lisp, employs a fully parenthesized prefix
notation for programs and (other) data; the grammar of Scheme generates
a sublanguage of the language used for data.  An important consequence
of this simple, uniform representation is the susceptibility of Scheme
programs and data to uniform treatment by other Scheme programs.  For
example, the ‘eval’ procedure evaluates a Scheme program expressed as
data.

The ‘read’ procedure performs syntactic as well as lexical decomposition
of the data it reads.  The ‘read’ procedure parses its input as data
(section *note External representation::), not as program.

The formal syntax of Scheme is described in section *note Formal
syntax::.

1.3 Notation and terminology
============================

1.3.1 Primitive; library; and optional features
-----------------------------------------------

It is required that every implementation of Scheme support all features
that are not marked as being “optional”.  Implementations are free to
omit optional features of Scheme or to add extensions, provided the
extensions are not in conflict with the language reported here.  In
particular, implementations must support portable code by providing a
syntactic mode that preempts no lexical conventions of this report.

To aid in understanding and implementing Scheme, some features are
marked as “library”.  These can be easily implemented in terms of the
other, primitive, features.  They are redundant in the strict sense of
the word, but they capture common patterns of usage, and are therefore
provided as convenient abbreviations.

1.3.2 Error situations and unspecified behavior
-----------------------------------------------

When speaking of an error situation, this report uses the phrase "an
error is signalled" to indicate that implementations must detect and
report the error.  If such wording does not appear in the discussion of
an error, then implementations are not required to detect or report the
error, though they are encouraged to do so.  An error situation that
implementations are not required to detect is usually referred to simply
as "an error."

For example, it is an error for a procedure to be passed an argument
that the procedure is not explicitly specified to handle, even though
such domain errors are seldom mentioned in this report.  Implementations
may extend a procedure's domain of definition to include such arguments.

This report uses the phrase "may report a violation of an implementation
restriction" to indicate circumstances under which an implementation is
permitted to report that it is unable to continue execution of a correct
program because of some restriction imposed by the implementation.
Implementation restrictions are of course discouraged, but
implementations are encouraged to report violations of implementation
restrictions.

For example, an implementation may report a violation of an
implementation restriction if it does not have enough storage to run a
program.

If the value of an expression is said to be "unspecified," then the
expression must evaluate to some object without signalling an error, but
the value depends on the implementation; this report explicitly does not
say what value should be returned.

1.3.3 Entry format
------------------

Chapters *note Expressions:: and *note Standard procedures:: are
organized into entries.  Each entry describes one language feature or a
group of related features, where a feature is either a syntactic
construct or a built-in procedure.  An entry begins with one or more
header lines of the form

 -- CATEGORY: TEMPLATE

for required, primitive features, or

 -- QUALIFIER CATEGORY: TEMPLATE

where QUALIFIER is either "library" or "optional" as defined in section
*note Primitive; library; and optional features::.

If CATEGORY is "syntax", the entry describes an expression type, and the
template gives the syntax of the expression type.  Components of
expressions are designated by syntactic variables, which are written
using angle brackets, for example, <expression>, <variable>.  Syntactic
variables should be understood to denote segments of program text; for
example, <expression> stands for any string of characters which is a
syntactically valid expression.  The notation

 <thing1> ...

indicates zero or more occurrences of a <thing>, and

 <thing1> <thing2> ...

indicates one or more occurrences of a <thing>.

If CATEGORY is "procedure", then the entry describes a procedure, and
the header line gives a template for a call to the procedure.  Argument
names in the template are ITALICIZED.  Thus the header line

 -- procedure: vector-ref VECTOR K

indicates that the built-in procedure vector-ref takes two arguments, a
vector VECTOR and an exact non-negative integer K (see below).  The
header lines

 -- procedure: make-vector K

 -- procedure: make-vector K FILL

indicate that the make-vector procedure must be defined to take either
one or two arguments.

It is an error for an operation to be presented with an argument that it
is not specified to handle.  For succinctness, we follow the convention
that if an argument name is also the name of a type listed in section
*note Disjointness of types::, then that argument must be of the named
type.  For example, the header line for vector-ref given above dictates
that the first argument to vector-ref must be a vector.  The following
naming conventions also imply type restrictions:

     OBJ
          any object
     LIST, LIST1, ... LISTJ, ...
          list (see section *note Pairs and lists::)
     Z, Z1, ... ZJ, ...
          complex number
     X, X1, ... XJ, ...
          real number
     Y, Y1, ... YJ, ...
          real number
     Q, Q1, ... QJ, ...
          rational number
     N, N1, ... NJ, ...
          integer
     K, K1, ... KJ, ...
          exact non-negative integer

1.3.4 Evaluation examples
-------------------------

The symbol "⇒" used in program examples should be read "evaluates to."
For example,


     (* 5 8)                                ==>  40


means that the expression (* 5 8) evaluates to the object 40.  Or, more
precisely: the expression given by the sequence of characters "(* 5 8)"
evaluates, in the initial environment, to an object that may be
represented externally by the sequence of characters "40".  See section
*note External representations:: for a discussion of external
representations of objects.

1.3.5 Naming conventions
------------------------

By convention, the names of procedures that always return a boolean
value usually end in "‘?’".  Such procedures are called predicates.

By convention, the names of procedures that store values into previously
allocated locations (see section *note Storage model::) usually end in
"‘!’".  Such procedures are called mutation procedures.  By convention,
the value returned by a mutation procedure is unspecified.

By convention, "‘->’" appears within the names of procedures that take
an object of one type and return an analogous object of another type.
For example, ‘list->vector’ takes a list and returns a vector whose
elements are the same as those of the list.

2 Lexical conventions
*********************

This section gives an informal account of some of the lexical
conventions used in writing Scheme programs.  For a formal syntax of
Scheme, see section *note Formal syntax::.

Upper and lower case forms of a letter are never distinguished except
within character and string constants.  For example, ‘Foo’ is the same
identifier as ‘FOO’, and #x1AB is the same number as #X1ab.

2.1 Identifiers
===============

Most identifiers allowed by other programming languages are also
acceptable to Scheme.  The precise rules for forming identifiers vary
among implementations of Scheme, but in all implementations a sequence
of letters, digits, and "extended alphabetic characters" that begins
with a character that cannot begin a number is an identifier.  In
addition, ‘+’, ‘-’, and ‘...’ are identifiers.  Here are some examples
of identifiers:


     lambda                   q
     list->vector             soup
     +                        V17a
     <=?                      a34kTMNs
     the-word-recursion-has-many-meanings


Extended alphabetic characters may be used within identifiers as if they
were letters.  The following are extended alphabetic characters:


     ! $ % & * + - . / : < = > ? @ ^ _ ~

See section *note Lexical structure:: for a formal syntax of
identifiers.

Identifiers have two uses within Scheme programs:

   • Any identifier may be used as a variable or as a syntactic keyword
     (see sections *note Variables; syntactic keywords; and regions::
     and *note Macros::).

   • When an identifier appears as a literal or within a literal (see
     section *note Literal expressions::), it is being used to denote a
     _symbol_ (see section *note Symbols::).

2.2 Whitespace and comments
===========================

“Whitespace” characters are spaces and newlines.  (Implementations
typically provide additional whitespace characters such as tab or page
break.)  Whitespace is used for improved readability and as necessary to
separate tokens from each other, a token being an indivisible lexical
unit such as an identifier or number, but is otherwise insignificant.
Whitespace may occur between any two tokens, but not within a token.
Whitespace may also occur inside a string, where it is significant.

A semicolon (;) indicates the start of a comment.  The comment continues
to the end of the line on which the semicolon appears.  Comments are
invisible to Scheme, but the end of the line is visible as whitespace.
This prevents a comment from appearing in the middle of an identifier or
number.


     ;;; The FACT procedure computes the factorial
     ;;; of a non-negative integer.
     (define fact
       (lambda (n)
         (if (= n 0)
             1        ;Base case: return 1
             (* n (fact (- n 1))))))


2.3 Other notations
===================

For a description of the notations used for numbers, see section *note
Numbers::.

. + -
     These are used in numbers, and may also occur anywhere in an
     identifier except as the first character.  A delimited plus or
     minus sign by itself is also an identifier.  A delimited period
     (not occurring within a number or identifier) is used in the
     notation for pairs (section *note Pairs and lists::), and to
     indicate a rest-parameter in a formal parameter list (section *note
     Procedures::).  A delimited sequence of three successive periods is
     also an identifier.

( )
     Parentheses are used for grouping and to notate lists (section
     *note Pairs and lists::).

'
     The single quote character is used to indicate literal data
     (section *note Literal expressions::).

`
     The backquote character is used to indicate almost-constant data
     (section *note Quasiquotation::).

, ,@
     The character comma and the sequence comma at-sign are used in
     conjunction with backquote (section *note Quasiquotation::).

"
     The double quote character is used to delimit strings (section
     *note Strings::).

\
     Backslash is used in the syntax for character constants (section
     *note Characters::) and as an escape character within string
     constants (section *note Strings::).

[ ] { } |
     Left and right square brackets and curly braces and vertical bar
     are reserved for possible future extensions to the language.

#
     Sharp sign is used for a variety of purposes depending on the
     character that immediately follows it:

#t #f
     These are the boolean constants (section *note Booleans::).

#\
     This introduces a character constant (section *note Characters::).

#(
     This introduces a vector constant (section *note Vectors::).
     Vector constants are terminated by ) .

#e #i #b #o #d #x
     These are used in the notation for numbers (section *note Syntax of
     numerical constants::).

3 Basic concepts
****************

3.1 Variables; syntactic keywords; and regions
==============================================

An identifier may name a type of syntax, or it may name a location where
a value can be stored.  An identifier that names a type of syntax is
called a _syntactic keyword_ and is said to be _bound_ to that syntax.
An identifier that names a location is called a _variable_ and is said
to be _bound_ to that location.  The set of all visible bindings in
effect at some point in a program is known as the _environment_ in
effect at that point.  The value stored in the location to which a
variable is bound is called the variable's value.  By abuse of
terminology, the variable is sometimes said to name the value or to be
bound to the value.  This is not quite accurate, but confusion rarely
results from this practice.

Certain expression types are used to create new kinds of syntax and bind
syntactic keywords to those new syntaxes, while other expression types
create new locations and bind variables to those locations.  These
expression types are called _binding constructs_.

Those that bind syntactic keywords are listed in section *note Macros::.
The most fundamental of the variable binding constructs is the ‘lambda’
expression, because all other variable binding constructs can be
explained in terms of ‘lambda’ expressions.  The other variable binding
constructs are ‘let’, ‘let*’, ‘letrec’, and ‘do’ expressions (see
sections *note Procedures::, *note Binding constructs::, and *note
Iteration::).

Like Algol and Pascal, and unlike most other dialects of Lisp except for
Common Lisp, Scheme is a statically scoped language with block
structure.  To each place where an identifier is bound in a program
there corresponds a “region” of the program text within which the
binding is visible.  The region is determined by the particular binding
construct that establishes the binding; if the binding is established by
a ‘lambda’ expression, for example, then its region is the entire
‘lambda’ expression.  Every mention of an identifier refers to the
binding of the identifier that established the innermost of the regions
containing the use.  If there is no binding of the identifier whose
region contains the use, then the use refers to the binding for the
variable in the top level environment, if any (chapters *note
Expressions:: and *note Standard procedures::); if there is no binding
for the identifier, it is said to be “unbound”.

3.2 Disjointness of types
=========================

No object satisfies more than one of the following predicates:


     boolean?          pair?
     symbol?           number?
     char?             string?
     vector?           port?
     procedure?


These predicates define the types _boolean_, _pair_, _symbol_, _number_,
_char_ (or _character_), _string_, _vector_, _port_, and _procedure_.
The empty list is a special object of its own type; it satisfies none of
the above predicates.

Although there is a separate boolean type, any Scheme value can be used
as a boolean value for the purpose of a conditional test.  As explained
in section *note Booleans::, all values count as true in such a test
except for #f.  This report uses the word "true" to refer to any Scheme
value except #f, and the word "false" to refer to #f.

3.3 External representations
============================

An important concept in Scheme (and Lisp) is that of the _external
representation_ of an object as a sequence of characters.  For example,
an external representation of the integer 28 is the sequence of
characters "28", and an external representation of a list consisting of
the integers 8 and 13 is the sequence of characters "(8 13)".

The external representation of an object is not necessarily unique.  The
integer 28 also has representations "#e28.000" and "#x1c", and the list
in the previous paragraph also has the representations "( 08 13 )" and
"(8 . (13 . ()))" (see section *note Pairs and lists::).

Many objects have standard external representations, but some, such as
procedures, do not have standard representations (although particular
implementations may define representations for them).

An external representation may be written in a program to obtain the
corresponding object (see ‘quote’, section *note Literal expressions::).

External representations can also be used for input and output.  The
procedure ‘read’ (section *note Input::) parses external
representations, and the procedure ‘write’ (section *note Output::)
generates them.  Together, they provide an elegant and powerful
input/output facility.

Note that the sequence of characters "(+ 2 6)" is _not_ an external
representation of the integer 8, even though it _is_ an expression
evaluating to the integer 8; rather, it is an external representation of
a three-element list, the elements of which are the symbol + and the
integers 2 and 6.  Scheme's syntax has the property that any sequence of
characters that is an expression is also the external representation of
some object.  This can lead to confusion, since it may not be obvious
out of context whether a given sequence of characters is intended to
denote data or program, but it is also a source of power, since it
facilitates writing programs such as interpreters and compilers that
treat programs as data (or vice versa).

The syntax of external representations of various kinds of objects
accompanies the description of the primitives for manipulating the
objects in the appropriate sections of chapter *note Standard
procedures::.

3.4 Storage model
=================

Variables and objects such as pairs, vectors, and strings implicitly
denote locations or sequences of locations.  A string, for example,
denotes as many locations as there are characters in the string.  (These
locations need not correspond to a full machine word.)  A new value may
be stored into one of these locations using the string-set! procedure,
but the string continues to denote the same locations as before.

An object fetched from a location, by a variable reference or by a
procedure such as ‘car’, ‘vector-ref’, or ‘string-ref’, is equivalent in
the sense of ‘eqv?’ (section *note Equivalence predicates::) to the
object last stored in the location before the fetch.

Every location is marked to show whether it is in use.  No variable or
object ever refers to a location that is not in use.  Whenever this
report speaks of storage being allocated for a variable or object, what
is meant is that an appropriate number of locations are chosen from the
set of locations that are not in use, and the chosen locations are
marked to indicate that they are now in use before the variable or
object is made to denote them.

In many systems it is desirable for constants (i.e.  the values of
literal expressions) to reside in read-only-memory.  To express this, it
is convenient to imagine that every object that denotes locations is
associated with a flag telling whether that object is mutable or
immutable.  In such systems literal constants and the strings returned
by ‘symbol->string’ are immutable objects, while all objects created by
the other procedures listed in this report are mutable.  It is an error
to attempt to store a new value into a location that is denoted by an
immutable object.

3.5 Proper tail recursion
=========================

Implementations of Scheme are required to be _properly tail-recursive_.
Procedure calls that occur in certain syntactic contexts defined below
are 'tail calls'.  A Scheme implementation is properly tail-recursive if
it supports an unbounded number of active tail calls.  A call is
_active_ if the called procedure may still return.  Note that this
includes calls that may be returned from either by the current
continuation or by continuations captured earlier by
‘call-with-current-continuation’ that are later invoked.  In the absence
of captured continuations, calls could return at most once and the
active calls would be those that had not yet returned.  A formal
definition of proper tail recursion can be found in
[propertailrecursion].

     _Rationale:_

     Intuitively, no space is needed for an active tail call because the
     continuation that is used in the tail call has the same semantics
     as the continuation passed to the procedure containing the call.
     Although an improper implementation might use a new continuation in
     the call, a return to this new continuation would be followed
     immediately by a return to the continuation passed to the
     procedure.  A properly tail-recursive implementation returns to
     that continuation directly.

     Proper tail recursion was one of the central ideas in Steele and
     Sussman's original version of Scheme.  Their first Scheme
     interpreter implemented both functions and actors.  Control flow
     was expressed using actors, which differed from functions in that
     they passed their results on to another actor instead of returning
     to a caller.  In the terminology of this section, each actor
     finished with a tail call to another actor.

     Steele and Sussman later observed that in their interpreter the
     code for dealing with actors was identical to that for functions
     and thus there was no need to include both in the language.

A _tail call_ is a procedure call that occurs in a _tail context_.  Tail
contexts are defined inductively.  Note that a tail context is always
determined with respect to a particular lambda expression.

   • The last expression within the body of a lambda expression, shown
     as <tail expression> below, occurs in a tail context.

     (lambda <formals>
       <definition>* <expression>* <tail expression>)



   • If one of the following expressions is in a tail context, then the
     subexpressions shown as <tail expression> are in a tail context.
     These were derived from rules in the grammar given in chapter *note
     Formal syntax and semantics:: by replacing some occurrences of
     <expression> with <tail expression>.  Only those rules that contain
     tail contexts are shown here.

     (if <expression> <tail expression> <tail expression>)
     (if <expression> <tail expression>)

     (cond <cond clause>+)
     (cond <cond clause>* (else <tail sequence>))

     (case <expression>
       <case clause>+)
     (case <expression>
       <case clause>*
       (else <tail sequence>))

     (and <expression>* <tail expression>)
     (or <expression>* <tail expression>)

     (let (<binding spec>*) <tail body>)
     (let <variable> (<binding spec>*) <tail body>)
     (let* (<binding spec>*) <tail body>)
     (letrec (<binding spec>*) <tail body>)

     (let-syntax (<syntax spec>*) <tail body>)
     (letrec-syntax (<syntax spec>*) <tail body>)

     (begin <tail sequence>)

     (do (<iteration spec>*)
         (<test> <tail sequence>)
       <expression>*)

     where

     <cond clause> --> (<test> <tail sequence>)
     <case clause> --> ((<datum>*) <tail sequence>)

     <tail body> --> <definition>* <tail sequence>
     <tail sequence> --> <expression>* <tail expression>



   • If a ‘cond’ expression is in a tail context, and has a clause of
     the form ‘(<expression1> => <expression2>)’ then the (implied) call
     to the procedure that results from the evaluation of <expression2>
     is in a tail context.  <expression2> itself is not in a tail
     context.

Certain built-in procedures are also required to perform tail calls.
The first argument passed to ‘apply’ and to
‘call-with-current-continuation’, and the second argument passed to
‘call-with-values’, must be called via a tail call.  Similarly, ‘eval’
must evaluate its argument as if it were in tail position within the
‘eval’ procedure.

In the following example the only tail call is the call to ‘f’.  None of
the calls to ‘g’ or ‘h’ are tail calls.  The reference to ‘x’ is in a
tail context, but it is not a call and thus is not a tail call.


     (lambda ()
       (if (g)
           (let ((x (h)))
             x)
           (and (g) (f))))


     _Note:_ Implementations are allowed, but not required, to recognize
     that some non-tail calls, such as the call to ‘h’ above, can be
     evaluated as though they were tail calls.  In the example above,
     the ‘let’ expression could be compiled as a tail call to ‘h’.  (The
     possibility of ‘h’ returning an unexpected number of values can be
     ignored, because in that case the effect of the ‘let’ is explicitly
     unspecified and implementation-dependent.)

4 Expressions
*************

Expression types are categorized as _primitive_ or _derived_.  Primitive
expression types include variables and procedure calls.  Derived
expression types are not semantically primitive, but can instead be
defined as macros.  With the exception of ‘quasiquote’, whose macro
definition is complex, the derived expressions are classified as library
features.  Suitable definitions are given in section *note Derived
expression type::.

4.1 Primitive expression types
==============================

4.1.1 Variable references
-------------------------

 -- syntax: <variable>

     An expression consisting of a variable (section *note Variables;
     syntactic keywords; and regions::) is a variable reference.  The
     value of the variable reference is the value stored in the location
     to which the variable is bound.  It is an error to reference an
     unbound variable.

     (define x 28)
     x                                      ==>  28


4.1.2 Literal expressions
-------------------------

 -- syntax: quote <datum>

 -- syntax: '<datum>

 -- syntax: <constant>

     ‘(quote <datum>)’ evaluates to <datum>.  <Datum> may be any
     external representation of a Scheme object (see section *note
     External representations::).  This notation is used to include
     literal constants in Scheme code.


     (quote a)                              ==>  a
     (quote #(a b c))                       ==>  #(a b c)
     (quote (+ 1 2))                        ==>  (+ 1 2)


     ‘(quote <datum>)’ may be abbreviated as '<datum>.  The two
     notations are equivalent in all respects.

     'a                                     ==>  a
     '#(a b c)                              ==>  #(a b c)
     '()                                    ==>  ()
     '(+ 1 2)                               ==>  (+ 1 2)
     '(quote a)                             ==>  (quote a)
     ''a                                    ==>  (quote a)


     Numerical constants, string constants, character constants, and
     boolean constants evaluate "to themselves"; they need not be
     quoted.

     '"abc"                                 ==>  "abc"
     "abc"                                  ==>  "abc"
     '145932                                ==>  145932
     145932                                 ==>  145932
     '#t                                    ==>  #t
     #t                                     ==>  #t


     As noted in section *note Storage model::, it is an error to alter
     a constant (i.e.  the value of a literal expression) using a
     mutation procedure like ‘set-car!’ or ‘string-set!’.

4.1.3 Procedure calls
---------------------

 -- syntax: <operator> <operand1> ...,

     A procedure call is written by simply enclosing in parentheses
     expressions for the procedure to be called and the arguments to be
     passed to it.  The operator and operand expressions are evaluated
     (in an unspecified order) and the resulting procedure is passed the
     resulting arguments.


     (+ 3 4)                                ==>  7
     ((if #f + *) 3 4)                      ==>  12


     A number of procedures are available as the values of variables in
     the initial environment; for example, the addition and
     multiplication procedures in the above examples are the values of
     the variables ‘+’ and ‘*’.  New procedures are created by
     evaluating lambda expressions (see section *note Procedures::).

     Procedure calls may return any number of values (see ‘values’ in
     section *note Control features::).  With the exception of ‘values’
     the procedures available in the initial environment return one
     value or, for procedures such as ‘apply’, pass on the values
     returned by a call to one of their arguments.

     Procedure calls are also called _combinations_.

          _Note:_ In contrast to other dialects of Lisp, the order of
          evaluation is unspecified, and the operator expression and the
          operand expressions are always evaluated with the same
          evaluation rules.

          _Note:_ Although the order of evaluation is otherwise
          unspecified, the effect of any concurrent evaluation of the
          operator and operand expressions is constrained to be
          consistent with some sequential order of evaluation.  The
          order of evaluation may be chosen differently for each
          procedure call.

          _Note:_ In many dialects of Lisp, the empty combination, (),
          is a legitimate expression.  In Scheme, combinations must have
          at least one subexpression, so () is not a syntactically valid
          expression.

4.1.4 Procedures
----------------

 -- syntax: lambda <formals> <body>

     _Syntax:_ <Formals> should be a formal arguments list as described
     below, and <body> should be a sequence of one or more expressions.

     _Semantics:_ A lambda expression evaluates to a procedure.  The
     environment in effect when the lambda expression was evaluated is
     remembered as part of the procedure.  When the procedure is later
     called with some actual arguments, the environment in which the
     lambda expression was evaluated will be extended by binding the
     variables in the formal argument list to fresh locations, the
     corresponding actual argument values will be stored in those
     locations, and the expressions in the body of the lambda expression
     will be evaluated sequentially in the extended environment.  The
     result(s) of the last expression in the body will be returned as
     the result(s) of the procedure call.

     (lambda (x) (+ x x))                   ==>  __a procedure
     ((lambda (x) (+ x x)) 4)               ==>  8

     (define reverse-subtract
       (lambda (x y) (- y x)))
     (reverse-subtract 7 10)                ==>  3

     (define add4
       (let ((x 4))
         (lambda (y) (+ x y))))
     (add4 6)                               ==>  10


     <Formals> should have one of the following forms:

        • (<variable1> ...,): The procedure takes a fixed number of
          arguments; when the procedure is called, the arguments will be
          stored in the bindings of the corresponding variables.

        • <variable>: The procedure takes any number of arguments; when
          the procedure is called, the sequence of actual arguments is
          converted into a newly allocated list, and the list is stored
          in the binding of the <variable>.

        • (<variable1> ..., <variable_n> . <variable_n+1>): If a
          space-delimited period precedes the last variable, then the
          procedure takes n or more arguments, where n is the number of
          formal arguments before the period (there must be at least
          one).  The value stored in the binding of the last variable
          will be a newly allocated list of the actual arguments left
          over after all the other actual arguments have been matched up
          against the other formal arguments.

     It is an error for a <variable> to appear more than once in
     <formals>.

     ((lambda x x) 3 4 5 6)                 ==>  (3 4 5 6)
     ((lambda (x y . z) z)
      3 4 5 6)                              ==>  (5 6)


     Each procedure created as the result of evaluating a lambda
     expression is (conceptually) tagged with a storage location, in
     order to make ‘eqv?’ and ‘eq?’ work on procedures (see section
     *note Equivalence predicates::).

4.1.5 Conditionals
------------------

 -- syntax: if <test> <consequent> <alternate>
 -- syntax: if <test> <consequent>

     _Syntax:_ <Test>, <consequent>, and <alternate> may be arbitrary
     expressions.

     _Semantics:_ An ‘if’ expression is evaluated as follows: first,
     <test> is evaluated.  If it yields a true value (see section *note
     Booleans::), then <consequent> is evaluated and its value(s)
     is(are) returned.  Otherwise <alternate> is evaluated and its
     value(s) is(are) returned.  If <test> yields a false value and no
     <alternate> is specified, then the result of the expression is
     unspecified.

     (if (> 3 2) 'yes 'no)                  ==>  yes
     (if (> 2 3) 'yes 'no)                  ==>  no
     (if (> 3 2)
         (- 3 2)
         (+ 3 2))                           ==>  1


4.1.6 Assignments
-----------------

 -- syntax: set! <variable> <expression>

     <Expression> is evaluated, and the resulting value is stored in the
     location to which <variable> is bound.  <Variable> must be bound
     either in some region enclosing the ‘set!’ expression or at top
     level.  The result of the ‘set!’ expression is unspecified.

     (define x 2)
     (+ x 1)                                ==>  3
     (set! x 4)                             ==>  _unspecified_
     (+ x 1)                                ==>  5


4.2 Derived expression types
============================

The constructs in this section are hygienic, as discussed in section
*note Macros::.  For reference purposes, section *note Derived
expression type:: gives macro definitions that will convert most of the
constructs described in this section into the primitive constructs
described in the previous section.

4.2.1 Conditionals
------------------

 -- library syntax: cond <clause1> <clause2> ...,

     _Syntax:_ Each <clause> should be of the form

     (<test> <expression1> ...,)


     where <test> is any expression.  Alternatively, a <clause> may be
     of the form

     (<test> => <expression>)


     The last <clause> may be an "else clause," which has the form

     (else <expression1> <expression2> ...,).


     _Semantics:_ A ‘cond’ expression is evaluated by evaluating the
     <test> expressions of successive <clause>s in order until one of
     them evaluates to a true value (see section *note Booleans::).
     When a <test> evaluates to a true value, then the remaining
     <expression>s in its <clause> are evaluated in order, and the
     result(s) of the last <expression> in the <clause> is(are) returned
     as the result(s) of the entire ‘cond’ expression.  If the selected
     <clause> contains only the <test> and no <expression>s, then the
     value of the <test> is returned as the result.  If the selected
     <clause> uses the ‘=>’ alternate form, then the <expression> is
     evaluated.  Its value must be a procedure that accepts one
     argument; this procedure is then called on the value of the <test>
     and the value(s) returned by this procedure is(are) returned by the
     ‘cond’ expression.  If all <test>s evaluate to false values, and
     there is no else clause, then the result of the conditional
     expression is unspecified; if there is an else clause, then its
     <expression>s are evaluated, and the value(s) of the last one
     is(are) returned.

     (cond ((> 3 2) 'greater)
           ((< 3 2) 'less))                 ==>  greater

     (cond ((> 3 3) 'greater)
           ((< 3 3) 'less)
           (else 'equal))                   ==>  equal

     (cond ((assv 'b '((a 1) (b 2))) => cadr)
           (else #f))                       ==>  2


 -- library syntax: case <key> <clause1> <clause2> ...,

     _Syntax:_ <Key> may be any expression.  Each <clause> should have
     the form

     ((<datum1> ...,) <expression1> <expression2> ...,),


     where each <datum> is an external representation of some object.
     All the <datum>s must be distinct.  The last <clause> may be an
     "else clause," which has the form

     (else <expression1> <expression2> ...,).


     _Semantics:_ A ‘case’ expression is evaluated as follows.  <Key> is
     evaluated and its result is compared against each <datum>.  If the
     result of evaluating <key> is equivalent (in the sense of ‘eqv?’;
     see section *note Equivalence predicates::) to a <datum>, then the
     expressions in the corresponding <clause> are evaluated from left
     to right and the result(s) of the last expression in the <clause>
     is(are) returned as the result(s) of the ‘case’ expression.  If the
     result of evaluating <key> is different from every <datum>, then if
     there is an else clause its expressions are evaluated and the
     result(s) of the last is(are) the result(s) of the ‘case’
     expression; otherwise the result of the ‘case’ expression is
     unspecified.

     (case (* 2 3)
       ((2 3 5 7) 'prime)
       ((1 4 6 8 9) 'composite))            ==>  composite
     (case (car '(c d))
       ((a) 'a)
       ((b) 'b))                            ==>  _unspecified_
     (case (car '(c d))
       ((a e i o u) 'vowel)
       ((w y) 'semivowel)
       (else 'consonant))                   ==>  consonant


 -- library syntax: and <test1> ...,

     The <test> expressions are evaluated from left to right, and the
     value of the first expression that evaluates to a false value (see
     section *note Booleans::) is returned.  Any remaining expressions
     are not evaluated.  If all the expressions evaluate to true values,
     the value of the last expression is returned.  If there are no
     expressions then #t is returned.

     (and (= 2 2) (> 2 1))                  ==>  #t
     (and (= 2 2) (< 2 1))                  ==>  #f
     (and 1 2 'c '(f g))                    ==>  (f g)
     (and)                                  ==>  #t


 -- library syntax: or <test1> ...,

     The <test> expressions are evaluated from left to right, and the
     value of the first expression that evaluates to a true value (see
     section *note Booleans::) is returned.  Any remaining expressions
     are not evaluated.  If all expressions evaluate to false values,
     the value of the last expression is returned.  If there are no
     expressions then #f is returned.

     (or (= 2 2) (> 2 1))                   ==>  #t
     (or (= 2 2) (< 2 1))                   ==>  #t
     (or #f #f #f)                          ==>  #f
     (or (memq 'b '(a b c))
         (/ 3 0))                           ==>  (b c)


4.2.2 Binding constructs
------------------------

The three binding constructs ‘let’, ‘let*’, and ‘letrec’ give Scheme a
block structure, like Algol 60.  The syntax of the three constructs is
identical, but they differ in the regions they establish for their
variable bindings.  In a ‘let’ expression, the initial values are
computed before any of the variables become bound; in a ‘let*’
expression, the bindings and evaluations are performed sequentially;
while in a ‘letrec’ expression, all the bindings are in effect while
their initial values are being computed, thus allowing mutually
recursive definitions.

 -- library syntax: let <bindings> <body>

     _Syntax:_ <Bindings> should have the form

     ((<variable1> <init1>) ...,),


     where each <init> is an expression, and <body> should be a sequence
     of one or more expressions.  It is an error for a <variable> to
     appear more than once in the list of variables being bound.

     _Semantics:_ The <init>s are evaluated in the current environment
     (in some unspecified order), the <variable>s are bound to fresh
     locations holding the results, the <body> is evaluated in the
     extended environment, and the value(s) of the last expression of
     <body> is(are) returned.  Each binding of a <variable> has <body>
     as its region.

     (let ((x 2) (y 3))
       (* x y))                             ==>  6

     (let ((x 2) (y 3))
       (let ((x 7)
             (z (+ x y)))
         (* z x)))                          ==>  35


     See also named ‘let’, section *note Iteration::.

 -- library syntax: let* <bindings> <body>

     _Syntax:_ <Bindings> should have the form

     ((<variable1> <init1>) ...,),


     and <body> should be a sequence of one or more expressions.

     _Semantics:_ ‘Let*’ is similar to ‘let’, but the bindings are
     performed sequentially from left to right, and the region of a
     binding indicated by ‘(<variable> <init>)’ is that part of the
     ‘let*’ expression to the right of the binding.  Thus the second
     binding is done in an environment in which the first binding is
     visible, and so on.

     (let ((x 2) (y 3))
       (let* ((x 7)
              (z (+ x y)))
         (* z x)))                          ==>  70


 -- library syntax: letrec <bindings> <body>

     _Syntax:_ <Bindings> should have the form

     ((<variable1> <init1>) ...,),


     and <body> should be a sequence of one or more expressions.  It is
     an error for a <variable> to appear more than once in the list of
     variables being bound.

     _Semantics:_ The <variable>s are bound to fresh locations holding
     undefined values, the <init>s are evaluated in the resulting
     environment (in some unspecified order), each <variable> is
     assigned to the result of the corresponding <init>, the <body> is
     evaluated in the resulting environment, and the value(s) of the
     last expression in <body> is(are) returned.  Each binding of a
     <variable> has the entire ‘letrec’ expression as its region, making
     it possible to define mutually recursive procedures.

     (letrec ((even?
               (lambda (n)
                 (if (zero? n)
                     #t
                     (odd? (- n 1)))))
              (odd?
               (lambda (n)
                 (if (zero? n)
                     #f
                     (even? (- n 1))))))
       (even? 88))
                                            ==>  #t


     One restriction on ‘letrec’ is very important: it must be possible
     to evaluate each <init> without assigning or referring to the value
     of any <variable>.  If this restriction is violated, then it is an
     error.  The restriction is necessary because Scheme passes
     arguments by value rather than by name.  In the most common uses of
     ‘letrec’, all the <init>s are lambda expressions and the
     restriction is satisfied automatically.

4.2.3 Sequencing
----------------

 -- library syntax: begin <expression1> <expression2> ...,

     The <expression>s are evaluated sequentially from left to right,
     and the value(s) of the last <expression> is(are) returned.  This
     expression type is used to sequence side effects such as input and
     output.

     (define x 0)

     (begin (set! x 5)
            (+ x 1))                        ==>  6

     (begin (display "4 plus 1 equals ")
            (display (+ 4 1)))              ==>  _unspecified_
               _and prints_  4 plus 1 equals 5


4.2.4 Iteration
---------------

 -- library syntax: do ((<variable1> <init1> <step1>) ...) (<test>
          <expression> ...) <command> ...

     ‘Do’ is an iteration construct.  It specifies a set of variables to
     be bound, how they are to be initialized at the start, and how they
     are to be updated on each iteration.  When a termination condition
     is met, the loop exits after evaluating the <expression>s.

     ‘Do’ expressions are evaluated as follows: The <init> expressions
     are evaluated (in some unspecified order), the <variable>s are
     bound to fresh locations, the results of the <init> expressions are
     stored in the bindings of the <variable>s, and then the iteration
     phase begins.

     Each iteration begins by evaluating <test>; if the result is false
     (see section *note Booleans::), then the <command> expressions are
     evaluated in order for effect, the <step> expressions are evaluated
     in some unspecified order, the <variable>s are bound to fresh
     locations, the results of the <step>s are stored in the bindings of
     the <variable>s, and the next iteration begins.

     If <test> evaluates to a true value, then the <expression>s are
     evaluated from left to right and the value(s) of the last
     <expression> is(are) returned.  If no <expression>s are present,
     then the value of the ‘do’ expression is unspecified.

     The region of the binding of a <variable> consists of the entire
     ‘do’ expression except for the <init>s.  It is an error for a
     <variable> to appear more than once in the list of ‘do’ variables.

     A <step> may be omitted, in which case the effect is the same as if
     ‘(<variable> <init> <variable>)’ had been written instead of
     ‘(<variable> <init>)’.

     (do ((vec (make-vector 5))
          (i 0 (+ i 1)))
         ((= i 5) vec)
       (vector-set! vec i i))               ==>  #(0 1 2 3 4)

     (let ((x '(1 3 5 7 9)))
       (do ((x x (cdr x))
            (sum 0 (+ sum (car x))))
           ((null? x) sum)))                ==>  25


 -- library syntax: let <variable> <bindings> <body>

     "Named ‘let’" is a variant on the syntax of ‘let’ which provides a
     more general looping construct than ‘do’ and may also be used to
     express recursions.  It has the same syntax and semantics as
     ordinary ‘let’ except that <variable> is bound within <body> to a
     procedure whose formal arguments are the bound variables and whose
     body is <body>.  Thus the execution of <body> may be repeated by
     invoking the procedure named by <variable>.

     (let loop ((numbers '(3 -2 1 6 -5))
                (nonneg '())
                (neg '()))
       (cond ((null? numbers) (list nonneg neg))
             ((>= (car numbers) 0)
              (loop (cdr numbers)
                    (cons (car numbers) nonneg)
                    neg))
             ((< (car numbers) 0)
              (loop (cdr numbers)
                    nonneg
                    (cons (car numbers) neg)))))
               ==>  ((6 1 3) (-5 -2))


4.2.5 Delayed evaluation
------------------------

 -- library syntax: delay <expression>

     The ‘delay’ construct is used together with the procedure ‘force’
     to implement “lazy evaluation” or “call by need”.  (delay
     <expression>) returns an object called a “promise” which at some
     point in the future may be asked (by the ‘force’ procedure) to
     evaluate <expression>, and deliver the resulting value.  The effect
     of <expression> returning multiple values is unspecified.

     See the description of ‘force’ (section *note Control features::)
     for a more complete description of ‘delay’.

4.2.6 Quasiquotation
--------------------

 -- syntax: quasiquote <qq template>

 -- syntax: `<qq template>

     "Backquote" or "quasiquote" expressions are useful for constructing
     a list or vector structure when most but not all of the desired
     structure is known in advance.  If no commas appear within the <qq
     template>, the result of evaluating `<qq template> is equivalent to
     the result of evaluating '<qq template>.  If a comma appears within
     the <qq template>, however, the expression following the comma is
     evaluated ("unquoted") and its result is inserted into the
     structure instead of the comma and the expression.  If a comma
     appears followed immediately by an at-sign (@), then the following
     expression must evaluate to a list; the opening and closing
     parentheses of the list are then "stripped away" and the elements
     of the list are inserted in place of the comma at-sign expression
     sequence.  A comma at-sign should only appear within a list or
     vector <qq template>.

     `(list ,(+ 1 2) 4)                     ==>  (list 3 4)
     (let ((name 'a)) `(list ,name ',name))
               ==>  (list a (quote a))
     `(a ,(+ 1 2) ,@(map abs '(4 -5 6)) b)
               ==>  (a 3 4 5 6 b)
     `((‘foo’ ,(- 10 3)) ,@(cdr '(c)) . ,(car '(cons)))
               ==>  ((foo 7) . cons)
     `#(10 5 ,(sqrt 4) ,@(map sqrt '(16 9)) 8)
               ==>  #(10 5 2 4 3 8)


     Quasiquote forms may be nested.  Substitutions are made only for
     unquoted components appearing at the same nesting level as the
     outermost backquote.  The nesting level increases by one inside
     each successive quasiquotation, and decreases by one inside each
     unquotation.

     `(a `(b ,(+ 1 2) ,(foo ,(+ 1 3) d) e) f)
               ==>  (a `(b ,(+ 1 2) ,(foo 4 d) e) f)
     (let ((name1 'x)
           (name2 'y))
       `(a `(b ,,name1 ,',name2 d) e))
               ==>  (a `(b ,x ,'y d) e)


     The two notations `<qq template> and (quasiquote <qq template>) are
     identical in all respects.  ‘,<expression>’ is identical to
     ‘(unquote <expression>)’, and ‘,@<expression>’ is identical to
     ‘(unquote-splicing <expression>)’.  The external syntax generated
     by ‘write’ for two-element lists whose car is one of these symbols
     may vary between implementations.

     (quasiquote (list (unquote (+ 1 2)) 4))
               ==>  (list 3 4)
     '(quasiquote (list (unquote (+ 1 2)) 4))
               ==>  `(list ,(+ 1 2) 4)
          __i.e., (quasiquote (list (unquote (+ 1 2)) 4))


     Unpredictable behavior can result if any of the symbols
     ‘quasiquote’, ‘unquote’, or ‘unquote-splicing’ appear in positions
     within a <qq template> otherwise than as described above.

4.3 Macros
==========

Scheme programs can define and use new derived expression types, called
_macros_.  Program-defined expression types have the syntax


     (<keyword> <datum> ...)


where <keyword> is an identifier that uniquely determines the expression
type.  This identifier is called the _syntactic keyword_, or simply
_keyword_, of the macro.  The number of the <datum>s, and their syntax,
depends on the expression type.

Each instance of a macro is called a _use_ of the macro.  The set of
rules that specifies how a use of a macro is transcribed into a more
primitive expression is called the _transformer_ of the macro.

The macro definition facility consists of two parts:

   • A set of expressions used to establish that certain identifiers are
     macro keywords, associate them with macro transformers, and control
     the scope within which a macro is defined, and

   • a pattern language for specifying macro transformers.

The syntactic keyword of a macro may shadow variable bindings, and local
variable bindings may shadow keyword bindings.  All macros defined using
the pattern language are "hygienic" and "referentially transparent" and
thus preserve Scheme's lexical scoping [Kohlbecker86], [ hygienic],
[Bawden88], [macrosthatwork], [syntacticabstraction]:

   • If a macro transformer inserts a binding for an identifier
     (variable or keyword), the identifier will in effect be renamed
     throughout its scope to avoid conflicts with other identifiers.
     Note that a ‘define’ at top level may or may not introduce a
     binding; see section *note Definitions::.

   • If a macro transformer inserts a free reference to an identifier,
     the reference refers to the binding that was visible where the
     transformer was specified, regardless of any local bindings that
     may surround the use of the macro.

4.3.1 Binding constructs for syntactic keywords
-----------------------------------------------

‘Let-syntax’ and ‘letrec-syntax’ are analogous to ‘let’ and ‘letrec’,
but they bind syntactic keywords to macro transformers instead of
binding variables to locations that contain values.  Syntactic keywords
may also be bound at top level; see section *note Syntax definitions::.

 -- syntax: let-syntax <bindings> <body>

     _Syntax:_ <Bindings> should have the form

     ((<keyword> <transformer spec>) ...,)


     Each <keyword> is an identifier, each <transformer spec> is an
     instance of ‘syntax-rules’, and <body> should be a sequence of one
     or more expressions.  It is an error for a <keyword> to appear more
     than once in the list of keywords being bound.

     _Semantics:_ The <body> is expanded in the syntactic environment
     obtained by extending the syntactic environment of the ‘let-syntax’
     expression with macros whose keywords are the <keyword>s, bound to
     the specified transformers.  Each binding of a <keyword> has <body>
     as its region.

     (let-syntax ((when (syntax-rules ()
                          ((when test stmt1 stmt2 ...)
                           (if test
                               (begin stmt1
                                      stmt2 ...))))))
       (let ((if #t))
         (when if (set! if 'now))
         if))                               ==>  now

     (let ((x 'outer))
       (let-syntax ((m (syntax-rules () ((m) x))))
         (let ((x 'inner))
           (m))))                           ==>  outer


 -- syntax: letrec-syntax <bindings> <body>

     _Syntax:_ Same as for ‘let-syntax’.

     _Semantics:_ The <body> is expanded in the syntactic environment
     obtained by extending the syntactic environment of the
     ‘letrec-syntax’ expression with macros whose keywords are the
     <keyword>s, bound to the specified transformers.  Each binding of a
     <keyword> has the <bindings> as well as the <body> within its
     region, so the transformers can transcribe expressions into uses of
     the macros introduced by the ‘letrec-syntax’ expression.

     (letrec-syntax
       ((my-or (syntax-rules ()
                 ((my-or) #f)
                 ((my-or e) e)
                 ((my-or e1 e2 ...)
                  (let ((temp e1))
                    (if temp
                        temp
                        (my-or e2 ...)))))))
       (let ((x #f)
             (y 7)
             (temp 8)
             (let odd?)
             (if even?))
         (my-or x
                (let temp)
                (if y)
                y)))                        ==>  7


4.3.2 Pattern language
----------------------

A <transformer spec> has the following form:

 -- : syntax-rules <literals> <syntax rule> ...,

     _Syntax:_ <Literals> is a list of identifiers and each <syntax
     rule> should be of the form

     (<pattern> <template>)


     The <pattern> in a <syntax rule> is a list <pattern> that begins
     with the keyword for the macro.

     A <pattern> is either an identifier, a constant, or one of the
     following

     (<pattern> ...)
     (<pattern> <pattern> ... . <pattern>)
     (<pattern> ... <pattern> <ellipsis>)
     #(<pattern> ...)
     #(<pattern> ... <pattern> <ellipsis>)


     and a template is either an identifier, a constant, or one of the
     following

     (<element> ...)
     (<element> <element> ... . <template>)
     #(<element> ...)


     where an <element> is a <template> optionally followed by an
     <ellipsis> and an <ellipsis> is the identifier "‘...’" (which
     cannot be used as an identifier in either a template or a pattern).

     _Semantics:_ An instance of ‘syntax-rules’ produces a new macro
     transformer by specifying a sequence of hygienic rewrite rules.  A
     use of a macro whose keyword is associated with a transformer
     specified by ‘syntax-rules’ is matched against the patterns
     contained in the <syntax rule>s, beginning with the leftmost
     <syntax rule>.  When a match is found, the macro use is transcribed
     hygienically according to the template.

     An identifier that appears in the pattern of a <syntax rule> is a
     _pattern variable_, unless it is the keyword that begins the
     pattern, is listed in <literals>, or is the identifier "‘...’".
     Pattern variables match arbitrary input elements and are used to
     refer to elements of the input in the template.  It is an error for
     the same pattern variable to appear more than once in a <pattern>.

     The keyword at the beginning of the pattern in a <syntax rule> is
     not involved in the matching and is not considered a pattern
     variable or literal identifier.

          _Rationale:_ The scope of the keyword is determined by the
          expression or syntax definition that binds it to the
          associated macro transformer.  If the keyword were a pattern
          variable or literal identifier, then the template that follows
          the pattern would be within its scope regardless of whether
          the keyword were bound by ‘let-syntax’ or by ‘letrec-syntax’.

     Identifiers that appear in <literals> are interpreted as literal
     identifiers to be matched against corresponding subforms of the
     input.  A subform in the input matches a literal identifier if and
     only if it is an identifier and either both its occurrence in the
     macro expression and its occurrence in the macro definition have
     the same lexical binding, or the two identifiers are equal and both
     have no lexical binding.

     A subpattern followed by ‘...’ can match zero or more elements of
     the input.  It is an error for ‘...’ to appear in <literals>.
     Within a pattern the identifier ‘...’ must follow the last element
     of a nonempty sequence of subpatterns.

     More formally, an input form F matches a pattern P if and only if:

        • P is a non-literal identifier; or

        • P is a literal identifier and F is an identifier with the same
          binding; or

        • P is a list ‘(P_1 ... P_n)’ and F is a list of n forms that
          match P_1 through P_n, respectively; or

        • P is an improper list ‘(P_1 P_2 ... P_n . P_n+1)’ and F is a
          list or improper list of n or more forms that match P_1
          through P_n, respectively, and whose nth "cdr" matches P_n+1;
          or

        • P is of the form ‘(P_1 ... P_n P_n+1 <ellipsis>)’ where
          <ellipsis> is the identifier ‘...’ and F is a proper list of
          at least n forms, the first n of which match P_1 through P_n,
          respectively, and each remaining element of F matches P_n+1;
          or

        • P is a vector of the form ‘#(P_1 ... P_n)’ and F is a vector
          of n forms that match P_1 through P_n; or

        • P is of the form ‘#(P_1 ... P_n P_n+1 <ellipsis>)’ where
          <ellipsis> is the identifier ‘...’ and F is a vector of n or
          more forms the first n of which match P_1 through P_n,
          respectively, and each remaining element of F matches P_n+1;
          or

        • P is a datum and F is equal to P in the sense of the ‘equal?’
          procedure.

     It is an error to use a macro keyword, within the scope of its
     binding, in an expression that does not match any of the patterns.

     When a macro use is transcribed according to the template of the
     matching <syntax rule>, pattern variables that occur in the
     template are replaced by the subforms they match in the input.
     Pattern variables that occur in subpatterns followed by one or more
     instances of the identifier ‘...’ are allowed only in subtemplates
     that are followed by as many instances of ‘...’.  They are replaced
     in the output by all of the subforms they match in the input,
     distributed as indicated.  It is an error if the output cannot be
     built up as specified.

     Identifiers that appear in the template but are not pattern
     variables or the identifier ‘...’ are inserted into the output as
     literal identifiers.  If a literal identifier is inserted as a free
     identifier then it refers to the binding of that identifier within
     whose scope the instance of ‘syntax-rules’ appears.  If a literal
     identifier is inserted as a bound identifier then it is in effect
     renamed to prevent inadvertent captures of free identifiers.

     As an example, if ‘let’ and ‘cond’ are defined as in section *note
     Derived expression type:: then they are hygienic (as required) and
     the following is not an error.

     (let ((=> #f))
       (cond (#t => 'ok)))                  ==> ok


     The macro transformer for ‘cond’ recognizes ‘=>’ as a local
     variable, and hence an expression, and not as the top-level
     identifier ‘=>’, which the macro transformer treats as a syntactic
     keyword.  Thus the example expands into

     (let ((=> #f))
       (if #t (begin => 'ok)))


     instead of

     (let ((=> #f))
       (let ((temp #t))
         (if temp ('ok temp))))


     which would result in an invalid procedure call.

5 Program structure
*******************

5.1 Programs
============

A Scheme program consists of a sequence of expressions, definitions, and
syntax definitions.  Expressions are described in chapter *note
Expressions::; definitions and syntax definitions are the subject of the
rest of the present chapter.

Programs are typically stored in files or entered interactively to a
running Scheme system, although other paradigms are possible; questions
of user interface lie outside the scope of this report.  (Indeed, Scheme
would still be useful as a notation for expressing computational methods
even in the absence of a mechanical implementation.)

Definitions and syntax definitions occurring at the top level of a
program can be interpreted declaratively.  They cause bindings to be
created in the top level environment or modify the value of existing
top-level bindings.  Expressions occurring at the top level of a program
are interpreted imperatively; they are executed in order when the
program is invoked or loaded, and typically perform some kind of
initialization.

At the top level of a program (begin <form1> ...,) is equivalent to the
sequence of expressions, definitions, and syntax definitions that form
the body of the ‘begin’.

5.2 Definitions
===============

Definitions are valid in some, but not all, contexts where expressions
are allowed.  They are valid only at the top level of a <program> and at
the beginning of a <body>.

A definition should have one of the following forms:

   • (define <variable> <expression>)

   • (define (<variable> <formals>) <body>)

     <Formals> should be either a sequence of zero or more variables, or
     a sequence of one or more variables followed by a space-delimited
     period and another variable (as in a lambda expression).  This form
     is equivalent to


          (define <variable>
            (lambda (<formals>) <body>)).


   • (define (<variable> . <formal>) <body>)

     <Formal> should be a single variable.  This form is equivalent to


          (define <variable>
            (lambda <formal> <body>)).


5.2.1 Top level definitions
---------------------------

At the top level of a program, a definition


     (define <variable> <expression>)


has essentially the same effect as the assignment expression


     (set! <variable> <expression>)


if <variable> is bound.  If <variable> is not bound, however, then the
definition will bind <variable> to a new location before performing the
assignment, whereas it would be an error to perform a ‘set!’ on an
unbound variable.


     (define add3
       (lambda (x) (+ x 3)))
     (add3 3)                               ==>  6
     (define first car)
     (first '(1 2))                         ==>  1


Some implementations of Scheme use an initial environment in which all
possible variables are bound to locations, most of which contain
undefined values.  Top level definitions in such an implementation are
truly equivalent to assignments.

5.2.2 Internal definitions
--------------------------

Definitions may occur at the beginning of a <body> (that is, the body of
a ‘lambda’, ‘let’, ‘let*’, ‘letrec’, ‘let-syntax’, or ‘letrec-syntax’
expression or that of a definition of an appropriate form).  Such
definitions are known as _internal definitions_ as opposed to the top
level definitions described above.  The variable defined by an internal
definition is local to the <body>.  That is, <variable> is bound rather
than assigned, and the region of the binding is the entire <body>.  For
example,


     (let ((x 5))
       (define foo (lambda (y) (bar x y)))
       (define bar (lambda (a b) (+ (* a b) a)))
       (foo (+ x 3)))                       ==>  45


A <body> containing internal definitions can always be converted into a
completely equivalent ‘letrec’ expression.  For example, the ‘let’
expression in the above example is equivalent to


     (let ((x 5))
       (letrec ((foo (lambda (y) (bar x y)))
                (bar (lambda (a b) (+ (* a b) a))))
         (foo (+ x 3))))


Just as for the equivalent ‘letrec’ expression, it must be possible to
evaluate each <expression> of every internal definition in a <body>
without assigning or referring to the value of any <variable> being
defined.

Wherever an internal definition may occur (begin <definition1> ...,) is
equivalent to the sequence of definitions that form the body of the
‘begin’.

5.3 Syntax definitions
======================

Syntax definitions are valid only at the top level of a <program>.

They have the following form:

(define-syntax <keyword> <transformer spec>)

<Keyword> is an identifier, and the <transformer spec> should be an
instance of ‘syntax-rules’.  The top-level syntactic environment is
extended by binding the <keyword> to the specified transformer.

There is no ‘define-syntax’ analogue of internal definitions.

Although macros may expand into definitions and syntax definitions in
any context that permits them, it is an error for a definition or syntax
definition to shadow a syntactic keyword whose meaning is needed to
determine whether some form in the group of forms that contains the
shadowing definition is in fact a definition, or, for internal
definitions, is needed to determine the boundary between the group and
the expressions that follow the group.  For example, the following are
errors:


     (define define 3)

     (begin (define begin list))

     (let-syntax
       ((foo (syntax-rules ()
               ((foo (proc args ...) body ...)
                (define proc
                  (lambda (args ...)
                    body ...))))))
       (let ((x 3))
         (foo (plus x y) (+ x y))
         (define foo x)
         (plus foo x)))


6 Standard procedures
*********************

This chapter describes Scheme's built-in procedures.  The initial (or
"top level") Scheme environment starts out with a number of variables
bound to locations containing useful values, most of which are primitive
procedures that manipulate data.  For example, the variable ‘abs’ is
bound to (a location initially containing) a procedure of one argument
that computes the absolute value of a number, and the variable ‘+’ is
bound to a procedure that computes sums.  Built-in procedures that can
easily be written in terms of other built-in procedures are identified
as "library procedures".

A program may use a top-level definition to bind any variable.  It may
subsequently alter any such binding by an assignment (see *note
Assignments::).  These operations do not modify the behavior of Scheme's
built-in procedures.  Altering any top-level binding that has not been
introduced by a definition has an unspecified effect on the behavior of
the built-in procedures.

6.1 Equivalence predicates
==========================

A “predicate” is a procedure that always returns a boolean value (#t or
#f).  An “equivalence predicate” is the computational analogue of a
mathematical equivalence relation (it is symmetric, reflexive, and
transitive).  Of the equivalence predicates described in this section,
‘eq?’ is the finest or most discriminating, and ‘equal?’ is the
coarsest.  ‘Eqv?’ is slightly less discriminating than ‘eq?’.

 -- procedure: eqv? obj1 obj2

     The ‘eqv?’ procedure defines a useful equivalence relation on
     objects.  Briefly, it returns #t if OBJ1 and OBJ2 should normally
     be regarded as the same object.  This relation is left slightly
     open to interpretation, but the following partial specification of
     ‘eqv?’ holds for all implementations of Scheme.

     The ‘eqv?’ procedure returns #t if:

        • OBJ1 and OBJ2 are both #t or both #f.

        • OBJ1 and OBJ2 are both symbols and

          (string=? (symbol->string obj1)
                    (symbol->string obj2))
                                            ==>  #t


               _Note:_ This assumes that neither OBJ1 nor OBJ2 is an
               "uninterned symbol" as alluded to in section *note
               Symbols::.  This report does not presume to specify the
               behavior of ‘eqv?’ on implementation-dependent
               extensions.

        • OBJ1 and OBJ2 are both numbers, are numerically equal (see
          ‘=’, section *note Numbers::), and are either both exact or
          both inexact.

        • OBJ1 and OBJ2 are both characters and are the same character
          according to the ‘char=?’ procedure (section *note
          Characters::).

        • both OBJ1 and OBJ2 are the empty list.

        • OBJ1 and OBJ2 are pairs, vectors, or strings that denote the
          same locations in the store (section *note Storage model::).

        • OBJ1 and OBJ2 are procedures whose location tags are equal
          (section *note Procedures::).

     The ‘eqv?’ procedure returns #f if:

        • OBJ1 and OBJ2 are of different types (section *note
          Disjointness of types::).

        • one of OBJ1 and OBJ2 is #t but the other is #f.

        • OBJ1 and OBJ2 are symbols but

          (string=? (symbol->string OBJ1)
                    (symbol->string OBJ2))
                                            ==>  #f


        • one of OBJ1 and OBJ2 is an exact number but the other is an
          inexact number.

        • OBJ1 and OBJ2 are numbers for which the ‘=’ procedure returns
          #f.

        • OBJ1 and OBJ2 are characters for which the ‘char=?’ procedure
          returns #f.

        • one of OBJ1 and OBJ2 is the empty list but the other is not.

        • OBJ1 and OBJ2 are pairs, vectors, or strings that denote
          distinct locations.

        • OBJ1 and OBJ2 are procedures that would behave differently
          (return different value(s) or have different side effects) for
          some arguments.

     (eqv? 'a 'a)                           ==>  #t
     (eqv? 'a 'b)                           ==>  #f
     (eqv? 2 2)                             ==>  #t
     (eqv? '() '())                         ==>  #t
     (eqv? 100000000 100000000)             ==>  #t
     (eqv? (cons 1 2) (cons 1 2))           ==>  #f
     (eqv? (lambda () 1)
           (lambda () 2))                   ==>  #f
     (eqv? #f 'nil)                         ==>  #f
     (let ((p (lambda (x) x)))
       (eqv? p p))                          ==>  #t


     The following examples illustrate cases in which the above rules do
     not fully specify the behavior of ‘eqv?’.  All that can be said
     about such cases is that the value returned by ‘eqv?’ must be a
     boolean.

     (eqv? "" "")                           ==>  _unspecified_
     (eqv? '#() '#())                       ==>  _unspecified_
     (eqv? (lambda (x) x)
           (lambda (x) x))                  ==>  _unspecified_
     (eqv? (lambda (x) x)
           (lambda (y) y))                  ==>  _unspecified_


     The next set of examples shows the use of ‘eqv?’ with procedures
     that have local state.  ‘Gen-counter’ must return a distinct
     procedure every time, since each procedure has its own internal
     counter.  ‘Gen-loser’, however, returns equivalent procedures each
     time, since the local state does not affect the value or side
     effects of the procedures.

     (define gen-counter
       (lambda ()
         (let ((n 0))
           (lambda () (set! n (+ n 1)) n))))
     (let ((g (gen-counter)))
       (eqv? g g))                          ==>  #t
     (eqv? (gen-counter) (gen-counter))
                                            ==>  #f
     (define gen-loser
       (lambda ()
         (let ((n 0))
           (lambda () (set! n (+ n 1)) 27))))
     (let ((g (gen-loser)))
       (eqv? g g))                          ==>  #t
     (eqv? (gen-loser) (gen-loser))
                                            ==>  _unspecified_

     (letrec ((f (lambda () (if (eqv? f g) 'both 'f)))
              (g (lambda () (if (eqv? f g) 'both 'g))))
       (eqv? f g))
                                            ==>  _unspecified_

     (letrec ((f (lambda () (if (eqv? f g) 'f 'both)))
              (g (lambda () (if (eqv? f g) 'g 'both))))
       (eqv? f g))
                                            ==>  #f


     Since it is an error to modify constant objects (those returned by
     literal expressions), implementations are permitted, though not
     required, to share structure between constants where appropriate.
     Thus the value of ‘eqv?’ on constants is sometimes
     implementation-dependent.

     (eqv? '(a) '(a))                       ==>  _unspecified_
     (eqv? "a" "a")                         ==>  _unspecified_
     (eqv? '(b) (cdr '(a b)))               ==>  _unspecified_
     (let ((x '(a)))
       (eqv? x x))                          ==>  #t


          _Rationale:_ The above definition of ‘eqv?’ allows
          implementations latitude in their treatment of procedures and
          literals: implementations are free either to detect or to fail
          to detect that two procedures or two literals are equivalent
          to each other, and can decide whether or not to merge
          representations of equivalent objects by using the same
          pointer or bit pattern to represent both.

 -- procedure: eq? obj1 obj2

     ‘Eq?’ is similar to ‘eqv?’ except that in some cases it is capable
     of discerning distinctions finer than those detectable by ‘eqv?’.

     ‘Eq?’ and ‘eqv?’ are guaranteed to have the same behavior on
     symbols, booleans, the empty list, pairs, procedures, and non-empty
     strings and vectors.  ‘Eq?’'s behavior on numbers and characters is
     implementation-dependent, but it will always return either true or
     false, and will return true only when ‘eqv?’ would also return
     true.  ‘Eq?’ may also behave differently from ‘eqv?’ on empty
     vectors and empty strings.

     (eq? 'a 'a)                            ==>  #t
     (eq? '(a) '(a))                        ==>  _unspecified_
     (eq? (list 'a) (list 'a))              ==>  #f
     (eq? "a" "a")                          ==>  _unspecified_
     (eq? "" "")                            ==>  _unspecified_
     (eq? '() '())                          ==>  #t
     (eq? 2 2)                              ==>  _unspecified_
     (eq? #\A #\A)                          ==>  _unspecified_
     (eq? car car)                          ==>  #t
     (let ((n (+ 2 3)))
       (eq? n n))                           ==>  _unspecified_
     (let ((x '(a)))
       (eq? x x))                           ==>  #t
     (let ((x '#()))
       (eq? x x))                           ==>  #t
     (let ((p (lambda (x) x)))
       (eq? p p))                           ==>  #t


          _Rationale:_ It will usually be possible to implement ‘eq?’
          much more efficiently than ‘eqv?’, for example, as a simple
          pointer comparison instead of as some more complicated
          operation.  One reason is that it may not be possible to
          compute ‘eqv?’ of two numbers in constant time, whereas ‘eq?’
          implemented as pointer comparison will always finish in
          constant time.  ‘Eq?’ may be used like ‘eqv?’ in applications
          using procedures to implement objects with state since it
          obeys the same constraints as ‘eqv?’.

 -- library procedure: equal? obj1 obj2

     ‘Equal?’ recursively compares the contents of pairs, vectors, and
     strings, applying ‘eqv?’ on other objects such as numbers and
     symbols.  A rule of thumb is that objects are generally ‘equal?’ if
     they print the same.  ‘Equal?’ may fail to terminate if its
     arguments are circular data structures.

     (equal? 'a 'a)                         ==>  #t
     (equal? '(a) '(a))                     ==>  #t
     (equal? '(a (b) c)
             '(a (b) c))                    ==>  #t
     (equal? "abc" "abc")                   ==>  #t
     (equal? 2 2)                           ==>  #t
     (equal? (make-vector 5 'a)
             (make-vector 5 'a))            ==>  #t
     (equal? (lambda (x) x)
             (lambda (y) y))                ==>  _unspecified_


6.2 Numbers
===========

Numerical computation has traditionally been neglected by the Lisp
community.  Until Common Lisp there was no carefully thought out
strategy for organizing numerical computation, and with the exception of
the MacLisp system [Pitman83] little effort was made to execute
numerical code efficiently.  This report recognizes the excellent work
of the Common Lisp committee and accepts many of their recommendations.
In some ways this report simplifies and generalizes their proposals in a
manner consistent with the purposes of Scheme.

It is important to distinguish between the mathematical numbers, the
Scheme numbers that attempt to model them, the machine representations
used to implement the Scheme numbers, and notations used to write
numbers.  This report uses the types number, complex, real, rational,
and integer to refer to both mathematical numbers and Scheme numbers.
Machine representations such as fixed point and floating point are
referred to by names such as fixnum and flonum.

6.2.1 Numerical types
---------------------

Mathematically, numbers may be arranged into a tower of subtypes in
which each level is a subset of the level above it:

         number
          complex
          real
          rational
          integer

For example, 3 is an integer.  Therefore 3 is also a rational, a real,
and a complex.  The same is true of the Scheme numbers that model 3.
For Scheme numbers, these types are defined by the predicates ‘number?’,
‘complex?’, ‘real?’, ‘rational?’, and ‘integer?’.

There is no simple relationship between a number's type and its
representation inside a computer.  Although most implementations of
Scheme will offer at least two different representations of 3, these
different representations denote the same integer.

Scheme's numerical operations treat numbers as abstract data, as
independent of their representation as possible.  Although an
implementation of Scheme may use fixnum, flonum, and perhaps other
representations for numbers, this should not be apparent to a casual
programmer writing simple programs.

It is necessary, however, to distinguish between numbers that are
represented exactly and those that may not be.  For example, indexes
into data structures must be known exactly, as must some polynomial
coefficients in a symbolic algebra system.  On the other hand, the
results of measurements are inherently inexact, and irrational numbers
may be approximated by rational and therefore inexact approximations.
In order to catch uses of inexact numbers where exact numbers are
required, Scheme explicitly distinguishes exact from inexact numbers.
This distinction is orthogonal to the dimension of type.

6.2.2 Exactness
---------------

Scheme numbers are either exact or inexact.  A number is exact if it was
written as an exact constant or was derived from exact numbers using
only exact operations.  A number is inexact if it was written as an
inexact constant, if it was derived using inexact ingredients, or if it
was derived using inexact operations.  Thus inexactness is a contagious
property of a number.

If two implementations produce exact results for a computation that did
not involve inexact intermediate results, the two ultimate results will
be mathematically equivalent.  This is generally not true of
computations involving inexact numbers since approximate methods such as
floating point arithmetic may be used, but it is the duty of each
implementation to make the result as close as practical to the
mathematically ideal result.

Rational operations such as ‘+’ should always produce exact results when
given exact arguments.  If the operation is unable to produce an exact
result, then it may either report the violation of an implementation
restriction or it may silently coerce its result to an inexact value.
See section *note Implementation restrictions::.

With the exception of ‘inexact->exact’, the operations described in this
section must generally return inexact results when given any inexact
arguments.  An operation may, however, return an exact result if it can
prove that the value of the result is unaffected by the inexactness of
its arguments.  For example, multiplication of any number by an exact
zero may produce an exact zero result, even if the other argument is
inexact.

6.2.3 Implementation restrictions
---------------------------------

Implementations of Scheme are not required to implement the whole tower
of subtypes given in section *note Numerical types::, but they must
implement a coherent subset consistent with both the purposes of the
implementation and the spirit of the Scheme language.  For example, an
implementation in which all numbers are real may still be quite useful.

Implementations may also support only a limited range of numbers of any
type, subject to the requirements of this section.  The supported range
for exact numbers of any type may be different from the supported range
for inexact numbers of that type.  For example, an implementation that
uses flonums to represent all its inexact real numbers may support a
practically unbounded range of exact integers and rationals while
limiting the range of inexact reals (and therefore the range of inexact
integers and rationals) to the dynamic range of the flonum format.
Furthermore the gaps between the representable inexact integers and
rationals are likely to be very large in such an implementation as the
limits of this range are approached.

An implementation of Scheme must support exact integers throughout the
range of numbers that may be used for indexes of lists, vectors, and
strings or that may result from computing the length of a list, vector,
or string.  The ‘length’, ‘vector-length’, and ‘string-length’
procedures must return an exact integer, and it is an error to use
anything but an exact integer as an index.  Furthermore any integer
constant within the index range, if expressed by an exact integer
syntax, will indeed be read as an exact integer, regardless of any
implementation restrictions that may apply outside this range.  Finally,
the procedures listed below will always return an exact integer result
provided all their arguments are exact integers and the mathematically
expected result is representable as an exact integer within the
implementation:


     +            -             *
     quotient     remainder     modulo
     max          min           abs
     numerator    denominator   gcd
     lcm          floor         ceiling
     truncate     round         rationalize
     expt


Implementations are encouraged, but not required, to support exact
integers and exact rationals of practically unlimited size and
precision, and to implement the above procedures and the ‘/’ procedure
in such a way that they always return exact results when given exact
arguments.  If one of these procedures is unable to deliver an exact
result when given exact arguments, then it may either report a violation
of an implementation restriction or it may silently coerce its result to
an inexact number.  Such a coercion may cause an error later.

An implementation may use floating point and other approximate
representation strategies for inexact numbers.

This report recommends, but does not require, that the IEEE 32-bit and
64-bit floating point standards be followed by implementations that use
flonum representations, and that implementations using other
representations should match or exceed the precision achievable using
these floating point standards [IEEE].

In particular, implementations that use flonum representations must
follow these rules: A flonum result must be represented with at least as
much precision as is used to express any of the inexact arguments to
that operation.  It is desirable (but not required) for potentially
inexact operations such as ‘sqrt’, when applied to exact arguments, to
produce exact answers whenever possible (for example the square root of
an exact 4 ought to be an exact 2).  If, however, an exact number is
operated upon so as to produce an inexact result (as by ‘sqrt’), and if
the result is represented as a flonum, then the most precise flonum
format available must be used; but if the result is represented in some
other way then the representation must have at least as much precision
as the most precise flonum format available.

Although Scheme allows a variety of written notations for numbers, any
particular implementation may support only some of them.  For example,
an implementation in which all numbers are real need not support the
rectangular and polar notations for complex numbers.  If an
implementation encounters an exact numerical constant that it cannot
represent as an exact number, then it may either report a violation of
an implementation restriction or it may silently represent the constant
by an inexact number.

6.2.4 Syntax of numerical constants
-----------------------------------

The syntax of the written representations for numbers is described
formally in section *note Lexical structure::.  Note that case is not
significant in numerical constants.

A number may be written in binary, octal, decimal, or hexadecimal by the
use of a radix prefix.  The radix prefixes are ‘#b’ (binary), ‘#o’
(octal), ‘#d’ (decimal), and ‘#x’ (hexadecimal).  With no radix prefix,
a number is assumed to be expressed in decimal.

A numerical constant may be specified to be either exact or inexact by a
prefix.  The prefixes are ‘#e’ for exact, and ‘#i’ for inexact.  An
exactness prefix may appear before or after any radix prefix that is
used.  If the written representation of a number has no exactness
prefix, the constant may be either inexact or exact.  It is inexact if
it contains a decimal point, an exponent, or a "#" character in the
place of a digit, otherwise it is exact.

In systems with inexact numbers of varying precisions it may be useful
to specify the precision of a constant.  For this purpose, numerical
constants may be written with an exponent marker that indicates the
desired precision of the inexact representation.  The letters ‘s’, ‘f’,
‘d’, and ‘l’ specify the use of SHORT, SINGLE, DOUBLE, and LONG
precision, respectively.  (When fewer than four internal inexact
representations exist, the four size specifications are mapped onto
those available.  For example, an implementation with two internal
representations may map short and single together and long and double
together.)  In addition, the exponent marker ‘e’ specifies the default
precision for the implementation.  The default precision has at least as
much precision as DOUBLE, but implementations may wish to allow this
default to be set by the user.


     3.14159265358979F0
            Round to single -- 3.141593
     0.6L0
            Extend to long -- .600000000000000


6.2.5 Numerical operations
--------------------------

The reader is referred to section *note Entry format:: for a summary of
the naming conventions used to specify restrictions on the types of
arguments to numerical routines.

The examples used in this section assume that any numerical constant
written using an exact notation is indeed represented as an exact
number.  Some examples also assume that certain numerical constants
written using an inexact notation can be represented without loss of
accuracy; the inexact constants were chosen so that this is likely to be
true in implementations that use flonums to represent inexact numbers.

 -- procedure: number? obj
 -- procedure: complex? obj
 -- procedure: real? obj
 -- procedure: rational? obj
 -- procedure: integer? obj

     These numerical type predicates can be applied to any kind of
     argument, including non-numbers.  They return #t if the object is
     of the named type, and otherwise they return #f.  In general, if a
     type predicate is true of a number then all higher type predicates
     are also true of that number.  Consequently, if a type predicate is
     false of a number, then all lower type predicates are also false of
     that number.

     If Z is an inexact complex number, then ‘(real? Z)’ is true if and
     only if ‘(zero? (imag-part Z))’ is true.  If X is an inexact real
     number, then ‘(integer? X)’ is true if and only if ‘(= X (round
     X))’.

     (complex? 3+4i)                        ==>  #t
     (complex? 3)                           ==>  #t
     (real? 3)                              ==>  #t
     (real? -2.5+0.0i)                      ==>  #t
     (real? #e1e10)                         ==>  #t
     (rational? 6/10)                       ==>  #t
     (rational? 6/3)                        ==>  #t
     (integer? 3+0i)                        ==>  #t
     (integer? 3.0)                         ==>  #t
     (integer? 8/4)                         ==>  #t


          _Note:_ The behavior of these type predicates on inexact
          numbers is unreliable, since any inaccuracy may affect the
          result.

          _Note:_ In many implementations the ‘rational?’ procedure will
          be the same as ‘real?’, and the ‘complex?’ procedure will be
          the same as ‘number?’, but unusual implementations may be able
          to represent some irrational numbers exactly or may extend the
          number system to support some kind of non-complex numbers.

 -- procedure: exact? Z
 -- procedure: inexact? Z

     These numerical predicates provide tests for the exactness of a
     quantity.  For any Scheme number, precisely one of these predicates
     is true.

 -- procedure: = z1 z2 z3 ...,
 -- procedure: < x1 x2 x3 ...,
 -- procedure: > x1 x2 x3 ...,
 -- procedure: <= x1 x2 x3 ...,
 -- procedure: >= x1 x2 x3 ...,

     These procedures return #t if their arguments are (respectively):
     equal, monotonically increasing, monotonically decreasing,
     monotonically nondecreasing, or monotonically nonincreasing.

     These predicates are required to be transitive.

          _Note:_ The traditional implementations of these predicates in
          Lisp-like languages are not transitive.

          _Note:_ While it is not an error to compare inexact numbers
          using these predicates, the results may be unreliable because
          a small inaccuracy may affect the result; this is especially
          true of ‘=’ and ‘zero?’.  When in doubt, consult a numerical
          analyst.

 -- library procedure: zero? Z
 -- library procedure: positive? X
 -- library procedure: negative? X
 -- library procedure: odd? N
 -- library procedure: even? N

     These numerical predicates test a number for a particular property,
     returning #t or #f.  See note above.

 -- library procedure: max x1 x2 ...,
 -- library procedure: min x1 x2 ...,

     These procedures return the maximum or minimum of their arguments.

     (max 3 4)                              ==>  4    ; exact
     (max 3.9 4)                            ==>  4.0  ; inexact


          _Note:_ If any argument is inexact, then the result will also
          be inexact (unless the procedure can prove that the inaccuracy
          is not large enough to affect the result, which is possible
          only in unusual implementations).  If ‘min’ or ‘max’ is used
          to compare numbers of mixed exactness, and the numerical value
          of the result cannot be represented as an inexact number
          without loss of accuracy, then the procedure may report a
          violation of an implementation restriction.

 -- procedure: + z1 ...,
 -- procedure: * z1 ...,

     These procedures return the sum or product of their arguments.

     (+ 3 4)                                ==>  7
     (+ 3)                                  ==>  3
     (+)                                    ==>  0
     (* 4)                                  ==>  4
     (*)                                    ==>  1


 -- procedure: - z1 z2
 -- procedure: - Z
 -- optional procedure: - z1 z2 ...,
 -- procedure: / z1 z2
 -- procedure: / Z
 -- optional procedure: / z1 z2 ...,

     With two or more arguments, these procedures return the difference
     or quotient of their arguments, associating to the left.  With one
     argument, however, they return the additive or multiplicative
     inverse of their argument.

     (- 3 4)                                ==>  -1
     (- 3 4 5)                              ==>  -6
     (- 3)                                  ==>  -3
     (/ 3 4 5)                              ==>  3/20
     (/ 3)                                  ==>  1/3


 -- library procedure: abs x

     ‘Abs’ returns the absolute value of its argument.

     (abs -7)                               ==>  7


 -- procedure: quotient n1 n2
 -- procedure: remainder n1 n2
 -- procedure: modulo n1 n2

     These procedures implement number-theoretic (integer) division.  N2
     should be non-zero.  All three procedures return integers.  If
     N1/N2 is an integer:

         (quotient N1 N2)                   ==> N1/N2
         (remainder N1 N2)                  ==> 0
         (modulo N1 N2)                     ==> 0


     If N1/N2 is not an integer:

         (quotient N1 N2)                   ==> N_Q
         (remainder N1 N2)                  ==> N_R
         (modulo N1 N2)                     ==> N_M


     where N_Q is N1/N2 rounded towards zero, 0 < |N_R| < |N2|, 0 <
     |N_M| < |N2|, N_R and N_M differ from N1 by a multiple of N2, N_R
     has the same sign as N1, and N_M has the same sign as N2.

     From this we can conclude that for integers N1 and N2 with N2 not
     equal to 0,

          (= N1 (+ (* N2 (quotient N1 N2))
                (remainder N1 N2)))
                                            ==>  #t


     provided all numbers involved in that computation are exact.

     (modulo 13 4)                          ==>  1
     (remainder 13 4)                       ==>  1

     (modulo -13 4)                         ==>  3
     (remainder -13 4)                      ==>  -1

     (modulo 13 -4)                         ==>  -3
     (remainder 13 -4)                      ==>  1

     (modulo -13 -4)                        ==>  -1
     (remainder -13 -4)                     ==>  -1

     (remainder -13 -4.0)                   ==>  -1.0  ; inexact


 -- library procedure: gcd n1 ...,
 -- library procedure: lcm n1 ...,

     These procedures return the greatest common divisor or least common
     multiple of their arguments.  The result is always non-negative.

     (gcd 32 -36)                           ==>  4
     (gcd)                                  ==>  0
     (lcm 32 -36)                           ==>  288
     (lcm 32.0 -36)                         ==>  288.0  ; inexact
     (lcm)                                  ==>  1


 -- procedure: numerator Q
 -- procedure: denominator Q

     These procedures return the numerator or denominator of their
     argument; the result is computed as if the argument was represented
     as a fraction in lowest terms.  The denominator is always positive.
     The denominator of 0 is defined to be 1.

     (numerator (/ 6 4))                    ==>  3
     (denominator (/ 6 4))                  ==>  2
     (denominator
       (exact->inexact (/ 6 4)))            ==> 2.0


 -- procedure: floor x
 -- procedure: ceiling x
 -- procedure: truncate x
 -- procedure: round x

     These procedures return integers.  ‘Floor’ returns the largest
     integer not larger than X.  ‘Ceiling’ returns the smallest integer
     not smaller than X.  ‘Truncate’ returns the integer closest to X
     whose absolute value is not larger than the absolute value of X.
     ‘Round’ returns the closest integer to X, rounding to even when X
     is halfway between two integers.

          _Rationale:_ ‘Round’ rounds to even for consistency with the
          default rounding mode specified by the IEEE floating point
          standard.

          _Note:_ If the argument to one of these procedures is inexact,
          then the result will also be inexact.  If an exact value is
          needed, the result should be passed to the ‘inexact->exact’
          procedure.

     (floor -4.3)                           ==>  -5.0
     (ceiling -4.3)                         ==>  -4.0
     (truncate -4.3)                        ==>  -4.0
     (round -4.3)                           ==>  -4.0

     (floor 3.5)                            ==>  3.0
     (ceiling 3.5)                          ==>  4.0
     (truncate 3.5)                         ==>  3.0
     (round 3.5)                            ==>  4.0  ; inexact

     (round 7/2)                            ==>  4    ; exact
     (round 7)                              ==>  7


 -- library procedure: rationalize x y

     ‘Rationalize’ returns the _simplest_ rational number differing from
     X by no more than Y.  A rational number r_1 is _simpler_ than
     another rational number r_2 if r_1 = p_1/q_1 and r_2 = p_2/q_2 (in
     lowest terms) and |p_1|<= |p_2| and |q_1| <= |q_2|.  Thus 3/5 is
     simpler than 4/7.  Although not all rationals are comparable in
     this ordering (consider 2/7 and 3/5) any interval contains a
     rational number that is simpler than every other rational number in
     that interval (the simpler 2/5 lies between 2/7 and 3/5).  Note
     that 0 = 0/1 is the simplest rational of all.

     (rationalize
       (inexact->exact .3) 1/10)            ==> 1/3    ; exact
     (rationalize .3 1/10)                  ==> #i1/3  ; inexact


 -- procedure: exp Z
 -- procedure: log Z
 -- procedure: sin Z
 -- procedure: cos Z
 -- procedure: tan Z
 -- procedure: asin Z
 -- procedure: acos Z
 -- procedure: atan Z
 -- procedure: atan Y X

     These procedures are part of every implementation that supports
     general real numbers; they compute the usual transcendental
     functions.  ‘log’ computes the natural logarithm of Z (not the base
     ten logarithm).  ‘asin’, ‘acos’, and ‘atan’ compute arcsine
     (sin^-1), arccosine (cos^-1), and arctangent (tan^-1),
     respectively.  The two-argument variant of ‘atan’ computes (angle
     (make-rectangular X Y)) (see below), even in implementations that
     don't support general complex numbers.

     In general, the mathematical functions log, arcsine, arccosine, and
     arctangent are multiply defined.  The value of log z is defined to
     be the one whose imaginary part lies in the range from -pi
     (exclusive) to pi (inclusive).  log 0 is undefined.  With log
     defined this way, the values of sin^-1 z, cos^-1 z, and tan^-1 z
     are according to the following formulae:

                sin^-1 z = -i log (i z + sqrt(1 - z^2))

                     cos^-1 z = pi / 2 - sin^-1 z

          tan^-1 z = (log (1 + i z) - log (1 - i z)) / (2 i)

     The above specification follows [CLtL], which in turn cites
     [Penfield81]; refer to these sources for more detailed discussion
     of branch cuts, boundary conditions, and implementation of these
     functions.  When it is possible these procedures produce a real
     result from a real argument.

 -- procedure: sqrt Z

     Returns the principal square root of Z.  The result will have
     either positive real part, or zero real part and non-negative
     imaginary part.

 -- procedure: expt z1 z2

     Returns Z1 raised to the power Z2.  For z_1 ~= 0

                        z_1^z_2 = e^z_2 log z_1

     0^z is 1 if z = 0 and 0 otherwise.

 -- procedure: make-rectangular x1 x2
 -- procedure: make-polar x3 x4
 -- procedure: real-part Z
 -- procedure: imag-part Z
 -- procedure: magnitude Z
 -- procedure: angle Z

     These procedures are part of every implementation that supports
     general complex numbers.  Suppose X1, X2, X3, and X4 are real
     numbers and Z is a complex number such that

                      Z = X1 + X2i = X3 . e^i X4

     Then

     (make-rectangular X1 X2)               ==> Z
     (make-polar X3 X4)                     ==> Z
     (real-part Z)                          ==> X1
     (imag-part Z)                          ==> X2
     (magnitude Z)                          ==> |X3|
     (angle Z)                              ==> x_angle


     where -pi < x_angle <= pi with x_angle = X4 + 2pi n for some
     integer n.

          _Rationale:_ ‘Magnitude’ is the same as ‘abs’ for a real
          argument, but ‘abs’ must be present in all implementations,
          whereas ‘magnitude’ need only be present in implementations
          that support general complex numbers.

 -- procedure: exact->inexact Z
 -- procedure: inexact->exact Z

     ‘Exact->inexact’ returns an inexact representation of Z.  The value
     returned is the inexact number that is numerically closest to the
     argument.  If an exact argument has no reasonably close inexact
     equivalent, then a violation of an implementation restriction may
     be reported.

     ‘Inexact->exact’ returns an exact representation of Z.  The value
     returned is the exact number that is numerically closest to the
     argument.  If an inexact argument has no reasonably close exact
     equivalent, then a violation of an implementation restriction may
     be reported.

     These procedures implement the natural one-to-one correspondence
     between exact and inexact integers throughout an
     implementation-dependent range.  See section *note Implementation
     restrictions::.




6.2.6 Numerical input and output
--------------------------------

 -- procedure: number->string z
 -- procedure: number->string z radix

     RADIX must be an exact integer, either 2, 8, 10, or 16.  If
     omitted, RADIX defaults to 10.  The procedure ‘number->string’
     takes a number and a radix and returns as a string an external
     representation of the given number in the given radix such that

     (let ((number NUMBER)
           (radix RADIX))
       (eqv? number
             (string->number (number->string number
                                             radix)
                             radix)))


     is true.  It is an error if no possible result makes this
     expression true.

     If Z is inexact, the radix is 10, and the above expression can be
     satisfied by a result that contains a decimal point, then the
     result contains a decimal point and is expressed using the minimum
     number of digits (exclusive of exponent and trailing zeroes) needed
     to make the above expression true [howtoprint], [howtoread];
     otherwise the format of the result is unspecified.

     The result returned by ‘number->string’ never contains an explicit
     radix prefix.

          _Note:_ The error case can occur only when Z is not a complex
          number or is a complex number with a non-rational real or
          imaginary part.

          _Rationale:_ If Z is an inexact number represented using
          flonums, and the radix is 10, then the above expression is
          normally satisfied by a result containing a decimal point.
          The unspecified case allows for infinities, NaNs, and
          non-flonum representations.

 -- procedure: string->number string
 -- procedure: string->number string radix

     Returns a number of the maximally precise representation expressed
     by the given STRING.  RADIX must be an exact integer, either 2, 8,
     10, or 16.  If supplied, RADIX is a default radix that may be
     overridden by an explicit radix prefix in STRING (e.g.  "#o177").
     If RADIX is not supplied, then the default radix is 10.  If STRING
     is not a syntactically valid notation for a number, then
     ‘string->number’ returns #f.

     (string->number "100")                 ==>  100
     (string->number "100" 16)              ==>  256
     (string->number "1e2")                 ==>  100.0
     (string->number "15##")                ==>  1500.0


          _Note:_ The domain of ‘string->number’ may be restricted by
          implementations in the following ways.  ‘String->number’ is
          permitted to return #f whenever STRING contains an explicit
          radix prefix.  If all numbers supported by an implementation
          are real, then ‘string->number’ is permitted to return #f
          whenever STRING uses the polar or rectangular notations for
          complex numbers.  If all numbers are integers, then
          ‘string->number’ may return #f whenever the fractional
          notation is used.  If all numbers are exact, then
          ‘string->number’ may return #f whenever an exponent marker or
          explicit exactness prefix is used, or if a # appears in place
          of a digit.  If all inexact numbers are integers, then
          ‘string->number’ may return #f whenever a decimal point is
          used.

6.3 Other data types
====================

This section describes operations on some of Scheme's non-numeric data
types: booleans, pairs, lists, symbols, characters, strings and vectors.

6.3.1 Booleans
--------------

The standard boolean objects for true and false are written as #t and
#f.  What really matters, though, are the objects that the Scheme
conditional expressions (‘if’, ‘cond’, ‘and’, ‘or’, ‘do’) treat as true
or false.  The phrase "a true value" (or sometimes just "true") means
any object treated as true by the conditional expressions, and the
phrase "a false value" (or "false") means any object treated as false by
the conditional expressions.

Of all the standard Scheme values, only #f counts as false in
conditional expressions.  Except for #f, all standard Scheme values,
including #t, pairs, the empty list, symbols, numbers, strings, vectors,
and procedures, count as true.

     _Note:_ Programmers accustomed to other dialects of Lisp should be
     aware that Scheme distinguishes both #f and the empty list from the
     symbol ‘nil’.

Boolean constants evaluate to themselves, so they do not need to be
quoted in programs.


     #t                                     ==>  #t
     #f                                     ==>  #f
     '#f                                    ==>  #f


 -- library procedure: not obj

     ‘Not’ returns #t if OBJ is false, and returns #f otherwise.

     (not #t)                               ==>  #f
     (not 3)                                ==>  #f
     (not (list 3))                         ==>  #f
     (not #f)                               ==>  #t
     (not '())                              ==>  #f
     (not (list))                           ==>  #f
     (not 'nil)                             ==>  #f


 -- library procedure: boolean? obj

     ‘Boolean?’ returns #t if OBJ is either #t or #f and returns #f
     otherwise.

     (boolean? #f)                          ==>  #t
     (boolean? 0)                           ==>  #f
     (boolean? '())                         ==>  #f


6.3.2 Pairs and lists
---------------------

A “pair” (sometimes called a “dotted pair”) is a record structure with
two fields called the car and cdr fields (for historical reasons).
Pairs are created by the procedure ‘cons’.  The car and cdr fields are
accessed by the procedures ‘car’ and ‘cdr’.  The car and cdr fields are
assigned by the procedures ‘set-car!’ and ‘set-cdr!’.

Pairs are used primarily to represent lists.  A list can be defined
recursively as either the empty list or a pair whose cdr is a list.
More precisely, the set of lists is defined as the smallest set X such
that

   • The empty list is in X.
   • If LIST is in X, then any pair whose cdr field contains LIST is
     also in X.

The objects in the car fields of successive pairs of a list are the
elements of the list.  For example, a two-element list is a pair whose
car is the first element and whose cdr is a pair whose car is the second
element and whose cdr is the empty list.  The length of a list is the
number of elements, which is the same as the number of pairs.

The empty list is a special object of its own type (it is not a pair);
it has no elements and its length is zero.

     _Note:_ The above definitions imply that all lists have finite
     length and are terminated by the empty list.

The most general notation (external representation) for Scheme pairs is
the "dotted" notation ‘(C1 . C2)’ where C1 is the value of the car field
and C2 is the value of the cdr field.  For example ‘(4 . 5)’ is a pair
whose car is 4 and whose cdr is 5.  Note that ‘(4 . 5)’ is the external
representation of a pair, not an expression that evaluates to a pair.

A more streamlined notation can be used for lists: the elements of the
list are simply enclosed in parentheses and separated by spaces.  The
empty list is written () .  For example,


     (a b c d e)


and


     (a . (b . (c . (d . (e . ())))))


are equivalent notations for a list of symbols.

A chain of pairs not ending in the empty list is called an “improper
list”.  Note that an improper list is not a list.  The list and dotted
notations can be combined to represent improper lists:


     (a b c . d)


is equivalent to


     (a . (b . (c . d)))


Whether a given pair is a list depends upon what is stored in the cdr
field.  When the ‘set-cdr!’ procedure is used, an object can be a list
one moment and not the next:


     (define x (list 'a 'b 'c))
     (define y x)
     y                                      ==>  (a b c)
     (list? y)                              ==>  #t
     (set-cdr! x 4)                         ==>  _unspecified_
     x                                      ==>  (a . 4)
     (eqv? x y)                             ==>  #t
     y                                      ==>  (a . 4)
     (list? y)                              ==>  #f
     (set-cdr! x x)                         ==>  _unspecified_
     (list? x)                              ==>  #f


Within literal expressions and representations of objects read by the
‘read’ procedure, the forms '<datum>, `<datum>, ,<datum>, and ,@<datum>
denote two-element lists whose first elements are the symbols ‘quote’,
‘quasiquote’, ‘unquote’, and ‘unquote-splicing’, respectively.  The
second element in each case is <datum>.  This convention is supported so
that arbitrary Scheme programs may be represented as lists.  That is,
according to Scheme's grammar, every <expression> is also a <datum> (see
section *note External representation::).  Among other things, this
permits the use of the ‘read’ procedure to parse Scheme programs.  See
section *note External representations::.

 -- procedure: pair? obj

     ‘Pair?’ returns #t if OBJ is a pair, and otherwise returns #f.

     (pair? '(a . b))                       ==>  #t
     (pair? '(a b c))                       ==>  #t
     (pair? '())                            ==>  #f
     (pair? '#(a b))                        ==>  #f


 -- procedure: cons obj1 obj2

     Returns a newly allocated pair whose car is OBJ1 and whose cdr is
     OBJ2.  The pair is guaranteed to be different (in the sense of
     ‘eqv?’) from every existing object.

     (cons 'a '())                          ==>  (a)
     (cons '(a) '(b c d))                   ==>  ((a) b c d)
     (cons "a" '(b c))                      ==>  ("a" b c)
     (cons 'a 3)                            ==>  (a . 3)
     (cons '(a b) 'c)                       ==>  ((a b) . c)


 -- procedure: car pair

     Returns the contents of the car field of PAIR.  Note that it is an
     error to take the car of the empty list.

     (car '(a b c))                         ==>  a
     (car '((a) b c d))                     ==>  (a)
     (car '(1 . 2))                         ==>  1
     (car '())                              ==>  _error_


 -- procedure: cdr pair

     Returns the contents of the cdr field of PAIR.  Note that it is an
     error to take the cdr of the empty list.

     (cdr '((a) b c d))                     ==>  (b c d)
     (cdr '(1 . 2))                         ==>  2
     (cdr '())                              ==>  _error_


 -- procedure: set-car! pair obj

     Stores OBJ in the car field of PAIR.  The value returned by
     ‘set-car!’ is unspecified.

     (define (f) (list 'not-a-constant-list))
     (define (g) '(constant-list))
     (set-car! (f) 3)                       ==>  _unspecified_
     (set-car! (g) 3)                       ==>  _error_


 -- procedure: set-cdr! pair obj

     Stores OBJ in the cdr field of PAIR.  The value returned by
     ‘set-cdr!’ is unspecified.

 -- library procedure: caar pair
 -- library procedure: cadr pair

 --  ...:  ...

 -- library procedure: cdddar pair
 -- library procedure: cddddr pair

     These procedures are compositions of ‘car’ and ‘cdr’, where for
     example ‘caddr’ could be defined by

     (define caddr (lambda (x) (car (cdr (cdr x))))).


     Arbitrary compositions, up to four deep, are provided.  There are
     twenty-eight of these procedures in all.

 -- library procedure: null? obj

     Returns #t if OBJ is the empty list, otherwise returns #f.

 -- library procedure: list? obj

     Returns #t if OBJ is a list, otherwise returns #f.  By definition,
     all lists have finite length and are terminated by the empty list.

             (list? '(a b c))               ==>  #t
             (list? '())                    ==>  #t
             (list? '(a . b))               ==>  #f
             (let ((x (list 'a)))
               (set-cdr! x x)
               (list? x))                   ==>  #f


 -- library procedure: list OBJ ...,

     Returns a newly allocated list of its arguments.

     (list 'a (+ 3 4) 'c)                   ==>  (a 7 c)
     (list)                                 ==>  ()


 -- library procedure: length list

     Returns the length of LIST.

     (length '(a b c))                      ==>  3
     (length '(a (b) (c d e)))              ==>  3
     (length '())                           ==>  0


 -- library procedure: append list ...,

     Returns a list consisting of the elements of the first LIST
     followed by the elements of the other LISTs.

     (append '(x) '(y))                     ==>  (x y)
     (append '(a) '(b c d))                 ==>  (a b c d)
     (append '(a (b)) '((c)))               ==>  (a (b) (c))


     The resulting list is always newly allocated, except that it shares
     structure with the last LIST argument.  The last argument may
     actually be any object; an improper list results if the last
     argument is not a proper list.

     (append '(a b) '(c . d))               ==>  (a b c . d)
     (append '() 'a)                        ==>  a


 -- library procedure: reverse list

     Returns a newly allocated list consisting of the elements of LIST
     in reverse order.

     (reverse '(a b c))                     ==>  (c b a)
     (reverse '(a (b c) d (e (f))))
               ==>  ((e (f)) d (b c) a)


 -- library procedure: list-tail list K

     Returns the sublist of LIST obtained by omitting the first K
     elements.  It is an error if LIST has fewer than K elements.
     ‘List-tail’ could be defined by

     (define list-tail
       (lambda (x k)
         (if (zero? k)
             x
             (list-tail (cdr x) (- k 1)))))


 -- library procedure: list-ref list K

     Returns the Kth element of LIST.  (This is the same as the car of
     (list-tail LIST K).)  It is an error if LIST has fewer than K
     elements.

     (list-ref '(a b c d) 2)                 ==>  c
     (list-ref '(a b c d)
               (inexact->exact (round 1.8)))
               ==>  c


 -- library procedure: memq obj list
 -- library procedure: memv obj list
 -- library procedure: member obj list

     These procedures return the first sublist of LIST whose car is OBJ,
     where the sublists of LIST are the non-empty lists returned by
     (list-tail LIST K) for K less than the length of LIST.  If OBJ does
     not occur in LIST, then #f (not the empty list) is returned.
     ‘Memq’ uses ‘eq?’ to compare OBJ with the elements of LIST, while
     ‘memv’ uses ‘eqv?’ and ‘member’ uses ‘equal?’.

     (memq 'a '(a b c))                     ==>  (a b c)
     (memq 'b '(a b c))                     ==>  (b c)
     (memq 'a '(b c d))                     ==>  #f
     (memq (list 'a) '(b (a) c))            ==>  #f
     (member (list 'a)
             '(b (a) c))                    ==>  ((a) c)
     (memq 101 '(100 101 102))              ==>  _unspecified_
     (memv 101 '(100 101 102))              ==>  (101 102)


 -- library procedure: assq obj alist
 -- library procedure: assv obj alist
 -- library procedure: assoc obj alist

     ALIST (for "association list") must be a list of pairs.  These
     procedures find the first pair in ALIST whose car field is OBJ, and
     returns that pair.  If no pair in ALIST has OBJ as its car, then #f
     (not the empty list) is returned.  ‘Assq’ uses ‘eq?’ to compare OBJ
     with the car fields of the pairs in ALIST, while ‘assv’ uses ‘eqv?’
     and ‘assoc’ uses ‘equal?’.

     (define e '((a 1) (b 2) (c 3)))
     (assq 'a e)                            ==>  (a 1)
     (assq 'b e)                            ==>  (b 2)
     (assq 'd e)                            ==>  #f
     (assq (list 'a) '(((a)) ((b)) ((c))))
                                            ==>  #f
     (assoc (list 'a) '(((a)) ((b)) ((c))))
                                            ==>  ((a))
     (assq 5 '((2 3) (5 7) (11 13)))
                                            ==>  _unspecified_
     (assv 5 '((2 3) (5 7) (11 13)))
                                            ==>  (5 7)


          _Rationale:_ Although they are ordinarily used as predicates,
          ‘memq’, ‘memv’, ‘member’, ‘assq’, ‘assv’, and ‘assoc’ do not
          have question marks in their names because they return useful
          values rather than just #t or #f.

6.3.3 Symbols
-------------

Symbols are objects whose usefulness rests on the fact that two symbols
are identical (in the sense of ‘eqv?’) if and only if their names are
spelled the same way.  This is exactly the property needed to represent
identifiers in programs, and so most implementations of Scheme use them
internally for that purpose.  Symbols are useful for many other
applications; for instance, they may be used the way enumerated values
are used in Pascal.

The rules for writing a symbol are exactly the same as the rules for
writing an identifier; see sections *note Identifiers:: and *note
Lexical structure::.

It is guaranteed that any symbol that has been returned as part of a
literal expression, or read using the ‘read’ procedure, and subsequently
written out using the ‘write’ procedure, will read back in as the
identical symbol (in the sense of ‘eqv?’).  The ‘string->symbol’
procedure, however, can create symbols for which this write/read
invariance may not hold because their names contain special characters
or letters in the non-standard case.

     _Note:_ Some implementations of Scheme have a feature known as
     "slashification" in order to guarantee write/read invariance for
     all symbols, but historically the most important use of this
     feature has been to compensate for the lack of a string data type.

     Some implementations also have "uninterned symbols", which defeat
     write/read invariance even in implementations with slashification,
     and also generate exceptions to the rule that two symbols are the
     same if and only if their names are spelled the same.

 -- procedure: symbol? obj

     Returns #t if OBJ is a symbol, otherwise returns #f.

     (symbol? 'foo)                         ==>  #t
     (symbol? (car '(a b)))                 ==>  #t
     (symbol? "bar")                        ==>  #f
     (symbol? 'nil)                         ==>  #t
     (symbol? '())                          ==>  #f
     (symbol? #f)                           ==>  #f


 -- procedure: symbol->string symbol

     Returns the name of SYMBOL as a string.  If the symbol was part of
     an object returned as the value of a literal expression (section
     *note Literal expressions::) or by a call to the ‘read’ procedure,
     and its name contains alphabetic characters, then the string
     returned will contain characters in the implementation's preferred
     standard case--some implementations will prefer upper case, others
     lower case.  If the symbol was returned by ‘string->symbol’, the
     case of characters in the string returned will be the same as the
     case in the string that was passed to ‘string->symbol’.  It is an
     error to apply mutation procedures like ‘string-set!’ to strings
     returned by this procedure.

     The following examples assume that the implementation's standard
     case is lower case:

     (symbol->string 'flying-fish)
                                            ==>  "flying-fish"
     (symbol->string 'Martin)               ==>  "martin"
     (symbol->string
        (string->symbol "Malvina"))
                                            ==>  "Malvina"


 -- procedure: string->symbol string

     Returns the symbol whose name is STRING.  This procedure can create
     symbols with names containing special characters or letters in the
     non-standard case, but it is usually a bad idea to create such
     symbols because in some implementations of Scheme they cannot be
     read as themselves.  See ‘symbol->string’.

     The following examples assume that the implementation's standard
     case is lower case:

     (eq? 'mISSISSIppi 'mississippi)
               ==>  #t
     (string->symbol "mISSISSIppi")
               ==>
       the symbol with name "mISSISSIppi"
     (eq? 'bitBlt (string->symbol "bitBlt"))
               ==>  #f
     (eq? 'JollyWog
          (string->symbol
            (symbol->string 'JollyWog)))
               ==>  #t
     (string=? "K. Harper, M.D."
               (symbol->string
                 (string->symbol "K. Harper, M.D.")))
               ==>  #t


6.3.4 Characters
----------------

Characters are objects that represent printed characters such as letters
and digits.  Characters are written using the notation #\<character> or
#\<character name>.  For example:

     #\a
          ; lower case letter
     #\A
          ; upper case letter
     #\(
          ; left parenthesis
     #\ 
          ; the space character
     #\space
          ; the preferred way to write a space
     #\newline
          ; the newline character

Case is significant in #\<character>, but not in #\<character name>.

If <character> in #\<character> is alphabetic, then the character
following <character> must be a delimiter character such as a space or
parenthesis.  This rule resolves the ambiguous case where, for example,
the sequence of characters "#\ space" could be taken to be either a
representation of the space character or a representation of the
character "#\ s" followed by a representation of the symbol "pace."

Characters written in the #\ notation are self-evaluating.  That is,
they do not have to be quoted in programs.

Some of the procedures that operate on characters ignore the difference
between upper case and lower case.  The procedures that ignore case have
"-ci" (for "case insensitive") embedded in their names.

 -- procedure: char? obj

     Returns #t if OBJ is a character, otherwise returns #f.

 -- procedure: char=? char1 char2
 -- procedure: char<? char1 char2
 -- procedure: char>? char1 char2
 -- procedure: char<=? char1 char2
 -- procedure: char>=? char1 char2

     These procedures impose a total ordering on the set of characters.
     It is guaranteed that under this ordering:

        • The upper case characters are in order.  For example, ‘(char<?
          #\A #\B)’ returns #t.
        • The lower case characters are in order.  For example, ‘(char<?
          #\a #\b)’ returns #t.
        • The digits are in order.  For example, ‘(char<? #\0 #\9)’
          returns #t.
        • Either all the digits precede all the upper case letters, or
          vice versa.
        • Either all the digits precede all the lower case letters, or
          vice versa.

     Some implementations may generalize these procedures to take more
     than two arguments, as with the corresponding numerical predicates.

 -- library procedure: char-ci=? char1 char2
 -- library procedure: char-ci<? char1 char2
 -- library procedure: char-ci>? char1 char2
 -- library procedure: char-ci<=? char1 char2
 -- library procedure: char-ci>=? char1 char2

     These procedures are similar to ‘char=?’ et cetera, but they treat
     upper case and lower case letters as the same.  For example,
     ‘(char-ci=? #\A #\a)’ returns #t.  Some implementations may
     generalize these procedures to take more than two arguments, as
     with the corresponding numerical predicates.

 -- library procedure: char-alphabetic? char
 -- library procedure: char-numeric? char
 -- library procedure: char-whitespace? char
 -- library procedure: char-upper-case? letter
 -- library procedure: char-lower-case? letter

     These procedures return #t if their arguments are alphabetic,
     numeric, whitespace, upper case, or lower case characters,
     respectively, otherwise they return #f.  The following remarks,
     which are specific to the ASCII character set, are intended only as
     a guide: The alphabetic characters are the 52 upper and lower case
     letters.  The numeric characters are the ten decimal digits.  The
     whitespace characters are space, tab, line feed, form feed, and
     carriage return.

 -- procedure: char->integer char
 -- procedure: integer->char N

     Given a character, ‘char->integer’ returns an exact integer
     representation of the character.  Given an exact integer that is
     the image of a character under ‘char->integer’, ‘integer->char’
     returns that character.  These procedures implement
     order-preserving isomorphisms between the set of characters under
     the ‘char<=?’ ordering and some subset of the integers under the
     ‘<=’ ordering.  That is, if

     (char<=? A B) ⇒ #t  and  (<= X Y) ⇒ #t


     and X and Y are in the domain of ‘integer->char’, then

     (<= (char->integer A)
         (char->integer B))                 ==>  #t

     (char<=? (integer->char X)
              (integer->char Y))            ==>  #t


 -- library procedure: char-upcase char
 -- library procedure: char-downcase char

     These procedures return a character CHAR2 such that ‘(char-ci=?
     CHAR CHAR2)’.  In addition, if CHAR is alphabetic, then the result
     of ‘char-upcase’ is upper case and the result of ‘char-downcase’ is
     lower case.

6.3.5 Strings
-------------

Strings are sequences of characters.  Strings are written as sequences
of characters enclosed within doublequotes (‘"’).  A doublequote can be
written inside a string only by escaping it with a backslash (\), as in


     "The word \"recursion\" has many meanings."


A backslash can be written inside a string only by escaping it with
another backslash.  Scheme does not specify the effect of a backslash
within a string that is not followed by a doublequote or backslash.

A string constant may continue from one line to the next, but the exact
contents of such a string are unspecified.

The _length_ of a string is the number of characters that it contains.
This number is an exact, non-negative integer that is fixed when the
string is created.  The “valid indexes” of a string are the exact
non-negative integers less than the length of the string.  The first
character of a string has index 0, the second has index 1, and so on.

In phrases such as "the characters of STRING beginning with index START
and ending with index END," it is understood that the index START is
inclusive and the index END is exclusive.  Thus if START and END are the
same index, a null substring is referred to, and if START is zero and
END is the length of STRING, then the entire string is referred to.

Some of the procedures that operate on strings ignore the difference
between upper and lower case.  The versions that ignore case have
"‘-ci’" (for "case insensitive") embedded in their names.

 -- procedure: string? obj

     Returns #t if OBJ is a string, otherwise returns #f.

 -- procedure: make-string K
 -- procedure: make-string K char

     ‘Make-string’ returns a newly allocated string of length K.  If
     CHAR is given, then all elements of the string are initialized to
     CHAR, otherwise the contents of the STRING are unspecified.

 -- library procedure: string char ...,

     Returns a newly allocated string composed of the arguments.

 -- procedure: string-length string

     Returns the number of characters in the given STRING.

 -- procedure: string-ref string K

     K must be a valid index of STRING.  ‘String-ref’ returns character
     K of STRING using zero-origin indexing.

 -- procedure: string-set! string k char

     K must be a valid index of STRING .  ‘String-set!’ stores CHAR in
     element K of STRING and returns an unspecified value.

     (define (f) (make-string 3 #\*))
     (define (g) "***")
     (string-set! (f) 0 #\?)                ==>  _unspecified_
     (string-set! (g) 0 #\?)                ==>  _error_
     (string-set! (symbol->string 'immutable)
                  0
                  #\?)                      ==>  _error_


 -- library procedure: string=? string1 string2
 -- library procedure: string-ci=? string1 string2

     Returns #t if the two strings are the same length and contain the
     same characters in the same positions, otherwise returns #f.
     ‘String-ci=?’ treats upper and lower case letters as though they
     were the same character, but ‘string=?’ treats upper and lower case
     as distinct characters.

 -- library procedure: string<? string1 string2
 -- library procedure: string>? string1 string2
 -- library procedure: string<=? string1 string2
 -- library procedure: string>=? string1 string2
 -- library procedure: string-ci<? string1 string2
 -- library procedure: string-ci>? string1 string2
 -- library procedure: string-ci<=? string1 string2
 -- library procedure: string-ci>=? string1 string2

     These procedures are the lexicographic extensions to strings of the
     corresponding orderings on characters.  For example, ‘string<?’ is
     the lexicographic ordering on strings induced by the ordering
     ‘char<?’ on characters.  If two strings differ in length but are
     the same up to the length of the shorter string, the shorter string
     is considered to be lexicographically less than the longer string.

     Implementations may generalize these and the ‘string=?’ and
     ‘string-ci=?’ procedures to take more than two arguments, as with
     the corresponding numerical predicates.

 -- library procedure: substring string start end

     STRING must be a string, and START and END must be exact integers
     satisfying

             0 <= START <= END <= (string-length STRING).

     ‘Substring’ returns a newly allocated string formed from the
     characters of STRING beginning with index START (inclusive) and
     ending with index END (exclusive).

 -- library procedure: string-append STRING ...,

     Returns a newly allocated string whose characters form the
     concatenation of the given strings.

 -- library procedure: string->list string
 -- library procedure: list->string list

     ‘String->list’ returns a newly allocated list of the characters
     that make up the given string.  ‘List->string’ returns a newly
     allocated string formed from the characters in the list LIST, which
     must be a list of characters.  ‘String->list’ and ‘list->string’
     are inverses so far as ‘equal?’ is concerned.

 -- library procedure: string-copy string

     Returns a newly allocated copy of the given STRING.

 -- library procedure: string-fill! string char

     Stores CHAR in every element of the given STRING and returns an
     unspecified value.

6.3.6 Vectors
-------------

Vectors are heterogeneous structures whose elements are indexed by
integers.  A vector typically occupies less space than a list of the
same length, and the average time required to access a randomly chosen
element is typically less for the vector than for the list.

The _length_ of a vector is the number of elements that it contains.
This number is a non-negative integer that is fixed when the vector is
created.  The _valid indexes_ of a vector are the exact non-negative
integers less than the length of the vector.  The first element in a
vector is indexed by zero, and the last element is indexed by one less
than the length of the vector.

Vectors are written using the notation #(OBJ ...,).  For example, a
vector of length 3 containing the number zero in element 0, the list ‘(2
2 2 2)’ in element 1, and the string ‘"Anna"’ in element 2 can be
written as following:


     #(0 (2 2 2 2) "Anna")


Note that this is the external representation of a vector, not an
expression evaluating to a vector.  Like list constants, vector
constants must be quoted:


     '#(0 (2 2 2 2) "Anna")
               ==>  #(0 (2 2 2 2) "Anna")


 -- procedure: vector? obj

     Returns #t if OBJ is a vector, otherwise returns #f.

 -- procedure: make-vector k
 -- procedure: make-vector k fill

     Returns a newly allocated vector of K elements.  If a second
     argument is given, then each element is initialized to FILL.
     Otherwise the initial contents of each element is unspecified.

 -- library procedure: vector obj ...,

     Returns a newly allocated vector whose elements contain the given
     arguments.  Analogous to ‘list’.

     (vector 'a 'b 'c)                      ==>  #(a b c)


 -- procedure: vector-length vector

     Returns the number of elements in VECTOR as an exact integer.

 -- procedure: vector-ref vector k

     K must be a valid index of VECTOR.  ‘Vector-ref’ returns the
     contents of element K of VECTOR.

     (vector-ref '#(1 1 2 3 5 8 13 21)
                 5)
               ==>  8
     (vector-ref '#(1 1 2 3 5 8 13 21)
                 (let ((i (round (* 2 (acos -1)))))
                   (if (inexact? i)
                       (inexact->exact i)
                       i)))
               ==> 13


 -- procedure: vector-set! vector k obj

     K must be a valid index of VECTOR.  ‘Vector-set!’ stores OBJ in
     element K of VECTOR.  The value returned by ‘vector-set!’ is
     unspecified.

     (let ((vec (vector 0 '(2 2 2 2) "Anna")))
       (vector-set! vec 1 '("Sue" "Sue"))
       vec)
               ==>  #(0 ("Sue" "Sue") "Anna")

     (vector-set! '#(0 1 2) 1 "doe")
               ==>  _error_  ; constant vector


 -- library procedure: vector->list vector
 -- library procedure: list->vector list

     ‘Vector->list’ returns a newly allocated list of the objects
     contained in the elements of VECTOR.  ‘List->vector’ returns a
     newly created vector initialized to the elements of the list LIST.

     (vector->list '#(dah dah didah))
               ==>  (dah dah didah)
     (list->vector '(dididit dah))
               ==>  #(dididit dah)


 -- library procedure: vector-fill! vector fill

     Stores FILL in every element of VECTOR.  The value returned by
     ‘vector-fill!’ is unspecified.

6.4 Control features
====================

This chapter describes various primitive procedures which control the
flow of program execution in special ways.  The ‘procedure?’ predicate
is also described here.

 -- procedure: procedure? obj

     Returns #t if OBJ is a procedure, otherwise returns #f.

     (procedure? car)                       ==>  #t
     (procedure? 'car)                      ==>  #f
     (procedure? (lambda (x) (* x x)))
                                            ==>  #t
     (procedure? '(lambda (x) (* x x)))
                                            ==>  #f
     (call-with-current-continuation procedure?)
                                            ==>  #t


 -- procedure: apply proc arg1 ... args

     PROC must be a procedure and ARGS must be a list.  Calls PROC with
     the elements of the list ‘(append (list ARG1 ...,) ARGS)’ as the
     actual arguments.

     (apply + (list 3 4))                   ==>  7

     (define compose
       (lambda (f g)
         (lambda args
           (f (apply g args)))))

     ((compose sqrt *) 12 75)               ==>  30


 -- library procedure: map proc list1 list2 ...,

     The LISTs must be lists, and PROC must be a procedure taking as
     many arguments as there are lists and returning a single value.  If
     more than one LIST is given, then they must all be the same length.
     ‘Map’ applies PROC element-wise to the elements of the LISTs and
     returns a list of the results, in order.  The dynamic order in
     which PROC is applied to the elements of the LISTs is unspecified.

     (map cadr '((a b) (d e) (g h)))
               ==>  (b e h)

     (map (lambda (n) (expt n n))
          '(1 2 3 4 5))
               ==>  (1 4 27 256 3125)

     (map + '(1 2 3) '(4 5 6))              ==>  (5 7 9)

     (let ((count 0))
       (map (lambda (ignored)
              (set! count (+ count 1))
              count)
            '(a b)))                        ==>  (1 2) OR (2 1)


 -- library procedure: for-each proc list1 list2 ...,

     The arguments to ‘for-each’ are like the arguments to ‘map’, but
     ‘for-each’ calls PROC for its side effects rather than for its
     values.  Unlike ‘map’, ‘for-each’ is guaranteed to call PROC on the
     elements of the LISTs in order from the first element(s) to the
     last, and the value returned by ‘for-each’ is unspecified.

     (let ((v (make-vector 5)))
       (for-each (lambda (i)
                   (vector-set! v i (* i i)))
                 '(0 1 2 3 4))
       v)                                   ==>  #(0 1 4 9 16)


 -- library procedure: force promise

     Forces the value of PROMISE (see ‘delay’, section *note Delayed
     evaluation::).  If no value has been computed for the promise, then
     a value is computed and returned.  The value of the promise is
     cached (or "memoized") so that if it is forced a second time, the
     previously computed value is returned.

     (force (delay (+ 1 2)))                ==>  3
     (let ((p (delay (+ 1 2))))
       (list (force p) (force p)))
                                            ==>  (3 3)

     (define a-stream
       (letrec ((next
                 (lambda (n)
                   (cons n (delay (next (+ n 1)))))))
         (next 0)))
     (define head car)
     (define tail
       (lambda (stream) (force (cdr stream))))

     (head (tail (tail a-stream)))
                                            ==>  2


     ‘Force’ and ‘delay’ are mainly intended for programs written in
     functional style.  The following examples should not be considered
     to illustrate good programming style, but they illustrate the
     property that only one value is computed for a promise, no matter
     how many times it is forced.

     (define count 0)
     (define p
       (delay (begin (set! count (+ count 1))
                     (if (> count x)
                         count
                         (force p)))))
     (define x 5)
     p                                      ==>  a promise
     (force p)                              ==>  6
     p                                      ==>  a promise, still
     (begin (set! x 10)
            (force p))                      ==>  6


     Here is a possible implementation of ‘delay’ and ‘force’.  Promises
     are implemented here as procedures of no arguments, and ‘force’
     simply calls its argument:

     (define force
       (lambda (object)
         (object)))


     We define the expression

     (delay <expression>)


     to have the same meaning as the procedure call

     (make-promise (lambda () <expression>))


     as follows

     (define-syntax delay
       (syntax-rules ()
         ((delay expression)
          (make-promise (lambda () expression))))),


     where ‘make-promise’ is defined as follows:

     (define make-promise
       (lambda (proc)
         (let ((result-ready? #f)
               (result #f))
           (lambda ()
             (if result-ready?
                 result
                 (let ((x (proc)))
                   (if result-ready?
                       result
                       (begin (set! result-ready? #t)
                              (set! result x)
                              result))))))))


          _Rationale:_ A promise may refer to its own value, as in the
          last example above.  Forcing such a promise may cause the
          promise to be forced a second time before the value of the
          first force has been computed.  This complicates the
          definition of ‘make-promise’.

     Various extensions to this semantics of ‘delay’ and ‘force’ are
     supported in some implementations:

        • Calling ‘force’ on an object that is not a promise may simply
          return the object.

        • It may be the case that there is no means by which a promise
          can be operationally distinguished from its forced value.
          That is, expressions like the following may evaluate to either
          #t or to #f, depending on the implementation:

          (eqv? (delay 1) 1)                ==>  _unspecified_
          (pair? (delay (cons 1 2)))        ==>  _unspecified_


        • Some implementations may implement "implicit forcing," where
          the value of a promise is forced by primitive procedures like
          ‘cdr’ and ‘+’:

          (+ (delay (* 3 7)) 13)            ==>  34


 -- procedure: call-with-current-continuation proc

     PROC must be a procedure of one argument.  The procedure
     ‘call-with-current-continuation’ packages up the current
     continuation (see the rationale below) as an "escape procedure" and
     passes it as an argument to PROC.  The escape procedure is a Scheme
     procedure that, if it is later called, will abandon whatever
     continuation is in effect at that later time and will instead use
     the continuation that was in effect when the escape procedure was
     created.  Calling the escape procedure may cause the invocation of
     BEFORE and AFTER thunks installed using ‘dynamic-wind’.

     The escape procedure accepts the same number of arguments as the
     continuation to the original call to
     call-with-current-continuation.  Except for continuations created
     by the ‘call-with-values’ procedure, all continuations take exactly
     one value.  The effect of passing no value or more than one value
     to continuations that were not created by call-with-values is
     unspecified.

     The escape procedure that is passed to PROC has unlimited extent
     just like any other procedure in Scheme.  It may be stored in
     variables or data structures and may be called as many times as
     desired.

     The following examples show only the most common ways in which
     ‘call-with-current-continuation’ is used.  If all real uses were as
     simple as these examples, there would be no need for a procedure
     with the power of ‘call-with-current-continuation’.

     (call-with-current-continuation
       (lambda (exit)
         (for-each (lambda (x)
                     (if (negative? x)
                         (exit x)))
                   '(54 0 37 -3 245 19))
         #t))                               ==>  -3

     (define list-length
       (lambda (obj)
         (call-with-current-continuation
           (lambda (return)
             (letrec ((r
                       (lambda (obj)
                         (cond ((null? obj) 0)
                               ((pair? obj)
                                (+ (r (cdr obj)) 1))
                               (else (return #f))))))
               (r obj))))))

     (list-length '(1 2 3 4))               ==>  4

     (list-length '(a b . c))               ==>  #f


          _Rationale:_

          A common use of ‘call-with-current-continuation’ is for
          structured, non-local exits from loops or procedure bodies,
          but in fact ‘call-with-current-continuation’ is extremely
          useful for implementing a wide variety of advanced control
          structures.

          Whenever a Scheme expression is evaluated there is a
          “continuation” wanting the result of the expression.  The
          continuation represents an entire (default) future for the
          computation.  If the expression is evaluated at top level, for
          example, then the continuation might take the result, print it
          on the screen, prompt for the next input, evaluate it, and so
          on forever.  Most of the time the continuation includes
          actions specified by user code, as in a continuation that will
          take the result, multiply it by the value stored in a local
          variable, add seven, and give the answer to the top level
          continuation to be printed.  Normally these ubiquitous
          continuations are hidden behind the scenes and programmers do
          not think much about them.  On rare occasions, however, a
          programmer may need to deal with continuations explicitly.
          ‘Call-with-current-continuation’ allows Scheme programmers to
          do that by creating a procedure that acts just like the
          current continuation.

          Most programming languages incorporate one or more
          special-purpose escape constructs with names like exit,
          ‘return’, or even goto.  In 1965, however, Peter Landin
          [Landin65] invented a general purpose escape operator called
          the J-operator.  John Reynolds [Reynolds72] described a
          simpler but equally powerful construct in 1972.  The ‘catch’
          special form described by Sussman and Steele in the 1975
          report on Scheme is exactly the same as Reynolds's construct,
          though its name came from a less general construct in MacLisp.
          Several Scheme implementors noticed that the full power of the
          ‘catch’ construct could be provided by a procedure instead of
          by a special syntactic construct, and the name
          ‘call-with-current-continuation’ was coined in 1982.  This
          name is descriptive, but opinions differ on the merits of such
          a long name, and some people use the name ‘call/cc’ instead.

 -- procedure: values obj ...

     Delivers all of its arguments to its continuation.  Except for
     continuations created by the ‘call-with-values’ procedure, all
     continuations take exactly one value.  Values might be defined as
     follows:

     (define (values . things)
       (call-with-current-continuation
         (lambda (cont) (apply cont things))))


 -- procedure: call-with-values producer consumer

     Calls its PRODUCER argument with no values and a continuation that,
     when passed some values, calls the CONSUMER procedure with those
     values as arguments.  The continuation for the call to CONSUMER is
     the continuation of the call to call-with-values.

     (call-with-values (lambda () (values 4 5))
                       (lambda (a b) b))
                                                        ==>  5

     (call-with-values * -)                             ==>  -1


 -- procedure: dynamic-wind before thunk after

     Calls THUNK without arguments, returning the result(s) of this
     call.  BEFORE and AFTER are called, also without arguments, as
     required by the following rules (note that in the absence of calls
     to continuations captured using ‘call-with-current-continuation’
     the three arguments are called once each, in order).  BEFORE is
     called whenever execution enters the dynamic extent of the call to
     THUNK and AFTER is called whenever it exits that dynamic extent.
     The dynamic extent of a procedure call is the period between when
     the call is initiated and when it returns.  In Scheme, because of
     ‘call-with-current-continuation’, the dynamic extent of a call may
     not be a single, connected time period.  It is defined as follows:

        • The dynamic extent is entered when execution of the body of
          the called procedure begins.

        • The dynamic extent is also entered when execution is not
          within the dynamic extent and a continuation is invoked that
          was captured (using ‘call-with-current-continuation’) during
          the dynamic extent.

        • It is exited when the called procedure returns.

        • It is also exited when execution is within the dynamic extent
          and a continuation is invoked that was captured while not
          within the dynamic extent.

     If a second call to ‘dynamic-wind’ occurs within the dynamic extent
     of the call to THUNK and then a continuation is invoked in such a
     way that the AFTERs from these two invocations of ‘dynamic-wind’
     are both to be called, then the AFTER associated with the second
     (inner) call to ‘dynamic-wind’ is called first.

     If a second call to ‘dynamic-wind’ occurs within the dynamic extent
     of the call to THUNK and then a continuation is invoked in such a
     way that the BEFOREs from these two invocations of ‘dynamic-wind’
     are both to be called, then the BEFORE associated with the first
     (outer) call to ‘dynamic-wind’ is called first.

     If invoking a continuation requires calling the BEFORE from one
     call to ‘dynamic-wind’ and the AFTER from another, then the AFTER
     is called first.

     The effect of using a captured continuation to enter or exit the
     dynamic extent of a call to BEFORE or AFTER is undefined.

     (let ((path '())
           (c #f))
       (let ((add (lambda (s)
                    (set! path (cons s path)))))
         (dynamic-wind
           (lambda () (add 'connect))
           (lambda ()
             (add (call-with-current-continuation
                    (lambda (c0)
                      (set! c c0)
                      'talk1))))
           (lambda () (add 'disconnect)))
         (if (< (length path) 4)
             (c 'talk2)
             (reverse path))))

               ==> (connect talk1 disconnect
                    connect talk2 disconnect)


6.5 Eval
========

 -- procedure: eval expression environment-specifier

     Evaluates EXPRESSION in the specified environment and returns its
     value.  EXPRESSION must be a valid Scheme expression represented as
     data, and ENVIRONMENT-SPECIFIER must be a value returned by one of
     the three procedures described below.  Implementations may extend
     ‘eval’ to allow non-expression programs (definitions) as the first
     argument and to allow other values as environments, with the
     restriction that ‘eval’ is not allowed to create new bindings in
     the environments associated with ‘null-environment’ or
     ‘scheme-report-environment’.

     (eval '(* 7 3) (scheme-report-environment 5))
                                                        ==>  21

     (let ((f (eval '(lambda (f x) (f x x))
                    (null-environment 5))))
       (f + 10))
                                                        ==>  20


 -- procedure: scheme-report-environment version
 -- procedure: null-environment version

     VERSION must be the exact integer ‘5’, corresponding to this
     revision of the Scheme report (the Revised^5 Report on Scheme).
     ‘Scheme-report-environment’ returns a specifier for an environment
     that is empty except for all bindings defined in this report that
     are either required or both optional and supported by the
     implementation.  ‘Null-environment’ returns a specifier for an
     environment that is empty except for the (syntactic) bindings for
     all syntactic keywords defined in this report that are either
     required or both optional and supported by the implementation.

     Other values of VERSION can be used to specify environments
     matching past revisions of this report, but their support is not
     required.  An implementation will signal an error if VERSION is
     neither ‘5’ nor another value supported by the implementation.

     The effect of assigning (through the use of ‘eval’) a variable
     bound in a ‘scheme-report-environment’ (for example ‘car’) is
     unspecified.  Thus the environments specified by
     ‘scheme-report-environment’ may be immutable.

 -- optional procedure: interaction-environment

     This procedure returns a specifier for the environment that
     contains implementation-defined bindings, typically a superset of
     those listed in the report.  The intent is that this procedure will
     return the environment in which the implementation would evaluate
     expressions dynamically typed by the user.

6.6 Input and output
====================

6.6.1 Ports
-----------

Ports represent input and output devices.  To Scheme, an input port is a
Scheme object that can deliver characters upon command, while an output
port is a Scheme object that can accept characters.

 -- library procedure: call-with-input-file string proc
 -- library procedure: call-with-output-file string proc

     STRING should be a string naming a file, and PROC should be a
     procedure that accepts one argument.  For ‘call-with-input-file’,
     the file should already exist; for ‘call-with-output-file’, the
     effect is unspecified if the file already exists.  These procedures
     call PROC with one argument: the port obtained by opening the named
     file for input or output.  If the file cannot be opened, an error
     is signalled.  If PROC returns, then the port is closed
     automatically and the value(s) yielded by the PROC is(are)
     returned.  If PROC does not return, then the port will not be
     closed automatically unless it is possible to prove that the port
     will never again be used for a read or write operation.

          _Rationale:_ Because Scheme's escape procedures have unlimited
          extent, it is possible to escape from the current continuation
          but later to escape back in.  If implementations were
          permitted to close the port on any escape from the current
          continuation, then it would be impossible to write portable
          code using both ‘call-with-current-continuation’ and
          ‘call-with-input-file’ or ‘call-with-output-file’.

 -- procedure: input-port? obj
 -- procedure: output-port? obj

     Returns #t if OBJ is an input port or output port respectively,
     otherwise returns #f.

 -- procedure: current-input-port
 -- procedure: current-output-port

     Returns the current default input or output port.

 -- optional procedure: with-input-from-file string thunk
 -- optional procedure: with-output-to-file string thunk

     STRING should be a string naming a file, and PROC should be a
     procedure of no arguments.  For ‘with-input-from-file’, the file
     should already exist; for ‘with-output-to-file’, the effect is
     unspecified if the file already exists.  The file is opened for
     input or output, an input or output port connected to it is made
     the default value returned by ‘current-input-port’ or
     ‘current-output-port’ (and is used by (read), (write OBJ), and so
     forth), and the THUNK is called with no arguments.  When the THUNK
     returns, the port is closed and the previous default is restored.
     ‘With-input-from-file’ and ‘with-output-to-file’ return(s) the
     value(s) yielded by THUNK.  If an escape procedure is used to
     escape from the continuation of these procedures, their behavior is
     implementation dependent.

 -- procedure: open-input-file filename

     Takes a string naming an existing file and returns an input port
     capable of delivering characters from the file.  If the file cannot
     be opened, an error is signalled.

 -- procedure: open-output-file filename

     Takes a string naming an output file to be created and returns an
     output port capable of writing characters to a new file by that
     name.  If the file cannot be opened, an error is signalled.  If a
     file with the given name already exists, the effect is unspecified.

 -- procedure: close-input-port port
 -- procedure: close-output-port port

     Closes the file associated with PORT, rendering the PORT incapable
     of delivering or accepting characters.

     These routines have no effect if the file has already been closed.
     The value returned is unspecified.

6.6.2 Input
-----------

 





 -- library procedure: read
 -- library procedure: read port

     ‘Read’ converts external representations of Scheme objects into the
     objects themselves.  That is, it is a parser for the nonterminal
     <datum> (see sections *note External representation:: and *note
     Pairs and lists::).  ‘Read’ returns the next object parsable from
     the given input PORT, updating PORT to point to the first character
     past the end of the external representation of the object.

     If an end of file is encountered in the input before any characters
     are found that can begin an object, then an end of file object is
     returned.  The port remains open, and further attempts to read will
     also return an end of file object.  If an end of file is
     encountered after the beginning of an object's external
     representation, but the external representation is incomplete and
     therefore not parsable, an error is signalled.

     The PORT argument may be omitted, in which case it defaults to the
     value returned by ‘current-input-port’.  It is an error to read
     from a closed port.

 -- procedure: read-char
 -- procedure: read-char port

     Returns the next character available from the input PORT, updating
     the PORT to point to the following character.  If no more
     characters are available, an end of file object is returned.  PORT
     may be omitted, in which case it defaults to the value returned by
     ‘current-input-port’.

 -- procedure: peek-char
 -- procedure: peek-char port

     Returns the next character available from the input PORT, _without_
     updating the PORT to point to the following character.  If no more
     characters are available, an end of file object is returned.  PORT
     may be omitted, in which case it defaults to the value returned by
     ‘current-input-port’.

          _Note:_ The value returned by a call to ‘peek-char’ is the
          same as the value that would have been returned by a call to
          ‘read-char’ with the same PORT.  The only difference is that
          the very next call to ‘read-char’ or ‘peek-char’ on that PORT
          will return the value returned by the preceding call to
          ‘peek-char’.  In particular, a call to ‘peek-char’ on an
          interactive port will hang waiting for input whenever a call
          to ‘read-char’ would have hung.

 -- procedure: eof-object? obj

     Returns #t if OBJ is an end of file object, otherwise returns #f.
     The precise set of end of file objects will vary among
     implementations, but in any case no end of file object will ever be
     an object that can be read in using ‘read’.

 -- procedure: char-ready?
 -- procedure: char-ready? port

     Returns #t if a character is ready on the input PORT and returns #f
     otherwise.  If ‘char-ready’ returns #t then the next ‘read-char’
     operation on the given PORT is guaranteed not to hang.  If the PORT
     is at end of file then ‘char-ready?’ returns #t.  PORT may be
     omitted, in which case it defaults to the value returned by
     ‘current-input-port’.

          _Rationale:_ ‘Char-ready?’ exists to make it possible for a
          program to accept characters from interactive ports without
          getting stuck waiting for input.  Any input editors associated
          with such ports must ensure that characters whose existence
          has been asserted by ‘char-ready?’ cannot be rubbed out.  If
          ‘char-ready?’ were to return #f at end of file, a port at end
          of file would be indistinguishable from an interactive port
          that has no ready characters.

6.6.3 Output
------------






 -- library procedure: write obj
 -- library procedure: write obj port

     Writes a written representation of OBJ to the given PORT.  Strings
     that appear in the written representation are enclosed in
     doublequotes, and within those strings backslash and doublequote
     characters are escaped by backslashes.  Character objects are
     written using the ‘#\’ notation.  ‘Write’ returns an unspecified
     value.  The PORT argument may be omitted, in which case it defaults
     to the value returned by ‘current-output-port’.

 -- library procedure: display obj
 -- library procedure: display obj port

     Writes a representation of OBJ to the given PORT.  Strings that
     appear in the written representation are not enclosed in
     doublequotes, and no characters are escaped within those strings.
     Character objects appear in the representation as if written by
     ‘write-char’ instead of by ‘write’.  ‘Display’ returns an
     unspecified value.  The PORT argument may be omitted, in which case
     it defaults to the value returned by ‘current-output-port’.

          _Rationale:_ ‘Write’ is intended for producing
          machine-readable output and ‘display’ is for producing
          human-readable output.  Implementations that allow
          "slashification" within symbols will probably want ‘write’ but
          not ‘display’ to slashify funny characters in symbols.

 -- library procedure: newline
 -- library procedure: newline port

     Writes an end of line to PORT.  Exactly how this is done differs
     from one operating system to another.  Returns an unspecified
     value.  The PORT argument may be omitted, in which case it defaults
     to the value returned by ‘current-output-port’.

 -- procedure: write-char char
 -- procedure: write-char char port

     Writes the character CHAR (not an external representation of the
     character) to the given PORT and returns an unspecified value.  The
     PORT argument may be omitted, in which case it defaults to the
     value returned by ‘current-output-port’.

6.6.4 System interface
----------------------

Questions of system interface generally fall outside of the domain of
this report.  However, the following operations are important enough to
deserve description here.

 -- optional procedure: load filename

     FILENAME should be a string naming an existing file containing
     Scheme source code.  The ‘load’ procedure reads expressions and
     definitions from the file and evaluates them sequentially.  It is
     unspecified whether the results of the expressions are printed.
     The ‘load’ procedure does not affect the values returned by
     ‘current-input-port’ and ‘current-output-port’.  ‘Load’ returns an
     unspecified value.

          _Rationale:_ For portability, ‘load’ must operate on source
          files.  Its operation on other kinds of files necessarily
          varies among implementations.

 -- optional procedure: transcript-on filename
 -- optional procedure: transcript-off

     FILENAME must be a string naming an output file to be created.  The
     effect of ‘transcript-on’ is to open the named file for output, and
     to cause a transcript of subsequent interaction between the user
     and the Scheme system to be written to the file.  The transcript is
     ended by a call to ‘transcript-off’, which closes the transcript
     file.  Only one transcript may be in progress at any time, though
     some implementations may relax this restriction.  The values
     returned by these procedures are unspecified.

7 Formal syntax and semantics
*****************************

This chapter provides formal descriptions of what has already been
described informally in previous chapters of this report.

7.1 Formal syntax
=================

This section provides a formal syntax for Scheme written in an extended
BNF.

All spaces in the grammar are for legibility.  Case is insignificant;
for example, ‘#x1A’ and ‘#X1a’ are equivalent.  <empty> stands for the
empty string.

The following extensions to BNF are used to make the description more
concise: <thing>* means zero or more occurrences of <thing>; and
<thing>+ means at least one <thing>.

7.1.1 Lexical structure
-----------------------

This section describes how individual tokens (identifiers, numbers,
etc.)  are formed from sequences of characters.  The following sections
describe how expressions and programs are formed from sequences of
tokens.

<Intertoken space> may occur on either side of any token, but not within
a token.

Tokens which require implicit termination (identifiers, numbers,
characters, and dot) may be terminated by any <delimiter>, but not
necessarily by anything else.

The following five characters are reserved for future extensions to the
language: [ ] { } |

<token> --> <identifier> | <boolean> | <number>
     | <character> | <string>
     | ( | ) | #( | ' | ` | , | ,@ | .
<delimiter> --> <whitespace> | ( | ) | " | ;
<whitespace> --> <space or newline>
<comment> --> ;  <all subsequent characters up to a
                 line break>
<atmosphere> --> <whitespace> | <comment>
<intertoken space> --> <atmosphere>*


<identifier> --> <initial> <subsequent>*
     | <peculiar identifier>
<initial> --> <letter> | <special initial>
<letter> --> a | b | c | ... | z

<special initial> --> ! | $ | % | & | * | / | : | < | =
     | > | ? | ^ | _ | ~
<subsequent> --> <initial> | <digit>
     | <special subsequent>
<digit> --> 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
<special subsequent> --> + | - | . | @
<peculiar identifier> --> + | - | ...
<syntactic keyword> --> <expression keyword>
     | else | => | define
     | unquote | unquote-splicing
<expression keyword> --> quote | lambda | if
     | set! | begin | cond | and | or | case
     | let | let* | letrec | do | delay
     | quasiquote

‘<variable> ⇒ <’any <identifier> that isn't
                also a <syntactic keyword>>

<boolean> --> #t | #f
<character> --> #\ <any character>
     | #\ <character name>
<character name> --> space | newline

<string> --> " <string element>* "
<string element> --> <any character other than " or \>
     | \" | \\


<number> --> <num 2>| <num 8>
     | <num 10>| <num 16>



The following rules for <num R>, <complex R>, <real R>, <ureal R>,
<uinteger R>, and <prefix R> should be replicated for R = 2, 8, 10, and
16.  There are no rules for <decimal 2>, <decimal 8>, and <decimal 16>,
which means that numbers containing decimal points or exponents must be
in decimal radix.

<num R> --> <prefix R> <complex R>
<complex R> --> <real R> | <real R> @ <real R>
    | <real R> + <ureal R> i | <real R> - <ureal R> i
    | <real R> + i | <real R> - i
    | + <ureal R> i | - <ureal R> i | + i | - i
<real R> --> <sign> <ureal R>
<ureal R> --> <uinteger R>
    | <uinteger R> / <uinteger R>
    | <decimal R>
<decimal 10> --> <uinteger 10> <suffix>
    | . <digit 10>+ #* <suffix>
    | <digit 10>+ . <digit 10>* #* <suffix>
    | <digit 10>+ #+ . #* <suffix>
<uinteger R> --> <digit R>+ #*
<prefix R> --> <radix R> <exactness>
    | <exactness> <radix R>



<suffix> --> <empty>
    | <exponent marker> <sign> <digit 10>+
<exponent marker> --> e | s | f | d | l
<sign> --> <empty>  | + |  -
<exactness> --> <empty> | #i | #e
<radix 2> --> #b
<radix 8> --> #o
<radix 10> --> <empty> | #d
<radix 16> --> #x
<digit 2> --> 0 | 1
<digit 8> --> 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
<digit 10> --> <digit>
<digit 16> --> <digit 10> | a | b | c | d | e | f


7.1.2 External representations
------------------------------

<Datum> is what the ‘read’ procedure (section *note Input::)
successfully parses.  Note that any string that parses as an
<expression> will also parse as a <datum>.

<datum> --> <simple datum> | <compound datum>
<simple datum> --> <boolean> | <number>
     | <character> | <string> |  <symbol>
<symbol> --> <identifier>
<compound datum> --> <list> | <vector>
<list> --> (<datum>*) | (<datum>+ . <datum>)
       | <abbreviation>
<abbreviation> --> <abbrev prefix> <datum>
<abbrev prefix> --> ' | ` | , | ,@
<vector> --> #(<datum>*)


7.1.3 Expressions
-----------------

<expression> --> <variable>
     | <literal>
     | <procedure call>
     | <lambda expression>
     | <conditional>
     | <assignment>
     | <derived expression>
     | <macro use>
     | <macro block>

<literal> --> <quotation> | <self-evaluating>
<self-evaluating> --> <boolean> | <number>
     | <character> | <string>
<quotation> --> '<datum> | (quote <datum>)
<procedure call> --> (<operator> <operand>*)
<operator> --> <expression>
<operand> --> <expression>

<lambda expression> --> (lambda <formals> <body>)
<formals> --> (<variable>*) | <variable>
     | (<variable>+ . <variable>)
<body> --> <definition>* <sequence>
<sequence> --> <command>* <expression>
<command> --> <expression>

<conditional> --> (if <test> <consequent> <alternate>)
<test> --> <expression>
<consequent> --> <expression>
<alternate> --> <expression> | <empty>

<assignment> --> (set! <variable> <expression>)

<derived expression> -->
       (cond <cond clause>+)
     | (cond <cond clause>* (else <sequence>))
     | (case <expression>
         <case clause>+)
     | (case <expression>
         <case clause>*
         (else <sequence>))
     | (and <test>*)
     | (or <test>*)
     | (let (<binding spec>*) <body>)
     | (let <variable> (<binding spec>*) <body>)
     | (let* (<binding spec>*) <body>)
     | (letrec (<binding spec>*) <body>)
     | (begin <sequence>)
     | (do (<iteration spec>*)
           (<test> <do result>)
         <command>*)
     | (delay <expression>)
     | <quasiquotation>

<cond clause> --> (<test> <sequence>)
      | (<test>)
      | (<test> => <recipient>)
<recipient> --> <expression>
<case clause> --> ((<datum>*) <sequence>)
<binding spec> --> (<variable> <expression>)
<iteration spec> --> (<variable> <init> <step>)
    | (<variable> <init>)
<init> --> <expression>
<step> --> <expression>
<do result> --> <sequence> | <empty>

<macro use> --> (<keyword> <datum>*)
<keyword> --> <identifier>

<macro block> -->
     (let-syntax (<syntax spec>*) <body>)
     | (letrec-syntax (<syntax spec>*) <body>)
<syntax spec> --> (<keyword> <transformer spec>)




7.1.4 Quasiquotations
---------------------

The following grammar for quasiquote expressions is not context-free.
It is presented as a recipe for generating an infinite number of
production rules.  Imagine a copy of the following rules for D = 1, 2,3,
....  D keeps track of the nesting depth.

<quasiquotation> --> <quasiquotation 1>
<qq template 0> --> <expression>
<quasiquotation D> --> `<qq template D>
       | (quasiquote <qq template D>)
<qq template D> --> <simple datum>
       | <list qq template D>
       | <vector qq template D>
       | <unquotation D>
<list qq template D> --> (<qq template or splice D>*)
       | (<qq template or splice D>+ . <qq template D>)
       | '<qq template D>
       | <quasiquotation D+1>
<vector qq template D> --> #(<qq template or splice D>*)
<unquotation D> --> ,<qq template D-1>
       | (unquote <qq template D-1>)
<qq template or splice D> --> <qq template D>
       | <splicing unquotation D>
<splicing unquotation D> --> ,@<qq template D-1>
       | (unquote-splicing <qq template D-1>)


In <quasiquotation>s, a <list qq template D> can sometimes be confused
with either an <unquotation D> or a <splicing unquotation D>.  The
interpretation as an <unquotation> or <splicing unquotation D> takes
precedence.

7.1.5 Transformers
------------------

<transformer spec> -->
    (syntax-rules (<identifier>*) <syntax rule>*)
<syntax rule> --> (<pattern> <template>)
<pattern> --> <pattern identifier>
     | (<pattern>*)
     | (<pattern>+ . <pattern>)
     | (<pattern>* <pattern> <ellipsis>)
     | #(<pattern>*)
     | #(<pattern>* <pattern> <ellipsis>)
     | <pattern datum>
<pattern datum> --> <string>
     | <character>
     | <boolean>
     | <number>
<template> --> <pattern identifier>
     | (<template element>*)
     | (<template element>+ . <template>)
     | #(<template element>*)
     | <template datum>
<template element> --> <template>
     | <template> <ellipsis>
<template datum> --> <pattern datum>
<pattern identifier> --> <any identifier except ‘...’>
<ellipsis> --> <the identifier ‘...’>



7.1.6 Programs and definitions
------------------------------

<program> --> <command or definition>*
<command or definition> --> <command>
    | <definition>
    | <syntax definition>
    | (begin <command or definition>+)
<definition> --> (define <variable> <expression>)
      | (define (<variable> <def formals>) <body>)
      | (begin <definition>*)
<def formals> --> <variable>*
      | <variable>* . <variable>
<syntax definition> -->
     (define-syntax <keyword> <transformer spec>)



7.2 Formal semantics
====================

This section provides a formal denotational semantics for the primitive
expressions of Scheme and selected built-in procedures.  The concepts
and notation used here are described in [STOY77].

     _Note:_ The formal semantics section was written in LaTeX which is
     incompatible with TeXinfo.  See the Formal semantics section of the
     original document from which this was derived.

7.3 Derived expression types
============================

This section gives macro definitions for the derived expression types in
terms of the primitive expression types (literal, variable, call,
‘lambda’, ‘if’, ‘set!’).  See section *note Control features:: for a
possible definition of ‘delay’.


     (define-syntax cond
       (syntax-rules (else =>)
         ((cond (else result1 result2 ...))
          (begin result1 result2 ...))
         ((cond (test => result))
          (let ((temp test))
            (if temp (result temp))))
         ((cond (test => result) clause1 clause2 ...)
          (let ((temp test))
            (if temp
                (result temp)
                (cond clause1 clause2 ...))))
         ((cond (test)) test)
         ((cond (test) clause1 clause2 ...)
          (let ((temp test))
            (if temp
                temp
                (cond clause1 clause2 ...))))
         ((cond (test result1 result2 ...))
          (if test (begin result1 result2 ...)))
         ((cond (test result1 result2 ...)
                clause1 clause2 ...)
          (if test
              (begin result1 result2 ...)
              (cond clause1 clause2 ...)))))



     (define-syntax case
       (syntax-rules (else)
         ((case (key ...)
            clauses ...)
          (let ((atom-key (key ...)))
            (case atom-key clauses ...)))
         ((case key
            (else result1 result2 ...))
          (begin result1 result2 ...))
         ((case key
            ((atoms ...) result1 result2 ...))
          (if (memv key '(atoms ...))
              (begin result1 result2 ...)))
         ((case key
            ((atoms ...) result1 result2 ...)
            clause clauses ...)
          (if (memv key '(atoms ...))
              (begin result1 result2 ...)
              (case key clause clauses ...)))))



     (define-syntax and
       (syntax-rules ()
         ((and) #t)
         ((and test) test)
         ((and test1 test2 ...)
          (if test1 (and test2 ...) #f))))



     (define-syntax or
       (syntax-rules ()
         ((or) #f)
         ((or test) test)
         ((or test1 test2 ...)
          (let ((x test1))
            (if x x (or test2 ...))))))



     (define-syntax let
       (syntax-rules ()
         ((let ((name val) ...) body1 body2 ...)
          ((lambda (name ...) body1 body2 ...)
           val ...))
         ((let tag ((name val) ...) body1 body2 ...)
          ((letrec ((tag (lambda (name ...)
                           body1 body2 ...)))
             tag)
           val ...))))



     (define-syntax let*
       (syntax-rules ()
         ((let* () body1 body2 ...)
          (let () body1 body2 ...))
         ((let* ((name1 val1) (name2 val2) ...)
            body1 body2 ...)
          (let ((name1 val1))
            (let* ((name2 val2) ...)
              body1 body2 ...)))))


The following ‘letrec’ macro uses the symbol ‘<undefined>’ in place of
an expression which returns something that when stored in a location
makes it an error to try to obtain the value stored in the location (no
such expression is defined in Scheme).  A trick is used to generate the
temporary names needed to avoid specifying the order in which the values
are evaluated.  This could also be accomplished by using an auxiliary
macro.


     (define-syntax letrec
       (syntax-rules ()
         ((letrec ((var1 init1) ...) body ...)
          (letrec "generate temp names"
            (var1 ...)
            ()
            ((var1 init1) ...)
            body ...))
         ((letrec "generate temp names"
            ()
            (temp1 ...)
            ((var1 init1) ...)
            body ...)
          (let ((var1 <undefined>) ...)
            (let ((temp1 init1) ...)
              (set! var1 temp1)
              ...
              body ...)))
         ((letrec "generate temp names"
            (x y ...)
            (temp ...)
            ((var1 init1) ...)
            body ...)
          (letrec "generate temp names"
            (y ...)
            (newtemp temp ...)
            ((var1 init1) ...)
            body ...))))



     (define-syntax begin
       (syntax-rules ()
         ((begin exp ...)
          ((lambda () exp ...)))))


The following alternative expansion for ‘begin’ does not make use of the
ability to write more than one expression in the body of a lambda
expression.  In any case, note that these rules apply only if the body
of the ‘begin’ contains no definitions.


     (define-syntax begin
       (syntax-rules ()
         ((begin exp)
          exp)
         ((begin exp1 exp2 ...)
          (let ((x exp1))
            (begin exp2 ...)))))


The following definition of ‘do’ uses a trick to expand the variable
clauses.  As with ‘letrec’ above, an auxiliary macro would also work.
The expression ‘(if #f #f)’ is used to obtain an unspecific value.


     (define-syntax do
       (syntax-rules ()
         ((do ((var init step ...) ...)
              (test expr ...)
              command ...)
          (letrec
            ((loop
              (lambda (var ...)
                (if test
                    (begin
                      (if #f #f)
                      expr ...)
                    (begin
                      command
                      ...
                      (loop (do "step" var step ...)
                            ...))))))
            (loop init ...)))
         ((do "step" x)
          x)
         ((do "step" x y)
          y)))


Notes
*****

Language changes
================

This section enumerates the changes that have been made to Scheme since
the "Revised^4 report" [R4RS] was published.

   • The report is now a superset of the IEEE standard for Scheme
     [IEEEScheme]: implementations that conform to the report will also
     conform to the standard.  This required the following changes:

        • The empty list is now required to count as true.

        • The classification of features as essential or inessential has
          been removed.  There are now three classes of built-in
          procedures: primitive, library, and optional.  The optional
          procedures are ‘load’, ‘with-input-from-file’,
          ‘with-output-to-file’, ‘transcript-on’, ‘transcript-off’, and
          ‘interaction-environment’, and ‘-’ and ‘/’ with more than two
          arguments.  None of these are in the IEEE standard.

        • Programs are allowed to redefine built-in procedures.  Doing
          so will not change the behavior of other built-in procedures.

   • _Port_ has been added to the list of disjoint types.

   • The macro appendix has been removed.  High-level macros are now
     part of the main body of the report.  The rewrite rules for derived
     expressions have been replaced with macro definitions.  There are
     no reserved identifiers.

   • ‘Syntax-rules’ now allows vector patterns.

   • Multiple-value returns, ‘eval’, and ‘dynamic-wind’ have been added.

   • The calls that are required to be implemented in a properly
     tail-recursive fashion are defined explicitly.

   • '‘@’' can be used within identifiers.  '‘|’' is reserved for
     possible future extensions.

Additional material
*******************

The Internet Scheme Repository at

            <http://www.cs.indiana.edu/scheme-repository/>

contains an extensive Scheme bibliography, as well as papers, programs,
implementations, and other material related to Scheme.

Example
*******

‘Integrate-system’ integrates the system

            y_k^^ = f_k(y_1, y_2, ..., y_n), k = 1, ..., n

of differential equations with the method of Runge-Kutta.

The parameter system-derivative is a function that takes a system state
(a vector of values for the state variables y_1, ..., y_n) and produces
a system derivative (the values y_1^^, ...,y_n^^).  The parameter
initial-state provides an initial system state, and h is an initial
guess for the length of the integration step.

The value returned by ‘integrate-system’ is an infinite stream of system
states.


     (define integrate-system
       (lambda (system-derivative initial-state h)
         (let ((next (runge-kutta-4 system-derivative h)))
           (letrec ((states
                     (cons initial-state
                           (delay (map-streams next
                                               states)))))
             states))))


‘Runge-Kutta-4’ takes a function, f, that produces a system derivative
from a system state.  ‘Runge-Kutta-4’ produces a function that takes a
system state and produces a new system state.


     (define runge-kutta-4
       (lambda (f h)
         (let ((*h (scale-vector h))
               (*2 (scale-vector 2))
               (*1/2 (scale-vector (/ 1 2)))
               (*1/6 (scale-vector (/ 1 6))))
           (lambda (y)
             ;; y is a system state
             (let* ((k0 (*h (f y)))
                    (k1 (*h (f (add-vectors y (*1/2 k0)))))
                    (k2 (*h (f (add-vectors y (*1/2 k1)))))
                    (k3 (*h (f (add-vectors y k2)))))
               (add-vectors y
                 (*1/6 (add-vectors k0
                                    (*2 k1)
                                    (*2 k2)
                                    k3))))))))

     (define elementwise
       (lambda (f)
         (lambda vectors
           (generate-vector
             (vector-length (car vectors))
             (lambda (i)
               (apply f
                      (map (lambda (v) (vector-ref  v i))
                           vectors)))))))

     (define generate-vector
       (lambda (size proc)
         (let ((ans (make-vector size)))
           (letrec ((loop
                     (lambda (i)
                       (cond ((= i size) ans)
                             (else
                              (vector-set! ans i (proc i))
                              (loop (+ i 1)))))))
             (loop 0)))))

     (define add-vectors (elementwise +))

     (define scale-vector
       (lambda (s)
         (elementwise (lambda (x) (* x s)))))


‘Map-streams’ is analogous to ‘map’: it applies its first argument (a
procedure) to all the elements of its second argument (a stream).


     (define map-streams
       (lambda (f s)
         (cons (f (head s))
               (delay (map-streams f (tail s))))))


Infinite streams are implemented as pairs whose car holds the first
element of the stream and whose cdr holds a promise to deliver the rest
of the stream.


     (define head car)
     (define tail
       (lambda (stream) (force (cdr stream))))








The following illustrates the use of ‘integrate-system’ in integrating
the system

                     C dv_C / dt = -i_L - v_C / R

                           L di_L / dt = v_C

which models a damped oscillator.


     (define damped-oscillator
       (lambda (R L C)
         (lambda (state)
           (let ((Vc (vector-ref state 0))
                 (Il (vector-ref state 1)))
             (vector (- 0 (+ (/ Vc (* R C)) (/ Il C)))
                     (/ Vc L))))))

     (define the-states
       (integrate-system
          (damped-oscillator 10000 1000 .001)
          '#(1 0)
          .01))


Bibliography
************

   • [SICP] Harold Abelson and Gerald Jay Sussman with Julie Sussman.
     _Structure and Interpretation of Computer Programs, second
     edition._  MIT Press, Cambridge, 1996.

   • [Bawden88] Alan Bawden and Jonathan Rees.  Syntactic closures.  In
     _Proceedings of the 1988 ACM Symposium on Lisp and Functional
     Programming_, pages 86-95.

   • [howtoprint] Robert G. Burger and R. Kent Dybvig.  Printing
     floating-point numbers quickly and accurately.  In _Proceedings of
     the ACM SIGPLAN '96 Conference on Programming Language Design and
     Implementation_, pages 108-116.

   • [RRRS] William Clinger, editor.  The revised revised report on
     Scheme, or an uncommon Lisp.  MIT Artificial Intelligence Memo 848,
     August 1985.  Also published as Computer Science Department
     Technical Report 174, Indiana University, June 1985.

   • [howtoread] William Clinger.  How to read floating point numbers
     accurately.  In _Proceedings of the ACM SIGPLAN '90 Conference on
     Programming Language Design and Implementation_, pages 92-101.
     Proceedings published as _SIGPLAN Notices_ 25(6), June 1990.

   • [R4RS] William Clinger and Jonathan Rees, editors.  The revised^4
     report on the algorithmic language Scheme.  In _ACM Lisp Pointers_
     4(3), pages 1-55, 1991.

   • [macrosthatwork] William Clinger and Jonathan Rees.  Macros that
     work.  In _Proceedings of the 1991 ACM Conference on Principles of
     Programming Languages_, pages 155-162.

   • [propertailrecursion] William Clinger.  Proper Tail Recursion and
     Space Efficiency.  To appear in _Proceedings of the 1998 ACM
     Conference on Programming Language Design and Implementation_, June
     1998.

   • [syntacticabstraction] R. Kent Dybvig, Robert Hieb, and Carl
     Bruggeman.  Syntactic abstraction in Scheme.  _Lisp and Symbolic
     Computation_ 5(4):295-326, 1993.

   • [Scheme311] Carol Fessenden, William Clinger, Daniel P. Friedman,
     and Christopher Haynes.  Scheme 311 version 4 reference manual.
     Indiana University Computer Science Technical Report 137, February
     1983.  Superseded by [Scheme84].

   • [Scheme84] D. Friedman, C. Haynes, E. Kohlbecker, and M. Wand.
     Scheme 84 interim reference manual.  Indiana University Computer
     Science Technical Report 153, January 1985.

   • [IEEE] _IEEE Standard 754-1985.  IEEE Standard for Binary
     Floating-Point Arithmetic._  IEEE, New York, 1985.

   • [IEEEScheme] _IEEE Standard 1178-1990.  IEEE Standard for the
     Scheme Programming Language._  IEEE, New York, 1991.

   • [Kohlbecker86] Eugene E. Kohlbecker Jr.  _Syntactic Extensions in
     the Programming Language Lisp._  PhD thesis, Indiana University,
     August 1986.

   • [hygienic] Eugene E. Kohlbecker Jr., Daniel P. Friedman, Matthias
     Felleisen, and Bruce Duba.  Hygienic macro expansion.  In
     _Proceedings of the 1986 ACM Conference on Lisp and Functional
     Programming_, pages 151-161.

   • [Landin65] Peter Landin.  A correspondence between Algol 60 and
     Church's lambda notation: Part I. _Communications of the ACM_
     8(2):89-101, February 1965.

   • [MITScheme] MIT Department of Electrical Engineering and Computer
     Science.  Scheme manual, seventh edition.  September 1984.

   • [Naur63] Peter Naur et al.  Revised report on the algorithmic
     language Algol 60.  _Communications of the ACM_ 6(1):1-17, January
     1963.

   • [Penfield81] Paul Penfield, Jr.  Principal values and branch cuts
     in complex APL. In _APL '81 Conference Proceedings,_ pages 248-256.
     ACM SIGAPL, San Francisco, September 1981.  Proceedings published
     as _APL Quote Quad_ 12(1), ACM, September 1981.

   • [Pitman83] Kent M. Pitman.  The revised MacLisp manual (Saturday
     evening edition).  MIT Laboratory for Computer Science Technical
     Report 295, May 1983.

   • [Rees82] Jonathan A. Rees and Norman I. Adams IV. T: A dialect of
     Lisp or, lambda: The ultimate software tool.  In _Conference Record
     of the 1982 ACM Symposium on Lisp and Functional Programming_,
     pages 114-122.

   • [Rees84] Jonathan A. Rees, Norman I. Adams IV, and James R. Meehan.
     The T manual, fourth edition.  Yale University Computer Science
     Department, January 1984.

   • [R3RS] Jonathan Rees and William Clinger, editors.  The revised^3
     report on the algorithmic language Scheme.  In _ACM SIGPLAN
     Notices_ 21(12), pages 37-79, December 1986.

   • [Reynolds72] John Reynolds.  Definitional interpreters for higher
     order programming languages.  In _ACM Conference Proceedings_,
     pages 717-740.  ACM, 1972.

   • [Scheme78] Guy Lewis Steele Jr.  and Gerald Jay Sussman.  The
     revised report on Scheme, a dialect of Lisp.  MIT Artificial
     Intelligence Memo 452, January 1978.

   • [Rabbit] Guy Lewis Steele Jr.  Rabbit: a compiler for Scheme.  MIT
     Artificial Intelligence Laboratory Technical Report 474, May 1978.

   • [CLtL] Guy Lewis Steele Jr.  _Common Lisp: The Language, second
     edition._  Digital Press, Burlington MA, 1990.

   • [Scheme75] Gerald Jay Sussman and Guy Lewis Steele Jr.  Scheme: an
     interpreter for extended lambda calculus.  MIT Artificial
     Intelligence Memo 349, December 1975.

   • [Stoy77] Joseph E. Stoy.  _Denotational Semantics: The
     Scott-Strachey Approach to Programming Language Theory._  MIT
     Press, Cambridge, 1977.

   • [TImanual85] Texas Instruments, Inc.  TI Scheme Language Reference
     Manual.  Preliminary version 1.0, November 1985.

Alphabetic index of definitions of concepts, keywords, and procedures
*********************************************************************

The principal entry for each term, procedure, or keyword is listed
first, separated from the other entries by a semicolon.







Concepts
========

* Menu:

* ,:                                     Quasiquotation.     (line 1400)
* ,@:                                    Quasiquotation.     (line 1403)
* ;:                                     Whitespace and comments.
                                                             (line  459)
* ':                                     Literal expressions.
                                                             (line  852)
* ' <1>:                                 Quasiquotation.     (line 1442)
* =>:                                    Conditional.        (line 1073)
* backquote:                             Quasiquotation.     (line 1395)
* binding:                               Variables; syntactic keywords; and regions.
                                                             (line  548)
* binding construct:                     Variables; syntactic keywords; and regions.
                                                             (line  560)
* bound:                                 Variables; syntactic keywords; and regions.
                                                             (line  581)
* call:                                  Procedure calls.    (line  899)
* call by need:                          Delayed evaluation. (line 1379)
* combination:                           Procedure calls.    (line  920)
* comma:                                 Quasiquotation.     (line 1398)
* comment:                               Whitespace and comments.
                                                             (line  459)
* comment <1>:                           Lexical structure.  (line 4385)
* constant:                              Storage model.      (line  674)
* continuation:                          Control features.   (line 3908)
* define:                                Definitions.        (line 1763)
* define-syntax:                         Syntax definitions. (line 1865)
* definition:                            Definitions.        (line 1763)
* do:                                    Iteration.          (line 1305)
* dotted pair:                           Pairs and lists.    (line 2881)
* else:                                  Conditional.        (line 1073)
* empty list:                            Disjointness of types.
                                                             (line  601)
* empty list <1>:                        Booleans.           (line 2843)
* empty list <2>:                        Pairs and lists.    (line 2888)
* empty list <3>:                        Pairs and lists.    (line 2902)
* empty list <4>:                        Pairs and lists.    (line 2916)
* empty list <5>:                        Pairs and lists.    (line 2999)
* empty list <6>:                        Pairs and lists.    (line 3052)
* equivalence predicate:                 Equivalence predicates.
                                                             (line 1925)
* error:                                 Error situations and unspecified behavior.
                                                             (line  259)
* escape procedure:                      Control features.   (line 3848)
* exact:                                 Equivalence predicates.
                                                             (line 1972)
* exactness:                             Exactness.          (line 2209)
* false:                                 Disjointness of types.
                                                             (line  605)
* false <1>:                             Booleans.           (line 2832)
* false <2>:                             Booleans.           (line 2834)
* hygienic:                              Macros.             (line 1486)
* identifier:                            Identifiers.        (line  412)
* identifier <1>:                        Variables; syntactic keywords; and regions.
                                                             (line  543)
* identifier <2>:                        Symbols.            (line 3195)
* identifier <3>:                        Lexical structure.  (line 4379)
* immutable:                             Storage model.      (line  678)
* implementation restriction:            Error situations and unspecified behavior.
                                                             (line  278)
* implementation restriction <1>:        Implementation restrictions.
                                                             (line 2241)
* improper list:                         Pairs and lists.    (line 2931)
* inexact:                               Equivalence predicates.
                                                             (line 1972)
* initial environment:                   Standard procedures.
                                                             (line 1904)
* internal definition:                   Internal definitions.
                                                             (line 1828)
* keyword:                               Macros.             (line 1465)
* keyword <1>:                           Macros.             (line 1481)
* keyword <2>:                           Lexical structure.  (line 4402)
* lazy evaluation:                       Delayed evaluation. (line 1379)
* library:                               Primitive; library; and optional features.
                                                             (line  252)
* library procedure:                     Standard procedures.
                                                             (line 1904)
* location:                              Storage model.      (line  655)
* macro:                                 Macros.             (line 1457)
* macro keyword:                         Macros.             (line 1465)
* macro transformer:                     Macros.             (line 1470)
* macro use:                             Macros.             (line 1468)
* mutable:                               Storage model.      (line  677)
* number:                                Numbers.            (line 2151)
* numerical types:                       Numerical types.    (line 2171)
* object:                                Semantics.          (line  166)
* optional:                              Primitive; library; and optional features.
                                                             (line  244)
* pair:                                  Pairs and lists.    (line 2881)
* port:                                  Ports.              (line 4096)
* predicate:                             Equivalence predicates.
                                                             (line 1924)
* procedure call:                        Procedure calls.    (line  899)
* promise:                               Delayed evaluation. (line 1381)
* promise <1>:                           Control features.   (line 3728)
* proper tail recursion:                 Proper tail recursion.
                                                             (line  688)
* referentially transparent:             Macros.             (line 1486)
* region:                                Variables; syntactic keywords; and regions.
                                                             (line  571)
* region <1>:                            Assignments.        (line 1034)
* region <2>:                            Binding constructs. (line 1179)
* region <3>:                            Binding constructs. (line 1203)
* region <4>:                            Binding constructs. (line 1227)
* region <5>:                            Binding constructs. (line 1256)
* region <6>:                            Iteration.          (line 1329)
* simplest rational:                     Numerical operations.
                                                             (line 2627)
* syntactic keyword:                     Identifiers.        (line  447)
* syntactic keyword <1>:                 Variables; syntactic keywords; and regions.
                                                             (line  545)
* syntactic keyword <2>:                 Macros.             (line 1465)
* syntactic keyword <3>:                 Lexical structure.  (line 4402)
* syntax definition:                     Syntax definitions. (line 1865)
* tail call:                             Proper tail recursion.
                                                             (line  723)
* token:                                 Lexical structure.  (line 4363)
* top level environment:                 Variables; syntactic keywords; and regions.
                                                             (line  581)
* top level environment <1>:             Standard procedures.
                                                             (line 1904)
* true:                                  Disjointness of types.
                                                             (line  605)
* true <1>:                              Conditionals.       (line 1013)
* true <2>:                              Conditional.        (line 1075)
* true <3>:                              Booleans.           (line 2832)
* type:                                  Disjointness of types.
                                                             (line  601)
* unbound:                               Variables; syntactic keywords; and regions.
                                                             (line  581)
* unbound <1>:                           Variable references.
                                                             (line  837)
* unbound <2>:                           Top level definitions.
                                                             (line 1806)
* unspecified:                           Error situations and unspecified behavior.
                                                             (line  287)
* valid indexes:                         Strings.            (line 3428)
* valid indexes <1>:                     Vectors.            (line 3553)
* variable:                              Identifiers.        (line  446)
* variable <1>:                          Variables; syntactic keywords; and regions.
                                                             (line  547)
* variable <2>:                          Variable references.
                                                             (line  833)
* variable <3>:                          Lexical structure.  (line 4410)
* Whitespace:                            Whitespace and comments.
                                                             (line  450)

Procedures
==========

* Menu:

*  ...:                                  Pairs and lists.    (line 3036)
* -:                                     Numerical operations.
                                                             (line 2486)
* - <1>:                                 Numerical operations.
                                                             (line 2487)
* - <2>:                                 Numerical operations.
                                                             (line 2488)
* '<datum>:                              Literal expressions.
                                                             (line  848)
* *:                                     Numerical operations.
                                                             (line 2475)
* /:                                     Numerical operations.
                                                             (line 2489)
* / <1>:                                 Numerical operations.
                                                             (line 2490)
* / <2>:                                 Numerical operations.
                                                             (line 2491)
* `<qq template>:                        Quasiquotation.     (line 1393)
* +:                                     Numerical operations.
                                                             (line 2474)
* <:                                     Numerical operations.
                                                             (line 2427)
* <=:                                    Numerical operations.
                                                             (line 2429)
* <constant>:                            Literal expressions.
                                                             (line  850)
* <operator>:                            Procedure calls.    (line  893)
* <variable>:                            Variable references.
                                                             (line  831)
* =:                                     Numerical operations.
                                                             (line 2426)
* >:                                     Numerical operations.
                                                             (line 2428)
* >=:                                    Numerical operations.
                                                             (line 2430)
* abs:                                   Numerical operations.
                                                             (line 2505)
* acos:                                  Numerical operations.
                                                             (line 2646)
* and:                                   Conditional.        (line 1143)
* angle:                                 Numerical operations.
                                                             (line 2697)
* append:                                Pairs and lists.    (line 3084)
* apply:                                 Control features.   (line 3669)
* asin:                                  Numerical operations.
                                                             (line 2645)
* assoc:                                 Pairs and lists.    (line 3161)
* assq:                                  Pairs and lists.    (line 3159)
* assv:                                  Pairs and lists.    (line 3160)
* atan:                                  Numerical operations.
                                                             (line 2647)
* atan <1>:                              Numerical operations.
                                                             (line 2648)
* begin:                                 Sequencing.         (line 1283)
* boolean?:                              Booleans.           (line 2868)
* caar:                                  Pairs and lists.    (line 3033)
* cadr:                                  Pairs and lists.    (line 3034)
* call-with-current-continuation:        Control features.   (line 3843)
* call-with-input-file:                  Ports.              (line 4098)
* call-with-output-file:                 Ports.              (line 4099)
* call-with-values:                      Control features.   (line 3952)
* car:                                   Pairs and lists.    (line 2996)
* case:                                  Conditional.        (line 1103)
* cdddar:                                Pairs and lists.    (line 3038)
* cddddr:                                Pairs and lists.    (line 3039)
* cdr:                                   Pairs and lists.    (line 3007)
* ceiling:                               Numerical operations.
                                                             (line 2589)
* char->integer:                         Characters.         (line 3377)
* char-alphabetic?:                      Characters.         (line 3362)
* char-ci<?:                             Characters.         (line 3351)
* char-ci<=?:                            Characters.         (line 3353)
* char-ci=?:                             Characters.         (line 3350)
* char-ci>?:                             Characters.         (line 3352)
* char-ci>=?:                            Characters.         (line 3354)
* char-downcase:                         Characters.         (line 3401)
* char-lower-case?:                      Characters.         (line 3366)
* char-numeric?:                         Characters.         (line 3363)
* char-ready?:                           Input.              (line 4236)
* char-ready? <1>:                       Input.              (line 4237)
* char-upcase:                           Characters.         (line 3400)
* char-upper-case?:                      Characters.         (line 3365)
* char-whitespace?:                      Characters.         (line 3364)
* char?:                                 Characters.         (line 3323)
* char<?:                                Characters.         (line 3328)
* char<=?:                               Characters.         (line 3330)
* char=?:                                Characters.         (line 3327)
* char>?:                                Characters.         (line 3329)
* char>=?:                               Characters.         (line 3331)
* close-input-port:                      Ports.              (line 4162)
* close-output-port:                     Ports.              (line 4163)
* complex?:                              Numerical operations.
                                                             (line 2379)
* cond:                                  Conditional.        (line 1055)
* cons:                                  Pairs and lists.    (line 2983)
* cos:                                   Numerical operations.
                                                             (line 2643)
* current-input-port:                    Ports.              (line 4127)
* current-output-port:                   Ports.              (line 4128)
* delay:                                 Delayed evaluation. (line 1376)
* denominator:                           Numerical operations.
                                                             (line 2575)
* display:                               Output.             (line 4274)
* display <1>:                           Output.             (line 4275)
* do:                                    Iteration.          (line 1303)
* dynamic-wind:                          Control features.   (line 3966)
* eof-object?:                           Input.              (line 4229)
* eq?:                                   Equivalence predicates.
                                                             (line 2086)
* equal?:                                Equivalence predicates.
                                                             (line 2128)
* eqv?:                                  Equivalence predicates.
                                                             (line 1931)
* eval:                                  Eval.               (line 4036)
* even?:                                 Numerical operations.
                                                             (line 2451)
* exact->inexact:                        Numerical operations.
                                                             (line 2723)
* exact?:                                Numerical operations.
                                                             (line 2419)
* exp:                                   Numerical operations.
                                                             (line 2640)
* expt:                                  Numerical operations.
                                                             (line 2684)
* floor:                                 Numerical operations.
                                                             (line 2588)
* for-each:                              Control features.   (line 3710)
* force:                                 Control features.   (line 3725)
* gcd:                                   Numerical operations.
                                                             (line 2561)
* if:                                    Conditionals.       (line 1006)
* if <1>:                                Conditionals.       (line 1007)
* imag-part:                             Numerical operations.
                                                             (line 2695)
* inexact->exact:                        Numerical operations.
                                                             (line 2724)
* inexact?:                              Numerical operations.
                                                             (line 2420)
* input-port?:                           Ports.              (line 4121)
* integer->char:                         Characters.         (line 3378)
* integer?:                              Numerical operations.
                                                             (line 2382)
* interaction-environment:               Eval.               (line 4080)
* lambda:                                Procedures.         (line  940)
* lcm:                                   Numerical operations.
                                                             (line 2562)
* length:                                Pairs and lists.    (line 3075)
* let:                                   Binding constructs. (line 1187)
* let <1>:                               Iteration.          (line 1348)
* let-syntax:                            Binding constructs for syntactic keywords.
                                                             (line 1505)
* let*:                                  Binding constructs. (line 1216)
* letrec:                                Binding constructs. (line 1238)
* letrec-syntax:                         Binding constructs for syntactic keywords.
                                                             (line 1538)
* list:                                  Pairs and lists.    (line 3067)
* list->string:                          Strings.            (line 3526)
* list->vector:                          Vectors.            (line 3631)
* list-ref:                              Pairs and lists.    (line 3126)
* list-tail:                             Pairs and lists.    (line 3113)
* list?:                                 Pairs and lists.    (line 3054)
* load:                                  System interface.   (line 4314)
* log:                                   Numerical operations.
                                                             (line 2641)
* magnitude:                             Numerical operations.
                                                             (line 2696)
* make-polar:                            Numerical operations.
                                                             (line 2693)
* make-rectangular:                      Numerical operations.
                                                             (line 2692)
* make-string:                           Strings.            (line 3446)
* make-string <1>:                       Strings.            (line 3447)
* make-vector:                           Entry format.       (line  333)
* make-vector <1>:                       Entry format.       (line  335)
* make-vector <2>:                       Vectors.            (line 3580)
* make-vector <3>:                       Vectors.            (line 3581)
* map:                                   Control features.   (line 3685)
* max:                                   Numerical operations.
                                                             (line 2456)
* member:                                Pairs and lists.    (line 3140)
* memq:                                  Pairs and lists.    (line 3138)
* memv:                                  Pairs and lists.    (line 3139)
* min:                                   Numerical operations.
                                                             (line 2457)
* modulo:                                Numerical operations.
                                                             (line 2514)
* negative?:                             Numerical operations.
                                                             (line 2449)
* newline:                               Output.             (line 4291)
* newline <1>:                           Output.             (line 4292)
* not:                                   Booleans.           (line 2855)
* null-environment:                      Eval.               (line 4058)
* null?:                                 Pairs and lists.    (line 3050)
* number->string:                        Numerical input and output.
                                                             (line 2749)
* number->string <1>:                    Numerical input and output.
                                                             (line 2750)
* number?:                               Numerical operations.
                                                             (line 2378)
* numerator:                             Numerical operations.
                                                             (line 2574)
* odd?:                                  Numerical operations.
                                                             (line 2450)
* open-input-file:                       Ports.              (line 4149)
* open-output-file:                      Ports.              (line 4155)
* or:                                    Conditional.        (line 1158)
* output-port?:                          Ports.              (line 4122)
* pair?:                                 Pairs and lists.    (line 2973)
* peek-char:                             Input.              (line 4211)
* peek-char <1>:                         Input.              (line 4212)
* positive?:                             Numerical operations.
                                                             (line 2448)
* procedure?:                            Control features.   (line 3655)
* quasiquote:                            Quasiquotation.     (line 1391)
* quote:                                 Literal expressions.
                                                             (line  846)
* quotient:                              Numerical operations.
                                                             (line 2512)
* rational?:                             Numerical operations.
                                                             (line 2381)
* rationalize:                           Numerical operations.
                                                             (line 2623)
* read:                                  Input.              (line 4180)
* read <1>:                              Input.              (line 4181)
* read-char:                             Input.              (line 4202)
* read-char <1>:                         Input.              (line 4203)
* real-part:                             Numerical operations.
                                                             (line 2694)
* real?:                                 Numerical operations.
                                                             (line 2380)
* remainder:                             Numerical operations.
                                                             (line 2513)
* reverse:                               Pairs and lists.    (line 3103)
* round:                                 Numerical operations.
                                                             (line 2591)
* scheme-report-environment:             Eval.               (line 4057)
* set-car!:                              Pairs and lists.    (line 3017)
* set-cdr!:                              Pairs and lists.    (line 3028)
* set!:                                  Assignments.        (line 1030)
* sin:                                   Numerical operations.
                                                             (line 2642)
* sqrt:                                  Numerical operations.
                                                             (line 2678)
* string:                                Strings.            (line 3453)
* string->list:                          Strings.            (line 3525)
* string->number:                        Numerical input and output.
                                                             (line 2788)
* string->number <1>:                    Numerical input and output.
                                                             (line 2789)
* string->symbol:                        Symbols.            (line 3259)
* string-append:                         Strings.            (line 3520)
* string-ci<?:                           Strings.            (line 3493)
* string-ci<=?:                          Strings.            (line 3495)
* string-ci=?:                           Strings.            (line 3481)
* string-ci>?:                           Strings.            (line 3494)
* string-ci>=?:                          Strings.            (line 3496)
* string-copy:                           Strings.            (line 3534)
* string-fill!:                          Strings.            (line 3538)
* string-length:                         Strings.            (line 3457)
* string-ref:                            Strings.            (line 3461)
* string-set!:                           Strings.            (line 3466)
* string?:                               Strings.            (line 3442)
* string<?:                              Strings.            (line 3489)
* string<=?:                             Strings.            (line 3491)
* string=?:                              Strings.            (line 3480)
* string>?:                              Strings.            (line 3490)
* string>=?:                             Strings.            (line 3492)
* substring:                             Strings.            (line 3509)
* symbol->string:                        Symbols.            (line 3234)
* symbol?:                               Symbols.            (line 3222)
* syntax-rules:                          Pattern language.   (line 1575)
* tan:                                   Numerical operations.
                                                             (line 2644)
* TEMPLATE:                              Entry format.       (line  298)
* TEMPLATE <1>:                          Entry format.       (line  302)
* transcript-off:                        System interface.   (line 4329)
* transcript-on:                         System interface.   (line 4328)
* truncate:                              Numerical operations.
                                                             (line 2590)
* values:                                Control features.   (line 3940)
* vector:                                Vectors.            (line 3587)
* vector->list:                          Vectors.            (line 3630)
* vector-fill!:                          Vectors.            (line 3643)
* vector-length:                         Vectors.            (line 3595)
* vector-ref:                            Entry format.       (line  327)
* vector-ref <1>:                        Vectors.            (line 3599)
* vector-set!:                           Vectors.            (line 3615)
* vector?:                               Vectors.            (line 3576)
* with-input-from-file:                  Ports.              (line 4132)
* with-output-to-file:                   Ports.              (line 4133)
* write:                                 Output.             (line 4263)
* write <1>:                             Output.             (line 4264)
* write-char:                            Output.             (line 4299)
* write-char <1>:                        Output.             (line 4300)
* zero?:                                 Numerical operations.
                                                             (line 2447)

References
==========

* Menu:

* Bawden88:                              Bibliography.       (line 5041)
* CLtL:                                  Bibliography.       (line 5145)
* howtoprint:                            Bibliography.       (line 5045)
* howtoread:                             Bibliography.       (line 5055)
* hygienic:                              Bibliography.       (line 5096)
* IEEE:                                  Bibliography.       (line 5086)
* IEEEScheme:                            Bibliography.       (line 5089)
* Kohlbecker86:                          Bibliography.       (line 5092)
* Landin65:                              Bibliography.       (line 5101)
* macrosthatwork:                        Bibliography.       (line 5064)
* MITScheme:                             Bibliography.       (line 5105)
* Naur63:                                Bibliography.       (line 5108)
* Penfield81:                            Bibliography.       (line 5112)
* Pitman83:                              Bibliography.       (line 5117)
* propertailrecursion:                   Bibliography.       (line 5068)
* R3RS:                                  Bibliography.       (line 5130)
* R4RS:                                  Bibliography.       (line 5060)
* Rabbit:                                Bibliography.       (line 5142)
* Rees82:                                Bibliography.       (line 5121)
* Rees84:                                Bibliography.       (line 5126)
* Reynolds72:                            Bibliography.       (line 5134)
* RRRS:                                  Bibliography.       (line 5050)
* Scheme311:                             Bibliography.       (line 5077)
* Scheme75:                              Bibliography.       (line 5148)
* Scheme78:                              Bibliography.       (line 5138)
* Scheme84:                              Bibliography.       (line 5082)
* SICP:                                  Bibliography.       (line 5037)
* Stoy77:                                Bibliography.       (line 5152)
* syntacticabstraction:                  Bibliography.       (line 5073)
* TImanual85:                            Bibliography.       (line 5156)

Revised(5) Report on the Algorithmic Language Scheme
Contents
Introduction
  Background
  Acknowledgements
1 Overview of Scheme
  1.1 Semantics
  1.2 Syntax
  1.3 Notation and terminology
    1.3.1 Primitive; library; and optional features
    1.3.2 Error situations and unspecified behavior
    1.3.3 Entry format
    1.3.4 Evaluation examples
    1.3.5 Naming conventions
2 Lexical conventions
  2.1 Identifiers
  2.2 Whitespace and comments
  2.3 Other notations
3 Basic concepts
  3.1 Variables; syntactic keywords; and regions
  3.2 Disjointness of types
  3.3 External representations
  3.4 Storage model
  3.5 Proper tail recursion
4 Expressions
  4.1 Primitive expression types
    4.1.1 Variable references
    4.1.2 Literal expressions
    4.1.3 Procedure calls
    4.1.4 Procedures
    4.1.5 Conditionals
    4.1.6 Assignments
  4.2 Derived expression types
    4.2.1 Conditionals
    4.2.2 Binding constructs
    4.2.3 Sequencing
    4.2.4 Iteration
    4.2.5 Delayed evaluation
    4.2.6 Quasiquotation
  4.3 Macros
    4.3.1 Binding constructs for syntactic keywords
    4.3.2 Pattern language
5 Program structure
  5.1 Programs
  5.2 Definitions
    5.2.1 Top level definitions
    5.2.2 Internal definitions
  5.3 Syntax definitions
6 Standard procedures
  6.1 Equivalence predicates
  6.2 Numbers
    6.2.1 Numerical types
    6.2.2 Exactness
    6.2.3 Implementation restrictions
    6.2.4 Syntax of numerical constants
    6.2.5 Numerical operations
    6.2.6 Numerical input and output
  6.3 Other data types
    6.3.1 Booleans
    6.3.2 Pairs and lists
    6.3.3 Symbols
    6.3.4 Characters
    6.3.5 Strings
    6.3.6 Vectors
  6.4 Control features
  6.5 Eval
  6.6 Input and output
    6.6.1 Ports
    6.6.2 Input
    6.6.3 Output
    6.6.4 System interface
7 Formal syntax and semantics
  7.1 Formal syntax
    7.1.1 Lexical structure
    7.1.2 External representations
    7.1.3 Expressions
    7.1.4 Quasiquotations
    7.1.5 Transformers
    7.1.6 Programs and definitions
  7.2 Formal semantics
  7.3 Derived expression types
Notes
  Language changes
Additional material
Example
Bibliography
Alphabetic index of definitions of concepts, keywords, and procedures
  Concepts
  Procedures
  References
