Abstraction: ratintvl
Version: 1
Date: 3 November, 2002; revised 4 November 2002
Author: Mike O'Donnell

Values in the ratintvl abstraction are closed intervals with rational
endpoints. They are intended to represent approximate information
about real numbers, but all procedures are defined in terms of their
effects on the intervals.

Constructors
------------

(make_ratintvl left right) takes rationals left and right, and returns
    a representation of the closed interval from left to right.

    If left, right are not both rationals, report an error.

    If left > right, report an error.

(rational->ratintvl rational) returns the closed interval containing
    only the given rational

    If rational is not a rational, report an error.

Predicates
----------

(ratintvl? expr) returns #t if expr is a ratintvl, #f otherwise.

(point_ratintvl? ri) returns #t if ri is a ratintvl containing
    precisely one real, #f otherwise.

    If ri is not a ratintvl, report an error.

(contained_ratintvl? ri1 ri2) returns #t if ri1 is a subinterval (possibly
    equal) of ri2, #f otherwise.

    If either of ri1, ri2 is not a ratintvl, report an error.

(equal_ratintvl? ri1 ri2) returns #t if ri1=ri2, #f otherwise.

    If either of ri1, ri2 is not a ratintvl, report an error.

(precedes_ratinvtl? ri1 ri2) returns #t if all reals in ri1 strictly
    precede all reals in ri2.

    If either of ri1, ri2 is not a ratintvl, report an error.

(overlaps_ratintvl? ri1 ri2) returns #t if ri1 overlaps ri2, #f otherwise.

    If either of ri1, ri2 is not a ratintvl, report an error.

(lbounds_ratintvl? ri1, ri2) returns #t if ri1 contains at least one
    real less than all reals in ri2, #f otherwise.

    If either of ri1, ri2 is not a ratintvl, report an error.

(ubounded_ratintvl? ri1, ri2) returns #t if ri2 contains at least one
    real greater than all reals in ri1, #f otherwise.

    If either of ri1, ri2 is not a ratintvl, report an error.

Operations
----------

(union_ratintvl ri1 ri2) returns the ratintvl containing all reals in
    ri1 and all reals in ri2.

    If ri1 and ri2 are not overlapping ratintvls, report an error.

(intersect_ratintvl ri1 ri2) returns the ratintvl containing all
    reals that are in both of ri1 and ri2.

    If ri1 and ri2 are not overlapping ratintvls, report an error.

(add_ratintvl ri1 ri2) returns the ratintvl containing all sums of one
    real in ri1 plus one real in ri2.

    If either of ri1, ri2 is not a ratintvl, report an error.

(sub_ratintvl ri1 ri2) returns the ratintvl containing all differences
    of one real in ri1 minus one real in ri2.

    If either of ri1, ri2 is not a ratintvl, report an error.

(mult_ratintvl ri1 ri2) returns the ratintvl containing all products
    of one real in ri1 times one real in ri2.

    If either of ri1, ri2 is not a ratintvl, report an error.

(div_ratintvl ri1 ri2) returns the ratintvl containing all ratios
    of one real in ri1 divided by one real in ri2.

    If either of ri1, ri2 is not a ratintvl, or if ri2 contains 0,
    report an error.

(power_ratintvl ri int) returns the ratintvl containing all values of
    one real in ri to the integer power int.

    If ri is not a ratintvl, or if int is not an integer, or if ri
    contains 0 and int=0, report an error.

(simplest_ratintvl ri size) returns the simplest ratintvl containing
    the ratintvl ri and no bigger than the rational size. The
    complexity of the ratintvl [nl/dl, nu/du], where nl/dl and nu/du
    are in lowest terms, is the absolute value of the product of the
    nonzero values of nl, dl, nu, du. That is, the complexity is
    usually nl*dl*nu*du, but if either or both of nl, nu =0, it is
    omitted from the product.

    If ri is larger than size, report an error.

Selectors
---------

(min_ratintvl->rational ri) returns the left rational endpoint of ri.

    If ri is not a ratintvl, report an error.

(max_ratintvl->rational ri) returns the right rational endpoint of ri.

    If ri is not a ratintvl, report an error.

(mid_ratintvl->rational ri) returns the rational midpoint of ri.

Printout
--------

(decimal_sn_ratintlv->string ri) returns a character string displaying
    the most accurate decimal representation of ri in scientific
    notation. Use the following format:

        <msgn><mdigit>.<mdigits>E<esgn><edigits>

    Each of <msgn> and <esgn> is either "-" or "". <mdigit> is one
    decimal digit from 0-9. <mdigits> is a sequence of 0 or more
    decimal digits. <edigits> is a sequence of 1 or more decimal
    digits. The first <mdigit>, and every digit in <mdigits>, must be
    significant (i.e., correct for every real in ri). <edigits> must
    have no leading 0s, except for a single 0 if the exponent value
    is 0.

    If ri is not a ratintvl, or if the most accurate decimal
    representation has no significant digits, report an error.

(decimal_ratintlv->string ri) returns a character string displaying

    the most accurate representation of ri in normal decimal
    notation. Use the following format:

        <sgn><idigits>.<ddigits>

    <sgn> is either "-" or "". <idigits> is a sequence of 1 or more
    decimal digits. <ddigits> is a sequence of 0 or more decimal
    digits. Every digit in <idigits> and <ddigits> must be significant
    (i.e., correct for every real in ri). <idigits> must have no
    leading 0s, except for a single 0 if the integer part
    is 0. <ddigits> must have trailing 0s if they are significant.

    If ri is not a ratintvl, or if the most accurate decimal
    representation has no significant digits, report an error.