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!standard A.20(0)          18-03-28 AI12-0208-1/02
!class Amendment 16-12-19
!status work item 16-12-19
!status received 16-09-27
!priority Low
!difficulty Medium
!subject Predefined Big numbers support
!summary
Define Big Numbers packages to support arbitrary precision mathematics.
!problem
Some applications need larger numbers than Standard.Integer. All Ada compilers have this capability in order to implement static expressions; shouldn't some such package be available to Ada users as well? (Yes.)
!proposal
(See Summary.)
!wording
A.5.5 Big Numbers
Support is provided for integer arithmetic involving values larger than than those supported by the target machine, and for arbitrary-precision rationals.
The package Ada.Numerics.Big_Numbers has the following declaration:
package Ada.Numerics.Big_Numbers is type Number is interface;
function "=" (L, R : Number) return Boolean is abstract; function "<" (L, R : Number) return Boolean is abstract; function "<=" (L, R : Number) return Boolean is abstract; function ">" (L, R : Number) return Boolean is abstract; function ">=" (L, R : Number) return Boolean is abstract;
function To_String (Arg : Number) return String is abstract; function From_String (Arg : String) return Number is abstract; function "+" (Arg : Integer) return Number is abstract;
function "-" (L : Number) return Number is abstract; function "abs" (L : Number) return Number is abstract; function "+" (L, R : Number) return Number is abstract; function "-" (L, R : Number) return Number is abstract; function "*" (L, R : Number) return Number is abstract; function "/" (L, R : Number) return Number is abstract; function Min (L, R : Number) return Number is abstract; function Max (L, R : Number) return Number is abstract; end Ada.Numerics.Big_Numbers;
[TBD: aspects specified for this package? Pure, Nonblocking, others? Same question applies to other packages declared in later sections. Would these aspects constrain implementations in undesirable ways?]
[TBD: It would be nice to use subtypes in parameter profiles (e.g., a Nonzero_Number subtype for second argument of "/", but this requires AI12-0243 and the future of that AI is very uncertain.]
A.5.5.1 Big Integers
The package Ada.Numerics.Big_Numbers.Big_Integers has the following definition:
package Ada.Numerics.Big_Numbers.Big_Integers is type Big_Integer is new Number with private with Default_Initial_Condition => Big_Integer = 0, Integer_Literal => From_String;
[TBD: Remove Integer_Literal aspect spec if AI12-0249-1 not approved. If Default_Initial_Condition AI12-0265-1 is approved and Integer_Literal AI is not then replace "0" with "+0" in the condition and as needed in subsequent conditions.]
overriding function "=" (L, R : Big_Integer) return Boolean; overriding function "<" (L, R : Big_Integer) return Boolean; overriding function "<=" (L, R : Big_Integer) return Boolean; overriding function ">" (L, R : Big_Integer) return Boolean; overriding function ">=" (L, R : Big_Integer) return Boolean;
overriding function "+" (Arg : Integer) return Big_Integer; function To_Big_Integer (Arg : Integer) return Big_Integer renames "+";
subtype Big_Positive is Big_Integer with Dynamic_Predicate => Big_Positive > 0, Predicate_Failure => (raise Constraint_Error);
subtype Big_Natural is Big_Integer with Dynamic_Predicate => Big_Natural >= 0, Predicate_Failure => (raise Constraint_Error);
function In_Range (Arg, Lo, Hi : Big_Integer) return Boolean is ((Lo <= Arg) and (Arg <= Hi));
[TBD: In_Range formal parameter names. "Lo & Hi" vs. "Low & High"?]
function To_Integer (Arg : Big_Integer) return Integer with Pre => In_Range (Arg, Lo => +Integer'First, Hi => +Integer'Last) or else (raise Constraint_Error);
generic type Int is range <>; package Signed_Conversions is function To_Big_Integer (Arg : Int) return Big_Integer; function From_Big_Integer (Arg : Big_Integer) return Int with Pre => In_Range (Arg, Lo => To_Big_Integer (Int'First), Hi => To_Big_Integer (Int'Last)) or else (raise Constraint_Error); end Signed_Conversions;
generic type Int is mod <>; package Unsigned_Conversions is function To_Big_Integer (Arg : Int) return Big_Integer; function From_Big_Integer (Arg : Big_Integer) return Int with Pre => In_Range (Arg, Lo => To_Big_Integer (Int'First), Hi => To_Big_Integer (Int'Last)) or else (raise Constraint_Error); end Unsigned_Conversions;
overriding function To_String (Arg : Big_Integer) return String; overriding function From_String (Arg : String) return Big_Integer;
overriding function "-" (L : Big_Integer) return Big_Integer; overriding function "abs" (L : Big_Integer) return Big_Integer; overriding function "+" (L, R : Big_Integer) return Big_Integer; overriding function "-" (L, R : Big_Integer) return Big_Integer; overriding function "*" (L, R : Big_Integer) return Big_Integer; overriding function "/" (L, R : Big_Integer) return Big_Integer; function "mod" (L, R : Big_Integer) return Big_Integer; function "rem" (L, R : Big_Integer) return Big_Integer; function "**" (L : Big_Integer; R : Natural) return Big_Integer; overriding function Min (L, R : Big_Integer) return Big_Integer; overriding function Max (L, R : Big_Integer) return Big_Integer;
function Greatest_Common_Divisor (L, R : Big_Integer) return Big_Integer with Pre => (L /= 0 and R /= 0) or else (raise Constraint_Error), Post => (Big_Integer'Result > 0);
private ... -- not specified by the language end Ada.Numerics.Big_Numbers.Big_Integers;
To_String and From_String behave like Integer'Image and Integer'Value except that more digits are produced and accepted. [In particular, the low bound of To_String'Result is 1, To_String'Result includes a leading blank in the case of a nonnegative argument, and Constraint_Error is raised if the argument to From_String is syntactically incorrect.]
The other functions have their usual mathematical meanings.
[TBD: GCD should return Big_Positive, not Big_Integer, if AI12-0243 somehow allows this.]
A.5.5.1.1 Bounded Big Integers
An instance of the language-defined generic package Numerics.Big_Numbers.Bounded_Big_Integers provides a Big_Integer type and operations corresponding to those declared in Numerics.Big_Numbers.Big_Integers, but with the difference that the maximum storage (and, consequently, the set of representable values) is bounded.
The generic package Ada.Numerics.Big_Numbers.Bounded_Big_Integers has the following definition:
generic Capacity : Natural; package Ada.Numerics.Big_Numbers.Bounded_Big_Integers is
<all the same visible declarations as those in Big_Numbers.Big_Integers, including declaration of Bounded_Big_Integers.Big_Integer type>
function Last return Big_Integer is ((+256) ** Capacity); function First return Big_Integer is (-Last); private ... -- not specified by the language end Ada.Numerics.Big_Numbers.Bounded_Big_Integers;
Each operation behaves like the corresponding Big_Numbers.Big_Integers operation except that Constraint_Error is raised if a result R would fail the test In_Range (R, First, Last). This includes the streaming operations Big_Integer'Read and Big_Integer'Input.
AARM Note: Roughly speaking, behavior is as if the type invariant for Bounded_Big_Integer is
In_Range (Bounded_Big_Integer, First, Last) or else (raise Constraint_Error)
although that is not specified explicitly because that would require introducing some awkward code in order to avoid infinite recursion.
Implementation Requirements
For each instance of Bounded_Big_Integers, the output generated by Big_Integer'Output or Big_Integer'Write shall be readable by the corresponding Input or Read operations of the Big_Integer type declared in either another instance of Bounded_Big_Integers or in the package Numerics.Big_Numbers.Big_Integers. [This is subject to the preceding requirement that Constraint_Error is raised in some cases.]
Implementation Advice
The implementation of (an instance of) Bounded_Big_Integers should not make use of controlled types or dynamic allocation.
[end of Implementation Advice]
The generic unit Ada.Numerics.Big_Numbers.Bounded_Big_Integers.Conversions provides operations for converting between the Big_Integer types declared in Big_Numbers.Big_Integers and in an instance of
Big_Numbers.Bounded_Big_Integers.
The generic package Ada.Numerics.Big_Numbers.Bounded_Big_Integers.Conversions has the following definition:
with Ada.Numerics.Big_Numbers.Big_Integers; generic package Ada.Numerics.Big_Numbers.Bounded_Big_Integers.Conversions is function From_Unbounded (Arg : Big_Integers.Big_Integer) return Big_Integer; function To_Unbounded (Arg : Big_Integer) return Big_Integers.Big_Integer; end Ada.Numerics.Big_Numbers.Bounded_Big_Integers.Conversions;
AARM Note:This unit is declared as a child unit because we don't want Big_Integers to be in the closure of Bounded_Big_Integers. It is a generic package because its parent cannot have a non-generic child unit.
[TBD: This could be done differently by using a formal instance instead of declaring the Conversions package as a child of Bounded_Big_Integers. Would there be any advantage to this approach? The advantage of the proposed approach is visibility of the private part, but it does seem awkward to have a generic with no generic formals and no local state.]
A.5.5.2 Big Rationals
The package Ada.Numerics.Big_Numbers.Big_Rationals has the following definition:
with Ada.Numerics.Big_Numbers.Big_Integers; package Ada.Numerics.Big_Numbers.Big_Rationals is use Big_Integers;
type Big_Rational is new Number with private Default_Initial_Condition => Big_Rational = 0.0, Real_Literal => From_String;
function "/" (Num, Den : Big_Integer) return Big_Rational with Pre => (Den /= 0) or else (raise Constraint_Error);
function Numerator (Arg : Big_Rational) return Big_Integer; function Denominator (Arg : Big_Rational) return Big_Positive;
overriding function "+" (Arg : Integer) return Big_Rational is ((+Arg) / 1);
function To_Big_Rational (Arg : Integer) return Big_Rational renames "+";
overriding function "=" (L, R : Big_Rational) return Boolean; overriding function "<" (L, R : Big_Rational) return Boolean; overriding function "<=" (L, R : Big_Rational) return Boolean; overriding function ">" (L, R : Big_Rational) return Boolean; overriding function ">=" (L, R : Big_Rational) return Boolean;
overriding function To_String (Arg : Big_Rational) return String is (To_String (Numerator (Arg)) & " /" & To_String (Denominator (Arg))); overriding function From_String (Arg : String) return Big_Rational;
overriding function "-" (L : Big_Rational) return Big_Rational; overriding function "abs" (L : Big_Rational) return Big_Rational; overriding function "+" (L, R : Big_Rational) return Big_Rational; overriding function "-" (L, R : Big_Rational) return Big_Rational; overriding function "*" (L, R : Big_Rational) return Big_Rational; overriding function "/" (L, R : Big_Rational) return Big_Rational; function "**" (L : Big_Rational; R : Integer) return Big_Rational; overriding function Min (L, R : Big_Rational) return Big_Rational; overriding function Max (L, R : Big_Rational) return Big_Rational; private ... -- not specified by the language end Ada.Numerics.Big_Numbers.Big_Rationals;
From_String implements the inverse function of To_String; Constraint_Error is propagated in error cases. The other functions have their usual mathematical meanings.
Any Big_Rational result R returned by any of these functions satisifies the condition
(R = 0.0) or else (Greatest_Common_Denominator (Numerator (R), Denominator (R)) = 1).
AARM Note: No Bounded_Big_Rationals generic package is provided.
!discussion
** None yet.
!ASIS
No ASIS effect (assuming this is ONLY a library).
!ACATS test
An ACATS C-Test is needed to check that the new capabilities are supported.
!appendix

From: Steve Baird
Sent: Tuesday, September 27, 2016  4:09 PM

professor at U. of Utah:
    blog.regehr.org/archives/1401

Regehr says:
   In most programming languages, the default integer type should be a
   bignum: an arbitrary-precision integer that allocates more space when
   needed. Efficient bignum libraries exist and most integers never end
   up needing more than one machine word anyway, except in domains like
   crypto.

Nobody is suggesting changing how Standard.Integer works for Ada, but a
language-defined Bignum package (presumably supporting Rationals as well as
Integers) would be a step in the right direction.

It seems like the same arguments which were used (correctly, IMO) to justify
adding predefined container packages to the language also apply here. As Tuck
phrased it in a private message: portability and more capability "out of the
box."

Does some de facto standard already exist?

****************************************************************

From: Bob Duff
Sent: Tuesday, September 27, 2016  4:32 PM

> Nobody is suggesting changing how Standard.Integer works

But somebody might suggest that things like "type T is range 1..10**100;"
should be supported by all Ada compilers.

> It seems like the same arguments which were used (correctly, IMO) to
> justify adding predefined container packages to the language also
> apply here. As Tuck phrased it in a private message:
>     portability and more capability "out of the box."

Plus the fact that all Ada compilers have to support that functionality at
compile time, but can't provide it to their users in a portable way at run time.

> Does some de facto standard already exist?

For C and C++, yes.  For Ada, no.

For Common Lisp, Java, C#, and many others, a de jure standard exists.

****************************************************************

From: Randy Brukardt
Sent: Wednesday, September 28, 2016  12:49 PM

> Does some de facto standard already exist?

No. I could be convinced to contribute RR's Univmath package as a starting point
for discussion.

****************************************************************

From: Jean-Pierre Rosen
Sent: Thursday, September 29, 2016  12:22 AM

There are several packages available, see http://bignumber.chez.com/index.html

****************************************************************

From: Randy Brukardt
Sent: Thursday, September 29, 2016  12:28 PM

Surely, like containers there are as many Bignum packages as there are Ada
programmers (much like containers - everybody has one). But is someone putting
them into RM format?? That's what it means to "contribute" a package here.

****************************************************************

From: John Barnes
Sent: Thursday, September 29, 2016  2:05 PM

I see there has been chatter on big number packages.

I wrote such a package many years ago. I was intending to write a book called
Fun with Ada using big examples of Ada 83 programs. But it got overtaken by
events such as having to write the book on Ada 95.

But I kept the package, used some child stuff from Ada 95 but otherwise left it
alone, I still use it for dabbling with large prime numbers and so on. I think
it is based on base 10,000 which will run on a 16 bit machine and is easy for
conversion for printing.

But I fear that agreeing on something might be tricky.

****************************************************************

From: Florian Schanda
Sent: Friday, September 30, 2016  2:35 AM

> But I kept the package, used some child stuff from Ada 95 but
> otherwise left  it alone, I still use it for dabbling with large prime
> numbers and so on. I think it is based on base 10,000 which will run
> on a 16 bit machine and is easy for conversion for printing.

Generally, these days, you would probably want to stick to largest power-of- two
as printing these is not a massive concern but performance is. :)

Anyway, I think whatever we come up with, it should be possible to implement it
via a binding to GMP [https://gmplib.org] which is more or less the gold
standard for arbitrary precision arithmetic. Of course, some runtime may wish to
have a more verifiable implementation... So, I think there are two requirements
we should make sure to fulfil:

   1. the api should be amenable to static analysis and formal verification
   2. the api should make it easy to bind to gmp

(Not saying this list is exhaustive.)

I just want to avoid people starting from various in-house and private projects;
its probably a good idea instead to start from established libraries.

****************************************************************

From: Steve Baird
Sent: Friday, September 30, 2016  12:27 PM

> So, I think there are two
> requirements we should make sure to fulfil:
>
>     1. the api should be amenable to static analysis and formal verification
>     2. the api should make it easy to bind to gm

It is also at least possible that we'll want something similar to what we have
with the containers, where we have one version for use in situations where
controlled types and dynamic storage allocation are ok and another for use in
other situations.

****************************************************************

From: Jean-Pierre Rosen
Sent: Friday, September 30, 2016  2:44 PM

Hmmm... bounded and unbounded bignums?

****************************************************************

From: Tucker Taft
Sent: Friday, September 30, 2016  3:52 PM

Perhaps: "big enough nums, already..."

****************************************************************

From: Steve Baird
Sent: Tuesday, December 12, 2017  7:20 PM

I thought I'd take a look at how Java and C++ do bignums to see if there are any
ideas there worth incorporating.

My going-in idea is to have two packages with similar specs; one has "Capacity"
discriminants and the other is implemented using dynamic storage allocation of
some sort (e.g., controlled types and allocators). Like the bounded/unbounded
versions of the containers.

C++ doesn't really have a standard for bignums, but the GCC/GMP stuff
looks pretty similar to what I expected.

Java, however, surprised me (note that I am far from a Java expert so it could
be that I am just confused here).

The Java big-real spec doesn't have Numerator and Denominator functions which
yield big-ints.

The Java type seems to be named BigDecimal.

BigDecimal is implemented as a single big-int value accompanied by two ints
(Scale and Precision), at least according to
    stackoverflow.com/questions/10655254/how-bigdecimal-is-implemented

Which leads to my question:
    If Ada defined a spec where the intended implementation for bigreal
    is clearly two bigints (one for numerator, one for denominator),
    would this result in lots of "I coded up the same algorithm in Ada
    and Java and performance was a lot worse in Ada" horror stories?

Apparently BigDecimal lets you have, in effect, a lot of decimal digits but the
value "one third" still cannot be represented exactly.

Why did the Java folks do it that way? It seems like you lose a lot of value if
you can't exactly represent, for example, one third.

But perhaps most folks don't care about that functionality and the
performance/functionality tradeoff chosen by Java is closer to what most folks
want.

Opinions? Opinions about Java are of some interest, but what I really want is
opinions about what we should do in Ada.

p.s. Note that the current plan for this AI is to add one or more new predefined
packages but no changes to language rules. In particular, numeric literals for a
non-numeric type is the topic of another AI.

****************************************************************

From: Tucker Taft
Sent: Wednesday, December 13, 2017  9:15 AM

We want rational, not decimal, computations, I believe.  So I would ignore
Java's BigDecimal.

A different and interesting capability is true "real" arithmetic, which works
for transcendentals, etc.  It is intriguing, but probably not what people really
want.

I'll send the PDF for an article by Hans Boehm about "real" arithmetic
separately, since it will probably not make it through the SPAM filter!

****************************************************************

From: Randy Brukardt
Sent: Wednesday, December 13, 2017  10:57 AM

> We want rational, not decimal, computations, I believe.  So I would
> ignore Java's BigDecimal.

Isn't that Steve's question? Ada compiler vendors use rational computations
since that is required by the ACATS (it's necessary that 1/3 /=
0.33333333333333333333333333333). But is that the best choice for the Ada
community? I don't know.

> A different and interesting capability is true "real"
> arithmetic, which works for transcendentals, etc.  It is intriguing,
> but probably not what people really want.
>
> I'll send the PDF for an article by Hans Boehm about "real"
> arithmetic separately, since it will probably not make it through the
> SPAM filter!

It might be too large for the list as well. If so, I can post it in the Grab Bag
if you send it directly to me.

****************************************************************

From: Edmond Schonberg
Sent: Wednesday, December 13, 2017  1:36 PM

> I'll send the PDF for an article by Hans Boehm about "real" arithmetic
> separately, since it will probably not make it through the SPAM filter!

Both the rational representation and Boehm’s approach require arbitrary
precision integer arithmetic, so the spec of that new package is
straightforward. The papers describing the implementation of Boehm’s approach
claim that it is much more efficient than working on rationals, where numerator
and denominator grow very rapidly, while the other method only computes required
bits. I have no idea whether numerical analysts use this method, From the
literature it seems to be of interest to number theorists.

****************************************************************

From: Randy Brukardt
Sent: Wednesday, December 13, 2017  4:49 PM

> > It might be too large for the list as well. If so, I can post it in
> > the Grab Bag if you send it directly to me.
>
> It was too big.

By about 4 Megabytes. :-)

Since the article is copyrighted, I put it in the private part of the website.
Find it at:

http://www.ada-auth.org/standards/private/real_arithmetic-boehm.pdf

Ed suggested privately:

> Additional details on the underlying model and its implementation in:
> http://keithbriggs.info/documents/xr-paper2.pdf

****************************************************************

From: Randy Brukardt
Sent: Wednesday, December 13, 2017  5:10 PM

...
> Both the rational representation and Boehm's approach require
> arbitrary precision integer arithmetic, so the spec of that new
> package is straightforward.
> The papers describing the implementation of Boehm's approach claim
> that it is much more efficient than working on rationals, where
> numerator and denominator grow very rapidly, while the other method
> only computes required bits. I have no idea whether numerical analysts
> use this method, From the literature it seems to be of interest to
> number theorists.

The problem with the Boehm method is that it requires specifying those "required
bits", which seems problematic in a programming environment. One could do it
with a form of type declaration (Ada's "digits" seems to be the right general
idea), but that doesn't make much sense in a library form. Boehm's actual use
gets those on the fly (by interacting with the user), and they also use a
rational representation as a backup. So it seems that a rational representation
is going to show up somewhere.

I've repeatedly had the fever dream of a class-wide bignum base type, something
like:

      package Root_Bignum is
          type Root_Bignum_Type is abstract tagged null record;

          function Bignum (Val : in Long_Float) return Root_Bignum_Type is abstract;
          -- To get the effect of literals and conversions.

          function "+" (Left, Right : in Root_Bignum_Type) return Root_Bignum_Type is abstract;
          -- And all of the rest.

          function Expected_Digits return Natural is abstract;
              -- The number of digits supported by this type; 0 is returned
              -- if the number is essentially infinite.

          -- And probably some other queries.
      end Root_Bignum;

And then there could be multiple specific implementations with different
performance characteristics, everything from Long_Float itself thru infinite
rational representations.

This would allow one to create most of the interesting algorithms as class-wide
operations (with implementations that could adjust to the characteristics of the
underlying type), for instance:

   function Sqrt (Val : in Root_Bignum_Type'Class;
                  Required_Digits : Natural := 0) return Root_Bignum_Type'Class;

Here with "Required_Digits" specifies how many digits of result are needed.
If 0, the value would be retrieved from the underlying representation.
(Probably have to raise an exception if that gives "infinite".)

Such a layout would also allow easy changing of representations, which probably
would be needed for tuning purposes (most of these maths being slow, at least by
conventional standards).

This would have the clear advantage of avoiding being locked into a single form
of Bignum math, when clearly there are other choices out there useful for
particular purposes.

****************************************************************

From: Randy Brukardt
Sent: Wednesday, December 13, 2017  5:25 PM

I said:
...
> The problem with the Boehm method is that it requires specifying those
> "required bits", which seems problematic in a programming environment.

but then also noted:

>           function Sqrt (Val : in Root_Bignum_Type'Class;
>                          Required_Digits : Natural := 0) return
> Root_Bignum_Type'Class;

essentially, recognizing that many non-terminating algorithms have to have some
sort of termination criteria.

For ease of use purposes, one would prefer to only specify numbers of digits if
they're really needed (as in Sqrt or PI, etc.). But if there are going to be a
lot of such operations, one would want to be able to specify that once. Hope
that explains my thinking here.

Also, a Bignum library needs a corresponding Text_IO library. And probably a
custom version of GEF. (The Janus/Ada compiler library has most of these
features, and they are used extensively.)

****************************************************************

From: Steve Baird
Sent: Friday, January 19, 2018  2:36 PM

We have agreed that we want bignum support in the form of one or more predefined
packages with no other language extensions (e.g., no new rules for numeric
literals) as part of this AI.

The general approach seems fairly clear, although there are a lot of details to
decide (not the least of which are the choices for names). I think we want two
forms, "vanilla" and "bounded" (analogous to, for example,
Ada.Containers.Vectors and Ada.Containers.Bounded_Vectors). In one form, the two
"big" numeric types (tentatively named Big_Integer and Big_Rational) are defined
as undiscriminated types. In the second form, these types are discriminated with
some sort of a capacity discriminant. The idea is that the first form is allowed
to use dynamic storage allocation and controlled types in its implementation
while the second form is not; the discriminant somehow indicates the set of
representable values via some mapping (should this mapping be implementation
dependent?).

At a high level, we might have something like

    package Ada.Big_Numbers is
      -- empty spec like Ada.Containers package
    end;

    package Ada.Big_Numbers.Big_Integers is
       type Big_Integer is private;

       function GCD (Left, Right : Big_Integer) return Integer;

       function "+" (Arg : Some_Concrete_Integer_Type_TBD)
         return Big_Integer;

       ... ops for Big_Integer ...
    end Ada.Big_Numbers.Big_Integers.

    with Ada.Big_Numbers.Big_Integers;
    package Ada.Big_Numbers.Big_Rationals is
      use type Big_Integers.Big_Integer;

      type Big_Rational is private with
        Type_Invariant =>
          Big_Rational = +0 or else
          Big_Integers.GCD
            (Big_Integers.Numerator (Big_Rational),
             Big_Integers.Denominator (Big_Rational)) = +1;

      function Numerator (Arg : Big_Rational) return Big_Integer;
      function Denominator (Arg : Big_Rational) return Big_Integer;

      function "/" (Num, Den : Big_Integer) return Big_Rational
        with Pre => Den /= +0;

      ... other ops for Big_Rational ...
    end Ada.Big_Numbers.Big_Rationals;

    package Ada.Big_Numbers.Bounded_Big_Integers is ... end;

    package Ada.Big_Numbers.Bounded_Big_Rationals is ... end;

Questions/observations include:

1) Do we declare deferred constants, parameterless functions, or neither
    for things like Zero, One, and Two?

2) Which ops do we include? It seems obvious that we define at least
    the arithmetic and relational ops that are defined for any
    predefined integer (respectively float) type for Big_Integer
    (respectively, Big_Rational).

    What Pre/Postconditions are specified for these ops?
    These might involve subtype predicates.
    For example (suggested by Bob), do we want

       subtype Nonzero_Integer is Big_Integer with
           Predicate => Nonzero_Integer /= Zero;
       function "/"
         (X: Big_Integer; Y: Nonzero_Integer) return Big_Integer;
       -- similar for "mod", "rem".

     ?

    What other operations should be provided?
      - Conversion between Big_Int and what concrete integer types?
        I'd say define a type with range Min_Int .. Max_Int
        and provide conversion functions for that type. Also provide
        two generic conversion functions that take a generic formal
        signed/modular type.

      - Conversion between Big_Rational and what concrete integer or
        float types? Same idea. Conversion between a maximal
        floating point type and then a pair of conversion generics
        with formal float/fixed parameters.

      - What shortcuts do we provide (i.e., ops that can easily be
        built out of other ops)? Assignment procedures like
          Add (X, Y); -- X := X + Y
        or mixed-type operators whose only purpose is to spare users
        from having to write explicit conversion?

3) It seems clear that we don't want the bounded form of either
    package to "with" the unbounded form but we do want conversion
    functions for going between corresponding bounded and unbounded
    types. Perhaps these go in child units of the two bounded packages
    (those child units could then "with" the corresponding unbounded
    packages). Should streaming of the two forms be compatible as with
    vectors and bounded vectors?

4) We need an Assign procedure. In the unbounded case it can be just
    a wrapper for predefined assignment, but in the bounded case it
    has to deal with the case where the two arguments have different
    capacities. It's fairly obvious what to do in most cases, but what
    about assigning a Big_Rational value which cannot be represented
    exactly given the capacity of the target. Raise an exception or
    round? In either case, we probably want to provide a Round function
    that deterministically finds an approximation to a given
    value which can be represented as a value having a given
    capacity. This can be useful in the unbounded case just to save
    storage. Should this Round function be implementation-dependent?
    If not, then we might end up talking about convergents and
    semi-convergents in the Ada RM (or at least in the AARM),
    which would be somewhat odd (see
shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximations-are-the-convergents-of-the-continued-fraction
    ). I do not think we want to define Succ/Pred functions which take
    a Big_Rational and a capacity value.

5) We want to be sure that a binding to GNU/GMP is straightforward in
    the unbounded case. [Fortunately, that does not require using the
    same identifiers used in GNU/GMP (mpz_t and mpq_t).]
    See gmplib.org/manual for the GNU/GMP interfaces.

6) Do we want functions to describe the mapping between Capacity
    discriminant values and the associated set of representable values?
    For example, a function from a value (Big_Integer or Big_Rational)
    to the smallest capacity value that could be used to represent it.
    For Big_Integer there could presumably be Min and Max functions
    that take a capacity argument. For Big_Rational, it's not so clear.
    We could require, for example, that a given capacity value allows
    representing a given Big_Rational value if it is >= the sum of
    the capacity requirements of the Numerator and the Denominator.

7) Bob feels (and I agree) that the ARG should not formally approve any
    changes until we have experience with an implementation. At this
    point the ARG should be focused on providing informal guidance on
    this topic.

Opinions?

****************************************************************

From: Randy Brukardt
Sent: Friday, January 19, 2018  10:18 PM

...
> Questions/observations include:

0) Should Big_Integer and (especially) Big_Rational be visibly tagged?

If so, then we can use prefix notation on functions like Numerator and
Denominator. We could also consider deriving both versions (usual and bounded)
from an abstract ancestor.

> 1) Do we declare deferred constants, parameterless functions,
>     or neither for things like Zero, One, and Two?

If tagged, I'll finally get an excuse to show why what I called "tag
propagation" is necessary to implement the dispatching rules in 3.9.2. :-) (One
has to consider a set of calls, not a single call, for determining the static or
dynamic tag for dispatching. That's demonstratably necessary to process tagged
expressions with constants or literals.)

Anyway, the answer to this depends on whether there is a sufficiently short
constructor -- and that really depends on whether Tucker invents a useful
"literals for private type" AI. So I don't think this can be answered until we
find out about that.

> 2) Which ops do we include? It seems obvious that we define at least
>     the arithmetic and relational ops that are defined for any
>     predefined integer (respectively float) type for Big_Integer
>     (respectively, Big_Rational).
>
>     What Pre/Postconditions are specified for these ops?
>     These might involve subtype predicates.
>     For example (suggested by Bob), do we want
>
>        subtype Nonzero_Integer is Big_Integer with
>            Predicate => Nonzero_Integer /= Zero;
>        function "/"
>          (X: Big_Integer; Y: Nonzero_Integer) return Big_Integer;
>        -- similar for "mod", "rem".
>
>      ?

Shouldn't this predicate raise Constraint_Error rather than defaulting to
Assertion_Error, to be more like the other numeric operations? Otherwise, I'm
all in favor of this formulation. Note, however, that since the underlying type
is likely to be controlled and thus tagged, this would require some changes to
other rules; there is already an AI about that (AI12-0243-1).

>     What other operations should be provided?
>       - Conversion between Big_Int and what concrete integer types?
>         I'd say define a type with range Min_Int .. Max_Int
>         and provide conversion functions for that type. Also provide
>         two generic conversion functions that take a generic formal
>         signed/modular type.

Sounds OK.

>       - Conversion between Big_Rational and what concrete integer or
>         float types? Same idea. Conversion between a maximal
>         floating point type and then a pair of conversion generics
>         with formal float/fixed parameters.

Sounds OK again.

>       - What shortcuts do we provide (i.e., ops that can easily be
>         built out of other ops)? Assignment procedures like
>           Add (X, Y); -- X := X + Y
>         or mixed-type operators whose only purpose is to spare users
>         from having to write explicit conversion?

The only reason for mixed type operators is to make literals available. But if
one does those, then we can't add literals properly in the future
(Ada.Strings.Unbounded is damaged by this). So I say no.

I wouldn't bother with any other routines until at least such time as Bob
:-) has built some ACATS tests.

> 3) It seems clear that we don't want the bounded form of either
>     package to "with" the unbounded form but we do want conversion
>     functions for going between corresponding bounded and unbounded
>     types. Perhaps these go in child units of the two bounded packages
>     (those child units could then "with" the corresponding unbounded
>     packages).

Alternatively, both could be derived from an abstract type, and a class-wide
conversion provided. That would get rid of the empty package in your proposal.
:-)

>     Should streaming of the two forms be compatible as with
>     vectors and bounded vectors?

Yes.

> 4) We need an Assign procedure. In the unbounded case it can be just
>     a wrapper for predefined assignment, but in the bounded case it
>     has to deal with the case where the two arguments have different
>     capacities. It's fairly obvious what to do in most cases, but what
>     about assigning a Big_Rational value which cannot be represented
>     exactly given the capacity of the target. Raise an exception or
>     round?

I think I'd raise Capacity_Error. (Isn't that what the containers do?) Having
exact math be silently non-exact seems like exactly (pun) the wrong thing to do.

>     In either case, we probably want to provide a Round function
>     that deterministically finds an approximation to a given
>     value which can be represented as a value having a given
>     capacity. This can be useful in the unbounded case just to save
>     storage. Should this Round function be implementation-dependent?
>     If not, then we might end up talking about convergents and
>     semi-convergents in the Ada RM (or at least in the AARM),
>     which would be somewhat odd (see
> shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-appr
> oximations-are-the-convergents-of-the-continued-fraction
>     ). I do not think we want to define Succ/Pred functions which take
>     a Big_Rational and a capacity value.

???

I don't think Round (or any other operation) ought to be
implementation-dependent, so I think it would need a real definition. Hopefully
with "semi-convergents" or other terms that no one has heard of. ;-)

> 5) We want to be sure that a binding to GNU/GMP is straightforward in
>     the unbounded case. [Fortunately, that does not require using the
>     same identifiers used in GNU/GMP (mpz_t and mpq_t).]
>     See gmplib.org/manual for the GNU/GMP interfaces.

Makes sense.

> 6) Do we want functions to describe the mapping between Capacity
>     discriminant values and the associated set of representable values?
>     For example, a function from a value (Big_Integer or Big_Rational)
>     to the smallest capacity value that could be used to represent it.
>     For Big_Integer there could presumably be Min and Max functions
>     that take a capacity argument. For Big_Rational, it's not so clear.
>     We could require, for example, that a given capacity value allows
>     representing a given Big_Rational value if it is >= the sum of
>     the capacity requirements of the Numerator and the Denominator.

It seems that the Capacity needs to mean something to the end user, not just the
compiler. So such functions seem necessary, but KISS for those!!

> 7) Bob feels (and I agree) that the ARG should not formally approve any
>     changes until we have experience with an implementation. At this
>     point the ARG should be focused on providing informal guidance on
>     this topic.

I agree that Bob should prototype these packages, including writing ACATS-style
tests for them, so that we can put them into the Ada 2020 Standard. I'll put it
on his action item list. ;-)

Seriously, we already have an ARG rule that all Amendment AIs are supposed to
include (some) ACATS tests, and we really should have a similar rule that
proposed packages are prototyped as well. This is the assumed responsibility of
an AI author, so if you can't get Bob to help, you're pretty much stuck, and
need to do that before the AI could be assumed complete.

OTOH, we haven't required that from any other AI author, so why start now??
(We really ought to, I don't have a very big budget to write Ada 2020 ACATS
tests. Topic to discuss during the call?)

****************************************************************

From: Jean-Pierre Rosen
Sent: Saturday, January 20, 2018  12:22 AM

> Questions/observations include:
> [...]
>
I'd add:
8) IOs
   Should an IO package be associated to each of these bignums?
   Note that the issue of IO may influence the representation of
   of bignums: I once knew an implementation where each super-digit
   was limited to 1_000_000_000 (instead of the natural 2_147_483_647),
   just to avoid terribly inefficient IOs.

****************************************************************

From: Tucker Taft
Sent: Saturday, January 20, 2018  11:08 AM

> ...
>
>> 1) Do we declare deferred constants, parameterless functions,
>>    or neither for things like Zero, One, and Two?
>
> If tagged, I'll finally get an excuse to show why what I called "tag
> propagation" is necessary to implement the dispatching rules in 3.9.2.
> :-) (One has to consider a set of calls, not a single call, for
> determining the static or dynamic tag for dispatching. That's
> demonstratably necessary to process tagged expressions with constants
> or literals.)

I agree that you have to do "tag propagation" to properly handle tag
indeterminate calls.  Has anyone claimed otherwise?

>
> Anyway, the answer to this depends on whether there is a sufficiently
> short constructor -- and that really depends on whether Tucker invents
> a useful "literals for private type" AI. So I don't think this can be
> answered until we find out about that.

I'm on it. ;-)

****************************************************************

From: Randy Brukardt
Sent: Saturday, January 20, 2018  7:29 PM

> I agree that you have to do "tag propagation" to properly handle tag
> indeterminate calls.  Has anyone claimed otherwise?

Not that I know of, but based on my compiler surveys, no one implements it other
than Janus/Ada. Admittedly, I haven't checked this recently.

I've long had a tagged Bignum-like package on my ACATS test to-construct list
(because one needs usage-orientation for such tests) in order to test this rule.
So far as I can tell, the ACATS doesn't currrently test cases like those that
arise in Bignum:

      procedure Something (Val : in out Num'Class) is
      begin
          Val := + Zero; -- Zero gets the tag of Val, propagated through "+".
          declare
              Org : Num'Class := Val + (- One); -- Org and One get the tag of Val.
          begin
              ...
          end;
      end Something;

I'll probably come up with more realistic-looking expressions for this test, but
the idea should be obvious. (I'll have to test both static and dynamic binding,
as well as tag indeterminate cases.)

****************************************************************

From: John Barnes
Sent: Monday, January 22, 2018  5:49 AM

I wrote a bignum package in Ada 83 some 30 years ago. I did make some updates to
use Ada 95, mainly child packages. I still use it for numerical stuff for
courses at Oxford.

Notable points perhaps.

I did use a power of 10 for the base to ease IO. It was originally on a 16 bit
machine.  (386 perhaps). It still works on this horrid Windows 10. Not much
faster than on my old XP laptop. I don't know what Windows 10 is doing.
Obviously playing with itself - ridiculous.

I provided constants Zero and One. I didn't think any others were necessary.
Others were provided by eg

Two: Number := Make-Number(2);

I provided a package for subprograms Add, Sub, Mul, Div, Neg, Compare, Length,
To_Number, To_Text, To_Integer.

And a package for functions +. -, abs, *, / rem, mod, <,  <=, >, >=, =

And other packages for I/O.

Long time ago. Certainly very useful.

****************************************************************

From: Steve Baird
Sent: Monday, January 22, 2018  12:33 PM

> I'd add:
> 8) IOs
>     Should an IO package be associated to each of these bignums?

Good question.

If we provide conversion functions to and from String then would any further IO
support be needed?

****************************************************************

From: Steve Baird
Sent: Monday, January 22, 2018  1:24 PM

> ...
>> Questions/observations include:
>
> 0) Should Big_Integer and (especially) Big_Rational be visibly tagged?
>
> If so, then we can use prefix notation on functions like Numerator and
> Denominator. We could also consider deriving both versions (usual and
> bounded) from an abstract ancestor.

If we go this way, then should this common ancestor be an interface type? I'd
say yes.

Does it then get all the same ops, so that the non-abstract ops declared for the
Bounded and Unbounded types would all be overriding?

Would this make the AI12-0243-ish issues any worse (consider the proposed
Nonzero_Integer parameter subtype mentioned earlier)? I know these problems are
bad enough already, but my question is whether this would make matters any
worse.

>> 2) Which ops do we include? It seems obvious that we define at least
>>      the arithmetic and relational ops that are defined for any
>>      predefined integer (respectively float) type for Big_Integer
>>      (respectively, Big_Rational).
>>
>>      What Pre/Postconditions are specified for these ops?
>>      These might involve subtype predicates.
>>      For example (suggested by Bob), do we want
>>
>>         subtype Nonzero_Integer is Big_Integer with
>>             Predicate => Nonzero_Integer /= Zero;
>>         function "/"
>>           (X: Big_Integer; Y: Nonzero_Integer) return Big_Integer;
>>         -- similar for "mod", "rem".
>>
>>       ?
>
> Shouldn't this predicate raise Constraint_Error rather than defaulting
> to Assertion_Error, to be more like the other numeric operations?

Good point; I agree.

>> 3) It seems clear that we don't want the bounded form of either
>>      package to "with" the unbounded form but we do want conversion
>>      functions for going between corresponding bounded and unbounded
>>      types. Perhaps these go in child units of the two bounded packages
>>      (those child units could then "with" the corresponding unbounded
>>      packages).
>
> Alternatively, both could be derived from an abstract type, and a
> class-wide conversion provided. That would get rid of the empty
> package in your proposal. :-)

Could you provide a more detailed spec? I don't see how this would work, but I
suspect that I'm misunderstanding your proposal.

>> 4) We need an Assign procedure. In the unbounded case it can be just
>>      a wrapper for predefined assignment, but in the bounded case it
>>      has to deal with the case where the two arguments have different
>>      capacities. It's fairly obvious what to do in most cases, but what
>>      about assigning a Big_Rational value which cannot be represented
>>      exactly given the capacity of the target. Raise an exception or
>>      round?
>
> I think I'd raise Capacity_Error. (Isn't that what the containers do?)
> Having exact math be silently non-exact seems like exactly (pun) the
> wrong thing to do.

Is it that simple? Suppose somebody wants large rationals (e.g., 2048-bit
numerators and denominators) with rounding. It's not that they require exact
arithmetic - they just want a lot more range/precision than what you get from
Ada's numeric types. It may be that this is an unimportant corner case and you
are right to dismiss it; I don't know.


>> 6) Do we want functions to describe the mapping between Capacity
>>      discriminant values and the associated set of representable values?
>>      For example, a function from a value (Big_Integer or Big_Rational)
>>      to the smallest capacity value that could be used to represent it.
>>      For Big_Integer there could presumably be Min and Max functions
>>      that take a capacity argument. For Big_Rational, it's not so clear.
>>      We could require, for example, that a given capacity value allows
>>      representing a given Big_Rational value if it is >= the sum of
>>      the capacity requirements of the Numerator and the Denominator.
>
> It seems that the Capacity needs to mean something to the end user,
> not just the compiler. So such functions seem necessary, but KISS for those!!

Am I right in guessing that you'd like these functions to be portable (as
opposed to being implementation-defined)?

****************************************************************

From: Randy Brukardt
Sent: Monday, January 22, 2018  3:41 PM

> > I'd add:
> > 8) IOs
> >     Should an IO package be associated to each of these bignums?
>
> Good question.
>
> If we provide conversion functions to and from String then would any
> further IO support be needed?

We currently have Text_IO nested packages or children for pretty much any type
for which it makes sense to have text input-output, despite the fact that every
such type has an Image function or the equivalent (To_String for unbounded
strings).

So I'd rather expect a Ada.Text_IO.BigNum_IO package. If we don't define it now,
we will the next time around.

(The Janus/Ada UnivMath package has a complete set of Text_IO packages, and they
are heavily used. I believe they can output both rational and decimal
representation for the universal_real type.)

****************************************************************

From: Randy Brukardt
Sent: Monday, January 22, 2018  3:36 PM

> > Steve Baird writes:
> > ...
> >> Questions/observations include:
> >
> > 0) Should Big_Integer and (especially) Big_Rational be visibly tagged?
> >
> > If so, then we can use prefix notation on functions like Numerator
> > and Denominator. We could also consider deriving both versions
> > (usual and
> > bounded) from an abstract ancestor.
>
> If we go this way, then should this common ancestor be an interface
> type? I'd say yes.

I suggested making it abstract so it could have some concrete operations if
those made sense. But perhaps they don't make sense.

> Does it then get all the same ops, so that the non-abstract ops
> declared for the Bounded and Unbounded types would all be overriding?

I would expect that the vast majority of operations are in the interface, so
dispatching can be used, and one can write class-wide algorithms that work with
any Bignum representation. Probably the capacity-specific operations would be
left out.

> Would this make the AI12-0243-ish issues any worse (consider the
> proposed Nonzero_Integer parameter subtype mentioned earlier)? I know
> these problems are bad enough already, but my question is whether this
> would make matters any worse.

It just makes a solution more urgent, but it doesn't change the issues any.

...
> >> 3) It seems clear that we don't want the bounded form of either
> >>      package to "with" the unbounded form but we do want conversion
> >>      functions for going between corresponding bounded and unbounded
> >>      types. Perhaps these go in child units of the two bounded packages
> >>      (those child units could then "with" the corresponding unbounded
> >>      packages).
> >
> > Alternatively, both could be derived from an abstract type, and a
> > class-wide conversion provided. That would get rid of the empty
> > package in your proposal. :-)
>
> Could you provide a more detailed spec? I don't see how this would
> work, but I suspect that I'm misunderstanding your proposal.

I was thinking about including cross-cut operations in the spec, something
like:

    type BigNum is abstract tagged with private;

    function Convert (Val : in Bignum'Class) return Bignum;

but thinking about it now, I can't figure out how one would implement one of
those.

You'd probably have to have a concrete universal representation to make that
work:

    function Convert (Val : in Bignum) return Universal_Big;

    function Convert (Val : in Universal_Big) return BigNum;

but of course that would bring in the memory allocation/finalization issues
that you are trying to avoid.

So at this moment I'm thinking that direct conversions would have to be left
out; you could generally do it through intermediary types like Max_Integer
using Numerator/Demomonator.

> >> 4) We need an Assign procedure. In the unbounded case it can be just
> >>      a wrapper for predefined assignment, but in the bounded case it
> >>      has to deal with the case where the two arguments have different
> >>      capacities. It's fairly obvious what to do in most cases, but what
> >>      about assigning a Big_Rational value which cannot be represented
> >>      exactly given the capacity of the target. Raise an exception or
> >>      round?
> >
> > I think I'd raise Capacity_Error. (Isn't that what the containers
> > do?) Having exact math be silently non-exact seems like exactly
> > (pun) the wrong thing to do.
>
> Is it that simple? Suppose somebody wants large rationals (e.g.,
> 2048-bit numerators and denominators) with rounding.
> It's not that they require exact arithmetic - they just want a lot
> more range/precision than what you get from Ada's numeric types.
> It may be that this is an unimportant corner case and you are right to
> dismiss it; I don't know.

We're not trying to be all things to all people. I'd consider these "exact"
math packages and treat them accordingly. If there is an abstract root, one
can "easily" make a clone version that uses rounding if someone needs that.
(Defining the rounding is hard, as you noted elsewhere.)

> >> 6) Do we want functions to describe the mapping between Capacity
> >>      discriminant values and the associated set of representable values?
> >>      For example, a function from a value (Big_Integer or Big_Rational)
> >>      to the smallest capacity value that could be used to represent it.
> >>      For Big_Integer there could presumably be Min and Max functions
> >>      that take a capacity argument. For Big_Rational, it's not so clear.
> >>      We could require, for example, that a given capacity value allows
> >>      representing a given Big_Rational value if it is >= the sum of
> >>      the capacity requirements of the Numerator and the Denominator.
> >
> > It seems that the Capacity needs to mean something to the end user,
> > not just the compiler. So such functions seem necessary, but KISS
> > for those!!
>
> Am I right in guessing that you'd like these functions to be portable
> (as opposed to being implementation-defined)?

I think so; otherwise it rather defeats the purpose of language-defined packages
(to provide the ultimate in portability).

****************************************************************

From: Bob Duff
Sent: Sunday, January 28, 2018  11:29 AM

> Steve Baird writes:
> ...
> > Questions/observations include:
> 
> 0) Should Big_Integer and (especially) Big_Rational be visibly tagged?

Surely not.  I think we want to be competetive (efficiency-wise) with all
sorts of other languages, and taggedness will destroy that.

Let's not have another "tampering" fiasco.

> If so, then we can use prefix notation on functions like Numerator and 
> Denominator.

I'm not a big fan of that feature, but if we want it, we should figure out
how to do it for untagged types.

>... We could also consider deriving both versions (usual and
> bounded) from an abstract ancestor.

Consider, ..., and reject.  ;-)

****************************************************************

From: Jeff Cousins
Sent: Sunday, January 28, 2018  12:21 PM

John Barnes wrote:

  I wrote a bignum package in Ada 83 some 30 years ago
 
Would you be able to let us see the spec for this?

****************************************************************

From: Randy Brukardt
Sent: Sunday, January 28, 2018  9:13 PM

> > 0) Should Big_Integer and (especially) Big_Rational be
> visibly tagged?
> 
> Surely not.  I think we want to be competetive
> (efficiency-wise) with all sorts of other languages, and taggedness 
> will destroy that.

??? Tags (as opposed to controlled types) add almost no overhead, especially
in a case like unbounded Bignum which probably will have to be controlled
anyway. (The only overhead of a tagged type is initializing the tag in the
object.) So long as one uses a single specific type, everything is statically
bound and the cost is essentially the same as an untagged type (again,
especially as the underlying specific type most likely will be tagged and
certainly will be large).

I wasn't suggesting that we define any class-wide operations other than
representation conversion (which should be rarely used in any case).
Class-wide operations are the only operations that add overhead.

> Let's not have another "tampering" fiasco.

I'm still waiting for an example program showing this supposed "fiasco". No
one has ever submitted one to the ARG. We've essentially been asked to believe
this issue by repeated assertion. (And most tampering checks can be done at
compile-time, with sufficient will.)

If there was a fiasco here, it was that the goals of the containers did not 
include making them particularly fast. If they are then misused for
high-performance code, one is going to get the expected disappointment.
Perhaps we started with the wrong set of goals.

> > If so, then we can use prefix notation on functions like Numerator 
> > and Denominator.
> 
> I'm not a big fan of that feature, but if we want it, we should figure 
> out how to do it for untagged types.

We've already discussed that in a different e-mail thread. It seems dangerous.

> >... We could also consider deriving both versions (usual and
> > bounded) from an abstract ancestor.
> 
> Consider, ..., and reject.  ;-)

Again, why? We have a request for a "universal" numeric type, and the only
sane way to provide that is with dispatching. Probably, we'll just forget 
that request, but it seems worth spending a bit of time to see if it makes
sense.

****************************************************************

From: John Barnes
Sent: Tuesday, January 30, 2018  4:22 AM

I am feverishly giving lectures on numbers at Oxford at the moment but I am 
trying to keep an eye on what the ARG is up to.

Did you know that a new Mersenne prime was discovered on Boxing Day (26
December) 2017. It is 2**77232917 - 1 and has only 23,249,425 digits. Will
the Bignum package cope with it?

****************************************************************

From: Tucker Taft
Sent: Tuesday, January 30, 2018  3:02 PM

Yes, I noticed that new Mersenne prime as well.  And 23 mega-digit is nothing
for a modern iPhone. ;-)  Just be sure to set aside a bit of extra time to
print it out using the Image function.  Except in base 2, of course, which
I could do right now.  Ready:  1111111111 [... 77,232,900 1's] 1111111!

****************************************************************

From: Jeff Cousins
Sent: Wednesday, January 31, 2018  6:31 AM

[This is John Barnes' Bignum package and some test programs - Editor.]

-- file books\fun\progs\numbers.ada

-- Restructured using children

-- Types and No_Of_Places in parent package 

-- 20-10-06

package Numbers is

   Max_Index: constant := 1000;
   subtype Index is Integer range 0 .. Max_Index;
   type Number(Max_Digits: Index := 1) is private;

   Zero, One: constant Number;

   Number_Error : exception;

private
   Base_Exp: constant := 4;
   Base: constant := 10 ** Base_Exp;

   type Base_Digit is range -Base .. 2 * Base - 1;

   type Base_Digit_Array is
      array(Index range <>) of Base_Digit;

   type Number(Max_Digits: Index := 1) is
      record
         Sign: Integer := +1;
         Length: Index := 0;
         D: Base_Digit_Array(1..Max_Digits);
      end record;

   Zero: constant Number := (0, +1, 0, (others => 0));
   One: constant Number := (1, +1, 1, (1 => 1));

   function No_Of_Places(N: Number) return Integer;

end Numbers;


package body Numbers is

   function No_Of_Places(N: Number) return Integer is
   begin
      if N.Length = 0 then 
         return 1;
      else
         return N.Length * Base_Exp;
      end if;
   end No_Of_Places;

end Numbers;

--------------------------------------------------------------------

package Numbers.Proc is

   subtype Index is Numbers.Index;
   subtype Number is Numbers.Number;
   Zero: Number renames Numbers.Zero;
   One: Number renames Numbers.One;
   Number_Error: exception renames Numbers.Number_Error;

   procedure Add(X, Y: Number; Z: out Number);
   procedure Sub(X, Y: Number; Z: out Number);
   procedure Mul(X, Y: Number; Z: out Number);
   procedure Div(X, Y: Number; Quotient,
                               Remainder: out Number);

   procedure Neg(X: in out Number);

   function Compare(X, Y: Number) return Integer;

   function Length(N: Number) return Index;

   procedure To_Number(S: String; N: out Number);
   procedure To_Number(I: Integer; N: out Number);
   procedure To_Text(N: Number; S: out String);
   procedure To_Integer(N: Number; I: out Integer);

end Numbers.Proc;

--------------------------------------------------------------------

package Numbers.IO is

   Default_Width: Natural := 0;
   procedure Put(Item: Number;
                Width: Natural := Default_Width);
   procedure Get(Item: out Number);

end Numbers.IO;

--------------------------------------------------------------------

package Numbers.Func is

   subtype Number is Numbers.Number;
   Zero: Number renames Numbers.Zero;
   One: Number renames Numbers.One;
   Number_Error: exception renames Numbers.Number_Error;

   function "+" (X: Number) return Number;
   function "-" (X: Number) return Number;
   function "abs" (X: Number) return Number;
   function "+" (X, Y: Number) return Number;
   function "-" (X, Y: Number) return Number;
   function "*" (X, Y: Number) return Number;
   function "/" (X, Y: Number) return Number;
   function "rem" (X, Y: Number) return Number;
   function "mod" (X, Y: Number) return Number;
   function "**" (X: Number; N: Natural) return Number;
   function "<" (X, Y: Number) return Boolean;
   function "<=" (X, Y: Number) return Boolean;
   function ">" (X, Y: Number) return Boolean;
   function ">=" (X, Y: Number) return Boolean;

   function "=" (X, Y: Number) return Boolean;

   function Make_Number(S: String) return Number;
   function Make_Number(I: Integer) return Number;
   function String_Of(N: Number) return String;
   function Integer_Of(N: Number) return Integer;

end Numbers.Func;

--------------------------------------------------------------------

package body Numbers.Proc is

   Base_Squared: constant := Base**2;
   subtype Single is Base_Digit;
   type Double is range -Base_Squared .. 2*Base_Squared - 1;


   function Unsigned_Compare(X, Y: Number) return Integer is
   -- ignoring signs
   -- returns +1, 0 or -1 according as X >, = or < Y
   begin
      if X.Length > Y.Length then return +1; end if;
      if X.Length < Y.Length then return -1; end if;
      for I in reverse 1 .. X.Length loop
         if X.D(I) > Y.D(I) then return +1; end if;
         if X.D(I) < Y.D(I) then return -1; end if;
      end loop;
      return 0;  -- the numbers are equal
   end Unsigned_Compare;

   function Compare(X, Y: Number) return Integer is
   -- returns +1, 0 or -1 according as X >, = or < Y
   begin
      if X.Sign /= Y.Sign then return X.Sign; end if;
      return Unsigned_Compare(X, Y) * X.Sign;
   end Compare;

   procedure Raw_Add(X, Y: Number; Z: out Number) is
   -- assumes X not smaller than Y
      Carry: Single := 0;
      Digit: Single;
      ZL: Index := X.Length;  -- length of answer
   begin
      if Z.Max_Digits < ZL then
         raise Number_Error;  -- Z not big enough to hold X
      end if;
      for I in 1 .. ZL loop
         Digit := X.D(I) + Carry;
         if I <= Y.Length then
            Digit := Digit + Y.D(I);
         end if;
         if Digit >= Base then
            Carry := 1;  Digit := Digit - Base;
         else
            Carry := 0;
         end if;
         Z.D(I) := Digit;
      end loop;
      if Carry /= 0 then
         if ZL = Z.Max_Digits then
            raise Number_Error;  --  too big to fit in Z
         end if;
         ZL := ZL + 1;
         Z.D(ZL) := Carry;
      end if;
      Z.Length := ZL;
   end Raw_Add;

   procedure Raw_Sub(X, Y: Number; Z: out Number) is
   -- assumes X not smaller than Y
      Carry: Single := 0;
      Digit: Single;
      ZL: Index := X.Length;  -- length of answer
   begin
      if Z.Max_Digits < ZL then
         raise Number_Error;  -- Z not big enough to hold X
      end if;
      for I in 1 .. ZL loop
         Digit := X.D(I) - Carry;
         if I <= Y.Length then
            Digit := Digit - Y.D(I);
         end if;
         if Digit < 0 then
            Carry := 1;  Digit := Digit + Base;
         else
            Carry := 0;
         end if;
         Z.D(I) := Digit;
      end loop;
      while Z.D(ZL) = 0 loop  -- SHOULD THIS NOT FAIL???
         ZL := ZL - 1;         -- remove leading zeroes
      end loop;
      Z.Length := ZL;
   end Raw_Sub;

   procedure Add(X, Y: Number; Z: out Number) is
      UCMPXY: Integer := Unsigned_Compare(X, Y);
   begin
      if X.Sign = Y.Sign then
         Z.Sign := X.Sign;
         if UCMPXY >= 0 then
            Raw_Add(X, Y, Z);
         else
            Raw_Add(Y, X, Z);  -- reverse if Y larger
         end if;
      else
         if UCMPXY > 0 then
            Raw_Sub(X, Y, Z);
            Z.Sign := X.Sign;
         elsif UCMPXY < 0 then
            Raw_Sub(Y, X, Z);
            Z.Sign := -X.Sign;
         else  --  answer is zero
            Z.Sign := +1; Z.Length := 0;
         end if;
      end if;
   end Add;

   procedure Sub(X, Y: Number; Z: out Number) is
      UCMPXY: Integer := Unsigned_Compare(X, Y);
   begin
      if X.Sign /= Y.Sign then
         Z.Sign := X.Sign;
         if UCMPXY >= 0 then
            Raw_Add(X, Y, Z);
         else
            Raw_Add(Y, X, Z);  -- reverse if Y larger
         end if;
      else
         if UCMPXY > 0 then
            Raw_Sub(X, Y, Z);
            Z.Sign := X.Sign;
         elsif UCMPXY < 0 then
            Raw_Sub(Y, X, Z);
            Z.Sign := -X.Sign;
         else  --  answer is zero
            Z.Sign := +1; Z.Length := 0;
         end if;
      end if;
   end Sub;

   procedure Neg(X: in out Number) is
   begin
      -- do nothing in zero case
      if X.Length = 0 then return; end if;
      X.Sign := -X.Sign;
   end Neg;

   function Length(N: Number) return Index is
   begin
      return N.Length;
   end Length;

   procedure Mul(X, Y: Number; Z: out Number) is

      Carry: Double;
      Digit: Double;
      ZL: Index;
   begin
      if Z.Max_Digits < X.Length + Y.Length then
         raise Number_Error;
      end if;
      if X.Length = 0 or Y.Length = 0 then  -- zero case
         Z.Sign := +1;  Z.Length := 0;
         return;
      end if;

      ZL := X.Length + Y.Length - 1;
                   -- lower possible length of answer

      -- copy X to top of Z; so X and Z can be same array
      for I in reverse 1 .. X.Length loop
         Z.D(I + Y.Length) := X.D(I);
      end loop;
      declare -- initialise limits and length of cycle
         Z_Index: Index;
         Y_Index: Index;
         Initial_Z_Index: Index := Y.Length + 1; 
         Initial_Y_Index: Index := 1;            
         Cycle_Length: Index := 1;               
      begin
         Carry := 0;
         for I in 1 .. ZL loop
            Digit := Carry;
            Carry := 0;
            Z_Index := Initial_Z_Index;          
            Y_Index := Initial_Y_Index;          
            for J in 1 .. Cycle_Length loop
               if Digit > Base_Squared then
                  Digit := Digit - Base_Squared;
                  Carry := Carry + Base;
               end if;
               Digit := Digit + Double(Z.D(Z_Index))
                              * Double(Y.D(Y_Index));
               Z_Index := Z_Index + 1;
               Y_Index := Y_Index - 1;
            end loop;
            -- now adjust limits and length of cycle
            if I < Y.Length then
               Cycle_Length := Cycle_Length + 1;
               Initial_Y_Index := Initial_Y_Index + 1;
            else
               Initial_Z_Index := Initial_Z_Index + 1;
            end if;
            if I < X.Length then
               Cycle_Length := Cycle_Length + 1;
            end if;
            Cycle_Length := Cycle_length - 1;
            Carry := Carry + Digit / Base;
            Z.D(I) := Single(Digit mod Base);
         end loop;
      end;
      if Carry /= 0 then -- one more digit in answer
         ZL := ZL + 1;
         Z.D(ZL) := Single(Carry);
      end if;
      Z.Length := ZL;
      Z.Sign := X.Sign * Y.Sign;
   end Mul;

   procedure Div(X, Y: Number; Quotient,
                               Remainder: out Number) is
      U: Number renames Quotient;
      V: Number renames Remainder;
      Digit, Scale, Carry: Double;
      U0, U1, U2: Double;
      V1, V2: Double;
      QD: Double;
      LOQ: constant Index := Y.Length;
      HIQ: constant Index := X.Length;
      QL: Index;
      RL: Index;
      QStart: Index;
      J : Index;
   begin
      if Y.Length = 0 then
         raise Number_Error;
      end if;
      if Quotient.Max_Digits < X.Length or
         Remainder.Max_Digits < Y.Length then
         raise Number_Error;
      end if;
      if X.Length < Y.Length then  -- Quotient is definitely zero
         Quotient.Sign := +1;
         Quotient.Length := 0;
         Remainder.Sign := X.Sign;
         Remainder.Length := X.Length;
         for I in 1 .. X.Length loop
            Remainder.D(I) := X.D(I);
         end loop;
         return;
      end if;

      QL := X.Length - Y.Length + 1;
      RL := Y.Length;
      QStart := QL;

      -- compute normalizing factor
      Scale := Base/Double(Y.D(Y.Length)+1);

      -- scale X and copy to U
      Carry := 0;
      for I in 1 .. X.Length loop
         Digit := Double(X.D(I)) * Scale + Carry;
         Carry := Digit / Base;
         U.D(I) := Single(Digit mod Base);
      end loop;
      U0 := Carry;  -- leading digit of dividend

      -- scale Y and copy to V
      Carry := 0;
      for I in 1 .. Y.Length loop
         Digit := Double(Y.D(I)) * Scale + Carry;
         Carry := Digit / Base;
         V.D(I) := Single(Digit mod Base);
      end loop;
      -- no further carry

      -- set V1 and V2 to first two digits of divisor
      V1 := Double(V.D(Y.Length));
      if Y.Length > 1 then
         V2 := Double(V.D(Y.Length-1));
      else
         V2 := 0;
      end if;

      -- now iterate over digits in answer
      -- with U0, U1 and U2 being first three digits of dividend
      for I in reverse LOQ .. HIQ loop
         U1 := Double(U.D(I));
         if Y.Length > 1 then
            U2 := Double(U.D(I-1));
         else
            U2 := 0;
         end if;

         -- now set initial estimate of digit in quotient
         if U0 = V1 then
            QD := Base - 1;
         else
            QD := (U0 * Base + U1) / V1;
         end if;

         -- now refine estimate by considering U2 also
         while V2*QD > (U0*Base+U1-QD*V1)*Base + U2 loop
            QD := QD - 1;
         end loop;

         -- QD is now correct digit or possibly one too big
         -- subtract QD times V from U
         Carry := 0;
         J := QStart;
         for I in 1 .. Y.Length loop
            Digit := Double(U.D(J)) - Carry
                        - QD * Double(V.D(I));
            if Digit < 0 then
               Carry := (-1-Digit) / Base + 1;
               Digit := Digit + Carry * Base;
            else
               Carry := 0;
            end if;
            U.D(J) := Single(Digit);
            J := J + 1;
         end loop;
         if Carry > U0 then -- estimate was too large
            declare
               Carry, Digit: Single;
            begin
               QD := QD - 1;
               Carry := 0;  J := QStart;
               for I in 1 .. Y.Length loop
                  Digit := U.D(J) + Carry + V.D(I);
                  if Digit >= Base then
                     Carry := 1;
                     Digit := Digit - Base;
                  else
                     Carry := 0;
                  end if;
                  U.D(J) := Digit;
                  J := J + 1;
               end loop;
            end;
         end if;
         -- QD is now the required digit
         U0 := Double(U.D(I));  U.D(I) := Single(QD);
         QStart := QStart - 1;
      end loop;
      
      -- delete possible leading zero in quotient
      if U.D(HIQ) = 0 then
         QL := QL - 1;
      end if;
   
      -- copy remainder into place and scale 
      -- top digit is in U0 still
      Digit := U0;
      for I in reverse 2 .. RL loop
         Remainder.D(I) := Single(Digit/Scale);
         Carry := Digit mod Scale;
         Digit := Double(U.D(I-1)) + Carry * Base;
      end loop;
      Remainder.D(1) := Single(Digit/Scale);
      -- delete leading zeroes in remainder
      while RL > 0 and then Remainder.D(RL) = 0 loop
         RL := RL - 1;
      end loop;
      Remainder.Length := RL;
      if Remainder.Length = 0 then
         Remainder.Sign := +1;
      else
         Remainder.Sign := X.Sign;
      end if;

      -- slide quotient into place
      -- Quotient.D(1 .. QL) := U.D(LOQ .. HIQ);
      for I in 1 .. QL loop
         Quotient.D(I) := U.D(I + LOQ - 1);
      end loop;
      Quotient.Length := QL;
      if Quotient.Length = 0 then
         Quotient.Sign := +1;
      else
         Quotient.Sign := X.Sign * Y.Sign;
      end if;

   end Div;

   procedure To_Number(S: String; N: out Number) is
      NL: Index := 0;
      Place: Integer := 0;
      Is_A_Number: Boolean := False;
      Digit: Single := 0;
      Ch: Character;
      Last_I: Positive;
      Dig_Of: constant array (Character range '0' .. '9') of
                   Single := (0, 1, 2, 3, 4, 5, 6, 7, 8, 9);
   begin
      N.Sign := +1;   -- set default sign
      -- scan string from end
      for I in reverse S'Range loop
         Last_I := I;   -- note how far we have got
         Ch := S(I);
         case Ch is
            when '0' .. '9' =>
               -- add digit to number so far
               if Place = 0 then
                  NL := NL + 1;
                  if NL > N.Max_Digits then
                     raise Number_Error;
                  end if;
               end if;
               Digit := Digit + Dig_Of(Ch) * 10**Place;
               Place := Place + 1;
               if Place = Base_Exp then
                  N.D(NL) := Digit;
                  Digit := 0;
                  Place := 0;
               end if;
               Is_A_Number := True;
            when '_' =>
               -- underscore must be embedded in digits
               if not Is_A_Number then
                  raise Number_Error;
               end if;
               Is_A_Number := False;
            when '+' | '-' | ' ' =>
               -- lump so far must be a valid number
               if not Is_A_Number then
                  raise Number_Error;
               end if;
               if Ch ='-' then N.Sign := -1; end if;
               exit;         -- leave loop
            when others =>
               raise Number_Error;
         end case;
      end loop;
      -- check we had a number
      if not Is_A_Number then
         raise Number_Error;
      end if;
      -- add the last digit if necessary
      if Place /= 0 then
         N.D(NL) := Digit;
      end if;
      -- check that any other characters are leading spaces
      for I in S'First .. Last_I - 1 loop
         if S(I) /= ' ' then
            raise Number_Error;
         end if;
      end loop;
      -- remove leading zeroes if any, beware zero case
      while NL > 0 and then N.D(NL) = 0 loop
         NL := NL - 1;
      end loop;
      N.Length := NL;
   end To_Number;

   procedure To_Number(I: Integer; N: out Number) is
      NL: Index := 0;
      II: Integer;
   begin
      if I = 0 then
         N.Sign := +1;  N.Length := 0;
         return;
      end if;
      if I > 0 then
         II := I;  N.Sign := +1;
      else
         II := -I;  N.Sign := -1;
      end if;
      while II /= 0 loop
         NL := NL + 1;
         if NL > N.Max_Digits then
            raise Number_Error;
         end if;
         N.D(NL) := Single(II mod Base);
         II := II / Base;
      end loop;
      N.Length := NL;
   end To_Number;

   procedure To_Text(N: Number; S: out String) is
      SI: Natural := S'Last;
      Digit: Single;
      Char_Of: constant array (Single range 0 .. 9) of
                              Character := "0123456789";
   begin
      if N.Length = 0 then   -- zero case
         if SI < 2 then
            raise Number_Error;
         end if;
         S(SI) := Char_Of(0);
         S(SI-1) := '+';
         for I in 1 .. SI-2 loop
            S(I) := ' ';
         end loop;
         return;
      end if;
      if SI < Base_Exp * N.Length + 1 then
         raise Number_Error;
      end if;
      for I in 1 .. N.Length loop
         Digit := N.D(I);
         for J in 1 .. Base_Exp loop
            S(SI) := Char_Of(Digit mod 10);
            Digit := Digit / 10;
            SI := SI - 1;
         end loop;
      end loop;
      while S(SI + 1) = '0' loop
         SI := SI + 1;  -- delete leading zeroes
      end loop;
      if N.Sign = +1 then
         S(SI) := '+';
      else
         S(SI) := '-';
      end if;
      for I in 1 .. SI - 1 loop
         S(I) := ' ';
      end loop;
   end To_Text;

   procedure To_Integer(N: Number; I: out Integer) is
      II: Integer := 0;
   begin
      for I in reverse 1 .. N.Length loop
         II := II * Base + Integer(N.D(I));
      end loop;
      if N.Sign = -1 then II := -II; end if;
      I := II;
   end To_Integer;

end Numbers.Proc;

--------------------------------------------------------------------

with Ada.Text_IO; use Ada;
with Numbers.Proc;
package body Numbers.IO is

   use Proc;

   procedure Put(Item: Number;
                 Width: Natural := Default_Width) is
      Block_Size: constant := 3;
      Places: Integer := No_Of_Places(Item);
      S: String(1 .. Places + 1);
      SP: Positive := 1;
      Before_Break: Integer;
   begin
      To_Text(Item, S);
      -- allow for leading spaces in S
      while S(SP) = ' ' loop
         SP := SP + 1;  Places := Places - 1;
      end loop;
      -- now output leading spaces for padding if any
      for I in 1 .. Width -
            (Places + 1 + (Places - 1) / Block_Size) loop
         Text_IO.Put(' ');
      end loop;
      if S(SP) = '+' then S(SP) := ' '; end if;
      Text_IO.Put(S(SP));  -- output minus or space
      -- output digits with underscores every "Blocksize"
      Before_Break := (Places - 1) rem Block_Size + 1;
      for I in SP + 1 .. S'Last loop
         if Before_Break = 0 then
            Text_IO.Put('_');
            Before_Break := Block_Size;
         end if;
         Text_IO.Put(S(I));
         Before_Break := Before_Break - 1;
      end loop;
   end Put;

   procedure Get(Item: out Number) is
      -- declare string large enough to hold maximum value
      -- allows every other character to be an underscore!
      S: String(1 .. Base_Exp * Max_Index * 2);
      SP: Positive := 1;
      Places: Integer := 0;
      Ch: Character;
      EOL: Boolean;       -- end of line
   begin
      -- loop for first digit or sign, skipping spaces
      loop
         Text_IO.Get(Ch);
         case Ch is
            when ' ' =>
               null;
            when '+'| '-' =>
               S(SP) := Ch;  SP := SP + 1;
               exit;
            when '0' .. '9' =>
               S(SP) := Ch;  SP := SP + 1; Places := 1;
               exit;
            when others =>
               raise Number_Error;
         end case;
      end loop;
      -- now accept only digits and underscores
      -- count the digits in Places
      -- stop on end of line or other character
      loop
         Text_IO.Look_Ahead(Ch, EOL);
         exit when EOL;
         Text_IO.Get(Ch);
         case Ch is
            when '0' .. '9' =>
               S(SP) := Ch;  SP := SP + 1;
               Places := Places + 1;
            when '_' =>
               S(SP) := Ch;  SP := SP + 1;
            when others =>
               exit;
         end case;
      end loop;
      -- now declare a Number big enough
      -- note Item assumed unconstrained
      declare
         Result: Number((Places - 1)/Base_Exp + 1);
      begin
         To_Number(S(1 .. SP - 1), Result);
         Item := Result;
      end;
   end Get;

end Numbers.IO;

--------------------------------------------------------------------

with Numbers.Proc;
package body Numbers.Func is

   use Proc;

   function "+" (X: Number) return Number is
   begin
      return X;
   end "+";

   function "-" (X: Number) return Number is
      N: Number(X.Max_Digits);
   begin
      N := X;  Neg(N);
      return N;
   end "-";

   function "abs" (X: Number) return Number is
   begin
      if X < Zero then return -X; else return X; end if;
   end "abs";

   function "+" (X, Y: Number) return Number is
      Z: Number(Index'Max(Length(X), Length(Y)) + 1);
   begin
      Add(X, Y, Z);
      return Z;
   end "+";

   function "-" (X, Y: Number) return Number is
      Z: Number(Index'Max(Length(X), Length(Y)) + 1);
   begin
      Sub(X, Y, Z);
      return Z;
   end "-";

   function "*" (X, Y: Number) return Number is
      Z: Number(Length(X) + Length(Y));
   begin
      Mul(X, Y, Z);
      return Z;
   end "*";

   function "/" (X, Y: Number) return Number is
      Q: Number(Length(X));
      R: Number(Length(Y));
   begin
      Div(X, Y, Q, R);
      return Q;
   end "/";

   function "rem" (X, Y: Number) return Number is
      Q: Number(Length(X));
      R: Number(Length(Y));
   begin
      Div(X, Y, Q, R);
      return R;
   end "rem";

   function "mod" (X, Y: Number) return Number is
      Q: Number(Length(X));
      R: Number(Length(Y));
   begin
      Div(X, Y, Q, R);
      if (X < Zero and Y > Zero) or (X > Zero and Y < Zero) then
         R := R + Y; 
      end if;
      return R;
   end "mod";

   function "**" (X: Number; N: Natural) return Number is
      Result: Number := One;
      Term: Number := X;
      M: Natural := N;
   begin
      loop
         if M rem 2 /= 0 then
            Result := Term * Result;
         end if;
         M := M / 2;
         exit when M = 0;
         Term := Term * Term;
      end loop;
      return Result;
   end "**";

   function "<" (X, Y: Number) return Boolean is
   begin
      return Compare(X, Y) < 0;
   end "<";

   function "<=" (X, Y: Number) return Boolean is
   begin
      return Compare(X, Y) <= 0;
   end "<=";

   function ">" (X, Y: Number) return Boolean is
   begin
      return Compare(X, Y) > 0;
   end ">";

   function ">=" (X, Y: Number) return Boolean is
   begin
      return Compare(X, Y) >= 0;
   end ">=";

   function "=" (X, Y: Number) return Boolean is
   begin
      return Compare(X, Y) = 0;
   end "=";

   function Make_Number(S: String) return Number is
      Result: Number((S'Length - 1) / Base_Exp + 1);
   begin
      To_Number(S, Result);
      return Result;
   end Make_Number;

   function Make_Number(I: Integer) return Number is
      Base_Digits: Index := 0;
      II: Integer := abs I;
      Base: constant := 10 ** Base_Exp;
   begin
      -- loop to determine discriminant for result
      while II /= 0 loop
         Base_Digits := Base_Digits + 1;
         II := II / Base;
      end loop;
      declare
         Result: Number(Base_Digits);
      begin
         To_Number(I, Result);
         return Result;
      end;
   end Make_Number;

   function String_Of(N: Number) return String is
      Places: Integer := No_Of_Places(N);
      S: String(1 .. Places + 1);
      SP: Positive := 1;
   begin
      To_Text(N, S);
      -- allow for leading spaces in S
      while S(SP) = ' ' loop
         SP := SP + 1;
      end loop;
      return S(SP .. S'Last);
   end String_Of;

   function Integer_Of(N: Number) return Integer is
      Result: Integer;
   begin
      To_Integer(N, Result);
      return Result;
   end Integer_Of;
      
end Numbers.Func;


--- Numbers calculator ---------------------------------------------

--with ID, Numbers.Func; use Numbers.Func;
--package Numbers_ID is new ID(Number, Zero, One);

--with ID.IO, Numbers.IO; use Numbers.IO;
--package Numbers_ID.The_IO is new Numbers_ID.IO;

--with Numbers_ID.The_IO;
--with Numbers.Func;
--with Calculator;
--procedure Numcalc is
  -- new Calculator.Run(Numbers_ID,
                     -- Numbers_ID.The_IO,
                     -- Numbers.Func."+");


--- test 1 - powers of 11 and 99 -----------------------------------

with Numbers.Proc; use Numbers.Proc;
with Ada.Text_IO; use Ada;
procedure Test_11_99 is

   U: Number(2);

   procedure P(X: Number) is
      S: String(1 .. 150);
   begin
      To_Text(X, S);
      Text_IO.Put(S); Text_IO.New_Line;
   end P;

   procedure Power(U: Number; V: Integer) is
      W: Number(50);
   begin
      To_Number(1, W);
      for I in 1 .. V loop
         Mul(W, U, W);
         P(W);
      end loop;
   end Power;

begin
   To_Number(11, U);
   Power(U, 50);
   To_Number(99, U);
   Power(U, 50);
end;

-- test 2 - Mersenne using procedural forms ------------------------

with Ada.Calendar; use Ada.Calendar;
with Numbers.Proc; use Numbers.Proc;
with Ada.Text_IO, Ada.Integer_Text_IO;
use Ada.Text_IO, Ada.Integer_Text_IO;
procedure Test2 is

   Nop: constant := 30;

   Loop_Start, Loop_End: Integer;
   T_Start, T_End: Time;

   Is_Prime: Boolean;
   MM: Number(50);

   Primes: array(1 .. Nop)  of Integer := 
   (3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,
    61,67,71,73,79,83,89,97,101,103,107,109,113,127);

   package Duration_IO is new Fixed_IO(Duration);
   use Duration_IO;


procedure Lucas_Lehmer (P: Integer;
                        Mersenne: out Number;
                        Is_Prime: out Boolean) is

   Two: Number(1);
   M: Number(Mersenne.Max_Digits); 
   L, W, Quotient: Number(M.Max_Digits*2);
begin
   To_Number(2, Two);
   To_Number(4, L);
   To_Number(1, M);
   for I in 1 .. P loop
      Mul(M, Two, M);
   end loop;
   Sub(M, One, M);
   for I in 1 .. P-2 loop
      Mul(L, L, W);
      Sub(W, Two, W);
      Div(W, M, Quotient, L);
    --  L := (L**2 - Two) mod M;
   end loop;
   Is_Prime := Compare(L, Zero) = 0;
   Mersenne := M;
end Lucas_Lehmer;

   procedure Put(X: Number) is
      S: String(1 .. 45);
   begin
      To_Text(X, S);
      Put(S); 
   end Put;

begin
   Put_Line("Start loop? ");  Get(Loop_Start);
   Put_Line("End loop? ");  Get(Loop_End);

   Put_Line("Mersenne Primes");
   Put_Line("    Time     P                           2**P-1");
   for I in Loop_Start .. Loop_End loop
      New_Line;
      T_Start := Clock;
      Lucas_Lehmer(Primes(I), MM, Is_Prime);
      T_End := Clock;
      New_Line;  Put(T_End - T_Start, 2, 1);
      Put(Primes(I), 4);  Put(" : ");
      Put(MM);
      if Is_Prime then
         Put(" is prime");
      else
         Put(" is not prime");
      end if;
   end loop;
end Test2;


--- test 5 - Mersenne using functional forms -----------------------

with Ada.Calendar; use Ada.Calendar;
with Numbers.Func; use Numbers.Func;
with Numbers.IO; use Numbers.IO;
with Ada.Text_IO, Ada.Integer_Text_IO;
use Ada.Text_IO, Ada.Integer_Text_IO;
procedure Test5 is

   Nop: constant := 30;

   Loop_Start, Loop_End: Integer;
   T_Start, T_End: Time;

   Is_Prime: Boolean;
   LL, MM: Number;

   Primes: array(1 .. Nop)  of Integer := 
   (3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,
    61,67,71,73,79,83,89,97,101,103,107,109,113,127);

   package Duration_IO is new Fixed_IO(Duration);
   use Duration_IO;


procedure Lucas_Lehmer (Q: Integer;
                        Mersenne: out Number;
						Lout: out Number;
                        Is_Prime: out Boolean) is

   Two: constant Number := Make_Number(2);
   M: constant Number := Two**Q - One; 
   L: Number := Make_Number(4);
begin
   for I in 1 .. Q-2 loop
      L := (L**2 - Two); -- mod M; -- mod M here is optional;
	  Put(L);  New_line(2);
   end loop;
   Is_Prime := L mod M = Zero;
   Lout := l;
   Mersenne := M;
end Lucas_Lehmer;

begin
   Put_Line("Start loop? ");  Get(Loop_Start);
   Put_Line("End loop? ");  Get(Loop_End);

   Put_Line("Mersenne Primes");
   Put_Line("   Time      P                          2**P-1");
   for I in Loop_Start .. Loop_End loop
      New_Line;
      T_Start := Clock;
      Lucas_Lehmer(Primes(I), MM, LL, Is_Prime);
      T_End := Clock;
      New_Line;  Put(T_End - T_Start, 2, 1);
      Put(Primes(I), 4);  Put(" : ");
      Put(MM, 20);
      if Is_Prime then
         Put(" is prime");
		New_line;
		 put((MM+One)/Make_number(2)*MM, 30);  Put(" is perfect");
		 New_Line;
      else
         Put(" is not prime");
      end if;
      New_line;
	  -- comment out next four lines to avoid detail
	  Put("L is "); Put(LL); Put(" equals "); Put(MM); Put(" times "); Put(LL/MM);
	  if not Is_prime then
	  	New_Line; Put(" remainder = "); Put(LL mod MM);
	  end if;
	-- end of comment
   end loop;
   skip_Line(2);
end Test5;

--- test 6 - GCD and Mersenne --------------------------------------

with Ada.Text_IO, Ada.Integer_Text_IO;
use Ada.Text_IO, Ada.Integer_Text_IO;
with Numbers.Proc; use Numbers.Proc;
procedure Test6 is
   subtype CNumber is Number(50);
   M1, M2, M3: CNumber;
   P1, P2, P3: Integer;
   G, H: CNumber;
   XX, YY, QQ, ZZ: CNumber;
   Two: CNumber;
   N: Cnumber; 
   Start, Stop: Integer;

procedure GCD(X, Y: CNumber; Z: out CNumber) is
begin
   XX := X;  YY := Y;
   while Compare(YY, Zero) /= 0 loop
      Div(XX, YY, QQ, ZZ);
      XX := YY;
      YY := ZZ;
   end loop;
   Z := XX;
end GCD;

procedure Mersenne(P: Integer; M:  in out CNumber) is
begin
   To_Number(2, Two);
   To_Number(1, N);
   for I in 1 .. P loop
      Mul(N, Two, N);
   end loop;
   Sub(N, One, N);
   M := N;
end Mersenne;

begin
   To_Number(2, Two);
   Put_Line("Start? "); Get(Start);
   Put_Line("Stop?  "); Get(Stop);
   for I in Start .. Stop loop
      P1 := 2*I + 1;
      Mersenne(P1, M1);
      for J in 1 .. I -1 loop
         P2 := 2*J + 1;
         Mersenne(P2, M2);
         GCD(M1, M2, G);
         for K in 1 .. I loop
            P3 := 2*K + 1;
            Mersenne(P3, M3);
            if Compare(M3, G) > 0 then
               exit;
            end if;
            if Compare(M3, G) =0 then
               New_Line;
               Put(P1); Put(P2); Put(P3);
               exit;
            end if;
         end loop;
      end loop;
   end loop;
end Test6;

------- test 7 multipication

with Ada.Text_IO, Ada.Integer_Text_IO;
use Ada.Text_IO, Ada.Integer_Text_IO;
with Numbers.Proc; use Numbers.Proc;
with Numbers.IO; use Numbers.IO;
procedure Test7 is
	X: Number;
	Y: Number;
 	Z: Number(50);
begin
	Put_line("Multiplier test");
	Put("X = ");  Get(X);
	Put("Y = ");  Get(Y);
	Mul(X, Y, Z);
	Put_Line("product is ");
	Put(Z);
	New_Line(2);
end Test7;
  
****************************************************************

From: Jeff Cousins
Sent: Wednesday, January 31, 2018  6:17 AM

[This some additional test programs for John Barnes' Bignum package; the
Numbers package was duplicated at the front, which I removed. - Editor.]


--package Monitor is
--	C1, C2, C3, C4: Integer;
--end;

package Primes is
	Max: constant := 20000;
	Prime: array (1..Max) of Integer;
	pragma Elaborate_Body;
end;


package body Primes is
	N: Integer:= 2;
	Index: Integer := Prime'First;
	Found: Boolean := False;
begin
	-- initialization part to build prime table
	Prime(Prime'First) := 2;
	loop
		N := N+1; Found := True;
		for I in Prime'First .. Index loop
			if n rem Prime(I) = 0 then -- divides by existing prime
				Found := False; exit;
			end if;
		end loop;
		if Found then

			-- found a new prime
			Index := Index+1; Prime(Index) := n;
			exit when Index = Max;
		end if;
	end loop;
end Primes;

Package Squares is
	Square_Digits: Integer := 4;
	Square_Ten_Power: Integer := 10**Square_Digits;
	Poss_Last_Digits: array(0..Square_Ten_Power-1) of Boolean := (others => False);
	pragma Elaborate_Body;
end;

package body Squares is
begin
	
	-- make Poss_Last_Digits array
	for I in 1 .. Square_Ten_Power loop
		Poss_Last_Digits(I*I mod Square_Ten_Power) := True;
	end loop;
end Squares;		
	


with Numbers.Func; use Numbers.Func;
-- with Monitor;
procedure Square_root(XX: in Number; Try: Number; X: out Number; R: out Number) is
	-- X is the largest X such that X*X + R equals XX with R >= zero
	-- Try is initial guess 
	K, KK: Number;
	DK: Number;
	Three: Number := Make_Number(3);
begin
	k := Try;
	loop
		KK := K*K;
		DK := (XX - KK)/(K+K);
		-- iterate until nearly there
		if abs DK < Three then -- nearly there
			if XX < KK then
				K := XX/K;		-- ensure K*K is less than XX
			end if; 
			exit;
		end if;
		-- do another iterate
		K := K+DK;
	--	Monitor.c2 := Monitor.C2 + 1;
	end loop;

	-- now loop from below

	loop
		KK := K*K;
	--	Monitor.C2 := Monitor.C2 + 1;
		if KK >= XX then 
			if KK = XX then
				X := K; R := Zero; return;
			end if;
			X := K - One; R := XX - X*X; return;
		end if;
		K := K+One;
	end loop;


end Square_Root;


with Square_Root;
with Numbers.Func; use Numbers.Func;
-- with Monitor;
with Squares; use Squares;
procedure Fermat_Long(N: in Number; Min_Prime: in Number; P, Q: out Number) is
	-- we know that factors up to Min_Prime have been removed, so max square to try is
	-- roughly (N/Min_Prime + Min_Prime)/2; we add 2 for luck
	Two: Number := Make_Number(2);
	Number_Square_Digits: Number := Make_Number(Square_Digits);
	Number_Square_Ten_Power: Number := Make_Number(Square_Ten_Power);
	X: Number;
	Y: Number;
	R: Number;
	K: Number;
	DK: Number;
	N2, X2, K2: Integer;
	Last_Digits: Integer;
	Try: Number := One;
	Max_square : Number := (N/Min_Prime + Min_Prime)/ Two + Two;

	
begin

	Square_Root(N, One, X, R);
	if R = Zero then
		-- N was a perfect square
		P := X; Q := X;
		return;
	end if;
	--	Monitor.C1 := Monitor.C2;
	--	Monitor.C2 := 0;

		K := X*X-n;
		DK := X+X+One;
		N2 := Integer_Of(N rem Number_Square_Ten_Power);
		X2 := Integer_Of(X rem Number_Square_Ten_Power);

	loop
	--	Monitor.C3 := Monitor.C3 + 1;
		X := X + One;
		if X > Max_Square then
			-- must be prime
			P := N; Q := One;
			return;
		end if;

		X2 := (X2 + 1) rem Square_Ten_Power;
		K := K + DK;	-- omit if DK not used
		DK := DK + Two;
	--	K := X*X-N;  -- omit if DK used
		K2 := (X2*X2-N2) mod Square_Ten_Power;
	--	Last_Digits := Integer_Of(K rem Number_Square_Ten_Power);
		Last_Digits := K2;
		if Poss_Last_Digits(Last_Digits) then
	--		Monitor.C4 := Monitor.C4 + 1;

			Square_Root(K, Try, Y, R);
			if R = Zero then
				-- X*X-N was a perfect square
				P := X+Y; Q := X-Y;
				return;
			end if;
			Try := Y;
		end if;
	end loop;
exception
	when others => p := Zero; Q := Zero;
end Fermat_long;

with Numbers.IO; use Numbers.IO;
with Numbers.Func; use Numbers.Func;
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
-- with Monitor;
with Primes; use Primes;
with Fermat_Long;
with Ada.Calendar; use Ada.Calendar;
procedure Do_Fermat_Long is
	NN, PP, QQ: Number;
	T_Start, T_End: Time;
	package Duration_IO is new Fixed_IO(Duration);
	use Duration_IO;
begin	
	Put_Line("Welcome to Fermat's method (multilength)");
	loop
<<Again>>
		Begin	
			New_Line;
			Put("Insert number N = "); Get(NN);
		exception
			when others =>
				Put_Line("Not a number or too big");  Skip_Line;
				goto Again;
		end;

		exit when NN = Zero;

			-- check to see if N divides by a known prime

		for i in Prime'Range loop
			loop
				if NN rem make_Number(Prime(I)) = Zero then
					Put("N divides by "); Put(Prime(I), 0); Put_Line(" so removing factor");
					NN := NN / Make_Number(Prime(I));
				else
					exit;
				end if;
			end loop;
		end loop;


		if nN rem Make_number(4) = Make_Number(2) then
			Put_Line("Algorithm fails on odd multiples of 2, so halving N"); 
			NN := NN/Make_Number(2);
			Put("New N = "); Put(NN, 0); New_Line;
		end if;
		if NN = One then
			Put_Line("all factors removed");
		else
			
	--	Monitor.C1 := 0;
	--	Monitor.C2 := 0;
	--	Monitor.C3 := 0;
	--	Monitor.C4 := 0;
		T_Start := Clock;
		Fermat_Long(NN, Make_Number(Prime(Max)), PP, QQ);
		T_End := Clock;
		if PP = Zero and QQ = zero then
			Put_Line("Failed internally");
			goto Again;
		end if;
		Put("Two factors are   "); Put(PP, 0); Put("   "); Put(QQ, 0); New_Line;
		Put(T_End-T_Start, 2, 1);  New_Line;
	 --	Put(Monitor.C1, 9);  Put(Monitor.C2, 9); Put(Monitor.C3, 9); Put(Monitor.C4, 9); New_Line;
	end if;
	end loop;
	Put_line("Goodbye");  Skip_Line(2);
end Do_Fermat_Long;

****************************************************************

From: Randy Brukardt
Sent: Wednesday, January 31, 2018  5:03 PM

>Sending in three parts as the original message was too big for the mail
>server.

Just so the rest of you know, "Part 3" was an executable Windows program which
we've decided not to distribute at all (it's too big for the list, and
executables for specific computers seem out-of-bounds for this list anyway,
given all of us know how to operate our favorite Ada compiler and probably
several non-favorite compilers as well).

So don't look for the non-existent part 3.

****************************************************************

From: Steve Baird
Sent: Wednesday, February 28, 2018  8:03 PM

I'm attaching some preliminary specs that reflect the feedback from the last
discussion of this set of predefined packages. [This is version /01 of the
AI - ED.]

There is still some polishing to do with respect to, for example, pre/post
conditions for the various operations, but this should give us a more concrete
framework for discussion.

I gave up on making Bounded_Integer a discriminated type and instead defined
the enclosing package Bounded_Integers as a generic package. That eliminates
questions such as "what is the discriminant of the result of adding together
two values whose discriminants differ?".

It is intended that the bounded and unbounded Big_Integer types should be
streaming-compatible as with Vectors and Bounded_Vectors

As discussed last meeting, there is no Bounded_Big_Rationals package (or
generic package).

The types are tagged now and descended from a common interface type.
This means that most parameter subtypes have to be first named subtypes.
This should not be considered an endorsement of this idea; we might (as Bob 
suggests) want these types to be untagged. It's just easier to see what this
looks like and then have to imagine what the untagged version might be than
the other way around. I agree with Randy's comments that taggedness by itself
doesn't imply performance problems.

Currently no declaration of a GCD function. Do we want that? If so, then in
which package? If we declare it in Big_Reals then it is not a primitive and so
parameters can be Big_Positives instead of Big_Integers. If we really want it
in Big_Integers package(s) then it would either need Big_Integer parameter
subtypes or it would need to be declared in a nested package to avoid
primitive status.

Comments?

****************************************************************

From: Randy Brukardt
Sent: Thursday, March  1, 2018  9:49 PM

> I'm attaching some preliminary specs that reflect the feedback from 
> the last discussion of this set of predefined packages.
...
> Comments?

Shouldn't this be some set of children of Ada.Numerics? These kinda seem like
numbers. ;-)

You don't have the numeric literal definitions (see AI12-0249-1) -- that seems
necessary for usability. (One wonders if the literals should be defined on the
interface, which suggests that AI12-0249-1 needs a bit of extension.)

Otherwise, I didn't notice anything that I would change.

****************************************************************

From: Steve Baird
Sent: Sunday, March  4, 2018  1:00 AM

> Shouldn't this be some set of children of Ada.Numerics? These kinda 
> seem like numbers.;-)
> 

Good point.
So the root package for this stuff becomes Ada.Numerics.Big_Numbers.


> You don't have the numeric literal definitions (see AI12-0249-1) -- 
> that seems necessary for usability. (One wonders if the literals 
> should be defined on g interface, which suggests that AI12-0249-1 
> needs a bit of
> extension.)

We decided earlier that we didn't want that inter-AI dependency.

But I agree that if we we are willing to introduce that dependency then of 
course support for literals would make sense.

Should it be conditional, as in "if AI12-0249 is approved, then this AI also
includes blah, blah"?

****************************************************************

From: Randy Brukardt
Sent: Sunday, March  4, 2018  1:16 AM

Yes, I'd write it that way. I'd probably stay away from other AI dependencies,
but literals are pretty fundamental - supporting them makes the package way
more usable.

****************************************************************

From: Steve Baird
Sent: Wednesday, March 28, 2018  6:50 PM

Attached is proposed wording for this AI. [This is version /02 of this
AI - ED].

There are some TBDs interspersed.

****************************************************************

From: Randy Brukardt
Sent: Thursday, March 29, 2018  7:48 PM

> Attached is proposed wording for this AI.
> 
> There are some TBDs interspersed.

Here's a few thoughts:

>[TBD: aspects specified for this package? Pure, Nonblocking, others?
>Same question applies to other packages declared in later sections.
>Would these aspects constrain implementations in undesirable ways?]

All of the packages should be nonblocking. I don't think any reasonable
implementation would need access to delay statements. ;-)

The interface package should be Pure (why not, it doesn't have any
implementation). The bounded package also should be pure (we do that for all
of the bounded forms elsewhere.

The others probably should be preelaborated (and the types having
preelaborable_initialization), lest we make it too hard to use the needed
dynamic allocation.

>[TBD: It would be nice to use subtypes in parameter profiles (e.g.,
>a Nonzero_Number subtype for second argument of "/", but this requires
>AI12-0243 and the future of that AI is very uncertain.]

You can always use a Pre'Class as an alternative to a subtype. It's not
quite as convenient, but it makes the same check, and presuming that
AI12-0112-1 stays are currently envisioned, that check would be suppressible
with "pragma Suppress (Numerics_Check);".

>[TBD: Remove Integer_Literal aspect spec if AI12-0249-1 not approved.
>If Default_Initial_Condition AI12-0265-1 is approved and Integer_Literal AI
is
>not then replace "0" with "+0" in the condition and as needed in
>subsequent conditions.]

I put the AI number in here for Default_Initial_Condition.

>[TBD: In_Range formal parameter names. "Lo & Hi" vs. "Low & High"?]

Ada usually doesn't use abbreviations, and saving one or two characters this
way isn't appealing. Use Low and High.

> A.5.5.1.1 Bounded Big Integers

Umm, please, no 5 level subclauses. Since there are currently no four level
items in the RM, we need to discuss that explicitly. I had to add a fourth
level for ASIS, but Ada only uses three levels. And the ACATS only uses two
levels in the annexes, which is already a problem for the containers (there
being only one set of sequence numbers for all of the containers tests).

>AARM Note: Roughly speaking, behavior is as if the type invariant for
> Bounded_Big_Integer is
>  In_Range (Bounded_Big_Integer, First, Last) or else (raise
Constraint_Error)
>although that is not specified explicitly because that would
>require introducing some awkward code in order to avoid infinite
>recursion.

Awkward code? Please explain. Type invariants are explicitly not enforced on
'in' parameters of functions specifically to avoid infinite recursion in the
type invariant expression. You'd probably need a function for this purpose
(to avoid the functions First and Last -- is that what you meant??), say:
     In_Base_Range (Bounded_Big_Integer) or else (raise Constraint_Error)
where In_Base_Range is equivalent to In_Range (Bounded_Big_Integer, First,
Last) with the expressions of First and Last substituted. One could also
make those expressions the defaults for Low and High.

>[TBD: This could be done differently by using a formal instance instead
>of declaring the Conversions package as a child of Bounded_Big_Integers.
>Would there be any advantage to this approach? The advantage of the
>proposed approach is visibility of the private part, but it does seem
>awkward to have a generic with no generic formals and no local state.]

Well, it would be easier to implement in Janus/Ada, where we never got
sprouting to work. But that's hardly a reason. I suspect it would be more
obvious what's going on than a generic child -- as a data point, all of the
extra operations of Ada.Strings.Bounded take formal packages rather than
being children -- but that may have been driven by other considerations.

One argument for making it a formal package is that this conversion package
really belongs to both big number packages -- it's somewhat artificial to
make it live in the hierarchy of one or the other.

>Any Big_Rational result R returned by any of these functions satisifies the
>condition
>   (R = 0.0) or else
>   (Greatest_Common_Denominator (Numerator (R), Denominator (R)) = 1).

Arguably, that should be a postcondition, since the implementation isn't
required to check it (it it required to *pass* it). Then a separate rule
isn't needed. You'd probably want to declare a function with this meaning,
'cause duplicating the above 2 dozen times would be annoying.

>AARM Note: No Bounded_Big_Rationals generic package is provided.

We've discussed why, but there needs to be a version of that discussion
either in this note or in the (sadly empty) !discussion section. Future
readers will be puzzled otherwise (including, most likely, us).

****************************************************************

Questions? Ask the ACAA Technical Agent