International Academy of Noosphere

the Baltic branch

V. Aladjev, M. Shishakov,

V. Vaganov

Functional and procedural programming in Mathematica

Second edition

Lulu Press – 2020

Functional and procedural programming in Mathematica.
V. Aladjev, M. Shishakov, V. Vaganov.– Lulu Press, 396 p., 2020.

Software presented in the book contains a number of useful and effective receptions of the procedural and functional programming in Mathematica that extend the system software and allow sometimes much more efficiently and easily to program the software for various purposes. Among them there are means that are of interest from the point of view of including of their or their analogs in Mathematica, at the same time they use approaches, rather useful in programming of various applications. In addition, it must be kept in mind that the classification of the presented tools by their appointment in a certain measure has a rather conditional character because these tools can be crossed substantially among themselves by the functionality.

The freeware package MathToolBox containing more 1420 tools is attached to the present book. The MathToolBox not only contains a number of useful procedures and functions, but can serve as a rather useful collection of programming examples using both standard and non–standard techniques of functional–procedural programming.

The book is oriented on a wide enough circle of the users from computer mathematics systems, researchers, teachers and students of universities for courses of computer science, physics, mathematics, and a lot of other natural disciplines. The book will be of interest also to the specialists of industry and technology which use the computer mathematics systems in own professional activity. At last, the book is a rather useful handbook with fruitful methods on the procedural and functional programming in the Mathematica system.

Mathematica 2, 5 ÷ 12.1.1.0 – trademarks of Wolfram Research Inc.

Printed by Lulu Press

December, 2020

© <2020> by V. Aladjev, M. Shishakov, V. Vaganov

Contents

Introduction 5

Chapter 1: The user functions in Mathematica 7

1.1. The user functions, defined intuitively 8

1.2. The user pure functions 12

1.3. Some useful tools for work with the user functions 20

Chapter 2: The user procedures in Mathematica 28

2.1. Definition of procedures in Mathematica software 29

2.2. Headings of procedures 38

2.3. Optional arguments of procedures and functions 57

2.4. Local variables in Mathematica procedures 71

2.5. Exit mechanisms of the user procedures 86

2.6. Tools for testing of procedures and functions 94

2.7. The nested blocks and modules 102

Chapter 3: Programming tools of the user procedure body 107

3.1. Branching control structures in the Mathematica 111

* Conditional branching structures 111

* Unconditional transitions 115

3.2. Cyclic control structures in the Mathematica 120

3.3. Some special types of cyclic control structures in

the Mathematica system 125

3.4. Mathematica tools for string expressions 130

* Replacements and extractions in strings 141

* Sub-strings processing in strings 143

* Expressions containing in strings 155

3.5. Mathematica tools for lists processing 160

3.6. Additional tools for the Mathematica 197

3.7. Attributes of procedures and functions 248

3.8. Additional tools expanding the built-in

Mathematica functions, or its software as a whole 254

3.9. Certain additional tools of expressions processing

in the Mathematica software 272

* Tools of testing of correctness of expressions 273

* Expressions processing at level of their components 277

* Replacement sub-expressions in expressions 285

* Expressions in strings 297

Chapter 4: Software for input-output in Mathematica 300

4.1. Tools of Mathematica for work with internal files 300

4.2. Tools of Mathematica for work with external files 305

4.3. Tools of Mathematica for attributes processing of

directories and data files 316

4.4. Additional tools for files and directories processing

of file system of the computer 323

4.5. Special tools for files and directories processing 333

Chapter 5: Organization of the user software 341

5.1. MathToolBox package for Mathematica system 345

5.2. Operating with the user packages in Mathematica 354

* The concept of context in Mathematica 355

* Interconnection of contexts and packages 363

5.3. Additional tools of operating with user packages 377

References 391

About the authors 395

Book completes our multi–year cycle of book publications on general theory of statistics, computer science, cellular automata, Mathematica and Maple, with creation of software for the latest two computer mathematics systems Maple and Mathematica [42] and [16] accordingly.

All rights reserved. This book or any portion thereof may not be reproduced or
used in any manner whatsoever without the express written permission of the
publisher except for the use of brief quotations in a book review or a scholarly
journal. This book completes a multi-year cycle of our publications in five areas:
General theory of statistics, Information science, Cellular automata, Maple and
Mathematica along with creation of software for Maple and Mathematica.

The authors express deep gratitude and appreciation to Misters Uwe Teubel and Dmitry Vassiliev – the representatives of the firms REAG Renewable Energy AG & Purwatt AG (Switzerland) – for essential assistance rendered by preparation of the present book.

For contacts: aladjev@europe.com, aladjev@yandex.ru, aladjev@mail.ru

Introduction

Systems of computer mathematics (SCM) find wide enough application in a whole number of natural, economic and social sciences such as: technologies, chemistry, informatics, mathematics, physics, education, economics, sociology, etc. Systems Mathematica and Maple, and some others are more and more demanded for learning of mathematically oriented disciplines, in scientifical researches and technologies. SCM are main tools for scientists, teachers, researchers, and engineers. The investigations on the basis of SCM, as a rule, well combine algebraical methods with advanced computing methods. In this sense, SCM are a certain interdisciplinary area between informatics and mathematics in which researches will be concentrated on the development of algorithms for algebraic (symbolical) and numerical calculations, data processing, and development of programming languages along with software for realization of algorithms of this kind and problems of different purpose basing on them.

It is possible to note that among modern SCM the leading positions are undoubtedly taken by the Maple and Matematica systems, alternately being ahead of each other on this or that indicator. The given statement is based on our long–term use in different purposes of both systems – Maple and Mathematica and also work on preparation and edition of the book series and some papers in Russian and English in this direction [1-42]. At the same time, as a result of expanded use of the above systems were created the package MathToolBox for Mathematica and library UserLib6789 for Maple [16,42]. All tools from the above library and package are supplied with freeware license and have open program code. Programming of many projects in Maple and Mathematica substantially promoted emergence of a lot of system tools from the above library and package. So, openness of MathToolBox package code allows both to modify the tools containing in it, and to program on their basis own tools, or to use their components in various appendices. Experience of use of library and package showed efficiency of such approach.

In particular, MathToolBox package contains more than 1420 tools of different purpose which eliminate restrictions of a number of standard tools of Mathematica system, and expand its software with new tools. In this context, this package can serve as a certain additional tool of procedural and functional programming, especially useful in the numerous appendices where certain nonstandard evaluations have to accompany programming. At that, tools represented in this package have a direct relationship to certain principal questions of procedural and functional programming in Mathematica system, not only for the decision of applied problems, but, above all, for creation of software extending frequently used facilities of the system, eliminating their defects or extending Mathematica with new facilities. The software presented in the package contains a lot of rather useful and effective receptions of programming in the Mathematica, and extends its software that allows to program the tasks of various purpose more simply and more effectively. A number of tools of the given package is used in the examples illustrating these or those provisions of the book.

In the book basic objects of programming in Mathematica – functions and procedures are considered that consist the base of functional–procedural programming. The given book considers some principal questions of procedural–functional programming in Mathematica, not only for decision of various applied tasks, but, as well, for creation of software expanding frequently used facilities of the system and/or eliminating their limitations, or expanding the system with new facilities. The important role of functions and procedures take place at creation of effective and intelligible user software in various fields. This also applies to software reliability.

The book is oriented on a wide enough circle of the users of the CMS, researchers, mathematicians, physicists, teachers and students of universities for courses of mathematics, computer science, physics, chemistry, etc. The book will be of interest to the specialists of industry and technology, economics, medicine and others too, that use the CMS in own professional activity.

Chapter 1: The user functions in Mathematica

Function is one of basic concepts of mathematics, being not less important object in programming systems, not excluding Mathematica system, with that difference that function – as a subject to programming – very significantly differs from strictly mathematical concept of function, considering specific features of programming in the Mathematica system.

Mathematica has a large number of functions of different purpose and operations over them. Each function is rather well documented and supplied with the most typical examples of its application. Unfortunately, unlike Maple, the program code of the Mathematica functions is closed that significantly narrows a possibility of deeper mastering the system and acquaintance with the receptions used by it and a technique of programming.

Truth, with rare exception, Mathematica allows to make assignments that override the standard built–in functions and meaning of Mathematica objects. However, program codes of the built–in functions of the system remain closed for the user. Names of built–in functions submit to some general properties, namely: names consist of complete English words, or standard mathematical abbreviations. In addition, the first letter of each name is capitalized. Functions whose names end with Q letter as a rule usually "ask a question", and return either True or False.

Many important functions of the Mathematica system can be found in our books [1-15], that provide detailed discussions of the built-in functions, the features of their implementation along with our tools [16] which expand or supplement them. A full arsenal of functionality can be found in the system manual. Here we will cover the basic issues of organizing user functions in Mathematica along with some tools that provide a number of important operations over user-defined functions. Moreover, it is possible to get acquainted with many of them in package MathToolBox [16] and in certain our books [1-15]. We pass to definition of the main functional user objects.

1.1. The classical functions, defined intuitively. Functions of this type are determines by expressions as follows:

(a) F[x_, y_, z_, …] := [x, y, z, …]

(b) F[x_ /; TestQ[x], y_, z_ /; TestQ[z], …] := [x, y, z, …]

Definition allows two formats (a) and (b) from which the first format is similar to the format of a classical mathematical function f(x, y, z, ...) = (x, y, z, ...) from n variables with that difference that for formal arguments x,y,z,... the object template are used “_” which can designate any admissible expression of the system while (x,y,z,…) designate an arbitrary expression from variables x,y,z ..., including constants and function calls, for example:

In[1425]:= F[x_, y_, z_] := a*x*b*y + N[Sin[2020]]

In[1426]:= F[42, 78, 2020]

Out[1426]= 0.044062 + 3276*a*b

Whereas the second format unlike the first allows to hold testing of actual arguments obtained at function call F[x, y, z] for validity. It does a testing expression TestQ[x] attributed to formal argument e.g. x – if at least one testing expression on a relevant argument returns False, then function call is returned unevaluated, otherwise the function call returns the required value, if at evaluation of expression  was no an erroneous or especial situation, for example:

In[8]:= F[x_ /; IntegerQ[x], y_, z_Symbol] := a*x*b*y + c*z

In[9]:= F[77, 90, G]

Out[9]= 6930*a*b + c*G

In[10]:= F[77, 90, 78]

Out[10]= F[77, 90, 78]

At the same time, for an actual argument its Head, e.g. List, Integer, String, Symbol, etc, as that illustrates the above example, can act as a test. At standard approach the concept of a function does not allow the use of local variables. As a rule Mathematica system assumes that all variables are global. This means that every time you use a name e.g. w, the Mathematica normally assumes that you are referring to the same object. However, at program writing, you may not want all variables to be global. In this case, you need the w in different points of a program to be treated as a local variable. At the same time, local variables in function definition can be set using modules. Within module you can give a list of variables which are to be treated as local. This goal can be achieved with help of blocks too. A simple example illustrates the told:

In[47]:= F[x_, y_] := x*y + Module[{a = 5, b = 6}, a*x + b*y]

In[48]:= F[42, 78]

Out[48]= 3954

In[49]:= {a, b} = {0, 0}

Out[49]= {0, 0}

In[50]:= F[42, 78]

Out[50]= 3954

In[51]:= S[x_, y_] := x*Block[{a = 5, b = 6}, a/b*y^2]

In[52]:= S[42, 48]

Out[52]= 80640

In[53]:= {a, b} = {0, 0}; S[42, 48]

Out[53]= 80640

In addition, pattern object "___" or "__" is used for formal arguments in a function definition can stand for any sequence of zero or more expressions or for any nonempty sequence of expressions accordingly, for example:

In[2257]:= G[x__] := Plus[x]

In[2258]:= G[500, 90, 46]

Out[2258]= 636

In[2259]:= G1[x___] := Plus[x]

In[2260]:= {G1[], G1[42, 47, 67]}

Out[2260]= {0, 156}

Lack of such definition of function is that the user cannot use local variables, for example, the coefficients of polynomials and other symbols other than calls of standard functions or local symbols determined by the artificial reception described above (otherwise function will depend on their current values in the current session with Mathematica system). So, at such function definition expression  should depend only on formal arguments. But in a number of cases it is quite enough. This function definition is used to give a more complete view of this object and in certain practical cases it is indeed used [9-12,16].

Meanwhile, using our procedure [16] with program code:

In[2274]:= Sequences[x__] := Module[{a = Flatten[{x}], b},

b = "Sequence[" <> ToString1[a] <> "]";

a = Flatten[StringPosition[b, {"{", "}"}]];

ToExpression[StringReplace[b, {StringTake[b, {a[[1]],

a[[1]]}] –> "", StringTake[b, {a[[–1]], a[[–1]]}] –> ""}]]]

that extends the standard built-in Sequence function and useful in many applications, it is possible to represent another way to define the user functions, modules, and blocks, more precisely their headers based on constructions of the format:

<Function name>@Sequences[formal args] := 

The following a rather simple example illustrates the told:

In[75]:= G@Sequences[a_Integer, b_Symbol, c_] := a+b+c

In[76]:= Definition[G]

Out[76]= G[a_Integer, b_Symbol, c_] := a + b + c

In[77]:= G[6.7, 5, 7]

Out[77]= G[6.7, 5, 7]

In[78]:= G[67, m, 7]

Out[78]= 74 + m

Naturally, in real programming, this definition of functions is not practical, serving only the purposes of possible methods of defining said objects. In the meantime, such extension of the built-in tool allows comparing the capabilities of both tools in determining the underlying objects of the system.

Along with the above types of functions, the Mathematica uses also the Compile function intended for compilation of the functions which calculate numeric expressions at quite certain assumptions. The Compile function has the 4 formats of coding, each of which is oriented on a certain compilation type:

Compile[{x1, x2,…}, G] – compiles a function for calculation of an expression G in the assumption that all values of arguments xj {j =

1, 2, …} have numerical character;

Compile[{{x1, t1}, {x2, t2}, {x3, t3}, …}, G] – compiles a function for calculation of an expression G in the assumption that all values of xj arguments have type tj {j = 1, 2, 3,…} accordingly;

Compile[{{x1, p1, w1}, {x2, p2, w2},…}, J] – compiles a function for calculation of an expression J in the assumption that values of xj arguments are wj ranks of array of objects, each of that corresponds to a type pj {j = 1, 2, 3,…};

Compile[s, W, {{p1, pw1}, {{p2, pw2}, …}] – compiles a function for calculation of a certain expression W in the assumption that its s sub-expressions that correspond to the pk templates have the pwj types accordingly {k = 1, 2, 3, …}.

The Compile function processes procedural and functional objects, matrix operations, numeric functions, functions for lists processing, etc. Each Compile function generates a special object CompiledFunction. The function call Compile[…, Evaluate[w]] is used to specify the fact that w expression should be evaluated symbolically before compilation. For testing of functions of this type a rather simple CompileFuncQ function is used [8], whose call CompileFuncQ[x] returns True if a x represents the Compile function, and False otherwise. The fragment represents source code of the CompileFuncQ function with examples of its use.

In[7]:= V := Compile[{{x, _Real}, {y, _Real}}, x*y]; K := (#1*#2) &;

A := Function[{x, y}, x/y]; H[x_] := Block[{}, x]; H[x_, y_] := x + y;

SetAttributes["H", Protected]; P[x__] := Plus[Sequences[{x}]];

GS[x_ /; IntegerQ[x], y_ /; IntegerQ[y]] := Sin[78] + Cos[42];

Sv[x_ /; IntegerQ[x], y_ /; IntegerQ[y]] := x^2 + y^2;

Sv = Compile[{{x, _Integer}, {y, _Real}}, (x + y)^6];

S := Compile[{{x, _Integer}, {y, _Real}}, (x + y)^3];

G = Compile[{{x, _Integer}, {y, _Real}}, (x/y)];

In[8]:= CompileFuncQ[x_] := If[SuffPref[ToString1[Definition3[x]],

"Definition3[CompiledFunction[{", 1], True, False]

In[9]:= Map[CompileFuncQ, {Sv, S, G, V, P, A, K, H, GS, Sv}]

Out[9]= {True, True, True, True, False, False, False, False, False, True}

The CompileFuncQ function expands testing possibilities of the functional objects in Mathematica, by representing a certain interest first of all for the system programming.

Thus, as a reasonably useful version of the built-in Compile function, one can represent the Compile1 procedure whose call Compile1[f, x, y] returns a function of the classical type whose body is defined by an algebraic y expression, whereas f defines a name of the generated function and x defines the list of types attributed to the parameters of an expression y. The factual x argument is specified in the format {{a, Ta}, b, {c, Tc},…}, where {a, b, c,…} defines parameters of the y expression, whereas {Ta, Tc,…} defines the testing Q-functions while single elements of x defines validity of any expression for corresponding parameter.

In[10]:= Compile1[f_ /; SymbolQ[f], x_ /; ListQ[x], y_] :=

Module[{a = ToString1[y], b, c, d, g, h, t}, b = FactualVarsStr1[a];

b = Select[b, ! SameQ[#[[1]], "System`"] &][[1]][[2 ;–1]];

SetAttributes[g, Listable]; g[t_] := ToString[t];

t = Map[ToString, Map[If[ListQ[#], #[[1]], #] &, x]];

c = ToString[f] <> "["; d = Map[g, x];

Do[h = d[[j]]; If[ListQ[h] && MemberQ[b, h[[1]]],

c = c <> h[[1]] <> "_ /;" <> h[[2]] <> "[" <> h[[1]] <> "],",

If[MemberQ[b, h], c = c <> h <> "_ ,", 77]], {j, Length[d]}];

t = Complement[b, t];

If[t != {}, Do[c = c <> t[[j]] <> "_ ,", {j, Length[t]}], 78];

c = StringTake[c, {1, –2}] <> "] := " <> a; ToExpression[c];

Definition[f]]

In[11]:= Compile1[f, {x, {n, IntegerQ}, {M, RationalQ}, {h, RealQ}},

(Sin[h] + a*x^b)/(c*M – Log[n])]

Out[11]= f[x_, n_/; IntegerQ[n], M_/; RationalQ[M], h_/; RealQ[h],

a_, b_, c_] := (a*x^b + Sin[h])/(c*M – Log[n])

In[12]:= N[f[t, 77, 7/8, 7.8, m, n, p], 3]

Out[12]= (0.998543 + m*t^n)/(–4.34 + 0.875*p)

Fragment above represents source code of the procedure.

1.2. The user pure functions. First of all, we will notice that so–called functional programming isn't any discovery of the Mathematica system, and goes back to a number of software that appeared long before the above system. In this regard, it is pertinently focused slightly more in details the attention on the concept of functional programming in historical aspect. While here we only will note certain moments characterizing specifics of the paradigm of functional programming. We will note only that the foundation of functional programming has been laid approximately at the same time, as imperative programming that is the most widespread now, i.e. in the thirties years of the last century. A. Church (USA) – the author of λ–calculus and one of founders of the concept of Cellular Automata in connection with his works in field of infinite automata and mathematical logic along with H. Curry (England) and M. Schönfinkel (Germany) that have developed the mathematical theory of combinators, with good reason can be considered as the founders of mathematical foundation of functional programming. In addition, functional programming languages, especially the purely functional ones such as the Hope and Rex, have largely been used in academical circles rather than in commercial software. Whereas prominent functional programming languages such as Lisp have been used in the industrial and commercial applications. Today, functional programming paradigm is also supported in a number of domain-specific programming languages, for example, by the Mathem–language of the Mathematica. From a rather large number of languages of functional programming it is possible to note the following languages that exerted a great influence on progress in this field, namely: Lisp, Scheme, ISWIM, family ML, Miranda, Haskell, etc. By and large, if the imperative languages are based on operations of assignment and cycle, the functional languages on recursions. The most important advantages of the functional languages are considered in our books in detail [8-14], namely:

– programs on functional languages as a rule are much shorter and simpler than their analogues on the imperative languages;

– almost all modern functional languages are strictly typified and ensure the safety of programs; at that, the strict typification allows to generate more effective code;

– in a functional language the functions can be transferred as an argument to other functions or are returned as result of their calls;

– in pure functional languages (which aren't allowing by–effects for the functions) there is no an operator of assigning, the objects of such language can't be modified or deleted, it is only possible to create new objects by decomposition and synthesis of the existing objects. In the pure functional languages all functions are free from by–effects.

Meanwhile, functional languages can imitate certain useful imperative properties. Not every functional language are a pure forasmuch in a lot of cases the admissibility of by–effects allows to essentially simplify programming. However, today the most developed functional languages are as a rule pure. With many interesting enough questions concerning a subject of functional programming, the reader can familiarize oneself, for example, in [15]. Whereas with an quite interesting critical remarks on functional languages and possible ways of their elimination it is possible to familiarize oneself in [8-15,22].

A number of concepts and paradigms are quite specific for functional programming and absent in imperative programming. Meanwhile, many programming languages, as a rule, are based on several paradigms of programming, in particular imperative programming languages can successfully use also concepts of functional programming. In particular, as an important enough concept are so–called the pure functions, whose results of run depends only on their actual arguments. Such functions possess certain useful properties a part of which it is possible to use for optimization of program code and parallelization of calculations. In principle, there are no special difficulties for programming in the functional style in languages that aren't the functional. The Mathematica language supports the mixed paradigm of functional and procedural programming. We will consider the elements of functional programming in Mathematica in whose basis the concept of the pure function lays. The pure functions – one of the basic concepts of functional programming which is a component of programming system in Mathematica in general.

Typically, when a function is called, its name is specified, whereas pure functions allow specifying functions that can be applied to arguments without having to define their explicit names. If a certain function G is used repeatedly, then can define the function using definition, considered above, and refer to the function by its name G. On the other hand, if some function is intend to use only once, then will probably find it better to give the function in format of pure function without ever naming it. The reader that is familiar with formal logic or the LISP programming language, will recognize Mathematica pure functions as being like λ expressions or not named functions. Note, that pure functions are also close to the pure mathematical notion of operators. Mathematica uses the following formats for definition of pure functions, namely:

(a) Function[x, body]

(b) Function[{x1, x2, …, xn}, body]

(c) body &

Format (a) determines pure function in which x in function body is replaced by any given argument. An arbitrary admissible expression from variable x, including also constants and calls of functions acts as a body. Format (b) determines a pure function in which variables x1, x2, …, xn in function body is replaced by the corresponding arguments. Whereas format (с) determines a function body containing formal arguments denoted as # or #1, #2, #3, etc. It is so-called short format. Note, the pure functions of formats (a), (b) do not allow to use the typification mechanism for formal arguments similarly to the classical functions, namely constructs of type "h_..." are not allowed. This is a very serious difference between pure function and classical function whereas otherwise, already when evaluating the definition of any pure function with typed formal arguments, Mathematica identifies the erroneous situations.

The calls of pure functions of the above formats are defined according to simple examples below, namely:

In[2162]:= Function[x, x^2 + 42][25]

Out[2162]= 667

In[2163]:= Function[{x, y, z}, x^2 + y^2 + z^2][42, 75, 78]

Out[2163]= 13473

In[2164]:= #1^2 + #2^2 &[900, 50]

Out[2164]= 812500

In[2165]:= Function[x_Integer, x^2 + 42]

Function::flpar: Parameter specification x_Integer in …

When Function[body] or body & is applied to a set of formal arguments, # (or #1) is replaced by the first argument, #2 by the second, and so on while #0 is replaced by the function itself. If there are more arguments than #j in the function supplied, the remaining arguments are ignored. At that, ## defines sequence of all given arguments, whereas ##n stands for arguments from number n onward. Function has attribute HoldAll and function body is evaluated only after the formal arguments have been replaced by actual arguments. Function construct can be nested in any way. Each is treated as a scoping construct, with named inner variables being renamed if necessary. Furthermore, in the call Function[args, body, attrs] as optional argument attrs can be a single attribute or a list of attributes, e.g. HoldAll, Protected and so on, while the call Function[Null, body, attrs] represents a function in which the arguments in function body are given using slot #, etc. Such format of a pure function is very useful when a one–time call of the function is required.

Note that unlike the format (c), the formats (a) and (b) of a pure function allow use as the body rather complex program constructions, including local variables, cycles and definitions of blocks, modules and functions. A simple enough example quite visually illustrates what has been said:

In[7]:= {a, b} = {0, 0}

Out[7]= {0, 0}

In[8]:= Function[{x, y, z}, Save["t", {a, b}]; {a, b} = {42, 47};

g[h_, g_] := h*g; If[x <= 6, Do[Print[{a, b, y^z}], z], x+y+z];

{(a+b)*(x+y+z) + g[42, 47], Get["t"], DeleteFile["t"]}[[1]]][1, 2, 3]

{42, 47, 8}

{42, 47, 8}

{42, 47, 8}

Out[8]= 2508

In[9]:= {a, b}

Out[9]= {0, 0}

In[10]:= g[42, 47]

Out[10]= 1974

So, in the given example the locality of variables a and b is provided with saving their initial (up to a function call) values at the time of function call by their saving in the temporary "tmp" file, their redefinitions in the course of execution of a function body with the subsequent uploading of initial values of a and b from the "tmp" file to the current session of Mathematica with removal of the "tmp" file from the file system of the computer. Such mechanism completely answers the principle of locality of variables in a function body. This example also illustrates the ability to define a new function in the body of a pure function with its use, both within the body of the source function and outside of the function.

At using of the pure functions, unlike traditional functions, there is no need to designate their names, allowing to code their definitions directly in points of their call that is caused by that the results of the calls of pure functions depend only on values of the actual arguments received by them. Selection from a list x of elements that satisfy certain conditions and elementwise application of a function to elements of the list x can be done by constructions of the view Select[x, test[#] &] and Map[F[#] &, x] respectively as illustrate the following examples, namely:

In[3336]:= Select[{a, 72, 77, 42, 67, 2019, s, 47, 500}, OddQ[#] &]

Out[3336]= {77, 67, 2019, 47}

In[3337]:= Map[(#^2 + #) &, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}]

Out[3337]= {2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156}

At use of the short form of a pure function it is necessary to be careful at its coding because the ampersand (&) has quite low priority. For example, expression #1 + #2 – #3 + #2*#4 & without parentheses is correct whereas, in general, they are obligatory, in particular, at use of a pure function as the right part of a rule as illustrate very simple examples [10-12]. The universal format of a call of pure f function is defined as f @@ {args}, for instance:

In[4214]:= OddQ[#] || PrimeQ[#] || IntegerQ[#] & @@ {78}

Out[4214]= True

In combination with a number of functions, in particular, Map, Select and some others the use of pure functions is rather convenient, therefore the question of converting the traditional functions into the pure functions seems an quite topical; for its decision various approaches, including creation of the program converters can be used. Thus, we used pure functions a rather widely at programming of a lot of problems of various types of the applied, and the system character [1-16,22].

Unlike a number of convertors of classical functions to pure function, procedure below allows to keep the testing functions of the general kind for arguments of the converted functions

W[x_ /; Test[x], y_H, z_, h__H1, t__, …] := [x, y, z, h, t, …]

by converting of a classical W function to pure function of short format, i.e. the procedure call FuncToPureFunction[W] returns the pure function in the following kind

If[Test[#1] && HQ[#2] && True && H1Q[#4] && True && …, [#1, #2, #3, #4, #5, …], IncorrectArgs[Sequential numbers of incorrect factual arguments]] &

where {heads H and H1} can absent. In addition, for arguments of type {h__H1, t__} the resultant pure function considers only their first factual value. At the same time, only one of the {Real, List, Complex, Integer, Rational, String, Symbol} list is allowed as a head H for a y expression. In a case of inadmissibility of at least one argument and/or its testing (for example, incorrect head H or H1) the call of the pure function returns the formal call of the kind IncorrectArgs[n1, n2,…, np], where {n1, n2,…, np} determine sequential numbers of incorrect arguments. With source code of the FuncToPureFunction procedure with typical examples of its application that a quite clearly illustrate the results returned by the procedure can be found below [8-12,16].

In[5]:= FuncToPureFunction[x_ /; FunctionQ[x]] :=

Module[{a, b = ProcBody[x], c = 1, d, g, h, p, s = Map[ToString,

{Complex, Integer, List, Rational, Real, String, Symbol}]},

a = ArgsBFM1[x]; g = Map[{Set[d, "#" <> ToString[c++]]; d,

StringReplaceVars[#[[2]], #[[1]] –> d]} &, a];

h = Map[If[! StringFreeQ[#[[2]], #[[1]]], #[[2]],

If[#[[2]] == "Arbitrary", "True", If[MemberQ[s, #[[2]]], #[[2]] <>

"Q[" <> #[[1]] <> "]", "False"]]] &, g];

p = Map["(" <> # <> ")" &, h]; p = Riffle[p, "&&"];

a = Map[#[[1]] &, a]; c = Map[#[[1]] &, g];

a = StringReplaceVars[b, GenRules[a, c]];

ToExpression["If[" <> p <> "," <> a <> "," <>

"IncorrectArgs@@Flatten[Position[" <>

ToString[h] <> ", False]]" <> "]&"]]

In[6]:= f[x_ /; IntegerQ[x], y_ /; y == 77, z_List, t_, p__, h___] :=

x*y*h + Length[z]*t

In[7]:= FuncToPureFunction[f]

Out[7]= If[IntegerQ[#1] && #2 == 77 && ListQ[#3] && True && True && True, #1*#2*#6 + Length[#3]*#4, IncorrectArgs @@ Flatten[Position[{IntegerQ[#1], #2 == 77, ListQ[#3], True, True, True}, False]]] &

It should be noted that pure functions allow for a kind of parameterization based on constructions of the following format:

<Name[params]> := <Pure function, containing params>

where Name – a name of pure function and params determines parameters entering the function body. Calls of pure functions parameterized in this way have the following format:

Name[params][actual args]

The following simple example illustrates the told:

In[2215]:= T[n_, m_] := Function[{x, y, z}, n*(x + m*y + z)]

In[2216]:= T[42, 47][1, 2, 3]

Out[2216]= 4116

In[2217]:= G[p_, n_] := p*(#1^2 + #2^2*#3^n) &

In[2218]:= G[5, 2][5, 6, 7]

Out[2218]= 8945

In[2219]:= With[{p=5, n=2}, p*(#1^2 + #2^2*#3^n) &][5,6,7]

Out[2219]= 8945

In a number of applications such parameterization is quite useful, allowing customizing the tool for application based on parameter values. Some useful tools for working with the user functions are presented below, whereas a wider range of such tools are available for classical functions which are suitable also for many user procedure processing. This should be borne in mind when discussing issues related to the procedures.

1.3. Some useful tools for work with the user functions. For work with functions Mathematica has a wide range of tools for working with functions, including user ones. Here we present a number of tools for working with user functions that do not go to the standard set. First of all, it is of particular interest to test a symbol or expression to be a function. Some of the tools in this direction are presented in our package MathToolBox [16,9-12]. The most common tool of this type is the function FunctionQ, whose definition in turn contains QFunction and QFunction1 procedures and function PureFuncQ of the same type, whose call on a symbol or expression x returns True, if x is a function and False otherwise. Program code of the FunctionQ function can be presented as follows:

In[25]:= FunctionQ[x_] := If[StringQ[x], QFunction1[x] ||

PureFuncQ[ToExpression[x]], PureFuncQ[x] || QFunction[x]]

In[26]:= y := #1^2 + #2^2 &; {FunctionQ[#1^2 + #2^2 &],

FunctionQ[y]}

Out[26]= {True, True}

In[27]:= z = Function[{x, y, z}, If[x <= 6, Do[Print[y^z], z],

x + y + z]];

In[28]:= FunctionQ[Function[{x, y, z}, If[x <= 6,

Do[Print[y^z], z], x + y + z]]]

Out[28]= True

In[29]:= FunctionQ[z]

Out[29]= True

In[30]:= PureFuncQ[Function[{x, y, z}, If[x <= 6,

Do[Print[y^z], z], x + y + z]]]

Out[30]= True

In[31]:= PureFuncQ[#1^2 + #2^2 &]

Out[31]= True

In[32]:= G[x_] := Plus[Sequences[x]]

In[33]:= {FunctionQ[G], PureFuncQ[G]}

Out[33]= {True, False}

At the same time, the function call PureFuncQ[w] allows to identify object w to be a pure function, returning True, if w is a pure function and False otherwise. This tool along with others provides strict differentiation of such basic element of functional and procedural programming, as a function [8-16].

The question of determining the formal arguments of user function is of quite certain interest. Built-in tools of the system do not directly solve this issue and we have created a number of tools for the purpose of determining the formal arguments of the function, module and block. For a function, this problem is solved e.g. by the ArgsF function with program code:

In[47]:= ArgsF[x_] := If[PureFuncQ[x], ArgsPureFunc[x],

If[FunctionQ[x], Args[x], ToString1[x] <> " – not function"]]

In[48]:= GS@Sequences[a_Integer, b_Symbol, c_] := a+b+c

In[49]:= ArgsF[GS]

Out[49]= {a_Integer, b_Symbol, c_}

In[50]:= H := #1^2 + #2^2*#3^4 &

In[51]:= ArgsF[H]

Out[51]= {"#1", "#2", "#3"}

In[52]:= T := Function[{x, y, z}, x + y + z]; ArgsF[T]

Out[52]= {"x", "y", "z"}

In[53]:= ArgsF[Agn]

Out[53]= "Agn – not function"

The call ArgsF[x] returns the list of formal arguments in the string format of function x, including pure function, if x is not a function the call returns the corresponding message. The ArgsF function uses the testing tools FunctionQ and PureFuncQ along with tools Args, ArgsPureFunc, and ToString1 with which it is possible to get acquainted in [1-12], while the interested reader can get acquainted here with a number of tools (that complement the built-in system tools) for work with the user functions.

For converting of a pure function of formats (a) and (b) to format (c) the PureFuncToShort procedure with the following program code serves:

In[4220]:= PureFuncToShort[x_ /; PureFuncQ[x]] :=

Module[{a, b, c, d},

If[ShortPureFuncQ[x], x, a = ArgsPureFunc[x]; d = a;

b = Map[ToExpression, a]; Map[ClearAll, a];

a = Map[ToExpression, a];

c = Quiet[GenRules[a, Range2[#, Length[a]]]];

{ToExpression[ToString1[ReplaceAll[x[[2]], c]] <> "&"],

ToExpression[ToString[d] <> "=" <> ToString1[b]]}[[1]]]]

In[4221]:= G := Function[{x, y, z}, p*(x + y + z)];

In[4222]:= PureFuncToShort[G]

Out[4222]= p*(#1 + #2 + #3) &

The procedure call PureFuncToShort[w] returns the result of converting of a pure function w to its short format. In turn, the PureFuncToFunction procedure is intended for converting of a pure function to classical. The following example presents its program code with a simple example of its application:

In[7]:= PureFuncToFunction[x_/; PureFuncQ[x], y_/; !HowAct[y]]:=

Module[{a = ArgsPureFunc[x], b, c},

If[ShortPureFuncQ[x], b = (StringReplace[#1, "#" –> "x"] &) /@ a;

c = StringReplace[ToString1[x], GenRules[a, b]];

b = ToExpression /@ Sort[(#1 <> "_" &) /@ b];

ToExpression[ToString1[y[Sequences[b]]] <> ":=" <>

StringTake[c, {1, –3}]], a = (#1 <> "_" &) /@ a;

ToExpression[ToString1[y[Sequences[ToExpression /@ a]]] <> ":="

<> ToString1[x[[2]]]]]]

In[8]:= t := p*(#1^2 + #2^2*#3^n) &

In[9]:= PureFuncToFunction[t, H]

In[10]:= Definition[H]

Out[10]= H[x1_, x2_, x3_] := p*(x1^2 + x2^2*x3^n)

The procedure call with two arguments x and H, where x – a pure function and H – an undefined symbol, converts the pure x function to a classical function named H.

We have programmed a number of procedures that ensure the mutual conversion of classical, pure and pure functions of the short format. In particular, the CfToPure procedure allows the classical functions to be converted into pure functions of the short format [12-16]. Calling the CfToPure[f] procedure returns the result of converting a classical function f to its equivalent in the form of a pure function of short format. The procedure uses a number of techniques useful in practical programming, which are recommended to the reader. The fragment below represents source code of the CfToPure procedure and examples of its use.

In[4]:= {x, y, z} = {42, 47, 67};

In[5]:= F[x_Integer, y_, z_ /; IntegerQ[z]] := (x^2 + y^3) +

(y^2 – x*y*z)/Sqrt[x^2 + y^2 + z^2]

In[6]:= CfToPure[f_ /; FunctionQ[f]] := Module[{a, b, c, d},

a = Map[ToString[#] <> "@" &, Args[f]];

a = Map[StringReplace[#, "_" ~~ ___ ~~ "@" –> ""] &, a];

d = "###"; Save2[d, a];

b = Map["#" <> ToString[#] &, Range[Length[a]]];

Map[Clear, a]; c = GenRules[a, b];

c = Map[ToExpression[#[[1]]] –> ToExpression[#[[2]]] &, c];

a = StringReplace[Definition2[f][[1]],

Headings[f][[2]] <> " := " –> ""]; a = ToExpression[a <> " &"];

{ReplaceAll[c][a], Get[d], DeleteFile[d]}[[1]]]

In[7]:= CfToPure[F]

Out[7]= (x^2 + y^3) + (y^2 – x*y*z)/Sqrt[x^2 + y^2 + z^2] &

In[8]:= {x, y, z}

Out[8]= {42, 47, 67}

In[9]:= G[x_Integer, y_List, z_ /; IntegerQ[z]] := N[Sin[x]/

Log[Length[y]] + (y^2 – x*y*z)^(x+z)/Sqrt[x^2+y^2+z^2]]

In[10]:= CfToPure[G]

Out[10]= N[Sin[#1]/Log[Length[#2]] + (#2^2 – #1*#2*#3)^

(#1 + #3)/Sqrt[#1^2 + #2^2 + #3^2]] &

As in a function body along with formal arguments global variables and calls of functions can enter, then the problem of determination them is of a certain interest to any function. In this regard the GlobalsInFunc function can be useful enough with the following program code:

In[2223]:= VarsInFunc[x_] := Complement[Select[VarsInExpr[

ToString2[Definition[x]]], ! SystemQ[#] &],

{ToString[x]}, ArgsBFM[x]];

GlobalsInFunc[x_] := If[PureFuncQ[x],

Complement[VarsInExpr[x], Flatten[{ArgsPureFunc[x],

"Function"}]], If[FunctionQ[x], VarsInFunc[x],

"Argument – not function"]]

In[2224]:= F[x_, y_, z_] := a*x*b*y + N[Sin[2020]]

In[2225]:= GlobalsInFunc[F]

Out[2225]= {"a", "b"}

In[2226]:= G := Function[{x, y, z}, p*(x + y + z)];

In[2227]:= GlobalsInFunc[G]

Out[2227]= {"p"}

In[2228]:= T := p*(#1^2 + #2^2*#3^n) &

In[2229]:= GlobalsInFunc[T]

Out[2229]= {"n", "p"}

The call GlobalsInFunc[x] returns the list of global symbols in string format entering a function x definition (pure or classical) and other than the built-in symbols of Mathematica, if x is not a function, than the call returns the corresponding message. The function uses the tools of our MathToolBox package [7,8,16], in particular the function whose call VarsInFunc[x] returns the list of global symbols in string format entering a classical function x definition and other than the built-in symbols of the system.

Generally, the user functions do not allow to use the local variables, however a simple artificial reception allows to do it. Sequential list structure of execution of operations when the list is evaluated element–wise from left to right is for this purpose used, i.e. function is represented in the following format:

Name[args] := {Save["#", {locals}], {locals = values}, function body, Get["#"], DeleteFile["#"]}[[3]]

The presented format is rather transparent and well illustrates essence of the method of use in user function of local variables. The following simple example illustrates told:

In[9]:= GS[x_, y_, z_] := {Save["#", {a, b, c}], {a=2, b=7, c=6},

a*x + b + y + c*z, Get["#"], DeleteFile["#"]}[[3]]

In[10]:= {a, b, c} = {1, 2, 3}

Out[10]= {1, 2, 3}

In[11]:= GS[42, 47, 67]

Out[11]= 540

In[12]:= {a, b, c}

Out[12]= {1, 2, 3}

In some cases, the given approach may prove to be quite useful in practical programming. A number of our software tools have effectively used this approach.

Meanwhile, a rather significant remark needs to be made regarding the objects multiplicity of definitions with the same name. As noted repeatedly, Mathematica system differentiates objects, above all, by their headers, if any. It is therefore quite real that we may deal with different definitions under the same name. This applies to all objects that have headers. In view of this circumstance, we have programmed a number of tools for various cases of application [1-16]. First of all, we need a tool of testing the presence of such multiplicity for a particular object (function, module, block). This function is successfully performed by the MultipleQ procedure with program code:

In[47]:= MultipleQ[x_ /; SymbolQ[x], j___] := Module[{a},

a = ToString[InputForm[Definition[x]]];

a = StringReplace[a, "\n \n"  "\[CircleDot]"];

a = StringSplit[a, "\[CircleDot]"];

If[{j} != {} && !ValueQ[j], j=a]; If[Length[a] > 1, True, False]]

In[48]:= G[x_, y_] := p*(x^2 + y^2); G[x_Integer, y_] := x*y

G[x_, y_, z_] := x^2 + y^2 + z^2; G[x_, y_List] := x*y

In[49]:= MultipleQ[G, gs]

Out[49]= True

In[50]:= gs

Out[50]= {"G[x_Integer, y_] := x*y", "G[x_, y_List] := x*y",

"G[x_, y_] := p*(x^2+y^2)", "G[x_, y_, z_] := x^2+y^2+z^2"}

In[53]:= MultipleQ1[x_ /; SymbolQ[x], j___] :=

Module[{a = Definition2[x]},

If[{j} != {} && ! HowAct[j], j = a[[1 ;–2]], 77];

If[Length[a[[1 ;–2]]] == 1, False, True]]

In[54]:= MultipleQ2[x_Symbol] := If[Length[ToExpression[

Unique1[Definition2[x], j]][[1 ;–2]]] > 1, True, False]

The call MultipleQ[x] returns True if symbol x has multiple definitions, and False otherwise. Through optional j argument the list of x definitions in string format is returned. MultipleQ1 and MultipleQ2 tools – functional analogues of the MultipleQ.

At last, quite natural interest represents the existence of the user procedures and functions activated in the current session. Solution of the question can be received by tool of a procedure whose call ActBFMuserQ[] returns True if such objects exist in the current session, and False otherwise; the call ActBFMuserQ[x] thru optional x argument – an indefinite variable – returns the 2–element nested list whose the first element contains name of the user object in the string format while the second defines list of its types in string format respectively. The fragment represents source code of the ActBFMuserQ with an example of its use.

In[7]:= ActBFMuserQ[x___ /; If[{x} == {}, True, If[Length[{x}] == 1

&& ! HowAct[x], True, False]]] := Module[{b = {}, c = 1, d, h,

a = Select[Names["`*"], ! UnevaluatedQ[Definition2, #] &]},

For[c, c <= Length[a], c++, h = Quiet[ProcFuncTypeQ[a[[c]]]];

If[h[[1]], AppendTo[b, {a[[c]], h[[–1]]}], Null]];

If[b == {}, False, If[{x} != {}, x = ReduceLists[b]]; True]]

In[8]:= V := Compile[{{x, _Real}, {y, _Real}}, (x^3 + y)^2];

Art := Function[{x, y}, x*Sin[y]]; K := (#1^2 + #2^4) &;

GS[x_ /; IntegerQ[x], y_ /; IntegerQ[y]] := Sin[90] + Cos[42];

G = Compile[{{x, _Integer}, {y, _Real}}, x*y];

H[x_] := Block[{}, x]; H[x_, y_] := x + y;

SetAttributes["H", Protected]; P[x_] := Module[{}, x];

P[y_] := Module[{}, y]; P[x__] := Plus[Sequences[{x}]];

T42[x_, y_, z_] := x*y*z; R[x_] := Module[{a = 590}, x*a];

GSV := (#1^2 + #2^4 + #3^6) &

In[9]:= ActBFMuserQ[]

Out[9]= True

In[10]:= {ActBFMuserQ[t77], t77}

Out[10]= {True, {{"Art", {"PureFunction"}},

{"G", {"CompiledFunction"}}, {"GS", {"Function"}},

{"H", {"Block", "Function"}}, {"P1", {"Function"}},

{"R", {"Module"}}, {"K", "ShortPureFunction"},

{"V", {"CompiledFunction"}},

{"P", {"Module", "Module", "Function"}, {"T42", {"Function"}},

{"GSV", "ShortPureFunction"}}}}

The ActBFMuserQ procedure is of interest as an useful tool first of all in the system programming.

Along with the presented tools a number of other useful tools for working with user functions and instructive examples of their application can be found in [1-16]. Here again it should be noted that the examples of tools presented in this book use the tools of the mentioned MathToolBox package [16] that has freeware license. Given that this package and its tools will be frequently mentioned below, we will briefly describe it.

The package contains more than 1420 tools which eliminate restrictions of a number of the standard Mathematica tools and expand its software with new tools. In this context, the package can serve as a certain additional tool of modular programming, especially useful in numerous applications where certain non–standard evaluations have to accompany programming. At the same time, tools presented in the package have the most direct relation to certain principal questions of procedure–functional programming in Mathematica, not only for decision of applied tasks, but, above all, for programming of software extending frequently used facilities of the system and/or eliminating their defects or extending the system with new facilities. Software presented in this package contains a number of useful enough and effective receptions of programming in Mathematica, and extends its software which allows in the system to programme the tasks of various purposes more simply and effectively. The additional tools composing the above package embrace a rather wide circle of sections of the Mathematica system [15,16].

The given package contains definitions of some functionally equivalent tools programmed in various ways useful for use in practical programming. Along with this, they illustrate a lot of both efficient and undocumented features of Math-language of the system. In our opinion, the detailed analysis of source code of the package software can be a rather effective remedy on the path of the deeper mastering of programming in Mathematica. Experience of holding of the master classes of various levels on the systems Mathematica and Maple in all evidence confirms expediency of joint use of both standard tools of the systems of computer mathematics, and the user tools created in the course of programming of various appendices. A lot of tools contained in the package and mentioned below were described in [1-15] in detail enough. We now turn to the procedural objects of the Mathematica system.

Chapter 2: The user procedures in Mathematica

Procedure is one of basic concepts of programming system of Mathematica, being not less important object in systems of programming as a whole. A procedure is an object implementing the well-known concept of "black box" when with known input (actual arguments) and output (returned result) while the internal structure of the object, generally speaking, is closed. Procedure forms the basis of the so-called "procedural programming" where in the future by the procedure we will understand such objects as module and block. Procedural programming is one of basic paradigms of Mathematica which in an essential degree differs from similar paradigm of the well-known traditional procedural programming languages. This circumstance is the cornerstone of certain system problems relating to procedural programming in Mathematica. Above all, similar problems arise in the field of distinctions in implementation of the above paradigms in the Mathematica and in the environment of traditional procedural languages. Along with that, unlike a number of traditional and built-in languages the built-in Math–language has no a number of useful enough tools for operating with procedural objects. In our books [1-15] and MathToolBox package [16] are presented some such tools.

A number of the problems connected with similar tools is considered in the present chapter with preliminary discussion of the concept of `procedure` in Mathematica system as bases of its procedural paradigm. In addition, tools of analysis of this section concern only the user procedures and functions because definitions of all built–in system functions (unlike, say, from the Maple system) from the user are hidden, i.e. are inaccessible for processing by the standard tools of Mathematica system. Note that the discussion of procedural objects will take place relative to the latest version of Mathematica 12.1, which can dissonance with Mathematica of earlier versions. However this should not cause any serious misunderstandings. Math-language possesses a rather high level of immutability in relation to its versions.

2.1. Definition of procedures in Mathematica software

Procedures in Mathematica system are formally represented by program objects of the following 2 simple formats, namely:

M[x_ /; Test_x, y_ /; Test_y, ...] {:= | =} Module[{locals}, Module Body]

B[x_ /; Test_x, y_ /; Test_y, ...] {:= | =} Block[{locals}, Block Body]

i.e., procedures of both types represent the functions from two arguments – the procedure body (Body) and local variables (locals). Local variables – the list of names, perhaps, with initial values which are attributed to them. These variables have the local character concerning the procedure, i.e. their values aren't crossed with values of the symbols of the same name outside of the procedure. All other variables in the procedure have global character, sharing field of variables of the Mathematica current session. In addition, in the procedure definition it is possible to distinguish six following component, namely:

– procedure name (M and B in the both procedures definitions);

– procedure heading ({M|B}[x_ /; Test_x, y_ /; Test_y, ...] in the both procedures definitions);

– procedural brackets (Module[…] and Block[…]);

– local variables (list of local variables {locals}; can be empty list);

– procedure (Module, Block) body; can be empty;

– testing Test_x function (the function call Test[x] returns True or False depend on permissibility of an actual x argument; can absent).

When programmed, we typically try to program tasks as modular programs as possible to make them more readable and independent, and more convenient to maintain. One of ways of a solution of the given problem is use of the scope mechanism for variables, defining for them separate scopes. Mathematica provides 2 basic mechanisms for limiting the scope of variables in form of modules and blocks, hereinafter referred to as general term – procedures. Procedures along with functions play a decisive role in solving the issue of optimal organization of program code. Note, real programming uses modules much more frequently, but blocks are often more convenient in interactive programming.

Most traditional programming languages use a so–called "lexical scoping" mechanism for variables, which is analogous to the mechanism of modules in Mathematica. Whereas symbolic programming languages e.g. LISP allow also "dynamic scoping", analogous to the mechanism used by blocks in Mathematica. When lexical scoping is used, variables are treated as local ones to a particular part of a program code. At a dynamic scoping, the values of the allocated variables are local only to a certain part of the program execution. A Module carries out processing of the body when it is carried out as a component of the general program code, any variable from locals in the module body is considered as local one. Then natural execution of the general program code continues. A Block, ignoring all expressions of a body, uses for it the current values of variables from locals. In the course of execution of body the block uses values of locals for variables then recovers their original values after completion of the body. Most vividly these differences an example illustrates:

In[2254]:= {b, c} = {m, n};

In[2255]:= B[x_] := Block[{a = 7, b, c}, x*(a + b + c)]; B[7]

Out[2255]= 7*(7 + m + n)

In[2256]:= {b, c}

Out[2256]= {m, n}

In[2257]:= LocVars[x_String] := ToExpression[x <> "$" <>

ToString[$ModuleNumber – 4]]

In[2258]:= M[x_] := Module[{a = 7, b, c}, x*(a + b + c)]; M[7]

Out[2258]= 7*(7 + b$21248 + c$21248)

In[2259]:= Map[LocVars, {"a", "b", "c"}]

Out[2259]= {a$21248, b$21248, c$21248}

Thus, Module is a scoping construct that implements lexical scoping, while Block is a scoping construct implements dynamic scoping of local variables. Module creates new symbols to name each of its local variables every time it is called. Call of LocVars function immediately after calling a module (not containing calls of other modules) with local variables {a, b, c,...} returns the list of variables Map[LocVars, {"a", "b", "c",...}] that were generated at the time the module was called.

In the context of testing blocks for procedural nature in the above sense, the LocVars1 procedure may be of interest whose call LocVars1[x] returns True if x is a module or a block x has no local variables without initial values, or the list of local variables is empty, otherwise False is returned. While calling LocVars1[x,y] with the 2^nd optional argument y – an undefined symbol – through y additionally returns the 2-element list whose the first element defines the list of all local variables of x, while the 2^nd element defines the list of local variables without initial values. Below, fragment represents source code of the procedure with its use.

In[4]:= LocVars1[x_ /; BlockModQ[x], y___] := Module[{a, b, d = {}},

a = Locals5[x, b];

Do[AppendTo[d, If[Length[b[[k]]] < 2, a[[k]], Nothing]],

{k, Length[a]}]; If[{y} != {} && SymbolQ[y], y = {a, d}, 7];

If[ModuleQ[x], True, If[a == d, True, False]]]

In[5]:= B[x_] := Block[{a = 7, b, c = 8, d}, x*a*b*c*d];

B1[x_] := Block[{a, b, c, d}, x*a*b*c*d]

In[6]:= {LocVars1[B, t1], t1}

Out[6]= {False, {{"a = 7", "b", "c = 8", "d"}, {"b", "d"}}}

In[7]:= {LocVars1[B1, t2], t2}

Out[7]= {True, {{"a", "b", "c", "d"}, {"a", "b", "c", "d"}}}

In[8]:= G[x_] := Block[{}, x^2]; {LocVars1[G, t72], t72}

Out[8]= {True, {{}, {}}}

Considering importance of modular approach to software organization when a program code consists of set of the linked independent objects and reaction of objects is defined only by their inputs, in this quality to us the most preferable the modules are presented. In general software code of the modular structure, the output of the module is the input for another module. For this reason the lion share of tools of our package MathToolBox [16] is programmed in the form of modules.

Once again, pertinently to pay attention to one moment. A number of tools represented in the package are focused on the solution of identical problems, but they use various algorithms programmed with usage of various approaches. They not only illustrate variety of useful enough receptions but also revealing their shortcomings and advantages useful both in practical and system programming. In our opinion, such approach opens a rather wide field for awakening of creative activity of the reader in respect of improvement of his skills in the programming in Mathematica system. Now let's consider module components in more detail.

Let's note that in certain cases duplication of definitions of blocks, modules and functions under new names on condition of saving their former definitions invariable is required. So, for assignment of a name that is unique in the current session to the new means, the procedure whose call DupDef[x] returns a name (unique in the current session) whose definition will be equivalent to definition of the x symbol can be used. In addition, the given procedure successfully works with the means having multiple definitions too. The following fragment represents source code of the procedure with an example of its application.

In[3331]:= DupDef[x_ /; BlockFuncModQ[x]] :=

Module[{a = Flatten[{Definition1[x]}],

b = Unique[ToString[x]]},

ToExpression[Map[StringReplace[#, ToString[x] <> "[" –>

ToString[b] <> "[", 1] &, a]]; b]

In[3332]:= Gs[x_, y_] := x + y

In[3333]:= Gs[x_] := Module[{a = 77}, a*x^2]

In[3334]:= DupDef[Gs]

Out[3334]= Gs78

In[3335]:= Definition[Gs78]

Out[3335]= Gs78[x_, y_] := x + y

Gs78[x_] := Module[{a = 77}, a*x^2]

In particular, the DupDef procedure is useful enough when debugging of the user software.

The call Definition[x] of the standard function in a number of cases returns the definition of some x object with the context corresponding to it what at a rather large definitions becomes badly foreseeable and less acceptable for subsequent program processing as evidently illustrates a number of examples [6-15]. Moreover, the name of an object or its string format also can act as an actual argument. For elimination of this shortcoming we defined a number of tools allowing to obtain definitions of the procedures or functions in a certain optimized format. As such means it is possible to note the following: DefOptimum, DefFunc, Definition1, Definition2, Definition3 ÷ Definition5, DefFunc1, DefOpt, DefFunc2 and DefFunc3. These means along with some others are represented in [4-9] and included in the MathToolBox package [16]. The following fragment represents the source code of the most used tool of them with an example of its application.

In[49]:= Definition2[x_ /; SameQ[SymbolQ[x], HowAct[x]]] :=

Module[{a, b = Attributes[x], c},

If[SystemQ[x], Return[{"System", Attributes[x]}],

Off[Part::partw]]; ClearAttributes[x, b];

Quiet[a = ToString[InputForm[Definition[x]]];

Mapp[SetAttributes, {Rule, StringJoin}, Listable];

c = StringReplace[a, Flatten[{Rule[StringJoin[Contexts1[],

ToString[x] <> "`"], ""]}]]; c = StringSplit[c, "\n \n"];

Mapp[ClearAttributes, {Rule, StringJoin}, Listable];

SetAttributes[x, b]; a = AppendTo[c, b];

If[SameQ[a[[1]], "Null"] && a[[2]] == {}, On[Part::partw];

{"Undefined", Attributes[x]}, If[SameQ[a[[1]], "Null"] &&

a[[2]] != {} && ! SystemQ[x], On[Part::partw]; {"Undefined",

Attributes[x]}, If[SameQ[a[[1]], "Null"] && a[[2]] != {} &&

a[[2]] != {}, On[Part::partw]; {"System", Attributes[x]},

On[Part::partw]; a]]]]]

In[50]:= a[x_] := x + 6; a[x_, y_] := Module[{}, x/y]; a[t_Integer] := 7

In[51]:= SetAttributes[a, {Protected, Listable}]; Definition2[a]

Out[51]= {"a[t_Integer] := 7", "a[x_] := x + 6",

"a[x_, y_] := Module[{}, x/y]", {Listable, Protected}}

The Definition2 call on system functions returns the nested list, whose first element – "System", while the second element – the list of attributes ascribed to a factual argument. On the user function or procedure x the call Definition2[x] also returns the nested list, whose first element – the optimized definitions of x (in the sense of absence of contexts in them, at a series definitions in a case of multiplicity of x definition), whereas the second element – the list of attributes ascribed to x; in their absence the empty list acts as the second element of the returned list. In a case of False value on a test ascribed to a formal x argument, the procedure call Definition2[x] will be returned unevaluated. Analogously to the above procedures, the procedure Definition2 processes the main both the erroneous and especial situations.

Using the Definition2 procedure and the list representation, we give an example of a simple enough Info procedure whose call Info[x] in a convenient form returns the complete definition of a symbol x, including its usage. While the call Info[x, y] with the second optional y argument – an indefinite symbol – through it additionally returns the usage for x symbol in string format. The following fragment represents the source code of the Info procedure with examples of its typical application.

In[3221]:= Info[x_ /; SymbolQ[x], y___] :=

If[! HowAct[x], $Failed, {If[StringQ[x::usage],

Print[If[{y} != {} && ! HowAct[y], y = x::usage, x::usage]],

Print["Usage on " <> ToString[x] <> " is absent"]],

ToExpression[Definition2[x][[1 ;–2]]]; Definition[x]}[[–1]]]

In[3222]:= Info[StrStr]

The call StrStr[x] returns an expression x in string format if x is different from string; otherwise, the double string obtained from an expression x is returned.

Out[3222]= StrStr[x_] := If[StringQ[x], "\"" <> x <> "\"",

ToString[x]]

In[3223]:= x[x_] := x; x[x_, y_] := x*y; x[x_, y_, z_] := x*y*z

In[3224]:= Info[x]

Usage on x is absent

Out[3224]= x[x_] := x

x[x_, y_] := x*y

x[x_, y_, z_] := x*y*z

In[3225]:= Info[StrStr, g47]

The call StrStr[x] returns an expression x in string format if x is different from string; otherwise, the double string obtained from an expression x is returned.

Out[3225]= StrStr[x_] := If[StringQ[x], "\"" <> x <> "\"", ToString[x]]

In[3226]:= g47

Out[3226]= "The call StrStr[x] returns an expression x in string format if x is different from the string; otherwise, the double string obtained from an expression x is returned."

In[3227]:= Info[agn]

Out[3227]= $Failed

If the first x argument – an indefined symbol, the procedure call Info[x] returns $Failed.

Another moment should be mentioned. In some cases, the above procedural objects can be represented in functional form based on the list structures of the format:

F[x_, …, z_] := {Save[w, a, b, c, …]; Clear[a, b, c, …];

Procedure body, Get[w]; DeleteFile[w]}[[–2]]

The Save function saves all values of local and global procedure variables in a certain w file, then clears all these variables, and then executes the procedure body. Finally, Get function loads the w file into the current session, restoring {a, b, c,...} variables, and then deletes the w file. The result of the function call of the type described is the second list element, beginning with its end in a case if the procedure returns the result of the call through the last sentence of its body, which is a fairly frequent case. We can use {;|,} as the list element separators, depending on need. Thus, on the basis of the above design it is possible to program functional equivalents of quite complex procedures. An example is the functional equivalent of the SubsDel procedure described in section 3.4.

In[77]:= SubsDel[S_ /; StringQ[S], x_ /; StringQ[x], y_ /; ListQ[y] &&

AllTrue[Map[StringQ, y], TrueQ] &&

Plus[Sequences[Map[StringLength, y]]] == Length[y],

p_ /; MemberQ[{–1, 1}, p]] := {Save["#$#", {"b", "c", "d", "h", "k"}];

Clear[b, c, d, h, k]; {b, c = x, d, h = StringLength[S], k};

If[StringFreeQ[S, x], Return[S], b = StringPosition[S, x][[1]]];

For[k = If[p == 1, b[[2]] + 1, b[[1]] – 1], If[p == 1, k <= h, k >= 1],

If[p == 1, k++, k––], d = StringTake[S, {k, k}];

If[MemberQ[y, d] || If[p == 1, k == 1, k == h], Break[],

If[p == 1, c = c <> d, c = d <> c]; Continue[]]];

StringReplace[S, c –> ""], Get["#$#"]; DeleteFile["#$#"]}[[–2]]

In[78]:= SubsDel["12345avz6789", "avz", {"8", "5"}, 1]

Out[78]= "1234589"

A comparative analysis of both implementations confirms aforesaid.

It is well known [8-12,16], that the user package uploaded into the current session can contains tools x whose definitions obtained by means of the call Definition[x] aren't completely optimized, i.e. contain constructions of the kind j <> "x`" where j – a context from the system list $Packages. We have created a number of tools [16] that solve various problems of processing unoptimized definitions of both individual means and means in the user packages located in files of {"m", "mx"} formats. So, the call DefFunc1[x] provides return of the optimized definition of an object x whose definition is located in the user package or nb–document and that has been loaded into the current session. At that, a name x should define an object without any attributes and options or with attributes and/or options. Fragment below represents source code of the DefFunc1 procedure along with an example of its application.

In[7]:= Definition[ListListQ]

Out[7]= ListListQ[Global`ListListQ`L_] :=

If[ListQ[Global`ListListQ`L] && Global`ListListQ`L != {} &&

Length[Global`ListListQ`L] >= 1 &&

Length[Select[Global`ListListQ`L, ListQ[#1] &&

Length[#1] == Length[Global`ListListQ`L[[1]]] &]] ==

Length[Global`ListListQ`L], True, False]

In[8]:= DefFunc1[x_ /; SymbolQ[x] || StringQ[x]] :=

Module[{a = GenRules[Map14[StringJoin, {"Global`",

Context[x]}, ToString[x] <> "`"], ""], b = Definition2[x][[1]]},

ToExpression[Map[StringReplace[#, a] &, b]]; Definition[x]]

In[9]:= DefFunc1[ListListQ]

Out[9]= ListListQ[L_] := If[ListQ[L] && L != {} &&

Length[L] >= 1 && Length[Select[L, ListQ[#1] &&

Length[#1] == Length[L[[1]]] &]] == Length[L], True, False]

So, the procedure call DefFunc1[x] in an optimized format returns the definition of an object x contained in a package or a notebook loaded into the current session. The object name is set in string format or symbol format depending on the object type (procedure, function, global variable, procedure variable, etc.). Thus, in the 2^nd case the definition is essentially more readably, above all, for rather large source codes of procedures, functions, along

with other objects. The TypesTools procedure is rather useful.

Calling TypesTools[x] procedure returns the three-element list whose elements are sub-lists containing names in the string format of tools with context x in the terms of the objects such as Functions, Procedures, and Others, accordingly. During the procedure run, lists of the form {n, "Name"} are intermediately output, where n is the number of the tool being tested from the tools list with context x, and Name is the name of the tool being tested. The following fragment represents the source code of the TypesTools procedure, followed by its application.

In[3325]:= TypesTools[x_ /; ContextQ[x]] := Module[{a = {},

b = {}, c = {}, d = CNames[x], t, j}, Monitor[j = 1;

While[j <= Length[d], Pause[0.1]; t = d[[j]]; If[QFunction[t],

AppendTo[a, t], If[ProcQ[t], AppendTo[b, t], AppendTo[c, t]]];

j++], {j, t}]; {a, b, c}]

In[3326]:= TypesTools["AladjevProcedures`"]

{1, "AcNb"}

==============

{1416, "$Version2"}

Out[3326]:= {{"AcNb", "ActBFM", …, "XOR1"},

{"ActCsProcFunc", …, "$SysContextsInM1"},

{"AddDelPosString", "Args", …, "$Version2"}}

Calling ContextMonitor[x] procedure during its execution outputs the lists of the form {n, N, Tp, PI} where n is the number of the tool being tested from the tools list with context x, N – is the name of the tool being tested, Tp – its type {Function, Block, Module, Other} and PI – a progress indicator; returning nothing. The source code of the procedure is represented below.

ContextMonitor[x_/; ContextQ[x]] := Module[{c, d = CNames[x], t, j}, Monitor[c = 0; Map[{t = #, c++, j = TypeBFM[#], Pause[0.5]} &, d], {c, t, If[! MemberQ[{"Block", "Function", "Module"}, j], "Other", j], ProgressIndicator[c, {1, Length[d]}]}];]

TypesTools & ContextMonitor procedures use the Monitor function whose call Monitor[x,y] generates a temporary monitor place in which the continually updated current y value will be displayed during the course of evaluation of x. Such approach may be a rather useful in many rather important applications.

2.2. Headings of procedures in Mathematica software

Procedures in Mathematica system are formally presented by Modules and Blocks whose program structure begins with a Heading, i.e. heading – the header part of a procedure definition, preceded to the sign {:= | =} of the delayed, as a rule, or instant assignment. The components of the heading are the procedure name and its formal arguments with patterns "_", possibly also with testing functions assigned to them.

It is necessary to highlight that in Mathematica system as correct procedural objects {Block, Module} only those objects are considered which contain the patterns of the formal arguments located in a certain order, namely:

1. The coherent group of formal arguments with patterns "_" has to be located at the very beginning of the tuple of formal arguments in headings of the above procedural objects;

2. Formal arguments with patterns "__" or "___" can to finish the tuple of formal arguments; at that, couples of adjacent arguments with patterns consisting from {"__", "___"} are inadmissible because of possible violation of correctness (in context of scheduled computing algorithm) of calls of the above procedural objects as a simple enough fragment illustrates:

In[1221]:= A[x_, y__, z__] := x*y^2*z^3

In[1222]:= {A[x, y, z], A[x, y, h, z], A[x, y, h, x, a, b]}

Out[1222]= {x*y^2*z^3, h^z^3*x*y^2, h^x^a^b^3*x*y^2}

In[1223]:= G[x_, y__, z___] := x + y + z

In[1224]:= {G[x, y, z], G[x, y, m, n, z], G[x, y, z, h]}

Out[1224]= {x + y + z, m + n + x + y + z, h + x + y + z}

Other rather simple examples illustrate told. Thus, the real correctness of tuple of formal arguments can be coordinated to their arrangement in the tuple in context of patterns {“_”, “__”, “___”}. At the same time, for all patterns for formal arguments the testing Test_x function of a rather complex kind can be used as the following evident fragment illustrates:

In[1567]:= Sg[x_ /; StringQ[x], y__ /; If[Length[{y}] == 1,

IntegerQ[y], MemberQ3[{Integer, List, String},

Map[Head, {y}]]]] := {x, y}

In[1568]:= Sg["agn", 72, 77, "sv", {a, b}]

Out[1568]= {"agn", 72, 77, "sv", {a, b}}

In[1569]:= Sg1[x_ /; StringQ[x], y___ /; If[{y} == {}, True,

If[Length[{y}] == 1, IntegerQ[y], MemberQ3[{List, String,

Integer}, Map[Head, {y}]]]]] := {x, y}

In[1570]:= Sg1["agn", 72, 77, "sv", {a, b}, a + b]

Out[1570]= Sg1["agn", 72, 77, "sv", {a, b}, a + b]

Furthermore, a procedure and function headings in testing functions are allowed to use procedure and function definitions (using the list structure) that are activated in the current session at the time the objects containing them are called, for example:

In[1620]:= M1[x_ /; {SetDelayed[h[a_], Module[{}, 42*a]],

h[x] < 2020}[[2]], y_] := Module[{a=5, b=7}, a*x^2 + b*y^2]

In[1621]:= M1[42, 47]

Out[1621]= 24283

In[1622]:= Definition[h]

Out[1622]= h[a_] := Module[{}, 42*a]

Right there once again it should be noted one an essential moment. Definition of the testing Test_x function can be directly included to the procedure or function heading, becoming active in the current session by the first procedure or function call:

In[2]:= P[x_, y_ /; {IntOddQ[t_] := IntegerQ[t] && OddQ[t],

IntOddQ[y]}[[–1]]] := x*y

In[3]:= P[72, 77]

Out[3]= 5544

In[4]:= Definition[IntOddQ]

Out[4]= IntOddQ[t_] := IntegerQ[t] && OddQ[t]

In[5]:= t = 77; Save["#", t]

In[6]:= P[x_ /; Module[{}, If[x^2 < Read["#"], Close["#"];

True, Close["#"]; False]], y_] := x*y

In[7]:= P[8, 7]

Out[7]= 56

Generally speaking, Mathematica's syntax and semantics, when defining procedures (modules and blocks), allow the use of procedure definitions in both the definition of test functions for formal arguments and the definitions of local variables, as the following fairly simple fragment illustrates quite clearly, while for classical functions similar assumptions are only possible for test functions, of course.

In[9]:= Avz[x_ /; If[SetDelayed[g[t_], Module[{a = 42}, t^2 + a]];

g[x] < 777, True, False]] := Module[{a = SetDelayed[v[t_],

Module[{b = 47, c = 67}, b*t^3 + c]]}, a = 77; a*v[x]]

In[10]:= Avz[7]

Out[10]= 1246476

In[11]:= Avz[100]

Out[11]= Avz[100]

In[12]:= Definition[g]

Out[12]= g[t_] := Module[{a = 42}, t^2 + a]

In[13]:= Definition[v]

Out[13]= v[t_] := Module[{b = 47, c = 67}, b*t^3 + c]

In[14]:= Context[g]

Out[14]= "AladjevProcedures`"

In[15]:= Context[v]

Out[15]= "Global`"

In[16]:= $Context

Out[16]= "Global`"

In[17]:= Contexts[]

Out[17]= {"AladjevProcedures`", "AladjevProcedures`BitGet1`",

"AladjevProcedures`CharacterQ`", …}

In[18]:= Length[%]

Out[18]= 1611

The procedures whose definitions are contained in the test functions or in the local variables domain are activated in the current session when the procedure contains them is called. In addition, the symbol v defined in the local area has the current context, while the symbol g defined in the test function obtains the context being the first in the list Contexts[] of all contexts of the current session. The feasibility of using the above approach to defining procedures is discussed in some detail in [8,11,15].

In a number of tasks, substantially including debugging of the user software, there is a necessity of dynamic change of the testing functions for formal arguments of blocks, functions and modules in the moment of their call. The ChangeLc procedure allowing dynamically monitoring influence of testing functions of formal arguments of tools for correctness of their calls is for this purpose programmed. The following fragment submits the source code of the procedure with examples of its application.

In[4478]:= ChangeLc[x_ /; StringQ[x] || ListQ[x],

P_ /; BlockFuncModQ[P], y___] :=

Module[{a = ArgsBFM2[P], b = Definition1[P],

p, c, d, g, n, m = Flatten[{x}], t = ToString[P]},

p = Map[StringTake[#, Flatten[StringPosition[#, "_"]][[1]]] &, m];

b = StringReplace[b, "Global`" <> t <> "`" –> ""];

c = StringReplace[b, t <> "[" –> ToString[n] <> "[", 1];

a = Select[a, SuffPref[#, p, 1] &];

If[a == {}, P @@ {y},

c = StringReplace[c, Rule1[Riffle[a, m]], 1];

ToExpression[c]; t = n @@ {y};

{If[SuffPref[ToString1[t], ToString[n] <> "[", 1],

"Updating testing functions for formal arguments of " <>

ToString[P] <> " is unsuitable", t], Remove[n]}[[1]]]]

In[4479]:= G[x_, y_, z_Integer] := x^2 + y^2 + z^2

In[4480]:= ChangeLc["y_ /;IntegerQ[y]", G, 42, 47, 67]

Out[4480]= 8462

In[4481]:= ChangeLc["y_ /;RationalQ[y]", G, 42, 47, 67]

Out[4481]= "Updating testing functions for formal

arguments of G is unsuitable"

In[4482]:= ChangeLc["y_ /;RationalQ[y]", G, 42, 47/72, 67]

Out[4482]= 32417761/5184

The procedure call ChangeLc[x, P, y] returns the result of a call P[y] on condition of replacement of the testing functions of arguments of a block, module or function P by the new functions determined by a string equivalent x or by their list. The string equivalent is coded in shape "x_ /; TestQ[x]", x – an argument.

The task of checking formal arguments of blocks, modules, functions for their validity to admissible values plays a rather important role and this problem is solved by means of the test functions assigned to the corresponding formal arguments of the above objects. The first part of the next fragment represents an example of how to organize a system to test the actual values obtained when the Art procedure is called. At that, if the formal arguments receive invalid actual values, then procedure call is returned unevaluated with the corresponding message printing for the first of such formal arguments. Note that built-in means Mathematica also do quite so. Meantime, the task of checking all invalid values for formal arguments when calling the above objects is more relevant. Below is a possible approach to that.

In[2185]:= Art[x_ /; If[IntegerQ[x], True,

Print["Argument x is not integer, but received: x = " <>

ToString1[x]]; False],

y_ /; If[ListQ[y], True,

Print["Argument y is not a list, but received: y = " <>

ToString1[y]]; False],

z_ /; If[StringQ[z], True,

Print["Argument z is not a string, but received: z = " <>

ToString1[z]]; False]] :=

Module[{a = 77}, N[x*(Length[y] + StringLength[z])/a]]

In[2186]:= Art[47, {a, b, c}, "RansIan"]

Out[2186]= 6.1039

In[2187]:= Art[47.42, 67, "abc"]

Argument x is not integer, but received: x = 47.42

Out[2187]= Art[47.42, 67, "abc"]

In[2188]:= Art[47.42, 67, "abc"]

Argument x is not integer, but received: x = 47.42

Out[2188]= Art[47.42, 67, "abc"]

In[2189]:= Art[500, 67, "abc"]

Argument y is not a list, but received: y = 67

Out[2189]= Art[500, 67, "abc"]

In[2190]:= Art[500, {a, b, c}, 77]

Argument z is not a string, but received: z = 77

Out[2190]= Art[500, {a, b, c}, 77]

The above approach is based on the following construction:

G[x_ /; If[Test_x, $x$ = True, Print[…]; $x$ = False; True],

y_ /; If[Test_y, $y$ = True, Print[…]; $y$ = False; True], …

h_ /; AllTrue[{$x$, $y$, …}, # == True &]] := Module[{}, …]

The Kr procedure is an example of using of the above approach.

In[2192]:= Kr[x_ /; If[IntegerQ[x], $x$ = True,

Print["Argument x is not integer, but received: x = " <>

ToString1[x]]; $x$ = False; True],

y_ /; If[ListQ[y], $y$ = True,

Print["Argument y is not a list, but received: y = " <>

ToString1[y]]; $y$ = False; True],

z_ /; If[StringQ[z], $z$ = True,

Print["Argument z is not a string, but received: z = " <>

ToString1[z]]; $z$ = False; True],

h_ /; AllTrue[{$x$, $y$, $z$}, # == True &]] :=

Module[{a = 77}, N[x*(Length[y] + StringLength[z])/a]]

In[2193]:= Kr[47, {a, b, c}, "RansIan", 78]

Out[2193]= 6.1039

In[2194]:= Kr[42.47, 67, "abc", 78]

Argument x is not integer, but received: x = 42.47

Argument y is not a list, but received: y = 67

Out[2194]= Kr[42.47, 67, "abc", agn]

In[2195]:= Kr[42.47, 67, 90, 78]

Argument x is not integer, but received: x = 42.47

Argument y is not a list, but received: y = 67

Argument z is not a string, but received: z = 90

Out[2195]= Kr[42.47, 67, 90, agn]

The final part of the above fragment represents the Kr procedure equipped with a system for testing the values of the actual arguments passed to the formal arguments when the procedure is called. At that, only three {x, y, z} arguments are used as the formal arguments used by the procedure body, whereas the 4^th additional argument h, when the procedure is called obtains an arbitrary expression, and in the Kr body is not used. Argument h at the procedure header level tests the validity of the actual values obtained by formal arguments when the procedure is called. The argument allows to obtain information on all inadmissible values for formal arguments at the procedure call. The fragment is fairly transparent and does not require any clarifying.

Based on the form of the heading shown above, it is possible to represent its rather useful extension, that allows to form the function body at the level of such heading. In general, the format of such header takes the following form:

G[x_ /; If[Test[x], Save["#", a, b, c, …]; a = {x, True}; True,

Save["#", a, b, c, …]; Print["Argument x is not integer, but received:

x = " <> ToString1[x]]; a = {x, False}; True],

y_ /; If[Test[y], b = {y, True}; True,

Print["Argument y is not integer, but received:

y = " <> ToString1[y]]; b = {y, False}; True],

z_ /; If[Test[z], c = {z, True}; True,

Print["Argument z is not integer, but received:

z = " <> ToString1[z]]; c = {z, False}; True], …

………………………………………………………………………

h_ /; If[AllTrue[{a[[2]],b[[2]], c[[2]]}, # == True &],

Result = Function_Body[a[[1]], b[[1]], c[[1]]]; Get["#"];

DeleteFile["#"]; True, Get["#"]; DeleteFile["#"]; False]] := Result

where Test[g] – a testing expression, testing an expression g on admissibility to be g as a valid value for the g argument and Result is the result of Function_body evaluation on formal arguments {x,y,z,…}. Let give a concrete filling of the above structural form as an example:

In[3338]:= ArtKr[x_ /; If[IntegerQ[x], Save["#", a, b, c];

a = {x, True}; True, Save["#", a, b, c];

Print["Argument x is not Integer, but received:

x = " <> ToString1[x]]; a = {x, False}; True],

y_ /; If[RealQ[y], b = {y, True}; True,

Print["Argument y is not Real, but received:

y = " <> ToString1[y]]; b = {y, False}; True],

z_ /; If[RationalQ[z], c = {z, True}; True,

Print["Argument z is not Rational, but received:

z = " <> ToString1[z]]; c = {z, False}; True],

h_ /; If[AllTrue[{a[[2]], b[[2]], c[[2]]}, # == True &],

$Result$ = N[a[[1]]*b[[1]]*c[[1]]]; Get["#"]; DeleteFile["#"]; True,

Get["#"]; DeleteFile["#"]; False]] := $Result$

In[3339]:= $Result$ = ArtKr[72, 77.8, 50/9, vsv]

Out[3339]= 31120.

In[3340]:= $Result$ =ArtKr[72, 77.8, 78, vsv]

Argument z is not Rational, but received: z = 78

Out[3340]= ArtKr[72, 77.8, 78, vsv]

In[3341]:= ArtKr[72, 778, 78, vsv]

Argument y is not Real, but received: y = 778

Argument z is not Rational, but received: z = 78

Out[3341]= ArtKr[72, 778, 78, vsv]

In[3342]:= ArtKr[7.2, 77.8, 78, vsv]

Argument x is not Integer, but received: x = 7.2

Argument z is not Rational, but received: z = 78

Out[3342]= ArtKr[7.2, 77.8, 78, vsv]

In[3343]:= ArtKr[7.2, 900, 50, vsv]

Argument x is not Integer, but received: x = 7.2

Argument y is not Real, but received: y = 900

Argument z is not Rational, but received: z = 50

Out[3343]= ArtKr[7.2, 900, 50, vsv]

The function call ArtKr[x, y, z, h] returns the result of evaluation of a Function_Body on actual arguments {x, y, z}. In addition, the call ArtKr[x, y, z, h] as a value for formal h argument obtains an arbitrary expression. Through the global variable $Result$, the call ArtKr[x, y, z, h] additionally returns the same value. At that, the above structural form is based on the processing principle of actual arguments passed to an object (block, function, module), namely – the values of actual arguments are processed by the corresponding testing functions from left to right until the first calculation False that is obtained by a testing function, after that the call of the object is returned as unevaluated, otherwise the result of the object calculation is returned.

Therefore, the test functions for the leading formal arguments of the form are formed in such way that they return True when fixing the real value of the test functions and the values passed to them. Whereas the test function for the auxiliary argument h determines both the evaluation of the object body and validity of the factual values for the principal arguments {x, y, z}, solving the task of returning the result of the calculation of the body of the object or returns the result of the call unevaluated. Fragment above is fairly transparent and does not require any clarifying.

According to the scheme described above, Mathematica allows a number of useful modifications. In particular, the following form of organizing headers {block, function, module} having the following format may well be of interest:

S[x_ /; {X₁; …; X_p; {Test_x|True}}[[1]],

y_ /; {Y₁; … ; Y_n; Test_y}[[1]],

………………………………………………………..

z___ /; {Z₁; …; Z_t; Test_z}[[1]] := Object

In addition as {X1,…,Xp; Y1,…,Yn; ...; Z1,…,Zt} act admissible offers of the Mathematica language, including calls of blocks, functions and modules whereas as {Testx,Testy,…,Testz} act the testing functions ascribed to the corresponding formal {x,y,…,z} arguments. In a case if there is no the test function for a formal argument (i.e. argument allows any type of actual value) instead of it is used True. Moreover, the test function for pattern "x___" or "x__" should refer to all formal arguments relating thereto. In a case of such patterns used in an object header should be used or True, or a special test function, for example, that is presented below. In a number of cases similar approach can be interesting enough. The procedure call TrueListQ[x, y] returns True if a list x has elements whose types match to the corresponding elements of a list y, otherwise False is returned. If lengths of both lists are different, then the first Min[x, y] their elements are analyzed. At that, the call with the third argument z – an indefinite symbol – additionally returns the list of elements positions of list x whose types different from the corresponding types from the y list. The source code of the TrueListQ procedure is represented below.

In[942]:= TrueListQ[x_ /; ListQ[x], y_ /; ListQ[y], z___] :=

Module[{a = Length[x], b = Length[y], c = {}, t = {}},

Do[If[SameQ[Head[x[[j]]], y[[j]]], AppendTo[c, True],

AppendTo[t, j]; AppendTo[c, False]], {j, Min[a, b]}];

If[{z} != {} && SymbolQ[z], z = t; If[t == {}, True, False],

If[t == {}, True, False]]]

In[943]:= TrueListQ[{77, 6, "abc"}, {Integer, Symbol, String}]

Out[943]= False

In[944]:= TrueListQ[{77, 7.8, "a"}, {Integer, Real, String}]

Out[944]= True

In[945]:= {TrueListQ[{a, 78, "2020"}, {Integer, Real, String}, g], g}

Out[945]= {False, {1, 2}}

The following simple GW procedure uses the TrueListQ in heading for definition of its optional formal y argument along with illustrative examples of the procedure calls.

In[63]:= GW[x_, y___ /; TrueListQ[{y}, {Integer, Real, Rational},

agn]] := Module[{a = 78}, x*(y + a)]

In[64]:= GW[77]

Out[64]= 6006

In[65]:= GW[77, 78, 7.8, 7/8]

Out[65]= 12680.

In[66]:= {GW[77, 78, 42, 47], agn}

Out[66]= {GW[77, 78, 42, 47], {2, 3}}

As some instructive information, an example of a procedure whose heading is based on the principles mentioned above is presented. The procedure call Mn[x, y, z] returns the list of all arguments passed to the procedure without reference to them.

In[3349]:= Mn[x_ /; {WriteLine["#", x]; IntegerQ[x]}[[1]],

y_ /; {WriteLine["#", y]; ListQ[y]}[[1]],

z___ /; {WriteLine["#", z]; True}[[1]]] :=

Module[{Arg}, Close["#"]; Arg = ReadString["#"]; DeleteFile["#"];

Arg = StringReplace[Arg, "\n" –> ","];

Arg = StringTake[Arg, {1, –2}];

Arg = ToExpression[StringToList[Arg]]; Arg]

In[3350]:= Mn[77, {42, 47, 67}]

Out[3350]= {77, {42, 47, 67}}

In[3351]:= Mn[77, {42, 47, 67}, a + b]

Out[3351]= {77, {42, 47, 67}, a + b}

In[3352]:= Mn[m, {42, 47, 67}, a + b]

Out[3352]= Mn[m, {42, 47, 67}, a + b]

In[3353]:= Mn[m, {42, 47, 67}, a + b, 77, 78]

Out[3353]= Mn[m, {42, 47, 67}, a + b, 77, 78]

Another point should be emphasized. As already mentioned, when calling an object (block, function or module), the factual arguments passed to it in its heading are processed in the list order, i.e., from left to right in the list of formal arguments. In this regard, and in view of the foregoing, there is a rather real possibility of setting in definition of some formal argument of a test function that can be used as test functions for subsequent formal arguments of the object heading. The following fragment represents a rather simple Av function whose heading contains definitions of two Boolean functions, used by a testing function for the last formal z argument of the Av function. At the same time, when you call function Av, both Boolean functions B and V become active in the current session, as it is well shown in the examples below.

In[3442]:= Av[x_ /; {SetDelayed[B[t_], If[t <= 90, True, False]];

NumberQ[x]}[[1]],

y_ /; {SetDelayed[V[t_], ! RealQ[t]]; RealQ[y]}[[1]],

z_ /; B[z] && V[z]||RationalQ[z]] := N[Sqrt[x*y/z]]

In[3443]:= {Av[77, 7.8`, 42/47], Av[77, 90, 500]}

Out[3443]= {25.9249, Av[77, 90, 500]}

In[3444]:= Definition[B]

Out[3444]= B[t_] := If[t <= 90, True, False]

In[3445]:= Definition[V]

Out[3445]= V[t_] := ! RealQ[t]

In[3446]:= {B[42], V[47]}

Out[3446]= {True, True}

In[3447]:= Ag[x_ /; {SetDelayed[W[t_], Module[{a = 77}, a*t^2]];

IntegerQ[x]}[[1]]] := W[x] + x^2

In[3448]:= {Ag[42], W[42]}

Out[3448]= {137592, 135828}

In[3449]:= Definition[W]

Out[3449]= W[t_] := Module[{a = 77}, a*t^2]

Thus, in particular, the last example of the previous fragment also shows that the object W (block, function, module) defined in the header of another Ag object allows a correct call in the body of the Ag object yet when it is first called. When Ag is called, W becomes active in the current session. The above fragments are quite transparent and do not require any clarifying. Technique, presented in them are of some practical interest in programming of the objects headings and objects as a whole.

On the assumption of the general definition of a procedure, in particular, of modular type

M[x_ /; Test_x, y_ /; Test_y, ...] := Module[{Locals}, Procedure body]

and of the fact that concrete definition of procedure is identified not by its name, but its heading we will consider a set of useful enough tools [16] that provide the various manipulations with the headings of procedures and functions and play a important part in the procedural and functional programming and, above all, of programming of problems of the system character.

Having defined such object useful in many appendices as the heading of a procedure or function in the form "Name[List of formal arguments with the testing tools ascribed to them]", naturally arises the question of creating of means for testing of the objects regarding their relation to the `Heading` type. We have created a number of tools [15,16] in the form of procedures HeadingQ ÷ HeadingQ3. At the same time, between pairs of the procedures {HeadingQ, HeadingQ1} and {HeadingQ2, HeadingQ3} principal distinctions exist. Meantime, considering standard along with some other unlikely encoding formats of the procedures and functions headings, the above four tools can be considered as rather useful testing tools in programming. In addition, from experience of their use and their time characteristics it became clear that it is quite enough to be limited oneself only by tools HeadingQ and HeadingQ1 which cover a rather wide range of erroneous coding of headings. Meantime, taking into account the mechanism of expressions parse for their correctness, that Mathematica uses, creation of comprehensive means of testing of the headings is a rather difficult. Naturally, it is possible to use the non–standard receptions for receiving the testing tools for the headings having a rather wide set of the deviations from the standard ones, but such outlay often do not pay off by the received benefits.

In particular, the possibilities of the HeadingQ ÷ HeadingQ3 procedures are overlapped by opportunities of the means whose call TestHeadingQ returns True if a x represents the heading in

string format of a block, module, function, and False otherwise. Whereas the procedure call TestHeadingQ[x, y] with the second optional y argument – an indefinite variable – thru it additionally returns information specifying the reasons of the incorrectness of a heading x. At that, on x objects of the same name (multiple objects) the procedure call returns $Failed; the extension of the procedure to case of the multiple objects does not cause much difficulty. The source code of the TestHeadingQ procedure and the comparative analysis of its use concerning the HeadingQ ÷ HeadingQ3 tools say about its preference at programming in Mathematica [8,9,15]. So, of the detailed analysis follows, that the TestHeadingQ procedure on the opportunities surpasses the above testing tools of the same purpose.

In the light of real correctness of formal arguments that has been defined above and is caused by mutual arrangements of formal arguments in the context of patterns {"_", "__", "___"} we can significantly generalize the above group of procedures HeadingQ ÷ HeadingQ3, intended for testing of correctness of headings of the user blocks, functions & modules, additionally using the CorrTupleArgs procedure. For this purpose the tool HeadingsUQ can be used.

Calling the HeadingsUQ[x] procedure returns True if all components composing an object x in the context of the call Definition[x] have correct headings, and False otherwise. While the call HeadingsUQ[x, y] in addition through the 2^nd optional argument – an undefined y symbol – returns the list the elements of which are True if the heading of the appropriate component of x object is correct, or the list whose first element determines incorrect heading of a component of the x object, whereas the second element determines of the component number with this heading concerning the call Definition[x]. The source code of the TestHeadingQ procedure with examples of its use can be found in [15,16]. The HeadingsUQ procedure rather exhaustively tests the correctness of the headings of blocks, functions and modules.

It is appropriate to make one quite substantial remark here. As you know, Mathematica does not allow to use the testing functions in block, function, and module headers that will use factual argument values that relate to other formal arguments as the following simple fragment rather visually illustrates.

In[7]:= V[x_ /; IntegerQ[x], y_ /; EvenQ[x] && IntegerQ[y]] := x/y

In[8]:= V[42, 77]

Out[8]= V[42, 77]

In the example shown, the testing function assigned to the second formal argument y cannot include the value passed to the first formal argument x, otherwise by causing the return of V function call unevaluated. We will offer here one interesting reception, allowing to solve similar and some other problems. Consider it with a rather simple example.

In[2296]:= V[x_ /; {Save3["j", x], IntegerQ[x]}[[–1]],

y_ /; {If[Get["j"]*y > 100, True, False], DeleteFile["j"]}[[1]]] := x*y

In[2297]:= V[42, 77]

Out[2297]= 3234

In[2298]:= V[7, 8]

Out[2298]= V[7, 8]

In order to pass an actual value (passed to the x argument when the V function is called) to a testing function of another y argument, by the function Save3 the value is stored in a сertain temporary j file. Then call Get[j] allows to read the stored value from the file j in the required point of the testing function of any formal argument of the header, after necessity deleting file j. To solve such problems, a Save3 function is of interest – a version of the built-in Save function whose Save3[f, x] call saves in the f of the txt–format an expression x without returning anything.

In[2310]:= Save3[f_, x_] := {Write[f, x], Close[f]}[[1]]

In[2311]:= Save3["###", b*S + c*V + G*x]

In[2312]:= Get["##"]

Out[2312]= b*S + c*V + G*x

After calling Save3[f, x], the file with expression x remains closed and is loaded into the current session by function Get[f], returning the x value. Note that the Write function used by the Save3 has a number of features discussed in detail in [8]. In the first place, this refers to preserving the sequence of expressions.

The following procedure serves as an useful enough tool at manipulating with functions and procedures, its call HeadPF[x] returns heading in string format of a block, module or function with a x name activated in the current session, i.e. the function in its traditional understanding with heading. Whereas on other values of the x argument the call is returned unevaluated. At the same time, the problem of definition of headings is actual also in a case of the objects of the above type of the same name which have more than one heading. In this case the procedure call HeadPF[w] returns the list of headings in string format of the subobjects composing a w object as a whole. The following fragment represents the source code of the HeadPF procedure along with a typical example of its application.

In[2225]:= M[x_ /; x == "avzagn"] := Module[{a}, a*x];

M[x_, y_, z_] := x*y*z; M[x_String] := x;

M[x_ /; IntegerQ[x], y_String] := Module[{a, b, c}, x];

M[x_, y_] := Module[{a, b, c}, "abc"; x + y]

In[2226]:= HeadPF[x_ /; BlockFuncModQ[x]] := Module[{b,

c = ToString[x], a = Select[Flatten[{PureDefinition[x]}],

! SuffPref[#, "Default[", 1] &]}, b = Map[StringTake[#,

{1, Flatten[StringPosition[#, {" := ", " = "}]][[1]] – 1}] &, a];

If[Length[b] == 1, b[[1]], b]]

In[2227]:= HeadPF[M]

Out[2227]= {"M[x_ /; x == \"avzagn\"]", "M[x_, y_, z_]",

"M[x_ /; IntegerQ[x], y_String]", "M[x_String]", "M[x_, y_]"}

Here, it is necessary to make one essential enough remark concerning the means dealing with the headings of procedures and functions. The matter is that as it was shown earlier, the testing tools ascribed to the formal arguments, as a rule, consist of the earlier defined testing functions or the reserved keywords defining the type, for example, Integer. While as a side effect the built–in Math–language allows to use the constructions that can contain definitions of the testing tools, directly in the headings of procedures and functions. Certain interesting examples of that have been represented above. Meanwhile, despite of such opportunity, the programmed means are oriented, as a rule, on use for testing of admissibility of the factual arguments or the predefined means, or including the testing algorithms in the body of the procedures and functions. For this reason, HeadPF procedure presented above in such cases can return incorrect results. Whereas the HeadPFU procedure covers such peculiar cases. The procedure call HeadPFU[x] correctly returns the heading in string format of a block, module or function with a name x activated in the current session regardless of a way of definition of testing of factual arguments. The fragment below presents the source code of the HeadPFU procedure along with typical examples of its application.

In[2230]:= P[x_, y_ /; {IntOddQ[t_] := IntegerQ[t] && OddQ[t],

IntOddQ[y]}[[–1]]] := x*y

In[2231]:= HeadPF[P]

Out[2231]= "P[x_, y_ /; {IntOddQ[t_]"

In[2232]:= HeadPFU[x_ /; BlockFuncModQ[x]] :=

Module[{a = Flatten[{PureDefinition[x]}], b = {}, c, g},

Do[c = Map[#[[1]] – 1 &, StringPosition[a[[j]], " := "]];

Do[If[SyntaxQ[Set[g, StringTake[a[[j]], {1, c[[k]]}]]],

AppendTo[b, g]; Break[], Null], {k, Length[c]}], {j, Length[a]}];

If[Length[b] == 1, b[[1]], b]]

In[2233]:= HeadPFU[P]

Out[2233]= "P[x_, y_ /; {IntOddQ[t_] := IntegerQ[t] && OddQ[t],

IntOddQ[y]}[[–1]]]"

In[2234]:= V[x_ /; {Save3["#", x], IntegerQ[x]}[[–1]],

y_ /; {If[Get["#"]*y > 77, True, False], DeleteFile["#"]}[[1]]] := x*y

In[2235]:= HeadPFU[V]

Out[2235]= "V[x_ /; {Save3[\"#\", x], IntegerQ[x]}[[–1]],

y_ /; {If[Get[\"#\"]*y > 77, True, False], DeleteFile[\"#\"]}[[1]]]"

In this regard the testing of a x object regarding to be of the same name is enough actually; the QmultiplePF procedure [16] solves the problem whose call QmultiplePF[x] returns True, if x is an object of the same name (Function, Block, Module), and False otherwise. Whereas the procedure call QmultiplePF[x, y] with the second optional y argument – an indefinite variable – through y returns the list of definitions of all sub–objects with a name x.

Right there pertinently to note some more tools linked with

the HeadPF procedure. So, the Headings procedure is a rather useful expansion of the HeadPF procedure in a case of blocks, functions and modules of the same name but with the different headings. Generally, the call Headings[x] returns the nested list whose elements are the sub–lists which determine respectively headings of sub-objects composing an object x; the first elements of such sub-lists defines the types of sub-objects, whereas others define the headings corresponding to them. In a number of the appendices that widely use procedural programming, a rather useful is the HeadingsPF procedure that is an expansion of the previous procedure. Generally, the call HeadingsPF[] returns the nested list, whose elements are sub–lists that respectively define the headings of functions, blocks and modules, whose definitions have been evaluated in the current session; the first element of each such sub–list determines an object type in the context {"Block", "Module", "Function"} while the others define the headings corresponding to it. The procedure call returns a simple list if any of sub–lists does not contain headings; at that, if in the current session the evaluations of definitions of objects of the specified three types weren't made, the call returns empty list. The call with any arguments is returned unevaluated. The following fragment represents source code of the HeadingsPF procedure along with an example of its typical application.

In[2310]:= HeadingsPF[x___ /; SameQ[x, {}]] := Module[{a = {},

d = {}, k = 1, b, c = {{"Block"}, {"Function"}, {"Module"}}, t},

Map[If[Quiet[Check[BlockFuncModQ[#], False]],

AppendTo[a, #], Null] &, Names["`*"]];

b = Map[Headings[#] &, a]; While[k <= Length[b], t = b[[k]];

If[NestListQ[t], d = Join[d, t], AppendTo[d, t]]; k++];

Map[If[#[[1]] == "Block", c[[1]] = Join[c[[1]], #[[2 ;–1]]],

If[#[[1]] == "Function", c[[2]] = Join[c[[2]], #[[2 ;–1]]],

c[[3]] = Join[c[[3]], #[[2 ;–1]]]]] &, d];

c = Select[c, Length[#] > 1 &]; If[Length[c] == 1, c[[1]], c]]

In[2311]:= M[x_ /; SameQ[x, "avz"], y_] := Module[{a, b, c}, y];

L1[x_] := x; M[x_, y_, z_] := x + y + z; L[x_, y_] := x + y;

M[x_ /; x == "avz"] := Module[{a, b, c}, x];

M[x_ /; IntegerQ[x], y_String] := Module[{a, b, c}, x];

M[x_, y_] := Module[{a, b, c}, "agn"; x + y]; M[x_String] := x;

M[x_ /; ListQ[x], y_] := Block[{a, b, c}, "agn"; Length[x] + y];

F[x_ /; SameQ[x, "avz"], y_] := {x, y}; F[x_ /; x == "avz"] := x

In[2312]:= HeadingsPF[]

Out[2312]= {{"Block", "M[x_ /; ListQ[x], y_]"},

{"Function", "F[x_ /; x === \"avz\", y_]", "F[x_ /; x == \"avz\"]",

"L[x_, y_]", "L1[x_]", "M[x_, y_, z_]", "M[x_String]"},

{"Module", "M[x_ /; x === \"avz\", y_]", "M[x_, y_]",

"M[x_ /; x == \"avz\"]", "M[x_ /; IntegerQ[x], y_String]"}}

In the light of possibility of existence in the current session of procedures and functions of the same name with different headings the problem of removal from the session of an object with the concrete heading is of a certain interest; this problem is solved by the procedure RemProcOnHead, whose the source code with an example of its application is represented below.

In[7]:= RemProcOnHead[x_ /; HeadingQ[x]||HeadingQ1[x]||

ListQ[x] && AllTrue[Map[HeadingQ[#] &, x], TrueQ]] :=

Module[{b, c, d, p, a = HeadName[If[ListQ[x], x[[1]], x]]},

If[! MemberQ[Names["`*"] || ! HowAct[a], a], $Failed,

b = Definition2[a]; c = b[[1 ;–2]]; d = b[[–1]];

ToExpression["ClearAttributes["<>a<> ","<>ToString[d]<>"]"];

y = Map[StandHead, Flatten[{x}]];

p = Select[c, ! SuffPref[#, x, 1] &];

ToExpression["Clear[" <> a <> "]"];

If[p == {}, "Done", ToExpression[p];

ToExpression["SetAttributes["<>a <> "," <> ToString[d] <> "]"];

"Done"]]]

In[8]:= V[x_] := Module[{}, x^6]; V[x_Integer] := x^2

In[9]:= {RemProcOnHead["V[x_Integer]"],

RemProcOnHead["V[x_]"]}

Out[9]= {"Done", "Done"}

In[10]:= Definition[V]

Out[10]= Null

The call RemProcOnHead[x] returns "Done" with removing from the current session a procedure, function or their list with heading or with list of x headings that are given in string format; headings in call RemProcOnHead[x] is coded according to the format ToString1[x].

The task, closely adjacent to the headings, is to determine the arity of objects. For this purpose, we have created a number of means [8,9]. In particular, the ArityBFM1 procedure defining arity of objects of the type {module, classical function, block} serves as quite useful addition to the procedure Arity and some others.

The procedure call ArityBFM1[x] returns the arity (number of the formal arguments) of a block, function or module x in the format of 4–element list whose the 1^st, the 2^nd, the 3^rd elements define quantity of formal arguments with the patterns "_", "__" and "___" accordingly. While the 4^th element defines common quantity of formal arguments which can be different from the quantity of actual arguments. In addition, the x heading should has classical type, i.e. it should not include non–standard, but acceptable methods of headings definition. The unsuccessful procedure call returns $Failed.

The following procedure serves for definition of arity of system functions. The procedure call AritySystemFunction[x] returns the 2–element list whose 1^st element defines number of formal arguments, while the second element – quantity of admissible actual arguments. The call with the second optional y argument – an indefinite variable – thru it returns the generalized template of formal arguments.

In[3341]:= AritySystemFunction[x_, y___] := Module[{a, b, c, d, g, p},

If[! SysFunctionQ[x], $Failed, a = SyntaxInformation[x];

If[a == {}, $Failed, c = {0, 0, 0, 0}; b = Map[ToString, a[[1]][[2]]];

b = Map[If[SuffPref[#, "{", 1], "_",

If[SuffPref[#, {"Optional", "OptionsPattern"}, 1], "_.", #]] &, b];

If[{y} != {} && ! HowAct[y],

y = Map[ToExpression[ToString[Unique["x"]] <> #] &, b], 77];

Map[If[# == "_", c[[1]]++, If[# == "_.", c[[2]]++,

If[# == "__", c[[3]]++, c[[4]]++]]] &, b]; {p, c, d, g} = c;

d = DeleteDuplicates[{p + d, If[d != 0||g != 0, Infinity, p + c]}];

If[d != {0, Infinity}, d, Quiet[x[Null]]; If[Set[p, Messages[x]] == {}, d,

p = Select[Map[ToString, Map[Part[#, 2] &, p]],

! StringFreeQ[#, " expected"] &]; p = Map[StringTake[#,

{Flatten[StringPosition[#, ";"]][[1]] + 1, –2}] &, p];

p = ToExpression[Flatten[StringCases[p, DigitCharacter ..]]];

If[p == {}, {1}, If[Length[p] > 1, {Min[p], Max[p]}, p]]]]]]]

In[3342]:= {AritySystemFunction[If, g72], g72}

Out[3342]= {{2, 4}, {x16_, x17_, x18_., x19_.}}

2.3. Optional arguments of procedures and functions

When programming procedures and functions often there is an expediency to define some of their formal arguments as optional. In addition, in case of explicit lack of such arguments by a call of tools, they receive values by default. System built–in Mathematica functions use two basic methods for dealing with optional arguments that are suitable for the user tools too. The first method consists in that that a value of each argument x that is coded by pattern "_:" in the form x_:b should be replaced by b, if the argument omitted. Almost all built–in system functions that use this method, omit arguments since the end of the list of formal arguments. So, simple function G[x_: 5, y_: 7] := {x, y} two optional arguments have, for which values by default – 5 and 7 respectively. As Mathematica system assumes that arguments are omitted since end of the list of formal arguments then we have not a possibility by positionally (in general selectively) do an argument as optional.

In[3376]:= G[x_: 5, y_: 6] := {x, y}

In[3377]:= {G[], G[42, 77], G[42]}

Out[3377]= {{5, 6}, {42, 77}, {42, 6}}

So, from a simple example follows that for function G from two optional arguments the call G[42] refers value by default to the second argument, but not to the first. Therefore this method has a rather limited field of application.

The second method of organization of optional arguments consists in use of explicit names for the optional arguments by allowing to give them values using transformation rules. This method is particularly convenient for means like Plot function which have a rather large number of optional arguments, only a few of which usually need to be set in a particular example. The typical method is that values for so–called named optional arguments can be specified by including the appropriate rules at the end of the arguments at a function call. At determining of the named optional arguments for a procedure or function g, it is conventional to store the default values for these arguments as a list of transformation rules assigned to Options[g]. So, as a typical example of definition for a tool with zero or more named optional arguments can be:

g[x_, y_, OptionsPattern[]] := Module[{}, x*y]

Then for this means a list of transformation rules assigned to Options[g] is determined

Options[g] = {val1 –> a, val2 –> b}

Whereas expression OptionValue[w] for a named optional argument w has to be included to the body of the tool, in order that this mechanism worked. Below is presented the definition of a function g that allows up to 2 named optional arguments:

In[2114]:= g[x_, y_, OptionsPattern[]] :=

(OptionValue[val1]*x + OptionValue[val2]*y)/

(OptionValue[val1]*OptionValue[val2])

In[2115]:= Options[g] = {val1 –> 42, val2 –> 47}

Out[2115]= {val1 –> 42, val2 –> 47}

In[2116]:= g[90, 500]

Out[2116]= 13640/987

In[2117]:= g[90, 500, val1 –> 77]

Out[2117]= 30430/3619

In[2118]:= g[90, 500, val1 –> 10, val2 –> 100]

Out[2118]= 509/10

From the represented example the second method of work with named optional arguments is rather transparent. You can to familiarize with optional arguments in more detail in [22].

Other way of definition of optional arguments of functions and procedures consists in use of the template "_.", representing an optional argument with a default value specified by built–in Default function – Default[w] = Value. Let's note, the necessary values for call Default[w] for a tool w must always precede an optional argument of the w tool. Values defined for Default[w] are saved in DefaultValues[w]. Fragment below illustrates told:

In[1346]:= Default[w] = 5;

In[1347]:= w[x_., y_., z_.] := {x, y, z}; {w[], w[42, 47], w[42]}

Out[1347]= {{5, 5, 5}, {42, 47, 5}, {42, 5, 5}}

More detailed information on this way can be found in [15] and by a call ?Default in the current session of Mathematica. At the same time all considered approaches to the organization of work with optional arguments do not give the chance to define values by default for an arbitrary tuple of formal arguments.

The reception described below on a simple example can be for this purpose used. Definition of a procedure G for which we going to make all three formal arguments x, y and z optional, is made out as follows: Res defines the list of formal arguments in string format, Def determines the list of values by default for all formal arguments, the procedure body is made out in the form "{Body}", moreover, of the G definition is complemented with an obligatory argument Opt which represents the binary list of signs whose elements correspond to Res list elements. The zero element of the Opt list determines non-obligation of the Res list element corresponding to it, and vice versa. After the specified definition of the procedure body (d) the StringReplaceVars [16] procedure which provides substitution in the procedure body of the actual arguments (sign 1) or their values by default (sign 0) is applied to it. So, the procedure call G[x, y, z, Opt] returns the simplified value of the source procedure body on the arguments transferred to it. The simple example visually illustrates told.

In[7]:= G[x_., y_., z_., Opt_] := Module[{Res = {"x", "y", "z"},

Def = Map[ToString1[#] &, {42, 47, 67}],

args = Map[ToString1[#] &, {x, y, z}], b, c, d},

d = "Simplify[{b = (x + y)/(x + z); c = b*x*y*z; (b + c)^2}]";

d = StringReplaceVars[d, Map[If[Opt[[#]] == 0, Res[[#]] –>

Def[[#]], Res[[#]] –> args[[#]]] &, Range[Length[Res]]]];

ToExpression[d][[1]]]

In[8]:= G[a, s, t, {0, 1, 1}]

Out[8]= ((42 + s)^2*(1 + 42*s*t)^2)/(42 + t)^2

In[9]:= G[a, s, t, {1, 1, 1}]

Out[9]= ((a + s)^2*(1 + a*s*t)^2)/(a + t)^2

In[10]:= G[a, s, t, {0, 0, 0}]

Out[10]= 138557641644601/11881

In certain cases the above reception can be rather useful.

The problem for testing of correctness of the tuple of formal arguments in the user modules, blocks or functions can be solved by procedure whose call CorrTupleArgs[x] returns True if tuple of formal arguments of a procedure or a function x is correct in the sense stated above, and False otherwise [8,16].

On the other hand, the problem of preliminary definition of correctness of the factual arguments passing to a procedure or a function is of a rather interesting. If tool has no mechanisms of testing of actual arguments passing to it at a call, then the call on incorrect actual arguments as a rule is returned unevaluated. The procedure TestActArgs is one of variants of solution of the problem, whose call TestActArgs[x, y] returns empty list if tuple of actual arguments meets the tuple of testing functions of the corresponding formal arguments of x, otherwise the procedure call trough y returns the integer list of numbers of the formal arguments to which inadmissible actual arguments are supposed to be passed [1-16].

Meantime, use of the above procedure assumes that testing for correctness of actual arguments for formal arguments with patterns {"__", "___"} is carried out only for the first elements of tuples of actual arguments passing to formal arguments with the appropriate patterns. Whereas further development of the procedure is left to the interested reader as an useful exercise. In addition, it should be noted the important circumstance. If in the traditional programming languages the identification of an arbitrary procedure or function is made according to its name, in case of Math–language the identification is made according to its heading. This circumstance is caused by that the definition of a procedure or function in the Math-language is made by the manner different from traditional [1]. Simultaneous existence of procedures of the same name with different headings in such situation is admissible as it illustrates a simple fragment:

In[2434]:= M[x_, y_] := Module[{}, x + y];

M[x_] := Module[{}, x^2]; M[y_] := Module[{}, y^3];

M[x___] := Module[{}, {x}]

In[2435]:= Definition[M]

Out[2435]= M[x_, y_] := Module[{}, x + y]

M[y_] := Module[{}, y^3]

M[x___] := Module[{}, {x}]

At the call of procedure or function of the same name from the list is chosen the one, whose formal arguments of heading correspond to actual arguments of the call, otherwise the call is returned unevaluated, excepting simplifications of arguments according to the standard system agreements. At compliance of formal arguments of heading with actual ones the component of x procedure or function is caused, whose definition is the first in the list returned at the call Definition[x]; particularly, whose definition has been evaluated in the Mathematica by the first. Therefore, in order to avoid possible misunderstandings in case of procedure override with a name x, which includes a change of its header also, the call Clear[x] must first undo the previous procedure definition.

The works [1-16] present a range of interesting software for dealing with the headings and formal arguments that make up them. In particular, the following tools can be mentioned:

ProcQ[x] – returns True if x is a procedure and False otherwise;

BlockModQ[x] – returns True if x is a name defining a block or module; otherwise False is returned. At that, thru optional argument y – an indefinite variable the call BlockModQ[x, y] returns the type of the object x in the context of {"Block", "Module"} on condition that main result is True;

ArgsP[x] – returns the list whose elements – formal arguments of Block or Module x – are grouped with the preceding words Block and Module. Groups arise in the case of multiple object definition x;

In[72]:= ArgsP[x_ /; BlockModQ[x]] := Module[{b, c = {}, j,

a = Headings[x]},

a = If[MemberQ3[a, {"Module", "Block"}], a, {a}];

b = Length[a];

Do[Map[AppendTo[c, If[StringFreeQ[#, "["], #,

StringReplace[StringTake[#, StringLength[#] – 1],

ToString[x] <> "[" –> "{"] <> "}"]] &, a[[j]]], {j, 1, b}];

ToExpression[c]]

In[73]:= M[x_ /; IntegerQ[x]] := Module[{}, x^2];

M[x_, y_] := Module[{}, x*y^3]; M[x__] := Module[{}, {x}];

M[x_, y__] := Block[{a = 42, b = 47}, {a*x, b*y}]

In[74]:= ArgsP[M]

Out[74]= {Block, {x_, y__}, Module, {x_ /; IntegerQ[x]},

{x_, y_}, {x__}}

RenameP[x, y] – returns the empty list, renaming a Function, Block or Module x to a name, defined by the second argument y with its activation in the current session; note, object x remains unchanged

In[32]:= RenameP[x_ /; BlockFuncModQ[x], y_Symbol] :=

ReplaceAll[ToExpression[Map[StringReplace[#,

ToString[x] <> "[" –> ToString[y] <> "[", 1] &,

DefToStrM[x]]], Null –> Nothing]

In[33]:= M[x_ /; IntegerQ[x]] := Module[{}, x^2];

M[x_, y_] := Module[{}, x*y^3]; M[x__] := Module[{}, {x}];

M[x_, y__] := Block[{}, {x, y}]

In[34]:= RenameP[M, Agn]

Out[34]= {}

In[35]:= Definition[Agn]

Out[35]= Agn[x_ /; IntegerQ[x]] := Module[{}, x^2]

Agn[x_, y_] := Module[{}, x*y^3]

Agn[x_, y__] := Block[{}, {x, y}]

Agn[x__] := Module[{}, {x}]

Note that we have programmed a lot of other useful tools for different processing of both the headings, as a whole, and their separate components, first of all, formal arguments [1-16]. Some of these are used in examples given in this book. Before further consideration, it should be noted once again that, the block and module headings handling tools are also functionally suitable for the classical functions determined by the headings according to a general mechanism used by the blocks, modules, and classical functions. Thus, what is said about the procedure headings fully applies to functions, making it easy to convert the second ones to the first.

One of the main tasks at working with formal arguments is

to define them for the user procedures (modules and blocks) and functions. In this direction, we have created a number of means of various purposes. At the same time, the definition of testing functions for formal arguments may itself contain definitions of procedures and functions, which implies consideration is given to this circumstance in the development of means of processing formal arguments. So, the ArgsProcFunc procedure considers this circumstance illustrating it on an example of its application to the M procedure presented below.

The procedure call ArgsProcFunc[x] generally returns the nested list whose single elements define formal arguments in the string format, and 2–element sub–lists by the first element define formal arguments, while by the second define the testing functions assigned to them (conditions of their admissibility) also in the string format of a procedure, block or function x.

In[1938]:= ArgsProcFunc[x_ /; BlockModQ[x]] :=

Module[{a = Definition1[x], b, c = "", d, j, f, g},

d = StringLength[g = ToString[x]]; c = g;

For[j = d + 1, j <= StringLength[a], j++,

If[! SyntaxQ[g = g <> StringTake[a, {j, j}]], Continue[], Break[]]];

b = StringReplace[g, c <> "[" –> ",", 1];

b = StringInsert[b, ",", –1]; c = "";

Do[c = c <> StringTake[b, {j, j}]; If[SyntaxQ[c], Break[],

Continue[]], {j, 1, StringLength[b]}];

d = StringDrop[c, –2] <> ","; c = {};

Label[Agn]; f = StringPosition[d, ","];

f = DeleteDuplicates[Flatten[f]];

For[j = 2, j <= Length[f], j++,

If[SyntaxQ[g = StringTake[d, {f[[1]] + 1, f[[j]] – 1}]],

AppendTo[c, g]; d = StringReplace[d, "," <> g –> "", 1];

Goto[Agn], Continue[]]];

c = Map[StringReplace[#, GenRules[{"___", "__", " /;", "_:",

"_.", "_"}, "@$@"], 1] &, c];

c = Map[StringTrim[#] &, Map[StringSplit[#, "@$@"] &, c]];

Map[If[Length[#] == 1, #[[1]], If[SuffPref[#[[2]], g = "/; ", 1],

{#[[1]], StringReplace[#[[2]], g –> "", 1]}, #]] &, c]]

In[1939]:= M[x_, y_ /; {g[t_] := Module[{}, t^2], If[g[y] > 50,

True, False]}[[2]], r_: 77, p_., z___] :=

Module[{}, x*y*r*p/g[z]]

In[1940]:= {M[77, 8, 42, 72, 7], M[77, 7, 42, 72, 7]}

Out[1940]= {38016, M[77, 7, 42, 72, 7]}

In[1941]:= Definition[g]

Out[1941]= g[t_] := Module[{}, t^2]

In[1942]:= ArgsProcFunc[M]

Out[1942]= {"x", {"y", "{g[t_] := Module[{}, t^2], If[g[y] > 50,

True, False]}[[2]]"}, {"r", "77"}, "p", "z"}

It follows from the fragment that for a tool defined in the testing function of a formal argument, the availability domain is the entire current session, including the procedure body that generated it. ArgsProcFunc1 and ArgsProcFunc2 procedures, unlike ArgsProcFunc, are based on other principles, returning the lists of formal arguments with testing functions ascribed to them, while through the second optional argument is returned the same list with elements in the string format.

In[19]:= ArgsProcFunc1[x_ /; BlockFuncModQ[x], y___] :=

Module[{a = Definition1[x], b, c = "", d, j, f, g},

d = StringLength[g = ToString[x]]; c = g;

For[j = d + 1, j <= StringLength[a], j++,

If[! SyntaxQ[g = g <> StringTake[a, {j, j}]], Continue[], Break[]]];

b = ToExpression["{" <> StringTrim[g, {c <> "[", "]"}] <> "}"];

If[{y} == {}, b, If[SymbolQ[y], b; y = Map[ToString1, b]]]]

In[20]:= ArgsProcFunc1[M, t]

Out[20]= {"x_", "y_ /; {g[t_] := Module[{}, t^2], If[g[y] > 50,

True, False]}[[2]]", "r_:77", "p_.", "z___"}

In[77]:= ArgsProcFunc2[x_ /; BlockFuncModQ[x], y___] :=

Module[{a = ToExpression[HeadPFU[x]], b = {}, c, k},

Do[c = Quiet[Check[Part[a, k], Error, Part::partw]];

If[c === Error, Return[If[{y} == {}, b, If[SymbolQ[y], b;

y = Map[ToString1, b], b]]], AppendTo[b, c]], {k, Infinity}]]

In[78]:= ArgsProcFunc2[M]

Out[78]= {x_, y_ /; {g[t_] := Module[{}, t^2], If[g[y] > 50,

True, False]}[[2]], r_ : 77, p_., z___}

The question of processing of the formal arguments with good reason can be considered as the first problem relating to the calculations of the tuples of formal arguments of the user functions, modules and blocks that have been activated in the current session directly or on the basis of download of packages containing their definitions. In our works [1-15] certain tools for the solution of this problem have been programmed in the form of procedures Args, Args0 ÷ Args2, then we presented similar tools in narrower assortment and with the improved some functional characteristics. Above all, as a rather useful tool, we present the Args3 procedure whose call Args3[x] returns the list of formal arguments of the user module, block or function x. At the same time, the argument x can present an multiple object of the same name. The following fragment represents the source code of the Args3 procedure with the most typical examples of its use.

In[3122]:= M[x_ /; SameQ[x, "avz"], y_] := Module[{a, b, c}, y];

M[x_ /; x == "avz"] := Module[{a, b, c}, x];

M[x_ /; IntegerQ[x], y_String] := Module[{a, b, c}, x];

M[x_, y_] := Module[{a, b, c}, "agn"; x + y]; M[x_String] := x;

M[x_ /; ListQ[x], y_] := Block[{a, b, c}, "agn"; Length[x] + y];

F[x_ /; SameQ[x, "avz"], y_] := {x, y}

In[3123]:= Args3[s_ /; BlockFuncModQ[s]] :=

Module[{a = Headings[s], b, c = 7},

If[SimpleListQ[a], a = {a}, Set[c, 77]]; a = Map[#[[2]] &, a];

b = Map[StringReplace[#, ToString[s] –> "", 1] &, a];

b = Map[StringToList[StringTake[#, {2, –2}]] &, b];

If[c == 77, b, Flatten[b, 1]]]

In[3124]:= Args3[M]

Out[3124]= {{"x_ /; ListQ[x]", "y_"}, {"x_", "y_", "z_"},

{"x_ /; x === \"avz\"", "y_"}}

In[3125]:= Args3[F]

Out[3125]= {"x_ /; x === \"avz\"", "y_"}

In[3126]:= Args3[L]

Out[3126]= {"x_", "y_"}

At that, in case of the returned nested list its sub-lists define tuples of formal arguments in the order that is defined by order of components of x object according to the call Definition[x].

The following procedure is an useful modification of our argument processing means. The call ArgsBFM1[x] generally returns the list of the ListList type whose two-element sub–lists in string format determine the name of a formal argument (1^st element) and its admissibility test (2^nd element) of block, function or module x. Lack of the test is coded as "Arbitrary"; in addition, under lack of test is understood not only its actual lack, but also the optional or default patterns ascribed to a formal argument. The procedure successfully processes the objects of the same name too, i.e. objects with several headings. The next fragment represents source code of the ArgsBFM1 procedure with typical examples of its application.

In[3336]:= ArgsBFM1[x_ /; BlockFuncModQ[x]] :=

Module[{b, c, d, g = {}, t, a = ToString2[Args[x]]},

a = If[QmultiplePF[x], a, {a}];

Do[{b, c, d} = {{}, {}, {}}; t = a[[j]];

Do[AppendTo[b, If[ListQ[t[[k]]],

Map[StringSplit[#, {"/;", "__", "___", "_", ":", "_:", "_."}] &, t[[k]]],

StringSplit[t[[k]], {"/;", "__", "___", "_", ":", "_:", "_."}]]], {k, Length[t]}];

Do[AppendTo[c, If[NestListQ[b[[k]]],

Map[Map[If[#1 == " " || #1 == "", Nothing,

StringTrim[#1]] &, #] &, b[[k]]],

Map[If[# == " "||#1 == "", Nothing, StringTrim[#]] &, b[[k]]]]],

{k, 1, Length[b]}];

Do[AppendTo[d, If[NestListQ[c[[k]]],

Map[If[Length[#1] == 1, {#1[[1]], "Arbitrary"}, #1] &, c[[k]]],

If[Length[c[[k]]] == 1, {c[[k]][[1]], "Arbitrary"}, c[[k]]]]],

{k, 1, Length[c]}]; d = If[Length[d] == 1, d[[1]], d];

AppendTo[g, d], {j, Length[a]}]; If[Length[g] == 1, g[[1]], g]]

In[3337]:= G[x_, y_List, z_ : 77, h___] := Module[{g = 72}, Body];

G[x_, y_List, z_ : 78, {x1_, x2_List, x3_ : 90}, h___] :=

Module[{v = 42, g = 47, a = 87, k = 96, s = 67}, Body]

In[3338]:= ArgsBFM1[G]

Out[3338]= {{{"x", "Arbitrary"}, {"y", "List"}, {"z", "78"},

{{"x1", "Arbitrary"}, {"x2", "List"}, {"x3", "90"}}, {"h", "Arbitrary"}},

{{"x", "Arbitrary"}, {"y", "List"}, {"z", "77"}, {"h", "Arbitrary"}}}

The procedure has a number of useful applications [8,16].

Several useful tools have been created to work with arguments of blocks, functions and modules [16]. One example is the procedure whose call Args2[x] returns the list of formal arguments with testing functions assigned to them in string format for a module, function, or block x. While calling Args2[x, y] with the second optional argument y – an undefined symbol – via it returns the list consisting of 2–element sub–lists whose elements in string format represent arguments and testing functions ascribed to them.

In[111]:= Default[Sg, 6, 42]; Sg[x_, y_Integer, z_ /; EvenQ[z], t__,

h_: 0, g_.] := x*y + z + t + h*g

In[112]:= Args2[x_ /; BlockFuncModQ[x], y___] :=

Module[{a = Definition2[x][[1]], b = "", c, d = {}, t},

a = StringReplace5[a, {ToString[x] <> "[" –> "{", "] := " –> "}"}, 1];

Do[If[SyntaxQ[b = b <> StringTake[a, {j}]], Break[], 7], {j, Infinity}];

b = StringToList[StringTake[b, {2, –2}]];

If[{y} != {} && SymbolQ[y], c = Map[StringSplit[#, "_"] &, b];

y = Map[If[Length[#] == 1, AppendTo[d, #],

If[SuffPref[#[[2]], t = " /; ", 1],

AppendTo[d, {#[[1]], StringReplace[#[[2]], t –> ""]}],

If[SuffPref[#[[2]], ":", 1],

AppendTo[d, {#[[1]], "Optional"}], If[SuffPref[#[[2]], ".", 1],

AppendTo[d, {#[[1]], "Default"}], AppendTo[d, #}]]]]] &, c];

y = DeleteDuplicates[Flatten[y, 1]];

y = Map[If[Length[#] == 1, #[[1]], #] &, y], 7]; b]

In[113]:= Args2[Sg, sv]

Out[113]= {"x_", "y_Integer", "z_ /; EvenQ[z]", "t__", "h_:0", "g_."}

In[114]:= sv

Out[114]= {"x", {"y", "Integer"}, {"z", "EvenQ[z]"}, "t",

{"h", "Optional"}, {"g", "Default"}}

In[115]:= StringReplace5[x_String, y_ /; ListRulesQ[y],

z_Integer] := Module[{a = x}, Do[a = StringReplace[a, y[[j]], z],

{j, Length[y]}]; a]

In[116]:= StringReplace5["12345612345", {"2" –> "a", "4" –> "b",

"c" –> "d"}, 1]

Out[116]= "1a3b5612345"

The call StringReplace5[x, y, z] unlike the StringReplace of the same call format returns result of replacements only the first z entries of left part of each rule from a list y onto its right part in a string x.

It is worth noting that the permissibility of W constructions that include modules, blocks and functions including built–in ones, provided that BooleanQ[W] = True, makes it possible to conveniently include various kinds of diagnostic information in such constructs, as illustrated by the following simple example:

In[3337]:= T[x_ /; If[x > 50, {Print["Invalid first argument:

it must be greater than 50"], False}, True],

y_ /; If[IntegerQ[y], True, Print["Invalid second argument:

it must be an integer"]; False],

z_ /; If[PrimeQ[z], {g[t_] := t^2, True}[[2]], {Print["Invalid

third argument: it must be a prime"], False}]] := g[x*y*z]

In[3338]:= T[100, 7, 7]

Invalid first argument: it must be greater than 50

Out[3338]= T[100, 7, 7]

In[3339]:= T[50, 42/77, 7]

Invalid second argument: it must be an integer

Out[3339]= T[50, 6/11, 7]

In[3340]:= T[50, 77, 4]

Invalid third argument: it must be a prime

Out[3340]= T[50, 77, 4]

In[3341]:= T[50, 77, 7]

Out[3341]= 72630250

In[3342]:= Definition[g]

Out[3342]= g[t_] := t^2

Thus, the testing function for a formal x argument can be an arbitrary expression w[x], for that the ratio BooleanQ[W[x]] = True will be true. Moreover, in addition to printing of diagnostic information, testing expressions w[x] may include definitions of modules, blocks and functions whose definitions are activated only if the actual argument x is valid, making their available in the body of an original procedure or function and in the current session as a whole. This possibility is illustrated in the above function T by the function g activated only at call T[x, y, z] with valid values for the formal arguments {x, y, z}. So, definitions of formal arguments of modules, blocks, functions allow a rather wide range of testing expressions to be used for them.

Here it is necessary to make an essential remark concerning the tools that deal with arguments of procedures and functions. The matter is that as it was shown earlier, testing tools ascribed to the formal arguments, as a rule, consist of the earlier defined testing functions or the reserved key words defining the type, for example, Integer. Whereas as a side effect the built-in Math–language allows to use the constructions that contain definitions of the testing means, directly in headings of the procedures and functions. Interesting examples of that have been represented above. Meanwhile, despite of such possibility, the programmed means are oriented, as a rule, on use for testing of admissibility of the factual arguments or the predefined means, or including the testing algorithms in the body of procedures and functions. For this reason, the Args procedure along with similar to it in such cases can return incorrect results. Whereas the procedure ArgsU covers such peculiar cases. The procedure call ArgsU[x] regardless of way of definition of testing of factual arguments in the general case returns the nested list whose elements are 3–element sub-lists of strings; the 1^st element of such sub-lists is a name of formal argument, the second element defines template ascribed to this formal argument, and the 3^rd element defines a testing function, construction including definition of the testing function, a key word, or an optional value, otherwise the third element is the empty string, i.e. "". The next fragment presents source code of the ArgsU procedure with examples of its use.

In[7]:= ArgsU[x_ /; BlockFuncModQ[x]] :=

Module[{a = HeadPFU[x], b, c, d = {}, g},

a = StringTake[a, {StringLength[ToString[x]] + 2, –2}];

c = StringPartition2[a, ","];

b[t_] := {StringTake[t, {1, Set[g, Flatten[StringPosition[t, "_"]][[1]]] – 1}],

StringTake[t, {g, –1}]}; c = Quiet[Map[StringTrim@b@# &, c]];

c = Quiet[Map[{#[[1]], StringReplace[#[[2]], " /; " –> "", 1]} &, c]];

Map[If[SuffPref[#[[2]], "___", 1],

AppendTo[d, {#[[1]], "___", StringTrim2[#[[2]], "_", 1]}],

If[SuffPref[#[[2]], "__", 1],

AppendTo[d, {#[[1]], "__", StringTrim2[#[[2]], "_", 1]}],

If[SuffPref[#[[2]], "_:", 1],

AppendTo[d, {#[[1]], "_:", StringReplace[#[[2]], "_:" –> "", 1]}],

If[SuffPref[#[[2]], "_.", 1],

AppendTo[d, {#[[1]], "_.", StringReplace[#[[2]], "_." –> "", 1]}],

If[SuffPref[#[[2]], "_", 1],

AppendTo[d, {#[[1]], "_", StringTrim2[#[[2]], "_", 1]}]]]]]] &, c];

If[Length[d] == 1, d[[1]], d]]

In[8]:= Ak[x_, h_ /; IntegerQ[h], y_ /; {IntOddQ[t_/; IntegerQ[t]] :=

Module[{}, IntegerQ[t] && OddQ[t]], IntOddQ[y]}[[–1]],

p_ : 77, t_.] := x*y*h*p*t

In[9]:= Args[Ak]

ToExpression::sntx: Invalid syntax in or before

"{x_, h_ /; IntegerQ[h], y_ /; {IntOddQ[t_ /; IntegerQ[t]}".

Out[9]= $Failed

In[10]:= ArgsU[Ak]

Out[10]= {{"x", "_", ""}, {"h", "_", "IntegerQ[h]"},

{"y", "_", "{IntOddQ[t_/; IntegerQ[t]] := Module[{}, IntegerQ[t] &&

OddQ[t]], IntOddQ[y]}[[–1]]"}, {"p", "_:", "77"}, {"t", "_.", ""}}

The ArgsTypes procedure [8] serves for testing of the formal arguments of block, function or module activated in the current session. So, the call ArgsTypes[x] returns the nested list, whose 2-element sub-lists in the string format define names of formal arguments and their admissible types (in the broader sense the tests for their admissibility with initial values by default) respectively. In lack of a type an argument it is defined as "Arbitrary" which is characteristic for arguments of pure functions and arguments without the tests and/or initial values ascribed to them and also which have format patterns {"__", "___"}. The fragment below represents source code of the ArgsTypes procedure along with typical examples of its application.

Moreover, the ArgsTypes procedure successfully processes the above cases "objects of the same name with various headings", returning the nested two-element lists of the formal arguments concerning the subobjects composing an object x, in the order defined by the Definition function. And in this case 2–element lists have the format, represented above while for objects with empty list of formal arguments the empty list is returned, i.e. {}. Unlike ArgsTypes of the same name this procedure processes objects, including pure functions and Compile functions.

2.4. Local variables in Mathematica procedures

The list of local variables of the procedure in its definition is located at once behind the open parenthesis "[". This list can be as empty, and to contain variables (perhaps with their initial values) which scope is limited to a framework of the procedure, i.e. values of variables from the mentioned list of locals in the current session up a procedure call match their values after exit from the procedure. All other variables of a procedure that are not entering the locals list rely as the global for the procedure in scope of the current session with the system.

At the same time initial values for local variables can be both simple and rather complex expressions. In particular, through local variables it is possible to define procedures and functions available in the current session beyond the procedure containing their definitions as illustrates a simple example:

In[85]:= P[x_, y_] := Module[{a = SetDelayed[b[c_],

Module[{}, c^2]], d = 7, c = 8}, b[x] + d*x + c*y]

In[86]:= P[42, 47]

Out[86]= 2434

In[87]:= Definition[b]

Out[87]= b[c_] := Module[{}, c^2]

In[88]:= P1[x_, y_] := Module[{a = b[c_] := Module[{}, c^2],

d = 7, c = 8}, b[x] + d*x + c*y]

In[89]:= P1[42, 47]

Out[89]= 2434

In[90]:= P1[x_, y_, b_ /; ! ValueQ[b]] := Module[{d = 7, c = 8,

a = b[c_] := Module[{}, c^2]}, b[x] + d*x + c*y]

In[91]:= P1[42, 47, Name]

Out[91]= 2434

In[92]:= Definition[Name1]

Out[92]= Name[c_] := Module[{}, c^2]

In[93]:= P[x_, y_, b_ /; ! ValueQ[b]] := Module[{d = 7,

a = SetDelayed[b[c_], Module[{}, c^2]],

c = If[b[x] < 100, 0, 7]}, b[x] + d*x + c*y]

In[93]:= P[200, 10, Name]

Out[93]= 41960

Variable b in this example is global while the local variable a can be used arbitrary in procedure body without influencing on the variable of the same name outside of the procedure. At last, a simple modification of the above example allows to define in addition in procedure P1 a name for procedure b generated in its local variables. Note that a procedure or function generated in the list of local variables can successfully be called in initial values of other local variables of the procedure as appears from the above example.

We have created a number of tools to work with local user procedure variables [8,16]. Among them, LocalsMod procedure is of some interest.

In[2441]:= Avz[x_] := Module[{t, a = 77, b = k + t, d = Jn[x],

c = {m, {h, g}}}, a[t_] := t^2 + a*d; a[x] + d^2]

In[2442]:= LocalsMod[x_ /; BlockModQ[x]] :=

Module[{a = Definition1[x], b, c = {}, d, j, f, g},

b = Flatten[StringPosition[a, {"Module[", "Block["}, 1]][[-1]];

b = StringTake[a, {1, b}]; b = StringReplace[a, b –> ""];

Do[c = c <> StringTake[b, {j, j}];

If[SyntaxQ[c], Break[], Continue[]], {j, 1, StringLength[b]}];

d = "," <> StringDrop[StringDrop[c, 1], –1] <> ","; c = {};

Label[Agn]; f = StringPosition[d, ","];

f = DeleteDuplicates[Flatten[f]];

For[j = 2, j <= Length[f], j++,

If[SyntaxQ[g = StringTake[d, {f[[1]] + 1, f[[j]] – 1}]],

AppendTo[c, g]; d = StringReplace[d, "," <> g –> "", 1];

Goto[Agn], Continue[]]];

c = Map[If[StringFreeQ[#, "="], {StringTrim[#], "Null"},

{g = Flatten[StringPosition[#, "=", 1]][[1]],

{StringTake[#, {1, g – 2}],

StringTake[#, {g + 2, StringLength[#]}]}}[[2]]] &, c];

SetAttributes[StringTrim, Listable];

Map[StringTrim[#] &, c]]

In[2443]:= LocalsMod[Avz]

Out[2443]= {{"t", "Null"}, {"a", "77"}, {"b", "k + t"},

{"d", "Sin[x]"}, {"c", "{m, {h, g}}"}}

The previous fragment represents the source code of the procedure with an example of its use. Thus, the procedure call LocalsMod[x] returns a nested list whose two-element sub-lists as the first element defines a local variable of a procedure x in the string format, whereas the second element defines the initial condition assigned to it in the string format. If condition is not presented, then the second element is "Null". If procedure x has no local variables, then the call LocalsMod[x] returns the empty list, i.e. {}. The procedure has a number of useful applications.

Between the main objects (function, block, module) there is a mutual converting of one object to another if not to take global and local variables into account. Because of an opportunity and expediency of use of mechanism of local variables is reasonable converting of both classical and pure functions to procedures (blocks, modules). The MathToolBox package [4-7,16] provides a number of tools for different conversion cases. As an example of certain practical interest, the FunctToProc procedure can be cited. The procedure call FuncToProc[x, N] returns nothing, at the same time transforming a classical or pure function x to the procedure with name N with the same set of formal arguments along with a set of local variables created from global variables of the function x. Simple examples illustrate the application of the FuncToProc procedure.

FuncToProc[x_, N_] := Module[{a, b=GlobalsInFunc[x], c},

If[PureFuncQ[x], PureFuncToFunction[x, N];

c = DefToStr[N], If[FunctionQ[x], c = DefToStr[x],

Return["The first argument – not function"]]];

c = StringInsert[c, "Module[" <>

ToString[ToExpression[b]] <> ", ",

StringPosition[c, ":=", 1][[1]][[2]] + 2];

ToExpression[StringReplacePart[c, ToString[N],

StringPosition[c, "[", 1][[1]][[1]] – 1] <> "]"]];

In[10]:= G := Function[{x, y, z}, p*(x + y + z)]

In[11]:= T := p*(#1^2 + #2^2*#3^n) &

In[12]:= F[x_, y_] := x*y + Module[{a = 5, b = 6}, a*x + b*y]

In[13]:= FuncToProc[G, H]

In[14]:= Definition[H]

Out[14]= H[x_, y_, z_] := Module[{p}, p*(x + y + z)]

In[15]:= FuncToProc[T, V]

In[16]:= Definition[V]

Out[16]= V[x1_, x2_, x3_] := Module[{n, p}, p*(x1^2 +

x2^2*x3^n)]

In[17]:= FuncToProc[F, W]

In[18]:= Definition[W]

Out[18]= W[x_, y_] := Module[{a, b}, x*y + Module[{a = 5,

b = 6}, a*x + b*y]]

DefToStr[x_ /; SymbolQ[x] || StringQ[x]] := Module[{a, c,

b = "*" <> ToString[x]}, c = Map[If[StringFreeQ[#, b], #,

StringTake[#, 1, Flatten[StringPosition[#, b]][[1]] – 1}]] &,

StringSplit[ToString[InputForm[Definition[x]]], "\n \n"]];

If[Length[c] == 1, c[[1]], c]]

In[19]:= DefToStr[F]

Out[19]= "F[x_, y_] := x*y + Module[{a = 5, b = 6}, a*x + b*y]"

Along with a number of tools from the package [3-5,16], the procedure uses a procedure whose call DefToStr[x] returns the Input-expression of a symbol x definition in string format, that, generally speaking, the built–in ToString function is not able to do. Meanwhile, a rather significant remark needs to be made regarding the DefToStr tool. As noted repeatedly, Mathematica system differentiates objects not by their names, but by their headers, if any. It is therefore quite real that we may deal with different definitions under the same name. This applies to all objects that have headers. In view of this circumstance, we have developed a number of funds for various cases of application [1-16]. This can be fully attributed to the above DefToStr tool, whose call on a symbol having multiple definitions produce a correct result, i.e. the DefToStr procedure solves this problem. In the case of multiplicity of a symbol definition the procedure call DefToStr[x] returns the list of Input-expressions of a symbol x definitions in string format, that the built-in ToString function generally speaking, is not able to do. In some cases, the method may be useful in practical programming. Note, that the actual x argument in DefToStr[x] call must be of type Symbol or String (if for x there are headers) or String (if for x there are no headers), while on built-in system functions its call returns attributes ascribed to them. This procedure is rather demanded when processing multiple definitions of a function, block or module.

In[11]:= G[x_, y_] := p*(x^2 + y^2)

In[12]:= G[x_, y_, z_] := (x^2 + y^2 + z^2)

In[13]:= G[x_, y_, z_ /; IntegerQ[z]] := Module[{}, x*y +

42*x^2 + 47*y^2 + z^3]

In[14]:= F[x_, y_] := x*y + 42*x^2 + 47*y^2

In[15]:= DefToStr[G]

Out[15]= {"G[x_, y_] := p*(x^2 + y^2)",

"G[x_, y_, z_ /; IntegerQ[z]] := Module[{}, x*y +

42*x^2 + 47*y^2 + z^3]",

"G[x_, y_, z_] := x^2 + y^2 + z^2"}

In[16]:= DefToStr[F]

Out[16]= "F[x_, y_] := x*y + 42*x^2 + 47*y^2"

Experience has shown that the use of the procedure is quite effective in many applications.

For work with local and global variables of procedures we created a number of enough useful tools that are absent among the built-in system tools [16]. Their detailed description can be found in [1-15]. In particular, the GlobalToLocalInit procedure provides expansion of the list of local variables of a procedure at the expense of its global variables. Moreover, for the local variables received from global variables the initial values rely those that at the time of the procedure call had global variables. In the absence of global variables the procedure call returns the corresponding message. The call GlobalToLocalInit[x] returns nothing, generating in the current session the procedure with name $$x with the updated list of local variables. The following program fragment illustrates told:

In[712]:= GS[x_, y_, z_ /; IntegerQ[z]] := Module[{a= 72,

b = 77}, t*(x^2 + n*y^2 + v*Sin[p]*T[z]*z^2)/g*W[a, b]]

In[713]:= {t, n, p, v} = {2, 7, 6, 9}; T[x_] := x^2; W[x_, y_] := x*y;

In[714]:= GlobalToLocalInit[x_ /; BlockModQ[x]] :=

Module[{a = Args[x], b, c, d, f}, b = DefToStr[x];

c = ArgsLocalsGlobals1[x];

Quiet[d = Select[Flatten[Select[c[[3]],

! "System`" == #[[1]] &]], ! ContextQ[#] &]];

d = Select[d, ! BlockFuncModQ[#] &];

If[d == {}, Return["No global variables"],

Map[If[! Definition[#] === Null, #,

ToExpression[# <> "= " <> #]] &, d];

ToExpression["Save[\"#\"," <> ToString1[d] <> "]"];

f = ReadString["#"]; DeleteFile["#"];

f = Select[StringSplit[f, "\r\n"], # != " " &];

c = StringReplace[b, HeadPF[x] –> "", 1];

c = SubStrSymbolParity1[c, "{", "}"][[1]];

f = StringReplace[StringReplace[ToString[c] <>

ToString[f], "}{" –> ", "], "{, " –> "{"];

ToExpression["$$" <> StringReplace[b, c –> f, 1]]]]

In[715]:= GlobalToLocalInit[GS]

In[716]:= Definition[$$GS]

Out[716]= $$GS[x_, y_, z_ /; IntegerQ[z]] := Module[{a = 72,

b = 77, n = 7, p = 6, t = 2, v = 9},

(t*(x^2 + n*y^2 + v*Sin[p]*T[z]*z^2)*W[a, b])/g]

In[4852]:= {m, n, p} = {42, 47, 67};

In[4853]:= A[x_, y_, z_] := Module[{a := 7, b := 5, c := 77},

(a*x + b*y + c*z)*m*n*p]

In[4854]:= A[1, 2, 3]

Out[4854]= 32799984

In[4855]:= GlobalToLocalInit[A]

In[4856]:= Definition[$$A]

Out[4856]= $$A[x_, y_, z_] := Module[{a := 7, b := 5, c := 77,

n = 47, p = 67, m = 42}, (a*x + b*y + c*z)*m*n*p]

In[4857]:= {m, n, p} = {77, 72, 52};

In[4858]:= $$A[1, 2, 3]

Out[4858]= 32799984

By the way, the last example with procedure A in addition illustrates a possibility of correct use of the delayed definitions instead of immediate for initial values of local variables.

A function or procedure defined in a procedure body with b name of local variable also is local whereas defined through a local variable becomes global tool in the current session of the system. The following example illustrates told:

In[2010]:= B[x_, y_] := Module[{a = 7, b, c := 77},

If[x < y, b[t_] := t^2, b[t_] := t^3]; (a + c)*b[x*y]]

In[2011]:= B[7, 3]

Out[2011]= 777924

In[2012]:= Definition[b]

Out[2012]= Null

In[2013]:= J[x_, y_] := Module[{a = 7, b = {b[t_, c_] :=

(t + c)^2, b}[[2]], c := 77}, (a + c)*b[x, y]]

In[2014]:= J[2, 3]

Out[2014]= 2100

In[2015]:= Definition[b]

Out[2015]= b[t_, c_] := (t + c)^2

The above mechanism works when determining headings and local variables of the functions, blocks and modules. At the same time, it must be kept in mind that presented mechanism with good reason can be carried to non–standard methods of programming which are admissible, correct and are a collateral consequence of the built–in Math–language. Meantime, taking into account this circumstance to programming of processing tools of elements of procedures and functions (headings, local variables, arguments, etc.) certain additional demands are made. This mechanism can be considered facultative, i.e. optional to programming of procedures and functions. At the same time it must be kept in mind that definitions of the tools coded in the formal arguments and local variables become active in the all domain of the current session that isn't always acceptable. A reception correcting such situation presents the first example of previous fragment. The vast majority of tools presented in our package [16] don't use this mechanism. However certain our tools, represented in [1-15], use this mechanism, representing certain ways of its processing. Other example of processing of the local variables – a procedure updating their initial values.

In[104]:= ReplaceLocalInit[x_ /; BlockModQ[x], y_List] :=

Module[{a = DefToStr[x], b, c, d, f = {}, k,

h = If[NestListQ[y], y, {y}]}, c = ArgsLocalsGlobals1[x];

b = Map[If[Length[#] == 1, #,

{StringReplace[#[[1]], " :" –> ""], #[[2]]}] &, Locals4[x]];

k = Map[#[[1]] &, b];

If[b == {}, ToExpression["$" <> a],

d = Map[{#[[1]], ToString[#[[2]]]} &, h];

f = Flatten[{b, d}, 1]; f = Gather[f, #1[[1]] === #2[[1]] &];

f = Map[If[Length[#] == 1, #[[1]], #] &, f];

f = Map[If[NestListQ[#] && Length[#] == 2,

#[[2]][[1]] <> "=" <> #[[2]][[2]],

If[Length[#] == 1 && MemberQ[k, #[[1]]], #[[1]],

If[Length[#] == 2 && MemberQ[k, #[[1]]],

#[[1]] <> "=" <> #[[2]], Nothing]]] &, f];

d = StringReplace[a, HeadPF[x] <> " := " –> "", 1];

d = SubStrSymbolParity1[d, "{", "}"][[1]];

ToExpression["$" <> StringReplace[a, d –> ToString[f], 1]]]]

In[105]:= B[x_] := Module[{a = 77, b, c := 72, d = 9}, Body]

In[106]:= ReplaceLocalInit[B, {{"a", 90}, {"b", 500},

{"c", 900}, {"p", 52}, {"g", 50}}]; Definition[$B]

Out[106]= $B[x_] := Module[{a=90, b=500, c=900, d=9}, Body]

In[107]:= H[x_, y_, z_] := Module[{}, y = x^2; z = x^3; x + 5]

In[108]:= ReplaceLocalInit[H, {{"a", 90}, {"b", 500},

{"c", 900}, {"p", 52}, {"g", 50}}]; Definition[$H]

Out[108]= $H[x_, y_, z_] := Module[{}, y = x^2; z = x^3; x + 5]

In[109]:= ReplaceLocalInit[B, {{"a", 90}, {"c", 900}, {"p", 52}}]

In[110]:= Definition[$B]

Out[110]= $B[x_] := Module[{a = 90, b, c = 900, d = 9}, Body]

The call ReplaceLocalInit[x, {{"a", a1}, …, {"p", p1}}] returns nothing, activating in the current session the procedure $x that relatively x has local variables {a,…,p} with the updated initial values {a1,…,p1}. The fragment illustrates the source code of the procedure with examples of its use. Other processing means of local variables of the user procedures can be found in [1-18,22]. They sometimes significantly complement the built-in software.

Also it is worth paying attention to one moment of rather local variables of procedures. The scope of local variables is the procedure in which they are determined and after a procedure call these variables are unavailable in the current Mathematica session. The local variables have Temporary attribute which is assigned to the variables which are created as local variables of a procedure; they are automatically removed when they are no longer needed. At that, local variables with attribute Temporary conventionally receive names of the format a$nnn, where a – a name of local variable that is defined at procedure creation and nnn – the number based on the current serial number defined by the system $ModuleNumber variable. Note, $ModuleNumber is incremented every time a procedure or built-in Unique function is called. A Mathematica session starts with $ModuleNumber set to 1, that can be redefined, but in order to avoid the possible conflict situations leading to decrease in efficiency, the user is not recommended to redefine $ModuleNumber variable. For the purpose of gaining access to local variables out of domain of the procedures containing them it is possible to use a reception that is based on redefinition of their Temporary attribute, namely on its cancelling. Let's illustrate told by a rather simple example:

In[37]:= Gs[x_, y_, z_] := Module[{a, b},

Print[{Map[Attributes[#] &, {a, b}], {a, b}}];

a[t_, n_, m_] := t*n*m; b := 78; b*x*y*z]

In[38]:= Gs[42, 47, 67]

{{{Temporary}, {Temporary}}, {a$2666, b$2666}}

Out[38]= 10316124

In[39]:= Definition1[a$2666]

Out[39]= Null

In[40]:= b$2666

Out[40]= b$2666

In[41]:= Gs1[x_, y_, z_] := Module[{a, b, c}, Print[{a, b, c}];

{a, b, c} = {x, y, z}; a*b*c]

In[42]:= Gs1[42, 47, 67]

{a$2863, b$2863, c$2863}

Out[42]= 132258

In[43]:= {a$2863, b$2863, c$2863}

Out[43]= {a$2863, b$2863, c$2863}

In[44]:= Gs2[x_, y_, z_] := Module[{a, b, c}, Print[{a, b, c}];

Map[ClearAttributes[#, Temporary] &, {a, b, c}];

{a, b, c} = {x, y, z}; a*b*c]

In[45]:= Gs2[42, 47, 67]

{a$2873, b$2873, c$2873}

Out[45]= 132258

In[46]:= {a$2873, b$2873, c$2873}

Out[46]= {42, 47, 67}

Procedure Gs uses two local variables {a, b}; the first of them defines a name in function definition while the second defines simple integer variable. The procedure call returns an integer number with output of the nested list with attributes of local variables {a, b} and their values. It turns out that out of domain of the procedure local variables are indefinite ones.

Procedure call Gs1[x, y, z] returns an integer number along with output of the list of the local variables that are generated by the procedure from {a, b, c}. It turns out that out of domain of the procedure local variables {a, b, c} are indefinite ones.

The Gs2 procedure differs from the Gs1 procedure only in what in its body contains cancelling of Temporary attribute for all local variables {a, b, c}. Procedure call Gs2[x, y, z] returns an integer number along with output of the list of local variables that are generated by the procedure from {a, b, c}. It turns out that out of domain of the procedure local variables {a, b, c} are definite ones, obtaining quite concrete values. So, for ensuring access to local variables of a procedure out of its domain it is quite enough to cancel Temporary attribute for all (or selected) its local variables allowing to monitor intermediate calculations in the course of performing procedures. Such approach allows to provide monitoring of values of local variables of procedures that is especially important when debugging a rather large and complex software by transferring the local variables to the level of variables, unique for the current session. In certain cases this approach to debugging has quite certain advantages. For such purpose the simple LocalsInProc procedure which is based on the predetermined variable $ModuleNumber which determines the current serial number to be used for local variables that are created can appear quite useful. Any Mathematica session starts with $ModuleNumber set to 1, receiving the increment for each new local variable in the current session. The fragment below represents source code of the procedure with examples of use.

In[47]:= LocalsInProc[t_List] := Module[{a, b = Length[t]},

a = Map[ToString, ($ModuleNumber – 5)*

Flatten[Tuples[{1}, b]]];

Map[t[[#]] <> "$" <> a[[#]] &, Range[b]]]

In[48]:= Agn2[x_] := Module[{a, b, c, d}, Print[{a, b, c, d}];

{a, b, c, d} = {42, 47, 67, 77}; x*a*b*c]

In[49]:= Agn2[77]

{a$13366, b$13366, c$13366, d$13366}

Out[49]= 10183866

In[50]:= LocalsInProc[{"a", "b", "c", "d"}]

Out[50]= {a$13366, b$13366, c$13366, d$13366}

In[51]:= Avz[x_] := Module[{a, b, c}, Print[{a, b, c}];

a[t_] := t^2; a[x]]; Avz[77]

{a$49432, b$49432, c$49432}

Out[51]= 5929

In[52]:= LocalsInProc[{"a", "b", "c"}]

Out[52]= {"a$49432", "b$49432", "c$49432"}

The procedure call LocalsInProc[g] should follow directly a P procedure call, where g – the list of local variables of the P procedure, with return of the list of representations in form of x$nn (x – a variable name and nn – an integer in string format) that are generated for local variables of the P procedure. If g defines the empty list, i.e. procedure P has no local variables, then the call LocalsInProc[{}] returns the empty list, i.e. {}.

Additionally, a number of Locals ÷ Locals5 tools have been programmed, using some useful programming techniques and returning local variables in various sections. A typical example is the procedure whose call Locals5[x] returns the list of local variables in string format with initial values assigned to them, for a module, or a block x. While calling Locals5[x, y] with the second optional argument y – an undefined symbol – through it additionally returns the list consisting of two-element sub-lists whose elements in string format represent local variables and initial values ascribed to them. If there are no local variables, the empty list is returned. The fragment below represents the source code of the procedure with examples of its application.

In[422]:= Locals5[x_ /; BlockModQ[x], y___] := Module[{a, b, c},

a = StringJoin[Insert[Reverse[Headings[x]], " := ", 2]] <> "[";

b = StringReplace[PureDefinition[x], a –> "", 1];

Do[c = StringTake[b, k]; If[SyntaxQ[c], Return[c], 7],

{k, StringLength[b]}];

If[c == "{}", c = {}, c = StrToList[StringTake[c, {2, –2}]]];

If[{y} == {}, c, If[SymbolQ[y],

y = Map[StringSplit[#, " = "] &, c]; c, c]]]

In[423]:= M3[x_] := Block[{a = 90, b = 49, c, h}, h = 77; x*h];

B1[x_] := Block[{a = 90, b = 50, c = {72, 77}, d = 42}, x*a*b*c*d];

B[x_] := Block[{}, x]; Locals5[M3]

Out[423]= {"a = 90", "b = 49", "c", "h"}

In[424]:= Locals5[B]

Out[424]= {}

In[425]:= Locals5[B1]

Out[425]= {"a = 90", "b = 50", "c = {72, 77}", "d = 42"}

In[426]:= Locals5[B1, agn]

Out[426]= {"a = 90", "b = 50", "c = {72, 77}", "d = 42"}

In[427]:= agn

Out[427]= {{"a", "90"}, {"b", "50"}, {"c", "{72, 77}"}, {"d", "42"}}

In[428]:= Locals5[B, vsv]; vsv

Out[428]= {}

In[429]:= Locals5[M3, art]; art

Out[429]= {{"a", "90"}, {"b", "49"}, {"c"}, {"h"}}

As an analogue of the above Locals5 procedure we present the Locals6 procedure which uses certain useful programming

technique of the strings. The procedure has analogous formal arguments and its call returns results analogous to the call of the above Locals5 procedure. The following fragment represents source code of the procedure with examples of its application.

In[3119]:= Locals6[x_ /; BlockModQ[x], y___] :=

Module[{a = Definition2[x][[1]], b = Headings[x], c = "", d},

a = StringReplace[a, b[[2]] <> " := " <> b[[1]] <> "[" –> ""];

Do[If[SyntaxQ[c = c <> StringTake[a, {j}]], Break[], 7], {j, Infinity}];

c = StringToList[StringTake[c, {2, –2}]];

If[{y} != {} && SymbolQ[y],

y = Map[{b = Quiet[Check[Flatten[StringPosition[#, "="]][[1]], {}]],

If[b === {}, #, {StringTake[#, {1, b – 2}],

StringTake[#, {b + 2, –1}]}]}[[2]] &, c], 7]; c]

In[3120]:= B[x_, y_] := Block[{a=5, b, c=7}, b = (a*x+b*c*y)/(x+y) +

Sin[x + y]; If[b, 45, b, a*c]]

In[3121]:= SetAttributes[B, {Listable, Protected}]

In[3122]:= Locals6[B]

Out[122]= "{a = 5, b, c = 7}"

In[3123]:= Locals6[B, h7]

Out[3123]= {"a = 5", "b", "c = 7"}

In[3124]:= h7

Out[3124]= {{"a", "5"}, "b", {"c", "7"}}

In[3125]:= ModuleQ[StringReplaceVars]

Out[3125]= True

In[3126]:= Locals6[StringReplaceVars, g7]

Out[3126]= {"a = StringJoin[\"(\", S, \")\"]",

"L = Characters[\"`!@#%^&*(){}:\\\"\\\\/|<>?~–=+[];:'.,

1234567890_\"]", "R = Characters[\"`!@#%^&*(){}:\\\"\\\\/|

<>?~–=+[];:'., _\"]", "b", "c", "g = If[RuleQ[r], {r}, r]"}

In[3127]:= g7

Out[3127]= {{"a", "StringJoin[\"(\", S, \")\"]"},

{"L", "Characters[\"`!@#%^&*(){}:\\\"\\\\/|<>?~–=+[];:'.,

1234567890_\"]"}, {"R", "Characters[\"`!@#%^&*(){}:\\\"\\\\/|

<>?~–=+[];:'., _\"]"}, "b", "c", {"g", "If[RuleQ[r], {r}, r]"}}

Local variable processing tools are of particular interest in tasks of procedural programming. A number of other rather useful means for operating with local variables of both modules and blocks can be found in our books [7,10-16].

At last, it makes sense to mention one method of localization of variables in procedures that in certain situations can be quite convenient. The structural organization of such procedures can be presented as follows, namely:

G[Args] := Module[{}, Save[f, {a, b, c, …}]; Procedure body;

{Res, Get[f], DeleteFile[f]}[[1]]]

A procedure can have a certain quantity of local variables or not have them in general whereas as other local variables any admissible variables are used, not excepting variables out of domain of the procedure, i.e. intersection of variables from the outside and in the procedure is allowed. For their temporary isolation (on runtime of the procedure) from external domain of the procedure their saving in a file f by Save function precedes the procedure body in which these variables are used like local variables. Whereas an exit from the procedure is made out by the three-element list where Res – a result of the procedure call, Get[f] – loading in the current session of file f with recovery of values of variables {a, b, c, …} that were until the procedure call with the subsequent deleting of file f. The following fragment illustrates the above mechanism of localization of variables.

In[19]:= {a, b} = {42, 77}

Out[19]= {42, 77}

In[20]:= Gg[x_, y_] := Module[{}, Save["#$", {a, b, c}];

{a, b} = {47, 72}; {x*a/y*b, Get["#$"], DeleteFile["#$"]}[[1]]]

In[21]:= Gg[30, 21]

Out[21]= 33840/7

In[22]:= {a, b, c}

Out[22]= {42, 77, c}

At the same time, if the procedure has several possible exits then each exit should be issued in analogous way. In principle, the reception used for localization of variables described above within the framework of the procedure can be used not only in connection with its local variables.

In [15] we defined so–called active global variables as global variables to which in a Block, Module the assignments are done, whereas we understand the global variables different from the arguments as the passive global variables whose values are only used in objects of the specified type. In this regard tools which allow to evaluate the passive global variables for the user blocks and modules are being presented as a very interesting. One of similar means – the BlockFuncModVars procedure that solves even more general problem. The fragment represents the source code of the procedure BlockFuncModVars along with the most typical examples of its application.

In[24]:= BlockFuncModVars[x_ /; BlockFuncModQ[x]] :=

Module[{d, t, c = Args[x, 90], a = If[QFunction[x], {},

LocalsGlobals1[x]], s = {"System"}, u = {"Users"},

b = Flatten[{PureDefinition[x]}][[1]], h = {}}, d = ExtrVarsOfStr[b, 2];

If[a == {}, t = Map[If[Quiet[SystemQ[#]], AppendTo[s, #],

If[BlockFuncModQ[#], AppendTo[u, #], AppendTo[h, #]]] &, d];

{s, u = Select[u, # != ToString[x] &], c, MinusList[d, Join[s, u, c,

{ToString[x]}]]}, Map[If[Quiet[SystemQ[#]], AppendTo[s, #],

If[BlockFuncModQ[#], AppendTo[u, #], AppendTo[h, #]]] &, d];

{Select[s, ! MemberQ[{"$Failed", "True", "False"}, #] &],

Select[u, # != ToString[x] && ! MemberQ[a[[1]], #] &], c, a[[1]],

a[[3]], Select[h, ! MemberQ[Join[a[[1]], a[[3]], c,

{"System", "Users"}], #] &]}]]

In[25]:= A[m_, n_, p_ /; IntegerQ[p], h_ /; PrimeQ[h]] :=

Module[{a = 77}, h*(m + n + p)/a +

StringLength[ToString1[z]]/(Cos[c] + Sin[d])]

In[26]:= BlockFuncModVars[A]

Out[26]= {{"System", "Cos", "IntegerQ", "Module", "PrimeQ",

"Sin", "StringLength"}, {"Users", "ToString1"}, {"m", "n", "p", "h"},

{"a"}, {}, {"c", "d", "z"}}

The call BlockFuncModVars[x] returns the 6–element list, whose the 1^st element – the list of system functions used by block or module x, whose 1^st element is "System", whereas other names are system functions in string format; the 2^nd element – the list of the user tools used by the block or module x, whose the 1^st element is "User" while the others define names of tools in the string format; the 3^rd element defines the list of formal arguments in the string format of the block or module x; the 4^th element – the list of local variables in the string format; the fifth element – the list of active global variables in string format; at last, the sixth element determines the list of passive global variables in the string format of the block or module x.

2.5. Exit mechanisms of the user procedures (return results)

The procedure body implements a particular calculation or processing algorithm using control sentences of Math-language and its built–in functions along with other program tools. The result of successful execution of the procedure body is the exit from the procedure with purpose of return of its results to the current session of Mathematica system. In general, the return of results of procedure call can be organized by three manners: (1) through the last sentence of the procedure body, (2) through the system built–in Return function and (3) through its formal arguments, for example, procedures A, B and H accordingly:

In[7]:= A[x_] := Module[{a = 77, b = 72, c}, c = a + b; c*x]; A[1]

Out[7]= 149

In[8]:= B[x_, y_] := Module[{a = 77, b = 72, c}, c = a + b;

Return[77*x]; c*x; a*x + b*y]; B[1]

Out[8]= 77

In[9]:= H[x_, y_ /; ! ValueQ[y], z_ /; ! ValueQ[z]] :=

Module[{a = 77, b = 72, c}, c = a + b; y = c*x;

z = a*x^2; (a^2 + b^2)*x]; H[3, m, n]

Out[9]= 33339

In[10]:= {m, n}

Out[10]= {447, 693}

Depending on the algorithm logic realized by a procedure body, the procedure allows the use of several Return functions that return results of the procedure (in general, the exit from the procedure) at the required algorithm points. In principle, the use of Return[] is allowed even in the list of local variables (starting values) as illustrates the following example:

In[27]:= B[x_, y_] := Module[{a = If[x < 10 && y > 10, x*y,

Return[{x, y}]], b = 5, c = 7}, a*b*c]: B[5, 15]

Out[27]= 2625

In[28]:= B[15, 15]

Out[28]= {525, 525}

Whereas the third case can be used for returning of some additional results of the procedure call (values of local variables, intermediate evaluations, etc). In the present book and in [1-15] a number of examples of such approach have been illustrated. At that, formal arguments in heading of a procedure thru that it is planned to return results has to be of the form {g_, g__, g___} (i.e. without a testing Test_g function) and be indefinite symbols on which ValueQ[g] call returns False, or be coded in the form g_/; !ValueQ[g]. For practical reasons, it is recommended to use the latter option to return the result thru formal g argument so as not to introduce unpredictability into the results of the current session: so, in a case of simple definite g variable the erroneous situation arises in attempt of assignment of value for g in the procedure while for g variable as a heading name in the current session arises the multiplicity of definitions for it as illustrates a rather simple fragment:

In[3]:= y[t_] := t^2; z[t_] := Module[{}, t^3]

In[4]:= H[x_, y_, z_] := Module[{}, y = x^2; z = x^3; x + 5];

H[3, y, z]

Out[4]= 8

In[5]:= Definition[y]

Out[5]= y = 9

y[t_] := t^2

In[6]:= Definition[z]

Out[6]= z = 27

z[t_] := Module[{}, t^3]

In[7]:= Clear[y, z]; {y, z} = {72, 77}; H[3, y, z]

Set::setraw: Cannot assign to raw object 72.

Set::setraw: Cannot assign to raw object 77.

Out[7]= 8

Note that it is possible to use a mechanism represented by a fairly clear example to return intermediate results through the selected formal arguments with {__, ___} patterns:

In[2233]:= V[x_, y__] := Module[{a = 5, b = 7, c = {y}},

If[Length[c] >= 2, ToExpression[ToString[c[[1]]] <> "=77"];

ToExpression[ToString[c[[2]]] <> "=72"],

ToExpression[ToString[c[[1]]] <> "=52"]]; (a + b)*x]

In[2234]:= Clear[b, c]; V[5, b, c]

Out[2234]= 60

In[2235]:= {b, c}

Out[2235]= {77, 72}

Mechanisms for returning the results of calling procedures through formal arguments, including updating them in situ, are of particular interest. Certain approaches have been presented above and in [30-38]. Meanwhile, for the Mathematica, certain difficulties represent, in particular, the cases when a list acts as a formal argument. As supportive tool in this case a procedure can be used, whose call SymbolValue[x, y] returns the list of the names in string format having a context y and a value x. While the call SymbolValue[x] returns the list of format {{"a'", {"a1",..., "an"}}, ..., {"j'", {"j1",...,"jm"}}} where the list {"a'",...,"j'"} defines contexts according to the variable $ContextPath and {"a1"...},..., {"j1"...} defines the names corresponding to them in the string format with value x. The fragment below represents the source code of the SymbolValue procedure with examples of its use.

In[42]:= L = {a, b, c, d, g, h, d, s, u, t}

Out[42]= {a, b, c, d, g, h, d, s, u, t}

In[43]:= M := {a, b, c, d, g, h, d, s, u, t}

In[44]:= SymbolValue[x_, y___] := Module[{b, c = {}, j,

a = Length[$ContextPath]},

If[{y} != {} && ContextQ[y],

Map[If[SameQ[ToExpression[#], x], #, Nothing] &,

Names[y <> "*"]],

b = Map[{#, Names[# <> "*"]} &, $ContextPath];

Do[AppendTo[c, {b[[j]][[1]],

Map[If[SameQ[ToExpression[#], x], #, Nothing] &,

b[[j]][[2]]]}], {j, a}];

Map[If[#[[2]] == {}, Nothing, #] &, c]]]

In[45]:= SymbolValue[{a, b, c, d, g, h, d, s, u, t}, "Global`"]

Out[45]= {"L", "M"}

In[46]:= SymbolValue[{a, b, c, d, g, h, d, s, u, t}]

Out[46]= {{"SveGal`", {"L", "M"}}, {"Global`", {"L", "M"}}}

Consider an illustration of this approach on a simple example.

Trying to update the contents of the Ls list onto its sorted content in situ in the body of a Mn procedure with subsequent call of the procedure Mn[Ls] causes an erroneous diagnostics, and the Ls list itself obtains an incorrect value with output of erroneous messages. Whereas the procedure Mn1 using the above noted procedure SymbolValue successfully solves this problem, that the following rather simple and visual fragment illustrates.

In[77]:= Ls = {a, b, c, d, f, g, h, j, k, d};

In[78]:= Mn[x_List] := Module[{}, x = Sort[x]]

In[79]:= Mn[Ls]

… $IterationLimit: Iteration limit of 4096 exceeded.

========================================

… General: Further output of $IterationLimit::itlim will be

suppressed during this calculation.

Out[79]= {a, b, c, Hold[g], Hold[g], Hold[h], Hold[j],

Hold[k], Hold[d], Hold[f]}

In[80]:= Ls

… $IterationLimit: Iteration limit of 4096 exceeded.

=======================================

… General: Further output of $IterationLimit::itlim will be

suppressed during this calculation.

Out[80]= {a, b, c, Hold[g], Hold[h], Hold[j], Hold[k],

Hold[d], Hold[f], Hold[g]}

Note that calling the built–in function ReleaseHold[x] that should removes Hold, HoldForm, HoldPattern, HoldComplete in x does not work in some cases, which illustrates an example

In[81]:= ReleaseHold[{a, b, c, Hold[g], Hold[h], Hold[j],

Hold[k], Hold[d], Hold[f], Hold[g]}]

… $IterationLimit: Iteration limit of 4096 exceeded.

=========================================

…General: Further output of $IterationLimit::itlim will be

suppressed during this calculation.

Out[81]= {a, b, c, Hold[j], Hold[k], Hold[d], Hold[f], Hold[g],

Hold[h], Hold[j]}

In[81]:= Mn1[x_List] := Module[{},

ToExpression[SymbolValue[x, "Global`"][[1]] <> "=" <>

ToString1[Sort[x]]]]

In[82]:= Mn1[Ls]

Out[82]= {a, b, c, d, d, f, g, h, j, k}

In[83]:= Ls

Out[83]= {a, b, c, d, d, f, g, h, j, k}

Further is being often mentioned about return of result of a function or a procedure as unevaluated that concerns both the standard built–in tools, and the user tools. In any case, the call of a procedure or a function on an inadmissible tuple of actual arguments is returned by unevaluated, except for the standard simplifications of the actual arguments. In this connection the UnevaluatedQ function providing testing of a procedure or a function regarding of the return of its call as unevaluated on a concrete tuple of actual arguments has been programmed. The function call UnevaluatedQ[F, x] returns True if the call F[x] will be returned unevaluated, and False otherwise; on an erroneous call F[x] "ErrorInNumArgs" is returned. The fragment presents source code of UnevaluatedQ function with examples of its use.

In[46]:= UnevaluatedQ[F_, x___] := If[! SymbolQ[F], True,

ToString1[F[x]] === ToString[F] <> "[" <>

StringTake[ToString1[{x}], {2, -2}] <> "]"]

In[47]:= Sin[1, 2]

Sin: Sin called with 2 arguments; 1 argument is expected

Out[47]= Sin[1, 2]

In[48]:= UnevaluatedQ[Sin, 77, 72]

Out[48]= "ErrorInArgs"

In[49]:= G[x_Integer, y_String] := y <> ToString[x]

In[50]:= UnevaluatedQ[G, 77, 72]

Out[50]= True

The procedure presents an interest for program processing of the results of procedures and functions calls. The procedure was used by a number of means from our package [16].

As the results of exits from procedures, it is appropriate to consider also the various kinds of messages generated by the procedures as a result of erroneous and exceptional situations (generated both by the system and programmed in the procedures). At the same time the issue of program processing of such messages is of a particular interest. In particular, the problem is solved by the procedure whose call MessagesOut[x] returns the message generated by a procedure or function G whose call given in the format x = "G[z]". Whereas the procedure call MessagesOut[x,y] with the 2^nd optional y argument – an indefinite symbol – thru it returns the messages in the format {MessageName, "Message"}. In the absence of any messages the empty list, i.e. {}, is returned. An unsuccessful procedure call returns $Failed.

All messages assigned to a symbol x (Function,Module,Block) along with its usage (if any) are returned by calling Messages[x]. Meantime, the returned result is not very convenient for further processing in program mode. Therefore, we have created some useful tools for this purpose [1-16]. In particular, the procedure call Usage[x] in the string format returns the usage ascribed to a symbol x (built–in or user tool), its source code and application examples are represented below, namely:

In[3107]:= Usage[x_Symbol] := Module[{a, b = Context[x],

c = ToString[x]}, ToExpression["?" <> c];

a = ToExpression[StringJoin["Messages[", b, c, "]"]];

If[a == {}, {}, b = StringSplit[ToString[a], "HoldPattern"];

b = Select[b, ! StringFreeQ[#, c <> "::usage"] &];

b = Flatten[StringSplit[b, " :> "]][[2]];

If[SystemQ[x], StringDrop[b, –3], StringTrim[b, "}"]]]]

In[3108]:= Usage[ProcQ]

Out[3108]= "The call ProcQ[x] returns True if x is a

procedure and False otherwise."

In[3109]:= Usage[While]

Out[3109]= "While[test, body] evaluates test, then body,

repetitively, until test first fails to give True."

In[3110]:= Usage[Agn]

Out[3110]= {}

In this context, a simple procedure can be noted whose call ToFile[x, 1] in the current session opens a file x to read, to that the system along with the screen writes all messages that occur. Whereas call ToFile[x, 1, z] with the third optional argument z (an arbitrary expression) closes the x file, allowing in future to do analysis of the received messages.

In[42]:= ToFile[x_String, y_ /; MemberQ[{1, 2}, y], z___] :=

Module[{a, b}, If[{z} == {}, b = OpenWrite[x]; Write["#", b];

Close["#"]; If[y == 1, $Messages = Append[$Messages, b],

$Output = Append[$Output, b]];, $Output =

ReplaceAll[$Output, Read["#"] –> Nothing]; Close["#"];

$Messages = ReplaceAll[$Messages, Read["#"] –> Nothing];

Close[x]; Quiet[DeleteFile[Close["#"]]; ]]]

In[43]:= MessageFile["C:\\temp\\Error.txt", 1]

====================================

In[145]:= MessageFile["C:\\temp\\ Error.txt", 1, 7]

Out[145]= "C:\\temp\\Error.txt"

Moreover, the procedure calls with the second argument 2 are performed similarly except that all messages output by the Print function are written to the w file until appear a procedure call with the third argument z (an arbitrary expression).

Right there appropriate to note the Print1 procedure which extends the capabilities of the built–in Print function, allowing additionally to output to the file the result of a call of the Print function. The call Print1[x, y, z] prints z as output additionally saving the z expression in the x file if y = 1 or closing the x file if y=2. A simple function IsOpenFile can be used in the procedure implementation, whose call IsOpenFile[x] returns True if a file x exists and is open for writing, and False otherwise.

In[2228]:= Print1[x_, y_ /; MemberQ[{1, 2}, y], z___] :=

Module[{a}, If[FileExistsQ[x], If[y == 1, Print[z], Close[x];

$Output = ReplaceAll[$Output, Read["#"] –> Nothing];

DeleteFile[Close["#"]]], a = OpenWrite[x]; Write["#", a];

Close["#"]; $Output = Append[$Output, a]; Print[z]]]

In[2229]:= IsOpenFile[x_ /; StringQ[x]] := ! StringFreeQ[

StringReplace[ToString[InputForm[Streams[]]],

"\\\\" –> "\\"], StringJoin["OutputStream[\"", x, "\","]]

In[2230]:= IsOpenFile["c:/print2.txt"]

Out[2230]= False

The following procedure generalizes the previous function for testing of openness of files on reading and writing. Namely, the procedure call IsInOutFile[x] returns False if a file x is closed or missing, whereas the list {True, In, Out}, {True, In}, {True, Out} otherwise, where In – the file for reading and Out – for writing.

In[3242]:= IsInOutFile[x_ /; StringQ[x]] := Module[{a =

StringReplace[ToString[InputForm[Streams[]]],"\\\\"->"\\"]},

If[! StringFreeQ[a, StringJoin["OutputStream[\"", x, "\","]] &&

! StringFreeQ[a, StringJoin["InputStream[\"", x, "\","]],

{True, In, Out},

If[! StringFreeQ[a, StringJoin["OutputStream[\"", x, "\","]],

{True, Out},

If[! StringFreeQ[a, StringJoin["InputStream[\"", x, "\","]],

{True, In}, False]]]]

In[3243]:= IsInOutFile["C:\\temp/Cinema_2020.doc"]

Out[3243]= {True, In}

However, note that (as shown in [1-15]) the coding of the file path is symbolic–dependent, for example, path "с:/tmp/kino" is not identical to "C:/Tmp\\Kino" i.e. the system will recognize them as different paths. This should be taken into account when programming file access tools. It is useful to use standardized files path view as an easy approach [1-15].

Therefore, the output of messages, including Print function messages, can be ascribed fully to the return of results allowing further program processing along with their rendering on the display. Naturally, all procedure calls that complete by $Failed or $Aborted are processed programmatically. To recognize this return type it is possible to use the system function whose call FailureQ[x] returns True if x is equal to $Failed or $Aborted.

For more successful and flexible program processing of the main possible erroneous and exception situations that may occur in the procedure execution (Block, Module), it is recommended that the messages processing about them be programmed in the procedure, making it more reliable and steady against mistakes.

It should be noted that our tools for processing procedures and functions are oriented to the absence of attributes for them. Otherwise, the tools should be adapted to existence of attributes for the means being processed, which is easy to do.

2.6. Tools for testing of procedures and functions

Having defined procedures of two types (Module and Block) and functions, including pure functions, at the same time we have no standard tools for identification of objects of the given types. In this regard we created a series of means that allow to identify objects of the specified types. In the present section non–standard means for testing of procedural and functional objects are considered. In addition, it should be noted that the Mathematica – a rather closed system in contradistinction, for example, to its main competitor – the Maple system in which it is admissible to browse of source codes of its software which is located both in the main and in the auxiliary libraries. Whereas the Mathematica has no similar opportunity. In this connection the means presented below concerns only to the user functions and procedures loaded into the current session from a package (m– or mx–file), or a document (nb–file; also may contain a package) and activated in it.

For testing of objects onto procedural type we proposed a number of means among which it is possible to mark out such as ProcQ, ProcQ1, ProcQ2. The procedure call ProcQ[x] provides testing of an object x be as a procedural object {"Module","Block"}, returning accordingly True or False; while the ProcQ1 procedure is a useful enough modification of the ProcQ procedure, its call ProcQ1[x,t] returns True, if x – an object of the type Block, Module or DynamicModule, and "Others" or False otherwise; at that, the type of x object is returned through the factual t argument – an indefinite variable. The source codes of the above procedures, their description along with the most typical examples of their application are presented in our books and in our MathToolBox package [12-16]. A number of receptions used at their creation can be useful enough in the practical programming. The ProcQ procedure is an quite fast, processes attributes and options, but has certain restrictions, first of all, in a case of multiple objects of the same name. The fragment below represents source code the ProcQ procedure along with typical examples of its use.

In[7]:= ProcQ[x_] := Module[{a, atr = Quiet[Attributes[x]], b,

c, d, h}, If[! SymbolQ[x], False, If[SystemQ[x], False,

If[UnevaluatedQ[Definition2, x], False,

If[ListQ[atr] && atr != {}, ClearAllAttributes[x]];

a = StringReplace[ToString[InputForm[Definition[x]]],

Context[x] <> ToString[x] <> "`" –> ""];

Quiet[b = StringTake[a, {1, First[First[StringPosition[a,

{" := Block[{", " :=Block[{"}] – 1]]}];

c = StringTake[a, {1, First[First[StringPosition[a,

{" := Module[{", " :=Module[{"}] – 1]]}];

d = StringTake[a, {1, First[First[StringPosition[a,

{" := DynamicModule[{", " :=DynamicModule[{"}] – 1]]}]];

If[b === ToString[HeadPF[x]], SetAttributes[x, atr]; True,

If[c === ToString[HeadPF[x]], SetAttributes[x, atr]; True,

If[d === ToString[HeadPF[x]], SetAttributes[x, atr]; True,

SetAttributes[x, atr]; False]]]]]]]

In[8]:= Map[ProcQ, {Sin, a + b, ProcQ1, ProcQ, 77, ToString1}]

Out[8]= {False, False, True, True, False, True}

Generally, supposing existence of procedures of the above two types (Module and Block) in the Mathematica, for ensuring of the reliability it is recommended to use the procedures of the Module type. From examples represented in [8,10-15] follows, if local variables of a modular object aren't crossed with domain of values of the variables of the same name which are external in relation to it, the absolutely other picture takes place in a case with the local variables of a block object, namely: if initial values or values are ascribed to all local variables of such object in its body, they save effect in the object body; those variables of an object to which such values weren't ascribed accept values of the variables of the same name which are the external in relation to the block object. So, at the listed conditions the modular and block objects relative to the local variables (and in general as procedural objects) can quite be considered as equivalent. Naturally, the told remains in force also for block objects with empty lists of the local variables. The specified reasons were used as a basis for the programming of the RealProcQ procedure presented by the following fragment.

In[2107]:= RealProcQ[x_ /; BlockModQ[x]] := Module[{a, b, c},

If[ModuleQ[x], True,

a = StringJoin[Insert[Reverse[Headings[x]], " := ", 2]] <> "[";

b = StringReplace[PureDefinition[x], a –> "", 1];

Do[c = StringTake[b, k]; If[SyntaxQ[c], Return[c], 7],

{k, StringLength[b]}];

If[c == "{}", {}, c = StrToList[StringTake[c, {2, –2}]]];

c = Map[StringSplit[#, " = "] &, c];

AllTrue[Map[If[Length[#] > 1, True, False] &, c], TrueQ]]]

In[2108]:= B[x_] := Block[{a = 90, b = 50, c = {72, 77}, d = 42},

x*(a*b*c*d)]; RealProcQ[B]

Out[2108]= True

In[2109]:= M[x_] := Block[{a = 90, b = 49, c, h}, h = 77; x*h];

In[2110]:= RealProcQ[M]

Out[2110]= False

Experience of use of the RealProcQ procedure confirmed its efficiency at testing objects of the type Block that are considered as real procedures. At that, we will understand an object of the type {Module, Block} as a real procedure that in the Mathematica software is functionally equivalent to a Module, i.e. is a certain procedure in its classical understanding. The call RealProcQ[x] returns True if a symbol x defines a Module, or a Block which is equivalent to a Module, and False otherwise. In addition, it is supposed that a certain block is equivalent to a module if it has no the local variables or all its local variables have initial values or some local variables have initial values, while others obtain values by means of the operator "=" in the block body. From all our means solving the testing problem for procedural objects, the RealProcQ procedure with the greatest possible reliability identifies the set procedure in its classical understanding; thus, the real procedure can be of the type {Module, Block}.

In the context of providing of an object of the type {Module, Block} to be a real procedure recognized by the testing RealProcQ procedure, the ToRealProc procedure is of certain interest whose call ToRealProc[x, y] returns nothing, converting a module or a block x into the object of the same type and of the same name with the empty list of local variables that are placed in the object body at once behind the empty list of the local variables. At the same time all local variables of an object x are replaced with the symbols, unique in the current session, which are generated by means of the Unique function. In addition, along with a certain specified purpose the returned procedure provides a possibility of more effective debugging in interactive mode of procedure, allowing to do dynamic control of change of values of its local variables in the procedure run on concrete actual arguments. In addition to our means for testing of procedural objects, we can note a simple procedure, whose call UprocQ[x] returns False if an object x isn’t a procedure or is an object of the same name [8].

Meanwhile, since the structural definitions of modules and blocks are identical, the ToRealProc1 procedure that is based on this circumstance has a simple implementation. The following fragment represents source code of the ToRealProc1 procedure with typical examples of its application.

In[2229]:= B[x_] := Block[{a = 7, b, c = 8, d}, x*a*b*c*d];

In[2230]:= SetAttributes[B, {Protected, Listable, Orderless}]

In[2231]:= ToRealProc1[x_] := Module[{a = Attributes[x]},

If[ModuleQ[x], x, If[BlockQ[x],

If[FreeQ[a, Protected], 7, Unprotect[x]];

ToExpression[StringReplace[Definition2[x][[1]],

":= Block[{" –> ":= Module[{", 1]]; SetAttributes[x, a]; x, $Failed]]]

In[2232]:= ToRealProc1[B]

Out[2232]= B

In[2233]:= Definition[B]

Out[2233]= Attributes[B] = {Flat, Listable, Orderless, Protected}

B[x_] := Module[{a = 7, b, c = 8, d}, x*a*b*c*d]

In[2234]:= ToRealProc1[vsv]

Out[2234]= $Failed

Calling the ToRealProc1[x] procedure returns the module name that is equivalent to a module x or a block x, retaining all attributes and structure of the original object x. The scope of the module created is the current session. As it was noted above, in general case between procedures of types "Module" and "Block" exist principal distinctions that do not allow a priori to consider a block structure as a procedure in its above meaning. Therefore the type of a procedure should be chosen rather circumspectly, giving preference to the procedures of the "Module" type [8,12].

As noted repeatedly, Mathematica does not distinguish the objects by names, but by headers, allowing objects of the same name with different headers and even types (modules, functions, blocks) to exist in the current session. First of all, it is useful to define the function whose call TypeBFM[x] returns the type of an object x ("Block", "Function", "Module"), or $Failed otherwise.

In[7]:= B[x_] := Block[{a=7, b}, x*a*b]; B[x_, y_] := Module[{a, b},

x/y]; B[x_, y_, z_] := x^2 + y^2 + z^2; G[x_] := Module[{}, x*a]

In[8]:= TypeBFM[x_] := If[ModuleQ[x], "Module",

If[FunctionQ[x], "Function", If[BlockQ[x], "Block", $Failed]]]

In[9]:= TypeBFM[G]

Out[9]= "Module"

In[10]:= TypeQ[x_, y___] := Module[{a, b, c = {}, d = {}, t},

If[! SameQ[Head[x], Symbol], $Failed, a = Definition2[x][[1 ; –2]];

Map[{b = ToString[Unique[]],

t = StringReplace[#, ToString[x] <> "[" –> b <> "[", 1],

AppendTo[c, b], AppendTo[d, t]} &, a]; Map[ToExpression, d];

Map[ToExpression, c];

If[{y} != {} && SymbolQ[y], b = Map[Headings, c];

y = Map[{#[[1]], StringReplace[#[[2]],

"$" ~~ Shortest[__] ~~ "[" –> ToString[x] <> "["]} &, b];

y = If[Length[y] == 1, Flatten[y], y], 7]; a = Map[TypeBFM, c];

If[Length[a] == 1, a[[1]], a]]]

In[11]:= TypeQ[B]

Out[11]= {"Block", "Module", "Function"}

In[12]:= TypeQ[B, gs]

Out[12]= {"Block", "Module", "Function"}

In[13]:= gs

Out[13]= {{"Block", "B[x_]"}, {"Module", "B[x_, y_]"},

{"Function", "B[x_, y_, z_]"}}

In[14]:= {TypeQ[G, gv], gv}

Out[14]= {"Module", {"Module", "G[x_]"}}

Whereas the TypeQ procedure is primarily intended to test procedural and functional objects having multiple definitions of

the same name. Calling TypeQ[w] returns $Failed if w is not a symbol, whereas on w objects which have multiple definitions of the same name, the list is returned containing the subobject types that make up the w object in the form {"Block", "Function", "Module"}. While the calling TypeQ[w, j] with the 2^nd optional argument j – an undefined symbol – through j additionally returns the list in which for each sub-object of object w corresponds the 2–element list whose first element defines its type, whereas the 2^nd element defines its header in the string format. The previous fragment represents the source code of the TypeQ procedure and typical examples of its application.

Meanwhile, procedures, being generally objects other than functions, allow relatively simply conversion of functions into procedures, whereas under certain assumptions it is possible to convert procedures into functions too. Particularly, on example of the modular objects consider a principal scheme for this type of conversion algorithm that is based on the list structure.

Heading := {Save2[f, {locals, args}], Map[Clear, {locals, args}], {locals}, Procedure body, {Get[f], DeleteFile[f]}}[[–2]]

An algorithm of the ProcToFunc procedure, the original text of which together with examples of applications is represented below, is based on the above scheme. The list view of a module is generated in the format containing the following components:

Heading – the module or block heading;

Save2[f, {locals, args}] – saving of values of the joint list of local variable names without initial values and arguments without testing functions in a file f;

Map[Clear, {locals, args}] – clearing of the above arguments and local variables in the current session;

{locals} – the list of local variables with initial values, if any;

Procedure body – body of module or block;

{Get[f], DeleteFile[f]} – loading the f file to the current session, which contains the values of local variable names and arguments that were previously stored in it.

With in mind the told, the procedure call ProcToFunc[j] does not return anything, converting a module or block j into a list format function of the same name with preserving of attributes of the module or block. The following fragment represents the source code of the procedure with examples of its application.

In[42]:= ProcToFunc[j_ /; ProcQ[j]] := Module[{a, b, c, d, p},

a = Map[ToString[#] <> "@" &, Args[j]];

a = Map[StringReplace[#, "_" ~~ ___ ~~ "@" –> ""] &, a];

If[ModuleQ[j], b = Map[#[[1]] &, Locals4[j]]; c = Join[a, b];

d = Definition2[j][[1]]; d = StringReplace[d, "Module[" –>

"{Save2[\"##\"," <> ToString1[c] <> "], Map[Clear, " <>

ToString1[c] <> "],", 1]; d = StringReplacePart[d, "", {–1, –1}];

d = d <> ", {Get[\"##\"], DeleteFile[\"##\"]}}[[–2]]";

If[FreeQ[Attributes[j], Protected], ToExpression[d],

Unprotect[j]; ToExpression[d]; SetAttributes[j, Protected]],

b = Map[#[[1]] &, Locals4[j]]; c = Join[a, b];

p = Map[If[Length[#] != 1, #[[1]], Nothing] &, Locals4[j]];

p = Join[a, p]; d = Definition2[j][[1]];

d = StringReplace[d, "Block[" –>

"{Save2[\"##\"," <> ToString1[c] <> "], Map[Clear, " <>

ToString1[p] <> "],", 1]; d = StringReplacePart[d, "", {–1, –1}];

d = d <> ", {Get[\"##\"], DeleteFile[\"##\"]}}[[–2]]";

If[FreeQ[Attributes[j], Protected], ToExpression[d], Unprotect[j]; ToExpression[d]; SetAttributes[j, Protected]]]]

In[43]:= {a, b, c, x, y} = {1, 2, 3, 4, 5};

In[44]:= M[x_ /; IntegerQ[x], y_] := Module[{a = 5, b = 7, c},

a = a*x + b*y; c = a*b/(x + y); If[x*y > 100, a, c]]

In[45]:= SetAttributes[M, {Protected, Listable, Flat}]

In[46]:= ProcToFunc[M]

In[47]:= Definition[M]

Out[47]= Attributes[M] = {Flat, Listable, Protected}

M[x_/; IntegerQ[x], y_] := {Save2["##", {"x", "y", "a", "b", "c"}],

Clear /@ {"x", "y", "a", "b", "c"}, {a = 5, b = 7, c},

a = a*x + b*y; c = (a*b)/(x + y);

If[x*y > 100, a, c], {<< "##", DeleteFile["##"]}}[[–2]]

In[48]:= M[77, 72]

Out[48]= 889

In[49]:= {a, b, c, x, y}

Out[49]= {1, 2, 3, 4, 5}

In[50]:= B[x_, y_] := Block[{a = 5, b, c = 7}, a*x + b*c*y]

In[51]:= {a, b, c} = {72, 77, 67};

In[52]:= ProcToFunc[B]

In[53]:= Definition[B]

Out[53]= B[x_, y_] := {Save2["##", {"x", "y", "a", "b", "c"}],

Clear /@ {"x", "y", "a", "c"}, {a = 5, b, c = 7},

a*x + b*c*y, {<< "##", DeleteFile["##"]}}[[–2]]

In[54]:= B[42, 47]

Out[54]= 25543

In[55]:= {a, b, c}

Out[55]= {72, 77, 67}

Meantime, a module or block that is converted to a function must satisfy some of the conditions resulting from the inability to use certain built-in functions, such as Return, in Input mode. Whereas Goto and Label are quite permissible, for example:

In[338]:= f[x_] := {n = x; Label[g]; n++; If[n < 100, Goto[g], n]}; f[5]

Out[338]= {100}

In[339]:= f[x_] := {n = x; Label[g]; n++; If[n < 100, Goto[g], Goto[j]];

Label[j]; n}; f[5]

Out[339]= {100}

As a rule, the primary limitation for such conversion is size of the procedure's source code, making the function obscure. In general, the representation of modules and blocks in list format is in many cases more efficient in time.

Note, that when programming the algorithm of converting of a block into function, it should be taken into account that due to the specifics of the local variables in the blocks (as opposed to the modules), the function should save in a file the values of all local variables of the block, while the values of local variables with initial values of the block are cleared before the block body is executed when the resulting function is called. For the rest, the step of completing the resulting function is identical for the simulation of the block and module. In certain cases the above converting can be rather useful.

2.7. The nested blocks and modules

In Mathematica along with simple procedures that aren't containing in its body of definitions of other procedures, the application of the so–called nested procedures, i.e. procedures whose definitions allow definitions of other procedures in their body. The nesting level of such procedures is defined by only a size of work field of the system. Therefore a rather interesting problem of determination of the list of sub-procedures whose definitions are in the body of an arbitrary procedure of the type {Block, Module} arises. In this regard the SubProcs ÷ SubProcs3 procedures have been programmed [16] that successfully solve the problem. Here consider the SubProcs3 procedure whose call SubProcs3[x] returns the nested two–element list of the ListList type whose 1^st element defines the sub-list of headings of blocks and modules composing a main procedure x, while the second element defines the sub-list of the generated names of the blocks and modules composing the main procedure x, including itself procedure x, and that are activated in the current session of the Mathematica. The next fragment represents source code of the SubProcs3 procedure with the most typical examples of its use.

In[20]:= G[x_] := Module[{Vg, H77}, Vg[y_] := Module[{}, y^3];

H77[z_] := Module[{}, z^4]; x + Vg[x] + H77[x]];

G[x_, z_] := Module[{Vt, H, P}, Vt[t_] := Module[{}, t^3 + t^2 + 7];

H[t_] := Module[{}, t^4]; P[h_] := Module[{a = 590}, a^2 + h^2];

x + Vt[x] + H[z]*P[x]]; SetAttributes[G, {Protected, Listable}]

In[21]:= SubProcs3[y_, z___] := Module[{u = {}, m = 1, Sv,

v = Flatten[{PureDefinition[y]}]}, If[BlockFuncModQ[y],

Sv[S_String] := Module[{a = ExtrVarsOfStr[S, 1], b, c = {}, d,

t = 2, k = 1, cc = {}, n, p, j,

h = {StringTake[S, {1, Flatten[StringPosition[S, " := "]][[1]] – 1}]}},

a = Select[a, ! SystemQ[Symbol[#]] &&

! MemberQ[{ToString[G]}, #] &];

b = StringPosition[S, Map[" " <> # <> "[" &, a]];

p = Select[a, ! StringFreeQ[S, " " <> # <> "["] &]; b = Flatten[

Map[SubStrSymbolParity1[StringTake[S, {#[[1]], –1}], "[", "]"] &,

b]]; For[j = 1, j <= Length[p], j++, n = p[[j]];

For[k = 1, k <= Length[b] – 1, k++, d = b[[k]];

If[! StringFreeQ[d, "_"] && StringTake[b[[k + 1]], {1, 1}] == "[",

AppendTo[c, Map[n <> d <> " := " <> # <> b[[k + 1]] &,

{"Block", "Module"}]]]]]; c = DeleteDuplicates[Flatten[c]];

For[k = 1, k <= Length[c], k++, d = c[[k]];

If[! StringFreeQ[S, d], AppendTo[h, d], AppendTo[cc,

StringTake[d, {1, Flatten[StringPosition[d, " := "]][[1]] – 1}]]]];

{h, cc} = Map[DeleteDuplicates, {h, cc}];

p = Map[StringTake[#, {1, Flatten[StringPosition[#, "["]][[1]]}] &, h];

cc = Select[Select[cc, ! SuffPref[#, p, 1] &], ! StringFreeQ[S, #] &];

If[cc == {}, h, For[k = 1, k <= Length[cc], k++, p = cc[[k]];

p = StringCases[S, p <> " := " ~~ __ ~~ "; "];

AppendTo[h, StringTake[p,

{1, Flatten[StringPosition[p, ";"]][[1]] – 1}]]]]; Flatten[h]];

For[m, m <= Length[v], m++, AppendTo[u, Sv[v[[m]]]]];

u = Select[u, ! SameQ[#, Null] &]; u = If[Length[u] == 1, u[[1]], u];

If[{z} != {}, ToExpression[u]]; u, $Failed]]

In[22]:= SubProcs3[G]

Out[22]= {{"G[x_]", "Vg[y_] := Module[{}, y^3]",

"H77[z_] := Module[{}, z^4]"}, {"G[x_, z_]",

"Vt[t_] := Module[{}, t^3 + t^2 + 590]", "H[t_] := Module[{}, t^4]",

"P[h_] := Module[{a = 590}, a^2 + h^2]"}}

The SubProcs3 procedure is a rather useful extension of the SubProcs ÷ SubProcs2 procedures; its call SubProcs3[x] differs from a call SubProcs2[x] by the following 2 moments, namely: (1) the user block, function or module can act as an argument x, and (2) the returned list as the first element contains heading of the x object while other elements of the list present definitions of functions, blocks and modules in string format entering into the x definition. In a case of the x object of the same name, the returned list will be the nested list whose sublists have the above mentioned format. In addition, the call SubProcs3[x, y] with the second optional y argument – an arbitrary expression – returns the above list and at the same time activates in the current session all objects of the above type, which enter into the x. If function with heading acts as an object x, only its heading is returned; the similar result takes place and in a case of the x object which doesn't contain sub-objects of the above type whereas on the x object different from the user block, function or module, the call of the SubProcs3 procedure returns $Failed.

In some cases there is a necessity of definition for a module or block of the sub-objects of the type {Block, Function, Module}. The procedure call SubsProcQ[x, y] returns True if y is a global active sub-object of an object x of the above-mentioned type, and False otherwise. However, as the Math objects of this type differ not by names as that is accepted in the majority of programming systems, but by headings then thru the third optional argument the procedure call returns the nested list whose sub-lists as first element contain headings with a name x, whereas the second element contain the headings of sub-objects corresponding to them with y name. On the 1^st two arguments {x, y} of the types, different from given in a procedure heading, the procedure call SubsProcQ[x, y] returns False. The fragment below represents source code of the SubsProcQ procedure along with an example of its typical application.

In[7]:= SubsProcQ[x_, y_, z___] := Module[{a, b, k=1, j=1, Res = {}},

If[BlockModQ[x] && BlockFuncModQ[y],

{a, b} = Map[Flatten, {{Definition4[ToString[x]]},

Definition4[ToString[y]]}}];

For[k, k <= Length[b], k++, For[j, j <= Length[a], j++,

If[! StringFreeQ[a[[j]], b[[k]]], AppendTo[Res, {StringTake[a[[j]],

{1, Flatten[StringPosition[a[[j]], " := "]][[1]] – 1}],

StringTake[b[[k]], {1, Flatten[StringPosition[b[[k]]," := "]][[1]] – 1}]}],

Continue[]]]]; If[Res != {}, If[{z} != {} && ! HowAct[z],

z = If[Length[Res] == 1, Res[[1]], Res]; True], False], False]]

In[8]:= V[x_] := Block[{a, b, c}, a[m_] := m^2; b[n_] := n + Sin[n];

c[p_] := Module[{}, p]; a[x]*b[x]*c[x]]; c[p_] := Module[{}, p];

V[x_, y_] := Module[{a, b, c}, a[m_] := m^2; b[n_] := n + Sin[n];

c[p_] := Module[{}, p]; a[x]*b[x]*c[x]]; c[p_] := Module[{}, p]; p[x_] := x

In[9]:= {SubsProcQ[V, c, g72], g72}

Out[9]= {True, {{"V[x_]", "c[p_]"}, {"V[x_, y_]", "c[p_]"}}}

Except the means considered in books [8-15], a number of means for operating with sub-procedures is presented, here we will represent an useful SubsProcs procedure that analyzes the blocks and modules regarding presence in their definitions of sub-objects of type {Block, Module}. The call SubsProcs[x] returns generally the nested list of definitions in the string format of all sub-objects of type {Block, Module} whose definitions are in the body of an object x of type {Block, Module}. In addition, the first sub-list defines sub-objects of Module type, the second sub-list defines sub-objects of the Block type. In the presence of only one sub-list the simple list is returned whereas in the presence of the 1–element simple list its element is returned. In a case of lack of sub-objects of the above type, the call SubsProcs[x] returns the empty list, i.e. {} whereas on an object x, different from Block or Module, the call SubsProcs[x] is returned unevaluated. Fragment represents source code of the SubsProcs procedure and a typical example of its application.

In[1125]:= P[x_, y_] := Module[{Art, Kr, Gs, Vg, a},

Art[c_, d_] := Module[{b}, c + d]; Kr[n_] := Module[{}, n^2];

Vg[h_] := Block[{p = 90}, h^3 + p]; Gs[z_] := Module[{}, x^3];

a = Art[x, y] + Kr[x*y]*Gs[x + y] + Vg[x*y]]

In[1126]:= SubsProcs[x_/; BlockModQ[x]] := Module[{d, s = {}, g,

k = 1, p, h = "", v = 1, R = {}, Res = {}, a = PureDefinition[x], j,

m = 1, n = 0, b = {" := Module[{", " := Block[{"}, c = ProcBody[x]},

For[v, v <= 2, v++, If[StringFreeQ[c, b[[v]]], Break[],

d = StringPosition[c, b[[v]]]];

For[k, k <= Length[d], k++, j = d[[k]][[2]];

While[m != n, p = StringTake[c, {j, j}];

If[p == "[", m++; h = h <> p, If[p == "]", n++; h = h <> p,

h = h <> p]]; j++]; AppendTo[Res, h]; m = 1; n = 0; h = ""];

Res = Map10[StringJoin, If[v == 1, " := Module[", " := Block["],

Res]; g = Res; {Res, m, n, h} = {{}, 1, 0, "]"};

For[k = 1, k <= Length[d], k++, j = d[[k]][[1]] – 2;

While[m != n, p = StringTake[c, {j, j}];

If[p == "]", m++; h = p <> h,

If[p == "[", n++; h = p <> h, h = p <> h]]; j ––];

AppendTo[Res, h]; s = Append[s, j]; m = 1; n = 0; h = "]"];

Res = Map9[StringJoin, Res, g]; {g, h} = {Res, ""}; Res = {};

For[k = 1, k <= Length[s], k++,

For[j = s[[k]], j >= 1, j ––, p = StringTake[c, {j, j}];

If[p == " ", Break[], h = p <> h]]; AppendTo[Res, h]; h = ""];

AppendTo[R, Map9[StringJoin, Res, g]];

{Res, m, n, k, h, s} = {{}, 1, 0, 1, "", {}}];

R = If[Length[R] == 2, R, Flatten[R]]; If[Length[R] == 1, R[[1]], R]]

In[1127]:= SubsProcs[P]

Out[1127]= {{"Art[c_, d_] := Module[{b}, c + d]",

"Kr[n_] := Module[{}, n^2]", "Gs[z_] := Module[{}, x^3]"},

{"Vg[h_] := Block[{p = 90}, h^3 + p]"}}

The SubsProcs procedure can be rather simply expanded, in particular, for definition of the nesting levels of sub-procedures, and also onto unnamed sub-procedures. Moreover, in connection with the problem of nesting of the blocks and modules essential enough distinction between definitions of the nested procedures in the Maple and Mathematica takes place. So, in the Maple the definitions of the sub-procedures allow to use the lists of formal arguments identical with the main procedure containing them, while in the Mathematica similar combination is inadmissible, causing in the course of evaluation of determination of the main procedure erroneous situations [4-12]. Generally speaking, the given circumstance causes certain inconveniences, demanding a special attentiveness in the process of programming of the nested procedures. In a certain measure the similar situation arises and in a case of crossing of lists of formal arguments of the main procedure and the local variables of its sub-procedures whereas that is quite admissible in the Maple [8,22-41]. In this context the SubsProcs procedure can be applied successfully to procedures containing sub-procedures of the type {Block, Module} on condition of nonempty crossing of list of formal arguments of the main procedure along with the list of local variables of its sub-procedures (see also books on the Maple and Mathematica in our collections of works [6,14,15]). There you can also familiarize yourself with examples of nested procedures, useful in practical programming of various problems.

In principle, on the basis of the above tools it is possible to program a number of useful enough means of operating with expressions using the nested objects of type Block and Module. Some from them can be found in [15,16].

Chapter 3: Programming tools of the user procedure body

So far, we have considered mainly questions of the external design of any procedure, practically without touching upon the questions of its internal content – the body of the procedure that in the definition of procedure (module, block) is located between the list (may be empty) of local variables and the closing bracket "]". The basic components of the tools used to program the body of user procedures are discussed below. To describe an arbitrary computational algorithm of structures that are based on purely sequential operations, completely insufficient means for control the computing process. The modern structural programming focuses on one of the most error–prone factors: program logic and includes 3 main сomponents, i.e. top–down engineering, modular programming and structural coding. In addition, the first two components are discussed in sufficient detail in [3,15]. We will briefly focus here only on the third component.

The purpose of structural coding is to obtain the correct program (module) based on simple control structures. Control structures of sequence, branching, organization of cycles and function calls are selected as such basic structures (procedures, programmes). Note here that all said structures allow only one input and one output. At the same time, the first of 3 specified control structures (consecution, branching and cycle organization) make the minimum basis, on the basis of which it is possible to create a correct program of any complexity with one input, one output, without cycles and unattainable commands. Detailed discussion of the bases of control programming structures can be found, in particular, in [15] and in other available literature on the basics of modern programming.

Consecution reflects the principle of consistent running of the proposals of either or another program until some proposal changing the sequence occurs. For example, a typical sequence control structure is a sequence of simple assignment sentences in a particular programming language. Branching characterizes choosing one of all possible paths for next calculations; typical offers providing this control structure are offers of "if A then B else C" format.

"Cycle" structure performs a repeat execution of the offers as long as the some logical condition is true; typical offers that provide this control structure, offers of Do, Do_while, Do_until are. Basic structures define, respectively, sequential (following), conditional (branching) and iterative (loop) control transfer in programs [15]. It is shown, that any correct structured program theoretically of any complexity can be written only with using of control sequential structures, While–cycles and if–branching offers. Meantime, the extension of the set of specified tools in primarily by providing function calls along with mechanism of procedures can make programming very easy without breaking the structure programmes with increasing their modularity and robustness.

At the same time, combinations (iterations, attachments) of correct structured programs obtained on a basis of the specified control structure do not violate the principle of their degree of structure and correctness. Thus, the program of an arbitrary complexity and size can be obtained on a basis of an appropriate combination of extended basis (function calls, cycle, consecution, procedure mechanism and branching) of control structures. This approach allows to eliminate the use of labels and unconditional transitions in programs. The structure of such programs can be clearly traced from the beginning (top) to the end (down) in the absence of control transfers to the upper layers. So, exactly in light of this, it is languages of this type that represent a rather convenient linguistic support in the development of effective structured programs, combining the best traditions of modular- structural technology, that in this case is oriented towards the mathematical field of applications and is sufficient large number of programmatically unprofessional users from a range of the application fields, including not entirely mathematical focus.

In the following, the "sentence" or "offer" of Math–language means the construction of the following simple form, namely: Math–expression ";" where any correct Mathematica language

expression is allowed. Sentences are coded one after the other, each in a single string or multiple in one string; they in general are ended with separators ";", are processed strictly sequentially if the control structures of branching or loop do not define of a different order. The most widely used definition of a sentence of assignment is based on the operator "=" or ":=" accordingly of an immediate or delayed assignment which allow multiple assignments on the basis of list structure, namely:

{a, b, c, d, …}{=|:=}{m, n, p, k, …}

The basic difference between these formats consists in that the value g is evaluated at the time when expression g=a will be coded. On the other hand, a value g is not evaluated when the assignment g:=a is made, but is instead evaluated each time the value of g is requested. However, in some cases, computational constructs do not allow these formats to be used, in which case the following system built-in functions can be successfully used accordingly: Set[g,a] is equivalent of "g=a" and SetDelayed[g,a] is equivalent of "g:=a". So, in the list structure, the elements can be both comma-separated expressions and semicolon-separated sentences, for example:

In[2226]:= {g[t_] := t, s[t_] := t^2, v[t_] := t^3};

In[2227]:= Map[#[5] &, {g, s, v}]

Out[2227]= {5, 25, 125}

In[2228]:= Clear[g, s, v]

In[2229]:= {g[t_] := t; s[t_] := t^2; v[t_] := t^3};

In[2230]:= Map[#[5] &, {g, s, v}]

Out[2230]= {5, 25, 125}

Particularly at procedures writing the necessity to modify the value of a particular variable repeatedly often arises. The Mathematica provides special notations for modification of the variables values, and for some other often used situations. Let us give a summary of such notations:

k++ – increment the value of k by 1

k–– – decrement the value of k by 1

++k – pre–increment value of k by 1

––k – pre–decrement value of k by 1

k+=h – add h to the value of k

k–=h – subtract h from k

x*=h – multiply x by h

x/=h – divide x by h

PrependTo[g, e] – prepend e to a list g

AppendTo[g, e] – append e to a list g

Let us give examples for all these cases, namely:

In[2230]:= k = 77; k++; k

Out[2230]= 78

In[2231]:= k = 77; k––; k

Out[2231]= 76

In[2232]:= k = 77; ++k; k

Out[2232]= 78

In[2233]:= k = 77; ––k; k

Out[2233]= 76

In[2234]:= k = 77; h = 7; k += h; k

Out[2234]= 84

In[2235]:= k = 77; h = 7; k –= h; k

Out[2235]= 70

In[2236]:= k = 77; h = 7; k *= h; k

Out[2236]= 539

In[2237]:= k = 77; h = 7; k /= h; k

Out[2237]= 11

In[2238]:= g = {a, b, c, d, e, h}; AppendTo[g, m]; g

Out[2238]= {a, b, c, d, e, h, m}

In[2239]:= g = {a, b, c, d, e, h}; PrependTo[g, m]; g

Out[2239]= {m, a, b, c, d, e, h}

Typically, a considerable or large part of a procedure body consists of sequential structures, i.e. blocks from sequences of assignment sentences of different types, possibly involving calls to built–in and user functions. In principle, the most of bodies of procedures can be implemented in the form of a sequential structure, excluding cycles. True, such organization can results in voluminous codes, so the following control structures are of particular interest, namely branching structures and cycles.

3.1. Branching control structures in Mathematica

Conditional branching structures. Complex computation algorithms and/or control (first of all) cannot has with a purely sequential scheme, however include various constructions that change the sequential order of the algorithm depending on the occurrence of certain conditions: conditional and unconditional transitions, loops and branching (structure of such type is called control one). Thus, for the organization of control structures of branching type the Mathematica system has a number of tools provided by so-called If-offer, having the following two coding formats, namely:

If[CE, a, b] – returns a, if conditional expression CE gives True, and b if conditional expression CE gives False;

If[CE, a, b, c] – returns a, if conditional expression CE gives True, b if expression CE gives False and c if expression CE gives to neither True nor False.

The meaning of the first format of If sentence is transparent, being represent in one form or another in many programming languages. Whereas the second format in the context of other Mathematica tools is not quite correct, namely. If a Boolean w expression does not make sense True or False, then If[w, a, b, c] (by definition) returns c, while call BooleanQ[w] returns False if an expression w does not equal neither True nor False and a call If[BooleanQ[w],a,b,c] will return b, defining a certain dissonance, which illustrates the following rather simple example:

In[3336]:= If[77, a, b, c]

Out[3336]= c

In[3337]:= If[BooleanQ[77], a, b, c]

Out[3337]= b

In[3338]:= BooleanQ[77]

Out[3338]= False

In our view, in order to resolve such situation, it is enough useful to define in Mathematica system a Boolean value other than True and False, similar to the FAIL value in Maple. In the Maple concept, the FAIL symbol is used by logic to define both an unknown and undefined value. At the same time, if it is used inside Boolean expressions with operators and, or and not, its behaviour is similar to the case of standard 3–digit logic. Many Maple procedures and functions, primarily of the testing type, return the FAIL symbol if they cannot clearly establish the truth or falsity of a Boolean expression. In addition, the If function allows an arbitrary level of nesting, whereas as a and b can be any sequences of valid Mathematica clauses. In the context of Mathematica syntax, both format of If function can participate in expressions forming.

The If clause is the most typical tool of providing branching algorithms. In this context, it should be noted that the if tool of the Maple language and If of the Mathematica language appear to be largely equivalent, however in the sense of readability it is somewhat easier to perceive rather complex branching algorithms implemented precisely by the if clauses of Maple. It is difficult to give preference to anyone in this regard.

Thus, the If function ris the most typical tool for ensuring of the branching algorithms. In this context it should be noted that If means of the Maple and Mathematica are considerably equivalent, however readability of difficult enough branching algorithms realized by means of if offers of the Maple is being perceived slightly more clearly. So, Maple allows conditional if offer of the following format, namely:

if l1 then v1 elif l2 then v2 elif l3 then v3 elif l4 then v4 … else vk end

where j–th lj – a logical condition and vj – an expression whose sense is rather transparent and considered, for example, in [38]. This offer is convenient at programming of a lot of conditional structures. For definition of a similar structure in Mathematica, the IFk procedure whose source code with examples of its use represents the following fragment can be used, namely:

In[3212]:= IFk[x__] := Module[{a = {x}, b, c = "", d = "If[", e = "]",

h = {}, k = 1}, b = Length[a];

If[For[k, k <= b – 1, k++,

AppendTo[h, b >= 2 && ListQ[a[[k]]] && Length[a[[k]]] == 2]];

DeleteDuplicates[h] != {True}, Return[Defer[IFk[x]]], k = 1];

For[k, k <= b – 1, k++,

c = c <> d <> ToString[a[[k]][[1]]] <> "," <>

ToString[a[[k]][[2]]] <> ","];

c = c <> ToString[a[[b]]] <> StringMultiple[e, b – 1];

ToExpression[c]]

In[3213]:= IFk[{a, b}, {c, d}, {g, s}, {m, n}, {q, p}, h]

Out[3213]= If[a, b, If[c, d, If[g, s, If[m, n, If[q, p, h]]]]]

The call of IFk procedure uses any number of the factual arguments more than one; the arguments use the two–element lists of the form {lj, vj}, except the last. Whereas the last factual argument is a correct expression; in addition, a testing of a lj on Boolean type isn’t done. The call of IFk procedure on a tuple of the correct factual arguments returns the result equivalent to execution of the corresponding Maple offer if.

The IFk1 procedure is an useful extension of the previous procedure which unlike IFk allows only Boolean expressions as factual arguments lj, otherwise returning the unevaluated call. In the rest, the IFk and IFk1 are functionally identical. Thus, similarly to the Maple offer if, the procedures IFk and IFk1 are quite useful at programming of the branching algorithms of the different types. With that said, the above procedures IFk, IFk1 are provided with a quite developed mechanism of testing of the factual arguments transferred at the procedure call whose algorithm is easily seen from the source code (see below).

In[7]:= IFk1[x__] := Module[{a = {x}, b, c = "", d = "If[", e = "]",

h = {}, k = 1}, b = Length[a];

If[For[k, k <= b – 1, k++,

AppendTo[h, b >= 2 && ListQ[a[[k]]] && Length[a[[k]]] == 2]];

DeleteDuplicates[h] != {True},

Return[Defer[IFk1[x]]], {h, k} = {{}, 1}];

If[For[k, k <= b – 1, k++, AppendTo[h, a[[k]][[1]]]];

Select[h, ! MemberQ[{True, False}, #] &] != {},

Return[Defer[IFk1[x]]], k = 1];

For[k = 1, k <= b – 1, k++,

c = c <> d <> ToString[a[[k]][[1]]] <> "," <>

ToString[a[[k]][[2]]] <> ","];

c = c <> ToString[a[[b]]] <> StringMultiple[e, b – 1];

ToExpression[c]]

In[8]:= IFk1[{False, b}, {False, d}, {False, s}, {True, G}, {False, p}, h]

Out[8]= G

Now, using the described approach fairly easy to program in Math–language an arbitrary Maple construction describing the branching algorithms [27–41].

In a number of cases the simple Iff procedure with number of arguments in quantity 1÷n which generalizes the standard If function is quite useful tool; it is very convenient at number of arguments, starting with 1, what is rather convenient in those cases when calls of the Iff function are generated in a certain procedure automatically, simplifying processing of erroneous and especial situations arising at the call of such procedure at number of arguments from range 2..4. The following fragment represents source code of the Iff procedure with an example. In addition, it must be kept in mind that all its factual arguments, since the second, are coded in string format in order to avoid their premature calculation at the call Iff[x, ...] when the factual arguments are being evaluated/simplified.

In[1114]:= Iff[x_, y__ /; StringQ[y]] := Module[{a = {x, y}, b},

b = Length[a];

If[b == 1 || b >= 5, Defer[Iff[x, y]],

If[b == 2, If[x, ToExpression[y]],

If[b == 3, If[x, ToExpression[y], ToExpression[a[[3]]]],

If[b == 4, If[x, ToExpression[a[[2]]], ToExpression[a[[3]]],

ToExpression[a[[4]]]], Null]]]]]

In[1115]:= a = {}; For[k = 1, k <= 77, k++, Iff[PrimeQ[k],

"AppendTo[a, k]"]]; a

Out[1115]= {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,

59, 61, 67, 71, 73}

To a certain degree to the If constructions adjoins the Which function of the following format:

Which[lc1, w1, lc2, w2, lc3, w3, ..., lck, wk]

which returns the result of evaluation of the first wj expression for which Boolean expression lcj (j = 1..k) accepts True value. If some of the evaluated conditions lcj doesn't return {True|False} the function call is returned unevaluated while in a case of False for all lcj conditions (j = 1..k) the function call returns Null, i.e. nothing. At dynamical generation of a Which object the simple procedure WhichN can be a rather useful that allows any even number of arguments similar to the Which function, otherwise returning result unevaluated. In the rest, the WhichN is similar to the Which function; the following fragment represents source code of the WhichN procedure with a typical example of its use.

In[47]:= WhichN[x__] := Module[{a = {x}, c = "Which[", d,

k = 1}, d = Length[a]; If[OddQ[d], Defer[WhichN[x]],

ToExpression[For[k, k <= d, k++, c = c <>

ToString[a[[k]]] <> ","]; StringTake[c, {1, –2}] <> "]"]]]

In[48]:= f = 90; WhichN[False, b, f == 90, SV, g, h, r, t]

Out[48]= SV

The above procedures Iff, IFk, IFk1 and Which represent a quite certain interest at programming a number of applications of various purpose, first of all, of the system character (similar tools are considered in our collection [15] and package MathToolBox).

Unconditional transitions. At that, control transfers of the unconditional type are defined in the language, usually, by the Goto constructions by which control is transferred to the Label point of the procedure specified by the corresponding Label. In Mathematica language, can to usage unconditional transitions based on the built-in Goto function encoded in the Goto[Label] format to organize the branching of algorithms along with the presented If clause. Meanwhile, in a number of cases, the use of this tool is very effective, in particular when it is necessary to load in Mathematica software the programs using unconditional transitions based on Goto offer. Particularly, Fortran programs, rather common in scientifical applications, are a fairly typical example. From our experience it should be noted that the use of Goto function greatly simplified immersion in Mathematica of Fortran–programs related to engineering and physical direction which rather widely use Goto–constructions. Note that, unlike Mathematica, the goto function in Maple only makes sense in the procedure body, allowing from the goto(Label) call point to go to a sentence preceded by the specified Label. Meanwhile, in Mathematica the Goto function is permissible also outside of procedures, although that makes little sense. There can be an arbitrary g expression as a Label, as the following rather simple fragment illustrates, namely:

In[1900]:= G = {}; p = 1; Label[agn]; AppendTo[G, p++];

If[p <= 10, Goto[agn], G]

Out[1900]= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

The above fragment illustrates a simple example of cyclic computations on a basis of Goto function without using built-in standard loop functions. However, such cyclic constructions (in the temporary relation) substantially concede to the cycles, for example, based on the built–in function Do, for example:

In[5]:= p = 0; Timing[Label[g]; If[p++ < 999999, Goto[g], p]]

Out[5]= {1.63801, 1000000}

In[6]:= p = 0; Timing[Do[p++, 1000000]; p]

Out[6]= {0.202801, 1000000}

Time differences are quite noticeable, starting with a cycle length greater than 4000 steps. However, cyclic constructions based on the unconditional Goto transition are in some cases quite effective at programming of the procedure bodies. Note another point is worth paying attention to, namely. It is known [20-41] that in Maple, an unconditional goto(Label) transition assumes a global variable as a Label, whereas the definition of Label as a local (that not test as an error in the procedure definition calculation stage) initiates an erroneous situation when calling the procedure. Whereas the Mathematica in an unconditional Goto[Label] transition as a Label allows both global variable and local along with an arbitrary expression, for example:

In[2254]:= Sv[k_Integer] := Module[{p = 0, g}, Label[g];

If[p++ <= k, Goto[g], {g, p}]]

In[2255]:= Sv[10000]

Out[2255]= {g$16657, 10002}

In[2256]:= Sv1[k_Integer, x_] := Module[{p = 0}, Label[x];

If[p++ <= k, Goto[x], {x, p}]]

In[2257]:= Sv1[10000, a*Sin[b] + c*Cos[d]]

Out[2257]= {c*Cos[d] + a*Sin[b], 10002}

Meanwhile, in order to avoid any misunderstandings, the Label is recommended to be defined as a local variable because the global Label calculated outside of a module will be always acceptable for the module, however calculated in the module body quite can distort calculations outside of the module. At that, multiplicity of occurrences of identical Goto functions into a procedure is a quite naturally and is defined by the realized algorithm while with the corresponding tags Label the similar situation, generally speaking, is inadmissible; in addition, it is not recognized at evaluation of a procedure definition and even at a stage of its performance, often substantially distorting the planned task algorithm. In this case only point of the module body that is marked by the first such Label receives the control. Moreover, it must be kept in mind that lack of a Label[j] for the corresponding call Goto[j] in a block or a module at a stage of evaluation of its definitions isn't recognized, however only at the time of performance with the real appeal to such Goto[j]. Interesting examples illustrating the told can be found in [6-15].

In this connection the GotoLabel procedure can represent a quite certain interest whose call GotoLabel[j] allows to analyse a procedure j on the subject of formal correctness of use of Goto functions and the Label tags corresponding to them. So, the call GotoLabel[j] returns the nested 3–element list whose the first element defines the list of all Goto functions used by a module j, the second element defines the list of all tags (excluding their multiplicity), the third element determines the list whose sublists determine Goto functions with the tags corresponding to them (in addition, as the first elements of these sub-lists the calls of the Goto function appear, whereas multiplicities of functions and tags remain). The fragment below represents source code of the GotoLabel procedure along with typical examples of its application.

In[7]:= GotoLabel[x_ /; BlockModQ[x]] := Module[{b, d, p, k = 1,

c = {{}, {}, {}}, j, h, v = {}, t, a = Flatten[{PureDefinition[x]}][[1]]},

b = ExtrVarsOfStr[a, 1];

b = DeleteDuplicates[Select[b, MemberQ[{"Label", "Goto"}, #] &]];

If[b == {}, c, d = StringPosition[a, Map[" " <> # <> "[" &,

{"Label", "Goto"}]]; t = StringLength[a];

For[k, k <= Length[d], k++, p = d[[k]]; h = ""; j = p[[2]];

While[j <= t, h = h <> StringTake[a, {j, j}];

If[StringCount[h, "["] == StringCount[h, "]"],

AppendTo[v, StringTake[a, {p[[1]] + 1, p[[2]] – 1}] <> h];

Break[]]; j++]]; h = DeleteDuplicates[v];

{Select[h, SuffPref[#, "Goto", 1] &],

Select[h, SuffPref[#, "Label", 1] &], Gather[Sort[v], #1 ==

StringReplace[#2, "Label[" –> "Goto[", 1] &]}]]

In[8]:= ArtKr[x_ /; IntegerQ[x]] := Module[{prime, agn},

If[PrimeQ[x], Goto[9; prime], If[OddQ[x], Goto[agn],

Goto[Sin]]]; Label[9; prime]; Print[x^2]; Goto[Sin];

Print[NextPrime[x]]; Goto[Sin]; Label[9; prime]; Null]

In[9]:= GotoLabel[ArtKr]

Out[9]= {{"Goto[9; prime]", "Goto[agn]", "Goto[Sin]"},

{"Label[9; prime]"}, {{"Goto[9; prime]", "Label[9; prime]",

"Label[9; prime]"}, {"Goto[agn]"}, {"Goto[Sin]", "Goto[Sin]",

"Goto[Sin]"}}}

In[10]:= Map[GotoLabel, {GotoLabel, TestArgsTypes, CallsInMean}]

Out[10]= {{{}, {}, {}}, {{}, {}, {}}, {{}, {}, {}}}

In[11]:= Map[GotoLabel, {SearchDir, StrDelEnds, OP, BootDrive}]

Out[11]= {{{}, {}, {}}, {{}, {}, {}}, {{"Goto[ArtKr]"}, {"Label[ArtKr]"},

{{"Goto[ArtKr]", "Label[ArtKr]"}}}, {{"Goto[avz]"}, {"Label[avz]"},

{{"Goto[avz]", "Goto[avz]", "Goto[avz]", "Label[avz]"}}}

We will note that existence of the nested list with the third sub-list containing Goto–functions without tags corresponding to them, in the result returned by means call GotoLabel[j] not necessarily speaks about existence of the function calls Goto[j] for which not exists any Label[j] tag. It can be, for example, in the case of generation of a value depending on some condition.

In[320]:= Av[x_Integer, y_Integer, p_ /; MemberQ[{1, 2, 3}, p]] :=

Module[{}, Goto[p]; Label[1]; Return[x + y]; Label[2];

Return[N[x/y]]; Label[3]; Return[x*y]]

In[321]:= Map[Av[590, 90, #] &, {1, 2, 3, 4}]

Out[321]= {680, 6.55556, 53100, Av[590, 90, 4]}

In[322]:= GotoLabel[Av]

Out3[22]= {{"Goto[p]"}, {"Label[1]", "Label[2]", "Label[3]"},

{{"Goto[p]"}, {"Label[1]"}, {"Label[2]"}, {"Label[3]"}}}

For example, according to simple example of the previous fragment the call GotoLabel[Av] contains {"Goto[p]"} in the third sub-list what, at first sight, it would be possible to consider as an impropriety of the corresponding call of Goto function. However, all the matter is that a value of actual p argument in the call Av[x,y,p] and defines a tag really existing in definition of the procedure, i.e. Label[p]. Therefore, the GotoLabel module only at the formal level analyzes existence of the Goto functions, "incorrect" from its point of view with "excess" tags. Whereas refinement of the results received on a basis of a procedure call GotoLabel[W] lies on the user, above all, by means of analysis of accordance of source code of the W to the correctness of the required algorithm.

As a Label may be a correct sentence, for example:

In[9]:= a[x_, y_] := Module[{a, b, c}, If[x/y > 7, Goto[a], Goto[b]];

Label[a[t_] := t^2; a]; c = a[x]; Goto[Fin]; Label[b[t_] := t^3; b];

c = b[y]; Label[Fin]; c]

In[10]:= {a[7, 12], a[77, 7]}

Out[10]= {1728, 5929}

The structured paradigm of programming doesn't assume application in programs of the Goto constructions allowing to transfer control from bottom to top. At the same time, in some cases the use of Goto function is rather effective, for example, at needing of embedding into Mathematica of programs which use unconditional transitions on a basis of the goto offer. Thus, Fortran programs can be adduced as a typical example that are very widespread in scientific appendices. From our experience follows, that the use of Goto function allowed significantly to simplify embedding into Mathematica of a number of rather large Fortran programs relating to the engineering and physical applications that very widely use the goto constructions. Right there it should be noted that from our standpoint the function Goto of Mathematica is more preferable, than goto function of Maple in respect of efficiency in the light of application in the procedural programming of various appendices, including also appendices of the system character.

3.2. Cyclic control structures in Mathematica

One of the basic cyclic structures of Mathematica is based on the For function having the following coding format:

For[start of a loop variable, test, increment, loop body]

Starting from a given initial value of loop variable, the loop body that contains language sentences is cyclically computed, with the loop variable cyclically incremented by an increment until the logical condition (test) remains True. The control words Continue[] (continuation) and {Break[], Return[]} (termination) respectively are used to control the For–cycle, as the following example well illustrates:

In[21]:= a := {7, 6, c, 8, d, f, g}; b := {a1, d, c1, 77, f1, m, n, t};

In[22]:= For[k = 1, k <= 1000, k++, p = a[[k]]; If[PrimeQ[p],

Break[p], If[IntegerQ[p], Return[], Continue[]]]]

Out[22]= 7

In[23]:= For[k = 1, k <= 1000, k++, p = b[[k]]; If[PrimeQ[p],

Break[p], If[IntegerQ[p], Return[p], Continue[]]]]

Out[23]= Return[77]

In[24]:= For[k = 1, k <= 1000, k++, p = b[[k]]; If[PrimeQ[p],

Break[], If[IntegerQ[p], Print[{k, p}]; Return[], Continue[]]]]

{4, 77}

Out[24]= Return[]

In[25]:= A[x___] := Module[{}, For[k = 1, k <= Infinity,

k++, p = b[[k]]; If[PrimeQ[p], Break[], If[IntegerQ[p],

Return[p], Continue[]]]]]

In[26]:= A[]

Out[26]= 77

In[27]:= G[x_] := (If[x > 10, Return[x]]; x^2)

In[28]:= G[90]

Out[28]= 90

Meanwhile, note in mind that Return[] terminates the loop both outside and inside the procedure or function. In addition, in the first case, the loop ends by Return[], while in the second case the termination of the loop by Return allows by means of the call Return[E] additionally to return a certain E expression, for example, a result at the end of the loop. The For-loop allows an arbitrary level of nesting, depending on the size of working area of the software environment. Moreover, in both cases by means of Break[gsv] can not only terminate the loop, but also return the required gsv expression.

In our books [1-3] the reciprocal functional equivalence of both systems is visually illustrated when the most important computing structures of Mathematica with this or that efficiency are simulated by Maple constructions and vice versa. Truly, in principle, it is a quite expected result because built-in languages of both systems are universal and in this regard with one or the other efficiency they can program an arbitrary algorithm. But in the time relation it is not so and at use of the loop structures of a rather large nesting level the Maple can have rather essential advantages before Mathematica. For confirmation we will give a simple example of a loop construction programmed both in the Maple, and the Mathematica system.

The results speak for themselves – if for the Maple 11 the execution of the following for–loop of the nesting 8 requires 7.8 s, the Mathematica 12.1 for execution of the same loop on base of For–function requires already 50.918 s, i.e. approximately 6.5 times more (estimations have been obtained on Dell OptiPlex 3020, i5-4570 3.2 GHz with 64-bit Windows 7 Professional 6.1.7601 service pack 1 Build 7601). In addition, with growth of the nesting depth and range of a loop variable at implementation of the loops this difference rather significantly grows.

> t := time(): for k1 to 10 do for k2 to 10 do for k3 to 10 do for k4 to 10 do for k5 to 10 do for k6 to 10 do for k7 to 10 do for k8 to 10 do 900 end do end do end do end do end do end do end do end do: time() – t; # (Maple 11)

7.800

In[8]= n = 10; t = TimeUsed[]; For[k1 = 1, k1 <= n, k1++,

For[k2 = 1, k2 <= n, k2++, For[k3 = 1, k3 <= n, k3++,

For[k4 = 1, k4 <= n, k4++, For[k5 = 1, k5 <= n, k5++,

For[k6 = 1, k6 <= n, k6++, For[k7 = 1, k7 <= n, k7++,

For[k8 = 1, k8 <= n, k8++, 500]]]]]]]]; TimeUsed[] - t

Out[8]= 50.918

Our long experience [1-42] of using Maple and Matematica systems of different versions has clearly shown that, in general, programming of loops in the first system is more efficient in the temporal relation.

As another fairly widely used tool in Mathematica for the organization of cyclic calculations is the Do-function, which has the following six coding formats, namely:

Do[E, n] – evaluates an expression E n times;

Do[E, {j, p}] – evaluates E with variable j successively taking

on the values 1 through k (in steps of 1);

Do[E, {j, k, p}] – starts with j = k;

Do[E, {j, k, p, dt}] – uses steps dt;

Do[E, {j, {k1, k2,…}}] – uses the successive values k1, k2,…;

Do[Exp, {j, j1, j2}, {k, k1, k2},…] – evaluates Exp looping over

different values of k etc. for each j.

In the above Do-constructions, there can be both individual expressions and sequences of language sentences as E and Exp (body). To exit the Do loop prematurely or continue it, the body of a loop uses functions Return, Break, Continue; in the case of the nested Do loop, control in such constructions is transferred to the external loop. Examples of Do-loops of the above formats

In[11]:= t := 0; Do[t = t + 1, 1000000]; t

Out[11]= 1000000

In[12]:= t := 0; Do[t = t + k^2, {k, 1000}]; t

Out[12]= 333833500

In[13]:= t := 0; Do[t = t + k^2, {k, 1000, 10000}]; t

Out[13]= 333050501500

In[14]:= t := 0; Do[t = t + k^2, {k, 1000, 10000, 20}]; t

Out[14]= 16700530000

In[15]:= t := 0; Do[t = t + k^2 + j^2 + h^2, {k, 10, 100},

{j, 10, 100}, {h, 10, 100}]; t

Out[15]= 8398548795

In[16]:= A[x_] := Module[{t = 0}, Do[If[t++ > x, Return[t],

Continue[]], Infinity]]; A[900]

Out[16]= 902

In[17]:= A1[x_] := Module[{t = 0}, Do[If[t++ > x, Break[t],

Continue[]], Infinity]]; A1[900]

Out[17]= 902

In[18]:= t = 0; Do[If[t++ > 100, Return[t], Continue[]], 1000]

Out[18]= 102

In[19]:= t = 0; Do[If[t++ > 100, Break[t], Continue[]], 1000]

Out[19]= 102

In[20]:= t := 0; k = 1; Do[t = t + k^2; If[k >= 30, Break[t],

Continue[]], {k, 1000}]; {t, k}

Out[20]= {9455, 1}

Note that Break and Return make sense for exit of Do loop both within a procedure containing this loop, and outside it. At the same time, the loop variable is local (see the last example of the previous fragment). Here is a list of several more Mathematica functions for cyclic computing:

While[E, body] – evaluates E, then repetitively evaluates body

until E to give True;

In[3338]:= {t, v} = {0, {}}; While[t++; t <= 100,

s = Total[AppendTo[v, t]]]; s

Out[3338]= 5050

Nest[F, E, n] – returns the result of application of a function F to

an E expression n times;

In[3339]:= Nest[F, (a + b + c), 6]

Out[3339]= F[F[F[F[F[F[a + b + c]]]]]]

FixedPoint[F, E] – starting with E, F is applied repeatedly until two nearby results will identical; in addition FixedPoint[F, E, n] stops after at most n steps. FixedPoint returns the last result it gets.

In[3340]:= FixedPoint[Sqrt, 19.42, 15]

Out[3340]= 1.00009

The FixedPoint1 procedure is a program equivalent of the FixedPoint based on Do loop:

In[5]:= FixedPoint1[f_, e_, n___] := Module[{a = e, b, p = 1},

Do[b = f @@ {a}; If[SameQ[b, a], Return[], a = b; p++;

Continue[]], If[{n} == {}, 10^10, n]]; {p – 1, b}]

In[6]:= FixedPoint1[Sqrt, 19.42, 15]

Out[6]= {15, 1.00009}

In[7]:= FixedPoint1[Sqrt, 19.42]

Out[7]= {52, 1.}

The procedure call FixedPoint1[f, e, n] returns the list of the form {p, FixedPoint[f, e, n]}, where p is the number of steps of the procedure required to obtain the desired result, whereas n is optional argument similar to n of FixedPoint[f, e, n].

NestWhile[F, E, Test] – starting with E, F is applied repeatedly

until applying Test to the result gives False;

In[1119]:= N[NestWhile[Sqrt, 90050, # > 7 &]]

Out[1119]= 4.16208

The NestWhile1 procedure is a program equivalent of the NestWhile based on Do loop:

In[1132]:= NestWhile1[f_, e_, t_] := Module[{a = e, b, p = 1},

Do[b = f @@ {a}; If[! t @@ {b}, Return[], a = b; p++;

Continue[]], Infinity]; {p, b}]

In[1133]:= N[NestWhile1[Sqrt, 90050, # > 7 &]]

Out[1133]= {3., 4.16208}

The procedure call NestWhile1[f, e, t] returns the list of the form {p, NestWhile[f, e, t]}, where p is the number of steps of the procedure required to obtain the desired result, whereas t is the t test analogous to t for NestWhile[f, e, t]. At that, in addition to the presented format the NestWhile has additional five formats, with which the reader can familiarize in the Mathematica help.

TakeWhile[L, t] – returns elements of a list L from the beginning

of the list, continuing so long as t for the L elements gives True.

In[1134]:= TakeWhile[{a, b, c, 7, e, r, 15, k, h, m, n, g, s, w},

SymbolQ[#] || PrimeQ[#] &]

Out[1134]= {a, b, c, 7, e, r}

The TakeWhile1 procedure is a program equivalent of the TakeWhile based on Do loop:

In[1135]:= TakeWhile1[L_List, t_] := Module[{p = {}},

Do[If[t @@ {L[[j]]}, AppendTo[p, L[[j]]], Return[{j, p}]],

{j, Length[L]}]]

In[1136]:= TakeWhile1[{a, b, c, 7, e, r, 15, k, h, m, n, k, h},

SymbolQ[#] || PrimeQ[#] &]

Out[1136]= {7, {a, b, c, 7, e, r}}

The procedure call TakeWhile1[L, T] returns the list of the form {p, TakeWhile[L, T]}, where p is the number of steps of the procedure required to obtain the desired result, whereas T is the test analogous to T in TakeWhile[L, T].

Catch[exp] – returns the argument of the first Throw call that is

generated in the evaluation of an exp expression.

In[4906]:= Catch[Do[If[j > 100 && PrimeQ[j], Throw[j]],

{j, Infinity}]]

Out[4906]= 101

At that, in addition to the presented format the Catch has additional two formats, with which the reader can familiarize in [15] or in the Mathematica help. The Break, Return, Goto and Continue functions are used to control whether the loop exits or continues, respectively. Other functions that make substantial use of cyclic computing can be found in the Mathematica help. In addition, most of them have rather simple program analogs using the basic functions Do, For of the loop along with function If of conditional transitions and functions of loop termination control Break, Return, Continue to organize cyclic calculations. Some instructive examples of similar program counterparts are presented above. The above largely applies to both simple and the nested loops whose attachments can alternating and shared different types of loops.

3.3. Special types of cyclic control structures in Mathematica

Along with the basic cyclic structures discussed above, the Mathematica has a number of useful special control structures of cyclic type, which allow to significantly simplify the solution of a number of important tasks. Such structures are realized on the basis of a number of built-in functions Map, MapAt, MapAll, MapIndexed, Select, Sum, Product, etc., allowing to describe the algorithms of mass tasks of processing and calculations in a very compact way. In addition, not only good visibility of programs is ensured, but also rather high efficiency of their execution. At the same time, first of all, the tools of the so–called Map group, among which the Map function is the most used, appear to be important. Consider this group a little more detailed.

The main Map function has two formats, namely:

Map[F, exp] or F/@exp – returns the result of applying of F to

each element of the first level of an expression exp;

Map[F, exp, levels] – returns the result of applying of F to parts

of an expression exp specified by levels; as the values for levels

argument can be:

n – levels from 1 through n

Infinity – levels from 1 through Infinity

{n} – level n only

{n1, n2} – levels from n1 through n2

Here are some examples of management by Map execution:

In[1]:= A[x_, y_] := Module[{t = {}}, Map[If[# <= x,

AppendTo[t, F[#]], Goto[g]] &, Range[y]]; Label[g]; t]

In[2]:= A[7, 90]

Out[2]= {F[1], F[2], F[3], F[4], F[5], F[6], F[7]}

In[3]:= A1[x_, y_] := Module[{t = {}}, CheckAbort[Map[

If[# <= x, AppendTo[t, F[#]], Abort[]] &, Range[y]], t]]

In[4]:= A1[7, 90]

Out[4]= {F[1], F[2], F[3], F[4], F[5], F[6], F[7]}

In[5]:= Map[g, Map[If[OddQ[#], #, Nothing] &, Range[18]]]

Out[5]= {g[1], g[3], g[5], g[7], g[9], g[11], g[13], g[15], g[17]}

In[6]:= t = {}; CheckAbort[Map[If[# <= 7, AppendTo[t, F[#]],

Abort[]] &, Range[90]], t]

Out[6]= {F[1], F[2], F[3], F[4], F[5], F[6], F[7]}

In[7]:= Map[j, (a + b)/(c – d)*m*n, Infinity]

Out[7]= j[m]*j[n]*j[j[a] + j[b]]*j[j[j[c] + j[j[–1]*j[d]]]^j[–1]]

MapAll[f, exp] is equivalent to Map[f, exp, {0, Infinity}]. At the same time, the call MapAll[f, exp, Heads –> True] returns the result of applying of f to heads in exp additionally. Whereas the MapIndexed function also have two formats, namely:

MapIndexed[F, exp] – returns the result of applying of F to the elements of an expression exp, giving the part number of each element as the second argument to F;

MapIndexed[F, exp, levels] – returns the result of applying of F to all parts of exp on levels specified by levels.

As a program implementation of the MapIndexed function of the first format can be given:

In[20]:= MapIndexed[F, (a + b)/(c – d) + m*n]

Out[20]= F[(a + b)/(c – d), {1}] + F[m*n, {2}]

In[21]:= MapIndexed1[F_, e_] := Module[{p = 1}, Part[e, 0]

@@ Map[F[#, {p++}] &, Op[e]]]

In[22]:= MapIndexed1[F, (a + b)/(c – d) + m*n]

Out[22]= F[(a + b)/(c – d), {1}] + F[m*n, {2}]

The program implementation of the second format of the MapIndexed function we leave to the reader as a useful exercise.

The SubsetMap function has two formats, namely:

SubsetMap[g, j, j1] – returns the result of replacing of elements of a j list by the corresponding elements (whose positions are defined by a list j1) of the list obtained by evaluating g[j].

SubsetMap[g, j, L] – returns the result of replacing of elements jn of an expression j at positions L on g[jn].

Here are some examples of the SubsetMap use:

In[6]:= SubsetMap[Power[#, 3] &, {a, b, c, d, g, h, t}, {1, 3, 5, 7}]

Out[6]= {a^3, b, c^3, d, g^3, h, t^3}

In[7]:= SubsetMap[Power[#, 7] &, {a, b, c, d, g, h, m, t}, 3;7]

Out[7]= {a, b, c^7, d^7, g^7, h^7, m^7, t}

As a program implementation of the SubsetMap1 function of the first format for case of lists as an argument j can be given:

In[9]:= SubsetMap1[F_, j_, L_] := Map[If[MemberQ[L,

Flatten[Position[j, #]][[1]]], g[#], #] &, j]

In[10]:= SubsetMap1[g, {a, b, c, d, g, h, m, t}, {3, 4, 5, 6, 7}]

Out[10]= {a, b, g[c], g[d], g[g], g[h], g[m], t}

The program realization of the more common cases of the SubsetMap function we leave to the reader as a useful exercise.

Frequently used functions such as Sum, Product, Delete and Select also may be carried to cyclic constructions. Function Sum has five formats of coding from which as an example give the following format:

Sum[F[j], {j, {j1, j2, j3,…}}] – returns the result of summarizing of an expression F[j] using successive values j1, j2, j3,… for j.

Here is function examples use and its program analogue:

In[3336]:= Sum[1/j^2, {j, {1, Infinity}}]

Out[3336]= 1

In[3337]:= Total[Map[1/#^2 &, {1, Infinity}]]

Out[3337]= 1

In[3338]:= Sum[(j + 1)/j^2, {j, {1, m}}]

Out[3338]= 2 + (1 + m)/m^2

In[3339]:= Total[Map[(# + 1)/#^2 &, {1, m}]]

Out[3339]= 2 + (1 + m)/m^2

Function Delete has three formats of coding do not causing any difficulties. Both functions allow multiple levels of nesting. Once again, the functions underlying cyclic constructions allow nesting. Function Select has two formats of coding, namely:

Select[L, test] – returns the list of elements Lj of L list for which

test gives True;

Select[L, test, n] – returns the list of the first n elements Lj of L

list for which test gives True.

Here is function examples use and its program analogue:

In[4272]:= Select[{a, b, 77, c, 7, d, 14, 5, n}, SymbolQ[#] &&

! SameQ[#, a] || PrimeQ[#] &]

Out[4272]= {b, c, 7, d, 5}

In[4275]:= Select5[L_, test_, n___] := Module[{t},

t = Map[If[test @@ {#}, #, Nothing] &, L];

If[{n} == {}, t, If[Length[t] >= n, t[[1 ;n]], "Invalid third

argument value"]]]

In[4276]:= Select5[{a, b, 77, c, 7, d, 14, 5}, SymbolQ[#] &&

! SameQ[#, a] || PrimeQ[#] &]

Out[4276]= {b, c, 7, d, 5}

In[4277]:= Select5[{a, b, 77, c, 7, d, 14, 5}, SymbolQ[#] &&

! SameQ[#, a] || PrimeQ[#] &, 3]

Out[4277]= {b, c, 7}

In[4285]:= Select5[L_, test_, n___] := {Save["#77#", t],

t = Map[If[test @@ {#}, #, Nothing] &, L], If[{n} == {}, t,

If[Length[t] >= n, t[[1 ; n]], "Invalid third argument value"]],

Get["#77#"], DeleteFile["#77#"]}[[–3]]

In[4286]:= t = 2020; Select5[{a, b, 77, c, 7, d, 14, 5},

SymbolQ[#] && ! SameQ[#, a] || PrimeQ[#] &, 3]

Out[4286]= {b, c, 7}

In[4287]:= t

Out[4287]= 2020

In[4288]:= Select5[{a, b, 77, c, 7, d, 14, 5}, SymbolQ[#] &&

! SameQ[#, a] || PrimeQ[#] &, 7]

Out[4288]= "Invalid third argument value"

In[8]:= Nest1[F_Symbol, x_, n_Integer] := {Save["#78#", a],

a = x, Do[a = F[a], n], a, Get["#78#"], DeleteFile["#78#"]}[[-3]]

In[9]:= a = 77; Nest1[G, m + n, 7]

Out[9]= G[G[G[G[G[G[G[m + n]]]]]]]

In[10]:= Nest2[F_List, x_] := {Save["#78#", a], a = x,

Do[a = Reverse[F][[j]][a], {j, Length[F]}], a, Get["#78#"],

DeleteFile["#78#"]}[[–3]]

In[11]:= {a, j} = {42, 77}; {Nest2[{F, G, H, S, V}, x], {42, 77}}

Out[11]= {F[G[H[S[V[x]]]]], {42, 77}}

The previous fragment also represents program realizations of the built-in Select (Select5) and Nest (Nest1) functions in the form of functions of the list format along with an extension of the Nest (Nest2). The function call Nest2[{a, b, c, d, …}, x] where {a, b, c, d,…} – the list of symbols and x – an expression, returns a[b[c[d[…[x]…]]]]. In these functions, the question of localizing variables by means of built–in Save, Get, DeleteFile functions may be of definite interest. It is shown [6] that for most built–in functions which provide the cyclic structures, it is possible to successfully program their analogues based on system Map, Do and If functions. Because of the importance of the Map function we have created a so-called Map-group consisting of the Map1÷ Map25 means which cover a wide range of applications [6,16].

3.4. Mathematica tools for string expressions

The Mathematica language provides a variety of functions for manipulating strings. Most of these functions are based on a viewing strings as a sequence of characters, and many of these functions are analogous to ones for manipulating lists. Without taking into account the fact that Mathematica has a rather large set of tools for work with string structures, the necessity of tools which are absent in it arises. Some of such tools are represented in the given section; among them are available both simple, and more difficult which appeared in the course of programming of tasks of various purpose as additional procedures and functions and simplifying or facilitating the programming. The section presents a number of tools providing usefulenough operations on strings which supplement and expand the standard system means. These means are widely used at programming of many tools in our package Mathematica, as well as in programming other applications. More fully these tools are presented in [16].

Examples throughout this book illustrate formalization of procedures in the Mathematica that reflects its basic elements and principles, allowing by taking into account the material to directly start creation, at the beginning, of simple procedures of different purpose that are based on processing of string objects. Here only procedures of so–called "system" character intended for processing of string structures are considered that, however, represent also the most direct applied interest for programming of various appendices. Moreover, the procedures and functions that have quite foreseeable volume of source code that allows to carry out their a rather simple analysis are represented here. Their analysis can serve as a rather useful exercise for the reader both who is beginning programming and already having rather serious experience in this direction. Later we will understand under "system" tools the actually built–in tools, and our tools oriented on mass application. In addition it should be noted that string structures are of special interest not only as basic structures with which the system and the user operate, but also as a basis, in particular, of dynamic generation of objects in Mathematica,

including procedures and functions. The mechanism of dynamic generation is quite simple and enough in detail is considered in [8-15], whereas examples of its use can be found in source codes of tools of the present book. Below we will present a number of useful enough means for strings processing in Mathematica.

The procedure call SuffPref[S, s, n] provides testing of a S string regarding to begin (n=1), to end (n=2) or both ends (n=3) be limited by a substring or substrings from a s list. In a case of establishment of such fact the SuffPref returns True, otherwise False is returned. Whereas the function call StrStr[x] provides return an expression x different from a string, in string format, and a double string otherwise. In a number of cases the StrStr function is a rather useful at working with strings, in particular, with the standard StringReplace function. The following below represents source codes of the above tools along with examples of their application.

In[7]:= SuffPref[S_ /; StringQ[S], s_ /; StringQ[s]||ListQ[s] &&

AllTrue[Map[StringQ, s], TrueQ], n_ /; MemberQ[{1, 2, 3}, n]] :=

Module[{a, b, c, k = 1}, If[StringFreeQ[S, s], False,

b = StringLength[S]; c = Flatten[StringPosition[S, s]];

If[n == 3 && c[[1]] == 1 && c[[–1]] == b, True,

If[n == 1 && c[[1]] == 1, True,

If[n == 2 && c[[–1]] == b, True, False]]]]]

In[8]:= StrStr[x_] := If[StringQ[x], "\"" <> x <> "\"", ToString[x]]

In[9]:= Map[StrStr, {"RANS", a + b, IAN, {72, 77, 67}, F[x, y]}]

Out[9]= {"\"RANS\"", "a + b", "IAN", "{72, 77, 67}", "F[x, y]"}

In[10]:= SuffPref["IAN_RANS_RAC_REA_90_500", "90_500", 2]

Out[10]= True

If the StrStr function is intended, first of all, for creation of double strings, then the following simple procedure converts double strings and strings of higher nesting level to the classical strings. The procedure call ReduceString[x] returns the result of converting of a string x to the usual classical string.

In[15]:= ReduceString[x_ /; StringQ[x]] := Module[{a = x},

Do[If[SuffPref[a, "\"", 3], a = StringTake[a, {2, –2}],

Return[a]], {j, 1, Infinity}]]

In[16]:= ReduceString["\"\"\"\"\"{a + b, \"g\", s}\"\"\"\"\""]

Out[16]= "{a + b, \"g\", s}"

The SuffPrefList procedure [8,16] is a rather useful version of the above SuffPref procedure concerning the lists. At n=1, the procedure call SuffPrefList[x, y, n] returns the maximal subset common for the lists x and y with their beginning, whereas at n=2, the procedure call SuffPrefList[x, y, n] returns the maximal subset common for the lists x and y since their end, otherwise the call returns two–element sub-list whose elements define the above limiting sub-lists of the lists x and y with the both ends; moreover, if the call SuffPrefList[x, y, n, z] has fourth optional z argument – an arbitrary function from one argument, then each element of lists x and y is previously processed by a z function. The fragment below represents source code of the SuffPrefList procedure along with typical example of its application.

In[7]:= SuffPrefList[x_ /; ListQ[x], y_ /; ListQ[y],

t_ /; MemberQ[{1, 2, 3}, t], z___] :=

Module[{a = Sort[Map[Length, {x, y}]], b = x, c = y, d = {{}, {}}, j},

If[{z} != {} && FunctionQ[z] || SystemQ[z],

{b, c} = {Map[z, b], Map[z, c]}, 6]; Goto[t]; Label[3]; Label[1];

Do[If[b[[j]] === c[[j]], AppendTo[d[[1]], b[[j]]], Break[]], {j, 1, a[[1]]}];

If[t == 1, Goto[Exit], 6]; Label[2];

Do[If[b[[j]] === c[[j]], AppendTo[d[[2]], b[[j]]], Break[]],

{j, –1, –a[[1]], –1}]; d[[2]] = Reverse[d[[2]]]; Label[Exit];

If[t == 1, d[[1]], If[t == 2, d[[2]], {d[[1]], d[[2]]}]]]

In[8]:= SuffPrefList[{x, y, x, y, a, b, c, x, y, x, y}, {x, y, x, y}, 3]

Out[8]= {{x, y, x, y}, {x, y, x, y}}

StringTrim1÷ StringTrim3 procedures are useful extensions to the built-in StringTrim function. In particular, the procedure call StringTrim3[S, s, s1, s2, n, m] returns the result of truncation of a S string by s symbols on the left (n=1), on the right (n=2) or both ends (n=3) for case of s1 = "" and s2 = "", at condition that truncating is done onto m depth. Whereas in a case of the s1 and s2 arguments different from empty string instead of truncating the corresponding inserting of strings s1 (at the left) and s2 (on the right) are done. Thus, the arguments s1 and s2 – two string arguments for processing of the ends of the initial S string at the left and on the right accordingly are defined. In addition, in a case of S = "" or s = "" the procedure call returns the S string; at that, a single character or their string can act as the s argument. The fragment below represents source code of the StringTrim3 procedure along with typical example of its application.

In[7]:= StringTrim3[S_String, s_String, s1_String, s2_String,

n_Integer, m_ /; MemberQ[{1, 2, 3}, m]] :=

Module[{a = S, b, c = "", p = 1, t = 1, h},

If[S == "" || s == "", S, Do[b = StringPosition[a, s];

If[b == {}, Break[], a = StringReplacePart[a, "",

If[m == 1, {If[b[[1]][[1]] == 1, p++; b[[1]], Nothing]},

If[m == 2, {If[b[[–1]][[2]] == StringLength[a], t++; b[[–1]],

Nothing]}, {If[b[[1]][[1]] == 1, p++; b[[1]], Nothing],

If[b[[–1]][[2]] == StringLength[a], t++; b[[–1]], Nothing]}]]]];

If[a == c, Break[], 6]; c = a, {j, 1, n}]; h = {p, t} – {1, 1};

If[m == 1, StringRepeat[s1, h[[1]]] <> c,

If[m == 2, c <> StringRepeat[s2, h[[2]]],

StringRepeat[s1, h[[1]]] <> c <> StringRepeat[s2, h[[2]]]]]]]

In[8]:= StringTrim3["dd1dd1ddaabcddd1d1dd1dd1dd1dd1dd1",

"dd1", "avz", "agn", 3, 3]

Out[8]= "avzavzddaabcddd1d1dd1dd1agnagnagn"

Thus, by varying of values of the actual arguments {s, s1, s2, n, m}, it is possible to processing of the ends of arbitrary strings in a rather wide range.

The call SequenceCases[x, y] of the built-in function returns the list of all sub-lists in a x list that match a sequence pattern y. In addition, the default option Overlaps –> False is assumed, therefore the SequenceCases call returns only sub-lists which do not overlap. In addition to the function the following procedure is presented as a rather useful tool. The call SequenceCases1[x,y] as a whole returns the nested list whose two–element sub-lists have the following format {n, h} where h – the sub-list of a list x which is formed by means of maximally admissible continuous concatenation of a list y, and n – an initial position in the list x of this sub-list. At that, the procedure call SequenceCases1[x,y,z] with 3^rd optional z argument – an indefinite symbol – through z returns the nested list whose the 1^st element defines the sub-lists maximal in length in format {m, n, p}, where m – the length of a sublist and n, p – its first and end position accordingly, whereas the 2^nd element determines the sub-lists minimal in length of the above format with obvious modification. In a case of erroneous situations the procedure call returns $Failed or empty list. The following fragment represents the source code of the procedure with typical examples of its application.

In[7]:= SequenceCases1[x_ /; ListQ[x], y_ /; ListQ[y], z___] :=

Module[{a = SequencePosition[x, y], b = 6, c, t},

If[Length[y] > Length[x], $Failed,

While[! SameQ[a, b], a = Gather[a, #1[[2]] == #2[[1]] – 1 &];

a = Map[Extract[#, {{1}, {–1}}] &, Map[Flatten, a]];

b = Gather[a, #1[[2]] == #2[[1]] – 1 &];

b = Map[Extract[#, {{1}, {–1}}] &, Map[Flatten, b]]];

b = Map[{#[[1]], Part[x, #[[1]] ;#[[–1]]]} &, b];

If[{z} == {}, b, c = Map[{#[[1]], Length[#[[2]]]} &, b];

c = Map8[Max, Min, Map[#[[2]] &, c]];

z = {Map[If[Set[t, Length[#[[2]]]] == c[[1]], {c[[1]], #[[1]], #[[1]] +

t – 1}, Nothing] &, b], Map[If[Set[t, Length[#[[2]]]] == c[[2]],

{c[[2]], #[[1]], #[[1]] + t – 1}, Nothing] &, b]};

z = Map[If[Length[#] == 1, #[[1]], #] &, z]; b]]]

In[8]:= SequenceCases1[{a, x, y, x, y, x, y, x, y, x, y, x, y, x, y, a, m,

n, x, y, b, c, x, y, x, y, h, x, y}, {x, y}]

Out[8]= {{2, {x, y, x, y, x, y, x, y, x, y, x, y, x, y}}, {19, {x, y}},

{23, {x, y, x, y}}, {28, {x, y}}}

In[9]:= SequenceCases1[{a, x, y, x, y, x, y, x, y, x, y, x, y, x, y, a, m,

n, x, y, b, c, x, y, x, y, h, x, y}, {x, y}, g]; g

Out[9]= {{14, 2, 15}, {{2, 19, 20}, {2, 28, 29}}}

Additionally to the built-in StringReplace function and the four our procedures StringReplace1÷StringReplace6, extending the first, the following procedure represents undoubted interest. The procedure call StringReplaceVars[S, r] returns the result of replacement in a string S of all occurrences of the left sides of a rule or their list r onto the right sides corresponding to them. In a case of absence of the above left sides entering in the S string the procedure call returns initial S string. Distinctive feature of the procedure is the fact, that it considers the left sides of r rules as separately located expressions in the S string, i.e. framed by the special characters. The procedure is of interest at processing of strings. The following fragment represents source code of the StringReplaceVars procedure with typical examples of its use.

In[7]:= StringReplaceVars[S_ /; StringQ[S], r_ /; RuleQ[r] ||

ListRulesQ[r]] := Module[{a = "(" <> S <> ")",

L = Characters["`!@#%^&*(){}:\"\\/|<>?~–=+[];:'., 1234567890_"],

R = Characters["`!@#%^&*(){}:\"\\/|<>?~–=+[];:'., _"], b, c,

g = If[RuleQ[r], {r}, r]},

Do[b = StringPosition[a, g[[j]][[1]]];

c = Select[b, MemberQ[L, StringTake[a, {#[[1]] – 1}]] &&

MemberQ[R, StringTake[a, {#[[2]] + 1}]] &];

a = StringReplacePart[a, g[[j]][[2]], c], {j, 1, Length[g]}];

StringTake[a, {2, –2}]]

In[8]:= StringReplaceVars["Sqrt[t*p] + t^t", "t" –> "(a + b)"]

Out[8]= "Sqrt[(a + b)*p] + (a + b)^(a + b)"

In[9]:= StringReplaceVars["(125 123 678 123 90)", {"123" –> "abc",

"678" –> "mn", "90" –> "avz"}]

Out[9]= "(125 abc mn abc avz)"

The following procedure – a version of the StringReplaceVars procedure that is useful in the certain cases. The procedure call StringReplaceVars1[S, x, y, r] returns the result of replacement in a string S of all occurrences of the left sides of a rule or their list r to the right sides corresponding to them. In a case of absence of the above left sides entering in the S string the procedure call returns the initial S string. Distinctive feature of this procedure with respect to the procedure StringReplaceVars is the fact that it considers the left sides of r rules which are separately located expressions in the S string, i.e. they are framed on the left by the characters from a string x and are framed on the right by means of characters from a string y. The procedure is of certain interest at strings processing, in particular, at processing of definitions in string format of blocks and modules. The following fragment represents an example of the StringReplaceVars1 procedure use.

In[3317]:= StringReplaceVars1["{a = \"ayb\", b = Sin[x],

c = {\"a\", 77}, d = {\"a\", \"b\"}}", "{( 1234567890", " })",

{"a" –> "m", "b" –> "n"}]

Out[3317]= "{m = \"ayb\", n = Sin[x], c = {\"a\", 77},

d = {\"a\", \"b\"}}"

Earlier it was already noted that certain functional facilities of the Mathematica need to be reworked both for purpose of expansion of domain of application, and elimination of certain shortcomings. It to the full extent concerns such an important enough function whose call ToString[x] returns the result of the converting of an arbitrary expression x to the string format. The standard function incorrectly converts expressions into string format that contain string sub-expressions if to code them in the standard way. By this reason we defined procedure ToString1 whose call ToString1[x] returns the result of correct converting of an arbitrary expression x in the string format. The fragment below represents source codes of the ToString1 procedure and ToString1 function with examples of their use. In a number of appendices these means is popular enough.

In[1]:= ToString1[x_] := Module[{a = "###", b = "", c, k = 1},

Write[a, x]; Close[a];

For[k, k < Infinity, k++, c = Read[a, String];

If[SameQ[c, EndOfFile], Return[DeleteFile[Close[a]]; b],

b = b <> StrDelEnds[c, " ", 1]]]]

In[2]:= K[x_] := Module[{a = "r", b = " = "}, a <> b <> ToString[x]]

In[3]:= ToString[Definition[K]]

Out[3]= "K[x_] := Module[{a = r, b = = }, a<>b<>ToString[x]]"

In[4]:= ToExpression[%]

ToExpression::sntx: Invalid syntax in or before

"K[x_] := Module[{a = r, b = = }, a<>b<>ToString[x]]".

Out[4]= $Failed

In[5]:= ToString1[Definition[K]]

Out[5]= "K[x_] := Module[{a = \"r\", b = \" = \"},

StringJoin[a, b, ToString[x]]]"

In[6]:= ToExpression[%]

In[7]:= ToString1[x_] := {Write["#$#", x], Close["#$#"],

ReadString["#$#"], DeleteFile["#$#"]}[[–2]]

In[8]:= ToString1[Definition[K]]

Out[8]= "K[x_] := Module[{a = \"r\", b = \" = \"},

StringJoin[a, b, ToString[x]]]"

In[9]:= ToString2[x_] := Module[{a},

If[ListQ[x], SetAttributes[ToString1, Listable]; a = ToString1[x];

ClearAttributes[ToString1, Listable]; a, ToString1[x]]]

In[10]:= ToString2[{{77, 7}, {4, {a, b, {x, y}, c}, 5}, {30, 23}}]

Out[10]= {{"77", "7"}, {"4", {"a", "b", {"x", "y"}, "c"}, "5"}, {"30", "23"}}

In[11]:= ToString5[x_] := Module[{a, b, c, d}, a[b_] := x;

c = Definition1[a]; d = Flatten[StringPosition[c, " := ", 1]][[–1]];

StringTake[c, {d + 1, –1}]]

In[12]:= ToString5[{{a, b, "c", "d + h"}, "m", "p + d"}]

Out[12]= "{{a, b, \"c\", \"d + h\"}, \"m\", \"p + d\"}"

In[13]:= ToString1[{{a, b, "c", "d + h"}, "m", "p + d"}]

Out[13]= "{{a, b, \"c\", \"d + h\"}, \"m\", \"p + d\"}"

In[26]:= {a, b} = {72, 77};

In[27]:= ToString6[x_] := {Save["#", "a"], ClearAll["a"],

a = Unique["$"], a[t_] := x, a = Definition1[a],

StringTake[a, {Flatten[StringPosition[a, " := ", 1]][[–1]] + 1, –1}],

{Get["#"], DeleteFile["#"]}}[[–2]]

In[28]:= ToString6[{{c, d, "c", "d + h"}, "m", "p + d", "m/n"}]

Out[28]= "{{c, d, \"c\", \"d + h\"}, \"m\", \"p + d\", \"m/n\"}"

In[29]:= ToString1[{{c, d, "c", "d + h"}, "m", "p + d", "m/n"}]

Out[29]= "{{c, d, \"c\", \"d + h\"}, \"m\", \"p + d\", \"m/n\"}"

In[30]:= {a, b}

Out[30]= {72, 77}

In[46]:= Unique3[x_, n_] :=

Module[{a = Characters[ToString[x]], b},

b = Map[StringJoin, {Select[a, LetterQ[#] &], Select[a, DigitQ[#] &]}];

ToExpression[b[[1]] <> ToString[ToExpression[b[[2]]] – n]]]

In[47]:= Unique3[Unique["avz"], 7]

Out[47]= avz204

In[48]:= ToString7[x_] := {Unique3[Unique["g"], 0][t_] := x;

Unique3[Unique["g"], 0] = Definition1[Unique3[Unique["g"], 0]];

StringTake[Definition1[Unique3[Unique["g"], 3]],

{Flatten[StringPosition[Definition1[Unique3[Unique["g"], 4]],

" := ", 1]][[–1]] + 1, –1}]}[[1]]

In[49]:= ToString7[{{c, d, "c", "d + h"}, "m", "p + d", "m/n"}]

Out[49]= "{{c, d, \"c\", \"d + h\"}, \"m\", \"p + d\", \"m/n\"}"

In[50]:= ToString1[{{c, d, "c", "d + h"}, "m", "p + d", "m/n"}]

Out[50]= "{{c, d, \"c\", \"d + h\"}, \"m\", \"p + d\", \"m/n\"}"

Immediate application of the ToString1 procedure allows to simplify rather significantly the programming of a lot of tasks. In addition, examples of the previous fragment rather visually illustrate application of both means on the concrete example which emphases the advantages of our procedure. Whereas the ToString2 procedure expands the previous procedure onto lists of any level of nesting. So, the call ToString2[x] on an argument x, different from list is equivalent to the call ToString1[x], while on a list x is equivalent to the procedure call ToString1[x] that is endowed with the Listable attribute. The call ToString3[j] serves for converting of an expression j into the string InputForm form. The function has a number of rather useful appendices. Whereas the call ToString4[x] is analogous to the call ToString1[x] if x is a symbol, otherwise string \"(\" <> ToString1[x] <> \")\" will be returned. At last, the ToString5 procedure is a certain functional analogue of the ToString1 procedure whose source code is based on a different algorithm that does not use the file access. While the ToString6 function is a functional analogue of the ToString5 procedure but with using of the file access. Meanwhile, using a procedure whose call Unique3[Unique[x], n] returns the name of a variable unique to the current session and which was earlier generated by a Unique[x] call chain n steps before the current time (the x argument is either a letter or a string from them in string format) it is possible to program the ToString7 function in form of the list which is equivalent to the above ToString1 procedure and does not use the file access. With source codes of the above 7 tools with examples of their application the interested reader can partially familiarize above and fully in [8,10-16]. Note, that the tools StrStr, ToString1 ÷ ToString8 are rather useful enough at processing of expressions in the string format.

A number of the important problems dealing with strings processing do the SubsString procedure as a rather useful tool, whose call SubsString[s, {a, b, c,…}] returns the list of substrings of a string s which are limited by substrings {a, b, c,…}, whereas the procedure call SubsString[s, {a, b, c, d, …}, p] with the third optional p argument – a pure function in short format – returns the list of substrings of the s string which are limited by substrings {a, b, c, …}, meeting the condition defined by a pure p function.

Furthermore, the procedure call SubsString[s, {a, b, c, …}, p] with the third optional p argument – any expression different from pure function – returns the list of substrings limited by substrings {a, b, c, …}, with removed prefixes and suffixes {a, b, c, d,…}[[1]] and {a, b, c, d,…}[[–1]] accordingly. In absence in the string s of at least one of substrings {a, b, c, d,…} the procedure call returns the empty list. The following fragment represents source code of the procedure with typical examples of its application.

In[80]:= SubsString[s_ /; StringQ[s], y_ /; ListQ[y], pf___] :=

Module[{a = "", b, c, k = 1},

If[Set[c, Length[y]] < 2, s, b = Map[ToString1, y];

While[k <= c – 1, a = a <> b[[k]] <> "~~ Shortest[__] ~~ "; k++];

a = a <> b[[–1]]; b = StringCases[s, ToExpression[a]];

If[{pf} != {} && PureFuncQ[pf], Select[b, pf],

If[{pf} != {}, Map[StringTake[#, {StringLength[y[[1]]] + 1,

–StringLength[y[[–1]]] – 1}] &, b], Select[b, StringQ[#] &]]]]]

In[81]:= SubsString["adfgbffgbavzgagngbArtggbKgrg",

{"b", "g"}, StringFreeQ[#, "f"] &]

Out[81]= {"bavzg", "bArtg", "bKg"}

In[82]:= SubsString["abcxx7xxx42345abcyy7yyy42345",

{"ab", "42"}, 590]

Out[82]= {"cxx7xxx", "cyy7yyy"}

On the other hand, the SubsString1 procedure is an useful enough SubsString procedure extension, being of interest at the programming of the problems connected with processing of the strings too. The procedure call SubsString1[s, y, f, t] returns the list of substrings of a string s that are limited by the substrings of a list y; at that, if a testing pure function acts as f argument, the returned list will contain only the substrings satisfying this test. At that, at t = 1 the returned substrings are limited to ultra substrings of the y list, whereas at t = 0 substrings are returned without the limiting ultra substrings of the y list. At that, in the presence of the 5^th optional r argument – any expression – search of substrings in the s string is done from right to left that as a whole simplifies algorithms of search of the required substrings. With source code of the SubsString1 procedure and examples of its use the interested reader can familiarize in [8,9,14-16].

For operating with strings the SubsDel procedure is a quite certain interest whose call SubsDel[S, x, y, p] returns the result of removal from a string S of all sub-strings that are limited on the right (at the left) by a sub-string x and at the left (on the right) by the first met symbol in string format from the y list; in addition, search of y symbol is done to the left (p=-1) or to the right (p=1). In addition, the deleted sub-strings will contain a sub-string x since one end and the first symbol met from y since other end. Moreover, if in the course of search the symbols from the y list weren't found until end of the S string, the rest of initial S string is removed. Fragment represents source code of the procedure SubsDel along with a typical example of its application [8,16].

In[2321]:= SubsDel[S_ /; StringQ[S], x_ /; StringQ[x],

y_ /; ListQ[y] && AllTrue[Map[StringQ, y], TrueQ] &&

Plus[Sequences[Map[StringLength, y]]] == Length[y],

p_ /; MemberQ[{–1, 1}, p]] := Module[{b, c = x, d,

h = StringLength[S], k},

If[StringFreeQ[S, x], Return[S], b = StringPosition[S, x][[1]]];

For[k = If[p == 1, b[[2]] + 1, b[[1]] – 1], If[p == 1, k <= h, k >= 1],

If[p == 1, k++, k––], d = StringTake[S, {k, k}];

If[MemberQ[y, d] || If[p == 1, k == 1, k == h], Break[],

If[p == 1, c = c <> d, c = d <> c]; Continue[]]];

StringReplace[S, c –> ""]]

In[2322]:= SubsDel["12345avz6789", "avz", {"8", "5"}, 1]

Out[2322]= "1234589"

While the procedure call SubDelStr[x, t] provides removal from a string x of all sub–strings which are limited by numbers of the positions set by a list t of the ListList type from 2–element sub-lists. On incorrect tuples of actual arguments the procedure call is returned unevaluated. The following fragment represents source code of the procedure and an example of its application.

In[28]:= SubDelStr[x_ /; StringQ[x], t_ /; ListListQ[t]] :=

Module[{k = 1, a = {}}, If[! t == Select[t, ListQ[#] &&

Length[#] == 2 &]||t[[–1]][[2]] > StringLength[x]||t[[1]][[1]] < 1,

Return[Defer[SubDelStr[x, t]]],

For[k, k <= Length[t], k++,AppendTo[a, StringTake[x, t[[k]]] –> ""]];

StringReplace[x, a]]]

In[29]:= SubDelStr["123456789abcdfdh", {{3, 5}, {7, 8}, {10, 12}}]

Out[29]= "1269dfdh"

Replacements and extractions in strings. In a number of problems of strings processing, there is a need of replacement not simply of sub-strings but sub-strings limited by the certain sub-strings. The procedure solves one of such problems, its call StringReplaceS[S, s1, s2] returns the result of substitution into a S string instead of entries into it of sub-strings s1 limited by "x" strings on the left and on the right from the specified lists L and R respectively, by s2 sub-strings (StringLength["x"]=1); in a case of absence of such entries the procedure call returns the S string. The following fragment represents source code of the procedure StringReplaceS with an example of its typical application.

In[3822]:= StringReplaceS[S_ /; StringQ[S], s1_ /; StringQ[s1],

s2_ /; StringQ[s2]] := Module[{a = StringLength[S],

L = Characters["`!@#%^&*(){}:\"\\/|<>?~–=+[];:'., 1234567890"],

R = Characters["`!@#%^&*(){}:\"\\/|<>?~=+[];:'., "], c = {}, k = 1,

p, b = StringPosition[S, s1]},

If[b == {}, S, While[k <= Length[b], p = b[[k]];

If[Quiet[(p[[1]] == 1 && p[[2]] == a) || (p[[1]] == 1 &&

MemberQ[R, StringTake[S, {p[[2]] + 1, p[[2]] + 1}]]) ||

(MemberQ[L, StringTake[S, {p[[1]] – 1, p[[1]] – 1}]] &&

MemberQ[R, StringTake[S, {p[[2]] + 1, p[[2]] + 1}]]) ||

(p[[2]] == a && MemberQ[L, StringTake[S, {p[[1]] – 1, p[[1]] – 1}]])],

c = Append[c, p]]; k++]; StringReplacePart[S, s2, c]]]

In[3823]:= S = "abc& c + bd6abc[abc] – abc77*xyz^abc&78";

StringReplaceS[S, "abc", "xyz"]

Out[3823]= "xyz& c + bd6xyz[xyz] – abc77*xyz^xyz&78"

The above procedure, in particular, is a rather useful tool at processing of definitions of blocks and modules in respect of the operating with their formal arguments and local variables [14].

In a number of cases at strings processing it is necessary to extract from them the sub-strings limited by the symbol {"}, i.e. "strings in strings". This problem is solved by the procedure, whose call StrFromStr[x] returns the list of such sub-strings that are in a string x; otherwise, the call StrFromStr[x] returns the empty list, i.e. {}. The fragment below represents source code of the procedure along with a typical example of its application.

In[11]:= StrFromStr[x_String] := Module[{a = "\"", b, c = {}, k = 1},

b = DeleteDuplicates[Flatten[StringPosition[x, a]]];

For[k, k <= Length[b] – 1, k++,

AppendTo[c, ToExpression[StringTake[x, {b[[k]], b[[k + 1]]}]]];

k = k + 1]; c]

In[12]:= StrFromStr["12345\"678abc\"xyz\"50090\"mnph"]

Out[12]= {"678abc", "50090"}

To the above procedure an useful UniformString function adjoins. The strings can contain the sub-strings bounded by "\"" in particular the cdf/nb–files in the string format. To make such strings by uniform, the simple function UniformString is used. The call UniformString[x] returns a string x in the uniform form, whereas the call UniformString[x, y], where y is an expression returns the list of sub-strings of the x string which are bounded by "\"". The following fragment represents source code of the UniformString function with examples of its application.

In[4]:= UniformString[x_ /; StringQ[x], y___] :=

If[{y} == {}, StringReplace[x, "\"" –> ""],

Map[StringReplace[#, "\"" –> ""] &,

StringCases[x, Shortest["\"" ~~ __ ~~ "\""]]]]

In[5]:= UniformString["{\"ClearValues\", \"::\", \"usage\"}"]

Out[5]= "{ClearValues, ::, usage}"

In[6]:= UniformString["{\"ClearValues\", \"::\", \"usage\"}", 7]

Out[6]= {"ClearValues", "::", "usage"}

The above function is a rather useful tool at processing of the string representation of cdf/nb–files which is based on their internal formats [14,22]. Unlike standard StringSplit function, the call StringSplit1[x, y] performs semantic splitting of a string x by symbol y onto elements of the returned list. The semantics is reduced to the point that in the returned list only sub-strings of the x string which contain the correct expressions are placed; in a case of lack of such substrings the procedure call returns the empty list. The StringSplit1 procedure appears as a rather useful tool, in particular at programming of tools of processing of the headings of blocks, functions and modules. The comparative analysis of the StringSplit and StringSplit1 tools speaks well for that. Fragment below represents source code of the StringSplit1 procedure along with typical examples of its application.

In[7]:= StringSplit1[x_ /; StringQ[x], y_ /; StringQ[y] ||

StringLength[y] == 1] :=

Module[{a = StringSplit[x, y], b, c = {}, d, p, k = 1, j = 1},

d = Length[a]; Label[G]; For[k = j, k <= d, k++, p = a[[k]];

If[! SameQ[Quiet[ToExpression[p]], $Failed], AppendTo[c, p],

b = a[[k]]; For[j = k, j <= d – 1, j++, b = b <> y <> a[[j + 1]];

If[! SameQ[Quiet[ToExpression[b]], $Failed], AppendTo[c, b];

Goto[G], Null]]]]; Map[StringTrim, c]]

In[8]:= StringSplit1["x_String, y_Integer, z_/; FreeQ[{1, 2, 3, 4}, z] ||OddQ[z], h_, s_String, c_ /; StringQ[c]||StringLength[c] == 1", ","]

Out[8]= {"x_String", "y_Integer", "z_/; FreeQ[{1, 2, 3, 4}, z] || OddQ[z]", "h_", "s_String", "c_ /; StringQ[c]||StringLength[c] == 1"}

Sub-strings processing in strings. At sub-strings processing in strings is often need of check of existence fact of sub-strings overlapping that enter to the strings. The call SubStrOverlapQ[x,y] returns True, if a string x contains overlapping sub-strings y or sub-strings from x matching the general string expression y, and False otherwise. While the call SubStrOverlapQ[x, y, z] through optional z argument – an indefinite symbol – additionally returns the list of consecutive quantities of overlapping of the y sub-list in the x string. The following fragment represents source code of the function with typical examples of its application.

In[7]:= SubStrOverlapQ[x_ /; StringQ[x], y_, z___] :=

MemberQ[{If[{z} != {} && NullQ[z], z = {}, 77],

Map[If[Length[#] == 1, True,

If[! NullQ[z], Quiet[AppendTo[z, Length[#]]], 78]; False] &,

Split[StringPosition[x, y],

MemberQ[Range[#1[[1]] + 1, #1[[2]]], #2[[1]]] &]]}[[2]], False]

In[8]:= {SubStrOverlapQ["dd1dd1ddaaaabcdd1d1dd1dd1dd1\

dd1aaaaaadd1", "aa", g1], g1}

Out[8]= {True, {3, 5}}

In[9]:= {SubStrOverlapQ["AAABBBBBAABABBBBCCCBAA\

AAAA", x_ ~~ x_, g2], g2}

Out[9]= {True, {2, 4, 3, 2, 5}}

The procedure SubStrSymbolParity represents undoubted interest at processing of definitions of functions or procedures given in the string format. The call SubStrSymbolParity[x,y,z,d] with four arguments returns the list of sub-strings of a string x that are limited by one–character strings y, z (y ≠ z); in addition, search of such substrings in the string x is done from left to right (d = 0), and from right to left (d = 1). Whereas the procedure call SubStrSymbolParity[x,y,z,d,t] with the 5^th optional argument – a positive number t > 0 – provides search in a substring of x which is limited by a position t and the end of x string at d = 0, and by the beginning of x string and t at d = 1. In a case of receiving of inadmissible arguments the call is returned unevaluated, while at impossibility of extraction of demanded sub-strings the call returns $Failed. This procedure is an useful tool, in particular, at solution of tasks of extraction from definitions of procedures of the list of local variables, headings, etc. The following fragment represents source code of the SubStrSymbolParity procedure with some typical examples of its application.

In[5]:= SubStrSymbolParity[x_ /; StringQ[x], y_ /; CharacterQ[y],

z_ /; CharacterQ[z], d_ /; MemberQ[{0, 1}, d], t___ /; t == {} ||

PosIntQ[{t}[[1]]]] := Module[{a, b = {}, c = {y, z}, k = 1, j, f,

m = 1, n = 0, p, h},

If[{t} == {}, f = x, f = StringTake[x,

If[d == 0, {t, StringLength[x]}, {1, t}]]];

If[Map10[StringFreeQ, f, c] != {False, False} || y == z, Return[],

a = StringPosition[f, If[d == 0, c[[1]], c[[2]]]]];

For[k, k <= Length[a], k++,

j = If[d == 0, a[[k]][[1]] + 1, a[[k]][[2]] – 1];

h = If[d == 0, y, z]; While[m != n,

p = Quiet[Check[StringTake[f, {j, j}], Return[$Failed]]];

If[p == y, If[d == 0, m++, n++];

If[d == 0, h = h <> p, h = p <> h], If[p == z, If[d == 0, n++, m++];

If[d == 0, h = h <> p, h = p <> h],

If[d == 0, h = h <> p, h = p <> h]]];

If[d == 0, j++, j––]]; AppendTo[b, h]; m = 1; n = 0; h = ""]; b]

In[6]:= SubStrSymbolParity["123{abcdf}7{ran}8{ian}9", "{", "}", 0]

Out[6]= {"{abcdf}", "{ran}", "{ian}"}

In[7]:= SubStrSymbolParity["12{abf}6{ran}8{ian}9", "{", "}", 1, 25]

Out[7]= {"{abf}", "{rans}"}

Meantime, in many cases it is possible to use a simpler and reactive version of the above procedure, whose procedure call SubStrSymbolParity1[x, y, z] with three arguments returns the list of sub-strings of a string x that are limited by one-character strings {y,z} (y ≠ z); in addition, search of such substrings is done from left to right. In the absence of the required sub-strings the procedure call returns the empty list, i.e. {}. A simple procedure is an useful enough modification of the SubStrSymbolParity1 procedure; its call StrSymbParity[S, s, x, y] returns the list whose elements are sub-strings of a string S which have format s1w on condition of parity of the minimum number of entries into a w substring of symbols x, y (x ≠ y). In the lack of such substrings or identity of symbols x, y, the call returns the empty list [16]. The SubStrSymbolParity, SubStrSymbolParity1 and StrSymbParity procedures are rather useful tools, for instance, at processing of definitions of modules and blocks given in string format. These procedures are used by a number of tools of our MathToolBox package [8-12,15,16].

The procedure below is a rather useful tool for ensuring of converting of strings of a certain structure into lists of strings. In particular, such problems arise at processing of arguments and local variables. The problem is solved a rather effectively by the StrToList procedure, providing converting of strings of the "{xxxxxxxxx … x}" format into the list of strings received from a "xxxxx … x" string parted by comma symbols. In absence in an initial string of both limiting symbols {"{", "}"} the string will be converted in list of symbols as a call Characters["xxx…x"]. The next fragment presents source code of the StrToList procedure with examples of its application.

In[8]:= StrToList[x_ /; StringQ[x]] := Module[{a, b = {}, c = {}, d,

h, k = 1, j, y = If[StringTake[x, {1}] == "{" &&

StringTake[x, {–1}] == "}", StringTake[x, {2, –2}], x]},

a = DeleteDuplicates[Flatten[StringPosition[y, "="]] + 2];

d = StringLength[y];

If[a == {}, Map[StringTrim, StringSplit[y, ","]],

While[k <= Length[a], c = ""; j = a[[k]];

For[j, j <= d, j++, c = c <> StringTake[y, {j}];

If[! SameQ[Quiet[ToExpression[c]], $Failed] &&

(j == d || StringTake[x, {j + 1}] == ","),

AppendTo[b, c –> ToString[Unique[$g$]]]; Break[]]]; k++];

h = Map[StringTrim, StringSplit[StringReplace[y, b], ","]];

Map14[StringReplace, h, RevRules1[b]]]]

In[9]:= StrToList["Kr, a = 90, b = {x, y, z}, c = {n, m, {42, 47, 67}}"]

Out[9]= {"Kr", "a = 90", "b = {x, y, z}", "c = {n, m, {42, 47, 67}}"}

In[10]:= StrToList["{a = 500, b = 90, c = {m, n}}"]

Out[10]= {"a = 500", "b = 90", "c = {m, n}"}

The above procedure is intended for converting of strings of format "{x…x}" or "x…x" into the list of strings received from strings of the specified format which are parted by symbols "=" and/or comma. Fragment examples an quite visually illustrate the basic principle of performance of the procedure along with formats of the returned results. At last, the fragment uses quite simple and useful procedure, whose call RevRules[x] returns the rule or list of rules that are reverse to the rules defined by a x argument – a rule of form a –> b or their list [16]. The procedure can be also represented in the functional form, namely:

In[3327]:= RevRules1[x_ /; RuleQ[x] || ListQ[x] &&

AllTrue[Map[RuleQ, x], TrueQ]] := Map[Rule[#[[2]], #[[1]]] &,

Flatten[{x}]][[If[Length[Flatten[{x}]] > 1, 1 ; Length[Flatten[{x}]], 1]]]

In[3328]:= RevRules1[{a –> b, c –> d, n –> m}]

Out[3328]= {b –> a, d –> c, m –> n}

Note, that the RevRules1 function is essentially used by the above StrToList procedure.

The following procedure is a rather useful means of strings processing in a case when it is required to identify in a string of occurrence of sub-strings of kind "abc…n" and "abc…np", p – a character. The procedure call SubsInString[x, y, z] returns 0, if substrings y and y<>z where y – a string and z – a character are absent in a string x; 1, if x contains y and not contain y <> z; 2, if x contains y <> z and not contain y, and three otherwise, i.e. the x string contains both y and y <> z [8,9]. Whereas the procedure call CorrSubStrings[x, n, y] returns the list of substrings of string x that contain all formally correct expressions, at the same time, search is done beginning with n–th position of the x string from left to right if the third optional y argument is absent, and from right to left if the y argument – any expression – exists. Fragment represents source code of the CorrSubStrings procedure and an example of its typical application.

In[7]:= CorrSubStrings[x_ /; StringQ[x], n_ /; PosIntQ[n], y___] :=

Module[{a = {}, b = StringLength[x], c = ""},

If[{y} != {}, Do[If[SyntaxQ[Set[c, StringTake[x, {j}] <> c]],

AppendTo[a, c], 7], {j, n, 1, –1}],

Do[If[SyntaxQ[Set[c, c <> StringTake[x, {j}]]],

AppendTo[a, c], 7], {j, n, b}]]; a]

In[8]:= CorrSubStrings["(a+b/x^2)/(c/x^3+d/y^2)+1/z^3", 29, 2]

Out[8]= {"3", "z^3", "1/z^3", "+1/z^3", "(c/x^3+d/y^2)+1/z^3",

"(a+b/x^2)/(c/x^3+d/y^2)+1/z^3"}

In a number of cases there is a necessity of reducing to the set number of the quantity of entries into a string of its adjacent sub-strings. That problem is solved by the ReduceAdjacentStr procedure presented by the following fragment. The procedure call ReduceAdjacentStr[x, y, n] returns the string – the result of reducing to an quantity n ≥ 0 of occurrences into a string x of its adjacent y substrings. If a string x not contain y substrings, then the call returns the initial x string; the call ReduceAdjacentStr[x, y, n, h] where h – an arbitrary expression, returns the above result on condition that at search of the y substrings in the x string the

lowercase and uppercase letters are identified.

In[41]:= ReduceAdjacentStr[x_ /; StringQ[x], y_ /; StringQ[y],

n_ /; IntegerQ[n], z___] := Module[{a = {}, b = {},

c = Append[StringPosition[x <> FromCharacterCode[0], y,

IgnoreCase –> If[{z} != {}, True, False]], {0, 0}], h, k},

If[c == {}, x, Do[If[c[[k]][[2]] + 1 == c[[k + 1]][[1]],

b = Union[b, {c[[k]], c[[k + 1]]}], b = Union[b, {c[[k]]}];

a = Union[a, {b}]; b = {}], {k, 1, Length[c] – 1}];

a = Select[a, Length[#] >= n &];

a = Map[Quiet[Check[{#[[1]], #[[–1]]}, Nothing]] &,

Map[Flatten, Map[#[[–Length[#] + n ;–1]] &, a]]];

StringReplacePart[x, "", a]]]

In[42]:= ReduceAdjacentStr["abababcdcdxmnabmabab", "ab", 3]

Out[42]= "abababcdcdxmnabmabab"

In contrast to the LongestCommonSubsequence function the procedure call LongestCommonSubsequence1[x, y, J] in a mode IgnoreCase –>J{True,False} finds the longest contiguous substrings that are common to the strings x and y. Whereas the procedure call LongestCommonSubsequence1[x,y,J,t] additionally through an indefinite t variable returns the list of all common contiguous substrings. The procedure essentially uses the procedure whose call Intersection1[x,y,z,…,J] returns the list of elements common to all lists of strings in the mode IgnoreCase –>J{True,False}. The fragment below represents source codes of both procedures and typical examples of their application.

In[6]:= LongestCommonSubsequence1[x_ /; StringQ[x],

y_ /; StringQ[y], Ig_ /; MemberQ[{False, True}, Ig], t___] :=

Module[{a = Characters[x], b = Characters[y], c, d, f},

f[z_, h_] := Map[If[StringFreeQ[h, #], Nothing, #] &,

Map[StringJoin[#] &, Subsets[z]][[2 ;–1]]];

c = Gather[d = Sort[If[Ig == True, Intersection1[f[a, x], f[b, y], Ig],

Intersection[f[a,x], f[b,y]]], StringLength[#1] <= StringLength[#2] &],

StringLength[#1] == StringLength[#2] &];

If[{t} != {} && ! HowAct[t], t = d, Null]; c = If[c == {}, {}, c[[–1]]];

If[c == {}, {}, If[Length[c] == 1, c[[1]], c]]]

In[7]:= {LongestCommonSubsequence1["AaAaBaCBbBaCaccccC",

"CacCCbbbAaABaBa", False, gs], gs}

Out[7]= {{"AaA", "aBa", "Cac"}, {"a", "A", "b", "B", "c", "C", "aA",

"Aa", "aB", "ac", "Ba", "Ca", "cC", "AaA", "aBa", "Cac"}}

In[8]:= {LongestCommonSubsequence1["Rans", "Ian", True, j], j}

Out[8]= {"an", {"a", "n", "an"}}

In[9]:= Intersection1[x__ /; AllTrue[Map[ListQ[#] &, {x}], TrueQ],

Ig_ /; MemberQ[{False, True}, Ig]] :=

Module[{b = Length[{x}], c = {}, d = {}},

Do[AppendTo[c, Map[StringQ, {x}[[j]]]], {j, 1, b}];

If[DeleteDuplicates[Flatten[c]] != {True}, $Failed,

If[Ig == False, Intersection[x], Do[AppendTo[d,

Map[{j, #, ToUpperCase[ToString[#]]} &, {x}[[j]]]], {j, 1, b}];

c = Map[DeleteDuplicates,

Gather[Flatten[Join[d], 1], #1[[3]] == #2[[3]] &]];

c = Flatten[Select[c, Length[#] >= b &], 1];

c = If[DeleteDuplicates[Map[#[[1]] &, c]] != Range[1, b], {},

DeleteDuplicates[Map[#[[2]] &, c]]]]]]

In[10]:= Intersection1[{"AB", "XY", "cd", "Mn"}, {"ab", "cD", "MN",

"pq", "mN"}, {"90", "mn", "Ag"}, {"500", "mn", "Av"}, True]

Out[10]= {"Mn", "MN", "mN", "mn"}

The LongestCommonSubsequence2 procedure is a certain extension of the LongestCommonSubsequence1 procedure for a case of a finite number of strings in which the search for longest common continuous sub-strings is done [8,11,14-16].

In[317]:= LongestCommonSubsequence2["ABxCDH", "ABxC",

"ABxCDCABCdc", "xyzABXC", "xyzABxC", "mnpABxC", True]

Out[317]= {"ABxC", "ABXC"}

For work with strings the following procedure is a rather useful, whose call InsertN[S, L, n] returns the result of inserting into a string S after its positions from a list n of substrings from a list L; in a case n = {< 1|≥ StringLength[S]} a sub-string will be located before S string or in its end respectively. It is supposed that the actual arguments L and n may contain various number of elements, in such case the excess n elements are ignored. At that, processing of a string S is carried out concerning the list of positions for m insertions defined according to the relation m = DeleteDuplicates[Sort[n]]. The procedure call InsertN[S, L, n]

with inadmissible arguments is returned as unevaluated. The procedure was used significantly in programming a number of MathToolBox package tools [16]. The next fragment represents source code of the procedure with examples of its application.

In[2220]:= InsertN[S_String, L_ /; ListQ[L], n_ /; ListQ[n] &&

Length[n] == Length[Select[n, IntegerQ[#] &]]] :=

Module[{a = Map[ToString, L], d = Characters[S], p, b, k = 1,

c = FromCharacterCode[2], m = DeleteDuplicates[Sort[n]]},

b = Map[c <> ToString[#] &, Range[1, Length[d]]];

b = Riffle[d, b]; p = Min[Length[a], Length[m]];

While[k <= p, If[m[[k]] < 1, PrependTo[b, a[[k]]],

If[m[[k]] > Length[d], AppendTo[b, a[[k]]],

b = ReplaceAll[b, c <> ToString[m[[k]]] –> a[[k]]]]]; k++];

StringJoin[Select[b, ! SuffPref[#, c, 1] &]]]

In[2221]:= InsertN["123456789Rans_Ian", {Ag, Vs, Art, Kr},

{6, 9, 3, 0, 3, 17}]

Out[2221]= "Ag123Vs456Art789KrRans_Ian"

In[2222]:= InsertN["123456789", {a, b, c, d, e, f, g, h, n, m},

{4, 2, 3, 0, 17, 9, 18}]

Out[2222]= "a12b3c4d56789efg"

Contrary to the InsertN procedure the call of the procedure DelSubStr[S, L] provides removal from a string S of substrings, whose positions are set by a list L; the L list has nesting 0 or 1, for example, {{3, 4}, {7}, {9}} or {1, 3, 5, 7, 9}, whereas the function call AddDelPosString[x, y, h, z] returns the result of truncation of a string x to the substring x[[1 ; y]] if z – an arbitrary expression and x[[y ; –1]] if {z} == {} with replacing of the deleted substring by a string h. In a case of an incorrect value y the call returns the initial x string or is returned unevaluated [16]. Both these tools are rather useful in a number of problems of strings processing of various structure and appointment.

The following procedure provides extraction from string of continuous substrings of length bigger than 1. The procedure call ContinuousSubs[x] returns the nested list whose elements have format {s, {p11, p12},…,{pn1, pn2}} where s – a substring and {pj1, pj2} (j = 1..n) determine the first and last positions of copies of a continuous s sub-string that compose a string x. The procedure

uses an auxiliary tool – the SplitIntegerList procedure whose call SplitIntegerList[x] returns the result of splitting of a strictly increasing x list of the IntegerList type into sub-lists consisting of strictly increasing tuples of integers. In addition, the procedure SplitIntegerList has other useful applications too [9,10-16].

In[6]:= ContinuousSubs[x_ /; StringQ[x]] := Module[{a},

a = Select[DeleteDuplicates[Map[StringPosition[x, #] &,

Characters[x]]], Length[#] > 1 &];

a = Map[DeleteDuplicates, Map[Flatten, a]];

a = Map[SplitIntegerList, a];

a = Select[Flatten[a, 1], Length[#] > 1 &];

a = Map[{StringTake[x, {#[[1]], #[[–1]]}], {#[[1]], #[[–1]]}} &, a];

a = Gather[Sort[a, #[[1]] &], #1[[1]] === #2[[1]] &];

a = Map[If[Length[#]==1, #[[1]], DeleteDuplicates[Flatten[#, 1]]] &, a];

SortBy[a, First]]

In[7]:= ContinuousSubs["abbbssccchggxxx66xxggazzzaaabbbssz7"]

Out[7]= {{"66", {16, 17}}, {"aaa", {26, 28}}, {"bbb", {2, 4}, {29, 31}},

{"ccc", {7, 9}}, {"gg", {11, 12}, {20, 21}}, {"ss", {5, 6}, {32, 33}},

{"xx", {18, 19}}, {"xxx", {13, 15}}, {"zzz", {23, 25}}}

The procedure call SolidSubStrings[x] returns the nested list whose the first elements of 2–element sub-lists define solid substrings of a string x which consist of identical characters in quantity > 1, while the second elements define their counts in the x string. In a case of lack of similar solid sub-strings the call returns the empty list, i.e. {}.

In[19]:= SolidSubStrings[x_ /; StringQ[x]] :=

Module[{a = DeleteDuplicates[Characters[x]], b, c={}, d=x, n=1},

b = Map["[" <> # <> "–" <> # <> "]+" &, a];

b = Map[Sort, Map[StringCases[x, RegularExpression[#]] &, b]];

b = Reverse[Sort[DeleteDuplicates[Flatten[Map[Select[#,

StringLength[#] > 1 &] &, b]]]]];

Do[AppendTo[c, StringCount[d, b[[j]]]];

d = StringReplace[d, b[[j]] –> ""], {j, 1, Length[b]}];

Sort[Map[{#, c[[n++]]} &, b], OrderedQ[{#1[[1]], #2[[1]]}] &]]

In[20]:= SolidSubStrings["cccabaacdaammddddpphaaaaaccaaa"]

Out[20]= {{"aa", 2}, {"aaa", 1}, {"aaaaa", 1}, {"cc", 1}, {"ccc", 1},

{"dddd", 1}, {"mm", 1}, {"pp", 1}}

In particular, the SubsStrLim procedure represents an quite certain interest for a number of appendices which significantly use procedure of extraction from the strings of substrings of an quite certain format. The next fragment represents source code of the SubsStrLim procedure and typical examples of its use.

In[1942]:= SubsStrLim[x_ /; StringQ[x], y_ /; StringQ[y] &&

StringLength[y] == 1, z_ /; StringQ[z] && StringLength[z] == 1] :=

Module[{a, b = x <> FromCharacterCode[6], c = y, d = {}, p, j,

k = 1, n, h}, If[! StringFreeQ[b, y] &&

! StringFreeQ[b, z], a = StringPosition[b, y];

n = Length[a]; For[k, k <= n, k++, p = a[[k]][[1]]; j = p;

While[h = Quiet[StringTake[b, {j + 1}]]; h != z, c = c <> h; j++];

c = c <> z; If[StringFreeQ[StringTake[c, {2, –2}], {y, z}],

AppendTo[d, c]]; c = y]];

Select[d, StringFreeQ[#, FromCharacterCode[6]] &]]

In[1944]:= SubsStrLim[DefOpt["SubsStrLim"], "{", "}"]

Out[1944]= {"{}", "{j + 1}", "{2, –2}", "{y, z}"}

In[1945]:= SubsStrLim[Definition2["DefOpt"][[1]], "{", "}"]

Out[1945]= {"{a = If[SymbolQ[x], If[SystemQ[x], b = \"Null\",

ToString1[Definition[x]], \"Null\"]], b, c}"}

The call SubsStrLim[x, y, z] returns the list of substrings of a string x that are limited by symbols {y, z} provided that these symbols don't belong to these substrings, excepting their ends. In particular, the SubsStrLim procedure is a quite useful means at need of extracting from definitions of the functions, blocks or modules given in the string format of certain components that compose them which are limited by certain symbols, at times, significantly simplifying a number of procedures of processing of such definitions. Whereas the procedure call SubsStrLim1[x, y, z] which is an useful enough modification of the SubsStrLim procedure, returns list of substrings of x string that are limited by symbols {y, z} provided that these symbols or don't enter in substrings, excepting their ends, or along with their ends have identical number of entries of pairs {y, z}, for example:

In[1947]:= SubsStrLim1["G[x_] := Block[{a = 7, b = 50, c = 900},

(a^2 + b^3 + c^4)*x]", "{", "}"]

Out[1947]= {"{a = 7, b = 50, c = 900}"}

The next means are useful at work with string structures. The procedure call StringPat[x, y] returns the string expression formed by strings of a list x and objects {"_", "__", "___"}; the call returns x if x – a string. The procedure call StringCases1[x, y, z] returns the list of the substrings in a string x that match a string expression, created by the call StringPat[x, y]. While a function call StringFreeQ1[x, y, z] returns True if no substring in x string matches a string expression, created by the call StringPat[x, y], and False otherwise. In the fragment below, source codes of the above tools with examples of their applications are presented.

In[87]:= StringPat[x_ /; StringQ[x] || ListStringQ[x],

y_ /; MemberQ[{"_", "__", "___"}, y]]:= Module[{a = "", b},

If[StringQ[x], x, b = Map[ToString1, x];

ToExpression[StringJoin[Map[# <> "~~" <> y <> "~~" &,

b[[1 ;–2]]], b[[–1]]]]]]

In[88]:= StringPat[{"ab", "df", "k"}, "__"]

Out[88]= "ab" ~~ __ ~~ "df" ~~ __ ~~ "k"

In[89]:= StringFreeQ1[x_ /; StringQ[x], y_ /; StringQ[y] ||

ListStringQ[y], z_ /; MemberQ[{"_", "__", "___"}, z]] :=

If[StringQ[y], StringFreeQ[x, y],

If[StringCases1[x, y, z] == {}, True, False]]

In[90]:= StringFreeQ1["abcfghkaactabcfghkt", {"ab", "df", "k"}, "___"]

Out[90]= True

In[91]:= StringCases2[x_ /; StringQ[x], y_ /; ListQ[y] && y != {}] :=

Module[{a, b = "", c = Map[ToString, y]},

Do[b = b <> ToString1[c[[k]]] <> "~~__~~", {k, 1, Length[c]}];

a = ToExpression[StringTake[b, {1, -7}]]; StringCases[x, Shortest[a]]]

In[92]:= StringCases2["a1234b7890000c5a b ca1b2c3", {a, b, c}]

Out[92]= {"a1234b7890000c", "a b c", "a1b2c"}

Whereas the call StringCases2[x, y] returns a list of disjoint substrings in a string x that match the string expression of the format Shortest[j1~~__~~j2 ~~__~~…~~__~~jk], where y = {j1, j2, …, jk} and y is different from the empty list, i.e. {}. The above tools are essentially used by a number of means of the package MathToolBox, enough frequently essentially improving the programming algorithms which deal with string expressions.

The following procedure along with independent interest appears as a rather useful tool, e.g., in a case of computer study of questions of the reproducibility of finite sub-configurations in 1-dimensional cellular automata (CA) models. The procedure call StringEquiQ[x,y,n] returns True if strings x and y differ no more than in n positions, and False otherwise. At that, in the presence of the fifth optional z argument – an indefinite symbol – through it the message "Strings lengths aren't equal", if lengths of strings x and y are different, or the nested list {{n1, {x1, y1}}, {n2, {x2, y2}}, …, {np, {xp, yp}}} are returned whose 2–element sub-lists define respectively numbers of positions (nj), characters of a string x (xj) and characters of a string y (yj) which differ in nj positions (j = 1..p). It must be kept in mind that the 4^th argument r defines the string comparison mode – lowercase and uppercase letters are considered as different (r=True), or lowercase and uppercase letters are considered as equivalent (r=False). The next fragment represents the source code of the StringEquiQ procedure with some typical examples of its application.

In[7]:= StringEquiQ[x_ /; StringQ[x], y_ /; StringQ[y],

n_ /; IntegerQ[n], r_: True, z___] := Module[{c = {}, d, m, p, x1, y1,

a = StringLength[x], b = StringLength[y]},

If[{z} != {} && ! HowAct[z], d = 77, Null];

If[r === True, x1 = x; y1 = y, x1 = ToLowerCase[x];

y1 = ToLowerCase[y]];

If[x1 == y1, True, If[a != b,

If[d == 77, z = "Strings are`t equal in lengths", Null]; False,

{m, p} = Map[Characters, {x1, y1}];

Do[If[m[[j]] != p[[j]], AppendTo[c, {j, {StringTake[x, {j}],

StringTake[y, {j}]}}], Null], {j, 1, a}];

If[d == 77, z = c, Null]; If[Length[c] > n, False, True]]]]

In[8]:= StringEquiQ["Grsu_2018", "Grsu_2019", 2, False, v42]

Out[8]= True

In[9]:= v42

Out[9]= {{9, {"8", "9"}}}

In[10]:= StringEquiQ["Grsu_2019", "Grsu_4247", 2, False, g47]

Out[10]= False

In[11]:= g47

Out[11]= {{6, {"2", "4"}}, {7, {"0", "2"}}, {8, {"1", "4"}}, {9, {"9", "7"}}}

At this point, we conclude the problem related to processing of string expressions in the Mathematica whereas a rather large number of other useful enough tools of this purpose developed by us can be found in [4-16]. Our experience of use of the Maple and Mathematica for programming of tools for operating with string structures showed that standard tools of Maple by many essential indicators yield to the tools of the same type of Math–language. For problems of this type the Math-language appears simpler not only in connection with more developed tools, but also because the procedural and functional paradigm allows to use the mechanism of the pure functions. The item represents a number of tools expanding the built–in tools of system that are oriented on work with string structures. These and other means of this type are located in our package [16]. At that, their correct use assumes that this package is loaded into the current session, the need for which is due to the fact that many tools of package together with built–in means make substantial use of the tools whose definitions are in the same package.

Expressions containing in strings. In a number of cases of expressions processing the problem of excretion of one or other type of expressions from strings is quite topical. In this relation a certain interest the ExprOfStr procedure represents, whose source code with examples of its use represents the following fragment. The procedure call ExprOfStr[w, n, m, L] returns the result of extraction from a string w limited by its n–th position and the end, of the first correct expression provided that search is done on the left (m = –1) / on the right (m = 1) from the given position; at that, a symbol, next or previous behind the found expression must belong to a list L. The call is returned in string format; in a case of the absence of a correct expression $Failed is returned while the procedure call on inadmissible arguments is returned unevaluated.

In[7]:= ExprOfStr[x_ /; StringQ[x], n_ /; IntegerQ[n] && n > 0,

p_ /; MemberQ[{–1, 1}, p], L_ /; ListQ[L]] := Module[{a = "", b, k},

If[n >= StringLength[x], Return[Defer[ExprOfStr[x,n,p,L]]], Null];

For[k = n, If[p == –1, k >= 1, k <= StringLength[x]],

If[p == –1, k––, k++], If[p == –1, a = StringTake[x, {k}] <> a,

a = a <> StringTake[x, {k}]]; If[! SyntaxQ[a], Null,

If[If[p == –1, k == 1, k == StringLength[x]] ||

MemberQ[L, Quiet[StringTake[x, If[p == –1, {k – 1}, {k + 1}]]]],

Return[a], Null]]]; $Failed]

In[8]:= ExprOfStr["12345;F[(a+b)/(c+d)]; A_2020", 7, 1, {"^", ";"}]

Out[8]= "F[(a+b)/(c+d)]"

The ExprOfStr1 represents an useful enough modification of the previous procedure; its call ExprOfStr1[x, n, p] returns a substring of string x, that is minimum on length and in that a boundary element is a symbol in n–th position of a string x that contains a correct expression. At that, search of such substring is done from n–th position to the right and until the end of the x string (p = 1), and from the left from n–th position of x string to the beginning of the string (p = –1). In a case of absence of such substring the call returns $Failed, while on inadmissible factual arguments the procedure call is returned unevaluated [10,16].

In a certain relation to the above procedures the ExtrExpr procedure adjoins, whose the call ExtrExpr[S, n, m] returns the omnifarious correct expressions in the string format which are contained in a substring of a string S limited by positions with numbers n and m. In a case of absence of correct expressions the empty list is returned. The fragment below represents source code of the ExprOfStr procedure with an example of its use.

In[6]:= ExtrExpr[S_ /; StringQ[S], n_ /; IntegerQ[n],

m_ /; IntegerQ[m]] := Module[{a = StringLength[S], b, c, h = {}, k, j},

If[! (1 <= m <= a && n <= m), {}, b = StringTake[S, {n, m}];

Do[Do[c = Quiet[StringTake[b, {j, m – n – k + 1}]];

If[ExpressionQ[c], AppendTo[h, c]; Break[], Null],

{k, 0, m – n + 1}], {j, 1, m – n}];

DeleteDuplicates[Select[Map[StringTrim2[#,

{"+", "–", " ", "_"}, 3] &, h], ! MemberQ[{"", " "}, #] &]]]]

In[7]:= ExtrExpr["z=(Sin[x*y]+Log[x])-F[x,y]; S[x_]:= x^2;", 4, 39]

Out[7]= {"Sin[x*y]+Log[x]", "in[x*y]+Log[x]", "n[x*y]+Log[x]",

"x*y", "y", "Log[x]", "og[x]", "g[x]", "x",

"F[x,y]; S[x_]:= x^2", "S[x_] := x^2", "x^2"}

Using procedures CorrSubStrings, RevRules, ExtrVarsOfStr, SubsInString, StringReplaceVars, ToStringRational [8,16], it is possible to offer one more tool of extraction from an expression of all formally correct sub-expressions that are contained in it. The procedure call SubExprsOfExpr[x] returns the sorted list of every possible formally correct sub-expressions of an expression x. Meantime, if x expression contains no indefinite symbols, the procedure call returns the evaluated initial expression x.

In[14]:= SubExprsOfExpr[x_] := Module[{b, c, c1, d = {}, h, f,

p = 7000, a = StringReplace[ToStringRational[x], " " –> ""]},

b = StringLength[a]; c = ExtrVarsOfStr[a, 1];

If[c == {}, x, f = Select[c, ! StringFreeQ[a, # <> "["] &&

StringFreeQ[a, # <> "[["] &];

c1 = Map[# –> FromCharacterCode[p++] &, c];

h = StringReplaceVars[a, c1];

Do[AppendTo[d, CorrSubStrings[h, j]], {j, 1, b}];

d = Map[StringTrim[#, ("+" | "–") ...] &, Flatten[d]];

d = Map[StringReplaceVars[#, RevRules[c1]] &, d];

Do[Set[d, Map[If[MemberQ[{0, 2}, SubsInString[#, f[[j]], "["]], #,

Nothing] &, d]], {j, 1, Length[f]}];

Sort[DeleteDuplicates[ToExpression[d]]]]]

In[15]:= SubExprsOfExpr[(a + bc)*Log[z]/(c + Sin[x])]

Out[15]= {a, bc, a + bc, c, x, z, Log[z], (a + bc)*Log[z], Sin[x],

((a + bc)*Log[z])/(c + Sin[x]), c + Sin[x]}

Three ExtrVarsOfStr ÷ ExtrVarsOfStr2 procedures are of interest in extracting variables from strings. In particular, the procedure call ExtrVarsOfStr2[S, t] returns the sorted list of variables in the string format that managed to extract from a string S; at the absence of the similar variables the empty list, i.e. {} is returned. While the procedure call ExtrVarsOfStr2[S,t,x] with the third optional x argument – the list of strings of length 1 – returns the list of the variables of S on condition that at the above extracting, the elements from x list are not ignored. The following fragment represents source code of the ExtrVarsOfStr2 procedure with some typical examples of its application.

In[7]:= ExtrVarsOfStr2[S_ /; StringQ[S], x___ /; If[{x} == {}, True, ListQ[x]]] := Module[{k, j, d = {}, p, a, q = Map[ToString, Range[0, 9]], h = 1, c = "", L = Characters["`!@#%^&*(){}:\"\\/|<>?~–=+[];:'., 1234567890_"], R = Characters["`!@#%^&*(){}:\"\\/|<>?~–=+[];:'., _"], s = "," <> S <> ","}, a = StringLength[s];

If[{x} == {}, 7, {L, R} = Map[Select[#, ! MemberQ[x, #] &] &, {L, R}]];

Label[Gs]; For[k = h, k <= a, k++, p = StringTake[s, {k}];

If[! MemberQ[L, p], c = c <> p; j = k + 1;

While[j <= a, p = StringTake[s, {j}];

If[! MemberQ[R, p], c = c <> p, AppendTo[d, c];

h = j; c = ""; Goto[Gs]]; j++]]];

AppendTo[d, c]; d = Select[d, ! MemberQ[q, #] &];

d = Select[Map[StringReplace[#, {"+" –> "", "-" –> "", "_" –> ","}] &,

d], # != "" &]; d = Flatten[Select[d, ! StringFreeQ[s, #] &]];

Sort[DeleteDuplicates[Flatten[Map[StringSplit[#, ", "] &, d]]]]]

In[8]:= ExtrVarsOfStr2["(a*#1 + b*#2)/(c*#3 - i*#4) + j*#1*#4", {"#"}]

Out[8]= {"#1", "#2", "#3", "#4", "a", "b", "c", "i", "j"}

In[9]:= ExtrVarsOfStr2["(a*#1 + b*#2)/(c*#3 – d*#4) + m*#1*#4"]

Out[9]= {"a", "b", "c", "d", "m"}

The VarsInExpr procedure presents a rather useful version of the ExtrVarsOfStr ÷ ExtrVarsOfStr2 procedures. Procedure call VarsInExpr[x] returns the sorted list of variables in string format, that managed to extract from an expression x; in a case of absence of such variables the empty list is returned. At that, it is supposed that the expression x can be encoded in the string format, i.e. in ToString[InputForm[x]]. The fragment represents source code of the VarsInExpr procedure and examples of its use.

In[151]:= VarsInExpr[x_] :=

Module[{a = StringReplace[ToStringRational[x], " " –> ""],

b = Join[Range5[33 ;35, 37 ;47], Range[58, 64], Range[91, 96],

Range[123, 126]]},

Sort[DeleteDuplicates[Select[StringSplit[a,

Map[FromCharacterCode[#] &, b]],

SymbolQ[#] && # != "" &]]]]

In[152]:= VarsInExpr["(a + b)/(c + 590*d$) + Sin[c]*Cos[d + h]"]

Out[152]= {"a", "b", "c", "Cos", "d", "d$", "h", "Sin"}

In[153]:= VarsInExpr[(a + b/x^2)/Sin[x*y] + 1/z^3 + Log[y]]

Out[153]= {"a", "b", "Csc", "Log", "x", "y", "z"}

In general, the procedures ExtrVarsOfStr ÷ ExtrVarsOfStr2

and VarsInExpr can be also rather useful for structural analysis of algebraical expressions. Whereas the SymbToExpr procedure applies symbols to specified sub–expressions located at certain levels of an expression x. The procedure call SymbToExpr[x, d] returns the result of applying of symbols to n–th sub–elements located on m–th levels of an expression x. The 2^nd argument d is a simple or nested list of format {{n,s1,n2,…,np},…,{r,sj,r1,…,rt}}, where the 1^st element n of sub-list {n,s1,n1,…,np} defines level of the expression x, the 2^nd element s1 defines an applied symbol, while its other elements {n1,…,np} define the positions numbers of sub-expressions at this level. Procedure handles the erroneous situations conditioned by inadmissible levels, and positions of elements at them, printing appropriate messages. The fragment represents source code of the procedure and examples of its use.

In[6]:= SymbToExpr[x_, d_ /; ListQ[d]] := Module[{agn, t = 0, res = x,

h = If[NestListQ[d], d, {d}], k},

Do[If[Level[x, {j}] != {}, t++, Break[]], {j, Infinity}];

agn[y_, s_, n_, m_] := Module[{a},

If[n > t, Print["Level number more than maximal " <> ToString[t]],

If[Set[a, Cases[Level[y, {n}], Except[_Integer]]]; Length[a] < m,

Print["Level " <> ToString[n] <> " no contain an element with

number " <> ToString[m]],

res = Replace[y, a[[m]] –> h[[j]][[2]] @@ {a[[m]]}, n]]]];

Do[Do[agn[res, c, h[[j]][[1]], h[[j]][[k]]], {k, 3, Length[h[[j]]]}],

{j, Length[h]}]; res]

In[7]:= SymbToExpr[h + (x + y)/f[x, y^n]^g – m/n, {{4, s, 2, 3}, {2, h, 1, 2}}]

Out[7]= h – h[m]/n + (x + y)*h[f[x, s[y^n]]^–s[g]]

In[8]:= SymbToExpr[h + (x + y)/f[x, y^n]^g - m/n, {{4, S, 2, 3}, {2, V, 2, 8}}]

"Level 2 no contain an element with number 8"

Out[8]= h + (x + y)*f[x, S[y^n]]^–S[g] – m*V[1/n]

In[9]:= SymbToExpr[h + (x + y)/f[x, y^n]^g – m/n, {{6, s, 3}, {5, T, 1, 2}}]

"Level number more than maximal 5"

Out[9]= h – m/T[n] + f[x, T[y]^T[n]]^–g*(x + T[y])

In addition to the tools listed in this section, we have created a large number of other useful tools for strings processing that can be found in [16]. The tools have proven their effectiveness in many applications.

3.5. Mathematica tools for lists processing

At programming of many problems the use not of separate expressions, but their sets made in the form lists is expedient. At such approach we can instead of calculations of the separate expressions to do the demanded operations as over lists in a whole – unified objects – and over their separate elements. Lists of various types represent important and one of the most often used structures in the Mathematica system.

The list is one of the underlying data structures supported by the Mathematica. The classic list structure has the following format, namely: List1 := {a, b, c,…,d}, where as elements of List1 can be arbitrary expressions. When a list definition is calculated, its elements are calculated from left to right, remaining at their positions relative to the original list. Thus, in a list we have the possibility to pass the result of evaluation of an element to the following element of the list, accumulating an information in the last list element. In particular, list structures of the form {a, b, c, ..., Test}[[–1]] can act as test elements for formal arguments of blocks, functions, and modules, for example:

In[3333]:= A[x_ /; {If[OddQ[x], Print["Odd number"],

Print["Even number"]], IntegerQ[x]}[[–1]]] := x^2

In[3334]:= A[77]

Odd number

Out[3334]= 5929

In the meantime, mention should also be made of the list structure of form {a; b; c; d; ...; h} along with the list structures with alternation of the delimiters {";", ","} of list elements which are of quite important too. The difference between this lists and the above list structure rather illustratively illustrates the next simple example:

In[2229]:= {a = b, b = c, c = d}

Out[2229]= {b, c, d}

In[2230]:= {a = b; b = c; c = d}

Out[2230]= {d}

In[2231]:= {a = b, b = c; c = d}

Out[2231]= {b, d}

In[2234]:= B[x_ /; {h = x^3, If[OddQ[x], Print["Odd number"],

Print["Even number"]]; IntegerQ[x]}[[2]]] := x^2

In[2235]:= {B[78], h}

Even number

Out[2235]= {6084, 474552}

The above considerations are quite transparent and do not require any clarifying. Techniques represented in them are of some practical interest in programming of the objects headings and objects as a whole.

In the Mathematica many functions have Listable attribute saying that an operator, block, function, module F with Listable attribute are automatically applicable to each element of the list used respectively as their operand or argument. The call of the function ListableQ[x] returns True if a block, operator, function, module x has Listable attribute, and False otherwise [4,10-16]. At the formal level for a block, function and module F of arity 1 it is possible to note the following defining relation, namely:

Map[F, {a, b, c, d, …}] ≡ {F[a], F[b], F[c], F[d], …}

where in the left part the block, function and module F can be both with the Listable attribute, and without it whereas in the right part the existence of Listable attribute for block, function or module F is supposed. In addition, for blocks, functions and modules without the Listable attribute for receiving such effect the Map function is used. Meanwhile, for the nested lists, Map does not produce the desired result, as an example illustrates:

In[1529]:= Map[F, {a, b, {c, d, {m, n}}, t}]

Out[1529]= {F[a], F[b], F[{c, d, {m, n}}], F[t]}

Moreover, attributing the Listable attribute to the Map, we not only do not solve the problem, but violate this function. On the other hand, a function extends the built–in function Scan onto lists. The call ScanList[x, y] evaluates x applied through each element of y (see the source code and examples below):

In[47]:= ScanList[x_, y_] := If[MemberQ[Attributes[x], Listable],

Map[x, y], SetAttributes[x, Listable]; {Map[x, y],

ClearAttributes[x, Listable]}[[1]]]

In[48]:= ScanList[f, {a, b, {c, d, {m, n}}, t, {g, s}}]

Out[48]= {f[a], f[b], {f[c], f[d], {f[m], f[n]}}, f[t], {f[g], f[s]}}

In[49]:= x[t_] := N[Sin[t], 3]; ScanList[x, {42, 47, {67, 87, {6, 7}}, 3}]

Out[49]= {–0.917, 0.124, {–0.856, –0.822, {–0.279, 0.657}}, 0.141}

The problem is solved only by attributing of Listable attribute to F, as the previous example illustrates. Using of Listable attribute allows to make processing of the nested lists, without breaking their internal structure. The procedure call ListProcessing[w, y] returns the list of the same internal structure as a list w whose elements are processed by means of an y function. A fragment below represents source code of the procedure with examples of its application.

In[2225]:= ListProcessing[x_ /; ListQ[x], y_ /; SystemQ[y] ||

ProcQ[y] || FunctionQ[y] || PureFuncQ[y]] :=

Module[{f = y, g}, g = Attributes[f]; Unprotect[f];

Attributes[f] = {}; SetAttributes[f, Listable];

If[PureFuncQ[f], f = Map[f[#] &, x],

If[FreeQ[Attributes[y], Listable], SetAttributes[y, Listable];

f = y[x]; ClearAttributes[y, Listable], f = y[x]]]; f]

In[2226]:= x = {a, 7, d, c, {c, 5, s, {g, h, 8, p, d}, d}, 4, n};

ListProcessing[x, #^3 &]

Out[2226]= {a^3, 343, d^3, c^3, {c^3, 125, s^3,

{g^3, h^3, 512, p^3, d^3}, d^3}, 64, n^3}

In[2227]:= ListProcessing[x, ToString]

Out[2227]= {"a", "7", "d", "c", {"c", "5", "s", {"g", "h", "8", "p", "d"},

"d"}, "4", "n"}

In[2228]:= gv[x_] := If[SymbolQ[x], gs, If[PrimeQ[x], s, x]];

ListProcessing[x, gv]

Out[2228]= {gs, s, gs, {gs, s, gs, {gs, gs, 8, gs, gs}, gs}, 4, gs}

The procedure call ClusterSubLists[x] returns generally the nested list, whose 3–element lists determine elements of a list x, their positions and lengths of the sub-lists (clusters) that contain more than 1 identical element and that begin with this element. At the same time, the grouping of such sub-lists on base of their first element is made. In a case of lack of clusters in the x list the procedure call ClusterSubLists[x] returns the empty list, i.e. {}. The following fragment represents source code of the procedure ClusterSubLists with an example of its application.

In[3314]:= ClusterSubLists[x_ /; ListQ[x]] :=

Module[{a, b = {}, c, d, h = {}, g = {}, n = 1},

a = Gather[Map[{#, n++} &, x], #1[[1]] == #2[[1]] &];

a = Select[a, Length[#] > 1 &];

Do[AppendTo[b, Map[#[[2]] &, a[[j]]]], {j, 1, Length[a]}];

b = Map[Differences[#] &, b];

Do[Do[If[b[[j]][[k]] == 1,

g = Join[g, {a[[j]][[k]], a[[j]][[k + 1]]}], AppendTo[h, g]; g = {}],

{k, 1, Length[b[[j]]]}]; AppendTo[h, g]; g = {}, {j, 1, Length[b]}];

h = Map[If[# == {}, Nothing, DeleteDuplicates[#]] &,

DeleteDuplicates[h]];

ReduceLists[Map[ReduceLists, Gather[Map[Join[#[[1]],

{Length[#]}] &, h], #1[[1]] === #2[[1]] &]]]]

In[3315]:= ClusterSubLists[{a, a, a, c, c, c, c, a, a, a, b, b, b, a, a, a,

a, a, a, g, g, g, g, p, p, p, v, v, v, v, v, 77, 77, 77, 77, 77, 77, 77}]

Out[3315]= {{{a, 1, 3}, {a, 8, 3}, {a, 14, 5}}, {c, 4, 4}, {b, 11, 3}, {g, 19, 4},

{p, 23, 3}, {v, 26, 5}, {77, 31, 7}}

In difference of standard Partition function the procedure PartitionCond, supplementing its, allows to partition the list into sub-lists by its elements satisfying some testing function. The procedure call PartitionCond[x, y] returns the nested list consisting of sub-lists received by means of partition of a list x by its elements satisfying a testing y function. At the same time, these elements don't include in sub-lists. At absence in the x list of the separative elements an initial x list is returned. At that, at existing the third optional z argument – an arbitrary expression – the call PartitionCond[x, y, z] returns the above nested list on condition that the sequences of elements in its sub-lists will be limited by the z element. In addition, if the call contains four arguments and the fourth argument is 1 or 2 than the procedure call PartitionCond[x, y, z, h = {1|2}] returns the above nested list on condition that the elements sequences in its sub-lists will be limited by the z element at the left (h = 1) or at the right (h = 2) accordingly, otherwise the procedure call is equivalent the call PartitionCond[x, y]. The fragment below represents the source code of the PartitionCond procedure with typical examples of its application [4-8,14-16].

In[2217]:= PartitionCond[x_ /; ListQ[x],

y_ /; Quiet[ProcFuncBlQ[y, Unique["g"]]], z___] :=

Module[{a, b, c, d},

If[Select[x, y] == {}, x, If[{z} != {} && Length[{z}] > 1,

If[{z}[[2]] == 1, c = {z}[[1]]; d = Nothing,

If[{z}[[2]] == 2, c = Nothing; d = {z}[[1]], c = d = Nothing]],

If[Length[{z}] == 1, c = d = {z}[[1]], c = d = Nothing]];

a = {1, Length[x]}; b = {};

Do[AppendTo[a, If[(y) @@ {x[[j]]}, j, Nothing]], {j, 1, Length[x]}];

a = Sort[DeleteDuplicates[a]];

Do[AppendTo[b, x[[a[[j]] ;a[[j + 1]]]]], {j, 1, Length[a] – 1}];

b = ReplaceAll[Map[Map[If[(y) @@ {#}, Nothing, #] &, #1] &, b],

{} –> Nothing]];

Map[{a = #, Quiet[AppendTo[PrependTo[a, c], d]]}[[2]] &, b]]

In[2218]:= PartitionCond[{8, a, b, 3, 6, c, 7, d, h, 77, 72, d, s, 8, k,

f, a, 9}, IntegerQ[#] &]

Out[2218]= {{a, b}, {c}, {d, h}, {d, s}, {k, f, a}}

In[2219]:= PartitionCond[{8, a, b, 3, 6, c, 7, d, h, 77, 72, d, s, 8, k,

f, a, 9}, ! IntegerQ[#] &, gs]

Out[2219]= {{gs, 8, gs}, {gs, 3, 6, gs}, {gs, 7, gs}, {gs, 77, 72, gs},

{gs, 8, gs}, {gs, 9, gs}}

The Mathematica at manipulation with the list structures has certain shortcomings among which impossibility of direct assignment to elements of a list of expressions is, for example:

In[36]:= {a, b, c, d, h, g, s, x, y, z}[[10]] = 590

…Set: {a,b,c,d,h,g,s,x,y,z} in the part assignment is not a symbol.

Out[36]= 590

In order to simplify the implementation of procedures that use similar direct assignments to the list elements, the following ListAssignP procedure is used, whose call ListAssignP[x, n, y] returns the updated value of a list x which is based on results of assignment of a value y or the list of values to n elements of the x list where n – one position or their list. Moreover, if the lists n and y have different lengths, their common minimum value is chosen. The ListAssignP procedure expands the functionality of the Mathematica, doing quite correct assignments to the list elements what the system fully doesn't provide. The ListAssign1 procedure is a certain modification of the ListAssignP procedure its call is equivalent to the ListAssignP call. The source code of the ListAssignP procedure along with some examples of its use are represented below.

In[2138]:= ListAssignP[x_ /; ListQ[x], n_ /; PosIntQ[n] ||

PosIntListQ[n], y_] :=

Module[{a = DeleteDuplicates[Flatten[{n}]], b = Flatten[{y}], c, k = 1},

If[a[[–1]] > Length[x], Return[Defer[ListAssignP[x, n, y]]],

c = Min[Length[a], Length[b]]];

While[k <= c, Quiet[Check[ToExpression[ToString[x[[a[[k]]]]] <>

" = " <> ToString1[If[ListQ[n], b[[k]], y]]], Null]]; k++];

If[NestListQ1[x], x[[–1]], x]]

In[2139]:= ListAssignP[{x, y, z}, {1, 2, 3}, {42, 78, 2020}]

Out[2139]= {42, 78, 2020}

In[2140]:= Clear[x, y, z]; ListAssignP[{x, y, z}, 2, 590]

Out[2140]= {x, 590, z}

The following procedure carries out exchange of elements of a list that are determined by the pairs of positions. The call SwapInList[x,y] returns the exchange of elements of a list x that are determined by pairs of positions or their list y. At the same time, if the x list is explicitly encoded at the procedure call, the updated list is returned, if the x is determined by a symbol (that determines a list) in the string format then along with returning of the updated list also value of the x symbol is updated in situ. The procedure processes the erroneous situation caused by the indication of non–existent positions in the x list with returning $Failed and the appropriate message. Note, it is recommended to consider a reception used for updating of the list in situ. The procedure is of a certain interest at the operating with lists. The fragment represents source code of the SwapInList procedure and examples of its use which rather visually illustrate the told.

In[716]:= SwapInList[x_ /; ListQ[x] || StringQ[x] &&

ListQ[ToExpression[x]], y_ /; SimpleListQ[y]||ListListQ[y] &&

Length[y[[1]]] == 2] := Module[{a, b, c = ToExpression[x], d,

h = If[SimpleListQ[y], {y}, y]},

d = Complement[DeleteDuplicates[Flatten[y]], Range[1, Length[c]]];

If[d != {}, Print["Positions " <> ToString[d] <> " are absent in list " <>

ToString[c]]; $Failed, Do[a = c[[h[[j]][[1]]]]; b = c[[h[[j]][[2]]]];

c = ReplacePart[c, h[[j]][[1]] –> b];

c = ReplacePart[c, h[[j]][[2]] –> a], {j, 1, Length[h]}];

If[StringQ[x], ToExpression[x <> "=" <> ToString[c]], c]]]

In[717]:= SwapInList[{1, 2, 3, 4, 5, 6, 7, 8, 9}, {{1, 3}, {4, 6}, {8, 10}}]

Positions {10} are absent in list {1, 2, 3, 4, 5, 6, 7, 8, 9}

Out[717]= $Failed

In[718]:= SwapInList[{1, 2, 3, 4, 5, 6, 7, 8, 9}, {{1, 3}, {4, 6}}]

Out[718]= {3, 2, 1, 6, 5, 4, 7, 8, 9}

In the Mathematica for grouping of expressions along with simple lists also more complex list structures in the form of the nested lists are used, whose elements are lists (sub-lists) too. In this regard the lists of the ListList type whose elements – sub-lists of identical length are of special interest. The Mathematica for simple lists has testing function whose call ListQ[j] returns True, if j – a list, and False otherwise. While for testing of the nested lists we defined the useful enough functions NestListQ, NestQL, NestListQ1, ListListQ [16]. These tools are quite often used as a part of the testing components of the headings of procedures and functions both from our MathToolBox package, and in the different blocks, functions and modules first of all that are used in different problems of the system character [4-15].

In addition to the above testing functions some functions of the same class that are useful enough in programming of tools to processing of the list structures of an arbitrary organization have been created [16]. Among them can be noted testing tools such as BinaryListQ, IntegerListQ, ListNumericQ, ListSymbolQ, PosIntListQ, ListExprHeadQ, PosIntListQ, PosIntQ. In particular, the call ListExprHeadQ[x,h] returns True if a list x contains only elements meeting the condition Head[a] = h, and False otherwise. At that, the testing means process all elements of the analyzed list, including all its sub-lists of any level of nesting. The above means are often used at programming of the problems oriented on the lists processing. These and other means of the given type are located in our MathToolBox package [16]. Their correct use assumes that the package is uploaded into the current session.

A number of problems dealing with the nested lists require to determine their maximum nesting level. In this direction, we have proposed a number of means. For example, the procedure call ListLevelMax[x] whose source code along with examples of its application are shown below, returns the maximum nesting level of a list x. Whereas the procedure call ListLevelMax[x, y] with the 2^nd optional argument y – an arbitrary symbol – through y additionally returns the nested list whose 2-elements sub-lists as the first argument define numbers of nesting levels of the x list, while as the 2^nd elements define the quantities of elements that are different from lists at corresponding levels.

In[1222]:= ListLevelMax[x_ /; ListQ[x], y___] :=

Module[{a = x, b = {}, t = 1},

Do[AppendTo[b, {t, Length[Select[a, ! ListQ[#] &]]}];

a = Select[a, ListQ[#] &]; If[a == {}, Break[], t++];

a = Flatten[a, 1], {j, Infinity}];

If[{y} != {} && SymbolQ[y], y = b, 77]; t]

In[1223]:= ListLevelMax[{a, b, {c, {m, {c, {{b}}, d}, n}, d}, c, d}]

Out[1223]= 6

In[1224]:= ListLevelMax[{a, b, {c, {m, {c, {{b}}, d}, n}, d}, c, d}, gs]

Out[1224]= 6

In[1225]:= gs

Out[1225]= {{1, 4}, {2, 2}, {3, 2}, {4, 2}, {5, 0}, {6, 1}}

In[1226]:= ListLevelMax[{a, b, c, m, n, d, c, d}, gv]

Out[1226]= 1

In[1227]:= gv

Out[1227]= {{1, 8}}

The procedure call MaxLevel[x] returns the maximal level of nesting of a list x; its source code is represented below:

In[1230]:= MaxLevel[x_ /; ListQ[x]] := Module[{k},

If[! NestListQ1[x], 1, For[k = 1, k <= Infinity, k++,

If[Level[x, {k}] == {}, Break[], Continue[]]]; k]]

In[1231]:= MaxLevel[{a, b, {c, {m, {c, {{b}}, d}, n}, d}, c, d}]

Out[1231]= 7

In[1231]:= MaxLevel[{a, b, {c, {m, {c, {{}}, d}, n}, d}, c, d}]

Out[1231]= 6

In[1232]:= MaxLevelList[x_ /; ListQ[x]] :=

Module[{b = 0, a = ReplaceAll[x, {} –> {77}]},

Do[If[Level[a, {j}] == {}, Return[b], b++], {j, 1, Infinity}]]

In[1233]:= MaxLevelList[{a, b, {c, {m, {c, {{}}, d}, n}, d}, c, d}]

Out[1233]= 6

In[1234]:= MaxLevelList[{a, b, {c, {m, {c, {{b}}, d}, n}, d}, c, d}]

Out[1234]= 6

Meanwhile, it is appropriate to make a significant comment here about our means that use the built–in Level function, and without its using. The function call Level[expr, level] returns a list of all sub-expressions of expr on levels specified by level. At the same time the function call Level[expr, {level}] returns a list of all sub-expressions of expr on a level. Meanwhile, using the built–in Level function when the nested lists are used as an expr requires a certain care. For example, from the fragment below easily follows, you can obtain the incorrect results at using of the Level function when calculating the maximum nesting level of a list. In particular, on lists of form {…{}…} and {…{a, b, c}…}.

In[3342]:= Do [Print[Level[{{{{a, b, c}}}}, {j}]], {j, 1, 5}]

{{{{a, b, c}}}}

{{{a, b, c}}}

{{a, b, c}}

{a, b, c}

{}

In[3343]:= MaxLevel[{{{{a, b, c}}}}]

Out[3343]= 4

In[3344]:= Do [Print[Level[{{{{}}}}, {j}]], {j, 1, 5}]

{{{{}}}}

{{{}}}

{{}}

{}

{}

In[3345]:= MaxLevel[{{{{}}}}]

Out[3345]= 3

In[3346]:= MaxLevel1[{{{{a, b, c}}}}]

Out[3346]= 4

In[3347]:= Map[#[{{}}] &, {MaxLevel, ListLevelMax}]

Out[3347]= {1, 2}

It has been found that it is the use of an empty list {} as the last nesting level in the list that results in, among other things, an incorrect calculation of the maximum nesting level of a list. Therefore, in the development of the built–in Level function for lists, a simple Mlist function has been programmed, whose call Mlist[x] returns the replacement result in a list x of sub–lists of view {}, i.e. empty lists, to lists of view {w}, where w is a symbol unique to the current session. The following fragment presents the source code of the function with an example of its use along witx its use in the MaxLevel2 procedure being a modification of the above MaxLevel procedure, eliminating said its drawback.

In[1240]:= Mlist[x_] := If[ListQ[x],

ReplaceAll[x, {} –> ToString[{Unique["v77" ]}]], x]

In[1241]:= Mlist[{{}, {{{}}}, {}}]

Out[1241]= {"{v7711}", {{"{v7711}"}}, "{v7711}"}

In[1242]:= MaxLevel2[x_ /; ListQ[x]] := Module[{a = Mlist[x], k},

If[! NestListQ1[a], 1, For[k = 1, k <= Infinity, k++,

If[Level[a, {k}] == {}, Break[], Continue[]]]]; k – 1]

In[1243]:= MaxLevel2[{a, b, {c, {m, {c, {{b}}, d}, n}, d}, c, d}]

Out[1243]= 6

In[1244]:= MaxLevel2[{a, b, {c, {m, {c, {{}}, d}, n}, d}, c, d}]

Out[1244]= 6

In[1245]:= MaxLevel4[x_ /; ListQ[x]] :=

Module[{c = SetDelayed[a[t_], Nothing], b},

SetAttributes[a, Listable]; b = Map[a, x]; c = 1;

While[b != {}, b = Flatten[b, 1]; c++]; c]

In[1246]:= MaxLevel4[{a, {b, {c, {h, {k}}}}}]

Out[1246]= 5

Meanwhile, it should be borne in mind that the use of the Mlist function has its limitations – its application may be quite appropriate in a case of tasks whose algorithm remains correct if the structural organization of a list is preserved, but without taking into account its content. This is a case when calculating the maximum nesting level of a list.

Note that in addition to the above means, several versions of similar means have been created: MaxLevel1, MaxLevelList, MaxNestLevel, MaxLevel4 using other approaches [16].

Finally, based on the pure structure of the list, it is possible successfully solve a number of nested list processing problems, including calculating their maximum nesting level. At the same time, the pure structure of a list will be understood as a list, all elements of which are deleted, forming the corresponding list consisting only of characters {"{", "}"}. The following procedure solves the converting problem of a list to its pure structure. The source code of the PureListStr function is represented below.

In[2242]:= PureListStr[x_ /; ListQ[x]] := Module[{f,

a = Map[ToString[#] &, Flatten[x]]}, f[t_] := Quiet[Set[t, Nothing]];

SetAttributes[f, Listable]; {Map[f[#] &, x], Map[ClearAll[#] &, a]}[[1]]]

In[2243]:= PureListStr[{a, b, c, d}]

Out[2243]= {}

In[2244]:= PureListStr[{{a, {a, b, c, {{{a, {}, b}}}}, b, {c, d}}}]

Out[2244]= {{{{{{{}}}}}, {}}}

In[2245]:= PureListStr[{{a, {a, {c, {h}}, b, c, {{{a, {n}, b}}}}, b, {c, d}}}]

Out[2245]= {{{{{}}, {{{{}}}}}, {}}}

The function call PureListStr[x] returns the result of converting of a list x to its pure structure. So, the previous fragment additionally represents the examples of the above function application.

This function is a rather useful for lists processing problems. In particular, it makes it possible to calculate the maximum nesting level of a list based not on the built-in Level function, but on pure structure, as illustrated by example of the MaxLevel3 procedure, whose source code along with examples of its use is represented below.

In[2249]:= MaxLevel3[x_ /; ListQ[x]] :=

Module[{a = PureListStr[x], p = 1},

Do[If[a == {}, Return[p], a = ReplaceAll[a, {} –> Nothing]; p++],

{j, Infinity}]]

In[2250]:= Map[MaxLevel3, {{}, {{{b}}}}]

Out[2250]= {1, 3}

In[2251]:= MaxLevel3[{a, b, {c, {m, {c, {{{{{}}}}}, d}, n}, d}, c, d}]

Out[2251]= 9

The procedure call MaxLevel3[x] returns the maximal nesting level of a list x; examples presented above illustrate aforesaid. By organization the algorithm, the above procedure MaxLevel4 closely adjoins the MaxLevel3 procedure [16].

Along with the task of determining the maximum nesting level of a list, the PureListStr procedure allows to calculate so-called isolated levels of the list, where as the isolated nesting levels are understood the nesting levels in a list of isolated its sub-lists. The procedure call IsolLevelsList[x] returns the list of the isolated nesting levels of a list x, while the call IsolLevelsList[x, y] through optional argument y – an arbitrary symbol – additionally returns the maximal nesting level of the x list. Fragment below represents the source code of the IsolLevelsList procedure along with some typical examples of its application.

In[2277]:= IsolLevelsList[x_ /; ListQ[x]] :=

Module[{a = PureListStr[x], b, c, d = {}, h = "", k},

b = StringTake[ToString[a], {2, –2}]; c = StringSplit[b, ","];

c = Map[StringTrim[#] &, c];

Do[h = h <> c[[j]];

If[SyntaxQ[h], AppendTo[d, MaxLevel3[ToExpression[h]]]; h = "",

Do[h = h <> c[[k]],

If[SyntaxQ[h], AppendTo[d, MaxLevel3[ToExpression[h]]]; h = "",

Break[]]], {k, j, Length[c]}], {j, Length[c]}];

If[{y} != {} && SymbolQ[y], y = Max[Map[# + 1 &, d]], 7];

Map[# + 1 &, d]]

In[2278]:= d = {{{}}, {}, {{{{}, {}}}}, {{{{{}, {}}}}}};

In[2279]:= IsolLevelsList[d]

Out[2279]= {3, 2, 5, 6}

In[2280]:= d1 = {{}, {{}}, {{{}, {}, {}}}, {{{{}, {}}}}};

In[2281]:= {IsolLevelsList[d1, gs], gs}

Out[2281]= {{2, 3, 4, 5}, 5}

In[2282]:= MaxLevel3[d1]

Out[2282]= 5

In[2283]:= {IsolLevelsList[{{{{{{{}}}}}}}, gs1], gs1}

Out[2283]= {{7}, 7}

In[2284]:= {IsolLevelsList[{{}, {c, {c, {{b, {m, {n}, d}, n, d}}, d}}}, j], j}

Out[2284]= {{2, 7}, 7}

In addition to the above-mentioned problems based on use of pure list structures, there are many other problems utilizing this methodology. A number of quite interesting tasks that use such an approach can be found, for example, in [6,8-15].

The procedure below gives an useful method of elements transposition of the list. The procedure call ListRestruct[x, y] returns the result of mutual exchanges of elements of a list x that are defined by the rule or their list y of the kind aj –> bj, where aj – positions of elements of the list x and bj – their positions in the returned list, and vice versa. At that, the procedure excludes unacceptable rules from y list. The fragment represents source code of the ListRestruct procedure and examples of its use.

In[21]:= ListRestruct[x_List, y_ /; RuleQ[y]||ListRulesQ[y]] :=

Module[{b = If[RuleQ[y], {y}, y], c, d},

b = DeleteDuplicates[DeleteDuplicates[b, #1[[1]] === #2[[1]] &],

#1[[1]] === #2[[1]] &];

c = Map[#[[1]] –> x[[#[[2]]]] &, b]; d = ReplacePart[x, c];

c = Map[#[[1]] –> x[[#[[2]]]] &, RevRules[b]]; ReplacePart[d, c]]

In[22]:= ListRestruct[{1, 3, 77, 4, 5, 28, 3, 2, 6, 40, 4, 3, 20, 3},

{1 –> 6, 8 –> 12, 1 –> 14, 8 –> 10, 5 –> 10}]

Out[22]= {28, 3, 77, 4, 40, 1, 3, 3, 6, 5, 4, 2, 20, 3}

In[23]:= ListRestruct[{b, {j, n}, c + f[t], d/(h + d)}, {4 –> 1, 3 –> 2}]

Out[23]= {d/(d + h), c + f[t], {j, n}, b}

Means of operating with levels of the nested list are of a special interest. In this context the following tools can be rather useful. As one of such means the MaxLevel procedure can be presented whose call MaxLevel[x] returns the maximum nesting level of a list x (at that, the nesting level of a simple x list is supposed equal to one). In addition, MaxLevel1 procedure is an equivalent version of the previous procedure. Whereas the call ListLevels[x] returns the list of nesting levels of a list x; for the simple list x or empty list the procedure call returns {1}. The following fragment represents source codes of the above three procedures including some typical examples of their applications.

In[1333]:= L = {a, b, a, {d, c, s}, a, b, {b, c, {x, y, {v, g, z,

{90, {500, {}, 77}}, a, k, a}}, b}, c, b};

In[1334]:= MaxLevel[L_ /; ListQ[L]] := Module[{k},

For[k = 1, k <= Infinity, k++, If[Level[L, {k}] == {},

Break[], Continue[]]]; k]

In[1335]:= MaxLevel1[x_ /; ListQ[x]] := Module[{a = x, k = 1},

While[NestListQ1[a], k++; a = Flatten[a, 1]; Continue[]]; k]

In[1336]:= Map[#[L] &, {MaxLevel, MaxLevel1}]

Out[1336]= {7, 7}

In[52]:= ListLevels[x_ /; ListQ[x]] := Module[{a = x, b, c = {}, k = 1},

If[! NestListQ1[x], {1}, While[NestListQ1[a], b = Flatten[a, 1];

If[Length[b] >= Length[a], AppendTo[c, k++],

AppendTo[c, k++]]; a = b; Continue[]]; c = AppendTo[c, k++]]]

In[53]:= Map[ListLevels[#] &, {L, {{{{}}}}, {}}]

Out[53]= {{1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4}, {1}}

Between the above procedures the defining relations take place:

Flatten[x] ≡ Flatten[x, MaxLevel[x]]; MaxLevel[x] ≡ ListLevels[x][[-1]]

The AllElemsMaxLevels procedure extends the procedure ElemsMaxLevels onto all expressions composing the analyzed list [8,16]. The call AllElemsMaxLevels[x] returns the list of the ListList type whose 2-element sub-lists have all possible elements of a list x as the first elements, and their maximal nesting levels in the x list as the second elements.

In[67]:= AllElemsMaxLevels[x_ /; ListQ[x]] :=

Module[{g = ToString1[x], c, Agn},

If[x == {}, {}, Agn[z_ /; ListQ[x], y_] :=

Module[{a = MaxLevelList[z], d = {}, h},

h = Map[{#, Level[z, {#}]} &, Range[1, a]];

Do[Do[AppendTo[d, If[FreeQ[h[[j]][[2]], y[[k]]], Nothing,

{y[[k]], j}]], {k, 1, Length[y]}], {j, 1, a}];

d = Gather[d, #1[[1]] == #2[[1]] &];

d = Map[Sort[#, #1[[2]] >= #2[[2]] &] &, d]; Map[#[[1]] &, d]];

c = AllSubStrings[g, 0, ExprQ[#] &];

g = DeleteDuplicates[ToExpression[c]]; g = Agn[x, g];

If[Length[g] == 1, g[[1]], g]]]

In[68]:= L := {a, b, {{a, b}, {{m, n, {p, {{{f[t]}}}}}}}, c, {}, m, n, p,

{{{{a + b}, g}}}}; AllElemsMaxLevels[L]

Out[69]= {{a, 6}, {b, 6}, {m, 4}, {n, 4}, {p, 5}, {f, 8}, {t, 9}, {c, 1}, {g, 4}, {{}, 1}, {f[t], 8}, {a + b, 5}, {{a, b}, 2}, {{f[t]}, 8}, {{a + b}, 4}, {{{f[t]}}, 7}, {{{{f[t]}}}, 5}, {{{a + b}, g}, 3}, {{{{a + b}, g}}, 2}, {{p, {{{f[t]}}}}, 4},
{{{{{a + b}, g}}}, 1}, {{m, n, {p, {{{f[t]}}}}}, 3}, {{{m, n, {p, {{{f[t]}}}}}}, 2}, {{{a, b}, {{m, n, {p, {{{f[t]}}}}}}}, 1}, {{a, b, {{a, b}, {{m, n, {p, {{{f[t]}}}}}}},
c, {}, m, n, p, {{{{a + b}, g}}}}, 1}}

The procedure call ElemsOnLevelList[x] returns the nested list whose elements – the nested two–element lists, whose the first elements are nesting levels of a list x, whereas the second elements – the lists of elements at these nesting levels. If list x is empty, then the list {} is returned. Below, the source code of the procedure with an example of its application are represented.

In[2370]:= ElemsOnLevelList[x_ /; ListQ[x]] :=

Module[{a, b, c = {}, d = {}, k = 1, t, f, h, g},

f = ReplaceAll[x, {{} –> Set[g, FromCharacterCode[3]], 2 –> h}];

a = LevelsOfList[f];

Do[AppendTo[c, {a[[t = k++]], Flatten[f][[t]]}], {Length[a]}];

c = Map[Flatten, Gather[c, #1[[1]] == #2[[1]] &]]; k = 1;

Do[AppendTo[d, {c[[t = k++]][[1]], Select[c[[t]],

EvenQ[Flatten[Position[c[[t]], #]][[1]]] &]}], {Length[c]}];

c = ReplaceAll[d, {Null –> Nothing, h –> 2}];

ReplaceAll[Sort[c, #1[[1]] < #2[[1]] &], g –> Nothing]]

In[2371]:= ElemsOnLevelList[L]

Out[2371]= {{1, {a, b, c, m, n, p}}, {3, {a, b}}, {4, {m, n, g}},

{5, {p, a + b}}, {8, {f[t]}}}

The following procedure represents a rather useful version of the above procedure. The procedure call ElemsOnLevel1[x, y] returns the result of distribution of elements on nesting levels of a list x. In addition, at a nesting level n of form ...{a, b, ..., {c}, ..., d}... only {a, b, ..., d} elements other than lists are considered as elements, whereas the elements of more higher nesting level p > n, such as {c}, are ignored. The source code of the procedure ElemsOnLevel1 and examples of its use are represented below.

In[2221]:= ElemsOnLevel1[x_ /; ListQ[x], y___] :=

Module[{a = ElemsOnLevel[x], b, c = {}, d = {}},

b[t_] := AppendTo[c, t]; SetAttributes[b, Listable]; Map[b, a];

c = Gather[Partition[c, 3], #1[[3]] == #2[[3]] &];

c = Map[Flatten, c];

Do[AppendTo[d, {c[[j]][[3]], Map[c[[j]][[#]] &,

Range[1, Length[c[[j]]], 3]]}], {j, Length[c]}];

c = Sort[d, #1[[1]] < #2[[1]] &];

If[{y} != {} && ! HowAct[y],

y = Map[{#[[1]], Length[#[[2]]]} &, c], 77]; c]

In[2222]:= t := {{p, b, x^2 + z^2, c^b, 7.7, 78, m/n}, u, c^2,

{{{n, {"h", f, {m, n, p}, c}, np^2}}}, {77, "x"}}

In[2223]:= ElemsOnLevel1[t]

Out[2223]= {{1, {u, c^2}}, {2, {p, b, x^2 + z^2, c^b, 7.7, 78, m/n,

77, "x"}}, {4, {n, np^2}}, {5, {"h", f, c}}, {6, {m, n, p}}}

In[2224]:= ElemsOnLevel1[t, agn]

Out[2224]= {{1, {u, c^2}}, {2, {p, b, x^2 + z^2, c^b, 7.7, 78, m/n,

77, "x"}}, {4, {n, np^2}}, {5, {"h", f, c}}, {6, {m, n, p}}}

In[2225]:= agn

Out[2225]= {{1, 2}, {2, 9}, {4, 2}, {5, 3}, {6, 3}}

The procedure call returns the result as the nested list of the form {{la, {a1,…,an}}, {lb, {b1,...,bm}},…,{lc, {c1,...,cp}}} that defines the distribution of elements {q1,...,qt} on a nesting level lq, while thru the optional argument y – an indefinite variable – is returned the nested list of the ListList type of the form {{la, na}, {lb, ba},…, {lc, ca}}, where lj – a nesting level and nj – number of elements on this level. The ElemsOnLevel1 procedure is of certain interest as in determining the structural organization of nested lists and at programming of a number tasks connected with nested lists.

Using the ElemsOnLevelList procedure, we can determine a tool of converting of any list into so-called canonical form, i.e. in the list of form {Level1, {Level2, {Level3, {Level4,…,Leveln}…}}}}; at the same time, all elements of a Levelj level are sorted {j=1..n}. The problem is solved by a tool, whose call RestructLevelsList[x] returns a list x in the above canonical form. The fragment below represents source code of the tool and an example of its use.

In[376]:= RestructLevelsList[x_ /; ListQ[x]] :=

Module[{a, b = FromCharacterCode[3], g = FromCharacterCode[4], c, d, h = {}}, d = b; a = ElemsOnLevelList[ReplaceAll[x, {} –> {g}]];

c = Map[{AppendTo[h, #[[1]]]; #[[1]], Join[Sort[#[[2]]], {b}]} &, a];

h = Flatten[{1, Differences[h]}];

Do[d = ReplaceAll[d, b –> If[c[[j]][[1]] – j == 1, c[[j]][[2]],

Nest[List, c[[j]][[2]], h[[j]] – 1]]], {j, 1, Length[h]}];

ReplaceAll[d, {b –> Nothing, g –> {}}]]

In[377]:= RestructLevelsList[L]

Out[377]= {a, b, c, m, n, p, {{}, {a, b, {g, m, n, {a + b, p, {{{f[t]}}}}}}}}

The following procedure has algorithm different from the previous procedure and defines a tool of converting of any list into the above canonical form; all elements of a Levelj level are sorted {j = 1..n}. The procedure call CanonListForm[x] returns a list x in the above canonical form. The fragment presents source code of the procedure and typical examples of its application.

In[7]:= CanonListForm[x_ /; ListQ[x]] :=

Module[{a = Map[{#[[1]], Sort[#[[2]]]} &, ElemsOnLevelList[x]],

b, c, h = {}, p, m, s, f, n = 2, d = FromCharacterCode[3]},

b = Max[Map[#[[1]] &, a]]; AppendTo[h, c = "{" <> d <> "1,"];

Do[Set[c, "{" <> d <> ToString[n++] <> ","];

AppendTo[h, c], {j, 2, b}];

p = StringJoin[h] <> StringMultiple["}", b];

m = Map[Rule[d <> ToString[#[[1]]],

StringTake[ToString1[#[[2]]], {2, –2}]] &, a];

m = Map[If[#[[2]] == "", Rule[#[[1]], "{}"], #] &, m];

s = StringReplace[p, m];

f[t_] := If[! StringFreeQ[ToString[t], d], Nothing, t];

SetAttributes[f, Listable];

Map[f, ToExpression[StringReplace[s, ",}" –> "}"]]]]

In[8]:= L = {s, "a", g, {{{{{m}}}}}, m, {{{77}}}, m, c, m, {f, d, {{72, s}},

h, m, {m + p, {{p}}, n, 52, p, a}}, {7, {90, 500}}}; CanonListForm[L]

Out[8]= {"a", c, g, m, m, m, s, {7, d, f, h, m, {52, 90, 500, a, n, p,

m + p, {72, 77, s, {p, {m}}}}}}

The CanonListForm uses a number of receptions useful in practical programming of the problems dealt of lists of various types. Whereas the following simple function allows to obtain a representation of a list x in a slightly different canonical format: {{L1}, {{L2}}, {{{L3}}},…,{{{…{Ln}…}}}} where Lj present the sorted list elements at appropriate nesting j level of the x list (j = 1..n):

In[4331]:= CanonListForm1[x_ /; ListQ[x]] :=

Map[Nest[List, #[[2]], #[[1]] – 1] &, Map[{#[[1]], Sort[#[[2]]]} &,

ElemsOnLevelList[x]]]

In[4332]:= CanonListForm1[L = {a, b, {c, d, {g, h, {x, {y, {z}}}}}}]

Out[4332]= {{a, b}, {{c, d}}, {{{g, h}}}, {{{{x}}}}, {{{{{y}}}}}, {{{{{{z}}}}}}}

In[4333]:= Map[LevelsList[#] &, CanonListForm1[L]]

Out[4333]= {1, 2, 3, 4, 5, 6}

At the same time, the nesting of elements of a list returned by the CanonListForm1 function can be defined, for example, by LevelsList procedure calls as illustrates the last example of the previous fragment. Incidentally, the list of elements on each level of the nested list, which in turn are different from the lists, is of a certain interest. This problem is solved by means of the procedure whose call ElemsOnLevels4[x] returns the list of 2–element sub–lists in ascending order of the nesting levels of a list x; the first element of each sub-list defines the list of elements (which are different from the list) of the j–th nesting level, whereas the second element defines the j–level. The fragment represents source code of the procedure with an example of its use.

In[1947]:= ElemsOnLevels4[x_ /; ListQ[x]] :=

Module[{a = {}, b = 78, c, g}, g[t_] := ToString1[t];

SetAttributes[g, Listable]; c = Map[g, x];

Do[b = Level[c, {j}];

If[b != {}, AppendTo[a, {Map[If[ListQ[#], Nothing, #] &, b], j}],

Return[a]], {j, 1, Infinity}]; Map[ToExpression, a]]

In[1948]:= ElemsOnLevels4[{a, b, {c + t, d, {g, {m, n}, h, {x^y,

{y, {z, g, h/g, {m, n*p}, g}}}}}}]

Out[1948]= {{{a, b}, 1}, {{c + t, d}, 2}, {{g, h}, 3}, {{m, n, x^y}, 4},

{{y}, 5}, {{z, g, h/g, g}, 6}, {{m, n*p}, 7}}

On the basis of the above procedure the ElemsOnLevels5 procedure useful enough in a number of appendices dealing with the nested lists can be quite easily programmed. Calling the procedure ElemsOnLevels5[x] returns the list of 3–element sub–lists in ascending order of the nesting levels of a list x; the first element of each sub-list determines a list element t (which is different from the list), the second element defines the j-th nesting level on which it locates, whereas the third element defines the sequential number of the t element on the nesting j level. At the same time, the sub-lists of the returned list relative to their first elements are arranged in the order of elements of the Flatten[x] list. The fragment represents source code of the procedure with examples of its use. In particular, based on the procedure, it is easy to get the number of elements other than the list located at a set nesting level.

In[1949]:= ElemsOnLevels5[x_ /; ListQ[x]] :=

Module[{a = ElemsOnLevels4[x], b = {}, c, d = {}, k, j},

For[k = 1, k <= Length[a], k++, c = 1;

AppendTo[b, Map[{#, a[[k]][[2]], c++} &, a[[k]][[1]]]]];

For[k = 1, k <= Length[b], k++,

For[j = 1, j <= Length[b[[k]]], j++,

AppendTo[d, b[[k]][[j]]]]]; d]

In[1950]:= ElemsOnLevels5[{a, b, {"c + t", d, {g, {m, n}, h,

{x^y, {y, {z, g, h/g, {m, n*p}, g}}}}}}]

Out[1950]= {{a, 1, 1}, {b, 1, 2}, {"c + t", 2, 1}, {d, 2, 2}, {g, 3, 1},

{h, 3, 2}, {m, 4, 1}, {n, 4, 2}, {x^y, 4, 3}, {y, 5, 1},

{z, 6, 1}, {g, 6, 2}, {h/g, 6, 3}, {g, 6, 4}, {m, 7, 1},

{n*p, 7, 2}}

In[1951]:= ElemsOnLevels5[{a, b, c + t, d, g, m, n, x^y, h/g}]

Out[1951]= {{a, 1, 1}, {b, 1, 2}, {c + t, 1, 3}, {d, 1, 4}, {g, 1, 5},

{m, 1, 6}, {n, 1, 7}, {x^y, 1, 8}, {h/g, 1, 9}}

Meanwhile, using the ElemsOnLevels5 procedure, it is easy to program a procedure whose call result StructNestList[x] is the list whose elements are identical to the elements of the list returned by the call ElemsOnLevels5[x], while preserving the internal structure of the list x.

In[2221]:= StructNestList[x_ /; ListQ[x]] :=

Module[{a = ElemsOnLevels2[x], b, c}, c = 1;

b[t_] := Join[{t}, a[[c++]][[2 ;3]]]; SetAttributes[b, Listable];

Map[b[#] &, x]]

In[2222]:= StructNestList[{a, b, {"c + t", d, {g, {m, n}, h, {x^y, {y,

{z, g, h/g, {m, n*p}, g}}}}}}]

Out[2222]= {{a, 1, 1}, {b, 1, 2}, {{"c + t", 2, 1}, {d, 2, 2}, {{g, 3, 1},

{{m, 3, 2}, {n, 4, 1}}, {h, 4, 2}, {{x^y, 4, 3}, {{y, 5, 1}, {{z, 6, 1},

{g, 6, 2}, {h/g, 6, 3}, {{m, 6, 4}, {n*p, 7, 1}}, {g, 7, 2}}}}}}}

Whereas procedure call SelectNestElems[x, y] returns a list of elements of a list x that are at the given nesting levels with sequential numbers on them, defined by a list y of ListList type. If there is no a nesting level and/or sequential number from y, the corresponding message is printed.

In[2227]:= w = {a, b, {"c + t", d, {g, {m, n}, h, {x^y, {y,

{z, g, h/g, {m, n*p}, g}}}}}};

In[2228]:= SelectNestElems[x_ /; ListQ[x], y_ /; ListListQ[y] &&

IntegerListQ[y]] := Module[{a = ElemsOnLevels5[x], b = {}, c, d},

c = Map[#[[2 ;3]] &, a];

Map[If[MemberQ[c, #], Nothing,

Print["Level/element " <> ToString[#] <> " is absent"]] &, y];

Map[If[MemberQ[y, Set[d, a[[#]][[2 ;3]]]],

AppendTo[b, a[[#]][[1]]], Nothing] &, Range[1, Length[a]]]; b]

In[2229]:= SelectNestElems[w, {{2, 1}, {4, 3}, {6, 3}, {7, 1}}]

Out[2229]= {"c + t", x^y, h/g, m}

In[2230]:= SelectNestElems[w, {{2, 1}, {4, 5}, {6, 3}, {8, 2}}]

Level/element {4, 5} is absent

Level/element {8, 2} is absent

Out[2230]= {"c + t", h/g}

The procedure below is used to replace/delete individual elements in a nested list while maintaining its structure. Calling the procedure ReplaceNestList[x, y] returns the list – the result of replacement/deletion of elements of a nested list x on a basis of replacements defined by a nested list y of the form {{j1/Null, m1, n1},…,{jp/Null, mp, np}} where {mk, nk} – a nesting level and sequential number accordingly on it whose element is replaced on jk (k=1..p). Null instead of jk deletes the appropriate element of the list x. The procedure handles main erroneous situations linked with invalid nesting levels and/or positions on them; its source code with examples is presented below.

In[8]:= ReplaceNestList[x_ /; ListQ[x], y_ /; ListListQ[y]] :=

Module[{a, c, d = {}, f, f1, f2, g, s, h}, f1[t_] := ToString1[t];

f[t_] := If[IntegerQ[t], Nothing, t]; Map[SetAttributes[#, Listable] &, {f, f1, f2}]; g = Map[f1, x]; a = ElemsOnLevels5[g];

h = Intersection[c = Map[#[[2 ; 3]] &, a], s = Map[#[[2 ; 3]] &, y]];

If[Set[h, Complement[s, h]] == {}, 78,

If[h == s, Print["Second argument is invalid"]; Return[],

Print[ToString1[h] <> " are invalid Levels/Positions"]]];

h = Sort[Map[If[MemberQ[c, #[[2 ;3]]], #, Nothing] &, y],

#1[[2]] <= #2[[2]] &];

c = Map[If[MemberQ[Map[#[[2 ;3]] &, h], #[[2 ;3]]], #,

Nothing] &, a]; s = Map[{ToString1[#[[1]]], #[[2 ;3]]} &, h];

s = Map[Flatten, s]; c = Map[c[[#]] –> s[[#]] &, Range[1, Length[c]]];

c = ReplaceAll[a, c]; c = Flatten[Map[f, c]];

Do[If[MemberQ[Range[1, Length[c], 3], k],

AppendTo[d, c[[k]]], 78], {k, Length[c]}]; a = 1; f2[t_] := c[[a++]];

ReplaceAll[ToExpression[Map[f2, x]], Null –> Nothing]]

In[9]:= ReplaceNestList[w, {{G, 2, 1}, {S, 3, 2}, {V, 4, 3}, {X, 1, 1}}]

Out[9]= {X, b, {G, d, {g, {S, m}, n, {V, {y, {z, g, h/g,

{g, {m, n p}, p}, q}}}}}}

In[10]:= ReplaceNestList[w, {{mn, 2, 1}, {ab, 3, 2}, {cd, 4, 3},

{Null, 1, 1}, {yyy, 1, 4}, {mmm, 2, 5}}]

{{1, 4}, {2, 5}} are invalid Levels/Positions

Out[10]= {b, {mn, d, {g, {ab, m}, n, {cd, {y, {z, g, h/g,

{g, {m, n p}, p}, q}}}}}}

In[11]:= ReplaceNestList[w, {{mn, 2, 3}, {ab, 3, 5}, {cd, 4, 5}}]

Second argument is invalid

It should be noted that: unlike the standard Mathematica interpretation, under a nested list, we everywhere in the book understand a list whose elements are different from lists, that is, we consider nesting to the full depth that is more natural.

On the basis of the above procedure the NumElemsOnLevels function useful enough in a number of appendices can be quite easily programmed. The call NumElemsOnLevels[x] returns the list of ListList type, whose two–element sub-lists define nesting levels and quantity of elements on them respectively. While the function call NumElemsOnLevel[x, p] returns the quantity of elements of a list x on its nesting level p. In a case of absence of the p level the function call returns $Failed with printing of the appropriate message. The source codes of both functions with examples of their application are represented in [8-16]. Whereas the problem of by–level sorting of elements of the nested list is solved by the procedure whose call SortOnLevels1[x, y] returns the default sort result of each nesting level of a list x, if optional y argument is missing, or according to a set ordering function y.

In[7]:= SortOnLevels1[x_ /; ListQ[x], y___] := Module[{a, b = x, c},

a = MaxLevel[x]; Do[c = Level[b, {j}];

b = ReplaceAll[b, c –> Sort[c, If[{y} != {}, y, Nothing]]], {j, a, 1, –1}]; b]

In[8]:= g = {3, 2, 1, {5, 77, 6, 8, {72, 8, 9, {0, 5, 6, 1}}, 78, 12, 67, 78}};

In[9]:= SortOnLevels1[g, #1 <= #2 &]

Out[9]= {1, 2, 3, {5, 6, 8, 77, {8, 9, 72, {0, 1, 5, 6}}, 12, 67, 78, 78}}

At the same time, unlike the SortOnLevels1 procedure, the SortOnLevel1 procedure allows to sort list elements at its fixed nesting level. Calling the procedure SortOnLevel1[x, n] returns the result of elements sorting of the n–th nesting level of a list x relative to their location (e.g. a permutation of their positions) at the n–th level. If the call contains the above two arguments, the sort is in canonical order, otherwise the sort is based on the sorting function determined by the third optional y argument – a pure function. Using index j = {3|1} in expressions #k[[j]] in the pure function y as a sorting function, we can sort the x list at the n-th nesting level basing on positions of its elements or themselves elements accordingly as examples below illustrate.

In[1942]:= SortOnLevel1[x_ /; ListQ[x], n_ /; PosIntQ[n], y___] :=

Module[{a, b, c, d, p, f, g},

If[! MemberQ[ListLevels[x], n],

"Level " <> ToString[n] <> " is absent in the list",

a = ElemsOnLevels2[x]; a = Select[a, #[[2]] == n &];

If[a == {}, "The " <> ToString[n] <> "–th level does no

contain elements", p = FromCharacterCode[7];

b[t_] := ToString1[t]; f[t_] := If[StringQ[t], t, Nothing];

g[t_] := p <> t <> p;

Map[SetAttributes[#, Listable] &, {b, f, g}]; c = Map[b, x];

d = StructNestList[c];

a = Map[Flatten[{ToString1[#[[1]]], #[[2 ;3]]}] &, a];

c = Sort[a, If[{y} != {} && PureFuncQ[y], y, Order]];

d = ReplaceAll[d, Map[a[[#]] –> c[[#]] &, Range[1, Length[a]]]];

d = Map[f, d]; d = Map[g, d];

ToExpression[StringReplace[ToString[d],

{"{" <> p –> "", p <> "}" –> ""}]]]]]

In[1943]:= g = {{a, b, {d, z, {m, {1, 2, 3, 4, 5}, n, t}, u, c, {x, y}}}};

In[1944]:= SortOnLevel1[g, 5, #1[[3]] > #2[[3]] &]

Out[1944]= {{a, b, {d, z, {m, {5, 4, 3, 2, 1}, n, t}, u, c, {x, y}}}}

In[1945]:= SortOnLevel1[g, 3, Order[#1[[1]], #2[[1]]] &]

Out[1945]= {{a, b, {c, d, {m, {1, 2, 3, 4, 5}, n, t}, u, z, {x, y}}}}

In[1946]:= SortOnLevel1[g, 4, ToCharacterCode[#1[[1]]][[1]] >=

ToCharacterCode[#2[[1]]][[1]] &]

Out[1946]= {{a, b, {d, z, {y, {1, 2, 3, 4, 5}, x, t}, u, c, {n, m}}}}

The procedure call LevelsOfList[x] returns the list of levels of elements of a list Flatten[x] of a source x list. In addition, in a case of the empty x list, {} is returned; in a case of a simple x list the single list of length Length[Flatten[x]], i.e. {1, 1,...,1, 1} will be returned, i.e. level of all elements of a simple list is equal 1. On lists of the kind {{...{{}}...}} the procedure call returns the empty list, i.e. {}. The following fragment represents source code of the procedure along with a typical example of its application.

In[10]:= LevelsOfList[x_ /; ListQ[x]] := Module[{L, L1, t, p, k, h,

g, j, s, w = ReplaceAll[x, {} –> {FromCharacterCode[2]}]},

{j, s} = ReduceLevelsList[w]; j = Prepend[j, 1];

Delete[If[j == {}, {}, If[! NestListQ1[j], Map[#^0 &, Range[1, Length[j]]],

If[FullNestListQ[j], Map[#^0 &, Range[1, Length[Flatten[j]]]], {p, h,

L, L1, g} = {1, FromCharacterCode[1], j, {}, {}}; ClearAll[t];

Do[For[k = 1, k <= Length[L], k++,

If[! ListQ[L[[k]]] && ! SuffPref[ToString[L[[k]]], h, 1],

AppendTo[g, 0]; AppendTo[L1, h <> ToString[p]],

If[! ListQ[L[[k]]] && ! SuffPref[ToString[L[[k]]], h, 1],

AppendTo[g, 0]; AppendTo[L1, L[[k]]], AppendTo[g, 1];

AppendTo[L1, L[[k]]]]]];

If[! MemberQ[g, 1], L1, L = Flatten[L1, 1]; L1 = {}; g = {}; p++],

{Levels[j, t]; t}];

ToExpression[Map[StringDrop[#, 1] &, L]]]]] + s – 1, 1]]

In[11]:= L = {a, m, n, {{b}, {c}, {{m, {{{gv}}}, n, {{{{{gs}}}}}}}}, d};

In[12]:= LevelsOfList[L]

Out[12]= {1, 1, 1, 3, 3, 4, 7, 4, 9, 1}

In[13]:= LevelsOfList[Flatten[L]]

Out[13]= {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

In[14]:= ElemsAtLevels[x_ /; ListQ[x]] :=

Module[{a = Flatten[x], b}, b = LevelsOfList[x];

Map[{a[[#]], b[[#]]} &, Range[1, Length[a]]]]

In[15]:= ElemsAtLevels[L]

Out[15]= {{a, 1}, {m, 1}, {n, 1}, {b, 3}, {c, 3}, {m, 4}, {gv, 7}, {n, 4},

{gs, 9}, {d, 1}}

Using the previous procedure, it is easy to program simple procedure whose call ElemsAtLevels[x] returns the list of ListList type, whose two–element sub–lists determine the elements of a Flatten[x] list and the nesting levels at which they are in x.

Calling the procedure InsertNestList[x, y], where x is a list and y is the list of format {{a1, n1, p1}, ..., {at, nt, pt}}, returns the result of inserting of aj expressions as (nj + 1)-th elements at the pt-th nesting levels. Expression lists can be used as nj (j=1..t). The procedure handles main erroneous situations.

In[78]:= InsertNestList[x_ /; ListQ[x], y_ /; ListQ[y]] :=

Module[{a, b, c, c1, c2, d, f, g, y1, d1, d2, t, k, j},

a[t_] := ToString1[t]; f[t_] := If[IntegerQ[t], Nothing, t];

SetAttributes[a, Listable]; SetAttributes[f, Listable];

a = Map[a, x]; b = StructNestList[a]; c = ElemsOnLevels2[a];

y1=If[MaxLevel1[y]==1, {y}, y];

y1 = Map[Flatten[{ToString1[#[[1]]], #[[2 ;3]]}] &, y1];

y1=Sort[y1, #1[[2]] < #2[[2]] &];

d = Map[#[[2 ;3]] &, c]; g = Map[#[[2 ;3]] &, y1];

d1 = Complement[g, Intersection[d, g]];

If[d1 == {}, 78, Print["Elements with nesting levels/sequential

numbers " <> ToString[d1] <> " are absent in the list"]];

If[d1 === y1, Return["All insertions are invalid"], x];

d1 = Complement[g, d1];

d2 = Select[y1, MemberQ[d1, #[[2 ;3]]] &];

d2 = MatchLists1[d1, d2, 2;3];

d1 = Select[c, MemberQ[d1, #[[2;3]]] &]; d = d1;

Map[AppendTo[d1[[#]], d2[[#]][[1]]] &, Range[1, Length[d1]]];

d1 = Map[Flatten[{#, Null}] &, d1];

c1 = Map[d[[#]] –> d1[[#]] &, Range[1, Length[d]]];

c2 = Map[If[MemberQ[d, #],

Nothing, # –> Quiet[AppendTo[Set[t, #], Null]]] &, c];

d2 = ReplaceAll[b, Join[c1, c2]]; d2 = ToExpression[Map[f, d2]];

d2 = ToString1[d2];

d1 = Map[#[[2]] + 1 &, StringPosition[d2, "Null"]]; g = {};

For[k = 1, k <= Length[d1], k++, a = StringTake[d2, {d1[[k]]}];

For[j = d1[[k]] – 1, j >= 1, j––, a = Set[b, StringTake[d2, {j}]] <> a;

If[b == "{" && SyntaxQ[a], AppendTo[g, j]; Break[],

Continue[]]]]; g = Reverse[Sort[Join[g, d1]]];

Do[d2 = StringDrop[d2, {g[[j]]}], {j, Length[g]}];

d2 = ToExpression[StringReplace[d2, "Null" –> "Nothing"]]]

In[79]:= g = {{a, b, c, {d, {m, a/b}, "n+p", d^2}}};

In[80]:= InsertNestList[g, {{m, 2, 1}, {{mn, pq, ab}, 3, 1},

{"c/d", 4, 2}, {hg, 4, 3}}]

Elements with nesting levels/sequential numbers {{4, 3}}

are absent in the list

Out[80]= {{a, m, b, c, {d, mn, pq, ab, {m, a/b, c/d}, "n+p", d^2}}}

Whereas the following procedure is an useful version of a previous procedure based on a different algorithm. Calling the procedure InsertNestList1[x, y], where x is a list and y is the list of form {{a1, n1, p1}, ..., {at, nt, pt}}, returns the result of inserting of aj expressions as (nj + 1)^th elements at the pt^th nesting levels that allows to expand the nesting levels of the x list. Expression lists can be used as nj (j=1..t). The procedure handles the main erroneous situations.

In[84]:= InsertNestList1[x_ /; ListQ[x], y_ /; ListQ[y]] :=

Module[{a, b, c, d, y1, f, f1, g, s, s1, v, k, t, j},

a[t_] := ToString1[t]; f[t_] := If[IntegerQ[t], Nothing, t];

f1[t_] := If[StringQ[t], ToExpression[t], t];

Map[SetAttributes[#, Listable] &, {a, f, f1}]; v = Map[a, x];

a = ElemsOnLevels2[v]; b = ToString1[StructNestList[v]];

y1 = If[MaxLevel1[y] == 1, {y}, y];

y1 = Sort[y1, #1[[2]] < #2[[2]] &]; d = Map[#[[2 ;3]] &, a];

g = Map[#[[2 ;3]] &, y1]; c = Complement[g, Intersection[d, g]];

If[c == {}, 78, Print["Elements with nesting levels/sequential

numbers " <> ToString[c] <> " are absent in the list"]];

If[c === y1, Return["All insertions are invalid"], x];

g = Complement[g, c];

d = Map[If[MemberQ[g, #[[2 ;3]]], #, Nothing] &, a];

s1 = Map[If[! MemberQ[d, #], ToString1[#] –>

StringTake[ToString1[#], {2, –2}], Nothing] &, a];

s = Map[If[MemberQ[g, #[[2 ;3]]],

Flatten[{ToString1[#[[1]]], #[[2 ;3]]}], Nothing] &, y1];

c = MatchLists1[d, s, 2 ;3]; s = c;

s = Map[Flatten[Join[d[[#]], s[[#]]], 1] &, Range[1, Length[d]]];

d = Map[ToString1, d]; s = Map[ToString1, s];

g = StringReplace[b, Join[s1, Map[d[[#]] –>

StringTake[s[[#]], {2, –2}] &, Range[1, Length[d]]]]];

Map[f1, Map[f, ToExpression[g]]]]

In[85]:= InsertNestList1[g, {{p, 2, 1}, {v, 2, 3}, {{p/q, n, j*n}, 3, 3},

{agn, 4, 1}, {hg, 4, 3}}]

Elements with nesting levels/sequential numbers {{4, 3}}

are absent in the list

Out[85]= {{a, p, b, c, v, {d, {m, agn, a/b}, "n+p", d^2, {p/q, n, j*n}}}}

The above procedures InsertNestList and InsertNestList1 essentially use a rather useful MatchLists procedure which is of independent interest that working with nested lists. Calling procedure MatchLists[x, y, S] returns the result of rearranging an y list elements of type ListList according to the order of the x list elements of type ListList that are defined by a S span of both lists. In turn, calling the procedure MatchLists[x, y, S, F] with the fourth optional argument F – a symbol that defines the user function that applies to tuples x[[k]][[S]] and y[[j]][[S]] of lists elements x and y. The procedure handles the main erroneous situations: in particular, the procedure tests for compliance of the function arity to the number of elements determined by the span S, whereas if the types of lists x and y are different from ListList, then the procedure call is returned unevaluated. The following fragment represents the source code of the procedure with its use. The MatchLists1 is a version of the above tool [16].

In[92]:= MatchLists[x_ /; ListListQ[x], y_ /; ListListQ[y], S_Span,

F___] := Module[{a, b, c = {}, d, k, j},

If[Set[a, Length[x]] != Length[y]||Length[x[[1]]] !=

Length[y[[1]]], Return["Lists do not match in format"],

If[{F} != {} && FunctionQ[F] &&

Arity[F] == Abs[Part[S, 2] – Part[S, 1]],

d[t_, k_, p___] := p @@ t[[k]][[S]],

d[t_, k_, p___] := t[[k]][[S]]]];

For[k = 1, k <= a, k++, For[j = 1, j <= a, j++,

If[d[x, k, F] == d[y, j, F], AppendTo[c, y[[j]]], 78]]]; c]

In[93]:= x = {{a, 3, 1}, {c, 1, 2}, {b, 1, 3}, {d, 3, 2}};

y = {{m, 1, 2}, {n, 3, 2}, {p, 3, 1}, {t, 1, 3}};

In[94]:= MatchLists[x, y, 2 ;3]

Out[94]= {{p, 3, 1}, {m, 1, 2}, {t, 1, 3}, {n, 3, 2}}

In[95]:= F[x_, y_] := If[x < y, y, x]; MatchLists[x, y, 2 ;3, F]

Out[95]= {{p, 3, 1}, {m, 1, 2}, {t, 1, 3}, {n, 3, 2}}

To illustrate a number of techniques that are rather useful in solving problems related to the nested lists, you can refer to the RestNestList procedure algorithm, which is as follows: the calling RestNestList[x, y] procedure returns the original nested list x, while through second argument y – an indefinite variable – the Flatten[x] list is returned. The procedure forms a distribution vector of brackets ("{", "}") in the x list in the form of a W list of the ListList type on the basis of which the initial list x is restored from the Flatten[x] list. The source code of the above procedure illustrates this quite clearly.

In[325]:= RestNestList[x_ /; NestListQ1[x], y_ /; SymbolQ[y]] :=

Module[{a = ToString1[x], b, W = {}, d}, b = StringLength[a];

Do[d = StringTake[a, {j}]; If[d == "{", AppendTo[W, {1, j}],

If[d == "}", AppendTo[W, {2, j}], AppendTo[W, Nothing]]], {j, b}];

d = ToString1[Flatten[x]]; y = ToExpression[d];

Do[d = StringInsert[d, If[W[[j]][[1]] == 1, "{", "}"], W[[j]][[2]] + 1],

{j, 1, Length[W]}]; ToExpression[d][[1]]]

In[326]:= L = {a, m, n, {{b}, {c}, {{m, {{{"g"}}}, n, {{{{{gs}}}}}}}}, d};

In[327]:= RestNestList[L, gsv]

Out[327]= {a, m, n, {{b}, {c}, {{m, {{{"g"}}}, n, {{{{{gs}}}}}}}}, d}

In[328]:= gsv

Out[328]= {a, m, n, b, c, m, "g", n, gs, d}

Using the ElemsOnLevels2 procedure [8,9,16], it is possible to program a procedure allowing to process by the set symbol the list elements which are determined by their arrangements on nesting levels and their location at the nested levels. The call SymbToElemList[x, {m, n}, s] returns the result of application of a symbol s to an element of a list x which is located on a nesting level m, and be n–th element at this nesting level. The nested list {{m1, n1}, …, {mp, np}} can be as the second argument of the SymbToElemList, allowing to execute processing by means of s symbol the elements of the x list whose locations are defined by pairs {mj, nj} (j = 1..p) of the above format. At that, the structure of the initial x list is preserved. The fragment below represents source code of the procedure and examples of its application.

In[1118]:= Lst = {s, "a", g, {{{{{}}}}}, b, 72, m, c, m, {f, d, 77, s, h, m,

{m + p, {{}}, n, 50, p, a}}};

In[1119]:= SymbToElemList[x_ /; ListQ[x],

y_ /; NonNegativeIntListQ[y] && Length[y] == 2 ||

ListListQ[y] && And1[Map[Length[#] == 2 &&

NonNegativeIntListQ[#] &, y]], G_Symbol] :=

Module[{a, b, c, d, h, g = If[ListListQ[y], y, {y}], f, n = 1, u},

SetAttributes[d, Listable];

f = ReplaceAll[x, {} –> Set[u, {FromCharacterCode[6]}]];

h = Flatten[f]; a = ElemsOnLevels2[f];

d[t_] := "g" <> ToString[n++]; c = Map[d[#] &, f];

h = GenRules[Map["g" <> ToString[#] &, Range[1, Length[h]]], h];

SetAttributes[b, Listable]; a = ElemsOnLevels2[c];

Do[b[t_] := If[MemberQ[a, {t, g[[j]][[1]], g[[j]][[2]]}], G[t], t];

Set[c, Map[b[#] &, c]], {j, 1, Length[g]}];

ReplaceAll[ReplaceAll[c, h], {G[u[[1]]] –> G[], u –> {}}]]

In[1120]:= SymbToElemList[Lst, {{4, 1}, {6, 1}, {3, 1}, {1, 1}, {5, 1},

{1, 6}, {2, 3}}, S]

Out[1120]= {S[s], "a", g, {{{{{S[]}}}}}, b, 72, S[m], c, m, {f, d, S[77], s,

h, m, {S[m + p], {{S[]}}, n, 50, p, a}}}

In[1121]:= SymbToElemList[{{}, {{{}}, {{{}}}}}, {{2, 1}, {4, 1}, {5, 1}}, G]

Out[1121]= {{G[]}, {{{G[]}}, {{{G[]}}}}}

Unlike the previous tool, the SymbToLevelsList procedure provides processing by a symbol of all elements of a list which are located at the set nesting levels [12-16]. Using the procedure ElemsOnLevels2, it is possible to program the ExpandLevelsList procedure allowing to expand a list by means of inserting into the set positions (nesting's levels and sequential numbers on them) of a certain list. Using the ElemOfLevels procedure the useful testing procedure can be programmed, namely. The procedure call DispElemInListQ[x,y,n,j] returns True if an y element – j-th on n nesting level of a nonempty x list, and False otherwise. In addition to the above tools of processing of the nested lists the following function is an useful enough tool. The function call ElemOnLevelListQ[x, y, n] returns True, if an element y belongs to the n–th nesting level of a list x, and False otherwise. When processing of sub-lists in the lists is often need of check of the existence fact of sub-lists overlapping that enter to the lists. The procedure call SubListOverlapQ[x, y, z] returns True, if a list x on a nesting level z contains overlapping sub-lists y, and False otherwise. Whereas the call SubStrOverlapQ[x, y, z, n] through optional n argument – an indefinite symbol – additionally returns the list of consecutive quantities of overlapping of the y sub-list on the z nesting level of the x list. In a case of the nesting level z non-existing for the x list, the procedure call returns $Failed with printing of the appropriate message [7,9,10-16].

In some programming problems that use the list structures arises a quite urgent need of replacement of values of a list that are located at the set nesting levels. The standard functions of the Mathematica don't give this opportunity. In this regard the procedure has been created whose call ReplaceLevelList[x,n,y,z] returns the result of the replacement of y element of a list x that is located at a nesting level n onto a value z. The lists that have identical length also can act as arguments y, n and z. In a case of violation of the set condition the procedure call returns $Failed. In a case of lack of the 4^th optional z argument the procedure call ReplaceLevelList[x, n, y] returns the result of removal of y elements which are located on a nesting level n of the x list. The following fragment represents source code of the procedure and some typical examples of its application.

In[7]:= ReplaceLevelList[x_ /; ListQ[x], n_ /; IntegerQ[n] &&

n > 0 || IntegerListQ[n], y_, z___] :=

Module[{a, c = Flatten[x], d, b = SetAttributes[ToString4, Listable],

h = FromCharacterCode[2], k, p = {}, g = FromCharacterCode[3],

u = FromCharacterCode[4], m, n1 = Flatten[{n}],

y1 = Flatten[{y}], z1},

If[{z} != {}, z1 = Flatten[{z}], z1 = y1];

If[Length[DeleteDuplicates[Map[Length, {n1, y1, z1}]]] != 1,

$Failed, a = ToString4[x]; SetAttributes[StringJoin, Listable];

b = ToExpression[ToString4[StringJoin[u, a, g <> h]]];

b = ToString[b]; m = StringPosition[b, h]; d = LevelsOfList[x];

b = StringReplacePart[ToString[b], Map[ToString, d], m];

For[k = 1, k <= Length[n1], k++, AppendTo[p, u <>

Map[ToString4, y1][[k]] <> g <> Map[ToString, n1][[k]] –>

If[{z} == {}, "", Map[ToString, z1][[k]]]]];

b = StringReplace[b, p]; d = Map[g <> # &, Map[ToString, d]];

Map[ClearAttributes[#, Listable] &, {StringJoin, ToString4}];

ToExpression[StringReplace[b, Flatten[{u –> "", ", }" –> "}",

"{," –> "{", ", ," –> ",", " , ," –> "", " , " –> "", GenRules[d, ""]}]]]]]

In[8]:= p := {a, j, n, {{b + d}, {c}, {{m, {{{g}}}, n, {{{{{gsv, vgs}}}}}}}}, d};

In[9]:= ReplaceLevelList[p, {3, 9}, {b + d, gsv}, {Art, Kr}]

Out[9]= {a, j, n, {{Art}, {c}, {{m, {{{g}}}, n, {{{{{Kr, vgs}}}}}}}}, d}

In[10]:= ReplaceLevelList[p, 9, gsv, Art]

Out[10]= {a, j, n, {{b + d}, {c}, {{m, {{{g}}}, n, {{{{{Art, vgs}}}}}}}}, d}

The call MapAt[f,e,n] of the built-in MapAt function applies f to an element at position n in an e expression; whereas other its formats allow to apply f to the elements at the set positions in the e expression. At the same time, the following procedure allows to apply f to elements located on the set nesting levels with the set sequential numbers on these nesting levels in a list. The procedure call MapAtLevelsList[f,x,y] returns the result of applying of f to the elements of a list x that are located on the set nesting levels L1,…,Lk with the set sequential numbers p1, …, pk on these nesting levels; at that, the second argument is a y ListList–list of the format {{L1, p1}, …, {Lk, pk}}, as examples below a rather visually illustrate. Whereas the procedure call MapAtLevelsList[f, x, y, z] with third optional z argument – an undefined symbol – through it additionally returns the ListList list whose elements are three–elements sub-lists for which the first elements define elements of the list Flatten[x], the 2^nd elements define their positions in the Flatten[x] list, and the 3^rd elements define their nesting levels in the x list. On a list x of the format {{{...{{...}}...}}} the procedure call returns {{{...{{f[]}}...}}}. Whereas the call MapAtLevelsList[f, x, y, z] additionally thru the optional z argument returns the list of format {{{}, 1, MaxLevel[x]}}. This procedure is of interest in the problems of the lists processing. The fragment below represents source code of the procedure along with typical examples of its application.

In[27]:= MapAtLevelsList[f_ /; SymbolQ[f], x_ /; ListQ[x],

y_ /; ListListQ[y], z___] :=

Module[{a, b, c = FromCharacterCode[2], d, g, p, n = 1, s},

s = ReplaceAll[x, {} –> {c}]; a = LevelsOfList[s]; d = ClusterList[a];

b = Map[c <> # &, Map[ToString, y]];

f[t_] := {"(" <> If[t === Null, "f[]", ToString1[t]] <> ")" <> c <>

ToString[d[[n++]]]}[[1]]; d = MapListAll[f, s]; Clear[f];

g[t_] := If[MemberQ[b, StringTake[t,

{Set[p, Flatten[StringPosition[t, c]][[–1]]], –1}]],

ToString[f] <> "[" <> StringTake[t, {1, p – 1}] <> "]",

StringTake[t, {1, p – 1}]]; SetAttributes[g, Listable];

If[{z} != {} && NullQ[z], z = ReplaceAll[Map[{Flatten[s][[#]], #,

a[[#]]} &, Range[1, Length[a]]], "\.02" –> {}], 77];

n = ToExpression[Map[g, d]]; n = ReplaceAll[n, "\.02" –> Null];

ReplaceAll[n, {f[Null] –> f[], {Null} –> {}}]]

In[28]:= MapAtLevelsList[f, {a, b, c, {{x, y, z}, {{m, n*t, p}}}, c, d},

{{1, 1}, {1, 5}, {3, 3}, {4, 2}}]

Out[28]= {f[a], b, c, {{x, y, f[z]}, {{m, f[n t], p}}}, c, f[d]}

In[29]:= MapAtLevelsList[f, {{{{{a, b, c}, {x, y, z}}}}}, {{5, 1}, {5, 5},

{5, 3}}, g72]

Out[29]= {{{{{f[a], b, f[c]}, {x, f[y], z}}}}}

In[30]:= g72

Out[30]= {{a, 1, 5}, {b, 2, 5}, {c, 3, 5}, {x, 4, 5}, {y, 5, 5}, {z, 6, 5}}

Unlike the MapAtLevelsList the MapAtLevelsList1 allows to apply a set of functions to the specified elements of a list that are depended on a nesting level at which they are located. The call MapAtLevelsList1[x, y, z] returns the result of applying of a function from a list z to the elements of a list x that are located on a set nesting levels L1,…,Lk with the set sequential numbers p1,…,pk on these nesting levels; in addition, the 2^nd argument is a list y of the form {{L1, p1},…,{Lk, pk}}. While the 3^rd argument is a list z of the form {{F1, L1},...,{Fp, Lp}} where a sub-list {Fj, Lj} defines applying of a Fj function on a nesting level Lj. In a case if in the list z there are different functions at the same level, then only the first of them is used. The given procedure is of a certain interest in problems of the lists processing [6,8,10-16]. At last, the SymbolToLevelsList procedure gives possibility to apply a function to all elements of the set nesting levels of a list x. The call SymbolToLevelsList[x, m, Sf] returns the list structurally similar to a list x and to its elements at the nesting levels m (list or separate integer) function Sf is applied. In a case of absence of real nesting levels in m (i.e. non-existing levels in the x list) the call returns $Failed with printing of the appropriate message [8,16].

The standard ReplaceList function attempts to transform an expression by applying a rule or list of rules in all possible ways, and returns the list of the results obtained. The following procedure is an useful enough modification of the ReplaceList function. The call ReplaceList1[x, y, z, p] returns the list that is the result of replacement in a list x of elements whose positions are determined by a list y at a nesting level p onto appropriate expressions that are defined by a list z. In addition, if a certain element should be deleted, in the z list the key word "Nothing" is coded in the appropriate position. The procedure processes the main especial situations. The following fragment represents source code of the procedure and examples of its application.

In[7]:= ReplaceList1[x_ /; ListQ[x], y_ /; IntegerListQ[y],

z_ /; ListQ[z], p_ /; IntegerQ[p]] :=

Module[{a = MaxLevelList[x], b, c = {}},

If[Length[y] != Length[z], Print["The 2nd and 3rd arguments

have different lengths"]; $Failed,

If[p > a, Print["The given nesting level is more than maximal"];

$Failed, b = Level[x, {p}];

Do[AppendTo[c, If[y[[j]] <= Length[b], b[[y[[j]]]] –> If[z[[j]] ===

"Nothing", Nothing, z[[j]]], Nothing]], {j, 1, Length[y]}];

If[c == {}, x, Replace[x, c, {p}]]]]]

In[8]:= L := {a,b, {{a,b}, {{m,n, {p, {{{}}}}}}}, c, {}, m,n,p, {{{{a*b}, g}}}}

In[9]:= ReplaceList1[L, {1, 2, 5}, {Avz, Agn, Vsv}, 4]

Out[9]= {a,b,{{a,b},{{Avz,Agn,{p,{{{}}}}}}},c,{},m,n,p,{{{{a*b},Vsv}}}}

In[10]:= ReplaceList1[L, {1, 2}, {{Avz, Agn}, {77, 72}}, 2]

Out[10]= {a, b, {{Avz, Agn}, {77, 72}}, c, {}, m, n, p, {{{{a*b}, g}}}}

The InsertToList procedure is based on the above procedure ReplaceList1 and Sequences [8,16] and attempts to transform a list by including of expressions in the tuples of its elements that are located on the set nesting level. The InsertToList procedure is a rather useful modification of the built-in Insert function. The procedure call InsertToList[x, y, z, h, p] returns the list that is the result of inserting into x list on a nesting level p of expressions from a list z; the positions for inserting are defined by means of the appropriate y list which defines the positions of elements of the x list on the p–th nesting level [7,10-16].

The following procedure uses a bit different algorithm that is different from the above-mentioned InsertToList procedure. The InsertToList1 procedure algorithm is based on procedures ElemsOnLevels2 and StructNestList described in this book and in [8,16], and attempts to transform a list by including of the set of expressions before (after) of its element that is located on a set nesting level with the given position on it. The procedure call InsertToList1[x, {l, p}, {a, b, c,…}, t] returns the list – the result of inserting in a list x at a nesting level l after (t is absent) or before (t is an arbitrary expression) of its position p of expressions from the list z. The procedure handles the erroneous situations that are conditioned by invalid the 2^nd argument with output of the appropriate messages, returning the initial x list that is based on standard evaluations. The following fragment represents the source code of the procedure along with typical examples of its application.

In[78]:= InsertToList1[x_ /; ListQ[x], y_ /; ListQ[y],

z_ /; ListQ[z], t___] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x], c, d},

c = Select[a, #[[2 ;3]] == y &];

If[c != {}, If[{t} == {}, d = ToString1[c[[1]]] <> "," <>

StringTake[ToString1[Map[{ToString1[#], 1, 1} &, z]], {2, -2}],

d = StringTake[ToString1[Map[{ToString1[#], 1, 1} &, z]],

{2, –2}] <> "," <> ToString1[c[[1]]]];

b = ToString[ReplaceAll[b, c[[1]] –> d]];

ToExpression[ToInitNestList[ToExpression[b]]],

Print["Second argument " <> ToString[y] <> " is invalid"]; x]]

In[79]:= p = {{b, a, b, {c, d, {j, {j, {t, j}, n}, u, n}, j, h}, z, y, x}};

In[80]:= InsertToList1[p, {3, 3}, {c, c, c}]

Out[80]= {{b, a, b, {c, d, {j, {j, {t, j}, n}, u, n}, j, c, c, c, h}, z, y, x}}

In[81]:= InsertToList1[p, {3, 3}, {c, c, c}, gsv]

Out[81]= {{b, a, b, {c, d, {j, {j, {t, j}, n}, u, n}, c, c, c, j, h}, z, y, x}}

In[82]:= InsertToList1[p, {7, 3}, {a, b, c}, gsv]

Second argument {7, 3} is invalid

Out[82]= {{b, a, b, {c, d, {j, {j, {t, j}, n}, u, n}, j, h}, z, y, x}}

Using the ElemsOnLevels2 procedure [16], it is possible to program the procedure allowing to expand a list by means of inserting into the given positions (nesting levels and sequential numbers on them) of the list. The call ExpandLevelsList[x, y, z] returns the result of inserting of the z list to a nesting level m relative to n–th position at this nesting level; at that, the second y argument in common case is a nested list {{m, n}, p},...,{mt, nt}, pt}}, allowing to execute inserting to positions, whose locations are defined by pairs {mj, nj} {j=1..t} (nesting levels and sequential numbers on them), whereas pt{1, –1, 0} defines insertion at the right, at the left and at situ relative to the set position. Thus, the argument y has the format {{m, n}, p}, or the kind of the above nested list {{m, n}, p},...,{mt, nt}, pt}}. At that, the structure of an initial x list is preserved.

In[2385]:= ExpandLevelsList[x_ /; ListQ[x], y_, z_ /; ListQ[z]] :=

Module[{a, b, c, d, h, g = If[Length[Flatten[y]] == 3, {y}, y], f,

n = 1, u, p = ToString[z]}, SetAttributes[d, Listable];

f = ReplaceAll[x, {} –> Set[u, {FromCharacterCode[2]}]];

h = Flatten[f]; a = ElemsOnLevels2[f];

d[t_] := "g" <> ToString[n++]; c = Map[d[#] &, f];

h = GenRules[Map["g" <> ToString[#] &, Range[1, Length[h]]], h];

SetAttributes[b, Listable]; a = ElemsOnLevels2[c];

Do[b[t_] := If[MemberQ[a, {t, g[[j]][[1]][[1]], g[[j]][[1]][[2]]}],

If[g[[j]][[2]] == –1, p <> "," <> t, If[g[[j]][[2]] == 1, t <> "," <> p, p, 7]], t];

Set[c, Map[b[#] &, c]], {j, 1, Length[g]}];

c = ToExpression[ToString[c]];

ReplaceAll[ReplaceAll[c, Map[ToExpression[#[[1]]] –>

#[[2]] &, h]], u[[1]] –> {}]]

In[2386]:= ExpandLevelsList[{a, d, {b, h, {c, d}}}, {{{1, 1}, 1},

{{2, 1}, 0}, {{3, 1}, –1}}, {{x, y, z}}]

Out[2386]= {a, {{x, y, z}}, d, {{{x, y, z}}, h, {{{x, y, z}}, c, d}}}

In[2387]:= ExpandLevelsList[{a, d, {b, h, {c, d}}}, {{{1, 1}, 1},

{{2, 1}, 0}, {{3, 1}, –1}}, {{}}]

Out[2387]= {a, {{}}, d, {{{}}, h, {{{}}, c, d}}}

In[2388]:= ExpandLevelsList[{}, {{1, 1}, 0}, MultiEmptyList[7]]

Out[2388]= {{{{{{{{}}}}}}}}

The procedure is widely used by tools processing list structures.

Unlike the SelectNestElems procedure the following simple procedure, based on the ElemsOnLevels5 procedure, calling the procedure ElemsLocation[x, y] returns a list of the form {{h1, m1, n1},...,{hp, mp, np}}, whose elements {hj, mj, nj} (j=1..p) define the elements hj (which are determined by the argument y), the nesting levels and the sequential numbers on them accordingly which determine the locations of the elements hj in the list x.

In[2222]:= ElemsLocation[x_ /; ListQ[x], y_] := Module[{a = {}, c,

b = Flatten[{y}]}, If[Set[c, Intersection[Flatten[x], b]] == b, 78,

Print["Elements " <> ToString1[Complement[b, c]] <>

" are absent in the first argument"]];

Do[AppendTo[a, Map[If[SameQ[#[[1]], b[[j]]], #, Nothing] &,

ElemsOnLevels5[x]]], {j, Length[b]}]; Flatten[a, 1]]

In[2223]:= ElemsLocation[w, {m, x^y}]

Out[2223]= {{m, 4, 1}, {m, 7, 1}, {x^y, 4, 3}}

In[2224]:= ElemsLocation[{a, b, c, d, f, d, g, g, f}, {d, f, m, n, s, v}]

Elements {m, n, s, v} are absent in the first argument

Out[2224]= {{d, 1, 4}, {d, 1, 6}, {f, 1, 5}, {f, 1, 8}}

Unlike previous processing procedures of nested lists, the following procedure generates a nested list of so-called canonical form considered above based on list of elements with nesting levels ascribed to them. The procedure call CanonList[x] where x – a nested list of form {{a1, p1},…,{an, pn}} (aj – elements and pj – nesting levels on which they located) return a nested canonical list with the aj elements on pj nesting levels (j=1..n). Elements order on levels in the returned list are defined of their order at the x.

In[78]:= CanonList[x_ /; ListListQ[x]] :=

Module[{a = Max[Map[#[[2]] &, x]], b = "{", c = "}",

b1 = FromCharacterCode[7], c1 = FromCharacterCode[8], d, g, t},

d = StringJoin[Map[StringRepeat[#, a] &, {b1, c1}]];

t = Map[{ToString1[#[[1]]], #[[2]]} &, x];

t = Gather[t, #1[[2]] == #2[[2]] &]; t = Flatten[Map[Reverse, t], 1];

Map[{g = Map[#[[1]] &, StringPosition[d, b1]],

d = StringInsert[d, #[[1]] <> ",", g[[#[[2]]]] + 1]} &, t];

ToExpression[StringReplace[d, {b1 -> b, c1 -> c, "," <> c1 -> c}]]]

In[79]:= CanonList[{{a, 2}, {b, 4}, {c^2, 6}, {m/n, 2}, {t, 4}, {j*2, 6}}]

Out[79]= {{a, m/n, {{b, t, {{c^2, 2*j}}}}}}

Like the previous procedure, the following procedure also generates a nested list of so–called canonical form considered above based on list of elements with nesting levels ascribed to them and on a slightly different algorithm. The procedure call ListToCanon[x] returns a nested canonical list. Elements order on levels in the returned list are defined of their order at x. At the same time, the procedure call MaxLevelList2[x] used by the procedure (being one of the options for such tools) returns maximal nesting level of a list x. The fragment below represents source codes the above procedures with examples of their application. The codes use interesting enough methods of programming.

In[90]:= MaxLevelList2[x_ /; ListQ[x]] :=

Module[{a = x, b, n = 1},

Do[If[x == Set[b, Flatten[x]]||{n++;

SameQ[a = Flatten[a, 1], b]}[[1]], Return[n], 78], Infinity]]

In[91]:= ListToCanon[x_ /; ListQ[x]] :=

Module[{a = MaxLevelList2[x], b, c, f, g},

If[a == 1, x, b = StringRepeat["{", a];

b = b <> StringRepeat["}", a];

c = Gather[ElemsOnLevels2[x], #1[[2]] == #2[[2]] &];

f[t_] := Module[{p = ""},

Do[p = p <> ToString1[t[[j]][[1]]] <> ",", {j, 1, Length[t]}];

{t[[1]][[2]], If[Length[t] == 1, StringTake[p, {1, –2}], p]}];

c = Reverse[Map[f, c]];

Do[b = StringInsert[b,

If[StringTake[Set[g, c[[j]][[2]]], {1, –2}] == "{}", "", g],

{c[[j]][[1]] + 1}], {j, 1, Length[c]}]; ToExpression[b]]]

In[92]:= gs = {{a, b, {}, c, {d, {j, {a, {t}, b}, b}, a/b}, "n + p",

{a, g, n}, {{v, s, v}}, d^2}}; MaxLevelList2[gs]

Out[92]= 6

In[93]:= ListToCanon[gs]

Out[93]= {{a, b, c, "n + p", d^2, {{}, d, a/b, a, g, n,

{j, b, v, s, v, {a, b, {t}}}}}}

In[94]:= ListToCanon[Flatten[gs]]

Out[94]= {a, b, c, d, j, a, t, b, b, a/b, "n + p", a, g, n, v, s, v, d^2}

The following rather simple procedure is intended for the grouping of elements of a list according to their multiplicities. The procedure call GroupIdentMult[x] returns the nested list of the following format:

{{{n₁}, {x₁, x₂,…,x_a}}, {{n₂}, {y₁, y₂,…,y_b}}, …, {{n_k}, {z₁, z₂,…,z_c}}}

where {x_i, y_j, …, z_p} are elements of a list x and {n₁, n₂, …, n_k} – multiplicities corresponding to them {i = 1..a, j = 1..b,…,p = 1..c}. The fragment below represents the source code of the procedure along with some typical examples of its application.

In[1287]:= GroupIdentMult[x_ /; ListQ[x]] :=

Module[{a = Gather[x], b},

b = Map[{DeleteDuplicates[#][[1]], Length[#]} &, a];

b = Map[DeleteDuplicates[#] &,

Map[Flatten, Gather[b, SameQ[#1[[2]], #2[[2]]] &]]];

b = Map[{{#[[1]]}, Sort[#[[2 ;–1]]]} &, Map[Reverse,

Map[If[Length[#] > 2, Delete[Append[#, #[[2]]], 2], #] &, b]]];

b = Sort[b, #1[[1]][[1]] > #2[[1]][[1]] &];

If[Length[b] == 1, Flatten[b, 1], b]]

In[1288]:= L = {a, c, b, a, a, c, g, d, a, d, c, a, c, c, h, h, h, h, h, g, g};

In[1289]:= GroupIdentMult[L]

Out[1289]= {{{5}, {a, c, h}}, {{3}, {g}}, {{2}, {d}}, {{1}, {b}}}

In[1290]:= GroupIdentMult[{a, a, a, a, a, a, a, a, a, a, a, a, a, a, a}]

Out[1290]= {{15}, {a}}

In[1291]:= GroupIdentMult[RandomInteger[2, 47]]

Out[1291]= {{{19}, {2}}, {{16}, {0}}, {{12}, {1}}}

In[1292]:= GroupIdentMult[RandomInteger[42, 77]]

Out[1292]= {{{4}, {9, 20, 24, 32, 34}}, {{3}, {0, 2, 5, 12, 17, 18}},

{{2}, {3, 4, 8, 15, 16, 19, 21, 22, 25, 31, 35, 40, 41}},

{{1}, {6, 7, 10, 11, 14, 26, 28, 29, 30, 37, 38, 39, 42}}}

In[1293]:= Sum[%[[k]][[1]]*Length[%[[k]][[2]]], {k, 1, Length[%]}][[1]]

Out[1293]= 77

In addition, elements of the returned nested list are sorted in decreasing order of multiplicities of elements groups of the initial x list. In a number of problems dealing with structures of list type the procedure is of a certain interest. That procedure is used in a number of tools of the MathToolBox package [16].

Note, along with above means we have programmed a lot of other means for processing lists, which are intended both for mass application and in special cases. Some are original, while others to varying degrees extend the functionality of the built-in means or extend their applicability. All these tools are included in the attached package MathToolBox [16].

3.6. Additional tools for the Mathematica

Here we will present some of the additional tools created at using of Mathematica for the programming of applications and in preparation of a series of books on computer algebra systems.

The following procedure expands the standard MemberQ function onto the nested lists, its call MemberQL[x, y] returns True if an expression y belongs to any nesting level of list x, and False otherwise. While the call with the 3^rd optional z argument – an indefinite variable – in addition through z returns the list of the ListList type, the first element of each its sublist determines a level of the x list, whereas the second determines quantity of y elements on this level provided that the main output is True, otherwise z remains indefinite. Below is represented the source code and examples of its use (see also function MemberQL2 [16]).

In[2144]:= MemberQL1[x_ /; ListQ[x], y_, z___] :=

Module[{a = MaxLevel[x], b = {}, c = {}, k = 0},

Do[AppendTo[b, Count[Level[x, {j}], y]], {j, a}];

If[Plus[Sequences[b]] >= 1, If[SymbolQ[z], Map[{k++,

If[# != 0, AppendTo[c, {k, #}], 72]} &, b], 77]; z = c; True, False]]

In[2145]:= MemberQL1[{a, b, {c, d, {f, d, h, d}, s, {p, w, {n, m, d,

d, r, u, d, {x, d, {d, m, n, d}, d, y}}, t}}, x, y, z}, d, agn]

Out[2145]= True

In[2146]:= agn

Out[2146]= {{2, 1}, {3, 2}, {4, 3}, {5, 2}, {6, 2}}

In[2147]:= MemberQL1[Flatten[{a, b, {c, d, {f, d, h, d}, s, {p, w,

{n, m, d, d, r, u, d}, t}}, x, y, z}], d, avz]

Out[2147]= True

In[2148]:= avz

Out[2148]= {{1, 6}}

At programming of certain procedures of access to files has been detected the expediency of creation of a procedure useful also in other appendices. Therefore, the procedure PosSubList has been created, whose call PosSubList[x, y] returns the nested list of initial and final elements for entrance into a simple x list of the tuple of elements set by a list y. The fragment represents the source code of the procedure and an example of its use.

In[7]:= PosSubList[x_ /; ListQ[x], y_ /; ListQ[y]] :=

Module[{d, a = ToString1[x], b = ToString1[y],

c = FromCharacterCode[16]}, d = StringTake[b, {2, –2}];

If[! StringFreeQ[a, d], b = StringReplace[a, d –> c <> "," <>

StringTake[ToString1[y[[2 ;–1]]], {2, –2}]];

Map[{#, # + Length[y] – 1} &,

Flatten[Position[ToExpression[b], ToExpression[c]]]], {}]]

In[8]:= PosSubList[{a, a, b, a, a, a, b, a, x, a, b, a, y, z, a, b, a}, {a, b, a}]

Out[8]= {{2, 4}, {6, 8}, {10, 12}, {15, 17}}

The above approach once again illustrates incentive motives and prerequisites for programming of the user tools expanding the Mathematica software. Many of tools of our MathToolBox package appeared exactly in this way. So, the both procedures Gather1 and Gather2 a little extend the built-in function Gather, being an useful in a number of applications [8,12-16].

The following group of means serves for sorting of lists of various type. Among them can be noted such as SortNL, SortLS, SortNL1, SortLpos, SortNestList. For example, the procedure call SortNestList[x,p,y] returns the sorting result of a nested numeric or symbolic list x by p–th element of its sub-lists according to the sorting functions Less, Greater for numeric lists, and SymbolLess, SymbolGreater for symbolic lists. In both cases a nested list with the nesting level <3 as actual x argument is supposed, otherwise an initial x list is returned. In addition, in a case of symbolic lists the comparison of elements is carried out on the basis of their character codes. The following fragment represents source code of the procedure with a typical example of its application.

In[76]:= SortNestList[x_ /; NestListQ[x], p_ /; PosIntQ[p], y_] :=

Module[{a = DeleteDuplicates[Map[Length, x]], b},

b = If[AllTrue[Map[ListNumericQ, x], TrueQ] &&

MemberQ[{Greater, Less}, y], y,

If[AllTrue[Map[ListSymbolQ, x], TrueQ] &&

MemberQ[{SymbolGreater, SymbolLess}, y], y],

Return[Defer[SortNestList[x, p, y]]]];

If[Min[a] <= p <= Max[a], Sort[x, b[#1[[p]], #2[[p]]] &],

Defer[SortNestList[x, p, y]]]]

In[77]:= SortNestList[{{42, 47, 72}, {77, 72, 52}, {30, 23}}, 2, Less]

Out[77]= {{30, 23}, {42, 47, 72}, {77, 72, 52}}

Sorting of a nested list at the nesting levels composing it is of quite certain interest. In this regard the following procedure can be a rather useful tool. The call SortOnLevel[x, m] returns the sorting result of a list x at its nesting level m [16]. Whereas the SortOnLevels procedure is a rather natural generalization of the previous procedure. The call SortOnLevels[x,m,sf] where optional sf argument – a sorting pure function analogously to the previous procedure returns the sorting result of x list at its nesting levels m. Furthermore, as an argument m can be used a single level, the levels list or "All" word which determines all nesting levels of the x list which are subject to sorting. If all m levels are absent then the call returns $Failed with printing of the appropriate message, otherwise the call prints the message concerning only levels absent in the x list. It must be kept in mind, that by default the sorting is made in symbolic mode, i.e. all elements of the sorted x list before sorting by means of the built-in Sort function are presented in the string format. While the call SortOnLevels[x, m, sf] returns the sorting result of a list x at its nesting levels m depending on a sorting pure sf function that is the optional argument. The following fragment represents source code of the procedure and examples of its application.

In[75]:= SortOnLevels[x_ /; ListQ[x], m_ /; IntegerQ[m] ||

IntegerListQ[m] || m === All, sf___] :=

Module[{a, b, c = ReplaceAll[x, {} –> {Null}], d, h, t, g, s},

a = LevelsOfList[c]; d = Sort[DeleteDuplicates[a]];

If[m === All || SameQ[d, Set[h,

Sort[DeleteDuplicates[Flatten[{m}]]]]], t = d,

If[Set[g, Intersection[h, d]] == {},

Print["Levels " <> ToString[h] <> " is empty"];

Return[$Failed], t = g; s = Select[h, ! MemberQ[d, #] &];

If[s == {}, Null, Print["Levels " <> ToString[s] <>

" are absent in the list; " <> "nesting levels must belong to " <>

ToString[Range[1, Max[d]]]]]]];

Map[Set[c, SortOnLevel[c, #, sf]] &, t];

ReplaceAll[c, Null –> Nothing]]

In[76]:= SortOnLevels[{b, t, "c", {{3, 2, 1, {g, s*t, a}, k, 0}, 2, 3, 1},

1, 2, {{m, n, f}}, {{2, 3, 6, {{{g, f, d, s, m}}}}}}, 7, 8]

Levels {7} are absent in the list;

nesting levels must belong to {1, 2, 3, 4, 5, 6}

Out[76]= $Failed

In[77]:= SortOnLevels[{b, t, "c", {{3, 2, 1, {g, s*t, a}, k, 0}, 2, 3, 1},

1, 2, {{m, n, f}}, {{2, 3, 6, {{{g, f, d, s, m}}}}}}, 6]

Out[77]= {b, t, "c", {{3, 2, 1, {g, s*t, a}, k, 0}, 2, 3, 1}, 1, 2, {{m, n, f}},

{{2, 3, 6, {{{d, f, g, m, s}}}}}}

The procedure call SortOnLevel[x, m] returns the sorting result of a list x at its nesting level m. If the m level is absent, the procedure call returns $Failed, printing the appropriate message. It must be kept in mind that by default the sorting is made in the symbolic format, i.e. all elements of the sorted x list will be represented in the string format. Whereas the procedure call SortOnLevel[x, m, Sf] returns the sorting result of the x list at its nesting level m depending on the sorting pure Sf function that is the optional argument. In addition, the cross-cutting sorting at the set nesting level is made. The fragment below represents source code of the procedure and an example of its application.

In[10]:= SortOnLevel[x_ /; ListQ[x], m_ /; IntegerQ[m], Sf___] :=

Module[{a, b = ReplaceAll[x, {} –> {Null}], c, d, f, g, h = {}, n = 1,

p, v}, a = LevelsOfList[b]; d = DeleteDuplicates[a];

If[! MemberQ[d, m],

Print["Level " <> ToString[m] <> " is absent in the list; " <>

"nesting level must belong to " <> ToString[Range[1, Max[d]]]];

$Failed, f[t_] := "(" <> ToString1[t] <> ")" <> "_" <>

ToString1[a[[n++]]];

SetAttributes[f, Listable]; c = Map[f, b];

g[t_] := If[SuffPref[t, "_" <> ToString[m], 2], AppendTo[h, t], Null];

SetAttributes[g, Listable]; Map[g, c];

p = Sort[Map[StringTake[#, {1, Flatten[StringPosition[#, "_"]]

[[–1]] – 1}] &, h], Sf]; n = 1;

v[t_] := If[ToExpression[StringTake[t, {Flatten[StringPosition[t,

"_"]][[–1]] + 1, –1}]] == m, p[[n++]],

StringTake[t, {1, Flatten[StringPosition[t, "_"]][[–1]] – 1}]];

SetAttributes[v, Listable]; c = ToExpression[Map[v, c]];

ReplaceAll[c, Null –> Nothing]]]

In[11]:= SortOnLevel[{p, b, a, u, c, {{{n, {h, f, c}, m}}}, 3, 2, 1}, 1]

Out[11]= {1, 2, 3, a, b, {{{n, {h, f, c}, m}}}, c, p, u}

Along with the above and other similar means [8-16,22], a procedure whose call ElemsOnLevel[x] returns the format of a list x with its internal structure preserved is of great interest but instead of its elements there are 3–element lists, where as their 1^st element – element of the x list, the 2^nd element – its position in the Flatten[x] list and the third element – its nesting level in the x list (see fragment below).

In[7]:= ElemsOnLevel[x_ /; ListQ[x]] := Module[{d, f, f1, f2, n = 1,

p, z = "\[InvisibleComma]"},

f[t_] := ToString1[t] <> z <> ToString[n++];

f1[t_] := ToString[t] <> z <> ToString[p[[n++]]];

f2[t_] := ToExpression[StringSplit[t, z]];

Map[SetAttributes[#, Listable] &, {f, f1, f2}];

p = LevelsOfList[x]; d = Map[f,x]; n=1; d = Map[f1,d]; Map[f2,d]]

In[8]:= ElemsOnLevel[{{{{a, b}}}, {c, d, m, {{n, d}}}, s, g, {a, k}}]

Out[8]= {{{{{a, 1, 4}, {b, 2, 4}}}}, {{c, 3, 2}, {d, 4, 2}, {m, 5, 2},

{{{n, 6, 4}, {d, 7, 4}}}}, {s, 8, 1}, {g, 9, 1}, {{a, 10, 2}, {k, 11, 2}}}

The procedure is quite useful in solving of the number the problems of processing nested lists and in combination with the above–mentioned tools it extends the functional component of Mathematica in this direction. In particular, the procedure is useful in solving the task of testing the continuous distribution of elements at a set nesting level of a list. The problem is solved by means of SolidLevelQ procedure, represented below.

In[432]:= SolidLevelQ[x_ /; ListQ[x], n_ /; IntegerQ[n], y___] :=

Module[{a = ElemsOnLevel[x], b, c, p, t = 0},

If[MemberQ[Range[1, c = MaxLevel[x]], n],

b = Partition[Flatten[a], 3]; b = Select[b, #[[3]] == n &];

If[b == {}, Goto[j], b = Map[#[[2]] &, b]; a = Range[b[[1]], b[[-1]]];

Do[If[b[[p]] + 1 == b[[p + 1]], Nothing, t++], {p, Length[b] – 1}];

If[a == b, p = True, p = False];

If[p === False && {y} != {} && ! HowAct[y], y = t + 1, 77]; p],

Label[j]; Print["The second argument should be in range " <>

ToString[{1, c}]]; $Failed]]

In[433]:= SolidLevelQ[{3, 2, 1, {{g, s + t, a}, k, 0, 2, 3, 1}, 1, 2, {a, b},

m, n, {c, {d}, t}, g, {d}, f}, 1]

Out[433]= False

In[434]:= SolidLevelQ[{3, 2, 1, {{g, s + t, a}, k, 0, 2, 3, 1}, 1, 2, {a, b},

m, n, {c, {d}, t}, g, {d}, f}, 1, g]

Out[434]= False

In[435]:= g

Out[435]= 5

In[436]:= SolidLevelQ[{5, 6, 7, 8, {9, 11}, 10, 1}, 3]

"The second argument should be in diapason {1, 2}"

Out[436]= $Failed

In[437]:= SolidLevelQ[{5, 6, 7, 8, {9, {c}, 11}, 10, 1}, 2, svg]

Out[437]= False

In[438]:= svg

Out[438]= 2

In[439]:= SolidLevelQ[{5, 6, 7, 8, 9, 11, 10, 1}, 2]

"The second argument should be in diapason {1, 1}"

Out[439]= $Failed

In[440]:= SolidLevelQ[{5, 6, 7, 8, 9, 11, 10, 1}, 1]

Out[440]= True

Calling the SolidLevelQ[x, n] procedure returns True if the elements of a level n of a list x are arranged sequentially, i.e. not separated by other nesting levels, and False otherwise. Whereas through optional 3^rd argument y – an indefinite variable – the call SolidLevelQ[x, n, y] returns the number of solid segments of the elements of the n–th nesting level if the main result is False. The procedure processes the erroneous situation in a case of invalid nesting level n with returning value $Failed and printing of the appropriate message. The previous fragment represents source code of procedure and the typical examples of its application.

The following procedure should be preceded by a procedure that allows to do that a list of an arbitrary structure and nesting whose all elements are of the form {h}, is result in the list of the same structure, but with elements of the form h. The procedure call IntSimplList[x] returns the result of the above restructuring of a list x of an arbitrary structure and nesting.

In[7]:= IntSimplList[x_ /; ListQ[x]] := Module[{a = ToString1[x],

b = {}, c = {}, d, k, j, p = ""}, d = StringLength[a];

Do[If[StringTake[a, {j}] == "{" && StringTake[a, {j + 1}] != "{",

AppendTo[b, j], 77], {j, d – 1}];

For[k = 1, k <= Length[b], k++, For[j = b[[k]], j <= Infinity, j++,

p = p <> StringTake[a, {j}]; If[SyntaxQ[p], AppendTo[c, j];

Break[], Continue[]]; p = ""]];

ToExpression[StringReplacePart[a, "", Map[{#, #} &, Join[b, c]]]]]

In[8]:= sv := {{{p}, {F[b]}, {a + b}}, {u^3}, {c}, {{{{n/m}, {{"h"}, {c^2},

{g[f]}}, {n^2 + t}}}}, {{y}, {"x"}}}

In[9]:= IntSimplList[sv]

Out[9]= {{p, F[b], a + b}, u^3, c, {{{n/m, {"h", c^2, g[f]}, n^2 + t}}},

{y, "x"}}

By substantially using now the previous procedure, we can more easily program a procedure that applies a certain symbol to the desired element of the nested list at a set nesting level.

In[7]:= SymbolToElemOnLevel[F_ /; SymbolQ[F], x_ /; ListQ[x],

n_Integer, m_Integer] :=

Module[{a = ToString1[ElemsOnLevel[x]], b, c, d, g, t, u, s, z},

s[t_] := If[IntegerQ[t], ToString[t], t]; SetAttributes[s, Listable];

z[t_] := If[StringQ[t] && IntegerQ[ToExpression[t]],

ToExpression[t], If[Quiet[Check[Part[t, 0] === F, False]] &&

StringQ[Part[t, 1]] && IntegerQ[Set[u, ToExpression[Part[t, 1]]]],

F[u], t]]; d = Map[s, x]; SetAttributes[z, Listable];

a = ToString1[ElemsOnLevel[d]];

If[n <= MaxLevel[x] – 1 && n > 0, If[Length[Level[x, {n}]] >= m,

b = ReduceAdjacentStr[ReduceAdjacentStr[a, "{", 1], "}", 1];

b = ToExpression["{" <> b <> "}"];

g[t_] := If[IntegerQ[t], Nothing, t]; SetAttributes[g, Listable];

b = Quiet[Check[Select[b, #[[3]] == n &][[m]][[1]],

Return[{Print["No element with number " <> ToString[m] <>

" at level " <> ToString[n]]; $Failed}[[1]]]]];

c = ToString1[b]; b = F @@ {b}; b = ToString1[b];

t = ToExpression[StringReplace[a, c –> b]];

t = IntSimplList[Map[g, t]]; Map[z, t],

Print["No element with number " <> ToString[m] <> " at level "

<> ToString[n]]; $Failed],

Print["No level with number " <> ToString[n]]; $Failed]]

In[8]:= t := {{p, b, x^2 + y^2, c + b, 7.7, 78, m/n}, u, c^2,

{{{n, {"h", f, {m, n, p}, c}, n + p^2}}}, {y, "x"}}

In[9]:= SymbolToElemOnLevel[F, t, 2, 3]

Out[9]= {{p, b, F[x^2 + y^2], b + c, 7.7, 78, m/n}, u, c^2,

{{{n, {"h", f, {m, n, p}, c}, n + p^2}}}, {y, "x"}}

In[10]:= SymbolToElemOnLevel[F, t, 4, 2]

Out[10]= {{p, b, x^2 + y^2, b + c, 7.7, 78, m/n}, u, c^2,

{{{n, {"h", f, {m, n, p}, c}, F[n + p^2]}}}, {y, "x"}}}

In[11]:= SymbolToElemOnLevel[F, t, 2, 12]

"No element with number 12 at level 2"

Out[11]= $Failed

In[12]:= SymbolToElemOnLevel[F, t, 7, 6]

"No level with number 7"

Out[12]= $Failed

The procedure call SymbolToElemOnLevel[F,x,n,m] returns the result of applying of a symbol F to an element with number m of nesting level n of the nested list x. The procedure handles the erroneous situations due to using of incorrect nesting levels and elements numbers at nesting levels with return $Failed and printing of appropriate message. Meantime, at a nesting level n of form ...{a, b,..., {c},..., d}... only {a, b,..., d} elements other than lists are considered as elements, and elements of more higher nesting level p > n, such as {c}, are ignored. The above fragment represents the source code of the procedure along with typical examples of its application.

The following procedure is a rather useful generalization of the previous procedure. The call SymbolToElemOnLevel1[F,x,n] returns the result of applying of a symbol F to elements of the nested list x that are defined by a nested list n, coded in the form of integer nested list {{l1, {p1 ,..., pn}}, …, {lt, {q1, ..., qm}}}. In this list for elements in form {l1, {p1 ,..., pn}} as the first element l1 is a nesting level whereas the 2^nd element-list are numbers elements on the set nesting level. In addition, elements of the Integer type

in the list x are coded in the string format. At the same time, the procedure handles the erroneous situations which are caused by the using of incorrect nesting levels and elements numbers at nesting levels with return $Failed and printing of appropriate messages. The fragment below represents the source code of the SymbolToElemOnLevel1 procedure with examples of its use.

In[5]:= SymbolToElemOnLevel1[F_ /; SymbolQ[F],

x_ /; ListQ[x], n_ /; ListQ[n]] :=

Module[{a = x, b, d, m = If[IntegerQ[n[[1]]], {n}, n], t = {}, h = {},

s = {}, v = {}, u, k, j},

d = Gather[LevelsOfList[x]]; d = Map[{#[[1]], Length[#]} &, d];

Map[{s = Flatten[Append[s, d[[#]][[1]]]],

v = Flatten[Append[v, d[[#]][[2]]]]} &, Range[1, Length[d]]];

Map[{AppendTo[t, m[[#]][[1]]], AppendTo[h, Max[m[[#]][[2]]]]} &,

Range[1, Length[m]]];

If[Set[u, Complement[t, s]] != {},

Return[{Print["Levels " <> ToString[u] <>

" are absent"]; $Failed}[[1]]],

For[k = 1, k <= Length[d], k++, For[j = 1, j <= Length[m], j++,

If[d[[k]][[1]] == m[[j]][[1]] && d[[k]][[2]] < Max[m[[j]][[2]]],

Return[{Print["The element numbers greater than the number

of elements on level " <> ToString[d[[k]][[1]]]]; $Failed}[[1]]], 7]]];

Do[Do[a = SymbolToElemOnLevel[F, a, k[[1]], j], {j, k[[2]]}], {k, m}]; a]]

In[6]:= h := {{a, b, c}, c, d, {{{g, v, {m, n, p}, f}}}, {x, y, z}}

In[7]:= SymbolToElemOnLevel1[F, h, {{2, {1, 2, 3, 6}}, {4, {1, 2}},

{5, {1, 2, 3}, {4, {1, 2}}}}]

Out[7]= {{F[a], F[b], F[c]}, F[c], d, {{{F[g], F[v], {F[m], F[n], F[p]}, f}}},

{x, y, F[z]}}

In[8]:= SymbolToElemOnLevel1[F, h, {{2, {1, 4, 6}}, {5, {1, 2, 3}}}]

Out[8]= {{F[a], b, c}, c, d, {{{g, v, {F[m], F[n], F[p]}, f}}},

{F[x], y, F[z]}}

In[9]:= SymbolToElemOnLevel1[F, h, {{2, {1, 4, 6, 7}}, {6, {1, 2, 3}},

{7, {1, 2, 3}}}]

"Levels {6, 7} are absent"

Out[9]= $Failed

In[10]:= SymbolToElemOnLevel1[F, h, {{2, {1, 6, 7}}, {5, {1, 2}}}]

"The element numbers greater than the number of

elements on level 2"

Out[10]= $Failed

Finally, the simple SymbolToListAll procedure returns the result of applying of the given S symbol to each element at all nesting levels of a list x. The following fragment represents the source code of the procedure with an example of its application.

In[147]:= SymbolToListAll[x_ /; ListQ[x], S_ /; SymbolQ[S]] :=

Module[{a}, SetAttributes[a, Listable]; a[t_] := S @@ {x}; Map[a, x]]

In[148]:= t := {{a, b}, c, m + n, 77, {{{m, {{c + d, n/p}}}}}, {h, "g"}}

In[149]:= SymbolToListAll[t, F]

Out[149]= {{F[a], F[b]}, F[c], F[m + n], F[77], {{{F[m], {{F[c + d],

F[n/p]}}}}}, {F[h], F["g"]}}

The following procedure is a useful version of the previous procedure. The call SymbolToNestList[x, y] returns the result of applying of symbols to elements of a list x that are defined by a sequence y in the form {f1, l1 , p1}, …, {ft, lt, pt} where {f1,…, ft} is symbols that should be applied, the elements {l1, …, lt} define nesting levels whereas the elements {p1, …, pt} define elements numbers on the corresponding nesting levels. In addition, the procedure handles the erroneous situations which are caused by the using of incorrect nesting levels or elements numbers at nesting levels with printing of appropriate messages.

In[378]:= SymbolToNestList[x_ /; ListQ[x], y__] :=

Module[{a = ElemsOnLevels2[x], b, c, d, g, t, f, v,

j = FromCharacterCode[7]}, b[t_] := ToString1[t];

f[t_] := If[IntegerQ[t], Nothing, t]; v[t_] := j <> t <> j;

Map[SetAttributes[#, Listable] &, {b, f, v}];

c = StructNestList[Map[b, x]]; d = ElemsOnLevels2[x];

t = Complement[Map[#[[2 ;3]] &, {y}], Map[#[[2 ;3]] &, d]];

If[t != {}, Print["Elements " <> "with conditions

/position> " <> ToString[t] <> " are invalid"];

t = Select[{y}, ! MemberQ[t, #[[2 ;3]]] &], t = {y}];

d = Select[d, MemberQ[Map[#[[2 ;3]] &, t], #[[2 ;3]]] &];

g = MatchLists[d, t, 2 ;3];

g = Map[Flatten[{g[[#]][[1]] @@ {d[[#]][[1]]}, d[[#]][[2 ;3]]}] &,

Range[1, Length[g]]];

d = Map[Flatten[{ToString1[#[[1]]], #[[2 ;3]]}] &, d];

g = Map[Flatten[{ToString1[#[[1]]], #[[2 ;3]]}] &, g];

c = ReplaceAll[c, Map[d[[#]] –> g[[#]] &, Range[1, Length[d]]]];

c = ToString[Map[v, Map[f, c]]];

ToExpression[StringReplace[c, {"{" <> j –> "", j <> "}" –> ""}]]]]]

In[379]:= p = {{a, b, {d, z, {m, {1, 2, 3, {{{v, g, s, d, s, h}}}, 4, 5}, n, t},

{{u}}, c, {x, y, h, w}}}};

In[380]:= SymbolToNestList[p, {V, 3, 2}, {G, 5, 1}, {S, 8, 1},

{Art, 8, 6}, {Kr, 2, 2}, {T, 4, 1}, {T, 6, 5}, {T, 4, 7}, {J, 1, 2}]

Elements with conditions {level, position} {{1, 2}, {6, 5}} are invalid

Out[380]= {{a, Kr[b], {d, V[z], {T[m], {G[1], 2, 3, {{{S[v], g, s, d, s,

Art[h]}}}, 4, 5}, n, t}, {{u}}, c, {x, y, h, T[w]}}}}

The SymbolToNestList1 procedure is analogue of the above procedure with ignoring of invalid values of y argument.

In[390]:= SymbolToNestList1[x_ /; ListQ[x], y__] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x], c = {}, d,

g = {}, j, s, t = DeleteDuplicates[{y}, #1[[2 ;3]] == #2[[2 ;3]] &]},

d = Map[#[[2 ;3]] &, t]; s = Length[t];

For[j = 1, j <= Length[a], j++, If[MemberQ[d, a[[j]][[2 ;3]]],

AppendTo[c, a[[j]]], AppendTo[g, a[[j]]]]];

d = MatchLists[c, t, 2 ;3];

Quiet[d = Map[c[[#]] –> d[[#]][[1]] @@ {c[[#]][[1]]} &, Range[1, s]];

g = Join[d, Map[# –> #[[1]] &, g]]; ReplaceAll[b, g]]]

The above procedures SolidLevelQ, SymbolToElemOnLevel, LevelsOfList, IntSimplList, SymbolToListAll, SymbolToNestList, SymbolToElemOnLevel1 and SymbolToNestList1 are of a certain interest in programming problems related to the nested lists and as examples with use of the sapiential methods at programming of the problems of a similar type [6-15].

We will present couple more of tools using useful methods of programming of problems linked with processing of nested lists. Particularly, calling the function ToInitNestList[x] returns an initial nested y list from that a list x was obtained as a result x = StructNestList[y] whereas the call ToInitNestList[x, j] where j is an arbitrary expression returns an initial nested y list from which a list x was obtained as a result ElemsOnLevels1[y]. Its source code is represented below.

In[420]:= ToInitNestList[x_ /; ListQ[x], y___] := ReplaceAll[x,

Map[# –> #[[1]] &, Partition[Flatten[x], If[{y} != {}, 2, 3]]]]

Together with the above two procedures, the function is of very specific interest in handling nested lists [8-16].

Calling the procedure DeleteDuplicatesNest[x, n, g] returns the result of deleting of all duplicates from the set nesting level n of a list x. As a value n can be an integer, an integer list or the word All, that provides deletion at a concrete nesting level, at their set or all levels accordingly. In addition, the call can use the 3^rd optional g argument that is a test to pairs of elements to determine whether they should be considered duplicates.

In[78]:= DeleteDuplicatesNest[x_ /; ListQ[x], n_ /; PosIntQ[n]||

IntegerListQ[n] || n === All, g___] := Module[{f, h}, f[t_, p_] :=

Module[{a = ElemsOnLevels2[t], b = StructNestList[t], c, d},

c = Map[If[#[[2]] == p, #, Nothing] &, a];

d = DeleteDuplicates[c, If[{g} != {}, g[#1[[1]]] == g[#2[[1]]] &,

#1[[1]] == #2[[1]] &]];

d = Complement[c, d];

ToInitNestList[ReplaceAll[b, Map[# –> Nothing &, d]]]];

If[PosIntQ[n], f[x, n],

If[IntegerListQ[n], h = x; Do[h = f[h, j], {j, n}]; h, h = x;

Do[h = f[h, j], {j, ListLevels[x]}]; h]]]

In[79]:= p = {{b, m, 42, b, {d, c, c, {m, m, {t, t, {{{v, s, 9, s, d, {m, c},

s, h, s, d, 9, 9, v}}}, 4, 5, 4}, t, n, t}, {x, y, y, x}}}};

In[80]:= DeleteDuplicatesNest[p, All]

Out[80]= {{b, m, 42, {d, c, {m, {t, {{{v, s, 9, d, {m, c}, h}}}, 4, 5}, t, n},

{x, y}}}}

Calling the procedure ReplaceCondList[x, y] returns the result of replacement of elements of a list x, that are at nesting levels with set positions on them and that satisfy set conditions that are defined by sequence y of the lists {s1, f1, l1, p1}, ..., {st, ft, lt, pt}, where list {sj, fj, lj, pj} (j = 1..t) defines replacement of an element ej that is located at nesting level lj with position pj on it with condition fj[ej]=True onto a sj element. If substitution is not possible because of absence of nesting level lj or/and position pj then it is ignored without any message.

In[90]:= ReplaceCondList[x_ /; ListQ[x], y__] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x], d, k, j,

c = DeleteDuplicates[{y}, #1[[3 ;4]] == #2[[3 ;4]] &], h = {}},

c = Map[If[MemberQ[Map[#[[2 ;3]] &, a], #[[3 ;4]]], #,

Nothing] &, c]; d = Map[If[MemberQ[Map[#[[3 ;4]] &, c],

#[[2 ;3]]], #, Nothing] &, a]; a = Length[d];

For[k = 1, k <= a, k++, For[j = 1, j <= a, j++,

If[c[[k]][[3 ;4]] == d[[j]][[2 ;3]], AppendTo[h, c[[j]]], 78]]];

h = Map[If[h[[#]][[2]] @@ {d[[#]][[1]]},

d[[#]] –> Flatten[{h[[#]][[1]], h[[#]][[3 ;4]]}], Nothing] &,

Range[1, a]]; ToInitNestList[ReplaceAll[b, h]]]

Calling the procedure ExchangeLevels2[x, k, j] returns the result of elements exchanging of nesting levels k and j of nested list x. The source code of the procedure is represented below.

In[1942]:= ExchangeLevels2[x_/; NestListQ1[x], n_/; PosIntQ[n],

m_ /; PosIntQ[m]] := Module[{a, b, c, d, p, g, s, t},

t = ReplaceAll[x, {} –> {s}]; a = ElemsOnLevels2[t];

b = StructNestList[t];

If[! MemberQ3[Map[#[[2]] &, a], {n, m}],

"Second or/and third arguments are invalid",

d = Select[a, #[[2]] == n &]; p = Select[a, #[[2]] == m &];

c = {d[[1]], StringTake[ToString1[p], {2, –2}]};

g = {p[[1]], StringTake[ToString1[d], {2, –2}]};

c = ReplaceAll[b, Flatten[{Map[# –> Nothing &, d[[2 ;–1]]],

Map[# –> Nothing &, p[[2 ;–1]]], c[[1]] –> c[[2]],

g[[1]] –> g[[2]]}]];

c = ToInitNestList[ToExpression[ToString[c]]];

c = ReplaceRepeated[c, {} –> Nothing];

ReplaceAll[c, s –> Nothing]]]

In[1943]:= p = {{a, b, c, {c, d, t, {m, z, g, u, n}, m, h}, x, y, z, v}};

In[1944]:= ExchangeLevels2[p, 2, 4]

Out[1944]= {{m, z, g, u, n, {c, d, t, {a, b, c, x, y, z, v}, m, h}}}

Furthermore, in addition to the previous procedure, calling the procedure ExchangeElemsOnLevels[x, y] returns the result of the interchange of elements located at the specified nesting levels of a list x, taking into account their position on them. In addition, the second argument y – the list of ListList type – has the form {{{l1, p1}, {l2, p2}},…, {{lt, pt}, {ln, pn}}}, where lists pairs of the form {{lt, pt}, {ln, pn}} define nesting levels {lt, ln} and the positions {pt, pn} of elements on them that are subject to mutual exchange. The ExchangeElemsOnLevels procedure handles the main erroneous situations conditioned by the errors in the 2^nd y argument with return or printing of the appropriate messages. The fragment below represents the source code (that contains a number of useful programming methods) of the procedure with a number of examples of its typical application.

In[42]:= ExchangeElemsOnLevels[x_ /; ListQ[x],

y_ /; ListListQ[y]] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x], c,

d = {}, j, h = If[Length[Flatten[y]] == 4, {y}, y], g},

c = Map[If[MemberQ3[Map[#[[2 ;3]] &, a], #], #,

Nothing] &, h];

If[c == {}, Return["The second argument is invalid"],

If[h != c, Print["Elements " <> ToString[Complement[h, c]]

<> " in the second argument are invalid"], 78];

g[t_] := {t[[1]] –> Flatten[{t[[2]][[1]], t[[1]][[2 ;3]]}],

t[[2]] –> Flatten[{t[[1]][[1]], t[[2]][[2 ;3]]}]};

Do[AppendTo[d, {Select[a, #[[2 ;3]] == c[[j]][[1]] &][[1]],

Select[a, #[[2 ;3]] == c[[j]][[2]] &][[1]]}], {j, 1, Length[c]}];

d = Flatten[Map[g, d], 1];

ToInitNestList[ReplaceAll[b, d]]]]

In[43]:= p = {{a, b, {c, d, {m, g, {m, {{"t"}}, n}, u, n}, m, h}, x, y, z}};

In[44]:= ExchangeElemsOnLevels[p, {{{3, 4}, {4, 4}}, {{2, 2}, {3, 2}}}]

Out[44]= {{a, d, {c, b, {m, g, {m, {{"t"}}, n}, u, h}, m, n}, x, y, z}}

In[45]:= ExchangeElemsOnLevels[p, {{7, 1}, {2, 5}}]

Out[45]= {{a, b, {c, d, {m, g, {m, {{z}}, n}, u, n}, m, h}, x, y, "t"}}

Calling the procedure SortOnLevels2[x, p, f] returns the default sort result of a nesting level p of a list x, if optional f argument is missing, or according to the set ordering function f.

In[9]:= SortOnLevels2[x_ /; ListQ[x], n_ /; PosIntQ[n], Sf___] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x], c, d},

c = Select[a, #[[2]] == n &]; d = If[{Sf} != {}, Sort[c, Sf],

Sort[c, Order[#1[[1]], #2[[1]]] &]]; ToInitNestList[ReplaceAll[b,

Map[c[[#]] –> d[[#]] &, Range[1, Length[c]]]]]]

In[10]:= SortOnLevels2[{a, b, {c, d, {j, h, d, e, r, t}}}, 3]

Out[10]= {a, b, {c, d, {d, e, h, j, r, t}}}

The following procedure is a version of previous procedure whose source code contains some useful technique for handling nested lists. Calling the procedure SortOnLevels3[x, n, f] returns the result of sorting of elements located on the nesting levels of a list x that are defined by the n argument – a single level (n), a list of levels {n1, …, np}, and all nesting levels (n = All). Whereas the third optional f argument defines the ordering function, at the same time at its absence the default function Order is used. Fragment below represents the source code of this procedure along with some typical examples of its application.

In[7]:= SortOnLevels3[x_ /; ListQ[x], n_ /; IntegerQ[n] ||

IntegerListQ[n] || n === All, f___] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x],

t = ListLevels1[x], c, d, p, s, j},

p = If[IntegerQ[n], {n}, If[n === All, t, n]];

s = MinusList1[p, c = Complement[p, t]];

If[c == p, Print["All nesting levels " <> ToString[p] <>

" are invalid"]; Return[x],

If[c != {}, Print["Nesting levels " <> ToString[c] <>

" are invalid"], 78]];

Do[c = Select[a, #[[2]] == j &];

d = Sort[c, If[{f} == {}, Order[#1[[1]], #2[[1]]] &, f]];

b = ReplaceAll[b, Map[c[[#]] –> d[[#]] &,

Range[1, Length[c]]]], {j, s}]; ToInitNestList[b]]

In[8]:= p = {{b, j, a, b, {c, d, {j, {j, {t, j}, n}, u, n}, j, h}, z, y, x}};

In[9]:= SortOnLevels3[p, {1, 2, 7, 3, 4, 9}]

Nesting levels {1, 7, 9} are invalid

Out[9]= {{a, b, b, j, {c, d, {j, {j, {t, j}, n}, n, u}, h, j}, x, y, z}}

In[10]:= SortOnLevels3[p, {1, 9}]

All nesting levels {1, 9} are invalid

Out[10]= {{b, j, a, b, {c, d, {j, {j, {t, j}, n}, u, n}, j, h}, z, y, x}}

In[11]:= SortOnLevels3[p, All]

Out[11]= {{a, b, b, j, {c, d, {j, {j, {j, t}, n}, n, u}, h, j}, x, y, z}}

In[12]:= SortOnLevels3[Flatten[p], All]

Out[12]= {a, b, b, c, d, h, j, j, j, j, j, n, n, t, u, x, y, z}

Calling the procedure InsertToNestList[x, z, y] returns the result of inserting expressions before (z = 2) or after (z = 1) aj of the list elements x located at the nesting levels lj at the positions pj on these levels that are determined by the sequence y of the lists {aj, lj, pj}. The invalid lists in y are ignored.

In[508]:= InsertToNestList[x_/; ListQ[x], z_/; ! FreeQ[{1, 2}, z], y__] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x], d, h, t,

g = {}, c = DeleteDuplicates[{y}, #1[[2 ;3]] == #2[[2 ;3]] &], s = {}},

c = Map[If[MemberQ[Map[#[[2 ; 3]] &, a], #[[-2 ; -1]]], #, Nothing] &, c];

d = Map[Flatten[{StringTake[ToString1[#[[1 ;Length[#] – 2]]],

{2, –2}], #[[–2 ;–1]]}] &, c]; h = Map[#[[2 ;3]] &, d];

Map[If[MemberQ[h, #[[2 ;3]]], AppendTo[g, #],

AppendTo[s, #]] &, a]; t = MatchLists1[d, g, 2 ;3];

s = Map[# –> #[[1]] &, s]; t = Map[t[[#]] –> If[z == 1,

ToString1[ t[[#]][[1]]] <> "," <> d[[#]][[1]],

d[[#]][[1]] <> "," <> ToString1[t[[#]][[1]]]] &, Range[1, Length[t]]];

ToExpression[ToString[ReplaceAll[b, Join[s, t]]]]]

In[509]:= p = {{a, b, {d, z, {m, {t, {{{v, g, s, d, s, h}}}, 4, 5}, n, t}, {x, y, h}}}};

In[510]:= InsertToNestList[p, 2, {x, y, z, 3, 2}, {ab, 2, 1}, {"agn", 8, 2}]

Out[510]= {{ab, a, b, {d, x, y, z, z, {m, {t, {{{v, "agn", g, s, d, s, h}}}, 4, 5},

n, t}, {x, y, h}}}}

The possibility of exchange by elements of the nested lists which are located at the different nesting levels is of a certain interest. The problem is successfully solved by the procedure SwapOnLevels. The procedure call SwapOnLevels[x, y] returns the result of exchange of elements of a list x, being at the nesting levels and with their sequential numbers at these levels that are defined by the nested list y of the format {{{s1, p1}, {s2, p2}}, ..., {{sk1, pk1}, {sk2, pk2}}} where a pair {{sj1, pj1}, {sj2, pj2}} defines replacement of an element which is at the nesting level sj1 with a sequential number pj1 at this level by an element that is at the nesting level sj2 with a sequential number pj2 at this level, and vice versa. In a case of impossibility of such exchange because of lack of a nesting level sj or/and a sequential number pj the procedure call prints the appropriate message. In addition, in a case of absence in the y list of the acceptable pairs for exchange the procedure call returns $Failed and prints the above message.

In[229]:= L := {b, t, c, {{3, 2, 1, {g, s, a}, k, 0}, 2, 3, 1}, 1, 2, {m, n, f}}

In[230]:= SwapOnLevels[L, {{{1, 5}, {4, 3}}, {{1, 1}, {3, 3}}}]

Out[230]= {1, t, c, {{3, 2, b, {g, s, 2}, k, 0}, 2, 3, 1}, 1, a, {m, n, f}}

In[231]:= SwapOnLevels[x_ /; ListQ[x], y_ /; ListListQ[y]] :=

Module[{a = ElemsOnLevels2[x], b, c, d, f, g, h, p, n, s, z, u = 0,

w = ReplaceAll[x, {} –> Null]},

g[t_] := "(" <> ToString1[t] <> ")"; SetAttributes[g, Listable];

f[t_] := h[[n++]]; SetAttributes[f, Listable];

s[t_] := ToExpression[t][[1]]; SetAttributes[s, Listable];

Do[z = y[[j]]; n = 1; If[MemberQ5[a, #[[2 ;3]] == z[[1]] &] &&

MemberQ5[a, #[[2 ;3]] == z[[2]] &],

b = Select[a, #[[2 ;3]] == z[[1]] &][[1]];

c = Select[a, #[[2 ;3]] == z[[2]] &][[1]];

a = ReplaceAll[a, b –> d]; a = ReplaceAll[a, c –> h];

a = ReplaceAll[a, d –> c]; a = ReplaceAll[a, h –> b];

p = Map[g, w]; h = Map[ToString1, a]; h = Map[s, Map[f, p]], u++;

Print["Pairs " <> ToString[z] <>

" are incorrect for elements swap"]], {j, 1, Length[y]}];

If[u == Length[y], $Failed, ReplaceAll[h, Null –> Nothing]]]

In[232]:= SwapOnLevels[L, {{{{7, 1}, {2, 3}}, {{8, 1}, {3, 3}}}}]

Pairs {{{7, 1}, {2, 3}}, {{8, 1}, {3, 3}}} are incorrect for elements swap

Out[232]= $Failed

A certain interest is the conditional sorting of lists when some elements in a list are not involved in the sorting, that is, they are defined as unmovable. The immobility of the elements of the sorted list is determined by some condition (position, type, etc.). The SortLcond procedure is one of means solving this problem. The procedure call SortLcond[x, y, z] returns the sorting result of a list x according to a sorting function defined by the optional argument z – a pure function (at absence of z argument the canonical sorting order is used), provided that the elements of the initial x list that are defined by a position or their list y are unmovable. The procedure processes the erroneous positions situation with returning $Failed with printing of the appropriate messages.

In[14]:= SortLcond[x_ /; ListQ[x], y_ /; IntegerQ[y] ||

IntegerListQ[y], z___ /; PureFuncQ[z]] :=

Module[{a = Flatten[{x}], b = {}, c = Length[x]},

Map[If[# < 1 || # > c, AppendTo[b, #], 7] &, y];

If[b != {}, Print["Positions " <> ToString[b] <> " are invalid"]; $Failed,

b = Map[{#, a[[#]]} &, y]; Map[{a[[#]] = Nothing} &, y];

a = Sort[a, z];

Do[a = Insert[a, b[[j]][[2]], b[[j]][[1]]], {j, Length[b]}]; a]]

In[15]:= SortLcond[Range[1, 19], {1, 5, 9, 13, 17}, #1 > #2 &]

Out[15]= {1, 19, 18, 16, 5, 15, 14, 12, 9, 11, 10, 8, 13, 7, 6, 4, 17, 3, 2}

In[16]:= SortLcond[Range[1, 19], {0, 5, 21, 27}, #1 > #2 &]

Positions {0, 21, 27} are invalid

Out[16]= $Failed

The natural version of the previous procedure is when the immobility of the elements of the sorted list is determined by their type. The SortLcond1 procedure is one of means solving this problem. The procedure call SortLcond1[x, y, z] returns the sorting result of a list x according to a sorting function defined by the optional argument z – a pure function (in a case of absence of z argument the canonical sorting order is used), provided that the elements of the initial x list that satisfy a testing pure function y are unmovable. The following fragment represents source code of the SortLcond1 procedure with an example of its application.

In[15]:= SortLcond1[x_ /; ListQ[x], y_ /; PureFuncQ[y],

z___ /; PureFuncQ[z]] :=

Module[{a = Flatten[{x}], b = {}, c = Length[x], k = 0},

Map[If[{k++, y @@ {#}}[[2]], AppendTo[b, {k, #}], 77] &, a];

Map[{a[[#]] = Nothing} &, Map[#[[1]] &, b]]; a = Sort[a, z];

Do[a = Insert[a, b[[j]][[2]], b[[j]][[1]]], {j, Length[b]}]; a]

In[16]:= SortLcond1[Range[1, 19], PrimeQ[#] &, #1 > #2 &]

Out[16]= {18, 2, 3, 16, 5, 15, 7, 14, 12, 10, 11, 9, 13, 8, 6, 4, 17, 1, 19}

Conditional sorting of types determined by SortLcond and SortLcond1 procedures it is simple to extend on the nested lists, both as a whole and relative to the set nesting levels. We leave this to the interested reader as a rather useful exercise.

An example is the SortOnLevels procedure, being a rather natural generalization of the SortOnLevel procedure [8,16]. The procedure call SortOnLevels[x, m, Sf] where the optional Sf argument determines a sorting pure function analogously to the SortOnLevel procedure returns the sorting result of a list x at its nesting levels m. Furthermore, as an argument m can be used a single level, the levels list or the "All" word that defines all nesting levels of the x list which are subject to sorting. If all m levels are absent then the procedure call returns $Failed with printing of appropriate message, otherwise the procedure call prints the message concerning only levels absent in the x list. It must be kept in mind, by default the sorting is made in symbolic format, i.e. all elements of a sorted x list before sorting by means of the standard Sort function are presented in the string format. Whereas the procedure call SortOnLevels[x, m, Sf] returns the sorting result of the x list at its nesting levels m depending on a sorting pure Sf function that is the optional argument. Fragment represents source code of the SortOnLevels procedure with an example of its typical application.

In[7]:= SortOnLevels[x_ /; ListQ[x], m_ /; IntegerQ[m] ||

IntegerListQ[m] || m === All, Sf___] :=

Module[{a, b, c = ReplaceAll[x, {} –> {Null}], d, h, t, g, s},

a = LevelsOfList[c]; d = Sort[DeleteDuplicates[a]];

If[m === All ||

SameQ[d, Set[h, Sort[DeleteDuplicates[Flatten[{m}]]]]], t = d,

If[Set[g, Intersection[h, d]] == {}, Print["Levels " <> ToString[h] <>

" are absent in the list; " <> "nesting levels must belong to " <>

ToString[Range[1, Max[d]]]];

Return[$Failed], t = g; s = Select[h, ! MemberQ[d, #] &];

If[s == {}, Null, Print["Levels " <> ToString[s] <>

" are absent in the list; " <> "nesting levels must belong to " <>

ToString[Range[1, Max[d]]]]]]];

Map[Set[c, SortOnLevel[c, #, Sf]] &, t];

ReplaceAll[c, Null –> Nothing]]

In[8]:= SortOnLevels[{{{3, 4, 5, 0, 8, 7, 9, 1}}, 9, 8, 7, {t, b, c, a, t},

{{6, 3, 3, 5, 2, 9, 8}}}, {1, 2, 3}]

Out[8]= {{{0, 1, 2, 3, 3, 3, 4, 5}}, 7, 8, 9, {a, b, c, t, t}, {{5, 6, 7, 8, 8, 9, 9}}}

The call Replace[e, r, lev] of the built-in function applies the rules r to parts of a list e specified by the nesting levels lev. In addition, all rules are applied to all identical e elements of the lev. An useful enough addition to the Replace function is the procedure, whose procedure call ReplaceOnLevels[x, y] returns the result of replacement of elements of a list x, that are at the nesting levels and with their sequential numbers at these levels which are defined by the nested list y of the format {{s1, p1, e1}, {s2, p2, e2},…,{sk, pk, ek}} where sub-list {sj, pj, ej} (j = 1..k) defines replacement of an element that is located at nesting level sj with sequential number pj at this level by an ej element. If some such substitution is not possible because of lack of a nesting level sj or/and a sequential number pj then the procedure call prints the appropriate message. At the same time, in a case of absence in the y list of the acceptable values {sj, pj, ej} for replacement the procedure call returns $Failed with printing of appropriate message. The fragment represents source code of the procedure with some typical examples of its application.

In[2242]:= ReplaceOnLevels[x_ /; ListQ[x], y_ /; ListQ[y]] :=

Module[{a, b, d = {}, f, g, h, p, n, s, z, u = 0,

w = ReplaceAll[x, {} –> Null], r = If[ListListQ[y], y, {y}]},

a = ElemsOnLevels2[w]; b = Map[#[[2 ;3]] &, a];

g[t_] := "(" <> ToString1[t] <> ")"; SetAttributes[g, Listable];

f[t_] := h[[n++]]; SetAttributes[f, Listable];

s[t_] := ToExpression[t][[1]]; SetAttributes[s, Listable];

Do[z = r[[j]]; n = 1; If[MemberQ[b, z[[1 ;2]]],

Do[If[a[[k]][[2 ;3]] == z[[1 ;2]],

AppendTo[d, PrependTo[a[[k]][[2 ;3]], z[[3]]]],

AppendTo[d, a[[k]]]], {k, 1, Length[a]}]; a = d; d = {};

p = Map[g, w]; h = Map[ToString1, a]; h = Map[s, Map[f, p]], u++;

Print["Data " <> ToString[z] <> " for replacement are

incorrect"]], {j, 1, Length[r]}];

If[u == Length[r], $Failed, ReplaceAll[h, Null –> Nothing]]]

In[2243]:= ReplaceOnLevels[{t, c, {{{a + b, h}}}, {m, n, f},

{{{{c/d, g}}}}}, {{4, 1, m + n}, {5, 1, Sin[x]}}]

Out[2243]= {t, c, {{{m + n, h}}}, {m, n, f}, {{{{Sin[x], g}}}}}

In[2244]:= ReplaceOnLevels[{t, c, {{{}}}, {m, n, f}, {{{{}}}}},

{{4, 1, m + n}, {3, 1, Sin[x]}}]

Out[2244]= {t, c, {{Sin[x]}}, {m, n, f}, {{{m + n}}}}

In turn, the ReplaceOnLevels1 procedure [16] works similar to the previous ReplaceOnLevels procedure with an important difference. The call ReplaceOnLevels1[x, y] returns the result of replacement of elements of a list x, that are at the nesting levels and satisfying set conditions that are defined by the nested list y of the format {{s1, c1, e1}, {s2, c2, e2},…,{sk, ck, ek}}, where sub-list {sj, cj, ej} (j = 1..k) determines replacement of an element that is located at nesting level sj and satisfies set condition cj by an ej element. If substitution is not possible because of absence of a nesting level sj or/and falsity of testing condition cj, the call prints the appropriate message. At the same time, in a case of lack in y list of any acceptable values {sj, cj, ej} for replacement the call returns $Failed with print of the appropriate messages.

A rather useful addition to the ReplaceOnLevels procedure is the procedure whose call ExchangeListsElems[x, y, z] returns the result of exchange by elements of a list x and a list y that are located at the nesting levels and with their sequential numbers at these levels. The receptions that were used at programming of the procedures LevelsOfList, ElemsOnLevels ÷ ElemsOnLevels5, SortOnLevel, SwapOnLevels, ReplaceOnLevels, ReplaceInNestList and ExchangeListsElems are of a certain interest to the problems dealing with the nested lists along with independent interest [8].

Unlike the built–in Insert function the following procedure is a rather convenient means for inserting of expressions to the given place of any nesting level of the list. The procedure call InsertInLevels[x, y, z] returns the result of inserting in a list x of expj expressions that are defined by the second y argument of the form {{lev1, ps1, exp1},…,{levk, psk, expk}} where levj defines the nesting level of x, psj defines the sequential number of this element on the level and expj defines an expression that will be inserted before (z = 1) or after (z = 2) this element (j = 1..k). At that, on inadmissible pairs {levj, psj} the procedure call prints the appropriate messages. Whereas, in a case of lack in the y list of any acceptable {levj, psj} for inserting, the call returns $Failed with print of the appropriate messages. At the same time, the 4^th optional g argument – an indefinite symbol – allows to insert in the list the required sub-lists, increasing nesting level of x list in general. In this case, the expression expj is coded in the form g[a, b, c, …] which admits different useful compositions as the examples of the fragment below along with source code of the

InsertInLevels procedure very visually illustrate.

In[5]:= InsertInLevels[x_ /; ListQ[x], y_ /; ListQ[y],

z_ /; MemberQ[{1, 2}, z], g___] :=

Module[{a, b, c, d, h, f = ReplaceAll[x, {} –> Null], p, u = 0,

r = If[ListListQ[y], y, {y}]}, a = ElemsOnLevels2[f];

b = Map[#[[2 ;3]] &, a];

d[t_] := If[StringQ[t] && SuffPref[t, "Sequences[", 1],

ToExpression[t], t]; SetAttributes[d, Listable];

Do[p = r[[j]]; If[Set[h, Select[a, #[[2 ;3]] == p[[1 ;2]] &]] == {},

u++; Print["Level and/or position " <> ToString[p[[1 ;2]]] <>

" are incorrect"], c = If[z == 2,

"Sequences[" <> ToString[{h[[1]][[1]], p[[3]]}] <> "]",

"Sequences[" <> ToString[{p[[3]], h[[1]][[1]]}] <> "]"];

f = Map[d, ReplaceOnLevels[f, {Flatten[{p[[1 ;2]], c}]}]]],

{j, 1, Length[r]}];

If[u == Length[r], $Failed, ReplaceAll[f, {If[{g} != {} &&

NullQ[g], g –> List, Nothing], Null –> Nothing}]]]

In[6]:= InsertInLevels[L = {t, c, {{{a + b, h}}}, {m, n, f}, {{{{77, g}}}},

77, 72}, {{4, 1, avz}, {5, 2, agn}}, 1]

Out[6]= {t, c, {{{avz, a + b, h}}}, {m, n, f}, {{{{77, agn, g}}}}, 77, 72}

In[7]:= InsertInLevels[L, {{7, 1, avz}, {5, 2, agn}}, 2]

Level and/or position {7, 1} are incorrect

Out[7]= {t, c, {{{a + b, h}}}, {m, n, f}, {{{{77, g, agn}}}}, 77, 72}

In[8]:= InsertInLevels[{1, 2, 3, 4, 5, 6, 7, 8, 9}, {{1, 3, F[{avz, a, b, c}]},

{1, 7, F[{agn, art, kr}]}}, 2, F]

Out[8]= {1, 2, 3, {avz, a, b, c}, 4, 5, 7, {agn, art, kr}, 7, 8, 9}

Calling the procedure InsertLevel[x, n, m, y, z] returns the result of inserting to a list x of a new nesting level defined by the list y that should be located on the n–th nesting level after (before) m–th position on it depending on absence or existence of the optional z argument accordingly.

In[942]:= InsertLevel[x_ /; ListQ[x], n_ /; PosIntQ[n],

m_ /; PosIntQ[m], y_ /; ListQ[y], z___] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x],

c, d, p = 1, g}, If[! MemberQ[Map[#[[2 ;3]] &, a], {n, m}],

"Second or/and third arguments are invalid",

g = Map[{#, n + 1, p++} &, y];

c = Select[a, #[[2 ;3]] == {n, m} &][[1]];

d = ReplaceAll[b, c –> If[{z} == {}, ToString1[c] <> "," <>

ToString1[g], ToString1[g] <> "," <> ToString1[c]]];

c = ToInitNestList[ToExpression[ToString[d]]]]]

In[943]:= p = {{a, b, {c, d, {m, g, {m, n}, u, n}, m, h}, x, y, z}};

In[944]:= InsertLevel[p, 5, 2, {A, V, Z}, gs]

Out[944]= {{a, b, {c, d, {m, g, {m, {A, V, Z}, n}, u, n}, m, h}, x, y, z}}

In[945]:= InsertLevel[p, 5, 2, {A, V, Z}]

Out[945]= {{a, b, {c, d, {m, g, {m, n, {A, V, Z}}, u, n}, m, h}, x, y, z}}

Unlike the built-in Delete function the following procedure is a rather convenient means for deleting of expressions on the given place of any nesting level of the list. The procedure call DeleteAtLevels[x, y] returns the result of deleting in a list x of its elements that are determined by the second y argument of the format {{lev1, ps1}, ..., {levk, psk}}, where levj determines the nesting level of x and psj defines the sequential number of this element on the level (j = 1..k). On inadmissible pairs {levj, psj} the procedure call prints the appropriate messages. Whereas in a case of lack in the y list of any acceptable {levj, psj} pair for the deleting, the procedure call returns $Failed with printing of the appropriate message. The following fragment represents source code of the procedure with typical examples of its application.

In[7]:= DeleteAtLevels[x_ /; ListQ[x], y_ /; ListQ[y]] :=

Module[{a, b, c = 0, d, h, f = ReplaceAll[x, {} –> Null],

r = If[ListListQ[y], y, {y}]},

a = ElemsOnLevels2[f]; b = Map[#[[2 ;3]] &, a];

d[t_] := If[t === "Nothing", ToExpression[t], t];

SetAttributes[d, Listable];

Do[h = r[[j]]; If[Select[b, # == h &] == {}, c++;

Print["Level and/or position " <> ToString[h] <> " are incorrect"],

f = ReplaceOnLevels[f, {Flatten[{h, "Nothing"}]}]], {j, 1, Length[r]}];

f = Map[d, f];

If[c == Length[r], $Failed, ReplaceAll[f, Null –> Nothing]]]

In[8]:= DeleteAtLevels[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {{1, 5}, {1, 8}, {1, 3}}]

Out[8]= {1, 2, 4, 6, 7, 9, 10}

In[9]:= DeleteAtLevels[{t, c, {{{a*b, h}}}, {m, n, f}, {{{{77, g}}}},

77, 72}, {{4, 1}, {4, 2}, {5, 2}, {2, 1}, {2, 3}}]

Out[9]= {t, c, {{{}}}, {n}, {{{{77}}}}, 77, 72}

In a number of cases exists a need of generation of the list of variables in the form Vk (k = 1..n), where V – a name and n – a positive integer. The standard functions CharacterRange and Range of the Mathematica don't solve the problem, therefore for these purposes it is rather successfully possible to use the procedures Range1 ÷ Range6, whose source codes with typical examples of their use can be found in [8,16]. The above tools of so–called Range group are useful at processing of the lists. So, the coder–decoder CodeEncode, operating on the principle of switch, uses 2 functions Range5 and Prime. The procedure call CodeEncode[x] returns the result of coding of a string x of Latin printable ASCII–symbols, and vice versa. The CodeEncode1 tool is a rather useful extension of the above CodeEncode procedure in the event of files of ASCII format. The use of this procedure for encoding/decoding text files confirmed its effectiveness.

The next group of tools serves for expansion of the built-in MemberQ function, and its tools are quite useful in work with list structures. This group is based on procedures MemberQ1 ÷ MemberQ10 that along with procedures MemberT, MemberLN, MemberQL and MemberQL1 are useful in the lists processing. In principle, these means allow interesting enough modifications significantly broadening the sphere of their application. Source codes of the above tools of so–called Member–group along with examples of their application can be found in [8,10-16].

On a basis of the Intersection2 and ElemsOnLevelList [6,12] procedures the next procedure can be programmed that allows to receive intersection of the elements of nesting levels of a list. The procedure call ElemsLevelsIntersect[x] returns the nested list whose two-element sub-lists has the format {{n, m}, {{e1, p1}, {e2, p2}, …}}, where {n, m} – the nesting levels of a list x (n ÷ m) and {ej, pj} determines common ej element for the nesting levels {n, m}, whereas pj is its common minimal multiplicity for both nesting levels. In a case of the only list as an argument x the call returns the empty list, i.e. {}. The following fragment represents source codes of the Intersection2 and ElemsOnLevelList along with typical examples of their application.

In[7]:= Intersection2[x__List] := Module[{a = Intersection[x], b = {}},

Do[AppendTo[b, DeleteDuplicates[Flatten[Map[{a[[j]],

Count[#, a[[j]]]} &, {x}]]]], {j, Length[a]}]; b]

In[8]:= Intersection2[{a + b, a, b, a + b}, {a, a + b, d, b, a + b},

{a + b, b, a, d, a + b}]

Out[8]= {{a, 1}, {b, 1}, {a + b, 2}}

In[9]:= ElemsLevelsIntersect[x_ /; ListQ[x]] :=

Module[{a = ElemsOnLevelList[x], b, c = {}, d, h = {}},

d = Length[a]; Do[Do[If[k == j, Null,

If[Set[b, Intersection2[a[[k]][[2]], a[[j]][[2]]]] != {},

AppendTo[c, {{a[[k]][[1]], a[[j]][[1]]}, b}], Null]], {j, k, d}], {k, d}];

Do[AppendTo[h, {c[[j]][[1]], Map[{#[[1]], Min[#[[2 ;–1]]]} &,

c[[j]][[2]]]}], {j, Length[c]}]; h]

In[9]:= ElemsLevelsIntersect[{a, c, b, b, c, {b, b, b, c, {c, c, b, b}}}]

Out[9]= {{{1, 2}, {{b, 2}, {c, 1}}}, {{1, 3}, {{b, 2}, {c, 2}}}, {{2, 3}, {{b, 2}, {c, 1}}}}

The procedure call ToCanonicList[x] converts a list x of the ListList type whose sub-lists have form {a, hj} and/or {{a1,…,ap}, hj}, into equivalent nested classical list where a, ak are elements of list (k = 1..p) and hj – the nesting levels on which they have to be in the resultant list. The nested list with any combination of sub–lists of the above forms is allowed as the x argument. The following fragment represents source code of the ToClassicList procedure along with two typical examples of its use that well illustrates the procedure essence, and illustrates certain useful methods of procedural programming too.

In[15]:= ToCanonicList[x_ /; ListQ[x] && ListListQ[x] &&

Length[x[[1]]] == 2] :=

Module[{a, b, c, d, y, v, h, f, m, g = FromCharacterCode[6]},

v[t_] := Join[Map[{{#, t[[2]]}} &, t[[1]]]];

h[t_] := If[ListQ[t[[1]]], Map[{#, t[[2]]} &, t[[1]]], t];

f[t_] := Module[{n = 1}, Map[If[OddQ[n++], #, Nothing] &, t]];

y = Partition[Flatten[Join[Map[If[ListQ[#[[1]]], v[#], {#}] &, x]]], 2];

a = Sort[y, #1[[2]] <= #2[[2]] &];

a = Map[{#[[1]][[-1]], Flatten[#]} &, Gather[a, #1[[2]] == #2[[2]] &]];

a = Map[{#[[1]], Sort[f[#[[2]]]]} &, a]; m = Length[a];

d = Map[#[[1]] &, a]; d = Flatten[{d[[1]], Differences[d]}];

a = Map[{#[[1]], b = #[[2]]; AppendTo[b, g]} &, a];

c = MultiList[a[[1]][[2]], d[[1]] – 1];

Do[c = Quiet[ReplaceAll[c, g –> MultiList[a[[j]][[2]], d[[j]] – 1]]],

{j, 2, m}]; ReplaceAll[c, g –> Nothing]]

In[16]:= ToCanonicList[{{a*b, 2}, {b, 2}, {{z, m, n}, 12}, {c, 4}, {d, 4},

{{n, y, w, z}, 5}, {b, 2}, {m*x, 5}, {n, 5}, {t, 7}, {g, 7}, {x, 12}, {y, 12}}]

Out[16]= {{b, b, a*b, {{c, d, {n, n, w, m*x, y, z, {{g, t,

{{{{{m, n, x, y, z}}}}}}}}}}}}

In[17]:= ToCanonicList[{{a, 1}, {b, 2}, {c, 3}, {d, 4}, {g, 5}, {h, 6}}]

Out[17]= {a, {b, {c, {d, {g, {h}}}}}}

Another approach to processing of the nested lists is the procedure whose call ToCanonicalList[x] returns the result of converting of a list x to the canonical form. The fragment below represents source code of the procedure with an example of its application. Its algorithm along with above procedure contains certain useful programming methods of the nested lists.

In[78]:= ToCanonicalList[x_ /; ListQ[x]] :=

Module[{a = ElemsOnLevels2[x], b, c, d, p = 0, j},

b = Reverse[Sort[DeleteDuplicates[Map[#[[2]] &, a]]]];

c = StringRepeat["{", b[[1]]] <> StringRepeat["}", b[[1]]];

d = Map[ToString1, Table[Map[If[#[[2]] == j,

#[[1]], Nothing] &, a], {j, b}]]; d = Map[StringTake[#, {1, –2}] <>

If[p++ == 0, ", Nothing", ","] &, d];

ToExpression[StringReplacePart[c, d, Map[{#, #} &, b]]]]

In[79]:= p = {{a, b, {c, d, {m, g, {m, n}, u, n}, m, h}, x, y, z}};

In[80]:= ToCanonicalList[p]

Out[80]= {{a, b, x, y, z, {c, d, m, h, {m, g, u, n, {m, n}}}}}

A useful addition to the built-in Level function is procedure whose call DispLevels[x] returns the list of ListList type from 2–element sub-lists whose the first element defines the level of the x list, while the second element determines the level itself of the x list. The following fragment represents the source code of the DispLevels procedure with some examples of its application.

In[29]:= DispLevels[x_List] := Module[{a = {{1, x}}, b = x, k = 2},

Do[b = Level[b, {1}]; b = Flatten[Map[If[! ListQ[#], Nothing, #] &, b], 1];

AppendTo[a, {k++, b}], MaxLevel[x] – 2]; a]

In[30]:= DispLevels[{a, b, c, {m, {h, g}}, m}]

Out[30]= {{1, {a, b, c, {m, {h, g}}, m}}, {2, {m, {h, g}}}, {3, {h, g}}}

Using the DispLevels procedure, it is easy to obtain function whose call ReplaceLL[x, n, y] returns the result of replacing the n–th level of a list x with an expression y, whereas at using the word "Nothing" instead of y removes the n–th level of the x list. The function processes the erroneous situation associated with an incorrect list nesting level with output of the corresponding message. The fragment below represents the source code of the ReplaceLL function with examples of its typical application.

In[27]:= ReplaceLL[x_ /; ListQ[x], n_Integer, y_] :=

If[0 < n && n <= MaxLevel[x],

ReplaceAll[x, DispLevels[x][[n]][[2]] –> y],

Print["Level " <> ToString[n] <> " is invalid"]]

In[28]:= g := {{a, b}, c, d, {{{m, {p, t}, n}}}, x, y}

In[29]:= ReplaceLL[g, 4, {a, b}]

Out[29]= {{a, b}, c, d, {{{a, b}}}, x, y}

In[30]:= ReplaceLL[g, 5, {a, b}]

Out[30]= {{a, b}, c, d, {{{m, {a, b}, n}}}, x, y}

In[31]:= ReplaceLL[g, 5, Nothing]

Out[31]= {{a, b}, c, d, {{{m, n}}}, x, y}

In[32]:= ReplaceLL[g, 7, Nothing]

Level 7 is invalid

A natural extension of the previous ReplaceLL function is the InsDelRepList procedure, that allows to remove elements, replace, and insert an element at a set nesting level and in a set position of the specified nested level of a list. The InsDelRepList procedure calls define the following:

InsDelRepList[x, n, m, "Del"] – returns the result of deleting of the m–th element at the n–th nesting level of a list x;

InsDelRepList[x, n, m, "Ins", y] – returns the result of inserting of an expression y into m–th position at the n–th nesting level of a list x; at the same time, if m<0 then the inserting is done to the beginning of the x list, whereas at m > length of a nesting level n the inserting is done into the end of the level n;

InsDelRepList[x, n, m, "Rep", y] – returns the result of replacing of an element at m–th position at the n–th nesting level of a list x with an expression y. The procedure processes the erroneous situation that is associated with the inadmissible nesting level with return $Failed and printing of the corresponding message. The fragment represents the source code of the procedure with examples of its application.

In[1942]:= InsDelRepList[x_ /; ListQ[x], n_Integer, m_Integer,

h_ /; MemberQ[{"Ins", "Del", "Rep"}, h], y___] :=

Module[{a = DispLevels[x], b, c, d},

If[n <= 0 || n > MaxLevel[x],

Print["Level " <> ToString[n] <> " is invalid"]; $Failed,

b = a[[n]][[2]]; d = ToString[m] <> "–th element is lack";

If[h === "Del", If[Length[b] < m, Print[d]; $Failed,

ReplaceAll[x, b –> Delete[b, m]]],

If[h === "Rep", If[Length[b] < m, Print[d]; $Failed,

ReplaceAll[x, b –> ListAssign2[b, m, y]]],

If[m < 1, c = Prepend[b, y], If[m > Length[b], c = Append[b, y],

c = Insert[b, y, m]]]; ReplaceAll[x, b –> c]]]]]

In[1943]:= g47 := {{a, b}, c, d, {{{m, {p, t}, n}}}, x, y, z}

In[1944]:= InsDelRepList[g47, 5, 1, "Del"]

Out[1944]= {{a, b}, c, d, {{{m, {t}, n}}}, x, y, z}

In[1945]:= InsDelRepList[g47, 5, 2, "Ins", agn]

Out[1945]= {{a, b}, c, d, {{{m, {p, agn, t}, n}}}, x, y, z}

In[1946]:= InsDelRepList[g47, 5, 2, "Rep", avz]

Out[1946]= {{a, b}, c, d, {{{m, {p, avz}, n}}}, x, y, z}

In[1947]:= InsDelRepList[g47, 7, 2, "Rep", avz]

Level 7 is invalid

Out[1947]= $Failed

In[1948]:= InsDelRepList[g47, 5, 0, "Ins", avz]

Out[1948]= {{a, b}, c, d, {{{m, {avz, p, t}, n}}}, x, y, z}

Due to the canonical representation of lists, the procedure ExchangeLevels is of a certain interest [8,16], returning a list in canonical form on condition of exchange of its nesting levels. At that, the procedure below is an useful version of the above tool.

In[47]:= ExchangeLevels1[x_ /; ListQ[x], i_Integer, j_Integer] :=

Module[{a = DispLevels[x]},

If[MemberQ[Map[(# <= 0||# > MaxLevel[x]) &, {i, j}], True],

Print["Levels " <> ToString[{i, j}] <> " are invalid"]; $Failed,

ReplaceAll[ReplaceAll[x, a[[i]][[2]] –> a[[j]][[2]]],

a[[j]][[2]] –> a[[i]][[2]]]]]

In[48]:= ExchangeLevels1[g47, 2, 5]

Out[48]= {{a, b}, c, d, {{{m, {a, b, {{m, {p, t}, n}}}, n}}}, x, y, z}

The procedure call ExchangeLevels1[x,i,j] returns the result of exchange of the nesting levels i and j of a nested list x. A quite naturally there is the problem of exchange of the nesting levels of 2 various lists that is solved by means of the ListsExchLevels procedure. Using the method on which the last procedures are based, it is simple to program the means focused on the typical manipulations with the nested lists, in particular, removing and adding of the nesting levels. On the basis of the method used in the last procedures it is possible to do different manipulations with the nested lists including also their subsequent converting in lists of the canonical form [8-10,16]. For example, it is easy to program a function whose call PosElemOnLevel[x, n, y] returns positions of an element y on the n–th nesting level of a nested x list. The function processes the erroneous situation, associated with the inadmissible nesting level with return $Failed and print of the corresponding message. Fragment below represents the source code of the function with examples of its application.

In[3342]:= PosElemOnLevel[x_ /; ListQ[x], n_Integer, y_] :=

If[n <= 0 || n > MaxLevel[x],

Print["Level " <> ToString[n] <> " is invalid"]; $Failed,

Flatten[Position[DispLevels[x][[n]][[2]], y]]]

In[3343]:= g47 := {{a, b}, c, d, {{{m, {g, t, h, g, t, g}, n}}}, x, y, z, q}

In[3344]:= PosElemOnLevel[g47, 5, g]

Out[3344]= {1, 4, 6}

In[3345]:= PosElemOnLevel[g47, 7, g]

Level 7 is invalid

Out[3345]= $Failed

In[3346]:= NumberOnLevels[x_ /; ListQ[x]] :=

Map[{#[[1]], Map[#[[1]] &, Length[#[[2]]]]} &, DispLevels[x]]

In[3347]:= NumberOnLevels[g47]

Out[3347]= {{1, 8}, {2, 3}, {3, 1}, {4, 3}, {5, 6}}

That fragment is ended with a simple function whose call NumberOnLevels[x] returns the list of ListList type whose first sub-lists elements define the nesting levels of a list x, while the second elements define the number of elements at each of these nesting levels. Means are of interest when working with lists.

The following procedure provides the sorting of elements of a list that are located on the set nesting level. The procedure call SortListOnLevel[x, n, y] returns the result of sorting of elements of a list x on its nesting level n according to an ordering function y (an optional argument); in the absence of y argument by default the sorting according in the standard order is done.

In[7]:= SortListOnLevel[x_ /; ListQ[x], n_Integer, y___] :=

Module[{a = DispLevels[x]},

If[MemberQ[Map[(# <= 0 || # > MaxLevel[x]) &, {n}], True],

Print["Level " <> ToString[n] <> " is invalid"]; $Failed,

ReplaceAll[x, a[[n]][[2]] –> Sort[a[[n]][[2]], y]]]]

In[8]:= SortListOnLevel[g47, 5, ToCharacterCode[ToString[#1]][[1]] >

ToCharacterCode[ToString[#2]][[1]] &]

Out[8]= {{a, b}, c, d, {{{m, {t, t, h, g, g, g}, n}}}, x, y, z, q}

Unlike the standard functions Split, SplitBy the procedure call Split1[x, y] splits a list x on sub-lists consisting of elements that are located between occurrences of an element or elements of list y. If y elements don't belong to the x list, the initial x list is returned. The following fragment represents source code of the Split1 procedure with the typical example of its application.

In[7]:= Split1[x_ /; ListQ[x], y_] := Module[{a, b, c = {}, d, h, k = 1},

If[MemberQ3[x, y] || MemberQ[x, y],

a = If[ListQ[y], Sort[Flatten[Map[Position[x, #] &, y]]],

Flatten[Position[x, y]]]; h = a; If[a[[1]] != 1, PrependTo[a, 1]];

If[a[[–1]] != Length[x], AppendTo[a, Length[x]]]; d = Length[a];

While[k <= d – 1, AppendTo[c, x[[a[[k]] ;If[k == d – 1, a[[k + 1]],

a[[k + 1]] – 1]]]]; k++];

If[h[[–1]] == Length[x], AppendTo[c, {x[[–1]]}]]; c, x]]

In[8]:= Split1[{a, b, a, b, c, d, a, b, a, b, c, d, a, b, d}, {a, c, d}]

Out[8]= {{a, b}, {a, b}, {c}, {d}, {a, b}, {a, b}, {c}, {d}, {a, b, d}, {d}}

At operating with the nested lists, a procedure whose call Lie[w] returns the result of converting of a source list w into a structurally similar list whose elements are in the string format and have length equal to the maximum length of all elements in the w list at all its nesting levels is of a certain interest. The following fragment represents the source code of the procedure with an example of its typical application.

In[369]:= Lie[x_ /; ListQ[x]] := Module[{a, b = Flatten[x], c, d, f},

SetAttributes[ToString1, Listable]; a = ToString1[b];

c = StringLength[a[[1]]];

Map[If[c < Set[d, StringLength[#]], c = d, 7] &, a];

f[t_, s_] := ToString1[t] <> StringRepeat[" ",

s – StringLength[ToString1[t]]]; SetAttributes[f, Listable];

ClearAttributes[ToString1, Listable]; f[x, c]]

In[370]:= t := {{m, bbbb, g}, {{{n, m, ddd}}}, n, mmmmmm}; Lie[t]

Out[370]= {{"m ", "bbbb ", "g "}, {{{"n ", "m ", "ddd "}}},

"n ", "mmmmmm"}

Using the Lie procedure, in particular, it is possible to easily program a procedure whose call ContinuousSL[x, y] returns the result of a сontinuous sort of all the elements of the nested list x (i.e. ignoring its nesting levels) with retention its internal structure. The sorting is done either by accepted conventions for the Sort function or on a basis of the optional y argument defining the ordering function. The following fragment represents the source code of the ContinuousSL procedure with examples of its use.

In[2430]:= ContinuousSL[x_ /; ListQ[x], y___] :=

Module[{a = ToString1[Lie[x]], b, c = {}, d},

b = StringLength[a]; Do[d = StringTake[a, {j}];

If[MemberQ[{"{", "}"}, d], AppendTo[c, {d, j}], 7], {j, b}];

d = ToString1[Lie[Sort[Flatten[x], y]]];

Do[d = StringInsert[d, c[[j]][[1]], c[[j]][[2]] + 1], {j, 1, Length[c]}];

Map[ToExpression, ToExpression[d]][[1]]]

In[2431]:= g47 := {{7, 122}, 14, 8, {{{500, {0, 90}, 6}}}, {47, 77, 42}}

In[2432]:= ContinuousSL[g47, #1 > #2 &]

Out[2432]= {{500, 122}, 90, 77, {{{47, {42, 14}, 8}}}, {7, 6, 0}}

In[2433]:= L = {c, 5, 9, {{sv}, {34}, {{agn, {{{500}}}, b, {{{90}}}}}}, a};

In[2434]:= ContinuousSL[L]

Out[2434]= {5, 9, 34, {{90}, {500}, {{a, {{{agn}}}, b, {{{c}}}}}}, sv}

The technique used in the continuous sorting algorithm of nested lists can be a rather useful in solving other types of work with nested lists. This is especially true for problems involving different permutations of elements of a file Flatten[x] with next restoring the internal structure of the nested x list. At the same time, it should be borne in mind that such technique implies the standartization of list elements along the length of their elements having a maximum length. As that is done in the ContinuousSL procedure by means of use the Lie procedure.

In a number of problems of processing of simple x lists on which the call SimpleListQ[x] returns True, the next procedure is of a certain interest. The procedure call ClusterList[x] returns the nested list of ListList type whose 2–element sub-lists by the first element determine element of the simple nonempty x list whereas by the second element define its sequential number in a cluster of identical elements to which this element belongs. In the following fragment source code of the ClusterList procedure with some typical examples of its application are represented.

In[2273]:= ClusterList[x_ /; x != {} && SimpleListQ[x]] :=

Module[{a = {}, b = {}, c = Join[x, {x[[–1]]}], d = Map[{#, 1} &,

DeleteDuplicates[x]], n},

Do[n = d[[Flatten[Position[d, Flatten[Select[d, #[[1]] ==

c[[j]] &]]]][[1]]]][[2]]++;

Flatten[Select[d, #[[1]] == c[[j]] &]][[2]];

If[c[[j]] === c[[j + 1]], AppendTo[a, {c[[j]], n}], AppendTo[b, a];

AppendTo[b, {c[[j]], n}]; a = {}], {j, 1, Length[c] – 1}];

AppendTo[b, a]; b = ToString1[Select[b, # != {} &]];

b = If[SuffPref[b, "{{{", 1], b, "{" <> b];

b = ToExpression[StringReplace[b, {"{{" –> "{", "}}" –> "}"}]]]

In[2274]:= ClusterList[{1, 1, 1, 3, 3, 3, 4, 4, 5, 5, 5, 4, 1, 1, 1}]

Out[2274]= {{1, 1}, {1, 2}, {1, 3}, {3, 1}, {3, 2}, {3, 3}, {4, 1},

{4, 2}, {5, 1}, {5, 2}, {5, 3}, {4, 3}, {1, 4}, {1, 5}, {1, 6}}

In[2275]:= ClusterList[{1, 1, 1, 3, 3, t + p, 3, 4, 4, 5, a, 5, 5, 4,

1, 1, a, 1, (a + b) + c^c}]

Out[2275]= {{1, 1}, {1, 2}, {1, 3}, {3, 1}, {3, 2}, {p + t, 1}, {3, 3}, {4, 1},

{4, 2}, {5, 1}, {a, 1}, {5, 2}, {5, 3}, {4, 3}, {1, 4}, {1, 5},

{a, 2}, {1, 6}, {a + b + c^c, 1}}

In[2276]:= ClusterList[{6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6}]

Out[2276]= {{6, 1}, {6, 2}, {6, 3}, {6, 4}, {6, 5}, {6, 6}, {6, 7},

{6, 8}, {6, 9}, {6, 10}, {6, 11}, {6, 12}}

In[2277]:= ClusterList[{1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4}]

Out[2277]= {{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2},

{3, 3}, {4, 1}, {4, 2}, {4, 3}}

In order to simplify of algorithm of a number of tools that are oriented on solving of processing of the contents of the user packages, in particular, the SubListsMin procedure can be a rather useful used, and also for lists processing as a whole. The call SubListsMin[L, x, y, t] returns the sub-lists of a list L which are limited by {x, y} elements and have the minimum length; at t = "r" selection is executed from left to right, and at t = "l" from right to left. While the procedure call SubListsMin[L, x, y, t, z] with optional fifth z argument – an arbitrary expression – returns sublists without the limiting {x, y} elements. The next fragment presents source code of the procedure and examples of its use.

In[1242]:= SubListsMin[L_ /; ListQ[L], x_, y_,

t_ /; MemberQ[{"r", "l"}, t], z___] :=

Module[{a, b, c, d = {}, k = 1, j},

{a, b} = Map[Flatten, Map3[Position, L, {x, y}]];

If[a == {} || b == {} || a == {} && b == {} || L == {}, {},

b = Select[Map[If[If[t == "r", Greater, Less][#, a[[1]]], #] &, b],

! SameQ[#, Null] &];

For[k, k <= Length[a], k++, j = 1; While[j <= Length[b],

If[If[t == "r", Greater, Less][b[[j]], a[[k]]], AppendTo[d,

If[t == "r", a[[k]] ;b[[j]], b[[j]] ;a[[k]]]]; Break[]]; j++]];

d = Sort[d, Part[#1, 2] – Part[#1, 1] <= Part[#2, 2] – Part[#2, 1] &];

d = Select[d, Part[#, 2] – Part[#, 1] == Part[d[[1]], 2] – Part[d[[1]], 1] &];

d = Map[L[[#]] &, d]; d = If[{z} != {}, Map[#[[2 ;–2]] &, d], d];

If[Length[d] == 1, Flatten[d], d]]]

In[1243]:= SubListsMin[{a, b, a, c, d, q, v, d, w, j, k, d, h, f, d, h},

a, h, "r", 90]

Out[1243]= {c, d, q, v, d, w, j, k, d}

A useful addition to built-in functions Replace, ReplaceList and our tools ReplaceList1, ReplaceList2, ReplaceListCond, is a procedure whose call RepInsList[I, x, y, z, n, r] returns the result as follows, namely:

At I = "Ins" the result is inserting of an argument z (may be a list or nested list) before (r = "l") or after (r = "r") of occurrences of a sub–list y into a list x; whereas at argument I = "Rep" the result is replacement in a list x of occurrences of a sub–list y on argument z (may be a list or nested list); at that, if n is an integer, then n first replacements in the x are done, if n is Infinity, then replacements in the list x of all occurrences of a sub–list y onto argument z are done; at last, if n – an integer list then it defines sequentional numbers of the occurrences of y in the list x that are exposed to the above replacements. What is said about the argument n also fully applies to the call mode at I = "Rep". The procedure call on unacceptable arguments returns $Failed with appropriate diagnostic messages or is returned unevaluated. The following fragment represents the source code of the RepInsList procedure with some examples of its typical application.

In[25]:= RepInsList[I_ /; MemberQ[{"Rep", "Ins"}, I],

x_ /; ListQ[x], y_ /; ListQ[y], z_, n_, r___] :=

Module[{a, b, c, d, t},

{a, b, c} = Map[StringTake[ToString1[#], {2, –2}] &,

{x, y, If[ListQ[z], z, {z}]}];

If[I == "Ins", If[{r} != {}, If[r === "l", c = c <> "," <> b,

If[r === "r", c = b <> "," <> c, Return[Print["Sixth optional

argument should be \"r\" or \"l\""]; $Failed]]]];

If[IntegerQ[n] || n === Infinity, t = StringReplace[a, b –> c, n],

If[PosIntListQ[n] === True, t = StringSplit2[a, b]; d = Length[t];

t = StringJoin[Map[If[MemberQ[n, #],

StringReplace[t[[#]], b –> c], t[[#]]] &, Range[1, d]]],

Return[Print["Fourth argument n should be an integer, integer

list or Infinity"]; $Failed]]],

If[IntegerQ[n] || n === Infinity, t = StringReplace[a, b –> c, n],

If[PosIntListQ[n] === True, t = StringSplit2[a, b]; d = Length[t];

t = StringJoin[Map[If[MemberQ[n, #],

StringReplace[t[[#]], b –> c], t[[#]]] &, Range[1, d]]],

Return[Print["Fourth argument n should be an integer, integer

list or Infinity"]; $Failed]]]]; ToExpression["{" <> t <> "}"]]

In[22]:= x:= {a, b, c, 1, 2, a, b, c, 3, a, b, c, 4, 5, 6, a, b, c, h, a, b, c, g};

y:= {a, b, c}; z:= {m, n, p};

In[27]:= RepInsList["Rep", x, y, z, {1, 3}]

Out[27]= {m, n, p, 1, 2, a, b, c, 3, m, n, p, 4, 5, 6, a, b, c, h, a, b, c, g}

In[28]:= RepInsList["Rep", {aa, bb, aa, cc}, {aa}, {{m, n}}, {2}]

Out[28]= {aa, bb, {m, n}, cc}

In[30]:= RepInsList["Rep", x, y, G + S, 3]

Out[30]= {G + S, 1, 2, G + S, 3, G + S, 4, 5, 6, a, b, c, h, a, b, c, g}

In[31]:= RepInsList["Ins", {aa, bb, aa, cc, aa}, {aa}, gsv, {1, 2}, "r"]

Out[31]= {aa, gsv, bb, aa, cc, aa, gsv}

In[31]:= RepInsList["Ins", {aa, bb, aa, cc, aa}, {aa}, H, Infinity, "l"]

Out[31]= {H, aa, bb, H, aa, cc, H, aa}

A useful addition to built-in functions Split, SplitBy along with our tools Split1, Split2, SplitIntegerList, SplitList, SplitList1 is a procedure (based on the above RepInsList procedure) whose call Split2[x, y, n] returns the result of splitting of a list x by sub-lists y; in addition, if n is an integer then n first splitting of the x are done, if n is Infinity then splitting of the list x by all occurrences of a sub-list y are done; at last, if n - an integer list then it defines sequential numbers of the occurrences of y in the list x by which the list x is exposed by the above splitting. The Split2 handles the main errors with printing of diagnostic messages. Fragment represents source code of the procedure and examples of it use.

In[90]:= Split2[x_ /; ListQ[x], y_ /; ListQ[y], n_] :=

Module[{a = ToString[Unique[g]], b, c},

b = RepInsList["Rep", x, y, a, n];

If[b === x, "Second argument is invalid",

b = "{" <> ToString[b] <> "}";

c = StringReplace[b, {"{" <> a <> "," –> "{", ", " <> a <> "," –> "},{",

", " <> a <> "}" –> "}"}]; ToExpression[c]]]

In[91]:= Split2[{a, b, c, v, g, a, b, c, h, a, b, g, j, a, b}, {a, b}, Infinity]

Out[91]= {{c, v, g}, {c, h}, {g, j}}

In[92]:= Split2[{a, b, c, v, g, a, b, c, h, a, b, g, j, a, b}, {a, b}, {1, 3}]

Out[92]= {{c, v, g, a, b, c, h}, {g, j, a, b}}

While the call StringReplace6[x, y –> z, n] returns the result of replacing in a list x of substrings y onto string z; at that, if n is an integer then n first replacements are done, if n is Infinity then replacements at all occurrences of a substring y are done; at last, if n – an integer list then it defines sequention numbers of the occurrences of y for which the replacements are done. In next fragment the source code of the procedure and an example of its typical application are represented.

In[95]:= StringReplace6[x_ /; StringQ[x], y_ /; RuleQ[y], n_] :=

Module[{a = StringSplit2[x, y[[1]]], b, c},

If[a === x, "Second argument is invalid", b = Length[a]];

If[IntegerQ[n] || n === Infinity, StringReplace[x, y, n],

StringJoin[Map[If[MemberQ[n, #], StringReplace[a[[#]],

y[[1]] –> y[[2]]], a[[#]]] &, Range[1, b]]]]]

In[96]:= StringReplace6["aabbccaahgaaffaa", "aa" –> "12", {2, 4}]

Out[96]= "aabbcc12hgaaff12"

At last, the following simple enough procedures MapList ÷ MapList2 are a some kind of generalization of the built-in Map function to the lists of an arbitrary structure. The procedure call MapList[f, x] returns the result of applying of a symbol, block, function, module or pure function f (except f symbol all remaining admissible objects shall have arity 1) to each element of a list x with maintaining its internal structure. It is supposed that sub-lists of the list x have the unary nesting levels, otherwise the procedure call is returned unevaluated. The following fragment represents source code of the procedure and an example of its application.

In[7]:= MapList[f_ /; SymbolQ[f] || BlockFuncModQ[f] ||

PureFuncQ[f], x_ /; ListQ[x] &&

And1[Map[Length[DeleteDuplicates[LevelsOfList[#]]] == 1 &, x]]] :=

Module[{g}, SetAttributes[g, Listable]; g[t_] := (f) @@ {t}; Map[g, x]]

In[8]:= F[g_] := g^3; MapList[F, {{a, b, c}, {{m, n}}, {}, {{{h, p}}}}]

Out[8]= {{a^3, b^3, c^3}, {{m^3, n^3}}, {}, {{{h^3, p^3}}}}

MapList1 and MapList2 are useful versions of the above tool [8].

In a number of nested lists processing tasks, it is of interest to distribute the same elements to the nesting levels of arbitrary list. The following AllonLevels procedure solves the problem in a certain sense. Calling AllonLevels[w] procedure returns the nested list of the following format

{{a, n_a1, ..., n_ap}, {b, n_b1, ..., n_bp}, …, {z, n_z1, ..., n_zp}}

where elements {a, b, c, ..., z} represent all elements of a list w without their duplication whereas elements {n_j1, ..., n_jp} submit nesting level numbers of the w list on which the corresponding elements j{a, b, c, ..., z} are located. The following fragment represents the source code of the AllonLevels procedure along with typical examples of its application.

In[77]:= AllonLevels[x_ /; ListQ[x]] := Module[{a = {}, c = 1,

b = MaxLevel1[x]},

While[c <= b, AppendTo[a, Map[{"_" <> ToString[c], #} &,

Level[x, {c}]]]; c++];

a = Map[If[IntegerQ[ToExpression[StringTake[#[[1]],

{2, –1}]]] && ! ListQ[#[[2]]], #, Nothing] &, Flatten[a, 1]];

a = Map[Flatten, Map[Reverse, a]];

a = Gather[a, #1[[1]] == #2[[1]] &];

a = Map[DeleteDuplicates, Map[Flatten, a]];

a = Map[If[Length[#] == 2,

{#[[1]], ToExpression[StringTake[#[[2]], {2, –1}]]},

Join[{#[[1]]}, Map[ToExpression[StringTake[#, {2, –1}]] &,

#[[2 ;–1]]]]] &, a]]

In[78]:= X:= {a, b, m, c, b, {m, b, n, a, p}, {{x, y, a, n, b, c, z}}}

In[79]:= AllonLevels[X]

Out[79]= {{a, 1, 2, 3}, {b, 1, 2, 3}, {m, 1, 2}, {c, 1, 3}, {n, 2, 3},

{p, 2}, {x, 3}, {y, 3}, {z, 3}}

In[80]:= AllonLevels[{a, b, c, d, f, g, h, s, d, r, t, u, u}]

Out[80]= {{a, 1}, {b, 1}, {c, 1}, {d, 1}, {f, 1}, {g, 1}, {h, 1}, {s, 1},

{r, 1}, {t, 1}, {u, 1}}

The procedure rather successfully is used at problems solving related to the nested lists processing.

While unlike the previous procedure the CountOnLevelList procedure defines distributes the number of occurrences of list elements to the nesting levels based on their multiplicity. The procedure call CountOnLevelList[x] returns the nested list which consists of 3-element sub-lists whose the 1^st element - an element of the list x, the 2^nd – a nesting level contained it and the third – number of occurrences of the element on this nesting level.

In[2225]:= CountOnLevelList[x_ /; ListQ[x]] := Module[{c = {},

j = 1, a = DeleteDuplicates[Flatten[x]], b = MaxLevel1[x]},

While[j <= Length[a], AppendTo[c, Map[{a[[j]], #,

Count[x, a[[j]], {#}]} &, Range[1, b]]]; j++];

Map[Partition[#, 3] &, Map[Flatten, c]]]

In[2226]:= X := {a, b, m, c, b, {m, b, n, a, p}, {{x, y, b, n, b, c, z}}}

In[2227]:= CountOnLevelList[X]

Out[2227]= {{{a, 1, 1}, {a, 2, 1}, {a, 3, 0}}, {{b, 1, 2}, {b, 2, 1}, {b, 3, 2}},

{{m, 1, 1}, {m, 2, 1}, {m, 3, 0}}, {{c, 1, 1}, {c, 2, 0}, {c, 3, 1}}, {{n, 1, 0}, {n, 2, 1},

{n, 3, 1}}, {{p, 1, 0}, {p, 2, 1}, {p, 3, 0}}, {{x, 1, 0}, {x, 2, 0}, {x, 3, 1}}, {{y, 1, 0},

{y, 2, 0}, {y, 3, 1}}, {{z, 1, 0}, {z, 2, 0}, {z, 3, 1}}}

In some cases, special cases occur when sorting lists, one of which the following procedure presents. So, the procedure call SSort[x, p, t, sf] returns the list – the result of a special sorting of a list x according to a sorting function sf (optional argument; in its absence the default function Order is used), when the resulting list at t=0 starts with sorted p list elements belonging to x and at t=1 ends with p elements. If elements p are not in the x list, then the Sort[x, sf] list is returned.

In[2221]:= SSort[x_ /; ListQ[x], p_ /; ListQ[p],

t_ /; ! FreeQ[{0, 1}, t], sf___] := Module[{a = Sort[x, sf], b, c},

b = Intersection[x, c = Sort[Select[x, ! FreeQ[p, #] &], sf]];

If[b == {}, a, If[t == 0, Join[c, Complement[x, b]],

Join[Complement[x, b], c]]]]

In[2222]:= SSort[{s, f, h, d, a, m, k, h, m, p, t, y, m}, {m, s, h}, 0]

Out[2222]= {h, h, m, m, m, s, a, d, f, k, p, t, y}

In[2223]:= SSort[{s, f, h, d, a, m, k, h, m, p, t, y, m}, {m, p, t}, 1]

Out[2223]= {a, d, f, h, k, s, y, m, m, m, p, t}

In a number of tasks dealing with handling the nested lists, there is a problem of binding to each specific element in the list of its nesting level while preserving the internal structure of an initial list L. This problem is solved by the procedure whose call StructNestLevels[L] returns the nested list with internal format identical to the list L, however instead of its elements there are lists of format {x, n}, where n is the nesting level of an arbitrary element x in the initial list L. The following fragment represents the source code of the StructNestLevels procedure with typical examples of its application.

In[221]:= StructNestLevels[x_ /; ListQ[x]] :=

Module[{a, b, s, c, d = {}, j, p = MaxLevel1[x], f, k, g, u = {}},

SetAttributes[a, Listable]; SetAttributes[f, Listable];

a[t_] := {AppendTo[d, {c = Unique["a"], t}]; c}[[1]];

b = Map[a, x]; s = Flatten[b]; j = Flatten[x]; s = GenRules[s, j];

For[k = 1, k <= p, k++, g = Level[b, {k}];

g = Map[If[ListQ[#], Nothing, #] &, g];

AppendTo[u, Map[{#, k} &, g]]]; u = Partition[Flatten[u], 2];

f[t_] := Map[If[! SameQ[t, #[[1]]], Nothing, {t, #[[2]]}] &,

u][[–1]]; b = Map[f, b]; ReplaceAll[b, s]]

In[222]:= L := {a, b, {a, a, b, d^2, t, {x, y, {m, n, m + p, {s, t}, p}, z}},

m, n, {x, y, {a, b/p, c}, z}, p}

In[223]:= StructNestLevels[L]

Out[223]= {{a, 1}, {b, 1}, {{a, 2}, {a, 2}, {b, 2}, {d^2, 2}, {t, 2},

{{x, 3}, {y, 3}, {{m, 4}, {n, 4}, {m + p, 4}, {{s, 5}, {t, 5}}, {p, 4}},

{z, 3}}}, {m, 1}, {n, 1}, {{x, 2}, {y, 2}, {{a, 3}, {b/p, 3}, {c, 3}},

{z, 2}}, {p, 1}}

In[224]:= L1 := {a, b, c, d, f, g, h, r, t, v, g, h, p, x, y, z}

In[225]:= StructNestLevels[L1]

Out[225]= {{a, 1}, {b, 1}, {c, 1}, {d, 1}, {f, 1}, {g, 1}, {h, 1}, {r, 1},

{t, 1}, {v, 1}, {g, 1}, {h, 1}, {p, 1}, {x, 1}, {y, 1}, {z, 1}}

In[226]:= StructNestLevels[{{}, {{{}}}, {}}]

Out[226]= {{}, {{{}}}, {}}

The procedure below is a natural extension of the previous procedure, its call StructNestLevels1[L] returns the nested list with internal format identical to the list L, however instead of its elements there are lists of view {x, n, m} where n is a nesting level of any element x in the initial list L and m – the sequential number of x element on its nesting level n. The fragment below represents the source code of the StructNestLevels1 procedure with some typical examples of its application.

In[78]:= StructNestLevels1[x_ /; ListQ[x]] := Module[{a, b,

s, c, d = {}, j, p = MaxLevel1[x], p1, f, k, g, u = {}, h = {}},

SetAttributes[a, Listable]; SetAttributes[f, Listable];

a[t_] := {AppendTo[d, {c = Unique["a"], t}]; c}[[1]];

b = Map[a, x]; s = Flatten[b]; j = Flatten[x];

s = GenRules[s, j]; For[k = 1, k <= p, k++, g = Level[b, {k}];

g = Map[If[ListQ[#], Nothing, #] &, g];

AppendTo[u, Map[{#, k} &, g]]]; u = Partition[Flatten[u], 2];

p1 = Gather[u, #1[[2]] == #2[[2]] &];

Map[For[k = 1, k <= Length[#], k++,

AppendTo[h, Join[#[[k]], {k}]]] &, p1]; h = SortBy[h, First];

u = h; f[t_] := Map[If[! SameQ[t, #[[1]]], Nothing, Join[{t},

#[[2 ;–1]]]] &, u][[–1]]; b = Map[f, b]; ReplaceAll[b, s]]

In[79]:= L := {a, b, {a, a, b, d^2, t, {x, y, {m, n, m + p, {s, t}, p}, z}},

m, n, {x, y, {a, b/p, c}, z}, p}

In[80]:= StructNestLevels1[L]

Out[80]= {{a, 1, 1}, {b, 1, 2}, {{a, 2, 1}, {a, 2, 2}, {b, 2, 3}, {d^2, 2, 4},

{t, 2, 5}, {{x, 3, 1}, {y, 3, 2}, {{m, 4, 1}, {n, 4, 2}, {m + p, 4, 3}, {{s, 5, 1},

{t, 5, 2}}, {p, 4, 4}}, {z, 3, 3}}}, {m, 1, 3}, {n, 1, 4}, {{x, 2, 6}, {y, 2, 7},

{{a, 3, 4}, {b/p, 3, 5}, {c, 3, 6}}, {z, 2, 8}}, {p, 1, 5}}

In[81]:= StructNestLevels1[{a, b, a, c, d, f, g, g, g}]

Out[81]= {{a, 1, 1}, {b, 1, 2}, {a, 1, 3}, {c, 1, 4}, {d, 1, 5}, {f, 1, 6},

{g, 1, 7}, {g, 1, 8}, {g, 1, 9}}

The result returned by the above procedure gives represents a rather detailed internal organization of nested lists.

Based on the previous procedure, a rather useful procedure can be obtained, which ensures the application of a symbol to elements or replacement of elements of a list which are at given levels of nesting and positions in them. Thus, the procedure call StructNestLevels2[x, f, y] returns the result of applying symbol f to the elements of a list x that are at the given nesting levels and positions in them, defined by a list y of the view {{n1, m1}, …, {np, mp}}, where nj is the nesting level and mj is the position of the element on it; in addition, all invalid pairs {nj, mj} are ignored without messages output. Furthermore, the procedure call StructNestLevels2[x, f, y, z] where optional z argument – an arbitrary expression – returns the result of replacement of the elements on f instead of applying to them of symbol f. Fragment represents the source code of the StructNestLevels2 procedure with typical examples of its application.

In[478]:= StructNestLevels2[x_ /; ListQ[x], F_ /; SymbolQ[F],

y_ /; ListQ[y], z___] := Module[{a, b, c, d = {}, p = MaxLevel1[x],

j, s, p1, f, k, g, u = {}, h = {}}, SetAttributes[a, Listable];

SetAttributes[f, Listable];

a[t_] := {AppendTo[d, {c = Unique["a"], t}]; c}[[1]];

b = Map[a, x]; s = Flatten[b]; j = Flatten[x]; s = GenRules[s, j];

For[k = 1, k <= p, k++, g = Level[b, {k}];

g = Map[If[ListQ[#], Nothing, #] &, g];

AppendTo[u, Map[{#, k} &, g]]]; u = Partition[Flatten[u], 2];

p1 = Gather[u, #1[[2]] == #2[[2]] &];

Map[For[k = 1, k <= Length[#], k++,

AppendTo[h, Join[#[[k]], {k}]]] &, p1]; h = SortBy[h, First];

u = h; h = Quiet[SortBy[If[ListListQ[y], y, {y}], First]]; c = {};

For[k = 1, k <= Length[h], k++, For[j = 1, j <= Length[u], j++,

If[h[[k]] == u[[j]][[2 ;–1]],

AppendTo[c, Join[{F @@ {u[[j]][[1]]}}, u[[j]][[2 ;–1]]]],

AppendTo[c, u[[j]]]]]];

c = DeleteDuplicates[c]; c = SortBy[c, #[[2 ;1]] &];

f[t_] := Map[If[! SameQ[t, #[[1]]], Nothing,

If[MemberQ[h, #[[2 ; –1]]], If[{z} != {}, F, F @@ {t}], t]] &, u][[–1]];

b = Map[f, b]; ReplaceAll[b, s]]

In[479]:= L := {a, b, {a, a, b, d^2, t, {x, y, {m, n, m + p, {s, t}, p}, z}},

m, n, {x, y, {a, b/p, c}, z}, p}

In[480]:= StructNestLevels2[L, G, {{2, 1}, {1, 2}, {3, 5}, {5, 1}, {5, 2},

{1, 5}, {4, 1}, {4, 3}, {2, 4}}]

Out[480]= {a, G[b], {G[a], a, b, G[d^2], t, {x, y, {G[m], n, G[m + p],

{G[s], G[t]}, p}, z}}, m, n, {x, y, {a, G[b/p], c}, z}, G[p]}

Based on the previous procedure, a rather useful procedure can be obtained, that ensures deletion of elements of a list that are at set levels of nesting and positions in them. The procedure call DelElemsOfList[x, y] returns the result of deletion of the set elements of a list x that are located at the set nesting levels and positions in them, defined by a list y of the format {{n1, m1}, …, {np, mp}}, where nj is the nesting level and mj is the position of element on it; in addition, all invalid pairs {nj, mj} are ignored without messages output. The following fragment represents the source code of the DelElemsOfList procedure with typical examples of its application.

In[42]:= DelElemsOfList[x_ /; ListQ[x], y_ /; ListQ[y]] :=

Module[{a, b, c, d}, a[t_] := ToString1[t];

d[t_] := If[SameQ[Head[t], c], Nothing, t];

SetAttributes[a, Listable]; SetAttributes[d, Listable];

b = Map[a, x]; a = StructNestLevels2[b, c, y];

ToExpression[Map[d, a]]]

In[43]:= L := {a, b + d, {a, a, b, d^2, t + u, {x, y, {m, n, m + p,

{s, t}, p}, z}}, m, n, {x, y, {a, b/p, c}, z}, p}

In[44]:= DelElemsOfList[L, {{2, 1}, {1, 2}, {3, 5}, {5, 1}, {5, 2},

{1, 5}, {4, 1}, {4, 3}, {2, 4}}]

Out[44]= {a, {a, b, t + u, {x, y, {n, {}, p}, z}}, m, n, {x, y, {a, c}, z}}

In[45]:= DelElemsOfList[{a, b, c, d, f, g, h, w + g, a + b},

{{1, 1}, {1, 2}, {3, 5}, {1, 5}, {5, 2}, {1, 8}, {4, 1}, {1, 9}, {2, 4}}]

Out[45]= {c, d, g, h}

The following procedure represents a version of the above procedure basing on a little bit different algorithm. Calling the procedure DelElemsOnLevels[x, y] returns the deletion result of the set elements of a list x that are located at the set nesting levels and positions in them, defined by the sequence y of the format {n1, m1}, …, {np, mp}, where nj is the nesting level and mj is the position of element on it; in addition, all invalid pairs {nj, mj} are ignored with output of the appropriate messages.

In[3347]:= DelElemsOnLevels[x_/; ListQ[x], y__List] :=

Module[{a, b, c, d, g = FromCharacterCode[7], s = {y}},

c = Map[Flatten[{ToString1[#[[1]]], #[[2 ;3]]}] &,

ElemsOnLevels2[x]];

c = Complement[s, Map[#[[2 ;3]] &, c]];

If[c != {}, Print["Elements " <> "with conditions

/position> " <> ToString[c] <> " are invalid"], 78];

a[t_] := g <> ToString1[t] <> g; SetAttributes[a, Listable];

b = StructNestList[Map[a, x]];

c = Map[Flatten[{ToString1[#[[1]]], #[[2 ;3]]}] &,

ElemsOnLevels2[x]];

d = Map[If[MemberQ[s, #[[2 ;3]]], #, Nothing] &, c];

d = Map[Flatten[{g <> #[[1]] <> g, #[[2 ;3]]}] &, d];

d = ReplaceAll[b, Map[# –> Nothing &, d]];

a[t_] := If[StringQ[t], t, Nothing]; b = ToString[Map[a, d]];

ToExpression[StringReplace[b, {"{" <> g –> "", g <> "}" –> ""}]]]

In[3348]:= p = {{a, b, {d, z, {m, {1, 3, {{{v, g, s, d, s, h}}}, 5}, n, t},

u, c, {x, y}}}};

In[3349]:= DelElemsOnLevels[p, {3, 2}, {8, 1}, {8, 2}, {8, 3}]

Out[3349]= {{a, b, {d, {m, {1, 3, {{{d, s, h}}}, 5}, n, t}, u, c, {x, y}}}}

In[3350]:= DelElemsOnLevels[p, {3, 2}, {8, 1}, {8, 2}, {8, 3},

{3, 7}, {1, 8}]

Elements with conditions {{1, 8}, {3, 7}} are invalid

Out[3350]= {{a, b, {d, {m, {1, 3, {{{d, s, h}}}, 5}, n, t}, u, c, {x, y}}}}

In[3351]:= DelElemsOnLevels[{a, b, c, d, f, g, h}, {1, 2}, {1, 3},

{1, 5}, {1, 7}, {8, 3}, {3, 7}, {1, 8}]

Elements with conditions {{1, 8}, {3, 7}, {8, 3}}

are invalid

Out[3351]= {a, d, g}

To the above tools the DeleteInNestList closely adjoins [16].

In addition to similar means [4,8-10,16], the following fairly simple procedure differentiates the elements of an arbitrary list by its nesting levels. The procedure call ElemsOnNestLevels[x] returns the result as the nested list of the form

{{la, {a1, …, an}}, {lb, {b1, ..., bm}}, …, {lc, {c1, ..., cp}}}

that defines the distribution of elements {q1, ..., qt} on a nesting level lq; in addition, only elements in the x list that are different from the lists are considered. The following fragment presents the source code, containing certain useful programme methods, along with typical examples of its application.

In[2220]:= ElemsOnNestLevels[x_ /; ListQ[x]] :=

Module[{j = a[t_] := ToString1[t]},

SetAttributes[a, Listable]; j = Map[a, x]; ClearAll[a];

ToExpression[Map[{#, Select[Level[j, {#}], ! ListQ[#1] &]} &,

Range[1, MaxLevel1[x]]]]]

In[2221]:= L = {a, b, {a, a, b, d^2, t, {x, y, {m, n, m + p, {s, t},

p}, z}}, m, n, {x, y, {a, b/p, c}, z}, p};

In[2222]:= ElemsOnNestLevels[L]

Out[2222]= {{1, {a, b, m, n, p}}, {2, {a, a, b, d^2, t, x, y, z}},

{3, {x, y, z, a, b/p, c}}, {4, {m, n, m + p, p}}, {5, {s, t}}}

In[2223]:= ElemsOnNestLevels[{a, b, c, d, f, g, h, r, t, y, p}]

Out[2223]= {{1, {a, b, c, d, f, g, h, r, t, y, p}}}

Using the previous procedure, we get a function whose call NumbersElemOnLevels[x] returns the list {{a1, c1}, ..., {at, ct}} of ListList type, where ak is the nesting level of a list x and ck is the number of elements (not lists) at this level.

In[2227]:= NumbersElemOnLevels[x_ /; ListQ[x]] :=

Map[{#[[1]], Length[#[[2]]]} &, ElemsOnNestLevels[x]]

In[2228]:= NumbersElemOnLevels[L]

Out[2228]= {{1, 5}, {2, 8}, {3, 6}, {4, 4}, {5, 2}}

At last, based on the previous procedures, a rather useful procedure can be obtained, which ensures the application of a set of symbol to elements of a list that are at set levels of nesting and positions in them. The procedure call StructNestLevels3[x, y] returns the result of applying of the symbols {f1,…,ft} to the elements of a list x that are at the set nesting levels and positions in them, defined by the y argument of the format {f1, {n1, m1},…, {np, mp}}, …, {ft, {q1,g1}, …, {qv,gv}} where {n1,…,np,…,q1,…, qv} is the nesting levels and {m1,…,mp,…,g1,…, gv} are positions that correspond to the elements on them; in addition, all inadmissible pairs {level, position} are ignored without messages output. The fragment represents the source code of the StructNestLevels3 procedure with some typical examples of its application.

In[2230]:= StructNestLevels3[x_ /; ListQ[x], y__] :=

Module[{a, RedSetLists, b, c = {}, j = 1},

If[Length[{y}] == 1, StructNestLevels2[x, y[[1]], y[[2]]],

RedSetLists[z_] := Module[{a = z, b, c, j},

For[j = 1, j <= Length[a] – 1, j++, a = Join[a[[1 ;j]],

ReplaceAll[a[[j + 1 ;–1]], GenRules[a[[j]], b]]]];

ReplaceAll[a, b –> Nothing]]; b = Map[#[[2 ;–1]] &, {y}];

While[j <= Length[b], AppendTo[c, Map[ToString1, b[[j]]]]; j++];

b = ToExpression[RedSetLists[c]]; c = x;

For[j = 1, j <= Length[b], j++,

c = StructNestLevels2[c, {y}[[j]][[1]], b[[j]]]]; c]]

In[2231]:= StructNestLevels3[L, {G, {1, 1}, {1, 2}, {3, 5}}, {F, {5, 1},

{3, 5}, {1, 1}, {2, 1}, {2, 2}}, {S, {4, 1}, {3, 5}, {5, 1}, {3, 1}}]

Out[2231]= {G[a], G[b], {F[a], F[a], b, d^2, t, {S[x], y, {S[m], n, …}

The call format StructNestLevels4[x, y] is similar to call format StructNestLevels3[x, y], however instead of applying symbols {f1,…,ft} to the elements of a list x that are at the given nesting levels and positions in them, the replacements of elements on {f1, …, ft} are done [16].

Note, the above procedure uses the procedure with source code

In[53]:= RedSetLists[z_ /; ListQ[z]] := Module[{a = z, b, c, j},

c[t_] := ToString1[t]; SetAttributes[c, Listable]; a = Map[c, a];

For[j = 1, j <= Length[a] – 1, j++, a = Join[a[[1 ;j]],

ReplaceAll[a[[j + 1 ;–1]], GenRules[a[[j]], b]]]];

ToExpression[ReplaceAll[a, b –> Nothing]]]

In[54]:= x = {a, b, c, d}; y = {h, a, b, g, d, s, r}; z = {b, t, c, k, u, g, h};

In[55]:= RedSetLists[{x, y, z}]

Out[55]= {{a, b, c, d}, {h, g, s, r}, {t, k, u}}

Calling the procedure RedSetLists[{x, y, ..., z}] returns a list of the form {x, y*,..., z*}, that differs from the original list in that its elements (lists) are pair–wise different in the composition of the elements that make up them.

Of particular interest is the variant of list sorting, provided that the elements in the specified list positions are stationary, i.e. are not sorted. The procedure call SortFixedPoints[x,y,z] returns the result of elements of sorting in a list x whose positions are different from those specified in a list y (the sorting is based on the ordering function z that is an optional argument, and if it is absent in canonical order), while elements in positions y remain unchanged without being sorted. In addition, if z is one ordering function, then the procedure call corresponds aforesaid. Whereas, if z is two element sequence of ordering functions, then elements of the x list in positions y are sorted by function {z}[[2]], whereas others elements of the x list are sorted by function {z}[[1]]. But if Length[{z}] > 2 the call finishes abnormally with return of the corresponding message. The following fragment represents the source code of the SortFixedPoints procedure with examples of its application. The procedure is of certain interest in a number of problems of modelling.

In[9]:= SortFixedPoints[x_/; ListQ[x], y_/; ListQ[y], z___] :=

Module[{a = Quiet[x[[y]]], b, c, d, g, t = {}, u, j, p = 1, s = 1},

If[! MemberQ3[Range[1, Length[x]], y],

Return["The second argument is invalid"]];

b = ReplacePart[x, GenRules[y, c]];

d = Sort[Select[b, ! SameQ[#, c] &],

If[{z} != {}, {z}[[1]], Order]]; If[MemberQ[{0, 1}, Length[{z}]],

For[j = 1, j <= Length[b], j++, If[SameQ[b[[j]], c],

AppendTo[t, x[[j]]], AppendTo[t, d[[p++]]]]]; t,

If[Length[{z}] == 2, g = Sort[x[[y]], {z}[[2]]];

For[j = 1, j <= Length[b], j++, If[SameQ[b[[j]], c],

AppendTo[t, g[[s++]]], AppendTo[t, d[[p++]]]]]; t,

Return["Number of arguments is more than allowed"]]]]

In[10]:= SortFixedPoints[{10, 9, 8, 7, 6, 5, 4, 3, 2, 1, g, d, a, c},

{1, 3, 5, 6, 7, 10, 12, 14}]

Out[10]= {10, 2, 8, 3, 6, 5, 4, 7, 9, 1, a, d, g, c}

In[11]:= SortFixedPoints[{10, 9, 8, 7, 6, 5, 4}, {1, 3, 5, 6},

Greater, Less]

Out[11]= {5, 9, 6, 7, 8, 10, 4}

The following fragment represents another useful type of list sorting provided by the tool whose call RepPartList[x, y, z] returns a list modification x, which consists of the element–by–element exchange of list elements standing at positions defined by the integer lists y and z. In addition, the successful procedure call assumes equality in the length of the lists y and z, as well as belonging of their to the list Range[1, Length[x]], otherwise the call returns the corresponding message, presented without any detail. The following fragment represents the source code of the procedure with a typical example of its application.

In[278]:= RepPartList[x_ /; ListQ[x], y_ /; ListQ[y],

z_ /; ListQ[z]] :=

Module[{a, b, a1, a2, b1, b2, c, d, p = 1, t = Length[y]},

If[MemberQ3[d = Range[1, Length[x]], y] &&

MemberQ3[d, z] && Length[y] == Length[z], a = x[[y]]; b = x[[z]];

a1 = Map[{#, y[[p++]]} &, a]; p = 1;

b1 = Map[{#, z[[p++]]} &, b]; p = 1;

c = Map[{#, p++} &, x]; p = 1; a2 = Map[#[[1]] &, a1];

b2 = Map[{a2[[#]], z[[#]]} &, Range[1, t]];

a2 = Map[{b1[[#]][[1]], y[[#]]} &, Range[1, t]];

c = DeleteDuplicates[Join[a2, b2, c], #1[[2]] == #2[[2]] &];

Map[#[[1]] &, Sort[c, #1[[2]] <= #2[[2]] &]],

"The second and/or third arguments are invalid"]]

In[279]:= RepPartList[{10, 9, 8, 7, 6, 5, 4, 3, 2, 1, g, d, a, c},

{1, 3, 5}, {7, 9, 13}]

Out[279]= {4, 9, 2, 7, a, 5, 10, 3, 8, 1, g, d, 6, c}

The following procedure represents another useful type of list sorting provided by the tool whose call RepSortList[x,y,z,f] returns a list modification x, which consists of the element–by–element exchange of list elements standing at positions defined by the integer lists y and z provided that the elements of the list x at the positions y and z are sorted by ordering functions that are defined by the optional argument sequence f. In addition, the successful procedure call assumes equality in the length of the lists y and z, as well as belonging of their to the list Range[1, Length[x]], otherwise the call returns the appropriate message, represented without detail. The following fragment represents the source code of the RepSortList procedure with examples of its typical application.

In[2378]:= RepSortList[x_ /; ListQ[x], y_ /; ListQ[y],

z_ /; ListQ[z], f___] :=

Module[{a, a1, a2, d, b1, b2, b, c, p = 1, t = Length[y]},

If[MemberQ3[d = Range[1, Length[x]], y] &&

MemberQ3[d, z] && Length[y] == Length[z],

a = x[[y]]; b = x[[z]]; a1 = Map[{#, y[[p++]]} &, a]; p = 1;

b1 = Map[{#, z[[p++]]} &, b]; p = 1;

c = Map[{#, p++} &, x]; p = 1; a2 = Map[#[[1]] &, a1];

b2 = Map[{a2[[#]], z[[#]]} &, Range[1, t]];

a2 = Map[{b1[[#]][[1]], y[[#]]} &, Range[1, t]];

b = Sort[Map[#[[1]] &, a2], If[{f} != {}, {f}[[1]], Order]];

a2 = Map[{b[[#]], a2[[#]][[2]]} &, Range[1, t]];

b = Sort[Map[#[[1]] &, b2], If[{f} != {} &&

Length[{f}] == 2, {f}[[2]], Order]];

b2 = Map[{b[[#]], b2[[#]][[2]]} &, Range[1, t]];

c = DeleteDuplicates[Join[a2, b2, c], #1[[2]] == #2[[2]] &];

Map[#[[1]] &, Sort[c, #1[[2]] <= #2[[2]] &]],

"The second and/or third arguments are invalid"]]

In[2379]:= RepSortList[{10, 9, 8, 7, 6, 5, 4, 3, 2, 1, g, d, a, c},

{1, 3, 5}, {7, 9, 13}]

Out[2379]= {2, 9, 4, 7, a, 5, 6, 3, 8, 1, g, d, 10, c}

In[2380]:= RepSortList[{10, 9, 8, 7, 6, 5, 4, 3, 2, 1, g, d, a, c},

{1, 3, 5}, {7, 9, 13}, Less, Greater]

Out[2380]= {2, 9, 4, 7, a, 5, 10, 3, 8, 1, g, d, 6, c}

Calling the list format function PartSortList[x, y, z] returns the sort result of a list x according to the ordering function z (z – the optional argument; defaults to Order) relative to its elements at positions defined by an y integer list. Fragment represents the source code of the function with an example of its application.

In[329]:= PartSortList[x_ /; ListQ[x], y_ /; ListQ[y], z___] :=

If[MemberQ3[Range[1, Length[x]], y], {Save["7#8", $8]; $8 = x;

$8[[y]] = Sort[$8[[y]], z]; $8, ClearAll["$8"]; Get["7#8"];

DeleteFile["7#8"]}[[1]], "The second argument is invalid"]

In[330]:= PartSortList[{10, 9, 8, 7, 6, 5, 4, 3, 2, 1}, {1, 5, 7, 10}, Less]

Out[330]= {1, 9, 8, 7, 4, 5, 6, 3, 2, 10}

Calling the procedure ReplaceElemsInList[x, y] returns the result of replacing the elements a list x based on the sequence of elements of the y argument of the format {j1, l1, p1},…,{jk, lk, pk}, where jk – new elements located at lk nesting levels and on pk positions on them, whereas at using the word "Nothing" instead replacing the deletions are done. The procedure call ignores all erroneous situations conditioned by invalid nesting levels and positions of the replaced elements in the x list without output of any messages. The fragment represents the source code of the ReplaceElemsInList procedure with an example of its use.

In[1650]:= ReplaceElemsInList[x_ /; ListQ[x], y__] :=

Module[{a = ElemsOnLevels2[x], b = StructNestList[x], c, d = {},

j, h = {y}}, Do[c = Select[a, #[[2 ;3]] == h[[j]][[2 ;3]] &];

If[c != {}, AppendTo[d, c[[1]] –> h[[j]]], 78], {j, 1, Length[h]}];

ToExpression[ToInitNestList[ReplaceAll[b, d]]]]

In[1651]:= p = {{b, j, a, b, {c, d, {j, {j, {t, j}, n}, u, n}, j, h}, z, y, x}};

In[1652]:= ReplaceElemsInList[p, {mn, 2, 1}, {gs, 6, 1}, {vg, 6, 2},

{"Nothing", 5, 1}, {78, 4, 1}, {"Nothing", 2, 7}]

Out[1652]= {{mn, j, a, b, {c, d, {78, {{gs, vg}, n}, u, n}, j, h}, z, y}}

Calling the list format function Replace7[x, y, z] returns the result of replacement in a list x of its elements being on positions defined by an integer list y onto appropriate elements of a list z. In addition, both lists y and z have identical length. Fragment represents the source code of the function with an example of its typical application.

In[341]:= Replace7[x_ /; ListQ[x], y_ /; IntegerListQ[y],

z_ /; ListQ[z]] := If[MemberQ3[Range[1, Length[x]], y] &&

Length[y] == Length[z], {Save["7#8", $78]; $78 = x; $78[[y]] = z;

$78, ClearAll["$78"]; Get["7#8"]; DeleteFile["7#8"]}[[1]],

"The second and/or third argument is invalid"]

In[342]:= Replace7[{10, 9, 8, 7, 6, 5, a, 4, 3, 2, 1}, {1, 3, 5, 7, 8, 10},

{A, B, C, D, G, H}]

Out3[342]= A, 9, B, 7, C, 5, D, G, 3, H, 1}

Calling the procedure CommonElemsLists[x, y] returns the nested list of the following form

{{a1, {l11, p11, j},…,{a1, lt1, pt1, j}},…,{ac, {lc1, pc1, j},…,

{lcp, pcp, j}},…,{ad, {ld1, pd1, j},…,{ldr, pdr, j}}}

where {a1,…,ac,…,ad} – elements common for both lists x and y, l and p with double indices define nesting levels and positions on them accordingly, while j {j = 1 or j = 2} defines accessory of a sub–list {lv1, pv1, j} to the list x or y accordingly. At the same time, the procedure handles basic erroneous situations with printing of the appropriate messages. The following fragment represents the source code of the procedure with some typical examples of its application.

In[1947]:= CommonElemsLists[x_ /; ListQ[x], y_ /; ListQ[y]] :=

Module[{a = Map[ElemsOnLevels2, {x, y}], b, c, d, t},

b = Intersection[Flatten[x], Flatten[y]];

If[b == {}, "Lists do not share any elements",

c = Select[a[[1]], MemberQ[b, #[[1]]] &];

c = Map[t = Append[#, 1] &, c];

d = Select[a[[2]], MemberQ[b, #[[1]]] &];

d = Map[t = Append[#, 2] &, d];

c = Gather[Join[c, d], #1[[1]] == #2[[1]] &];

Map[{#[[1]][[1]], Map[#[[2 ;4]] &, #1]} &, c]]]

In[1948]:= p = {{a, b, {c, d, {m, {m, {{t}}, n}, u, n}, m, h}, x, y}};

In[1949]:= CommonElemsLists[p, {a, m, x, t}]

Out[1949]= {{a, {{2, 1, 1}, {1, 1, 2}}}, {m, {{4, 1, 1}, {5, 1, 1}, {3, 3, 1},

{1, 2, 2}}}, {t, {{7, 1, 1}, {1, 4, 2}}}, {x, {{2, 3, 1}, {1, 3, 2}}}}

In[1950]:= CommonElemsLists[p, {sv, gs, z, w, v}]

Out[1950]= "Lists do not share any elements"

Calling the list format function RandSortList[w] returns the result of a pseudorandom sorting of elements of a list w. The used method can be useful in other appendices. The fragment represents the source code of the function with an example of its typical application.

In[345]:= RandSortList[w_ /; ListQ[w]] := {Save["7#8", $78];

$78 = w; $78[[RandomSample[Range[1, Length[w]]]]] = w;

$78, ClearAll["$78"]; Get["7#8"]; DeleteFile["7#8"]}[[1]]

In[346]:= RandSortList[{a, b, c, d, g, h}]

Out[346]= {a, g, d, c, h, b}

When programming algorithms in the form of list–format functions, it is sometimes necessary to use global variables as local variables, with the preliminary saving of their previous values in the file so that before exiting the function they can be restored. This is the approach used in the previous examples of functions. Meanwhile, for these purposes, you can use as local variables the variables generated by the built-in Unique function, whose call, in particular, Unique["w"] generates a new symbol with a name of the form wnnn. To ensure efficient application of the Unique function, a procedure is programmed whose call PrevUnique[x, n] returns the name of a variable that is the n–th from the end in the sequence generated by the Unique[x] calls of the current session; in addition, the x argument is a name in the string format. The fragment represents the source code of the function with a typical example of its application.

In[7]:= PrevUnique[x_ /; StringQ[x] && SymbolQ[x], n_Integer] :=

Module[{a = ToString[Unique[x]], b},

b = StringCases[a, b_ ~~ DigitCharacter ..];

b = StringTake[b[[1]], {2, –1}];

ToExpression[x <> ToString[FromDigits[b] – n]]]

In[8]:= {Unique["j"], Unique["j"], Unique["j"], PrevUnique["j", 3]}

Out[8]= {j11, j12, j13, j11}

In view of the above, the following list format function can be considered as a rather useful extension of the built-in Unique function. Calling the function ToUnique[x, y] returns the two–element list whose first element defines the name of a variable in the string format generated by the Unique[x] function while the second element specifies a value y assigned to the variable. The fragment below represents the source code of the function with some typical examples of its application

In[1942]:= ToUnique[x_ /; StringQ[x] && SymbolQ[x], y_] :=

{Write["#78", Unique[x]]; Close["#78"]; {ToString[Get["#78"]],

ToExpression[ToString[Get["#78"]] <> "=" <> ToString1[y]]},

DeleteFile["#78"]}[[1]]

In[1943]:= ToUnique["agn", m*Sin[p] + n*Cos[t]]

Out[1943]= {"agn73", n*Cos[t] + m*Sin[p]}

In[1944]:= agn73

Out[1944]= n*Cos[t] + m*Sin[p]

In[1945]:= P[x_, y_] := ToExpression[ToUnique["a", m][[1]]]*x +

ToExpression[ToUnique["b", n][[1]]]*y

In[1946]:= P[78, 73]

Out[1946]= 78*m + 73*n

In[1947]:= P[x_, y_] := Module[{a = m, b = n}, a*x + b*y]

In[1948]:= P[78, 73]

Out[1948]= 78*m + 73*n

Some above procedures is of certain interest at programming of a number of problems of modelling in particular of cellular automata dynamics.

A lot of the additional tools expanding the Mathematica, in particular, for effective programming of a number of problems of manipulation with the list structures of various organization are presented a quite in details in [1-16]. Being additional tools for work with lists – basic structures in Mathematica – they are rather useful in a number of applications of various purposes. A number of means were already considered, while others will be considered along with tools that are directly not associated with lists, but quite accepted for work with separate formats of lists. Many means represented here and in [1-16], arose in the process of programming a lot of applications as tools wearing a rather mass character. Some of them use techniques which are rather useful in the practical programming in Mathematica. In particular, the above relates to a computer study of the Cellular Automata (CA; Homogeneous Structures – HS).

3.7. Attributes of procedures and functions

In addition to definition of procedures (blocks, modules) and functions, the Mathematica system allows you to define general properties that are inherent in them. The system provides a set of attributes that can be used to specify different properties of procedures and functions irrespectively their definitions. At the same time, attributes can be assigned not only to functions and procedures, but also to symbols.

Attributes[G] – returns the attributes ascribed to a G symbol;

Attributes[G] = {a1, a2,…} – the function call sets the attributes

a1, a2,… for a G symbol;

Attributes[G] = {} – the function call sets G to have no attributes;

SetAttributes[G, atr] – the function call adds the atr attribute to

the attributes of a G symbol;

ClearAttributes[G, attr] – the function call removes a attr from

the attributes of a G symbol.

Here are the most used attributes with their descriptions:

Orderless – attribute defines orderless, commutative tools (their

formal arguments are sorted into standard order);

Flat – attribute defines flat, associative tools (their arguments are

"flattened out");

Listable – this attribute defines automatical element-by-element

application of a symbol over lists that appear as arguments;

Constant – this attribute defines that all derivatives of a symbol

are zero;

NumericFunction – the attribute defines that a tool is assumed to have a numerical value when its arguments are numeric quantities;

Protected – this attribute defines that values of a symbol cannot

be changed;

Locked – this attribute defines that attributes of a symbol cannot

be changed;

ReadProtected – attribute defines that values of a symbol cannot

be read;

Temporary – the attribute defines that a symbol is local variable,

removed when no longer used;

Stub – this attribute defines that a symbol needs is automatically

called if it is ever explicitly input.

The remaining admissible attributes can be found in detail in [2-6,9,10,15], here some useful examples of the application of the attributes presented above are given. For example, we can use the attribute Flat to specify that a function is "flat", so that nested invocations are automatically flattened, and it behaves as if it were associative:

In[2222]:= SetAttributes[F, Flat]

In[2223]:= F[F[x, y, z], F[p, F[m, n], t]]

Out[2223]= F[x, y, z, p, m, n, t]

The Listable attribute is of particular interest at operating with list structures, its use well illustrates the following rather simple example:

In[2224]:= Attributes[ToString]

Out[2224]= {Protected}

In[2225]:= L = {a, b, {c, d}, {{m, p, n}}};

In[2226]:= ToString[L]

Out[2226]= "{a, b, {c, d}, {{m, p, n}}}"

In[2227]:= SetAttributes[ToString, Listable]

In[2228]:= ToString[L]

Out[2228]= {"a", "b", {"c", "d"}, {{"m", "p", "n"}}}

As an useful and illustrative example of application of the Listable attribute, we give an example of a procedure whose call ListExtension[x,y,z,t] returns an extension of the list x elements (including nested lists) onto value z on the left, which for a pure function y return True if the optional argument t – an arbitrary expression – is absent and on the right otherwise. The fragment represents the source code of the ListExtension procedure with some examples of its application.

In[2240]:= ListExtension[x_ /; ListQ[x], y_ /; PureFuncQ[y], z_,

t___] := Module[{a}, SetAttributes[a, Listable];

a[w_] := If[y @@ {w} && {t} == {}, Flatten[Prepend[{w}, z]],

If[y @@ {w} && {t} != {}, Flatten[Append[{w}, z]], w]]; Map[a, x]]

In[2241]:= ListExpansion[{1, 2, {3, 4, {5, 6, 7, {8, 9, 10}}}},

EvenQ[#] &, {a, b, c}]

Out[2241]= {1, {a, b, c, 2}, {3, {a, b, c, 4}, {5, {a, b, c, 6}, 7,

{{a, b, c, 8}, 9, {a, b, c, 10}}}}}

In[2242]:= ListExpansion[{1, 2, {3, 4, {5, 6, 7, {8, 9, 10}}}},

OddQ[#] &, GS, 8]

Out[2242]= {{1, GS}, 2, {{3, GS}, 4, {{5, GS}, 6, {7, GS},

{8, {9, GS}, 10}}}}

The ListExpansion procedure is of particular interest when dealing primarily with nested lists. The procedure easily allows a number of generalizations, in particular, deleting or replacing list elements on which a pure y function returns True.

Specifically, calling the procedure ListHand[x, y, z] returns the result of applying of a pure function or a symbol z to the elements of the nested x list which on a pure function y return True. The fragment represents the source code of the procedure ListHand with typical examples of its application.

In[77]:= ListHand[x_ /; ListQ[x], y_ /; PureFuncQ[y],

z_ /; PureFuncQ[z] || SymbolQ[z]] :=

Module[{a}, SetAttributes[a, Listable];

a[w_] := If[y @@ {w}, z @@ {w}, w]; Map[a, x]]

In[78]:= ListHand[{2, {{3, 4}, {5, {6, 7, {8, 9}}}}}, EvenQ[#] &, #^3 &]

Out[78]= {8, {{3, 64}, {5, {216, 7, {512, 9}}}}}

In[79]:= s[j_] := j^3; ListHand[{2, {{3}, {5, {6, {9}}}}}, OddQ[#] &, s]

Out[79]= {2, {{27}, {125, {6, {729}}}}}

Note that the reception used in the procedures on the basis of a[w] function or its appropriate analogues with the Listable attribute allows to efficiently processes elements of nested lists, in addition saving their internal structure.

In this regard, a procedure whose call ListableS[F, x] returns the result of in one step, element–by–element application of a F symbol to a list x that acts as the second argument (wherein, after the procedure call, the attributes of the F symbol remain the same as before the procedure call):

In[2229]:= ListableS[F_Symbol, x_] := Module[{s, t,

g = Attributes[F]}, If[MemberQ[g, Listable], s = F[x]; t = 1,

SetAttributes[F, Listable]; s = F[x]];

If[t === 1, s, ClearAttributes[F, Listable]; s]]

In[2230]:= ListableS[ToString, L]

Out[2230]= {"a", "b", {"c", "d"}, {{"m", "p", "n"}}}

The ListableS procedure can be represented by equivalent function ListableS1 of the list format as follows:

In[2231]:= ListableS1[F_Symbol, x_] := {Save["#", {g, s, t}],

g = Attributes[F], If[MemberQ[g, Listable], s = F[x]; t = 1,

SetAttributes[F, Listable]; s = F[x]]; If[t === 1, s,

ClearAttributes[F, Listable]; s], Get["#"], DeleteFile["#"]}[[3]]

In[2232]:= {g, s, t} = {42, 47, 67}; ListableS1[F, L]

Out[2232]= {F[a], F[b], {F[c], F[d]}, {{F[m], F[p], F[n]}}}

In[2233]:= {g, s, t}

Out[2233]= {42, 47, 67}

The Temporary attribute determines that a symbol has local character, removed when no longer used. For instance, the local procedure variables have the Temporary attribute.

In[90]:= A[x__] := Module[{a = 77}, {a*x, Attributes[a]}]

In[91]:= A[5]

Out[91]= {385, {Temporary}}

In[92]:= A1[x_] := Module[{a = 7}, a = {a*x, Attributes[a], a};

ClearAttributes[a, Temporary]; a]

In[93]:= A1[5]

Out[93]= {35, {Temporary}, 7}

By managing the Temporary attribute, it is possible make the values of the local variables of the procedures available in the current session outside of these procedures, as the previous example illustrates. At that, you can both save the Temporary attribute for them or selectively cancel it.

The ReadProtected attribute prevents expression associated with a W symbol from being seen. In addition, this expression cannot be obtained not only by the function call Definition[W], but also be written to a file, for example, by the function Save, for example:

In[2129]:= G[x_, y_, z_] := x^2 + y^2 + z^2;

In[2130]:= SetAttributes[G, ReadProtected]

In[2131]:= Definition[G]

Out[2131]= Attributes[G] = {ReadProtected}

In[2132]:= Save["###", G]

In[2133]:= Read["###"]

Out[2133]= {ReadProtected}

This attribute is typically used to hide the values of certain program objects, especially procedure and function definitions. While this attribute is cancelled by means of the ClearAttributes function, for example:

In[2141]:= ClearAttributes[G, ReadProtected]

In[2142]:= Definition[G]

Out[2142]= G[x_, y_, z_] := x^2 + y^2 + z^2

The Constant attribute defines all derivatives of a g symbol to be zero, for example:

In[2555]:= SetAttributes[g, Constant]

In[2556]:= g[x_] := a*x^9 + b*x^10 + c

In[2557]:= Definition[g]

Out[2557]= Attributes[g] = {Constant} + c

g[x_] := a*x^9 + b*x^10

In[2558]:= Map[D[g, {x, #}] &, Range[10]]

Out[2558]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

The Locked attribute prohibits attributes change of a symbol:

In[2654]:= ClearAttributes[Vs, {Locked}]

… Attributes: Symbol Vs is locked.

In[2655]:= Attributes[Vs]

Out[2655]= {Listable, Locked}

In[2656]:= Clear[Vs]; Attributes[Vs]

Out[2656]= {Listable, Locked}

In[2657]:= ClearAll[Vs]; Attributes[Vs]

… ClearAll: Symbol Vs is locked.

Out[2657]= {Listable, Locked}

In[2658]:= ClearAll[Gs]

…ClearAll: Symbol Gs is Protected.

In[2659]:= Attributes[Gs]

Out[2659]= {Protected}

Note, that unlike the declaration by the Mathematica the function ClearAll to override a symbol attributes, this method does not work if the symbol has {Protected, Locked} attributes, as illustrated by the fragment above. The Mathematica claims, the call ClearAll[g1, g2,…] clears all values, definitions, defaults, attributes, messages, associated with {g1, g2,…} symbols whereas in reality it is not quite so. In addition, the Remove function also does not affect symbols with the {Protected, Locked} attributes. On the other hand, a symbol x with attributes {Protected, Locked} is saved in a w file by means of the functions call Save[w, x] and Write[w, Definition[x]] with saving of all attributes ascribed to the x symbol. At the same time, attributes assigned to a symbol x are relevant only within the current session, being lost in the new session. Calling Unlocked[f] deletes the current document nb by returning the updated nb document (by replacing the Locked attribute with the Stub attribute) and saving it to a nb–file f.

In[2759]:= Unlocked[fn_] := Module[{a = NotebookGet[]},

a = ReplaceAll[a, "Locked" –> "Stub"];

NotebookClose[]; NotebookSave[a, fn]]

In[2760]:= Unlocked["C:\\Temp\\Exp.nb"]

The NumericFunction attribute of a procedure or function F indicates that F[x1, x2, x3, …] should be considered a numerical quantity whenever all its arguments are numerical quantities:

In[2858]:= Attributes[Sin]

Out[2858]= {Listable, NumericFunction, Protected}

The Orderless attribute of a procedure or function indicates that its arguments are automatically sorted in canonical order. This property is accounted for in pattern matching.

In[2953]:= SetAttributes[F, Orderless]

In[2954]:= F[n, m, p, 3, t, 7, a]

Out[2954]= F[3, 7, a, m, n, p, t]

In[2955]:= F[z_, y_, x_] := x + y + z

In[2956]:= Definition[F]

Out[2956]= Attributes[F] = {Orderless}

F[x_, y_, z_] := x + y + z

In[2957]:= SetAttributes[g, Orderless]

In[2958]:= L = {n, m, u, 3, t, 7, a, c, u, d, s, j, a, p}; v := g @@ L;

Map[Part[v, #] &, Range[Length[L]]]

Out[2958]= {3, 7, a, a, c, d, j, m, n, p, s, t, u, u}

The last example of the previous fragment illustrates using of the Orderless attribute to sort lists in lexicographical order.

3.8. Additional tools expanding the built–in Mathematica functions, or its software as a whole

The string and list structures are of the most important in the Mathematica, they both are considered in the previous two chapters in the context of means, additional to the built-in tools, without regard to a large number of the standard functions of processing of structures of this type. Quite naturally, here it isn't possible to consider all range of standard functions of this type, sending the interested reader to the reference information on the Mathematica or to the appropriate numerous publications. It is possible to find many of these publications on the website http://www.wolfram.com/books. Having presented the means expanding the built–in Mathematica software in the context of processing of string and list structures in the present item we will represent the means expanding the Mathematica that are oriented on processing of other types of objects. Above all, we will present some tools of bit–by–bit processing of symbols.

The Bits procedure significantly uses function BinaryListQ, providing a number of useful operations during of operating with symbols. On the tuple of factual arguments <x, p>, where x – a 1–symbol string (character) and p – an integer in the range 0 .. 8, the call Bits[x, p] returns binary representation of the x in the form of the list, if p = 0, and p–th bit of such representation of a symbol x otherwise. While on a tuple of actual arguments <x, p>, where x – a nonempty binary list of length no more than 8 and p = 0, the procedure call returns a symbol corresponding to the set binary x list; in other cases the call Bits[x,p] is returned as unevaluated. The fragment below represents source code of the procedure Bits along with an example of its application.

In[7]:= Bits[x_, P_ /; IntegerQ[P]] := Module[{a, k},

If[StringQ[x] && StringLength[x] == 1, If[1 <= P <= 8,

PadLeft[IntegerDigits[ToCharacterCode[x][[1]], 2], 8][[P]],

If[P == 0, PadLeft[IntegerDigits[ToCharacterCode[x][[1]], 2], 8],

Defer[Bits[x, P]]]],

If[BinaryListQ[x] && 1 <= Length[Flatten[x]] <= 8, a = Length[x];

FromCharacterCode[Sum[x[[k]]*2^(a – k), {k, a}]], Defer[Bits[x, P]]]]]

In[8]:= Map9[Bits, {"A", "A", {1, 0, 0, 0, 0, 0, 1}, "A", {1, 1, 1, 1, 0, 1}},

{0, 2, 0, 9, 0}]

Out[8]= {{0, 1, 0, 0, 0, 0, 0, 1}, 1, "A", Bits["A", 9], "="}

If the previous Bits procedure provides a simple enough processing of symbols, the two procedures BitSet1 and BitGet1 provide the expanded bit–by–bit information processing like our Maple procedures [16]. In the Maple we created a number of procedures (Bit, Bit1, xNB, xbyte1, xbyte) that provide bit–by–bit information processing [22-28,42]; the Mathematica has similar means too, in particular, the call BitSet[n, k] returns the result of setting of 1 to the k-th position of binary representation of an integer n. The procedure call BitSet1[n,p] returns result of setting to positions of the binary representation of an integer n that are determined by the first elements of sub-lists of a nested p list, {0|1} – values; in addition, in a case of non-nested p list a value replacement only in a single position of n integer is made. Procedures BitSet1 and BitGet1 are included in our package [16].

It should be noted that the BitSet1 procedure functionally expands both the standard function BitSet, and BitClear of the Mathematica while the procedure BitGet1 functionally extands the standard functions BitGet and BitLength of the system. The call BitGet1[n, p] returns the list of bits in the positions of binary representation of an integer n that are defined by a p list; at the same time, in a case of an integer p the bit in position p of binary representation of n integer is returned. While the procedure call BitGet1[x, n, p] through a symbol x in addition returns number of bits in the binary representation of an integer n. The above means rather wide are used at bit–by–bit processing.

In the Mathematica the transformation rules are generally determined by the Rule function, whose call Rule[a, b] returns the transformation rule in the form a –> b. These rules are used in transformations of expressions by the functions ReplaceAll, Replace, ReplaceRepeated, StringReplace, StringReplaceList, ReplacePart, StringCases, that use either one rule, or their list as simple list, and a list of ListList type. For dynamic generation of such rules the GenRules procedure can be useful, whose call GenRules[x, y] depending on a type of its arguments returns a single rule or list of rules; the procedure call GenRules[x, y, z] with the third optional z argument – any expression – returns the list with single transformation rule or the nested list of ListList type. Depending on the coding format, the call returns result in the following formats, namely:

(1) GenRules[{x, y, z,…}, a]  {x –> a, y –> a, z –> a,…}

(2) GenRules[{x, y, z,…}, a, h]  {{x –> a}, {y –> a}, {z –> a},…}

(3) GenRules[{x, y, z,…}, {a, b, c,…}]  {x –> a, y –> b, z –> c,…}

(4) GenRules[{x, y, z,…}, {a, b, c,…}, h]  {{x –> a}, {y –> b}, {z –> c},…}

(5) GenRules[x, {a, b, c,…}]  {x –> a}

(6) GenRules[x, {a, b, c,…}, h]  {x –> a}

(7) GenRules[x, a]  {x –> a}

(8) GenRules[x, a, h]  {x –> a}

The GenRules procedure is useful, in particular, when in a procedure or function it is necessary to dynamically generate the transformation rules depending on conditions. The following fragment represents source code of the GenRules procedure and the most typical examples of its application onto all above cases of coding of the procedure call.

In[7]:= GenRules[x_, y_, z___] := Module[{a, b = Flatten[{x}],

c = Flatten[If[ListQ /@ {x, y} == {True, False},

PadLeft[{}, Length[x], y], {y}]]},

a = Min[Length /@ {b, c}]; b = Map9[Rule, b[[1 ; a]], c[[1 ; a]]];

If[{z} == {}, b, b = List /@ b; If[Length[b] == 1, Flatten[b], b]]]

In[8]:= {GenRules[{x, y, z}, {a, b, c}], GenRules[x, {a, b, c}],

GenRules[{x, y}, {a, b, c}], GenRules[x, a], GenRules[{x, y}, a]}

Out[8]= {{x  a, y  b, z  c}, {x  a}, {x  a, y  b}, {x  a},

{x  a, y  a}}

In[9]:= {GenRules[{x, y, z}, {a, b, c}, 7], GenRules[x, {a, b, c}, 42], GenRules[x, a, 6], GenRules[{x, y}, {a,b,c}, 7], GenRules[{x, y}, a, 72]}

Out[9]= {{{x  a}, {y  b}, {z  c}}, {x  a}, {x  a},

{{x  a}, {y  b}}, {{x  a}, {y  a}}}

Modifications GenRules1÷GenRules3 of the above procedure are quite useful in many applications [1-15]. A number of tools of the MathToolBox package essentially use means of so–called GenRules–group [16].

Considering the importance of the map function, starting with Maple 10, the option `inplace`, admissible only at use of the map function with rectangular rtable–objects at renewing these objects in situ was defined. While for objects of other type this mechanism isn't supported as certain examples from [18,21–42] illustrate. For the purpose of disposal of this shortcoming was offered a quite simple MapInSitu procedure [42]. Along with it the similar tools for Mathematica in the form of two functions MapInSitu, MapInSitu1 and the MapInSitu2 procedure have been offered. With mechanisms used by the Maple procedure MapInSitu and Math–tools MapInSitu÷MapInSitu2 the reader can familiarize in [6-16,27,42]. Tools of the MapInSitu-group for both systems are characterized by the prerequisite, the second argument at their call points out on an identifier in the string format to which a certain value has been ascribed earlier and that is updated "in situ" after its processing by the {map|Map} tool. Anyway, the calls of the tools return Map[x, y] as a result.

The following procedure makes grouping of expressions that are set by x argument according to their types defined by the Head2 procedure; in addition, a separate expression or their list is coded as a x argument. The procedure call GroupNames[x] returns simple list or nested list whose elements are lists whose first element – an object type according to the Head2 procedure, whereas the others – expressions of this type. The next fragment represents source code of the GroupNames procedure with an example of its typical application.

In[7]:= GroupNames[x_] := Module[{a = If[ListQ[x], x, {x}], c,

d, p, t, b = {{"Null", "Null"}}, k = 1},

For[k, k <= Length[a], k++, c = a[[k]]; d = Head2[c];

t = Flatten[Select[b, #[[1]] === d &]];

If[t == {} && (d === Symbol && Attributes[c] === {Temporary}),

AppendTo[b, {Temporary, c}],

If[t == {}, AppendTo[b, {d, c}], p = Flatten[Position[b, t]][[1]];

AppendTo[b[[p]], c]]]]; b = b[[2 ;-1]];

b = Gather[b, #1[[1]] == #2[[1]] &];

b = Map[DeleteDuplicates[Flatten[#]] &, b];

If[Length[b] == 1, Flatten[b], b]]

In[8]:= GroupNames[{Sin, Cos, ProcQ, Locals2, 90, Map1, StrStr,

72/77, Avz, Nvalue1, a + b, Avz, 77, Agn, If, Vsv}]

Out[8]= {{"System", Sin, Cos, If}, {"Module", ProcQ, Locals2,

Nvalue1, Agn}, {Integer, 90, 77}, {"Function", Map1, StrStr},

{Rational, 72/77}, {Temporary, Avz, Vsv}, {Plus, a + b}}

The procedure call RemoveNames[] without any arguments provides removal from the current Mathematica session of the names, whose types are other than procedures and functions, and whose definitions have been evaluated in the current session; moreover, the names are removed so that aren't recognized by the Mathematica any more. The procedure call RemoveNames[] along with removal of the above names from the current session returns the nested 2-element list whose first element defines the list of names of procedures, while the second element – the list of names of functions whose definitions have been evaluated in the current session. The following fragment represents source code of the RemoveNames procedure and an example of its use.

In[7]:= RemoveNames[x___] := Module[{a = Select[Names["`*"],

ToString[Definition[#]] != "Null" &], b},

ToExpression["Remove[" <> StringTake[ToString[MinusList[a,

Select[a, ProcQ[#] || ! SameQ[ToString[Quiet[DefFunc[#]]],

"Null"]||Quiet[Check[QFunction[#], False]] &]]], {2, -2}] <> "]"];

b = Select[a, ProcQ[#] &]; {b, MinusList[a, b]}]

In[8]:= RemoveNames[]

Out[8]= {{"Art", "Kr", "Rans"}, {"Ian"}}

The above procedure is an useful tool in some appendices connected with cleaning of the working Mathematica field from definitions of the non–used symbols. The procedure confirmed a certain efficiency in management of random access memory.

In the context of use of the built-in functions Nest and Map for definition of new pure functions on a basis of the available ones, we can offer the procedure as an useful generalization of the built-in Map function, whose call Mapp[F, E, x] returns the result of application of a function/procedure F to an expression E with transfer to it of the factual arguments defined by a tuple of x expressions which can be empty. In a case of the empty x tuple the identity Map[F, E] ≡ Mapp[F, E] takes place. As formal arguments of the built-in function Map[w, g] act the f name of a procedure/function while as the second argument – an arbitrary g expression, to whose operands of the 1^st level the w is applied. The following fragment represents the source code of the Mapp procedure and a typical example of its application.

In[308]:= Mapp[f_ /; ProcQ[f] || SysFuncQ[f] ||

SymbolQ[f], Ex_, x___] :=

Module[{a = Level[Ex, 1], b = {x}, c = {}, h, g = Head[Ex], k = 1},

If[b == {}, Map[f, Ex], h = Length[a]; For[k, k <= h, k++,

AppendTo[c, ToString[f] <> "[" <> ToString1[a[[k]]] <> ", " <>

ListStrToStr[Map[ToString1, {x}]] <> "]"]];

g @@ Map[ToExpression, c]]]

In[309]:= Mapp[F, {a, b, c}, x, y, z]

Out[309]= {F[a, "F", y, z], F[b, "F", y, z], F[c, "F", y, z]}

In[310]:= Mapp[ProcQ, {Sin, ProcQ, Mapp, Definition2, StrStr}]

Out[310]= {False, True, True, True, False}

Note that algorithm of the Mapp procedure is based on the following relation, namely:

Map[F, Exp] ≡ Head[Exp][Sequences[Map[F, Level[Exp, 1]]]]

whose rightness follows from definition of the standard Level, Map, Head functions and our Sequences procedure considered in [6,8-16]. The relation can be used and at realization of cyclic structures for the solution of problems of other directionality, including programming on a basis of use of mechanism of the pure functions. While the Mapp procedure in the definite cases rather significantly simplifies programming of various tasks. The Listable attribute for a function f determines that it will be automatically applied to elements of the list which acts as its argument. Such approach can be used rather successfully in a number of cases of programming of blocks, functions, modules. In this context a rather simple Mapp1 procedure is of interest, whose call Mapp1[x,y] unlike the call Map[x,y] returns the result of applying of a block, function or module x to all elements of y list, regardless of their location on list levels. Meanwhile, for a number of functions and expressions the Listable attribute does not work, and in this case the system provides special functions Map and Thread that in a certain relation can quite be referred to the structural means that provide application of functions to parts of expressions. In this regard we programmed a group of a rather simple and at the same time useful enough procedures and functions, so-called Map–group which rather significantly expand the built–in Map function. The Map–group consists of 25 tools Map1÷Map25 different complexity levels. These tools are considered in detail in [5-14], they included in our package MathToolBox [16]. The above means from Map–group rather significantly expand functionality of the built–in Map function. In a number of cases the means allow to simplify programming of procedures and functions.

An useful enough modification of standard Map function is the MapEx function whose call returns the list of application of symbol f to those elements of a list that satisfy a testing function w (block, function, pure function, or module from one argument) or whose positions are defined by the list of positive integers w. The built-in function call MapAll[f, exp] applies f symbol to all subexpressions of an exp expression which are on all its levels. Whereas the procedure call MapToExpr[f, y] returns the result of application of a symbol f or their list only to all symbols of y expression. Furthermore, from the list f are excluded symbols entering into the y expression. For example, in a case if f = {G, S} then applying of f to a symbol w gives G[S[w]]. At absence of acceptable f symbols the call returns the initial y expression. The call MapToExpr[f,y,z] with the 3^rd optional z argument – a symbol or their list – allows to exclude the z symbols from applying to them f symbols. In certain cases the procedure is a rather useful means at expressions processing.

The procedure MapListAll substantially adjoins the above Map–group. The call MapAt[F, expr, n] of the standard MapAt function provides applying F to the elements of an expression expr at its positions n. While the procedure call MapListAll[F, j] returns the result of applying F to all elements of a list j. At that, a simple analogue of the above procedure, basing on mechanism of temporary change of attributes for factual F argument in the call MapListAll1[F,j] solves the problem similarly to MapListAll procedure. On the list j of a kind {{…{{…}}…}} both procedures returns {{…{{F[]}}…}}. The following useful procedure MapExpr also adjoins the above Map–group.

Calling the MapExpr[x, y] procedure returns the result of applying a symbol x to each variable of an expression y, while calling the MapExpr[x, y, z] through optional argument z – an undefined symbol – additionally returns the list of all variables of the y expression in the string format. The following fragment represents the source code of the MapExpr procedure along with examples of its typical application.

In[1942]:= MapExpr[x_ /; SymbolQ[x], y_, z___] :=

Module[{a, b, c, d, p, g},

If[SameQ[Head[y], Symbol], ToExpression[ToString[x @@ {y}]],

a = ToString[MathMLForm[y]];

b = StringSplit[a, "\n" –> ""]; g = ToString[x];

b = Map[If[# == "", Nothing, StringTrim[#]] &, b];

c = Map[If[! StringFreeQ[#, ""] &&

! StringFreeQ[#, ""],

StringReplace[#, {"" –> "", "" –> ""}], Nothing] &, b];

c = Sort[Map[If[Quiet[SystemQ[Name1[#]]], Nothing, #] &, c]];

If[{z} != {} && ! HowAct[z], z = DeleteDuplicates[c], 77];

p = Flatten[Map[{"[" <> # <> "," –> "[" <> g <> "[" <> # <> "],",

" " <> # <> "]" –> " " <> g <> "[" <> # <> "]]",

" " <> # <> "," –> " " <> g <> "[" <> # <> "],",

"," <> # <> " " –> ", " <> g <> "[" <> # <> "] ",

"[" <> # <> "]" –> "[" <> g <> "[" <> # <> "]]"} &, c]];

ToExpression[StringReplace[ToString[FullForm[y]], p]]]]

In[1943]:= exp = (a*y/(m + n) + b^z)/(cg – d*Log[h])*Sin[d + bc];

In[1944]:= MapExpr["F", exp, avz]

Out[1944]= ((F[b]^F[z] + (F[a]*F[y])/(F[m] + F[n]))*Sin[F[bc] +

F[d]])/(F[cg] – F[d]*Log[F[h]])

In[1945]:= avz

Out[1945]= {"a", "b", "bc", "cg", "d", "h", "m", "n", "y", "z"}

In[1946]:= MapExpr[F, {a, b, {m, n, p}, g, s, Sin[y], c}, vsv]

Out[1946]= {F[a], F[b], {F[m], F[n], F[p]}, F[g], F[s], Sin[F[y]], F[c]}

In[1947]:= vsv

Out[1947]= {"a", "b", "c", "g", "m", "n", "p", "s", "y"}

In[1948]:= MapExpr[F, {a/b + c/d, n, {{h}}, Sin[p]}, avz]

Out[1948]= {F[a]/F[b] + F[c]/F[d], F[n], {{F[h]}}, Sin[F[p]]}

In[1949]:= avz

Out[1949]= {"a", "b", "c", "d", "h", "n", "p"}

In the context of structural analysis of the expressions, the VarsExpr procedure whose call VarsExpr[x] returns the five–element list, the 1^st element of which defines the list of standard tool names in the string format, the 2^nd element defines the list of the user tools names in the string format, the 3^rd element defines the list of the built–in operators in the string format, the fourth element defines the list of the variables in the string format, at last the firth element defines the list of the arithmetic operations in the string format which compose an expression x is a rather useful tool. The following fragment represents the source code of the VarsExpr procedure with an example of its typical use.

In[21]:= VarsExpr[x_] := Module[{a, b, d = {}, d1 = {}, d2 = {},

d3 = {}, d4 = {}, f},

a = PressStr[StringReplace[ToString[MathMLForm[x]], "\n" –> ""]];

b = Map[SubsString[a, #, 7] &,

{{"", "</mi>"}, {"", "</mo>"}}];

If[! StringFreeQ[a, "</msqrt>"], AppendTo[b, "sqrt"], 7];

b = Sort[DeleteDuplicates[Flatten[b]]];

Map[If[Quiet[SystemQ[Name1[#]]], AppendTo[d, Name1[#]],

If[Quiet[BlockFuncModQ[Name1[#]]], AppendTo[d1, #],

If[PrefixQ["&#", #], AppendTo[d2, FromCharacterCode[

ToExpression[StringTrim[SubsString[#, {"&#", ";"}, 7][[1]]]]]],

If[SymbolQ[#], AppendTo[d3, #], AppendTo[d4, #]]]]] &, b];

Map[Select[#, ! NullStrQ[#] &] &, {d, d1, d2, d3, d4}]]

In[22]:= VarsExpr[(f[1/b] –> Cos[a + Sqrt[c + d]])/(Tan[1/b] <–> 1/c^2)];

Out[22]= {{"Cos", "Sqrt", "Tan"}, {"f"}, {"<–>", "–>"},

{"a", "b", "c", "d"}, {"(", ")", "+"}}}

Meantime, one point should be highlighted. In some cases, in the Mathematica's procedures and functions, the result may contain a special space character that has Unicode F360 or appears as "\[InvisibleSpace]" in text format. That character by default is not visible on display, but is interpreted on input as an ordinary space. The space character can be used as an invisible separator between variables that are being multiplied together, as in A*B. In connection with the above character, when programming a number of tasks, it is necessary to distinguish between ordinary strings and strings containing the above space characters. This task is solved by means of function whose call StringIsItFreeQ[x] returns True if x don't contain the characters "\[InvisibleSpace]" and "\[InvisibleTimes]", and False otherwise. The next fragment represents the source codes of some useful tools, processing the strings that contain the above characters along with a number of exemplifying examples.

In[7]:= {a = "aaaaa", b = "a\[InvisibleSpace]\[InvisibleSpace]aa

\[InvisibleSpace]aa"}

Out[7]= {"aaaaa", "aaaaa"}

In[8]:= Map[StringLength[#] &, {a, b}]

Out[8]= {5, 8}

In[9]:= Map[StringFreeQ[#, "\[InvisibleSpace]"] &, {a, b}]

Out[9]= {True, False}

In[10]:= Map[Characters[#] &, {a, b}]

Out[10]= {{"a", "a", "a", "a", "a"}, {"a", "\[InvisibleSpace]",

"\[InvisibleSpace]", "a", "a", "\[InvisibleSpace]", "a", "a"}}

In[11]:= StringIsItFreeQ[x_] := If[StringQ[x], StringFreeQ[x,

{"\[InvisibleSpace]", "\[InvisibleTimes]"}], False]

In[12]:= StringIsFreeQ[b]

Out[12]= False

In[13]:= NullStrQ[x_ /; StringQ[x]] := SameQ[x, ""] ||

MemberQ[{"\[InvisibleSpace]", "\[InvisibleTimes]", "⁡"},

DeleteDuplicates[Characters[x]][[1]]]

In[14]:= NullStrQ["\[InvisibleSpace]\[InvisibleTimes]"]

Out[14]= True

In[15]:= PackStr[x_String] := StringReplace[x, {" " –> "",

"\[InvisibleSpace]" –> "", "\[InvisibleTimes]" –> ""}]

In[16]:= a = "aaa\[InvisibleTimes]\[InvisibleSpace]a aa \

\[InvisibleSpace] aa\[InvisibleTimes]aa";

In[17]:= StringLength[a]

Out[17]= 16

In[18]:= a = PackStr[a]

Out[18]= "aaaaaaaaaa"

In[19]:= StringLength[a]

Out[19]= 10

The function call PackStr[x] returns the result of packing of a string x by deleting from it of all characters occurrences of the form {" ", "\[InvisibleSpace]", "\[InvisibleTimes]"} to character "". The function call NullStrQ[x] returns True if a string x has the form "" or is composed of an arbitrary number of characters {"\[InvisibleSpace]", "\[InvisibleTimes]"} in any combinations, and False otherwise. The examples represented above, certain features that should be taken into account when programming tasks involving the using of such symbols are illustrated quite clearly. Indeed, Indeed, processing of visually identical strings and containing characters with Unicode F360,760, 765, 2082 and like them, and not containing them, by means of some built–in means can cause unpredictable and even unexpected results, which requires increased attention.

Similarly to Maple, Mathematica doesn't give any possibility to test inadmissibility of all actual arguments in block, function or module in a point of its call, interrupting its call already on the first inadmissible actual argument. Meanwhile, in view of importance of revealing of all inadmissible actual arguments only for one pass, the three procedures group TestArgsTypes ÷ TestArgsTypes2 solving this important enough problem and presented in our MathToolBox package [16] has been created. In contrast to the above TestArgsTypes procedures that provide the differentiated testing of actual arguments received by a tested object for their admissibility, a rather simple function TrueCallQ provides testing of the call correctness of an object of type {block, function, module} as a whole; the function call TrueCallQ[x, args] returns True if the call x[args] is correct, and False otherwise.

To the TestArgsTypes ÷ TestArgsTypes2 and TrueCallQ in a certain degree the TestArgsCall and TestFactArgs procedures adjoins whose call allows to allocate definitions of a function, block or module on which the call on the set actual arguments is quite correct. The source codes of the above tools are in [16].

In[7]:= TestActualArgs[x_/; BlockFuncModQ[x], a_/; ListQ[a]] :=

Module[{a = ArgsBFM1[x], b = 0, c, d, d1, t},

c = Map[{b++; If[#[[2]] == "Arbitrary", True,

ToExpression["ReplaceAll[" <> "Hold[" <> #[[2]] <> "]" <>

"," <> #[[1]] <> "–>" <> ToString1[args[[b]]] <> "]"]]} &, a];

d = Map[ReleaseHold, Flatten[c]]; d1 = AllTrue[d, TrueQ];

If[d1, d1, b = 1; t = {};

Map[If[#, b++, AppendTo[t, b++]] &, d]; {False, t}]]

In[8]:= G[x_, y_ /; StringQ[y], z__ /; IntegerQ[z],

t_ /; RationalQ[t]] := {x, y, z, t}

In[9]:= TestActualArgs[G, {a, bs, 77.78, 90}]

Out[9]= {False, {2, 3, 4}}

In[10]:= TestActualArgs[G, {a, "avz", 90, 77/78}]

Out[10]= True

The call TestActualArgs[x, y] of the above procedure returns True if actual arguments tuple y for an object x is admissible and {False, d} where d – the positions list of non-admissible actual y arguments. Supposed that only test functions and patterns types {"_","__","___"} are used in the x header (block, function, module).

In a number of cases exists an urgent need of determination of the program objects along with their types activated directly in the current session. That problem is solved by means of the TypeActObj procedure whose call TypeActObj[] without any arguments returns the nested list, whose sub–lists in the string format by the first element contain types of active objects of the current session, while other elements of the sub–lists are names corresponding to this type; in addition, the types recognized by the Mathematica, or types defined by us, in particular, {Function, Procedure} can protrude as a type. In some sense the TypeActObj procedure supplements the ObjType procedure [8,16]. The next fragment represents the source code of the TypeActObj with a typical example of its application.

In[2227]:= TypeActObj[] := Module[{a = Names["`*"], b = {}, c,

d, h, p, k = 1},

Quiet[For[k, k <= Length[a], k++, h = a[[k]]; c = ToExpression[h];

p = StringJoin["0", ToString[Head[c]]];

If[! StringFreeQ[h, "$"] || (p === Symbol &&

"Definition"[c] === Null), Continue[],

b = Append[b, {h, If[ProcQ[c], "0Procedure",

If[Head1[c] === Function, "0Function", p]]}]]]];

a = Quiet[Gather1[Select[b, ! #1[[2]] === Symbol &], 2]];

a = ToExpression[StringReplace[ToString1[DeleteDuplicates /@

Sort /@ Flatten /@ a], "AladjevProcedures`TypeActObj`" –> ""]];

Append[{}, Do[a[[k]][[1]] = StringTake[a[[k]][[1]], {2, –1}],

{k, Length[a]}]]; a]

In[2228]:= TypeActObj[]

Out[2228]= {{"Symbol", "args", "bc", "Bc", "cg", "Cg", "dims", "h",

"tan", "u", "v"}, {"List", "avz", "avz4"}, {"Times", "exp"},

{"Function", "F", "G"}, {"Procedure", "Sv", "VarsExpr"}}

The TypeActObj procedure is quite interesting in terms of analyzing of software and other objects activated in the current Mathematica session. The above procedure uses the Gather1 and Gather2 procedures; the second a little extend the function Gather1, being an useful in a number of applications. The call Gather1[L, n] returns the nested list formed on a basis of a list L of the ListList type by means of grouping of sub-lists of the L by its n–th element. Whereas the call Gather2[L] returns either the simple list, or the list of ListList type that defines only multiple elements of the L list with their multiplicities. At absence of the multiple elements in L the procedure call returns the empty list, i.e. {} [7,8,16]. In general, we have programmed much software which extend, complement, or correct some aspects of the built-in Mathematica functions, or its software as a whole. Because of the aforesaid the above means along with numerous means represented by us in [8-16] can be considered as useful enough tools at processing of the objects of various types in the current session at procedural–functional programming of the various problems. At that, in our opinion, the detailed analysis of their source codes can be a rather effective remedy on the path of the deeper mastering of programming in Mathematica. Experience of holding of the master classes on Mathematica system with all evidence confirms expediency of joint use of both standard tools, and the user tools that have been created in the course of programming of various appendices.

The analysis of the user blocks, functions or modules for content in their definition of calls of other user tools having the usage, is of certain interest. Calling the CallsInDef[x] procedure returns the list of names in the string format of the user tools whose calls are used in the definition of the user block, function or module x. While calling the CallsInDef[x, y] procedure with the 2^nd optional y argument – an arbitrary expression – returns the list of ListList type whose sub–lists determines names of the above user tools and their types (Block, Function, Module). The following fragment represents the source code of the procedure with typical examples of its application.

In[7]:= CallsInDef[x_ /; BlockFuncModQ[x], y___] :=

Module[{a, b = "7#8"}, Save[b, x];

a = StringCases[ReadString[b],

Shortest["/:" ~~ __ ~~ "::usage"]]; DeleteFile[b];

a = Map[StringReplace[StringReplace[#, "/: " –> "", 1],

"::usage" –> ""] &, a];

a = Map[If[SyntaxQ[#], #, Nothing] &, a][[2 ;–1]];

Map[If[{y} != {}, {#, TypeBFM[#]}, #] &, a]]

In[8]:= CallsInDef[Definition2]

Out[8]= {"SymbolQ", "HowAct", "ProtectedQ",

"Attributes1", "ReduceLists", "ToString1", "StrDelEnds",

"SuffPref", "SystemQ", "Mapp", "ProcQ", "UnevaluatedQ",

"ListStrToStr", "ClearAllAttributes", "BlockFuncModQ",

"HeadPF", "Map3", "MinusList", "SysFuncQ", "Contexts1"}

In[9]:= CallsInDef[HowAct, gs]

Out[9]= {{"Attributes1","Function"},{"ToString1","Module"},

{"StrDelEnds", "Module"}, {"ProtectedQ","Function"},

{"ReduceLists", "Module"}, {"SuffPref", "Module"}}

The following procedure, in a certain sense, complements the above procedure by allowing you to obtain call point names not having the usages. Calling the CallsWusage[x] procedure returns the list of names in the string format of the user tools without usage whose calls are used in the definition of the user block, function or module x. While calling the CallsWusage[x,y] procedure with the second optional y argument – an expression – returns the list of ListList type whose sub–lists defines names of the above user tools together with their types (Block, Function, Module). For calls of objects whose names in x are local, or not defined in the current session, calling CallsWusage[x, y] returns $Failed as the type. Fragment below represents the source code of the CallsWusage procedure with examples of its application.

In[42]:= gs[x_] := x^2 + 78; Va[x_] := x^3 + Sin[x+73]

In[43]:= Vz[x_] := Module[{}, Va[x]; a = ToString1[a]; gs[x]]

In[44]:= CallsWusage[x_ /; BlockFuncModQ[x], y___] :=

Module[{a, b, c, d, g, v, j, k, s = {}},

a = StringReplace[Definition2[x][[1]], "[[" –> ""];

b = StringPosition[a, "["]; b = DeleteDuplicates[Flatten[b]];

g = Join[Map[ToString, Range[0, 9]], d = Alphabet[],

Map[ToUpperCase, d]];

For[j = 1, j <= Length[b], j++, c = "";

For[k = b[[j]] – 1, k >= 1, k––,

If[!FreeQ[g, v = StringTake[a, {k}]], c = v <> c; Continue[],

AppendTo[s, c]; Break[]]]]; s = DeleteDuplicates[s];

s = Map[If[UsageQ[#], Nothing, #] &, s];

If[{y} != {}, Map[{#, TypeBFM[#]} &, s], s]]

In[45]:= CallsWusage[Vz]

Out[45]= {"gs", "Va"}

In[46]:= CallsWusage[Vz, gs]

Out[46]= {{"Va", "Function"}, {"gs", "Function"}}

As an auxiliary tool, the procedure uses a simple function whose call UsageQ[x] returns True, if x has an usage and False otherwise. The source code of the function is presented below.

In[110]:= UsageQ[x_] := If[SymbolQ[x],

! SuffixQ["`" <> ToString[x], Information[x, "Usage"]], False]

In[111]:= UsageQ[agn]

Out[111]= False

In[112]:= UsageQ["ProcQ"]

Out[112]= True

In contrast CallsInDef the following procedure gives more detailed analysis of the calls constituting a block, function, or module. Calling procedure CallsInBFM[x] returns a nested list of format {{tp1, {a11,...}},...,{tpn, {an1,...}}} where tpj define the type of calls defined by a list {aj1,...} whereas the call CallsInBFM[x, y] with the 2^nd optional argument y – an indefinite symbol – throu it additionally returns a list of format {{a11,…,a1p, n1}, {{a21,…,a2t, n2},…} where {{aj1,…,ajp} – the names of call points whereas nj – their number in full definition of a block, function or module x. Return value $Failed as tpj indicates that its corresponding call names are local or undefined in the current session. Fragment below represents the source code of the CallsInBFM procedure with typical examples of its application.

In[2242]:= CallsInBFM[x_ /; BlockFuncModQ[x], y___] :=

Module[{a, b = "7#8", c, d, k, j, v, f, g, s = {}, t}, Save[b, x];

a = ReadFullFile[b]; DeleteFile[b];

a = StringReplace[a, "[[" –> ""]; b = StringPosition[a, "["];

b = DeleteDuplicates[Flatten[b]];

g = Join[Map[ToString, Range[0, 9]], d = Alphabet[],

Map[ToUpperCase, d]];

For[j = 1, j <= Length[b], j++, c = "";

For[k = b[[j]] – 1, k >= 1, k––,

If[! FreeQ[g, v = StringTake[a, {k}]], c = v <> c; Continue[],

AppendTo[s, c]; Break[]]]];

g = Sort[Complement[DeleteDuplicates[s], {TypeBFM[x]}]];

v = Quiet[Map[If[# === "" || SameQ[Definition[#], Null],

Nothing, If[SystemQ[#], {#, "System"}, {#, TypeBFM[#]}]] &, g]];

v = Map[Flatten, Gather[v, #1[[2]] == #2[[2]] &]];

f[t_] := {t[[2]], t[[Range[1, Length[t], 2]]]}; v = Map[f[#] &, v];

If[{y} != {} && ! HowAct[y], y = Map[{#, Count[s, #]} &, g]; v, v]]

In[2243]:= CallsInBFM[StrStr]

Out[2243]= {{"System", {"If", "StringJoin", "StringQ", "ToString"}},

{"Function", {"StrStr"}}}

In[2243]:= CallsInBFM[StrStr, g73]; g73

Out[2243]= {{"If", 1}, {"StringJoin", 1}, {"StringQ", 1}, {"StrStr", 1},

{"ToString", 1}}

On one very significant point, it makes sense to stop our attention. When working with Mathematica, it is quite real that there is a lack of memory in its workspace. In such cases, it is quite real that the evaluation of new procedures and functions in the current session becomes impossible in the absence of any diagnosis by the system. Let us give one fairly simple example in this regard. In the current session an attempt of the evaluation of the definition of a rather simple procedure whose source code is presented below had been done.

In[3347]:= NamesInMfile[x_ /; FileExistsQ[x] &&

FileExtension[x] == "m"] := Module[{a = ReadString[x], b, c},

b = StringReplace[a, {"\n" –> "", "\r" –> ""}];

c = StringCases[b, Shortest["(*Begin[\"`" ~~ __ ~~ "`\"]*)"]];

a = Map[StringTake[#, {11, –6}] &, c];

b = Map[If[SymbolQ[#], #, Nothing] &, a]]

Calling NamesInMfile[x] procedure returns a list of names in string format that are contained in a file x of m–format which contains the user package in the format described above. At the same time, the names of the package means, without assigned usages, are also returned. In order to resolve the given situation without leaving the current session, we can propose an useful approach that is based on the list format of the procedures and functions. We represent a similar approach on the basis of the previous procedure. The procedure, together with the factual argument – a m–format file – is presented in the form of a list, whose elements separator is semicolons, as the next fragment easily illustrates.

In[3377]:= {x = "C:\\Mathematica\\MathToolBox.m";

b = StringReplace[ReadString[x], {"\n" –> "", "\r" –> ""}];

b = StringCases[b, Shortest["(*Begin[\"`" ~~ __ ~~ "`\"]*)"]];

b = Map[StringTake[#, {11, –6}] &, b];

Sort[Map[If[SymbolQ[#], #, Nothing] &, b]]}[[1]]

Out[3377]:= {"AcNb", "ActBFM", …, "$Version1", "$Version2"}

In[3378]:= Length[%]

Out[3378]= 1421

The procedure call NamesInMBfiles[x] returns a list of the names in string format that are contained in a file x of format "nb" which contains the user package in the format described above. In addition, the names of the package means, without assigned usages, are also returned. The fragment represents the source code of the procedure with an example of its application.

In[3388]:= NamesInNBfiles[x_ /; FileExistsQ[x]] :=

Module[{c = FileConvert[x –> "#.m"], b, h, g = {}, s = {}, k},

a = ReadString[c]; DeleteFile[c];

b = "`" <> StringReplace[a, {"RowBox[{" –> "", "\n" –> "",

"\r" –> ""}] <> "`";

c = Map[#[[1]] &, StringPosition[b, "`"]];

For[k = 1, k <= Length[c] – 1, k++,

h = StringTake[b, {c[[k]], c[[k + 1]]}];

h = StringReplace[h, "`" –> ""];

If[SymbolQ[h], AppendTo[g, h], 7]]; g = DeleteDuplicates[g];

Map[If[NameQ[#] && ! SystemQ[#] && StringLength[#] > 1,

AppendTo[s, #], 7] &, g]; Sort[s]]

In[3389]:= NamesInNBfiles["C:\\math\\mathtoolbox.nb"]

Out[3389]= {"AcNb", "ActBFM", "ActBFMuserQ", …, "$Version2"}

In[3390]:= Length[%]

Out[3390]= 1418

The above mentioned procedure for solving a task uses the FileConvert function, whose call FileConvert[F1 –> "File2.ext"] converts the contents of source file F1 to the format defined by the extension ext and saves the result to the file "File2.ext". In addition, files can be converted from formats supported by the function Import to formats supported by Export. Given that the internal content of a certain file plays a rather significant role in solving programming problems related to the structure of the file, the question of converting the file into a more acceptable format seems to be quite important. In particular, the above procedure uses the FileConvert function for converting of the nb–files to m–files, facilitating the task solution. This approach has been used in a number of applications [8-15].

3.9. Certain additional tools of expressions processing in the Mathematica software

Analogously to the most software systems the Mathematica understands everything with what it manipulates as expression (graphics, lists, formulas, strings, modules, functions, numbers, etc.). And although all these expressions, at first sight, significantly differ, Mathematica presents them in so–called full format. And only the postponed assignment ":=" has no full format. For the purpose of definition of the heading of an e expression (the type defining it) the built-in Head function is used whose call Head[e] returns the heading of an e expression:

In[3331]:= G := S; Z[x_] := Block[{}, x]; F[x_] := x; M[x_] := x;

M[x_, y_] := x + y; Map[Head, {ProcQ, Sin, 77, a + b, # &, G, Z,

Function[{x}, x], x*y, x^y, F, M}]

Out[3332]= {Symbol, Symbol, Integer, Plus, Function, Symbol,

Symbol, Function, Times, Power, Symbol, Symbol}

For more exact definition of headings we created an useful modification of built-in Head function in the form of the Head1 procedure expanding its opportunities, for example, it concerns testing of blocks, system functions, the user functions, modules, etc. Thus, the call Head1[x] returns the heading of an expression x in the context {Block, Function, Module, System, Symbol, Head[x], PureFunction}. In addition, on the objects of the same name that have one name with several definitions the procedure call will return $Failed. The fragment below represents source code of the Head1 procedure with examples of its use comparatively with the Head function as it is illustrated by certain examples of the following fragment on which the functional distinctions of both tools are rather evident.

In[3333]:= Head1[x_] := Module[{a = PureDefinition[x]},

If[ListQ[a], $Failed, If[a === "System", System,

If[BlockQ[x], Block, If[ModuleQ2[x], Module,

f[PureFuncQ[x], PureFunction, If[Quiet[Check[FunctionQ[x],

False]], Function, Head[x]]]]]]]]

In[3334]:= G := S; Z[x_] := Block[{}, x]; F[x_] := x; M[x_] := x;

M[x_, y_] := x + y; Map[Head, {ProcQ, Sin, 6, a + b, # &, G, Z,

Function[{x}, x], x*y, x^y, F, M}]

Out[3334]= {Symbol, Symbol, Integer, Plus, Function, Symbol,

Symbol, Function, Times, Power, Symbol, Symbol}

In[3335]:= Map[Head1, {ProcQ, Sin, 6, a + b, # &, G, Z,

Function[{x}, x], x*y, x^y, F, M}]

Out[3335]= {Module, System, Integer, Plus, PureFunction,

Symbol, Block, PureFunction, Times, Power, Function, $Failed}

The Head1 procedure has a quite certain meaning for more exact (relatively to system standard) classification of expressions according to their headings. On many expressions the calls of Head1 procedure and Head function are identical, whereas on certain their calls significantly differ. In [2,4-6,8-16], two useful modifications of the Head1: Head2 and Head3 are represented. The expression concept is the important unifying principle in the system having identical internal structure which allows to confine a rather small amount of the basic operations. Meantime, despite identical basic structure of expressions, Mathematica provides a set of various functions for work as with expression, and its separate components.

Tools of testing of correctness of expressions. The system Mathematica has a number of the means providing the testing of correctness of syntax of expressions among which only two functions are available to the user, namely:

SyntaxQ["x"] – returns True, if x – a syntactic correct expression; otherwise False is returned;

SyntaxLength["x"] – returns the quantity w of symbols, since the beginning of a "x" string that determines syntactic correct expression StringTake["x", {1, w}]; in a case w > StringLength["x"] the system declares that whole "x" string is correct, demanding continuation.

In our opinion, it isn't very conveniently in case of software processing of the expressions. Therefore extension in the form of SyntaxLength1 procedure is of certain interest.

In[4447]:= SyntaxLength1[x_ /; StringQ[x], y___] :=

Module[{a = "", b = 1, d, h = {}, c = StringLength[x]},

While[b <= c, d = SyntaxQ[a = a <> StringTake[x, {b}]];

If[d, AppendTo[h, StringTrim2[a, {"+", "–", " "}, 3]]]; b++];

h = DeleteDuplicates[h];

If[{y} != {} && ! HowAct[{y}[[1]]], {y} = {h}];

If[h == {}, 0, StringLength[h[[–1]]]]]

In[4448]:= {SyntaxLength1["d[a[1]] + b[2]", g], g}

Out[4448]= {14, {"d", "d[a[1]]", "d[a[1]] + b", "d[a[1]] + b[2]"}}

The call SyntaxLength1[x] returns the maximum number w of position in a string x such that the next condition is carried out ToExpression[StringTake[x, {1, w}]] – a syntactically correct expression, otherwise 0 (zero) is will be returned; whereas the call SyntaxLength1[x, y] through the 2^nd optional y argument – an indefinite variable – additionally returns the list of substrings of the string x representing correct expressions.

Unlike the SyntaxLength1 procedure, the procedure call SyntaxLength2[j] gathers correct sub-expressions extracted from a string j into lists of expressions identical on the length. The function call ExtrVarsOfStr1[j] returns the sorted list of possible symbols in string format successfully extracted from a string j; if symbols are absent, the empty list – {} is returned. Unlike the ExtrVarsOfStr procedure the ExtrVarsOfStr1 function provides the more exhaustive extraction of all possible symbols from the strings. Whereas, the procedure call FactualVarsStr1[j] returns the list of all factual variables extracted from a string j; source code of the last tool with an example are represented below.

In[6]:= FactualVarsStr1[x_ /; StringQ[x]] := Module[{b = "",

c = {}, h, t, k, a = StringLength[x] + 1, d = x <> "[", j = 1},

While[j <= a, For[k = j, k <= a, k++,

If[SymbolQ[t = StringTake[d, {k, k}]] || t == "`", b = b <> t,

If[! MemberQ[{"", "`"}, b],

AppendTo[c, {b, If[MemberQ[CNames["AladjevProcedures`"],

b], "AladjevProcedures`", h = If[ContextQ[b], "contexts",

Quiet[ToExpression["Context[" <> b <> "]"]]];

If[h == "AladjevProcedures`", $Context, h]]}], 7]; b = ""]; j = k + 1]];

c = Map[DeleteDuplicates,

Map[Flatten, Gather[c, #1[[2]] == #2[[2]] &]]];

c = Map[Sort[#, ContextQ[#1] &] &, c];

c = Map[If[MemberQ[#, "contexts"] &&

! MemberQ[#, "Global`"], Flatten[{"contexts",

ReplaceAll[#, "contexts" –> Nothing]}], #] &, c];

c = Map[Flatten[{#[[1]], Sort[#[[2 ;–1]]]}] &, c];

Map[If[#[[1]] != "Global`" && MemberQ[#, "contexts"],

Flatten[{"contexts", ReplaceAll[#, "contexts" –> Nothing]}], #] &, c]]

In[7]:= FactualVarsStr1[PureDefinition[StrStr]]

Out[7]= {{"AladjevProcedures`", "StrStr"}, {"Global`", "x"},

{"System`", "If", "StringJoin", "StringQ", "ToString"}}

The procedure call FactualVarsStr1[x] on the whole returns the nested list whose sub-lists have contexts as the first element whereas the others define the symbols extracted from a string x which have these contexts. If the string x contains contexts then "contexts" element precedes their sorted tuple in sub-list. The above means is useful enough in practical programming in the Mathematica and their codes contain a number of rather useful programming receptions [16].

Call SyntaxQ[x] of standard Mathematica function returns True if a string x corresponds to syntactically correct input for a single expression, and returns False otherwise. In addition, the function tests only syntax of expression, ignoring its semantics at its evaluation. Whereas the tools ExpressionQ, ExprQ, Expr1Q along with the syntax provide testing of expressions regarding their semantic correctness. The calls of all these tools on a string x returns True if string x contains a syntactically and semantically correct single expression, and False otherwise. Fragment below represents source code of the ExprQ1 function and examples of its use in comparison with the built–in SyntaxQ function.

In[2214]:= Expr1Q[x_ /; StringQ[x]] :=

Quiet[Check[SameQ[ToExpression[x], ToExpression[x]], False]]

In[2215]:= {z, a, c, d} = {500, 90, 77, 42}; SyntaxQ["z=(c+d)*a/0"]

Out[2215]= True

In[2216]:= Expr1Q["z=(c+d)*a/0"]

Out[2216]= False

In[2217]:= {SyntaxQ["77 = 72"], Expr1Q["77 = 72"]}

Out[2217]= {True, False}

Thus, the function call Expr1Q[x] returns False only if the call ToExpression[x] causes an erroneous situation for a string x that contains some expression.

It is possible to apply the Cases function for definition of the expressions coinciding with a given pattern, however not all problems of expressions comparison with patterns are solved by the standard means. For solution of the problem in broader aspect the EquExprPatt procedure can be rather useful whose call EquExprPatt[x, p] returns True if expression x corresponds to the given p pattern, and False otherwise. The fragment below presents source code of the procedure and an example of its use.

In[1395]:= EquExprPatt[x_, y_ /; ExprPatternQ[y]] :=

Module[{c, d = {}, j, t, v = {}, k = 1, p, g = {}, s = {},

a = Map[FullForm, Map[Expand, {x, y}]],

b = Mapp[MinusList, Map[OP, Map[Expand, {x, y}]],

{FullForm}], z = SetAttributes[ToString, Listable], w},

{b, c} = ToString[{b, a}]; p = StringPosition[c[[2]], {"Pattern[",

"Blank[]]"}];

While[k = 2*k–1; k <= Length[p], AppendTo[d, StringTake[c[[2]],

{p[[k]][[1]], p[[k + 1]][[2]]}]]; k++]; {t, k} = {ToExpression[d], 1};

While[k <= Length[t], AppendTo[v, StringJoin[ToString[Op[t[[k]]]]]];

k++]; v = ToString[v]; v = Map13[Rule, {d, v}];

v = StringReplace[c[[2]], v]; b = Quiet[Mapp[Select, b, ! SystemQ[#]||

BlockFuncModQ[ToString[#]] &]];

{b, k, j} = {ToString[b], 1, 1};

While[k <= Length[b[[1]]], z = b[[1]][[k]];

AppendTo[g, {"[" <> z <> "," –> "[w", " " <> z <> "," –> " w",

"[" <> z <> "]" –> "[w]", " " <> z <> "]" –> " w]"}]; k++];

While[j <= Length[b[[2]]], z = b[[2]][[j]];

AppendTo[s, {"[" <> z <> "," –> "[w", " " <> z <> "," –> " w",

"[" <> z <> "]" –> "[w]", " " <> z <> "]" –> " w]"}]; j++];

ClearAttributes[ToString, Listable];

z = Map9[StringReplace, {c[[1]], v}, Map[Flatten, {g, s}]];

SameQ[z[[1]], StringReplace[z[[2]], Join[GenRules[Flatten[Map[# <>

"," &, Map[ToString, t]]], "w"], GenRules[Flatten[Map[# <> "]" &,

Map[ToString, t]]], "w]"], GenRules[Flatten[Map[# <> ")" &,

Map[ToString, t]]], "w)"]]]]]

In[1396]:= Mapp[EquExprPatt, {a + b*c^5, 5 + 6*y^7, a + b*p^m,

a + b*m^p}, a + b*x_^n_]

Out[1396]= {True, True, True, True}

Expressions processing at level of their components. Means of this group provide a quite effective differentiated processing of expressions. Because of combined symbolical architecture the Mathematica gives a possibility of direct generalization of the element–oriented list operations to arbitrary expressions, that allows to support operations as on separate terms, and on sets of terms at the given levels in trees of the expressions. Without going into details to all tools supporting work with components of expressions, we will give only the main from them that have been complemented by our means. Whereas with more detailed description of built–in tools of this group, including admissible formats of coding, it is possible to get acquainted in the Help, or in the corresponding literature on the Mathematica system.

The call Variables[p] of built–in function returns the list of all independent variables of a polynomial p, at the same time, its application to an arbitrary expression has some limitations. Meantime for receiving all independent variables of a certain expression x it is quite possible to use a simple function whose call UnDefVars[x] returns the list of all independent variables of an expression x. Unlike the UnDefVars the call UnDefVars1[x] returns the list of all independent variables in string format of an expression x. In certain cases the mentioned functions have certain preferences relative to the built–in Variables function.

The call Replace[x, r {, j}] of built–in function returns result of application of a rule r of the form a → b or list of such rules for transformation of x expression as a whole; application of the 3^rd optional j argument defines application of r rules to parts of j level of a x expression. Meantime, the built-in Replace function has a number of restrictions some of which a simple procedure considerably obviates, whose call Replace1[x, r] returns result of application of r rules to all or selective independent variables of x expression. In a case of detection by the procedure Replace1 of empty rules the appropriate message will be printed with the indication of the list of those r rules that were empty, i.e. whose left parts aren't entered into the list of independent variables of x expression. Fragment below presents source code of Replace1 with examples of its use; in addition, comparison with result of use of the Replace function on the same expression is done.

In[33]:= Replace1[x_, y_ /; ListQ[y] &&

DeleteDuplicates[Map[Head, y]] == {Rule}||Head[y] == Rule] :=

Module[{a = x // FullForm // ToString, b = UnDefVars[x], c, p, l,

h = {}, r, k = 1, d = ToStringRule[DeleteDuplicates[Flatten[{y}]]]},

p = Map14[RhsLhs, d, "Lhs"];

c = Select[p, ! MemberQ[Map[ToString, b], #] &];

If[c != {}, Print["Rules " <> ToString[Flatten[Select[d, MemberQ[c,

RhsLhs[#, "Lhs"]] &]]] <> " are vacuous"]];

While[k <= Length[d], l = RhsLhs[d[[k]], "Lhs"];

r = RhsLhs[d[[k]], "Rhs"];

h = Append[h, {"[" <> l –> "[" <> r, " " <> l –> " " <> r,

l <> "]" –> r <> "]"}]; k++];

Simplify[ToExpression[StringReplace[a, Flatten[h]]]]]

In[34]:= X = (x^2 – y^2)/(Sin[x] + Cos[y]) + a*Log[x + y];

Replace[X, {x –> a + b, a –> 90, y –> Cos[a], z –> Log[t]}]

Out[34]= a*Log[x + y] + (x^2 – y^2)/(Cos[y] + Sin[x])

In[35]:= Replace1[X, {x –> a + b, a –> 90, y –> Cos[a], z –> Log[t],

t –> c + d}]

Rules {ComplexInfinity –> (Log[t]), t –> (c + d)} are vacuous

Out[35]= 90*Log[a + b + Cos[a]] + ((a + b)^2 – Cos[a]^2)/

(Cos[Cos[a]] + Sin[a + b])

Due to quite admissible impossibility of performance of the replacements of sub-expressions of an expression at the required its levels, there are questions as of belonging of sub-expressions of expression to its levels, and of belonging of sub-expression to the given expression level. The following two procedures in a certain extent solve these problems. A call SubExprOnLevels[x] returns the nested list whose 2–element sub-lists contain levels numbers as the first elements, and lists of sub-expressions on these levels as the second elements of an expression x. Whereas the call ExprOnLevelQ[x, y, z] returns True if a y sub-expression belongs to the z–th level of x expression, and False otherwise. In addition, in a case of False return, the call ExprOnLevelQ[x, y, z, t] through the optional t argument – an indefinite symbol – returns additionally the list of levels numbers of the x expression which contain the y as a sub-expression. The fragment below represents the source codes of both procedures with examples of their use.

In[1137]:= SubExprOnLevels[x_] := Module[{a, b, c = {}},

Do[If[Set[a, DeleteDuplicates[Level[x, {j}]]] ===

Set[b, DeleteDuplicates[Level[x, {j + 1}]]], Return[c],

AppendTo[c, {j, a}]], {j, Infinity}]]

In[1138]:= SubExprOnLevels[(x^2 + y^2)/(x + y)]

Out[1138]= {{1, {1/(x + y), x^2 + y^2}}, {2, {x + y, –1, x^2, y^2}},

{3, {x, y, 2}}}

In[1140]:= ExprOnLevelQ[x_, y_, z_Integer, t___] :=

Module[{a = SubExprOnLevels[x], b = {}, c = {}},

If[! MemberQ[Map[#[[1]] &, a], z], False,

If[MemberQ[a[[z]][[2]], y], True, If[{t} != {} && NullQ[t],

Do[If[MemberQ[a[[j]][[2]], y], AppendTo[b, a[[j]][[1]]], 7],

{j, Length[a]}]; t = b; False, 7]]]]

In[1141]:= ExprOnLevelQ[(x + y)/(x + a*x^2/b^2), x^2, 1, agn]

Out[1141]= False

In[1142]:= agn

Out[1142]= {4}

At structural analysis of expressions, a quite certain interest is the CompOfExpr procedure, which allows to determine the component composition of an algebraical expression.

In[7]:= CompOfExpr[x_] :=

Module[{a = ToString[FullForm[x]],

b = Map[ToString, FullFormF[]],

Num = {}, Str = {}, Sys = {}, Vars = {}, User = {}, av},

ClearAll[av]; a = StringReplace4[a, GenRules[b, ""], av];

a = "{" <> StringReplace[a, {"[" –> ",", "]" –> ","}] <> "}";

a = Select[ToExpression[a], ! SameQ[#, Null] &];

a = Sort[DeleteDuplicates[a]];

Map[If[StringQ[#], AppendTo[Str, #],

If[NumericQ[#], AppendTo[Num, #],

If[SystemQ[#], AppendTo[Sys, #],

If[BlockFuncModQ[#], AppendTo[User, #],

AppendTo[Vars, #]]]]] &, a];

{{"Sys", Sys}, {"User", User}, {"Vars", Vars}, {"Str", Str},

{"Num", Num}, {"Op", ToExpression[Complement[b, av[[2]]]]}}]

In[8]:= X = {(x^2 – y^2)/(Sin[x]+Cos[y])/a^Log[x+y], {"a", "b", "c"},

ToExpression["ToString1"], ToExpression["CompOfExpr"]};

In[9]:= CompOfExpr[X]

Out[9]= {{"Sys", {Cos, List, Log, Sin}}, {"User", {CompOfExpr,

ToString1}}, {"Vars", {a, x, y}}, {"Str", {"a", "b", "c"}},

{"Num", {–1, 2}}, {"Op", {Plus, Power, Times}}}

In[10]:= CompOfExpr[{(Sin[x] + b)/(Cos[y] + c), ToString1,

ProcQ, {73, 78}, "agn"}]

Out[10]= {{"Sys", {Cos, List, Sin}}, {"User", {ProcQ, ToString1}},

{"Vars", {b, c, x, y}}, {"Str", {"agn"}},

{"Num", {–1, 73, 78}}, {"Op", {Plus, Power, Times}}}

Calling CompOfExpr[x] procedure returns the nested list of two–element sub–lists; each sub–list contains one of the words "Sys" (system tools), "User" (the user tools), "Vars" (variables), "Str" (strings), "Num" (numbers), "Op" (operations) as the first element, while the second element represents the list of components (type of which is defined by the corresponding first word) of an algebraical x expression.

In the certain cases exists necessity to execute the exchange of values of variables with the corresponding exchange of all their attributes. So, variables x and y having values 77 and 72 should receive the values 42 and 47 accordingly along with the appropriate exchange of all their attributes. The procedure call VarExch[x, y] solves this problem, returning Null, i.e. nothing. The list of two names of variables in string format for exchange by values and attributes or the nested list of ListList type acts as the actual argument; anyway all elements of pairs of the list have to be definite, otherwise the call returns Null with printing of the appropriate diagnostic message, for example:

In[7]:= x = a + b; y = m – n; SetAttributes[x, {Protected, Listable}]

In[8]:= {x, y}

Out[8]= {a + b, m – n}

In[9]:= VarExch[{"x", "y"}]

In[10]:= Definition[x]

Out[10]= x = m – n

In[11]:= Definition[y]

Out[11]= Attributes[y] = {Listable, Protected}

y = a + b

On the other hand, the procedure call Rename[x, y] in the regular mode returns Null, providing replacement of x name of some defined object on y name with saving of all attributes of this object. In addition, the x name is removed from the current session by the Remove function. But if y argument defines the name of a defined object or an undefined name with attributes, the call is returned unevaluated. If the first x argument is illegal for renaming, the procedure call returns Null; in addition, the Rename procedure successfully processes also objects of the same name of type "Block", "Function", "Module". The Rename1 procedure is an useful version of the above procedure, being based on our procedure Definition2. The call Rename1[x, y] is similar to the call Rename[x, y] whereas the call Rename1[x, y, z] with the third optional z argument – an arbitrary expression – performs the same functions as the call Rename1[x, y] without change of an initial x object.

The VarExch1 procedure is a version of the above VarExch procedure and is based on use of the Rename procedure with global variables; it admits the same type of actual argument, but unlike the second procedure the call VarExch1[w] in a case of detection of indefinite elements of a list w or its sublists will be returned unevaluated without print of any diagnostic message. In the fragment below, source code of the Rename1 procedure along with typical examples of its application is represented.

In[7]:= Rename1[x_String /; HowAct[x], y_ /; ! HowAct[y], z___] :=

Module[{a = Attributes[x], b = Definition2[x][[1 ;–2]],

c = ToString[y]},

b = Map[StringReplacePart[#, c, {1, StringLength[x]}] &, b];

ToExpression[b];

ToExpression["SetAttributes[" <> c <> ", " <> ToString[a] <> "]"];

If[{z} == {}, ToExpression["ClearAttributes[" <> x <> ", " <>

ToString[a] <> "]; Remove[" <> x <> "]"], Null]]

In[8]:= x := 500; y = 500; SetAttributes[x, {Listable, Protected}]

In[9]:= Rename1["x", Trg42]

In[10]:= {x, Trg42, Attributes["Trg42"]}

Out[10]= {x, 500, {Listable, Protected}}

In[11]:= Rename1["y", Trg47, 90]

In[12]:= {y, Trg47, Attributes["Trg47"]}

Out[12]= {500, 500, {}}

Use in procedures of global variables, in a lot of cases will allow to simplify programming, sometimes significantly. This mechanism sufficient in detail is considered in [15]. Meanwhile, mechanism of global variables in Mathematica isn't universal, quite correctly working in a case of evaluation of definitions of procedures containing global variables in the current session in the Input–paragraph; whereas in general case it isn't supported when the loading in the current session of the procedures that contain global variables, in particular, from nb–files with the subsequent activation of their contents.

For the purpose of exclusion of similar situation a tool has been offered, whose call NbCallProc[x] reactivates a block, a function or a module x in the current session, whose definition was in a nb–file loaded into the current session with returning of Null, i.e. nothing. The call NbCallProc[x] reactivates in the current session all definitions of blocks, functions and modules with the same name x and with different headings. All these definitions have to be loaded previously from some nb–file into the current session and activated by means of function "Evaluate Notebook" of the GUI. The following fragment represents source code of the NbCallProc procedure with example of its use for the above VarExch1 procedure that uses the global variables.

In[2442]:= NbCallProc[x_ /; BlockFuncModQ[x]] :=

Module[{a = SubsDel[StringReplace[ToString1[DefFunc[x]],

"\n \n" –> ";"], "`" <> ToString[x] <> "`", {"[", ","}, –1]}, Clear[x];

ToExpression[a]]

In[2443]:= NbCallProc[VarExch1]

The performed verification convincingly demonstrates, that the VarExch1 that contains the global variables and uploaded from the nb–file with subsequent its activation (by the "Evaluate Notebook"), is carried out absolutely correctly and with correct functioning of mechanism of global variables restoring values after an exit from the VarExch1 procedure. The NbCallProc has a number of rather interesting appendices above all if necessity of application of procedures activated in the Input–paragraph of the current session arises.

Maple has 2 useful tools of manipulation with expressions of the type {range, equation, inequality, relation}, whose calls lhs(Exp) and rhs(Exp) return the left and the right parts of an expression Exp respectively. More precisely, the call lhs(Exp), rhs(Exp) returns a value op(1, Exp) and op(2, Exp) respectively. Whereas Mathematica has no similar useful means. The given deficiency is compensated by the RhsLhs procedure, whose the source code with examples of application are given below. The call RhsLhs[w, y] depending on a value {"Rhs", "Lhs"} of the second y argument returns right or left part of w expressions respectively relatively to operator Head[w], whereas the call RhsLhs[x,y,t] in addition through a undefined t variable returns operator Head[x] concerning whom splitting of the x expression onto left and right parts was made. RhsLhs procedure can be rather easily modified in the light of expansion of the analyzed operators Head[x]. RhsLhs1 procedure is a certain functional equivalent to the previous procedure [8-16].

In[7]:= RhsLhs[x__] := Module[{a = Head[{x}[[1]]],

b = ToString[InputForm[{x}[[1]]]], d, h = {x},

c = {{Greater, ">"}, {Or, "||"}, {GreaterEqual, ">="}, {Span, ";"},

{And, "&&"}, {LessEqual, "<="}, {Unequal, "!="}, {Rule, "–>"},

{Less, "<"}, {Plus, {"+", "–"}}, {Power, "^"}, {Equal, "=="},

{NonCommutativeMultiply, "**"}, {Times, {"*", "/"}}}},

If[Length[h] < 2 || ! MemberQ[{"Lhs", "Rhs"}, h[[2]]],

Return[Defer[RhsLhs[x]]], Null];

If[! MemberQ[Select[Flatten[c], ! StringQ[#] &], a] ||

a == Symbol, Return[Defer[RhsLhs[x]]], Null];

d = StringPosition[b, Flatten[Select[c, #[[1]] == a &], 1][[2]]];

a = Flatten[Select[c, #[[1]] == a &]];

If[Length[h] >= 3 && ! HowAct[h[[3]]],

ToExpression[ToString[h[[3]]] <> "=" <> ToString1[a]], Null];

ToExpression[If[h[[2]] == "Lhs",

StringTrim[StringTake[b, {1, d[[1]][[1]] – 1}]],

StringTrim[StringTake[b, {d[[1]][[2]] + 1, –1}]]]]]

In[8]:= Mapp[RhsLhs, {a^b, a*b, a –> b, a <= b, a||b, a && b}, "Rhs"]

Out[8]= {b, b, b, b, b, b}

In[9]:= {{RhsLhs[7 ; 42, "Rhs", s], s}, {RhsLhs[a && b, "Lhs", v], v}}

Out[9]= {{42, {Span, ";"}}, {a, {And, "&&"}}}

In a number of appendices the undoubted interest presents an analog of the Maple-procedure whattype(x) that returns the type of an expression x which is one of basic Maple–types. The procedure of the same name acts as a similar analog in system Mathematica whose call WhatType[x] returns type of an object x of one of basic types {"Module", "DynamicModule", "Complex", "Block", "Real", "Integer", "Rational", "Times", "Rule", "Power", "And", "Alternatives", "List", "Plus", "Condition", "StringJoin", "UndirectedEdge", …}. The fragment represents source code of the procedure and examples of its application for identification of the types of various objects.

In[7]:= WhatType[x_ /; StringQ[x]] := Module[{b = t, d,

c = $Packages, a = Quiet[Head[ToExpression[x]]]},

If[a === Symbol, Clear[t]; d = Context[x];

If[d == "Global`", d = Quiet[ProcFuncBlQ[x, t]];

If[d === True, Return[{t, t = b}[[1]]],

Return[{"Undefined", t = b}[[1]]]],

If[d == "System`", Return[{d, t = b}[[1]]], Null]],

Return[{ToString[a], t = b}[[1]]]];

If[Quiet[ProcFuncBlQ[x, t]],

If[MemberQ[{"Module", "DynamicModule", "Block"}, t],

Return[{t, t = b}[[1]]], t = b;

ToString[Quiet[Head[ToExpression[x]]]]], t = b; "Undefined"]]

In[8]:= Map[WhatType, {"a^b", "a**b", "3+7*I", "{42, 47}", "a&&b"}]

Out[8]= {"Power", "NonCommutativeMultiply", "Complex",

"List", "And"}

In[9]:= Map[WhatType, {"a_/; b", "a <> b", "a <–> b", "a|b"}]

Out[9]= {"Condition", "StringJoin", "TwoWayRule", "Alternatives"}

However, it should be noted that the WhatType procedure don`t support exhaustive testing of types, meantime on its basis it is simple to expand the class of the tested types.

The functions Replace and ReplaceAll of Mathematica have essential restrictions in relation to replacement of sub-expressions relatively of very simple expressions as it will illustrated below.

The reason of it can be explained by the following circumstance, using the procedure useful enough also as independent means. The call ExpOnLevels[x, y, z] returns the list of an expression x levels on which a sub-expression y is located. While procedure call with the third optional z argument – an indefinite symbol – through it additionally returns the nested list of ListList type of all sub-expressions of expression x on all its levels. In addition, the first element of each sub-list of such ListList list determines a level of the x expression while the second element defines the list of sub-expressions located on this level. If sub-expression y is absent on the levels identified by Level function, calling the procedure call returns the appropriate diagnostic message. The fragment below represents the source code of the ExpOnLevels procedure with some typical examples of its application.

In[7]:= ExpOnLevels[x_, y_, z___] := Module[{a = {}, b},

Do[AppendTo[a, {j, Set[b, Level[x, {j}]]}];

If[b == {}, Break[], Continue[]], {j, 1, Infinity}];

a = a[[1 ;–2]]; If[{z} != {} && NullQ[z], z = a, 77];

b = Map[If[MemberQ[#[[2]], y], #[[1]], Nothing] &, a];

If[b == {}, "Sub-expression " <> ToString1[y] <>

" can't be identified", b]]

In[8]:= ExpOnLevels[a + b^3 + 1/x^(3/(a + 2)) + b[t] + 1/x^2, x^2]

Out[8]= "Sub-expression x^2 can't be identified"

In[9]:= ExpOnLevels[a + b^3 + 1/x^(3/(a + 2)) + b[t] + a/x^2, x^2, g]

Out[9]= "Sub-expression x^2 can't be identified"

In[10]:= g

Out[10]= {{1, {a, b^3, a/x^2, x^(–(3/(2 + a))), b[t]}},

{2, {b, 3, a, 1/x^2, x, –(3/(2 + a)), t}},

{3, {x, –2, –3, 1/2 + a)}}, {4, {2 + a, –1}}, {5, {2, a}}}

Replacement sub-expressions in expressions. The procedure ExpOnLevels can determine admissible replacements carrying out by means of the standard functions Replace and ReplaceAll as they are based on the Level function that evaluates the list of all sub-expressions of an expression on the set levels. The call of the built–in Replace function on unused rules don't return any diagnostical information, at the same time, if all rules were not used, then the function call is returned as unevaluated. In turn, the procedure below that is based on the previous ExpOnLevels procedure gives full diagnostics concerning the unused rules. The procedure call ReplaceInExpr[x, y, z] is analogous to the call Replace[x, y, All] where y is a rule or their list whereas thru the 3^rd optional z argument – an undefined symbol – additionally is returned the message identifying the list of the unused rules.

The procedure call ReplaceInExpr1[x, y, z] is analogous to the call ReplaceInExpr[x, y, z] but not its result where y is a rule or their list whereas through the third optional z argument – an undefined symbol – additionally the message is returned which identifies a list of the unused rules. If unused rules are absent, then the z symbol remains undefined. The following fragment represents source code of the ReplaceInExpr1 procedure with typical examples of its application. Of the presented examples one can clearly see the difference between both procedures.

In[7]:= ReplaceInExpr1[x_, r_/; RuleQ[r]||ListRulesQ[r], y___] :=

Module[{a = ToString1 @@ {ToBoxes @@ {x}}, b,

c = If[RuleQ[r], {r}, r], d, h = {}},

Do[b = ToString1 @@ {ToBoxes @@ {c[[j]][[1]]}};

d = ToString1 @@ {ToBoxes @@ {c[[j]][[2]]}};

If[StringFreeQ[a, b], AppendTo[h, j],

a = StringReplace[a, b –> d]], {j, 1, Length[c]}];

a = ToExpression[a]; If[{y} != {} && NullQ[y], y = h, 77];

ReleaseHold[MakeExpression[a, StandardForm]]]

In[8]:= ReplaceInExpr1[x^2 + 1/x^2 + c[t], {x^2 –> m^3, t –> j[x]}]

Out[8]= 1/m^3 + m^3 + c[2[x]]

In[9]:= ReplaceInExpr1[(a + b[t]) + 1/x^(3/(b[t] + 2)) + b[t] +

a/x^2, {x^2 –> mp, b[t] –> gsv, x^3 –> tag, a –> 77}, gs]

Out[9]= 77 + 2*gsv + 77/mp + x^(–(3/(2 + gsv)))

In[10]:= gs

Out[10]= {3}

In[11]:= ReplaceInExpr[(a + b[t]) + 1/x^(3/(b[t] + 2)) + b[t] +

a/x^2, {x^2 –> mp, b[t] –> gsv, x^3 –> tag, a –> 77}, gs1]

Out[11]= 77 + 2*gsv + 77/x^2 + x^(–(3/(2 + gsv)))

In[12]:= gs1

Out[12]= "Rules {1, 3} were not used"

Using the built-in ToBoxes function that creates the boxes corresponding to the printed form of expressions in the form

StandardForm, we can allocate the sub-expressions composing an expression. The procedure call SubExpressions[x] returns the list of sub-expressions composing an expression x, in a case of impossibility the empty list is returned, i.e. {}. The fragment below represents source code of the SubExpressions procedure with a typical example of its application.

In[12]:= SubExpressions[x_] := Module[{b, c = {}, d, h, t,

a = ToString1[ToBoxes[x]]},

b = Select[ExtrVarsOfStr[a, 2], SystemQ[#] || UserQ[#] &];

Do[h = b[[j]]; d = StringPosition[a, h];

Do[t = "["; Do[If[StringCount[t, "["] == StringCount[t, "]"],

AppendTo[c, h <> t]; Break[],

t = t <> StringTake[a, {d[[p]][[2]] + k + 1}]], {k, 1, Infinity}],

{p, 1, Length[d]}], {j, 1, Length[b]}];

c = Map[Quiet[Check[ToExpression[#], Nothing]] &, c];

Map[ReleaseHold[MakeExpression[#, StandardForm]] &, c]]

In[13]:= SubExpressions[a*b + 1/x^(3/(b[t] + 2)) + J[t] + d/x^2]

Out[13]= {d/x^2, 3/(2 + b[t]), a*b + d/x^2 + x^(–(3/(2 + b[t]))) + J[t],

a*b, –(3/(2 + b[t])), 2 + b[t], b[t], Sin[t], x^2, x^(–(3/(2 + b[t])))}

Unlike the above SubExpressions procedure, the procedure SubExpressions1 is based on the standard FullForm function. The call SubExpressions1[x] returns the list of sub–expressions that compose an expression x, while in a case of impossibility the empty list is returned, i.e. {}. The system Level function and our SubExpressions ÷ SubExpressions2 means allow to outline possibilities of the Mathematica concerning the replacements of the sub–expressions in expressions [8,12-16].

As an alternative to the above tools can be offered the Subs procedure that is functionally equivalent to the above standard ReplaceAll function, however which is relieved of a number of its shortcomings. Procedure call Subs[x, y, z] returns the result of substitutions to an expression x of all occurrences of y sub–expressions onto z expressions. In addition, if x – an arbitrary correct expression, then as the 2^nd and 3^rd arguments defining substitutions of format y –> z, an unary substitution or their list coded in form y ≡ {y1, y2,…,yn} and z ≡ {z1, z2,…,zn} act, defining the list of substitutions {y1 –> z1, y2 –> z2, …, yn –> zn} which are carried out consistently in the order defined at the Subs call. A number of bright examples of its use on those expressions and with those types of substitutions where the Subs surpasses the standard ReplaceAll function is represented, in particular, in [8-15]. These examples rather clearly illustrate advantages of the Subs procedure before the similar system software.

At last, Substitution is an integrated procedure of the tools Subs ÷ Subs4. The procedure call Substitution[x, y, z] returns the result of substitutions into an arbitrary x expression of an expression z instead of occurrences in it of all y sub-expressions. In addition, if as x expression any correct expression admitted in Math–language is used, whereas as a single substitution or their set are coded y ≡ {y1,y2,...,yn} and z ≡ {z1,z2,...,zn} as the 2^nd and 3^rd arguments defining substitutions of the format y –> z by defining the set of substitutions {y1 –> z1, y2 –> z2, ..., yn –> zn} carried out consistently. The Substitution allows essentially to extend possibilities of expressions processing. The following fragment represents source code of the Substitution procedure and examples of its application, well illustrating its advantages over the standard means.

In[7]:= Substitution[x_, y_, z_] := Module[{d, k = 2, subs, subs1},

subs[m_, n_, p_] := Module[{a, b, c, h, t},

If[! HowAct[n], m /. n –> p, {a, b, c, h} =

First[{Map[ToString, Map[InputForm, {m, n, p, 1/n}]]}];

Simplify[ToExpression[StringReplace[StringReplace[a,

b –> "(" <> c <> ")"], h –> "1/" <> "(" <> c <> ")"]]];

If[t === m, m /. n –> p, t]]];

subs1[m_, n_, p_] := ToExpression[StringReplace[ToString[

FullForm[m]],

ToString[FullForm[n]] –> ToString[FullForm[p]]]];

If[! ListQ[y] && ! ListQ[z], If[Numerator[y] == 1 &&

! SameQ[Denominator[y], 1], subs1[x, y, z],

subs[x, y, z]], If[ListQ[y] && ListQ[z] && Length[y] ==

Length[z], If[Numerator[y[[1]]] == 1 &&

! SameQ[Denominator[y[[1]]], 1],

d = subs1[x, y[[1]], z[[1]]], d = subs[x, y[[1]], z[[1]]]];

For[k, k <= Length[y], k++, If[Numerator[y[[k]]] == 1 &&

! SameQ[Denominator[y[[k]]], 1], d = subs1[d, y[[k]], z[[k]]],

d = subs[d, y[[k]], z[[k]]]]]; d, Defer[Substitution[x, y, z]]]]]

In[8]:= Replace[1/x^2 + 1/y^3, {{x^2 –> a + b}, {y^3 –> c + d}}]

Out[8]= {1/x^2 + 1/y^3, 1/x^2 + 1/y^3}

In[9]:= Substitution[1/x^2 + 1/y^3, {x^2, y^3}, {a + b, c + d}]

Out[9]= 1/(a + b) + 1/(c + d)

In[10]:= Replace[1/x^2*1/y^3, {{1/x^2 –> a + b}, {1/y^3 –> c + d}}]

Out[10]= {1/(x^2*y^3), 1/(x^2*y^3)}

In[11]:= Substitution[1/x^2*1/y^3, {x^2, y^3}, {a + b, c + d}]

Out[11]= 1/((a + b)*(c + d))

It should be noted that the Substitution procedure rather significantly extends the standard tools intended for ensuring replacements of sub-expressions in expressions as the following examples rather visually illustrate. The Substitution1 procedure can be considered as an analogue of the Substitution procedure. Syntax of the procedure call Substitution1[x, y, z] is identical to the procedure call Substitution[x, y, z], returning the result of substitutions into arbitrary x expression of z sub-expression(s) instead of all occurrences of y sub-expression(s). A quite simple Substitution2 function is the simplified version of the previous Substitution procedure, its call Substitution2[x, y, z] returns the result of substitutions into an expression x of a sub-expression z instead of all occurrences of sub-expression y. The Substitution2 function significantly uses expressions in the FullForm form and supports a rather wide range of substitutions.

In[2211]:= Substitution2[x_, y_, z_] :=

ToExpression[StringReplace[ToString[FullForm[x]],

ToString[FullForm[y]] –> ToString[FullForm[z]]]]

In[2212]:= Substitution2[Sin[x^2]*Cos[1/b^3], 1/b^3, x^2]

Out[2212]= Cos[x^2]*Sin[x^2]

In[2213]:= Substitution2[1 + 1/x^2, x^2, a + b]

Out[2213]= 1 + 1/x^2

Meantime, already the second example of application of the function reveals its shortcomings, forcing on the same basis to complicate its algorithm. The following version in the procedure form somewhat enhances the function's possibilities.

In[2242]:= Substitution3[x_, y_, z_] := Module[{c,

a = ToString[FullForm[Simplify[x]]], b = ToString[FullForm[z]]},

c = Map[ToString[FullForm[#]] &, {y, 1/y}];

c = GenRules[c, b]; ToExpression[StringReplace[a, c]]]

In[2243]:= Substitution3[1 + 1/x^2, x^2, a + b]

Out[2243]= 1 + a + b

In[2244]:= Substitution3[x^2 + 1/(a + 1/x^2), x^2, a + b]

Out[2244]= a + b + 1/(2*a + b)

In[2245]:= Substitution3[x^2 + 1/(1/x^2 + x^2), x^2, a + b]

Out[2245]= a + b + 1/(2*a + 2*b)

In[2246]:= Substitution[x^2 + 1/(1/x^2 + x^2), x^2, a + b]

Out[2246]= a + b + 1/(a + b + 1/(a + b))

Meantime, and the above version of the Substitution2 don`t fully solve the problem, as illustrated by the 3^rd example of the previous fragment, making it impractical to further complicate such FullForm–based means. And only the above Substitution procedure solves the set problem.

The following procedure allows to eliminate restrictions inherent in means of the above so–called Subs–group. The call SubsInExpr[x,r] returns the result of substitution in an arbitrary expression x set in the form Hold[x] of the right parts of a rule or their list r instead of occurrences of the left parts corresponding them. In addition, all the left and the right parts of r rules should be coded in the Hold-form, otherwise the procedure call returns $Failed with printing of the appropriate message. The fragment below represents source code of the SubsInExpr procedure with some typical examples of its application.

In[2261]:= SubsInExpr[x_ /; Head[x] == Hold, r_ /; RuleQ[r] ||

ListRulesQ[r]] := Module[{a, b, c, f},

c = Map[Head[(#)[[1]]] === Hold && Head[(#)[[2]]] === Hold &,

Set[b, Flatten[{r}]]]; If[! And @@ c,

Print["Incorrect tuple of factual arguments"]; $Failed,

f[t_] := StringTake[ToString1[t], {6, –2}]; a = f[x];

c = Map[Rule[f[#[[1]]], "(" <> f[#[[2]]] <> ")"] &, b];

ToExpression[StringReplaceVars[a, c]]]]

In[2262]:= SubsInExpr[Hold[(a + b^3)/(c + d^(–2))],

Hold[d^–2] –> Hold[Sin[t]]]

Out[2262]= (a + b^3)/(c + Sin[t])

In[2263]:= SubsInExpr[Hold[1/(a + 1/x^2)],

Hold[1/x^2] –> Hold[Sin[t]]]

Out[2263]= 1/(a + Sin[t])

The function call SubsInExpr1[x,r] analogously to the above SubsInExpr procedure also returns the substitution result in an expression x set in the string format of the right parts of a rule or their list r instead of occurrences of the left parts corresponding them. At that, in contrast to the above SubsInExpr procedure all the left and the right parts of the rules r should be also coded in string format like its first argument, otherwise the function call is returned unevaluated. Fragment below represents the source code of the function SubsInExpr1 with examples of its use.

In[7]:= SubsInExpr1[x_ /; StringQ[x], r_ /; (RuleQ[r] ||

ListRulesQ[r]) && (And @@ Map[StringQ[(#)[[1]]] &&

StringQ[(#)[[2]]] &, Flatten[{r}]])] :=

ToExpression[StringReplaceVars[x, r]]

In[8]:= SubsInExpr1["1/x^2", "1/x^2" –> "Sin[t]"]

Out[8]= Sin[t]

In[9]:= SubsInExpr1["(a + b^3)/(c + d^(–2))", "d^(–2)" –> "Sin[t]"]

Out[9]= (a + b^3)/(c + Sin[t])

On a basis of our means of replacement of subexpressions in expressions, the means of differentiation and integration of expressions are programmed. For example, the Subs procedure is used in realization of the Df procedure [8] whose call Df[x, y] provides differentiation of an expression x on its arbitrary y sub-expression, and rather significantly extends the built-in function D. Our ReplaceAll1 procedure is functionally equivalent to the built-in ReplaceAll function by relieving a number of shortages of the second. Based on the ReplaceAll1 procedure a number of variants of the Df procedure, namely the Df1 and Df2, using a number of useful methods of programming were programmed. The Df1 and Df2 procedures in some cases are more useful than the Df procedure what rather visually illustrate examples given in [7-12,14]. In addition, the Df, Df1 and Df2 rather significantly extend the built-in D function.

The receiving of similar expansion as well for the standard Integrate function which has rather essential restrictions on use of arbitrary expressions as integration variables is presented an quite natural to us. Two variants of such expansion in the form of the simple procedures Int and Int1 which are based on the above Subs procedure have been proposed for the set purpose, whose source codes and examples of application can be found in [6-16]. Meanwhile, a simple enough Subs1 procedure, being as a certain extension and complement of the Subs procedure, can be used for realization of procedures Integrate2 and Diff1 of integration and differentiation accordingly.

In[3331]:= Integrate2[x_, y__] := Module[{a, b, d,

c = Map[Unique["gs"] &, Range[1, Length[{y}]]]},

a = Riffle[{y}, c]; a = If[Length[{y}] == 1, a, Partition[a, 2]];

d = Integrate[Subs1[x, a], Sequences[c]];

{Simplify[Subs1[d, If[ListListQ[a], Map[Reverse, a],

Reverse[a]]]], Map[Remove, c]}[[1]]]

In[3332]:= Integrate2[(a/b + d)/(c/b + h/b), 1/b, d]

Out[3332]= (d*((2*a)/b + d*Log[1/b]))/(2*(c + h))

In[3333]:= Integrate2[Sqrt[a + Sqrt[c + d]*b], Sqrt[c + d]]

Out[3333]= (2*(a + b*Sqrt[c + d])^(3/2))/(3*b)

In[3334]:= Diff1[x_, y__] := Module[{a, b, d,

c = Map[Unique["s"] &, Range[1, Length[{y}]]]},

a = Riffle[{y}, c]; a = If[Length[{y}] == 1, a, Partition[a, 2]];

d = D[Subs1[x, a], Sequences[c]];

{Simplify[Subs1[d, If[ListListQ[a], Map[Reverse, a],

Reverse[a]]]], Map[Remove, c]}[[1]]]

In[3335]:= Diff1[(c + a/b)*Sin[b + 1/x^2], a/b, 1/x^2]

Out[3335]= Cos[b + 1/x^2]

In[3336]:= Diff1[(c + a/b)*Sin[d/c + 1/x^2], 1/c, a/b, 1/x^2]

Out[3336]= –d*Sin[d/c + 1/x^2]

The procedure call Integrate2[x, y] provides integrating of an expression x on the sub-expressions defined by a sequence y. In addition, the procedure with the returned result by means of Simplify function performs a sequence of algebraical and other transformations and returns the simplest form it finds. While the procedure call Diff1[x, y] which is also realized on a basis of

the Subs1 returns the differentiation result of an expression x on the generalized {y, z, g ...} variables which can be as arbitrary expressions. The result is returned in the simplified form on the basis of the Simplify function. The above fragment represents source codes of the Diff1 and Integrate2 procedures along with examples of their application. The variants of realization of the means Df ÷ Df2, Diff1, Diff2, Int, Int1, Integrate2, ReplaceAll1, Subs, Subs1, SubsInExpr and SubsInExpr1 illustrate different receptions enough useful in a lot of problems of programming in the Mathematica and, first of all, in problems of the system character. Moreover, the above means essentially extend the appropriate system means [7-16].

The previous means, expanding similar built-in tools, at the same time have certain restrictions. The next means eliminate the defects inherent in the above and built-in means, being the most universal in this class for today. As a basis of the following means is the ToStringRational procedure. The procedure call ToStringRational[x] returns an expression x in the string format that is rather convenient for expressions processing, including replacements of sub-expressions, integrating and differentiation on the sub-expressions other than simple symbols. Whereas the procedure call SubsExpr[x, y] returns the result of substitutions in a x expression of right parts instead of occurrences of left parts defined by a rule or their list y into the x expression. Fragment below represents source codes of the above two procedures and some typical examples of their application.

In[7]:= ToStringRational[x_] := Module[{b, c = {}, h, p, t = "", n,

a = "(" <> StringReplace[ToString1[x] <> ")", " " –> ""]},

b = Map[#[[1]] – 1 &, StringPosition[a, "^(–"]];

Do[h = ""; p = SyntaxQ[StringTake[a, {b[[j]]}]];

If[p, Do[h = StringTake[a, {j}];

If[! SyntaxQ[h], AppendTo[c, j + 1]; Break[], 7], {j, b[[j]], 1, –1}],

Do[h = StringTake[a, {j}]; t = h <> t;

If[SyntaxQ[t], AppendTo[c, j]; Break[], 7], {j, b[[j]], 1, –1}]],

{j, 1, Length[b]}]; h = StringInsert[a, "1/", c];

h = StringTake[StringReplace[h, "^(–" –> "^("], {2, –2}];

h = StringReplace[h, "1/" –> ToString[n] <> "/"];

h = ToString1[ToExpression[h]];

h = StringReplace[h, ToString[n] –> "1"]]

In[8]:= ToStringRational[1/x^2 + 1/(1 + 1/(a + b)^2) + 1/x^3]

Out[8]= "1/(1 + 1/(a + b)^2) + 1/x^3 + 1/x^2"

In[13]:= SubsExpr[x_, y_ /; RuleQ[y] || ListRulesQ[y]] :=

Module[{a = If[RuleQ[y], {y}, y], b = ToStringRational[x], c},

c = Map[Rule[#[[1]], #[[2]]] &, Map[Map[ToStringRational,

{#[[1]], #[[2]]}] &, a]]; ToExpression[StringReplace[b, c]]]

In[14]:= SubsExpr[a*b/x^3 + 1/x^(3/(b[t] + 2)) + 1/f[a + 1/x^3]^2 +

1/(x^2 + m) – 1/x^3, {x^2 –> av, x^3 –> gs}]

Out[14]= –(1/gs) + (a*b)/gs + 1/(av + m) + x^(–(3/(2 + b[t]))) +

1/f[a + 1/gs]^2

Unlike the ToString1 tool the ToStringRational procedure provides convenient representation of rational expressions in the string format, that allows to process the expressions more effectively by means of strings processing tools. The procedures ToStringRational and SubsExpr, generalizing tools of the same orientation, meanwhile, don't belittle them illustrating different useful enough methods of programming with their advantages and shortcomings.

At last, the following two procedures Differentiation and Integrate3 provide the differentiation and integrating on the variables as which can be sub-expressions different from simple symbols. The call Differentiation[x, y, z, t, …] returns result of differentiation of an expression x on sub-expressions which can be differ from simple symbols defined by the {y, z, t, …} tuple of arguments starting from the second. Whereas the procedure call Integrate3[x, y, z, t, …] returns the result of integrating of an expression x on sub-expressions that can be differ from simple symbols defined by the {y, z, t, …} tuple of arguments starting from the second. It should be noted that the Differentiation and Integrate3 procedures return the result processed by means of built-in Simplify function that performs a sequence of algebraic and other transformations returning the simplest form it finds. The fragment below represents source codes of the above two procedures with some typical examples of their application.

In[7]:= Differentiation[x_, y__] := Module[{a = {y}, b = x, c},

c = Map[Unique["g"] &, Range[1, Length[a]]];

Do[b = SubsExpr[b, a[[j]] –> c[[j]]], {j, 1, Length[a]}];

b = D[b, Sequences[c]]; Simplify[SubsExpr[b, Map[c[[#]] –>

a[[#]] &, Range[1, Length[a]]]]]]

In[8]:= Differentiation[1/(1 + 1/x^2 + 1/x^3) + 1/(1 + 1/(a + b +

1/x^3)^2) + 1/(1 + 1/x^3), 1/x^3, x^2]

Out[8]= –((2*x^5)/(1 + x + x^3)^3)

In[15]:= Integrate3[x_, y__] := Module[{a = {y}, b = x, c},

c = Map[Unique["g"] &, Range[1, Length[a]]];

Do[b = SubsExpr[b, a[[j]] –> c[[j]]], {j, 1, Length[a]}];

b = Integrate[b, Sequences[c]];

Simplify[SubsExpr[b, Map[c[[#]] -> a[[#]] &, Range[1, Length[a]]]]]]

In[16]:= Integrate3[1/x^2 + 1/(x^2 + x^3), x^2, x^3]

Out[16]= x^2*(–1 + x*Log[x^2] + (1 + x)*Log[x^2*(1 + x)])

The procedures Diff and Integral1 have certain limitations that at use demand corresponding wariness; some idea of such restrictions is illustrated by the following very simple example:

In[2221]:= Diff[(a + b*m)/(c + d*n), a + b, c + d]

Out[2221]= –(m/((c + d)^2*n))

In[2222]:= Integral1[(a + b*m)/(c + d*n), a + b, c + d]

Out[2222]= ((a + b)^2 m*Log[c + d])/(2*n)

For the purpose of an exception of these shortcomings two modifications of the built–in Replace and ReplaceAll functions in the form of the procedures Replace4 and ReplaceAll2 have been programmed accordingly. These procedures expand the standard means and allow to code the previous two procedures Integral1 and Diff with wider range of correct applications in the context of use of the generalized variables of differentiation and integrating. The call Replace4[x, a –> b] returns the result of application to an expression x of a substitution a –> b, when as its left part an arbitrary expression is allowed. At absence in the x expression of occurrences of the sub-expression a an initial x expression is returned. Unlike previous the call ReplaceAll2[x,r] returns the result of application to an expression x of a rule r or consecutive application of rules from the r list; as the left parts of rules any expressions are allowed (see also ReplaceAll3 [16]).

Fragment below presents source codes of both procedures with some typical examples of their application.

In[2358]:= Replace4[x_, r_ /; RuleQ[r]] := Module[{a, b, c, h},

{a, b} = {ToString[x // InputForm],

Map[ToString, Map[InputForm, r]]};

c = StringPosition[a, Part[b, 1]];

If[c == {}, x, If[Head[Part[r, 1]] === Plus,

h = Map[If[(#[[1]] === 1 || MemberQ[{" ", "(", "[", "{"},

StringTake[a, {#[[1]] – 1, #[[1]] – 1}]]) && (#[[2]] ===

StringLength[a] || MemberQ[{" ", ")", "]", "}", ","},

StringTake[a, {#[[2]] + 1, #[[2]] + 1}]]), #] &, c],

h = Map[If[(#[[1]] === 1 || ! Quiet[SymbolQ[

StringTake[a, {#[[1]] – 1, #[[1]] – 1}]]]) && (#[[2]] ===

StringLength[a] || ! Quiet[SymbolQ[

StringTake[a, {#[[2]] + 1, #[[2]] + 1}]]]), #] &, c]];

h = Select[h, ! SameQ[#, Null] &];

ToExpression[StringReplacePart[a, "(" <> Part[b, 2] <> ")", h]]]]

In[2359]:= Replace4[(c + d*x)/(c + d + x), c + d –> a + b]

Out[2359]= (c + d*x)/(a + b + x)

In[2360]:= Replace4[Sqrt[a + b*x^2*d + c], x^2 –> a + b]

Out[2360]= Sqrt[a + c + b*(a + b)*d]

In[2364]:= ReplaceAll2[x_, r_ /; RuleQ[r] || ListRulesQ[r]] :=

Module[{a = x, k = 1}, If[RuleQ[r], Replace4[x, r],

While[k <= Length[r], a = Replace4[a, r[[k]]]; k++]; a]]

In[2365]:= ReplaceAll2[Sin[a + b*x^2*d + c*x^2], x^2 –> a + b]

Out[2365]= Sin[a + (a + b)*c + b*(a + b)*d]

On the basis of the Replace4 procedure the procedures Diff and Integral1 can be expanded in the form of procedures Difff and Integral2. The procedure call Difff[x,y,z,t,…] returns result of differentiation of an expression x on the generalized variables {y, z, t,…} which are any expressions. The result is returned in the simplified form on a basis of the built–in Simplify function. While the procedure call Integral2[x, y, z, …] returns the result of integrating of an expression x on the generalized variables {y, z, t, …} that are arbitrary expressions. The result is returned in the simplified form on the basis of the Simplify function. The fragment below represents the source codes of the above means

and examples of their application (see Replace5 ÷ Replace7 [16]).

In[53]:= Difff[x_, y__] := Module[{a = x, a1, a2, a3, c = {}, d, k = 1,

n = g, b = Length[{y}]}, Clear[g];

While[k <= b, d = {y}[[k]]; AppendTo[c, Unique[g]];

a1 = Replace4[a, d –> c[[k]]]; a2 = D[a1, c[[k]]];

a3 = Replace4[a2, c[[k]] –> d]; a = a3; k++]; g = n; Simplify[a3]]

In[54]:= Difff[(a + b)/(c + d + x), a + b, c + d]

Out[54]= –(1/(c + d + x)^2)

In[65]:= Integral2[x_, y__] := Module[{a = x, a1, a2, a3, c = {}, d,

k = 1, n = g, b = Length[{y}]}, Clear[g];

While[k <= b, d = {y}[[k]]; AppendTo[c, Unique[g]];

a1 = Replace4[a, d –> c[[k]]]; a2 = Integrate[a1, c[[k]]];

a3 = Replace4[a2, c[[k]] –> d]; a = a3; k++]; g = n; Simplify[a3]]

In[66]:= Integral2[(a + c + h*g + b*d)/(c + h), c + h, b*d]

Out[66]= 1/2*b*d*(2*a + 2*c + b*d + 2*g*h)*Log[c + h]

Along with the above tools of so-called Replace-group, the Replace5 function (basing on the built–in InputString function) can be mentioned, providing wide possibilities on replacement of sub-expressions in expressions in interactive mode. At last, the Replace6 procedure is intended for replacement of elements of a list that are located on the set nesting levels and meet certain conditions [8,16]. The above means are useful in many cases of processing of algebraic expressions that are based on a set of rules, including symbolic differentiation and integrating on the generalized variables that are arbitrary algebraic expressions.

Expressions in strings. The tools that highlight expressions from strings are of unquestionable interest. Meanwhile, in this direction, Mathematica has quite limited facilities and we have programmed a number of facilities of this type [8-16]. Above all, the procedure call ExprsInStrQ[w, y] returns True, if a string w contains correct expressions, and False otherwise; while through the second optional y argument – an indefinite variable – a list of expressions which are in x is returned. The fragment represents source code of the ExprsInStrQ procedure along with a typical example of its application.

In[77]:= ExprsInStrQ[x_ /; StringQ[x], y___] := Module[{a = {}, d,

c = 1, d, j, b = StringLength[x], k = 1},

For[k = c, k <= b, k++, For[j = k, j <= b, j++,

d = StringTake[x, {k, j}];

If[! SymbolQ[d] && SyntaxQ[d], AppendTo[a, d]]]; c++];

a = Select[Map[StringTrim, Map[StringTrim2[#,

{"–", "+", " "}, 3] &, a]], ExpressionQ[#] &];

If[a == {}, False, If[{y} != {} && ! HowAct[{y}[[1]]],

y = DeleteDuplicates[a]]; True]]

In[78]:= {ExprsInStrQ["a (c + d) – b^2 = Sin[x] h*/+", t], t}

Out[78]= {True, {"a*(c + d)", "a*(c + d) – b", "a*(c + d) – b^2",

"(c + d)", "(c + d) – b", "(c + d) – b^2", "c + d", "d", "b", "b^2",

"2", "Sin[x]", "Sin[x]*h", "in[x]", "in[x]*h", "n[x]", "n[x]*h"}}

In a number of problems of manipulation with expressions, including differentiation and integrating on the generalized variables, the question of definition of structure of expression through sub-expressions entering in it including any variables is topical enough. The given problem is solved by the ExprComp procedure, whose the call ExprComp[x] returns the list of all sub-expressions composing x expression, while the procedure call ExprComp[x, z], where the second optional z argument – an undefined variable – through z additionally returns the nested list of sub-expressions of an arbitrary expression x on nesting levels, since the first level. Fragment below represents source code of the ExprComp procedure and an example of its application.

In[7]:= ExprComp[x_, z___] := Module[{a = {x}, b, h = {}, F, q, t=1},

F[y_ /; ListQ[y]] := Module[{c = {}, d, p, k, j = 1},

For[j = 1, j <= Length[y], j++, k = 1;

While[k < Infinity, p = y[[j]];

a = Quiet[Check[Part[p, k], $Failed]];

If[a === $Failed, Break[], If[! SameQ[a, {}], AppendTo[c, a]]];

k++]]; c]; q = F[a];

While[q != {}, AppendTo[h, q]; q = Flatten[Map[F[{#}] &, q]]];

If[{z} != {} && ! HowAct[z],

z = Map[Select[#, ! NumberQ[#] &] &, h]];

Sort[Select[DeleteDuplicates[Flatten[h],

Abs[#1] === Abs[#2] &], ! NumberQ[#] &]]]

In[8]:= ExprComp[(1/b Cos[a+Sqrt[c+d]])/(Tan[1/b] – 1/c^2), g]

Out[8]= {a, 1/b, b, –(1/c^2), c, d, Sqrt[c + d], c + d, a + Sqrt[c + d],

Cos[a + Sqrt[c + d]], 1/b + Cos[a + Sqrt[c + d]], Tan[1/b],

1/(–(1/c^2) + Tan[1/b]), –(1/c^2) + Tan[1/b]}

In[9]:= g

Out[9]= {{1/b + Cos[a + Sqrt[c + d]], 1/(–(1/c^2) + Tan[1/b])},

{1/b, Cos[a + Sqrt[c + d]], –(1/c^2) + Tan[1/b]}, {b, a + Sqrt[c + d],

–(1/c^2), Tan[1/b]}, {a, Sqrt[c + d], 1/c^2, 1/b}, {c + d, c, b}, {c, d}}

The procedure below is of a certain interest and completes this chapter. The procedure call FuncToExpr[f, x] returns the result of applying of a symbol, block, function (including pure functions) or module f (except the f symbol all remaining admissible objects shall have arity 1) to each variable (excluding the standard symbols) of an expression x. While the call FuncToExpr[f, x, y] with the 3^rd optional y argument – a list of symbols – returns the result of applying of a symbol, block, function (including a pure function) or a module f (excepting f symbol all remaining admissible objects shall have arity 1) to each variable (excluding the standard symbols and symbols from y list) of an expression x. The following fragment represents source code of the FuncToExpr procedure along with some typical examples of its application.

In[7]:= FuncToExpr[f_ /; SymbolQ[f] || BlockFuncModQ[f] ||

PureFuncQ[f], x_, y___List] := Module[{a = ToString1[x], b, c},

b = ExtrVarsOfStr[a, 2];

b = Select[b, If[{y} == {}, ! SystemQ[#] &, (! SystemQ[#] &&

! MemberQ[Map[ToString, y], #]) &]];

c = Map[# -> ToString[(f)] <> "@@{" <> # <> "}" &, b];

ReplaceAll[x, Map[ToExpressionRule, c]]]

In[8]:= FuncToExpr[f, Sin[m + n]/(x*G[x, y] + S[y, t])]

Out[8]= Sin[f[m] + f[n]]/(f[x]*f[G][f[x], f[y]] + f[S][f[y], f[t]])

In[9]:= FuncToExpr[G, {a, b, c, d, 77*Sin[y + 72]}]

Out[9]= {G[a], G[b], G[c], G[d], 77*Sin[72 + G[y]]}

In[10]:= FuncToExpr[f, Sin[m + n]/(x*G[x, y] + S[y, t]), {G, S}]

Out[10]= Sin[f[m] + f[n]]/(f[x]*G[f[x], f[y]] + S[f[y], f[t]])

The means presented in the chapter along with other means from our package [8,16], extend, sometimes even substantially, the built–in Mathematica means.

Chapter 4: Software for input–output in Mathematica

The Mathematica language being the built-in programming language that first of all is oriented onto symbolic calculations and processing has rather limited facilities for data processing that first of all are located in external memory of the computer. In this regard the language significantly concedes to traditional programming languages. At the same time, being oriented, first of all, to solution of tasks in symbolic view, the Mathematica language provides a set of tools for access to files that can quite satisfy a wide range of the users of mathematical applications of the Mathematica. In this chapter the tools of access to files are considered on rather superficial level owing to the limited volume, extensiveness of this theme and purpose of the present book. The reader, interested in tools of access to datafiles of the Mathematica can appeal to documentation delivered with the system. At that, for the purpose of development of methods of access to file system of the computer we programmed a number of effective tools that are represented in the MathToolBox [16]. Whereas in the present chapter the attention is oriented on the means which expands standard tools of the Mathematica for ensuring of work with files of the computer. Some of them are useful to practical application in Mathematica. Here it is also appropriate to note that a number of tools of our MathToolBox package [16] subsequently served as analogues in subsequent versions of Mathematica; it applies primarily to file access tools.

4.1. Tools of the Mathematica for work with internal files

Means of Math–language provide access of the user to files of several types that can be conditionally divided into two large groups, namely: internal and external files. During the routine work the system deals with three various types of internal files from which we will note the files having extensions {"nb", "m", "mx"}, their structure is distinguished by the standard system tools and that are important enough already on the first stages of work with system. Before further consideration we will note that the concept of file qualifier (FQ) defining the full path to the required file in file system of the computer or to its subdirectory, practically, completely coincides with similar concept for Maple system excepting that if in the Maple for FQ the format of type {symbol, string} is allowed whereas in the Mathematica for FQ the string format is admissible only.

The function call Directory[] returns an active directory of the current session whereas the call SetDirectory[x] returns a x directory, doing it active in the current session; in addition, as an active (current) directory is understood the directory whose datafiles are processed by means of tools of access if only their names, but not full paths to them are specified. Meanwhile, the SetDirectory function allows only real–life subdirectories as an argument, causing on nonexistent subdirectories an erroneous situation with returning $Failed. On the other hand, the SetDir2 procedure allows to sets the current working directory; at that, as an argument can be also nonexistent subdirectories even on inactive I/O devices as the current subdirectories. The fragment represents source code of the procedure SetDir2 and its use.

In[25]:= SetDir2[x_ /; StringQ[x]] := Module[{a, b, c, t, y},

a = Map[# <> ":" &, Adrive[]];

b = Map[ToUpperCase, StringCases[x, t_ ~~ ":"]];

If[FreeQ[a, b], y = StringReplace[x, b –> a[[1]], 1], y = x];

If[Set[c, Quiet[CreateDirectory[y]]] === $Failed, y,

SetDirectory[c]]]

In[26]:= SetDirectory["F:\\Temp\\Grodno\\78"]

… SetDirectory: Cannot set current directory to

F:\\Temp\\Grodno\\78.

Out[26]= $Failed

In[27]:= SetDir2["F:\\Temp\\Grodno\\78"]

Out[27]= "C:\\Temp\\Grodno\\78"

In[28]:= Adrive[] := Module[{a, b, c, d}, {b, a} = {{},

CharacterRange["A", "Z"]}; Do[d = Directory[]; c = a[[k]];

AppendTo[b, If[Quiet[SetDirectory[c <> ":\\"]] === $Failed,

Nothing, SetDirectory[d]; c]], {k, 1, Length[a]}]; Sort[b]]

In[29]:= Adrive[]

Out[29]= {"C", "D", "E", "F", "G"}

The SetDir2 procedure essentially uses a procedure whose call Adrive[] returns the list of the current active I/O devices of the computer. MathToolBox package [16] provides a number of means along with their certain modifications (SetDir, SetDir1, Adrive1, CopyDir, CopyFileToDir, etc.) that use different useful programming techniques in Mathematica and extending the built–in file access tools useful from a practical standpoint. At the same time, it should be borne in mind that for the reason that the MathToolBox package was developed (truth, with rather large intervals) from 2013 to November 2020, different versions of the Mathematica system were used, so in number of cases is necessary to re–debugging some package tools under the current version of the system. As a rule, this is not particularly difficult.

We programmed a number of rather interesting procedures for ensuring work with files of the Mathematica Input–format whose names have extensions {".nb", ".m", ".txt"}, etc. All such tools are based on analysis of structure of the contents of files returned by access functions, in particular, ReadFullFile. Some of them give a possibility to create rather effective user libraries containing definitions of the Mathematica objects. These and certain other tools have been implemented as a part of a special package supporting the releases 8 ÷ 12.1.1 of Mathematica [16].

Some remarks should be made concerning the system Save function which saves the objects in a file in the Append mode; in addition, indefinite symbols in this file are not saved without of any messages, i.e. the Save call returns Null, i.e. nothing. At the same time, at saving of procedure or function with a name Avz in a file by means of the Save in the file all active objects of the same Avz name in the current session with different headings – the identifiers of their originality – are saved too. For elimination of this situation a generalization of the Save function concerning the saving of Mathematica objects with concrete headings is offered. The Save1 procedure solves the problem whose source code with typical examples of its application are represented by

means of the following fragment.

In[1326]:= Save1[x_String, y_ /; DeleteDuplicates[Map[StringQ,

Flatten[{y}]]][[1]]] := Module[{Rs, t = Flatten[{y}], k = 1},

Rs[n_, m_] := Module[{b, c = ToString[Unique[b]],

a = If[SymbolQ[m], Save[n, m], If[StringFreeQ[m, "["], $Failed,

StringTake[m, {1, Flatten[StringPosition[m, "["]][[1]] – 1}]]]},

If[a === Null, Return[], If[a === $Failed, Return[$Failed],

If[SymbolQ[a], b = DefFunc3[a], Return[$Failed]]]];

If[Length[b] == 1, Save[n, a], b = Select[b, SuffPref[#, m, 1] &]];

If[b != {}, b = c <> b[[1]], Return[$Failed]]; ToExpression[b];

a = c <> a; ToExpression["Save[" <> ToString1[n] <> "," <>

ToString1[a] <> "]"]; BinaryWrite[n, StringReplace[

ToString[StringJoin[Map[FromCharacterCode,

BinaryReadList[n]]]], c –> ""]]; Close[n];];

For[k, k <= Length[t], k++, Rs[x, t[[k]]]]]

In[1327]:= A[x_] := x^2; A[x_, y_] := x+y; A[x_, y_, z_] := x+y+z;
A[x__] := {x}; DefFunc3[A]

Out[1327]= {"A[x_] := x^2", "A[x_, y_] := x + y",

"A[x_, y_, z_] := x + y + z", "A[x__] := {x}"}

In[1328]:= Save1["rans.m", {"A[x_, y_, z_]", "A[x__]"}]

In[1329]:= Clear[A]; << "rans.m"

In[1330]:= DefFunc3[A]

Out[1330]= {"A[x_, y_, z_] := x + y + z", "A[x__] := {x}"}

The call Save1[x, y] saves in a file x the definitions of the objects defined by the second factual y argument – the name of an active object in the current session or its heading in string format, or their combinations in the list format. The Save1 procedure can be used as built-in Save function, and solving a saving problem of the chosen objects activated in the current session in the file differentially on the basis of their headings. A successful call returns Null, doing the demanded savings; otherwise, $Failed or unevaluated call are returned. The previous fragment presents the result of application of the Save1 procedure for a selective saving of the objects activated in the current session in file. In a number of cases the procedure Save1 has undoubted interest.

The function call Save2[x, y], where Save2 – a modification of the Save function, appends to a file x the definitions of the means set by a name or by their list y in the format convenient for subsequent processing by a number of means, particularly, by the CallSave procedure. Thus, the function is a rather useful extension of the built–in Save function [7,11-16].

In a number of cases there is an urgent need of saving in a file of state of the current session with subsequent restoration by means of loading of the file to the current session different from the previous session. In this context, SaveCurrentSession and RestoreCS procedures are useful for saving and restoration of state of the current session respectively. Thus, the procedure call SaveCurrentSession[] saves the state of the Mathematica current session in the m–file "SaveCS.m" with returning of the name of the target file. Whereas the call SaveCurrentSession[x] saves the state of the Mathematica current session in a m–file x with returning of the name of the target x file; in addition, if a x file has not "m" extension then the extension is added to the x string. The call RestoreCS[] restores the Mathematica current session that has been previously stored by means of procedure SaveCurrentSession in "SaveCS.m" file with returning Null, i.e. nothing. While the call RestoreCS[x] restores the Mathematica current session that has been previously stored by means of the procedure SaveCurrentSession in a m-file x with returning Null. In absence of the above m-file the procedure call returns $Failed. The fragment represents source codes of the above procedures along with some typical examples of their application.

In[47]:= SaveCurrentSession[x___String] :=

Module[{a = Names["*"], b = If[{x} == {}, "SaveCS.m",

If[SuffPref[x, ".m", 2], x, x <> ".m"]]}, Save1[b, a]; b]

In[48]:= RestoreCS[x___String] := Module[{a = If[{x} == {},

"SaveCS.m", If[FileExistsQ[x] && FileExtension[x] == "m",

x, $Failed]]}, If[a === $Failed, $Failed, On[General];

Quiet[Get[a]]; Off[General]]]

In[49]:= SaveCurrentSession["C:\\temp\\SaveCS.m"]

Out[49]= "C:\\Temp\\SaveCS.m"

In[50]:= RestoreCS["C:\\Temp\\SaveCS.m"]

So, the represented means are rather useful in a case when is required to create copies of the current Mathematica sessions at certain moments of operating with the system.

4.2. Tools of the Mathematica for work with external files

According to such a quite important indicator as means of access to files, the Mathematica, in our opinion, possesses a number of advantages in comparison with Maple. First of all, Mathematica does automatic processing of hundreds of data formats and their sub-formats on the basis of the unified use of symbolic expressions. For each specific format the correspondence between internal and external presentation of a format is defined using the general mechanism of data elements of Mathematica. For today Mathematica supports many various formats of files for various purposes, their list can be received by means of the variables $ImportFormats (the imported files) and $ExportFormats (the exported files) in quantities 226 and 188 respectively (version Mathematica 12.1), while the basic formats of files for versions 8–12.1.1 are considered rather in details in [1-15].

By the function call FileFormat[x] an attempt to define an input format for a file given by a name x in the string format is made. In a case of existence for the file x of name extension the FileFormat function is, as a whole, similar to the FileExtension function, returning the available extension, except for a case of packages (m–datafiles) when instead of extension the file type "Package" is returned. In addition, in certain cases the format identification is done incorrectly, in particular, the attempt to test a doc–file without an extension returns "XLS", ascribing it to the files created by means of Excel 95/97/2000/XP/2003 that is generally incorrect. The FileFormat function also incorrectly tests I/O device base directories, for example:

In[3331]:= Map[FileFormat, {"C:/", "C:\\"}]

… General: Further output of General::cdir will be

suppressed during this calculation.

… General: Cannot get deeper in directory tree:

C:\\Documents and Settings.

… General: Cannot get deeper in directory tree:

C:\\ProgramData\\Application Data.

… General: Cannot get deeper in directory tree:

C:\\ProgramData\\Desktop.

… General: Further output of General::dirdep will be

suppressed during this calculation.

Out[3331]= {"KML", "KML"}

In[3332]:= FileFormat["C:/Temp\\Burthday"]

Out[3332]= "XLS"

In this regard, the procedures FileFormat1 ÷ FileFormat3 have been programmed that extend the capabilities of the built-in FileFormat function and eliminate the above disadvantages. A version of the FileFormat function attempts to identify file type without its extension, being based on information of the creator of file that is contained in the contents of the file. The FileFormat3 procedure rather accurately identifies data files of the following often used types {DOC, DOCX, PDF, ODT, TXT, HTML}. In addition, concerning the TXT type the verification of a datafile is made in the latter case, believing that the datafile of this type has to consist only of symbols with the decimal codes:

0 ÷ 127 – ASCII symbols

1 ÷ 31 – the control ASCII symbols

32 ÷ 126 – the printed ASCII symbols

97 ÷ 122 – letters of the Latin alphabet in the lower register

129 ÷ 255 – Latin–1 symbols of ISO

192 ÷ 255 – letters of the European languages

The procedure call FileFormat3[x] returns the type of a file given by a name or a classifier x; in addition, if the data file has an extension, it relies as the extension of the data file. Whereas the call FileFormat3[x, y] with the second optional argument – an arbitrary y expression – in a case of file without extension returns its full name with extension defined for it, at the same time renaming the x data file, taking into account the calculated format. The following fragment represents source code of the FileFormat3 procedure and the most typical examples of its use.

In[3332]:= FileFormat3[x_ /; FileExistsQ[x], t___] := Module[{b,

c, a = FileExtension[x]}, If[a != "", ToUpperCase[a],

c = If[Quiet[StringTake[Read[x, String], {1, 5}]] === "%PDF-",

{Close[x], "PDF"}[[–1]], Close[x]; b = ReadFullFile[x];

c = If[! StringFreeQ[b, {"MSWordDoc", "Microsoft Office Word"}], "DOC",

If[! StringFreeQ[b, "docProps"], "DOCX"],

If[! StringFreeQ[b, ".opendocument.textPK"], "ODT",

If[! StringFreeQ[b, {"!DOCTYPE HTML", "text/html"}], "HTML",

If[MemberQ3[Range[0, 255], DeleteDuplicates[Flatten[

Map[ToCharacterCode[#] &, DeleteDuplicates[Characters[b]]]]]],

"TXT", Undefined]]]]];

If[{t} != {}, Quiet[Close[x]]; RenameFile[x, x <> "." <> c], c]]]

In[3333]:= Map[FileFormat2, {"C:/", "C:\\", "E:\\", "E:/",

"C:/Temp", "C:\\Temp"}]

Out[3333]= {"Directory", "Directory", "Directory",

"Directory", "Directory", "Directory"}

In[3334]:= FileFormat3["C:/Temp/Kiri"]

Out[3334]= "DOCX"

In[3335]:= FileFormat3["C:/Temp/Kiri", Agn]

Out[3335]= "C:\\Temp\\Kiri.DOCX"

In[3336]:= Map[FileFormat3, {"C:\\Temp.Burthday",

"c:\\Temp/cinema", "c:/Temp/ransian",

"c:/Temp/Grodno", "c:/Temp/Math_Trials"}]

Out[3336]= {"DOC", "TXT", "HTML", "PDF", "DOCX"}

Now, using the algorithm implemented by FileFormat3 it is rather simple to modify it for testing of other types of data files whose full names have no extension. That can be rather useful in the processing problems of data files. In a certain relation the FileFormat3 procedure complements the built–in FileFormat function along with procedures FileFormat1 and FileFormat2.

The Mathematica provides effective system–independent access to all aspects of data files of any size. At that, the opening and closing of files the following built-in functions of access are used: Close, OpenRead, OpenWrite, OpenAppend. Moreover, a name or full path to a file in the string format acts as a formal argument of the first three functions; at that, the function call OpenWrite[] without factual arguments is allowed, opening a new file located in a subdirectory intended for temporary files for writing. Whereas the Close function closes a file given by its name, full path or a Stream–object. In attempt to close a closed or nonexistent file the system causes an erroneous situation. For elimination of such situation, undesirable in many cases, it is possible to use the simple Closes function doing the closing of any file including a closed or nonexistent file without output of any erroneous messages with returning Null, i.e. nothing, but, perhaps, the name or full path to the closed file:

In[2331]:= Close["C:/Temp/Math/test77.txt"]

…General: C:/Temp/Math/test77.txt is not open.

Out[2331]= Close["C:/Temp/Math/test77.txt"]

In[2332]:= Closes[x_] := Quiet[Check[Close[x], Null]]

In[2333]:= Closes["C:/Temp/Math/test77.txt"]

An object of the following format is understood as a Stream object of the functions of access such as OpenRead, OpenWrite, OpenAppend, Read, BinaryWrite,Write, WriteString, etc:

{OutputStream|InputStream}[, ]

The function call Streams[] returns the list of Stream objects of files opened in the current session including system files. For obtaining the list of all Stream objects of files different from the system files it is possible to use the function call StreamsU[]:

In[2214]:= StreamsU[] := Streams[][[3 ;–1]]

In[2215]:= StreamsU[]

Out[2215]= {OutputStream["C:/temp/galina.txt", 3]}

In[2216]:= CloseAll[] := Map[Close, StreamsU[]]

In[2217]:= CloseAll[]; StreamsU[]

Out[2217]= {}

It must be kept in mind that after the termination of work with an opened file, it remains opened up to its explicit closing by means of the Close function. For closing of all channels and files opened in the current session, excepting system files, it is possible to apply a quite simple CloseAll function, whose call CloseAll[] closes all open both files and channels with return of the list of the closed files.

Similar to the Maple the Mathematica also has opportunity to open the same file on different streams and in various modes, using different coding of its name or path by using alternative registers for letters or/and replacement of "\\" separators of the subdirectories on "/", and vice versa at opening of files. The fragment below illustrates application of a similar approach for opening of the same file on two different channels on reading with the subsequent alternating reading of records from it.

In[2222]:= g = "C:\\temp\\cinema_2020.txt"; {S, S1} = {OpenRead[g], OpenRead[If[UpperCaseQ[StringTake[g, 1]], ToLowerCase[g], ToUpperCase[g]]]}

Out[2222]= {InputStream["C:/temp/cinema_2020.doc", 3],

InputStream["c:\\temp\\cinema_2020.txt", 4]}

In[2223]:= t = {}; For[k = 1, k <= 3, k++,

AppendTo[t, {Read[S], Read[S1]}]]

Out[2223]= {"ran1", "ran1", "ran2", "ran2", "ran3", "ran3"}

Meantime it must be kept in mind that the special attention at opening of the same file on different channels is necessary and, above all, at various modes of access to the file in order to avoid of the possible especial situations, including distortion of data in a file. Whereas in certain cases this approach at operating with large enough files can give a quite notable temporal effect with simplification of certain algorithms of data processing that are in files. The interesting enough examples of application of such approach can be found in our books [1-15].

The Mathematica has useful tools for operating with the pointer defining the current position of scanning of a file. The following functions provide such work: StreamPosition, Skip, SetStreamPosition and Find. So, functions StreamPosition and SetStreamPosition allow to monitor of the current position of the pointer of an open file and to establish for it a new position respectively. Moreover, on the closed or nonexistent datafiles the calls of these functions cause erroneous situations.

Reaction of the Skip function to the status of a file is similar, whereas the function call Find opens a stream to reading from a file. The sense of the presented functions is a rather transparent and in more detail it is possible to familiarize oneself with them, for instance, in books [12-14]. In connection with these tools the question of definition of the status of a file – opened, closed or doesn't exist – arises. In this regard FileOpenQ procedure can be as a rather useful tool [16], whose call FileOpenQ[F] returns the nested list {{R, F, Channel},…} if a F file is open on reading/ writing (R = {"read"|"write"}), F defines actually the F file in the stylized format (LowerCase + all "\\" are replaced on "/") whereas Channel defines the logical channel on which the F file in the mode specified by the first element of the R list was open; if F file is closed, the empty list is returned, i.e. {}, if F file is absent, then $Failed is returned. At that, the nested list is used with the purpose, that the F file can be opened according to syntactically various file specifiers, for example, "Agn47" and "AGN47", that allows to do its processing in the various modes simultaneously.

The FileOpenQ1 is a useful extension of FileOpenQ that was considered above. The call FileOpenQ1[f] returns the nested list of the format {{R, x, y,...,z}, {{R, x1, y1,...,z1}} if a f file is open for reading or writing (R = {"in"|"out"}), and f determines the file in any format (Register + "/" and/or "\\"); if the f file is closed or is absent, the empty list is returned, i.e. {}. Moreover, sub-lists {x, y,...,z} and {x1, y1,...,z1} define files or full paths to them that are open for reading and writing respectively. Moreover, if in the current session all user files are closed, except system files, the call FileOPenQ1[x] on an arbitrary x string returns $Failed. The files and paths to them are returned in formats which are determined in the list returned by the function call Streams[], irrespective of format of f file.

At last, the procedure call FileOpenQ2[x] returns True if a x file is open, and False otherwise (file is closed or absent). Whereas the call FileOpenQ2[x, y] also returns True or False depending on the openness of the file x, returning thru the second optional argument – a symbol y – the list in the string format of the file mode of x {"in", "out"} where x file is open for reading ("in") or writing ("out") accordingly. In addition, the x file can be opened in both modes at the same time. The fragment below represents source code of the FileOpenQ2 procedure and the most typical examples of its application.

In[2186]:= FileOpenQ2[x_String, y___Symbol] :=

Module[{a, b, c}, a = FileExistsQ[b = Stilized[x]];

If[a, c = Map[Stilized, Map[ToString, Streams[][[3 ; –1]]]];

c = Select[c, ! StringFreeQ[#, b] &];

c = Map[StringCases[#, ("o" | "i") ~~ __ ~~ "["] &, c];

c = Flatten[Map[StringTrim[#, "["] &, c]];

c = DeleteDuplicates[Map[StringReplace[#,

"putstream" –> ""] &, Flatten[c]]];

If[{y} == {}, If[c != {}, True, False],

If[c == {}, y = c; False, y = c; True]], False]]

In[2197]:= Write["c:\\temp/cinema_2020.doc", 424767];

Read["C:/temp/cinema_2020.doc"];

In[2198]:= FileOpenQ2["C:\\temp\\cinemA_2020.DOC"]

Out[2198]= True

In[2199]:= {FileOpenQ2["c:/temp/cinema_2020.doc", g], g}

Out[2199]= {True, {"in", "out"}}

In[2200]:= Stilized[x_String] :=

ToLowerCase[StringReplace[x, "/" –> "\\"]]

In[2201]:= Stilized["outputstream[C:/Temp/aAa.txt, 3]"]

Out[2201]= "outputstream[c:\\temp\\aaa.txt, 3]"

The fragment also represents a simple Stilized function that styles an arbitrary character string by replacing all instances of letters to the lower case and "/" to "\\". The function is used by the FileOpenQ2 procedure, making it easier to operate with files that use the same ID in different encoding.

Functions Skip, Find, StreamPosition, SetStreamPosition provide rather effective means for a rather thin manipulation with files and in combination with a number of other functions of access they provide the user with a standard set of functions for files processing, and give opportunity on their base to create own tools allowing how to solve specific problems of operating with files, and in a certain degree to extend standard means of the system. A number of similar means is presented and in our books [1-15], and in our MathToolBox package [16]. In addition to the presented standard operations of files processing, some other tools of the package rather significantly facilitates effective programming of higher level at the solution of many problems of files processing and management of Mathematica. Naturally, the consideration rather in details of earlier represented tools of access to files and the subsequent tools doesn't enter purposes of the present book therefore we will represent relatively them only short excursus in the form of brief information along with some comments on the represented means (see also books on the Mathematica system in [15]).

Among means of processing, the following functions can be noted: FileNames – depending on the coding format returns the list of full paths to the files and/or directories contained in the given directory onto arbitrary nesting depth in file system of the computer. Whereas the functions CopyFile, RenameFile, DeleteFile serve for copying, renaming and removal of the files accordingly. Except the listed means for work with data files the Mathematica has a number of rather useful functions that here aren't considered however with them the reader can familiarize in reference base of the Mathematica or in the corresponding literature [8]. Along with the above functions OpenRead, Read, OpenWrite, Read, Write, Skip and Streams of the lowest level of access to files, the functions Get, Put, Export, Import, ReadList, BinaryReadList, BinaryWrite, BinaryRead are too important in problems of access to files which support operations of reading and writing of data of the required formats. With these means along with a number of quite interesting examples and features of their use, at last with certain critical remarks to their address the reader can familiarize in [1-15]. Meanwhile, these means of access together with the considered means and tools remaining without our attention form a rather developed basis of effective processing of files of various formats.

On the other hand, along with actually processing of the internal contents of files, Mathematica has a number of means for search of files, their testing; work with their names, etc. We will list only some of them, namely: FindFile, FileNameDepth, FileExistsQ, FileBaseName, FileNameSplit, ExpandFileName, FileNameJoin, FileNameTake. With these means along with a range of interesting enough examples and features of their use, at last with some critical remarks to their address the reader can familiarize oneself in our books [1-15].

In particular, as it was noted earlier [1-15], the FileExistsQ function like certain other functions of access during the search is limited only to the directories defined in the predetermined $Path variable. For the purpose of elimination of the deficiency a rather simple FileExistsQ1 procedure has been offered. The following fragment represents source code of the FileExistsQ1 procedure along with a typical example of its application.

In[2226]:= FileExistsQ1[x__ /; StringQ[{x}[[1]]]] :=

Module[{b = {x}, a = SearchFile[{x}[[1]]]},

If[a == {}, False, If[Length[b] == 2 && ! HowAct[b[[2]]],

ToExpression[ToString[b[[2]]] <> " = " <> ToString1[a]], 1]; True]]

In[2227]:= FileExistsQ1["KDP_Book_2020.doc", t42]

Out[2227]= True

In[2228]:= t42

Out[2228]= {"C:\\Mathematica\\KDP_Book_2020.doc",

"E:\\Mathematica\\KDP_Book_2020.doc"}

The procedure call FileExistsQ1[x] returns True if x defines a real data file in system of directories of the computer and False otherwise; whereas the call FileExistsQ1[x, y], in addition thru the second actual argument – an indefinite variable – returns the list of full paths to the found data file x if the basic result of the call is True. That procedure substantially uses the procedure SearchFile, whose call SearchFile[F] returns the list containing full paths to a data file F found in file system of the computer; if the data file F has not been found, the empty list is returned. We will note, the procedure SearchFile uses the Run function that is used by a number of tools of our MathToolBox package [16]. The following fragment presents source code of the SearchFile procedure along with a typical example of its application.

In[2322]:= SearchFile[F_ /; StringQ[F]] :=

Module[{a, b, f, dir, h = Stilized[F], k},

{a, b, f} = {Map[ToUpperCase[#] <> ":\\" &, Adrive[]], {}, "###.txt"};

dir[y_ /; StringQ[y]] := Module[{a, b, c, v},

Run["Dir " <> "/A/B/S " <> y <> " > " <> f];

c = {}; Label[b]; a = Stilized[ToString[v = Read[f, String]]];

If[a == "endoffile", Close[f]; DeleteFile[f]; Return[c],

If[SuffPref[a, h, 2], If[FileExistsQ[v], AppendTo[c, v]];

Goto[b], Goto[b]]]];

For[k = 1, k <= Length[a], k++, AppendTo[b, dir[a[[k]]]]]; Flatten[b]]

In[2323]:= SearchFile["KDP_Book_2020.doc"]

Out[2323]= {"C:\\Mathematica\\KDP_Book_2020.doc",

"E:\\Mathematica\\KDP_Book_2020.doc"}

The SearchFile1 procedure is a functional analogue of the above SearchFile procedure. Unlike the previous procedure the SearchFile1 seeks out a file in three stages: (1) if the required file is set by a full path only existence of the concrete file is checked, at detection the full path to it is returned, (2) search is done in the list of directories determined by the predetermined $Path variable, (3) a search is done within file system of the computer. Note, that speed of both procedures essentially depends on the sizes of file system of the computer, first of all, if a required file isn't defined by the full path and isn't in the directories defined by the $Path variable. In addition, in this case the search is done if possible even in directory "c:\\$recycle.bin" of the Windows 7.

Along with tools of processing of external files the system has also the set of useful enough means for manipulation with directories of the Mathematica, and file system of the computer in general. We will list only some of these important functions:

DirectoryQ[j] – the call returns True if a j string determines an existing directory, and False otherwise; unfortunately, the standard function at coding "/" at the end of j string returns False irrespective of existence of the tested directory; a quite simple DirQ procedure [8] eliminates that defect of the standard function;

DirectoryName[d] – the call returns the path to a directory that contains a d file; if d is a real subdirectory then chain of subdirectories to it is returned; in addition, taking into account the file concept that identifies files and subdirectories, and the circumstance that the call DirectoryName[d] doesn't consider actual existence of d, the similar approach in a certain measure could be considered justified, however on condition of taking into account of reality of the tested d path such approach causes certain questions. Therefore, from this standpoint a rather simple DirName procedure [6] which returns "None" if d is a subdirectory, the path to a subdirectory containing d file, and $Failed otherwise is offered. At that, the search is done within full file system of the computer, but only not within system of subdirectories defined by means of the predetermined $Path variable;

CreateDirectory[d] – the call creates the given d directory with return of the path to it; meanwhile, such means doesn't work in a case of designation of the nonexistent device of external memory (flash card, disk, etc.) therefore we created a rather simple CDir procedure [7] that resolves this problem: the procedure call CDir[d] creates the given d directory with return of the full path to it; in the absence or inactivity of the device of external memory the directory is created on a device from the list of all active devices of external memory that has maximal volume of available memory with returning of the full path to it;

CopyDirectory[j1, j2] – the call completely copies a j1 directory into a j2 directory, but in the presence of the accepting j2 directory the call CopyDirectory[j1, j2] causes an erroneous situation with return of $Failed that in a number of cases is undesirable. For the purpose of elimination of such situation a rather simple CopyDir procedure can be offered, which in general is similar to the standard CopyDirectory function, but with the difference that in the presence of the accepting j2 directory the j1 directory is copied as a subdirectory of the j2 with returning of the full path to it;

DeleteDirectory[d] – the function call deletes from file system of the computer the given d directory with returning Null, i.e. nothing, regardless of attributes of the directory (Archive, Read–only, Hidden, and System). Meanwhile, such approach, in our opinion, is not quite justified, relying only on the circumstance that the user is precisely sure what he does. While in general there has to be an insurance from removal, in particular, of files and directories having such attributes as read-only (R), hidden (H) and system (S). To this end, particularly, it is possible before removing of an element of file system to previously check up its attributes what the useful Attrib procedure considered in the following section provides.

4.3. Tools of Mathematica for attributes processing of directories and data files

The Mathematica has no tools of operating with attributes of files and directories what, in our opinion, is a rather essential shortcoming at creation on its basis of various data processing systems. By the way, the similar means are absent in the Maple also, therefore we created for it a set of procedures {Atr, F_atr, F_atr1, F_atr2} that have solved this problem [41]. These tools presented below solve this problem for the Mathematica. The fragment below represents the Attrib procedure that provides processing of attributes of files and directories.

In[3327]:= Attrib[F_ /; StringQ[F], x_ /; ListQ[x] &&

AllTrue[Map3[MemberQ, {"–A", "–H", "–S", "–R", "+A", "+H",

"+S", "+R"}, x], TrueQ] || x == {} || x == "Attr"] :=

Module[{a, b = "attrib ", c, d = " > ", h = "attrib.exe", p, f, g, t, v},

a = ToString[v = Unique["ArtKr"]];

If[Set[t, LoadExtProg["attrib.exe"]] === $Failed,

Return[$Failed], Null];

If[StringLength[F] == 3 && DirQ[F] &&

StringTake[F, {2, 2}] == ":", Return["Drive " <> F],

If[StringLength[F] == 3 && DirQ[F], f = StandPath[F],

If[FileExistsQ1[StrDelEnds[F, "\\", 2], v], g = v;

f = StandPath[g[[1]]]; Clear[v], Return["<" <> F <> "> is not a

directory or a datafile"]]]];

If[x === "Attr", Run[b <> f <> d <> a],

If[x === {}, Run[b <> " –A –H –S –R " <> f <> d <> a],

Run[b <> StringReplace[StringJoin[x], {"+" –> " +",

"–" –> " –"}] <> " " <> f <> d <> a]]];

If[FileByteCount[a] == 0, Return[DeleteFile[a]],

d = Read[a, String]; DeleteFile[Close[a]]];

h = StringSplit[StringTrim[StringTake[d, {1, StringLength[d] –

StringLength[f]}]]]; Quiet[DeleteFile[t]];

h = Flatten[h /. {"HR" -> {"H", "R"}, "SH" –> {"S", "H"},

"SHR" –> {"S", "H", "R"}, "SRH" –> {"S", "R", "H"},

"HSR" –> {"H", "S", "R"}, "HRS" –> {"H", "R", "S"},

"RSH" –> {"R", "S", "H"}, "RHS" –> {"R", "H", "S"}}];

If[h === {"File", "not", "found", "–"} ||

MemberQ[h, "C:\\Documents"], "Drive " <> f, {h, g[[1]]}]]

In[3328]:= Attrib["C:\\Mathematica", "Attr"]

Out[3328]= {{}, "C:\\Mathematica"}

In[3329]:= Attrib["c:/temp\\", {"+A", "+R"}]

In[3330]:= Attrib["c:/temp\\", "Attr"]

Out[3330]= {{"A", "R"}, "C:\\Temp"}

In[3331]:= Attrib["c:/temp\\", {"–R"}]

In[3332]:= Attrib["c:/temp\\", "Attr"]

Out[3332]= {{"A"}, "C:\\Temp"}

The successful procedure call Attrib[w, "Attr"] returns the list of attributes of a file or directory w in the context Archive ("A"), Read–only ("R"), Hidden ("H") and System ("S"). At that, also other attributes inherent to the system files and directories are possible; in particular, on the main directories of devices of external memory "Drive w" whereas on a nonexistent directory or file the message "w isn't a directory or file" is returned. At that, the call is returned in the form of list of the format {x, y, …, z, Pt} where the last element determines a full path to a file or directory w; the files and subdirectories of the same name can be in various directories, however processing of attributes is made only concerning the first file/directory from the list of objects of the same name. If the full path to a file or directory w is defined as the first argument of the Attrib procedure, only this object is processed. The elements of the returned list that precede its last element define attributes of the processed file or directory.

The procedure call Attrib[w, {}] returns Null, i.e. nothing, cancelling all attributes for a processed file or directory w while the call Attrib[w, {"x","y",…,"z"}] where x,y,z{"-A","-H","-S", "–R", "+A", "+H", "+S", "+R"}, also returns Null, i.e. nothing, setting/cancelling the attributes of the processed directory or file w determined by the second argument. At impossibility to execute processing of attributes the call Attrib[w, x] returns the corresponding messages. Procedure Attrib allows to carry out processing of attributes of a file or a directory that is located in any place of file system of the computer. The Attrib procedure is represented to us as a rather useful means for operating with file system of the computer.

In turn, the Attrib1 procedure in many respects is similar to the Attrib procedure both in the functional, and in descriptive relation, but the Attrib1 procedure has also certain differences. The successful procedure call Attrib1[w, "Attr"] returns the list of attributes in the string format of a directory or a file w in the context Archive ("A"), Hidden ("H"), System ("S"), Read–only ("R"). The call Attrib1[w, {}] returns Null, cancelling all attributes for the processed file/directory w whereas the call Attrib1[w, {"x", "y",…,"z"}] where x, y, z{"–A", "–H", "–S", "–R", "+A", "+H", "+S", "+R"}, also returns Null, i.e. nothing, setting/cancelling the attributes of the processed file or directory w defined by the second argument, whereas the call Attrib1[w, x, y] with the 3^rd optional y argument – any expression – in addition deletes the program file "attrib.exe" from the directory determined by the call Directory[]. The source code of the Attrib1 procedure with the most typical examples of its use can be found in [8,10,16].

At the same time, the possible message "cmd.exe – Corrupt File" that arises at operating of the above procedures Attrib and Attrib1 should be ignored. Both procedures essentially use our procedures LoadExtProg, StrDelEnds, StandPath, FileExistsQ1 and DirQ along with use of the standard Run function and the Attrib command of the MS DOS. In addition, the possibility of removal of the "attrib.exe" program file from the directory that is determined by the call Directory[] after the call of the Attrib1 leaves the Mathematica unchanged. So, in implementation of both procedures the Run function was enough essentially used, which has the following coding format:

Run[s1,…,sn] – in the basic operational system (for example, MS DOS) executes a command that is formed from expressions sj (j=1..n) that are parted by blank symbols with return of code of success of the command completion in the form of an integer. As a rule, the Run function does not demand of an interactive input, however on some operational platforms it generates text messages. In [1-15] rather interesting examples of application of the Run function at programming in the Mathematica with the MS DOS commands are represented.

Note that using of the Run function illustrates one of rather useful methods of interface with the basic operational platform, but here two very essential moments take place. Above all, the Run function on some operational platforms (e.g., Windows XP Professional) demands certain external reaction of the user at an exit from the Mathematica into the operational platform, and secondly, the call by means of the Run function of functions or MS DOS commands assumes their existence in the directories system determined by the $Path variable since, otherwise the Mathematica doesn't recognize them.

For instance, similar situation takes place in the case of use of the external DOS commands, for this reason in realization of the procedures Attrib, Attrib1 which through the Run use the external command "attrib" of DOS, the connection to system of directories of $Path of the directories containing the "attrib.exe" utility has been provided whereas for internal commands of the DOS it isn't required. Once again, possible messages "cmd.exe – Corrupt File" which can arise at execution of the Run function should be ignored.

Thus, at using of the internal command Dir of MS DOS of an extension of the list of directories defined by the $Path is not required. At the same time, on the basis of standard reception on the basis of extension of a list defined by the $Path variable the Mathematica doesn't recognize the external commands of MS DOS. In this regard we programmed procedure whose call LoadExtProg[x] provides search in file system of the computer of a program x set by the full name with its subsequent copying into a directory defined by the call Directory[]. The procedure call LoadExtProg[x] searches out x data file in file system of the computer and copies it to the directory determined by the call Directory[], returning Directory[] <> "\\" <> x if the file already was in this directory or has been copied into this directory.

The first directory containing the found x file supplements the list of the directories defined by the predetermined $Path variable. Whereas the procedure call LoadExtProg[x, y] with the second optional y argument – an indefinite variable – in addition through y returns the list of all full paths to the found x data file without a modification of the directories list determined by the predetermined $Path variable. In case of absence of a possibility to find the required x file $Failed is returned. The next fragment presents source code of the LoadExtProg procedure along with an example of its application, in particular, for loading into the directory determined by the call Directory[] of a copy of external command "attrib.exe" of MS DOS with check of the result.

In[2232]:= LoadExtProg[x_ /; StringQ[x], y___] :=

Module[{a = Directory[], b = Unique["agn"], c, d, h},

If[PathToFileQ[x] && FileExistsQ[x],

CopyFileToDir[x, Directory[]], If[PathToFileQ[x] &&

! FileExistsQ[x], $Failed, d = a <> "\\" <> x;

If[FileExistsQ[d], d, h = FileExistsQ1[x, b];

If[h, CopyFileToDir[b[[1]], a];

If[{y} == {}, AppendTo[$Path, FileNameJoin[

FileNameSplit[b[[1]]][[1 ;–2]]]], y = b]; d, $Failed]]]]]

In[2233]:= LoadExtProg["C:\\attrib.exe"]

Out[2233]= $Failed

In[2234]:= LoadExtProg["attrib.exe"]

Out[2234]= "c:\\users\\aladjev\\documents\\attrib.exe"

In[2235]:= FileExistsQ[Directory[] <> "\\" <> "attrib.exe"]

Out[2235]= True

In[2236]:= Attrib1["c:\\Mathem\\kdp_book_2020.doc", "Attr"]

Out[2236]= {"A", "S", "R"}

Therefore, in advance by means of the call LoadExtProg[x] it is possible to provide access to a necessary x file if of course it exists in file system of the computer. So, using the LoadExtProg in total with the system Run function, it is possible to execute a number of very useful {.exe|.com} programs in the Mathematica software – the programs of different purpose that are absent in file system of the Mathematica thereby significantly extending the functionality of Mathematica that is demanded by a rather wide range of various appendices.

The above LoadExtProg and FileExistsQ1 procedures use the CopyFileToDir procedure whose call CopyFileToDir[x, y] provides copying of a file or directory x into a y directory with return of the full path to the copied file/directory. If the copied file already exists, it isn't updated if the target directory already exists, the x directory is copied into its subdirectory of the same name. The fragment presents source code of the CopyFileToDir procedure with some typical examples of its application:

In[5]:= CopyFileToDir[x_ /; PathToFileQ[x], y_ /; DirQ[y]] :=

Module[{a, b}, If[DirQ[x], CopyDir[x, y], If[FileExistsQ[x],

a = FileNameSplit[x][[–1]];

If[FileExistsQ[b = y <> "\\" <> a], b, CopyFile[x, b]], $Failed]]]

In[6]:= CopyFileToDir["c:/math/book_2020.doc", Directory[]]

Out[6]= "C:\\Users\\Aladjev\\Documents\\book_2020.doc"

In[7]:= CopyFileToDir["C:\\Temp", "E:\\Temp"]

Out[7]= "E:\\Temp\\Temp"

The CopyFileToDir procedure has a variety of appendices in processing problems of file system of the computer.

In conclusion of the section an useful enough procedure is represented which provides only two functions – (1) obtaining of the list of attributes ascribed to a file or directory, and (2) the removal of all ascribed attributes. The call Attribs[x] returns the list of attributes in string format which are ascribed to a file or a directory x. On the main directories of volumes of direct access the procedure call Attribs returns $Failed, while the procedure call Attribs[x, y] with the 2^nd optional y argument – an arbitrary expression – deletes all attributes which are ascribed to a file or a directory x with returning 0 at a successful call. The following fragment represents source code of the Attribs procedure with the most typical examples of its application:

In[3330]:= Attribs[x_ /; FileExistsQ[x] || DirectoryQ[x], y___] :=

Module[{b, a = StandPath[x], d = ToString[Unique["g"]],

c = "attrib.exe", g},

If[DirQ[x] && StringLength[x] == 3 &&

StringTake[x, {2, 2}] == ":", $Failed,

g[] := Quiet[DeleteFile[Directory[] <> "\\" <> c]];

If[! FileExistsQ[c], LoadExtProg[c]];

If[{y} == {}, Run[c <> " " <> a <> " > ", d]; g[];

b = Characters[StringReplace[StringTake[Read[d, String],

{1, –StringLength[a] – 1}], " " –> ""]]; DeleteFile[Close[d]]; b,

a = Run[c <> " –A –H –R –S " <> a]; g[]; a]]]

In[3331]:= Map[Attribs, {"c:\\", "e:/", "c:/tm", "c:/tm/bu.doc"}]

Out[3331]= {$Failed, $Failed, {"A"}, {"A", "R", "S"}}

In[3332]:= Attribs["c:/tm/bu.doc", 77]

Out[3332]= 0

In[3333]:= Attribs["c:/tm/bu.doc"]

Out[3333]= {}

Note, that as x argument the usage of an existing file, full path to a file or a directory is supposed. At the same time, the file "attrib.exe" is removed from the directory defined by the call Directory[] after a procedure call. The Attribs procedure is rather fast–acting, supplementing procedures Attrib, Attrib1. The Attribs procedure is effectively applied in programming of means of access to elements of file system of the computer at processing their attributes. Thus, it should be noted once again that the Mathematica has no means for processing of attributes of files and directories therefore the offered procedures Attrib, Attrib1 and Attribs in a certain measure fill this niche.

So, the declared possibility of extension of the Mathematica directories that is determined by the $Path variable, generally doesn't operate already concerning the external commands of MS DOS what well illustrates both consideration of the above procedures Attrib, LoadExtProg and Attrib1, and an example with the external "tasklist" command which is provided display of all active processes of the current session with Windows 7:

In[2565]:= Run["tasklist", " > ", "C:\\Temp\\tasklist.txt"]

Out[2565]= 1

In[2566]:= LoadExtProg["tasklist.exe"];

Run["tasklist" <> " > " <> "C:\\Temp\\tasklist.txt"]

Out[2566]= 0

: System Idle Process

: 0

: Services

: 0

===============

: Skype.exe

: 2396

: Console

: 1

===============

: Mathematica.exe

: 4120

: Console

: 1

: WolframKernel.exe

: 5888

: Console

: 1

===============

The first example of the previous fragment illustrates that attempt by means of the Run to execute the external command "tasklist" of DOS completes unsuccessfully (return code 1), while result of the call LoadExtProg["tasklist.exe"] with search and upload into the directory defined by the call Directory[] of the "tasklist.exe" file, allows to successfully execute by means of the Run function the external command "tasklist" with preserving of result of its performance in file of txt-format whose contents is represented by the shaded area. Meanwhile, use of external software on the basis of the Run function along with possibility of extension of functionality of the Mathematica causes a rather serious portability question. So, the means developed by means of this method with use of the external commands of MS DOS are subject to influence of variability of the commands of DOS depending on version of basic operating system. In a number of cases it demands a certain adaptation of the software according to the basic operating system.

4.4. Additional tools for files and directories processing of file system of the computer

This item presents tools of files and directories processing of file system of the computer which supplement and in certain cases also extend the above means. Unlike the DeleteDirectory and DeleteFile functions the DelDirFile1 procedure removes a directory or file from file system of the computer.

Similarly to functions DeleteFile and DeleteDirectory, the call DelDirFile[w] removes from file system of the computer a file or directory w, returning Null, i.e. nothing. Whereas the call DelDirFile[w, y] with the 2^nd optional argument y – an arbitrary expression – removes from file system of the computer a file or a directory w, irrespectively from existence of attribute Read–only for it. The DelDirFile1 procedure is a rather useful extension of the DelDirFile procedure [2] on a case of open files in addition to the Read–only attribute of both the separate files, and the files being in the deleted directory. The procedure call DelDirFile1[x] is equivalent to the call DelDirFile[w, y], providing removal of a file or directory x irrespective of openness of a separate w file and the Read–only attribute ascribed to it, or existence of similar files in w directory. The fragment represents source code of the DelDirFile1 procedure along with typical examples of its use.

In[2242]:= DelDirFile1[x_ /; StringQ[x] && FileExistsQ[x] ||

DirQ[x] && If[StringLength[x] == 3 &&

StringTake[x, {2, 2}] == ":", False, True]] :=

Module[{a = {}, b = "", c = Stilized[x], d, f = "#$#", k = 1},

If[DirQ[x], Run["Dir " <> c <> " /A/B/OG/S > " <> f]; Attribs[c, 7];

For[k, k < Infinity, k++, b = Read[f, String];

If[SameQ[b, EndOfFile], DeleteFile[Close[f]]; Break[],

Attribs[b, 7]; Close2[b]]];

DeleteDirectory[x, DeleteContents –> True], Close2[x];

Attribs[x, 7]; DeleteFile[x]]]

In[2243]:= Attribs["c:\\temp\\paper_201120.docx"]

Out[2243]= {"A", "S", "H", "R"}

In[2244]:= DelDirFile1["c:\\temp\\paper_201120.docx"]

In[2245]:= FileExistsQ["c:\\temp\\paper_201120.docx"]

Out[2245]= False

In[2246]:= DirectoryQ["c:\\temp\\Rans_IAN"]

Out[2246]= True

In[2247]:= Attribs["c:\\temp\\Rans_IAN"]

Out[2247]= {"A", "S", "H", "R"}

In[2248]:= DelDirFile1["c:\\temp\\Rans_IAN"]

In[2249]:= DirectoryQ["c:\\temp\\Rans_IAN"]

Out[2249]= False

Meanwhile, before representation of the following means it is expedient to determine one a rather useful procedure whose essence is as follows. As noted above, the file qualifier depends both on register of symbols, and the used dividers of directories. So, the same file with different qualifiers "c:\\Tmp\\cinem.txt" and "c:/tmp/cinem.txt" opens in 2 various streams. Therefore its closing by means of the built-in Close function doesn't close the "cinem.txt" file, demanding closing of all streams on which it was earlier open. For solution of this problem the Close1 and Close2 procedures have been programmed.

In[3]:= Read["C:/Temp\\paper_201120.docx"];

Read["C:/Temp/paper_201120.docx"];

Read["C:/Temp\\Paper_201120.docx"];

Read["c:/temp/paper_201120.Docx"]; StreamsU[]

Out[3]= {InputStream["C:/Temp\\paper_201120.docx", 3],

InputStream["C:/Temp/paper_201120.docx", 4],

InputStream["C:/Temp\\Paper_201120.docx", 5],

InputStream["c:/temp/paper_201120.Docx", 6]}

In[4]:= Close["C:\\Temp\\paper_201120.docx"]

…General: C:\Temp\paper_201120.docx is not open.

Out[4]= Close["C:\\Temp\\paper_201120.docx"]

In[5]:= StreamsU[]

Out[5]= {InputStream["C:/Temp\\paper_201120.docx", 3],

InputStream["C:/Temp/paper_201120.docx", 4],

InputStream["C:/Temp\\Paper_201120.docx", 5],

InputStream["c:/temp/paper_201120.Docx", 6]}

In[5]:= Close1[x___String] := Module[{a = Streams[][[3 ;–1]],

b = {x}, c = {}, k = 1, j},

If[a == {} || b == {}, {}, b = Select[{x}, FileExistsQ[#] &];

While[k <= Length[a], j = 1; While[j <= Length[b],

If[Stilized[a[[k]][[1]]] == Stilized[b[[j]]],

AppendTo[c, a[[k]]]]; j++]; k++];

Map[Close, c]; If[Length[b] == 1, b[[1]], b]]]

In[7]:= Close1["C:\\Temp\paper_201120.docx"]

Out[7]= "C:\\Temp\\paper_201120.docx"

In[13]:= StreamsU[]

Out[13]= {}

In[21]:= Close2[x___String] := Module[{a = Streams[][[3 ;–1]],

b = {}, c, d = Select[{x}, StringQ[#] &]},

If[d == {}, {}, c[y_] := Stilized[y];

Map[AppendTo[b, Part[#, 1]] &, a];

d = DeleteDuplicates[Map[c[#] &, d]];

Map[Close, Select[b, MemberQ[d, c[#]] &]]]]

In[22]:= Close["C:\\Temp\\paper_201120.docx"]

…General: C:\Temp\paper_201120.docx is not open.

Out[22]= Close["C:\\Temp\\paper_201120.docx"]

In[23]:= Close2["C:\\Temp\\paper_201120.docx"]

Out[23]= {"C:/Temp\\paper_201120.docx",

"C:/Temp/paper_201120.docx",

"C:/Temp\\Paper_201120.docx",

"c:/temp/paper_201120.Docx"}

In[24]:= StreamsU[]

Out[24]= {}

The call Close1[x, y, z, …] closes all off really–existing files in a {x, y, z, …} list irrespective of quantity of streams on which they have been opened by means of various files qualifiers with returning their list. In other cases the call on admissible factual arguments returns the empty list, while on inadmissible factual arguments a call is returned unevaluated. The procedure Close2 is a functional analogue of the above procedure Close1. The call Close2[x, y, z,…] closes all off really–existing files in a {x, y, z,…} list irrespective of quantity of streams on which they have been opened by various files qualifiers with returning their list. The call on any admissible factual arguments returns the empty list, whereas on inadmissible actual arguments the call is returned unevaluated. The previous fragment represents source codes of both procedures along with examples of their use. In a number of appendices Close1 and Close2 are quite useful means.

The tools representing a quite certain interest at operating with file system of the computer as independently and as a part of tools of files processing and directories complete the section. They are used and by a number of means of our MathToolBox package [16]. In particular, at operating with files the OpenFiles procedure can be rather useful, whose call OpenFiles[] returns the two–element nested list, whose the first sub-list with the 1^st "read" element contains full paths to the files opened on reading whereas the 2^nd sub-list with the first "write" element contains full paths to files opened on writing in the current session. In a case of absence of such files the call returns the empty list, i.e. {}. Whereas the call OpenFiles[x] with one factual x argument – a file classifier – returns the result of the above format relative to an open x file irrespective of a format of coding of its qualifier. If x defines a closed or a nonexistent file then the procedure call returns the empty list, i.e. {}. The following fragment represents source code of the procedure with a typical example of its use.

In[333]:= OpenFiles[x___String] := Module[{a = StreamsU[],

b, c, d, h1 = {"read"}, h2 = {"write"}},

If[a == {}, {}, d = Map[{Part[#, 0], Part[#, 1]} &, a];

b = Select[d, #[[1]] == InputStream &];

c = Select[d, #[[1]] == OutputStream &];

b = Map[DeleteDuplicates,

Map[Flatten, Gather[Join[b, c], #1[[1]] == #2[[1]] &]]];

b = Map[Flatten, Map[If[SameQ[#[[1]], InputStream],

AppendTo[h1, #[[2 ;–1]]], AppendTo[h2, #[[2 ;–1]]]] &, b]];

If[{x} == {}, b, If[SameQ[FileExistsQ[x], True], c = Map[Flatten,

Map[{#[[1]], Select[#, Stilized[#] === Stilized[x] &]} &, b]];

If[c == {{"read"}, {"write"}}, {}, c = Select[c, Length[#] > 1 &];

If[Length[c] > 1, c, c[[1]]], {}]]]]]

In[334]:= Write["c:/temp\\paper_201120.docx", 211120];

Read["C:/Temp/paper_201120.docx"];

In[335]:= OpenFiles["C:\\temp\\paper_201120.docx"]

Out[335]= {{"read", "C:/Temp/paper_201120.docx"},

{"write", "c:/temp\\paper_201120.docx"}}

As a procedure similar to the OpenFiles, the procedure can be used, whose call StreamFiles[] returns the nested list from 2 sub-lists, the first sub-list with the first "in" element contains full paths/names of the files opened on the reading whereas the 2^nd sublist with the first "out" element contains full paths/names of the files opened on the recording. Whereas in the absence of the open files the call StreamFiles[] returns "AllFilesClosed".

In[2336]:= StreamFiles[]

Out[2336]= {{"in", "C:/Temp/paper_201120.docx"},

{"out", "c:/temp\\paper_201120.docx"}}

In[2337]:= CloseAll[]; StreamFiles[]

Out[2337]= "AllFilesClosed"

The following procedure is represented as a rather useful tool at operating with file system of the computer, whose call DirEmptyQ[w] returns True if w directory is empty, otherwise False is returned. At that, the procedure call DirEmptyQ[w] is returned unevaluated, if w isn't a real directory. At the same time, the call DirEmptyQ[w, y] with optional argument y – an indefinite variable – through y returns the contents of w directory in format of Dir command of MS DOS. The following fragment presents source code of the DirEmptyQ procedure with typical enough examples of its application.

In[3128]:= DirEmptyQ[d_ /; DirQ[d], y___] :=

Module[{a, b = {"$$1", "$$2"}, c, p = Stilized[d], t = 1},

Map[Run["Dir " <> p <> If[SuffPref[p, "\\", 2], "", "\\"] <> # <>

" > " <> b[[t++]]] &, {" /S/B ", " /S "}];

c = If[ReadString[b[[1]]] === EndOfFile, True, False];

If[{y} != {} && SymbolQ[y],

y = StringReplace[ReadString[b[[2]]], "\n" –> ""], 7];

DeleteFile[b]; c]

In[3129]:= DirEmptyQ["c:\\Galina47"]

Out[3129]= DirEmptyQ["c:\\Galina47"]

In[3130]:= DirEmptyQ["c:\\Galina", gs]

Out[3130]= False

In[3131]:= gs

Out[3131]= "Volume in drive C is OS

Volume Serial Number is E22E-D2C5

Directory of c:\\galina

21.11.2020 15:27

.

21.11.2020 15:27

..

21.11.2020 17:46

Victor

0 File(s) 0 bytes

Directory of c:\\galina\\Victor

21.11.2020 17:46

.

21.11.2020 17:46

..

0 File(s) 0 bytes

Total Files Listed:

0 File(s) 0 bytes

5 Dir(s) 288 095 285 248 bytes free"

In addition to the previous DirEmptyQ procedure the call DirFD[j] returns the 2-element nested list whose the 1^st element determines the list of subdirectories of the first nesting level of a j directory whereas the second element – the list of files of a j directory; if the j directory is empty, the procedure call returns the empty list, i.e. {}. The fragment below presents source code of the DirFD procedure with an example of its application.

In[320]:= DirFD[d_ /; DirQ[d]] := Module[{a = "$#$", b = {{}, {}},

c, h, t, p = StandPath[StringReplace[d, "/" –> "\\"]]},

If[DirEmptyQ[p], Return[{}], Null];

c = Run["Dir " <> p <> " /B " <> If[SuffPref[p, "\\", 2], "", "\\"]

<> "*.* > " <> a];

t = Map[ToString, ReadList[a, String]]; DeleteFile[a];

Map[{h = d <> "\\" <> #; If[DirectoryQ[h], AppendTo[b[[1]], #],

If[FileExistsQ[h], AppendTo[b[[2]], #], Null]]} &, t]; b]

In[321]:= DirFD["C:\\Program Files\\Wolfram Research\\

Mathematica\\12.1\\Documentation\\English\\Packages"]

Out[321]= {{"ANOVA", "Audio", "AuthorTools", …, {}}}

In[322]:= Length[%[[1]]]

Out[322]= 48

In particular, in the previous example the list of directories with records on the packages delivered with the Mathematica 12.1 is returned. So, with Mathematica 48 packages of different purpose that is simply seen from the name of the subdirectories containing them are being delivered.

In addition to the DirFD procedure, the DirFull procedure presents a certain interest whose call DirFull[j] returns list of all full paths to subdirectories and files contained in a j directory and its subdirectories; the 1^st element of the list is the j directory. Whereas on an empty j directory the call returns the empty list.

At last, the procedure call TypeFilesD[d] returns the sorted list of types of files located in a d directory with returning word "undefined" on files without of name extension. In addition, the files located in the d directory and in all its subdirectories of an arbitrary nesting level are considered. Moreover, on the empty d directory the call TypeFilesD[d] returns the empty list, i.e. {}.

The FindFile1 procedure serves as a useful extension of the standard FindFile function, providing search of a file within file system of the computer. The call FindFile1[x] returns a full path to the found file x, or the list of full paths (if the x file is located in different directories of file system of the computer), otherwise the call returns the empty list. Whereas the call FindFile1[x, y] with the 2^nd optional y argument – full path to directory – returns full path to the found x file, or list of full paths located in the y directory and its subdirectories. At the same time, the FindFile2 function serves as other useful extension of standard FindFile function, providing search of a y file within a x directory. The function call FindFile2[w, y] returns the list of full paths to the found y file, otherwise the call returns the empty list. Search is done in the w directory and all its subdirectories. The fragment below presents source codes of the FindFile1 procedure and FindFile2 function with some typical examples of their application.

In[7]:= FindFile1[x_ /; StringQ[x], y___] := Module[{c, d = {}, k = 1,

a = If[{y} != {} && PathToFileQ[y], {y},

Map[# <> ":\\" &, Adrive[]]], b = "\\" <> ToLowerCase[x]},

For[k, k <= Length[a], k++,

c = Map[ToLowerCase, Quiet[FileNames["*", a[[k]], Infinity]]];

d = Join[d, Select[c, SuffPref[#, b, 2] && FileExistsQ[#] &]]];

If[Length[d] == 1, d[[1]], d]]

In[8]:= FindFile1["Book_2020.doc"]

Out[8]= {"c:\\ma\\book_2020.doc", "e:\\ma\\book_2020.doc"}

In[9]:= FindFile2[x_ /; DirectoryQ[x], y_ /; StringQ[y]] :=

Select[FileNames["*.*", x, Infinity],

Stilized[FileNameSplit[#][[–1]]] == Stilized[y] &]

In[10]:= FindFile2["e:\\", "book_2020.doc"]

Out[10]= {"e:\\Mathematica\\Book_2020.doc"}

The FindFile1 in many respects is functionally similar to the FileExistsQ1 procedure however in the time relation is partially less fast-acting in the same file system of the computer (a number of rather interesting questions of files and directories processing can be found in the collection [15]).

It is possible to represent the SearchDir procedure as one more indicative example, whose call SearchDir[d] returns the list of all paths in file system of the computer that are completed by a d subdirectory; in case of lack of such paths the procedure call SearchDir[d] returns the empty list. In a combination with the procedures FindFile1, FileExistsQ1 the SearchDir procedure is a rather useful at operating with file system of the computer, that confirms their use for the solution of problems of similar type. The fragment below presents source code of the SearchDir procedure with some typical examples of its application.

In[9]:= SearchDir[d_ /; StringQ[d]] := Module[{a = Adrive[], c,

t = {}, p, b = "\\" <> ToLowerCase[StringTrim[d, ("\\"|"/") ...]]

<> "\\", g = {}, k = 1, v}, For[k, k <= Length[a], k++, p = a[[k]];

c = Map[ToLowerCase, Quiet[FileNames["*", p <>

":\\", Infinity]]];

Map[If[! StringFreeQ[#, b] || SuffPref[#, b, 2] &&

DirQ[#], AppendTo[t, #], Null] &, c]];

For[k = 1, k <= Length[t], k++, p = t[[k]] <> "\\";

a = StringPosition[p, b];

If[a == {}, Continue[], a = Map[#[[2]] &, a];

Map[If[DirectoryQ[v = StringTake[p, {1, # – 1}]],

AppendTo[g, v], Null] &, a]]]; DeleteDuplicates[g]]

In[10]:= SearchDir["\\Temp/"]

Out[11]= {"c:\\temp", …, "e:\\temp", "e:\\temp\\temp"}

By operating time these tools yield to our procedures isDir and isFile for the Maple system, providing testing of files and directories respectively [39]. So, the isFile procedure not only tests the existence of a file, but also the mode of its openness, what in certain cases is a rather important. There are also other rather interesting tools for testing of the state of directories and files, including their types [8-10,15,16].

The Mathematica has two standard functions RenameFile and RenameDirectory for renaming of files and directories of file system of the computer respectively. Meanwhile, from the point of view of the file concept these functions would be very expedient to be done by uniform tools because in this concept directories and files are in many respects are identical and their processing can be executed by the same tools. At the same time the mentioned standard functions and on restrictions are quite identical: for renaming of a name x of an element of file system of the computer onto a new y name the element with name y has to be absent in the system, otherwise $Failed with a certain diagnostic message are returned. Furthermore, if as a y only a new name without full path to a new y element is coded, then it copying to the current directory is done; in a case of directory x it with all contents is copied into the current directory under a new y name. Therefore, similar organization is inconvenient in many respects, what stimulated us to determine for renaming of the directories and files the uniform RenDirFile procedure that provides renaming of an element x (directory or file) in situ with preservation of its type and all its attributes; at that, as an argument y a new name of the x element is used. Therefore, the successful procedure call RenDirFile[x, y] returns the full path to the renamed x element. In a case of existence of an element y the message "Directory/datafile already exists" is returned. In other unsuccessful cases the procedure call returns $Failed or is returned unevaluated. With source code of the RenDirFile the reader can familiarized, for example, in [10,15,16].

Meanwhile, at the same time to the means represented, we have created a whole series of means for computer file system access, that not only extend the capabilities of system functions focused on similar work, but also significantly complement the system capabilities, providing advanced facilities for processing elements of the computer file system. Note that MathToolBox tools [15,16] greatly enhance Mathematica's capabilities in the context of the computer file system access. Below, some special tools of files and directories processing are considered.

4.5. Special tools for files and directories processing

In this section some special means of processing of files and directories are represented; in certain cases they can be useful. Removal of a file in the current session is made by means of the standard DeleteFile function whose call DeleteFile[{x, y, z, ...}] returns Null, i.e. nothing in a case of successful removal of the given file or their list, and $Failed otherwise. At the same time, in the list of files only those are deleted that have no Protected–attribute. Moreover, this operation doesn't save the deleted files in the system $Recycle.Bin directory that in a number of cases is extremely undesirable, first of all, in the light of possibility of their subsequent restoration. The fact that the system function DeleteFile is based on the MS DOS command Del that according to specifics of this operating system immediately deletes a file from file system of the computer without its preservation, that significantly differs from similar operation of Windows system that by default saves the deleted file in the special $Recycle.Bin directory. For elimination of this shortcoming the DeleteFile1 procedure has been offered, whose source code with examples of its application are represented by the fragment below.

In[57]:= DeleteFile1[x_ /; StringQ[x] || ListQ[x], t___] :=

Module[{a = Flatten[{x}], b, c, p = {}},

b = SysDir[] <> If[! StringFreeQ[Ver[], " XP "],

"recycler", "$recycle.bin"];

Map[{Quiet[Close[#]], c = Attribs[#],

If[{t} != {}, CopyFileToDir[#, b];

If[MemberQ[c, "R"], Attribs[#, y]; DeleteFile[#],

AppendTo[p, #]; DeleteFile[#]],

If[MemberQ[c, "R"], AppendTo[p, #],

CopyFileToDir[#, b]; DeleteFile[#]]]} &, a]; p]

In[58]:= DeleteFile1[{"c:/temp/Avz.doc", "c:/temp/rans_ian.txt",

"c:/temp/description.doc", "c:/temp/Agn.doc"}, gs]

Out[58]= {"c:/temp/rans_ian.txt", "c:/temp/description.doc"}

In[59]:= DeleteFile1[{"c:/temp/Avz.doc", "c:/temp/rans_ian.txt",

"c:/temp/description.doc", "c:/temp/Agn.doc"}]

Out[59]= {"c:/temp/Avz.doc", "c:/temp/Agn.doc"}

The procedure call DeleteFile1[x] where x – a separate file or their list, returns the list of files x that have Protected attribute with deleting files given by an argument x with saving them in the $Recycle.Bin directory of the Windows system if they have not Protected attribute. Meanwhile, the files removed by the call DeleteFile1[x] are saved in $Recycle.bin directory however they are invisible to viewing by the system tools, for example, by Ms Explorer, complicating cleaning of the given system directory. While the call DeleteFile1[x, t] with the 2^nd optional t argument – an expression – returns the list of files x that have not Protected attribute with deleting files set by the argument x regardless of whether they have Protected attribute with saving them in the $recycle.bin directory of the Windows system. At the same time, in the $recycle.bin directory a copy only of the last deleted file always turns out. This procedure is oriented to Windows XP, 7 and above operational platforms however it can be spread and to other operational platforms.

Note, the procedure substantially uses our SysDir function, whose call SysDir[] returns the root directory of the DSS device on which Windows is installed:

In[17]:= SysDir[] := StringTake[GetEnvironment["windir"][[2]],

{1, 3}]; SysDir[]

Out[17]= "C:\\"

in combination with the Attribs and CopyFileToDir procedures. The first procedure was mentioned above, whereas the second procedure is used by our other files and directories processing tools too, extending the capabilities of the built-in access tools.

The procedure call CopyFileToDir[x, y] provides copying of a file or directory x into a y directory with return of the full path to the copied file or directory. In a case if the copied file x already exists, it isn't updated; if the target directory already exists, the x directory is copied into its subdirectory of the same name. This procedure has a variety of appendices in problems of processing of file system of the computer. The next fragment represents source code of the CopyFileToDir procedure with an example of its typical application.

In[1361]:= CopyFileToDir[x_ /; PathToFileQ[x], y_ /; DirQ[y]] :=

Module[{a, b}, If[DirQ[x], CopyDir[x, y],

If[FileExistsQ[x], a = FileNameSplit[x][[–1]];

If[FileExistsQ[b = y <> "\\" <> a], b, CopyFile[x, b]], $Failed]]]

In[1362]:= CopyFileToDir["c:/temp/Avz.doc", "c:/mathematica"]

Out[1362]= "C:\\mathematica\\Avz.doc"

For restoration from the system directory $Recycler.Bin of the packages which were removed by means of the DeleteFile1 procedure in Windows XP Professional, the RestoreDelPackage procedure providing restoration from $recycler.bin directory of such packages has been offered [1,8,12,16]. The successful call RestoreDelPackage[F, "Context’"], where the first argument F defines the name of a file of the format {"cdf", "m", "mx", "nb"} that is subject to restoration whereas the second argument – the context associated with a package returns the list of full paths to the restored files, at the same time by deleting from directory $Recycler.Bin the restored files with the necessary package. At that, this tools is supported on the Windows XP platform, while on the Windows 7 platform the RestoreDelFile procedure is of a certain interest, restoring files from the directory $Recycler.Bin that earlier were removed by the call of DeleteFile1 procedure.

The procedure call RestoreDelFile[f, r], where the 1^st actual argument f defines the name of a file or their list that are subject to restoration, while the second r argument defines the name of a target directory or full path to it for the restored files, returns the directory for the restored files; at the same time the delete of the restored files from the $recycle.bin directory is made in any case. In a case of absence of the requested files in the $recycle.bin directory the procedure call returns the empty list, i.e. {}, with printing of the appropriate messages (such messages are print for each file absent in the above Windows directory). It should be noted, that the nonempty files are restored too. If the 2^nd r argument defines a directory name in string format, but not the full path to it, a target r directory is created by means of call Directory[]; i.e. the current work directory. The fragment below represents source code of the procedure with examples of its application.

In[2216]:= RestoreDelFile[f_ /; StringQ[f] || ListQ[f],

r_ /; StringQ[r]] := Module[{a = Flatten[{f}], b, c, d, p, t = 0},

b = SysDir[] <> "$Recycle.bin";

Map[{c = b <> "\\" <> #,

If[! FileExistsQ[c], Print["File " <> # <> " can't be restored"];

t++, CopyFileToDir[c, p = If[DirQ[r], r, Directory[]]];

d = Attrib[c, "Attr"];

If[MemberQ[d[[1]], "R"], Attrib[c, {}], 7]; DeleteFile[c]]} &, a];

If[Length[a] == t, {}, b]]

In[2217]:= RestoreDelFile[{"description.doc", "Shishakov.doc",

"rans_ian.txt", "rans.txt"}, "c:\\restore"]

File rans.txt can't be restored

Out[2217]= "C:\\Users\\Aladjev\\Documents"

In[2218]:= RestoreDelFile[{"description7.doc", "Vaganov.doc",

"rans.txt", "ian2020.txt"}, "c:\\restore"]

File description7.doc can't be restored

File Vaganov.doc can't be restored

File rans.txt can't be restored

File ian2020.txt can't be restored

Out[2218]= {}

On the other hand, for removal from $Recycle.bin directory of the files saved by the DeleteFile1 procedure on the Windows XP platform, the procedure is used whose call ClearRecycler[] returns 0, deleting files of specified type from the $Recycle.bin directory with saving in it of the files removed by of Windows XP or its appendices. In addition, the Dick Cleanup command in Windows XP in some cases completely does not clear $Recycler from files what successfully does the call ClearRecycler["ALL"], returning 0 and providing removal of all files from the system Recycler directory. In [8,10,15] it is possible to familiarize with source code of the ClearRecycler procedure and examples of its use. For Windows 7 platform and above the ClearRecyclerBin procedure provides removal from the Recycler directory of all directories and files or only of those that are conditioned by the DeleteFile1 procedure. The procedure call ClearRecyclerBin[] returns Null, and provides removal from the system directory $Recycle.Bin of directories and files which are deleted by the DeleteFile1 procedure. Whereas the call ClearRecyclerBin[w], where w – an arbitrary expression – also returns Null, i.e. nothing, and removes from the $Recycle.Bin directory of all directories and files whatever the cause of their emergence in the directory. In addition, the procedure call on the empty Recycler directory returns $Failed or nothing, depending on the operating platform. The following fragment represents source code of the procedure with some typical examples of its application.

In[88]:= ClearRecyclerBin[x___] := Module[{c, d = {}, p, b = "#$#"},

c = SysDir[] <> If[! StringFreeQ[Ver[], " XP "],

"Recycler", "$Recycle.Bin"];

Run["Dir " <> c <> "/A/B/S/L > " <> b]; p = ReadList[b, String];

DeleteFile[b]; If[p == {}, Return[$Failed],

Map[{If[{x} != {}, Attrib[#, {}];

AppendTo[d, If[DirectoryQ[#], #, Nothing]];

Quiet[DeleteFile[#]],

If[SuffPref[FileNameSplit[#][[–1]], "$", 1] ||

SuffPref[#, "desktop.ini", 2], Null, Attrib[#, {}];

AppendTo[d, If[DirectoryQ[#], #, Nothing]];

Quiet[DeleteFile[#]]]]} &, p]]; Quiet[Map[DeleteDirectory, d]]; ]

In[89]:= ClearRecyclerBin[]

In[90]:= ClearRecyclerBin[All]

The following useful procedure has the general character at operating with the devices of direct access and is rather useful in a number of applications, above all, of the system character. The procedure below to a great extent is an analogue of Maple Vol_Free_Space procedure [41,42] that returns a volume of free memory on devices of direct access. The call FreeSpaceVol[w] depending on type of an actual w argument that should define the logical name in string format of a device, returns simple or the nested list; elements of its sub-lists determine a device name, volume of free memory for device of direct access, and unit of its measurement respectively. In a case of absence or inactivity of the w device the procedure call returns the message "Device is not ready". The following fragment represents source code of the FreeSpaceVol procedure with a typical example of its use.

In[9]:= FreeSpaceVol[x_ /; MemberQ3[Join[CharacterRange["a",

"z"], CharacterRange["A", "Z"]], Flatten[{x}]]] :=

Module[{a = "#$#", c, d = Flatten[{x}], f},

f[y_] := Module[{n, b}, n = Run["Dir " <> y <> ":\\ > " <> a];

If[n != 0, {y, "Device is not ready"}, b = StringSplit[

ReduceAdjacentStr[ReadFullFile[a], " ", 1]][[–3 ;–2]];

{y, ToExpression[StringReplace[b[[1]], "ÿ" –> ""]], b[[2]]}]];

c = Map[f, d]; DeleteFile[a]; If[Length[c] == 1, c[[1]], c]]

In[9]:= FreeSpaceVol[{"a", "c", "d", "e"}]

Out[9]= {{"a", "Device is not ready"}, {"c", 270305714176, "bytes"},

{"d", 0, "bytes"}, {"e", 2568826880, "bytes"}}

The procedure below facilitates solution of the problem of use of external Mathematica programs or operational platform. The procedure call ExtProgExe[x, y, h] provides a search in file system of the computer of a {exe|com} file with program with its subsequent execution on y arguments of the command string. Both arguments x and y should be encoded in the string format. Successful performance of this procedure returns the full path to “$TempFile$” file of ASCII format that contains the result of execution of a x program, and this file can be processed by the standard tools on the basis of its structure. At that, in a case of absence of the file with the demanded x program the procedure call returns $Failed while using of the third h optional argument – an arbitrary expression – the file with the x program uploaded in the current directory defined by the function call Directory[], is removed from this directory; also the “$TempFile$” datafile is removed if it is empty or implementation of the x program was terminated abnormally. The fragment below represents source code of the ExtProgExe procedure with examples of its use.

In[310]:= ExtProgExe[x_ /; StringQ[x], y_ /; StringQ[y], h___] :=

Module[{a = "$TempFile$", b = Directory[] <> "\\" <> x, c, d},

d = x <> " " <> y <> " > " <> a;

Empty::file = "File $TempFile$ is empty; the file had been deleted.";

If[FileExistsQ[b], c = Run[d], c = LoadExtProg[x];

If[c === $Failed, Return[$Failed]]; c = Run[d];

If[{h} != {}, DeleteFile[b]]];

If[c != 0, DeleteFile[a]; $Failed, If[EmptyFileQ[a], DeleteFile[a];

Message[Empty::file], Directory[] <> "\\" <> a]]]

In[311]:= ExtProgExe["tasklist.exe", " /svc ", 1]

Out[311]= "C:\\Users\\Aladjev\\Documents\\$TempFile$"

In[312]:= ExtProgExe["systeminfo.exe", "", 1]

Out[312]= "C:\\Users\\Aladjev\\Documents\\$TempFile$"

In[313]:= ExtProgExe["Rans_Ian.com", "a", b]

Out[In[313]:= $Failed

The following procedure is a rather useful generalization of built-in CopyFile function whose call CopyFile[a, b] returns the full name of the file to it is copied and $Failed if it cannot do the copy; in addition, file a must already exist, whereas b file must not. Thus, in the program mode the standard CopyFile function is insufficiently convenient. The procedure call CopyFile1[a, b] returns the full path to the file that had been copied. In contrast to CopyFile function, the procedure call CopyFile1[a, b] returns the list of format {b, b*} where b – the full path to the copied file and b* – the full path to the previously existing copy of a "xy.a" file in the format "$xy$.a", if the target directory for b already contained a file. In a case a ≡ b (where identity is considered up to standardized paths of both files) the call returns the standardized path to the a file, doing nothing. In other successful cases the call CopyFile1[a, b] returns the full path to the copied file. Even if the nonexistent path to the target directory for the copied file is defined, then such path taking into account available devices of direct access will be created. The fragment below represents source code of the CopyFile1 procedure with typical examples of its application which illustrate quite clearly the aforesaid.

In[81]:= CopyFile1[x_ /; FileExistsQ[x], y_ /; StringQ[y]] :=

Module[{a, b, c, d, p = Stilized[x], j = Stilized[y]},

b = Stilized[If[Set[a, DirectoryName[j]] === "", Directory[],

If[DirectoryQ[a], a, CDir[a]]]];

If[p == j, p, If[FileExistsQ[j], Quiet[Close[j]];

Quiet[RenameFile[j, c = StringReplace[j,

Set[d, FileBaseName[j]] –> "$" <> d <> "$"]]];

CopyFileToDir[p, b];

{b <> "\\" <> FileNameTake[p], j = b <> "\\" <> c},

CopyFileToDir[p, b]]]]

In[82]:= CopyFile1["C:/temp/СКА.doc", "СКА.DOC"]

Out[82]= {"c:\\users\\aladjev\\documents\\ска.doc",

"c:\\users\\aladjev\\documents\\$ска$.doc"}

In[83]:= CopyFile1["c:/temp/cka.doc", "g:\\Grodno\\Сka.doc"]

Out[83]= "C:\\grodno\\ска.doc"

So, the tools represented in the chapter sometimes a rather significantly simplify programming of problems dealing with file system of the computer. Along with that, these tools extend built-in means of access, illustrating a number of useful enough methods of programming of tasks of similar type. These means in a number of cases significantly supplement the access tools supported by system, facilitating programming of a number of rather important appendices that deal with the files of various format. Our experience of programming of the access tools that extend the similar means of the Mathematica allows to notice that basic access tools of Mathematica in combination with its global variables allow to program more simply and effectively the user original access tools. Moreover, the created access tools possess sometimes by the significantly bigger performance in relation to the similar means developed in the Maple software. Thus, in the Mathematica it is possible to solve the tasks linked with rather complex algorithms of processing of files, while in the Maple, first of all, in a case of rather large files the efficiency of such algorithms leaves much to be desired. In a number of appendices the means, represented in the present chapter along with other similar means from package [16] are represented as rather useful, by allowing, at times, to essentially simplify the programming. At that, it must be kept in mind, a lot of means which are based on the Run function and the DOS commands generally can be nonportable to other versions of the system and operational platform demanding the appropriate debugging to appropriate new conditions. In general, our access tools [1-16] are usefully expand and supplement the built-in Mathematica tools of the same purpose, making it easier to solve a number of tasks of processing elements of the computer's file system. At present our access tools [16] are used in various applications.

Chapter 5: Organization of the user software

The Mathematica no possess comfortable enough tools for organization of the user libraries similarly to the case of Maple, creating certain difficulties at organization of the user software developed in its environment. For saving of definitions objects and results of calculations Mathematica uses files of different organization. At that, files of text format that not only are easily loaded into the current session, in general are most often used, but also are convenient enough for processing by other known means, for example, the word processors. Moreover, the text format provides the portability on other computing platforms. One of the main prerequisites of saving in files is possibility of use of definitions along with their usage for the Mathematica objects in the subsequent sessions of the system. At that, with questions of standard saving of objects (blocks, modules, functions etc.) the interested reader can familiarize in details in [1-15,17], some of them were considered in this book in the context of organization of packages while here we present simple tools of organization of the user libraries in the Mathematica system.

Meanwhile, here it is expedient to make a number of very essential remarks on use of the above system means. First, the mechanism of processing of erroneous and especial situations represents a rather powerful mean of programming practically of each quite complex algorithm. However, in the Mathematica such mechanism is characterized by essential shortcomings, for example, successfully using in the Input mode the mechanism of messages print concerning erroneous {Off, On} situations, in body of procedures such mechanism generally doesn't work as illustrates the following a rather simple fragment:

In[1117]:= Import["C:\\Mat\\A.m"]

… Import::nffil: File C:\Mat\A.m not found during Import.

Out[17]= $Failed

In[1118]:= Off[Import::nffil]

In[1119]:= Import["D:\\Mat\\A.m"]

Out[1119]= $Failed

In[1123]:= On[Import::nffil]

In[24]:= F[x_] := Module[{a}, Off[Import::nffil];

a := Import[x]; On[Import::nffil]; a]

In[1125]:= F["C:\\Mat\\A.m"]

… Import::nffil: File C:\Mat\A.m not found during Import.

Out[1125]= $Failed

In a case of creation of rather complex procedures in which is required to solve questions of the suppressing of output of a number of erroneous messages, tools of Mathematica system are presented to us as insufficiently developed. The interested reader can familiarize with other peculiarities of the specified system tools in [1-15]. Now, we will present certain approaches concerning creation of the user libraries in Mathematica. Some of them can be useful in practical work with Mathematica.

In view of the scheme of organization of library considered in Maple [26] with organization different from the main library, we will present realization of the similar user library for a case of the Mathematica. On the first step in a directory, let us say, "c:/Lib" the txt–files with definitions of the user tools with their usage are placed. In principle, you can to place any number of definitions in the txt–files, in this case it is previously necessary to read a txt-file whose name coincides with name of the required tool, whereupon in the current session tools whose definitions are located in the file with usage are available. It is convenient in a case when in a single file are located the main tools along with tools accompanying them, excluding system tools. On the second step the tools together with their usage are created and debugged with their subsequent saving in the required file of a library subdirectory "C:\\Lib".

To save the definitions and usage in txt-files perhaps in two ways, namely: (1) by the function call Save, saving previously evaluated definitions and their usage in a txt–file given by its first argument; at that, saving is made in the append–mode, or (2) by creating txt–files with names of tools and usage whose contents are formed by means of a simple word processor, for example, Notepad. At that, by means of the Save function we have possibility to create libraries of the user tools, located in an arbitrary directory of file system of the computer.

For operating with txt-files of such organization was created the procedure CallSave whose call CallSave[x,y,z] returns the result of a call y[z] of a tool y on the z list of factual arguments passed to y provided that the tool definition y with usage are located in a x txt–file that has been earlier created by means of Save function. If a tool with the given y name is absent in the x file, the procedure call returns $Failed. If a x data-file contains definitions of several tools of the same name y, the procedure call CallSave[x,y,z] is executed relative to their definition whose formal arguments correspond to a z list of actual arguments. If y determines the list, the call returns the names list of all tools located in the x file [1,16,24]. More detailed information on the capabilities of such organization of user libraries can be found in [5-10]. In our opinion, the represented approach quite can be used for the organization of simple and effective user libraries of traditional type.

As another an useful enough approach, based on the use of mx–format files to store tool definitions and their usage, we will introduce the CALLmx procedure whose the call serves to save means definitions in the library directory in files of mx–format with their subsequent uploading into the current session. The possibilities of the procedure can be found in [1,5,10,11,16]. It is possible to represent the UserLib procedure that supports some useful enough functions as one more rather simple example of maintaining the user libraries. The call UserLib[W, g] provides a number of important functions on maintaining of a simple user library located in a W file of txt–format. As the second actual g argument the 2–element list acts that defines such functions as: (1) return of the list of tools names, contained in W, (2) print to the screen of full contents of a library file W or a separate tool, (3) saving in the library file W in the append-mode of a tool with the given name, (4) loading into the current session of all tools whose definitions are in the W, (5) uploading into the current session of a tool with the given name whose definition is in the W library file. There is a rather good opportunity to extend the procedure with a number of useful enough functions such as deletion from a library of definitions of the specified means or their obsolete versions, etc. With the procedure the reader can familiarize more in details, for example, in [4-9,15,16].

The list structure of the Mathematica allows to rather easily simulate the operating with structures of a number of systems of computer mathematics, for example, Maple. So, in Maple the tabular structure as one of the most important structures is used that is rather widely used both for the organization of structures of data, and for organization of the libraries of software. Similar tabular organization is widely used for organization of package modules of Maple along with a number of tools of our UserLib library [42]. For simulation of main operations with the tabular organization similar to the Maple, in Mathematica the Table1 procedure can be used. On a basis of such tabular organization supported by the Table1 procedure it is rather simply possible to determine the user libraries. As one of such approaches we presented an example of the library LibBase whose structural organization has format of the ListList–type [14,16]. The main operations with the library organized thus are supported by the procedure TabLib whose source code with examples of its use are represented in [8-10,12,15,16].

The interested reader can develop own means of the library organization in Mathematica, using approaches offered by us along with others. However, the problem of organization of convenient help bases for the user libraries exists. A number of approaches in this direction can be found in [14]. In particular, on a basis of the list structure supported by Mathematica, it is rather simply to define help bases for the user libraries. On this basis as one of possible approaches an example of the BaseHelp procedure has been represented, whose structural organization has the list format [10,16]. Meanwhile, it is possible to create the help bases on a basis of packages containing usage on means of the user library which are saved in files of mx–format. At that, for complete library it is possible to create only one help mx-file, uploading it as required into the current session by means of the Get function with receiving in the subsequent of access to all usage that are in the file. The Usage procedure can represent an quite certain interest for organization of a help database for the user libraries [8,9,12,16]. There too the interested reader can find and other useful tools for organizing user libraries.

5.1. MathToolBox package for Mathematica system

However, allowing the above approaches to organization of the user software, in our opinion the most convenient approach is the organization on the basis of packages. The packages are one of the main mechanisms of Mathematica extension which contain definitions of the new symbols with their usage that are intended for use both outside of package and in it. The symbols can correspond, in particular, to the definitions of new means defined in the package that extend the functional Mathematica possibilities. At that, according to the adopted system agreement all new symbols that entered in a certain package are placed in a context whose name is connected with name of the package.

At uploading of a package in the current session, the given context is added into the beginning of the list defined by global $ContextPath variable. As a rule, for ensuring of association of a package with context a construction BeginPackage["x'"] coded at its beginning is used. At uploading of package in the current session the "x'" context will update the current values of global variables $Context and $ContextPath. The closing bracket of the package is construction EndPackage[]. The package body itself, as a rule, consists of 2 parts – reference information (usage) for all symbols included in the package and definitions of symbols constituting the package. So, the N symbol help is framed in the form N::usage = "Description of N symbol". In turn, the symbols definitions are framed with constructs of the following format Begin["`N`"] and End[], for example, for the above symbol N. So, as a matter of experience works with Mathematica system, as a basis of the organization of the user software we offer a scheme.

At the first stage, in a new nb–document a program object of the following organization is formed, namely:

General information on a package

BeginPackage["Context`"]

N1::usage = "Description of N1 tool"

============================

Np::usage = "Description of Np tool"

Begin["`N1`"]

Definition of N1 tool

End[]

=============================

Begin["`Np`"]

Definition of Np tool

End[]

EndPackage[]

Before the main body of the object the general information about the package is placed, followed by the opening bracket of the package with a context assigned to it. The opening bracket is followed by a set of the usage–sentences containing reference information for all package tools named N1, ..., Np. At the same time, it should be noted that each tool included in the package must have the appropriate usage–sentence; otherwise, if you upload the package into the current session, you will not have access to such tool. The definition of each tool to be included in the given package is then placed in special block, for example, Begin["`N1`"] ... End[], ending all such blocks with the closing bracket EndPackage[]. The definitions of means included in the package should be previously, whenever possible, are debugged including processing of the main special and error situations.

At the second stage by a chain of commands “Evaluation  Evaluation Notebook” of the graphic user interface (GUI) is made evaluation of the Notebook, i.e. actually the package with return of results of the following form:

In[3212]:= BeginPackage["Agn`"]

Name::usage = "Description of Name tool"

Begin["`Name`"]

Name[x_] := x^2

End[]

Begin["`Name1`"]

Name1[x_] := x^3

End[]

EndPackage[]

Out[3212]= "Agn`"

Out[3213]= "Description of Name tool"

Out[3214]= "Agn`Name`"

Out[3216]= "Agn`Name`"

Out[3217]= "Agn`Name1`"

Out[3219]= "Agn`Name1`"

In[3220]:= CNames["Agn`"]

Out[3220]= {"Name"}

The procedure call CNames["Agn`"] returns the name list in string format of all symbols whose definitions are located in the package with context "Agn`".

The evaluation of the above package with context "Agn`" in the current session activates all definitions and usage contained in the package. Therefore the following stage consists in saving of the package in files of formats {nb, mx}. The first saving is made by a chain of commands of GUI "File  Save", whereas by a call of the built-in system function DumpSave of the format

In[15]:= DumpSave["c:\\Math\\FileName.mx", "Agn`"]

Out[15]= {"Agn`"}

saving of this package in a file "FileName.mx" of the Math directory is provided (names of directory and file to the discretion of the user) with return of list of contexts ascribed to the package. All subsequent loadings of the mx–file with the package to the current session are made at any time, when demanding use of the tools that are contained in it, by a call of the built–in system function Get of the format Get["C:\\Math\\FileName.mx"].

As noted above the built-in DumpSave function writes out definitions in a binary format that is optimized for input by the Mathematica. Meantime, at the call DumpSave[f, x] where f is a file of mx–format a symbol, list of symbols or a context can be used as the 2^nd x argument. In order to enhance the call format of the DumpSave function some tools have been proposed [15].

In particular, the procedure call DumpSave1[x, y] returns the nested list whose first element defines the full path to a file x of mx–format (if necessary to the file is assigned the mx–extension) whereas the second element defines the list of objects and/or contexts from the list defined by argument y whose definitions were unloaded into the x file of mx–format.

Evaluation of definition of a symbol x in the current session it will associate with the context of "Global'" which remains at its unloading into a mx–file by the DumpSave function. While in some cases there is a need of saving of symbols in the files of mx–format with other contexts. This problem is solved by the procedure whose call DumpSave2[f, x, y] returns nothing, at the same time unloading into mx–file f the definition of symbol or the list of symbols x that have context "Global'", with context y. So, in the current session the x symbol receives the y context.

Finally, a simple DumpSave3 procedure allows an arbitrary expression to be used as the second argument of the procedure. Calling the DumpSave3[f, exp] procedure returns a variable in a string format while saving an exp expression in a mx-file f. Then the subsequent call Get[f] returns nothing with activating the variable returned by the previous call DumpSave3[f, exp] with the exp value assigned to it. The fragment below represent the source code of the procedure with an example of it application.

In[2267]:= DumpSave3[f_ /; StringQ[f], exp_] := Module[{a},

{ToString[a], a := exp, DumpSave[f, a]}[[1]]]

In[2268]:= DumpSave3["dump.mx", a*77 + b*78]

Out[2268]= "a$33218"

In[2269]:= Clear["a$33218"]

In[2270]:= Get["dump.mx"]; ToExpression["a$33218"]

Out[2270]= 77*a + 78*b

Fundamental differences between files of two formats mx and nb assume to save packages at least in files of these types, and that is why. Files of mx–format are optimized for input in Mathematica, each such file has a plain text header identifying its type and contexts ascribed to it. In turn, the rest of a mx–file containing definitions of tools is presented in the dump binary format that is not allowing to edit it. Files saved by DumpSave function can be load by Get function. Meantime, files of format mx can`t be exchanged between operating systems that differ in compliance with $SystemWordLength predefined variable.

Whereas files of the format nb (notebooks) are structured interactive documents that can contain calculations, text, typeset expressions, user interface elements, etc. Notebooks are typical method of interacting with Mathematica front end. Files of this format are automatically associated with the Mathematica on computers which have the Mathematica installed. In addition, mx–files contain only printable ASCII characters, are viewable and largely readable in any text editor. In addition, notebooks are cross–platform, meaning that a notebook created on any supported platform can be read by Mathematica on any other platform. Therefore to edit a package it is reasonable to do on the basis of the nb–file containing it while most effective to do the current work with the package on the basis of its mx–file.

It is this approach to software saving that we have chosen and implemented in the package MathToolBox. Our package MathToolBox [16] that is attached to the present book can as a bright example of the above organization of the user software. Package contains more than 1420 tools eliminating restrictions of a number of standard tools of the Mathematica, and expand its software with new tools. In this context, MathToolBox can serve as a certain additional tool of procedural programming, especially useful in the numerous applications where certain non–standard evaluations have to accompany programming. In addition, tools presented in MathToolBox have a direct relation to principal questions of procedural–functional programming in Mathematica, not only for the decision of applied problems, but, above all, for creation of software that extends frequently used facilities of the system and/or eliminating their defects or extending the system with new facilities. Software presented in this package contains a number of rather useful and effective receptions of programming in the Mathematica, and extends its software which allows in the system to programme the tasks of various purposes more simply and effectively. Additional tools composing the above package embrace the following sections of the Mathematica system, namely:

- additional tools in interactive mode of the system

- additional tools of processing of expressions

- additional tools of processing of symbols and strings

- additional tools of processing of sequences and lists

- additional tools expanding standard built-in functions or the system

software as a whole (control structures branching and loop, etc.)

- determination of procedures in the Mathematica software

- determination of the user functions and pure functions

- means of testing of procedures and functions

- headings of procedures and function

- formal arguments of procedures and functions

- local variables of modules and blocks; means of their processing

- global variables of modules and blocks; means of their processing

- attributes, options and values by default for arguments of the user

blocks, functions and modules; additional means of their processing

- useful additional means for processing of procedures and functions

- additional means of the processing of internal Mathematica files

- additional means of the processing of external Mathematica files

- additional tools of the processing of attributes of directories and files

- additional and special means of processing of directories and files

- additional tools of work with packages and contexts ascribed to them

- organization of the user software in the Mathematica system

Archive Archive76.ZIP with this package (Freeware license) can be freely downloaded here [16]. Archive contains 5 files of formats {nb, mx, cdf, m, txt}. Such approach allows to satisfy the user on different operating platforms. Memory size demanded for the MathToolBox package in Mathematica of version 12.1.1 (on platform Windows 7 Professional) yields the following result:

In[1]:= MemoryInUse[]

Out[1]= 83421352

In[2]:= Get["C:\\Mathematica\\MathToolBox.mx"]

In[3]:= MemoryInUse[]

Out[3]= 95711392

In[4]:= N[(% – %%%)/1024^2]

Out[4]= 11.7207

i.e. in Mathematica 12.1.1.0 the MathToolBox package requires a little more 11.72 Mb while amount of tools whose definitions are located in the package, at the moment of its loading to the current session of the Mathematica system is available on the basis of the following simple evaluation:

In[5]:= Length[CNames["AladjevProcedures`"]]

Out[5]= 1424

where CNames – tool from the package, "AladjevProcedures`" – context ascribed to it. More detailed information on the means contained in the package can be found in [1-16].

Because the MathToolBox package contains definitions and usage for a large number of frequently used tools that often use each other, it is useful to define uploading of the package when you load Mathematica itself. To this end, you can to update the Init.m file of txt-format from $UserBaseDirectory <> "\\kernel" directory adding to it the entry Get["Dir\\MathToolBox.mx"] where Dir determines the full path to directory with mx–file of the package. It's easy to do with a simple text editor or in the current session of Mathematica. Furthermore, for this purpose it is possible to use a simple Init procedure whose source code with a typical example of its application is represented below.

In[317]:= Init[x_ /; StringQ[x]] := Module[{a, b, c},

a[d_] := StringReplace[d, "\\" –> "/"];

b = $UserBaseDirectory <> "\\kernel\\Init.m";

c = ReadString[b]; OpenWrite[b];

WriteString[b, c, "\n", a["Get[" <> "\"" <> x <> "\"]"]]; Close[b]]

In[318]:= Init["C:\\Mathematica\\MathToolBox.mx"]

Out[318]= "C:\\Users\\Aladjev\\AppData\\Roaming\\

Mathematica\\kernel\\Init.m"

The procedure call Init[x] where actual argument x defines the full path to a file of format mx, nb or m with the user package in addition mode updates the above file "Init.m" which locates in the user directory defined by the predefined $UserBaseDirectory variable according to the corresponding condition. In particular, if "Init.m" file had content before the procedure call

(** User Mathematica initialization file **),

then as a result of the procedure call the file gets the content

(** User Mathematica initialization file **)

Get["C:/Mathematica/MathToolBox.mx"]

After that, as a result of uploading Mathematica all means of the package become available in the current session along with updating of the lists of contexts defined by predetermined variables $Packages and $ContextPath by the package context. While the function call Contexts[] returns the list of all contexts supplemented by contexts "AladjevProcedures'Name'", where Name is the name of a package tool. The above scheme of the organization of the user software provides access to means of a package on an equal basis with built–in system means at once after uploading of the Mathematica except for two moments: (1) help on a tool N is available by ?N or by ??N only, and (2) all source codes of the package means are open.

The above technique allows to program a simple function, whose call JoinTxtFiles[x, y] returns the path to a x file that is updated by adding of a file y to the contents of a text file x.

In[35]:= JoinTxtFiles[x_String, y_String] := {WriteString[x,

ReadString[x], "\n", ReadString[y]], Close[x]}[[2]]

In[36]:= JoinTxtFiles["C:/Temp/In.txt", "C:/Temp/In1.txt"]

Out[36]= "C:/Temp/In.txt"

The above fragment represents source code of the function JoinTxtFiles with a typical example of its application.

The above method loads the desired file at the beginning of the current session, while on the function call Needs[Context, f] the desired file f (including the mx–file) with the package can be loaded at any time of the current session and its context placed in $Packages and $ContextPath, if it was not already there.

Meanwhile, it is appropriate to make one very significant comment here. The MathToolBox package [16] was created for a rather long period of time, albeit with a number of intervals. Therefore, Mathematica releases 8.0 – 12.1.1 are responsible for creating the tools that make it possible additional debugging of tools to the current Mathematica release. The stability of built-in Math–language of the Mathematica system is significantly higher than the corresponding component of the Maple system, but it is also not 100%. That is why, in some cases the necessary additional debugging of some package tools is required, that in the vast majority of cases is not essentially difficult. Naturally, due to a rather large number of package means, such additional debugging was carried out by us in a limited enough amount, leaving it to the reader who uses certain package means in his activities. In most cases, the means features of the package are compatible with the Mathematica 12.1.1 software environment. Note that some package tools are functionally fully or partially duplicate each other, illustrating different useful programming approaches and techniques, or certain undocumented (hidden) capabilities of the built–in the Mathematica language.

In any case, as demonstrated by many years of experience in teaching different courses in Mathematica at universities in Belarus and the Baltic States, this package not only represents the tools for applied and system programming in Mathematica, but also proved to be a rather useful collection of tasks aimed both at mastering programming in Mathematica system and at increasing its level as a whole. Being a rather useful reference and training material describing a lot of useful programming techniques in Mathematica, including the non–standard ones, the MathToolBox package is attached as an auxiliary material to our books [8-15].

5.2. Operating with the user packages in Mathematica

Similarly to the well–developed software the Mathematica is an extendable system, i.e. in addition to the built–in tools that quite cover requirements of quite wide range of the users, the system allows to program those tools that absent for the specific user of built–in language along with extending and correcting standard software. At that, the user can find the missing tools which are not built–in in numerous packages delivered with the Mathematica, and existing packages for various applied fields. The question consists only in finding of a package necessary for a concrete case containing definitions of functions, modules and other objects demanded for applications that are programmed in the Mathematica. A package has standard organization and contains definitions of various objects, these or those functions, procedures, variables, etc., that solve well–defined problems.

Mathematica provides the standard set of packages whose list is defined by a concrete version of the system. For receiving of packages list which are delivered with the current release of the Mathematica it is possible to use the procedure, whose call MathPackages[] returns list of packages names with a certain confidence speaking about their basic purpose. While the call MathPackages[j] with optional j argument (an indefinite variable) returns through it in addition the three–element list whose the 1^st element defines the current release of Mathematica, the 2^nd element – the type of the license and the 3^rd element – deadline of action of the license [16]. The following fragment represents the typical examples of its application.

In[91]:= MathPackages[]

Out[91]= {"AbelianGroup", "AbortHandling", …, "ZTransform"}

In[92]:= Length[%]

Out[92]= 3734

In[93]:= MathPackages[gs]; gs

Out[93]= {"12.1.1 for Microsoft Windows (64–bit)

(June 9, 2020)", "Professional", "Mon30Nov"}

From the fragment follows that the Mathematica of version 12.1.1 contains 3734 packages oriented on different appendices, including the packages of strictly system purpose. Before use of means contained in an applied package, the package should be previously loaded to the current Mathematica session by means of the function call Get[Package].

The concept of context in Mathematica. In Mathematica this concept was entered for organization of work with symbols that present various objects (modules, functions, variables, packages and so on), in particular, in order to avoid the possible conflicts with symbols of the same name. Main idea consists in that that full name of an arbitrary symbol consists of two parts, namely: a context and a short name, i.e. the full name of a certain object has format: "context'short name" where the sign <'> (backquote) carries out the role of some marker, identifying context in the system. For example, Agn'Vs represents a symbol with the Agn context and with short Vs name. At that, with such symbols it is possible to execute various operations as with usual names; furthermore, the system considers aaa'xy and bbb'xy as various symbols. So, the most widespread use of context consists in its assignment to functionally identical or semantically connected symbols. For example, Agn`StandPath, Agn`MathPackages define that procedures StandPath and MathPackages belong to the same group of the means which associate with "Agn'" context which is ascribed to a certain package. The current context of a session is in the global variable $Context:

In[2223]:= $Context

Out[2223]= "Global`"

In the current Mathematica session the current context by default is determined as "Global'". Whereas the global variable $ContextPath determines the list of contexts after the $Context variable for search of a symbol entered into the current session. It is possible to refer to the symbols from the current context simply by their short names; thus, if this symbol intersects with a symbol from the list determined by the $ContextPath variable, the second symbol will be used instead of the symbol from the current context, for example:

In[2224]:= $ContextPath

Out[2224]= {"DocumentationSearch`", "ResourceLocator`",

"URLUtilities`", "AladjevProcedures`", "PacletManager`",

"System`", "Global`"}

While the calls Context[x] and Contexts[] return a context ascribed to a x symbol and the list of all contexts of the current session respectively:

In[2225]:= Context[DeleteFile1]

Out[2225]= "AladjevProcedures`"

In[2226]:= Contexts[]

Out[2226]= {"AladjevProcedures`", …, "$CellContext`"}

In addition, by analogy with file system of the computer, contexts quite can be compared with directories. It is possible to determine the path to a file, specifying a directory containing it and a name of the file. At the same time, the current context can be quite associated with the current directory to the files of which can be referenced simply by their names. Furthermore, like file system the contexts can have hierarchical structure, in particular, "Visualization`VectorFields`VectorFieldsDump`". So, the path of search of a context of symbols in the Mathematica is similar to a path of search of program files. At the beginning of the session the current context by default is "Global'", and all symbols entered in the session are associated with this context, excepting the built-in symbols, e.g., Do, that are associated with "System'" context. Path of search of contexts by default includes the contexts for system–defined symbols. While for the symbols removed by means of the Remove function, the context can not be determined. At using of contexts there is no guarantee that 2 symbols of the same name are available in different contexts. Therefore the Mathematica defines as a maximum priority the priority of choice of that symbol with this name, whose context is the first in the list which is determined by the global variable $ContextPath. Therefore, for placement of such context at the beginning of the specified list you can use the construction:

In[30]:= $ContextPath

Out[30]= {"DocumentationSearch`", "ResourceLocator`",

"URLUtilities`", "PacletManager`", "System`", "Global`"}

In[31]:= PrependTo[$ContextPath, "Ian`"]

Out[31]= {"Ian`", "DocumentationSearch`", "URLUtilities`",

"ResourceLocator`", "PacletManager`", "System`", "Global`"}

In[32]:= $ContextPath

Out[32]= {"Ian`", "DocumentationSearch`", "URLUtilities`", "ResourceLocator`", "PacletManager`", "System`", "Global`"}

An useful procedure provides assignment of a context to a definite or indefinite symbol. The call ContextToSymbol1[x,y,z] returns Null, i.e. nothing, providing assignment of a y context to a x symbol while the third optional z argument – the string – defines for x an usage; at its absence for an indefinite x symbol the usage – empty string, i.e. "", whereas for a definite x symbol the usage has view "Help on x". With the ContextToSymbol1 procedure and examples of its use can familiarize in [8,15,16].

A rather useful procedure in a certain relation expands the above procedure and provides assignment of the set context to a definite symbol. The call ContextToSymbol2[x,y] returns two-element list of format {x,y} where x – the name of the processed definite x symbol and y – its new context, providing assignment of a y context to the definite x symbol; while the procedure call ContextToSymbol2[x, y, z] with the optional third z argument – an arbitrary expression – also returns 2–element list of the above format {x, y}, providing assignment of a certain y context to the definite x symbol with saving of symbol definition x and all its attributes along with usage in file "x.mx".

Thence, before a procedure call ContextToSymbol2[x, y] for change of the existing context of a symbol x on a new y symbol in the current session it is necessary to evaluate definition of x symbol and its usage in the format x::usage = "Help on x object." (if an usage exists, of course) with assignment to the x symbol of necessary attributes.

In addition, along with possibility of assignment of the set context to symbols the procedure ContextToSymbol2 is a rather useful means for extension by new means of the user package contained in a mx-file. The technology of similar updating is as follows. On the first step the definition of a new x tool with its usage describing these tools is evaluated along with ascribing of the necessary attributes. On the second step, by means of the call ContextToSymbol2[x, y, j] the assignment of a y context to x symbol along with its usage with saving of the updated x object in "x.mx" file is provided. At last, the call Get[w] loads into the current session the revised user package located in a mx–file w with an y context, whereas the subsequent call DumpSave[p, y] saves in the updated mx–file w all objects which have y context, including the objects that earlier have been obtained by means of the procedure call ContextToSymbol2[x, y] or Get["x.mx"]. Such approach provides a rather effective method of updating of the user packages located in mx–files [8-10,12-16].

The ContextToSymbol3 procedure is a useful modification of the above procedure ContextToSymbol2. The procedure call ContextToSymbol3[x, y] returns 2–element list of format {x, y}, where x – the name in string format of a definite x symbol, y – its new context, providing assignment of a certain y context to the definite x symbol; on the inadmissible arguments the call is returned as unevaluated. The fragment below represents source code of the ContextToSymbol3 procedure with example its use.

In[2214]:= ContextToSymbol3[x_ /; StringQ[x] &&

SymbolQ[x] && ! NullQ[x] && ! SameQ[x, "System`"],

y_ /; ContextQ[y]] :=

Module[{a = Flatten[{PureDefinition[x]}], b, c, d, h},

b = StringSplit[a[[1]], {"[", " = ", " := "}][[1]];

h = Help[Symbol[b]]; c = Attributes[x];

ToExpression["Unprotect[" <> b <> "]"];

d = "BeginPackage[\"" <> y <> "\"]" <> "\n" <>

If[NullQ[h], "", b <> "::usage=" <> ToString1[h]] <> "\n" <>

StringJoin[Map[# <> "\n" &, a]] <> "EndPackage[]";

Remove[x]; Quiet[ToExpression[d]];

ToExpression["SetAttributes[" <> b <> "," <> ToString[c] <> "]"]; {b, y}]

In[2215]:= V77[x_] := Module[{a = 42}, a*x];

V77[x_, y_] := Module[{}, x*y]; V77::usage = "Help on V77.";

SetAttributes[V77, {Listable, Protected}]

In[2216]:= ContextToSymbol3["V77", "AvzAgnVsv`"]

Out[2216]= {"V77", "AvzAgnVsv`"}

In[2217]:= Attributes[V77]

Out[2217]= {Listable, Protected}

In[2218]:= Definition[V77]

Out[2218]= Attributes[V77] = {Listable, Protected}

V77[x_] := Module[{a = 42}, a*x]

V77[x_, y_] := Module[{}, x*y]

In[2219]:= Context[V77]

Out[2219]= "AvzAgnVsv`"

In[2220]:= ?V77

Out[2220]= "Help on V77."

On the basis of the previous procedure, RenameMxContext procedure was programmed whose call RenameMxContext[x,y] returns the name of a mx–file x with replacement of its context by a new y context; at that, the package located in the existing x file can be uploaded into current session or not, while at use of the third optional argument – an arbitrary expression – the call RenameMxContext[x, y, z] additionally removes the x package from the current session. The fragment presents source code of the RenameMxContext procedure with examples of its use.

In[3368]:= BeginPackage["RansIan`"]

GSV::usage = "Help on GSV."

ArtKr::usage = "Help on ArtKr."

Begin["`ArtKr`"]

ArtKr[x_, y_, z_] := Module[{a = 74}, a*x*y*z]

End[]

Begin["`GSV`"]

GSV[x_, y_] := Module[{a = 69}, a*x*y]

End[]

EndPackage[];

In[3378]:= DumpSave["avzagn.mx", "RansIan`"]

Out[3378]= {"RansIan`"}

In[3379]:= CNames["RansIan`"]

Out[3379]= {"ArtKr", "GSV"}

In[3380]:= RenameMxContext[x_ /; FileExistsQ[x] &&

FileExtension[x] == "mx", y_ /; ContextQ[y], z___] :=

Module[{a = ContextInMxFile[x], b = Quiet[Check[Get[x], "Er"]], c},

If[b === "Er", "File " <> x <> " is corrupted", b = CNames[a];

c = Quiet[AppendTo[Map[a <> # <> "`" &, b], a]];

Map[Quiet[ContextToSymbol3[#, y]] &, b];

ToExpression["DumpSave[" <> ToString1[x] <> "," <> ToString1[y]

<> "]"]; DeletePackage[a]; If[{z} != {}, x, DeletePackage[y]; x]]]

In[3381]:= RenameMxContext["avzagn.mx", "Tampere`", 77]

Out[3381]= "avzagn.mx"

In[3382]:= ContextInMxFile["avzagn.mx"]

Out[3382]= "Tampere`"

The ReplaceContextInMx1 procedure – an extension of the ReplaceContextInMx procedure on a case of absence of context for a mx–file w. The procedure call ReplaceContextInMx1[x, w] returns the 3-element list of the form {a, x, w}, where a – an old context, x – a new context and w – updated w file with a new x context, if its previous version had context or did not have it (in this case the absence of context is identified as $Failed). In a case if w file is corrupted, the message "File w is corrupted" is returned. Since as a result of its execution the procedure clears all symbols in the current session that have the "Global'" context, then it is desirable to make calling this procedure at the beginning of the current session to avoid possible misunderstandings. The next fragment represents source code of the procedure and examples of its typical application.

In[3312]:= ReplaceContextInMx1[x_ /; ContextQ[x],

f_ /; FileExistsQ[f] && FileExtension[f] == "mx"] :=

Module[{a = ContextInMxFile[f], b, c, d},

If[ContextQ[a], RenameMxContext[f, x]; {a, x, f},

If[SameQ[a, $Failed], Map[ClearAll[#] &, CNames["Global`"]];

Get[f]; b = CNames["Global`"]; Map[ContextToSymbol3[#, x] &, b];

DumpSave[f, x]; {a, x, f}, a]]]

In[3313]:= ReplaceContextInMx1["Tampere`", "agn47.mx"]

Out[3313]= {"Grodno`", "Tampere`", "agn47.mx"}

In[3314]:= ReplaceContextInMx1["Tallinn`", "rans.mx"]

Out[3314]= "File rans.mx is corrupted"

In[3315]:= ReplaceContextInMx1["Grodno`", "avzagn.mx"]

Out[3315]= {$Failed, "Grodno`", "avzagn.mx"}

Presently, using the above procedure ContextToSymbol2 or ContextToSymbol3, the procedure that executes replenishment of mx-file f contained a package by new tools can be presented. The procedure call AdjunctionToMx[y, f] returns full path to a mx–file with a context updated by new y tools. At that, a single y symbol or their list can be as the first argument whereas their definitions and usage should be previously evaluated. Besides, if the mx–file had been already uploaded, it remains, otherwise it will be unloaded of the current session. The procedure call on a mx–file f without context returns $Failed.

In[2216]:= AdjunctionToMx[y_ /; SymbolQ[y] || ListQ[y] &&

DeleteDuplicates[Map[SymbolQ[#] &, Flatten[{y}]]] == {True},

f_ /; FileExistsQ[f] && FileExtension[f] == "mx"] :=

Module[{a = ContextInMxFile[f], b},

If[a === $Failed, $Failed, If[! FreeQ[$Packages, a], b = 77, Get[f]];

Map[Quiet[ContextToSymbol2[#, a]] &, Flatten[{y}]];

ToExpression["DumpSave[" <> ToString1[f] <> "," <>

ToString1[a] <> "]"];

If[! SameQ[b, 77], RemovePackage[a], 72]; f]]

In[2217]:= G72[x_, y_] := x^2 + y^2; G72::usage = "Help on G72

function."; V77[x_, y_] := x^3 + y^3;

V77::usage = "Help on V77 function."; Get["avzagn.mx"]

In[2218]:= ContextInMxFile["avzagn.mx"]

Out[2218]= "Tampere`"

In[2219]:= CNames["Tampere`"]

Out[2219]= {"ArtKr", "GSV"}

In[2220]:= AdjunctionToMx[{G72, V77}, "avzagn.mx"]

Out[2220]= "avzagn.mx"

In[2221]:= ClearAll[ArtKr, GSV, G72, V77]

In[2222]:= Get["avzagn.mx"]; CNames["Tampere`"]

Out[2222]= {"ArtKr", "G72", "GSV", "V77"}

In addition to the previous procedure the procedure call CreationMx[x, y, f] creates a new mx–file f with tools defined by the y argument (a single symbol or their list) and with a context x, returning full path to the f file. At that, definitions and usage of the y symbols should be previously evaluated. Furthermore, if mx–file f already exists, it remains, but instead of it a new mx–file is created. The procedure call for a context x that exists in the $Packages variable returns $Failed [8-10,15,16].

Sometimes, it is expedient to replace a context of means of the user package uploaded into the current session. This task is solved by means of the RemoveContext procedure, whose call RemoveContext[j, x, y, z, h,…] returns nothing, replacing in the current session a context j ascribed to the means {x, y, z, h,…} of the user package by the "Global'" context. In the absence of the {x, y, z, h,…} means with the j context the procedure call returns $Failed. In addition, a mx–file contained the means {x, y, z, h,…} remains without change. The fragment represents source code of the procedure RemoveContext with an example of its use.

In[18]:= RemoveContext[at_ /; ContextQ[at], x__] :=

Module[{a = {}, b, c = {}, d = Map[ToString, {x}], f = "$$$", Attr},

d = Intersection[CNames[at], d];

b = Flatten[Map[PureDefinition, d]];

If[b == {}, $Failed, Attr := Map[{#, Attributes[#]} &, d];

Do[AppendTo[a, StringReplace[b[[k]], at <>

d[[k]] <> "`" –> ""]], {k, 1, Length[d]}];

Write[f, a]; Close[f];

Do[AppendTo[c, at <> d[[k]]], {k, 1, Length[d]}]; c = Flatten[c];

Map[{ClearAttributes[#, Protected], Remove[#]} &, d];

Map[ToExpression, Get[f]]; DeleteFile[f];

Map[ToExpression["SetAttributes[" <> #[[1]] <> "," <>

ToString[#[[2]]] <> "]"] &, Attr]; ]]

In[19]:= Get["avzagn.mx"]

In[20]:= CNames["Tampere`"]

Out[20]= {"Art", "G72", "G721", "GSV", "V77"}

In[21]:= RemoveContext["Tampere`", Art, G72, G721, GSV, V77]

In[22]:= Map[Context, {Art, G72, G721, GSV, V77}]

Out[22]= {"Global`", "Global`", "Global`", "Global`", "Global`"}

In connection with the context processing means discussed above, it quite is reasonably note a procedure for testing of the correctness of mx-files. Calling CorrectMxQ[f] procedure returns True if the existing a mx-file is correctly loaded into the current session by means of the call Get[f], and False otherwise. While the call CorrectMxQ[f, t] additionally through the 2^nd optional t argument – an undefined variable – returns two-element list of the format {c, "Yes"} where c – a context of the f file if the f file was already loaded into the current session, {c, "No"} if f file is not loaded and the message "File f is corrupted" if f file is corrupted. Following fragment represents source code of the CorrectMxQ procedure with some examples of its typical application.

CorrectMxQ[f_ /; FileExistsQ[f] && FileExtension[f] == "mx", t___] :=

Module[{a = ContextInMxFile[f], b, c, d},

{If[ContextQ[a],

If[FreeQ[$ContextPath, a], d = {a, "No"}, d = {a, "Yes"}]; True,

If[! SameQ[a, $Failed], d = "File " <> f <> " is corrupted"; False,

b = CNames["Global`"]; DumpSave["##.mx", b];

Map[ClearAll, b]; Get[f]; c = CNames["Global`"];

If[MemberQ3[b, c], d = {a, "Yes"}; True, d = {a, "No"};

Map[ClearAll, CNames["Global`"]]; Get["##.mx"]; True]]],

Quiet[DeleteFile["##.mx"]],

If[{t} != {} && SymbolQ[t], t = d, Null]}[[1]]]

In[2331]:= CorrectMxQ["c:\\mathematica\\mathtoolbox.mx"]

Out[2331]= True

In[2332]:= CorrectMxQ["c:/mathematica/mathtoolbox.mx", h]

Out[2332]= True

In[2333]:= h

Out[2333]= {"AladjevProcedures`", "Yes"}

In[2334]:= CorrectMxQ["gsv.mx", g]

Out[2334]= False

In[2335]:= g

Out[2335]= "File gsv.mx is corrupted"

In[2336]:= CorrectMxQ["vsv.mx", t]

Out[2336]= True

In[2337]:= t

Out[2337]= {$Failed, "No"}

Interconnection of contexts and packages in Mathematica. Because of the importance of the relationships of contexts and packages, the necessity of defining of contexts of symbols arises. Meanwhile, in the current session, there may be symbols of the same name with different contexts whereas the built-in function Context returns only the 1^st context located in the $ContextPath list. Meantime the complete result is provided by the procedure whose call ContextDef[x] returns the list of contexts assigned to a symbol x. Following fragment represents source code of the procedure with some examples of its typical application.

In[7]:= BeginPackage["RansIan`"]

GSV::usage = "help on GSV."

Begin["`GSV`"]

GSV[x_, y_, z_] := Module[{a = 72}, x*y*z*a]

End[]

EndPackage[];

In[8]:= BeginPackage["Tampere`"]

GSV::usage = "help on GSV."

Begin["`GSV`"]

GSV[x_, y_, z_] := Module[{a = 77}, x*y*z*a]

End[]

EndPackage[];

… GSV: Symbol GSV appears in multiple contexts {Tampere`, RansIan`}; definitions in context Tampere` may shadow or be … .

In[13]:= GSV[x_Integer, z_Integer] := Module[{a = 42}, (x + z)*a]

In[14]:= Context[GSV]

Out[14]= "Tampere`"

In[15]:= $ContextPath

Out[15]= {"Tampere`", "RansIan`", "DocumentationSearch`",

"ResourceLocator`", "URLUtilities`", "AladjevProcedures`",

"PacletManager`", "System`", "Global`"}

In[16]:= ContextDef[x_ /; SymbolQ[x]] :=

Module[{a = $ContextPath, b = ToString[x], c = {}},

Do[If[! SameQ[ToString[Quiet[Check[ToExpression[

"Definition1[" <> ToString1[a[[k]] <> b] <> "]"], "Null"]]],

"Null"], AppendTo[c, {a[[k]] <> b, If[ToString[Definition[b]] !=

"Null", "Global`" <> b, Nothing]}]], {k, 1, Length[a]}];

Map[StringReplace[#, "`" <> ToString[x] –> "`"] &,

DeleteDuplicates[Flatten[c]]]]

In[17]:= ContextDef[GSV]

Out[17]= {"Tampere`", "Global`", "RansIan`"}

In[78]:= DefContext[x_ /; ContextQ[x], y_Symbol] :=

Module[{a = $ContextPath, b, c, d},

If[Set[d, ContextDef[y]] == {}||d == {"Global`"}, Definition[y],

If[FreeQ[d, x] || FreeQ[a, x], $Failed, a = PrependTo[a, x];

b = ToExpression[c = x <> ToString[y]];

If[x == "Global`" && ContextDef[y] != {},

Quiet[Definition2[y][[–2]]], StringReplace[Definition2[b][[1]],

{x –> "", ToString[y] <> "`" –> ""}]], $Failed]]]

In[79]:= DefContext["RansIan`", GSV]

Out[79]= "GSV[x_, y_, z_] := Module[{a = 72}, x*y*z*a]"

In[80]:= DefContext["Tampere`", GSV]

Out[80]= "GSV[x_, y_, z_] := Module[{a = 77}, x*y*z*a]"

In[81]:= DefContext["Global`", GSV]

Out[81]= "GSV[x_Integer, z_Integer] := Module[{a = 42}, (x+z)*a]"

Because of the multiplicity admissibility of contexts for a symbol, it is advantageous to have a means for determining the symbol depending on its context. The task is solved by means of a procedure whose source code is represented in the previous fragment and whose call DefContext[c, s] returns the definition of a symbol s having a context c. If s is an undefined symbol, the procedure call returns Null; if c is not a context of the s symbol, then the procedure call returns $Failed; if s is not a symbol, then the procedure call is returned as unevaluated. Thus, at using of the objects of the same name, to avoid misunderstandings it is necessary, generally, to associate them with the contexts which have been ascribed to them.

For receiving access to the package tools it is necessary that a package containing them was uploaded to the current session, and the list defined by means of the $ContextPath variable has to include the context corresponding to the package. A package can be loaded in any place of the current document by means of the call Get["context'"] or by means of the call Needs["context'"] to define uploading of a package if the context associated with the package is absent in the list defined by $Packages variable. In a case if package begins with BeginPackage["Package'"], at its uploading to the lists defined by the variables $ContextPath and $Packages only context "Package'" is placed, providing the access to exports of the package and system means.

If a package uses means of other packages, then the package should begin with the construction BeginPackage["Package'", {"Package1'", …, "Package2'"}] with indication of the list of the contexts associated with such packages. It allows to include, in addition, in the system lists $ContextPath and $Packages the demanded contexts. With features of uploading of packages the

reader can familiarize oneself, in particular, in [6-8,10-15].

When operating with contexts, the question often arises of testing a string expression as a potentially possible context. The system does not have such a tool and it is possible use a simple function whose call ContextQ[j] returns True if j is a potentially possible context and False otherwise.

In[78]:= ContextQ[j_] := StringQ[j] && StringLength[j] > 1 &&

StringFreeQ[j, "``"] && SymbolQ[StringReplace[j, "`" –> ""]] &&

StringTake[j, –1] == "`" && ! StringTake[j, 1] === "`"

In[79]:= ContextQ["a2a`b1bb`ccc1`"]

Out[79]= True

In[80]:= ContextQ["aa``bbb`ccc`"]

Out[80]= False

A package similarly the procedure allows a nesting; at that, in Mathematica all sub-packages composing it are distinguished and registered. The means defined in the main package and in its sub-packages are fully accessible in the current session after uploading of the nested package as quite visually illustrates a number of simple examples [8-10]. Meanwhile, for performance of the aforesaid it is necessary to redefine system $ContextPath variable after uploading of the nested package, having added all contexts of sub-packages of the main package to the list defined by the $ContextPath variable. In this context the ToContextPath procedure automatizes this task, whose call ToContextPath[x] provides updating of contents of the current list determined by $ContextPath variable by adding to its end of all contexts of m–file x containing in a simple or nested package. Fragment below represents source code of the procedure with an example of use.

In[3333]:= ToContextPath[x_ /; FileExistsQ[x] &&

FileExtension[x] == "m"] := Module[{a = ReadString[x], b},

b = StringReplace[a, "\n" –> ""];

b = StringCases[b, "["~~ Shortest[__] ~~"]"];

b = Map[StringTrim[#, ("[" | "]")] &, b];

b = Select[b, ContextQ[Quiet[ToExpression[#]]] &];

b = ToExpression[DeleteDuplicates[b]];

$ContextPath = Flatten[Insert[$ContextPath, b, –3]];

DeleteDuplicates[$ContextPath]]

In[3334]:= ToContextPath[Directory[] <> "\\init.m"]

Out[3334]= {"DocumentationSearch`", "ResourceLocator`",

"URLUtilities`", "AladjevProcedures`", "PacletManager`",

"NeuralNetworks`", "NeuralNetworks`Bootstrap`Private`",

"GeneralUtilities`", "MXNetLink`", "System`", "Global`"}

However, because of certain features the use of the nested packages doesn't make a special sense.

Thus, if the call Context[x] of built-in function returns the context associated with a symbol x, then an interesting enough question of detecting of a m–file with a package containing the given context arises. The call FindFileContext[x] returns the list of full paths to m–files with packages containing the x context; in a case of absence of such files the call returns the empty list. In addition, the call FindFileContext[x, y, z, …] with optional {y, z, …} arguments – the names in string format of devices of external memory of direct access – provides search of required files on the specified devices instead of all file system of the computer in a case of the procedure call with one actual argument. The search of the required m-files is done also in the $Recycle.bin directory of Windows 7 system [12-16]. However, it must be kept in mind, that search within all file system of the computer can demand a rather essential temporal expenditure.

We have created a number of tools [8] for processing of the user packages located in files of type {cdf, nb, m, mx}. At that, of particular interest is the procedure for determining whether a file of type mx with user package contains the specified tools.

In[3332]:= ToolsInMxQ[x_, y_ /; FileExistsQ[y] &&

FileExtension[y] == "mx", t___] :=

Module[{a = Map[ToString, Flatten[{x}]],

b = ContextsInFiles[y], c, d},

If[MemberQ[$ContextPath, b], MemberQ6[CNames[b], a, t],

Quiet[Get[y]]; c = MemberQ6[Set[d, CNames[b]], a, t];

Map[ClearAll[#] &, d]; $ContextPath =

ReplaceAll[$ContextPath, b –> Nothing];

Unprotect[$Packages]; $Packages =

Complement[$Packages, Contexts[StringTake[b, {1, –2}] <> "*"]];

SetAttributes[$Packages, Protected]; c]]

In[3333]:= ToolsInMxQ[{Vz, mm, Gn, nn}, "Agn.mx", t42]

Out[3333]= False

In[3334]:= t42

Out[3334]= {mm, nn}

The procedure call ToolsInMxQ[x, y, t] returns True if a tool x or their list belong to a file y of mx-type with the user package, and False otherwise. At that, through the 3^rd optional argument t – an undefined symbol – the list of elements of x that are absent in the y file are returned. The procedure uses an useful enough MemberQ6 procedure whose source code is represented below.

In[2]:= MemberQ6[x_ /; ListQ[x], y_ /; ListQ[y], t___] :=

Module[{a = {}, b = {}}, Do[If[MemberQ[x, y[[j]]],

AppendTo[a, True], AppendTo[a, False];

AppendTo[b, y[[j]]]], {j, Length[y]}];

a = AllTrue[a, # == True &];

If[{t} != {} && ! HowAct[t], t = b, 77]; a]

In[3]:= {MemberQ6[{a, b, c, d, g, h}, {a, d, h, u, t, s, w, g, p}, t], t}

Out[3]= {False, {u, t, s, w, p}}

The call MemberQ6[x, y, t] returns True if all elements of a list y belong to a list x, and False otherwise. In addition, through the third optional argument t – an undefined symbol – the list of elements of the y list that are not in the x list are returned.

In a sense the procedures ContextMfile and ContextNBfile are inverse to the procedures FindFileContext, FindFileContext1 and ContextInFile, their successful calls ContextMfile[w] and ContextNBfile[w] return the context associated with a package located in a file w of formats ".m" and {".nb", ".cdf"} accordingly; the w file is set by means of name or full path to it [8,10-16].

Unlike of the procedure DeletePackage [16] the procedure RemovePackage is that the symbols determined by the package, removes completely so that their names are no longer recognized in the current session. The call DeletePackage[x] returns Null, i. e. nothing, providing removal from the current session of the package, determined by a context x, including all its exported symbols and accordingly updating the lists determined by the $Packages, $ContextPath and Contexts[] variables. Fragment below represents source code of the DeletePackage procedure along with accompanying examples.

In[3331]:= BeginPackage["Package7`"]

W::usage = "Help on W."

Begin["`W`"]

W[x_Integer, y_Integer] := x^2 + y^2

End[]

EndPackage[];

In[3337]:= $Packages

Out[3337]= {"Package7`", "DocumentationSearch`", …, "Global`"}

In[3338]:= $ContextPath

Out[3338]= {"Package7`", "DocumentationSearch`", …, "Global`"}

In[3339]:= MemberQ[Contexts[], "Package7`"]

Out[3339]= True

In[3340]:= CNames["Package7`"]

Out[3340]= {"W"}

In[3341]:= ? W

Out[3341]= "Help on W."

In[3342]:= DeletePackage[x_] := Module[{a},

If[! MemberQ[$Packages, x], $Failed, a = Names[x <> "*"];

Map[ClearAttributes[#, Protected] &,

Flatten[{"$Packages", "Contexts", a}]]; Quiet[Map[Remove, a]];

$Packages = Select[$Packages, # != x &];

$ContextPath = Select[$ContextPath, # != x &];

Contexts[] = Select[Contexts[], StringCount[#, x] == 0 &];

Quiet[Map[Remove, Names[x <> "*"]]];

Map[SetAttributes[#, Protected] &,

{"$Packages", "Contexts"}]; ]]

In[3343]:= DeletePackage["Package7`"]

In[3344]:= $Packages

Out[3344]= {"DocumentationSearch`", …,"System`", "Global`"}

In[3345]:= $ContextPath

Out[3345]= {"DocumentationSearch`", …, "System`", "Global`"}

In[3346]:= CNames["Package7`"]

Out[3346]= {}

In[3347]:= ?W

Out[3347]= Missing["UnknownSymbol", "W"]

In[3348]:= MemberQ[Contexts[], "Package7`"]

Out[3348]= False

In certain cases, it makes sense to change the context that is assigned to the user's package contained in the mx–format file. This problem is solved by the procedure presented below. The procedure call ChangeContextInMx[x, y] returns the list of the form {dir, y}, where dir is the path to the $Name file formed in the same directory as the user's original Name mx–file in path x with the package that will have a new y context. If a file x does not have a context, then the procedure call returns $Failed with printing of the corresponding message. In addition, if the Name mx–file is already loaded into the Mathematica current session, it remains loaded (whereas the $Name file is not loaded), otherwise both the Name and $Name files are not loaded into the current session. The fragment represents source code of the procedure along with typical examples of its application.

In[2336]:= ChangeContextInMx[x_ /; FileExistsQ[x] &&

FileExtension[x] == "mx", y_ /; ContextQ[y]] :=

Module[{a, b, c, d}, b = ContextInMxFile[x];

If[SameQ[b, $Failed], Print["File " <> x <> " has no a context"];

$Failed, If[! MemberQ[$Packages, b], Get[x]; c = 78, c = 80];

a = CNames[b]; Map[ContextToSymbol3[#, y] &, a];

d = FileNameSplit[x]; d[[–1]] = "$" <> d[[–1]];

d = FileNameJoin[Flatten[{d[[1 ;–2]], d[[–1]]}]];

DumpSave[d, y]]; Unprotect[Contexts];

Contexts[] = ReplaceAll[Contexts[], Flatten[{b –> Nothing,

y –> Nothing, Map[b <> # –> Nothing &, a],

Map[y <> # –> Nothing &, a]}]]; Protect[Contexts];

Unprotect[$Packages]; $Packages = Complement[$Packages, {b, y}];

$ContextPath = Complement[$ContextPath, {b, y}];

Protect[$Packages]; Map[ClearAll[#] &, Map[y <> # &, a]];

If[c == 80, Get[x], Null]; {d, y}]

In[2337]:= ContextInMxFile["avzagn.mx"]

Out[2337]= "RansIan`"

In[2338]:= ChangeContextInMx["avzagn.mx", "NewContext`"]

Out[2338]= {"$avzagn.mx", "NewContext`"}

In[2339]:= Get["$avzagn.mx"]

In[2340]:= CNames["NewContext`"]

Out[2340]= {"ArtKr", "GS", "GSV"}

Regarding contexts in the user packages, it makes sense to make one rather significant comment, the essence of which is as follows. In principle, the Mathematica allows to use the blocks, functions, and modules of the same name only with different definitions in packages with different contexts. When loading such packages, the appropriate message is displayed, while in predefined variables $Packages and $ContextPath, the contexts of the packages are placed according to the order of their loading.

BeginPackage["IAN`"]

Vgs::usage = "Help on Vgs 1."

Begin["`Vgs`"]

Vgs[x_, y_] := Module[{a = 77}, a*x/y]

End[]

EndPackage[];

BeginPackage["Cont`"]

Vgs::usage = "Help on Vgs 2."

Begin["`Vgs`"]

Vgs[x_, y_] := Module[{a = 78}, a*(x + y)]

End[]

EndPackage[];

In[2441]:= DumpSave["avzagn.mx", "IAN`"];

In[2442]= DumpSave["$avzagn.mx", "Cont`"];

In[2443]:= Get["avzagn.mx"]; Get["$avzagn.mx"];

… Symbol Vgs appears in multiple contexts {Cont`, IAN`…

In[2444]:= $ContextPath

Out[2444]= {"Cont`", "Ian`", "DocumentationSearch`", …

In[2445]:= $Packages

Out[2445]= {"Cont`", "Ian`", "DocumentationSearch`", …

In[2446]:= DumpSave["$avzagn.mx", "Cont`"]

Out[2446]= {"Cont`"}

In[2447]:= {IAN`Vgs[78, 73], Cont`Vgs[78, 73]}

Out[2447]= {6006/73, 11778}

From the above fragment follows that two small packages with different contexts contain different definitions of the Vgs function of the same name. Loading the mx–files in the current session results in two Vgs functions with different contexts that you can be accessed by {Cont'|IAN'}Vgs. So, a block, function, or module is identified by its name and a context assigned to it.

To define contexts assigned to a symbol whose definitions are loaded from mx–files with different contexts, a fairly simple procedure can be used, whose call Context1[x] returns the list of contexts assigned to the x symbol. The source code of this procedure and examples of its use are represented below.

BeginPackage["IAN`"]

Vgs::usage = "Help on Vgs."

Begin["`Vgs`"]

Vgs[x_, y_] := Module[{a = 77}, a*x/y]

End[]

EndPackage[];

BeginPackage["RANS`"]

Vgs::usage = "Help on Vgs1."

Begin["`Vgs`"]

Vgs[x_, y_] := Module[{a = 78}, a*(x + y)]

End[]

EndPackage[];

BeginPackage["AVZ`"]

Vgs::usage = "Help on Vgs2."

Begin["`Vgs`"]

Vgs[x_, y_] := Module[{a = 80}, a*(x^2 + y^2)]

End[]

EndPackage[];

In[2542]:= DumpSave["avag.mx", "AVZ`"];

In[2543]= DumpSave["$avag.mx", "RANS`"];

In[2544]= DumpSave["$$avag.mx", "IAN`"];

In[2545]= Get["avag.mx"]; Get["$avag.mx"]; Get["$$avag.mx"]

In[2547]:= Context1[x_ /; SymbolQ[x]] := Module[{a, b = {}},

Map[If[MemberQ[CNames[#], ToString[x]], AppendTo[b, #],

Nothing] &, $Packages]; b]

In[2548]:= Context1[Vgs]

Out[2548]= {"IAN`", "RANS`", "AVZ`"}

In[2549]:= ContextToSymbol4[Vgs, "AGN`"]

Out[2549]= {"Vgs", "AGN`", "AVZ`", "RANS`", "IAN`"}

In[2550]:= Context1[Vgs]

Out[2550]= {"AGN`", "RANS`", "AVZ`"}

Note that calling ContextToSymbol3["Vgs", "AGN'"] changes the higher priority context of the Vgs symbol to "AGN'" without changing its other contexts and their priority order.

Context management is discussed in sufficient detail in [4, 6,8-15] along with a number of software means focused on the operating with contexts and associated with them objects. In particular, the procedure below is quite interesting whose call AllContexts[] returns list of contexts that contain in the system packages of the current Mathematica version, whereas the call AllContexts[x] returns True, if x is a context of the above type, and False otherwise. The fragment represents source code of the procedure along with typical examples of its application.

In[2247]:= AllContexts[y___] := Module[{a = Directory[], c, h,

b = SetDirectory[$InstallationDirectory]},

b = FileNames[{"*.m", "*.tr"}, {"*"}, Infinity];

h[x_] := StringReplace[StringJoin[Map[FromCharacterCode,

BinaryReadList[x]]], {"\n" –> "", "\r" –> "", "\t" –> "", " " –> "",

"{" –> "", "}" –> ""}];

c = Flatten[Select[Map[StringCases[h[#],

{"BeginPackage[\"" ~~ Shortest[W__] ~~ "`\"]",

"Needs[\"" ~~ Shortest[W__] ~~ "`\"]",

"Begin[\"" ~~ Shortest[W__] ~~ "`\"]",

"Get[\"" ~~ Shortest[W__] ~~ "`\"]",

"Package[\"" ~~ Shortest[W__] ~~ "`\"]"}] &, b], # != {} &]];

c = DeleteDuplicates[Select[c, StringFreeQ[#, {"*", "="}] &]];

SetDirectory[a]; c = Flatten[Append[Select[Map[

StringReplace[StringTake[#, {1, –2}], {"Get[" –> "",

"Begin[" –> "", "Needs[" –> "", "BeginPackage[" –> "",

"Package[" –> ""}, 1] &, c], # != "\"`Private`\"" &],

{"\"Global`\"", "\"CloudObjectLoader`\""}]];

c = Map[ToExpression, Sort[Select[DeleteDuplicates[Flatten[

Map[If[StringFreeQ[#, ","], #, StringSplit[#, ","]] &, c]]],

StringFreeQ[#, {"<>", "]", "["}] && StringTake[#, {1, 2}] != "\"`" &]]];

f[{y} != {}, MemberQ[c, y], c]]

In[2248]:= AllContexts[]

Out[2248]= "AnatomyGraphics3D`", …, "ZeroMQLink`"}

In[2249]:= Length[%]

Out[2249]= 1634

In[2250]:= AllContexts["PresenterTools`PresenterTools`"]

Out[2250]= True

As it was noted in [15], for each exported object of a certain package for it it is necessary to determine an usage. As a result of loading of such package into the current session all its exports will be available whereas the local objects, located in a section, in particular Private, will be inaccessible in the current session. For testing of a package loaded to the current session or loaded package which is located in a m–file regarding existence in it of global and local objects the procedure DefInPackage can be used, whose the call DefInPackage[x], where x defines a file or full path to it, or context associated with the package returns the nested list, whose the first element defines the package context, the second element – the list of local variables whereas the third element – the list of global variables of the x package. If the x argument doesn't define a package or context, the procedure call DefInPackage[x] is returned unevaluated. In a case of an unusable x context the procedure call returns $Failed. Fragment represents source code of the DefInPackage procedure with the most typical examples of its application.

In[7]:= DefInPackage[x_ /; MfilePackageQ[x]||ContextQ[x]] :=

Module[{a, b = {"Begin[\"`", "`\"]"}, c = "BeginPackage[\"", d,

p, g, t, k = 1, f, n = x},

Label[Avz];

If[ContextQ[n] && Contexts[n] != {}, f = "#"; Save[f, x];

g = FromCharacterCode[17]; t = n <> "Private`";

a = ReadFullFile[f, g]; DeleteFile[f]; d = CNames[n];

p = SubsString[a, {t, g}];

p = DeleteDuplicates[Map[StringCases[#, t ~~ Shortest[___] ~~ "["

<> t ~~ Shortest[___] ~~ " := "] &, p]];

p = Map[StringTake[#, {StringLength[t] + 1,

Flatten[StringPosition[#, "["]][[1]] – 1}] &, Flatten[p]];

{n, DeleteDuplicates[p], d}, If[FileExistsQ[n], a = ReadFullFile[n];

f = StringTake[SubsString[a, {c, "`\"]"}], {15, –3}][[1]];

If[MemberQ[$Packages, f], n = f; Goto[Avz]];

b = StringSplit[a, "*)(*"];

d = Select[b, ! StringFreeQ[StringReplace[#, " " –> ""], "::usage="] &];

d = Map[StringTake[#, {1, Flatten[StringPosition[#, "::"]][[1]] – 1}] &, d];

p = DeleteDuplicates[Select[b, StringDependAllQ[#,

{"Begin[\"`", "`\"]"}] &]];

p = MinusList[Map[StringTake[#, {9, –4}] &, p], {"Private"}];

t = Flatten[StringSplit[SubsString[a, {"Begin[\"`Private`\"]",

"End[]"}], "*)(*"]]; If[t == {}, {f, MinusList[d, p], p},

g = Map[StringReplace[#, " " –> ""] &, t[[2 ;–1]]];

g = Select[g, ! StringFreeQ[#, ":="] &];

g = Map[StringTake[#, {1, Flatten[StringPosition[#, ":"]][[1]] – 1}] &, g];

g = Map[Quiet[Check[StringTake[#, {1, Flatten[StringPosition[#,

"["]][[1]] – 1}], #]] &, g]; {f, g, d}], $Failed]]]

In[8]:= DefInPackage["C:\\Mathematica\\MathToolBox.m"]

Out[8]:= {"AladjevProcedures`", {}, {"AcNb", "ActBFM", …,

"$UserContexts1", "$UserContexts2", "$Version1","$Version2"}}

In[9]:= Length[%[[3]]]

Out[9]= 1426

In a number of cases there is a need of full removal from the current session of a package loaded to it. Partially the given problem is solved by the standard functions Clear and Remove, however they don't clear the lists that are defined by variables $ContextPath, $Packages along with the call Contexts[] off the package information. This problem is solved by means of the DeletePackage procedure whose the call DeletePackage[x] will return Null, in addition, completely removing from the current session a package defined by x context, including all exports of the x package and respectively updating the above system lists. Fragment below represents source code of the DeletePackage procedure with the most typical example of its application.

In[2247]:= DeletePackage[x_] := Module[{a},

If[! MemberQ[$Packages, x], $Failed, a = Names[x <> "*"];

Map[ClearAttributes[#, Protected] &,

Flatten[{"$Packages", "Contexts", a}]];

Quiet[Map[Remove, a]]; $Packages = Select[$Packages, # != x &];

$ContextPath = Select[$ContextPath, # != x &];

Contexts[] = Select[Contexts[], StringCount[#, x] == 0 &];

Quiet[Map[Remove, Names[x <> "*"]]];

Map[SetAttributes[#, Protected] &, {"$Packages", "Contexts"}];]]

In[2248]:= DeletePackage["AladjevProcedures`"]

In[2249]:= ?DeletePackage

Out[2249]= Missing["UnknownSymbol", "DeletePackage"]

Natural addition to the DeletePackage and RemovePackage procedures is the procedure DelOfPackage providing removal from the current session of the given tools of a loaded package. Its call DelOfPackage[x, y] returns the y list of tools names of a package given by its x context which have been removed from the current session. Whereas the call DelOfPackage[x, y, z] with the third optional z argument – a mx–file – returns 2–element list whose the first element defines mx–file z, while the second element defines the y list of tools of a package x that have been removed from the current session and have been saved in mx–file z. At that, only tools of y which are contained in x package will be removed. The fragment below represents source code of the DelOfPackage procedure with typical examples of its use.

In[7]:= DelOfPackage[x_ /; ContextQ[x], y_ /; SymbolQ[y] ||

(ListQ[y] && AllTrue[Map[SymbolQ, y], TrueQ]), z___] :=

Module[{a, b, c}, If[! MemberQ[$Packages, x], $Failed,

If[Set[b, Intersection[Names[x <> "*"],

a = Map[ToString, Flatten[{y}]]]] == {}, $Failed,

If[{z} != {} && StringQ[z] && SuffPref[z, ".mx", 2],

ToExpression["DumpSave[" <> ToString1[z] <> "," <>

ToString[b] <> "]"]; c = {z, b}, c = b];

ClearAttributes[b, Protected]; Map[Remove, b]; c]]]

In[8]:= DelOfPackage["AladjevProcedures`", {Mapp, Map1}, "#.mx"]

Out[8]= {"#.mx", {"Map1", "Mapp"}}

The IsPackageQ procedure [8] is intended for testing of any mx–file regarding existence of the user package in it along with upload of such package into the current session. The call of the IsPackageQ[x] procedure returns $Failed if the mx–file doesn't contain a package, True if the package that is in the x mx–file is uploaded to the current session, and False otherwise. Moreover, the procedure call IsPackageQ[x, y] through the second optional y argument – an indefinite variable - returns the context associated with the package uploaded to the current session. In addition, is supposed that the x file is recognized by the testing standard function FileExistsQ, otherwise the procedure call is returned unevaluated. The use examples the reader can found in [8-15].

5.3. Additional tools of operating with the user packages

Tools of the Mathematica for operating with datafiles can be subdivided to two groups conditionally: the tools supporting the work with files which are automatically recognized at addressing to them, and the tools supporting the work with files as a whole. This theme is quite extensive and is in more detail considered in [8], here some additional tools of working with files containing the user packages will be considered. Since tools of access to files of formats, even automatically recognized by Mathematica, don't solve a number of important problems, the user is compelled to program own means on the basis of standard tools and perhaps with use of own means. Qua of a rather useful example we will represent the procedure whose call DefFromPackage[x] returns 3–element list, whose first element is definition in string format of some symbol x whose context is different from {"System'", "Global'"}, the second element defines its usage while the third element defines attributes of the x symbol. In addition, on the symbols associated with 2 specified contexts, the procedure call returns only list of their attributes. The fragment represents the source code of the DefFromPackage procedure with examples of its typical application.

In[327]:= DefFromPackage[x_ /; SymbolQ[x]] := Module[{b = "",

a = Context[x], c = "", p, d = ToString[x], k = 1, h},

If[MemberQ[{"Global`", "System`"}, a], Return[Attributes[x]],

h = a <> d; ToExpression["Save[" <> ToString1[d] <> "," <>

ToString1[h] <> "]"]];

For[k, k < Infinity, k++, c = Read[d, String];

If[c === " ", Break[], b = b <> c]];

p = StringReplace[RedSymbStr[b, " ", " "], h <> "`" –> ""];

{c, k, b} = {"", 1, ""};

For[k, k < Infinity, k++, c = Read[d, String];

If[c === " " || c === EndOfFile, Break[], b = b <>

If[StringTake[c, {–1, –1}] == "\\", StringTake[c, {1, –2}], c]]];

DeleteFile[Close[d]]; {p, StringReplace[b, " /: " <> d –> ""],

Attributes[x]}]

In[328]:= DefFromPackage[StrStr]

Out[328]= {"StrStr[x_] := If[StringQ[x], StringJoin["\"", x, "\""],

ToString[x]]", "StrStr::usage = "The call StrStr[x] returns an

expression x in string format if x is different from string;

otherwise, the double string obtained from an expression

is returned.", {}}

The DefFromPackage procedure serves for obtaining of full information on x symbol whose definition is located in the user package uploaded into the current session. Unlike the standard FilePrint and Definition functions this procedure, first, doesn't print, but returns specified information completely available for subsequent processing, and, secondly, this information is returned in an optimum format. At that, in a number of cases the output of definition of a symbol that is located in an active package by the standard means is accompanied with a context associated with the package which complicates its viewing, and also its subsequent processing. Result of the DefFromPackage call also obviates this problem. The algorithm realized by the procedure is based on analysis of structure of a file received in result of saving of a "y'x" context, where x – a symbol at the call DefFromPackage[x] and "y'" – a context, associated with the uploaded package containing the definition of x symbol. In more detail the algorithm realized by the DefFromPackage is visible from its source code.

As the 2^nd example extending the algorithm of the previous procedure in the light of application of functions of access it is possible to represent an useful FullCalls procedure whose the call FullCalls[x] returns the list whose 1^st element is the context associated with a package loaded in the current session, while its other elements – the symbols of this package which are used by the user procedure or function x, or nested list of sub-lists of the above type at using by the x of symbols (names of procedures or functions) from several packages. Source code of the FullCalls procedure and an example of its application are represented in the following fragment.

In[2227]:= FullCalls[x_ /; ProcQ[x] || FunctionQ[x]] :=

Module[{a = {}, b, d, c = "::usage = ", k = 1},

Save[b = ToString[x], x];

For[k, k < Infinity, k++, d = Read[b, String];

If[d === EndOfFile, Break[], If[StringFreeQ[d, c], Continue[],

AppendTo[a, StringSplit[StringTake[d,

{1, Flatten[StringPosition[d, c]][[1]] – 1}], " /: "][[1]]]]]];

a = Select[a, SymbolQ[#] &]; DeleteFile[Close[b]];

a = Map[{#, Context[#]} &, DeleteDuplicates[a]];

a = If[Length[a] == 1, a, Map[DeleteDuplicates, Map[Flatten,

Gather[a, #1[[2]] === #2[[2]] &]]]]; {d, k} = {{}, 1};

While[k <= Length[a], b = Select[a[[k]], ContextQ[#] &];

c = Select[a[[k]], ! ContextQ[#] &];

AppendTo[d, Flatten[{b, Sort[c]}]]; k++];

d = MinusList[If[Length[d] == 1, Flatten[d], d], {ToString[x]}];

If[d == {Context[x]}, {}, d]]

In[2228]:= FullCalls[DefFromPackage]

Out[2228]= {"AladjevProcedures`", "GenRules", "ListListQ",

"Map3","Map13","Map9","RedSymbStr","StrDelEnds","SuffPref",

"StringMultiple", "SymbolQ", "SymbolQ1", "ToString1"}

Thus, the procedure call FullCalls[x] provides possibility of testing of the user procedure or function, different from built-in means, regarding using by it of means whose definitions are in packages loaded into the current session. In development of the procedure the FullCalls1 procedure can be offered with whose source code and typical examples of its use the interested reader can familiarize in [8,16]. In addition, both procedures FullCalls and FullCalls1 are rather useful at programming a number of appendices with which can familiarize in [7-12,14,15].

In a number of cases there is a necessity for uploading into the current session of the system Mathematica not entirely of a package, but only separate tools contained in it, for example, of a procedure or function, or their list. In the following fragment a procedure is represented, whose call ExtrOfMfile[x, y] returns Null, i.e. nothing, loading in the current session the definitions only of those tools that are defined by an y argument and are located in a file x of m–format. In addition, at existence in the m–file of several tools of the same name, the last is uploaded in the current session. While the call ExtrOfMfile[x, y, z] with the third optional z argument – an indefinite variable – in addition, through z returns the list of definitions of y tools which will be located in the m–file x. In a case of absence in the m–file x of y tools the procedure call returns $Failed. The following fragment represents source code of the ExtrOfMfile procedure along with some typical examples of its application.

In[6]:= ExtrOfMfile[f_/; FileExistsQ[f] && FileExtension[f] == "m",

s_ /; StringQ[s]||ListQ[s], z___] := Module[{Vsv, p = {}, v, m},

m = ReadFullFile[f];

If[StringFreeQ[m, Map["(*Begin[\"`" <> # <> "`\"]*)" &,

Map[ToString, s]]], $Failed,

Vsv[x_, y_] := Module[{a = m, b = FromCharacterCode[17],

c = FromCharacterCode[24], d = "(*Begin[\"`" <> y <> "`\"]*)",

h = "(*End[]*)", g = {}, t}, a = StringReplace[a, h –> c];

If[StringFreeQ[a, d], $Failed,

While[! StringFreeQ[a, d], a = StringReplace[a, d –> b, 1];

t = StringTake[SubStrSymbolParity1[a, b, c][[1]], {4, –4}];

t = StringReplace[t, {"(*" –> "", "*)" –> ""}];

AppendTo[g, t]; a = StringReplace[a, b –> "", 1]; Continue[]];

{g, ToExpression[g[[–1]]]}]];

If[StringQ[s], v = Quiet[Check[Vsv[f, s][[1]], $Failed]],

Map[{v = Quiet[Check[Vsv[f, #][[1]], $Failed]], AppendTo[p, v]} &,

Map[ToString, s]]];

If[{z} != {} && ! HowAct[z], z = If[StringQ[s], v, p]]; ]]

In[7]:= Clear[StrStr]

In[8]:= Definition[StrStr]

Out[8]= Null

In[9]:= ExtrOfMfile["c:/mathematica/MathToolBox.m", "StrStr"]

In[10]:= Definition[StrStr]

Out[10]= StrStr[x_] := If[StringQ[x], " <> x <> ", ToString[x]]

It should be noted that this procedure can be quite useful in a case of need of recovery in the current session of the damaged means without uploading of the user packages containing their definitions. In a certain measure, to the previous procedure the ExtrDefFromM procedure adjoins whose call ExtrDefFromM[x, y] in tabular form returns an usage and definition of a means y contained in a m–file x with the user package. This tool doesn't demand loading of the m–file x in the current session, allowing in it selectively to activate the means of the user package.

At last, the procedure ContextsInFiles provides evaluation of the context ascribed to a file of the format {"m", "mx", "cdf", "nb"}. The procedure call ContextsInFiles[w] returns the single context ascribed to a file w of the above formats. At absence of a context the call returns $Failed. In addition, it must be kept in mind that the context in files of the specified format is sought relative to the key word "BeginPackage" that is typically used at beginning of a package. A return of list of format {"Context1", …, "Contextp"} is equivalent to existence in a m–file of a certain construction of format BeginPackage["context1'", {"context2'", ..., "contextp'"}] where {"context2'", ..., "contextp'"} determines uploadings of the appropriate files if their contexts aren't in the $Packages variable. The next fragment represents source code of the procedure with the most typical examples of its use.

In[47]:= ContextsInFiles[x_ /; FileExistsQ[x] &&

MemberQ[{"m", "mx", "cdf", "nb"}, FileExtension[x]]] :=

Module[{a = StringReplace[ReadFullFile[x], {" " –> "",

"\n" –> "", "\t" –> ""}], b, b1,

d = Flatten[{Range[65, 90], Range[96, 122]}]},

If[FileExtension[x] == "m", b = StringCases[a,

Shortest["BeginPackage[\"" ~~ __ ~~ "`\"]"]];

If[b === {}, b = $Failed, b = Map[StringTake[#, {15, –3}] &, b][[1]];

b1 = StringCases[a, Shortest["BeginPackage[\"" ~~ __ ~~ "`\"}]"]];

If[b1 === {}, b1 = $Failed, If[! StringFreeQ[b1[[1]], {"{", "}"}],

b1 = StringReplace[b1[[1]], {"{" –> "", "}" –> "", "\"" –> ""}];

b1 = StringSplit[StringTake[b1, {14, –2}], ","], b1 = $Failed]]];

b = DeleteDuplicates[Flatten[{b, b1}]];

SetAttributes[ContextQ, Listable];

b = Select[b, AllTrue[Flatten[ContextQ[{#}]], TrueQ] &];

If[ListQ[b] && Length[DeleteDuplicates[b]] == 1, b[[1]], b],

If[FileExtension[x] == "mx", Quiet[Check[b = StringCases[a,

Shortest["CONT" ~~ __ ~~ "ENDCONT"]][[1]];

b = Flatten[Map[ToCharacterCode, Characters[StringReplace[b,

{"CONT" –> "", "ENDCONT" –> ""}]]]];

StringJoin[Map[FromCharacterCode[#] &,

Select[b, MemberQ[d, #] &]]], $Failed]],

Quiet[Check[a = StringCases[a, Shortest["BeginPackage\",

\"[\",\"\\\"\\<" ~~ __ ~~ "`\\>\\\"\",\"]\"}]"]][[1]];

a = StringReplace[a, "BeginPackage" –> ""];

b = Flatten[Map[ToCharacterCode, Characters[a]]];

StringJoin[Map[FromCharacterCode[#] &,

Select[b, MemberQ[d, #] &]]], $Failed]]]]]

In[48]:= ContextsInFiles["C:\\mathematica\\mathtoolbox.mx"]

Out[48]= "AladjevProcedures`"

In[49]:= ContextsInFiles[$InstallationDirectory <>\\SystemFiles\\

components\\ccodegenerator\\CCodeGenerator.m"]

Out[49]= {"CCodeGenerator`", "CompiledFunctionTools`",

"CompiledFunctionTools`Opcodes`", "SymbolicC`",

"CCompilerDriver`"}

In addition to the previous ContextsInFiles procedure the ContextsFromFiles procedure allows to extract contexts from files of any format that are located in the set directories or are given by own full or short names. The call ContextsFromFiles[] without arguments returns the sorted list of contexts from all files located in the catalog $InstallationDirectory and in all its subdirectories whereas the call ContextsFromFiles[x] returns the sorted list of contexts from all files located in a directory x and in all its sub-directories, at last the call ContextsFromFiles[x] on an existing x file returns the sorted list of contexts from the x file. The unsuccessful means call returns $Failed or is returned unevaluated. The next fragment represents source code of the procedure with the most typical examples of its application.

In[2226]:= ContextsFromFiles[x___String] :=

Module[{a = If[{x} == {}, FileNames["*", $InstallationDirectory,

Infinity], If[DirQ[x], FileNames["*", x, Infinity],

If[FileExistsQ[x], {x}, $Failed]]], b}, If[a === $Failed, $Failed,

b = Select[Map[Quiet[StringReplace[#, {": " -> "", " " -> "", "\"" -> ""}]] &,

Flatten[Map[StringCases[Quiet[Check[ReadFullFile[#], ""]],

(Shortest[": " ~~ __ ~~ "` "] | Shortest["\"" ~~ __ ~~ "`\""]) ..] &, a]]],

ContextQ[#] &]; Sort[DeleteDuplicates[b]]]]

In[2227]:= ContextsFromFiles[$InstallationDirectory <>

"\\AddOns\\Applications\\AuthorTools\\Kernel"]

Out[2227]= {"AuthorTools`", "AuthorTools`MakeBilateralCells`"}

Note, the above procedure operates on platforms Windows XP Professional and Windows 7 Professional. The performance of the procedure is higher if it is applied to a mx–file.

The binary mx–files are optimized for fast loading. Wherein, they cannot be exchanged between different operating systems or versions of the Mathematica. Any mx–file contains version of Mathematica, the type of operating system in which it has been created and the context if it will exist. The procedure call ContextMathOsMx[x] returns the three–element list whose the first element determines the context if it exists (otherwise, $Failed is returned), the second element defines the list whose elements define version, releases of the Mathematica and the 3^rd element defines operating system in which a mx–file x has been created.

In view of distinctions of the mx–files created on different platforms there is a natural expediency of creation of the tools testing a mx–file regarding the platform in which it was created in virtue of the DumpSave function. The TypeWinMx procedure is one of such tools. The procedure call TypeWinMx[x] in string format returns type of operating platform on which a mx–file x was created; correct result is returned for a case of the Windows platform, whereas on other platforms $Failed is returned. This is conditioned at absence of a possibility to carry out debugging on other platforms. The next fragment represents source code of the TypeWinMx procedure with examples of its application.

In[3339]:= TypeWinMx[x_ /; FileExistsQ[x] &&

FileExtension[x] == "mx"] := Module[{a, b, c, d},

If[StringFreeQ[$OperatingSystem, "Windows"], $Failed,

a = StringJoin[Select[Characters[ReadString[x]],

SymbolQ[#]||Quiet[IntegerQ[ToExpression[#]]]||# == "–" &]];

d = Map[FromCharacterCode, Range[2, 27]];

b = StringPosition[a, {"CONT", "ENDCONT"}];

If[b[[1]][[2]] == b[[2]][[2]],

c = StringCases1[a, {"Windows", "ENDCONT"}, "___"],

b = StringPosition[a, {"Windows", "CONT"}];

c = StringTake[a, {b[[1]][[1]], b[[2]][[2]]}]];

c = StringReplace[c, Flatten[{GenRules[d, ""],

"ENDCONT" –> "", "CONT" –> ""}]]; If[ListQ[c], c[[1]], c]]]

In[3340]:= TypeWinMx["c:\\mathematica\\mathToolBox.mx"]

Out[3340]= "Windows–x86–64"

Along with the problem of determining the context in files of format {m, mx, nb}, the problem of determining files of this type containing the given context is of particular interest. This task is solved by the procedure whose call FilesWithContext[x, y] returns the list containing full paths to files from a directory y and all its sub-directories of the format {m,mx,nb} that contain a context x. Argument y is optional and in case of its absence instead of it the directory Directory[] is used. The procedure call with incorrect directory y returns $Failed, whereas the absence in the directory y files with demanded context x returns empty list. Note that the procedure uses our procedure, whose call StandPath[w] returns the result of standardizing the path to a file or directory w, which ensures that the FilesWithContext procedure runs correctly on directories and files whose names contain space symbol. In the following fragment the source code of the procedure and examples of its application is represented.

In[42]:= FilesWithContext[x_ /; ContextQ[x], y___] :=

Module[{a, b, c, d, g, h},

If[{y} == {}, a = Directory[], If[! DirQ[y], Return[$Failed], a = y]];

b = Run["Dir " <> a <> "\\*.mx /B/S > " <> Directory[] <> "\\#"];

a = StandPath[a];

If[b == 0, c = ReadList["#", String], c = {}];

b = Run["Dir " <> a <> "\\*.m /B/S > " <> Directory[] <> "\\#"];

If[b == 0, d = ReadList["#", String], d = {}];

b = Run["Dir " <> a <> "\\*.nb /B/S > " <> Directory[] <> "\\#"];

If[b == 0, g = ReadList["#", String], g = {}]; DeleteFile["#"];

If[Set[h, Join[c, d]] == {}, {},

c = Map[StringReplace[#, " . " –> "."] &, h]];

Map[If[ContextFromFile[#] === x, #, Nothing] &, c]]

In[43]:= FilesWithContext["AVZ`", Directory[]]

Out[43]= {"C:\\Users\\Aladjev\\Documents\\$$avzagn.mx",

"C:\\Users\\Aladjev\\Documents\\Agn\\$$avzagn.m"}

In[44]:= FilesWithContext["RANS`"]

Out[44]= {"C:\\Users\\Aladjev\\Documents\\$avzagn.mx",

"C:\\Users\\Aladjev\\Documents\\Agn\\$avzagn.m"}

This procedure naturally extends to files of certain other types. A quite certain interest is the question of occurrence in the definition of the user tools of the tools calls other than built-in tools. The FullContent procedure solves this problem. Calling FullContent[x] procedure returns the sorted list containing all names in string format of the user tools whose calls are in definition of a tool x. Whereas the call FullContent[x, y] with the 2^nd optional y argument – an indefinite symbol – additionally through it returns the nested list of the above tools with contexts ascribed to them; the list is gathered according to the contexts. In addition, a context is the first in the list of the tools names with this context. Indeed, the name x is included in the returned list too. The following fragment represents the source code of the FullContent procedure with examples of its application.

In[42]:= FullContent[x_, y___] := Module[{a, b, c, n, m},

If[MemberQ[{FullContent, "FullContent"}, x],

c = FullContent[ProcQ], a = Save["###", x];

a = StringReplace[ReadString["###"], {"\n" –> "", "\r" –> ""}];

b = StringCases[a, Shortest[" /: " ~~ __ ~~ "::usage ="]];

DeleteFile["#"]; c = Map[{n = StringReplace[#, "::usage =" –> ""];

m = Flatten[StringPosition[n, " /: "]];

StringTake[n, {m[[–1]] + 1, –1}]} &, b];

c = Sort[DeleteDuplicates[Map[StringTrim[#, "\\"] &, Flatten[c]]]]];

If[{y} != {} && ! HowAct[y], y = Map[{#, Context[#]} &, c];

y = Gather[y, #1[[2]] == #2[[2]] &];

y = Sort[Map[Sort[DeleteDuplicates[Flatten[#]]] &, y],

If[ContextQ[#1], #1, #2] &]; c, c]]

In[43]:= FullContent[ProcQ, gs]

Out[43]= {"Attributes1", "BlockFuncModQ", "ClearAllAttributes", "Contexts1", "Definition2", "HeadPF", "HowAct", "ListStrToStr", "Map3", "Mapp", "MinusList", "ProcQ", "ReduceLists", "ProtectedQ", "PureDefinition", "StrDelEnds", "SuffPref", "SymbolQ", "SysFuncQ", "SystemQ", "ToString1", "UnevaluatedQ"}

In[44]:= gs

Out[44]= {{"AladjevProcedures`", "Attributes1", …, "Map3", …, "SymbolQ", "SysFuncQ", "SystemQ", "ToString1", "UnevaluatedQ"}}

In[45]:= {FullContent[LoadMxQ, vg], vg}

Out[45]= {{"LoadMxQ", "MxOnOpSys", "StrDelEnds", "StringCases2", "SuffPref", "ToString1"}, {{"AladjevProcedures`", "LoadMxQ", "MxOnOpSys", "StrDelEnds", "StringCases2", "SuffPref", "ToString1"}}

The m–format is typical package format. As a rule it is used for the distribution of both system and user packages. The next rather simple procedure allows to retrieving the contents of such user files without loading them into the current session. Calling the ContentsM[x] procedure returns the list of names in string format that compose the user m–format file x without loading it into the current session. The following fragment represents the source code of the ContentsM procedure and examples of its use.

In[1940]:= ContentsM[x_ /; FileExistsQ[x] &&

FileExtension[x] == "m"] := Module[{a = ReadFullFile[x], j},

j = StringCases[a, Shortest["(*Begin[\"`" ~~ __ ~~ "`\"]*)"]];

j = Map[StringReplace[#, {"(*Begin[\"`"-> "", "`\"]*)" -> ""}] &, j];

Sort[Map[If[SymbolQ[#], #, Nothing] &, j]]]

In[1941]:= ContentsM["C:\\Math\\mathtoolbox.m"]

Out[1941]= {"AcNb", "ActBFM", "ActBFMuserQ", "ActCsProcFunc", "ActivateMeansFromCdfNb", "ActRemObj", "ActUcontexts",…,"$UserContexts2", "$Version1", "$Version2"}

In[1942]:= Length[%]

Out[1942]= 1421

It is known, Mathematica language code and expressions can be stored in a variety of formats. In particular, {.wl, .m} file formats are used to store packages. As a rule they are used for the distribution of system and user packages. A rather simple procedure allows to retrieving the contents of the user wl–files without loading them into the current Mathematica session. Calling the ContentsWl[x] procedure returns the nested list of names in string format of blocks, functions and modules that compose the user wl–format file x without loading it into the current session. The elements of the returned list – 2–elements sub–lists whose the first element is a context whereas the other elements are names corresponding it. The following fragment represents the source code of the ContentsWl procedure with some typical examples of its application.

In[78]:= ContentsWl[x_ /; FileExistsQ[x] && FileExtension[x] ==

"wl"] := Module[{s, a, b, c, g, v}, s = ReadString[x];

a = StringSplit[s, {"(* ::Input:: *)", "(* ::Package:: *)"}];

b = StringReplace[a, {"(*" –> "", "*)" –> "", "\r" –> "", "\n" –> ""}];

c = Map[If[# != "" && SyntaxQ[#], #, Nothing] &, b];

g = Map[{g = Flatten[StringPosition[#, ":=", 1]]; If[g == {},

Nothing, If[HeadingQ4[v = StringTake[#, {1, g[[1]] – 1}]],

v, Nothing]]} &, c]; g = DeleteDuplicates[Flatten[g]];

g = Map[HeadName1[#] &, g]; g = Map[{#, Context[#]} &, g];

g = Gather[g, #1[[2]] == #2[[2]] &]; g = Map[Flatten, g];

g = Map[Sort[DeleteDuplicates[#]] &, g];

Map[Sort[#, ContextQ[#] &] &, g]]

In[79]:= ContentsWl["C:\\Mathematica\\Exp79.wl"]

Out[79]= {{"AladjevProcedures`", "ToUnique", "Replace7",

"RandSortList", "PrevUnique", "PartSortList", "Agn"},

{"Global`", "ReplaceInSitu", "P", "Mn", "MM", "G"}}

As an auxiliary tool, the procedure uses a procedure whose call HeadingQ4[x] returns True if x is a header in string format of a block, function, or module, and False otherwise. The source code of the HeadingQ4 procedure along with an example of its application are represented below.

In[84]:= HeadingQ4[x_ /; StringQ[x]] := Module[{i, j},

Quiet[ToExpression[j = ToString[i] <> x; j <> ":=Module[{j},j]"]];

BlockFuncModQ[HeadName1[j]]]

In[85]:= HeadingQ4["DefFromMfile[x_ /; SymbolQ[x],

y_ /; FileExistsQ[y] && FileExtension[y] == \"wl\"]"]

Out[85]= True

Unlike the previous procedure, calling the next procedure DefFromMfile[x, y] returns the definition in the string format of the x symbol contained in the user m–file y, if it is not present, $Failed is returned.

In[47]:= DefFromMfile[x_ /; SymbolQ[x],

y_ /; FileExistsQ[y] && FileExtension[y] == "m"] :=

Module[{a, b, c, d, j, g = ""}, a = ReadFullFile[y];

b = "(*Begin[\"`" <> ToString[x] <> "`\"]*)";

If[StringFreeQ[a, b], $Failed, c = StringPosition[a, b];

c = Flatten[c][[2]]; Do[For[j = c + 1, j <= Infinity, j++,

If[SyntaxQ[d = StringTake[a, {c + 1, j}]],

g = g <> StringTake[d, {3, –3}]; Break[], Continue[]]];

If[SyntaxQ[g], Return[g], c = j; Continue[]], Infinity]]]

In[48]:= DefFromMfile[StrStr, "C:\\Math\\MathToolBox.m"]

Out[48]= "StrStr[x_] := If[StringQ[x], \"\\\"\"<>x<>\"\\\"\",

ToString[x]]"

Unlike the ContentsM procedure, calling the following procedure ContentsNb[x] returns the nested list of names in string format that compose the user nb–format file x without loading it into the current session. In addition, the first element of the returned list – the list of the system names, the second element – the list whose sub–lists contains contexts with names of blocks, functions or modules corresponding them, and the third element – the list of other names in string format with preceding them contexts. The fragment below represents the source code of the ContentsNb procedure with typical examples of its application.

In[54]:= ContentsNb[x_ /; FileExistsQ[x] &&

FileExtension[x] == "nb"] :=

Module[{a = Quiet[ToString1[Get[x]]], b, c, d = {{}, {}, {}}},

b = StringCases[a, Shortest["RowBox[{" ~~ __ ~~ ", "]];

b = Map[StringReplace[#, {"RowBox[{"–>"", " "–>"", ", "–>""}] &, b];

c = Sort[Select[b, SymbolQ[#] &]];

c = DeleteDuplicates[Map[ToExpression, c]];

Map[If[SystemQ[#], AppendTo[d[[1]], #],

If[BlockFuncModQ[#], AppendTo[d[[2]], #],

AppendTo[d[[3]], #]]] &, c];

{d[[1]], Map[Sort[DeleteDuplicates[Flatten[Gather[#,

#1[[2]] == #2[[2]] &]]], ContextQ[#] &] &, {d[[2]], d[[3]]}]}]

In[55]:= ContentsNb["C:\\Mathematica\\Exp78.nb"]

Out[55]= {{"AppendTo", "Break", "Continue", "DeleteDuplicates", "Do", "FileExistsQ", "FileExtension", "Flatten", "For", "Get", "If", "Map", "Module", "Return", "Select", "Shortest", "Sort", "StringCases", "StringFreeQ", "StringPosition", "StringReplace", "StringTake", "SyntaxQ", "ToExpression", "ToString"}, {{"AladjevProcedures`", "ToString1", "SystemQ", "SymbolQ", "ReadFullFile", "ProcQ", "NbToString", "DefFromMfile", "ContentsNb", "BlockFuncModQ"}, {"AladjevProcedures`", "Global`", "j", "s", "g", "d", "c", "b", "a"}}}

Similarly to the procedure ContentsNb, the procedure call ContentsMx1[x] returns the nested list of names in string format that compose the user mx–format file x without loading it into the current session. In addition, the first element of the returned list – the list of the system names, the second element – the list that contains names in string format of functions, modules and blocks, and the third element – the list of other names in string format. It is assumed that the mx–file contains the context and definitions of blocks, functions, and/or modules along with their usage.

In[10]:= ContentsMx1[x_/; FileExistsQ[x] && FileExtension[x] ==

"mx"] := Module[{a = BinaryReadList[x], b, c, d = {{}, {}, {}}},

b = Flatten[{34, 37, Map[Range5[#] &, {0 ;31, 40 ;47, 58 ;64,

92 ;95, 123 ;126}]}];

c = Map[If[MemberQ[b, #] || # >= 127, Nothing, #] &, a];

c = StringJoin[Map[If[# == 32, "|", FromCharacterCode[#]] &, c]];

c = StringCases[c, Shortest["|" ~~ __ ~~ "["]];

c = Map[StringTake[#, {Flatten[StringPosition[#, "|"]][[-1]]+1, -2}] &, c];

c = DeleteDuplicates[Sort[Select[c, SymbolQ[#] &&

StringFreeQ[#, {"`", "$"}] &]]];

c = Select[c, Context[#] != "Global`" &];

Map[If[SystemQ[#], AppendTo[d[[1]], #],

If[BlockFuncModQ[#], AppendTo[d[[2]], #],

AppendTo[d[[3]], #]]] &, c]; d]

In[11]:= ContentsMx1["C:\\Mathem\\mathtoolbox.mx"]

Out[11]= {{"Abs", "And", "Append",…,"Unique", "Xnor", "Xor"},

{"AcNb", "ActBFM", "ActBFMuserQ",…,"Xnor1", "Xor1", "XOR1"},

{"a", "f", "g", "x", "y", "z"}}

At last, the procedure call ContentsTxt1[x] returns the list of names in string format that composes the user txt–format file x with a package without loading it into the current session.

In[77]:= ContentsTxt1[x_/; FileExistsQ[x] && FileExtension[x] ==

"txt"] := Module[{a = ReadFullFile[x]},

a = StringCases[a, Shortest["Begin[" ~~ __ ~~ "]"]];

a = Map[StringReplace[#, {"Begin[" -> "", "`" -> "", "]" -> ""}] &, a];

Select[Map[ToExpression, Sort[Select[a, SyntaxQ[#] &]]],

Quiet[SyntaxQ[#]] &]]

In[78]:= Length[ContentsTxt1["C:\\Math\\mathtoolbox.txt"]]

Out[78]= 1433

The mx–format is serialized package format. As a rule it is used for the distribution of both system and user packages. The binary file format is the optimized for fast loading. Meantime, as noted above, the mx–format files can't be exchanged between operating platforms which differ from the predefined variable $SystemWordLength which gives the number of bits in machine words on the computer system where Mathematica is running. Furthermore, the mx–format files created by newer versions of Mathematica may not be usable by older versions. This raises the issue of testing the ability to load an arbitrary file of the mx-format into the current session. The following procedure solves this problem. The fragment below represents the source code of the LoadMxQ procedure with examples of its application.

In[7]:= LoadMxQ[x_ /; FileExistsQ[x] && FileExtension[x] == "mx"] :=

Module[{a, b, c, MxOnOpSys},

MxOnOpSys[y_/; FileExistsQ[y] && FileExtension[y] == "mx"] :=

Module[{a = ReadString[y], b = "Get.*)", c = "CONT", d},

d = StringCases2[a, {b, c}][[1]];

d = StringReplace[d, {b –> "", c –> "", "\n" –> "", "END" –> ""}];

d = StringJoin[Map[If[MemberQ[Range[32, 127],

ToCharacterCode[#][[1]]], #, Nothing] &, Characters[d]]]];

g[t_] := t; DumpSave["###.mx", g];

If[MxOnOpSys[x] == MxOnOpSys["###.mx"],

DeleteFile["###.mx"]; True, DeleteFile["###.mx"]; False]]

In[8]:= LoadMxQ["C:\\Mathematica\\MathToolBox.mx"]

Out[8]= True

In[9]:= LoadMxQ["dump.mx"]

Out[9]= True

Calling procedure LoadMxQ[x] returns True if a mx–type file x was created on the current operating platform and can be loaded into the current Mathematica session by the Get[x] call, and False otherwise. In our works [8-16] a number of means for various processing of mx–files with interesting examples using in programming of various applications is presented. However, these tools have been debugged on Windows XP and Windows 7 platforms, in other cases requiring sometimes re–debugging.

References

1. Aladjev V.Z., Vaganov V.A. Modular programming: Mathematica vs Maple, and vice versa.– USA: Palo Alto, Fultus Corporation, 2011, ISBN 9781596822689, 418 p.

2. Aladjev V.Z. et al. Programming: System Maple or Mathematica?– Ukraine: Kherson, Oldi–Plus Press, 2011, ISBN 9789662393460, 474 p.

3. Aladjev V.Z., Boiko V.K., Rovba E.A. Programming in systems Mathematica and Maple: A Comparative Aspect.– Grodno, Grodno State University, 2011, 517 p.

4. Aladjev V.Z., Grinn D.S., Vaganov V.A. The extended functional tools for system Mathematica.– Kherson: Oldi–Plus Press, 2012, ISBN 9789662393590, 404 p.

5. Aladjev V.Z., Grinn D.S. Extension of functional environment of system Mathematica.– Kherson: Oldi–Plus Press, 2012, ISBN 9789662393729, 552 p.

6. Aladjev V.Z., Grinn D.S., Vaganov V.A. The selected system problems in software environment of system Mathematica.– Ukraine: Kherson: Oldi–Plus Press, 2013, 556 p.

7. Aladjev V.Z., Vaganov V.A. Extension of Mathematica system functionality.– USA: CreateSpace, An Amazon.com Company, 2015, ISBN 9781514237823, 590 p.

8. Aladjev V.Z., Vaganov V. Toolbox for the Mathematica programmers.– USA, CreateSpace, An Amazon.com Company, 2016, ISBN 1532748833, 630 p.

9. Aladjev V.Z., Boiko V.K., Shishakov M.L. The Art of programming in Mathematica System.– Estonia: Tallinn, TRG Press, 2016, ISBN 9789985950890, 735 p.

10. Aladjev V.Z., Boiko V.K., Shishakov M.L. The Art of programming in Mathematica system. 2^nd edition.– USA, Lulu Press, 2016, ISBN 9781365560736, 735 p.

11. Aladjev V.Z., Shishakov M.L. Software etudes in the Mathematica.– CreateSpace, An Amazon.com Company, 2017, 614 p., ISBN 1979037272

12. Aladjev V.Z., Shishakov M.L. Software etudes in the

Mathematica.– Amazon Digital Services, ASIN: B0775YSH72, 2017, www.amazon.com/dp/B0775YSH72

13. Aladjev V.Z., Shishakov M.L., Vaganov V.A. Practical programming in Mathematica. Fifth edition.– USA: Lulu Press, 2017, 613 p.

14. Aladjev V.Z., Boiko V.K., Grinn D.S., Haritonov V.A., Hunt Ü., Rouba Y.A., Shishakov M.L., Vaganov V.A. Books miscellany on Cellular Automata, General Statistics Theory, Maple and Mathematica.– Estonia: Tallinn, TRG Press, CD–edition, 2016.

15. Aladjev V.Z., Boiko V.K., Gostev A., Hunt Ü., Rouba E., Shishakov M.L., Grinn D.S., Vaganov V.A. Selected Books and software published by members of Baltic branch of the Intern. Academy of Noosphere.– Tallinn: TRG press, 2019, CD–edition, ISBN 978–9949–9876–3–4

16. Aladjev V.Z. Package of Procedures and Functions for Mathematica.– TRG: Tallinn, 2018; this package can be freely downloaded from sites https://yadi.sk/d/oC5lXLWa3PVEhi, https://yadi.sk/d/2GyQU2pQ3ZETZT.

17. Aladjev V.Z. Modular programming: Mathematica or Maple – A subjective standpoint / Int. school «Mathematical and computer modeling of fundamental objects and phenomena in systems of computer mathematics».– Kazan: Kazan Univ. Press, 2014.

18. V.Z. Aladjev, V.K. Boiko. Tools for computer research of cellular automata dynamics / Intern. School «Mathematical modelling of fundamental objects and phenomena in the systems of computer mathematics»; Int. scientific and practical conference «Information Technologies in Science and Education» (ITON–2017), November 4 – 6, 2017, Kazan, pp. 7 – 16.

19. https://bbian.webs.com/Our_publications_2019.pdf

20. Aladjev V.Z., Shishakov M.L. Introduction into mathematical package Mathematica 2.2.– Moscow: Filin Press, 1997, 363 p., ISBN 5895680046 (in Russian with English summary)

21. Aladjev V.Z., Vaganov V.A., Hunt Ü.J., Shishakov M.L. Introduction into environment of mathematical package Maple V.– Minsk: International Academy of Noosphere, the Baltic Branch, 1998, 452 p., ISBN 1406425698 (in Russian)

22. Aladjev V.Z., Vaganov V.A., Hunt Ü., Shishakov M.L. Programming in environment of mathematical package Maple V.– Minsk–Moscow: Russian Ecology Academy, 1999, 470 p., ISBN 4101212982 (in Russian with English summary)

23. Aladjev V.Z., Bogdevicius M.A. Solution of physical, technical and mathematical problems with Maple V.– Tallinn–Vilnius, TRG, 1999, 686 p., ISBN 9986053986 (in Russian)

24. Aladjev V.Z., Vaganov V.A., Hunt Ü.J., Shishakov M.L. Workstation for Mathematicians.– Minsk–Moscow: Russian Academy of Natural Sciences, 1999, 608 p., ISBN 3420614023

25. Aladjev V.Z., Shishakov M.L. Workstation of Mathematician.– Moscow: Laboratory of Basic Knowledge, 2000, 752 p. + CD, ISBN 5932080523 (in Russian)

26. Aladjev V.Z., Bogdevicius M.A. Maple 6: Solution of Mathematical, Statistical, Engineering and Physical Problems.– Moscow: Laboratory of Basic Knowledge, 2001, 850 p. + CD, ISBN 593308085X (in Russian with English summary)

27. Aladjev V.Z., Bogdevicius M.A. Special questions of work in environment of the mathematical package Maple.– Vilnius: International Academy of Noosphere, the Baltic Branch & Vilnius Gediminas Technical University, 2001, 208 p. + CD with Library, ISBN 9985927729 (in Russian with English summary)

28. Aladjev V.Z., Bogdevicius M.A. Interactive Maple: Solution of mathematical, statistical, engineering and physical problems.– Tallin: International Academy of Noosphere, the Baltic Branch, 2001–2002, CD with Booklet, ISBN 9985927710

29. Aladjev V.Z., Vaganov V.A., Grishin E.P. Additional software of mathematical package Maple of releases 6 and 7.– Tallinn: Intern. Academy of Noosphere, the Baltic Branch, 2002, 314 p. + CD with Library, ISBN 9985927737 (in Russian)

30. Aladjev V.Z. Effective work in mathematical package Maple.– Moscow: BINOM Press, 2002, 334 p. + CD, ISBN 593208118Х (in Russian with English summary)

31. Aladjev V.Z., Liopo V.A., Nikitin A.V. Mathematical package Maple in physical modeling.– Grodno: Grodno State University, 2002, 416 p., ISBN 3093318313 (in Russian)

32. Aladjev V.Z., Vaganov V.A. Computer Algebra System

Maple: A New Software Library.– Tallinn: Intern. Academy of Noosphere, the Baltic Branch, 2002, CD, ISBN 9985927753

33. Aladjev V.Z., Bogdevicius M.A., Prentkovskis O.V. A New Software for mathematical package Maple of releases 6, 7 and 8.– Vilnius: Vilnius Technical University and International Academy of Noosphere, the Baltic Branch, 2002, 404 p.

34. Aladjev V.Z., Vaganov V.A. Systems of computer algebra: A New Software Toolbox for Maple.– Tallinn: Intern. Academy of Noosphere, the Baltic Branch, 2003, 270 p. + CD, ISBN 9985927761

35. Aladjev V.Z., Bogdevicius M.A., Vaganov V.A. Systems of Computer Algebra: A New Software Toolbox for Maple. 2^nd edition.– Tallinn: Intern. Academy of Noosphere, 2004, 462 p.

36. Aladjev V.Z. Computer algebra systems: A new software toolbox for Maple.– USA: Palo Alto: Fultus Corporation, 2004, 575 p., ISBN 1596820004

37. Aladjev V.Z. Computer algebra systems: A new software toolbox for Maple.– USA: Palo Alto: Fultus Corporation, 2004, Adobe Acrobat eBook, ISBN 1596820152

38. Aladjev V.Z., Bogdevicius M.A. Maple: Programming, physical and engineering problems.– USA: Palo Alto: Fultus Corporation, 2006, 404 p., ISBN 1596820802, Adobe Acrobat eBook (pdf), ISBN 1596820810

39. Aladjev V.Z. Computer Algebra Systems. Maple: Art of Programming.– Moscow: BINOM Press, 2006, 792 p., ISBN 5932081899 (in Russian with English summary)

40. Aladjev V.Z. Foundations of programming in Maple: Textbook.– Tallinn: International Academy of Noosphere, 2006, 300 p., (pdf), ISBN 998595081X, 9789985950814 (in Russian)

41. Aladjev V.Z., Boiko V.K., Rovba E.A. Programming and applications elaboration in Maple: Monograph.– Grodno: GRSU, Tallinn: International Academy of Noosphere, 2007, 456 p., ISBN 9789854178912, ISBN 9789985950821 (in Russian)

42. Aladjev V.Z. A Library for Maple system: The Library can be freely downloaded from https://yadi.sk/d/UjQwqt1GFYsweA

About the authors

Aladjev V.Z.

Professor dr. Aladjev V.Z. was born on June 14, 1942 in the town Grodno (West Belarus). He is the First vice–president of the International Academy of Noosphere (IAN, 1998) and academician–secretary of Baltic branch of the IAN whose scientific results have received international recognition, first, in the field of cellular automata (homogeneous structures) theory. Dr. Aladjev V.Z. is known for the works on cellular automata theory and computer mathematical systems. Dr. Aladjev V.Z. is full member of the Russian Academy of Cosmonautics (1994), the Russian Academy of Natural Sciences (1995), International Academy of Noosphere (1998), and honorary member of the Russian Ecological Academy (1998).

He is the author of more than 500 scientifical publications, including 90 books and monographs in fields such as: general statistics, mathematics, informatics, Maple and Mathematica that have been published in many countries. Aladjev V.Z. established Estonian School of mathematical theory of homogeneous structures (cellular automata theory), whose fundamental results received international recognition and formed the basis of a new section of modern mathematical cybernetics.

Many applied works of Dr. Aladiev V.Z. relate to computer science, among which are widely known books on computer mathematics systems. Along with these original editions, he has developed a rather large library UserLib of new software tools (over 850 tools), awarded with the web–based Smart Award, and the unified package MathToolBox (over 1420 tools) for Maple and Mathematica, respectively, to enhance the functionality of these computer mathematics systems.

Aladjev V.Z. repeatedly participates as a guest lecturer in the different international scientific forums in mathematics and cybernetics or a member of the organizing committee of them. In 2015 Dr. Aladjev V.Z. was awarded by Gold medal "European Quality" of the European scientific and industrial consortium for works of scientific and applied character.

Shishakov M.L.

Dr. Shishakov M.L. was born on October 21, 1957 in Gomel area (Belarus). For its scientific activity Shishakov M.L. has more than 68 publications on information technology in various application fields, including more 26 books and monographs. Fields of his main interests are cybernetics, theory of statistics, problems of CAD and designing of mobile software, artificial intelligence, computer telecommunication, along with applied software for solution of different tasks of technical along with manufacturing nature. In 1998 M.L. Shishakov was elected as a full member of the IAN on section of the information science and information technologies.

At present, M.L. Shishakov is the director of the Belarusian–Swiss company "TDF Ecotech". TDF Ecotech company builds and operates the appropriate facilities for production of renewable energy through its subsidiary branches (with a focus on the Belarus region). Above all, the company works in the field of so–called "green" energy, combining the essential manufacturing activity with active scientific researches.

Vaganov V.A.

Dr. Vaganov V.A. was born on February 2, 1946 in Primorye Territory (Russia). Now Dr. Vaganov V.A. is the proprietor of the firms Fortex and Sinfex engaging of problems of delivery of the industrial materials to different firms of the Estonian republic. Simultaneously, dr. V.A. Vaganov is executive director of the Baltic branch of the IAN. In addition, Dr. Vaganov V.A. is well–known for the investigations on automation of economic and statistical works. V.A. Vaganov is the honorary member of the IANSD and the author of more than 62 applied and scientifical publications, at the same time including 10 books. The sphere of his scientific interests includes statistics, economics, problems the family institute, political sciences, cosmonautics, theory of sustainable development and other natural-scientific directions.