aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/indexedp.spad.pamphlet
blob: e4dd5805036211f2a3ad70c0b1f1f7b1605817ce (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra indexedp.spad}
\author{James Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category IDPC IndexedDirectProductCategory}
<<category IDPC IndexedDirectProductCategory>>=
)abbrev category IDPC IndexedDirectProductCategory
++ Author: James Davenport, Gabriel Dos Reis
++ Date Created:
++ Date Last Updated: May 19, 2013.
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category represents the direct product of some set with
++ respect to an ordered indexing set.

IndexedDirectProductCategory(A:BasicType,S:OrderedType): Category ==
  Join(BasicType,Functorial A,ConvertibleFrom List IndexedProductTerm(A,S)) with
    if A has SetCategory and S has SetCategory then SetCategory
    monomial:         (A, S) -> %
       ++ monomial(a,s) constructs a direct product element with the s
       ++ component set to \spad{a}
    leadingCoefficient:   % -> A
       ++ leadingCoefficient(z) returns the coefficient of the leading
       ++ (with respect to the ordering on the indexing set)
       ++ monomial of z.
       ++ Error: if z has no support.
    leadingSupport:   % -> S
       ++ leadingSupport(z) returns the index of leading
       ++ (with respect to the ordering on the indexing set) monomial of z.
       ++ Error: if z has no support.
    reductum:      % -> %
       ++ reductum(z) returns a new element created by removing the
       ++ leading coefficient/support pair from the element z.
       ++ Error: if z has no support.
    terms: % -> List IndexedProductTerm(A,S)
      ++ \spad{terms x} returns the list of terms in \spad{x}.
      ++ Each term is a pair of a support (the first component)
      ++ and the corresponding value (the second component).

@

\section{domain IDPT IndexedProductTerm}

<<domain IDPT IndexedProductTerm>>=
)abbrev domain IDPT IndexedProductTerm
++ Author: Gabriel Dos Reis
++ Date Last Updated: May 7, 2013
++ Description:
++  An indexed product term is a utility domain used in the
++  representation of indexed direct product objects.
IndexedProductTerm(A,S): Public == Private where
   A: BasicType
   S: OrderedType
   Public == Join(BasicType,CoercibleTo Pair(S,A)) with
     term : (S, A) -> %
       ++ \spad{term(s,a)} constructs a term with index \spad{s}
       ++ and coefficient \spad{a}.
     index : % -> S
       ++ \spad{index t} returns the index of the term \spad{t}.
     coefficient : % -> A
       ++ \spad{coefficient t} returns the coefficient of the tern \spad{t}.
   Private == Pair(S,A) add
     term(s,a) == per [s,a]
     index t == first rep t
     coefficient t == second rep t
     coerce(t: %): Pair(S,A) == rep t
@

\section{domain IDPO IndexedDirectProductObject}
<<domain IDPO IndexedDirectProductObject>>=
)abbrev domain IDPO IndexedDirectProductObject
++ Author: James Davenport, Gabriel Dos Reis
++ Date Created:
++ Date Last Updated: June 28, 2010
++ Description:
++   Indexed direct products of objects over a set \spad{A}
++   of generators indexed by an ordered set S. All items have finite support.
IndexedDirectProductObject(A,S): Public == Private where
  A: BasicType
  S: OrderedType
  Public == IndexedDirectProductCategory(A,S) with
    combineWithIf: (%,%, (A,A) -> A, (A,A) -> Boolean) -> %
      ++ \spad{combineWithIf(u,v,f,p)} returns the result of combining
      ++ index-wise, coefficients of \spad{u} and \spad{u} if when
      ++ satisfy the predicate \spad{p}.  Those pairs of coefficients
      ++ which fail\spad{p} are implicitly ignored.
  Private == add
    Rep == List IndexedProductTerm(A,S)
    if A has CoercibleTo OutputForm and S has CoercibleTo OutputForm then
      coerce(x:%):OutputForm ==
         bracket [rarrow(index(t)::OutputForm, coefficient(t)::OutputForm)
                   for t in rep x]

    x = y == rep x = rep y
    monomial(r,s) == per [term(s,r)]
    map(f,x) == per [term(index tm,f coefficient tm) for tm in rep x]
    reductum x == per rest rep x
    leadingCoefficient x  ==
       null rep x =>
         error "Can't take leadingCoefficient of empty product element"
       coefficient first rep x
    leadingSupport x  ==
       null rep x =>
         error "Can't take leadingCoefficient of empty product element"
       index first rep x
    terms x == rep x
    convert l == per l
    combineWithIf(u, v, f, p) ==
      x := rep u
      y := rep v
      empty? x => v
      empty? y => u
      z: Rep := nil
      prev: Rep := nil
      while not empty? x and not empty? y repeat
        xt := first x
        yt := first y
        index xt > index yt =>
          t := [xt]
          if empty? z then z := t
          else setrest!(prev,t)
          prev := t
          x := rest x
        index xt < index yt =>
          t := [yt]
          if empty? z then z := t
          else setrest!(prev,t)
          prev := t
          y := rest y
        not p(coefficient xt, coefficient yt) => iterate
        t := [term(index xt, f(coefficient xt, coefficient yt))]
        if empty? z then z := t
        else setrest!(prev,t)
        prev := t
        x := rest x
        y := rest y
      if empty? x then setrest!(prev,y)
      else if empty? y then setrest!(prev,x)
      per z

@
\section{domain IDPAM IndexedDirectProductAbelianMonoid}
<<domain IDPAM IndexedDirectProductAbelianMonoid>>=
)abbrev domain IDPAM IndexedDirectProductAbelianMonoid
++ Indexed direct products of abelian monoids over an abelian monoid \spad{A} of
++ generators indexed by the ordered set S. All items have finite support.
++ Only non-zero terms are stored.
IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedType):
    Join(AbelianMonoid,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductObject(A,S) add
       Term == IndexedProductTerm(A,S)
       import Term

       r: A
       n: NonNegativeInteger
       f: A -> A
       s: S
       0  == convert nil$List(Term)
       zero? x ==  null terms x

       import %tail: List Term -> List Term from Foreign Builtin

       qsetrest!: (List Term, List Term) -> List Term
       qsetrest!(l, e) ==
         %store(%tail l,e)$Foreign(Builtin)

       -- PERFORMANCE CRITICAL; Should build list up
       --  by merging 2 sorted lists.   Doing this will
       -- avoid the recursive calls (very useful if there is a
       -- large number of vars in a polynomial.
       x + y ==
         x' := terms x
         y' := terms y
         null x' => y
         null y' => x
         endcell: List Term := nil
         res: List Term := nil
         while not empty? x' and not empty? y' repeat 
           newcell: List Term := nil
           if index x'.first = index y'.first then
             r := coefficient x'.first + coefficient y'.first
             if not zero? r then 
               newcell := [term(index x'.first, r)]
             x' := rest x'
             y' := rest y'
           else if index x'.first > index y'.first then
             newcell := [x'.first]
             x' := rest x'
           else
             newcell := [y'.first]
             y' := rest y'
           if not empty? newcell then 
             if not empty? endcell then
               qsetrest!(endcell, newcell)
               endcell := newcell
             else
               res     := newcell;
               endcell := res
         end := 
           empty? x' => y'
           x'
         if empty? res then res := end
         else qsetrest!(endcell, end)
         convert res

       n * x  ==
         zero? n => 0
         one? n => x
         convert [term(index u,a) for u in terms x
                    | not zero?(a:=n * coefficient u)]

       monomial(r,s) ==
         zero? r => 0
         convert [term(s,r)]

       map(f,x) ==
         convert [term(index tm,a) for tm in terms x
                    | not zero?(a:=f coefficient tm)]

       reductum x ==
         null terms x => 0
         convert rest(terms x)

       leadingCoefficient x  ==
         null terms x => 0
         coefficient terms(x).first

       opposite?(x,y) ==
         u := terms x
         v := terms y
         repeat
           empty? u => return empty? v
           empty? v => return empty? u
           index u.first ~= index v.first => return false
           not opposite?(coefficient u.first,coefficient v.first) => return false
           u := rest u
           v := rest v

@
\section{domain IDPOAM IndexedDirectProductOrderedAbelianMonoid}
<<domain IDPOAM IndexedDirectProductOrderedAbelianMonoid>>=
)abbrev domain IDPOAM IndexedDirectProductOrderedAbelianMonoid
++ Indexed direct products of ordered abelian monoids \spad{A} of
++ generators indexed by the ordered set S.
++ The inherited order is lexicographical.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductOrderedAbelianMonoid(A:OrderedAbelianMonoid,S:OrderedType):
    Join(OrderedAbelianMonoid,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductAbelianMonoid(A,S) add
       x<y ==
         u := terms x
         v := terms y
         repeat
           -- note careful order of next two lines
           empty? v => return false
           empty? u => return true
           xt := first u
           yt := first v
           index xt < index yt => return true
           index yt < index xt => return false
           coefficient xt < coefficient yt => return true
           coefficient yt < coefficient xt => return false
           u := rest u
           v := rest v

@
\section{domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup}
<<domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup>>=
)abbrev domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup
++ Indexed direct products of ordered abelian monoid sups \spad{A},
++ generators indexed by the ordered set S.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductOrderedAbelianMonoidSup(A:OrderedAbelianMonoidSup,S:OrderedSet):
    Join(OrderedAbelianMonoidSup,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductOrderedAbelianMonoid(A,S) add
    --representations
       Term:=  Record(k:S,c:A)
       Rep:=  List Term
       x,y: %
       r: A
       s: S

       subtractIfCan(x,y) ==
         empty? y => just x
         empty? x => nothing
         x.first.k < y.first.k => nothing
         x.first.k > y.first.k =>
             t:= subtractIfCan(x.rest, y)
             t case nothing => nothing
             just cons( x.first, t)
         u := subtractIfCan(x.first.c, y.first.c)
         u case nothing => nothing
         zero? u => subtractIfCan(x.rest, y.rest)
         t := subtractIfCan(x.rest, y.rest)
         t case nothing => nothing
         just cons([x.first.k,u],t)

       sup(x,y) ==
         empty? y => x
         empty? x => y
         x.first.k < y.first.k => cons(y.first,sup(x,y.rest))
         x.first.k > y.first.k => cons(x.first,sup(x.rest,y))
         u:=sup(x.first.c, y.first.c)
         cons([x.first.k,u],sup(x.rest,y.rest))

@
\section{domain IDPAG IndexedDirectProductAbelianGroup}
<<domain IDPAG IndexedDirectProductAbelianGroup>>=
)abbrev domain IDPAG IndexedDirectProductAbelianGroup
++ Indexed direct products of abelian groups over an abelian group \spad{A} of
++ generators indexed by the ordered set S.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedType):
    Join(AbelianGroup,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductAbelianMonoid(A,S) add
    --representations
       Term == IndexedProductTerm(A,S)

       -x == convert [term(index u,-coefficient u) for u in terms x]
       n:Integer * x:% ==
         zero? n => 0
         one? n => x
         convert [term(index u,a) for u in terms x
                    | not zero?(a := n * coefficient u)]

       import %tail: List Term -> List Term from Foreign Builtin

       qsetrest!: (List Term, List Term) -> List Term
       qsetrest!(l, e) ==
         %store(%tail l,e)$Foreign(Builtin)

       x - y ==
         x' := terms x
         y' := terms y
         null x' => -y
         null y' => x
         endcell: List Term := nil
         res: List Term := nil
         while not empty? x' and not empty? y' repeat
           newcell: List Term := nil
           if index x'.first = index y'.first then
             r := coefficient x'.first - coefficient y'.first
             if not zero? r then
               newcell := [term(index x'.first, r)]
             x' := rest x'
             y' := rest y'
           else if index x'.first > index y'.first then
             newcell := [x'.first]
             x' := rest x'
           else
             newcell := [term(index y'.first,-coefficient y'.first)]
             y' := rest y'
           if not empty? newcell then
             if not empty? endcell then
               qsetrest!(endcell, newcell)
               endcell := newcell
             else
               res     := newcell;
               endcell := res
         end := 
           empty? x' => terms(-convert y')
           x'
         if empty? res then res := end
         else qsetrest!(endcell, end)
         convert res

@
\section{License}
<<license>>=
--Copyright (C) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2013, Gabriel Dos Reis.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<category IDPC IndexedDirectProductCategory>>
<<domain IDPT IndexedProductTerm>>
<<domain IDPO IndexedDirectProductObject>>
<<domain IDPAM IndexedDirectProductAbelianMonoid>>
<<domain IDPOAM IndexedDirectProductOrderedAbelianMonoid>>
<<domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup>>
<<domain IDPAG IndexedDirectProductAbelianGroup>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}