forked from fricas/fricas
-
Notifications
You must be signed in to change notification settings - Fork 0
/
card.spad
174 lines (158 loc) · 6.75 KB
/
card.spad
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
)abbrev domain CARD CardinalNumber
++ Author: S.M. Watt
++ Date Created: June 1986
++ Basic Operations: Aleph, +, -, *, ^
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: cardinal number, transfinite arithmetic
++ Examples:
++ References:
++ Goedel, "The consistency of the continuum hypothesis",
++ Ann. Math. Studies, Princeton Univ. Press, 1940
++ Description:
++ Members of the domain CardinalNumber are values indicating the
++ cardinality of sets, both finite and infinite. Arithmetic operations
++ are defined on cardinal numbers as follows.
++
++ If \spad{x = #X} and \spad{y = #Y} then
++ \spad{x+y = #(X+Y)} \tab{30}disjoint union
++ \spad{x-y = #(X-Y)} \tab{30}relative complement
++ \spad{x*y = #(X*Y)} \tab{30}cartesian product
++ \spad{x^y = #(X^Y)} \tab{30}\spad{X^Y = \{g| g: Y->X\}}
++
++ The non-negative integers have a natural construction as cardinals
++ \spad{0 = #\{\}}, \spad{1 = \{0\}}, \spad{2 = \{0, 1\}}, ..., \spad{n = \{i| 0 <= i < n\}}.
++
++ That \spad{0} acts as a zero for the multiplication of cardinals is
++ equivalent to the axiom of choice.
++
++ The generalized continuum hypothesis asserts
++ \centerline{\spad{2^Aleph i = Aleph(i+1)}}
++ and is independent of the axioms of set theory [Goedel 1940].
++
++ Three commonly encountered cardinal numbers are
++ \spad{a = #Z} \tab{30}countable infinity
++ \spad{c = #R} \tab{30}the continuum
++ \spad{f = #\{g| g: [0, 1]->R\}}
++
++ In this domain, these values are obtained using
++ \spad{a := Aleph 0}, \spad{c := 2^a}, \spad{f := 2^c}.
++
CardinalNumber : Join(OrderedSet, AbelianMonoid, Monoid,
RetractableTo NonNegativeInteger, CommutativeStar) with
"-": (%,%) -> Union(%,"failed")
++ \spad{x - y} returns an element z such that \spad{z+y=x} or "failed"
++ if no such element exists.
"^": (%, %) -> %
++ \spad{x^y} returns \spad{#(X^Y)} where \spad{X^Y} is defined
++ as \spad{\{g| g: Y->X\}}.
Aleph : NonNegativeInteger -> %
++ Aleph(n) provides the named (infinite) cardinal number.
finite? : % -> Boolean
++ finite?(\spad{a}) determines whether
++ \spad{a} is a finite cardinal,
++ i.e. an integer.
countable? : % -> Boolean
++ countable?(\spad{a}) determines
++ whether \spad{a} is a countable cardinal,
++ i.e. an integer or \spad{Aleph 0}.
generalizedContinuumHypothesisAssumed? : () -> Boolean
++ generalizedContinuumHypothesisAssumed?()
++ tests if the hypothesis is currently assumed.
generalizedContinuumHypothesisAssumed : Boolean -> Boolean
++ generalizedContinuumHypothesisAssumed(bool)
++ is used to dictate whether the hypothesis is to be assumed.
== add
NNI ==> NonNegativeInteger
FINord ==> -1
DUMMYval ==> -1
Rep := Record(order : Integer, ival : Integer)
import from Integer
GCHypothesis : Boolean := false
-- Creation
0 == [FINord, 0]
1 == [FINord, 1]
coerce(n : NonNegativeInteger) : % == [FINord, n]
Aleph n == [n, DUMMYval]
-- Output
ALEPHexpr := 'Aleph::OutputForm
coerce(x : %) : OutputForm ==
x.order = FINord => (x.ival)::OutputForm
prefix(ALEPHexpr, [(x.order)::OutputForm])
-- Manipulation
x = y ==
x.order ~= y.order => false
finite? x => x.ival = y.ival
true -- equal transfinites
x < y ==
x.order < y.order => true
x.order > y.order => false
finite? x => x.ival < y.ival
false -- equal transfinites
x : % + y : % ==
finite? x and finite? y => [FINord, x.ival+y.ival]
max(x, y)
x - y ==
x < y => "failed"
finite? x => [FINord, x.ival-y.ival]
x > y => x
"failed" -- equal transfinites
x : % * y : % ==
finite? x and finite? y => [FINord, x.ival*y.ival]
x = 0 or y = 0 => 0
max(x, y)
n : NonNegativeInteger * x : % ==
finite? x => [FINord, n*x.ival]
n = 0 => 0
x
x^(y : %) ==
y = 0 => 1
finite? y =>
not finite? x => x
[FINord, x.ival^(y.ival)::NNI]
x = 0 => 0
x = 1 => 1
GCHypothesis => [max(x.order-1, y.order) + 1, DUMMYval]
error "Transfinite exponentiation only implemented under GCH"
finite? x == x.order = FINord
countable? x == x.order < 1
retract(x : %) : NonNegativeInteger ==
finite? x => (x.ival)::NNI
error "Not finite"
retractIfCan(x:%):Union(NonNegativeInteger, "failed") ==
finite? x => (x.ival)::NNI
"failed"
-- State manipulation
generalizedContinuumHypothesisAssumed?() == GCHypothesis
generalizedContinuumHypothesisAssumed b == GCHypothesis := b
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--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.