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complet.spad
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complet.spad
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)abbrev domain ORDCOMP OrderedCompletion
++ Completion with +infinity and -infinity.
++ Author: Manuel Bronstein
++ Description: Adjunction of two real infinites quantities to a set.
++ Date Created: 4 Oct 1989
OrderedCompletion(R : SetCategory) : Exports == Implementation where
B ==> Boolean
Exports ==> Join(SetCategory, FullyRetractableTo R) with
plusInfinity : () -> %
++ plusInfinity() returns +infinity.
minusInfinity : () -> %
++ minusInfinity() returns -infinity.
finite? : % -> B
++ finite?(x) tests if x is finite.
infinite? : % -> B
++ infinite?(x) tests if x is +infinity or -infinity,
whatInfinity : % -> SingleInteger
++ whatInfinity(x) returns 0 if x is finite,
++ 1 if x is +infinity, and -1 if x is -infinity.
if R has AbelianMonoid then
_+ : (%, %) -> %
if R has AbelianGroup then
_- : % -> %
if R has OrderedSet then OrderedSet
if R has IntegerNumberSystem then
rational? : % -> Boolean
++ rational?(x) tests if x is a finite rational number.
rational : % -> Fraction Integer
++ rational(x) returns x as a finite rational number.
++ Error: if x cannot be so converted.
rationalIfCan : % -> Union(Fraction Integer, "failed")
++ rationalIfCan(x) returns x as a finite rational number if
++ it is one and "failed" otherwise.
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
Implementation ==> add
Rep := Union(fin : R, inf : B) -- true = +infinity, false = -infinity
if R has ConvertibleTo InputForm then
convert(x : %) : InputForm ==
x case fin => convert(x.fin)@InputForm
x.inf => convert([convert('plusInfinity)$InputForm])$InputForm
convert([convert('minusInfinity)$InputForm])$InputForm
coerce(r : R) : % == [r]
retract(x:%):R == (x case fin => x.fin; error "Not finite")
finite? x == x case fin
infinite? x == x case inf
plusInfinity() == [true]
minusInfinity() == [false]
retractIfCan(x:%):Union(R, "failed") ==
x case fin => x.fin
"failed"
coerce(x : %) : OutputForm ==
x case fin => (x.fin)::OutputForm
e := 'infinity::OutputForm
x.inf => empty() + e
- e
whatInfinity x ==
x case fin => 0
x.inf => 1
-1
x = y ==
x case inf =>
y case inf => not xor(x.inf, y.inf)
false
y case inf => false
x.fin = y.fin
if R has AbelianGroup then
- x ==
x case inf => [not(x.inf)]
[- (x.fin)]
if R has AbelianMonoid then
x + y ==
x case inf =>
y case fin => x
xor(x.inf, y.inf) => error "Undefined sum"
x
y case inf => y
[x.fin + y.fin]
if R has OrderedSet then
x < y ==
x case inf =>
y case inf =>
xor(x.inf, y.inf) => y.inf
false
not(x.inf)
y case inf => y.inf
x.fin < y.fin
if R has IntegerNumberSystem then
rational? x == finite? x
rational x == rational(retract(x)@R)
rationalIfCan x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" =>"failed"
rational(r@R)
)abbrev package ORDCOMP2 OrderedCompletionFunctions2
++ Lifting of maps to ordered completions
++ Author: Manuel Bronstein
++ Description: Lifting of maps to ordered completions.
++ Date Created: 4 Oct 1989
OrderedCompletionFunctions2(R, S) : Exports == Implementation where
R, S : SetCategory
ORR ==> OrderedCompletion R
ORS ==> OrderedCompletion S
Exports ==> with
map : (R -> S, ORR) -> ORS
++ map(f, r) lifts f and applies it to r, assuming that
++ f(plusInfinity) = plusInfinity and that
++ f(minusInfinity) = minusInfinity.
map : (R -> S, ORR, ORS, ORS) -> ORS
++ map(f, r, p, m) lifts f and applies it to r, assuming that
++ f(plusInfinity) = p and that f(minusInfinity) = m.
Implementation ==> add
map(f, r) == map(f, r, plusInfinity(), minusInfinity())
map(f, r, p, m) ==
zero?(n := whatInfinity r) => (f retract r)::ORS
(n = 1) => p
m
)abbrev domain ONECOMP OnePointCompletion
++ Completion with infinity
++ Author: Manuel Bronstein
++ Description: Adjunction of a complex infinity to a set.
++ Date Created: 4 Oct 1989
OnePointCompletion(R : SetCategory) : Exports == Implementation where
B ==> Boolean
Exports ==> Join(SetCategory, FullyRetractableTo R) with
infinity : () -> %
++ infinity() returns infinity.
finite? : % -> B
++ finite?(x) tests if x is finite.
infinite? : % -> B
++ infinite?(x) tests if x is infinite.
if R has IntegerNumberSystem then
rational? : % -> Boolean
++ rational?(x) tests if x is a finite rational number.
rational : % -> Fraction Integer
++ rational(x) returns x as a finite rational number.
++ Error: if x is not a rational number.
rationalIfCan : % -> Union(Fraction Integer, "failed")
++ rationalIfCan(x) returns x as a finite rational number if
++ it is one, "failed" otherwise.
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
Implementation ==> add
Rep := Union(R, "infinity")
if R has ConvertibleTo InputForm then
convert(x : %) : InputForm ==
-- Using @ instead of :: triggers compiler bug
x case R => convert(x::R)@InputForm
convert([convert('infinity)$InputForm])$InputForm
coerce(r : R) : % == r
retract(x:%):R == (x case R => x; error "Not finite")
finite? x == x case R
infinite? x == x case "infinity"
infinity() == "infinity"
retractIfCan(x:%):Union(R, "failed") == (x case R => x; "failed")
coerce(x : %) : OutputForm ==
x case "infinity" => 'infinity::OutputForm
x@R::OutputForm
x = y ==
x case "infinity" => y case "infinity"
y case "infinity" => false
-- Using @ instead of :: triggers compiler bug
x::R = y::R
if R has IntegerNumberSystem then
rational? x == finite? x
rational x == rational(retract(x)@R)
rationalIfCan x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" =>"failed"
-- Using @ instead of :: triggers compiler bug
rational(r::R)
)abbrev package ONECOMP2 OnePointCompletionFunctions2
++ Lifting of maps to one-point completions
++ Author: Manuel Bronstein
++ Description: Lifting of maps to one-point completions.
++ Date Created: 4 Oct 1989
OnePointCompletionFunctions2(R, S) : Exports == Implementation where
R, S : SetCategory
OPR ==> OnePointCompletion R
OPS ==> OnePointCompletion S
Exports ==> with
map : (R -> S, OPR) -> OPS
++ map(f, r) lifts f and applies it to r, assuming that
++ f(infinity) = infinity.
map : (R -> S, OPR, OPS) -> OPS
++ map(f, r, i) lifts f and applies it to r, assuming that
++ f(infinity) = i.
Implementation ==> add
map(f, r) == map(f, r, infinity())
map(f, r, i) ==
(u := retractIfCan r) case R => (f(u::R))::OPS
i
)abbrev package INFINITY Infinity
++ Top-level infinity
++ Author: Manuel Bronstein
++ Description: Default infinity signatures for the interpreter;
++ Date Created: 4 Oct 1989
Infinity() : with
infinity : () -> OnePointCompletion Integer
++ infinity() returns infinity.
plusInfinity : () -> OrderedCompletion Integer
++ plusInfinity() returns plusInfinity.
minusInfinity : () -> OrderedCompletion Integer
++ minusInfinity() returns minusInfinity.
== add
infinity() == infinity()$OnePointCompletion(Integer)
plusInfinity() == plusInfinity()$OrderedCompletion(Integer)
minusInfinity() == minusInfinity()$OrderedCompletion(Integer)
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
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--
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--
-- - Redistributions in binary form must reproduce the above copyright
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-- 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
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--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.