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--All rights reserved.
--Copyright (C) 2007-2009, 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.
import Type
import Integer
)abbrev category SEGCAT SegmentCategory
++ Author: Stephen M. Watt
++ Date Created: December 1986
++ Date Last Updated: June 3, 1991
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: range, segment
++ Examples:
++ References:
++ Description:
++ This category provides operations on ranges, or {\em segments}
++ as they are called.
SegmentCategory(S:Type): Category == ConvertibleFrom S with
SEGMENT: (S, S) -> %
++ \spad{l..h} creates a segment with l and h as the endpoints.
BY: (%, Integer) -> %
++ \spad{s by n} creates a new segment in which only every \spad{n}-th
++ element is used.
lo: % -> S
++ lo(s) returns the first endpoint of s.
++ Note: \spad{lo(l..h) = l}.
hi: % -> S
++ hi(s) returns the second endpoint of s.
++ Note: \spad{hi(l..h) = h}.
low: % -> S
++ low(s) returns the first endpoint of s.
++ Note: \spad{low(l..h) = l}.
high: % -> S
++ high(s) returns the second endpoint of s.
++ Note: \spad{high(l..h) = h}.
incr: % -> Integer
++ incr(s) returns \spad{n}, where s is a segment in which every
++ \spad{n}-th element is used.
++ Note: \spad{incr(l..h by n) = n}.
segment: (S, S) -> %
++ segment(i,j) is an alternate way to create the segment \spad{i..j}.
import OrderedRing
import StreamAggregate
import List
)abbrev category SEGXCAT SegmentExpansionCategory
++ Author: Stephen M. Watt
++ Date Created: June 5, 1991
++ Date Last Updated:
++ Basic Operations:
++ Related Domains: Segment, UniversalSegment
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description:
++ This category provides an interface for expanding segments to
++ a stream of elements.
SegmentExpansionCategory(S: OrderedRing, L: StreamAggregate(S)): Category ==
SegmentCategory(S) with
expand: List % -> L
++ expand(l) creates a new value of type L in which each segment
++ \spad{l..h by k} is replaced with \spad{l, l+k, ... lN},
++ where \spad{lN <= h < lN+k}.
++ For example, \spad{expand [1..4, 7..9] = [1,2,3,4,7,8,9]}.
expand: % -> L
++ expand(l..h by k) creates value of type L with elements
++ \spad{l, l+k, ... lN} where \spad{lN <= h < lN+k}.
++ For example, \spad{expand(1..5 by 2) = [1,3,5]}.
map: (S -> S, %) -> L
++ map(f,l..h by k) produces a value of type L by applying f
++ to each of the succesive elements of the segment, that is,
++ \spad{[f(l), f(l+k), ..., f(lN)]}, where \spad{lN <= h < lN+k}.
)abbrev domain RNGBIND RangeBinding
++ Author: Gabriel Dos Reis
++ Date Created: October 29, 2009
++ Date Last Updated: October 29, 2009
++ Related Constructors: SegmentCategory, SegmentBinding
++ Description:
++ This domain represents the notion of binding a variable to range
++ over a specific segment (either bounded, or half bounded).
RangeBinding(S, T): Public == Private where
T: Type
S: SegmentCategory T
Public == Type with
equation: (Symbol, S) -> %
++ \spad{equation(v,s)} creates a segment binding value with variable
++ \spad{v} and segment \spad{s}. Note that the interpreter parses
++ \spad{v=s} to this form.
variable: % -> Symbol
++ \spad{variable(x)} returns the variable from the left hand side of
++ the \spadtype{RangeBinding}. For example, if \spad{x} is
++ \spad{v=s}, then \spad{variable(x)} returns \spad{v}.
segment: % -> S
++ \spad{segment(x)} returns the segment from the right hand side of
++ the \spadtype{RangeBinding}. For example, if \spad{x} is
++ \spad{v=s}, then \spad{segment(x)} returns \spad{s}.
if S has SetCategory then SetCategory
Private == add
Rep == Record(var: Symbol, seg: S)
equation(v,s) == per [v,s]
variable x == rep(x).var
segment x == rep(x).seg
if S has SetCategory then
x = y ==
variable x = variable y and segment x = segment y
coerce(x: %): OutputForm ==
variable(x)::OutputForm = segment(x)::OutputForm
import Type
import SetCategory
import SegmentCategory
import SegmentExpansionCategory
import Integer
import List
)abbrev domain SEG Segment
++ Author: Stephen M. Watt
++ Date Created: December 1986
++ Date Last Updated: June 3, 1991
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: range, segment
++ Examples:
++ References:
++ Description:
++ This type is used to specify a range of values from type \spad{S}.
Segment(S:Type): SegmentCategory(S) with
if S has SetCategory then SetCategory
if S has OrderedRing then SegmentExpansionCategory(S, List S)
== add
Rep := Record(low: S, high: S, incr: Integer)
a..b == [a,b,1]
lo s == s.low
low s == s.low
hi s == s.high
high s == s.high
incr s == s.incr
segment(a,b) == [a,b,1]
BY(s, r) == [lo s, hi s, r]
if S has SetCategory then
(s1:%) = (s2:%) ==
s1.low = s2.low and s1.high=s2.high and s1.incr = s2.incr
coerce(s:%):OutputForm ==
seg := s.low::OutputForm..s.high::OutputForm
s.incr = 1 => seg
infix(" by "::OutputForm, seg, s.incr::OutputForm)
convert a == [a,a,1]
if S has OrderedRing then
expand(ls: List %):List S ==
lr := nil()$List(S)
for s in ls repeat
l := lo s
h := hi s
inc := (incr s)::S
zero? inc => error "Cannot expand a segment with an increment of zero"
if positive? inc then
while l <= h repeat
lr := concat(l, lr)
l := l + inc
else
while l >= h repeat
lr := concat(l, lr)
l := l + inc
reverse! lr
expand(s : %) == expand([s]$List(%))$%
map(f : S->S, s : %): List S ==
lr := nil()$List(S)
l := lo s
h := hi s
inc := (incr s)::S
if positive? inc then
while l <= h repeat
lr := concat(f l, lr)
l := l + inc
else
while l >= h repeat
lr := concat(f l, lr)
l := l + inc
reverse! lr
import Type
import OrderedRing
import Segment
import List
)abbrev package SEG2 SegmentFunctions2
++ Author:
++ Date Created:
++ Date Last Updated: June 4, 1991
++ Basic Operations:
++ Related Domains: Segment, UniversalSegment
++ Also See:
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This package provides operations for mapping functions onto segments.
SegmentFunctions2(R:Type, S:Type): public == private where
public ==> with
map: (R -> S, Segment R) -> Segment S
++ map(f,l..h) returns a new segment \spad{f(l)..f(h)}.
if R has OrderedRing then
map: (R -> S, Segment R) -> List S
++ map(f,s) expands the segment s, applying \spad{f} to each
++ value. For example, if \spad{s = l..h by k}, then the list
++ \spad{[f(l), f(l+k),..., f(lN)]} is computed, where
++ \spad{lN <= h < lN+k}.
private ==> add
map(f : R->S, r : Segment R): Segment S ==
(f(lo r)..f(hi r))$Segment(S)
if R has OrderedRing then
map(f : R->S, r : Segment R): List S ==
lr := nil()$List(S)
l := lo r
h := hi r
inc := (incr r)::R
if positive? inc then
while l <= h repeat
lr := concat(f(l), lr)
l := l + inc
else
while l >= h repeat
lr := concat(f(l), lr)
l := l + inc
reverse! lr
import Type
import SetCategory
import Segement
)abbrev domain SEGBIND SegmentBinding
++ Author:
++ Date Created:
++ Date Last Updated: June 4, 1991
++ Basic Operations:
++ Related Domains: Equation, Segment, Symbol
++ Also See:
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This domain is used to provide the function argument syntax \spad{v=a..b}.
++ This is used, for example, by the top-level \spadfun{draw} functions.
SegmentBinding(S:Type) == RangeBinding(Segment S, S)
import Type
import SegmentBinding
)abbrev package SEGBIND2 SegmentBindingFunctions2
++ Author:
++ Date Created:
++ Date Last Updated: June 4, 1991
++ Basic Operations:
++ Related Domains: SegmentBinding, Segment, Equation
++ Also See:
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This package provides operations for mapping functions onto
++ \spadtype{SegmentBinding}s.
SegmentBindingFunctions2(R:Type, S:Type): with
map: (R -> S, SegmentBinding R) -> SegmentBinding S
++ map(f,v=a..b) returns the value given by \spad{v=f(a)..f(b)}.
== add
map(f, b) ==
equation(variable b, map(f, segment b)$SegmentFunctions2(R, S))
import Type
import SegmentCategory
import Segment
)abbrev domain UNISEG UniversalSegment
++ Author: Robert S. Sutor
++ Date Created: 1987
++ Date Last Updated: June 4, 1991
++ Basic Operations:
++ Related Domains: Segment
++ Also See:
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This domain provides segments which may be half open.
++ That is, ranges of the form \spad{a..} or \spad{a..b}.
UniversalSegment(S: Type): SegmentCategory(S) with
SEGMENT: S -> %
++ \spad{l..} produces a half open segment,
++ that is, one with no upper bound.
segment: S -> %
++ segment(l) is an alternate way to construct the segment \spad{l..}.
coerce : Segment S -> %
++ coerce(x) allows \spadtype{Segment} values to be used as %.
hasHi: % -> Boolean
++ hasHi(s) tests whether the segment s has an upper bound.
if S has SetCategory then SetCategory
if S has OrderedRing then
SegmentExpansionCategory(S, Stream S)
-- expand : (List %, S) -> Stream S
-- expand : (%, S) -> Stream S
== add
Rec ==> Record(low: S, high: S, incr: Integer)
Rec2 ==> Record(low: S, incr: Integer)
SEG ==> Segment S
Rep := Union(Rec2, Rec)
a,b : S
s : %
i: Integer
ls : List %
segment a == [a, 1]$Rec2 :: Rep
segment(a,b) == [a,b,1]$Rec :: Rep
BY(s,i) ==
s case Rec => [lo s, hi s, i]$Rec ::Rep
[lo s, i]$Rec2 :: Rep
lo s ==
s case Rec2 => (s :: Rec2).low
(s :: Rec).low
low s ==
s case Rec2 => (s :: Rec2).low
(s :: Rec).low
hasHi s == s case Rec
hi s ==
not hasHi(s) => error "hi: segment has no upper bound"
(s :: Rec).high
high s ==
not hasHi(s) => error "high: segment has no upper bound"
(s :: Rec).high
incr s ==
s case Rec2 => (s :: Rec2).incr
(s :: Rec).incr
a.. == segment a
a..b == segment(a,b)
coerce(sg : SEG): % == segment(lo sg, hi sg)
convert a == [a,a,1]
if S has SetCategory then
(s1:%) = (s2:%) ==
s1 case Rec2 =>
s2 case Rec2 =>
s1.low = s2.low and s1.incr = s2.incr
false
s1 case Rec =>
s2 case Rec =>
s2.low = s2.low and s1.high=s2.high and s1.incr=s2.incr
false
false
coerce(s: %): OutputForm ==
seg :=
e := (lo s)::OutputForm
hasHi s => e..(hi s)::OutputForm
e..
inc := incr s
inc = 1 => seg
infix(" by "::OutputForm, seg, inc::OutputForm)
if S has OrderedRing then
expand(s:%) == expand([s])
map(f:S->S, s:%) == map(f, expand s)
plusInc(t: S, a: S): S == t + a
expand(ls: List %):Stream S ==
st:Stream S := empty()
null ls => st
lb:List(Segment S) := nil()
while not null ls and hasHi first ls repeat
s := first ls
ls := rest ls
ns := BY(lo(s)..hi(s), incr s)$Segment(S)
lb := concat!(lb,ns)
if not null ls then
s := first ls
st: Stream S := generate(#1 + incr(s)::S, lo s)
else
st: Stream S := empty()
concat(construct expand(lb), st)
import Type
import OrderedRing
import UniversalSegment
)abbrev package UNISEG2 UniversalSegmentFunctions2
++ Author:
++ Date Created:
++ Date Last Updated: June 4, 1991
++ Basic Operations:
++ Related Domains: Segment, UniversalSegment
++ Also See:
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This package provides operations for mapping functions onto segments.
UniversalSegmentFunctions2(R:Type, S:Type): with
map: (R -> S, UniversalSegment R) -> UniversalSegment S
++ map(f,seg) returns the new segment obtained by applying
++ f to the endpoints of seg.
if R has OrderedRing then
map: (R -> S, UniversalSegment R) -> Stream S
++ map(f,s) expands the segment s, applying \spad{f} to each value.
== add
map(f:R -> S, u:UniversalSegment R):UniversalSegment S ==
s := f lo u
hasHi u => segment(s, f hi u)
segment s
if R has OrderedRing then
map(f:R -> S, u:UniversalSegment R): Stream S ==
map(f, expand u)$StreamFunctions2(R, S)
import Monoid
import AbelianSemiGroup
)abbrev package INCRMAPS IncrementingMaps
++ Author:
++ Date Created:
++ Date Last Updated: June 4, 1991
++ Basic Operations:
++ Related Domains: UniversalSegment
++ Also See:
++ AMS Classifications:
++ Keywords: equation
++ Examples:
++ References:
++ Description:
++ This package provides operations to create incrementing functions.
IncrementingMaps(R:Join(Monoid, AbelianSemiGroup)): with
increment: () -> (R -> R)
++ increment() produces a function which adds \spad{1} to whatever
++ argument it is given. For example, if {f := increment()} then
++ \spad{f x} is \spad{x+1}.
incrementBy: R -> (R -> R)
++ incrementBy(n) produces a function which adds \spad{n} to whatever
++ argument it is given. For example, if {f := increment(n)} then
++ \spad{f x} is \spad{x+n}.
== add
increment() == 1 + #1
incrementBy n == n + #1
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