\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra array1.spad}
\author{Gabriel Dos Reis, Michael Monagan, Stephen Watt}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain PRIMARR PrimitiveArray}
<<domain PRIMARR PrimitiveArray>>=
)abbrev domain PRIMARR PrimitiveArray
++ This provides a fast array type with no bound checking on elt's.
++ Minimum index is 0 in this type, cannot be changed
PrimitiveArray(S:Type): OneDimensionalArrayAggregate S == add
   macro NNI == NonNegativeInteger
   import %icst0: Integer                    from Foreign Builtin
   import %vlength: % -> NNI                 from Foreign Builtin
   import %vcopy: % -> %                     from Foreign Builtin
   import %vfill: (%,S) -> %                 from Foreign Builtin
   import %aref: (%,Integer) -> S            from Foreign Builtin
   import %emptyArray: Type -> %             from Foreign Builtin
   import %list2array: (List S,Type) -> %    from Foreign Builtin
   import %simpleArray: (Type,NNI,S) -> %    from Foreign Builtin

   #x == %vlength x
   minIndex x == %icst0
   empty() == %emptyArray S
   construct l == %list2array(l,S)
   new(n, x) == %simpleArray(S,n,x)
   qelt(x, i) == %aref(x,i)
   elt(x:%, i:Integer) ==  %aref(x,i)
   qsetelt!(x, i, s) == %store(%aref(x,i),s)$Foreign(Builtin)
   setelt(x:%, i:Integer, s:S) == %store(%aref(x,i),s)$Foreign(Builtin)
   fill!(x, s) == %vfill(x,s)
   copy x == %vcopy x

@


\section{package PRIMARR2 PrimitiveArrayFunctions2}

<<package PRIMARR2 PrimitiveArrayFunctions2>>=
)abbrev package PRIMARR2 PrimitiveArrayFunctions2
++ This package provides tools for operating on primitive arrays
++ with unary and binary functions involving different underlying types
PrimitiveArrayFunctions2(A, B): Exports == Implementation where
  A, B: Type

  VA ==> PrimitiveArray A
  VB ==> PrimitiveArray B
  O2 ==> FiniteLinearAggregateFunctions2(A, VA, B, VB)
  Exports ==> with
    scan   : ((A, B) -> B, VA, B) -> VB
	++ scan(f,a,r) successively applies
	++ \spad{reduce(f,x,r)} to more and more leading sub-arrays
	++ x of primitive array \spad{a}.
	++ More precisely, if \spad{a} is \spad{[a1,a2,...]}, then
	++ \spad{scan(f,a,r)} returns
	++ \spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}.
    reduce : ((A, B) -> B, VA, B) -> B
	++ reduce(f,a,r) applies function f to each
	++ successive element of the
	++ primitive array \spad{a} and an accumulant initialized to r.
	++ For example,
	++ \spad{reduce(_+$Integer,[1,2,3],0)}
	++ does \spad{3+(2+(1+0))}. Note: third argument r
	++ may be regarded as the
	++ identity element for the function f.
    map    : (A -> B, VA) -> VB
	++ map(f,a) applies function f to each member of primitive array
	++ \spad{a} resulting in a new primitive array over a
	++ possibly different underlying domain.

  Implementation ==> add
    map(f, v)       == map(f, v)$O2
    scan(f, v, b)   == scan(f, v, b)$O2
    reduce(f, v, b) == reduce(f, v, b)$O2

@
\section{domain TUPLE Tuple}
<<domain TUPLE Tuple>>=
)abbrev domain TUPLE Tuple
++ This domain is used to interface with the interpreter's notion
++ of comma-delimited sequences of values.
Tuple(S:Type): HomotopicTo (PrimitiveArray S) with
  select: (%, NonNegativeInteger) -> S
	++ select(x,n) returns the n-th element of tuple x.
	++ tuples are 0-based
  length: % -> NonNegativeInteger
	++ length(x) returns the number of elements in tuple x
  if S has CoercibleTo(OutputForm) then CoercibleTo(OutputForm)
  if S has SetCategory then SetCategory
 == add
  Rep := Record(len : NonNegativeInteger, elts : PrimitiveArray S)

  coerce(x: PrimitiveArray S): %  == [#x, x]
  coerce(x:%): PrimitiveArray(S) == x.elts
  length x == x.len

  select(x, n) ==
    n >= x.len => error "Index out of bounds"
    x.elts.n

  if S has SetCategory then
    x = y == (x.len = y.len) and (x.elts =$PrimitiveArray(S) y.elts)
  if S has CoercibleTo(OutputForm) then
    coerce(x : %): OutputForm ==
      paren [(x.elts.i)::OutputForm
             for i in minIndex x.elts .. maxIndex x.elts]$List(OutputForm)

@
\section{domain IFARRAY IndexedFlexibleArray}
<<domain IFARRAY IndexedFlexibleArray>>=
)abbrev domain IFARRAY IndexedFlexibleArray
++ Author: Michael Monagan July/87, modified SMW June/91
++ A FlexibleArray is the notion of an array intended to allow for growth
++ at the end only.  Hence the following efficient operations
++   \spad{append(x,a)} meaning append item x at the end of the array \spad{a}
++   \spad{delete(a,n)} meaning delete the last item from the array \spad{a}
++ Flexible arrays support the other operations inherited from
++ \spadtype{ExtensibleLinearAggregate}. However, these are not efficient.
++ Flexible arrays combine the \spad{O(1)} access time property of arrays
++ with growing and shrinking at the end in \spad{O(1)} (average) time.
++ This is done by using an ordinary array which may have zero or more
++ empty slots at the end.  When the array becomes full it is copied
++ into a new larger (50% larger) array.  Conversely, when the array
++ becomes less than 1/2 full, it is copied into a smaller array.
++ Flexible arrays provide for an efficient implementation of many
++ data structures in particular heaps, stacks and sets.

IndexedFlexibleArray(S:Type, mn: Integer): Exports == Implementation where
  A ==> PrimitiveArray S
  I ==> Integer
  N ==> NonNegativeInteger
  U ==> UniversalSegment Integer
  Exports ==
    Join(OneDimensionalArrayAggregate S,ExtensibleLinearAggregate S) with
      flexibleArray : List S -> %
	++ flexibleArray(l) creates a flexible array from the list of elements l
      physicalLength : % -> NonNegativeInteger
   	++ physicalLength(x) returns the number of elements x can accomodate before growing
      physicalLength!: (%, I) -> %
	++ physicalLength!(x,n) changes the physical length of x to be n and returns the new array.
      shrinkable: Boolean -> Boolean
	++ shrinkable(b) sets the shrinkable attribute of flexible arrays to b and returns the previous value
  Implementation == add
    Rep := Record(physLen:I, logLen:I, f:A)
    shrinkable? : Boolean := true
    growAndFill : (%, I, S) -> %
    growWith    : (%, I, S) -> %
    growAdding  : (%, I, %) -> %
    shrink: (%, I)    -> %
    newa  : (N, A) -> A

    physicalLength(r) == (r.physLen) pretend NonNegativeInteger
    physicalLength!(r, n) ==
       r.physLen = 0  => error "flexible array must be non-empty"
       growWith(r, n, r.f.0)

    empty()      == [0, 0, empty()]
    #r           == (r.logLen)::N
    fill!(r, x) == (fill!(r.f, x); r)
    maxIndex r   == r.logLen - 1 + mn
    minIndex r   == mn
    new(n, a)    == [n, n, new(n, a)]

    shrinkable(b) ==
      oldval := shrinkable?
      shrinkable? := b
      oldval

    flexibleArray l ==
       n := #l
       n = 0 => empty()
       x := l.1
       a := new(n,x)
       for i in mn + 1..mn + n-1 for y in rest l repeat a.i := y
       a

    -- local utility operations
    newa(n, a) ==
       zero? n => empty()
       new(n, a.0)

    growAdding(r, b, s) ==
       b = 0 => r
       positive?(#r) => growAndFill(r, b, (r.f).0)
       positive?(#s) => growAndFill(r, b, (s.f).0)
       error "no default filler element"

    growAndFill(r, b, x) ==
       (r.logLen := r.logLen + b) <= r.physLen => r
       -- enlarge by 50% + b
       n := r.physLen + r.physLen quo 2 + 1
       if r.logLen > n then n := r.logLen
       growWith(r, n, x)

    growWith(r, n, x) ==
       y := new(n::N, x)$PrimitiveArray(S)
       a := r.f
       for k in 0 .. r.physLen-1 repeat y.k := a.k
       r.physLen := n
       r.f := y
       r

    shrink(r, i) ==
       r.logLen := r.logLen - i
       negative?(n := r.logLen) => error "internal bug in flexible array"
       2*n+2 > r.physLen => r
       not shrinkable? => r
       if n < r.logLen then error "cannot shrink flexible array to indicated size"
       n = 0 => empty()
       r.physLen := n
       y := newa(n::N, a := r.f)
       for k in 0 .. n-1 repeat y.k := a.k
       r.f := y
       r

    copy r ==
       n := #r
       a := r.f
       v := newa(n, a := r.f)
       for k in 0..n-1 repeat v.k := a.k
       [n, n, v]


    elt(r:%, i:I) ==
       i < mn or i >= r.logLen + mn =>
           error "index out of range"
       r.f.(i-mn)

    setelt(r:%, i:I, x:S) ==
       i < mn or i >= r.logLen + mn =>
           error "index out of range"
       r.f.(i-mn) := x

    -- operations inherited from extensible aggregate
    merge(g, a, b)   == merge!(g, copy a, b)
    concat(x:S, r:%) == insert!(x, r, mn)

    concat!(r:%, x:S) ==
       growAndFill(r, 1, x)
       r.f.(r.logLen-1) := x
       r

    concat!(a:%, b:%) ==
       if eq?(a, b) then b := copy b
       n := #a
       growAdding(a, #b, b)
       copyInto!(a, b, n + mn)

    remove!(g:(S->Boolean), a:%) ==
       k:I := 0
       for i in 0..maxIndex a - mn repeat
          if not g(a.i) then (a.k := a.i; k := k+1)
       shrink(a, #a - k)

    delete!(r:%, i1:I) ==
       i := i1 - mn
       negative? i or i > r.logLen => error "index out of range"
       for k in i..r.logLen-2 repeat r.f.k := r.f.(k+1)
       shrink(r, 1)

    delete!(r:%, i:U) ==
       l := lo i - mn; m := maxIndex r - mn
       h := (hasHi i => hi i - mn; m)
       negative? l or h > m => error "index out of range"
       for j in l.. for k in h+1..m repeat r.f.j := r.f.k
       shrink(r, max(0,h-l+1))

    insert!(x:S, r:%, i1:I):% ==
       i := i1 - mn
       n := r.logLen
       negative? i or i > n => error "index out of range"
       growAndFill(r, 1, x)
       for k in n-1 .. i by -1 repeat r.f.(k+1) := r.f.k
       r.f.i := x
       r

    insert!(a:%, b:%, i1:I):% ==
       i := i1 - mn
       if eq?(a, b) then b := copy b
       m := #a; n := #b
       negative? i or i > n => error "index out of range"
       growAdding(b, m, a)
       for k in n-1 .. i by -1 repeat b.f.(m+k) := b.f.k
       for k in m-1 .. 0 by -1 repeat b.f.(i+k) := a.f.k
       b

    merge!(g, a, b) ==
       m := #a; n := #b; growAdding(a, n, b)
       for i in m-1..0 by -1 for j in m+n-1.. by -1 repeat a.f.j := a.f.i
       i := n; j := 0
       k : Integer := 0
       while i < n+m and j < n repeat
          if g(a.f.i,b.f.j) then (a.f.k := a.f.i; i := i+1)
          else (a.f.k := b.f.j; j := j+1)
          k := k + 1
       for j' in j..n-1 repeat
         a.f.k := b.f.j'
         k := k + 1
       a

    select!(g:(S->Boolean), a:%) ==
       k:I := 0
       for i in 0..maxIndex a - mn repeat if g(a.f.i) then (a.f.k := a.f.i;k := k+1)
       shrink(a, #a - k)

    if S has SetCategory then
      removeDuplicates! a ==
         ct := #a
         ct < 2 => a

         i     := mn
         nlim  := mn + ct
         nlim0 := nlim
         while i < nlim repeat
            j := i+1
            for k in j..nlim-1 | a.k ~= a.i repeat
                a.j := a.k
                j := j+1
            nlim := j
            i := i+1
         nlim ~= nlim0 => delete!(a, i..)
         a

@
\section{domain FARRAY FlexibleArray}
<<domain FARRAY FlexibleArray>>=
)abbrev domain FARRAY FlexibleArray
++ A FlexibleArray is the notion of an array intended to allow for growth
++ at the end only.  Hence the following efficient operations
++   \spad{append(x,a)} meaning append item x at the end of the array \spad{a}
++   \spad{delete(a,n)} meaning delete the last item from the array \spad{a}
++ Flexible arrays support the other operations inherited from
++ \spadtype{ExtensibleLinearAggregate}. However, these are not efficient.
++ Flexible arrays combine the \spad{O(1)} access time property of arrays
++ with growing and shrinking at the end in \spad{O(1)} (average) time.
++ This is done by using an ordinary array which may have zero or more
++ empty slots at the end.  When the array becomes full it is copied
++ into a new larger (50% larger) array.  Conversely, when the array
++ becomes less than 1/2 full, it is copied into a smaller array.
++ Flexible arrays provide for an efficient implementation of many
++ data structures in particular heaps, stacks and sets.

FlexibleArray(S: Type) == Implementation where
  ARRAYMININDEX ==> 1       -- if you want to change this, be my guest
  Implementation ==> IndexedFlexibleArray(S, ARRAYMININDEX)
-- Join(OneDimensionalArrayAggregate S, ExtensibleLinearAggregate S)

@
\section{domain IARRAY1 IndexedOneDimensionalArray}
<<domain IARRAY1 IndexedOneDimensionalArray>>=
)abbrev domain IARRAY1 IndexedOneDimensionalArray
++ Author Micheal Monagan Aug/87
++ This is the basic one dimensional array data type.

IndexedOneDimensionalArray(S:Type, mn:Integer):
 OneDimensionalArrayAggregate S == add
   macro Qmax == maxIndexOfSimpleArray$Foreign(Builtin)
   macro Qsetelt == setSimpleArrayEntry$Foreign(Builtin)
   macro I == Integer

   import %icst0: I                         from Foreign Builtin
   import %icst1: I                         from Foreign Builtin
   import %ilt: (I,I) -> Boolean            from Foreign Builtin
   import %vlength: % -> NonNegativeInteger from Foreign Builtin
   import %aref: (%,Integer) -> S           from Foreign Builtin

   Rep == PrimitiveArray S

   newArray(n: Integer): % ==
     makeSimpleArray(getVMType(S)$Foreign(Builtin),n)$Foreign(Builtin)

   #x == # rep x
   copy x == per copy rep x
   fill!(x, s) == per fill!(rep x, s)
   minIndex x == mn

   empty() == per empty()$Rep
   new(n, s) == per new(n,s)$Rep
   construct l == per construct(l)$Rep

   map!(f, s1)  ==
      n: Integer := Qmax(s1)
      negative? n => s1
      for i in %icst0..n repeat Qsetelt(s1,i,f %aref(s1,i))
      s1

   map(f, s1)       ==
      n:Integer := Qmax(s1)
      negative? n => s1
      ss2:% := newArray(n+1)
      for i in %icst0..n repeat Qsetelt(ss2,i,f %aref(s1,i))
      ss2

   map(f, a, b)   ==
      maxind:Integer := min(Qmax a, Qmax b)
      negative? maxind => empty()
      c:% := newArray(maxind + %icst1)
      for i in %icst0..maxind repeat
        Qsetelt(c,i,f(%aref(a,i),%aref(b,i)))
      c

   if zero? mn then
     qelt(x, i)       == %aref(x, i)
     qsetelt!(x, i, s) == Qsetelt(x, i, s)

     elt(x:%, i:I) ==
       negative? i or i > maxIndex(x) => error "index out of range"
       qelt(x, i)

     setelt(x:%, i:I, s:S) ==
       negative? i or i > maxIndex(x) => error "index out of range"
       qsetelt!(x, i, s)

   else if one? mn then
     maxIndex x       == %vlength x
     qelt(x, i)       == %aref(x, i - %icst1)
     qsetelt!(x, i, s) == Qsetelt(x, i - %icst1, s)

     elt(x:%, i:I) ==
       %ilt(i,%icst1) or %ilt(%vlength x,i) =>
         error "index out of range"
       %aref(x, i - %icst1)

     setelt(x:%, i:I, s:S) ==
       %ilt(i,%icst1) or %ilt(%vlength x,i) =>
         error "index out of range"
       Qsetelt(x, i - %icst1, s)

    else
       qelt(x, i)       == %aref(x, i - mn)
       qsetelt!(x, i, s) == Qsetelt(x, i - mn, s)

       elt(x:%, i:I) ==
         i < mn or i > maxIndex(x) => error "index out of range"
         qelt(x, i)

       setelt(x:%, i:I, s:S) ==
         i < mn or i > maxIndex(x) => error "index out of range"
         qsetelt!(x, i, s)

@
\section{domain ARRAY1 OneDimensionalArray}
<<domain ARRAY1 OneDimensionalArray>>=
)abbrev domain ARRAY1 OneDimensionalArray
++ This is the domain of 1-based one dimensional arrays

OneDimensionalArray(S:Type): Exports == Implementation where
  Exports == OneDimensionalArrayAggregate S with
    oneDimensionalArray: List S -> %
	++ oneDimensionalArray(l) creates an array from a list of elements l
    oneDimensionalArray: (NonNegativeInteger, S) -> %
	++ oneDimensionalArray(n,s) creates an array from n copies of element s
  Implementation == IndexedOneDimensionalArray(S, 1) add
    oneDimensionalArray(u) == construct u
    oneDimensionalArray(n,s) == new(n,s)

@
\section{package ARRAY12 OneDimensionalArrayFunctions2}
<<package ARRAY12 OneDimensionalArrayFunctions2>>=
)abbrev package ARRAY12 OneDimensionalArrayFunctions2
++ This package provides tools for operating on one-dimensional arrays
++ with unary and binary functions involving different underlying types
OneDimensionalArrayFunctions2(A, B): Exports == Implementation where
  A, B: Type

  VA ==> OneDimensionalArray A
  VB ==> OneDimensionalArray B
  O2 ==> FiniteLinearAggregateFunctions2(A, VA, B, VB)

  Exports ==> with
    scan   : ((A, B) -> B, VA, B) -> VB
	++ scan(f,a,r) successively applies
	++ \spad{reduce(f,x,r)} to more and more leading sub-arrays
	++ x of one-dimensional array \spad{a}.
	++ More precisely, if \spad{a} is \spad{[a1,a2,...]}, then
	++ \spad{scan(f,a,r)} returns
	++ \spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}.
    reduce : ((A, B) -> B, VA, B) -> B
	++ reduce(f,a,r) applies function f to each
	++ successive element of the
	++ one-dimensional array \spad{a} and an accumulant initialized to r.
	++ For example,
	++ \spad{reduce(_+$Integer,[1,2,3],0)}
	++ does \spad{3+(2+(1+0))}. Note: third argument r
	++ may be regarded as the
	++ identity element for the function f.
    map    : (A -> B, VA) -> VB
	++ map(f,a) applies function f to each member of one-dimensional array
	++ \spad{a} resulting in a new one-dimensional array over a
	++ possibly different underlying domain.

  Implementation ==> add
    map(f, v)       == map(f, v)$O2
    scan(f, v, b)   == scan(f, v, b)$O2
    reduce(f, v, b) == reduce(f, v, b)$O2

@
\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>>

<<domain PRIMARR PrimitiveArray>>
<<package PRIMARR2 PrimitiveArrayFunctions2>>
<<domain TUPLE Tuple>>
<<domain IFARRAY IndexedFlexibleArray>>
<<domain FARRAY FlexibleArray>>
<<domain IARRAY1 IndexedOneDimensionalArray>>
<<domain ARRAY1 OneDimensionalArray>>
<<package ARRAY12 OneDimensionalArrayFunctions2>>

--%% TupleFunctions2
--TupleFunctions2(A:Type, B:Type): with
--  map: (A -> B, Tuple A) -> Tuple B
-- == add
--  map(f, t) ==
--    p:PrimitiveArray(B) := new length t
--    for i in minIndex p .. maxIndex p repeat
--      p.i := f select(t, i)
--    p::Tuple(B)

@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}