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SandBoxKernel

Edit detail for SandBoxKernel revision 3 of 6

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Editor: Bill Page Time: 2015/03/20 22:01:58 GMT+0
Note: equivalence relation

added:

)abbrev category PARTSET PartitionedSet
++ Sets whose elements are grouped into equivalence classes by a mapping
++ Author: Bill Page
++ Date Created: 20 March 2015
++ Description:
++   A partitioned set is a set whose elements have an integer as part
++   of their structure. This integer assigns each element to a "bin".
PartitionedSet : Category == SetCategory with
  position   : % -> NonNegativeInteger
    ++ position(x) returns the integer n associated to x.
  setPosition : (%, NonNegativeInteger) -> Void
    ++ setPosition(x, n) associates the integer n to x.

changed:
-  Exports ==> Join(CachableSet, OrderedSet, Patternable S) with
  Exports ==> Join(PartitionedSet, OrderedSet, Patternable S) with

removed:
-    import from SortedCache(%)

changed:
-
-    clearCache()
-
-    B2Z   : Boolean -> Integer
-    triage : (%, %)  -> Integer
    import from XHashTable(List N,Boolean)
    cache:XHashTable(List N,Boolean):=table()
    bin:N:=0


removed:
-    B2Z flag           == (flag => -1; 1)

changed:
-    kerEqual(k1 : %, k2 : %, f : (%, %) -> Boolean) : Boolean ==
-        height(k1)   ~= height(k2)   => false
-        operator(k1) ~= operator(k2) => false
-        (n1 := #(argument k1)) ~= (n2 := #(argument k2)) => false
-        f(k1, k2)
-
-    kernelEnterInCache(k : %) : % ==
-        if (f0 := property(operator k, SPECIALEQUAL)) case None then
-            f1 := (f0::None)  pretend ((%, %) -> Boolean)
-            (res := linearSearch(k, y +-> kerEqual(k, y, f1))) case % =>
-                return res::%
-        enterInCache(k, triage)
    kerEqual(k1:%,k2:%):Boolean ==
      height(k1)   ~= height(k2)   => false
      operator(k1) ~= operator(k2) => false
      (n1 := #(argument k1)) ~= (n2 := #(argument k2)) => false
      ((func := property(operator k1, SPECIALEQUAL)) case None) =>
        (((func::None) pretend ((%, %) -> Boolean)) (k1, k2))
      for x1 in argument(k1) for x2 in argument(k2) repeat
        x1 ~= x2 => false
      true

changed:
-      if position(k1) = 0 then k1 := kernelEnterInCache(k1)
-      if position(k2) = 0 then k2 := kernelEnterInCache(k2)
-      position(k1) = position(k2)
      p1:=position(k1); p2:=position(k2)
      p1=p2 => true
      if p1<p2 then
        eq:=search([p1,p2],cache)
        if eq case "failed" then
          eq:=kerEqual(k1,k2)
          if (cache([p1,p2]):=eq::Boolean) then setPosition(k2,p1)
      else
        eq:=search([p2,p1],cache)
        if eq case "failed" then
          eq:=kerEqual(k2,k1)
          if (cache([p2,p1]):=eq::Boolean) then setPosition(k1,p2)
      eq::Boolean

changed:
-      if position(k1) = 0 then k1 := kernelEnterInCache(k1)
-      if position(k2) = 0 then k2 := kernelEnterInCache(k2)
-      position(k1) < position(k2)
      -- We have to do this the hard way
      height(k1)   ~= height(k2)   => height(k1)   < height(k2)
      operator(k1) ~= operator(k2) => operator(k1) < operator(k2)
      (n1 := #(argument k1)) ~= (n2 := #(argument k2)) => n1 < n2
      ((func := property(operator k1, SPECIALEQUAL)) case None) and
        (((func::None) pretend ((%, %) -> Boolean)) (k1, k2)) => false
      for x1 in argument(k1) for x2 in argument(k2) repeat
        x1 ~= x2 => return smaller?(x1, x2)
      false

changed:
-      kernelEnterInCache(mkKer(fn, x, n))
      k:=mkKer(fn, x, n)
      setPosition(k,bin:=bin+1)
      k

removed:
-    triage(k1, k2) ==
-      height(k1)   ~= height(k2)   => B2Z(height(k1)   < height(k2))
-      operator(k1) ~= operator(k2) => B2Z(operator(k1) < operator(k2))
-      (n1 := #(argument k1)) ~= (n2 := #(argument k2)) => B2Z(n1 < n2)
-      -- Handled by linear search earlier
-      -- ((func := property(operator k1, SPECIALEQUAL)) case None) and
-      --  (((func::None) pretend ((%, %) -> Boolean)) (k1, k2)) => 0
-      for x1 in argument(k1) for x2 in argument(k2) repeat
-        x1 ~= x2 => return B2Z(smaller?(x1, x2))
-      0
-

added:

\begin{axiom}
kernel(operator 'test,[x1,x2],2)$Kernel(EXPR INT)
\end{axiom}

spad

)abbrev category CACHSET CachableSet
++ Sets whose elements can cache an integer
++ Author: Manuel Bronstein
++ Date Created: 31 Oct 1988
++ Date Last Updated: 14 May 1991
++ Description:
++   A cachable set is a set whose elements keep an integer as part
++   of their structure.
CachableSet : Category == SetCategory with
  position   : % -> NonNegativeInteger
    ++ position(x) returns the integer n associated to x.
  setPosition : (%, NonNegativeInteger) -> Void
    ++ setPosition(x, n) associates the integer n to x.

)abbrev package SCACHE SortedCache
++ Cache of elements in a set
++ Author: Manuel Bronstein
++ Date Created: 31 Oct 1988
++ Date Last Updated: 14 May 1991
++   A sorted cache of a cachable set S is a dynamic structure that
++   keeps the elements of S sorted and assigns an integer to each
++   element of S once it is in the cache. This way, equality and ordering
++   on S are tested directly on the integers associated with the elements
++   of S, once they have been entered in the cache.
SortedCache(S : CachableSet) : Exports == Implementation where
  N    ==> NonNegativeInteger
  DIFF ==> 1024

  Exports ==> with
    clearCache  : () -> Void
      ++ clearCache() empties the cache.
    enterInCache : (S, S -> Boolean) -> S
      ++ enterInCache(x, f) enters x in the cache, calling \spad{f(y)} to
      ++ determine whether x is equal to y. It returns x with an integer
      ++ associated with it.
    linearSearch : (S, S -> Boolean) -> Union(S, "failed")
      ++ linearSearch(x, f) searches x in the cache, calling \spad{f(y)}
      ++ to determine whether x is equal to y.  It returns y from cache
      ++ such that f(y) or failed is no such y exists.
    enterInCache : (S, (S, S) -> Integer) -> S
      ++ enterInCache(x, f) enters x in the cache, calling \spad{f(x, y)} to
      ++ determine whether \spad{x < y (f(x, y) < 0), x = y (f(x, y) = 0)}, or
      ++ \spad{x > y (f(x, y) > 0)}.
      ++ It returns x with an integer associated with it.

  Implementation ==> add
    shiftCache   : (N, N) -> Void
    insertInCache : (N, S, N) -> S
    expandCache : (S) -> Void
    insertBefore : (N, S) -> Void

    cache : PrimitiveArray S := empty()$(PrimitiveArray S)
    cache_size : N := 0
    cache_use : N := 0

    expandCache(x) ==
        if cache_size = cache_use then
            ocache := cache
            cache_size := 2*cache_size + 10
            cache := new(cache_size, x)$(PrimitiveArray S)
            for k in 0..(cache_use - 1) repeat
                cache(k) := ocache(k)
        void

    insertBefore(l, x) ==
        k : Integer
        expandCache(x)
        vscan := cache
        for k in 0..(cache_use - l - 1) repeat
            vscan(cache_use - k) := vscan(cache_use - k - 1)
        vscan(l) := x
        cache_use := cache_use + 1
        void

    shiftCache(l, n) ==
        k : Integer
        vscan := cache
        for k in l..(cache_use - 1) repeat
            x := vscan(k)
            setPosition(x, n + position x)
        void

    clearCache() ==
        k : Integer
        vscan := cache
        for k in 0..(cache_use - 1) repeat
            x := vscan(k)
            setPosition(x, 0)
        cache := empty()$(PrimitiveArray S)
        cache_size := 0
        cache_use := 0
        void

    insertAtEnd(x : S) : Void ==
        expandCache(x)
        cache(cache_use) := x
        cache_use := cache_use + 1
        void

    linearSearch(x : S, equal? : S -> Boolean) ==
        k : Integer := 0
        -- Can not use for loop because equal? can insert new elements
        -- and change cache_use
        while k < cache_use repeat
            vscan := cache
            y := vscan(k)
            equal?(y) =>
                setPosition(x, position y)
                return y
            vscan := cache
            -- skip over elements possibly inserted by equal?
            while not(EQ(y, vscan(k))$Lisp) repeat k := k + 1
            k := k + 1
        return "failed"

    enterInCache(x : S, equal? : S -> Boolean) ==
        (res := linearSearch(x, equal?)) case S =>
            res::S
        setPosition(x, 1 + cache_use)
        insertAtEnd(x)
        x

    enterInCache(x : S, triage : (S, S) -> Integer) ==
        vscan := cache
        l : Integer := -1
        m : Integer := cache_use
        m0 := m
        zero?(cache_use) =>
            setPosition(x, DIFF)
            insertAtEnd(x)
            return x
        while (l + 1) < m repeat
            vl : S
            vm : S
            m0 := cache_use
            if not(l < 0) then
                vl := qelt(vscan, l)
            has_vm := false
            if m < m0 then
                vm := qelt(vscan, m)
                has_vm := true
            i := shift(l + m, -1)
            cp := triage(x, y := qelt(vscan, i))
            zero?(cp) =>
                setPosition(x, position y)
                return y
            vscan := cache
            if not(l < 0) then
                if not(EQ(vl, qelt(vscan, l))$Lisp) then
                    l0 := l
                    while not(EQ(vl, qelt(vscan, l))$Lisp) repeat
                        l := l + 1
                    i := i + l - l0
                    m := m + l - l0
            if not(EQ(y, qelt(vscan, i))$Lisp) then
                i0 := i
                while not(EQ(y, qelt(vscan, i))$Lisp) repeat
                    i := i + 1
                m := m + i - i0
            if has_vm then
                if not(EQ(vm, qelt(vscan, m))$Lisp) then
                    while not(EQ(vm, qelt(vscan, m))$Lisp) repeat
                        m := m + 1
            if cp < 0 then
                m := i
            else
                l := i
        m = cache_use =>
            setPosition(x, (position qelt(vscan, m - 1)) + DIFF)
            insertAtEnd(x)
            return x
        pos : N :=
                l < 0 => 0
                position qelt(vscan, l)
        insertInCache((l+1)::N, x, pos)

    insertInCache(before, x, pos) ==
        y := cache(before)
        if ((pos+1) = position y) then shiftCache(before, DIFF)
        setPosition(x, pos + (((position y) - pos)::N quo 2))
        insertBefore(before, x)
        x

)abbrev domain MKCHSET MakeCachableSet
++ Make a cachable set from any set
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++   MakeCachableSet(S) returns a cachable set which is equal to S as a set.
MakeCachableSet(S : SetCategory) : Exports == Implementation where
  Exports ==> Join(CachableSet, CoercibleTo S) with
    coerce : S -> %
      ++ coerce(s) returns s viewed as an element of %.

  Implementation ==> add
    import from SortedCache(%)

    Rep := Record(setpart : S, pos : NonNegativeInteger)

    clearCache()

    position x             == x.pos
    setPosition(x, n)      == (x.pos := n; void)
    coerce(x : %) : S          == x.setpart
    coerce(x : %) : OutputForm == x::S::OutputForm
    coerce(s : S) : %          == enterInCache([s, 0]$Rep,
                                 (x1 +-> (s = x1::S))@(% -> Boolean))

--    x < y ==
--      if position(x) = 0 then enterInCache(x, x1+->(x::S = x1::S))
--      if position(y) = 0 then enterInCache(y, x1+->(y::S = x1::S))
--      position(x) < position(y)

    x = y ==
      if position(x) = 0 then enterInCache(x,
                                 (x1 +-> (x::S = x1::S))@(% -> Boolean))
      if position(y) = 0 then enterInCache(y,
                                 (x1 +-> (y::S = x1::S))@(% -> Boolean))
      position(x) = position(y)

)abbrev category PARTSET PartitionedSet
++ Sets whose elements are grouped into equivalence classes by a mapping
++ Author: Bill Page
++ Date Created: 20 March 2015
++ Description:
++   A partitioned set is a set whose elements have an integer as part
++   of their structure. This integer assigns each element to a "bin".
PartitionedSet : Category == SetCategory with
  position   : % -> NonNegativeInteger
    ++ position(x) returns the integer n associated to x.
  setPosition : (%, NonNegativeInteger) -> Void
    ++ setPosition(x, n) associates the integer n to x.

)abbrev domain KERNEL Kernel
++ Operators applied to elements of a set
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Date Last Updated: 10 August 1994
++ Description:
++ A kernel over a set S is an operator applied to a given list
++ of arguments from S.
Kernel(S : Comparable) : Exports == Implementation where
  O  ==> OutputForm
  N  ==> NonNegativeInteger
  OP ==> BasicOperator

  Exports ==> Join(PartitionedSet, OrderedSet, Patternable S) with
    name    : % -> Symbol
      ++ name(op(a1, ..., an)) returns the name of op.
    operator : % -> OP
      ++ operator(op(a1, ..., an)) returns the operator op.
    argument : % -> List S
      ++ argument(op(a1, ..., an)) returns \spad{[a1, ..., an]}.
    height  : % -> N
      ++ height(k) returns the nesting level of k.
    kernel  : (OP, List S, N) -> %
      ++ kernel(op, [a1, ..., an], m) returns the kernel \spad{op(a1, ..., an)}
      ++ of nesting level m.
      ++ Error: if op is k-ary for some k not equal to n.
    kernel  : Symbol -> %
      ++ kernel(x) returns x viewed as a kernel.
    symbolIfCan : % -> Union(Symbol, "failed")
      ++ symbolIfCan(k) returns k viewed as a symbol if k is a symbol, and
      ++ "failed" otherwise.
    is?     : (%, OP) -> Boolean
      ++ is?(op(a1, ..., an), f) tests if op = f.
    is?     : (%, Symbol) -> Boolean
      ++ is?(op(a1, ..., an), s) tests if the name of op is s.
    if S has ConvertibleTo InputForm then ConvertibleTo InputForm

  Implementation ==> add

    operator(k : %) : OP == SPAD_-KERNEL_-OP(k)$Lisp
    argument(k : %) : List S == SPAD_-KERNEL_-ARG(k)$Lisp
    height(k) == SPAD_-KERNEL_-NEST(k)$Lisp
    position(k : %) : N == SPAD_-KERNEL_-POSIT(k)$Lisp
    setPosition(k, n) == SET_-SPAD_-KERNEL_-POSIT(k, n)$Lisp
    mkKer(o : OP, a : List S, n : N) : % == makeSpadKernel(o, a, n)$Lisp

    SYMBOL  := '%symbol
    PMPRED  := '%pmpredicate
    PMOPT   := '%pmoptional
    PMMULT  := '%pmmultiple
    PMCONST := '%pmconstant
    SPECIALDISP  := '%specialDisp
    SPECIALEQUAL := '%specialEqual
    SPECIALINPUT := '%specialInput

    import from XHashTable(List N,Boolean)
    cache:XHashTable(List N,Boolean):=table()
    bin:N:=0

    preds : OP      -> List Any

    is?(k : %, s : Symbol) == is?(operator k, s)
    is?(k : %, o : OP)     == (operator k) = o
    name k             == name operator k
    kernel s           == kernel(assert(operator(s, 0), SYMBOL), nil(), 1)

    preds o ==
      (u := property(o, PMPRED)) case "failed" => nil()
      (u::None) pretend List(Any)

    symbolIfCan k ==
      has?(operator k, SYMBOL) => name operator k
      "failed"

    kerEqual(k1:%,k2:%):Boolean ==
      height(k1)   ~= height(k2)   => false
      operator(k1) ~= operator(k2) => false
      (n1 := #(argument k1)) ~= (n2 := #(argument k2)) => false
      ((func := property(operator k1, SPECIALEQUAL)) case None) =>
        (((func::None) pretend ((%, %) -> Boolean)) (k1, k2))
      for x1 in argument(k1) for x2 in argument(k2) repeat
        x1 ~= x2 => false
      true

    k1 = k2 ==
      p1:=position(k1); p2:=position(k2)
      p1=p2 => true
      if p1<p2 then
        eq:=search([p1,p2],cache)
        if eq case "failed" then
          eq:=kerEqual(k1,k2)
          if (cache([p1,p2]):=eq::Boolean) then setPosition(k2,p1)
      else
        eq:=search([p2,p1],cache)
        if eq case "failed" then
          eq:=kerEqual(k2,k1)
          if (cache([p2,p1]):=eq::Boolean) then setPosition(k1,p2)
      eq::Boolean

    k1 < k2 ==
      -- We have to do this the hard way
      height(k1)   ~= height(k2)   => height(k1)   < height(k2)
      operator(k1) ~= operator(k2) => operator(k1) < operator(k2)
      (n1 := #(argument k1)) ~= (n2 := #(argument k2)) => n1 < n2
      ((func := property(operator k1, SPECIALEQUAL)) case None) and
        (((func::None) pretend ((%, %) -> Boolean)) (k1, k2)) => false
      for x1 in argument(k1) for x2 in argument(k2) repeat
        x1 ~= x2 => return smaller?(x1, x2)
      false

    kernel(fn, x, n) ==
      ((u := arity fn) case N) and (#x ~= u::N)
                                    => error "Wrong number of arguments"
      k:=mkKer(fn, x, n)
      setPosition(k,bin:=bin+1)
      k

    -- SPECIALDISP contains a map List S -> OutputForm
    -- it is used when the converting the arguments first is not good,
    -- for instance with formal derivatives.
    coerce(k : %) : OutputForm ==
      (v := symbolIfCan k) case Symbol => v::Symbol::OutputForm
      (f := property(o := operator k, SPECIALDISP)) case None =>
        ((f::None) pretend (List S -> OutputForm)) (argument k)
      l := [x::OutputForm for x in argument k]$List(OutputForm)
      (u := display o) case "failed" => prefix(name(o)::OutputForm, l)
      (u::(List OutputForm -> OutputForm)) l

    if S has ConvertibleTo InputForm then
      convert(k : %) : InputForm ==
        (v := symbolIfCan k) case Symbol => convert(v::Symbol)@InputForm
        (f := property(o := operator k, SPECIALINPUT)) case None =>
          ((f::None) pretend (List S -> InputForm)) (argument k)
        l := [convert x for x in argument k]$List(InputForm)
        (u := input operator k) case "failed" =>
          convert concat(convert name operator k, l)
        (u::(List InputForm -> InputForm)) l

    if S has ConvertibleTo Pattern Integer then
      convert(k : %) : Pattern(Integer) ==
        o := operator k
        (v := symbolIfCan k) case Symbol =>
          s  := patternVariable(v::Symbol,
                      has?(o, PMCONST), has?(o, PMOPT), has?(o, PMMULT))
          empty?(l := preds o) => s
          setPredicates(s, l)
        o [convert x for x in argument(k)]$List(Pattern Integer)

    if S has ConvertibleTo Pattern Float then
      convert(k : %) : Pattern(Float) ==
        o := operator k
        (v := symbolIfCan k) case Symbol =>
          s  := patternVariable(v::Symbol,
                      has?(o, PMCONST), has?(o, PMOPT), has?(o, PMMULT))
          empty?(l := preds o) => s
          setPredicates(s, l)
        o [convert x for x in argument(k)]$List(Pattern Float)

)abbrev package KERNEL2 KernelFunctions2
++ Description:
++ This package exports some auxiliary functions on kernels
KernelFunctions2(R : Comparable, S : Comparable) : with
  constantKernel : R -> Kernel S
        ++ constantKernel(r) \undocumented
  constantIfCan : Kernel S -> Union(R, "failed")
        ++ constantIfCan(k) \undocumented

 == add
  import from BasicOperatorFunctions1(R)

  constantKernel r == kernel(constantOperator r, nil(), 1)
  constantIfCan k  == constantOpIfCan operator k

--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.

-- SPAD files for the functional world should be compiled in the
-- following order:
--
--   op  KL  expr function

spad

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6970269522029584696-25px001.spad
      using old system compiler.
   CACHSET abbreviates category CachableSet 
------------------------------------------------------------------------
   initializing NRLIB CACHSET for CachableSet 
   compiling into NRLIB CACHSET 

;;;     ***       |CachableSet| REDEFINED
Time: 0 SEC.

   finalizing NRLIB CACHSET 
   Processing CachableSet for Browser database:
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
; compiling file "/var/aw/var/LatexWiki/CACHSET.NRLIB/CACHSET.lsp" (written 20 MAR 2015 10:01:56 PM):

; /var/aw/var/LatexWiki/CACHSET.NRLIB/CACHSET.fasl written
; compilation finished in 0:00:00.005
------------------------------------------------------------------------
   CachableSet is now explicitly exposed in frame initial 
   CachableSet will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CACHSET.NRLIB/CACHSET

   SCACHE abbreviates package SortedCache 
------------------------------------------------------------------------
   initializing NRLIB SCACHE for SortedCache 
   compiling into NRLIB SCACHE 
   compiling local expandCache : S -> Void
Time: 0.01 SEC.

   compiling local insertBefore : (NonNegativeInteger,S) -> Void
Time: 0 SEC.

   compiling local shiftCache : (NonNegativeInteger,NonNegativeInteger) -> Void
Time: 0.01 SEC.

   compiling exported clearCache : () -> Void
Time: 0 SEC.

   compiling local insertAtEnd : S -> Void
Time: 0 SEC.

   compiling exported linearSearch : (S,S -> Boolean) -> Union(S,failed)
Time: 0 SEC.

   compiling exported enterInCache : (S,S -> Boolean) -> S
Time: 0 SEC.

   compiling exported enterInCache : (S,(S,S) -> Integer) -> S
Time: 0.01 SEC.

   compiling local insertInCache : (NonNegativeInteger,S,NonNegativeInteger) -> S
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |SortedCache| REDEFINED

;;;     ***       |SortedCache| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] enterInCache:  vl has no value
      [2] enterInCache:  m has no value
      [3] enterInCache:  vm has no value
      [4] enterInCache:  l has no value

   Cumulative Statistics for Constructor SortedCache
      Time: 0.03 seconds

   finalizing NRLIB SCACHE 
   Processing SortedCache for Browser database:
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--------constructor---------
--------(clearCache ((Void)))---------
--------(enterInCache (S S (Mapping (Boolean) S)))---------
--------(linearSearch ((Union S failed) S (Mapping (Boolean) S)))---------
--------(enterInCache (S S (Mapping (Integer) S S)))---------
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/SCACHE.spad-->SortedCache(): Spurious comments: \indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \indented{2}{A sorted cache of a cachable set \spad{S} is a dynamic structure that} \indented{2}{keeps the elements of \spad{S} sorted and assigns an integer to each} \indented{2}{element of \spad{S} once it is in the cache. This way,{} equality and ordering} \indented{2}{on \spad{S} are tested directly on the integers associated with the elements} \indented{2}{of \spad{S},{} once they have been entered in the cache.}
; compiling file "/var/aw/var/LatexWiki/SCACHE.NRLIB/SCACHE.lsp" (written 20 MAR 2015 10:01:56 PM):

; /var/aw/var/LatexWiki/SCACHE.NRLIB/SCACHE.fasl written
; compilation finished in 0:00:00.110
------------------------------------------------------------------------
   SortedCache is now explicitly exposed in frame initial 
   SortedCache will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SCACHE.NRLIB/SCACHE

   MKCHSET abbreviates domain MakeCachableSet 
------------------------------------------------------------------------
   initializing NRLIB MKCHSET for MakeCachableSet 
   compiling into NRLIB MKCHSET 
   importing SortedCache $
   compiling exported position : $ -> NonNegativeInteger
      MKCHSET;position;$Nni;1 is replaced by QCDR 
Time: 0 SEC.

   compiling exported setPosition : ($,NonNegativeInteger) -> Void
Time: 0 SEC.

   compiling exported coerce : $ -> S
      MKCHSET;coerce;$S;3 is replaced by QCAR 
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.

   compiling exported coerce : S -> $
Time: 0 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |MakeCachableSet| REDEFINED

;;;     ***       |MakeCachableSet| REDEFINED
Time: 0.01 SEC.

   Warnings: 
      [1] position:  pos has no value
      [2] setPosition:  pos has no value
      [3] coerce:  setpart has no value

   Cumulative Statistics for Constructor MakeCachableSet
      Time: 0.01 seconds

   finalizing NRLIB MKCHSET 
   Processing MakeCachableSet for Browser database:
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--------constructor---------
--------(clearCache ((Void)))---------
--------(enterInCache (S S (Mapping (Boolean) S)))---------
--------(linearSearch ((Union S failed) S (Mapping (Boolean) S)))---------
--------(enterInCache (S S (Mapping (Integer) S S)))---------
--------constructor---------
--------(coerce (% S))---------
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/MKCHSET.spad-->MakeCachableSet(): Spurious comments: \indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \indented{2}{A sorted cache of a cachable set \spad{S} is a dynamic structure that} \indented{2}{keeps the elements of \spad{S} sorted and assigns an integer to each} \indented{2}{element of \spad{S} once it is in the cache. This way,{} equality and ordering} \indented{2}{on \spad{S} are tested directly on the integers associated with the elements} \indented{2}{of \spad{S},{} once they have been entered in the cache.}
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/MKCHSET.spad-->MakeCachableSet(): Spurious comments: MakeCachableSet(\spad{S}) returns a cachable set which is equal to \spad{S} as a set.
; compiling file "/var/aw/var/LatexWiki/MKCHSET.NRLIB/MKCHSET.lsp" (written 20 MAR 2015 10:01:56 PM):

; /var/aw/var/LatexWiki/MKCHSET.NRLIB/MKCHSET.fasl written
; compilation finished in 0:00:00.028
------------------------------------------------------------------------
   MakeCachableSet is now explicitly exposed in frame initial 
   MakeCachableSet will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MKCHSET.NRLIB/MKCHSET

   PARTSET abbreviates category PartitionedSet 
------------------------------------------------------------------------
   initializing NRLIB PARTSET for PartitionedSet 
   compiling into NRLIB PARTSET 

;;;     ***       |PartitionedSet| REDEFINED
Time: 0 SEC.

   finalizing NRLIB PARTSET 
   Processing PartitionedSet for Browser database:
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--------constructor---------
--------(clearCache ((Void)))---------
--------(enterInCache (S S (Mapping (Boolean) S)))---------
--------(linearSearch ((Union S failed) S (Mapping (Boolean) S)))---------
--------(enterInCache (S S (Mapping (Integer) S S)))---------
--------constructor---------
--------(coerce (% S))---------
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--->-->PartitionedSet(): Spurious comments: \indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \indented{2}{A sorted cache of a cachable set \spad{S} is a dynamic structure that} \indented{2}{keeps the elements of \spad{S} sorted and assigns an integer to each} \indented{2}{element of \spad{S} once it is in the cache. This way,{} equality and ordering} \indented{2}{on \spad{S} are tested directly on the integers associated with the elements} \indented{2}{of \spad{S},{} once they have been entered in the cache.}
--->-->PartitionedSet(): Spurious comments: MakeCachableSet(\spad{S}) returns a cachable set which is equal to \spad{S} as a set.
--->-->PartitionedSet(): Spurious comments: A partitioned set is a set whose elements have an integer as part of their structure. This integer assigns each element to a "bin".
; compiling file "/var/aw/var/LatexWiki/PARTSET.NRLIB/PARTSET.lsp" (written 20 MAR 2015 10:01:56 PM):

; /var/aw/var/LatexWiki/PARTSET.NRLIB/PARTSET.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
   PartitionedSet is now explicitly exposed in frame initial 
   PartitionedSet will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/PARTSET.NRLIB/PARTSET

   KERNEL abbreviates domain Kernel 
------------------------------------------------------------------------
   initializing NRLIB KERNEL for Kernel 
   compiling into NRLIB KERNEL 
   compiling exported operator : $ -> BasicOperator
      KERNEL;operator;$Bo;1 is replaced by SPAD-KERNEL-OP 
Time: 0 SEC.

   compiling exported argument : $ -> List S
      KERNEL;argument;$L;2 is replaced by SPAD-KERNEL-ARG 
Time: 0.01 SEC.

   compiling exported height : $ -> NonNegativeInteger
      KERNEL;height;$Nni;3 is replaced by SPAD-KERNEL-NEST 
Time: 0 SEC.

   compiling exported position : $ -> NonNegativeInteger
      KERNEL;position;$Nni;4 is replaced by SPAD-KERNEL-POSIT 
Time: 0 SEC.

   compiling exported setPosition : ($,NonNegativeInteger) -> Void
      KERNEL;setPosition;$NniV;5 is replaced by SET-SPAD-KERNEL-POSIT 
Time: 0 SEC.

   compiling local mkKer : (BasicOperator,List S,NonNegativeInteger) -> $
      KERNEL;mkKer is replaced by makeSpadKernel 
Time: 0 SEC.

   importing XHashTable(List NonNegativeInteger,Boolean)
Local variable cache type redefined: (XHashTable (List (NonNegativeInteger)) (Boolean)) to (PrimitiveArray S)
   compiling exported is? : ($,Symbol) -> Boolean
Time: 0.01 SEC.

   compiling exported is? : ($,BasicOperator) -> Boolean
Time: 0 SEC.

   compiling exported name : $ -> Symbol
Time: 0 SEC.

   compiling exported kernel : Symbol -> $
Time: 0 SEC.

   compiling local preds : BasicOperator -> List Any
Time: 0 SEC.

   compiling exported symbolIfCan : $ -> Union(Symbol,failed)
Time: 0 SEC.

   compiling local kerEqual : ($,$) -> Boolean
Time: 0.02 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0 SEC.

   compiling exported < : ($,$) -> Boolean
Time: 0.01 SEC.

   compiling exported kernel : (BasicOperator,List S,NonNegativeInteger) -> $
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.02 SEC.

****** Domain: S already in scope
augmenting S: (ConvertibleTo (InputForm))
   compiling exported convert : $ -> InputForm
Time: 0.03 SEC.

****** Domain: S already in scope
augmenting S: (ConvertibleTo (Pattern (Integer)))
   compiling exported convert : $ -> Pattern Integer
Time: 0.02 SEC.

****** Domain: S already in scope
augmenting S: (ConvertibleTo (Pattern (Float)))
   compiling exported convert : $ -> Pattern Float
Time: 0.01 SEC.

****** Domain: S already in scope
augmenting S: (ConvertibleTo (InputForm))
****** Domain: S already in scope
augmenting S: (ConvertibleTo (Pattern (Float)))
****** Domain: S already in scope
augmenting S: (ConvertibleTo (Pattern (Integer)))
(time taken in buildFunctor:  0)

;;;     ***       |Kernel| REDEFINED

;;;     ***       |Kernel| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor Kernel
      Time: 0.13 seconds

   finalizing NRLIB KERNEL 
   Processing Kernel for Browser database:
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--------constructor---------
--------(clearCache ((Void)))---------
--------(enterInCache (S S (Mapping (Boolean) S)))---------
--------(linearSearch ((Union S failed) S (Mapping (Boolean) S)))---------
--------(enterInCache (S S (Mapping (Integer) S S)))---------
--------constructor---------
--------(coerce (% S))---------
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--------constructor---------
--------(name ((Symbol) %))---------
--------(operator ((BasicOperator) %))---------
--------(argument ((List S) %))---------
--------(height ((NonNegativeInteger) %))---------
--------(kernel (% (BasicOperator) (List S) (NonNegativeInteger)))---------
--------(kernel (% (Symbol)))---------
--------(symbolIfCan ((Union (Symbol) failed) %))---------
--------(is? ((Boolean) % (BasicOperator)))---------
--------(is? ((Boolean) % (Symbol)))---------
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL.spad-->Kernel(): Spurious comments: \indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \indented{2}{A sorted cache of a cachable set \spad{S} is a dynamic structure that} \indented{2}{keeps the elements of \spad{S} sorted and assigns an integer to each} \indented{2}{element of \spad{S} once it is in the cache. This way,{} equality and ordering} \indented{2}{on \spad{S} are tested directly on the integers associated with the elements} \indented{2}{of \spad{S},{} once they have been entered in the cache.}
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL.spad-->Kernel(): Spurious comments: MakeCachableSet(\spad{S}) returns a cachable set which is equal to \spad{S} as a set.
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL.spad-->Kernel(): Spurious comments: A partitioned set is a set whose elements have an integer as part of their structure. This integer assigns each element to a "bin".
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL.spad-->Kernel(): Spurious comments: A kernel over a set \spad{S} is an operator applied to a given list of arguments from \spad{S}.
; compiling file "/var/aw/var/LatexWiki/KERNEL.NRLIB/KERNEL.lsp" (written 20 MAR 2015 10:01:56 PM):

; /var/aw/var/LatexWiki/KERNEL.NRLIB/KERNEL.fasl written
; compilation finished in 0:00:00.117
------------------------------------------------------------------------
   Kernel is now explicitly exposed in frame initial 
   Kernel will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/KERNEL.NRLIB/KERNEL

   KERNEL2 abbreviates package KernelFunctions2 
------------------------------------------------------------------------
   initializing NRLIB KERNEL2 for KernelFunctions2 
   compiling into NRLIB KERNEL2 
   importing BasicOperatorFunctions1 R
   compiling exported constantKernel : R -> Kernel S
Time: 0 SEC.

   compiling exported constantIfCan : Kernel S -> Union(R,failed)
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |KernelFunctions2| REDEFINED

;;;     ***       |KernelFunctions2| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor KernelFunctions2
      Time: 0 seconds

   finalizing NRLIB KERNEL2 
   Processing KernelFunctions2 for Browser database:
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--------constructor---------
--------(clearCache ((Void)))---------
--------(enterInCache (S S (Mapping (Boolean) S)))---------
--------(linearSearch ((Union S failed) S (Mapping (Boolean) S)))---------
--------(enterInCache (S S (Mapping (Integer) S S)))---------
--------constructor---------
--------(coerce (% S))---------
--------constructor---------
--------(position ((NonNegativeInteger) %))---------
--------(setPosition ((Void) % (NonNegativeInteger)))---------
--------constructor---------
--------(name ((Symbol) %))---------
--------(operator ((BasicOperator) %))---------
--------(argument ((List S) %))---------
--------(height ((NonNegativeInteger) %))---------
--------(kernel (% (BasicOperator) (List S) (NonNegativeInteger)))---------
--------(kernel (% (Symbol)))---------
--------(symbolIfCan ((Union (Symbol) failed) %))---------
--------(is? ((Boolean) % (BasicOperator)))---------
--------(is? ((Boolean) % (Symbol)))---------
--------constructor---------
--------(constantKernel ((Kernel S) R))---------
--------(constantIfCan ((Union R failed) (Kernel S)))---------
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL2.spad-->KernelFunctions2(): Spurious comments: \indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \indented{2}{A sorted cache of a cachable set \spad{S} is a dynamic structure that} \indented{2}{keeps the elements of \spad{S} sorted and assigns an integer to each} \indented{2}{element of \spad{S} once it is in the cache. This way,{} equality and ordering} \indented{2}{on \spad{S} are tested directly on the integers associated with the elements} \indented{2}{of \spad{S},{} once they have been entered in the cache.}
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL2.spad-->KernelFunctions2(): Spurious comments: MakeCachableSet(\spad{S}) returns a cachable set which is equal to \spad{S} as a set.
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL2.spad-->KernelFunctions2(): Spurious comments: A partitioned set is a set whose elements have an integer as part of their structure. This integer assigns each element to a "bin".
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL2.spad-->KernelFunctions2(): Spurious comments: A kernel over a set \spad{S} is an operator applied to a given list of arguments from \spad{S}.
--->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/KERNEL2.spad-->KernelFunctions2(): Spurious comments: This package exports some auxiliary functions on kernels
; compiling file "/var/aw/var/LatexWiki/KERNEL2.NRLIB/KERNEL2.lsp" (written 20 MAR 2015 10:01:56 PM):

; /var/aw/var/LatexWiki/KERNEL2.NRLIB/KERNEL2.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
   KernelFunctions2 is now explicitly exposed in frame initial 
   KernelFunctions2 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/KERNEL2.NRLIB/KERNEL2

fricas

kernel(operator 'test,[x1,x2],2)$Kernel(EXPR INT)

$\label{eq1}test \left({x 1, \: x 2}\right)$

(1)

Type: Kernel(Expression(Integer))