\documentclass{article}
\usepackage{axiom}

\title{\File{src/interp/i-coerce.boot} Pamphlet}
\author{The Axiom Team}

\begin{document}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject

\section{Coercion conventions}

\begin{verbatim}
Coercion conventions

Coercion involves the  changing of the datatype of an  object.  This
   can be  done for conformality of  operations or, for  example, to
   change the structure of an object  into one that is understood by
   the printing routines.

The actual coercion  is controlled  by the  function "coerce"  which
   takes  and delivers  wrapped operands.   Also  see the  functions
   interpCoerce and coerceInteractive.

Sometimes one  does not  want to  actually change  the datatype  but
   rather wants to determine  whether it is possible to do  so.  The
   controlling function  to do this  is "canCoerceFrom".   The value
   passed   to  specific   coercion  routines   in   this  case   is
   "$fromCoerceable$".   The value returned is  true or false.   See
   specific examples for more info.

The special routines that  do the coercions typically  involve a "2"
   in their  names.   For example, G2E  converts type  "Gaussian" to
   type  "Expression".   These  special  routines take  and  deliver
   unwrapped operands.   The determination of which  special routine
   to  use  is  often  made  by  consulting  the  list  $CoerceTable
   (currently in COT BOOT) and  this is controlled by coerceByTable.
   Note that the special routines are in the file COERCEFN BOOT.
\end{verbatim}
\section{Function getConstantFromDomain}
[[getConstantFromDomain]] is used to look up the constants $0$ and $1$
from the given [[domainForm]].
\begin{enumerate}
\item if [[isPartialMode]] (see i-funsel.boot) returns true then the
domain modemap contains the constant [[$EmptyMode]] which indicates
that the domain is not fully formed. In this case we return [[NIL]].
\end{enumerate}
<<getConstantFromDomain>>=
getConstantFromDomain(form,domainForm) ==
    isPartialMode domainForm => NIL
    opAlist := getOperationAlistFromLisplib first domainForm
    key := opOf form
    entryList := LASSOC(key,opAlist)
    entryList isnt [[sig, ., ., .]] =>
        key = "One" => getConstantFromDomain(["1"], domainForm)
        key = "Zero" => getConstantFromDomain(["0"], domainForm)
        throwKeyedMsg("S2IC0008",[form,domainForm])
    -- i.e., there should be exactly one item under this key of that form
    domain := evalDomain domainForm
    SPADCALL compiledLookupCheck(key,sig,domain)

@
\section{License}
<<license>>=
-- 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.

@
<<*>>=
<<license>>
--%  Algebraic coercions using interactive code

algCoerceInteractive(p,source,target) ==
  -- now called in some groebner code
  $useConvertForCoercions : local := true
  source := devaluate source
  target := devaluate target
  u := coerceInteractive(objNewWrap(p,source),target)
  u => objValUnwrap(u)
  error ['"can't convert",p,'"of mode",source,'"to mode",target]

spad2BootCoerce(x,source,target) ==
  -- x : source and we wish to coerce to target
  -- used in spad code for Any
  null isValidType source => throwKeyedMsg("S2IE0004",[source])
  null isValidType target => throwKeyedMsg("S2IE0004",[target])
  x' := coerceInteractive(objNewWrap(x,source),target) =>
    objValUnwrap(x')
  throwKeyedMsgCannotCoerceWithValue(wrap x,source,target)

--%  Functions for Coercion or Else We'll Get Rough

coerceOrFail(triple,t,mapName) ==
  -- some code generated for this is in coerceInt0
  t = $NoValueMode => triple
  t' := coerceInteractive(triple,t)
  t' => objValUnwrap(t')
  sayKeyedMsg("S2IC0004",[mapName,objMode triple,t])
  '"failed"

coerceOrCroak(triple, t, mapName) ==
  -- this does the coercion and returns the value or dies
  t = $NoValueMode => triple
  t' := coerceOrConvertOrRetract(triple,t)
  t' => objValUnwrap(t')
  mapName = 'noMapName =>
    throwKeyedMsgCannotCoerceWithValue(objVal triple,objMode triple, t)
  sayKeyedMsg("S2IC0005",[mapName])
  throwKeyedMsgCannotCoerceWithValue(objVal triple,objMode triple, t)

coerceOrThrowFailure(value, t1, t2) ==
  (result := coerceOrRetract(objNewWrap(value, t1), t2)) or
    coercionFailure()
  objValUnwrap(result)

--%  Retraction functions

retract object ==
  type := objMode object
  STRINGP type => 'failed
  type = $EmptyMode => 'failed
  val := objVal object
  not isWrapped val and val isnt ['MAP,:.] => 'failed
  type' := equiType(type)
  (ans := retract1 objNew(val,equiType(type))) = 'failed => ans
  objNew(objVal ans,eqType objMode ans)

retract1 object ==
  -- this function is the new version of the old "pullback"
  -- it first tries to change the datatype of an object to that of
  -- largest contained type. Examples: P RN -> RN, RN -> I
  -- This is mostly for cases such as constant polynomials or
  -- quotients with 1 in the denominator.
  type := objMode object
  STRINGP type => 'failed
  val := objVal object
  type = $PositiveInteger =>    objNew(val,$NonNegativeInteger)
  type = $NonNegativeInteger => objNew(val,$Integer)
  type = $Integer and SINTP unwrap val => objNew(val, $SingleInteger)
  type' := equiType(type)
  if not EQ(type,type') then object := objNew(val,type')
  (1 = #type') or (type' is ['Union,:.]) or
    (type' is ['FunctionCalled,.])
     or (type' is ['OrderedVariableList,.]) or (type is ['Variable,.]) =>
      (object' := retract2Specialization(object)) => object'
      'failed
  null (underDomain := underDomainOf type') => 'failed
  -- try to retract the "coefficients"
  -- think of P RN -> P I or M RN -> M I
  object' := retractUnderDomain(object,type,underDomain)
  object' ^= 'failed => object'
  -- see if we can use the retract functions
  (object' := coerceRetract(object,underDomain)) => object'
  -- see if we have a special case here
  (object' := retract2Specialization(object)) => object'
  'failed

retractUnderDomain(object,type,underDomain) ==
  null (ud := underDomainOf underDomain) => 'failed
  [c,:args] := deconstructT type
  1 ^= #args => 'failed
  1 ^= #c => 'failed
  type'' := constructT(c,[ud])
  (object' := coerceInt(object,type'')) => object'
  'failed

retract2Specialization object ==
  -- handles some specialization retraction cases, like matrices
  val := objVal object
  val' := unwrap val
  type := objMode object

  type = $Any =>
    [dom,:obj] := val'
    objNewWrap(obj,dom)
  type is ['Union,:unionDoms] => coerceUnion2Branch object
  type = $Symbol =>
    objNewWrap(1,['OrderedVariableList,[val']])
  type is ['OrderedVariableList,var] =>
    coerceInt(objNewWrap(var.(val'-1),$Symbol), '(Polynomial (Integer)))
-- !! following retract seems wrong and breaks ug13.input
--  type is ['Variable,var] =>
--    coerceInt(object,$Symbol)
  type is ['Polynomial,D] =>
    val' is [ =1,x,:.] =>
      vl := REMDUP reverse varsInPoly val'
      1 = #vl => coerceInt(object,['UnivariatePolynomial,x,D])
      NIL
    val' is [ =0,:.] => coerceInt(object, D)
    NIL
  type is ['Matrix,D] =>
    n := # val'
    m := # val'.0
    n = m => objNew(val,['SquareMatrix,n,D])
    objNew(val,['RectangularMatrix,n,m,D])
  type is ['RectangularMatrix,n,m,D] =>
    n = m => objNew(val,['SquareMatrix,n,D])
    NIL
  (type is [agg,D]) and (agg in '(Vector Segment UniversalSegment)) =>
    D = $PositiveInteger => objNew(val,[agg,$NonNegativeInteger])
    D = $NonNegativeInteger => objNew(val,[agg,$Integer])
    NIL
  type is ['Array,bds,D] =>
    D = $PositiveInteger => objNew(val,['Array,bds,$NonNegativeInteger])
    D = $NonNegativeInteger => objNew(val,['Array,bds,$Integer])
    NIL
  type is ['List,D] =>
    D isnt ['List,D'] =>
      -- try to retract elements
      D = $PositiveInteger => objNew(val,['List,$NonNegativeInteger])
      D = $NonNegativeInteger => objNew(val,['List,$Integer])
      null val' => nil
--        null (um := underDomainOf D) => nil
--        objNewWrap(nil,['List,um])
      vl := nil
      tl := nil
      bad := nil
      for e in val' while not bad repeat
        (e' := retract objNewWrap(e,D)) = 'failed => bad := true
        vl := [objValUnwrap e',:vl]
        tl := [objMode e',:tl]
      bad => NIL
      (m := resolveTypeListAny tl) = D => NIL
      D = equiType(m) => NIL
      vl' := nil
      for e in vl for t in tl repeat
        t = m => vl' := [e,:vl']
        e' := coerceInt(objNewWrap(e,t),m)
        null e' => return NIL
        vl' := [objValUnwrap e',:vl']
      objNewWrap(vl',['List,m])
    D' = $PositiveInteger =>
      objNew(val,['List,['List,$NonNegativeInteger]])
    D' = $NonNegativeInteger =>
      objNew(val,['List,['List,$Integer]])
    D' is ['Variable,.] or D' is ['OrderedVariableList,.] =>
        coerceInt(object,['List,['List,$Symbol]])

    n := # val'
    m := # val'.0
    null isRectangularList(val',n,m) => NIL
    coerceInt(object,['Matrix,D'])
  type is ['Expression,D] =>
    [num,:den] := val'
    -- coerceRetract already handles case where den = 1
    num isnt [0,:num] => NIL
    den isnt [0,:den] => NIL
    objNewWrap([num,:den],[$QuotientField, D])
  type is ['SimpleAlgebraicExtension,k,rep,.] =>
    -- try to retract as an element of rep and see if we can get an
    -- element of k
    val' := retract objNew(val,rep)
    while (val' ^= 'failed) and
      (equiType(objMode val') ^= k) repeat
        val' := retract val'
    val' = 'failed => NIL
    val'

  type is ['UnivariatePuiseuxSeries, coef, var, cen] =>
    coerceInt(object, ['UnivariateLaurentSeries, coef, var, cen])
  type is ['UnivariateLaurentSeries, coef, var, cen] =>
    coerceInt(object, ['UnivariateTaylorSeries, coef, var, cen])

  type is ['FunctionCalled,name] =>
    null (m := get(name,'mode,$e)) => NIL
    isPartialMode m => NIL
    objNew(val,m)
  NIL

coerceOrConvertOrRetract(T,m) ==
  $useConvertForCoercions : local := true
  coerceOrRetract(T,m)

coerceOrRetract(T,m) ==
  (t' := coerceInteractive(T,m)) => t'
  t := T
  ans := nil
  repeat
    ans => return ans
    t := retract t   -- retract is new name for pullback
    t = 'failed => return ans
    ans := coerceInteractive(t,m)
  ans

coerceRetract(object,t2) ==
  -- tries to handle cases such as P I -> I
  (val := objValUnwrap(object)) = "$fromCoerceable$" => NIL
  t1 := objMode object
  t2 = $OutputForm => NIL
  isEqualOrSubDomain(t1,$Integer) and typeIsASmallInteger(t2) and SMINTP(val) =>
    objNewWrap(val,t2)
  t1 = $Integer    => NIL
  t1 = $Symbol     => NIL
  t1 = $OutputForm => NIL
  (c := retractByFunction(object, t2)) => c
  t1 is [D,:.] =>
    fun := GETL(D,'retract) or
           INTERN STRCONC('"retract",STRINGIMAGE D)
    functionp fun =>
      PUT(D,'retract,fun)
      c := CATCH('coerceFailure,FUNCALL(fun,object,t2))
      (c = $coerceFailure) => NIL
      c
    NIL
  NIL

retractByFunction(object,u) ==
  -- tries to retract by using function "retractIfCan"
  -- if the type belongs to the correct category.
  $reportBottomUpFlag: local := NIL
  t := objMode object
  -- JHD/CRF not ofCategory(t,['RetractableTo,u]) => NIL
  val := objValUnwrap object

  -- try to get and apply the function "retractable?"
  target := ['Union,u,'"failed"]
  funName := 'retractIfCan
  if $reportBottomUpFlag then
    sayFunctionSelection(funName,[t],target,NIL,
      '"coercion facility (retraction)")
  -- JHD/CRF if (mms := findFunctionInDomain(funName,t,target,[t],[t],'T,'T))
  -- MCD: changed penultimate variable to NIL.
  if (mms := append(findFunctionInDomain(funName,t,target,[t],[t],NIL,'T),
                    findFunctionInDomain(funName,u,target,[t],[t],NIL,'T)))
-- The above two lines were:      (RDJ/BMT 6/95)
--  if (mms := append(findFunctionInDomain(funName,t,target,[t],[t],'T,'T),
--                    findFunctionInDomain(funName,u,target,[t],[t],'T,'T)))
    then mms := orderMms(funName,mms,[t],[t],target)
  if $reportBottomUpFlag then
    sayFunctionSelectionResult(funName,[t],mms)
  null mms => NIL

  -- [[dc,:.],slot,.]:= CAR mms
  dc := CAAAR mms
  slot := CADAR mms
  dcVector:= evalDomain dc
  fun :=
--+
    compiledLookup(funName,[target,t],dcVector)
  NULL fun => NIL
  CAR(fun) = function Undef => NIL
--+
  $: fluid := dcVector
  object' := coerceUnion2Branch objNewWrap(SPADCALL(val,fun),target)
  u' := objMode object'
  u = u' => object'
  NIL

--% Coercion utilities

-- The next function extracts the structural definition of constants
-- from a given domain. For example, getConstantFromDomain('(One),S)
-- returns the representation of 1 in the domain S.

constantInDomain?(form,domainForm) ==
    opAlist := getOperationAlistFromLisplib first domainForm
    key := opOf form
    entryList := LASSOC(key,opAlist)
    entryList is [[., ., ., type]] and type in '(CONST ASCONST) => true
    key = "One" => constantInDomain?(["1"], domainForm)
    key = "Zero" => constantInDomain?(["0"], domainForm)
    false

<<getConstantFromDomain>>

domainOne(domain) == getConstantFromDomain('(One),domain)

domainZero(domain) == getConstantFromDomain('(Zero),domain)

equalOne(object, domain) ==
  -- tries using constant One and "=" from domain
  -- object should not be wrapped
  algEqual(object, getConstantFromDomain('(One),domain), domain)

equalZero(object, domain) ==
  -- tries using constant Zero and "=" from domain
  -- object should not be wrapped
  algEqual(object, getConstantFromDomain('(Zero),domain), domain)

algEqual(object1, object2, domain) ==
  -- sees if 2 objects of the same domain are equal by using the
  -- "=" from the domain
  -- objects should not be wrapped
--  eqfunc := getFunctionFromDomain("=",domain,[domain,domain])
  eqfunc := compiledLookupCheck("=",[$Boolean,'$,'$],evalDomain domain)
  SPADCALL(object1,object2, eqfunc)

--%  main algorithms for canCoerceFrom and coerceInteractive

-- coerceInteractive and canCoerceFrom are the two coercion functions
-- for $InteractiveMode. They translate RN, RF and RR to QF I, QF P
-- and RE RN, respectively, and call coerceInt or canCoerce, which
-- both work in the same way (e.g. coercion from t1 to t2):

-- 1. they try to coerce t1 to t2 directly (tower coercion), and, if
--   this fails, to coerce t1 to the last argument of t2 and embed
--   this last argument into t2. These embedding functions are now only
--   defined in the algebra code. (RSS 2-27-87)

-- 2. the tower coercion looks whether there is any applicable local
--   coercion, which means, one defined in boot or in algebra code.
--   If there is an applicable function from a constructor, which is
--   inside the type tower of t1, to the top level constructor of t2,
--   then this constructor is bubbled up inside t1. This means,
--   special coercion functions (defined in boot) are called, which
--   commute two constructors in a tower. Then the local coercion is
--   called on these constructors, which both are on top level now.

-- example:
-- let t1 = A B C D E (short for (A (B (C (D (E))))), where A ... E are
--   type constructors), and t2 = F D G H I J
-- there is no coercion from t1 to t2 directly, so we try to coerce
--   t1 to s1 = D G H I J, the last argument of t2
-- we create the type s2 = A D B C E and call a local coercion A2A
--   from t1 to s2, which, by recursively calling coerce, bubbles up
--   the constructor D
-- then we call a commute coerce from s2 to s3 = D A B C E and a local
--   coerce D2D from s3 to s1
-- finally we embed s1 into t2, which completes the coercion t1 to t2

-- the result of canCoerceFrom is TRUE or NIL
-- the result of coerceInteractive is a object or NIL (=failed)
-- all boot coercion functions have the following result:
-- 1. if u=$fromCoerceable$, then TRUE or NIL
-- 2. if the coercion succeeds, the coerced value (this may be NIL)
-- 3. if the coercion fails, they throw to a catch point in
--      coerceByFunction

--% Interpreter Coercion Query Functions

canCoerce1(t1,t2) ==
  -- general test for coercion
  -- the result is NIL if it fails
  t1 = t2 => true
  absolutelyCanCoerceByCheating(t1,t2) or t1 = '(None) or t2 = '(Any) or
    t1 in '((Mode)  (Domain) (SubDomain (Domain))) =>
      t2 = $OutputForm => true
      NIL
    -- next is for tagged union selectors for the time being
    t1 is ['Variable,=t2] or t2 is ['Variable,=t1] => true
    STRINGP t1 =>
      t2 = $String => true
      t2 = $OutputForm => true
      t2 is ['Union,:.] => canCoerceUnion(t1,t2)
      t2 is ['Variable,v] and (t1 = PNAME(v)) => true
      NIL
    STRINGP t2 =>
      t1 is ['Variable,v] and (t2 = PNAME(v)) => true
      NIL
    atom t1 or atom t2 => NIL
    null isValidType(t2) => NIL

    absolutelyCannotCoerce(t1,t2) => NIL

    nt1 := CAR t1
    nt2 := CAR t2

    EQ(nt1,'Mapping) => EQ(nt2,'Any)
    EQ(nt2,'Mapping) =>
      EQ(nt1,'Variable) or EQ(nt1,'FunctionCalled) =>
        canCoerceExplicit2Mapping(t1,t2)
      NIL
    EQ(nt1,'Union) or EQ(nt2,'Union) => canCoerceUnion(t1,t2)

    -- efficiency hack
    t1 is ['Segment, s1] and t2 is ['UniversalSegment, s2] and
        (isEqualOrSubDomain(s1, s2) or canCoerce(s1, s2)) => true

    t1 is ['Tuple,S] and t2 ^= '(OutputForm) => canCoerce(['List, S], t2)

    isRingT2 := ofCategory(t2,'(Ring))
    isRingT2 and isEqualOrSubDomain(t1,$Integer) => true
    (ans := canCoerceTopMatching(t1,t2,nt1,nt2)) ^= 'maybe => ans
    t2 = $Integer => canCoerceLocal(t1,t2)   -- is true
    ans := canCoerceTower(t1,t2) or
      [.,:arg]:= deconstructT t2
      arg and
        t:= last arg
        canCoerce(t1,t) and canCoerceByFunction(t,t2) and 'T
    ans or (t1 in '((PositiveInteger) (NonNegativeInteger))
      and canCoerce($Integer,t2))

canCoerceFrom0(t1,t2) ==
-- top level test for coercion, which transfers all RN, RF and RR into
-- equivalent types
  startTimingProcess 'querycoerce
  q :=
    isEqualOrSubDomain(t1,t2) or t1 = '(None) or t2 = '(Any) or
      if t2 = $OutputForm then (s1 := t1; s2 := t2)
      else (s1:= equiType(t1); s2:= equiType(t2))

      -- make sure we are trying to coerce to a legal type
      -- in particular, polynomials are repeated, etc.
      null isValidType(t2) => NIL
      null isLegitimateMode(t2,nil,nil) => NIL

      t1 = $RationalNumber =>
        isEqualOrSubDomain(t2,$Integer) => NIL
        canCoerce(t1,t2) or canCoerce(s1,s2)
      canCoerce(s1,s2)
  stopTimingProcess 'querycoerce
  q

isSubTowerOf(t1,t2) ==
  -- assumes RF and RN stuff has been expanded
  -- tests whether t1 is somewhere inside t2
  isEqualOrSubDomain(t1,t2) => true
  null (u := underDomainOf t2) => nil
  isSubTowerOf(t1,u)

canCoerceTopMatching(t1,t2,tt1,tt2) ==
  -- returns true, nil or maybe
  -- for example, if t1 = P[x] D1 and t2 = P[y] D2 and x = y then
  -- canCoerce will only be true if D1 = D2
  not EQ(tt1,tt2) => 'maybe
  doms := '(Polynomial List Matrix FiniteSet Vector Stream Gaussian)
  MEMQ(tt1,doms) => canCoerce(CADR t1, CADR t2)
  not (MEMQ(tt1,$univariateDomains) or MEMQ(tt2,$multivariateDomains)) =>
    'maybe
  u2 := deconstructT t2
  1 = #u2 => NIL
  u1 := deconstructT t1
  1 = #u1 => NIL                             -- no under domain
  first(u1) ^= first(u2) => 'maybe
  canCoerce(underDomainOf t1, underDomainOf t2)

canCoerceExplicit2Mapping(t1,t is ['Mapping,target,:argl]) ==
  -- determines if there a mapping called var with the given args
  -- and target
  $useCoerceOrCroak: local := nil
  t1 is ['Variable,var] =>
    null (mms :=selectMms1(var,target,argl,[NIL for a in argl],true)) => NIL
    mm := CAAR mms
    mm is [., targ, :.] =>
      targ = target => true
      false
    false
  t1 is ['FunctionCalled,fun] =>
    funNode := mkAtreeNode fun
    transferPropsToNode(fun,funNode)
    mms := CATCH('coerceOrCroaker, selectLocalMms(funNode,fun,argl,target))
    CONSP mms =>
      mms is [[['interpOnly,:.],:.]] => nil
      mm := CAAR mms
      mm is [., targ, :.] =>
        targ = target => true
        false
      false
    NIL
  NIL

canCoerceUnion(t1,t2) ==
  -- sees if one can coerce to or from a Union Domain
  -- assumes one of t1 and t2 is one

  -- get the domains in the union, checking for tagged unions
  if (isUnion1 := t1 is ['Union,:uds1]) then
    unionDoms1 :=
      uds1 and first uds1 is [":",:.] => [t for [.,.,t] in uds1]
      uds1
  if (isUnion2 := t2 is ['Union,:uds2]) then
    unionDoms2 :=
      uds2 and first uds2 is [":",:.] => [t for [.,.,t] in uds2]
      uds2

  isUnion2 =>
    member(t1,unionDoms2) => true
    isUnion1 =>
      and/[or/[canCoerce(ud1,ud2) for ud2 in unionDoms2]
        for ud1 in unionDoms1]
    or/[canCoerce(t1,ud) for ud in unionDoms2]
  -- next, a little lie
  t1 is ['Union,d1, ='"failed"] and t2 = d1 => true
  isUnion1 =>
    and/[canCoerce(ud,t2) for ud in unionDoms1]
  keyedSystemError("S2GE0016",['"canCoerceUnion",
     '"called with 2 non-Unions"])

canCoerceByMap(t1,t2) ==
  -- idea is this: if t1 is D U1 and t2 is D U2, then look for
  -- map: (U1 -> U2, D U1) -> D U2.  If it exists, then answer true
  -- if canCoerceFrom(t1,t2).
  u2 := deconstructT t2
  1 = #u2 => NIL
  u1 := deconstructT t1
  1 = #u1 => NIL                             -- no under domain
  CAR(u1) ^= CAR(u2) => NIL
  top := CAAR u1
  u1 := underDomainOf t1
  u2 := underDomainOf t2

  absolutelyCannotCoerce(u1,u2) => NIL

  -- save some time for those we know about
  know := '(List Vector Segment Stream UniversalSegment Array
    Polynomial UnivariatePolynomial SquareMatrix Matrix)
  top in know => canCoerce(u1,u2)

  null selectMms1('map,t2,[['Mapping,u2,u1],t1],
    [['Mapping,u2,u1],u1],NIL) => NIL
  -- don't bother checking for Undef, so avoid instantiation
  canCoerce(u1,u2)

canCoerceTower(t1,t2) ==
-- tries to find a coercion between top level t2 and somewhere inside t1
-- builds new bubbled type, for which coercion is called recursively
  canCoerceByMap(t1,t2) or newCanCoerceCommute(t1,t2) or
   canCoerceLocal(t1,t2) or canCoercePermute(t1,t2) or
    [c1,:arg1]:= deconstructT t1
    arg1 and
      TL:= NIL
      arg:= arg1
      until x or not arg repeat x:=
        t:= last arg
        [c,:arg]:= deconstructT t
        TL:= [c,arg,:TL]
        arg and coerceIntTest(t,t2) and
          CDDR TL =>
            s:= constructT(c1,replaceLast(arg1,bubbleConstructor TL))
            canCoerceLocal(t1,s) and
              [c2,:arg2]:= deconstructT last s
              s1:= bubbleConstructor [c2,arg2,c1,arg1]
              canCoerceCommute(s,s1) and canCoerceLocal(s1,t2)
          s:= bubbleConstructor [c,arg,c1,arg1]
          newCanCoerceCommute(t1,s) and canCoerceLocal(s,t2)
      x

canCoerceLocal(t1,t2) ==
  -- test for coercion on top level
  p:= ASSQ(CAR t1,$CoerceTable)
  p and ASSQ(CAR t2,CDR p) is [.,:[tag,fun]] =>
    tag='partial => NIL
    tag='total   => true
    (functionp(fun) and
       (v:=CATCH('coerceFailure,FUNCALL(fun,'_$fromCoerceable_$,t1,t2)))
         and v ^= $coerceFailure)  or  canCoerceByFunction(t1,t2)
  canCoerceByFunction(t1,t2)

canCoerceCommute(t1,t2) ==
-- THIS IS OUT-MODED AND WILL GO AWAY SOON  RSS 2-87
-- t1 is t2 with the two top level constructors commuted
-- looks for the existence of a commuting function
  CAR(t1) in (l := [$QuotientField, 'Gaussian]) and
    CAR(t2) in l => true
  p:= ASSQ(CAR t1,$CommuteTable)
  p and ASSQ(CAR t2,CDR p) is [.,:['commute,.]]

newCanCoerceCommute(t1,t2) ==
  coerceIntCommute(objNewWrap("$fromCoerceable$",t1),t2)

canCoercePermute(t1,t2) ==
  -- try to generate a sequence of transpositions that will convert
  -- t1 into t2
  t2 in '((Integer) (OutputForm)) => NIL
  towers := computeTTTranspositions(t1,t2)
  -- at this point, CAR towers = t1 and last towers should be similar
  -- to t2 in the sense that the components of t1 are in the same order
  -- as in t2. If length towers = 2 and t2 = last towers, we quit to
  -- avoid an infinte loop.
  NULL towers or NULL CDR towers => NIL
  NULL CDDR towers and t2 = CADR towers => NIL
  -- do the coercions successively, quitting if any fail
  ok := true
  for t in CDR towers while ok repeat
    ok := canCoerce(t1,t)
    if ok then t1 := t
  ok

canConvertByFunction(m1,m2) ==
  null $useConvertForCoercions => NIL
  canCoerceByFunction1(m1,m2,'convert)

canCoerceByFunction(m1,m2) == canCoerceByFunction1(m1,m2,'coerce)

canCoerceByFunction1(m1,m2,fun) ==
  -- calls selectMms with $Coerce=NIL and tests for required target=m2
  $declaredMode:local:= NIL
  $reportBottomUpFlag:local:= NIL
  -- have to handle cases where we might have changed from RN to QF I
  -- make 2 lists of expanded and unexpanded types
  l1 := REMDUP [m1,eqType m1]
  l2 := REMDUP [m2,eqType m2]
  ans  := NIL
  for t1 in l1 while not ans repeat
    for t2 in l2 while not ans repeat
      l := selectMms1(fun,t2,[t1],[t1],NIL)
      ans := [x for x in l | x is [sig,:.] and CADR sig=t2 and
       CADDR sig=t1 and
        CAR(sig) isnt ['TypeEquivalence,:.]] and true
  ans

absolutelyCanCoerceByCheating(t1,t2) ==
  -- this typically involves subdomains and towers where the only
  -- difference is a subdomain
  isEqualOrSubDomain(t1,t2) => true
  typeIsASmallInteger(t1) and t2 = $Integer => true
  ATOM(t1) or ATOM(t2) => false
  [tl1,:u1] := deconstructT t1
  [tl2,:u2] := deconstructT t2
  tl1 = '(Stream) and tl2 = '(InfiniteTuple) =>
    #u1 ^= #u2 => false
    "and"/[absolutelyCanCoerceByCheating(x1,x2) for x1 in u1 for x2 in u2]
  tl1 ^= tl2 => false
  #u1 ^= #u2 => false
  "and"/[absolutelyCanCoerceByCheating(x1,x2) for x1 in u1 for x2 in u2]

absolutelyCannotCoerce(t1,t2) ==
  -- response of true means "definitely cannot coerce"
  -- this is largely an efficiency hack
  ATOM(t1) or ATOM(t2) => NIL
  t2 = '(None) => true
  n1   := CAR t1
  n2   := CAR t2
  QFI  := [$QuotientField, $Integer]
  int2 := isEqualOrSubDomain(t2,$Integer)
  scalars := '(BigFloat NewFloat Float DoubleFloat RationalNumber)

  MEMQ(n1,scalars) and int2 => true
  (t1 = QFI) and int2       => true

  num2 := int2 or MEMQ(n2,scalars) or (t2 = QFI)
  isVar1 := MEMQ(n1,'(Variable Symbol))

  num2 and isVar1 => true
  num2 and MEMQ(n1,$univariateDomains) => true
  num2 and MEMQ(n1,$multivariateDomains) => true
  miscpols :=  '(Polynomial ElementaryFunction SimpleAlgebraicExtension)
  num2 and MEMQ(n1,miscpols) => true

  aggs :=  '(
    Matrix List Vector Stream Array RectangularMatrix FiniteSet
       )
  u1 := underDomainOf t1
  u2 := underDomainOf t2
  MEMQ(n1,aggs) and (u1 = t2) => true
  MEMQ(n2,aggs) and (u2 = t1) => true

  algs :=  '(
    SquareMatrix Gaussian RectangularMatrix Quaternion
       )
  nonpols := append(aggs,algs)
  num2 and MEMQ(n1,nonpols) => true
  isVar1 and MEMQ(n2,nonpols) and
    absolutelyCannotCoerce(t1,u2) => true

  (MEMQ(n1,scalars) or (t1 = QFI)) and (t2 = '(Polynomial (Integer))) =>
    true

  v2 := deconstructT t2
  1 = #v2 => NIL
  v1 := deconstructT t1
  1 = #v1 => NIL
  CAR(v1) ^= CAR(v2) => NIL
  absolutelyCannotCoerce(u1,u2)

typeIsASmallInteger x == (x = $SingleInteger)


--% Interpreter Coercion Functions

coerceInteractive(triple,t2) ==
  -- bind flag for recording/reporting instantiations
  -- (see recordInstantiation)
  t1 := objMode triple
  val := objVal triple
  null(t2) or t2 = $EmptyMode => NIL
  t2 = t1 => triple
  t2 = '$NoValueMode => objNew(val,t2)
  if t2 is ['SubDomain,x,.] then t2:= x
  -- JHD added category Aug 1996 for BasicMath
  t1 in '((Category) (Mode) (Domain) (SubDomain (Domain))) =>
    t2 = $OutputForm => objNew(val,t2)
    NIL
  t1 = '$NoValueMode =>
    if $compilingMap then clearDependentMaps($mapName,nil)
    throwKeyedMsg("S2IC0009",[t2,$mapName])
  $insideCoerceInteractive: local := true
  expr2 := EQUAL(t2,$OutputForm)
  if expr2 then startTimingProcess 'print
  else startTimingProcess 'coercion
  -- next 2 lines handle cases like '"failed"
  result :=
    expr2 and (t1 = val) => objNew(val,$OutputForm)
    expr2 and t1 is ['Variable,var] => objNewWrap(var,$OutputForm)
    coerceInt0(triple,t2)
  if expr2 then stopTimingProcess 'print
  else stopTimingProcess 'coercion
  result

coerceInt0(triple,t2) ==
  -- top level interactive coercion, which transfers all RN, RF and RR
  -- into equivalent types
  val := objVal triple
  t1  := objMode triple

  val='_$fromCoerceable_$ => canCoerceFrom(t1,t2)
  t1 = t2 => triple
  if t2 = $OutputForm then
    s1 := t1
    s2 := t2
  else
    s1 := equiType(t1)
    s2 := equiType(t2)
    s1 = s2 => return objNew(val,t2)
  -- t1 is ['Mapping,:.] and t2 ^= '(Any) => NIL
  -- note: may be able to coerce TO mapping
  -- treat Exit like Any
  -- handle case where we must generate code
  null(isWrapped val) and
    (t1 isnt ['FunctionCalled,:.] or not $genValue)=>
      intCodeGenCOERCE(triple,t2)
  t1 = $Any and t2 ^= $OutputForm and ([t1',:val'] := unwrap val) and
    (ans := coerceInt0(objNewWrap(val',t1'),t2)) => ans
  if not EQ(s1,t1) then triple := objNew(val,s1)
  x := coerceInt(triple,s2) =>
    EQ(s2,t2) => x
    objSetMode(x,t2)
    x
  NIL

coerceInt(triple, t2) ==
  val := coerceInt1(triple, t2) => val
  t1 := objMode triple
  t1 is ['Variable, :.] =>
    newMode := getMinimalVarMode(unwrap objVal triple, nil)
    newVal := coerceInt(triple, newMode)
    coerceInt(newVal, t2)
  nil

coerceInt1(triple,t2) ==
  -- general interactive coercion
  -- the result is a new triple with type m2 or NIL (= failed)
  $useCoerceOrCroak: local := true
  t2 = $EmptyMode => NIL
  t1 := objMode triple
  t1=t2 => triple
  val := objVal triple
  absolutelyCanCoerceByCheating(t1,t2) => objNew(val,t2)
  isSubDomain(t2, t1) => coerceSubDomain(val, t1, t2)

  if typeIsASmallInteger(t1) then
    (t2 = $Integer) or typeIsASmallInteger(t2) => return objNew(val,t2)
    sintp := SINTP val
    sintp and (t2 = $PositiveInteger) and val > 0 => return objNew(val,t2)
    sintp and (t2 = $NonNegativeInteger) and val >= 0 => return objNew(val,t2)

  typeIsASmallInteger(t2) and isEqualOrSubDomain(t1, $Integer) and INTP val =>
    SINTP val => objNew(val,t2)
    NIL

  t2 = $Void => objNew(voidValue(),$Void)
  t2 = $Any => objNewWrap([t1,:unwrap val],'(Any))

  t1 = $Any and t2 ^= $OutputForm and ([t1',:val'] := unwrap val) and
    (ans := coerceInt(objNewWrap(val',t1'),t2)) => ans

  -- next is for tagged union selectors for the time being
  t1 is ['Variable,=t2] or t2 is ['Variable,=t1] => objNew(val,t2)

  STRINGP t2 =>
    t1 is ['Variable,v] and (t2 = PNAME(v)) => objNewWrap(t2,t2)
    val' := unwrap val
    (t2 = val') and ((val' = t1) or (t1 = $String)) => objNew(val,t2)
    NIL
  --  t1 is ['Tuple,S] and t2 ^= '(OutputForm) =>
  t1 is ['Tuple,S]  =>
    coerceInt1(objNewWrap(asTupleAsList unwrap val, ['List, S]), t2)
  t1 is ['Union,:.] => coerceIntFromUnion(triple,t2)
  t2 is ['Union,:.] => coerceInt2Union(triple,t2)
  (STRINGP t1) and (t2 = $String) => objNew(val,$String)
  (STRINGP t1) and (t2 is ['Variable,v]) =>
    t1 = PNAME(v) => objNewWrap(v,t2)
    NIL
  (STRINGP t1) and (t1 = unwrap val) =>
    t2 = $OutputForm => objNew(t1,$OutputForm)
    NIL
  atom t1 => NIL

  if t1 = $AnonymousFunction and (t2 is ['Mapping,target,:margl]) then
    $useCoerceOrCroak := nil
    [.,vars,:body] := unwrap val
    vars :=
      atom vars => [vars]
      vars is ['Tuple,:.] => rest vars
      vars
    #margl ^= #vars => 'continue
    tree := mkAtree ['ADEF,vars,[target,:margl],[NIL for x in rest t2],:body]
    CATCH('coerceOrCroaker, bottomUp tree) = 'croaked => nil
    return getValue tree

  (t1 = $Symbol) and (t2 is ['Mapping,target,:margl]) =>
    null (mms := selectMms1(unwrap val,nil,margl,margl,target)) => NIL
    [dc,targ,:argl] := CAAR mms
    targ ^= target => NIL
    $genValue =>
      fun := getFunctionFromDomain(unwrap val,dc,argl)
      objNewWrap(fun,t2)
    val := NRTcompileEvalForm(unwrap val, CDR CAAR mms, evalDomain dc)
    objNew(val, t2)
  (t1 is ['Variable,sym]) and (t2 is ['Mapping,target,:margl]) =>
    null (mms := selectMms1(sym,target,margl,margl,NIL)) => 
       null (mms := selectMms1(sym,target,margl,margl,true)) => NIL
    [dc,targ,:argl] := CAAR mms
    targ ^= target => NIL
    dc is ["__FreeFunction__",:freeFun] => objNew( freeFun, t2 )
    $genValue => objNewWrap( getFunctionFromDomain(sym,dc,argl), t2 )
    val := NRTcompileEvalForm(sym, CDR CAAR mms, evalDomain dc)
    objNew(val, t2)
  (t1 is ['FunctionCalled,sym]) and (t2 is ['Mapping,target,:margl]) =>
    symNode := mkAtreeNode sym
    transferPropsToNode(sym,symNode)
    null (mms := selectLocalMms(symNode,sym,margl,target)) => NIL
    [dc,targ,:argl] := CAAR mms
    targ ^= target => NIL
    ml := [target,:margl]
    intName :=
      or/[mm for mm in mms | (mm is [[., :ml1],oldName,:.]
        and compareTypeLists(ml1,ml))] => [oldName]
      NIL
    null intName => NIL
    objNewWrap(intName,t2)
  (t1 is ['FunctionCalled,sym]) =>
    (t3 := get(sym,'mode,$e)) and t3 is ['Mapping,:.] =>
      (triple' := coerceInt(triple,t3)) => coerceInt(triple',t2)
      NIL
    NIL

  EQ(CAR(t1),'Variable) and PAIRP(t2) and
    (isEqualOrSubDomain(t2,$Integer) or
      (t2 = [$QuotientField, $Integer]) or MEMQ(CAR(t2),
        '(RationalNumber BigFloat NewFloat Float DoubleFloat))) => NIL

  ans := coerceRetract(triple,t2) or coerceIntTower(triple,t2) or
    [.,:arg]:= deconstructT t2
    arg and
      t:= coerceInt(triple,last arg)
      t and coerceByFunction(t,t2)
  ans or (isSubDomain(t1,$Integer) and
    coerceInt(objNew(val,$Integer),t2)) or
      coerceIntAlgebraicConstant(triple,t2) or
        coerceIntX(val,t1,t2)

coerceSubDomain(val, tSuper, tSub) ==
  -- Try to coerce from a sub domain to a super domain
  val = '_$fromCoerceable_$ => nil
  super := GETDATABASE(first tSub, 'SUPERDOMAIN)
  superDomain := first super
  superDomain = tSuper =>
    coerceImmediateSubDomain(val, tSuper, tSub, CADR super)
  coerceSubDomain(val, tSuper, superDomain) =>
    coerceImmediateSubDomain(val, superDomain, tSub, CADR super)
  nil

coerceImmediateSubDomain(val, tSuper, tSub, pred) ==
  predfn := getSubDomainPredicate(tSuper, tSub, pred)
  FUNCALL(predfn, val, nil) => objNew(val, tSub)
  nil

getSubDomainPredicate(tSuper, tSub, pred) ==
  $env: local := $InteractiveFrame
  predfn := HGET($superHash, CONS(tSuper, tSub)) => predfn
  name := GENSYM()
  decl := ['_:, name, ['Mapping, $Boolean, tSuper]]
  interpret(decl, nil)
  arg := GENSYM()
  pred' := SUBST(arg, "#1", pred)
  defn := ['DEF, [name, arg], '(NIL NIL), '(NIL NIL), removeZeroOne pred']
  interpret(defn, nil)
  op := mkAtree name
  transferPropsToNode(name, op)
  predfn := CADAR selectLocalMms(op, name, [tSuper],$Boolean)
  HPUT($superHash, CONS(tSuper, tSub), predfn)
  predfn

coerceIntX(val,t1, t2) ==
  -- some experimental things
  t1 = '(List (None)) =>
    -- this will almost always be an empty list
    null unwrap val =>
      -- try getting a better flavor of List
      null (t0 := underDomainOf(t2)) => NIL
      coerceInt(objNewWrap(val,['List,t0]),t2)
    NIL
  NIL

compareTypeLists(tl1,tl2) ==
  -- returns true if every type in tl1 is = or is a subdomain of
  -- the corresponding type in tl2
  for t1 in tl1 for t2 in tl2 repeat
    null isEqualOrSubDomain(t1,t2) => return NIL
  true

coerceIntAlgebraicConstant(object,t2) ==
  -- should use = from domain, but have to check on defaults code
  t1 := objMode object
  val := objValUnwrap object
  ofCategory(t1,'(Monoid)) and ofCategory(t2,'(Monoid)) and
    val = getConstantFromDomain('(One),t1) =>
      objNewWrap(getConstantFromDomain('(One),t2),t2)
  ofCategory(t1,'(AbelianMonoid)) and ofCategory(t2,'(AbelianMonoid)) and
    val = getConstantFromDomain('(Zero),t1) =>
      objNewWrap(getConstantFromDomain('(Zero),t2),t2)
  NIL

stripUnionTags doms ==
  [if dom is [":",.,dom'] then dom' else dom for dom in doms]

isTaggedUnion u ==
  u is ['Union,:tl] and tl and first tl is [":",.,.] and true

getUnionOrRecordTags u ==
  tags := nil
  if u is ['Union, :tl] or u is ['Record, :tl] then
      for t in tl repeat
         if t is [":",tag,.] then tags := cons(tag, tags)
  tags

coerceUnion2Branch(object) ==
  [.,:unionDoms] := objMode object
  doms := orderUnionEntries unionDoms
  predList:= mkPredList doms
  doms := stripUnionTags doms
  val' := objValUnwrap object
  predicate := NIL
  targetType:= NIL
  for typ in doms for pred in predList while ^targetType repeat
    evalSharpOne(pred,val') =>
      predicate := pred
      targetType := typ
  null targetType => keyedSystemError("S2IC0013",NIL)
  predicate is ['EQCAR,.,p] => objNewWrap(CDR val',targetType)
  objNew(objVal object,targetType)

coerceBranch2Union(object,union) ==
  -- assumes type is a member of unionDoms
  unionDoms := CDR union
  doms := orderUnionEntries unionDoms
  predList:= mkPredList doms
  doms := stripUnionTags doms
  p := position(objMode object,doms)
  p = -1 => keyedSystemError("S2IC0014",[objMode object,union])
  val := objVal object
  predList.p is ['EQCAR,.,tag] =>
    objNewWrap([removeQuote tag,:unwrap val],union)
  objNew(val,union)

coerceInt2Union(object,union) ==
  -- coerces to a Union type, adding numeric tags
  -- first cut
  unionDoms := stripUnionTags CDR union
  t1 := objMode object
  member(t1,unionDoms) => coerceBranch2Union(object,union)
  val := objVal object
  val' := unwrap val
  (t1 = $String) and member(val',unionDoms) =>
    coerceBranch2Union(objNew(val,val'),union)
  noCoerce := true
  val' := nil
  for d in unionDoms while noCoerce repeat
    (val' := coerceInt(object,d)) => noCoerce := nil
  val' => coerceBranch2Union(val',union)
  NIL

coerceIntFromUnion(object,t2) ==
  -- coerces from a Union type to something else
  coerceInt(coerceUnion2Branch object,t2)

coerceIntByMap(triple,t2) ==
  -- idea is this: if t1 is D U1 and t2 is D U2, then look for
  -- map: (U1 -> U2, D U1) -> D U2.  If it exists, then create a
  -- function to do the coercion on the element level and call the
  -- map function.
  t1 := objMode triple
  t2 = t1 => triple
  u2 := deconstructT t2    -- compute t2 first because of Expression
  1 = #u2 => NIL           -- no under domain
  u1 := deconstructT t1
  1 = #u1 => NIL
  CAAR u1 ^= CAAR u2 => nil  -- constructors not equal
  ^valueArgsEqual?(t1, t2) => NIL
--  CAR u1 ^= CAR u2 => NIL
  top := CAAR u1
  u1 := underDomainOf t1
  u2 := underDomainOf t2

  -- handle a couple of special cases for subdomains of Integer
  top in '(List Vector Segment Stream UniversalSegment Array)
    and isSubDomain(u1,u2) => objNew(objVal triple, t2)

  args := [['Mapping,u2,u1],t1]
  if $reportBottomUpFlag then
    sayFunctionSelection('map,args,t2,NIL,
      '"coercion facility (map)")
  mms := selectMms1('map,t2,args,args,NIL)
  if $reportBottomUpFlag then
    sayFunctionSelectionResult('map,args,mms)
  null mms => NIL

  [[dc,:sig],slot,.]:= CAR mms
  fun := compiledLookup('map,sig,evalDomain(dc))
  NULL fun => NIL
  [fn,:d]:= fun
  fn = function Undef => NIL
  -- now compile a function to do the coercion
  code := ['SPADCALL,['CONS,["function","coerceIntByMapInner"],MKQ [u1,:u2]],
    wrapped2Quote objVal triple,MKQ fun]
  -- and apply the function
  val := CATCH('coerceFailure,timedEvaluate code)
  (val = $coerceFailure) => NIL
  objNewWrap(val,t2)

coerceIntByMapInner(arg,[u1,:u2]) == coerceOrThrowFailure(arg,u1,u2)
-- [u1,:u2] gets passed as the "environment", which is why we have this
-- slightly clumsy locution  JHD 31.July,1990

valueArgsEqual?(t1, t2) ==
  -- returns true if the object-valued arguments to t1 and t2 are the same
  -- under coercion
  coSig := CDR GETDATABASE(CAR t1, 'COSIG)
  constrSig := CDR getConstructorSignature CAR t1
  tl1 := replaceSharps(constrSig, t1)
  tl2 := replaceSharps(constrSig, t2)
  not MEMQ(NIL, coSig) => true
  done := false
  value := true
  for a1 in CDR t1 for a2 in CDR t2 for cs in coSig
    for m1 in tl1 for m2 in tl2 while not done repeat
          ^cs =>
            trip := objNewWrap(a1, m1)
            newVal := coerceInt(trip, m2)
            null newVal => (done := true; value := false)
            ^algEqual(a2, objValUnwrap newVal, m2) =>
              (done := true; value := false)
  value

coerceIntTower(triple,t2) ==
  -- tries to find a coercion from top level t2 to somewhere inside t1
  -- builds new argument type, for which coercion is called recursively
  x := coerceIntByMap(triple,t2) => x
  x := coerceIntCommute(triple,t2) => x
  x := coerceIntPermute(triple,t2) => x
  x := coerceIntSpecial(triple,t2) => x
  x := coerceIntTableOrFunction(triple,t2) => x
  t1 := objMode triple
  [c1,:arg1]:= deconstructT t1
  arg1 and
    TL:= NIL
    arg:= arg1
    until x or not arg repeat
      t:= last arg
      [c,:arg]:= deconstructT t
      TL:= [c,arg,:TL]
      x := arg and coerceIntTest(t,t2) =>
        CDDR TL =>
          s := constructT(c1,replaceLast(arg1,bubbleConstructor TL))
          (null isValidType(s)) => (x := NIL)
          x := (coerceIntByMap(triple,s) or
            coerceIntTableOrFunction(triple,s)) =>
              [c2,:arg2]:= deconstructT last s
              s:= bubbleConstructor [c2,arg2,c1,arg1]
              (null isValidType(s)) => (x := NIL)
              x:= coerceIntCommute(x,s) =>
                x := (coerceIntByMap(x,t2) or
                  coerceIntTableOrFunction(x,t2))
        s:= bubbleConstructor [c,arg,c1,arg1]
        (null isValidType(s)) => (x := NIL)
        x:= coerceIntCommute(triple,s) =>
          x:= (coerceIntByMap(x,t2) or
            coerceIntTableOrFunction(x,t2))
    x

coerceIntSpecial(triple,t2) ==
  t1 := objMode triple
  t2 is ['SimpleAlgebraicExtension,R,U,.] and t1 = R =>
    null (x := coerceInt(triple,U)) => NIL
    coerceInt(x,t2)
  NIL

coerceIntTableOrFunction(triple,t2) ==
  -- this function does the actual coercion to t2, but not to an
  -- argument type of t2
  null isValidType t2 => NIL  -- added 9-18-85 by RSS
  null isLegitimateMode(t2,NIL,NIL) => NIL  -- added 6-28-87 by RSS
  t1 := objMode triple
  p:= ASSQ(CAR t1,$CoerceTable)
  p and ASSQ(CAR t2,CDR p) is [.,:[tag,fun]] =>
    val := objVal triple
    fun='Identity => objNew(val,t2)
    tag='total =>
      coerceByTable(fun,val,t1,t2,'T) or coerceByFunction(triple,t2)
    coerceByTable(fun,val,t1,t2,NIL) or coerceByFunction(triple,t2)
  coerceByFunction(triple,t2)

coerceCommuteTest(t1,t2) ==
  null isLegitimateMode(t2,NIL,NIL) => NIL

  -- sees whether t1 = D1 D2 R and t2 = D2 D1 S
  null (u1 := underDomainOf t1) => NIL
  null (u2 := underDomainOf t2) => NIL

  -- must have underdomains (ie, R and S must be there)

  null (v1 := underDomainOf u1) => NIL
  null (v2 := underDomainOf u2) => NIL

  -- now check that cross of constructors is correct
  (CAR(deconstructT t1) = CAR(deconstructT u2)) and
    (CAR(deconstructT t2) = CAR(deconstructT u1))

coerceIntCommute(obj,target) ==
  -- note that the value in obj may be $fromCoerceable$, for canCoerce
  source := objMode obj
  null coerceCommuteTest(source,target) => NIL
  S := underDomainOf source
  T := underDomainOf target
  source = T => NIL      -- handle in other ways

  source is [D,:.] =>
    fun := GETL(D,'coerceCommute) or
           INTERN STRCONC('"commute",STRINGIMAGE D)
    functionp fun =>
      PUT(D,'coerceCommute,fun)
      u := objValUnwrap obj
      c := CATCH('coerceFailure,FUNCALL(fun,u,source,S,target,T))
      (c = $coerceFailure) => NIL
      u = "$fromCoerceable$" => c
      objNewWrap(c,target)
    NIL
  NIL

coerceIntPermute(object,t2) ==
  t2 in '((Integer) (OutputForm)) => NIL
  t1 := objMode object
  towers := computeTTTranspositions(t1,t2)
  -- at this point, CAR towers = t1 and last towers should be similar
  -- to t2 in the sense that the components of t1 are in the same order
  -- as in t2. If length towers = 2 and t2 = last towers, we quit to
  -- avoid an infinte loop.
  NULL towers or NULL CDR towers => NIL
  NULL CDDR towers and t2 = CADR towers => NIL
  -- do the coercions successively, quitting if any fail
  ok := true
  for t in CDR towers while ok repeat
    null (object := coerceInt(object,t)) => ok := NIL
  ok => object
  NIL

computeTTTranspositions(t1,t2) ==
  -- decompose t1 into its tower parts
  tl1 := decomposeTypeIntoTower t1
  tl2 := decomposeTypeIntoTower t2
  -- if not at least 2 parts, don't bother working here
  null (rest tl1 and rest tl2) => NIL
  -- determine the relative order of the parts of t1 in t2
  p2 := [position(d1,tl2) for d1 in tl1]
  member(-1,p2) => NIL            -- something not present
  -- if they are all ascending, this function will do nothing
  p2' := MSORT p2
  p2 = p2' => NIL
  -- if anything is repeated twice, leave
  p2' ^= MSORT REMDUP p2' => NIL
  -- create a list of permutations that transform the tower parts
  -- of t1 into the order they are in in t2
  n1 := #tl1
  p2 := LIST2VEC compress(p2,0,# REMDUP tl1) where
    compress(l,start,len) ==
      start >= len => l
      member(start,l) => compress(l,start+1,len)
      compress([(i < start => i; i - 1) for i in l],start,len)
  -- p2 now has the same position numbers as p1, we need to determine
  -- a list of permutations that takes p1 into p2.
  -- them
  perms := permuteToOrder(p2,n1-1,0)
  towers := [tl1]
  tower := LIST2VEC tl1
  for perm in perms repeat
    t := tower.(CAR perm)
    tower.(CAR perm) := tower.(CDR perm)
    tower.(CDR perm) := t
    towers := CONS(VEC2LIST tower,towers)
  towers := [reassembleTowerIntoType tower for tower in towers]
  if CAR(towers) ^= t2 then towers := cons(t2,towers)
  NREVERSE towers

decomposeTypeIntoTower t ==
  ATOM t => [t]
  d := deconstructT t
  NULL rest d => [t]
  rd := REVERSE t
  [reverse QCDR rd,:decomposeTypeIntoTower QCAR rd]

reassembleTowerIntoType tower ==
  ATOM tower => tower
  NULL rest tower => CAR tower
  [:top,t,s] := tower
  reassembleTowerIntoType [:top,[:t,s]]

permuteToOrder(p,n,start) ==
  -- p is a vector of the numbers 0..n. This function returns a list
  -- of swaps of adjacent elements so that p will be in order. We only
  -- begin looking at index start
  r := n - start
  r <= 0 => NIL
  r = 1 =>
    p.r < p.(r+1) => NIL
    [[r,:(r+1)]]
  p.start = start => permuteToOrder(p,n,start+1)
  -- bubble up element start to the top. Find out where it is
  stpos := NIL
  for i in start+1..n while not stpos repeat
    if p.i = start then stpos := i
  perms := NIL
  while stpos ^= start repeat
    x := stpos - 1
    perms := [[x,:stpos],:perms]
    t := p.stpos
    p.stpos := p.x
    p.x := t
    stpos := x
  APPEND(NREVERSE perms,permuteToOrder(p,n,start+1))

coerceIntTest(t1,t2) ==
  -- looks whether there exists a table entry or a coercion function
  -- thus the type can be bubbled before coerceIntTableOrFunction is called
  t1=t2 or
    b:=
      p:= ASSQ(CAR t1,$CoerceTable)
      p and ASSQ(CAR t2,CDR p)
    b or coerceConvertMmSelection('coerce,t1,t2) or
      ($useConvertForCoercions and
        coerceConvertMmSelection('convert,t1,t2))

coerceByTable(fn,x,t1,t2,isTotalCoerce) ==
  -- catch point for 'failure in boot coercions
  t2 = $OutputForm and ^(newType? t1) => NIL
  isWrapped x =>
    x:= unwrap x
    c:= CATCH('coerceFailure,FUNCALL(fn,x,t1,t2))
    c=$coerceFailure => NIL
    objNewWrap(c,t2)
  isTotalCoerce => objNew([fn,x,MKQ t1,MKQ t2],t2)
  objNew(['catchCoerceFailure,MKQ fn,x,MKQ t1,MKQ t2],t2)

catchCoerceFailure(fn,x,t1,t2) ==
  -- compiles a catchpoint for compiling boot coercions
  c:= CATCH('coerceFailure,FUNCALL(fn,x,t1,t2))
  c = $coerceFailure =>
    throwKeyedMsgCannotCoerceWithValue(wrap unwrap x,t1,t2)
  c

coercionFailure() ==
  -- does the throw on coercion failure
  THROW('coerceFailure,$coerceFailure)

coerceByFunction(T,m2) ==
  -- using the new modemap selection without coercions
  -- should not be called by canCoerceFrom
  x := objVal T
  x = '_$fromCoerceable_$ => NIL
  m2 is ['Union,:.] => NIL
  m1 := objMode T
  m2 is ['Boolean,:.] and m1 is ['Equation,ud] =>
    dcVector := evalDomain ud
    fun :=
      isWrapped x =>
        NRTcompiledLookup("=", [$Boolean, '$, '$], dcVector)
      NRTcompileEvalForm("=", [$Boolean, '$, '$], dcVector)
    [fn,:d]:= fun
    isWrapped x =>
      x:= unwrap x
      mkObjWrap(SPADCALL(CAR x,CDR x,fun),m2)
    x isnt ['SPADCALL,a,b,:.] => keyedSystemError("S2IC0015",NIL)
    code := ['SPADCALL, a, b, fun]
    objNew(code,$Boolean)
  -- If more than one function is found, any should suffice, I think -scm
  if not (mm := coerceConvertMmSelection(funName := 'coerce,m1,m2)) then
    mm := coerceConvertMmSelection(funName := 'convert,m1,m2)
  mm =>
    [[dc,tar,:args],slot,.]:= mm
    dcVector := evalDomain(dc)
    fun:=
--+
      isWrapped x =>
        NRTcompiledLookup(funName,slot,dcVector)
      NRTcompileEvalForm(funName,slot,dcVector)
    [fn,:d]:= fun
    fn = function Undef => NIL
    isWrapped x =>
--+
      $: fluid := dcVector
      val := CATCH('coerceFailure, SPADCALL(unwrap x,fun))
      (val = $coerceFailure) => NIL
      objNewWrap(val,m2)
    env := fun
    code := ['failCheck, ['SPADCALL, x, env]]
--  tar is ['Union,:.] => objNew(['failCheck,code],m2)
    objNew(code,m2)
  -- try going back to types like RN instead of QF I
  m1' := eqType m1
  m2' := eqType m2
  (m1 ^= m1') or (m2 ^= m2') => coerceByFunction(objNew(x,m1'),m2')
  NIL

hasCorrectTarget(m,sig is [dc,tar,:.]) ==
  -- tests whether the target of signature sig is either m or a union
  -- containing m. It also discards TEQ as it is not meant to be
  -- used at top-level
  dc is ['TypeEquivalence,:.] => NIL
  m=tar => 'T
  tar is ['Union,t,'failed] => t=m
  tar is ['Union,'failed,t] and t=m

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