Initial population.

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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/interp nrungo.boot}
+\author{The Axiom Team}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\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>>
+
+--=======================================================
+--             Lookup From Interpreter
+--=======================================================
+
+NRTevalDomain form ==
+  form is ['SETELT,:.] => eval form
+  evalDomain form
+
+--------------------> NEW DEFINITION (see interop.boot.pamphlet)
+compiledLookup(op,sig,dollar) ==
+--called by coerceByFunction, evalForm, findEqualFun, findUniqueOpInDomain,
+--  getFunctionFromDomain, optDeltaEntry, retractByFunction
+  if not VECP dollar then dollar := NRTevalDomain dollar
+  basicLookup(op,sig,dollar,dollar)
+
+--------------------> NEW DEFINITION (see interop.boot.pamphlet)
+basicLookup(op,sig,domain,dollar) ==
+  domain.1 is ['lookupInDomain,:.] => lookupInDomainVector(op,sig,domain,dollar)
+  ----------new world code follows------------
+  $lookupDefaults : local := nil -- new world
+  u := lookupInDomainVector(op,sig,domain,dollar) => u
+  $lookupDefaults := true
+  lookupInDomainVector(op,sig,domain,dollar)
+
+compiledLookupCheck(op,sig,dollar) ==
+  fn := compiledLookup(op,sig,dollar)
+
+  -- NEW COMPILER COMPATIBILITY ON
+
+  if      (fn = nil)  and (op = "^") then
+    fn := compiledLookup("**",sig,dollar)
+  else if (fn = nil)  and (op = "**") then
+    fn := compiledLookup("^",sig,dollar)
+
+  -- NEW COMPILER COMPATIBILITY OFF
+
+  fn = nil =>
+    keyedSystemError("S2NR0001",[op,formatSignature sig,dollar.0])
+  fn
+
+--=======================================================
+--                 Lookup From Compiled Code
+--=======================================================
+goGet(:l) ==
+  [:arglist,env] := l
+  arglist is ['goGet,:.] => stop()
+  [[.,[op,initSig,:code]],thisDomain] := env
+  domainSlot := QSQUOTIENT(code,8192)
+  code1 := QSREMAINDER(code,8192)
+  if QSODDP code1 then isConstant := true
+  code2 := QSQUOTIENT(code1,2)
+  if QSODDP code2 then explicitLookupDomainIfTrue := true
+  index := QSQUOTIENT(code2,2)
+  kind := (isConstant = true => 'CONST; 'ELT)
+  sig := [NRTreplaceLocalTypes(s,thisDomain) for s in initSig]
+  sig := substDomainArgs(thisDomain,sig)
+  lookupDomain :=
+     domainSlot = 0 => thisDomain
+     thisDomain.domainSlot -- where we look for the operation
+  if PAIRP lookupDomain then lookupDomain := NRTevalDomain lookupDomain
+  dollar :=                             -- what matches $ in signatures
+    explicitLookupDomainIfTrue => lookupDomain
+    thisDomain
+  if PAIRP dollar then dollar := NRTevalDomain dollar
+  fn:= basicLookup(op,sig,lookupDomain,dollar)
+  fn = nil => keyedSystemError("S2NR0001",[op,sig,lookupDomain.0])
+  val:= APPLY(first fn,[:arglist,rest fn])
+  SETELT(thisDomain,index,fn)
+  val
+
+NRTreplaceLocalTypes(t,dom) ==
+   atom t =>
+     not INTEGERP t => t
+     t:= dom.t
+     if PAIRP t then t:= NRTevalDomain t
+     t.0
+   MEMQ(CAR t,'(Mapping Union Record _:)) =>
+      [CAR t,:[NRTreplaceLocalTypes(x,dom) for x in rest t]]
+   t
+
+substDomainArgs(domain,object) ==
+    form := devaluate domain
+    SUBLISLIS([form,:rest form],["$$",:$FormalMapVariableList],object)
+
+--=======================================================
+--       Lookup Function in Slot 1 (via SPADCALL)
+--=======================================================
+domainTableLookup(op,sig,dollar,env) == lookupInTable(op,sig,dollar,env)
+lookupInTable(op,sig,dollar,[domain,table]) ==
+  EQ(table,'derived) => lookupInAddChain(op,sig,domain,dollar)
+  success := false
+  someMatch := false
+  while not success for [sig1,:code] in LASSQ(op,table) repeat
+    success :=
+      null compareSig(sig,sig1,dollar.0,domain) => false
+      code is ['subsumed,a] =>
+            subsumptionSig :=
+               EQSUBSTLIST(rest(domain.0),$FormalMapVariableList,a)
+            someMatch:=true
+            false
+      predIndex := QSQUOTIENT(code,8192)
+      predIndex ^= 0 and null lookupPred($predVector.predIndex,dollar,domain)
+        => false
+      loc := QSQUOTIENT(QSREMAINDER(code,8192),2)
+      loc = 0 =>
+        someMatch := true
+        nil
+      slot := domain.loc
+      EQCAR(slot,'goGet) =>
+        lookupDisplay(op,sig,domain,'" !! goGet found, will ignore")
+        lookupInAddChain(op,sig,domain,dollar) or 'failed
+      NULL slot =>
+        lookupDisplay(op,sig,domain,'" !! null slot entry, continuing")
+        lookupInAddChain(op,sig,domain,dollar) or 'failed
+      lookupDisplay(op,sig,domain,'" !! found in NEW table!!")
+      slot
+  NE(success,'failed) and success => success
+  subsumptionSig and (u:= SPADCALL(op,subsumptionSig,dollar,domain.1)) => u
+  someMatch => lookupInAddChain(op,sig,domain,dollar)
+  nil
+
+--=======================================================
+--       Lookup Addlist (from lookupInDomainTable or lookupInDomain)
+--=======================================================
+lookupInAddChain(op,sig,addFormDomain,dollar) ==
+  addFunction:=lookupInDomain(op,sig,addFormDomain,dollar,5)
+  defaultingFunction addFunction =>
+     lookupInCategories(op,sig,addFormDomain,dollar) or addFunction
+  addFunction or lookupInCategories(op,sig,addFormDomain,dollar)
+
+
+defaultingFunction op ==
+  not(op is [.,:dom]) => false
+  not VECP dom => false
+  not (#dom > 0) => false
+  not (dom.0 is [packageName,:.]) => false
+  not IDENTP packageName => false
+  pname := PNAME packageName
+  pname.(MAXINDEX pname) = char "&"
+
+--=======================================================
+--   Lookup In Domain (from lookupInAddChain)
+--=======================================================
+lookupInDomain(op,sig,addFormDomain,dollar,index) ==
+  addFormCell := addFormDomain.index =>
+    INTEGERP KAR addFormCell =>
+      or/[lookupInDomain(op,sig,addFormDomain,dollar,i) for i in addFormCell]
+    if null VECP addFormCell then addFormCell := eval addFormCell
+    lookupInDomainVector(op,sig,addFormCell,dollar)
+  nil
+
+--------------------> NEW DEFINITION (see interop.boot.pamphlet)
+lookupInDomainVector(op,sig,domain,dollar) ==
+  slot1 := domain.1
+  SPADCALL(op,sig,dollar,slot1)
+
+--=======================================================
+--       Category Default Lookup (from goGet or lookupInAddChain)
+--=======================================================
+lookupInCategories(op,sig,dom,dollar) ==
+  catformList := dom.4.0
+  varList := ['$,:$FormalMapVariableList]
+  valueList := [dom,:[dom.(5+i) for i in 1..(# rest dom.0)]]
+  valueList := [MKQ val for val in valueList]
+  nsig := MSUBST(dom.0,dollar.0,sig)
+  r := or/[lookupInDomainVector(op,nsig,
+                   eval EQSUBSTLIST(valueList,varList,catform),dollar)
+        for catform in catformList | pred] where pred ==
+   (table := HGET($Slot1DataBase,first catform)) and
+      (u := LASSQ(op,table))  --compare without checking predicates
+         and (v := or/[rest x for x in u | #sig = #x.0])
+        -- following lines commented out because compareSig needs domain
+        -- and (v := or/[rest x for x in u |
+        --             compareSig(sig,x.0,dollar.0, catform)])
+  r or lookupDisplay(op,sig,'"category defaults",'"-- not found")
+
+--=======================================================
+--                       Predicates
+--=======================================================
+lookupPred(pred,dollar,domain) ==
+  pred = true => true
+  pred = 'asserted => false
+  pred is ['AND,:pl] or pred is ['and,:pl] =>
+    and/[lookupPred(p,dollar,domain) for p in pl]
+  pred is ['OR,:pl] or pred is ['or,:pl] =>
+    or/[lookupPred(p,dollar,domain) for p in pl]
+  pred is ['NOT,p] or pred is ['not,p] => not lookupPred(p,dollar,domain)
+  pred is ['is,dom1,dom2] => domainEqual(dom1,dom2)
+  pred is ['has,a,b] =>
+    VECP a =>
+      keyedSystemError("S2GE0016",['"lookupPred",
+        '"vector as  first argument to has"])
+    a := eval mkEvalable substDollarArgs(dollar,domain,a)
+    b := substDollarArgs(dollar,domain,b)
+    HasCategory(a,b)
+  keyedSystemError("S2NR0002",[pred])
+
+substDollarArgs(dollar,domain,object) ==
+    form := devaluate domain
+    SUBLISLIS([devaluate dollar,:rest form],
+                ["$",:$FormalMapVariableList],object)
+
+compareSig(sig,tableSig,dollar,domain) ==
+  not (#sig = #tableSig) => false
+  null (target := first sig)
+   or lazyCompareSigEqual(target,first tableSig,dollar,domain) =>
+     and/[lazyCompareSigEqual(s,t,dollar,domain)
+              for s in rest sig for t in rest tableSig]
+
+--------------------> NEW DEFINITION (override in xrun.boot.pamphlet)
+lazyCompareSigEqual(s,tslot,dollar,domain) ==
+  tslot = '$ => s = devaluate dollar  --needed for browser
+  INTEGERP tslot and PAIRP(lazyt:=domain.tslot) and PAIRP s =>
+      lazyt is [.,.,.,[.,item,.]] and
+        item is [.,[functorName,:.]] and functorName = CAR s =>
+          compareSigEqual(s,(NRTevalDomain lazyt).0,dollar,domain)
+      nil
+  compareSigEqual(s,NRTreplaceLocalTypes(tslot,domain),dollar,domain)
+
+
+compareSigEqual(s,t,dollar,domain) ==
+  EQUAL(s,t) => true
+  ATOM t =>
+    u :=
+      EQ(t,'$) => dollar
+      isSharpVar t =>
+        VECP domain => ELT(rest domain.0,POSN1(t,$FormalMapVariableList))
+        ELT(rest domain,POSN1(t,$FormalMapVariableList))
+      STRINGP t and IDENTP s => (s := PNAME s; t)
+      nil
+    s = '$ => compareSigEqual(dollar,u,dollar,domain)
+    u => compareSigEqual(s,u,dollar,domain)
+    EQUAL(s,u)
+  EQ(s,'$) => compareSigEqual(dollar,t,dollar,domain)
+  ATOM s => nil
+  #s ^= #t => nil
+  match := true
+  for u in s for v in t repeat
+    not compareSigEqual(u,v,dollar,domain) => return(match:=false)
+  match
+
+-----------------------Compiler for Interpreter---------------------------------
+NRTcompileEvalForm(opName,sigTail,dcVector) ==
+  u := NRTcompiledLookup(opName,sigTail,dcVector)
+  not ($insideCompileBodyIfTrue = true) => MKQ u
+  k := NRTgetMinivectorIndex(u,opName,sigTail,dcVector)
+  ['ELT,"$$$",k]  --$$$ denotes minivector
+
+--------------------> NEW DEFINITION (see interop.boot.pamphlet)
+NRTcompiledLookup(op,sig,dom) ==
+  if CONTAINED('_#,sig) then
+      sig := [NRTtypeHack t for t in sig]
+  compiledLookupCheck(op,sig,dom)
+
+NRTtypeHack t ==
+  ATOM t => t
+  CAR t = '_# => # CADR t
+  [CAR t,:[NRTtypeHack tt for tt in CDR t]]
+
+NRTgetMinivectorIndex(u,op,sig,domVector) ==
+  s := # $minivector
+  k := or/[k for k in 0..(s-1)
+        for x in $minivector | EQ(x,u)] => k
+  $minivector := [:$minivector,u]
+  if $compilingInputFile then
+    $minivectorCode := [:$minivectorCode,[op,sig,devaluate domVector]]
+--  pp '"-- minivectorCode -->"
+--  pp $minivectorCode
+  s
+
+NRTisRecurrenceRelation(op,body,minivectorName) ==
+  -- returns [body p1 p2 ... pk] for a k-term recurrence relation
+  -- where the n-th term is computed using the (n-1)st,...,(n-k)th
+  -- whose values are initially computed using the expressions
+  -- p1,...,pk respectively; body has #2,#3,... in place of
+  -- f(k-1),f(k-2),...
+
+  body isnt ['COND,:pcl] => false
+  -- body should have a conditional expression which
+  -- gives k boundary values, one general term plus possibly an
+  -- "out of domain" condition
+--pcl is [:.,[ ''T,:mess]] and not (CONTAINED('throwMessage,mess) or
+--  CONTAINED('throwKeyedMsg,mess)) => NIL
+  pcl := [x for x in pcl | not (x is [''T,:mess] and
+    (CONTAINED('throwMessage,mess) or
+      CONTAINED('throwKeyedMsg,mess)))]
+  integer := EVALFUN $Integer
+  iequalSlot:=compiledLookupCheck("=",'((Boolean) $ $),integer)
+  lesspSlot:=compiledLookupCheck("<",'((Boolean) $ $),integer)
+  bf := '(Boolean)
+  notpSlot:= compiledLookupCheck("not",'((Boolean)(Boolean)),EVALFUN bf)
+  for [p,c] in pcl repeat
+    p is ['SPADCALL,sharpVar,n1,['ELT,=minivectorName,slot]]
+      and EQ(iequalSlot,$minivector.slot) =>
+        initList:= [[n1,:c],:initList]
+        sharpList := insert(sharpVar,sharpList)
+        n:=n1
+    miscList:= [[p,c],:miscList]
+  miscList isnt [[generalPred,generalTerm]] or sharpList isnt [sharpArg] =>
+      return false
+    --first general term starts at n
+
+  --Must have at least one special value; insist that they be consecutive
+  null initList => false
+  specialValues:= MSORT ASSOCLEFT initList
+  or/[null INTEGERP n for n in specialValues] => false
+  minIndex:= "MIN"/specialValues
+  not (and/[i=x for i in minIndex..(minIndex+n-1) for x in specialValues]) =>
+    sayKeyedMsg("S2IX0005",
+      ["append"/[['" ",sv]  for sv in specialValues]])
+    return nil
+
+  --Determine the order k of the recurrence and index n of first general term
+  k:= #specialValues
+  n:= k+minIndex
+  --Check general predicate
+  predOk :=
+    generalPred is '(QUOTE T) => true
+    generalPred is ['SPADCALL,m,=sharpArg,['ELT,=minivectorName,slot]]
+      and EQ(lesspSlot,$minivector.slot)=> m+1
+    generalPred is ['SPADCALL,['SPADCALL,=sharpArg,m,
+      ['ELT,=minivectorName,slot]], ['ELT,=minivectorName,notSlot]]
+        and EQ(lesspSlot,$minivector.slot)
+          and EQ(notpSlot,$minivector.notSlot) => m
+    generalPred is ['NOT,['SPADCALL,=sharpArg,m,['ELT,=minivectorName, =lesspSlot]]]
+      and EQ(lesspSlot,$minivector.slot) => m
+    return nil
+  INTEGERP predOk and predOk ^= n =>
+    sayKeyedMsg("S2IX0006",[n,m])
+    return nil
+
+  --Check general term for references to just the k previous values
+  diffCell:=compiledLookupCheck("-",'($ $ $),integer)
+  diffSlot := or/[i for i in 0.. for x in $minivector | EQ(x,diffCell)]
+                or return nil
+  --Check general term for references to just the k previous values
+  sharpPosition := PARSE_-INTEGER SUBSTRING(sharpArg,1,nil)
+  al:= mkDiffAssoc(op,generalTerm,k,sharpPosition,sharpArg,diffSlot,minivectorName)
+  null al => false
+  '$failed in al => false
+  body:= generalTerm
+  for [a,:b] in al repeat
+    body:= substitute(b,a,body)
+  result:= [body,sharpArg,n-1,:NREVERSE [LASSOC(i,initList) or
+      systemErrorHere('"NRTisRecurrenceRelation")
+        for i in minIndex..(n-1)]]
+
+mkDiffAssoc(op,body,k,sharpPosition,sharpArg,diffSlot,vecname) ==
+  -- returns alist which should not have any entries = $failed
+  -- form substitution list of the form:
+  -- ( ((f (,DIFFERENCE #1 1)) . #2) ((f (,DIFFERENCE #1 2)) . #3) ...)
+  --   but also checking that all difference values lie in 1..k
+  atom body => nil
+  body is ['COND,:pl] =>
+    "union"/[mkDiffAssoc(op,c,k,sharpPosition,sharpArg,diffSlot,vecname) for [p,c] in pl]
+  body is [fn,:argl] =>
+    (fn = op) and argl.(sharpPosition-1) is
+      ['SPADCALL,=sharpArg,n,['ELT,=vecname,=diffSlot]] =>
+          NUMP n and n > 0 and n <= k =>
+            [[body,:$TriangleVariableList.n]]
+          ['$failed]
+    "union"/[mkDiffAssoc(op,x,k,sharpPosition,sharpArg,diffSlot,vecname) for x in argl]
+  systemErrorHere '"mkDiffAssoc"
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}