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(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     24710,        419]*)
(*NotebookOutlinePosition[     25799,        456]*)
(*  CellTagsIndexPosition[     25755,        452]*)
(*WindowFrame->Normal*)



Notebook[{


Cell[CellGroupData[{Cell[TextData[
"(*\n *\t\tGlobal controls.\n *)\n\nshowproof\t\t=\tFalse;\nshowreason\t\t=\t\
True;\nshowHeuristics\t=\tTrue;\n\n(*\n *\t\tWe represent the simple \
symmetric sum, [a,b,c],\n *\t\tby the function SS[a,b,c].\n *\t\tThis \
function can have any number of arguments\n *\t\t(but at least 2).\n *\t\tWe \
always print this function as \"[a,b,c]\".\n *)\n\n\
Format[SS[x__]]:=SequenceForm[\"[\",Infix[{x},\",\"],\"]\"];\n\n(*\n *\t\tFix \
the builtin \"Variables\" function to work properly\n *\t\twith inequalities.\
\n *)\n\nUnprotect[Variables];\nVariables[x_> \
y_]:=Union[Variables[x],Variables[y]];\n\
Variables[x_>=y_]:=Union[Variables[x],Variables[y]];\nVariables[x_< \
y_]:=Union[Variables[x],Variables[y]];\n\
Variables[x_<=y_]:=Union[Variables[x],Variables[y]];\n\
Variables[x_==y_]:=Union[Variables[x],Variables[y]];\nProtect[Variables];\n\n\
(*\n *\tCreate a function that finds the apparent sign of a term.\n *)\n\n\
apparentSign[x_?NumberQ] := If[x<0,-1,1,1];\napparentSign[x_?AtomQ] := 1;\n\
apparentSign[x_+y_] :=\n\tIf[apparentSign[x]===apparentSign[y],\n\t\t\
apparentSign[x],\n\t\tunknownSign\n\t  ];\napparentSign[x_ * y_] :=\n\t\
If[IntegerQ[apparentSign[x]]\n\t &&IntegerQ[apparentSign[y]],\n\t\t\
apparentSign[x]*apparentSign[y],\n\t\tunknownSign\n\t  ];\n\
apparentSign[x_^n_Integer] :=\n\tIf[IntegerQ[apparentSign[x]],\n\t\t\
apparentSign[x]^n,\n\t\tunknownSign\n\t  ];\napparentSign[Sqrt[x_]] := 1;\n\n\
(*\n *\tMove a term to the lhs or rhs depending on its sign.\n *)\n\n\
Sep[term_]:=\nBlock[{numf},\n\tnumf = numfactor[term];\n\tIf[!NumberQ[numf],\n\
\t\tnumf = Factor[numf];\n\t\tnumf = First[numf];\n\t  ];\n\t\n\t\
If[!NumberQ[numf],\n\t\tIf[apparentSign[term]==1, numf=1];\n\t\t\
If[apparentSign[term]==-1, numf=-1];\n\t  ];\n\n\tIf[numf<0,\n\t\tright -= \
term,\n\t\tleft += term,\n\t\tleft += term\n\t\t];\n\t];\n\n\
checksymm[expr_]:=\nBlock[ {vars},\n vars = Variables[expr];\n \
Return[symm[expr,vars,checkit,expr]];\n\t];\n\ncheckit[expr1_,expr2_] :=\n\
Block[{temp},\n\tIf[Together[expr1-expr2]==0,Return[True]];\n\tReturn[False];\
\n\t];\n\n(*\n *\tMove all positive terms to the left\n *\tand negative terms \
to the right.\n *)\n\nsignseparate[]:=\n(* Implicit arguments: left, right, \
op. *)\nBlock[{expr},\n expr = Expand[Together[left-right]];\n\n left = expr;\
\n right = 0;\n\n If[NumberQ[expr] && expr<0,\n\tright =\t-left;\n\tleft =\t\
0;\n\tReturn[];\n   ];\n\n If[AtomQ[expr],Return[]];\n\n If[Head[expr] =!= \
Plus,\n  \t\tIf[apparentSign[expr]==(-1),\n\t\t\tright =\t-left;\n\t\t\tleft \
=\t0;\n  \t\t  ];\n  \t\tReturn[];\n    ];\n\n (*\n \tThe expression is a sum \
of terms.\n    Scan these terms and move the\n    positive ones to the left\n \
   and the negative ones to the right.\n *)\n\n left =  0;\n right = 0;\n \
Map[Sep,expr];\n Return[];\n \t];\n\n(*\n *\tCollect together terms of the \
same sign.\n *)\n\ncollect[]:=\n (* Implicit arguments: left, right, op. *)\n \
Block[ {oldleft,oldright},\n\n oldleft =\tleft;\n oldright =\tright;\n\n \
signseparate[];\n\n If[showproof &&\n \t(Together[left-oldleft]=!=0 ||\n \t \
Together[right-oldright]=!=0),\n    \tPrint[\"Expanding and collecting terms \
of the same sign gives:\"];\n\t\tPrint[op[left,right]];\n    ];\n\n];\n\n(*\n \
*\tBreak an inequality into 3 parts:\n *\tthe lhs, the operator, and the rhs.\
\n *\tStore in \"left\", \"op\", and \"right\" respectively.\n *)\n\n\
getop[inequal_]:=\nBlock[{op},\n op = Head[inequal];\n\n left =  \
Part[inequal,1];\n right = Part[inequal,2];\n\n If[op =!= Less  &&\n    op \
=!= LessEqual &&\n    op =!= Greater  &&\n    op =!= GreaterEqual &&\n    op \
=!= Equal,\n\t\tPrint[\"Expression, \",inequal,\" is not an inequality.\"];\n \
       Print[\"Main operator is: \", op];\n        Error[inequal];\n   ];\n\n \
Return[op];\n\t];\n\nhomogineq[inequal_,varlist_]:=\n\
Block[{left,right,op,temp,vleft,vright,vl,vr,result},\n\n \
If[showproof,showreason=True];\n\n op = getop[inequal];\n\n normalorder[];\n\n\
 clearfractions[];\n\n result = checkobvious[];\n If[result==True || \
result==False,Return[result]];\n\n gcdize[];\n\n collect[];\n\n vleft =\t\
vec[left,varlist];\n vright =\tvec[right,varlist];\n\n (* clear fractions by \
multiplying both\n    sides by the lcm of the denominators *)\n\n vl =\t\
Apply[Plus,vleft];\n vl =\tDenominator[Together[vl]];\n vr =\t\
Apply[Plus,vright];\n vr =\tDenominator[Together[vr]];\n temp =\t\
PolynomialLCM[vl,vr];\n vleft =\ttemp*vleft;\n vright =\ttemp*vright;\n\n \
If[showreason,\n    Print[\"Expressing in terms of symmetric polynomials \
gives:\"];\n\tPrint[op[Apply[Plus,vleft],Apply[Plus,vright]]];\n   ];\n\n \
result = majorize[vleft,vright];\n\n If[supressbanner=!=True && \
result=!=True,\n    Print[\"left  side: \",vleft];\n\tPrint[\"right side: \
\",vright];\n   ];\n\n Return[result];\n\n\t];"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"right\"\n   \
  is similar to existing symbol \"Right\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"left\"\n    \
 is similar to existing symbol \"Left\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"collect\"\n \
    is similar to existing symbol \"Collect\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"vleft\"\n   \
  is similar to existing symbol \"left\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"vright\"\n  \
   is similar to existing symbol \"right\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{Cell[TextData[
"ApplyMuirhead[reason_,Lindex_,Rindex_]:=\n(* implicit inputs and outputs:\n\t\
leftsum, rightsum, left, right, change\n*)\nBlock[{lentry,rentry,con,\n\t\
rawleft,rawright,lfactor,rfactor},\n\tlentry =\tleft[[Lindex]];\n\trentry =\t\
right[[Rindex]];\n\tlfactor =\tnumfactor[lentry];\n\trawleft =\t\
lentry/lfactor;\n    rfactor =\tnumfactor[rentry];\n    rawright =\t\
rentry/rfactor;\n\tcon =\t\tMin[lfactor,rfactor];\n\tIf[showreason,\n\t    \
Print[con*rawleft>=con*rawright];\n\t\tIf[showHeuristics,\n\t\t\tPrint[\"... \
by Heuristic \",reason,\".\"]];\n\t  ];\n\trightsum =\trightsum - \
con*rawright;\n\tleftsum =\tleftsum - con*rawleft;\n\tleft =\t\t\
listerize[leftsum];\n\tright =\t\tlisterize[rightsum];\n\tchange =\tTrue;\n\t\
];\n\n(*\n\tmajorize takes two vectors of symmetric sums [a,b,c],\n\tand \
returns true if the left one majorizes the right one\n\tor if the inequality \
can be proven false.\n\tIt returns false if right majorizes left and returns\n\
\tnotsure if it is not sure.\n*)\n\nmajorize[lefty_,righty_] :=\n\
Block[{lfactor,rfactor,rawright,rawleft,len,result,termsize,p,d,m1,m2,con,\n\t\
 leftsum,rightsum,whileresult,change,temp,lcount,lentry,\n\t \
rcount,rentry,res2,every,some,Li,Ri,applied},\n\n left = lefty;\n right = \
righty;\n\n \t(*\n \t Heuristic 0:  If there is one term on each side,\n\t\t  \
then the result is determined by Muirhead's Theorem.\n\t *)\n\n \
If[Length[left]==1 && Length[right]==1,\n \tleft =\t\tFirst[left];\n\tright =\
\t\tFirst[right];\n\n\tresult =\tmajor[left,right];\n\n\tIf[showreason && \
result==True,\n\t    Print[\"This result follows from the majorization \
theorem.\"];\n\t  ];\n\n\tReturn[result];\n\t];\n\n If[showreason,\n    \
Print[\"This follows from the following majorizations:\"];\n\t];\n\n\
While[Length[right]>0,\n leftsum =\tApply[Plus,left];\n rightsum =\t\
Apply[Plus,right];\n\n change = False;\n \n (* Heuristic 1:\n    See if there \
is some term on the left that does not majorize\n    anything on the right.  \
If that is the case, then see if we\n    can apply Schur's theorem, either\n\n\
\t\t\t[p+2d,0,0]  + [p,d,d]  > 2 [p+d,d,0]\t\tcase a\n\n    or\t\t[p+d,p+d,0] \
+ [2p,d,d] > 2 [p+d,d,p]\t\tcase b, d<=2p\n\n    or\t\t[p+d,p+d,0] + [2p,d,d] \
> 2 [p+d,d,p]\t\tcase c, d>2p\n *)\n\n temp =\t\tFirst[left];\n temp =\t\t\
temp/numfactor[temp];\n termsize =\tLength[temp];\n\n If[termsize==3,\n  \
For[i=1,i<=Length[left],i++,\n\tlfactor =\tnumfactor[left[[i]]];\n    rawleft \
=\tleft[[i]]/lfactor;\n\n    For[j=1, j<=Length[right],j++,\n\t\trfactor =\t\
numfactor[right[[j]]];\n\t\trawright =\tright[[j]]/rfactor;\n\t\t\
If[major[rawleft,rawright],Break[]];\n       ];\n    \
If[j<=Length[right],Continue[]];\n\n    If[Part[rawleft,2]==Part[rawleft,3], \
(* i.e. [x,d,d] *)\n\t\td =\tPart[rawleft,2];\n\t    p =\tPart[rawleft,1];\n\t\
    m1 =\tCoefficient[leftsum,SS[p+2*d,0,0]];\n\t    m2 =\t\
Coefficient[rightsum,SS[p+d,d,0]];\n\t    If[m1=!=0 && m2>1,\n\t    \tcon = \
Min[lfactor,m1,Floor[m2/2]];\n\t\t\tleftsum =  \
leftsum-con*SS[p+2*d,0,0]-con*SS[p,d,d];\n\t\t\trightsum = \
rightsum-2*con*SS[p+d,d,0];\n\t\t\tIf[showreason,\n\t\t    \t\
Print[con*SS[p+2*d,0,0]+con*SS[p,d,d]>=2*con*SS[p+d,d,0]];\n\t\t\t\t\
If[showHeuristics,Print[\"... by Heuristic 1a.\"]];\n\t\t\t  ];\n\t\t\tleft =\
\tlisterize[leftsum];\n\t\t\tright =\tlisterize[rightsum];\n\t\t\tchange = \
True;\n\t\t\tBreak[];\n\t    (* else *),\n\t    If[EvenQ[p],\n\t   \t\tp =\t\
p/2;\n\t\t\tm1 =\tCoefficient[leftsum,SS[p+d,p+d,0]];\n\t\t\tm2 =\t\
Coefficient[rightsum,SS[p+d,d,p]];\n\t\t\tIf[m1!=0 && m2>1,\n\t\t\t\tcon = \
Min[lfactor,m1,Floor[m2/2]];\n\t\t\t\tleftsum = \
leftsum-con*SS[p+d,p+d,0]-con*SS[2p,d,d];\n\t\t\t\trightsum = \
rightsum-2*con*SS[p+d,d,p];\n\t\t\t\tIf[showreason,\n\t\t\t \t \t\
Print[con*SS[p+d,p+d,0]+con*SS[2p,d,d]>=2*con*SS[p+d,d,p]];\n\t\t\t\t\t\
If[showHeuristics,Print[\"... by Heuristic 1b.\"]];\n\t\t\t \t  ];\n\t\t\t\t\
left =\tlisterize[leftsum];\n\t\t\t\tright =\tlisterize[rightsum];\n\t\t\t\t\
change = True;\n\t\t\t\tBreak[];\n\t\t\t  ];\n\t      ]; (* end if EvenQ *)\n\
\t   ]; (* end if m1!=0 *)\n     ];\t(* end if [x,d,d] *)\n    \
If[Part[rawleft,1]==Part[rawleft,2], (* i.e. [d,d,x] *)\n\t\td =\t\
Part[rawleft,1];\n\t    p =\tPart[rawleft,3];\n\t    If[EvenQ[p],\n\t    \tp \
= p/2;\n\t\t    m1 =\tCoefficient[leftsum,SS[p+d,p+d,0]];\n\t\t    m2 =\t\
Coefficient[rightsum,SS[p+d,d,p]];\n\t\t    If[m1=!=0 && m2>1,\n\t\t    \tcon \
= Min[lfactor,m1,Floor[m2/2]];\n\t\t\t\tleftsum =  \
leftsum-con*SS[p+d,p+d,0]-con*SS[d,d,2p];\n\t\t\t\trightsum = \
rightsum-2*con*SS[p+d,d,p];\n\t\t\t\tIf[showreason,\n\t\t\t    \t\
Print[con*SS[p+d,p+d,0]+con*SS[d,d,2p]>=2*con*SS[p+d,d,p]];\n\t\t\t\t\t\
If[showHeuristics,Print[\"... by Heuristic 1c.\"]];\n\t\t\t\t  ];\n\t\t\t\t\
left =\tlisterize[leftsum];\n\t\t\t\tright =\tlisterize[rightsum];\n\t\t\t\t\
change = True;\n\t\t\t\tBreak[];\n\t\t     ]; (* end if m1!=0 *)\n\t\t   ]; \
(* end if EvenQ *)\n     ];\t(* end if [d,d,x] *)\n    ];\t(* end for i *)\n  \
If[i<=Length[left],Continue[]];\n\n  ]; \t(* end if termsize==3 *)\n\n(*\n\t \
Heuristic 2: If there is only one term on the right,\n\t\t then the terms on \
the left must majorize it away.\n *)\n\n If[Length[right]==1,\n\tlen =\t\t\
Length[left];\n\n\tFor[i=len, i>0, i--,\n\t\t\
If[Length[right]==0,Return[True]];\n\t\trfactor =\tnumfactor[right[[1]]];\n\t\
\tIf[rfactor<=0,Return[True]];\n\t\trawright =\tright[[1]]/rfactor;\n\t    \
lfactor =\tnumfactor[left[[i]]];\n\t    rawleft =\tleft[[i]]/lfactor;\n\t    \
If[major[rawleft,rawright],\n\t    \tApplyMuirhead[\"2\",i,1];\n\t      ];\n\t\
   ];\n\t];\n\n If[leftsum==0 && rightsum=!=0,Return[False]];\n \
If[rightsum==0,Return[True]];\n\n (*\n \tHeuristic 3:\n    If there is a term \
on the right that is majorized by\n    precisely one term on the left, then \
we may as well\n    remove that term now.\n *)\n\n  applied = False;\n  \
For[j=1, j<=Length[right], j++,\n    rfactor =\tnumfactor[right[[j]]];\n    \
rawright =\tright[[j]]/rfactor;\n    lcount =\t0;\n\n    For[i=1, \
i<=Length[left], i++,\n\t\tlfactor =\tnumfactor[left[[i]]];\n\t\trawleft =\t\
left[[i]]/lfactor;\n\t\tIf[major[rawleft,rawright],\n\t    \tlcount++;\n\t\t\t\
Li = i;\n\t      ];\n       ];\n\n    If[lcount==1,\n    \t\
ApplyMuirhead[\"3\",Li,j];\n    \tapplied = True;\n\t\tBreak[];\n      ];\n   \
];\n If[applied,Continue[]];\n\n If[leftsum==0 && \
rightsum=!=0,Return[False]];\n If[rightsum==0,Return[True]];\n\n (* Heuristic \
4:\n    If there is a term on the left that majorizes\n    precisely one term \
on the right, then we may as well\n    remove that term now.\n *)\n\n  \
applied = False;\n  For[j=1, j<=Length[left], j++,\n    lfactor =\t\
numfactor[left[[j]]];\n    rawleft =\tleft[[j]]/lfactor;\n    rcount =\t0;\n\n\
    For[i=1, i<=Length[right], i++,\n\t\trfactor =\tnumfactor[right[[i]]];\n\t\
\trawright =\tright[[i]]/rfactor;\n\t\tIf[major[rawleft,rawright],\n\t    \t\
rcount++;\n\t    \tRi = i;\n\t      ];\n       ];\n\n    If[rcount==1,\n    \t\
ApplyMuirhead[\"4\",j,Ri];\n    \tapplied = True;\n\t\tBreak[];\n      ];\n   \
];\n If[applied,Continue[]];\n\n If[leftsum==0 && \
rightsum=!=0,Return[False]];\n If[rightsum==0,Return[True]];\n\n (* Heuristic \
5:\n    If every term on the left majorizes every term on the right,\n    \
then we can pair them off any way we want.\n *)\n\n  every = True;\n  \
For[j=1, j<=Length[right], j++,\n    rfactor =\tnumfactor[right[[j]]];\n    \
rawright =\tright[[j]]/rfactor;\n\n    For[i=1, i<=Length[left], i++,\n\t\t\
lfactor =\tnumfactor[left[[i]]];\n\t\trawleft =\tleft[[i]]/lfactor;\n\t\t\
If[!major[rawleft,rawright],\n\t\t\tevery = False;\n\t\t\tBreak[];\n\t\t  ];\n\
       ];\n     If[!every,Break[]];\n     ];\n\n  If[every,\n  \t\
ApplyMuirhead[\"5\",1,Length[right]];\n    ];\n\n If[leftsum==0 && \
rightsum=!=0,Return[False]];\n If[rightsum==0,Return[True]];\n\n (* Heuristic \
6:\n    If some term on the left majorizes some term on the right,\n    then \
we can remove this pair.\n *)\n\n  For[j=Length[right], j>0, j--,\n    \
rfactor =\tnumfactor[right[[j]]];\n    rawright =\tright[[j]]/rfactor;\n\n\t\
some = False;\n    For[i=1, i<=Length[left], i++,\n\t\tlfactor =\t\
numfactor[left[[i]]];\n\t\trawleft =\tleft[[i]]/lfactor;\n\t\t\
If[major[rawleft,rawright],\n\t\t\tApplyMuirhead[\"6\",i,j];\n\t\t\tsome = \
True;\n\t\t\tBreak[];\n\t\t  ];\n   \t   ];\n   If[some, Break[]];\n   ];\n\n\
If[leftsum==0 && rightsum!=0,Return[False]];\nIf[rightsum==0,Return[True]];\n\
\nIf[!change, Break[]];\n\n];\n\n If[Length[right]==0,Return[True]];\n \
If[Length[left]>1 && Length[right]>1,Return[notsure]];\n \
Return[notyetimplemented];\n\n];\n \n(*\n *\tTurn \
n1[a1,b1,c1]+n2[a2,b2,c3]+n3[a3,b3,c3] into the list\n *\t{n1[a1,b1,c1] , \
n2[a2,b2,c3] , n3[a3,b3,c3]}.\n *)\n\nlisterize[expr_]:=\nBlock[{temp},\n \t\
If[AtomQ[expr],\n\t\tIf[expr===0, Return[{}], Return[{expr}]];\n\t  ];\n \
If[Head[expr]===Plus,\n \tReturn[ReplacePart[expr,List,0]];\n   ];\n   \t\n \
Return[{expr}];\n ];\n"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"rentry\"\n  \
   is similar to existing symbol \"lentry\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"rfactor\"\n \
    is similar to existing symbol \"lfactor\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"rcount\"\n  \
   is similar to existing symbol \"lcount\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData[
"(* major takes two symmetric sums, such as [a,b,c] and [d,e,f]\n   and \
returns true if the left one majorizes the right one.\n*)\n\n\
major[left_,right_] :=\n\
Block[{lfactor,rfactor,lterm,rterm,lsum,rsum,len,result,ineqList},\n\t(* \
Extract a and {x,y,z} from a{x,y,z} *)\n\t{lfactor,lterm} =\t\
FactorTermsList[left];\n\t{rfactor,rterm} =\tFactorTermsList[right];\n\t\
If[lfactor < rfactor,Return[funnyratio]];\n\n\t(* turn SS[a,b,c] into normal \
list {a,b,c} *)\n\n \tlterm = lterm/.{SS[x__]->{x}};\n \trterm = \
rterm/.{SS[x__]->{x}};\n\n\tIf[Length[lterm] != Length[rterm],\n\t\t\
Print[\"Length error.\"];\n\t \tPrint[lterm,\" \",rterm,\" \",lfactor,\" \
\",rfactor];\n\t\tReturn[lengthsdiffer];\n\t  ];\n\tlen = Length[lterm];\n\n\t\
(* compute the partial sums of each one\n\tand check for majorization *)\n\n\t\
lsum = Drop[FoldList[Plus,0,lterm],1];\n\trsum = \
Drop[FoldList[Plus,0,rterm],1];\n\n\t(* form sequence of inequalities *)\n\t\
ineqList = MapThread[GreaterEqual,{lsum,rsum}];\n\t\n\t(* \"and\" them all \
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"(*\n\tCall a function \"fn\" once for each\n\tpermutation of the variables \
in \"expr\".\n\tThe list of variables is given in \"varlist\".\n\tThe \
function is called as\n\t\tfn(permutedExpr,arg)\n\tand returns True to \
continue, False to abort.\n*)\nsymm[expr_,varlist_,fn_,arg_]:=\n\
Block[{val,permList,len,i,perm,permExpr,permRule},\n\tpermList = \
Permutations[varlist];\n\tlen = Length[permList];\n\tFor[i=1,i<=len,i++,\n\t\t\
perm = permList[[i]];\n\t\tpermRule = MapThread[Rule,{varlist,perm}];\n\t\t\
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 \tIf[isterm[expr],\n\t    \tsize = 1,\n\t    \tsize = Length[expr]\n\t      \
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\t\" expr=\",expr,\" coef=\",coef];\n\t  \t  ];\n\t\tratio =\tcoef/co;\n\n\t\t\
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PrependTo[resultvec,vector];\n\t\texpr =\t\tExpand[expr - ratio*sym];\n\t\t\
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\t  ];\n\t\tIf[newsize >= size && expr =!= 0,\n\t\t\tPrint[\"internal error, \
expression got larger\"];\n\t\t\tPrint[\"term=\",term,\" ratio=\",ratio,\" \
coef=\",coef,\" co=\",co];\n\t\t\tPrint[\"size=\",size,\" newsize=\",newsize,\
\" sym=\",sym];\n\t\t\tPrint[expr];\n\t\t\terror[expr];\n\t\t  ];\n    \t];\n\
\n    Return[resultvec];\n\t];\n\nisterm[expr_] :=\nBlock[{},\n\t\
If[AtomQ[expr],Return[True]];\n\tIf[Head[expr]=!=Plus,Return[True]];\n\t\
Return[False];\n\t];\n\nsymmetrize[term_,varlist_]:=\nBlock[ {result},\n\n\t\
cumulate[tem_,arg_]:= {result+=tem;Return[True];};\n\n\tresult = 0;\n\t\
symm[term,varlist,cumulate,0];\n\tReturn[result];\n\t];\n\ntest[expr_] :=\n\
Block[ {varlist,result},\n\tvarlist = Variables[expr];\n\tPrint[\" \"];\n\t\
Print[\"----------\"];\n\tPrint[expr];\n\tresult = homogineq[expr,varlist];\n\
\tIf[result=!=True,\n\t\tPrint[\"********* COULD NOT PROVE THIS INEQUALITY \
*********\"];\n\t  ];\n\t];\n\nclearfractions[] :=\n(* Implicit arguments: \
left, right, op. *)\nBlock[ {dl,dr,prod,pr},\n\tleft =\tTogether[left];\n\t\
right =\tTogether[right];\n\tdl = Denominator[left];\n\tdr = \
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\tleft =\tNumerator[left]*(prod/dl);\n\t\tright =\t\
Numerator[right]*(prod/dr);\n\t\tleft =\tExpand[Together[left]];\n\t\tright =\
\tExpand[Together[right]];\n\t\tIf[showproof,\n\t \t\tpr = Factor[prod];\n\t  \
 \t\tPrint[\"Multiplying both sides by \",pr,\" gives:\"];\n\t   \t\t\
Print[op[left,right]];\n\t   \t  ];\n\t  ];\n\t];\n\ngcdize[] :=\n(* Implicit \
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right==0, Return[]];\n\n\ttemp = PolynomialGCD[left,right];\n\n\t\
If[NumberQ[temp] && temp<0,\n \t\tPrint[\"Oops: gcd was negative.\"];\n\t  ];\
\n\n \t(* Should really confirm that temp is positive. *)\n\n \t\
If[temp==1,Return[]];\n\n\tleft =\tExpand[Together[left/temp]];\n\tright =\t\
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right=!=0 && op=!=Equal,\n    If[showreason,\n      Print[\"This is false \
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