BackendEquation.mo

/*
 * This file is part of OpenModelica.
 *
 * Copyright (c) 1998-2014, Open Source Modelica Consortium (OSMC),
 * c/o Linköpings universitet, Department of Computer and Information Science,
 * SE-58183 Linköping, Sweden.
 *
 * All rights reserved.
 *
 * THIS PROGRAM IS PROVIDED UNDER THE TERMS OF GPL VERSION 3 LICENSE OR
 * THIS OSMC PUBLIC LICENSE (OSMC-PL) VERSION 1.2.
 * ANY USE, REPRODUCTION OR DISTRIBUTION OF THIS PROGRAM CONSTITUTES
 * RECIPIENT'S ACCEPTANCE OF THE OSMC PUBLIC LICENSE OR THE GPL VERSION 3,
 * ACCORDING TO RECIPIENTS CHOICE.
 *
 * The OpenModelica software and the Open Source Modelica
 * Consortium (OSMC) Public License (OSMC-PL) are obtained
 * from OSMC, either from the above address,
 * from the URLs: http://www.ida.liu.se/projects/OpenModelica or
 * http://www.openmodelica.org, and in the OpenModelica distribution.
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 *
 * This program is distributed WITHOUT ANY WARRANTY; without
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encapsulated package BackendEquation
" file:        BackendEquation.mo
  package:     BackendEquation
  description: BackendEquation contains functions that do something with
               BackendDAE.Equation data type."

import Absyn;
import BackendDAE;
import DAE;

protected
import Algorithm;
import Array;
import BackendDAEUtil;
import BackendDump;
import BackendVariable;
import BaseHashTable;
import AvlSetInt;
import ClassInf;
import ComponentReference;
import DAEUtil;
import Debug;
import ElementSource;
import Error;
import ExpandableArray;
import Expression;
import ExpressionDump;
import ExpressionSimplify;
import ExpressionSolve;
import Flags;
import HashTable;
import List;

public function emptyEqns "author: lochel
  Returns an empty expandable equation array."
  output BackendDAE.EquationArray equationArray = emptyEqnsSized(0);
end emptyEqns;

public function emptyEqnsSized "author: lochel
  Returns an empty expandable equation array with a given size."
  input Integer size;
  output BackendDAE.EquationArray outEquationArray = ExpandableArray.new(size, BackendDAE.DUMMY_EQUATION());
end emptyEqnsSized;

public function add "author: lochel
  Adds an equation to an EquationArray."
  input BackendDAE.Equation inEquation;
  input output BackendDAE.EquationArray equationArray;
algorithm
  ExpandableArray.add(inEquation, equationArray);
end add;

public function addList "author: hkiel
  Adds a list of BackendDAE.Equation to BackendDAE.EquationArray."
  input list<BackendDAE.Equation> eqnlst;
  input output BackendDAE.EquationArray equationArray;
algorithm
  ExpandableArray.expandToSize(ExpandableArray.getLastUsedIndex(equationArray) + listLength(eqnlst), equationArray);
  for e in eqnlst loop
    equationArray := add(e, equationArray);
  end for;
end addList;

public function delete
  input Integer inPos "one-based indexing";
  input output BackendDAE.EquationArray equationArray;
algorithm
  ExpandableArray.delete(inPos, equationArray);
end delete;

public function deleteList
  "Deletes all equations from the given list of indices."
  input output BackendDAE.EquationArray equationArray;
  input list<Integer> inIndices "one-based indexing";
algorithm
  for index in inIndices loop
    ExpandableArray.delete(index, equationArray);
  end for;
end deleteList;

public function merge "author: lochel
  This function returns an EquationArray containing all the equations from both
  inputs."
  input BackendDAE.EquationArray inEqns1;
  input BackendDAE.EquationArray inEqns2;
  output BackendDAE.EquationArray outEqns;
algorithm
  outEqns := copyEquationArray(inEqns2);
  outEqns := addList(equationList(inEqns1), outEqns);
end merge;

public function listEquation "author: PA
  Transform a list of equations into an expandable BackendDAE.Equation array."
  input list<BackendDAE.Equation> inEquationList;
  output BackendDAE.EquationArray outEquationArray;
algorithm
  outEquationArray := ExpandableArray.new(listLength(inEquationList), BackendDAE.DUMMY_EQUATION());
  for eq in inEquationList loop
    ExpandableArray.add(eq, outEquationArray);
  end for;
end listEquation;

public function equationList "author: lochel
  Transform the expandable BackendDAE.Equation array to a list of Equations."
  input BackendDAE.EquationArray equationArray;
  output list<BackendDAE.Equation> outEquationLst = ExpandableArray.toList(equationArray);
end equationList;

public function copyEquationArray
  "Performs a deep copy of an expandable equation array."
  input BackendDAE.EquationArray inEquationArray;
  output BackendDAE.EquationArray outEquationArray = ExpandableArray.copy(inEquationArray, BackendDAE.DUMMY_EQUATION());
end copyEquationArray;

public function setAtIndex "author: lochel
  Sets the n-th array element of an EquationArray.
  Please note: one-based indexing"
  input output BackendDAE.EquationArray equationArray;
  input Integer inPos "one-based indexing";
  input BackendDAE.Equation inEquation;
algorithm
  if ExpandableArray.occupied(inPos, equationArray) then
    ExpandableArray.update(inPos, inEquation, equationArray);
  else
    ExpandableArray.set(inPos, inEquation, equationArray);
  end if;
end setAtIndex;

public function setAtIndexFirst "author: waurich
  Sets the n-th array element of an EquationArray but with index at first argument.
  Please note: one-based indexing"
  input Integer inPos "one-based indexing";
  input BackendDAE.Equation inEquation;
  input BackendDAE.EquationArray inEquationArray;
  output BackendDAE.EquationArray outEquationArray = setAtIndex(inEquationArray, inPos, inEquation);
end setAtIndexFirst;

public function get
  "Return the n-th equation from the expandable equation array
  indexed from 1..N."
  input BackendDAE.EquationArray inEquationArray;
  input Integer inPos "one-based indexing";
  output BackendDAE.Equation outEquation = ExpandableArray.get(inPos, inEquationArray);
end get;

public function getList "author: Frenkel TUD 2011-05
  returns the equations given by the list of indexes"
  input list<Integer> inIndices "one-based indexing";
  input BackendDAE.EquationArray inEquationArray;
  output list<BackendDAE.Equation> outEqns;
algorithm
  outEqns := list(get(inEquationArray, index) for index in inIndices);
end getList;

public function equationArraySize "author: lochel
  Returns the size of the equations in an BackendDAE.EquationArray, which
  should correspond to the number of variables."
  input BackendDAE.EquationArray equationArray;
  output Integer outSize;
algorithm
  outSize := 0;
  for i in 1:ExpandableArray.getLastUsedIndex(equationArray) loop
    if ExpandableArray.occupied(i, equationArray) then
      outSize := outSize + equationSize(ExpandableArray.get(i, equationArray));
    end if;
  end for;
end equationArraySize;

public function getNumberOfEquations "author: lochel
  Returns the number of equations in an BackendDAE.EquationArray, which not
  corresponds to the number of equations in the system."
  input BackendDAE.EquationArray inEquationArray;
  output Integer outSize = ExpandableArray.getNumberOfElements(inEquationArray);
end getNumberOfEquations;

public function traverseEquationArray<T> "author: lochel
  Traverses all equations of a BackendDAE.EquationArray."
  input BackendDAE.EquationArray equationArray;
  input Func inFunc;
  input output T extraArg;

  partial function Func
    input output BackendDAE.Equation eq;
    input output T extraArg;
  end Func;
protected
  BackendDAE.Equation eqn;
algorithm
  for i in 1:ExpandableArray.getLastUsedIndex(equationArray) loop
    if ExpandableArray.occupied(i, equationArray) then
      eqn := ExpandableArray.get(i, equationArray);
      (_, extraArg) := inFunc(eqn, extraArg);
    end if;
  end for;
end traverseEquationArray;

public function traverseEquationArray_WithStop<T> "author: lochel
  Traverses all equations of a BackendDAE.EquationArray."
  input BackendDAE.EquationArray equationArray;
  input FuncWithStop inFuncWithStop;
  input output T extraArg;

  partial function FuncWithStop
    input output BackendDAE.Equation eq;
    input T inExtraArg;
    output Boolean cont;
    output T outExtraArg;
  end FuncWithStop;
protected
  Boolean continue_;
  BackendDAE.Equation eqn;
algorithm
  for i in 1:ExpandableArray.getLastUsedIndex(equationArray) loop
    if ExpandableArray.occupied(i, equationArray) then
      eqn := ExpandableArray.get(i, equationArray);
      (_, continue_, extraArg) := inFuncWithStop(eqn, extraArg);

      if not continue_ then
        break;
      end if;
    end if;
  end for;
end traverseEquationArray_WithStop;

public function traverseEquationArray_WithUpdate<T>
  "Traverses all equations of a BackendDAE.EquationArray."
  input output BackendDAE.EquationArray equationArray;
  input FuncWithUpdate inFuncWithUpdate;
  input output T extraArg;

  partial function FuncWithUpdate
    input output BackendDAE.Equation inoutEq;
    input output T inoutA;
  end FuncWithUpdate;
protected
  BackendDAE.Equation e, new_e;
algorithm
  for i in 1:ExpandableArray.getLastUsedIndex(equationArray) loop
    if ExpandableArray.occupied(i, equationArray) then
      e := ExpandableArray.get(i, equationArray);
      (new_e, extraArg) := inFuncWithUpdate(e, extraArg);
      if not referenceEq(e, new_e) then
        ExpandableArray.update(i, new_e, equationArray);
      end if;
    end if;
  end for;
end traverseEquationArray_WithUpdate;

public function getWhenEquationExpr "Get the left and right hand parts from an equation appearing in a when clause"
  input BackendDAE.WhenEquation inWhenEquation;
  output DAE.ComponentRef outComponentRef;
  output DAE.Exp outExp;
algorithm
  try
    BackendDAE.WHEN_STMTS(whenStmtLst={BackendDAE.ASSIGN(left=DAE.CREF(componentRef = outComponentRef), right=outExp)}) := inWhenEquation;
else
    Error.addInternalError("BackendEquation.getWhenEquationExpr failed\n", sourceInfo());
  end try;
end getWhenEquationExpr;

public function setWhenElsePart
"Set the else part of an whenEquation"
  input BackendDAE.WhenEquation inWhenEquation;
  input BackendDAE.WhenEquation inElseWhenEquation;
  output BackendDAE.WhenEquation outWhenEquation;
algorithm
  outWhenEquation := match(inWhenEquation)
  local
    DAE.Exp cond;
    BackendDAE.WhenEquation elsewhenPart;
    list<BackendDAE.WhenOperator> whenStmtLst;

    case (BackendDAE.WHEN_STMTS(condition=cond, whenStmtLst = whenStmtLst, elsewhenPart=NONE()))
    then BackendDAE.WHEN_STMTS(cond, whenStmtLst, SOME(inElseWhenEquation));

    case (BackendDAE.WHEN_STMTS(condition=cond, whenStmtLst = whenStmtLst, elsewhenPart=SOME(elsewhenPart)))
    then BackendDAE.WHEN_STMTS(condition=cond, whenStmtLst = whenStmtLst, elsewhenPart=SOME(setWhenElsePart(elsewhenPart, inElseWhenEquation)));

    else fail();
  end match;
end setWhenElsePart;

public function equationsLstVars "author: Frenkel TUD 2011-05
  From the equations and a variable array return all
  occurring variables from the array."
  input list<BackendDAE.Equation> inEquationLst;
  input BackendDAE.Variables inVars;
  output list<BackendDAE.Var> outVars;
protected
  AvlSetInt.Tree indexes;
  list<Integer> keys;
algorithm
  if listEmpty(inEquationLst) then
    outVars := {};
    return;
  end if;
  (_, indexes) := traverseExpsOfEquationList(inEquationLst, function checkEquationsVarsExpTopDownTraverseHelper(vars=inVars),  AvlSetInt.new());
  keys := AvlSetInt.listKeys(indexes);
  outVars := List.map1r(keys, BackendVariable.getVarAt, inVars);
end equationsLstVars;

public function equationsVars "author: Frenkel TUD 2011-05
  From the equations and a variable array return all
  occurring variables from the array."
  input BackendDAE.EquationArray inEquations;
  input BackendDAE.Variables inVars;
  output list<BackendDAE.Var> outVars;
protected
  AvlSetInt.Tree indexes;
  list<Integer> keys;
algorithm
  indexes := BackendDAEUtil.traverseBackendDAEExpsEqns(inEquations, function checkEquationsVarsExpTopDownTraverseHelper(vars=inVars),  AvlSetInt.new());
  keys := AvlSetInt.listKeys(indexes);
  outVars := List.map1r(keys, BackendVariable.getVarAt, inVars);
end equationsVars;

public function equationVars "author: Frenkel TUD 2012-03
  From the equation and a variable array return all
  variables in the equation."
  input BackendDAE.Equation inEquation;
  input BackendDAE.Variables inVars;
  output list<BackendDAE.Var> outVars;
protected
  AvlSetInt.Tree indexes;
  list<Integer> keys;
algorithm
  (_, indexes) := traverseExpsOfEquation(inEquation, function checkEquationsVarsExpTopDownTraverseHelper(vars=inVars),  AvlSetInt.new());
  keys := AvlSetInt.listKeys(indexes);
  outVars := List.map1r(keys, BackendVariable.getVarAt, inVars);
end equationVars;

public function expressionVars "author: Frenkel TUD 2012-03
  From the expression and a variable array return all
  variables in the expression."
  input DAE.Exp inExp;
  input BackendDAE.Variables vars;
  output list<BackendDAE.Var> outVars;
protected
  AvlSetInt.Tree indexes;
  list<Integer> keys;
algorithm
  (_, indexes) := Expression.traverseExpTopDown(inExp, function checkEquationsVarsExpTopDown(vars=vars), AvlSetInt.new());
  keys := AvlSetInt.listKeys(indexes);
  outVars := List.map1r(keys, BackendVariable.getVarAt, vars);
end expressionVars;

public function expressionVarsIndexes
  "From the expression and a variable array return all variables in the expression."
  input DAE.Exp exp;
  input output AvlSetInt.Tree indexes;
  input CheckEquationsVarsExpTopDownFunc func;

  partial function CheckEquationsVarsExpTopDownFunc
    input output DAE.Exp exp;
    output Boolean cont;
    input output AvlSetInt.Tree tree;
  end CheckEquationsVarsExpTopDownFunc;
algorithm
  (_, indexes) := Expression.traverseExpTopDown(exp, func, indexes);
end expressionVarsIndexes;

public function checkEquationsVarsExpTopDownTraverseHelper
"This function traverses expssions to-down to collect all
 variables that are involed into the expression."
  input output DAE.Exp exp;
  input output AvlSetInt.Tree tree;
  input BackendDAE.Variables vars;
algorithm
  (exp, tree) := Expression.traverseExpTopDown(exp,function checkEquationsVarsExpTopDown(vars=vars),tree);
end checkEquationsVarsExpTopDownTraverseHelper;

public function checkEquationsVarsExpTopDown
  input output DAE.Exp exp;
  output Boolean cont;
  input output AvlSetInt.Tree tree;
  input BackendDAE.Variables vars;
algorithm
  (cont,tree) := match exp
    local
      DAE.ComponentRef cr;
      list<Integer> ilst;
      String idn;

    // special case for time, it is never part of the equation system
    case DAE.CREF(componentRef=DAE.CREF_IDENT(ident="time"))
    then (true, tree);

    // case for function pointers
    case DAE.CREF(ty=DAE.T_FUNCTION_REFERENCE_FUNC())
    then (true, tree);

    // case for pre and previous vars
    case DAE.CALL(path = Absyn.IDENT(name = idn)) guard idn=="pre" or idn=="previous"
    then (false, tree);

    // add it
    case DAE.CREF(componentRef = cr)
      algorithm
        try
        (_, ilst) := BackendVariable.getVar(cr, vars);
        tree := AvlSetInt.addList(tree, ilst);
        else
        end try;
      then (true, tree);

    else (true, tree);
  end match;
end checkEquationsVarsExpTopDown;

public function assertWithCondTrue "author: Frenkel TUD 2012-12"
  input BackendDAE.Equation inEqn;
  output Boolean b;
algorithm
  b := match inEqn
    case BackendDAE.ALGORITHM(alg=DAE.ALGORITHM_STMTS({DAE.STMT_ASSERT(cond=DAE.BCONST(true))})) then false;
    else true;
  end match;
end assertWithCondTrue;

public function equationsParams "author: marcusw
  From a list of equations return all occurring parameter variables. Duplicates are removed."
  input list<BackendDAE.Equation> inEquationLst;
  input BackendDAE.Variables inVars;
  output list<BackendDAE.Var> outParamVars;
  output list<Integer> outParamVarsIdc;
algorithm
  (_, (_,(outParamVars,outParamVarsIdc,_))) := traverseExpsOfEquationList(inEquationLst, Expression.traverseSubexpressionsHelper, (traversingParamRefFinder, ({}, {}, inVars)));
end equationsParams;

protected function traversingParamRefFinder "author: marcusw
  Checks if the given expression is a parameter variable and adds it to the result list on true. Duplicates are removed."
  input DAE.Exp inExp;
  input tuple<list<BackendDAE.Var>, list<Integer>, BackendDAE.Variables> inTpl; //<result list, result idc, all variables>
  output DAE.Exp outExp;
  output tuple<list<BackendDAE.Var>, list<Integer>, BackendDAE.Variables> outTpl;
algorithm
  outExp := inExp;
  outTpl := match(inExp,inTpl)
    local
      BackendDAE.Variables allVars;
      list<BackendDAE.Var> vars, foundVars;
      list<Integer> varIdc, foundVarsIdc;
      DAE.ComponentRef cr;

    case (DAE.CREF(componentRef=DAE.CREF_IDENT(ident="time")), _)
      then inTpl;
    case ( DAE.CREF(componentRef=cr), (vars, varIdc, allVars)) equation
      //print("traversingParamRefFinder: try to get Variable " + ComponentReference.crefStr(cr) + "\n");
      (foundVars, foundVarsIdc) = BackendVariable.getVar(cr, allVars);
      (vars, varIdc) = traversingParamRefFinder0(foundVars, foundVarsIdc, vars, varIdc);
      then (vars, varIdc, allVars);
    else inTpl;
  end match;
end traversingParamRefFinder;

protected function traversingParamRefFinder0
  input list<BackendDAE.Var> iVars;
  input list<Integer> iVarIdc;
  input list<BackendDAE.Var> iParamVarsList;
  input list<Integer> iParamVarsIdc;
  output list<BackendDAE.Var> oParamVarsList;
  output list<Integer> oParamVarIdc;
protected
  BackendDAE.Var var;
  list<BackendDAE.Var> vars = iParamVarsList;
  list<Integer> varIdc = iParamVarsIdc;
  Integer varIdx;
  list<Integer> rest = iVarIdc;
algorithm
  for var in iVars loop
    varIdx::rest := rest;
    if(BackendVariable.isParam(var)) then
      vars := List.unionEltOnTrue(var, vars, BackendVariable.varEqual);
      varIdc := List.unionEltOnTrue(varIdx, varIdc, intEq);
    end if;
  end for;
  oParamVarsList := vars;
  oParamVarIdc := varIdc;
end traversingParamRefFinder0;

public function iterationVarsinRelations
 "From a list of equations return all variables, which used inside
  a relation, then this equation system is marked as mixed and also
  the indexes of the relations may be used in future."
  input list<BackendDAE.Equation> inEquationLst;
  input BackendDAE.Variables inVars;
  output Boolean mixedSystem;
  output list<Integer> indexes;
algorithm
  (_, (_,(indexes, _))) := traverseExpsOfEquationList(inEquationLst, Expression.traverseSubexpressionsHelper, (traversingRelationsforIterationVars, ({}, inVars)));
  mixedSystem := not listEmpty(indexes);
end iterationVarsinRelations;

protected function traversingRelationsforIterationVars
  input DAE.Exp inExp;
  input tuple<list<Integer>, BackendDAE.Variables> inTpl;
  output DAE.Exp outExp;
  output tuple<list<Integer>, BackendDAE.Variables> outTpl;
algorithm
  outExp := inExp;
  outTpl := match (inExp,inTpl)
    local
      BackendDAE.Variables vars;
      list<Integer> indexes;
      DAE.Exp e1, e2;
      Integer index;
      list<BackendDAE.Var> vlst1,vlst2;

    case (DAE.RELATION(exp1=e1, exp2=e2, index=index), (indexes, vars))
      equation
        vlst1 = expressionVars(e1, vars);
        vlst2 = expressionVars(e2, vars);
        //if lists are not empty continue
        if not(listEmpty(vlst1) and listEmpty(vlst2)) then
          outTpl = (index::indexes, vars);
        else
          outTpl = inTpl;
        end if;
    then (outTpl);

    else (inTpl);
  end match;
end traversingRelationsforIterationVars;

public function equationsCrefs "author: PA
  From a list of equations return all occurring variables/component references."
  input list<BackendDAE.Equation> inEquationLst;
  output list<DAE.ComponentRef> outExpComponentRefLst;
algorithm
  (_, (_,outExpComponentRefLst)) := traverseExpsOfEquationList(inEquationLst, Expression.traverseSubexpressionsHelper, (Expression.traversingComponentRefFinder, {}));
end equationsCrefs;

public function equationCrefs "author: PA
  From one equation return all occurring variables/component references."
  input BackendDAE.Equation inEquation;
  output list<DAE.ComponentRef> outExpComponentRefLst;
algorithm
  (_, (_, outExpComponentRefLst)) := traverseExpsOfEquation(inEquation, Expression.traverseSubexpressionsHelper, (Expression.traversingComponentRefFinder, {}));
end equationCrefs;

public function getAllCrefFromEquations
  input BackendDAE.EquationArray inEqns;
  output list<DAE.ComponentRef> cr_lst;
algorithm
  cr_lst := traverseEquationArray(inEqns, traversingEquationCrefFinder, {});
end getAllCrefFromEquations;

protected function traversingEquationCrefFinder "author: Frenkel TUD 2010-11"
  input BackendDAE.Equation inEq;
  input list<DAE.ComponentRef> inCrefs;
  output BackendDAE.Equation e;
  output list<DAE.ComponentRef> cr_lst;
algorithm
  e := inEq;
  cr_lst := inCrefs;
  (_, (_,cr_lst)) := traverseExpsOfEquation(e, Expression.traverseSubexpressionsHelper, (Expression.traversingComponentRefFinder, cr_lst));
end traversingEquationCrefFinder;

public function equationUnknownCrefs "author: Frenkel TUD 2012-05
  From the equation and a variable array return all
  variables in the equation an not in the variable array."
  input list<BackendDAE.Equation> inEquationLst;
  input BackendDAE.Variables inVars;
  input BackendDAE.Variables inKnVars;
  output list<DAE.ComponentRef> cr_lst;
protected
  HashTable.HashTable ht;
algorithm
  ht := HashTable.emptyHashTable();
  (_, (_, (_, _, ht))) := traverseExpsOfEquationList(inEquationLst, Expression.traverseSubexpressionsHelper, (checkEquationsUnknownCrefsExp, (inVars, inKnVars, ht)));
  cr_lst := BaseHashTable.hashTableKeyList(ht);
end equationUnknownCrefs;

protected function checkEquationsUnknownCrefsExp
  input DAE.Exp inExp;
  input tuple<BackendDAE.Variables, BackendDAE.Variables, HashTable.HashTable> inTuple;
  output DAE.Exp outExp;
  output tuple<BackendDAE.Variables, BackendDAE.Variables, HashTable.HashTable> outTuple;
algorithm
  (outExp,outTuple) := matchcontinue (inExp,inTuple)
    local
      DAE.Exp e, e1;
      BackendDAE.Variables vars, knvars;
      HashTable.HashTable ht;
      DAE.ComponentRef cr;
      list<DAE.Exp> expl;
      list<DAE.Var> varLst;

    // special case for time, it is never part of the equation system
    case (DAE.CREF(componentRef = DAE.CREF_IDENT(ident="time")), _)
      then (inExp, inTuple);

    // special case for records
    case (e as DAE.CREF(componentRef = cr, ty= DAE.T_COMPLEX(varLst=varLst, complexClassType=ClassInf.RECORD(_))), _)
      equation
        expl = List.map1(varLst, Expression.generateCrefsExpFromExpVar, cr);
        (_, outTuple) = Expression.traverseExpList(expl, checkEquationsUnknownCrefsExp, inTuple);
      then (e, outTuple);

    // special case for arrays
    case (e as DAE.CREF(ty = DAE.T_ARRAY()), _)
      equation
        (e1, true) = Expression.extendArrExp(e, false);
        (_, outTuple) = Expression.traverseExpBottomUp(e1, checkEquationsUnknownCrefsExp, inTuple);
      then (e, outTuple);

    // case for function pointers
    case (DAE.CREF(ty=DAE.T_FUNCTION_REFERENCE_FUNC()), _)
      then (inExp, inTuple);

    // already there
    case (DAE.CREF(componentRef = cr), (_, _, ht)) guard BaseHashTable.hasKey(cr, ht)
      then (inExp, inTuple);

    // known
    case (DAE.CREF(componentRef = cr), (vars, _, _))
      equation
       (_, _) = BackendVariable.getVar(cr, vars);
      then (inExp, inTuple);

    case (DAE.CREF(componentRef = cr), (_, knvars, _))
      equation
        (_, _) = BackendVariable.getVar(cr, knvars);
      then (inExp, inTuple);

    // add it
    case (DAE.CREF(componentRef = cr), (vars, knvars, ht))
      equation
        ht = BaseHashTable.add((cr, 0), ht);
      then (inExp, (vars, knvars, ht));

    else (inExp,inTuple);
  end matchcontinue;
end checkEquationsUnknownCrefsExp;

public function traverseExpsOfEquationList<ArgT> "author: Frenkel TUD 2010-11
  Traverse all expressions of a list of Equations. It is possible to change the
  equations and the multidim equations and the algorithms."
  input list<BackendDAE.Equation> inEquations;
  input FuncExpType func;
  input ArgT inArg;
  output list<BackendDAE.Equation> outEquations = {};
  output ArgT outArg = inArg;

  partial function FuncExpType
    input output DAE.Exp inoutExp;
    input output ArgT inoutArg;
  end FuncExpType;
algorithm
  for eq in inEquations loop
    (eq, outArg) := traverseExpsOfEquation(eq, func, outArg);
    outEquations := eq::outEquations;
  end for;

  outEquations := MetaModelica.Dangerous.listReverseInPlace(outEquations);
end traverseExpsOfEquationList;

public function traverseExpsOfEquationList_WithStop "author: Frenkel TUD 2012-09
  Traverses all expressions of a list of equations.
  It is possible to change the equations."
  replaceable type Type_a subtypeof Any;
  input list<BackendDAE.Equation> inEquations;
  input FuncExpType inFunc;
  input Type_a inTypeA;
  output Boolean outBoolean = true;
  output Type_a outTypeA = inTypeA;
  partial function FuncExpType
    input DAE.Exp inExp;
    input Type_a inTypeA;
    output DAE.Exp outExp;
    output Boolean cont;
    output Type_a outA;
  end FuncExpType;
algorithm
  for eqn in inEquations loop
    (outBoolean, outTypeA) := traverseExpsOfEquation_WithStop(eqn, inFunc, outTypeA);
    if not outBoolean then
      break;
    end if;
  end for;
end traverseExpsOfEquationList_WithStop;

public function traverseExpsOfEquationList_WithoutChange<ArgT> "
  Traverse all expressions of a list of Equations"
  input BackendDAE.Equation inEquation;
  input FuncExpType func;
  input ArgT inArg;
  output ArgT outArg = inArg;

  partial function FuncExpType
    input output DAE.Exp inoutExp;
    input output ArgT inoutArg;
  end FuncExpType;
algorithm
  (_, outArg) := traverseExpsOfEquation(inEquation, func, outArg);
end traverseExpsOfEquationList_WithoutChange;


protected function traverseExpsOfEquationListList_WithStop<T> "author: Frenkel TUD 2012-09
  Traverses all expressions of a list of equations.
  It is possible to change the equations."
  input list<list<BackendDAE.Equation>> inEquations;
  input FuncExpType func;
  input T inTypeA;
  output Boolean outBoolean = true;
  output T outTypeA = inTypeA;

  partial function FuncExpType
    input DAE.Exp inExp;
    input T inTypeA;
    output DAE.Exp outExp;
    output Boolean cont;
    output T outA;
  end FuncExpType;
algorithm

  for eqn in inEquations loop
    (outBoolean, outTypeA) := traverseExpsOfEquationList_WithStop(eqn, func, outTypeA);
    if not outBoolean then
      break;
    end if;
  end for;

end traverseExpsOfEquationListList_WithStop;

public function traverseExpsOfEquation<T> "author: Frenkel TUD 2010-11
  Traverses all expressions of an equation.
  It is possible to change the equation."
  input BackendDAE.Equation inEquation;
  input FuncExpType inFunc;
  input T inTypeA;
  output BackendDAE.Equation outEquation;
  output T outTypeA;

  partial function FuncExpType
    input output DAE.Exp inoutExp;
    input output T inoutTypeA;
  end FuncExpType;
algorithm
  (outEquation, outTypeA) := match(inEquation)
    local
      DAE.Exp e1, e2, e_1, e_2, start, stop, iter;
      list<DAE.Exp> expl;
      DAE.Type tp;
      DAE.ComponentRef cr, cr1;
      BackendDAE.WhenEquation we, we_1;
      DAE.ElementSource source;
      Integer size;
      T extArg;
      list<Integer> dimSize;
      DAE.Algorithm alg;
      list<DAE.Statement> stmts, stmts1;
      list<BackendDAE.Equation> eqns;
      list<list<BackendDAE.Equation>> eqnslst;
      DAE.Expand crefExpand;
      BackendDAE.EquationAttributes attr;

    case BackendDAE.EQUATION(exp=e1, scalar=e2, source=source, attr=attr) equation
      (e_1, extArg) = inFunc(e1, inTypeA);
      (e_2, extArg) = inFunc(e2, extArg);
    then (if referenceEq(e1,e_1) and referenceEq(e2,e_2) then inEquation else BackendDAE.EQUATION(e_1, e_2, source, attr), extArg);

    case BackendDAE.ARRAY_EQUATION(dimSize=dimSize, left=e1, right=e2, source=source, attr=attr) equation
      (e_1, extArg) = inFunc(e1, inTypeA);
      (e_2, extArg) = inFunc(e2, extArg);
    then (if referenceEq(e1,e_1) and referenceEq(e2,e_2) then inEquation else BackendDAE.ARRAY_EQUATION(dimSize, e_1, e_2, source, attr), extArg);

    case BackendDAE.SOLVED_EQUATION(componentRef=cr, exp=e2, source=source, attr=attr) equation
      tp = Expression.typeof(e2);
      e1 = Expression.makeCrefExp(cr, tp);
      (DAE.CREF(cr1, _), extArg) = inFunc(e1, inTypeA);
      (e_2, extArg) = inFunc(e2, extArg);
    then (if referenceEq(cr,cr1) and referenceEq(e2,e_2) then inEquation else BackendDAE.SOLVED_EQUATION(cr1, e_2, source, attr), extArg);

    case BackendDAE.RESIDUAL_EQUATION(exp=e1, source=source, attr=attr) equation
      (e_1, extArg) = inFunc(e1, inTypeA);
    then (if referenceEq(e1,e_1) then inEquation else BackendDAE.RESIDUAL_EQUATION(e_1, source, attr), extArg);

    case BackendDAE.WHEN_EQUATION(size=size, whenEquation= we, source=source, attr=attr) equation
      (we_1, extArg) = traverseExpsOfWhenEquation(we, inFunc, inTypeA);
    then (if referenceEq(we,we_1) then inEquation else BackendDAE.WHEN_EQUATION(size, we_1, source, attr), extArg);

    case BackendDAE.ALGORITHM(size=size, alg=DAE.ALGORITHM_STMTS(statementLst = stmts), source=source, expand=crefExpand, attr=attr) equation
      (stmts1, extArg) = DAEUtil.traverseDAEEquationsStmts(stmts, inFunc, inTypeA);
    then (if referenceEq(stmts,stmts1) then inEquation else BackendDAE.ALGORITHM(size, DAE.ALGORITHM_STMTS(stmts1), source, crefExpand, attr), extArg);

    case BackendDAE.COMPLEX_EQUATION(size=size, left=e1, right=e2, source=source, attr=attr) equation
      (e_1, extArg) = inFunc(e1, inTypeA);
      (e_2, extArg) = inFunc(e2, extArg);
    then (if referenceEq(e1,e_1) and referenceEq(e2,e_2) then inEquation else BackendDAE.COMPLEX_EQUATION(size, e_1, e_2, source, attr), extArg);

    case BackendDAE.IF_EQUATION(conditions=expl, eqnstrue=eqnslst, eqnsfalse=eqns, source=source, attr=attr) equation
      (expl, extArg) = traverseExpsOfExpList(expl, inFunc, inTypeA);
      (eqnslst, extArg) = List.map1Fold(eqnslst, traverseExpsOfEquationList, inFunc, extArg);
      (eqns, extArg) = List.map1Fold(eqns, traverseExpsOfEquation, inFunc, extArg);
    then (BackendDAE.IF_EQUATION(expl, eqnslst, eqns, source, attr), extArg);

    case BackendDAE.FOR_EQUATION(iter=iter,start=start,stop=stop,left=e1, right=e2, source=source, attr=attr) equation
      (e_1, extArg) = inFunc(e1, inTypeA);
      (e_2, extArg) = inFunc(e2, extArg);
    then (if referenceEq(e1,e_1) and referenceEq(e2,e_2) then inEquation else BackendDAE.FOR_EQUATION(iter,start,stop,e_1,e_2,source,attr), extArg);
  end match;
end traverseExpsOfEquation;


public function traverseExpsOfEquation_WithStop<T> "author: Frenkel TUD 2010-11
  Traverses all expressions of a equation.
  It is not possible to change the equation."
  input BackendDAE.Equation inEquation;
  input FuncExpType func;
  input T inTypeA;
  output Boolean outBoolean;
  output T outTypeA;

  partial function FuncExpType
    input DAE.Exp inExp;
    input T inTypeA;
    output DAE.Exp outExp;
    output Boolean cont;
    output T outA;
  end FuncExpType;
algorithm
  (outBoolean, outTypeA) := match (inEquation, func, inTypeA)
    local
      DAE.Exp e1, e2, cond;
      list<DAE.Exp> expl;
      DAE.Type tp;
      DAE.ComponentRef cr;
      BackendDAE.WhenEquation we;
      DAE.ElementSource source;
      Integer size;
      T ext_arg, ext_arg_1, ext_arg_2, ext_arg_3;
      list<Integer> dimSize;
      DAE.Algorithm alg;
      list<DAE.Statement> stmts;
      list<BackendDAE.Equation> eqns;
      list<list<BackendDAE.Equation>> eqnslst;
      Boolean b, b1, b2, b3, b4;
      BackendDAE.EquationAttributes eqAttr;

    case (BackendDAE.EQUATION(exp=e1, scalar=e2), _, _)
      equation
        (_, b, ext_arg) = func(e1, inTypeA);
        if b then
          (_, b, ext_arg) = func(e2, ext_arg);
        end if;
      then (b, ext_arg);

    case (BackendDAE.ARRAY_EQUATION(left=e1, right=e2), _, _)
      equation
        (_, b, ext_arg) = func(e1, inTypeA);
        if b then
          (_, b, ext_arg) = func(e2, ext_arg);
        end if;
      then (b, ext_arg);

    case (BackendDAE.SOLVED_EQUATION(componentRef=cr, exp=e2), _, _)
      equation
        tp = Expression.typeof(e2);
        e1 = Expression.makeCrefExp(cr, tp);
        (_, b, ext_arg) = func(e1, inTypeA);
        if b then
          (_, b, ext_arg) = func(e2, ext_arg);
        end if;
      then (b, ext_arg);

    case (BackendDAE.RESIDUAL_EQUATION(exp=e1), _, _)
      equation
        (_, b, ext_arg) = func(e1, inTypeA);
      then (b, ext_arg);

    case (BackendDAE.WHEN_EQUATION(whenEquation= we), _, _)
      equation
        (b, ext_arg) = traverseExpsOfWhenEquation_WithStop(we, func, inTypeA);
    then (b, ext_arg);

    case (BackendDAE.ALGORITHM(alg=DAE.ALGORITHM_STMTS()), _, _)
      equation
        true = Flags.isSet(Flags.FAILTRACE);
        Debug.traceln("not implemented error - BackendDAE.ALGORITHM - BackendEquation.traverseExpsOfEquation_WithStop\n");
        //(_, ext_arg_1) = DAEUtil.traverseDAEEquationsStmts(statementLst, func, inTypeA);
      then fail();

    //(true, inTypeA);
    case (BackendDAE.COMPLEX_EQUATION(left=e1, right=e2), _, _)
      equation
        (_, b, ext_arg) = func(e1, inTypeA);
        if b then
          (_, b, ext_arg) = func(e2, ext_arg);
        end if;
      then (b, ext_arg);

    case (BackendDAE.IF_EQUATION(conditions=expl, eqnstrue=eqnslst, eqnsfalse=eqns), _, _)
      equation
        (b, ext_arg) = traverseExpsOfExpList_WithStop(expl, func, inTypeA);
        if b then
          (b, ext_arg) = traverseExpsOfEquationListList_WithStop(eqnslst, func, ext_arg);
        end if;
        if b then
          (b, ext_arg) = traverseExpsOfEquationList_WithStop(eqns, func, ext_arg);
        end if;
      then (b, ext_arg);
  end match;
end traverseExpsOfEquation_WithStop;

protected function traverseExpsOfWhenEquation<T>
"Traverses all expressions of a when equation.
  Helper function of traverseExpsOfEquation."
  input BackendDAE.WhenEquation inWhenEquation;
  input FuncExpType inFunc;
  input T inTypeA;
  output BackendDAE.WhenEquation outWhenEquation;
  output T outTypeA;

  partial function FuncExpType
    input output DAE.Exp inoutExp;
    input output T inoutTypeA;
  end FuncExpType;
algorithm
  (outWhenEquation, outTypeA) := match(inWhenEquation)
  local
    BackendDAE.WhenEquation we, elsewe;
    Option<BackendDAE.WhenEquation> oelsewe;
    DAE.Exp cond;
    list<BackendDAE.WhenOperator> whenStmtLst;
    T extArg;

    case BackendDAE.WHEN_STMTS(condition=cond, whenStmtLst=whenStmtLst, elsewhenPart = oelsewe)
      equation
        (cond, extArg) = inFunc(cond, inTypeA);
        (whenStmtLst, extArg) = traverseExpsOfWhenOps(whenStmtLst, inFunc, extArg, {});

        if isSome(oelsewe) then
          SOME(elsewe) = oelsewe;
          (elsewe, extArg) = traverseExpsOfWhenEquation(elsewe, inFunc, extArg);
          oelsewe = SOME(elsewe);
        else
          oelsewe = NONE();
        end if;
      then (BackendDAE.WHEN_STMTS(cond, whenStmtLst, oelsewe), extArg);
  end match;
end traverseExpsOfWhenEquation;

protected function traverseExpsOfWhenOps<T>
"Traverses all expressions of a when equation.
  Helper function of traverseExpsOfEquation."
  input list<BackendDAE.WhenOperator> inWhenOps;
  input FuncExpType inFunc;
  input T inTypeA;
  input list<BackendDAE.WhenOperator> inAccum;
  output list<BackendDAE.WhenOperator> outWhenOps;
  output T outTypeA;

  partial function FuncExpType
    input output DAE.Exp inExp;
    input output T inTypeA;
  end FuncExpType;
algorithm
  (outWhenOps, outTypeA) := match (inWhenOps)
    local
      DAE.Type tp;
      DAE.Exp e1, e2, level;
      DAE.ComponentRef cr, cr1;
      list<BackendDAE.WhenOperator> rest;
      T extArg;
      DAE.ElementSource source;

    case {} then (listReverse(inAccum),inTypeA);
    case (BackendDAE.ASSIGN(left = e1, right = e2, source = source)::rest)
      equation
        _ = Expression.typeof(e2);
        (e1, extArg) = inFunc(e1, inTypeA);
        (e2, extArg) = inFunc(e2, extArg);
        (outWhenOps, extArg) = traverseExpsOfWhenOps(rest, inFunc, extArg,  BackendDAE.ASSIGN(e1, e2, source)::inAccum);
      then (outWhenOps, extArg);
    case (BackendDAE.REINIT(stateVar = cr, value = e2,  source = source)::rest)
      equation
        e1 = Expression.crefExp(cr);
        (e1, extArg) = inFunc(e1, inTypeA);
        if Expression.isCref(e1) then
          DAE.CREF(cr1, _) = e1;
        else
          cr1=cr;
        end if;
        (e2, extArg) = inFunc(e2, extArg);
        (outWhenOps, extArg) = traverseExpsOfWhenOps(rest, inFunc, extArg,  BackendDAE.REINIT(cr1, e2, source)::inAccum);
    then (outWhenOps, extArg);
    case (BackendDAE.ASSERT(condition = e1, message = e2, level = level,  source = source)::rest)
      equation
        (e1, extArg) = inFunc(e1, inTypeA);
        (e2, extArg) = inFunc(e2, extArg);
        (outWhenOps, extArg) = traverseExpsOfWhenOps(rest, inFunc, extArg,  BackendDAE.ASSERT(e1, e2, level, source)::inAccum);
    then (outWhenOps, extArg);
    case (BackendDAE.TERMINATE(message = e1,  source = source)::rest)
      equation
        (e1, extArg) = inFunc(e1, inTypeA);
        (outWhenOps, extArg) = traverseExpsOfWhenOps(rest, inFunc, extArg,  BackendDAE.TERMINATE(e1, source)::inAccum);
    then (outWhenOps, extArg);
    case (BackendDAE.NORETCALL(exp = e1,  source = source)::rest)
      equation
        (e1, extArg) = inFunc(e1, inTypeA);
        (outWhenOps, extArg) = traverseExpsOfWhenOps(rest, inFunc, extArg,  BackendDAE.NORETCALL(e1, source)::inAccum);
    then (outWhenOps, extArg);
  end match;
end traverseExpsOfWhenOps;

protected function traverseExpsOfWhenEquation_WithStop<T>
"Traverses all expressions of a when equation.
  Helper function of traverseExpsOfEquation."
  input BackendDAE.WhenEquation inWhenEquation;
  input FuncExpType inFunc;
  input T inTypeA;
  output Boolean outCont;
  output T outTypeA;

  partial function FuncExpType
    input DAE.Exp inExp;
    input T inTypeA;
    output DAE.Exp outExp;
    output Boolean cont;
    output T outTypeA;
  end FuncExpType;
algorithm
  (outCont, outTypeA) := match(inWhenEquation)
  local
    BackendDAE.WhenEquation elsewe;
    Option<BackendDAE.WhenEquation> oelsewe;
    DAE.Exp cond;
    list<BackendDAE.WhenOperator> whenStmtLst;
    T extArg;
    Boolean b;

    case BackendDAE.WHEN_STMTS(condition=cond, whenStmtLst=whenStmtLst, elsewhenPart = oelsewe)
      equation
        (_, b, extArg) = inFunc(cond, inTypeA);
        if b then
          (b, extArg) = traverseExpsOfWhenOps_WithStop(whenStmtLst, inFunc, extArg, b);
        end if;

        if b then
          if isSome(oelsewe) then
            SOME(elsewe) = oelsewe;
            (b, extArg) = traverseExpsOfWhenEquation_WithStop(elsewe, inFunc, extArg);
          end if;
        end if;
      then (b, extArg);
  end match;
end traverseExpsOfWhenEquation_WithStop;

public function statementEq
"Creates equation from one statement."
  input DAE.Statement iStmts;
  output BackendDAE.Equation oEq;
algorithm
  (oEq) := match(iStmts)
    local
      DAE.ComponentRef cr;
      DAE.Exp exp;
      list<DAE.Exp> explst;

    case (DAE.STMT_ASSIGN(exp1 = DAE.CREF(componentRef = cr), exp = exp))
      then generateEquation(Expression.crefExp(cr), exp);

    case (DAE.STMT_ASSIGN_ARR(lhs = DAE.CREF(componentRef = cr), exp = exp))
      then generateEquation(Expression.crefExp(cr), exp);

    case (DAE.STMT_TUPLE_ASSIGN(expExpLst = explst, exp = exp))
      then generateEquation(Expression.makeTuple(explst), exp);

  end match;
end statementEq;

protected function traverseExpsOfWhenOps_WithStop<T>
"Traverses all expressions of a when equation.
  Helper function of traverseExpsOfEquation."
  input list<BackendDAE.WhenOperator> inWhenOps;
  input FuncExpType inFunc;
  input T inTypeA;
  input Boolean inCont;
  output Boolean outCont;
  output T outTypeA;

  partial function FuncExpType
    input DAE.Exp inExp;
    input T inTypeA;
    output DAE.Exp outExp;
    output Boolean cont;
    output T outTypeA;
  end FuncExpType;
algorithm
  (outCont, outTypeA) := match (inWhenOps)
    local
      DAE.Type tp;
      DAE.Exp e1, e2, level;
      DAE.ComponentRef cr, cr1;
      list<BackendDAE.WhenOperator> rest;
      T extArg;
      DAE.ElementSource source;
      Boolean b = false;

    case {} then (inCont,inTypeA);
    case (BackendDAE.ASSIGN(left = e1, right = e2)::rest)
      equation
        _ = Expression.typeof(e2);
        if inCont then
         (_, b, extArg) = inFunc(e1, inTypeA);
        end if;
        if b then
         (_, b, extArg) = inFunc(e2, extArg);
        end if;
        (b, extArg) = traverseExpsOfWhenOps_WithStop(rest, inFunc, extArg,  b);
      then (b, extArg);

    case (BackendDAE.REINIT(stateVar = cr, value = e2)::rest)
      equation
        tp = Expression.typeof(e2);
        e1 = Expression.makeCrefExp(cr, tp);
        if inCont then
         (_, b, extArg) = inFunc(e1, inTypeA);
        end if;
        if b then
         (_, b, extArg) = inFunc(e2, extArg);
        end if;
        (b, extArg) = traverseExpsOfWhenOps_WithStop(rest, inFunc, extArg,  b);
      then (b, extArg);

    case (BackendDAE.ASSERT(condition = e1, message = e2)::rest)
      equation
        if inCont then
         (_, b, extArg) = inFunc(e1, inTypeA);
        end if;
        if b then
         (_, b, extArg) = inFunc(e2, extArg);
        end if;
        (b, extArg) = traverseExpsOfWhenOps_WithStop(rest, inFunc, extArg,  b);
      then (b, extArg);

    case (BackendDAE.TERMINATE(message = e1)::rest)
      equation
        if inCont then
         (_, b, extArg) = inFunc(e1, inTypeA);
        end if;
        (b, extArg) = traverseExpsOfWhenOps_WithStop(rest, inFunc, extArg,  b);
      then (b, extArg);

    case (BackendDAE.NORETCALL(exp = e1)::rest)
      equation
        if inCont then
         (_, b, extArg) = inFunc(e1, inTypeA);
        end if;
        (b, extArg) = traverseExpsOfWhenOps_WithStop(rest, inFunc, extArg,  b);
      then (b, extArg);
  end match;
end traverseExpsOfWhenOps_WithStop;

protected function traverseExpsOfExpList<T> "author Frenkel TUD:
  Calls user function for each element of list."
  input list<DAE.Exp> inExpl;
  input FuncExpType rel;
  input T inExtArg;
  output list<DAE.Exp> outExpl = {};
  output T outTypeA = inExtArg;

  partial function FuncExpType
    input output DAE.Exp inoutExp;
    input output T inoutTypeA;
  end FuncExpType;
algorithm

  for e in inExpl loop
    (e, outTypeA) := rel(e, outTypeA);
    outExpl := e :: outExpl;
  end for;

   outExpl := MetaModelica.Dangerous.listReverseInPlace(outExpl);
end traverseExpsOfExpList;

protected function traverseExpsOfExpList_WithStop<T> "author Frenkel TUD
  Calls user function for each element of list."
  input list<DAE.Exp> inExpl;
  input FuncExpType rel;
  input T inExtArg;
  output Boolean outBoolean = true;
  output T outTypeA = inExtArg;
  partial function FuncExpType
    input DAE.Exp inExp;
    input T inTypeA;
    output DAE.Exp outExp;
    output Boolean cont;
    output T outA;
  end FuncExpType;
algorithm
  for e in inExpl loop
    (_, outBoolean, outTypeA) := rel(e, outTypeA);
    if not outBoolean then
      break;
    end if;
  end for;
end traverseExpsOfExpList_WithStop;

public function equationEqual
  "Returns true if two equations are equal"
  input BackendDAE.Equation e1;
  input BackendDAE.Equation e2;
  output Boolean res = true;
algorithm
  if referenceEq(e1, e2) then return; end if;
  res := match (e1, e2)
    local
      DAE.Exp e11, e12, e21, e22, exp1, exp2;
      DAE.ComponentRef cr1, cr2;
      DAE.Algorithm alg1, alg2;
      list<DAE.Exp> explst1, explst2;

    case (BackendDAE.EQUATION(exp=e11, scalar=e12), BackendDAE.EQUATION(exp=e21, scalar=e22)) equation
      res = boolAnd(Expression.expEqual(e11, e21), Expression.expEqual(e12, e22));
    then res;

    case (BackendDAE.ARRAY_EQUATION(left=e11, right=e12), BackendDAE.ARRAY_EQUATION(left=e21, right=e22)) equation
      res = boolAnd(Expression.expEqual(e11, e21), Expression.expEqual(e12, e22));
    then res;

    case (BackendDAE.COMPLEX_EQUATION(left=e11, right=e12), BackendDAE.COMPLEX_EQUATION(left=e21, right=e22)) equation
      res = boolAnd(Expression.expEqual(e11, e21), Expression.expEqual(e12, e22));
    then res;

    case (BackendDAE.SOLVED_EQUATION(componentRef=cr1, exp=exp1), BackendDAE.SOLVED_EQUATION(componentRef=cr2, exp=exp2)) equation
      res = boolAnd(ComponentReference.crefEqualNoStringCompare(cr1, cr2), Expression.expEqual(exp1, exp2));
    then res;

    case (BackendDAE.RESIDUAL_EQUATION(exp=exp1), BackendDAE.RESIDUAL_EQUATION(exp=exp2)) equation
      res = Expression.expEqual(exp1, exp2);
    then res;

    case (BackendDAE.ALGORITHM(alg=alg1), BackendDAE.ALGORITHM(alg=alg2)) equation
      explst1 = Algorithm.getAllExps(alg1);
      explst2 = Algorithm.getAllExps(alg2);
      res = List.isEqualOnTrue(explst1, explst2, Expression.expEqual);
    then res;

    case (BackendDAE.WHEN_EQUATION(whenEquation = BackendDAE.WHEN_STMTS(whenStmtLst={BackendDAE.ASSIGN(left=e11, right=e12)})), BackendDAE.WHEN_EQUATION(whenEquation = BackendDAE.WHEN_STMTS(whenStmtLst={BackendDAE.ASSIGN(left=e21, right=e22)}))) equation
      res = boolAnd(Expression.expEqual(e11, e21), Expression.expEqual(e12, e22));
    then res;

    else false;
  end match;
end equationEqual;

public function equationAddDAE "author: Frenkel TUD 2011-05"
  input BackendDAE.Equation inEquation;
  input BackendDAE.EqSystem inEqSystem;
  output BackendDAE.EqSystem outEqSystem;
algorithm
  outEqSystem := BackendDAEUtil.setEqSystEqs(inEqSystem, add(inEquation, inEqSystem.orderedEqs));
  outEqSystem.matching := BackendDAE.NO_MATCHING();
end equationAddDAE;

public function equationsAddDAE "author: Frenkel TUD 2011-05"
  input list<BackendDAE.Equation> inEquations;
  input BackendDAE.EqSystem inEqSystem;
  output BackendDAE.EqSystem outEqSystem = inEqSystem;
algorithm
  outEqSystem.orderedEqs := addList(inEquations, outEqSystem.orderedEqs);
  outEqSystem.matching := BackendDAE.NO_MATCHING();
end equationsAddDAE;

public function requationsAddDAE "author: Frenkel TUD 2012-10
  Add a list of equations to removed equations of a BackendDAE.
  If the variable already exists, the function updates the variable."
  input list<BackendDAE.Equation> inEquations;
  input BackendDAE.EqSystem inSyst;
  output BackendDAE.EqSystem outSyst;
algorithm
  outSyst := if listEmpty(inEquations) then inSyst
    else BackendDAEUtil.setEqSystRemovedEqns(inSyst, addList(inEquations, inSyst.removedEqs));
end requationsAddDAE;

public function removeRemovedEqs
  "Remove all removedEqs."
  input output BackendDAE.EqSystem eqSystem;
algorithm
  ExpandableArray.clear(eqSystem.removedEqs);
end removeRemovedEqs;

public function equationToScalarResidualForm "author: Frenkel TUD 2012-06
  This function transforms an equation to its scalar residual form.
  For instance, a=b is transformed to a-b=0, and the instance {a[1], a[2]}=b to a[1]=b[1] and a[2]=b[2]"
  input BackendDAE.Equation inEquation;
  input DAE.FunctionTree funcTree;
  output list<BackendDAE.Equation> outEquations;
algorithm

  outEquations := match (inEquation)
    local
      DAE.Exp e, e1, e2, exp;
      DAE.ComponentRef cr;
      DAE.ElementSource source;
      BackendDAE.Equation backendEq;
      list<Integer> ds;
      Integer size;
      list<DAE.Exp> explst, explst2;
      list<BackendDAE.Equation> eqns;
      list<list<DAE.Subscript>> subslst;
      Real r;
      BackendDAE.EquationAttributes attr;
      list<DAE.ComponentRef> crlst, crlst2;

    case (BackendDAE.EQUATION(exp=DAE.TUPLE(explst), scalar=e2, source=source, attr=attr)) equation
      ((_, eqns)) = List.fold3(explst,equationTupleToScalarResidualForm, e2, source, attr, (1, {}));
    then eqns;

    case (BackendDAE.EQUATION(exp= DAE.RCONST(real= 0.0), scalar=e2, source=source, attr=attr)) equation
      (e, _) = ExpressionSimplify.simplify(e2);
      eqns ={BackendDAE.RESIDUAL_EQUATION(e, source, attr)};
    then eqns;

    case (BackendDAE.EQUATION(exp=e1, scalar= DAE.RCONST(real=0.0), source=source, attr=attr)) equation
      (e, _) = ExpressionSimplify.simplify(e1);
      eqns ={BackendDAE.RESIDUAL_EQUATION(e, source, attr)};
    then eqns;

    case (BackendDAE.EQUATION(exp=e1, scalar=e2, source=source, attr=attr)) equation
      exp = Expression.createResidualExp(e1, e2);
    then {BackendDAE.RESIDUAL_EQUATION(exp, source, attr)};

    case (BackendDAE.SOLVED_EQUATION(componentRef=cr, exp=e2, source=source, attr=attr)) equation
      e1 = Expression.crefExp(cr);
      exp = Expression.createResidualExp(e1, e2);
    then {BackendDAE.RESIDUAL_EQUATION(exp, source, attr)};

    case (BackendDAE.ARRAY_EQUATION(dimSize=ds, left=e1, right=e2, source=source, attr=attr)) equation
      exp = Expression.createResidualExp(e1, e2);
      subslst = Expression.dimensionSizesSubscripts(ds);
      subslst = Expression.rangesToSubscripts(subslst);
      explst = List.map1r(subslst, Expression.applyExpSubscripts, exp);
      explst = ExpressionSimplify.simplifyList(explst);
      eqns = List.map2(explst, generateRESIDUAL_EQUATION, source, attr);
    then eqns;

    case (BackendDAE.COMPLEX_EQUATION(left=e1 as DAE.CALL(expLst=explst),
          right=e2, source=source, attr=attr))
    guard(Expression.isRecordCall(e1, funcTree) and Expression.isCref(e2))
    equation
      crlst = ComponentReference.expandCref(Expression.expCref(e2),true);
      explst2 = list(Expression.crefExp(c) for c in crlst);
      explst = List.threadMap(explst, explst2, Expression.createResidualExp);
      eqns = List.map2(explst, generateRESIDUAL_EQUATION, source, attr);
    then eqns;

    case (BackendDAE.COMPLEX_EQUATION(right=e1 as DAE.CALL(expLst=explst),
          left=e2, source=source, attr=attr))
    guard(Expression.isRecordCall(e1, funcTree) and Expression.isCref(e2))
    equation
      crlst = ComponentReference.expandCref(Expression.expCref(e2),true);
      explst2 = list(Expression.crefExp(c) for c in crlst);
      explst = List.threadMap(explst, explst2, Expression.createResidualExp);
      eqns = List.map2(explst, generateRESIDUAL_EQUATION, source, attr);
    then eqns;

    case (BackendDAE.COMPLEX_EQUATION(left=e1, right=e2, source=source, attr=attr))
    guard(Expression.isCref(e1) and Expression.isCref(e2))
    equation
      crlst = ComponentReference.expandCref(Expression.expCref(e1),true);
      crlst2 = ComponentReference.expandCref(Expression.expCref(e2),true);
      explst = list(Expression.crefExp(c) for c in crlst);
      explst2 = list(Expression.crefExp(c) for c in crlst2);
      explst = List.threadMap(explst, explst2, Expression.createResidualExp);
      eqns = List.map2(explst, generateRESIDUAL_EQUATION, source, attr);
    then eqns;

    case (BackendDAE.COMPLEX_EQUATION(left=e1, right=e2, source=source, attr=attr)) equation
      exp = Expression.createResidualExp(e1, e2);
    then {BackendDAE.RESIDUAL_EQUATION(exp, source, attr)};

    case (backendEq as BackendDAE.RESIDUAL_EQUATION())
    then {backendEq};

    case (backendEq as BackendDAE.ALGORITHM())
    then {backendEq};

    case (backendEq as BackendDAE.WHEN_EQUATION())
    then {backendEq};

    else
      equation
        true = Flags.isSet(Flags.FAILTRACE);
        BackendDump.printEquation(inEquation);
        Debug.trace("- BackendDAE.equationToScalarResidualForm failed\n");
      then fail();
  end match;
end equationToScalarResidualForm;

protected function equationTupleToScalarResidualForm "
  Tuple-expressions (function calls) that need to be converted to residual form
  are scalarized in a stupid, straight-forward way."
  input DAE.Exp cr;
  input DAE.Exp exp;
  input DAE.ElementSource inSource;
  input BackendDAE.EquationAttributes inEqAttr;
  input tuple<Integer, list<BackendDAE.Equation>> inTpl;
  output tuple<Integer, list<BackendDAE.Equation>> outTpl;
algorithm
  outTpl := match (cr, exp, inSource, inEqAttr, inTpl)
    local
      Integer i;
      list<BackendDAE.Equation> eqs;
      String str;
      DAE.Exp e;

    // wild-card does not produce a residual
    case (DAE.CREF(componentRef=DAE.WILD()), _, _, _, (i, eqs))
    then ((i+1, eqs));

    // 0-length arrays do not produce a residual
    case (DAE.ARRAY(array={}), _, _, _, (i, eqs))
    then ((i+1, eqs));

    // a scalar real
    case (DAE.CREF(ty=DAE.T_REAL()), _, _, _, (i, eqs)) equation
      eqs = BackendDAE.RESIDUAL_EQUATION(DAE.TSUB(exp, i, DAE.T_REAL_DEFAULT), inSource, inEqAttr)::eqs;
    then ((i+1, eqs));

    // create a sum for arrays...
    case (DAE.CREF(ty=DAE.T_ARRAY(ty=DAE.T_REAL())), _, _, _, (i, eqs)) equation
      e = Expression.makePureBuiltinCall("sum", {DAE.TSUB(exp, i, DAE.T_REAL_DEFAULT)}, DAE.T_REAL_DEFAULT);
      eqs = BackendDAE.RESIDUAL_EQUATION(e, inSource, inEqAttr)::eqs;
    then ((i+1, eqs));

    case (_, _, _, _, (i, _)) equation
      str = "BackendEquation.equationTupleToScalarResidualForm failed: " + intString(i) + ": " + ExpressionDump.printExpStr(cr);
      Error.addSourceMessage(Error.INTERNAL_ERROR, {str}, ElementSource.getElementSourceFileInfo(inSource));
    then fail();
  end match;
end equationTupleToScalarResidualForm;

public function equationToResidualForm "author: PA
  This function transforms an equation to its residual form.
  For instance, a=b is transformed to a-b=0"
  input BackendDAE.Equation inEquation;
  output BackendDAE.Equation outEquation;
algorithm
  outEquation := matchcontinue (inEquation)
    local
      DAE.Exp e, e1, e2, exp;
      DAE.ComponentRef cr;
      DAE.ElementSource source;
      BackendDAE.Equation backendEq;
      BackendDAE.EquationAttributes eqAttr;

    case (BackendDAE.EQUATION(exp=e1, scalar=e2, source=source, attr=eqAttr)) equation
      //ExpressionDump.dumpExpWithTitle("equationToResidualForm 1\n", e2);
      exp = Expression.createResidualExp(e1, e2);
      (e, _) = ExpressionSimplify.simplify(exp);
    then BackendDAE.RESIDUAL_EQUATION(e, source, eqAttr);

    case (BackendDAE.SOLVED_EQUATION(componentRef=cr, exp=e2, source=source, attr=eqAttr)) equation
      e1 = Expression.crefExp(cr);
      exp = Expression.createResidualExp(e1, e2);
      (e, _) = ExpressionSimplify.simplify(exp);
    then BackendDAE.RESIDUAL_EQUATION(e, source, eqAttr);

    case (BackendDAE.ARRAY_EQUATION(left=e1, right=e2, source=source, attr=eqAttr)) equation
      exp = Expression.createResidualExp(e1, e2);
      (e, _) = ExpressionSimplify.simplify(exp);
    then BackendDAE.RESIDUAL_EQUATION(e, source, eqAttr);

    case (BackendDAE.COMPLEX_EQUATION(left=e1, right=e2, source=source, attr=eqAttr)) equation
      exp = Expression.createResidualExp(e1, e2);
      (e, _) = ExpressionSimplify.simplify(exp);
    then BackendDAE.RESIDUAL_EQUATION(e, source, eqAttr);

    case (backendEq as BackendDAE.RESIDUAL_EQUATION())
    then backendEq;

    case (backendEq as BackendDAE.ALGORITHM())
    then backendEq;

    case (backendEq as BackendDAE.WHEN_EQUATION())
    then backendEq;

    else
      equation
        true = Flags.isSet(Flags.FAILTRACE);
        Debug.trace("- BackendDAE.equationToResidualForm failed\n");
      then fail();
  end matchcontinue;
end equationToResidualForm;

public function traverseEquationToScalarResidualForm
"author: Frenkel TUD 2010-11"
  input BackendDAE.Equation inEq;
  input tuple<DAE.FunctionTree, list<BackendDAE.Equation>> inEqs;
  output BackendDAE.Equation outEq;
  output tuple<DAE.FunctionTree, list<BackendDAE.Equation>> outEqs;
algorithm
  (outEq,outEqs) := matchcontinue(inEq,inEqs)
    local
      list<BackendDAE.Equation> eqns,reqn;
      BackendDAE.Equation eqn;
      DAE.FunctionTree funcs;

    case (eqn, (funcs, eqns))
      equation
        reqn = equationToScalarResidualForm(eqn, funcs);
        eqns = listAppend(reqn,eqns);
      then (eqn, (funcs, eqns));

    else (inEq,inEqs);
  end matchcontinue;
end traverseEquationToScalarResidualForm;

public function convertResidualsIntoSolvedEquations
"This function converts residuals into solved equations of the following form:
    e.g.: 0 = a+b -> $res1 = a+b"
  input list<BackendDAE.Equation> inResidualList;
  input String inName;
  input BackendDAE.Var inTemplateVar;
  input Integer inIndex;
  output list<BackendDAE.Equation> outEquationList = {};
  output list<BackendDAE.Var> outVariableList = {};
  output Integer outVarIndex = inIndex;
algorithm
  for eq in inResidualList loop
    _ := match eq
      local
        DAE.Exp exp;
        DAE.ElementSource source "origin of equation";
        BackendDAE.EquationAttributes eqAttr;
        String varName;
        DAE.ComponentRef componentRef;
        DAE.Exp expVarName;
        BackendDAE.Equation currEquation;
        BackendDAE.Var currVariable;

      case BackendDAE.RESIDUAL_EQUATION(exp=exp,source=source,attr=eqAttr)
        equation
          componentRef = DAE.CREF_IDENT(inName + intString(outVarIndex), DAE.T_REAL_DEFAULT, {});
          currEquation = BackendDAE.SOLVED_EQUATION(componentRef, exp, source, eqAttr);
          currVariable = BackendVariable.copyVarNewName(componentRef, inTemplateVar);

          outVarIndex = outVarIndex + 1;
          outEquationList = currEquation::outEquationList;
          outVariableList = currVariable::outVariableList;
        then ();
      else
        equation
          true = Flags.isSet(Flags.FAILTRACE);
          Error.addInternalError("function convertResidualsIntoSolvedEquations failed", sourceInfo());
        then fail();
    end match;
  end for;
  outEquationList := MetaModelica.Dangerous.listReverseInPlace(outEquationList);
  outVariableList := MetaModelica.Dangerous.listReverseInPlace(outVariableList);
end convertResidualsIntoSolvedEquations;

public function equationInfo "
  Retrieve the line number information from a BackendDAE.BackendDAEequation"
  input BackendDAE.Equation eq;
  output SourceInfo info;
algorithm
  info := ElementSource.getElementSourceFileInfo(equationSource(eq));
end equationInfo;

public function markedEquationSource
  input BackendDAE.EqSystem inEqSystem;
  input Integer inPos "one-based indexing";
  output DAE.ElementSource outSource;
algorithm
  outSource := equationSource(get(inEqSystem.orderedEqs, inPos));
end markedEquationSource;

public function equationSource "
  Retrieve the source from a BackendDAE.BackendDAEequation"
  input BackendDAE.Equation eq;
  output DAE.ElementSource source;
algorithm
  source := match eq
    case BackendDAE.EQUATION(source=source) then source;
    case BackendDAE.ARRAY_EQUATION(source=source) then source;
    case BackendDAE.SOLVED_EQUATION(source=source) then source;
    case BackendDAE.RESIDUAL_EQUATION(source=source) then source;
    case BackendDAE.WHEN_EQUATION(source=source) then source;
    case BackendDAE.ALGORITHM(source=source) then source;
    case BackendDAE.COMPLEX_EQUATION(source=source) then source;
    case BackendDAE.IF_EQUATION(source=source) then source;
    else equation
      Error.addInternalError("BackendEquation.equationSource failed!", sourceInfo());
    then fail();
  end match;
end equationSource;

public function equationSizeKeepAlgorithmAsOne
  "Same as equationSize but keeps algorithms as one equation"
  input BackendDAE.Equation eq;
  output Integer osize;
algorithm
  osize := match(eq)
  case BackendDAE.ALGORITHM(_)
    then 1;
  else
  then equationSize(eq);
  end match;
end equationSizeKeepAlgorithmAsOne;

public function equationSize
  "Retrieve the size from a BackendDAE.BackendDAEequation"
  input BackendDAE.Equation eq;
  output Integer osize;
algorithm
  osize := match eq
    local
      list<Integer> ds;
      Integer size, start, stop;
      list<BackendDAE.Equation> eqnsfalse;

    case BackendDAE.EQUATION()
    then 1;

    case BackendDAE.ARRAY_EQUATION(dimSize=ds) equation
      size = List.fold(ds, intMul, 1);
    then size;

    case BackendDAE.SOLVED_EQUATION()
    then 1;

    case BackendDAE.RESIDUAL_EQUATION()
    then 1;

    case BackendDAE.WHEN_EQUATION(size=size)
    then size;

    case BackendDAE.ALGORITHM(size=size)
    then size;

    case BackendDAE.COMPLEX_EQUATION(size=size)
    then size;

    case BackendDAE.IF_EQUATION(eqnsfalse=eqnsfalse) equation
      size = equationLstSize(eqnsfalse);
    then size;

    case BackendDAE.FOR_EQUATION(start=DAE.ICONST(start), stop=DAE.ICONST(stop)) equation
      size = stop-start+1;
    then size;

    else equation
      Error.addInternalError("BackendEquation.equationSize failed!", sourceInfo());
    then fail();
  end match;
end equationSize;

public function isInitialEquation
  input BackendDAE.Equation inEquation;
  output Boolean outBool;
protected
  BackendDAE.EquationKind eqKind;
algorithm
  eqKind := equationKind(inEquation);
  outBool := isInitialEqKind(eqKind);
end isInitialEquation;

public function isInitialEqKind
  input BackendDAE.EquationKind inEqKind;
  output Boolean outBool;
algorithm
  outBool := match (inEqKind)
    case (BackendDAE.INITIAL_EQUATION()) then true;
    else false;
  end match;
end isInitialEqKind;

public function equationKind "Retrieve the kind from a BackendDAE.BackendDAEequation"
  input BackendDAE.Equation inEquation;
  output BackendDAE.EquationKind outEqKind;
algorithm
  outEqKind := match inEquation
    local
      BackendDAE.EquationKind kind;

    case BackendDAE.EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    case BackendDAE.ARRAY_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    case BackendDAE.SOLVED_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    case BackendDAE.RESIDUAL_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    case BackendDAE.WHEN_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    case BackendDAE.ALGORITHM(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    case BackendDAE.COMPLEX_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    case BackendDAE.IF_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(kind=kind)) then kind;
    else equation
      Error.addInternalError("BackendEquation.equationKind failed!", sourceInfo());
    then fail();
  end match;
end equationKind;

public function equationLstSize
  input list<BackendDAE.Equation> inEqns;
  output Integer size = 0;
algorithm
  for eqn in inEqns loop
    size := size + equationSize(eqn);
  end for;
end equationLstSize;

public function equationLstSizeKeepAlgorithmAsOne "author: vwaurich
Gets the scalar size of the equations. Algorithms are handled as single equations."
  input list<BackendDAE.Equation> inEqns;
  output Integer size = 0;
algorithm
  for eqn in inEqns loop
    size := size + equationSizeKeepAlgorithmAsOne(eqn);
  end for;
end equationLstSizeKeepAlgorithmAsOne;

public function generateEquation "author Frenkel TUD 2012-12
  helper to generate an equation from lhs and rhs.
  the type of this function is determined by the lhs.
  This function is called if an equation is found which is not simple"
  input DAE.Exp lhs;
  input DAE.Exp rhs;
  input DAE.ElementSource source = DAE.emptyElementSource;
  input BackendDAE.EquationAttributes inEqAttr = BackendDAE.EQ_ATTR_DEFAULT_UNKNOWN;
  output BackendDAE.Equation outEqn;
protected
  DAE.Type ty;
algorithm
  ty := Expression.typeof(lhs);
  outEqn := match ()
    local
      Integer size;
      DAE.Dimensions dims;
      list<Integer> ds;
      Boolean b1, b2;

    // complex types to complex equations
    case () guard
      DAEUtil.expTypeComplex(ty) or DAEUtil.expTypeTuple(ty)
    equation
      size = Expression.sizeOf(ty);
    then BackendDAE.COMPLEX_EQUATION(size, lhs, rhs, source, inEqAttr);

    // array types to array equations
    case () guard
      DAEUtil.expTypeArray(ty)
    equation
      dims = Expression.arrayDimension(ty);
      ds = Expression.dimensionsSizes(dims);
    then BackendDAE.ARRAY_EQUATION(ds, lhs, rhs, source, inEqAttr);

    // other types
    case () guard
      not DAEUtil.expTypeComplex(ty) and
      not DAEUtil.expTypeArray(ty)
    then BackendDAE.EQUATION(lhs, rhs, source, inEqAttr);

    else equation
      // show only on failtrace!
      true = Flags.isSet(Flags.FAILTRACE);
      Debug.traceln("- BackendEquation.generateEquation failed on: " + ExpressionDump.printExpStr(lhs) + " = " + ExpressionDump.printExpStr(rhs) + "\n");
    then fail();
  end match;
end generateEquation;

public function generateEQUATION "author: Frenkel TUD 2010-05"
  input DAE.Exp iLhs;
  input DAE.Exp iRhs;
  input DAE.ElementSource Source;
  input BackendDAE.EquationKind inEqKind;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := BackendDAE.EQUATION(iLhs, iRhs, Source, BackendDAE.EQUATION_ATTRIBUTES(false, inEqKind));
end generateEQUATION;

public function getEquationAttributes
  input BackendDAE.Equation inEqn;
  output BackendDAE.EquationAttributes outAttr;
algorithm
  outAttr := match inEqn
    local
      BackendDAE.EquationAttributes attr;

    case BackendDAE.EQUATION(attr=attr) then attr;
    case BackendDAE.ARRAY_EQUATION(attr=attr) then attr;
    case BackendDAE.SOLVED_EQUATION(attr=attr) then attr;
    case BackendDAE.RESIDUAL_EQUATION(attr=attr) then attr;
    case BackendDAE.ALGORITHM(attr=attr) then attr;
    case BackendDAE.WHEN_EQUATION(attr=attr) then attr;
    case BackendDAE.COMPLEX_EQUATION(attr=attr) then attr;
    case BackendDAE.IF_EQUATION(attr=attr) then attr;
    case BackendDAE.FOR_EQUATION(attr=attr) then attr;

    else equation
      Error.addInternalError("function getEquationAttributes failed", sourceInfo());
    then fail();
  end match;
end getEquationAttributes;

public function setEquationAttributes
  input BackendDAE.Equation inEqn;
  input BackendDAE.EquationAttributes inAttr;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := match (inEqn, inAttr)
    local
      DAE.ElementSource source;
      list<Integer> dimSize;
      DAE.Exp lhs;
      DAE.Exp rhs;
      DAE.ComponentRef componentRef;
      Integer size;
      DAE.Algorithm alg;
      DAE.Expand expand;
      BackendDAE.WhenEquation whenEquation;
      list< .DAE.Exp> conditions;
      list<list<BackendDAE.Equation>> eqnstrue;
      list<BackendDAE.Equation> eqnsfalse;

    case (BackendDAE.EQUATION(exp=lhs, scalar=rhs, source=source), _)
    then BackendDAE.EQUATION(lhs, rhs, source, inAttr);

    case (BackendDAE.ARRAY_EQUATION(dimSize=dimSize, left=lhs, right=rhs, source=source), _)
    then BackendDAE.ARRAY_EQUATION(dimSize, lhs, rhs, source, inAttr);

    case (BackendDAE.SOLVED_EQUATION(componentRef=componentRef, exp=rhs, source=source), _)
    then BackendDAE.SOLVED_EQUATION(componentRef, rhs, source, inAttr);

    case (BackendDAE.RESIDUAL_EQUATION(exp=rhs, source=source), _)
    then BackendDAE.RESIDUAL_EQUATION(rhs, source, inAttr);

    case (BackendDAE.ALGORITHM(size=size, alg=alg, source=source, expand=expand), _)
    then BackendDAE.ALGORITHM(size, alg, source, expand, inAttr);

    case (BackendDAE.WHEN_EQUATION(size=size, whenEquation=whenEquation, source=source), _)
    then BackendDAE.WHEN_EQUATION(size, whenEquation, source, inAttr);

    case (BackendDAE.COMPLEX_EQUATION(size=size, left=lhs, right=rhs, source=source), _)
    then BackendDAE.COMPLEX_EQUATION(size, lhs, rhs, source, inAttr);

    case (BackendDAE.IF_EQUATION(conditions=conditions, eqnstrue=eqnstrue, eqnsfalse=eqnsfalse, source=source), _)
    then BackendDAE.IF_EQUATION(conditions, eqnstrue, eqnsfalse, source, inAttr);

    else equation
      Error.addInternalError("function setEquationAttributes failed", sourceInfo());
    then fail();
  end match;
end setEquationAttributes;

public function setEquationLHS
"
  sets the left hand side expression of an equation.
"
  input BackendDAE.Equation inEqn;
  input DAE.Exp lhs;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := match inEqn
    local
      DAE.ElementSource source;
      list<Integer> dimSize;
      DAE.Exp rhs;
      DAE.ComponentRef componentRef;
      Integer size;
      DAE.Algorithm alg;
      DAE.Expand expand;
      BackendDAE.WhenEquation whenEquation;
      list< .DAE.Exp> conditions;
      list<list<BackendDAE.Equation>> eqnstrue;
      list<BackendDAE.Equation> eqnsfalse;
      BackendDAE.EquationAttributes attr;

    case BackendDAE.EQUATION(scalar=rhs, source=source, attr=attr)
    then BackendDAE.EQUATION(lhs, rhs, source, attr);

    case BackendDAE.ARRAY_EQUATION(dimSize=dimSize, right=rhs, source=source, attr=attr)
    then BackendDAE.ARRAY_EQUATION(dimSize, lhs, rhs, source, attr);

    else equation
      Error.addInternalError("function setEquationLHS failed", sourceInfo());
    then fail();
  end match;
end setEquationLHS;

public function setEquationRHS
"
  sets the right hand side expression of an equation.
"
  input BackendDAE.Equation inEqn;
  input DAE.Exp rhs;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := match inEqn
    local
      DAE.ElementSource source;
      list<Integer> dimSize;
      DAE.Exp lhs;
      DAE.ComponentRef componentRef;
      Integer size;
      DAE.Algorithm alg;
      DAE.Expand expand;
      BackendDAE.WhenEquation whenEquation;
      list< .DAE.Exp> conditions;
      list<list<BackendDAE.Equation>> eqnstrue;
      list<BackendDAE.Equation> eqnsfalse;
      BackendDAE.EquationAttributes attr;

    case BackendDAE.EQUATION(exp=lhs, source=source, attr=attr)
    then BackendDAE.EQUATION(lhs, rhs, source, attr);

    case BackendDAE.ARRAY_EQUATION(dimSize=dimSize, left=lhs, source=source, attr=attr)
    then BackendDAE.ARRAY_EQUATION(dimSize, lhs, rhs, source, attr);

    case BackendDAE.SOLVED_EQUATION(componentRef=componentRef, source=source, attr=attr)
    then BackendDAE.SOLVED_EQUATION(componentRef, rhs, source, attr);

    case BackendDAE.RESIDUAL_EQUATION(source=source, attr=attr)
    then BackendDAE.RESIDUAL_EQUATION(rhs, source, attr);

    else equation
      Error.addInternalError("function setEquationRHS failed", sourceInfo());
    then fail();
  end match;
end setEquationRHS;

public function generateSolvedEqnsfromOption "author: Frenkel TUD 2010-05"
  input DAE.ComponentRef inLhs;
  input Option<DAE.Exp> inRhs;
  input DAE.ElementSource inSource;
  input BackendDAE.EquationKind inEqKind;
  output list<BackendDAE.Equation> outEqn;
algorithm
  outEqn :=  match (inLhs, inRhs, inSource, inEqKind)
  local
    DAE.Exp rhs;

    case (_, SOME(rhs), _, _)
    then {BackendDAE.SOLVED_EQUATION(inLhs, rhs, inSource, BackendDAE.EQUATION_ATTRIBUTES(false, inEqKind))};

    else {};
  end match;
end generateSolvedEqnsfromOption;

public function generateResidualFromRelation "author: vitalij"
  input String conCrefName;
  input DAE.Exp iRhs;
  input DAE.ElementSource Source;
  input BackendDAE.Variables inVars;
  input BackendDAE.Variables knvars;
  input BackendDAE.VarKind conKind;

  output list<BackendDAE.Equation> outEqn;
  output BackendDAE.Var vout;
algorithm
  (outEqn, vout) :=  match (conCrefName, iRhs)
    local
      DAE.Exp rhs, e1, e2 , expNull, lowBound;
      DAE.ComponentRef lhs, cr;
      BackendDAE.Var dummyVar, v;
      BackendDAE.Equation eqn;

    case (_, DAE.RELATION(e1, DAE.LESS(_), e2, _, _)) equation
      lhs = ComponentReference.makeCrefIdent(conCrefName, DAE.T_REAL_DEFAULT, {});
      dummyVar = BackendDAE.VAR(lhs, conKind, DAE.OUTPUT(), DAE.NON_PARALLEL(), DAE.T_REAL_DEFAULT, NONE(), NONE(), {}, DAE.emptyElementSource, NONE(), NONE(), DAE.BCONST(false), NONE(), DAE.NON_CONNECTOR(), DAE.NOT_INNER_OUTER(), false);
      rhs = Expression.expSub(e1,e2);
      (rhs, _) = ExpressionSimplify.simplify1(rhs);
      expNull = DAE.RCONST(0.0);
      lowBound = DAE.RCONST(-1e21);
      dummyVar = BackendVariable.setVarMinMax(dummyVar, SOME(lowBound), SOME(expNull));
    then ({BackendDAE.SOLVED_EQUATION(lhs, rhs, Source, BackendDAE.EQ_ATTR_DEFAULT_UNKNOWN)}, dummyVar);

    case (_, DAE.RELATION(e1, DAE.LESSEQ(_), e2, _, _)) equation
      lhs = ComponentReference.makeCrefIdent(conCrefName, DAE.T_REAL_DEFAULT, {});
      dummyVar = BackendDAE.VAR(lhs, conKind, DAE.OUTPUT(), DAE.NON_PARALLEL(), DAE.T_REAL_DEFAULT, NONE(), NONE(), {}, DAE.emptyElementSource, NONE(), NONE(), DAE.BCONST(false), NONE(), DAE.NON_CONNECTOR(), DAE.NOT_INNER_OUTER(), false);
      rhs = Expression.expSub(e1,e2);
      (rhs, _) = ExpressionSimplify.simplify1(rhs);
      expNull = DAE.RCONST(0.0);
      lowBound = DAE.RCONST(-1e21);
      dummyVar = BackendVariable.setVarMinMax(dummyVar, SOME(lowBound), SOME(expNull));
    then ({BackendDAE.SOLVED_EQUATION(lhs, rhs, Source, BackendDAE.EQ_ATTR_DEFAULT_UNKNOWN)}, dummyVar);

    case (_, DAE.RELATION(e1, DAE.GREATER(_), e2, _, _)) equation
      lhs = ComponentReference.makeCrefIdent(conCrefName, DAE.T_REAL_DEFAULT, {});
      dummyVar = BackendDAE.VAR(lhs, conKind, DAE.OUTPUT(), DAE.NON_PARALLEL(), DAE.T_REAL_DEFAULT, NONE(), NONE(), {}, DAE.emptyElementSource, NONE(), NONE(), DAE.BCONST(false), NONE(), DAE.NON_CONNECTOR(), DAE.NOT_INNER_OUTER(), false);
      rhs =  Expression.expSub(e2,e1);
      (rhs, _) = ExpressionSimplify.simplify1(rhs);
      expNull = DAE.RCONST(0.0);
      lowBound = DAE.RCONST(-1e21);
      dummyVar = BackendVariable.setVarMinMax(dummyVar, SOME(lowBound), SOME(expNull));
    then ({BackendDAE.SOLVED_EQUATION(lhs, rhs, Source, BackendDAE.EQ_ATTR_DEFAULT_UNKNOWN)}, dummyVar);

    case (_, DAE.RELATION(e1, DAE.GREATEREQ(_), e2, _, _)) equation
      lhs = ComponentReference.makeCrefIdent(conCrefName, DAE.T_REAL_DEFAULT, {});
      dummyVar = BackendDAE.VAR(lhs, conKind, DAE.OUTPUT(), DAE.NON_PARALLEL(), DAE.T_REAL_DEFAULT, NONE(), NONE(), {}, DAE.emptyElementSource, NONE(), NONE(), DAE.BCONST(false), NONE(), DAE.NON_CONNECTOR(), DAE.NOT_INNER_OUTER(), false);
      rhs =  Expression.expSub(e2,e1);
      (rhs, _) = ExpressionSimplify.simplify(rhs);
      expNull = DAE.RCONST(0.0);
      lowBound = DAE.RCONST(-1e21);
      dummyVar = BackendVariable.setVarMinMax(dummyVar, SOME(lowBound), SOME(expNull));
    then ({BackendDAE.SOLVED_EQUATION(lhs, rhs, Source, BackendDAE.EQ_ATTR_DEFAULT_UNKNOWN)}, dummyVar);

    case (_, DAE.RELATION(e1, DAE.EQUAL(_), e2, _, _)) equation
      lhs = ComponentReference.makeCrefIdent(conCrefName, DAE.T_REAL_DEFAULT, {});
      dummyVar = BackendDAE.VAR(lhs, conKind, DAE.OUTPUT(), DAE.NON_PARALLEL(), DAE.T_REAL_DEFAULT, NONE(), NONE(), {}, DAE.emptyElementSource, NONE(), NONE(), DAE.BCONST(false), NONE(), DAE.NON_CONNECTOR(), DAE.NOT_INNER_OUTER(), false);
      rhs =  Expression.expSub(e2,e1);
      (rhs, _) = ExpressionSimplify.simplify(rhs);
      expNull = DAE.RCONST(0.0);
      dummyVar = BackendVariable.setVarMinMax(dummyVar, SOME(expNull), SOME(expNull));
    then ({BackendDAE.SOLVED_EQUATION(lhs, rhs, Source, BackendDAE.EQ_ATTR_DEFAULT_UNKNOWN)}, dummyVar);

    case(_,e1 as DAE.CREF(componentRef = cr)) algorithm
      try
        (v, _) := BackendVariable.getVarSingle(cr, inVars);
      else
        (v, _) := BackendVariable.getVarSingle(cr, knvars);
      end try;

      lhs := ComponentReference.makeCrefIdent(conCrefName, DAE.T_REAL_DEFAULT, {});
      dummyVar := BackendDAE.VAR(lhs, conKind, DAE.OUTPUT(), DAE.NON_PARALLEL(), DAE.T_REAL_DEFAULT, NONE(), NONE(), {}, DAE.emptyElementSource, NONE(), NONE(), DAE.BCONST(false), NONE(), DAE.NON_CONNECTOR(), DAE.NOT_INNER_OUTER(), false);
      dummyVar := BackendVariable.mergeAliasVars(dummyVar, v, false, knvars);
      eqn := BackendDAE.SOLVED_EQUATION(lhs, e1, Source, BackendDAE.EQ_ATTR_DEFAULT_UNKNOWN);

    then ({eqn}, dummyVar);
    else fail();
  end match;
end generateResidualFromRelation;

public function makeTmpEqnForExp
"
  author: Vitalij Ruge
  make and append an eqn and var for expression
"
  input DAE.Exp iExp;
  input String name "var name";
  input Integer offset;
  input BackendDAE.EquationArray ieqns;
  input BackendDAE.Variables ivars;
  input BackendDAE.Shared ishared;
  input Boolean noPara = false;
  output DAE.Exp oExp;
  output BackendDAE.EquationArray oeqns = ieqns;
  output BackendDAE.Variables ovars = ivars;
  output BackendDAE.Shared oshared = ishared;
  //output BackendDAE.StrongComponent
  output Boolean update;
  output Boolean para = false;
protected
  DAE.ComponentRef cr;
  BackendDAE.Var tmpvar;
  String name_ = "__OMC__" + intString(offset) + "$" + name;
  DAE.Exp y;
  BackendDAE.Equation eqn;
  list<BackendDAE.Var> eqnVars, eqnKnVars, inputsKnVars;
  BackendDAE.Variables knowVars;
  Boolean b;

algorithm

  (y, _) := ExpressionSimplify.simplify(iExp);
  if makeTmpEqnForExp_rule(y) then
    update := true;

    cr  := ComponentReference.makeCrefIdent(name_, DAE.T_REAL_DEFAULT , {});
    oExp := Expression.crefExp(cr);

    tmpvar := BackendVariable.makeVar(cr);
    tmpvar := BackendVariable.setVarTS(tmpvar,SOME(BackendDAE.AVOID()));

    eqn := BackendDAE.EQUATION(oExp, y, DAE.emptyElementSource, BackendDAE.EQ_ATTR_DEFAULT_DYNAMIC);
    if Flags.isSet(Flags.DUMP_SIMPLIFY_LOOPS) then
      print(BackendDump.equationString(eqn) + " -- new eqn--\n");
    end if;
    eqnVars := equationVars(eqn, ivars);
    b := listEmpty(eqnVars) and not Expression.expHasCref(y, DAE.crefTime);
    if b then
      knowVars := BackendVariable.daeGlobalKnownVars(oshared);
      eqnKnVars := equationVars(eqn, knowVars);
      (inputsKnVars,_) := List.splitOnTrue(eqnKnVars, BackendVariable.isInput);
       b := listEmpty(inputsKnVars);
    end if;

    b := false "hack";
    if b then
      if noPara then
        oExp := ExpressionSimplify.simplify(iExp);
        update := false;
      else
        tmpvar := BackendVariable.setBindExp(tmpvar, SOME(y));
        tmpvar := BackendVariable.setVarKind(tmpvar, BackendDAE.PARAM());
        oshared := BackendVariable.addGlobalKnownVarDAE(tmpvar, oshared);
        para := true;
      end if;
    else
      oeqns := BackendEquation.add(eqn, oeqns);
      ovars := BackendVariable.addVar(tmpvar, ovars);
    end if;

  else
   oExp := y;
   update := false;
  end if;

end makeTmpEqnForExp;

protected function makeTmpEqnForExp_rule
  input DAE.Exp inExp;
  output Boolean allowed;
algorithm

  if Expression.isCref(inExp) or Expression.isConst(inExp) or Expression.isUnaryCref(inExp) then
    allowed := false;
    return;
  end if;

  allowed := match inExp
             local DAE.Exp e1, e2;
             case DAE.BINARY(e1,DAE.DIV(),e2)
             guard (Expression.isOne(e1) or Expression.isConstMinusOne(e1)) and (Expression.isCref(e2) or Expression.isUnaryCref(e2))
              then false;
             case DAE.CAST(exp=e1) then makeTmpEqnForExp_rule(e1);
             else true;
             end match;

end makeTmpEqnForExp_rule;

public function normalizationVec
"
  author: Vitalij Ruge
  normalization of vector
"
  input array<DAE.Exp> vec;
  input String name "var name";
  input Integer offset;
  input BackendDAE.EquationArray ieqns;
  input BackendDAE.Variables ivars;
  input BackendDAE.Shared ishared;
  output array<DAE.Exp> nvec;
  output BackendDAE.EquationArray oeqns;
  output BackendDAE.Variables ovars;
  output BackendDAE.Shared oshared;
protected
  DAE.Exp len = Expression.lenVec(vec);
algorithm
  (len,oeqns,ovars,oshared) := makeTmpEqnForExp(len, name, offset, ieqns, ivars,ishared);
  if Expression.isZero(len) then
    fail();
  end if;
  nvec := Array.map1(vec, Expression.makeDiv, len);
end normalizationVec;

public function solveEquation "author: wbraun
  Solves an equation w.r.t. a component reference. All equations are transformed
  to a EQUATION(cref, exp).
  Algorithm, when and if-equation are left as they are."
  input BackendDAE.Equation eqn;
  input DAE.Exp crefExp;
  input Option<DAE.FunctionTree> functions;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := matchcontinue (eqn, crefExp, functions)
    local
      DAE.Exp e1, e2;
      DAE.Exp res;
      DAE.ComponentRef cref, cr;
      list<DAE.ComponentRef> crefLst;
      BackendDAE.Equation eq;
      DAE.ElementSource source;
      BackendDAE.EquationAttributes eqAttr;

    case (BackendDAE.EQUATION(exp=e1, scalar=e2, source=source, attr=eqAttr), _, _) equation
      (res, _, {}, {}) = ExpressionSolve.solve2(e1, e2, crefExp, functions, NONE());
    then (BackendDAE.EQUATION(crefExp, res, source, eqAttr));

    case (BackendDAE.ARRAY_EQUATION(left=e1, right=e2, source=source, attr=eqAttr), _, _) equation
      (res, _, {}, {}) = ExpressionSolve.solve2(e1, e2, crefExp, functions, NONE());
    then (BackendDAE.EQUATION(crefExp, res, source, eqAttr));

    case (BackendDAE.SOLVED_EQUATION(componentRef=cref, exp=e2, source=source, attr=eqAttr), _,_) equation
      cr = Expression.expCref(crefExp);
      true = ComponentReference.crefEqual(cref, cr);
    then (BackendDAE.EQUATION(crefExp, e2, source, eqAttr));

    case (BackendDAE.SOLVED_EQUATION(componentRef=cref, exp=e2, source=source, attr=eqAttr), _, _) equation
      // already checked in rule above:
      //cr = Expression.expCref(crefExp);
      //false = ComponentReference.crefEqual(cref, cr);
      e1 = Expression.crefExp(cref);
      (res, _, {}, {}) = ExpressionSolve.solve2(e1, e2, crefExp, functions, NONE());
    then (BackendDAE.EQUATION(crefExp, res, source, eqAttr));

    case (BackendDAE.RESIDUAL_EQUATION(exp=e2, source=source, attr=eqAttr), _, _) equation
      e1 = Expression.makeConstZero(Expression.typeof(e2));
      (res, _, {}, {}) = ExpressionSolve.solve2(e2, e1, crefExp, functions, NONE());
    then (BackendDAE.EQUATION(crefExp, res, source, eqAttr));

    case (BackendDAE.COMPLEX_EQUATION(left=e1, right=e2, source=source, attr=eqAttr), _, _) equation
      (res, _, {}, {}) = ExpressionSolve.solve2(e1, e2, crefExp, functions, NONE());
    then (BackendDAE.EQUATION(crefExp, res, source, eqAttr));
/*
    case (eq as BackendDAE.ALGORITHM(alg=_), _)
    then eq;

    case (eq as BackendDAE.WHEN_EQUATION(size=_), _)
    then eq;

    case (eq as BackendDAE.IF_EQUATION(conditions=_), _)
    then eq;
*/
    else equation
      true = Flags.isSet(Flags.FAILTRACE);
      BackendDump.dumpBackendDAEEqnList({eqn}, "function BackendEquation.solveEquation failed w.r.t " + ExpressionDump.printExpStr(crefExp), true);
      Error.addInternalError("function solveEquation failed", sourceInfo());
    then fail();
  end matchcontinue;
end solveEquation;

public function generateRESIDUAL_EQUATION "author: Frenkel TUD 2010-05"
  input DAE.Exp inExp;
  input DAE.ElementSource inSource;
  input BackendDAE.EquationAttributes inEqAttr;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := BackendDAE.RESIDUAL_EQUATION(inExp, inSource, inEqAttr);
end generateRESIDUAL_EQUATION;

public function generateRESIDUAL_EQUATION1
  input tuple<DAE.Exp, DAE.Exp> inTpl;
  input DAE.ElementSource source;
  input BackendDAE.EquationAttributes inEqAttr;
  output BackendDAE.Equation outEqn;
protected
  DAE.Exp e1, e2, e;
algorithm
  (e1, e2) := inTpl;
  e := Expression.createResidualExp(e1, e2);
  outEqn := BackendDAE.RESIDUAL_EQUATION(e, source, inEqAttr);
end generateRESIDUAL_EQUATION1;

public function equationSystemsEqnsLst
  input BackendDAE.EqSystems systs;
  output list<BackendDAE.Equation> outEqns = {};
protected
  list<BackendDAE.Equation> eqns;
  BackendDAE.EquationArray eq;
algorithm
  for es in systs loop
    BackendDAE.EQSYSTEM(orderedEqs=eq) := es;
    eqns := equationList(eq);
    outEqns := List.append_reverse(eqns, outEqns);
  end for;
  outEqns := MetaModelica.Dangerous.listReverseInPlace(outEqns);
end equationSystemsEqnsLst;

public function getEqnsFromEqSystems "
  Extracts the orderedEqs attribute from an equation system."
  input BackendDAE.EqSystems inEqSystems;
  output BackendDAE.EquationArray outOrderedEqs;
algorithm
  outOrderedEqs := listEquation(equationSystemsEqnsLst(inEqSystems));
end getEqnsFromEqSystems;

public function getEqnsFromEqSystem "
  Extracts the orderedEqs attribute from an equation system."
  input BackendDAE.EqSystem inEqSystem;
  output BackendDAE.EquationArray outOrderedEqs = inEqSystem.orderedEqs;
end getEqnsFromEqSystem;

public function getInitialEqnsFromShared "
  Extracts the initial equations from a shared object."
  input BackendDAE.Shared inShared;
  output BackendDAE.EquationArray outInitialEqs = inShared.initialEqs;
end getInitialEqnsFromShared;

public function aliasEquation "author Frenkel TUD 2011-04
  Returns the two sides of an alias equation as expressions and cref.
  If the equation is not simple, this function will fail."
  input BackendDAE.Equation inEqn;
  output list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> outTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
algorithm
  outTpls := match (inEqn)
    local
      DAE.Exp e, e1, e2;
      DAE.ComponentRef cr;

    case (BackendDAE.EQUATION(exp=e1, scalar=e2))
    then aliasEquation1(e1, e2, {});

    case (BackendDAE.ARRAY_EQUATION(left=e1, right=e2))
    then aliasEquation1(e1, e2, {});

    case (BackendDAE.SOLVED_EQUATION(componentRef=cr, exp=e2)) equation
      e = Expression.crefExp(cr);
    then aliasEquation1(e, e2, {});

    case (BackendDAE.RESIDUAL_EQUATION(exp=e1))
    then aliasExpression(e1, {});

    case (BackendDAE.COMPLEX_EQUATION(left=e1, right=e2))
    then aliasEquation1(e1, e2, {});
  end match;
end aliasEquation;

protected function aliasEquation1 "author Frenkel TUD 2011-04
  helper for aliasEquation"
  input DAE.Exp lhs;
  input DAE.Exp rhs;
  input list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> inTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
  output list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> outTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
algorithm
  outTpls := match (lhs, rhs, inTpls)
    local
      DAE.ComponentRef cr1, cr2;
      DAE.Exp e1, e2;
      DAE.Type ty;
      list<DAE.Exp> elst1, elst2;
      list<list<DAE.Exp>> elstlst1, elstlst2;
      list<DAE.Var> varLst1, varLst2;
      Absyn.Path patha, patha1, pathb, pathb1;

    // a = b;
    case (DAE.CREF(componentRef = cr1), DAE.CREF(componentRef = cr2), _)
    then (cr1, cr2, lhs, rhs, false)::inTpls;

    // a = -b;
    case (DAE.CREF(componentRef = cr1), DAE.UNARY(DAE.UMINUS(ty), DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, DAE.UNARY(DAE.UMINUS(ty), lhs), rhs, true)::inTpls;

    case (DAE.CREF(componentRef = cr1), DAE.UNARY(DAE.UMINUS_ARR(ty), DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, DAE.UNARY(DAE.UMINUS_ARR(ty), lhs), rhs, true)::inTpls;

    // -a = b;
    case (DAE.UNARY(DAE.UMINUS(ty), DAE.CREF(componentRef = cr1)), DAE.CREF(componentRef = cr2), _)
    then (cr1, cr2, lhs, DAE.UNARY(DAE.UMINUS(ty), rhs), true)::inTpls;

    case (DAE.UNARY(DAE.UMINUS_ARR(ty), DAE.CREF(componentRef = cr1)), DAE.CREF(componentRef = cr2), _)
    then (cr1, cr2, lhs, DAE.UNARY(DAE.UMINUS_ARR(ty), rhs), true)::inTpls;

    // -a = -b;
    case (DAE.UNARY(DAE.UMINUS(_), e1 as DAE.CREF(componentRef = cr1)), DAE.UNARY(DAE.UMINUS(_), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, e2, false)::inTpls;

    case (DAE.UNARY(DAE.UMINUS_ARR(_), e1 as DAE.CREF(componentRef = cr1)), DAE.UNARY(DAE.UMINUS_ARR(_), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, e2, false)::inTpls;

    // a = not b;
    case (DAE.CREF(componentRef = cr1), DAE.LUNARY(DAE.NOT(ty), DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, DAE.LUNARY(DAE.NOT(ty), lhs), rhs, true)::inTpls;

    // not a = b;
    case (DAE.LUNARY(DAE.NOT(ty), DAE.CREF(componentRef = cr1)), DAE.CREF(componentRef = cr2), _)
    then (cr1, cr2, lhs, DAE.LUNARY(DAE.NOT(ty), rhs), true)::inTpls;

    // not a = not b;
    case (DAE.LUNARY(DAE.NOT(_), e1 as DAE.CREF(componentRef = cr1)), DAE.LUNARY(DAE.NOT(_), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, e2, false)::inTpls;

    // {a1, a2, a3, ..} = {b1, b2, b3, ..};
    case (DAE.ARRAY(array = elst1), DAE.ARRAY(array = elst2), _)
    then List.threadFold(elst1, elst2, aliasEquation1, inTpls);

    case (DAE.MATRIX(matrix = elstlst1), DAE.MATRIX(matrix = elstlst2), _)
    then List.threadFold(elstlst1, elstlst2, aliasEquationLst, inTpls);

    // a = {b1, b2, b3, ..}
    //case (DAE.CREF(componentRef = cr1), DAE.ARRAY(array = elst2, dims=dims), _)
    //then //    aliasArray(cr1, false, elst2, dims, inTpls);

    // -a = {b1, b2, b3, ..}
    //case (DAE.UNARY(DAE.UMINUS_ARR(_), e1 as DAE.CREF(componentRef = cr1)), DAE.ARRAY(array = elst2, dims=dims), _)
    //then //    aliasArray(cr1, true, elst2, dims, inTpls);

    // a = -{b1, b2, b3, ..}
    //case (DAE.CREF(componentRef = cr1), DAE.UNARY(DAE.UMINUS_ARR(_), DAE.ARRAY(array = elst2, ty=ty)), _)

    // -a = -{b1, b2, b3, ..}
    //case (DAE.UNARY(DAE.UMINUS_ARR(_), e1 as DAE.CREF(componentRef = cr1)), DAE.UNARY(DAE.UMINUS_ARR(_), DAE.ARRAY(array = elst2, ty=ty)), _)

    // {a1, a2, a3, ..} = b
    //case (DAE.ARRAY(array = elst1), DAE.CREF(componentRef = cr2), _)

    // -{a1, a2, a3, ..} = b
    //case (DAE.UNARY(DAE.UMINUS_ARR(_), DAE.ARRAY(array = elst1, ty=ty)), DAE.CREF(componentRef = cr2), _)

    // {a1, a2, a3, ..} = -b
    //case (DAE.ARRAY(array = elst1), DAE.UNARY(DAE.UMINUS_ARR(_), e2 as DAE.CREF(componentRef = cr2)), _)

    // -{a1, a2, a3, ..} = -b
    //case (DAE.UNARY(DAE.UMINUS_ARR(_)DAE.ARRAY(array = elst1, ty=ty)), DAE.UNARY(DAE.UMINUS_ARR(_), e2 as DAE.CREF(componentRef = cr2)), _)

    // not a = {b1, b2, b3, ..}
    //case (DAE.LUNARY(DAE.NOT(_), e1 as DAE.CREF(componentRef = cr1)), DAE.ARRAY(array = elst2, ty=ty), _)

    // a = not {b1, b2, b3, ..}
    //case (DAE.CREF(componentRef = cr1), DAE.LUNARY(DAE.NOT(_), DAE.ARRAY(array = elst2, ty=ty)), _)

    // not a = not {b1, b2, b3, ..}
    //case (DAE.LUNARY(DAE.NOT(_), e1 as DAE.CREF(componentRef = cr1)), DAE.LUNARY(DAE.NOT(_), DAE.ARRAY(array = elst2, ty=ty)), _)

    // {a1, a2, a3, ..} = not b
    //case (DAE.ARRAY(array = elst1, ty=ty), DAE.LUNARY(DAE.NOT(_), e2 as DAE.CREF(componentRef = cr2)), _)

    // not {a1, a2, a3, ..} = b
    //case (DAE.LUNARY(DAE.NOT(_), DAE.ARRAY(array = elst1, ty=ty)), DAE.CREF(componentRef = cr2), _)

    // not {a1, a2, a3, ..} = not b
    //case (DAE.LUNARY(DAE.NOT(_), DAE.ARRAY(array = elst1, ty=ty)), DAE.LUNARY(DAE.NOT(_), e2 as DAE.CREF(componentRef = cr2)), _)

    // a = Record(b1, b2, b3, ..)
    case (DAE.CREF(componentRef = cr1), DAE.CALL(path=pathb, expLst=elst2, attr=DAE.CALL_ATTR(ty=DAE.T_COMPLEX(varLst=varLst2, complexClassType=ClassInf.RECORD(pathb1)))), _) equation
      true = Absyn.pathEqual(pathb, pathb1);
    then aliasRecord(cr1, varLst2, elst2, inTpls);

    // Record(a1, a2, a3, ..) = b
    case (DAE.CALL(path=patha, expLst=elst1, attr=DAE.CALL_ATTR(ty=DAE.T_COMPLEX(varLst=varLst1, complexClassType=ClassInf.RECORD(patha1)))), DAE.CREF(componentRef = cr2), _) equation
      true = Absyn.pathEqual(patha, patha1);
    then aliasRecord(cr2, varLst1, elst1, inTpls);

    // Record(a1, a2, a3, ..) = Record(b1, b2, b3, ..)
    case (DAE.CALL(path=patha, expLst=elst1, attr=DAE.CALL_ATTR(ty=DAE.T_COMPLEX(complexClassType=ClassInf.RECORD(patha1)))),
      DAE.CALL(path=pathb, expLst=elst2, attr=DAE.CALL_ATTR(ty=DAE.T_COMPLEX(complexClassType=ClassInf.RECORD(pathb1)))), _) equation
      true = Absyn.pathEqual(patha, patha1);
      true = Absyn.pathEqual(pathb, pathb1);
    then List.threadFold(elst1, elst2, aliasEquation1, inTpls);

    // matchcontinue part
    else aliasEquation2(lhs, rhs, inTpls);
  end match;
end aliasEquation1;

protected function aliasEquationLst "author Frenkel TUD 2011-04
  helper for aliasEquation"
  input list<DAE.Exp> elst1;
  input list<DAE.Exp> elst2;
  input list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> inTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
  output list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> outTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
algorithm
  outTpls := List.threadFold(elst1, elst2, aliasEquation1, inTpls);
end aliasEquationLst;

protected function aliasEquation2 "author Frenkel TUD 2011-04
  helper for aliasEquation"
  input DAE.Exp lhs;
  input DAE.Exp rhs;
  input list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> inTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
  output list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> outTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
algorithm
  outTpls := match (lhs, rhs, inTpls)
    local
      list<DAE.Exp> elst1, elst2;

    // {a1+b1, a2+b2, a3+b3, ..} = 0;
    case (DAE.ARRAY(array = elst1), _, _) guard
      Expression.isZero(rhs)
    then List.fold(elst1, aliasExpression, inTpls);

    // 0 = {a1+b1, a2+b2, a3+b3, ..};
    case (_, DAE.ARRAY(array = elst2), _) guard
      Expression.isZero(lhs)
    then List.fold(elst2, aliasExpression, inTpls);

    // lhs = 0
    case (_, _, _) guard
      Expression.isZero(rhs)
    then aliasExpression(lhs, inTpls);

    // 0 = rhs
    case (_, _, _) guard
      Expression.isZero(lhs)
    then aliasExpression(rhs, inTpls);
  end match;
end aliasEquation2;

protected function aliasRecord "author Frenkel TUD 2011-04
  helper for aliasEquation"
  input DAE.ComponentRef cr;
  input list<DAE.Var> varLst;
  input list<DAE.Exp> explst;
  input list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> inTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
  output list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> outTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
algorithm
  outTpls := match (cr, varLst, explst, inTpls)
    local
      DAE.ComponentRef cr1, cr2;
      DAE.Exp e1, e2;
      DAE.Type ty, ty1;
      list<DAE.Exp> elst;
      list<DAE.Var> vlst;
      DAE.Ident ident;

    case (_, {}, {}, _)
    then inTpls;

    // a = b
    case (_, DAE.TYPES_VAR(name=ident, ty=ty)::vlst, (e2 as DAE.CREF(componentRef = cr2))::elst, _) equation
      cr1 = ComponentReference.crefPrependIdent(cr, ident, {}, ty);
      e1 = DAE.CREF(cr1, ty);
    then aliasRecord(cr, vlst, elst, (cr1, cr2, e1, e2, false)::inTpls);

    // a = -b
    case (_, DAE.TYPES_VAR(name=ident, ty=ty)::vlst, (e2 as DAE.UNARY(DAE.UMINUS(ty1), DAE.CREF(componentRef = cr2)))::elst, _) equation
      cr1 = ComponentReference.crefPrependIdent(cr, ident, {}, ty);
      e1 = DAE.UNARY(DAE.UMINUS(ty1), DAE.CREF(cr1, ty));
    then aliasRecord(cr, vlst, elst, (cr1, cr2, e1, e2, true)::inTpls);

    case (_, DAE.TYPES_VAR(name=ident, ty=ty)::vlst, (e2 as DAE.UNARY(DAE.UMINUS_ARR(ty1), DAE.CREF(componentRef = cr2)))::elst, _) equation
      cr1 = ComponentReference.crefPrependIdent(cr, ident, {}, ty);
      e1 = DAE.UNARY(DAE.UMINUS_ARR(ty1), DAE.CREF(cr1, ty));
    then aliasRecord(cr, vlst, elst, (cr1, cr2, e1, e2, true)::inTpls);

    // a = not b
    case (_, DAE.TYPES_VAR(name=ident, ty=ty)::vlst, (e2 as DAE.LUNARY(DAE.NOT(ty1), DAE.CREF(componentRef = cr2)))::elst, _) equation
      cr1 = ComponentReference.crefPrependIdent(cr, ident, {}, ty);
      e1 = DAE.LUNARY(DAE.NOT(ty1), DAE.CREF(cr1, ty));
    then aliasRecord(cr, vlst, elst, (cr1, cr2, e1, e2, true)::inTpls);

    // a = {b1, b2, b3}
  end match;
end aliasRecord;

protected function aliasExpression "author Frenkel TUD 2011-11
  Returns the two sides of an alias expression as expressions and cref.
  If the expression is not simple, this function will fail."
  input DAE.Exp exp;
  input list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> inTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
  output list<tuple<DAE.ComponentRef, DAE.ComponentRef, DAE.Exp, DAE.Exp, Boolean>> outTpls "(cr1, cr2, cr1=e2, cr2=e1, true if negated alias)";
algorithm
  outTpls := match (exp, inTpls)
    local
      DAE.ComponentRef cr1, cr2;
      DAE.Exp e1, e2;
      DAE.Type ty;

    // a + b
    case (DAE.BINARY(e1 as DAE.CREF(componentRef = cr1), DAE.ADD(ty=ty), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, DAE.UNARY(DAE.UMINUS(ty), e1), DAE.UNARY(DAE.UMINUS(ty), e2), true)::inTpls;

    case (DAE.BINARY(e1 as DAE.CREF(componentRef = cr1), DAE.ADD_ARR(ty=ty), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, DAE.UNARY(DAE.UMINUS_ARR(ty), e1), DAE.UNARY(DAE.UMINUS_ARR(ty), e2), true)::inTpls;

    // a - b
    case (DAE.BINARY(e1 as DAE.CREF(componentRef = cr1), DAE.SUB(), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, e2, false)::inTpls;

    case (DAE.BINARY(e1 as DAE.CREF(componentRef = cr1), DAE.SUB_ARR(), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, e2, false)::inTpls;

    // -a + b
    case (DAE.BINARY(DAE.UNARY(DAE.UMINUS(_), e1 as DAE.CREF(componentRef = cr1)), DAE.ADD(), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, e2, false)::inTpls;

    case (DAE.BINARY(DAE.UNARY(DAE.UMINUS_ARR(_), e1 as DAE.CREF(componentRef = cr1)), DAE.ADD_ARR(), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, e2, false)::inTpls;

    // -a - b = 0
    case (DAE.BINARY(e1 as DAE.UNARY(DAE.UMINUS(_), DAE.CREF(componentRef = cr1)), DAE.SUB(ty=ty), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, DAE.UNARY(DAE.UMINUS(ty), e2), true)::inTpls;

    case (DAE.BINARY(e1 as DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CREF(componentRef = cr1)), DAE.SUB_ARR(ty=ty), e2 as DAE.CREF(componentRef = cr2)), _)
    then (cr1, cr2, e1, DAE.UNARY(DAE.UMINUS_ARR(ty), e2), true)::inTpls;
  end match;
end aliasExpression;

public function derivativeEquation "author Frenkel TUD 2011-04
  Returns the two sides of an derivative equation as expressions and cref.
  If the equation is not a derivative equation, this function will fail."
  input BackendDAE.Equation eqn;
  output DAE.ComponentRef cr;
  output DAE.ComponentRef dcr "the derivative of cr";
  output DAE.Exp e;
  output DAE.Exp de "der(cr)";
  output Boolean negate;
algorithm
  (cr, dcr, e, de, negate) := match (eqn)
    local
      DAE.Exp ne, ne2;

    // a = der(b);
    case (BackendDAE.EQUATION(exp=e as DAE.CREF(componentRef = dcr), scalar=de as  DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})))
    then (cr, dcr, e, de, false);

    // der(a) = b;
    case (BackendDAE.EQUATION(exp=de as  DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)}), scalar=e as DAE.CREF(componentRef = dcr)))
    then (cr, dcr, e, de, false);

    // a = -der(b);
    case (BackendDAE.EQUATION(exp=e as DAE.CREF(componentRef = dcr), scalar=de as  DAE.UNARY(DAE.UMINUS(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})))) equation
      ne = Expression.negate(e);
    then (cr, dcr, ne, de, true);

    case (BackendDAE.EQUATION(exp=e as DAE.CREF(componentRef = dcr), scalar=de as  DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})))) equation
      ne = Expression.negate(e);
    then (cr, dcr, ne, de, true);

    // -der(a) = b;
    case (BackendDAE.EQUATION(exp=de as  DAE.UNARY(DAE.UMINUS(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})), scalar=e as DAE.CREF(componentRef = dcr))) equation
      ne = Expression.negate(e);
    then (cr, dcr, ne, de, true);

    case (BackendDAE.EQUATION(exp=de as  DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})), scalar=e as DAE.CREF(componentRef = dcr))) equation
      ne = Expression.negate(e);
    then (cr, dcr, ne, de, true);

    // -a = der(b);
    case (BackendDAE.EQUATION(exp=e as DAE.UNARY(DAE.UMINUS(_), DAE.CREF(componentRef = dcr)), scalar=de as  DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)}))) equation
      ne = Expression.negate(de);
    then (cr, dcr, e, ne, true);

    case (BackendDAE.EQUATION(exp=e as DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CREF(componentRef = dcr)), scalar=de as  DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)}))) equation
      ne = Expression.negate(de);
    then (cr, dcr, e, ne, true);

    // der(a) = -b;
    case (BackendDAE.EQUATION(exp=de as  DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)}), scalar=e as DAE.UNARY(DAE.UMINUS(_), DAE.CREF(componentRef = dcr)))) equation
      ne = Expression.negate(de);
    then (cr, dcr, e, ne, true);

    case (BackendDAE.EQUATION(exp=de as  DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)}), scalar=e as DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CREF(componentRef = dcr)))) equation
      ne = Expression.negate(de);
    then (cr, dcr, e, ne, true);

    // -a = -der(b);
    case (BackendDAE.EQUATION(exp=e as DAE.UNARY(DAE.UMINUS(_), DAE.CREF(componentRef = dcr)), scalar=de as  DAE.UNARY(DAE.UMINUS(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})))) equation
      ne = Expression.negate(e);
      ne2 = Expression.negate(de);
    then (cr, dcr, ne, ne2, false);

    case (BackendDAE.EQUATION(exp=e as DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CREF(componentRef = dcr)), scalar=de as  DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})))) equation
      ne = Expression.negate(e);
      ne2 = Expression.negate(de);
    then (cr, dcr, ne, ne2, false);

    // -der(a) = -b;
    case (BackendDAE.EQUATION(exp=de as  DAE.UNARY(DAE.UMINUS(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})), scalar=e as DAE.UNARY(DAE.UMINUS(_), DAE.CREF(componentRef = dcr)))) equation
      ne = Expression.negate(e);
      ne2 = Expression.negate(de);
    then (cr, dcr, ne, ne2, false);

    case (BackendDAE.EQUATION(exp=de as  DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CALL(path = Absyn.IDENT(name = "der"), expLst = {DAE.CREF(componentRef = cr)})), scalar=e as DAE.UNARY(DAE.UMINUS_ARR(_), DAE.CREF(componentRef = dcr)))) equation
      ne = Expression.negate(e);
      ne2 = Expression.negate(de);
    then (cr, dcr, ne, ne2, false);
  end match;
end derivativeEquation;

public function addOperation "
  Adds symbolic transformation information to equation's source."
  input BackendDAE.Equation inEqn;
  input DAE.SymbolicOperation inSymOp;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := match (inEqn, inSymOp)
    local
      Integer size;
      DAE.Exp e1, e2;
      list<DAE.Exp> conditions;
      DAE.ElementSource source;
      BackendDAE.WhenEquation whenEquation;
      DAE.ComponentRef cr1;
      list<BackendDAE.Equation> eqnsfalse;
      list<list<BackendDAE.Equation>> eqnstrue;
      list<Integer> ds;
      DAE.Algorithm alg;
      DAE.Expand crefExpand;
      BackendDAE.EquationAttributes eqAttr;

    case (BackendDAE.EQUATION(e1, e2, source, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.EQUATION(e1, e2, source, eqAttr);

    case (BackendDAE.ARRAY_EQUATION(ds, e1, e2, source, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.ARRAY_EQUATION(ds, e1, e2, source, eqAttr);

    case (BackendDAE.SOLVED_EQUATION(cr1, e1, source, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.SOLVED_EQUATION(cr1, e1, source, eqAttr);

    case (BackendDAE.RESIDUAL_EQUATION(e1, source, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.RESIDUAL_EQUATION(e1, source, eqAttr);

    case (BackendDAE.ALGORITHM(size, alg, source, crefExpand, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.ALGORITHM(size, alg, source, crefExpand, eqAttr);

    case (BackendDAE.WHEN_EQUATION(size, whenEquation, source, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.WHEN_EQUATION(size, whenEquation, source, eqAttr);

    case (BackendDAE.COMPLEX_EQUATION(size, e1, e2, source, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.COMPLEX_EQUATION(size, e1, e2, source, eqAttr);

    case (BackendDAE.IF_EQUATION(conditions, eqnstrue, eqnsfalse, source, eqAttr), _) equation
      source = ElementSource.addSymbolicTransformation(source, inSymOp);
    then BackendDAE.IF_EQUATION(conditions, eqnstrue, eqnsfalse, source, eqAttr);

    else equation
      Error.addInternalError("BackendEquation.addOperation failed", sourceInfo());
    then fail();
  end match;
end addOperation;

public function isEquationsSystem
  input BackendDAE.StrongComponent comp;
  output Boolean res;
algorithm
  res := match comp
         case BackendDAE.EQUATIONSYSTEM() then true;
         else false;
         end match;
end isEquationsSystem;

public function isTornSystem
  input BackendDAE.StrongComponent comp;
  output Boolean res;
algorithm
  res := match comp
         case BackendDAE.TORNSYSTEM() then true;
         else false;
         end match;
end isTornSystem;

public function isWhenEquation
  input BackendDAE.Equation inEqn;
  output Boolean b;
algorithm
  b := match (inEqn)
    case BackendDAE.WHEN_EQUATION() then true;
    else false;
  end match;
end isWhenEquation;

public function isWhenEquationOrDiscreteAlgorithm
  input BackendDAE.Equation inEqn;
  input BackendDAE.Variables vars;
  output Boolean b;
algorithm
  b := matchcontinue(inEqn)
    local
      Boolean b1;
      list<DAE.Statement> stmts;
      list<DAE.ComponentRef> lhsCrefs;
    case BackendDAE.WHEN_EQUATION() then true;
    case BackendDAE.ALGORITHM(alg = DAE.ALGORITHM_STMTS(stmts))
      algorithm
        b1 := true;
        for s in stmts loop
          (lhsCrefs,_) := Expression.extractCrefsStatment(s);
          for c in lhsCrefs loop
            b1 := b1 and BackendVariable.isDiscrete(c,vars);
          end for;
        end for;
      then b1;
    else false;
  end matchcontinue;
end isWhenEquationOrDiscreteAlgorithm;

public function isArrayEquation
  input BackendDAE.Equation inEqn;
  output Boolean b;
algorithm
  b := match (inEqn)
    case BackendDAE.ARRAY_EQUATION() then true;
    else false;
  end match;
end isArrayEquation;

public function isAlgorithm
  input BackendDAE.Equation inEqn;
  output Boolean b;
algorithm
  b := match (inEqn)
    case BackendDAE.ALGORITHM() then true;
    else false;
  end match;
end isAlgorithm;

public function isComplexEquation
  input BackendDAE.Equation inEqn;
  output Boolean b;
algorithm
  b := match (inEqn)
    case BackendDAE.COMPLEX_EQUATION() then true;
    else false;
  end match;
end isComplexEquation;

public function isEquation
  input BackendDAE.Equation inEqn;
  output Boolean b;
algorithm
  b := match (inEqn)
    case BackendDAE.EQUATION() then true;
    else false;
  end match;
end isEquation;

public function isNotAlgorithm
  input BackendDAE.Equation inEqn;
  output Boolean b;
algorithm
  b := not isAlgorithm(inEqn);
end isNotAlgorithm;

public function markDifferentiated"sets differentiated=true in EquationAttributes"
  input BackendDAE.Equation inEqn;
  output BackendDAE.Equation outEqn;
algorithm
  outEqn := match (inEqn)
    local
      DAE.Exp e1, e2;
      DAE.ComponentRef cr;
      DAE.ElementSource source;
      Integer size;
      list<Integer> dimSize;
      list<DAE.Exp> conditions;
      list<list<BackendDAE.Equation>> eqnstrue;
      list<BackendDAE.Equation> eqnsfalse;
      BackendDAE.EquationAttributes eqAttr;

    case BackendDAE.EQUATION(exp=e1, scalar=e2, source=source, attr=eqAttr) equation
      eqAttr = markDifferentiated2(eqAttr);
    then BackendDAE.EQUATION(e1, e2, source, eqAttr);

    case BackendDAE.ARRAY_EQUATION(dimSize=dimSize, left=e1, right=e2, source=source, attr=eqAttr) equation
      eqAttr = markDifferentiated2(eqAttr);
    then BackendDAE.ARRAY_EQUATION(dimSize, e1, e2, source, eqAttr);

    case BackendDAE.SOLVED_EQUATION(componentRef=cr, exp=e2, source=source, attr=eqAttr) equation
      eqAttr = markDifferentiated2(eqAttr);
    then BackendDAE.SOLVED_EQUATION(cr, e2, source, eqAttr);

    case BackendDAE.RESIDUAL_EQUATION(exp=e2, source=source, attr=eqAttr) equation
      eqAttr = markDifferentiated2(eqAttr);
    then BackendDAE.RESIDUAL_EQUATION(e2, source, eqAttr);

    case BackendDAE.COMPLEX_EQUATION(size=size, left=e1, right=e2, source=source, attr=eqAttr) equation
      eqAttr = markDifferentiated2(eqAttr);
    then BackendDAE.COMPLEX_EQUATION(size, e1, e2, source, eqAttr);

    case BackendDAE.ALGORITHM()
    then inEqn;

    case BackendDAE.WHEN_EQUATION()
    then inEqn;

    case BackendDAE.IF_EQUATION(conditions=conditions, eqnstrue=eqnstrue, eqnsfalse=eqnsfalse, source=source, attr=eqAttr) equation
      eqAttr = markDifferentiated2(eqAttr);
      eqnstrue = List.mapList(eqnstrue, markDifferentiated);
      eqnsfalse = List.map(eqnsfalse, markDifferentiated);
    then BackendDAE.IF_EQUATION(conditions, eqnstrue, eqnsfalse, source, eqAttr);
  end match;
end markDifferentiated;

protected function markDifferentiated2
  input output BackendDAE.EquationAttributes attr;
algorithm
  attr.differentiated := true;
end markDifferentiated2;

public function isDifferentiated
  input BackendDAE.Equation inEqn;
  output Boolean diffed;
algorithm
  diffed := match (inEqn)
    local
      Boolean b;
      BackendDAE.Equation eqn;
    case BackendDAE.EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(differentiated=b)) then b;
    case BackendDAE.ARRAY_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(differentiated=b)) then b;
    case BackendDAE.SOLVED_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(differentiated=b)) then b;
    case BackendDAE.RESIDUAL_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(differentiated=b)) then b;
    case BackendDAE.COMPLEX_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(differentiated=b)) then b;
    case BackendDAE.ALGORITHM(attr=BackendDAE.EQUATION_ATTRIBUTES(differentiated=b)) then b;
    case BackendDAE.WHEN_EQUATION(attr=BackendDAE.EQUATION_ATTRIBUTES(differentiated=b))then b;
    case BackendDAE.IF_EQUATION(eqnsfalse=eqn::_) then isDifferentiated(eqn);
  end match;
end isDifferentiated;

public function replaceDerOpInEquationList
  "Replaces all der(cref) with $DER.cref in a list of equations."
  input list<BackendDAE.Equation> inEqns;
  output list<BackendDAE.Equation> outEqns;
algorithm
  (outEqns, ) := traverseExpsOfEquationList(inEqns, Expression.replaceDerOpInExpCond, NONE());
end replaceDerOpInEquationList;

public function getEquationRHS"gets the right hand side expression of an equation.
author:Waurich TUD 2014-10"
  input BackendDAE.Equation eq;
  output DAE.Exp rhs;
algorithm
  rhs := match(eq)
    local
      DAE.Exp exp1;
    case(BackendDAE.EQUATION(scalar = exp1))
      then exp1;
    case(BackendDAE.ARRAY_EQUATION(right = exp1))
      then exp1;
    case(BackendDAE.SOLVED_EQUATION(exp = exp1))
      then exp1;
    case(BackendDAE.COMPLEX_EQUATION(right = exp1))
      then exp1;
    case(BackendDAE.WHEN_EQUATION(whenEquation=BackendDAE.WHEN_STMTS(condition=DAE.BCONST(bool=true),whenStmtLst={BackendDAE.ASSIGN(right=exp1)})))
      then exp1;
    else
      //equation print("BackendEquation.getEquationRHS failed!\n!");
      then fail();
  end match;
end getEquationRHS;

public function getEquationLHS"gets the left hand side expression of an equation.
author:Waurich TUD 2014-10"
  input BackendDAE.Equation eq;
  output DAE.Exp lhs;
algorithm
  lhs := match(eq)
    local
      DAE.Exp exp1;
      DAE.ComponentRef cref;
    case(BackendDAE.EQUATION(exp = exp1))
      then exp1;
    case(BackendDAE.ARRAY_EQUATION(left = exp1))
      then exp1;
    case(BackendDAE.SOLVED_EQUATION(componentRef = cref))
      then Expression.crefExp(cref);
    case(BackendDAE.COMPLEX_EQUATION(left = exp1))
      then exp1;
    case(BackendDAE.WHEN_EQUATION(whenEquation=BackendDAE.WHEN_STMTS(condition=DAE.BCONST(bool=true),whenStmtLst={BackendDAE.ASSIGN(left=exp1)})))
      then exp1;
    else
      //equation print("BackendEquation.getEquationLHS failed!\n");
      then fail();
  end match;
end getEquationLHS;

annotation(__OpenModelica_Interface="backend");
end BackendEquation;