src/window.c

/*
** 2018 May 08
**
** The author disclaims copyright to this source code.  In place of
** a legal notice, here is a blessing:
**
**    May you do good and not evil.
**    May you find forgiveness for yourself and forgive others.
**    May you share freely, never taking more than you give.
**
*************************************************************************
*/
#include "sqliteInt.h"

#ifndef SQLITE_OMIT_WINDOWFUNC

/*
** SELECT REWRITING
**
**   Any SELECT statement that contains one or more window functions in
**   either the select list or ORDER BY clause (the only two places window
**   functions may be used) is transformed by function sqlite3WindowRewrite()
**   in order to support window function processing. For example, with the
**   schema:
**
**     CREATE TABLE t1(a, b, c, d, e, f, g);
**
**   the statement:
**
**     SELECT a+1, max(b) OVER (PARTITION BY c ORDER BY d) FROM t1 ORDER BY e;
**
**   is transformed to:
**
**     SELECT a+1, max(b) OVER (PARTITION BY c ORDER BY d) FROM (
**         SELECT a, e, c, d, b FROM t1 ORDER BY c, d
**     ) ORDER BY e;
**
**   The flattening optimization is disabled when processing this transformed
**   SELECT statement. This allows the implementation of the window function
**   (in this case max()) to process rows sorted in order of (c, d), which
**   makes things easier for obvious reasons. More generally:
**
**     * FROM, WHERE, GROUP BY and HAVING clauses are all moved to 
**       the sub-query.
**
**     * ORDER BY, LIMIT and OFFSET remain part of the parent query.
**
**     * Terminals from each of the expression trees that make up the 
**       select-list and ORDER BY expressions in the parent query are
**       selected by the sub-query. For the purposes of the transformation,
**       terminals are column references and aggregate functions.
**
**   If there is more than one window function in the SELECT that uses
**   the same window declaration (the OVER bit), then a single scan may
**   be used to process more than one window function. For example:
**
**     SELECT max(b) OVER (PARTITION BY c ORDER BY d), 
**            min(e) OVER (PARTITION BY c ORDER BY d) 
**     FROM t1;
**
**   is transformed in the same way as the example above. However:
**
**     SELECT max(b) OVER (PARTITION BY c ORDER BY d), 
**            min(e) OVER (PARTITION BY a ORDER BY b) 
**     FROM t1;
**
**   Must be transformed to:
**
**     SELECT max(b) OVER (PARTITION BY c ORDER BY d) FROM (
**         SELECT e, min(e) OVER (PARTITION BY a ORDER BY b), c, d, b FROM
**           SELECT a, e, c, d, b FROM t1 ORDER BY a, b
**         ) ORDER BY c, d
**     ) ORDER BY e;
**
**   so that both min() and max() may process rows in the order defined by
**   their respective window declarations.
**
** INTERFACE WITH SELECT.C
**
**   When processing the rewritten SELECT statement, code in select.c calls
**   sqlite3WhereBegin() to begin iterating through the results of the
**   sub-query, which is always implemented as a co-routine. It then calls
**   sqlite3WindowCodeStep() to process rows and finish the scan by calling
**   sqlite3WhereEnd().
**
**   sqlite3WindowCodeStep() generates VM code so that, for each row returned
**   by the sub-query a sub-routine (OP_Gosub) coded by select.c is invoked.
**   When the sub-routine is invoked:
**
**     * The results of all window-functions for the row are stored
**       in the associated Window.regResult registers.
**
**     * The required terminal values are stored in the current row of
**       temp table Window.iEphCsr.
**
**   In some cases, depending on the window frame and the specific window
**   functions invoked, sqlite3WindowCodeStep() caches each entire partition
**   in a temp table before returning any rows. In other cases it does not.
**   This detail is encapsulated within this file, the code generated by
**   select.c is the same in either case.
**
** BUILT-IN WINDOW FUNCTIONS
**
**   This implementation features the following built-in window functions:
**
**     row_number()
**     rank()
**     dense_rank()
**     percent_rank()
**     cume_dist()
**     ntile(N)
**     lead(expr [, offset [, default]])
**     lag(expr [, offset [, default]])
**     first_value(expr)
**     last_value(expr)
**     nth_value(expr, N)
**   
**   These are the same built-in window functions supported by Postgres. 
**   Although the behaviour of aggregate window functions (functions that
**   can be used as either aggregates or window funtions) allows them to
**   be implemented using an API, built-in window functions are much more
**   esoteric. Additionally, some window functions (e.g. nth_value()) 
**   may only be implemented by caching the entire partition in memory.
**   As such, some built-in window functions use the same API as aggregate
**   window functions and some are implemented directly using VDBE 
**   instructions. Additionally, for those functions that use the API, the
**   window frame is sometimes modified before the SELECT statement is
**   rewritten. For example, regardless of the specified window frame, the
**   row_number() function always uses:
**
**     ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW
**
**   See sqlite3WindowUpdate() for details.
**
**   As well as some of the built-in window functions, aggregate window
**   functions min() and max() are implemented using VDBE instructions if
**   the start of the window frame is declared as anything other than 
**   UNBOUNDED PRECEDING.
*/

/*
** Implementation of built-in window function row_number(). Assumes that the
** window frame has been coerced to:
**
**   ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW
*/
static void row_numberStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  i64 *p = (i64*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ) (*p)++;
  UNUSED_PARAMETER(nArg);
  UNUSED_PARAMETER(apArg);
}
static void row_numberValueFunc(sqlite3_context *pCtx){
  i64 *p = (i64*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  sqlite3_result_int64(pCtx, (p ? *p : 0));
}

/*
** Context object type used by rank(), dense_rank(), percent_rank() and
** cume_dist().
*/
struct CallCount {
  i64 nValue;
  i64 nStep;
  i64 nTotal;
};

/*
** Implementation of built-in window function dense_rank(). Assumes that
** the window frame has been set to:
**
**   RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW 
*/
static void dense_rankStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct CallCount *p;
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ) p->nStep = 1;
  UNUSED_PARAMETER(nArg);
  UNUSED_PARAMETER(apArg);
}
static void dense_rankValueFunc(sqlite3_context *pCtx){
  struct CallCount *p;
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    if( p->nStep ){
      p->nValue++;
      p->nStep = 0;
    }
    sqlite3_result_int64(pCtx, p->nValue);
  }
}

/*
** Implementation of built-in window function nth_value(). This
** implementation is used in "slow mode" only - when the EXCLUDE clause
** is not set to the default value "NO OTHERS".
*/
struct NthValueCtx {
  i64 nStep;
  sqlite3_value *pValue;
};
static void nth_valueStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct NthValueCtx *p;
  p = (struct NthValueCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    i64 iVal;
    switch( sqlite3_value_numeric_type(apArg[1]) ){
      case SQLITE_INTEGER:
        iVal = sqlite3_value_int64(apArg[1]);
        break;
      case SQLITE_FLOAT: {
        double fVal = sqlite3_value_double(apArg[1]);
        if( ((i64)fVal)!=fVal ) goto error_out;
        iVal = (i64)fVal;
        break;
      }
      default:
        goto error_out;
    }
    if( iVal<=0 ) goto error_out;

    p->nStep++;
    if( iVal==p->nStep ){
      p->pValue = sqlite3_value_dup(apArg[0]);
      if( !p->pValue ){
        sqlite3_result_error_nomem(pCtx);
      }
    }
  }
  UNUSED_PARAMETER(nArg);
  UNUSED_PARAMETER(apArg);
  return;

 error_out:
  sqlite3_result_error(
      pCtx, "second argument to nth_value must be a positive integer", -1
  );
}
static void nth_valueFinalizeFunc(sqlite3_context *pCtx){
  struct NthValueCtx *p;
  p = (struct NthValueCtx*)sqlite3_aggregate_context(pCtx, 0);
  if( p && p->pValue ){
    sqlite3_result_value(pCtx, p->pValue);
    sqlite3_value_free(p->pValue);
    p->pValue = 0;
  }
}
#define nth_valueInvFunc noopStepFunc
#define nth_valueValueFunc noopValueFunc

static void first_valueStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct NthValueCtx *p;
  p = (struct NthValueCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p && p->pValue==0 ){
    p->pValue = sqlite3_value_dup(apArg[0]);
    if( !p->pValue ){
      sqlite3_result_error_nomem(pCtx);
    }
  }
  UNUSED_PARAMETER(nArg);
  UNUSED_PARAMETER(apArg);
}
static void first_valueFinalizeFunc(sqlite3_context *pCtx){
  struct NthValueCtx *p;
  p = (struct NthValueCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p && p->pValue ){
    sqlite3_result_value(pCtx, p->pValue);
    sqlite3_value_free(p->pValue);
    p->pValue = 0;
  }
}
#define first_valueInvFunc noopStepFunc
#define first_valueValueFunc noopValueFunc

/*
** Implementation of built-in window function rank(). Assumes that
** the window frame has been set to:
**
**   RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW 
*/
static void rankStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct CallCount *p;
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    p->nStep++;
    if( p->nValue==0 ){
      p->nValue = p->nStep;
    }
  }
  UNUSED_PARAMETER(nArg);
  UNUSED_PARAMETER(apArg);
}
static void rankValueFunc(sqlite3_context *pCtx){
  struct CallCount *p;
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    sqlite3_result_int64(pCtx, p->nValue);
    p->nValue = 0;
  }
}

/*
** Implementation of built-in window function percent_rank(). Assumes that
** the window frame has been set to:
**
**   GROUPS BETWEEN CURRENT ROW AND UNBOUNDED FOLLOWING
*/
static void percent_rankStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct CallCount *p;
  UNUSED_PARAMETER(nArg); assert( nArg==0 );
  UNUSED_PARAMETER(apArg);
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    p->nTotal++;
  }
}
static void percent_rankInvFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct CallCount *p;
  UNUSED_PARAMETER(nArg); assert( nArg==0 );
  UNUSED_PARAMETER(apArg);
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  p->nStep++;
}
static void percent_rankValueFunc(sqlite3_context *pCtx){
  struct CallCount *p;
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    p->nValue = p->nStep;
    if( p->nTotal>1 ){
      double r = (double)p->nValue / (double)(p->nTotal-1);
      sqlite3_result_double(pCtx, r);
    }else{
      sqlite3_result_double(pCtx, 0.0);
    }
  }
}
#define percent_rankFinalizeFunc percent_rankValueFunc

/*
** Implementation of built-in window function cume_dist(). Assumes that
** the window frame has been set to:
**
**   GROUPS BETWEEN 1 FOLLOWING AND UNBOUNDED FOLLOWING
*/
static void cume_distStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct CallCount *p;
  UNUSED_PARAMETER(nArg); assert( nArg==0 );
  UNUSED_PARAMETER(apArg);
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    p->nTotal++;
  }
}
static void cume_distInvFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct CallCount *p;
  UNUSED_PARAMETER(nArg); assert( nArg==0 );
  UNUSED_PARAMETER(apArg);
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  p->nStep++;
}
static void cume_distValueFunc(sqlite3_context *pCtx){
  struct CallCount *p;
  p = (struct CallCount*)sqlite3_aggregate_context(pCtx, 0);
  if( p ){
    double r = (double)(p->nStep) / (double)(p->nTotal);
    sqlite3_result_double(pCtx, r);
  }
}
#define cume_distFinalizeFunc cume_distValueFunc

/*
** Context object for ntile() window function.
*/
struct NtileCtx {
  i64 nTotal;                     /* Total rows in partition */
  i64 nParam;                     /* Parameter passed to ntile(N) */
  i64 iRow;                       /* Current row */
};

/*
** Implementation of ntile(). This assumes that the window frame has
** been coerced to:
**
**   ROWS CURRENT ROW AND UNBOUNDED FOLLOWING
*/
static void ntileStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct NtileCtx *p;
  assert( nArg==1 ); UNUSED_PARAMETER(nArg);
  p = (struct NtileCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    if( p->nTotal==0 ){
      p->nParam = sqlite3_value_int64(apArg[0]);
      if( p->nParam<=0 ){
        sqlite3_result_error(
            pCtx, "argument of ntile must be a positive integer", -1
        );
      }
    }
    p->nTotal++;
  }
}
static void ntileInvFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct NtileCtx *p;
  assert( nArg==1 ); UNUSED_PARAMETER(nArg);
  UNUSED_PARAMETER(apArg);
  p = (struct NtileCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  p->iRow++;
}
static void ntileValueFunc(sqlite3_context *pCtx){
  struct NtileCtx *p;
  p = (struct NtileCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p && p->nParam>0 ){
    int nSize = (p->nTotal / p->nParam);
    if( nSize==0 ){
      sqlite3_result_int64(pCtx, p->iRow+1);
    }else{
      i64 nLarge = p->nTotal - p->nParam*nSize;
      i64 iSmall = nLarge*(nSize+1);
      i64 iRow = p->iRow;

      assert( (nLarge*(nSize+1) + (p->nParam-nLarge)*nSize)==p->nTotal );

      if( iRow<iSmall ){
        sqlite3_result_int64(pCtx, 1 + iRow/(nSize+1));
      }else{
        sqlite3_result_int64(pCtx, 1 + nLarge + (iRow-iSmall)/nSize);
      }
    }
  }
}
#define ntileFinalizeFunc ntileValueFunc

/*
** Context object for last_value() window function.
*/
struct LastValueCtx {
  sqlite3_value *pVal;
  int nVal;
};

/*
** Implementation of last_value().
*/
static void last_valueStepFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct LastValueCtx *p;
  UNUSED_PARAMETER(nArg);
  p = (struct LastValueCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p ){
    sqlite3_value_free(p->pVal);
    p->pVal = sqlite3_value_dup(apArg[0]);
    if( p->pVal==0 ){
      sqlite3_result_error_nomem(pCtx);
    }else{
      p->nVal++;
    }
  }
}
static void last_valueInvFunc(
  sqlite3_context *pCtx, 
  int nArg,
  sqlite3_value **apArg
){
  struct LastValueCtx *p;
  UNUSED_PARAMETER(nArg);
  UNUSED_PARAMETER(apArg);
  p = (struct LastValueCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( ALWAYS(p) ){
    p->nVal--;
    if( p->nVal==0 ){
      sqlite3_value_free(p->pVal);
      p->pVal = 0;
    }
  }
}
static void last_valueValueFunc(sqlite3_context *pCtx){
  struct LastValueCtx *p;
  p = (struct LastValueCtx*)sqlite3_aggregate_context(pCtx, 0);
  if( p && p->pVal ){
    sqlite3_result_value(pCtx, p->pVal);
  }
}
static void last_valueFinalizeFunc(sqlite3_context *pCtx){
  struct LastValueCtx *p;
  p = (struct LastValueCtx*)sqlite3_aggregate_context(pCtx, sizeof(*p));
  if( p && p->pVal ){
    sqlite3_result_value(pCtx, p->pVal);
    sqlite3_value_free(p->pVal);
    p->pVal = 0;
  }
}

/*
** Static names for the built-in window function names.  These static
** names are used, rather than string literals, so that FuncDef objects
** can be associated with a particular window function by direct
** comparison of the zName pointer.  Example:
**
**       if( pFuncDef->zName==row_valueName ){ ... }
*/
static const char row_numberName[] =   "row_number";
static const char dense_rankName[] =   "dense_rank";
static const char rankName[] =         "rank";
static const char percent_rankName[] = "percent_rank";
static const char cume_distName[] =    "cume_dist";
static const char ntileName[] =        "ntile";
static const char last_valueName[] =   "last_value";
static const char nth_valueName[] =    "nth_value";
static const char first_valueName[] =  "first_value";
static const char leadName[] =         "lead";
static const char lagName[] =          "lag";

/*
** No-op implementations of xStep() and xFinalize().  Used as place-holders
** for built-in window functions that never call those interfaces.
**
** The noopValueFunc() is called but is expected to do nothing.  The
** noopStepFunc() is never called, and so it is marked with NO_TEST to
** let the test coverage routine know not to expect this function to be
** invoked.
*/
static void noopStepFunc(    /*NO_TEST*/
  sqlite3_context *p,        /*NO_TEST*/
  int n,                     /*NO_TEST*/
  sqlite3_value **a          /*NO_TEST*/
){                           /*NO_TEST*/
  UNUSED_PARAMETER(p);       /*NO_TEST*/
  UNUSED_PARAMETER(n);       /*NO_TEST*/
  UNUSED_PARAMETER(a);       /*NO_TEST*/
  assert(0);                 /*NO_TEST*/
}                            /*NO_TEST*/
static void noopValueFunc(sqlite3_context *p){ UNUSED_PARAMETER(p); /*no-op*/ }

/* Window functions that use all window interfaces: xStep, xFinal,
** xValue, and xInverse */
#define WINDOWFUNCALL(name,nArg,extra) {                                   \
  nArg, (SQLITE_UTF8|SQLITE_FUNC_WINDOW|extra), 0, 0,                      \
  name ## StepFunc, name ## FinalizeFunc, name ## ValueFunc,               \
  name ## InvFunc, name ## Name, {0}                                       \
}

/* Window functions that are implemented using bytecode and thus have
** no-op routines for their methods */
#define WINDOWFUNCNOOP(name,nArg,extra) {                                  \
  nArg, (SQLITE_UTF8|SQLITE_FUNC_WINDOW|extra), 0, 0,                      \
  noopStepFunc, noopValueFunc, noopValueFunc,                              \
  noopStepFunc, name ## Name, {0}                                          \
}

/* Window functions that use all window interfaces: xStep, the
** same routine for xFinalize and xValue and which never call
** xInverse. */
#define WINDOWFUNCX(name,nArg,extra) {                                     \
  nArg, (SQLITE_UTF8|SQLITE_FUNC_WINDOW|extra), 0, 0,                      \
  name ## StepFunc, name ## ValueFunc, name ## ValueFunc,                  \
  noopStepFunc, name ## Name, {0}                                          \
}


/*
** Register those built-in window functions that are not also aggregates.
*/
void sqlite3WindowFunctions(void){
  static FuncDef aWindowFuncs[] = {
    WINDOWFUNCX(row_number, 0, 0),
    WINDOWFUNCX(dense_rank, 0, 0),
    WINDOWFUNCX(rank, 0, 0),
    WINDOWFUNCALL(percent_rank, 0, 0),
    WINDOWFUNCALL(cume_dist, 0, 0),
    WINDOWFUNCALL(ntile, 1, 0),
    WINDOWFUNCALL(last_value, 1, 0),
    WINDOWFUNCALL(nth_value, 2, 0),
    WINDOWFUNCALL(first_value, 1, 0),
    WINDOWFUNCNOOP(lead, 1, 0),
    WINDOWFUNCNOOP(lead, 2, 0),
    WINDOWFUNCNOOP(lead, 3, 0),
    WINDOWFUNCNOOP(lag, 1, 0),
    WINDOWFUNCNOOP(lag, 2, 0),
    WINDOWFUNCNOOP(lag, 3, 0),
  };
  sqlite3InsertBuiltinFuncs(aWindowFuncs, ArraySize(aWindowFuncs));
}

static Window *windowFind(Parse *pParse, Window *pList, const char *zName){
  Window *p;
  for(p=pList; p; p=p->pNextWin){
    if( sqlite3StrICmp(p->zName, zName)==0 ) break;
  }
  if( p==0 ){
    sqlite3ErrorMsg(pParse, "no such window: %s", zName);
  }
  return p;
}

/*
** This function is called immediately after resolving the function name
** for a window function within a SELECT statement. Argument pList is a
** linked list of WINDOW definitions for the current SELECT statement.
** Argument pFunc is the function definition just resolved and pWin
** is the Window object representing the associated OVER clause. This
** function updates the contents of pWin as follows:
**
**   * If the OVER clause refered to a named window (as in "max(x) OVER win"),
**     search list pList for a matching WINDOW definition, and update pWin
**     accordingly. If no such WINDOW clause can be found, leave an error
**     in pParse.
**
**   * If the function is a built-in window function that requires the
**     window to be coerced (see "BUILT-IN WINDOW FUNCTIONS" at the top
**     of this file), pWin is updated here.
*/
void sqlite3WindowUpdate(
  Parse *pParse, 
  Window *pList,                  /* List of named windows for this SELECT */
  Window *pWin,                   /* Window frame to update */
  FuncDef *pFunc                  /* Window function definition */
){
  if( pWin->zName && pWin->eFrmType==0 ){
    Window *p = windowFind(pParse, pList, pWin->zName);
    if( p==0 ) return;
    pWin->pPartition = sqlite3ExprListDup(pParse->db, p->pPartition, 0);
    pWin->pOrderBy = sqlite3ExprListDup(pParse->db, p->pOrderBy, 0);
    pWin->pStart = sqlite3ExprDup(pParse->db, p->pStart, 0);
    pWin->pEnd = sqlite3ExprDup(pParse->db, p->pEnd, 0);
    pWin->eStart = p->eStart;
    pWin->eEnd = p->eEnd;
    pWin->eFrmType = p->eFrmType;
    pWin->eExclude = p->eExclude;
  }else{
    sqlite3WindowChain(pParse, pWin, pList);
  }
  if( (pWin->eFrmType==TK_RANGE)
   && (pWin->pStart || pWin->pEnd) 
   && (pWin->pOrderBy==0 || pWin->pOrderBy->nExpr!=1)
  ){
    sqlite3ErrorMsg(pParse, 
      "RANGE with offset PRECEDING/FOLLOWING requires one ORDER BY expression"
    );
  }else
  if( pFunc->funcFlags & SQLITE_FUNC_WINDOW ){
    sqlite3 *db = pParse->db;
    if( pWin->pFilter ){
      sqlite3ErrorMsg(pParse, 
          "FILTER clause may only be used with aggregate window functions"
      );
    }else{
      struct WindowUpdate {
        const char *zFunc;
        int eFrmType;
        int eStart;
        int eEnd;
      } aUp[] = {
        { row_numberName,   TK_ROWS,   TK_UNBOUNDED, TK_CURRENT }, 
        { dense_rankName,   TK_RANGE,  TK_UNBOUNDED, TK_CURRENT }, 
        { rankName,         TK_RANGE,  TK_UNBOUNDED, TK_CURRENT }, 
        { percent_rankName, TK_GROUPS, TK_CURRENT,   TK_UNBOUNDED }, 
        { cume_distName,    TK_GROUPS, TK_FOLLOWING, TK_UNBOUNDED }, 
        { ntileName,        TK_ROWS,   TK_CURRENT,   TK_UNBOUNDED }, 
        { leadName,         TK_ROWS,   TK_UNBOUNDED, TK_UNBOUNDED }, 
        { lagName,          TK_ROWS,   TK_UNBOUNDED, TK_CURRENT }, 
      };
      int i;
      for(i=0; i<ArraySize(aUp); i++){
        if( pFunc->zName==aUp[i].zFunc ){
          sqlite3ExprDelete(db, pWin->pStart);
          sqlite3ExprDelete(db, pWin->pEnd);
          pWin->pEnd = pWin->pStart = 0;
          pWin->eFrmType = aUp[i].eFrmType;
          pWin->eStart = aUp[i].eStart;
          pWin->eEnd = aUp[i].eEnd;
          pWin->eExclude = 0;
          if( pWin->eStart==TK_FOLLOWING ){
            pWin->pStart = sqlite3Expr(db, TK_INTEGER, "1");
          }
          break;
        }
      }
    }
  }
  pWin->pFunc = pFunc;
}

/*
** Context object passed through sqlite3WalkExprList() to
** selectWindowRewriteExprCb() by selectWindowRewriteEList().
*/
typedef struct WindowRewrite WindowRewrite;
struct WindowRewrite {
  Window *pWin;
  SrcList *pSrc;
  ExprList *pSub;
  Table *pTab;
  Select *pSubSelect;             /* Current sub-select, if any */
};

/*
** Callback function used by selectWindowRewriteEList(). If necessary,
** this function appends to the output expression-list and updates 
** expression (*ppExpr) in place.
*/
static int selectWindowRewriteExprCb(Walker *pWalker, Expr *pExpr){
  struct WindowRewrite *p = pWalker->u.pRewrite;
  Parse *pParse = pWalker->pParse;
  assert( p!=0 );
  assert( p->pWin!=0 );

  /* If this function is being called from within a scalar sub-select
  ** that used by the SELECT statement being processed, only process
  ** TK_COLUMN expressions that refer to it (the outer SELECT). Do
  ** not process aggregates or window functions at all, as they belong
  ** to the scalar sub-select.  */
  if( p->pSubSelect ){
    if( pExpr->op!=TK_COLUMN ){
      return WRC_Continue;
    }else{
      int nSrc = p->pSrc->nSrc;
      int i;
      for(i=0; i<nSrc; i++){
        if( pExpr->iTable==p->pSrc->a[i].iCursor ) break;
      }
      if( i==nSrc ) return WRC_Continue;
    }
  }

  switch( pExpr->op ){

    case TK_FUNCTION:
      if( !ExprHasProperty(pExpr, EP_WinFunc) ){
        break;
      }else{
        Window *pWin;
        for(pWin=p->pWin; pWin; pWin=pWin->pNextWin){
          if( pExpr->y.pWin==pWin ){
            assert( pWin->pOwner==pExpr );
            return WRC_Prune;
          }
        }
      }
      /* Fall through.  */

    case TK_AGG_FUNCTION:
    case TK_COLUMN: {
      Expr *pDup = sqlite3ExprDup(pParse->db, pExpr, 0);
      p->pSub = sqlite3ExprListAppend(pParse, p->pSub, pDup);
      if( p->pSub ){
        assert( ExprHasProperty(pExpr, EP_Static)==0 );
        ExprSetProperty(pExpr, EP_Static);
        sqlite3ExprDelete(pParse->db, pExpr);
        ExprClearProperty(pExpr, EP_Static);
        memset(pExpr, 0, sizeof(Expr));

        pExpr->op = TK_COLUMN;
        pExpr->iColumn = p->pSub->nExpr-1;
        pExpr->iTable = p->pWin->iEphCsr;
        pExpr->y.pTab = p->pTab;
      }

      break;
    }

    default: /* no-op */
      break;
  }

  return WRC_Continue;
}
static int selectWindowRewriteSelectCb(Walker *pWalker, Select *pSelect){
  struct WindowRewrite *p = pWalker->u.pRewrite;
  Select *pSave = p->pSubSelect;
  if( pSave==pSelect ){
    return WRC_Continue;
  }else{
    p->pSubSelect = pSelect;
    sqlite3WalkSelect(pWalker, pSelect);
    p->pSubSelect = pSave;
  }
  return WRC_Prune;
}


/*
** Iterate through each expression in expression-list pEList. For each:
**
**   * TK_COLUMN,
**   * aggregate function, or
**   * window function with a Window object that is not a member of the 
**     Window list passed as the second argument (pWin).
**
** Append the node to output expression-list (*ppSub). And replace it
** with a TK_COLUMN that reads the (N-1)th element of table 
** pWin->iEphCsr, where N is the number of elements in (*ppSub) after
** appending the new one.
*/
static void selectWindowRewriteEList(
  Parse *pParse, 
  Window *pWin,
  SrcList *pSrc,
  ExprList *pEList,               /* Rewrite expressions in this list */
  Table *pTab,
  ExprList **ppSub                /* IN/OUT: Sub-select expression-list */
){
  Walker sWalker;
  WindowRewrite sRewrite;

  assert( pWin!=0 );
  memset(&sWalker, 0, sizeof(Walker));
  memset(&sRewrite, 0, sizeof(WindowRewrite));

  sRewrite.pSub = *ppSub;
  sRewrite.pWin = pWin;
  sRewrite.pSrc = pSrc;
  sRewrite.pTab = pTab;

  sWalker.pParse = pParse;
  sWalker.xExprCallback = selectWindowRewriteExprCb;
  sWalker.xSelectCallback = selectWindowRewriteSelectCb;
  sWalker.u.pRewrite = &sRewrite;

  (void)sqlite3WalkExprList(&sWalker, pEList);

  *ppSub = sRewrite.pSub;
}

/*
** Append a copy of each expression in expression-list pAppend to
** expression list pList. Return a pointer to the result list.
*/
static ExprList *exprListAppendList(
  Parse *pParse,          /* Parsing context */
  ExprList *pList,        /* List to which to append. Might be NULL */
  ExprList *pAppend,      /* List of values to append. Might be NULL */
  int bIntToNull
){
  if( pAppend ){
    int i;
    int nInit = pList ? pList->nExpr : 0;
    for(i=0; i<pAppend->nExpr; i++){
      Expr *pDup = sqlite3ExprDup(pParse->db, pAppend->a[i].pExpr, 0);
      if( bIntToNull && pDup && pDup->op==TK_INTEGER ){
        pDup->op = TK_NULL;
        pDup->flags &= ~(EP_IntValue|EP_IsTrue|EP_IsFalse);
      }
      pList = sqlite3ExprListAppend(pParse, pList, pDup);
      if( pList ) pList->a[nInit+i].sortFlags = pAppend->a[i].sortFlags;
    }
  }
  return pList;
}

/*
** If the SELECT statement passed as the second argument does not invoke
** any SQL window functions, this function is a no-op. Otherwise, it 
** rewrites the SELECT statement so that window function xStep functions
** are invoked in the correct order as described under "SELECT REWRITING"
** at the top of this file.
*/
int sqlite3WindowRewrite(Parse *pParse, Select *p){
  int rc = SQLITE_OK;
  if( p->pWin && p->pPrior==0 && (p->selFlags & SF_WinRewrite)==0 ){
    Vdbe *v = sqlite3GetVdbe(pParse);
    sqlite3 *db = pParse->db;
    Select *pSub = 0;             /* The subquery */
    SrcList *pSrc = p->pSrc;
    Expr *pWhere = p->pWhere;
    ExprList *pGroupBy = p->pGroupBy;
    Expr *pHaving = p->pHaving;
    ExprList *pSort = 0;

    ExprList *pSublist = 0;       /* Expression list for sub-query */
    Window *pMWin = p->pWin;      /* Master window object */
    Window *pWin;                 /* Window object iterator */
    Table *pTab;

    pTab = sqlite3DbMallocZero(db, sizeof(Table));
    if( pTab==0 ){
      return SQLITE_NOMEM;
    }

    p->pSrc = 0;
    p->pWhere = 0;
    p->pGroupBy = 0;
    p->pHaving = 0;
    p->selFlags &= ~SF_Aggregate;
    p->selFlags |= SF_WinRewrite;

    /* Create the ORDER BY clause for the sub-select. This is the concatenation
    ** of the window PARTITION and ORDER BY clauses. Then, if this makes it
    ** redundant, remove the ORDER BY from the parent SELECT.  */
    pSort = sqlite3ExprListDup(db, pMWin->pPartition, 0);
    pSort = exprListAppendList(pParse, pSort, pMWin->pOrderBy, 1);
    if( pSort && p->pOrderBy && p->pOrderBy->nExpr<=pSort->nExpr ){
      int nSave = pSort->nExpr;
      pSort->nExpr = p->pOrderBy->nExpr;
      if( sqlite3ExprListCompare(pSort, p->pOrderBy, -1)==0 ){
        sqlite3ExprListDelete(db, p->pOrderBy);
        p->pOrderBy = 0;
      }
      pSort->nExpr = nSave;
    }

    /* Assign a cursor number for the ephemeral table used to buffer rows.
    ** The OpenEphemeral instruction is coded later, after it is known how
    ** many columns the table will have.  */
    pMWin->iEphCsr = pParse->nTab++;
    pParse->nTab += 3;

    selectWindowRewriteEList(pParse, pMWin, pSrc, p->pEList, pTab, &pSublist);
    selectWindowRewriteEList(pParse, pMWin, pSrc, p->pOrderBy, pTab, &pSublist);
    pMWin->nBufferCol = (pSublist ? pSublist->nExpr : 0);

    /* Append the PARTITION BY and ORDER BY expressions to the to the 
    ** sub-select expression list. They are required to figure out where 
    ** boundaries for partitions and sets of peer rows lie.  */
    pSublist = exprListAppendList(pParse, pSublist, pMWin->pPartition, 0);
    pSublist = exprListAppendList(pParse, pSublist, pMWin->pOrderBy, 0);

    /* Append the arguments passed to each window function to the
    ** sub-select expression list. Also allocate two registers for each
    ** window function - one for the accumulator, another for interim
    ** results.  */
    for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
      ExprList *pArgs = pWin->pOwner->x.pList;
      if( pWin->pFunc->funcFlags & SQLITE_FUNC_SUBTYPE ){
        selectWindowRewriteEList(pParse, pMWin, pSrc, pArgs, pTab, &pSublist);
        pWin->iArgCol = (pSublist ? pSublist->nExpr : 0);
        pWin->bExprArgs = 1;
      }else{
        pWin->iArgCol = (pSublist ? pSublist->nExpr : 0);
        pSublist = exprListAppendList(pParse, pSublist, pArgs, 0);
      }
      if( pWin->pFilter ){
        Expr *pFilter = sqlite3ExprDup(db, pWin->pFilter, 0);
        pSublist = sqlite3ExprListAppend(pParse, pSublist, pFilter);
      }
      pWin->regAccum = ++pParse->nMem;
      pWin->regResult = ++pParse->nMem;
      sqlite3VdbeAddOp2(v, OP_Null, 0, pWin->regAccum);
    }

    /* If there is no ORDER BY or PARTITION BY clause, and the window
    ** function accepts zero arguments, and there are no other columns
    ** selected (e.g. "SELECT row_number() OVER () FROM t1"), it is possible
    ** that pSublist is still NULL here. Add a constant expression here to 
    ** keep everything legal in this case. 
    */
    if( pSublist==0 ){
      pSublist = sqlite3ExprListAppend(pParse, 0, 
        sqlite3Expr(db, TK_INTEGER, "0")
      );
    }

    pSub = sqlite3SelectNew(
        pParse, pSublist, pSrc, pWhere, pGroupBy, pHaving, pSort, 0, 0
    );
    p->pSrc = sqlite3SrcListAppend(pParse, 0, 0, 0);
    if( p->pSrc ){
      Table *pTab2;
      p->pSrc->a[0].pSelect = pSub;
      sqlite3SrcListAssignCursors(pParse, p->pSrc);
      pSub->selFlags |= SF_Expanded;
      pTab2 = sqlite3ResultSetOfSelect(pParse, pSub, SQLITE_AFF_NONE);
      if( pTab2==0 ){
        rc = SQLITE_NOMEM;
      }else{
        memcpy(pTab, pTab2, sizeof(Table));
        pTab->tabFlags |= TF_Ephemeral;
        p->pSrc->a[0].pTab = pTab;
        pTab = pTab2;
      }
      sqlite3VdbeAddOp2(v, OP_OpenEphemeral, pMWin->iEphCsr, pSublist->nExpr);
      sqlite3VdbeAddOp2(v, OP_OpenDup, pMWin->iEphCsr+1, pMWin->iEphCsr);
      sqlite3VdbeAddOp2(v, OP_OpenDup, pMWin->iEphCsr+2, pMWin->iEphCsr);
      sqlite3VdbeAddOp2(v, OP_OpenDup, pMWin->iEphCsr+3, pMWin->iEphCsr);
    }else{
      sqlite3SelectDelete(db, pSub);
    }
    if( db->mallocFailed ) rc = SQLITE_NOMEM;
    sqlite3DbFree(db, pTab);
  }

  return rc;
}

/*
** Unlink the Window object from the Select to which it is attached,
** if it is attached.
*/
void sqlite3WindowUnlinkFromSelect(Window *p){
  if( p->ppThis ){
    *p->ppThis = p->pNextWin;
    if( p->pNextWin ) p->pNextWin->ppThis = p->ppThis;
    p->ppThis = 0;
  }
}

/*
** Free the Window object passed as the second argument.
*/
void sqlite3WindowDelete(sqlite3 *db, Window *p){
  if( p ){
    sqlite3WindowUnlinkFromSelect(p);
    sqlite3ExprDelete(db, p->pFilter);
    sqlite3ExprListDelete(db, p->pPartition);
    sqlite3ExprListDelete(db, p->pOrderBy);
    sqlite3ExprDelete(db, p->pEnd);
    sqlite3ExprDelete(db, p->pStart);
    sqlite3DbFree(db, p->zName);
    sqlite3DbFree(db, p->zBase);
    sqlite3DbFree(db, p);
  }
}

/*
** Free the linked list of Window objects starting at the second argument.
*/
void sqlite3WindowListDelete(sqlite3 *db, Window *p){
  while( p ){
    Window *pNext = p->pNextWin;
    sqlite3WindowDelete(db, p);
    p = pNext;
  }
}

/*
** The argument expression is an PRECEDING or FOLLOWING offset.  The
** value should be a non-negative integer.  If the value is not a
** constant, change it to NULL.  The fact that it is then a non-negative
** integer will be caught later.  But it is important not to leave
** variable values in the expression tree.
*/
static Expr *sqlite3WindowOffsetExpr(Parse *pParse, Expr *pExpr){
  if( 0==sqlite3ExprIsConstant(pExpr) ){
    if( IN_RENAME_OBJECT ) sqlite3RenameExprUnmap(pParse, pExpr);
    sqlite3ExprDelete(pParse->db, pExpr);
    pExpr = sqlite3ExprAlloc(pParse->db, TK_NULL, 0, 0);
  }
  return pExpr;
}

/*
** Allocate and return a new Window object describing a Window Definition.
*/
Window *sqlite3WindowAlloc(
  Parse *pParse,    /* Parsing context */
  int eType,        /* Frame type. TK_RANGE, TK_ROWS, TK_GROUPS, or 0 */
  int eStart,       /* Start type: CURRENT, PRECEDING, FOLLOWING, UNBOUNDED */
  Expr *pStart,     /* Start window size if TK_PRECEDING or FOLLOWING */
  int eEnd,         /* End type: CURRENT, FOLLOWING, TK_UNBOUNDED, PRECEDING */
  Expr *pEnd,       /* End window size if TK_FOLLOWING or PRECEDING */
  u8 eExclude       /* EXCLUDE clause */
){
  Window *pWin = 0;
  int bImplicitFrame = 0;

  /* Parser assures the following: */
  assert( eType==0 || eType==TK_RANGE || eType==TK_ROWS || eType==TK_GROUPS );
  assert( eStart==TK_CURRENT || eStart==TK_PRECEDING
           || eStart==TK_UNBOUNDED || eStart==TK_FOLLOWING );
  assert( eEnd==TK_CURRENT || eEnd==TK_FOLLOWING
           || eEnd==TK_UNBOUNDED || eEnd==TK_PRECEDING );
  assert( (eStart==TK_PRECEDING || eStart==TK_FOLLOWING)==(pStart!=0) );
  assert( (eEnd==TK_FOLLOWING || eEnd==TK_PRECEDING)==(pEnd!=0) );

  if( eType==0 ){
    bImplicitFrame = 1;
    eType = TK_RANGE;
  }

  /* Additionally, the
  ** starting boundary type may not occur earlier in the following list than
  ** the ending boundary type:
  **
  **   UNBOUNDED PRECEDING
  **   <expr> PRECEDING
  **   CURRENT ROW
  **   <expr> FOLLOWING
  **   UNBOUNDED FOLLOWING
  **
  ** The parser ensures that "UNBOUNDED PRECEDING" cannot be used as an ending
  ** boundary, and than "UNBOUNDED FOLLOWING" cannot be used as a starting
  ** frame boundary.
  */
  if( (eStart==TK_CURRENT && eEnd==TK_PRECEDING)
   || (eStart==TK_FOLLOWING && (eEnd==TK_PRECEDING || eEnd==TK_CURRENT))
  ){
    sqlite3ErrorMsg(pParse, "unsupported frame specification");
    goto windowAllocErr;
  }

  pWin = (Window*)sqlite3DbMallocZero(pParse->db, sizeof(Window));
  if( pWin==0 ) goto windowAllocErr;
  pWin->eFrmType = eType;
  pWin->eStart = eStart;
  pWin->eEnd = eEnd;
  if( eExclude==0 && OptimizationDisabled(pParse->db, SQLITE_WindowFunc) ){
    eExclude = TK_NO;
  }
  pWin->eExclude = eExclude;
  pWin->bImplicitFrame = bImplicitFrame;
  pWin->pEnd = sqlite3WindowOffsetExpr(pParse, pEnd);
  pWin->pStart = sqlite3WindowOffsetExpr(pParse, pStart);
  return pWin;

windowAllocErr:
  sqlite3ExprDelete(pParse->db, pEnd);
  sqlite3ExprDelete(pParse->db, pStart);
  return 0;
}

/*
** Attach PARTITION and ORDER BY clauses pPartition and pOrderBy to window
** pWin. Also, if parameter pBase is not NULL, set pWin->zBase to the
** equivalent nul-terminated string.
*/
Window *sqlite3WindowAssemble(
  Parse *pParse, 
  Window *pWin, 
  ExprList *pPartition, 
  ExprList *pOrderBy, 
  Token *pBase
){
  if( pWin ){
    pWin->pPartition = pPartition;
    pWin->pOrderBy = pOrderBy;
    if( pBase ){
      pWin->zBase = sqlite3DbStrNDup(pParse->db, pBase->z, pBase->n);
    }
  }else{
    sqlite3ExprListDelete(pParse->db, pPartition);
    sqlite3ExprListDelete(pParse->db, pOrderBy);
  }
  return pWin;
}

/*
** Window *pWin has just been created from a WINDOW clause. Tokne pBase
** is the base window. Earlier windows from the same WINDOW clause are
** stored in the linked list starting at pWin->pNextWin. This function
** either updates *pWin according to the base specification, or else
** leaves an error in pParse.
*/
void sqlite3WindowChain(Parse *pParse, Window *pWin, Window *pList){
  if( pWin->zBase ){
    sqlite3 *db = pParse->db;
    Window *pExist = windowFind(pParse, pList, pWin->zBase);
    if( pExist ){
      const char *zErr = 0;
      /* Check for errors */
      if( pWin->pPartition ){
        zErr = "PARTITION clause";
      }else if( pExist->pOrderBy && pWin->pOrderBy ){
        zErr = "ORDER BY clause";
      }else if( pExist->bImplicitFrame==0 ){
        zErr = "frame specification";
      }
      if( zErr ){
        sqlite3ErrorMsg(pParse, 
            "cannot override %s of window: %s", zErr, pWin->zBase
        );
      }else{
        pWin->pPartition = sqlite3ExprListDup(db, pExist->pPartition, 0);
        if( pExist->pOrderBy ){
          assert( pWin->pOrderBy==0 );
          pWin->pOrderBy = sqlite3ExprListDup(db, pExist->pOrderBy, 0);
        }
        sqlite3DbFree(db, pWin->zBase);
        pWin->zBase = 0;
      }
    }
  }
}

/*
** Attach window object pWin to expression p.
*/
void sqlite3WindowAttach(Parse *pParse, Expr *p, Window *pWin){
  if( p ){
    assert( p->op==TK_FUNCTION );
    assert( pWin );
    p->y.pWin = pWin;
    ExprSetProperty(p, EP_WinFunc);
    pWin->pOwner = p;
    if( (p->flags & EP_Distinct) && pWin->eFrmType!=TK_FILTER ){
      sqlite3ErrorMsg(pParse,
          "DISTINCT is not supported for window functions"
      );
    }
  }else{
    sqlite3WindowDelete(pParse->db, pWin);
  }
}

/*
** Possibly link window pWin into the list at pSel->pWin (window functions
** to be processed as part of SELECT statement pSel). The window is linked
** in if either (a) there are no other windows already linked to this
** SELECT, or (b) the windows already linked use a compatible window frame.
*/
void sqlite3WindowLink(Select *pSel, Window *pWin){
  if( pSel!=0
   && (0==pSel->pWin || 0==sqlite3WindowCompare(0, pSel->pWin, pWin, 0))
  ){
    pWin->pNextWin = pSel->pWin;
    if( pSel->pWin ){
      pSel->pWin->ppThis = &pWin->pNextWin;
    }
    pSel->pWin = pWin;
    pWin->ppThis = &pSel->pWin;
  }
}

/*
** Return 0 if the two window objects are identical, or non-zero otherwise.
** Identical window objects can be processed in a single scan.
*/
int sqlite3WindowCompare(Parse *pParse, Window *p1, Window *p2, int bFilter){
  if( p1->eFrmType!=p2->eFrmType ) return 1;
  if( p1->eStart!=p2->eStart ) return 1;
  if( p1->eEnd!=p2->eEnd ) return 1;
  if( p1->eExclude!=p2->eExclude ) return 1;
  if( sqlite3ExprCompare(pParse, p1->pStart, p2->pStart, -1) ) return 1;
  if( sqlite3ExprCompare(pParse, p1->pEnd, p2->pEnd, -1) ) return 1;
  if( sqlite3ExprListCompare(p1->pPartition, p2->pPartition, -1) ) return 1;
  if( sqlite3ExprListCompare(p1->pOrderBy, p2->pOrderBy, -1) ) return 1;
  if( bFilter ){
    if( sqlite3ExprCompare(pParse, p1->pFilter, p2->pFilter, -1) ) return 1;
  }
  return 0;
}


/*
** This is called by code in select.c before it calls sqlite3WhereBegin()
** to begin iterating through the sub-query results. It is used to allocate
** and initialize registers and cursors used by sqlite3WindowCodeStep().
*/
void sqlite3WindowCodeInit(Parse *pParse, Window *pMWin){
  Window *pWin;
  Vdbe *v = sqlite3GetVdbe(pParse);

  /* Allocate registers to use for PARTITION BY values, if any. Initialize
  ** said registers to NULL.  */
  if( pMWin->pPartition ){
    int nExpr = pMWin->pPartition->nExpr;
    pMWin->regPart = pParse->nMem+1;
    pParse->nMem += nExpr;
    sqlite3VdbeAddOp3(v, OP_Null, 0, pMWin->regPart, pMWin->regPart+nExpr-1);
  }

  pMWin->regOne = ++pParse->nMem;
  sqlite3VdbeAddOp2(v, OP_Integer, 1, pMWin->regOne);

  if( pMWin->eExclude ){
    pMWin->regStartRowid = ++pParse->nMem;
    pMWin->regEndRowid = ++pParse->nMem;
    pMWin->csrApp = pParse->nTab++;
    sqlite3VdbeAddOp2(v, OP_Integer, 1, pMWin->regStartRowid);
    sqlite3VdbeAddOp2(v, OP_Integer, 0, pMWin->regEndRowid);
    sqlite3VdbeAddOp2(v, OP_OpenDup, pMWin->csrApp, pMWin->iEphCsr);
    return;
  }

  for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
    FuncDef *p = pWin->pFunc;
    if( (p->funcFlags & SQLITE_FUNC_MINMAX) && pWin->eStart!=TK_UNBOUNDED ){
      /* The inline versions of min() and max() require a single ephemeral
      ** table and 3 registers. The registers are used as follows:
      **
      **   regApp+0: slot to copy min()/max() argument to for MakeRecord
      **   regApp+1: integer value used to ensure keys are unique
      **   regApp+2: output of MakeRecord
      */
      ExprList *pList = pWin->pOwner->x.pList;
      KeyInfo *pKeyInfo = sqlite3KeyInfoFromExprList(pParse, pList, 0, 0);
      pWin->csrApp = pParse->nTab++;
      pWin->regApp = pParse->nMem+1;
      pParse->nMem += 3;
      if( pKeyInfo && pWin->pFunc->zName[1]=='i' ){
        assert( pKeyInfo->aSortFlags[0]==0 );
        pKeyInfo->aSortFlags[0] = KEYINFO_ORDER_DESC;
      }
      sqlite3VdbeAddOp2(v, OP_OpenEphemeral, pWin->csrApp, 2);
      sqlite3VdbeAppendP4(v, pKeyInfo, P4_KEYINFO);
      sqlite3VdbeAddOp2(v, OP_Integer, 0, pWin->regApp+1);
    }
    else if( p->zName==nth_valueName || p->zName==first_valueName ){
      /* Allocate two registers at pWin->regApp. These will be used to
      ** store the start and end index of the current frame.  */
      pWin->regApp = pParse->nMem+1;
      pWin->csrApp = pParse->nTab++;
      pParse->nMem += 2;
      sqlite3VdbeAddOp2(v, OP_OpenDup, pWin->csrApp, pMWin->iEphCsr);
    }
    else if( p->zName==leadName || p->zName==lagName ){
      pWin->csrApp = pParse->nTab++;
      sqlite3VdbeAddOp2(v, OP_OpenDup, pWin->csrApp, pMWin->iEphCsr);
    }
  }
}

#define WINDOW_STARTING_INT  0
#define WINDOW_ENDING_INT    1
#define WINDOW_NTH_VALUE_INT 2
#define WINDOW_STARTING_NUM  3
#define WINDOW_ENDING_NUM    4

/*
** A "PRECEDING <expr>" (eCond==0) or "FOLLOWING <expr>" (eCond==1) or the
** value of the second argument to nth_value() (eCond==2) has just been
** evaluated and the result left in register reg. This function generates VM
** code to check that the value is a non-negative integer and throws an
** exception if it is not.
*/
static void windowCheckValue(Parse *pParse, int reg, int eCond){
  static const char *azErr[] = {
    "frame starting offset must be a non-negative integer",
    "frame ending offset must be a non-negative integer",
    "second argument to nth_value must be a positive integer",
    "frame starting offset must be a non-negative number",
    "frame ending offset must be a non-negative number",
  };
  static int aOp[] = { OP_Ge, OP_Ge, OP_Gt, OP_Ge, OP_Ge };
  Vdbe *v = sqlite3GetVdbe(pParse);
  int regZero = sqlite3GetTempReg(pParse);
  assert( eCond>=0 && eCond<ArraySize(azErr) );
  sqlite3VdbeAddOp2(v, OP_Integer, 0, regZero);
  if( eCond>=WINDOW_STARTING_NUM ){
    int regString = sqlite3GetTempReg(pParse);
    sqlite3VdbeAddOp4(v, OP_String8, 0, regString, 0, "", P4_STATIC);
    sqlite3VdbeAddOp3(v, OP_Ge, regString, sqlite3VdbeCurrentAddr(v)+2, reg);
    sqlite3VdbeChangeP5(v, SQLITE_AFF_NUMERIC|SQLITE_JUMPIFNULL);
    VdbeCoverage(v);
    assert( eCond==3 || eCond==4 );
    VdbeCoverageIf(v, eCond==3);
    VdbeCoverageIf(v, eCond==4);
  }else{
    sqlite3VdbeAddOp2(v, OP_MustBeInt, reg, sqlite3VdbeCurrentAddr(v)+2);
    VdbeCoverage(v);
    assert( eCond==0 || eCond==1 || eCond==2 );
    VdbeCoverageIf(v, eCond==0);
    VdbeCoverageIf(v, eCond==1);
    VdbeCoverageIf(v, eCond==2);
  }
  sqlite3VdbeAddOp3(v, aOp[eCond], regZero, sqlite3VdbeCurrentAddr(v)+2, reg);
  VdbeCoverageNeverNullIf(v, eCond==0); /* NULL case captured by */
  VdbeCoverageNeverNullIf(v, eCond==1); /*   the OP_MustBeInt */
  VdbeCoverageNeverNullIf(v, eCond==2);
  VdbeCoverageNeverNullIf(v, eCond==3); /* NULL case caught by */
  VdbeCoverageNeverNullIf(v, eCond==4); /*   the OP_Ge */
  sqlite3MayAbort(pParse);
  sqlite3VdbeAddOp2(v, OP_Halt, SQLITE_ERROR, OE_Abort);
  sqlite3VdbeAppendP4(v, (void*)azErr[eCond], P4_STATIC);
  sqlite3ReleaseTempReg(pParse, regZero);
}

/*
** Return the number of arguments passed to the window-function associated
** with the object passed as the only argument to this function.
*/
static int windowArgCount(Window *pWin){
  ExprList *pList = pWin->pOwner->x.pList;
  return (pList ? pList->nExpr : 0);
}

typedef struct WindowCodeArg WindowCodeArg;
typedef struct WindowCsrAndReg WindowCsrAndReg;

/*
** See comments above struct WindowCodeArg.
*/
struct WindowCsrAndReg {
  int csr;                        /* Cursor number */
  int reg;                        /* First in array of peer values */
};

/*
** A single instance of this structure is allocated on the stack by 
** sqlite3WindowCodeStep() and a pointer to it passed to the various helper
** routines. This is to reduce the number of arguments required by each
** helper function.
**
** regArg:
**   Each window function requires an accumulator register (just as an
**   ordinary aggregate function does). This variable is set to the first
**   in an array of accumulator registers - one for each window function
**   in the WindowCodeArg.pMWin list.
**
** eDelete:
**   The window functions implementation sometimes caches the input rows
**   that it processes in a temporary table. If it is not zero, this
**   variable indicates when rows may be removed from the temp table (in
**   order to reduce memory requirements - it would always be safe just
**   to leave them there). Possible values for eDelete are:
**
**      WINDOW_RETURN_ROW:
**        An input row can be discarded after it is returned to the caller.
**
**      WINDOW_AGGINVERSE:
**        An input row can be discarded after the window functions xInverse()
**        callbacks have been invoked in it.
**
**      WINDOW_AGGSTEP:
**        An input row can be discarded after the window functions xStep()
**        callbacks have been invoked in it.
**
** start,current,end
**   Consider a window-frame similar to the following:
**
**     (ORDER BY a, b GROUPS BETWEEN 2 PRECEDING AND 2 FOLLOWING)
**
**   The windows functions implmentation caches the input rows in a temp
**   table, sorted by "a, b" (it actually populates the cache lazily, and
**   aggressively removes rows once they are no longer required, but that's
**   a mere detail). It keeps three cursors open on the temp table. One
**   (current) that points to the next row to return to the query engine
**   once its window function values have been calculated. Another (end)
**   points to the next row to call the xStep() method of each window function
**   on (so that it is 2 groups ahead of current). And a third (start) that
**   points to the next row to call the xInverse() method of each window
**   function on.
**
**   Each cursor (start, current and end) consists of a VDBE cursor
**   (WindowCsrAndReg.csr) and an array of registers (starting at
**   WindowCodeArg.reg) that always contains a copy of the peer values 
**   read from the corresponding cursor.
**
**   Depending on the window-frame in question, all three cursors may not
**   be required. In this case both WindowCodeArg.csr and reg are set to
**   0.
*/
struct WindowCodeArg {
  Parse *pParse;             /* Parse context */
  Window *pMWin;             /* First in list of functions being processed */
  Vdbe *pVdbe;               /* VDBE object */
  int addrGosub;             /* OP_Gosub to this address to return one row */
  int regGosub;              /* Register used with OP_Gosub(addrGosub) */
  int regArg;                /* First in array of accumulator registers */
  int eDelete;               /* See above */

  WindowCsrAndReg start;
  WindowCsrAndReg current;
  WindowCsrAndReg end;
};

/*
** Generate VM code to read the window frames peer values from cursor csr into
** an array of registers starting at reg.
*/
static void windowReadPeerValues(
  WindowCodeArg *p,
  int csr,
  int reg
){
  Window *pMWin = p->pMWin;
  ExprList *pOrderBy = pMWin->pOrderBy;
  if( pOrderBy ){
    Vdbe *v = sqlite3GetVdbe(p->pParse);
    ExprList *pPart = pMWin->pPartition;
    int iColOff = pMWin->nBufferCol + (pPart ? pPart->nExpr : 0);
    int i;
    for(i=0; i<pOrderBy->nExpr; i++){
      sqlite3VdbeAddOp3(v, OP_Column, csr, iColOff+i, reg+i);
    }
  }
}

/*
** Generate VM code to invoke either xStep() (if bInverse is 0) or 
** xInverse (if bInverse is non-zero) for each window function in the 
** linked list starting at pMWin. Or, for built-in window functions
** that do not use the standard function API, generate the required
** inline VM code.
**
** If argument csr is greater than or equal to 0, then argument reg is
** the first register in an array of registers guaranteed to be large
** enough to hold the array of arguments for each function. In this case
** the arguments are extracted from the current row of csr into the
** array of registers before invoking OP_AggStep or OP_AggInverse
**
** Or, if csr is less than zero, then the array of registers at reg is
** already populated with all columns from the current row of the sub-query.
**
** If argument regPartSize is non-zero, then it is a register containing the
** number of rows in the current partition.
*/
static void windowAggStep(
  WindowCodeArg *p,
  Window *pMWin,                  /* Linked list of window functions */
  int csr,                        /* Read arguments from this cursor */
  int bInverse,                   /* True to invoke xInverse instead of xStep */
  int reg                         /* Array of registers */
){
  Parse *pParse = p->pParse;
  Vdbe *v = sqlite3GetVdbe(pParse);
  Window *pWin;
  for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
    FuncDef *pFunc = pWin->pFunc;
    int regArg;
    int nArg = pWin->bExprArgs ? 0 : windowArgCount(pWin);
    int i;

    assert( bInverse==0 || pWin->eStart!=TK_UNBOUNDED );

    /* All OVER clauses in the same window function aggregate step must
    ** be the same. */
    assert( pWin==pMWin || sqlite3WindowCompare(pParse,pWin,pMWin,0)==0 );

    for(i=0; i<nArg; i++){
      if( i!=1 || pFunc->zName!=nth_valueName ){
        sqlite3VdbeAddOp3(v, OP_Column, csr, pWin->iArgCol+i, reg+i);
      }else{
        sqlite3VdbeAddOp3(v, OP_Column, pMWin->iEphCsr, pWin->iArgCol+i, reg+i);
      }
    }
    regArg = reg;

    if( pMWin->regStartRowid==0
     && (pFunc->funcFlags & SQLITE_FUNC_MINMAX) 
     && (pWin->eStart!=TK_UNBOUNDED)
    ){
      int addrIsNull = sqlite3VdbeAddOp1(v, OP_IsNull, regArg);
      VdbeCoverage(v);
      if( bInverse==0 ){
        sqlite3VdbeAddOp2(v, OP_AddImm, pWin->regApp+1, 1);
        sqlite3VdbeAddOp2(v, OP_SCopy, regArg, pWin->regApp);
        sqlite3VdbeAddOp3(v, OP_MakeRecord, pWin->regApp, 2, pWin->regApp+2);
        sqlite3VdbeAddOp2(v, OP_IdxInsert, pWin->csrApp, pWin->regApp+2);
      }else{
        sqlite3VdbeAddOp4Int(v, OP_SeekGE, pWin->csrApp, 0, regArg, 1);
        VdbeCoverageNeverTaken(v);
        sqlite3VdbeAddOp1(v, OP_Delete, pWin->csrApp);
        sqlite3VdbeJumpHere(v, sqlite3VdbeCurrentAddr(v)-2);
      }
      sqlite3VdbeJumpHere(v, addrIsNull);
    }else if( pWin->regApp ){
      assert( pFunc->zName==nth_valueName
           || pFunc->zName==first_valueName
      );
      assert( bInverse==0 || bInverse==1 );
      sqlite3VdbeAddOp2(v, OP_AddImm, pWin->regApp+1-bInverse, 1);
    }else if( pFunc->xSFunc!=noopStepFunc ){
      int addrIf = 0;
      if( pWin->pFilter ){
        int regTmp;
        assert( pWin->bExprArgs || !nArg ||nArg==pWin->pOwner->x.pList->nExpr );
        assert( pWin->bExprArgs || nArg  ||pWin->pOwner->x.pList==0 );
        regTmp = sqlite3GetTempReg(pParse);
        sqlite3VdbeAddOp3(v, OP_Column, csr, pWin->iArgCol+nArg,regTmp);
        addrIf = sqlite3VdbeAddOp3(v, OP_IfNot, regTmp, 0, 1);
        VdbeCoverage(v);
        sqlite3ReleaseTempReg(pParse, regTmp);
      }
      
      if( pWin->bExprArgs ){
        int iStart = sqlite3VdbeCurrentAddr(v);
        VdbeOp *pOp, *pEnd;

        nArg = pWin->pOwner->x.pList->nExpr;
        regArg = sqlite3GetTempRange(pParse, nArg);
        sqlite3ExprCodeExprList(pParse, pWin->pOwner->x.pList, regArg, 0, 0);

        pEnd = sqlite3VdbeGetOp(v, -1);
        for(pOp=sqlite3VdbeGetOp(v, iStart); pOp<=pEnd; pOp++){
          if( pOp->opcode==OP_Column && pOp->p1==pWin->iEphCsr ){
            pOp->p1 = csr;
          }
        }
      }
      if( pFunc->funcFlags & SQLITE_FUNC_NEEDCOLL ){
        CollSeq *pColl;
        assert( nArg>0 );
        pColl = sqlite3ExprNNCollSeq(pParse, pWin->pOwner->x.pList->a[0].pExpr);
        sqlite3VdbeAddOp4(v, OP_CollSeq, 0,0,0, (const char*)pColl, P4_COLLSEQ);
      }
      sqlite3VdbeAddOp3(v, bInverse? OP_AggInverse : OP_AggStep, 
                        bInverse, regArg, pWin->regAccum);
      sqlite3VdbeAppendP4(v, pFunc, P4_FUNCDEF);
      sqlite3VdbeChangeP5(v, (u8)nArg);
      if( pWin->bExprArgs ){
        sqlite3ReleaseTempRange(pParse, regArg, nArg);
      }
      if( addrIf ) sqlite3VdbeJumpHere(v, addrIf);
    }
  }
}

/*
** Values that may be passed as the second argument to windowCodeOp().
*/
#define WINDOW_RETURN_ROW 1
#define WINDOW_AGGINVERSE 2
#define WINDOW_AGGSTEP    3

/*
** Generate VM code to invoke either xValue() (bFin==0) or xFinalize()
** (bFin==1) for each window function in the linked list starting at
** pMWin. Or, for built-in window-functions that do not use the standard
** API, generate the equivalent VM code.
*/
static void windowAggFinal(WindowCodeArg *p, int bFin){
  Parse *pParse = p->pParse;
  Window *pMWin = p->pMWin;
  Vdbe *v = sqlite3GetVdbe(pParse);
  Window *pWin;

  for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
    if( pMWin->regStartRowid==0
     && (pWin->pFunc->funcFlags & SQLITE_FUNC_MINMAX) 
     && (pWin->eStart!=TK_UNBOUNDED)
    ){
      sqlite3VdbeAddOp2(v, OP_Null, 0, pWin->regResult);
      sqlite3VdbeAddOp1(v, OP_Last, pWin->csrApp);
      VdbeCoverage(v);
      sqlite3VdbeAddOp3(v, OP_Column, pWin->csrApp, 0, pWin->regResult);
      sqlite3VdbeJumpHere(v, sqlite3VdbeCurrentAddr(v)-2);
    }else if( pWin->regApp ){
      assert( pMWin->regStartRowid==0 );
    }else{
      int nArg = windowArgCount(pWin);
      if( bFin ){
        sqlite3VdbeAddOp2(v, OP_AggFinal, pWin->regAccum, nArg);
        sqlite3VdbeAppendP4(v, pWin->pFunc, P4_FUNCDEF);
        sqlite3VdbeAddOp2(v, OP_Copy, pWin->regAccum, pWin->regResult);
        sqlite3VdbeAddOp2(v, OP_Null, 0, pWin->regAccum);
      }else{
        sqlite3VdbeAddOp3(v, OP_AggValue,pWin->regAccum,nArg,pWin->regResult);
        sqlite3VdbeAppendP4(v, pWin->pFunc, P4_FUNCDEF);
      }
    }
  }
}

/*
** Generate code to calculate the current values of all window functions in the
** p->pMWin list by doing a full scan of the current window frame. Store the
** results in the Window.regResult registers, ready to return the upper
** layer.
*/
static void windowFullScan(WindowCodeArg *p){
  Window *pWin;
  Parse *pParse = p->pParse;
  Window *pMWin = p->pMWin;
  Vdbe *v = p->pVdbe;

  int regCRowid = 0;              /* Current rowid value */
  int regCPeer = 0;               /* Current peer values */
  int regRowid = 0;               /* AggStep rowid value */
  int regPeer = 0;                /* AggStep peer values */

  int nPeer;
  int lblNext;
  int lblBrk;
  int addrNext;
  int csr;

  VdbeModuleComment((v, "windowFullScan begin"));

  assert( pMWin!=0 );
  csr = pMWin->csrApp;
  nPeer = (pMWin->pOrderBy ? pMWin->pOrderBy->nExpr : 0);

  lblNext = sqlite3VdbeMakeLabel(pParse);
  lblBrk = sqlite3VdbeMakeLabel(pParse);

  regCRowid = sqlite3GetTempReg(pParse);
  regRowid = sqlite3GetTempReg(pParse);
  if( nPeer ){
    regCPeer = sqlite3GetTempRange(pParse, nPeer);
    regPeer = sqlite3GetTempRange(pParse, nPeer);
  }

  sqlite3VdbeAddOp2(v, OP_Rowid, pMWin->iEphCsr, regCRowid);
  windowReadPeerValues(p, pMWin->iEphCsr, regCPeer);

  for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
    sqlite3VdbeAddOp2(v, OP_Null, 0, pWin->regAccum);
  }

  sqlite3VdbeAddOp3(v, OP_SeekGE, csr, lblBrk, pMWin->regStartRowid);
  VdbeCoverage(v);
  addrNext = sqlite3VdbeCurrentAddr(v);
  sqlite3VdbeAddOp2(v, OP_Rowid, csr, regRowid);
  sqlite3VdbeAddOp3(v, OP_Gt, pMWin->regEndRowid, lblBrk, regRowid);
  VdbeCoverageNeverNull(v);

  if( pMWin->eExclude==TK_CURRENT ){
    sqlite3VdbeAddOp3(v, OP_Eq, regCRowid, lblNext, regRowid);
    VdbeCoverageNeverNull(v);
  }else if( pMWin->eExclude!=TK_NO ){
    int addr;
    int addrEq = 0;
    KeyInfo *pKeyInfo = 0;

    if( pMWin->pOrderBy ){
      pKeyInfo = sqlite3KeyInfoFromExprList(pParse, pMWin->pOrderBy, 0, 0);
    }
    if( pMWin->eExclude==TK_TIES ){
      addrEq = sqlite3VdbeAddOp3(v, OP_Eq, regCRowid, 0, regRowid);
      VdbeCoverageNeverNull(v);
    }
    if( pKeyInfo ){
      windowReadPeerValues(p, csr, regPeer);
      sqlite3VdbeAddOp3(v, OP_Compare, regPeer, regCPeer, nPeer);
      sqlite3VdbeAppendP4(v, (void*)pKeyInfo, P4_KEYINFO);
      addr = sqlite3VdbeCurrentAddr(v)+1;
      sqlite3VdbeAddOp3(v, OP_Jump, addr, lblNext, addr);
      VdbeCoverageEqNe(v);
    }else{
      sqlite3VdbeAddOp2(v, OP_Goto, 0, lblNext);
    }
    if( addrEq ) sqlite3VdbeJumpHere(v, addrEq);
  }

  windowAggStep(p, pMWin, csr, 0, p->regArg);

  sqlite3VdbeResolveLabel(v, lblNext);
  sqlite3VdbeAddOp2(v, OP_Next, csr, addrNext);
  VdbeCoverage(v);
  sqlite3VdbeJumpHere(v, addrNext-1);
  sqlite3VdbeJumpHere(v, addrNext+1);
  sqlite3ReleaseTempReg(pParse, regRowid);
  sqlite3ReleaseTempReg(pParse, regCRowid);
  if( nPeer ){
    sqlite3ReleaseTempRange(pParse, regPeer, nPeer);
    sqlite3ReleaseTempRange(pParse, regCPeer, nPeer);
  }

  windowAggFinal(p, 1);
  VdbeModuleComment((v, "windowFullScan end"));
}

/*
** Invoke the sub-routine at regGosub (generated by code in select.c) to
** return the current row of Window.iEphCsr. If all window functions are
** aggregate window functions that use the standard API, a single
** OP_Gosub instruction is all that this routine generates. Extra VM code
** for per-row processing is only generated for the following built-in window
** functions:
**
**   nth_value()
**   first_value()
**   lag()
**   lead()
*/
static void windowReturnOneRow(WindowCodeArg *p){
  Window *pMWin = p->pMWin;
  Vdbe *v = p->pVdbe;

  if( pMWin->regStartRowid ){
    windowFullScan(p);
  }else{
    Parse *pParse = p->pParse;
    Window *pWin;

    for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
      FuncDef *pFunc = pWin->pFunc;
      if( pFunc->zName==nth_valueName
       || pFunc->zName==first_valueName
      ){
        int csr = pWin->csrApp;
        int lbl = sqlite3VdbeMakeLabel(pParse);
        int tmpReg = sqlite3GetTempReg(pParse);
        sqlite3VdbeAddOp2(v, OP_Null, 0, pWin->regResult);
  
        if( pFunc->zName==nth_valueName ){
          sqlite3VdbeAddOp3(v, OP_Column,pMWin->iEphCsr,pWin->iArgCol+1,tmpReg);
          windowCheckValue(pParse, tmpReg, 2);
        }else{
          sqlite3VdbeAddOp2(v, OP_Integer, 1, tmpReg);
        }
        sqlite3VdbeAddOp3(v, OP_Add, tmpReg, pWin->regApp, tmpReg);
        sqlite3VdbeAddOp3(v, OP_Gt, pWin->regApp+1, lbl, tmpReg);
        VdbeCoverageNeverNull(v);
        sqlite3VdbeAddOp3(v, OP_SeekRowid, csr, 0, tmpReg);
        VdbeCoverageNeverTaken(v);
        sqlite3VdbeAddOp3(v, OP_Column, csr, pWin->iArgCol, pWin->regResult);
        sqlite3VdbeResolveLabel(v, lbl);
        sqlite3ReleaseTempReg(pParse, tmpReg);
      }
      else if( pFunc->zName==leadName || pFunc->zName==lagName ){
        int nArg = pWin->pOwner->x.pList->nExpr;
        int csr = pWin->csrApp;
        int lbl = sqlite3VdbeMakeLabel(pParse);
        int tmpReg = sqlite3GetTempReg(pParse);
        int iEph = pMWin->iEphCsr;
  
        if( nArg<3 ){
          sqlite3VdbeAddOp2(v, OP_Null, 0, pWin->regResult);
        }else{
          sqlite3VdbeAddOp3(v, OP_Column, iEph,pWin->iArgCol+2,pWin->regResult);
        }
        sqlite3VdbeAddOp2(v, OP_Rowid, iEph, tmpReg);
        if( nArg<2 ){
          int val = (pFunc->zName==leadName ? 1 : -1);
          sqlite3VdbeAddOp2(v, OP_AddImm, tmpReg, val);
        }else{
          int op = (pFunc->zName==leadName ? OP_Add : OP_Subtract);
          int tmpReg2 = sqlite3GetTempReg(pParse);
          sqlite3VdbeAddOp3(v, OP_Column, iEph, pWin->iArgCol+1, tmpReg2);
          sqlite3VdbeAddOp3(v, op, tmpReg2, tmpReg, tmpReg);
          sqlite3ReleaseTempReg(pParse, tmpReg2);
        }
  
        sqlite3VdbeAddOp3(v, OP_SeekRowid, csr, lbl, tmpReg);
        VdbeCoverage(v);
        sqlite3VdbeAddOp3(v, OP_Column, csr, pWin->iArgCol, pWin->regResult);
        sqlite3VdbeResolveLabel(v, lbl);
        sqlite3ReleaseTempReg(pParse, tmpReg);
      }
    }
  }
  sqlite3VdbeAddOp2(v, OP_Gosub, p->regGosub, p->addrGosub);
}

/*
** Generate code to set the accumulator register for each window function
** in the linked list passed as the second argument to NULL. And perform
** any equivalent initialization required by any built-in window functions
** in the list.
*/
static int windowInitAccum(Parse *pParse, Window *pMWin){
  Vdbe *v = sqlite3GetVdbe(pParse);
  int regArg;
  int nArg = 0;
  Window *pWin;
  for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
    FuncDef *pFunc = pWin->pFunc;
    sqlite3VdbeAddOp2(v, OP_Null, 0, pWin->regAccum);
    nArg = MAX(nArg, windowArgCount(pWin));
    if( pMWin->regStartRowid==0 ){
      if( pFunc->zName==nth_valueName || pFunc->zName==first_valueName ){
        sqlite3VdbeAddOp2(v, OP_Integer, 0, pWin->regApp);
        sqlite3VdbeAddOp2(v, OP_Integer, 0, pWin->regApp+1);
      }

      if( (pFunc->funcFlags & SQLITE_FUNC_MINMAX) && pWin->csrApp ){
        assert( pWin->eStart!=TK_UNBOUNDED );
        sqlite3VdbeAddOp1(v, OP_ResetSorter, pWin->csrApp);
        sqlite3VdbeAddOp2(v, OP_Integer, 0, pWin->regApp+1);
      }
    }
  }
  regArg = pParse->nMem+1;
  pParse->nMem += nArg;
  return regArg;
}

/* 
** Return true if the current frame should be cached in the ephemeral table,
** even if there are no xInverse() calls required.
*/
static int windowCacheFrame(Window *pMWin){
  Window *pWin;
  if( pMWin->regStartRowid ) return 1;
  for(pWin=pMWin; pWin; pWin=pWin->pNextWin){
    FuncDef *pFunc = pWin->pFunc;
    if( (pFunc->zName==nth_valueName)
     || (pFunc->zName==first_valueName)
     || (pFunc->zName==leadName)
     || (pFunc->zName==lagName)
    ){
      return 1;
    }
  }
  return 0;
}

/*
** regOld and regNew are each the first register in an array of size
** pOrderBy->nExpr. This function generates code to compare the two
** arrays of registers using the collation sequences and other comparison
** parameters specified by pOrderBy. 
**
** If the two arrays are not equal, the contents of regNew is copied to 
** regOld and control falls through. Otherwise, if the contents of the arrays
** are equal, an OP_Goto is executed. The address of the OP_Goto is returned.
*/
static void windowIfNewPeer(
  Parse *pParse,
  ExprList *pOrderBy,
  int regNew,                     /* First in array of new values */
  int regOld,                     /* First in array of old values */
  int addr                        /* Jump here */
){
  Vdbe *v = sqlite3GetVdbe(pParse);
  if( pOrderBy ){
    int nVal = pOrderBy->nExpr;
    KeyInfo *pKeyInfo = sqlite3KeyInfoFromExprList(pParse, pOrderBy, 0, 0);
    sqlite3VdbeAddOp3(v, OP_Compare, regOld, regNew, nVal);
    sqlite3VdbeAppendP4(v, (void*)pKeyInfo, P4_KEYINFO);
    sqlite3VdbeAddOp3(v, OP_Jump, 
      sqlite3VdbeCurrentAddr(v)+1, addr, sqlite3VdbeCurrentAddr(v)+1
    );
    VdbeCoverageEqNe(v);
    sqlite3VdbeAddOp3(v, OP_Copy, regNew, regOld, nVal-1);
  }else{
    sqlite3VdbeAddOp2(v, OP_Goto, 0, addr);
  }
}

/*
** This function is called as part of generating VM programs for RANGE
** offset PRECEDING/FOLLOWING frame boundaries. Assuming "ASC" order for
** the ORDER BY term in the window, and that argument op is OP_Ge, it generates
** code equivalent to:
**
**   if( csr1.peerVal + regVal >= csr2.peerVal ) goto lbl;
**
** The value of parameter op may also be OP_Gt or OP_Le. In these cases the
** operator in the above pseudo-code is replaced with ">" or "<=", respectively.
**
** If the sort-order for the ORDER BY term in the window is DESC, then the
** comparison is reversed. Instead of adding regVal to csr1.peerVal, it is
** subtracted. And the comparison operator is inverted to - ">=" becomes "<=",
** ">" becomes "<", and so on. So, with DESC sort order, if the argument op
** is OP_Ge, the generated code is equivalent to:
**
**   if( csr1.peerVal - regVal <= csr2.peerVal ) goto lbl;
**
** A special type of arithmetic is used such that if csr1.peerVal is not
** a numeric type (real or integer), then the result of the addition addition
** or subtraction is a a copy of csr1.peerVal.
*/
static void windowCodeRangeTest(
  WindowCodeArg *p, 
  int op,                         /* OP_Ge, OP_Gt, or OP_Le */
  int csr1,                       /* Cursor number for cursor 1 */
  int regVal,                     /* Register containing non-negative number */
  int csr2,                       /* Cursor number for cursor 2 */
  int lbl                         /* Jump destination if condition is true */
){
  Parse *pParse = p->pParse;
  Vdbe *v = sqlite3GetVdbe(pParse);
  ExprList *pOrderBy = p->pMWin->pOrderBy;  /* ORDER BY clause for window */
  int reg1 = sqlite3GetTempReg(pParse);     /* Reg. for csr1.peerVal+regVal */
  int reg2 = sqlite3GetTempReg(pParse);     /* Reg. for csr2.peerVal */
  int regString = ++pParse->nMem;           /* Reg. for constant value '' */
  int arith = OP_Add;                       /* OP_Add or OP_Subtract */
  int addrGe;                               /* Jump destination */

  assert( op==OP_Ge || op==OP_Gt || op==OP_Le );
  assert( pOrderBy && pOrderBy->nExpr==1 );
  if( pOrderBy->a[0].sortFlags & KEYINFO_ORDER_DESC ){
    switch( op ){
      case OP_Ge: op = OP_Le; break;
      case OP_Gt: op = OP_Lt; break;
      default: assert( op==OP_Le ); op = OP_Ge; break;
    }
    arith = OP_Subtract;
  }

  /* Read the peer-value from each cursor into a register */
  windowReadPeerValues(p, csr1, reg1);
  windowReadPeerValues(p, csr2, reg2);

  VdbeModuleComment((v, "CodeRangeTest: if( R%d %s R%d %s R%d ) goto lbl",
      reg1, (arith==OP_Add ? "+" : "-"), regVal,
      ((op==OP_Ge) ? ">=" : (op==OP_Le) ? "<=" : (op==OP_Gt) ? ">" : "<"), reg2
  ));

  /* Register reg1 currently contains csr1.peerVal (the peer-value from csr1).
  ** This block adds (or subtracts for DESC) the numeric value in regVal
  ** from it. Or, if reg1 is not numeric (it is a NULL, a text value or a blob),
  ** then leave reg1 as it is. In pseudo-code, this is implemented as:
  **
  **   if( reg1>='' ) goto addrGe;
  **   reg1 = reg1 +/- regVal
  **   addrGe:
  **
  ** Since all strings and blobs are greater-than-or-equal-to an empty string,
  ** the add/subtract is skipped for these, as required. If reg1 is a NULL,
  ** then the arithmetic is performed, but since adding or subtracting from
  ** NULL is always NULL anyway, this case is handled as required too.  */
  sqlite3VdbeAddOp4(v, OP_String8, 0, regString, 0, "", P4_STATIC);
  addrGe = sqlite3VdbeAddOp3(v, OP_Ge, regString, 0, reg1);
  VdbeCoverage(v);
  sqlite3VdbeAddOp3(v, arith, regVal, reg1, reg1);
  sqlite3VdbeJumpHere(v, addrGe);

  /* If the BIGNULL flag is set for the ORDER BY, then it is required to 
  ** consider NULL values to be larger than all other values, instead of 
  ** the usual smaller. The VDBE opcodes OP_Ge and so on do not handle this
  ** (and adding that capability causes a performance regression), so
  ** instead if the BIGNULL flag is set then cases where either reg1 or
  ** reg2 are NULL are handled separately in the following block. The code
  ** generated is equivalent to:
  **
  **   if( reg1 IS NULL ){
  **     if( op==OP_Ge ) goto lbl;
  **     if( op==OP_Gt && reg2 IS NOT NULL ) goto lbl;
  **     if( op==OP_Le && reg2 IS NULL ) goto lbl;
  **   }else if( reg2 IS NULL ){
  **     if( op==OP_Le ) goto lbl;
  **   }
  **
  ** Additionally, if either reg1 or reg2 are NULL but the jump to lbl is 
  ** not taken, control jumps over the comparison operator coded below this
  ** block.  */
  if( pOrderBy->a[0].sortFlags & KEYINFO_ORDER_BIGNULL ){
    /* This block runs if reg1 contains a NULL. */
    int addr = sqlite3VdbeAddOp1(v, OP_NotNull, reg1); VdbeCoverage(v);
    switch( op ){
      case OP_Ge: 
        sqlite3VdbeAddOp2(v, OP_Goto, 0, lbl); 
        break;
      case OP_Gt: 
        sqlite3VdbeAddOp2(v, OP_NotNull, reg2, lbl); 
        VdbeCoverage(v); 
        break;
      case OP_Le: 
        sqlite3VdbeAddOp2(v, OP_IsNull, reg2, lbl); 
        VdbeCoverage(v); 
        break;
      default: assert( op==OP_Lt ); /* no-op */ break;
    }
    sqlite3VdbeAddOp2(v, OP_Goto, 0, sqlite3VdbeCurrentAddr(v)+3);

    /* This block runs if reg1 is not NULL, but reg2 is. */
    sqlite3VdbeJumpHere(v, addr);
    sqlite3VdbeAddOp2(v, OP_IsNull, reg2, lbl); VdbeCoverage(v);
    if( op==OP_Gt || op==OP_Ge ){
      sqlite3VdbeChangeP2(v, -1, sqlite3VdbeCurrentAddr(v)+1);
    }
  }

  /* Compare registers reg2 and reg1, taking the jump if required. Note that
  ** control skips over this test if the BIGNULL flag is set and either
  ** reg1 or reg2 contain a NULL value.  */
  sqlite3VdbeAddOp3(v, op, reg2, lbl, reg1); VdbeCoverage(v);
  sqlite3VdbeChangeP5(v, SQLITE_NULLEQ);

  assert( op==OP_Ge || op==OP_Gt || op==OP_Lt || op==OP_Le );
  testcase(op==OP_Ge); VdbeCoverageIf(v, op==OP_Ge);
  testcase(op==OP_Lt); VdbeCoverageIf(v, op==OP_Lt);
  testcase(op==OP_Le); VdbeCoverageIf(v, op==OP_Le);
  testcase(op==OP_Gt); VdbeCoverageIf(v, op==OP_Gt);
  sqlite3ReleaseTempReg(pParse, reg1);
  sqlite3ReleaseTempReg(pParse, reg2);

  VdbeModuleComment((v, "CodeRangeTest: end"));
}

/*
** Helper function for sqlite3WindowCodeStep(). Each call to this function
** generates VM code for a single RETURN_ROW, AGGSTEP or AGGINVERSE 
** operation. Refer to the header comment for sqlite3WindowCodeStep() for
** details.
*/
static int windowCodeOp(
 WindowCodeArg *p,                /* Context object */
 int op,                          /* WINDOW_RETURN_ROW, AGGSTEP or AGGINVERSE */
 int regCountdown,                /* Register for OP_IfPos countdown */
 int jumpOnEof                    /* Jump here if stepped cursor reaches EOF */
){
  int csr, reg;
  Parse *pParse = p->pParse;
  Window *pMWin = p->pMWin;
  int ret = 0;
  Vdbe *v = p->pVdbe;
  int addrContinue = 0;
  int bPeer = (pMWin->eFrmType!=TK_ROWS);

  int lblDone = sqlite3VdbeMakeLabel(pParse);
  int addrNextRange = 0;

  /* Special case - WINDOW_AGGINVERSE is always a no-op if the frame
  ** starts with UNBOUNDED PRECEDING. */
  if( op==WINDOW_AGGINVERSE && pMWin->eStart==TK_UNBOUNDED ){
    assert( regCountdown==0 && jumpOnEof==0 );
    return 0;
  }

  if( regCountdown>0 ){
    if( pMWin->eFrmType==TK_RANGE ){
      addrNextRange = sqlite3VdbeCurrentAddr(v);
      assert( op==WINDOW_AGGINVERSE || op==WINDOW_AGGSTEP );
      if( op==WINDOW_AGGINVERSE ){
        if( pMWin->eStart==TK_FOLLOWING ){
          windowCodeRangeTest(
              p, OP_Le, p->current.csr, regCountdown, p->start.csr, lblDone
          );
        }else{
          windowCodeRangeTest(
              p, OP_Ge, p->start.csr, regCountdown, p->current.csr, lblDone
          );
        }
      }else{
        windowCodeRangeTest(
            p, OP_Gt, p->end.csr, regCountdown, p->current.csr, lblDone
        );
      }
    }else{
      sqlite3VdbeAddOp3(v, OP_IfPos, regCountdown, lblDone, 1);
      VdbeCoverage(v);
    }
  }

  if( op==WINDOW_RETURN_ROW && pMWin->regStartRowid==0 ){
    windowAggFinal(p, 0);
  }
  addrContinue = sqlite3VdbeCurrentAddr(v);

  /* If this is a (RANGE BETWEEN a FOLLOWING AND b FOLLOWING) or
  ** (RANGE BETWEEN b PRECEDING AND a PRECEDING) frame, ensure the 
  ** start cursor does not advance past the end cursor within the 
  ** temporary table. It otherwise might, if (a>b).  */
  if( pMWin->eStart==pMWin->eEnd && regCountdown
   && pMWin->eFrmType==TK_RANGE && op==WINDOW_AGGINVERSE
  ){
    int regRowid1 = sqlite3GetTempReg(pParse);
    int regRowid2 = sqlite3GetTempReg(pParse);
    sqlite3VdbeAddOp2(v, OP_Rowid, p->start.csr, regRowid1);
    sqlite3VdbeAddOp2(v, OP_Rowid, p->end.csr, regRowid2);
    sqlite3VdbeAddOp3(v, OP_Ge, regRowid2, lblDone, regRowid1);
    VdbeCoverage(v);
    sqlite3ReleaseTempReg(pParse, regRowid1);
    sqlite3ReleaseTempReg(pParse, regRowid2);
    assert( pMWin->eStart==TK_PRECEDING || pMWin->eStart==TK_FOLLOWING );
  }

  switch( op ){
    case WINDOW_RETURN_ROW:
      csr = p->current.csr;
      reg = p->current.reg;
      windowReturnOneRow(p);
      break;

    case WINDOW_AGGINVERSE:
      csr = p->start.csr;
      reg = p->start.reg;
      if( pMWin->regStartRowid ){
        assert( pMWin->regEndRowid );
        sqlite3VdbeAddOp2(v, OP_AddImm, pMWin->regStartRowid, 1);
      }else{
        windowAggStep(p, pMWin, csr, 1, p->regArg);
      }
      break;

    default:
      assert( op==WINDOW_AGGSTEP );
      csr = p->end.csr;
      reg = p->end.reg;
      if( pMWin->regStartRowid ){
        assert( pMWin->regEndRowid );
        sqlite3VdbeAddOp2(v, OP_AddImm, pMWin->regEndRowid, 1);
      }else{
        windowAggStep(p, pMWin, csr, 0, p->regArg);
      }
      break;
  }

  if( op==p->eDelete ){
    sqlite3VdbeAddOp1(v, OP_Delete, csr);
    sqlite3VdbeChangeP5(v, OPFLAG_SAVEPOSITION);
  }

  if( jumpOnEof ){
    sqlite3VdbeAddOp2(v, OP_Next, csr, sqlite3VdbeCurrentAddr(v)+2);
    VdbeCoverage(v);
    ret = sqlite3VdbeAddOp0(v, OP_Goto);
  }else{
    sqlite3VdbeAddOp2(v, OP_Next, csr, sqlite3VdbeCurrentAddr(v)+1+bPeer);
    VdbeCoverage(v);
    if( bPeer ){
      sqlite3VdbeAddOp2(v, OP_Goto, 0, lblDone);
    }
  }

  if( bPeer ){
    int nReg = (pMWin->pOrderBy ? pMWin->pOrderBy->nExpr : 0);
    int regTmp = (nReg ? sqlite3GetTempRange(pParse, nReg) : 0);
    windowReadPeerValues(p, csr, regTmp);
    windowIfNewPeer(pParse, pMWin->pOrderBy, regTmp, reg, addrContinue);
    sqlite3ReleaseTempRange(pParse, regTmp, nReg);
  }

  if( addrNextRange ){
    sqlite3VdbeAddOp2(v, OP_Goto, 0, addrNextRange);
  }
  sqlite3VdbeResolveLabel(v, lblDone);
  return ret;
}


/*
** Allocate and return a duplicate of the Window object indicated by the
** third argument. Set the Window.pOwner field of the new object to
** pOwner.
*/
Window *sqlite3WindowDup(sqlite3 *db, Expr *pOwner, Window *p){
  Window *pNew = 0;
  if( ALWAYS(p) ){
    pNew = sqlite3DbMallocZero(db, sizeof(Window));
    if( pNew ){
      pNew->zName = sqlite3DbStrDup(db, p->zName);
      pNew->zBase = sqlite3DbStrDup(db, p->zBase);
      pNew->pFilter = sqlite3ExprDup(db, p->pFilter, 0);
      pNew->pFunc = p->pFunc;
      pNew->pPartition = sqlite3ExprListDup(db, p->pPartition, 0);
      pNew->pOrderBy = sqlite3ExprListDup(db, p->pOrderBy, 0);
      pNew->eFrmType = p->eFrmType;
      pNew->eEnd = p->eEnd;
      pNew->eStart = p->eStart;
      pNew->eExclude = p->eExclude;
      pNew->regResult = p->regResult;
      pNew->pStart = sqlite3ExprDup(db, p->pStart, 0);
      pNew->pEnd = sqlite3ExprDup(db, p->pEnd, 0);
      pNew->pOwner = pOwner;
      pNew->bImplicitFrame = p->bImplicitFrame;
    }
  }
  return pNew;
}

/*
** Return a copy of the linked list of Window objects passed as the
** second argument.
*/
Window *sqlite3WindowListDup(sqlite3 *db, Window *p){
  Window *pWin;
  Window *pRet = 0;
  Window **pp = &pRet;

  for(pWin=p; pWin; pWin=pWin->pNextWin){
    *pp = sqlite3WindowDup(db, 0, pWin);
    if( *pp==0 ) break;
    pp = &((*pp)->pNextWin);
  }

  return pRet;
}

/*
** Return true if it can be determined at compile time that expression 
** pExpr evaluates to a value that, when cast to an integer, is greater 
** than zero. False otherwise.
**
** If an OOM error occurs, this function sets the Parse.db.mallocFailed 
** flag and returns zero.
*/
static int windowExprGtZero(Parse *pParse, Expr *pExpr){
  int ret = 0;
  sqlite3 *db = pParse->db;
  sqlite3_value *pVal = 0;
  sqlite3ValueFromExpr(db, pExpr, db->enc, SQLITE_AFF_NUMERIC, &pVal);
  if( pVal && sqlite3_value_int(pVal)>0 ){
    ret = 1;
  }
  sqlite3ValueFree(pVal);
  return ret;
}

/*
** sqlite3WhereBegin() has already been called for the SELECT statement 
** passed as the second argument when this function is invoked. It generates
** code to populate the Window.regResult register for each window function 
** and invoke the sub-routine at instruction addrGosub once for each row.
** sqlite3WhereEnd() is always called before returning. 
**
** This function handles several different types of window frames, which
** require slightly different processing. The following pseudo code is
** used to implement window frames of the form:
**
**   ROWS BETWEEN <expr1> PRECEDING AND <expr2> FOLLOWING
**
** Other window frame types use variants of the following:
**
**     ... loop started by sqlite3WhereBegin() ...
**       if( new partition ){
**         Gosub flush
**       }
**       Insert new row into eph table.
**       
**       if( first row of partition ){
**         // Rewind three cursors, all open on the eph table.
**         Rewind(csrEnd);
**         Rewind(csrStart);
**         Rewind(csrCurrent);
**       
**         regEnd = <expr2>          // FOLLOWING expression
**         regStart = <expr1>        // PRECEDING expression
**       }else{
**         // First time this branch is taken, the eph table contains two 
**         // rows. The first row in the partition, which all three cursors
**         // currently point to, and the following row.
**         AGGSTEP
**         if( (regEnd--)<=0 ){
**           RETURN_ROW
**           if( (regStart--)<=0 ){
**             AGGINVERSE
**           }
**         }
**       }
**     }
**     flush:
**       AGGSTEP
**       while( 1 ){
**         RETURN ROW
**         if( csrCurrent is EOF ) break;
**         if( (regStart--)<=0 ){
**           AggInverse(csrStart)
**           Next(csrStart)
**         }
**       }
**
** The pseudo-code above uses the following shorthand:
**
**   AGGSTEP:    invoke the aggregate xStep() function for each window function
**               with arguments read from the current row of cursor csrEnd, then
**               step cursor csrEnd forward one row (i.e. sqlite3BtreeNext()).
**
**   RETURN_ROW: return a row to the caller based on the contents of the 
**               current row of csrCurrent and the current state of all 
**               aggregates. Then step cursor csrCurrent forward one row.
**
**   AGGINVERSE: invoke the aggregate xInverse() function for each window 
**               functions with arguments read from the current row of cursor
**               csrStart. Then step csrStart forward one row.
**
** There are two other ROWS window frames that are handled significantly
** differently from the above - "BETWEEN <expr> PRECEDING AND <expr> PRECEDING"
** and "BETWEEN <expr> FOLLOWING AND <expr> FOLLOWING". These are special 
** cases because they change the order in which the three cursors (csrStart,
** csrCurrent and csrEnd) iterate through the ephemeral table. Cases that
** use UNBOUNDED or CURRENT ROW are much simpler variations on one of these
** three.
**
**   ROWS BETWEEN <expr1> PRECEDING AND <expr2> PRECEDING
**
**     ... loop started by sqlite3WhereBegin() ...
**       if( new partition ){
**         Gosub flush
**       }
**       Insert new row into eph table.
**       if( first row of partition ){
**         Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**         regEnd = <expr2>
**         regStart = <expr1>
**       }else{
**         if( (regEnd--)<=0 ){
**           AGGSTEP
**         }
**         RETURN_ROW
**         if( (regStart--)<=0 ){
**           AGGINVERSE
**         }
**       }
**     }
**     flush:
**       if( (regEnd--)<=0 ){
**         AGGSTEP
**       }
**       RETURN_ROW
**
**
**   ROWS BETWEEN <expr1> FOLLOWING AND <expr2> FOLLOWING
**
**     ... loop started by sqlite3WhereBegin() ...
**     if( new partition ){
**       Gosub flush
**     }
**     Insert new row into eph table.
**     if( first row of partition ){
**       Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**       regEnd = <expr2>
**       regStart = regEnd - <expr1>
**     }else{
**       AGGSTEP
**       if( (regEnd--)<=0 ){
**         RETURN_ROW
**       }
**       if( (regStart--)<=0 ){
**         AGGINVERSE
**       }
**     }
**   }
**   flush:
**     AGGSTEP
**     while( 1 ){
**       if( (regEnd--)<=0 ){
**         RETURN_ROW
**         if( eof ) break;
**       }
**       if( (regStart--)<=0 ){
**         AGGINVERSE
**         if( eof ) break
**       }
**     }
**     while( !eof csrCurrent ){
**       RETURN_ROW
**     }
**
** For the most part, the patterns above are adapted to support UNBOUNDED by
** assuming that it is equivalent to "infinity PRECEDING/FOLLOWING" and
** CURRENT ROW by assuming that it is equivilent to "0 PRECEDING/FOLLOWING".
** This is optimized of course - branches that will never be taken and
** conditions that are always true are omitted from the VM code. The only
** exceptional case is:
**
**   ROWS BETWEEN <expr1> FOLLOWING AND UNBOUNDED FOLLOWING
**
**     ... loop started by sqlite3WhereBegin() ...
**     if( new partition ){
**       Gosub flush
**     }
**     Insert new row into eph table.
**     if( first row of partition ){
**       Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**       regStart = <expr1>
**     }else{
**       AGGSTEP
**     }
**   }
**   flush:
**     AGGSTEP
**     while( 1 ){
**       if( (regStart--)<=0 ){
**         AGGINVERSE
**         if( eof ) break
**       }
**       RETURN_ROW
**     }
**     while( !eof csrCurrent ){
**       RETURN_ROW
**     }
**
** Also requiring special handling are the cases:
**
**   ROWS BETWEEN <expr1> PRECEDING AND <expr2> PRECEDING
**   ROWS BETWEEN <expr1> FOLLOWING AND <expr2> FOLLOWING
**
** when (expr1 < expr2). This is detected at runtime, not by this function.
** To handle this case, the pseudo-code programs depicted above are modified
** slightly to be:
**
**     ... loop started by sqlite3WhereBegin() ...
**     if( new partition ){
**       Gosub flush
**     }
**     Insert new row into eph table.
**     if( first row of partition ){
**       Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**       regEnd = <expr2>
**       regStart = <expr1>
**       if( regEnd < regStart ){
**         RETURN_ROW
**         delete eph table contents
**         continue
**       }
**     ...
**
** The new "continue" statement in the above jumps to the next iteration
** of the outer loop - the one started by sqlite3WhereBegin().
**
** The various GROUPS cases are implemented using the same patterns as
** ROWS. The VM code is modified slightly so that:
**
**   1. The else branch in the main loop is only taken if the row just
**      added to the ephemeral table is the start of a new group. In
**      other words, it becomes:
**
**         ... loop started by sqlite3WhereBegin() ...
**         if( new partition ){
**           Gosub flush
**         }
**         Insert new row into eph table.
**         if( first row of partition ){
**           Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**           regEnd = <expr2>
**           regStart = <expr1>
**         }else if( new group ){
**           ... 
**         }
**       }
**
**   2. Instead of processing a single row, each RETURN_ROW, AGGSTEP or 
**      AGGINVERSE step processes the current row of the relevant cursor and
**      all subsequent rows belonging to the same group.
**
** RANGE window frames are a little different again. As for GROUPS, the 
** main loop runs once per group only. And RETURN_ROW, AGGSTEP and AGGINVERSE
** deal in groups instead of rows. As for ROWS and GROUPS, there are three
** basic cases:
**
**   RANGE BETWEEN <expr1> PRECEDING AND <expr2> FOLLOWING
**
**     ... loop started by sqlite3WhereBegin() ...
**       if( new partition ){
**         Gosub flush
**       }
**       Insert new row into eph table.
**       if( first row of partition ){
**         Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**         regEnd = <expr2>
**         regStart = <expr1>
**       }else{
**         AGGSTEP
**         while( (csrCurrent.key + regEnd) < csrEnd.key ){
**           RETURN_ROW
**           while( csrStart.key + regStart) < csrCurrent.key ){
**             AGGINVERSE
**           }
**         }
**       }
**     }
**     flush:
**       AGGSTEP
**       while( 1 ){
**         RETURN ROW
**         if( csrCurrent is EOF ) break;
**           while( csrStart.key + regStart) < csrCurrent.key ){
**             AGGINVERSE
**           }
**         }
**       }
**
** In the above notation, "csr.key" means the current value of the ORDER BY 
** expression (there is only ever 1 for a RANGE that uses an <expr> FOLLOWING
** or <expr PRECEDING) read from cursor csr.
**
**   RANGE BETWEEN <expr1> PRECEDING AND <expr2> PRECEDING
**
**     ... loop started by sqlite3WhereBegin() ...
**       if( new partition ){
**         Gosub flush
**       }
**       Insert new row into eph table.
**       if( first row of partition ){
**         Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**         regEnd = <expr2>
**         regStart = <expr1>
**       }else{
**         while( (csrEnd.key + regEnd) <= csrCurrent.key ){
**           AGGSTEP
**         }
**         while( (csrStart.key + regStart) < csrCurrent.key ){
**           AGGINVERSE
**         }
**         RETURN_ROW
**       }
**     }
**     flush:
**       while( (csrEnd.key + regEnd) <= csrCurrent.key ){
**         AGGSTEP
**       }
**       while( (csrStart.key + regStart) < csrCurrent.key ){
**         AGGINVERSE
**       }
**       RETURN_ROW
**
**   RANGE BETWEEN <expr1> FOLLOWING AND <expr2> FOLLOWING
**
**     ... loop started by sqlite3WhereBegin() ...
**       if( new partition ){
**         Gosub flush
**       }
**       Insert new row into eph table.
**       if( first row of partition ){
**         Rewind(csrEnd) ; Rewind(csrStart) ; Rewind(csrCurrent)
**         regEnd = <expr2>
**         regStart = <expr1>
**       }else{
**         AGGSTEP
**         while( (csrCurrent.key + regEnd) < csrEnd.key ){
**           while( (csrCurrent.key + regStart) > csrStart.key ){
**             AGGINVERSE
**           }
**           RETURN_ROW
**         }
**       }
**     }
**     flush:
**       AGGSTEP
**       while( 1 ){
**         while( (csrCurrent.key + regStart) > csrStart.key ){
**           AGGINVERSE
**           if( eof ) break "while( 1 )" loop.
**         }
**         RETURN_ROW
**       }
**       while( !eof csrCurrent ){
**         RETURN_ROW
**       }
**
** The text above leaves out many details. Refer to the code and comments
** below for a more complete picture.
*/
void sqlite3WindowCodeStep(
  Parse *pParse,                  /* Parse context */
  Select *p,                      /* Rewritten SELECT statement */
  WhereInfo *pWInfo,              /* Context returned by sqlite3WhereBegin() */
  int regGosub,                   /* Register for OP_Gosub */
  int addrGosub                   /* OP_Gosub here to return each row */
){
  Window *pMWin = p->pWin;
  ExprList *pOrderBy = pMWin->pOrderBy;
  Vdbe *v = sqlite3GetVdbe(pParse);
  int csrWrite;                   /* Cursor used to write to eph. table */
  int csrInput = p->pSrc->a[0].iCursor;     /* Cursor of sub-select */
  int nInput = p->pSrc->a[0].pTab->nCol;    /* Number of cols returned by sub */
  int iInput;                               /* To iterate through sub cols */
  int addrNe;                     /* Address of OP_Ne */
  int addrGosubFlush = 0;         /* Address of OP_Gosub to flush: */
  int addrInteger = 0;            /* Address of OP_Integer */
  int addrEmpty;                  /* Address of OP_Rewind in flush: */
  int regNew;                     /* Array of registers holding new input row */
  int regRecord;                  /* regNew array in record form */
  int regRowid;                   /* Rowid for regRecord in eph table */
  int regNewPeer = 0;             /* Peer values for new row (part of regNew) */
  int regPeer = 0;                /* Peer values for current row */
  int regFlushPart = 0;           /* Register for "Gosub flush_partition" */
  WindowCodeArg s;                /* Context object for sub-routines */
  int lblWhereEnd;                /* Label just before sqlite3WhereEnd() code */
  int regStart = 0;               /* Value of <expr> PRECEDING */
  int regEnd = 0;                 /* Value of <expr> FOLLOWING */

  assert( pMWin->eStart==TK_PRECEDING || pMWin->eStart==TK_CURRENT 
       || pMWin->eStart==TK_FOLLOWING || pMWin->eStart==TK_UNBOUNDED 
  );
  assert( pMWin->eEnd==TK_FOLLOWING || pMWin->eEnd==TK_CURRENT 
       || pMWin->eEnd==TK_UNBOUNDED || pMWin->eEnd==TK_PRECEDING 
  );
  assert( pMWin->eExclude==0 || pMWin->eExclude==TK_CURRENT
       || pMWin->eExclude==TK_GROUP || pMWin->eExclude==TK_TIES
       || pMWin->eExclude==TK_NO
  );

  lblWhereEnd = sqlite3VdbeMakeLabel(pParse);

  /* Fill in the context object */
  memset(&s, 0, sizeof(WindowCodeArg));
  s.pParse = pParse;
  s.pMWin = pMWin;
  s.pVdbe = v;
  s.regGosub = regGosub;
  s.addrGosub = addrGosub;
  s.current.csr = pMWin->iEphCsr;
  csrWrite = s.current.csr+1;
  s.start.csr = s.current.csr+2;
  s.end.csr = s.current.csr+3;

  /* Figure out when rows may be deleted from the ephemeral table. There
  ** are four options - they may never be deleted (eDelete==0), they may 
  ** be deleted as soon as they are no longer part of the window frame
  ** (eDelete==WINDOW_AGGINVERSE), they may be deleted as after the row 
  ** has been returned to the caller (WINDOW_RETURN_ROW), or they may
  ** be deleted after they enter the frame (WINDOW_AGGSTEP). */
  switch( pMWin->eStart ){
    case TK_FOLLOWING:
      if( pMWin->eFrmType!=TK_RANGE
       && windowExprGtZero(pParse, pMWin->pStart)
      ){
        s.eDelete = WINDOW_RETURN_ROW;
      }
      break;
    case TK_UNBOUNDED:
      if( windowCacheFrame(pMWin)==0 ){
        if( pMWin->eEnd==TK_PRECEDING ){
          if( pMWin->eFrmType!=TK_RANGE
           && windowExprGtZero(pParse, pMWin->pEnd)
          ){
            s.eDelete = WINDOW_AGGSTEP;
          }
        }else{
          s.eDelete = WINDOW_RETURN_ROW;
        }
      }
      break;
    default:
      s.eDelete = WINDOW_AGGINVERSE;
      break;
  }

  /* Allocate registers for the array of values from the sub-query, the
  ** samve values in record form, and the rowid used to insert said record
  ** into the ephemeral table.  */
  regNew = pParse->nMem+1;
  pParse->nMem += nInput;
  regRecord = ++pParse->nMem;
  regRowid = ++pParse->nMem;

  /* If the window frame contains an "<expr> PRECEDING" or "<expr> FOLLOWING"
  ** clause, allocate registers to store the results of evaluating each
  ** <expr>.  */
  if( pMWin->eStart==TK_PRECEDING || pMWin->eStart==TK_FOLLOWING ){
    regStart = ++pParse->nMem;
  }
  if( pMWin->eEnd==TK_PRECEDING || pMWin->eEnd==TK_FOLLOWING ){
    regEnd = ++pParse->nMem;
  }

  /* If this is not a "ROWS BETWEEN ..." frame, then allocate arrays of
  ** registers to store copies of the ORDER BY expressions (peer values) 
  ** for the main loop, and for each cursor (start, current and end). */
  if( pMWin->eFrmType!=TK_ROWS ){
    int nPeer = (pOrderBy ? pOrderBy->nExpr : 0);
    regNewPeer = regNew + pMWin->nBufferCol;
    if( pMWin->pPartition ) regNewPeer += pMWin->pPartition->nExpr;
    regPeer = pParse->nMem+1;       pParse->nMem += nPeer;
    s.start.reg = pParse->nMem+1;   pParse->nMem += nPeer;
    s.current.reg = pParse->nMem+1; pParse->nMem += nPeer;
    s.end.reg = pParse->nMem+1;     pParse->nMem += nPeer;
  }

  /* Load the column values for the row returned by the sub-select
  ** into an array of registers starting at regNew. Assemble them into
  ** a record in register regRecord. */
  for(iInput=0; iInput<nInput; iInput++){
    sqlite3VdbeAddOp3(v, OP_Column, csrInput, iInput, regNew+iInput);
  }
  sqlite3VdbeAddOp3(v, OP_MakeRecord, regNew, nInput, regRecord);

  /* An input row has just been read into an array of registers starting
  ** at regNew. If the window has a PARTITION clause, this block generates 
  ** VM code to check if the input row is the start of a new partition.
  ** If so, it does an OP_Gosub to an address to be filled in later. The
  ** address of the OP_Gosub is stored in local variable addrGosubFlush. */
  if( pMWin->pPartition ){
    int addr;
    ExprList *pPart = pMWin->pPartition;
    int nPart = pPart->nExpr;
    int regNewPart = regNew + pMWin->nBufferCol;
    KeyInfo *pKeyInfo = sqlite3KeyInfoFromExprList(pParse, pPart, 0, 0);

    regFlushPart = ++pParse->nMem;
    addr = sqlite3VdbeAddOp3(v, OP_Compare, regNewPart, pMWin->regPart, nPart);
    sqlite3VdbeAppendP4(v, (void*)pKeyInfo, P4_KEYINFO);
    sqlite3VdbeAddOp3(v, OP_Jump, addr+2, addr+4, addr+2);
    VdbeCoverageEqNe(v);
    addrGosubFlush = sqlite3VdbeAddOp1(v, OP_Gosub, regFlushPart);
    VdbeComment((v, "call flush_partition"));
    sqlite3VdbeAddOp3(v, OP_Copy, regNewPart, pMWin->regPart, nPart-1);
  }

  /* Insert the new row into the ephemeral table */
  sqlite3VdbeAddOp2(v, OP_NewRowid, csrWrite, regRowid);
  sqlite3VdbeAddOp3(v, OP_Insert, csrWrite, regRecord, regRowid);
  addrNe = sqlite3VdbeAddOp3(v, OP_Ne, pMWin->regOne, 0, regRowid);
  VdbeCoverageNeverNull(v);

  /* This block is run for the first row of each partition */
  s.regArg = windowInitAccum(pParse, pMWin);

  if( regStart ){
    sqlite3ExprCode(pParse, pMWin->pStart, regStart);
    windowCheckValue(pParse, regStart, 0 + (pMWin->eFrmType==TK_RANGE?3:0));
  }
  if( regEnd ){
    sqlite3ExprCode(pParse, pMWin->pEnd, regEnd);
    windowCheckValue(pParse, regEnd, 1 + (pMWin->eFrmType==TK_RANGE?3:0));
  }

  if( pMWin->eFrmType!=TK_RANGE && pMWin->eStart==pMWin->eEnd && regStart ){
    int op = ((pMWin->eStart==TK_FOLLOWING) ? OP_Ge : OP_Le);
    int addrGe = sqlite3VdbeAddOp3(v, op, regStart, 0, regEnd);
    VdbeCoverageNeverNullIf(v, op==OP_Ge); /* NeverNull because bound <expr> */
    VdbeCoverageNeverNullIf(v, op==OP_Le); /*   values previously checked */
    windowAggFinal(&s, 0);
    sqlite3VdbeAddOp2(v, OP_Rewind, s.current.csr, 1);
    VdbeCoverageNeverTaken(v);
    windowReturnOneRow(&s);
    sqlite3VdbeAddOp1(v, OP_ResetSorter, s.current.csr);
    sqlite3VdbeAddOp2(v, OP_Goto, 0, lblWhereEnd);
    sqlite3VdbeJumpHere(v, addrGe);
  }
  if( pMWin->eStart==TK_FOLLOWING && pMWin->eFrmType!=TK_RANGE && regEnd ){
    assert( pMWin->eEnd==TK_FOLLOWING );
    sqlite3VdbeAddOp3(v, OP_Subtract, regStart, regEnd, regStart);
  }

  if( pMWin->eStart!=TK_UNBOUNDED ){
    sqlite3VdbeAddOp2(v, OP_Rewind, s.start.csr, 1);
    VdbeCoverageNeverTaken(v);
  }
  sqlite3VdbeAddOp2(v, OP_Rewind, s.current.csr, 1);
  VdbeCoverageNeverTaken(v);
  sqlite3VdbeAddOp2(v, OP_Rewind, s.end.csr, 1);
  VdbeCoverageNeverTaken(v);
  if( regPeer && pOrderBy ){
    sqlite3VdbeAddOp3(v, OP_Copy, regNewPeer, regPeer, pOrderBy->nExpr-1);
    sqlite3VdbeAddOp3(v, OP_Copy, regPeer, s.start.reg, pOrderBy->nExpr-1);
    sqlite3VdbeAddOp3(v, OP_Copy, regPeer, s.current.reg, pOrderBy->nExpr-1);
    sqlite3VdbeAddOp3(v, OP_Copy, regPeer, s.end.reg, pOrderBy->nExpr-1);
  }

  sqlite3VdbeAddOp2(v, OP_Goto, 0, lblWhereEnd);

  sqlite3VdbeJumpHere(v, addrNe);

  /* Beginning of the block executed for the second and subsequent rows. */
  if( regPeer ){
    windowIfNewPeer(pParse, pOrderBy, regNewPeer, regPeer, lblWhereEnd);
  }
  if( pMWin->eStart==TK_FOLLOWING ){
    windowCodeOp(&s, WINDOW_AGGSTEP, 0, 0);
    if( pMWin->eEnd!=TK_UNBOUNDED ){
      if( pMWin->eFrmType==TK_RANGE ){
        int lbl = sqlite3VdbeMakeLabel(pParse);
        int addrNext = sqlite3VdbeCurrentAddr(v);
        windowCodeRangeTest(&s, OP_Ge, s.current.csr, regEnd, s.end.csr, lbl);
        windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
        windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 0);
        sqlite3VdbeAddOp2(v, OP_Goto, 0, addrNext);
        sqlite3VdbeResolveLabel(v, lbl);
      }else{
        windowCodeOp(&s, WINDOW_RETURN_ROW, regEnd, 0);
        windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
      }
    }
  }else
  if( pMWin->eEnd==TK_PRECEDING ){
    int bRPS = (pMWin->eStart==TK_PRECEDING && pMWin->eFrmType==TK_RANGE);
    windowCodeOp(&s, WINDOW_AGGSTEP, regEnd, 0);
    if( bRPS ) windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
    windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 0);
    if( !bRPS ) windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
  }else{
    int addr = 0;
    windowCodeOp(&s, WINDOW_AGGSTEP, 0, 0);
    if( pMWin->eEnd!=TK_UNBOUNDED ){
      if( pMWin->eFrmType==TK_RANGE ){
        int lbl = 0;
        addr = sqlite3VdbeCurrentAddr(v);
        if( regEnd ){
          lbl = sqlite3VdbeMakeLabel(pParse);
          windowCodeRangeTest(&s, OP_Ge, s.current.csr, regEnd, s.end.csr, lbl);
        }
        windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 0);
        windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
        if( regEnd ){
          sqlite3VdbeAddOp2(v, OP_Goto, 0, addr);
          sqlite3VdbeResolveLabel(v, lbl);
        }
      }else{
        if( regEnd ){
          addr = sqlite3VdbeAddOp3(v, OP_IfPos, regEnd, 0, 1);
          VdbeCoverage(v);
        }
        windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 0);
        windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
        if( regEnd ) sqlite3VdbeJumpHere(v, addr);
      }
    }
  }

  /* End of the main input loop */
  sqlite3VdbeResolveLabel(v, lblWhereEnd);
  sqlite3WhereEnd(pWInfo);

  /* Fall through */
  if( pMWin->pPartition ){
    addrInteger = sqlite3VdbeAddOp2(v, OP_Integer, 0, regFlushPart);
    sqlite3VdbeJumpHere(v, addrGosubFlush);
  }

  addrEmpty = sqlite3VdbeAddOp1(v, OP_Rewind, csrWrite);
  VdbeCoverage(v);
  if( pMWin->eEnd==TK_PRECEDING ){
    int bRPS = (pMWin->eStart==TK_PRECEDING && pMWin->eFrmType==TK_RANGE);
    windowCodeOp(&s, WINDOW_AGGSTEP, regEnd, 0);
    if( bRPS ) windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
    windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 0);
  }else if( pMWin->eStart==TK_FOLLOWING ){
    int addrStart;
    int addrBreak1;
    int addrBreak2;
    int addrBreak3;
    windowCodeOp(&s, WINDOW_AGGSTEP, 0, 0);
    if( pMWin->eFrmType==TK_RANGE ){
      addrStart = sqlite3VdbeCurrentAddr(v);
      addrBreak2 = windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 1);
      addrBreak1 = windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 1);
    }else
    if( pMWin->eEnd==TK_UNBOUNDED ){
      addrStart = sqlite3VdbeCurrentAddr(v);
      addrBreak1 = windowCodeOp(&s, WINDOW_RETURN_ROW, regStart, 1);
      addrBreak2 = windowCodeOp(&s, WINDOW_AGGINVERSE, 0, 1);
    }else{
      assert( pMWin->eEnd==TK_FOLLOWING );
      addrStart = sqlite3VdbeCurrentAddr(v);
      addrBreak1 = windowCodeOp(&s, WINDOW_RETURN_ROW, regEnd, 1);
      addrBreak2 = windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 1);
    }
    sqlite3VdbeAddOp2(v, OP_Goto, 0, addrStart);
    sqlite3VdbeJumpHere(v, addrBreak2);
    addrStart = sqlite3VdbeCurrentAddr(v);
    addrBreak3 = windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 1);
    sqlite3VdbeAddOp2(v, OP_Goto, 0, addrStart);
    sqlite3VdbeJumpHere(v, addrBreak1);
    sqlite3VdbeJumpHere(v, addrBreak3);
  }else{
    int addrBreak;
    int addrStart;
    windowCodeOp(&s, WINDOW_AGGSTEP, 0, 0);
    addrStart = sqlite3VdbeCurrentAddr(v);
    addrBreak = windowCodeOp(&s, WINDOW_RETURN_ROW, 0, 1);
    windowCodeOp(&s, WINDOW_AGGINVERSE, regStart, 0);
    sqlite3VdbeAddOp2(v, OP_Goto, 0, addrStart);
    sqlite3VdbeJumpHere(v, addrBreak);
  }
  sqlite3VdbeJumpHere(v, addrEmpty);

  sqlite3VdbeAddOp1(v, OP_ResetSorter, s.current.csr);
  if( pMWin->pPartition ){
    if( pMWin->regStartRowid ){
      sqlite3VdbeAddOp2(v, OP_Integer, 1, pMWin->regStartRowid);
      sqlite3VdbeAddOp2(v, OP_Integer, 0, pMWin->regEndRowid);
    }
    sqlite3VdbeChangeP1(v, addrInteger, sqlite3VdbeCurrentAddr(v));
    sqlite3VdbeAddOp1(v, OP_Return, regFlushPart);
  }
}

#endif /* SQLITE_OMIT_WINDOWFUNC */