Index: src/wherecode.c
==================================================================
--- src/wherecode.c
+++ src/wherecode.c
@@ -1127,10 +1127,12 @@
   int addrBrk;                    /* Jump here to break out of the loop */
   int addrHalt;                   /* addrBrk for the outermost loop */
   int addrCont;                   /* Jump here to continue with next cycle */
   int iRowidReg = 0;        /* Rowid is stored in this register, if not zero */
   int iReleaseReg = 0;      /* Temp register to free before returning */
+  Index *pIdx = 0;          /* Index used by loop (if any) */
+  int loopAgain;            /* True if constraint generator loop should repeat */
 
   pParse = pWInfo->pParse;
   v = pParse->pVdbe;
   pWC = &pWInfo->sWC;
   db = pParse->db;
@@ -1452,11 +1454,10 @@
     WhereTerm *pRangeEnd = 0;    /* Inequality constraint at range end */
     int startEq;                 /* True if range start uses ==, >= or <= */
     int endEq;                   /* True if range end uses ==, >= or <= */
     int start_constraints;       /* Start of range is constrained */
     int nConstraint;             /* Number of constraint terms */
-    Index *pIdx;                 /* The index we will be using */
     int iIdxCur;                 /* The VDBE cursor for the index */
     int nExtraReg = 0;           /* Number of extra registers needed */
     int op;                      /* Instruction opcode */
     char *zStartAff;             /* Affinity for start of range constraint */
     char *zEndAff = 0;           /* Affinity for end of range constraint */
@@ -1703,10 +1704,11 @@
     if( (pLoop->wsFlags & WHERE_CONSTRAINT)==0 ){
       pLevel->p5 = SQLITE_STMTSTATUS_FULLSCAN_STEP;
     }else{
       assert( pLevel->p5==0 );
     }
+    if( omitTable ) pIdx = 0;
   }else
 
 #ifndef SQLITE_OMIT_OR_OPTIMIZATION
   if( pLoop->wsFlags & WHERE_MULTI_OR ){
     /* Case 5:  Two or more separately indexed terms connected by OR
@@ -2020,47 +2022,60 @@
   pLevel->addrVisit = sqlite3VdbeCurrentAddr(v);
 #endif
 
   /* Insert code to test every subexpression that can be completely
   ** computed using the current set of tables.
+  **
+  ** This loop may run either once (pIdx==0) or twice (pIdx!=0). If
+  ** it is run twice, then the first iteration codes those sub-expressions
+  ** that can be computed using columns from pIdx only (without seeking
+  ** the main table cursor). 
   */
-  for(pTerm=pWC->a, j=pWC->nTerm; j>0; j--, pTerm++){
-    Expr *pE;
-    int skipLikeAddr = 0;
-    testcase( pTerm->wtFlags & TERM_VIRTUAL );
-    testcase( pTerm->wtFlags & TERM_CODED );
-    if( pTerm->wtFlags & (TERM_VIRTUAL|TERM_CODED) ) continue;
-    if( (pTerm->prereqAll & pLevel->notReady)!=0 ){
-      testcase( pWInfo->untestedTerms==0
-               && (pWInfo->wctrlFlags & WHERE_OR_SUBCLAUSE)!=0 );
-      pWInfo->untestedTerms = 1;
-      continue;
-    }
-    pE = pTerm->pExpr;
-    assert( pE!=0 );
-    if( pLevel->iLeftJoin && !ExprHasProperty(pE, EP_FromJoin) ){
-      continue;
-    }
-    if( pTerm->wtFlags & TERM_LIKECOND ){
-      /* If the TERM_LIKECOND flag is set, that means that the range search
-      ** is sufficient to guarantee that the LIKE operator is true, so we
-      ** can skip the call to the like(A,B) function.  But this only works
-      ** for strings.  So do not skip the call to the function on the pass
-      ** that compares BLOBs. */
+  do{
+    loopAgain = 0;
+    for(pTerm=pWC->a, j=pWC->nTerm; j>0; j--, pTerm++){
+      Expr *pE;
+      int skipLikeAddr = 0;
+      testcase( pTerm->wtFlags & TERM_VIRTUAL );
+      testcase( pTerm->wtFlags & TERM_CODED );
+      if( pTerm->wtFlags & (TERM_VIRTUAL|TERM_CODED) ) continue;
+      if( (pTerm->prereqAll & pLevel->notReady)!=0 ){
+        testcase( pWInfo->untestedTerms==0
+            && (pWInfo->wctrlFlags & WHERE_OR_SUBCLAUSE)!=0 );
+        pWInfo->untestedTerms = 1;
+        continue;
+      }
+      pE = pTerm->pExpr;
+      assert( pE!=0 );
+      if( pLevel->iLeftJoin && !ExprHasProperty(pE, EP_FromJoin) ){
+        continue;
+      }
+      if( pIdx && !sqlite3ExprCoveredByIndex(pE, pLevel->iTabCur, pIdx) ){
+        loopAgain = 1;
+        continue;
+      }
+      if( pTerm->wtFlags & TERM_LIKECOND ){
+        /* If the TERM_LIKECOND flag is set, that means that the range search
+        ** is sufficient to guarantee that the LIKE operator is true, so we
+        ** can skip the call to the like(A,B) function.  But this only works
+        ** for strings.  So do not skip the call to the function on the pass
+        ** that compares BLOBs. */
 #ifdef SQLITE_LIKE_DOESNT_MATCH_BLOBS
-      continue;
+        continue;
 #else
-      u32 x = pLevel->iLikeRepCntr;
-      assert( x>0 );
-      skipLikeAddr = sqlite3VdbeAddOp1(v, (x&1)? OP_IfNot : OP_If, (int)(x>>1));
-      VdbeCoverage(v);
+        u32 x = pLevel->iLikeRepCntr;
+        assert( x>0 );
+        skipLikeAddr = sqlite3VdbeAddOp1(v, (x&1)?OP_IfNot:OP_If, (int)(x>>1));
+        VdbeCoverage(v);
 #endif
+      }
+      sqlite3ExprIfFalse(pParse, pE, addrCont, SQLITE_JUMPIFNULL);
+      if( skipLikeAddr ) sqlite3VdbeJumpHere(v, skipLikeAddr);
+      pTerm->wtFlags |= TERM_CODED;
     }
-    sqlite3ExprIfFalse(pParse, pE, addrCont, SQLITE_JUMPIFNULL);
-    if( skipLikeAddr ) sqlite3VdbeJumpHere(v, skipLikeAddr);
-    pTerm->wtFlags |= TERM_CODED;
-  }
+    pIdx = 0;
+  }while( loopAgain );
 
   /* Insert code to test for implied constraints based on transitivity
   ** of the "==" operator.
   **
   ** Example: If the WHERE clause contains "t1.a=t2.b" and "t2.b=123"

ADDED   test/pushdown.test
Index: test/pushdown.test
==================================================================
--- /dev/null
+++ test/pushdown.test
@@ -0,0 +1,59 @@
+# 2017 April 29
+#
+# 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.
+#
+#***********************************************************************
+
+set testdir [file dirname $argv0]
+source $testdir/tester.tcl
+set testprefix pushdown
+
+do_execsql_test 1.0 {
+  CREATE TABLE t1(a, b, c);
+  INSERT INTO t1 VALUES(1, 'b1', 'c1');
+  INSERT INTO t1 VALUES(2, 'b2', 'c2');
+  INSERT INTO t1 VALUES(3, 'b3', 'c3');
+  INSERT INTO t1 VALUES(4, 'b4', 'c4');
+  CREATE INDEX i1 ON t1(a, c);
+}
+
+proc f {val} {
+  lappend ::L $val
+  return 0
+}
+db func f f 
+
+do_test 1.1 {
+  set L [list]
+  execsql { SELECT * FROM t1 WHERE a=2 AND f(b) AND f(c) }
+  set L
+} {c2}
+
+do_test 1.2 {
+  set L [list]
+  execsql { SELECT * FROM t1 WHERE a=3 AND f(c) AND f(b) }
+  set L
+} {c3}
+
+do_execsql_test 1.3 {
+  DROP INDEX i1;
+  CREATE INDEX i1 ON t1(a, b);
+}
+do_test 1.4 {
+  set L [list]
+  execsql { SELECT * FROM t1 WHERE a=2 AND f(b) AND f(c) }
+  set L
+} {b2}
+
+do_test 1.5 {
+  set L [list]
+  execsql { SELECT * FROM t1 WHERE a=3 AND f(c) AND f(b) }
+  set L
+} {b3}
+  
+finish_test