I wrote:
> Unless somebody's got a better idea, I'll push forward with making
> this happen.

Here's a proposed patch for that.  I ended up reverting the code
changes of 4ace45677 in favor of an alternative I'd considered
previously, which is to mark the indextlist elements as resjunk
if they're non-retrievable, and then make setrefs.c deal with
not relying on those elements.  This avoids the EXPLAIN breakage
I showed, since now we still have the indextlist elements needed
to interpret the indexqual and indexorderby expressions.

0001 is what I propose to back-patch (modulo putting the new
IndexOnlyScan.recheckqual field at the end, in the back branches).

In addition, it seems to me that we can simplify check_index_only()
by reverting b5febc1d1's changes, because we've now been forced to
put in a non-half-baked solution for the problem it addressed.
That's 0002 attached.  I'd be inclined not to back-patch that though.

                        regards, tom lane

diff --git a/src/backend/commands/explain.c b/src/backend/commands/explain.c
index 09f5253abb..60d0d4ad0f 100644
--- a/src/backend/commands/explain.c
+++ b/src/backend/commands/explain.c
@@ -1746,7 +1746,7 @@ ExplainNode(PlanState *planstate, List *ancestors,
 		case T_IndexOnlyScan:
 			show_scan_qual(((IndexOnlyScan *) plan)->indexqual,
 						   "Index Cond", planstate, ancestors, es);
-			if (((IndexOnlyScan *) plan)->indexqual)
+			if (((IndexOnlyScan *) plan)->recheckqual)
 				show_instrumentation_count("Rows Removed by Index Recheck", 2,
 										   planstate, es);
 			show_scan_qual(((IndexOnlyScan *) plan)->indexorderby,
diff --git a/src/backend/executor/nodeIndexonlyscan.c b/src/backend/executor/nodeIndexonlyscan.c
index 0754e28a9a..8fee958135 100644
--- a/src/backend/executor/nodeIndexonlyscan.c
+++ b/src/backend/executor/nodeIndexonlyscan.c
@@ -214,13 +214,11 @@ IndexOnlyNext(IndexOnlyScanState *node)
 
 		/*
 		 * If the index was lossy, we have to recheck the index quals.
-		 * (Currently, this can never happen, but we should support the case
-		 * for possible future use, eg with GiST indexes.)
 		 */
 		if (scandesc->xs_recheck)
 		{
 			econtext->ecxt_scantuple = slot;
-			if (!ExecQualAndReset(node->indexqual, econtext))
+			if (!ExecQualAndReset(node->recheckqual, econtext))
 			{
 				/* Fails recheck, so drop it and loop back for another */
 				InstrCountFiltered2(node, 1);
@@ -555,8 +553,8 @@ ExecInitIndexOnlyScan(IndexOnlyScan *node, EState *estate, int eflags)
 	 */
 	indexstate->ss.ps.qual =
 		ExecInitQual(node->scan.plan.qual, (PlanState *) indexstate);
-	indexstate->indexqual =
-		ExecInitQual(node->indexqual, (PlanState *) indexstate);
+	indexstate->recheckqual =
+		ExecInitQual(node->recheckqual, (PlanState *) indexstate);
 
 	/*
 	 * If we are just doing EXPLAIN (ie, aren't going to run the plan), stop
diff --git a/src/backend/nodes/copyfuncs.c b/src/backend/nodes/copyfuncs.c
index df0b747883..18e778e856 100644
--- a/src/backend/nodes/copyfuncs.c
+++ b/src/backend/nodes/copyfuncs.c
@@ -519,6 +519,7 @@ _copyIndexOnlyScan(const IndexOnlyScan *from)
 	 */
 	COPY_SCALAR_FIELD(indexid);
 	COPY_NODE_FIELD(indexqual);
+	COPY_NODE_FIELD(recheckqual);
 	COPY_NODE_FIELD(indexorderby);
 	COPY_NODE_FIELD(indextlist);
 	COPY_SCALAR_FIELD(indexorderdir);
diff --git a/src/backend/nodes/outfuncs.c b/src/backend/nodes/outfuncs.c
index 91a89b6d51..6c0979ec35 100644
--- a/src/backend/nodes/outfuncs.c
+++ b/src/backend/nodes/outfuncs.c
@@ -580,6 +580,7 @@ _outIndexOnlyScan(StringInfo str, const IndexOnlyScan *node)
 
 	WRITE_OID_FIELD(indexid);
 	WRITE_NODE_FIELD(indexqual);
+	WRITE_NODE_FIELD(recheckqual);
 	WRITE_NODE_FIELD(indexorderby);
 	WRITE_NODE_FIELD(indextlist);
 	WRITE_ENUM_FIELD(indexorderdir, ScanDirection);
diff --git a/src/backend/nodes/readfuncs.c b/src/backend/nodes/readfuncs.c
index d79af6e56e..2a699c216b 100644
--- a/src/backend/nodes/readfuncs.c
+++ b/src/backend/nodes/readfuncs.c
@@ -1884,6 +1884,7 @@ _readIndexOnlyScan(void)
 
 	READ_OID_FIELD(indexid);
 	READ_NODE_FIELD(indexqual);
+	READ_NODE_FIELD(recheckqual);
 	READ_NODE_FIELD(indexorderby);
 	READ_NODE_FIELD(indextlist);
 	READ_ENUM_FIELD(indexorderdir, ScanDirection);
diff --git a/src/backend/optimizer/plan/createplan.c b/src/backend/optimizer/plan/createplan.c
index ff2b14e880..4b7347bc0e 100644
--- a/src/backend/optimizer/plan/createplan.c
+++ b/src/backend/optimizer/plan/createplan.c
@@ -20,7 +20,6 @@
 #include <math.h>
 
 #include "access/sysattr.h"
-#include "catalog/pg_am.h"
 #include "catalog/pg_class.h"
 #include "foreign/fdwapi.h"
 #include "miscadmin.h"
@@ -189,10 +188,10 @@ static IndexScan *make_indexscan(List *qptlist, List *qpqual, Index scanrelid,
 								 ScanDirection indexscandir);
 static IndexOnlyScan *make_indexonlyscan(List *qptlist, List *qpqual,
 										 Index scanrelid, Oid indexid,
-										 List *indexqual, List *indexorderby,
+										 List *indexqual, List *recheckqual,
+										 List *indexorderby,
 										 List *indextlist,
 										 ScanDirection indexscandir);
-static List *make_indexonly_tlist(IndexOptInfo *indexinfo);
 static BitmapIndexScan *make_bitmap_indexscan(Index scanrelid, Oid indexid,
 											  List *indexqual,
 											  List *indexqualorig);
@@ -623,7 +622,7 @@ create_scan_plan(PlannerInfo *root, Path *best_path, int flags)
 		if (best_path->pathtype == T_IndexOnlyScan)
 		{
 			/* For index-only scan, the preferred tlist is the index's */
-			tlist = copyObject(make_indexonly_tlist(((IndexPath *) best_path)->indexinfo));
+			tlist = copyObject(((IndexPath *) best_path)->indexinfo->indextlist);
 
 			/*
 			 * Transfer sortgroupref data to the replacement tlist, if
@@ -2934,7 +2933,8 @@ create_indexscan_plan(PlannerInfo *root,
 	List	   *indexclauses = best_path->indexclauses;
 	List	   *indexorderbys = best_path->indexorderbys;
 	Index		baserelid = best_path->path.parent->relid;
-	Oid			indexoid = best_path->indexinfo->indexoid;
+	IndexOptInfo *indexinfo = best_path->indexinfo;
+	Oid			indexoid = indexinfo->indexoid;
 	List	   *qpqual;
 	List	   *stripped_indexquals;
 	List	   *fixed_indexquals;
@@ -3064,6 +3064,24 @@ create_indexscan_plan(PlannerInfo *root,
 		}
 	}
 
+	/*
+	 * For an index-only scan, we must mark indextlist entries as resjunk if
+	 * they are columns that the index AM can't return; this cues setrefs.c to
+	 * not generate references to those columns.
+	 */
+	if (indexonly)
+	{
+		int			i = 0;
+
+		foreach(l, indexinfo->indextlist)
+		{
+			TargetEntry *indextle = (TargetEntry *) lfirst(l);
+
+			indextle->resjunk = !indexinfo->canreturn[i];
+			i++;
+		}
+	}
+
 	/* Finally ready to build the plan node */
 	if (indexonly)
 		scan_plan = (Scan *) make_indexonlyscan(tlist,
@@ -3071,8 +3089,9 @@ create_indexscan_plan(PlannerInfo *root,
 												baserelid,
 												indexoid,
 												fixed_indexquals,
+												stripped_indexquals,
 												fixed_indexorderbys,
-												make_indexonly_tlist(best_path->indexinfo),
+												indexinfo->indextlist,
 												best_path->indexscandir);
 	else
 		scan_plan = (Scan *) make_indexscan(tlist,
@@ -5444,6 +5463,7 @@ make_indexonlyscan(List *qptlist,
 				   Index scanrelid,
 				   Oid indexid,
 				   List *indexqual,
+				   List *recheckqual,
 				   List *indexorderby,
 				   List *indextlist,
 				   ScanDirection indexscandir)
@@ -5458,6 +5478,7 @@ make_indexonlyscan(List *qptlist,
 	node->scan.scanrelid = scanrelid;
 	node->indexid = indexid;
 	node->indexqual = indexqual;
+	node->recheckqual = recheckqual;
 	node->indexorderby = indexorderby;
 	node->indextlist = indextlist;
 	node->indexorderdir = indexscandir;
@@ -5465,53 +5486,6 @@ make_indexonlyscan(List *qptlist,
 	return node;
 }
 
-/*
- * make_indexonly_tlist
- *
- * Construct the indextlist for an IndexOnlyScan plan node.
- * We must replace any column that can't be returned by the index AM
- * with a null Const of the appropriate datatype.  This is necessary
- * to prevent setrefs.c from trying to use the value of such a column,
- * and anyway it makes the indextlist a better representative of what
- * the indexscan will really return.  (We do this here, not where the
- * IndexOptInfo is originally constructed, because earlier planner
- * steps need to know what is in such columns.)
- */
-static List *
-make_indexonly_tlist(IndexOptInfo *indexinfo)
-{
-	List	   *result;
-	int			i;
-	ListCell   *lc;
-
-	/* We needn't work hard for the common case of btrees. */
-	if (indexinfo->relam == BTREE_AM_OID)
-		return indexinfo->indextlist;
-
-	result = NIL;
-	i = 0;
-	foreach(lc, indexinfo->indextlist)
-	{
-		TargetEntry *indextle = (TargetEntry *) lfirst(lc);
-
-		if (indexinfo->canreturn[i])
-			result = lappend(result, indextle);
-		else
-		{
-			TargetEntry *newtle = makeNode(TargetEntry);
-			Node	   *texpr = (Node *) indextle->expr;
-
-			memcpy(newtle, indextle, sizeof(TargetEntry));
-			newtle->expr = (Expr *) makeNullConst(exprType(texpr),
-												  exprTypmod(texpr),
-												  exprCollation(texpr));
-			result = lappend(result, newtle);
-		}
-		i++;
-	}
-	return result;
-}
-
 static BitmapIndexScan *
 make_bitmap_indexscan(Index scanrelid,
 					  Oid indexid,
diff --git a/src/backend/optimizer/plan/setrefs.c b/src/backend/optimizer/plan/setrefs.c
index 6ccec759bd..9c2aba45a6 100644
--- a/src/backend/optimizer/plan/setrefs.c
+++ b/src/backend/optimizer/plan/setrefs.c
@@ -1142,8 +1142,26 @@ set_indexonlyscan_references(PlannerInfo *root,
 							 int rtoffset)
 {
 	indexed_tlist *index_itlist;
+	List	   *stripped_indextlist;
+	ListCell   *lc;
+
+	/*
+	 * Vars in the plan node's targetlist, qual, and recheckqual must only
+	 * reference columns that the index AM can actually return.  To ensure
+	 * this, remove non-returnable columns (which are marked as resjunk) from
+	 * the indexed tlist.  We can just drop them because the indexed_tlist
+	 * machinery pays attention to TLE resnos, not physical list position.
+	 */
+	stripped_indextlist = NIL;
+	foreach(lc, plan->indextlist)
+	{
+		TargetEntry *indextle = (TargetEntry *) lfirst(lc);
+
+		if (!indextle->resjunk)
+			stripped_indextlist = lappend(stripped_indextlist, indextle);
+	}
 
-	index_itlist = build_tlist_index(plan->indextlist);
+	index_itlist = build_tlist_index(stripped_indextlist);
 
 	plan->scan.scanrelid += rtoffset;
 	plan->scan.plan.targetlist = (List *)
@@ -1160,6 +1178,13 @@ set_indexonlyscan_references(PlannerInfo *root,
 					   INDEX_VAR,
 					   rtoffset,
 					   NUM_EXEC_QUAL((Plan *) plan));
+	plan->recheckqual = (List *)
+		fix_upper_expr(root,
+					   (Node *) plan->recheckqual,
+					   index_itlist,
+					   INDEX_VAR,
+					   rtoffset,
+					   NUM_EXEC_QUAL((Plan *) plan));
 	/* indexqual is already transformed to reference index columns */
 	plan->indexqual = fix_scan_list(root, plan->indexqual,
 									rtoffset, 1);
diff --git a/src/backend/optimizer/plan/subselect.c b/src/backend/optimizer/plan/subselect.c
index c9f7a09d10..aee5295183 100644
--- a/src/backend/optimizer/plan/subselect.c
+++ b/src/backend/optimizer/plan/subselect.c
@@ -2336,6 +2336,8 @@ finalize_plan(PlannerInfo *root, Plan *plan,
 		case T_IndexOnlyScan:
 			finalize_primnode((Node *) ((IndexOnlyScan *) plan)->indexqual,
 							  &context);
+			finalize_primnode((Node *) ((IndexOnlyScan *) plan)->recheckqual,
+							  &context);
 			finalize_primnode((Node *) ((IndexOnlyScan *) plan)->indexorderby,
 							  &context);
 
diff --git a/src/include/nodes/execnodes.h b/src/include/nodes/execnodes.h
index ddc3529332..4ff98f4040 100644
--- a/src/include/nodes/execnodes.h
+++ b/src/include/nodes/execnodes.h
@@ -1490,7 +1490,7 @@ typedef struct IndexScanState
 /* ----------------
  *	 IndexOnlyScanState information
  *
- *		indexqual		   execution state for indexqual expressions
+ *		recheckqual		   execution state for recheckqual expressions
  *		ScanKeys		   Skey structures for index quals
  *		NumScanKeys		   number of ScanKeys
  *		OrderByKeys		   Skey structures for index ordering operators
@@ -1509,7 +1509,7 @@ typedef struct IndexScanState
 typedef struct IndexOnlyScanState
 {
 	ScanState	ss;				/* its first field is NodeTag */
-	ExprState  *indexqual;
+	ExprState  *recheckqual;
 	struct ScanKeyData *ioss_ScanKeys;
 	int			ioss_NumScanKeys;
 	struct ScanKeyData *ioss_OrderByKeys;
diff --git a/src/include/nodes/plannodes.h b/src/include/nodes/plannodes.h
index 13e07ead31..aa11dcedab 100644
--- a/src/include/nodes/plannodes.h
+++ b/src/include/nodes/plannodes.h
@@ -420,15 +420,28 @@ typedef struct IndexScan
  * index-only scan, in which the data comes from the index not the heap.
  * Because of this, *all* Vars in the plan node's targetlist, qual, and
  * index expressions reference index columns and have varno = INDEX_VAR.
- * Hence we do not need separate indexqualorig and indexorderbyorig lists,
- * since their contents would be equivalent to indexqual and indexorderby.
+ *
+ * We could almost use indexqual directly against the index's output tuple
+ * when rechecking lossy index operators, but that won't work for quals on
+ * index columns that are not retrievable.  Hence, recheckqual is needed
+ * for rechecks: it expresses the same condition as indexqual, but using
+ * only index columns that are retrievable.  (We will not generate an
+ * index-only scan if this is not possible.  An example is that if an
+ * index has table column "x" in a retrievable index column "ind1", plus
+ * an expression f(x) in a non-retrievable column "ind2", an indexable
+ * query on f(x) will use "ind2" in indexqual and f(ind1) in recheckqual.
+ * Without the "ind1" column, an index-only scan would be disallowed.)
+ *
+ * We don't currently need a recheckable equivalent of indexorderby,
+ * because we don't support lossy operators in index ORDER BY.
  *
  * To help EXPLAIN interpret the index Vars for display, we provide
  * indextlist, which represents the contents of the index as a targetlist
  * with one TLE per index column.  Vars appearing in this list reference
  * the base table, and this is the only field in the plan node that may
- * contain such Vars.  Note however that index columns that the AM can't
- * reconstruct are replaced by null Consts in indextlist.
+ * contain such Vars.  Also, for the convenience of setrefs.c, TLEs in
+ * indextlist are marked as resjunk if they correspond to columns that
+ * the index AM cannot reconstruct.
  * ----------------
  */
 typedef struct IndexOnlyScan
@@ -436,6 +449,7 @@ typedef struct IndexOnlyScan
 	Scan		scan;
 	Oid			indexid;		/* OID of index to scan */
 	List	   *indexqual;		/* list of index quals (usually OpExprs) */
+	List	   *recheckqual;	/* index quals in recheckable form */
 	List	   *indexorderby;	/* list of index ORDER BY exprs */
 	List	   *indextlist;		/* TargetEntry list describing index's cols */
 	ScanDirection indexorderdir;	/* forward or backward or don't care */
diff --git a/src/test/regress/expected/gist.out b/src/test/regress/expected/gist.out
index 6c6ced91e9..4ba2e7f29e 100644
--- a/src/test/regress/expected/gist.out
+++ b/src/test/regress/expected/gist.out
@@ -340,6 +340,37 @@ where p <@ box(point(5, 5), point(5.3, 5.3));
  <(5.3,5.3),1>
 (7 rows)
 
+-- Similarly, test that index rechecks involving a non-returnable column
+-- are done correctly.
+explain (verbose, costs off)
+select p from gist_tbl where circle(p,1) @> circle(point(0,0),0.95);
+                                      QUERY PLAN                                       
+---------------------------------------------------------------------------------------
+ Index Only Scan using gist_tbl_multi_index on public.gist_tbl
+   Output: p
+   Index Cond: ((circle(gist_tbl.p, '1'::double precision)) @> '<(0,0),0.95>'::circle)
+(3 rows)
+
+select p from gist_tbl where circle(p,1) @> circle(point(0,0),0.95);
+   p   
+-------
+ (0,0)
+(1 row)
+
+-- This case isn't supported, but it should at least EXPLAIN correctly.
+explain (verbose, costs off)
+select p from gist_tbl order by circle(p,1) <-> point(0,0) limit 1;
+                                     QUERY PLAN                                     
+------------------------------------------------------------------------------------
+ Limit
+   Output: p, ((circle(p, '1'::double precision) <-> '(0,0)'::point))
+   ->  Index Only Scan using gist_tbl_multi_index on public.gist_tbl
+         Output: p, (circle(p, '1'::double precision) <-> '(0,0)'::point)
+         Order By: ((circle(gist_tbl.p, '1'::double precision)) <-> '(0,0)'::point)
+(5 rows)
+
+select p from gist_tbl order by circle(p,1) <-> point(0,0) limit 1;
+ERROR:  lossy distance functions are not supported in index-only scans
 -- Clean up
 reset enable_seqscan;
 reset enable_bitmapscan;
diff --git a/src/test/regress/sql/gist.sql b/src/test/regress/sql/gist.sql
index c6bac320b1..cf1069c207 100644
--- a/src/test/regress/sql/gist.sql
+++ b/src/test/regress/sql/gist.sql
@@ -153,6 +153,17 @@ where p <@ box(point(5, 5), point(5.3, 5.3));
 select circle(p,1) from gist_tbl
 where p <@ box(point(5, 5), point(5.3, 5.3));
 
+-- Similarly, test that index rechecks involving a non-returnable column
+-- are done correctly.
+explain (verbose, costs off)
+select p from gist_tbl where circle(p,1) @> circle(point(0,0),0.95);
+select p from gist_tbl where circle(p,1) @> circle(point(0,0),0.95);
+
+-- This case isn't supported, but it should at least EXPLAIN correctly.
+explain (verbose, costs off)
+select p from gist_tbl order by circle(p,1) <-> point(0,0) limit 1;
+select p from gist_tbl order by circle(p,1) <-> point(0,0) limit 1;
+
 -- Clean up
 reset enable_seqscan;
 reset enable_bitmapscan;
diff --git a/contrib/btree_gist/expected/inet.out b/contrib/btree_gist/expected/inet.out
index c323d903da..f15f1435f0 100644
--- a/contrib/btree_gist/expected/inet.out
+++ b/contrib/btree_gist/expected/inet.out
@@ -83,13 +83,13 @@ SELECT count(*) FROM inettmp WHERE a  = '89.225.196.191'::inet;
 
 DROP INDEX inetidx;
 CREATE INDEX ON inettmp USING gist (a gist_inet_ops, a inet_ops);
--- likewise here (checks for core planner bug)
+-- this can be an index-only scan, as long as the planner uses the right column
 EXPLAIN (COSTS OFF)
 SELECT count(*) FROM inettmp WHERE a  = '89.225.196.191'::inet;
-                     QUERY PLAN                     
-----------------------------------------------------
+                       QUERY PLAN                        
+---------------------------------------------------------
  Aggregate
-   ->  Index Scan using inettmp_a_a1_idx on inettmp
+   ->  Index Only Scan using inettmp_a_a1_idx on inettmp
          Index Cond: (a = '89.225.196.191'::inet)
 (3 rows)
 
diff --git a/contrib/btree_gist/sql/inet.sql b/contrib/btree_gist/sql/inet.sql
index 4b8d354b00..249e8085c3 100644
--- a/contrib/btree_gist/sql/inet.sql
+++ b/contrib/btree_gist/sql/inet.sql
@@ -42,7 +42,7 @@ DROP INDEX inetidx;
 
 CREATE INDEX ON inettmp USING gist (a gist_inet_ops, a inet_ops);
 
--- likewise here (checks for core planner bug)
+-- this can be an index-only scan, as long as the planner uses the right column
 EXPLAIN (COSTS OFF)
 SELECT count(*) FROM inettmp WHERE a  = '89.225.196.191'::inet;
 
diff --git a/src/backend/optimizer/path/indxpath.c b/src/backend/optimizer/path/indxpath.c
index 0e4e00eaf0..e2def356f6 100644
--- a/src/backend/optimizer/path/indxpath.c
+++ b/src/backend/optimizer/path/indxpath.c
@@ -1807,7 +1807,6 @@ check_index_only(RelOptInfo *rel, IndexOptInfo *index)
 	bool		result;
 	Bitmapset  *attrs_used = NULL;
 	Bitmapset  *index_canreturn_attrs = NULL;
-	Bitmapset  *index_cannotreturn_attrs = NULL;
 	ListCell   *lc;
 	int			i;
 
@@ -1847,11 +1846,7 @@ check_index_only(RelOptInfo *rel, IndexOptInfo *index)
 
 	/*
 	 * Construct a bitmapset of columns that the index can return back in an
-	 * index-only scan.  If there are multiple index columns containing the
-	 * same attribute, all of them must be capable of returning the value,
-	 * since we might recheck operators on any of them.  (Potentially we could
-	 * be smarter about that, but it's such a weird situation that it doesn't
-	 * seem worth spending a lot of sweat on.)
+	 * index-only scan.
 	 */
 	for (i = 0; i < index->ncolumns; i++)
 	{
@@ -1868,21 +1863,13 @@ check_index_only(RelOptInfo *rel, IndexOptInfo *index)
 			index_canreturn_attrs =
 				bms_add_member(index_canreturn_attrs,
 							   attno - FirstLowInvalidHeapAttributeNumber);
-		else
-			index_cannotreturn_attrs =
-				bms_add_member(index_cannotreturn_attrs,
-							   attno - FirstLowInvalidHeapAttributeNumber);
 	}
 
-	index_canreturn_attrs = bms_del_members(index_canreturn_attrs,
-											index_cannotreturn_attrs);
-
 	/* Do we have all the necessary attributes? */
 	result = bms_is_subset(attrs_used, index_canreturn_attrs);
 
 	bms_free(attrs_used);
 	bms_free(index_canreturn_attrs);
-	bms_free(index_cannotreturn_attrs);
 
 	return result;
 }

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