Use partial indices on SQLIte.

synapse/storage/background_updates.py

@@ -581,22 +581,20 @@ def create_index_psql(conn: Connection) -> None:
         def create_index_sqlite(conn: Connection) -> None:
             # Sqlite doesn't support concurrent creation of indexes.
             #
-            # We don't use partial indices on SQLite as it wasn't introduced
-            # until 3.8, and wheezy and CentOS 7 have 3.7
-            #
             # We assume that sqlite doesn't give us invalid indices; however
             # we may still end up with the index existing but the
             # background_updates not having been recorded if synapse got shut
             # down at the wrong moment - hance we use IF NOT EXISTS. (SQLite
             # has supported CREATE TABLE|INDEX IF NOT EXISTS since 3.3.0.)
             sql = (
                 "CREATE %(unique)s INDEX IF NOT EXISTS %(name)s ON %(table)s"
-                " (%(columns)s)"
+                " (%(columns)s) %(where_clause)s"
             ) % {
                 "unique": "UNIQUE" if unique else "",
                 "name": index_name,
                 "table": table,
                 "columns": ", ".join(columns),
+                "where_clause": "WHERE " + where_clause if where_clause else "",
             }
 
             c = conn.cursor()