To understand how the rule system works it is necessary to know when it is invoked and what its input and results are.
The rule system is located between the parser and the planner. It takes the output of the parser, one query tree, and the user-defined rewrite rules, which are also query trees with some extra information, and creates zero or more query trees as result. So its input and output are always things the parser itself could have produced and thus, anything it sees is basically representable as an SQL statement.
    Now what is a query tree? It is an internal representation of an
    SQL statement where the single parts that it is
    built from are stored separately. These query trees can be shown
    in the server log if you set the configuration parameters
    debug_print_parse,
    debug_print_rewritten, or
    debug_print_plan.  The rule actions are also
    stored as query trees, in the system catalog
    pg_rewrite.  They are not formatted like
    the log output, but they contain exactly the same information.
Reading a raw query tree requires some experience. But since SQL representations of query trees are sufficient to understand the rule system, this chapter will not teach how to read them.
When reading the SQL representations of the query trees in this chapter it is necessary to be able to identify the parts the statement is broken into when it is in the query tree structure. The parts of a query tree are
        This is a simple value telling which command
        (SELECT, INSERT,
        UPDATE, DELETE) produced
        the query tree.
    
        The range table is a list of relations that are used in the query.
        In a SELECT statement these are the relations given after
        the FROM key word.
    
Every range table entry identifies a table or view and tells by which name it is called in the other parts of the query. In the query tree, the range table entries are referenced by number rather than by name, so here it doesn't matter if there are duplicate names as it would in an SQL statement. This can happen after the range tables of rules have been merged in. The examples in this chapter will not have this situation.
This is an index into the range table that identifies the relation where the results of the query go.
        SELECT queries don't have a result
        relation. (The special case of SELECT INTO is
        mostly identical to CREATE TABLE followed by
        INSERT ... SELECT, and is not discussed
        separately here.)
    
        For INSERT, UPDATE, and
        DELETE commands, the result relation is the table
        (or view!) where the changes are to take effect.
    
        The target list is a list of expressions that define the
        result of the query.  In the case of a
        SELECT, these expressions are the ones that
        build the final output of the query.  They correspond to the
        expressions between the key words SELECT
        and FROM.  (* is just an
        abbreviation for all the column names of a relation.  It is
        expanded by the parser into the individual columns, so the
        rule system never sees it.)
    
        DELETE commands don't need a normal target list
        because they don't produce any result.  Instead, the planner
        adds a special CTID entry to the empty target list,
        to allow the executor to find the row to be deleted.
        (CTID is added when the result relation is an ordinary
        table.  If it is a view, a whole-row variable is added instead, by
        the rule system, as described in Section 41.2.4.)
    
        For INSERT commands, the target list describes
        the new rows that should go into the result relation. It consists of the
        expressions in the VALUES clause or the ones from the
        SELECT clause in INSERT
        ... SELECT.  The first step of the rewrite process adds
        target list entries for any columns that were not assigned to by
        the original command but have defaults.  Any remaining columns (with
        neither a given value nor a default) will be filled in by the
        planner with a constant null expression.
    
        For UPDATE commands, the target list
        describes the new rows that should replace the old ones. In the
        rule system, it contains just the expressions from the SET
        column = expression part of the command.  The planner will
        handle missing columns by inserting expressions that copy the values
        from the old row into the new one.  Just as for DELETE,
        a CTID or whole-row variable is added so that
        the executor can identify the old row to be updated.
    
Every entry in the target list contains an expression that can be a constant value, a variable pointing to a column of one of the relations in the range table, a parameter, or an expression tree made of function calls, constants, variables, operators, etc.
        The query's qualification is an expression much like one of
        those contained in the target list entries. The result value of
        this expression is a Boolean that tells whether the operation
        (INSERT, UPDATE,
        DELETE, or SELECT) for the
        final result row should be executed or not. It corresponds to the WHERE clause
        of an SQL statement.
    
        The query's join tree shows the structure of the FROM clause.
        For a simple query like SELECT ... FROM a, b, c, the join tree is just
        a list of the FROM items, because we are allowed to join them in
        any order.  But when JOIN expressions, particularly outer joins,
        are used, we have to join in the order shown by the joins.
        In that case, the join tree shows the structure of the JOIN expressions.  The
        restrictions associated with particular JOIN clauses (from ON or
        USING expressions) are stored as qualification expressions attached
        to those join-tree nodes.  It turns out to be convenient to store
        the top-level WHERE expression as a qualification attached to the
        top-level join-tree item, too.  So really the join tree represents
        both the FROM and WHERE clauses of a SELECT.
    
        The other parts of the query tree like the ORDER BY
        clause aren't of interest here. The rule system
        substitutes some entries there while applying rules, but that
        doesn't have much to do with the fundamentals of the rule
        system.