[section Expressions]

_yap_ consists of expressions and functions that operate on them.  A function
that takes an expression will accept any type that models the _Expr_ concept.

For a type `T` to model the _Expr_ concept, `T` must contain at least an
_kind_ (terminal, plus-operation, etc.) and a _tuple_ of values.  That's it.

[note The _tuple_ of values is constrained, based on the kind of the
expression; see the full _Expr_ documentation for details.]

Here's an example of an expression:

[minimal_template]

That's a template that models _ExprTmpl_.  Instantiated with the proper
template parameters, it produces _Exprs_.

Ok, so it's not that interesting by itself _emdash_ `minimal_expr` has no
operations defined for it.  But we can still use it with the _yap_ functions
that take an _Expr_.  Let's make a _yap_ plus-expression manually:

[minimal_template_manual_construction]

If we evaluate it using _eval_, it does what you would expect:

[minimal_template_evaluation]

One more thing.  It is important to remember that _yap_ expressions are
all-lazy, all the time.  There is no auto-evaluation of a _yap_ expression
like there is with normal C++ expressions.  If you want your expressions to be
evaluated, you must call _eval_, or define non-lazy operations that force
evaluation where and when you want it.  This last approach is usually the
right one, and there are lots of examples of how to do this in the _examples_
section.  In particular, checkout the _lazy_vector_ and _tarray_ examples.

[endsect]


[section Mix-and-Match Expression Templates]

Because _yap_ operates on _Exprs_, it is possible to mix and match _Exprs_
that are instantiations of different templates.

Here's why that's important.  Say we have two types in a library.  `S` is a
string type, and `M` is a matrix type.  In the code here, `s` and `m` are
objects of types `S` and `M` respectively.  Say we also have typical operator
overloads for these types, so `m * m` and `s[0]` are well-formed expressions,
but `m[0]` and `s * s` are not.

To use these with _yap_ I might write an expression template for each:

    template <...>
    struct m_expr
    {
        // ...
    };

    BOOST_YAP_USER_BINARY_OPERATOR(times, m_expr, m_expr)

    template <...>
    struct s_expr
    {
        // ...
        BOOST_YAP_USER_SUBSCRIPT_OPERATOR(::s_expr)
    };

With this, I might write a _yap_ expression like:

    some_expr_producing_func(S("my_matrix")) * some_matrix

I can transform this expression however I like, and do not have to worry about
the fact that it contains expressions instantiated from different templates.

If _yap_ required an expression to be instantiated from a single expression
template `expr<>`, `expr<>` would have to have both operators.  This means
that all of a sudden `s * s` and `m[0]` would be well-formed expressions
within a _yap_ expression, but *not* for the real types `S` and `M`
respectively.  That would be super weird.

[endsect]


[section Kinds of Expressions]

Most of the expression kinds are the overloadable operators (`operator!()`,
`operator<<=()`, etc.), See _kind_ for the full list.

There are three special kinds of expressions:

[variablelist
    [[_terminal_] [A terminal contains a non-Expression value, and represents a leaf-node in an
expression tree.  A terminal may have a _placeholder_ value, in which case it acts as a placeholder. ]]
    [[_if_else_] [An `if_else` expression is analogous to the C++ ternary operator (`?:`).  It's up to you to make sure that the conditional expression given to `if_else` can be converted to `bool`; _yap_ does not check this.]]
    [[_expr_ref_] [An `expr_ref` expression is one that acts as a (possibly `const`) lvalue reference to another expression.  It exists to prevent unnecessary copies of expressions.]]
]

[endsect]


[section Operators]

Let's see an expression template type with some operators:

[lazy_vector_decl]

Those macros are used to define operator overloads that return _Exprs_.  As
shown here, that sort of operator can be mixed with normal, non-lazy ones
_emdash_ the `operator[]` is a normal eager function.

Use of the macros is not necessary (you can write your own operators that
return _Exprs_ if you like), but it is suitable 99% of the time.

Making the operators easy to define like this allows you to define custom
expression templates that have only the operators defined that are appropriate
for your use case.

Detailed documentation on all the available macros can be found later in the
_operator_macros_ section.

[endsect]


[section Transforming Expressions]

Transformations in _yap_ are done using the _xform_ function.

Let's take another look at the example expression from the intro:

[$yap/img/expr.png]

Consider a call to _xform_, operating on that expression:

    auto result = boost::yap::transform(expr, xform);

_yap_'s _xform_ first looks at the top level expression, which in this case is
a `+` expression.  If the transform object `xform` matches the `+` expression,
_xform_ is done; it just returns `xform(expr)`.  If `xform` does not match the
`+` expression, _xform_ transforms all its operands (which for `operator+()`
is just the left and right operands), and returns a new `+` expression with
those transformed operands.  What I mean by "match" is covered in detail
below.

The overall effect of this is that _xform_ effectively copies an `expr` node
that *does not* match `xform`, and returns a transformed node for an `expr`
node that *does* match `xform`.

_xform_ can also take multiple transform objects.  If you call it with N
transform objects, it will attempt to match each of the N transforms to a
given expression, one at a time and in their given order.  Only if no
transform matches an expression does the copy-and-recurse behavior kick in.

[note There's another form of _xform_, _xform_strict_.  _xform_strict_ is
identical to _xform_ except that it does not copy or recurse into an unmatched
expression.  Instead, a failed match is a hard error.  This is useful when you
have written a transform that you expect to completely transform an
expression, and you want the compiler to tell you if you've made a mistake.]

One common result of calling _xform_ is that you create a copy of `expr`, with
a few matching nodes transformed.  But this does not have to be the result of
calling _xform_, because a _yap_ transformation is free-form; it must return a
value, but may do just about anything else.  It can transform an expression
into anything _emdash_ a new expression of any kind, or even a non-expression
value (effectively evaluating the expression).  As before, here is the
`get_arity` transform from the _calc3_ example.  It returns a value, not an
_Expr_:

[calc3_get_arity_xform]

Also, note that in this case the transform is stateless, but you could also
give your transform objects data members containing contextual state:

[vector_take_nth_xform]

[tip Often when you create an expression, you will want to evaluate it in
different contexts, changing its evaluation _emdash_ or even entire meaning
_emdash_ in each context.  _eval_ is wrong for this task, since it only takes
values for substitution into placeholders.  In these situations, you should
instead use multiple transforms that evaluate your expression in different
ways.]

[heading When _xform_ Recurses]

As described above, _xform_ only recurses when it *does not* find a match.
This means that if you want to transform a nonterminal, say an
`expr_kind::call` expression we'll call `C`, and *also* `C`'s subexpressions,
you must explicitly call _xform_ yourself in your transform that matches `C`.
You can see this kind of explicit _xform_ call in the recursive case of
`get_arity` in the example code above.

[note The code you write with _yap_ is likely going to be very generic,
especially when you're writing a transform.  _xform_ requires an _Expr_ as its
first parameter.  In situations when you want to make sure that the first
parameter you pass to _xform_ is always a _yap_ expression, use the _as_expr_
function.  This is commonly needed when writing a transform in which you
manually recurse by calling _xform_ inside one of your transform overloads.]

[heading Transform Matching]

In _yap_ a _XForm_ is a _Callable_ that has *zero or more* overloads that
model the _ExprXForm_ or _TagXForm_ concepts.

An _ExprXForm_ overload takes a single parameter whose type is the expression
to be transformed.  Here's one from a transform object in the _future_group_
example:

[expr_xform]

_ExprXForms_ are most useful when you want to transform a narrow set of
expression types (perhaps only one).  In particular, you can distinguish
between `const` and non-`const`, reference and non-reference, etc., in the
expression and its operands in a way that you have less control over with the
other kind of transform.

A _TagXForm_ overload takes a tag that indicates the _kind_ of the expression
to be transformed, and then (loosely) the value of each operand of the
expression to be transformed.  This looseness prevents you from needing to
write out the full type of the matched expression.  Here's one from the
_pipable_algorithms_ example:

[tag_xform]

_TagXForms_ are most useful when the transform needs to match an expression
without regard to whether its operands are _expr_ref_ expressions, or _emdash_
if they are terminals _emdash_ whether they contain or refer to their values.
_TagXForms_ tend to be far more concise.

[heading A More Rigorous Description of TagTransform Parameters]

That "(loosely)" before probably bothered you, right?  Me too.  Each non-tag
parameter is passed to a _TagXForm_ by calling an operand accessor appropriate
to `expr`'s kind, and then calling a terminal-specific version of _value_
(`terminal_value()`) on the result.  For example, consider a plus expression
`expr`.  The _TagXForm_ on a transform object `xform` would be called like
this:

    xform(plus_tag, terminal_value(left(expr)), terminal_value(right(expr)))

The operand accessors (_left_ and _right_ in this example) all dereference
_expr_ref_ expressions before operating on them, and `terminal_value()` does
the same.

`terminal_value()` works much like _value_, except that it does not take the
value of a *nonterminal* unary expression; it just forwards a nonterminal
through.  It still takes values out of terminals and unwraps _expr_ref_
expressions, though.

The auto-unwrapping of terminals means that you can effectively ignore the
presence of _expr_ref_ expressions when writing a _TagXForm_.  You can also
just deal with the values inside terminals, and not the terminals
themselves. Also, you can match all terminal value qualifiers (`const` or not,
lvalue or rvalue) uniformly with a `T const &` parameter.  Finally, you can
write _TagXForm_ parameter types that can catch conversions; for instance, you
can match any negation expression containing a terminal, *or a reference to
one*, containing a value convertible to `double` like this:

     struct xform
     {
         auto operator() (boost::yap::negate_tag, double x)
         { return /* ... */; }
     }

That will match a negation of a terminal containing an `unsigned int`,
`unsigned int &`, `int const &`, `float &&`, etc.  It will also match a
negation of a reference to such a terminal.

[heading Mixing the Two Kinds of Transforms]

You can have two overloads in your transform that match an expression, one an
_ExprXForm_ and one a _TagXForm_, and there will not be any ambiguity.  The
_TagXForm_ is matched first, and the _ExprXForm_ is matched only if the
_TagXForm_ did not.  You don't have to worry about ambiguity, but save
yourself some confusion and mix the two kinds of overloads as little as
possible.

[note The above only applies when you have an _ExprXForm_ and a _TagXForm_
that match *the same kind of expression*.  Having unrelated _ExprXForms_ and
_TagXForms_ within the same transform object is often quite useful.]

[heading Multiple Transform Objects]

In the case that multiple transform objects are being used in _xform_, the
above logic applies to each one independently before the next one is used.  In
other words, in the call `boost::yap::transform(expr, a, b)`, _xform_ tries to
match any _TagXForm_ from `a` to an expression first, then any _ExprXForm_
from `a`, then any _TagXForm_ from `b`, and finally any _ExprXForm_ from `b`.
Only the first matching overload in this sequence is used; all overloads later
in the sequence or in later transforms, whether they match or not, are simply
ignored.

[heading YAP-Supplied Transforms]

_yap_ comes with a couple of functions that return ready-made transforms,
_replacements_ and _evaluation_.

The transforms returned by _replacements_ replace only placeholder terminals.
Placeholder `I` is replaced by the `I-1`-th argument passed to _replacements_.
Placeholders are `1`-based for consistency with other Boost and `std`
placeholders.

There are also a couple of specialty transform functions,
_replace_placeholders_ and _eval_.  These are convenience functions that just
call _xform_ on an expression using _replacements_ or _evaluation_ as the
transform, respectively.

The behavior of _evaluation_ is covered in the next section, [link
boost_yap.manual.evaluating_expressions Evaluating Expressions].

[endsect]


[section Evaluating Expressions]

_yap_ expressions are evaluated explicitly, by calling the _eval_ function or
calling _xform_ using a transform object returned from _evaluation_.  The
former is a convenince function that does the latter.

_eval_ simply removes all the _yap_ machinery from an expression and evaluates
it exactly as it would have been if _yap_ were not used.  This means that
functions are called, operators evaluated, etc. all as normal.  To illustrate
this, take a look at the implementation of `operator,()` used in _eval_:

[evaluation_transform_comma]

What this transformation does is transform the left and right expressions, and
then use the built-in `operator,()` on the result.  The evaluation
transformations for the other operators do the same thing _emdash_ evaluate
the operands, then return the result of applying the built-in operator to the
operands.

Function calls are done in a similar way, except that the callable is also a
subexpression that needs to be evaluated before being called:

[evaluation_transform_call]

[endsect]


[section Operator Macros]

If you got here without reading the _operators_ section, go read that first.
Here are the operator macros and their uses:

[table Unary and Binary Operator-Defining Macros
    [[Macro] [Use] [First/Left Operand Type] [Right Operand Type] [Notes]]

    [[_unary_m_] [Unary operators.] [An _Expr_ instantiated from _ExprTmpl_ macro parameter `expr_template`.] [--] []]
    [[_binary_m_] [Binary operators.] [Any type.] [Any type.] [At least one parameter must be an _Expr_ instantiated from _ExprTmpl_ macro parameter `expr_template`.]]
    [[_udt_unary_m_] [Free operators defined over non-_Expr_ types constrained by a type trait (e.g. all `std::map<>`s).] [Any non-_Expr_ that satisfies the given type trait.] [--] []]
    [[_udt_udt_binary_m_] [Free operators defined over non-_Expr_ types constrained by a pair of type traits (e.g. a `std::map<>` on the left, and a `std::vector<>` on the right).  Useful for type-asymmetric operators.] [Any non-_Expr_ that satisfies the left-hand type trait.] [Any non-_Expr_ that satisfies the right-hand type trait.] []]
    [[_udt_any_binary_m_] [Free operators defined over pairs of non-_Expr_ types, one constrained by a type trait and one not (e.g. a `std::list<>` on either side, and anything on the other).] [Any non-_Expr_.] [--] [At least one parameter must satisfy the given type trait.]]
]

Some operators may only be defined as member functions, and so are not covered
by general-purpose the unary and binary operator macros above:

[table The Member-Only Operator Macros
    [[Macro] [Use] [Operands] [Notes]]

    [[_member_assign_m_] [Assignment operator.] [Any type except `decltype(*this)`.] [Does not conflict with the assignment or move assignment operators.]]
    [[_member_subscript_m_] [Subscript operator.] [Any type.] []]
    [[_member_call_m_] [Call operator taking any number of parameters.] [Any type.] []]
    [[_member_call_n_m_] [Call operator taking exactly N parameters.] [Any type.] []]
]

[table if_else Psuedo-Operator Macros
    [[Macro] [Use] [Operands] [Notes]]

    [[_expr_if_else_m_] [Free `if_else()` function that requires at least one parameter to be an expression.] [Any type.] [At least one parameter must be an _Expr_.]]
    [[_udt_any_if_else_m_] [Free `if_else()` function for non-_Expr_ types that requires at least one parameter to satisfy the given type trait.] [Any non-_Expr_.] [At least one parameter must satisfy the given type trait.]]
]

[note Operands are handled in a uniform way across all functions defined by
all the macros listed here.  See _how_treated_ for details.]

[endsect]


[section How Expression Operands Are Treated]

For any _expr_ operator overload, or any function defined using one of the
function definition macros, operands are treated in a uniform way.

The guiding design principle here is that an expression built using _yap_
should match the semantics of a builtin C++ expression as closely as possible.
This implies that an rvalue be treated as if it were a temporary (as it may in
fact have initially been) throughout the building and transformation of an
expression, and that an lvalue should retain its connection to the underlying
named entity to which it refers.

For example, if you see

    auto expr = a + 1;

you should expect that `a` will be an lvalue reference to some object of type
`decltype(a)`, regardless of whether `a` is a _yap_ _Expr_ or a builtin type.
Similarly, you should expect the `1` to be an rvalue, whether wrapped in a
terminal or not.

Let's take a quick look at _make_term_.  If you call it with a `T` rvalue, the
terminal's value type is a `T`, and the rvalue gets moved into it.  If you
call it with a `T [const]` lvalue, the value type is `T [const] &`, and the
reference refers to the lvalue (read `[const]` as "possibly
`const`-qualified").  This is important because you might write through the
terminal later in an assignment operation.  You don't want to lose the ability
to do this, or be forced to write some Baroque pile of code to do so _emdash_
it should be natural and easy.

And it is:

[assign_through_terminal]

Now, there is a wrinkle.  _yap_'s lazy expressions can be built piecemeal:

    auto subexpr = boost::yap::make_terminal(1) + 2;
    // This is fine, and acts more-or-less as if you wrote "1 / (1 + 2)".
    auto expr = 1 / subexpr;

whereas C++'s eager builtin expressions cannot:

    auto subexpr = 1 + 2;    // Same as "int subexpr = 3;".  Hm.
    auto expr = 1 / subexpr; // Same as "int expr = 0;" Arg.

Ok, so since you can build these lazy _yap_ expressions up from
subexpressions, how do we treat the subexpressions?  We treat them in exactly
the same way as _make_term_ treats its parameter.  Rvalues are moved in, and
lvalues are captured by (possibly `const`) reference.

[note If you want to subvert the capture-by-reference semantics of using
subexpressions, just `std::move()` them.  That will force a move _emdash_ or
copy of values for which move is not defined.]

The capture-by-reference behavior is implemented via a special _kind_,
_expr_ref_.  An `expr_ref` expression has a single data element: a (possibly
`const` (Can I stop saying that every time?  You get it, right? Ok, good.))
reference to an expression.  This additional level of indirection causes some
complications at times, as you can see in the examples.  Fortunately, the
complications are not overly cumbersome.

So, given the rules above, here is a comprehensive breakdown of what happens
when an operand is passed to a _yap_ operator.  In this table, `expr_tmpl` is
an _ExprTmpl_, and `T` is a non-_Expr_ type.  `E` refers to any non-`expr_ref`
_Expr_.  _yap_ does a partial decay on non-_Expr_ operands, in which `cv` and
reference qualifiers are left unchanged, but arrays are decayed to pointers
and functions are decayed to function pointers. `PARTIAL_DECAY(T)` indicates
such a partial decay of `T`.

[table Operand Handling
    [[Operand] [Captured As] [Notes]]

    [[`T const &`] [`expr_tmpl<expr_kind::terminal, boost::hana::tuple<PARTIAL_DECAY(T)>>`] []]
    [[`T &`] [`expr_tmpl<expr_kind::terminal, boost::hana::tuple<PARTIAL_DECAY(T)>>`] []]
    [[`T &&`] [`expr_tmpl<expr_kind::terminal, boost::hana::tuple<PARTIAL_DECAY(T)>>`] [Operand moved.]]

    [[`E const &`] [`expr_tmpl<expr_kind::expr_ref, boost::hana::tuple<E const &>>`] []]
    [[`E &`] [`expr_tmpl<expr_kind::expr_ref, boost::hana::tuple<E &>>`] []]
    [[`E &&`] [`E`] [Operand moved.]]

    [[`expr_tmpl<expr_kind::expr_ref, ...> const &`] [`expr_tmpl<expr_kind::expr_ref, ...>`] []]
    [[`expr_tmpl<expr_kind::expr_ref, ...> &`] [`expr_tmpl<expr_kind::expr_ref, ...>`] []]
    [[`expr_tmpl<expr_kind::expr_ref, ...> &&`] [`expr_tmpl<expr_kind::expr_ref, ...>`] [Operand moved.]]
]

The partial decay of non-_Expr_ operands is another example of how _yap_
attempts to create expression trees that are as semantically close to builtin
expressions as possible.

[endsect]


[section Printing]

_yap_ has a convenient _print_ function, that prints an expression tree to a
stream.  It is not intended for production work (for instance, it has no
formatting options), but it is excellent for debugging and instrumentation.

Since it is only a debugging aid, _print_ is found in a separate header not
included when you include _yap_ with

    #include <boost/yap/yap.hpp>

You must include `<boost/yap/print.hpp>` explicitly.

_print_ handles several patterns of expression specially, to allow a concise
representation of a given expression tree.  For example, given this
definition:

[print_decl]

and this expression:

[print_expr]

_print_ produces this output:

[pre
expr<->
    expr<+>
        term<boost::yap::placeholder<4ll>>[=4\]
        expr<*>
            term<double &>[=1\]
            term<thing>[=<<unprintable-value>>\] &
    term<char const*>[=lvalue terminal\] const &
]

As you can see, _print_ shows one node per line, and represents the tree
structure with indentation.  It abbreviates non-terminal nodes in the tree
`expr<op>`, where `op` is an operator symbol.  Terminal nodes are abbreviated
`term<T>`, where `T` is the type of value contained in the terminal; this may
be a reference type or a value.

A `term` node may not be a terminal node at all, but an _expr_ref_ expression
containing a terminal.  Such a _expr_ref_ node has a `&` or `const &` suffix,
to indicate that it is a mutable or `const` reference, respectively.

Each `term` node has a bracketed value near the end.  The format is `[=X]`
where `X` is the value the terminal contains.  If the terminal contains a
value for which no `operator<<(std::ostream &, ...)` overload exists (such as
the `thing` type above), `X` will be `<<unprintable-value>>`.

[endsect]
