lean2/src/kernel/expr.h

/*
Copyright (c) 2013-2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Author: Leonardo de Moura
*/
#pragma once
#include <algorithm>
#include <iostream>
#include <limits>
#include <utility>
#include <tuple>
#include <string>
#include "util/thread.h"
#include "util/lua.h"
#include "util/rc.h"
#include "util/name.h"
#include "util/hash.h"
#include "util/buffer.h"
#include "util/list_fn.h"
#include "util/optional.h"
#include "util/serializer.h"
#include "util/sexpr/format.h"
#include "kernel/level.h"
#include "kernel/extension_context.h"

namespace lean {
// Tags are used by frontends to mark expressions. They are automatically propagated by
// procedures such as update_app, update_binder, etc.
typedef unsigned tag;
constexpr tag nulltag = std::numeric_limits<unsigned>::max();
class expr;
/* =======================================
   Expressions
   expr ::=   Var           idx
          |   Sort          level
          |   Constant      name [levels]
          |   Meta          name expr
          |   Local         name expr
          |   App           expr expr
          |   Lambda        name expr expr
          |   Pi            name expr expr
          |   Let           name expr expr expr

          |   Macro         macro
*/
enum class expr_kind { Var, Sort, Constant, Meta, Local, App, Lambda, Pi, Let, Macro };
class expr_cell {
protected:
    // The bits of the following field mean:
    //    0-1  - term is an arrow (0 - not initialized, 1 - is arrow, 2 - is not arrow)
    // Remark: we use atomic_uchar because these flags are computed lazily (i.e., after the expression is created)
    atomic_uchar       m_flags;
    unsigned           m_kind:8;
    unsigned           m_has_mv:1;         // term contains metavariables
    unsigned           m_has_local:1;      // term contains local constants
    unsigned           m_has_param_univ:1; // term constains parametric universe levels
    unsigned           m_hash;             // hash based on the structure of the expression (this is a good hash for structural equality)
    unsigned           m_hash_alloc;       // hash based on 'time' of allocation (this is a good hash for pointer-based equality)
    atomic_uint        m_tag;
    MK_LEAN_RC(); // Declare m_rc counter
    void dealloc();

    optional<bool> is_arrow() const;
    void set_is_arrow(bool flag);
    friend bool is_arrow(expr const & e);

     static void dec_ref(expr & c, buffer<expr_cell*> & todelete);
public:
    expr_cell(expr_kind k, unsigned h, bool has_mv, bool has_local, bool has_param_univ);
    expr_kind kind() const { return static_cast<expr_kind>(m_kind); }
    unsigned  hash() const { return m_hash; }
    unsigned  hash_alloc() const { return m_hash_alloc; }
    bool has_metavar() const { return m_has_mv; }
    bool has_local() const { return m_has_local; }
    bool has_param_univ() const { return m_has_param_univ; }
    void set_tag(tag t);
    tag get_tag() const { return m_tag; }
};

class macro_definition;
class binder_info;

/**
   \brief Exprs for encoding formulas/expressions, types and proofs.
*/
class expr {
private:
    expr_cell * m_ptr;
    explicit expr(expr_cell * ptr):m_ptr(ptr) { if (m_ptr) m_ptr->inc_ref(); }
    friend class expr_cell;
    expr_cell * steal_ptr() { expr_cell * r = m_ptr; m_ptr = nullptr; return r; }
    friend class optional<expr>;
public:
    /**
      \brief The default constructor creates a reference to a "dummy"
      expression.  The actual "dummy" expression is not relevant, and
      no procedure should rely on the kind of expression used.

      We have a default constructor because some collections only work
      with types that have a default constructor.
    */
    expr();
    expr(expr const & s):m_ptr(s.m_ptr) { if (m_ptr) m_ptr->inc_ref(); }
    expr(expr && s):m_ptr(s.m_ptr) { s.m_ptr = nullptr; }
    ~expr() { if (m_ptr) m_ptr->dec_ref(); }

    friend void swap(expr & a, expr & b) { std::swap(a.m_ptr, b.m_ptr); }

    expr & operator=(expr const & s) { LEAN_COPY_REF(s); }
    expr & operator=(expr && s) { LEAN_MOVE_REF(s); }

    expr_kind kind() const { return m_ptr->kind(); }
    unsigned  hash() const { return m_ptr ? m_ptr->hash() : 23; }
    unsigned  hash_alloc() const { return m_ptr ? m_ptr->hash_alloc() : 23; }
    bool has_metavar() const { return m_ptr->has_metavar(); }
    bool has_local() const { return m_ptr->has_local(); }
    bool has_param_univ() const { return m_ptr->has_param_univ(); }

    void set_tag(tag t) { m_ptr->set_tag(t); }
    tag get_tag() const { return m_ptr->get_tag(); }


    expr_cell * raw() const { return m_ptr; }

    friend expr mk_var(unsigned idx);
    friend expr mk_sort(level const & l);
    friend expr mk_constant(name const & n, levels const & ls);
    friend expr mk_metavar(name const & n, expr const & t);
    friend expr mk_local(name const & n, name const & pp_n, expr const & t);
    friend expr mk_app(expr const & f, expr const & a);
    friend expr mk_pair(expr const & f, expr const & s, expr const & t);
    friend expr mk_proj(bool fst, expr const & p);
    friend expr mk_binding(expr_kind k, name const & n, expr const & t, expr const & e, binder_info const & i);
    friend expr mk_let(name const & n, expr const & t, expr const & v, expr const & e);
    friend expr mk_macro(macro_definition const & m, unsigned num, expr const * args);

    friend bool is_eqp(expr const & a, expr const & b) { return a.m_ptr == b.m_ptr; }
    // Overloaded operator() can be used to create applications
    expr operator()(expr const & a1) const;
    expr operator()(expr const & a1, expr const & a2) const;
    expr operator()(expr const & a1, expr const & a2, expr const & a3) const;
    expr operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4) const;
    expr operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5) const;
    expr operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5, expr const & a6) const;
    expr operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5, expr const & a6,
                    expr const & a7) const;
    expr operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5, expr const & a6,
                    expr const & a7, expr const & a8) const;
};

// =======================================
// Structural equality
/** \brief Binder information is ignored in the following predicate */
bool operator==(expr const & a, expr const & b);
inline bool operator!=(expr const & a, expr const & b) { return !operator==(a, b); }
/** \brief Similar to ==, but it also compares binder information */
bool is_bi_equal(expr const & a, expr const & b);
// =======================================

SPECIALIZE_OPTIONAL_FOR_SMART_PTR(expr)

inline optional<expr> none_expr() { return optional<expr>(); }
inline optional<expr> some_expr(expr const & e) { return optional<expr>(e); }
inline optional<expr> some_expr(expr && e) { return optional<expr>(std::forward<expr>(e)); }

inline bool is_eqp(optional<expr> const & a, optional<expr> const & b) {
    return static_cast<bool>(a) == static_cast<bool>(b) && (!a || is_eqp(*a, *b));
}

/** \brief Bounded variables. They are encoded using de Bruijn's indices. */
class expr_var : public expr_cell {
    unsigned m_vidx; // de Bruijn index
public:
    expr_var(unsigned idx);
    unsigned get_vidx() const { return m_vidx; }
};

/** \brief (parametric) Constants. */
class expr_const : public expr_cell {
    name       m_name;
    levels     m_levels;
public:
    expr_const(name const & n, levels const & ls);
    name const & get_name() const { return m_name; }
    levels const & get_levels() const { return m_levels; }
};

/** \brief Metavariables and local constants */
class expr_mlocal : public expr_cell {
protected:
    name   m_name;
    expr   m_type;
    friend expr_cell;
    void dealloc(buffer<expr_cell*> & todelete);
public:
    expr_mlocal(bool is_meta, name const & n, expr const & t);
    name const & get_name() const { return m_name; }
    expr const & get_type() const { return m_type; }
};

/** \brief expr_mlocal subclass for local constants. */
class expr_local : public expr_mlocal {
    // The name used in the binder that generate this local,
    // it is only used for pretty printing. This field is ignored
    // when comparing expressions.
    name m_pp_name;
    friend expr_cell;
    void dealloc(buffer<expr_cell*> & todelete);
public:
    expr_local(name const & n, name const & pp_name, expr const & t);
    name const & get_pp_name() const { return m_pp_name; }
};

/** \brief Composite expressions */
class expr_composite : public expr_cell {
    unsigned m_depth;
    unsigned m_free_var_range;
    friend unsigned get_depth(expr const & e);
    friend unsigned get_free_var_range(expr const & e);
public:
    expr_composite(expr_kind k, unsigned h, bool has_mv, bool has_local, bool has_param_univ, unsigned d, unsigned fv_range);
};

/** \brief Applications */
class expr_app : public expr_composite {
    expr     m_fn;
    expr     m_arg;
    friend expr_cell;
    void dealloc(buffer<expr_cell*> & todelete);
public:
    expr_app(expr const & fn, expr const & arg);
    expr const & get_fn() const { return m_fn; }
    expr const & get_arg() const { return m_arg; }
};

/**
   \brief Auxiliary annotation for binders (Lambda and Pi). This information
   is only used for elaboration.
*/
class binder_info {
    unsigned m_implicit:1;   //! if true, binder argument is an implicit argument
    unsigned m_cast:1;       //! if true, binder argument is a target for using cast
    unsigned m_contextual:1; //! if true, binder argument is assumed to be part of the context, and may be argument for metavariables.
public:
    binder_info(bool implicit = false, bool cast = false, bool contextual = true):
        m_implicit(implicit), m_cast(cast), m_contextual(contextual) {}
    bool is_implicit() const { return m_implicit; }
    bool is_cast() const { return m_cast; }
    bool is_contextual() const { return m_contextual; }
};

bool operator==(binder_info const & i1, binder_info const & i2);
inline bool operator!=(binder_info const & i1, binder_info const & i2) { return !(i1 == i2); }

class binder {
    friend class expr_binding;
    name             m_name;
    expr             m_type;
    binder_info      m_info;
public:
    binder(name const & n, expr const & t, binder_info const & bi = binder_info()):
        m_name(n), m_type(t), m_info(bi) {}
    name const & get_name() const { return m_name; }
    expr const & get_type() const { return m_type; }
    binder_info const & get_info() const { return m_info; }
    binder update_type(expr const & t) const { return binder(m_name, t, m_info); }
};

/** \brief Lambda and Pi expressions */
class expr_binding : public expr_composite {
    binder           m_binder;
    expr             m_body;
    friend class expr_cell;
    void dealloc(buffer<expr_cell*> & todelete);
public:
    expr_binding(expr_kind k, name const & n, expr const & t, expr const & e, binder_info const & i = binder_info());
    name const & get_name() const   { return m_binder.get_name(); }
    expr const & get_domain() const { return m_binder.get_type(); }
    expr const & get_body() const   { return m_body; }
    binder_info const & get_info() const { return m_binder.get_info(); }
    binder const & get_binder() const { return m_binder; }
};

/** \brief Let expressions */
class expr_let : public expr_composite {
    name    m_name;
    expr    m_type;
    expr    m_value;
    expr    m_body;
    friend class expr_cell;
    void dealloc(buffer<expr_cell*> & todelete);
public:
    expr_let(name const & n, expr const & t, expr const & v, expr const & b);
    ~expr_let();
    name const & get_name() const   { return m_name; }
    expr const & get_type() const   { return m_type; }
    expr const & get_value() const  { return m_value; }
    expr const & get_body() const   { return m_body; }
};

/** \brief Sort */
class expr_sort : public expr_cell {
    level    m_level;
public:
    expr_sort(level const & l);
    ~expr_sort();
    level const & get_level() const { return m_level; }
};

class formatter;

/** \brief Abstract class for macro_definitions */
class macro_definition_cell {
protected:
    void dealloc() { delete this; }
    MK_LEAN_RC();
    /**
       \brief Auxiliary method used for implementing a total order on macro
       attachments. It is invoked by operator<, and it is only invoked when
       <tt>get_name() == other.get_name()</tt>
    */
    virtual bool lt(macro_definition_cell const &) const;
public:
    macro_definition_cell():m_rc(0) {}
    virtual ~macro_definition_cell() {}
    virtual name get_name() const = 0;
    virtual expr get_type(expr const & m, expr const * arg_types, extension_context & ctx) const = 0;
    virtual optional<expr> expand(expr const & m, extension_context & ctx) const = 0;
    virtual optional<expr> expand1(expr const & m, extension_context & ctx) const { return expand(m, ctx); }
    virtual unsigned trust_level() const;
    virtual bool operator==(macro_definition_cell const & other) const;
    virtual void display(std::ostream & out) const;
    virtual format pp(formatter const & fmt, options const & opts) const;
    virtual bool is_atomic_pp(bool unicode, bool coercion) const;
    virtual unsigned hash() const;
    virtual void write(serializer & s) const = 0;
    typedef std::function<expr(deserializer&, unsigned, expr const *)> reader;
};

/** \brief Smart pointer for macro definitions */
class macro_definition {
public:
    macro_definition_cell * m_ptr;
public:
    explicit macro_definition(macro_definition_cell * ptr);
    macro_definition(macro_definition const & s);
    macro_definition(macro_definition && s);
    ~macro_definition();

    macro_definition & operator=(macro_definition const & s);
    macro_definition & operator=(macro_definition && s);

    name get_name() const { return m_ptr->get_name(); }
    expr get_type(expr const & m, expr const * arg_types, extension_context & ctx) const { return m_ptr->get_type(m, arg_types, ctx); }
    optional<expr> expand(expr const & m, extension_context & ctx) const { return m_ptr->expand(m, ctx); }
    optional<expr> expand1(expr const & m, extension_context & ctx) const { return m_ptr->expand1(m, ctx); }
    unsigned trust_level() const { return m_ptr->trust_level(); }
    bool operator==(macro_definition const & other) const { return m_ptr->operator==(*other.m_ptr); }
    bool operator!=(macro_definition const & other) const { return !operator==(other); }
    bool operator<(macro_definition const & other) const;
    void display(std::ostream & out) const { return m_ptr->display(out); }
    format pp(formatter const & fmt, options const & opts) const { return m_ptr->pp(fmt, opts); }
    bool is_atomic_pp(bool unicode, bool coercion) const { return m_ptr->is_atomic_pp(unicode, coercion); }
    unsigned hash() const { return m_ptr->hash(); }
    void write(serializer & s) const { return m_ptr->write(s); }
};

/** \brief Macro attachments */
class expr_macro : public expr_composite {
    macro_definition m_definition;
    unsigned         m_num_args;
    expr *           m_args;
    friend class expr_cell;
    friend expr copy(expr const & a);
    friend expr update_macro(expr const & e, unsigned num, expr const * args);
    void dealloc(buffer<expr_cell*> & todelete);
public:
    expr_macro(macro_definition const & v, unsigned num, expr const * args);
    ~expr_macro();

    macro_definition const & get_def() const { return m_definition; }
    expr const * get_args() const { return m_args; }
    expr const & get_arg(unsigned idx) const { lean_assert(idx < m_num_args); return m_args[idx]; }
    unsigned get_num_args() const { return m_num_args; }
};

// =======================================
// Testers
inline bool is_var(expr_cell * e)         { return e->kind() == expr_kind::Var; }
inline bool is_constant(expr_cell * e)    { return e->kind() == expr_kind::Constant; }
inline bool is_local(expr_cell * e)       { return e->kind() == expr_kind::Local; }
inline bool is_metavar(expr_cell * e)     { return e->kind() == expr_kind::Meta; }
inline bool is_macro(expr_cell * e)       { return e->kind() == expr_kind::Macro; }
inline bool is_app(expr_cell * e)         { return e->kind() == expr_kind::App; }
inline bool is_lambda(expr_cell * e)      { return e->kind() == expr_kind::Lambda; }
inline bool is_pi(expr_cell * e)          { return e->kind() == expr_kind::Pi; }
inline bool is_sort(expr_cell * e)        { return e->kind() == expr_kind::Sort; }
inline bool is_let(expr_cell * e)         { return e->kind() == expr_kind::Let; }
inline bool is_binding(expr_cell * e)     { return is_lambda(e) || is_pi(e); }
inline bool is_mlocal(expr_cell * e)      { return is_metavar(e) || is_local(e); }

inline bool is_var(expr const & e)        { return e.kind() == expr_kind::Var; }
inline bool is_constant(expr const & e)   { return e.kind() == expr_kind::Constant; }
inline bool is_local(expr const & e)      { return e.kind() == expr_kind::Local; }
inline bool is_metavar(expr const & e)    { return e.kind() == expr_kind::Meta; }
inline bool is_macro(expr const & e)      { return e.kind() == expr_kind::Macro; }
inline bool is_app(expr const & e)        { return e.kind() == expr_kind::App; }
inline bool is_lambda(expr const & e)     { return e.kind() == expr_kind::Lambda; }
inline bool is_pi(expr const & e)         { return e.kind() == expr_kind::Pi; }
inline bool is_sort(expr const & e)       { return e.kind() == expr_kind::Sort; }
inline bool is_let(expr const & e)        { return e.kind() == expr_kind::Let; }
inline bool is_binding(expr const & e)    { return is_lambda(e) || is_pi(e); }
inline bool is_mlocal(expr const & e)     { return is_metavar(e) || is_local(e); }

bool is_atomic(expr const & e);
bool is_arrow(expr const & t);
/** \brief Return true iff \c e is a metavariable or an application of a metavariable */
bool is_meta(expr const & e);
// =======================================

// =======================================
// Constructors
inline expr mk_var(unsigned idx) { return expr(new expr_var(idx)); }
inline expr Var(unsigned idx) { return mk_var(idx); }
inline expr mk_constant(name const & n, levels const & ls) { return expr(new expr_const(n, ls)); }
inline expr mk_constant(name const & n) { return mk_constant(n, levels()); }
inline expr Const(name const & n) { return mk_constant(n); }
inline expr mk_macro(macro_definition const & m, unsigned num = 0, expr const * args = nullptr) { return expr(new expr_macro(m, num, args)); }
inline expr mk_metavar(name const & n, expr const & t) { return expr(new expr_mlocal(true, n, t)); }
inline expr mk_local(name const & n, name const & pp_n, expr const & t) { return expr(new expr_local(n, pp_n, t)); }
inline expr mk_app(expr const & f, expr const & a) { return expr(new expr_app(f, a)); }
       expr mk_app(expr const & f, unsigned num_args, expr const * args);
       expr mk_app(unsigned num_args, expr const * args);
inline expr mk_app(std::initializer_list<expr> const & l) { return mk_app(l.size(), l.begin()); }
template<typename T> expr mk_app(T const & args) { return mk_app(args.size(), args.data()); }
template<typename T> expr mk_app(expr const & f, T const & args) { return mk_app(f, args.size(), args.data()); }
       expr mk_rev_app(expr const & f, unsigned num_args, expr const * args);
       expr mk_rev_app(unsigned num_args, expr const * args);
template<typename T> expr mk_rev_app(T const & args) { return mk_rev_app(args.size(), args.data()); }
template<typename T> expr mk_rev_app(expr const & f, T const & args) { return mk_rev_app(f, args.size(), args.data()); }
inline expr mk_binding(expr_kind k, name const & n, expr const & t, expr const & e, binder_info const & i = binder_info()) {
    return expr(new expr_binding(k, n, t, e, i));
}
inline expr mk_lambda(name const & n, expr const & t, expr const & e, binder_info const & i = binder_info()) {
    return mk_binding(expr_kind::Lambda, n, t, e, i);
}
inline expr mk_pi(name const & n, expr const & t, expr const & e, binder_info const & i = binder_info()) {
    return mk_binding(expr_kind::Pi, n, t, e, i);
}
inline expr mk_let(name const & n, expr const & t, expr const & v, expr const & e) { return expr(new expr_let(n, t, v, e)); }
inline expr mk_sort(level const & l) { return expr(new expr_sort(l)); }

/** \brief Return <tt>Pi(x.{sz-1}, domain[sz-1], ..., Pi(x.{0}, domain[0], range)...)</tt> */
expr mk_pi(unsigned sz, expr const * domain, expr const & range);
inline expr mk_pi(buffer<expr> const & domain, expr const & range) { return mk_pi(domain.size(), domain.data(), range); }

expr mk_Bool();
expr mk_Type();
extern expr Type;
extern expr Bool;

bool is_default_var_name(name const & n);
expr mk_arrow(expr const & t, expr const & e);
inline expr operator>>(expr const & t, expr const & e) { return mk_arrow(t, e); }

// Auxiliary
inline expr mk_app(expr const & e1, expr const & e2, expr const & e3) { return mk_app({e1, e2, e3}); }
inline expr mk_app(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({e1, e2, e3, e4}); }
inline expr mk_app(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5) {
    return mk_app({e1, e2, e3, e4, e5});
}
inline expr expr::operator()(expr const & a1) const { return mk_app({*this, a1}); }
inline expr expr::operator()(expr const & a1, expr const & a2) const { return mk_app({*this, a1, a2}); }
inline expr expr::operator()(expr const & a1, expr const & a2, expr const & a3) const { return mk_app({*this, a1, a2, a3}); }
inline expr expr::operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4) const {
    return mk_app({*this, a1, a2, a3, a4});
}
inline expr expr::operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5) const {
    return mk_app({*this, a1, a2, a3, a4, a5});
}
inline expr expr::operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5,
                             expr const & a6) const {
    return mk_app({*this, a1, a2, a3, a4, a5, a6});
}
inline expr expr::operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5,
                             expr const & a6, expr const & a7) const {
    return mk_app({*this, a1, a2, a3, a4, a5, a6, a7});
}
inline expr expr::operator()(expr const & a1, expr const & a2, expr const & a3, expr const & a4, expr const & a5,
                             expr const & a6, expr const & a7, expr const & a8) const {
    return mk_app({*this, a1, a2, a3, a4, a5, a6, a7, a8});
}

/** \brief Return application (...((f x_{n-1}) x_{n-2}) ... x_0) */
expr mk_app_vars(expr const & f, unsigned n);
// =======================================

// =======================================
// Casting (these functions are only needed for low-level code)
inline expr_var *         to_var(expr_cell * e)        { lean_assert(is_var(e));         return static_cast<expr_var*>(e); }
inline expr_const *       to_constant(expr_cell * e)   { lean_assert(is_constant(e));    return static_cast<expr_const*>(e); }
inline expr_app *         to_app(expr_cell * e)        { lean_assert(is_app(e));         return static_cast<expr_app*>(e); }
inline expr_binding *     to_binding(expr_cell * e)    { lean_assert(is_binding(e));     return static_cast<expr_binding*>(e); }
inline expr_let *         to_let(expr_cell * e)        { lean_assert(is_let(e));         return static_cast<expr_let*>(e); }
inline expr_sort *        to_sort(expr_cell * e)       { lean_assert(is_sort(e));        return static_cast<expr_sort*>(e); }
inline expr_mlocal *      to_mlocal(expr_cell * e)     { lean_assert(is_mlocal(e));      return static_cast<expr_mlocal*>(e); }
inline expr_local *       to_local(expr_cell * e)      { lean_assert(is_local(e));       return static_cast<expr_local*>(e); }
inline expr_mlocal *      to_metavar(expr_cell * e)    { lean_assert(is_metavar(e));     return static_cast<expr_mlocal*>(e); }
inline expr_macro *       to_macro(expr_cell * e)      { lean_assert(is_macro(e));       return static_cast<expr_macro*>(e); }

inline expr_var *         to_var(expr const & e)         { return to_var(e.raw()); }
inline expr_const *       to_constant(expr const & e)    { return to_constant(e.raw()); }
inline expr_app *         to_app(expr const & e)         { return to_app(e.raw()); }
inline expr_binding *     to_binding(expr const & e)     { return to_binding(e.raw()); }
inline expr_let *         to_let(expr const & e)         { return to_let(e.raw()); }
inline expr_sort *        to_sort(expr const & e)        { return to_sort(e.raw()); }
inline expr_mlocal *      to_mlocal(expr const & e)      { return to_mlocal(e.raw()); }
inline expr_mlocal *      to_metavar(expr const & e)     { return to_metavar(e.raw()); }
inline expr_local *       to_local(expr const & e)       { return to_local(e.raw()); }
inline expr_macro *       to_macro(expr const & e)       { return to_macro(e.raw()); }
// =======================================


// =======================================
// Accessors
inline unsigned       get_rc(expr_cell * e)                { return e->get_rc(); }
inline bool           is_shared(expr_cell * e)             { return get_rc(e) > 1; }
inline unsigned       var_idx(expr_cell * e)               { return to_var(e)->get_vidx(); }
inline bool           is_var(expr_cell * e, unsigned i)    { return is_var(e) && var_idx(e) == i; }
inline name const &   const_name(expr_cell * e)            { return to_constant(e)->get_name(); }
inline levels const & const_levels(expr_cell * e)          { return to_constant(e)->get_levels(); }
inline macro_definition const & macro_def(expr_cell * e)   { return to_macro(e)->get_def(); }
inline expr const *   macro_args(expr_cell * e)            { return to_macro(e)->get_args(); }
inline expr const &   macro_arg(expr_cell * e, unsigned i) { return to_macro(e)->get_arg(i); }
inline unsigned       macro_num_args(expr_cell * e)        { return to_macro(e)->get_num_args(); }
inline expr const &   app_fn(expr_cell * e)                { return to_app(e)->get_fn(); }
inline expr const &   app_arg(expr_cell * e)               { return to_app(e)->get_arg(); }
inline name const &   binding_name(expr_cell * e)          { return to_binding(e)->get_name(); }
inline expr const &   binding_domain(expr_cell * e)        { return to_binding(e)->get_domain(); }
inline expr const &   binding_body(expr_cell * e)          { return to_binding(e)->get_body(); }
inline binder_info const & binding_info(expr_cell * e)     { return to_binding(e)->get_info(); }
inline binder const & binding_binder(expr_cell * e)        { return to_binding(e)->get_binder(); }
inline level const &  sort_level(expr_cell * e)            { return to_sort(e)->get_level(); }
inline name const &   let_name(expr_cell * e)              { return to_let(e)->get_name(); }
inline expr const &   let_value(expr_cell * e)             { return to_let(e)->get_value(); }
inline expr const &   let_type(expr_cell * e)              { return to_let(e)->get_type(); }
inline expr const &   let_body(expr_cell * e)              { return to_let(e)->get_body(); }
inline name const &   mlocal_name(expr_cell * e)           { return to_mlocal(e)->get_name(); }
inline expr const &   mlocal_type(expr_cell * e)           { return to_mlocal(e)->get_type(); }

inline unsigned       get_rc(expr const & e)                { return e.raw()->get_rc(); }
inline bool           is_shared(expr const & e)             { return get_rc(e) > 1; }
inline unsigned       var_idx(expr const & e)               { return to_var(e)->get_vidx(); }
inline bool           is_var(expr const & e, unsigned i)    { return is_var(e) && var_idx(e) == i; }
inline name const &   const_name(expr const & e)            { return to_constant(e)->get_name(); }
inline levels const & const_levels(expr const & e)          { return to_constant(e)->get_levels(); }
inline macro_definition const & macro_def(expr const & e)   { return to_macro(e)->get_def(); }
inline expr const *   macro_args(expr const & e)            { return to_macro(e)->get_args(); }
inline expr const &   macro_arg(expr const & e, unsigned i) { return to_macro(e)->get_arg(i); }
inline unsigned       macro_num_args(expr const & e)        { return to_macro(e)->get_num_args(); }
inline expr const &   app_fn(expr const & e)                { return to_app(e)->get_fn(); }
inline expr const &   app_arg(expr const & e)               { return to_app(e)->get_arg(); }
inline name const &   binding_name(expr const & e)          { return to_binding(e)->get_name(); }
inline expr const &   binding_domain(expr const & e)        { return to_binding(e)->get_domain(); }
inline expr const &   binding_body(expr const & e)          { return to_binding(e)->get_body(); }
inline binder_info const & binding_info(expr const & e)     { return to_binding(e)->get_info(); }
inline binder const & binding_binder(expr const & e)        { return to_binding(e)->get_binder(); }
inline level const &  sort_level(expr const & e)            { return to_sort(e)->get_level(); }
inline name const &   let_name(expr const & e)              { return to_let(e)->get_name(); }
inline expr const &   let_value(expr const & e)             { return to_let(e)->get_value(); }
inline expr const &   let_type(expr const & e)              { return to_let(e)->get_type(); }
inline expr const &   let_body(expr const & e)              { return to_let(e)->get_body(); }
inline name const &   mlocal_name(expr const & e)           { return to_mlocal(e)->get_name(); }
inline expr const &   mlocal_type(expr const & e)           { return to_mlocal(e)->get_type(); }
inline name const &   local_pp_name(expr const & e)         { return to_local(e)->get_pp_name(); }

inline bool is_constant(expr const & e, name const & n) { return is_constant(e) && const_name(e) == n; }
inline bool has_metavar(expr const & e) { return e.has_metavar(); }
inline bool has_local(expr const & e) { return e.has_local(); }
inline bool has_param_univ(expr const & e) { return e.has_param_univ(); }
unsigned get_depth(expr const & e);
/**
   \brief Return \c R s.t. the de Bruijn index of all free variables
   occurring in \c e is in the interval <tt>[0, R)</tt>.
*/
unsigned get_free_var_range(expr const & e);
/** \brief Return true iff the given expression has free variables. */
inline bool has_free_vars(expr const & e) { return get_free_var_range(e) > 0; }
/** \brief Return true iff the given expression does not have free variables. */
inline bool closed(expr const & e) { return !has_free_vars(e); }
/** \brief Return true iff \c e contains a free variable >= low. */
inline bool has_free_var_ge(expr const & e, unsigned low) { return get_free_var_range(e) > low; }
/**
    \brief Given \c e of the form <tt>(...(f a1) ... an)</tt>, store a1 ... an in args.
    If \c e is not an application, then nothing is stored in args.

    It returns the f.
*/
expr const & get_app_args(expr const & e, buffer<expr> & args);
/**
   \brief Similar to \c get_app_args, but arguments are stored in reverse order in \c args.
   If e is of the form <tt>(...(f a1) ... an)</tt>, then the procedure stores [an, ..., a1] in \c args.
*/
expr const & get_app_rev_args(expr const & e, buffer<expr> & args);
/**
   \brief Given of the form <tt>(...(f a1) ... an)</tt>, return \c f. If \c e is not an application,
   then return \c e.
*/
expr const & get_app_fn(expr const & e);
// =======================================


// =======================================
// Expression+Offset
typedef std::pair<expr, unsigned>       expr_offset;
typedef std::pair<expr_cell*, unsigned> expr_cell_offset;
// =======================================

// =======================================
// Auxiliary functionals
/** \brief Functional object for hashing kernel expressions. */
struct expr_hash { unsigned operator()(expr const & e) const { return e.hash(); } };
/**
    \brief Functional object for hashing (based on allocation time) kernel expressions.

    This hash is compatible with pointer equality.
    \warning This hash is incompatible with structural equality (i.e., std::equal_to<expr>)
*/
struct expr_hash_alloc { unsigned operator()(expr const & e) const { return e.hash_alloc(); } };
/** \brief Functional object for testing pointer equality between kernel expressions. */
struct expr_eqp { bool operator()(expr const & e1, expr const & e2) const { return is_eqp(e1, e2); } };
/** \brief Functional object for hashing kernel expression cells. */
struct expr_cell_hash { unsigned operator()(expr_cell * e) const { return e->hash_alloc(); } };
/** \brief Functional object for testing pointer equality between kernel cell expressions. */
struct expr_cell_eqp { bool operator()(expr_cell * e1, expr_cell * e2) const { return e1 == e2; } };
/** \brief Functional object for hashing a pair (n, k) where n is a kernel expressions, and k is an offset. */
struct expr_offset_hash { unsigned operator()(expr_offset const & p) const { return hash(p.first.hash_alloc(), p.second); } };
/** \brief Functional object for comparing pairs (expression, offset). */
struct expr_offset_eqp { unsigned operator()(expr_offset const & p1, expr_offset const & p2) const { return is_eqp(p1.first, p2.first) && p1.second == p2.second; } };
/** \brief Functional object for hashing a pair (n, k) where n is a kernel cell expressions, and k is an offset. */
struct expr_cell_offset_hash { unsigned operator()(expr_cell_offset const & p) const { return hash(p.first->hash_alloc(), p.second); } };
/** \brief Functional object for comparing pairs (expression cell, offset). */
struct expr_cell_offset_eqp { unsigned operator()(expr_cell_offset const & p1, expr_cell_offset const & p2) const { return p1 == p2; } };
// =======================================

// =======================================
// Update
expr update_app(expr const & e, expr const & new_fn, expr const & new_arg);
expr update_rev_app(expr const & e, unsigned num, expr const * new_args);
template<typename C> expr update_rev_app(expr const & e, C const & c) { return update_rev_app(e, c.size(), c.data()); }
expr update_binding(expr const & e, expr const & new_domain, expr const & new_body);
expr update_let(expr const & e, expr const & new_type, expr const & new_val, expr const & new_body);
expr update_mlocal(expr const & e, expr const & new_type);
expr update_sort(expr const & e, level const & new_level);
expr update_constant(expr const & e, levels const & new_levels);
expr update_macro(expr const & e, unsigned num, expr const * args);
// =======================================

std::ostream & operator<<(std::ostream & out, expr const & e);
}