def.h

Go to the documentation of this file.

#pragma once
 
#include <optional>
 
#include <fe/assert.h>
#include <fe/cast.h>
#include <fe/enum.h>
 
#include "mim/config.h"
 
#include "mim/util/dbg.h"
#include "mim/util/hash.h"
#include "mim/util/sets.h"
#include "mim/util/util.h"
#include "mim/util/vector.h"
 
// clang-format off
#define MIM_NODE(m)                                                                                                                             \
    m(Type,   type,   Judge::Meta) m(Univ,  univ,  Judge::Meta)  m(UMax,    umax,    Judge::Meta) m(UInc, uinc, Judge::Meta)                    \
    m(Pi,     pi,     Judge::Form) m(Lam,   lam,   Judge::Intro) m(App,     app,     Judge::Elim)                                               \
    m(Sigma,  sigma,  Judge::Form) m(Tuple, tuple, Judge::Intro) m(Extract, extract, Judge::Elim) m(Insert, insert, Judge::Intro | Judge::Elim) \
    m(Arr,    arr,    Judge::Form) m(Pack,  pack,  Judge::Intro)                                                                                \
    m(Join,   join,   Judge::Form) m(Vel,   vel,   Judge::Intro) m(Test, test,       Judge::Elim) m(Top,    top,    Judge::Intro)               \
    m(Meet,   meet,   Judge::Form) m(Ac,    ac,    Judge::Intro) m(Pick, pick,       Judge::Elim) m(Bot,    bot,    Judge::Intro)               \
    m(Uniq,   Uniq,   Judge::Form)                                                                                                              \
    m(Nat,    nat,    Judge::Form)                                                                                                              \
    m(Idx,    idx,    Judge::Intro) m(Lit,  lit,  Judge::Intro)                                                                                 \
    m(Axiom,  axiom,  Judge::Intro)                                                                                                             \
    m(Var,    var,    Judge::Intro)                                                                                                             \
    m(Global, global, Judge::Intro)                                                                                                             \
    m(Proxy,  proxy,  Judge::Intro)                                                                                                             \
    m(Hole,   hole,   Judge::Hole)
// clang-format on
 
namespace mim {
 
class App;
class Axiom;
class Var;
class Def;
class World;
 
/// @name Def
/// GIDSet / GIDMap keyed by Def::gid of `conset Def*`.
///@{
template<class To> using DefMap = GIDMap<const Def*, To>;
using DefSet                    = GIDSet<const Def*>;
using Def2Def                   = DefMap<const Def*>;
using Defs                      = View<const Def*>;
using DefVec                    = Vector<const Def*>;
///@}
 
/// @name Def (Mutable)
/// GIDSet / GIDMap keyed by Def::gid of `Def*`.
///@{
template<class To> using MutMap = GIDMap<Def*, To>;
using MutSet                    = GIDSet<Def*>;
using Mut2Mut                   = MutMap<Def*>;
using Muts                      = Sets<Def>::Set;
///@}
 
/// @name Var
/// GIDSet / GIDMap keyed by Var::gid of `const Var*`.
///@{
template<class To> using VarMap = GIDMap<const Var*, To>;
using VarSet                    = GIDSet<const Var*>;
using Var2Var                   = VarMap<const Var*>;
using Vars                      = Sets<const Var>::Set;
///@}
 
using NormalizeFn = const Def* (*)(const Def*, const Def*, const Def*);
 
using fe::operator&;
using fe::operator|;
using fe::operator^;
using fe::operator<=>;
using fe::operator==;
using fe::operator!=;
 
/// @name Enums that classify certain aspects of Def%s.
///@{
 
enum class Node : node_t {
#define CODE(node, name, _) node,
    MIM_NODE(CODE)
#undef CODE
};
 
#define CODE(node, name, _) +size_t(1)
static constexpr size_t Num_Nodes = size_t(0) MIM_NODE(CODE);
#undef CODE
 
/// TODO remove or fix this
enum class Sort { Term, Type, Kind, Space, Univ, Level };
 
/// Tracks a dependency to certain Def%s transitively through the Def::deps() up to but excliding *mutables*.
enum class Dep : unsigned {
    None  = 0,
    Mut   = 1 << 0,
    Var   = 1 << 1,
    Hole  = 1 << 2,
    Proxy = 1 << 3,
};
 
/// [Judgement](https://ncatlab.org/nlab/show/judgment).
enum class Judge : u32 {
    // clang-format off
    Form  = 1 << 0, ///< [Type Formation](https://ncatlab.org/nlab/show/type+formation) like `T -> T`.
    Intro = 1 << 1, ///< [Term Introduction](https://ncatlab.org/nlab/show/natural+deduction) like `λ(x: Nat): Nat = x`.
    Elim  = 1 << 2, ///< [Term Elimination](https://ncatlab.org/nlab/show/term+elimination) like `f a`.
    Meta  = 1 << 3, ///< Meta rules for Univ%erse and Type levels.
    Hole  = 1 << 4, ///< Special rule for Hole.
    // clang-format on
};
///@}
 
} // namespace mim
 
#ifndef DOXYGEN
template<> struct fe::is_bit_enum<mim::Dep> : std::true_type {};
template<> struct fe::is_bit_enum<mim::Judge> : std::true_type {};
#endif
 
namespace mim {
 
/// Use as mixin to wrap all kind of Def::proj and Def::projs variants.
#define MIM_PROJ(NAME, CONST)                                                                                          \
    nat_t num_##NAME##s() CONST noexcept { return ((const Def*)NAME())->num_projs(); }                                 \
    nat_t num_t##NAME##s() CONST noexcept { return ((const Def*)NAME())->num_tprojs(); }                               \
    const Def* NAME(nat_t a, nat_t i) CONST noexcept { return ((const Def*)NAME())->proj(a, i); }                      \
    const Def* NAME(nat_t i) CONST noexcept { return ((const Def*)NAME())->proj(i); }                                  \
    const Def* t##NAME(nat_t i) CONST noexcept { return ((const Def*)NAME())->tproj(i); }                              \
    template<nat_t A = std::dynamic_extent, class F> auto NAME##s(F f) CONST noexcept {                                \
        return ((const Def*)NAME())->projs<A, F>(f);                                                                   \
    }                                                                                                                  \
    template<class F> auto t##NAME##s(F f) CONST noexcept { return ((const Def*)NAME())->tprojs<F>(f); }               \
    template<nat_t A = std::dynamic_extent> auto NAME##s() CONST noexcept { return ((const Def*)NAME())->projs<A>(); } \
    auto t##NAME##s() CONST noexcept { return ((const Def*)NAME())->tprojs(); }                                        \
    template<class F> auto NAME##s(nat_t a, F f) CONST noexcept { return ((const Def*)NAME())->projs<F>(a, f); }       \
    auto NAME##s(nat_t a) CONST noexcept { return ((const Def*)NAME())->projs(a); }
 
/// CRTP-based Mixin to declare setters for Def::loc \& Def::name using a *covariant* return type.
template<class P, class D = Def> class Setters { // D is only needed to make the resolution `D::template set` lazy
private:
    P* super() { return static_cast<P*>(this); }
    const P* super() const { return static_cast<const P*>(this); }
 
public:
    // clang-format off
    template<bool Ow = false> const P* set(Loc l               ) const { super()->D::template set<Ow>(l); return super(); }
    template<bool Ow = false>       P* set(Loc l               )       { super()->D::template set<Ow>(l); return super(); }
    template<bool Ow = false> const P* set(       Sym s        ) const { super()->D::template set<Ow>(s); return super(); }
    template<bool Ow = false>       P* set(       Sym s        )       { super()->D::template set<Ow>(s); return super(); }
    template<bool Ow = false> const P* set(       std::string s) const { super()->D::template set<Ow>(std::move(s)); return super(); }
    template<bool Ow = false>       P* set(       std::string s)       { super()->D::template set<Ow>(std::move(s)); return super(); }
    template<bool Ow = false> const P* set(Loc l, Sym s        ) const { super()->D::template set<Ow>(l, s); return super(); }
    template<bool Ow = false>       P* set(Loc l, Sym s        )       { super()->D::template set<Ow>(l, s); return super(); }
    template<bool Ow = false> const P* set(Loc l, std::string s) const { super()->D::template set<Ow>(l, std::move(s)); return super(); }
    template<bool Ow = false>       P* set(Loc l, std::string s)       { super()->D::template set<Ow>(l, std::move(s)); return super(); }
    template<bool Ow = false> const P* set(Dbg d               ) const { super()->D::template set<Ow>(d); return super(); }
    template<bool Ow = false>       P* set(Dbg d               )       { super()->D::template set<Ow>(d); return super(); }
    // clang-format on
};
 
/// Base class for all Def%s.
///
/// These are the most important subclasses:
/// | Type Formation    | Term Introduction | Term Elimination  |
/// | ----------------- | ----------------- | ----------------- |
/// | Pi                | Lam               | App               |
/// | Sigma / Arr       | Tuple / Pack      | Extract           |
/// |                   | Insert            | Insert            |
/// | Uniq              | Wrap              | Unwrap            |
/// | Join              | Vel               | Test              |
/// | Meet              | Ac                | Pick              |
/// | Nat               | Lit               |                   |
/// | Idx               | Lit               |                   |
/// In addition there is:
/// * Var: A variable. Currently the following Def%s may be binders:
///     * Pi, Lam, Sigma, Arr, Pack
/// * Axiom: To introduce new entities.
/// * Proxy: Used for intermediate values during optimizations.
/// * Hole: Hole in the presentation filled by type inference (always mutable as the holes are filled in later).
/// * Type, Univ, UMax, UInc: To keep track of type levels.
///
/// The data layout (see World::alloc and Def::deps) looks like this:
/// ```
/// Def| type | op(0) ... op(num_ops-1) |
///           |-----------ops-----------|
///                  deps
///    |--------------------------------| if type() != nullptr &&  is_set()
///           |-------------------------| if type() == nullptr &&  is_set()
///    |------|                           if type() != nullptr && !is_set()
///    ||                                 if type() == nullptr && !is_set()
/// ```
/// @attention This means that any subclass of Def **must not** introduce additional members.
/// @see @ref mut
class Def : public fe::RuntimeCast<Def> {
private:
    Def& operator=(const Def&) = delete;
    Def(const Def&)            = delete;
 
protected:
    /// @name C'tors and D'tors
    ///@{
    Def(World*, Node, const Def* type, Defs ops, flags_t flags); ///< Constructor for an *immutable* Def.
    Def(Node, const Def* type, Defs ops, flags_t flags);         ///< As above but World retrieved from @p type.
    Def(Node, const Def* type, size_t num_ops, flags_t flags);   ///< Constructor for a *mutable* Def.
    virtual ~Def() = default;
    ///@}
 
public:
    /// @name Getters
    ///@{
    World& world() const noexcept;
    constexpr flags_t flags() const noexcept { return flags_; }
    constexpr u32 gid() const noexcept { return gid_; }   ///< Global id - *unique* number for this Def.
    constexpr u32 tid() const noexcept { return tid_; }   ///< Trie id - only used in Trie.
    constexpr u32 mark() const noexcept { return mark_; } ///< Used internally by free_vars().
    constexpr size_t hash() const noexcept { return hash_; }
    constexpr Node node() const noexcept { return node_; }
    std::string_view node_name() const;
    ///@}
 
    /// @name Judgement
    /// What kind of Judge%ment represents this Def?
    ///@{
    u32 judge() const noexcept;
    // clang-format off
    bool is_form()  const noexcept { return judge() & Judge::Form;  }
    bool is_intro() const noexcept { return judge() & Judge::Intro; }
    bool is_elim()  const noexcept { return judge() & Judge::Elim;  }
    bool is_meta()  const noexcept { return judge() & Judge::Meta;  }
    // clang-format on
    ///@}
 
    /// @name type
    ///@{
 
    /// Yields the "raw" type of this Def (maybe `nullptr`).
    /// @see Def::unfold_type.
    const Def* type() const noexcept { return type_; }
    /// Yields the type of this Def and builds a new `Type (UInc n)` if necessary.
    const Def* unfold_type() const;
    bool is_term() const;
    ///@}
 
    /// @name arity
    ///@{
    const Def* arity() const;
    std::optional<nat_t> isa_lit_arity() const;
    nat_t as_lit_arity() const {
        auto a = isa_lit_arity();
        assert(a.has_value());
        return *a;
    }
    ///@}
 
    /// @name ops
    ///@{
    template<size_t N = std::dynamic_extent> constexpr auto ops() const noexcept {
        return View<const Def*, N>(ops_ptr(), num_ops_);
    }
    const Def* op(size_t i) const noexcept { return ops()[i]; }
    constexpr size_t num_ops() const noexcept { return num_ops_; }
    ///@}
 
    /// @name Setting Ops (Mutables Only)
    /// @anchor set_ops
    /// You can set and change the Def::ops of a mutable after construction.
    /// However, you have to obey the following rules:
    /// 1. If Def::is_set() is ...
    ///     1. ... `false`, [set](@ref Def::set) the [operands](@ref Def::ops) from
    ///         * left (`i == 0`) to
    ///         * right (`i == num_ops() - 1`).
    ///     2. ... `true`, [reset](@ref Def::reset) the operands from left to right as in 1a.
    /// 2. In addition, you can invoke Def::unset() at *any time* to start over with 1a:
    /// ```
    /// mut->unset()->set({a, b, c}); // This will always work, but should be your last resort.
    /// ```
    ///
    /// MimIR assumes that a mutable is *final*, when its last operand is set.
    /// Then, Def::check() will be invoked.
    ///@{
    Def* set(size_t i, const Def*);                                        ///< Successively   set from left to right.
    Def* reset(size_t i, const Def* def) { return unset(i)->set(i, def); } ///< Successively reset from left to right.
    Def* set(Defs ops);                                                    ///< Def::set @p ops all at once.
    Def* reset(Defs ops);                                                  ///< Def::reset @p ops all at once.
    Def* unset();        ///< Unsets all Def::ops; works even, if not set at all or partially.
    bool is_set() const; ///< Yields `true` if empty or the last op is set.
    ///@}
 
    /// @name deps
    /// All *dependencies* of a Def and includes:
    /// * Def::type() (if not `nullptr`) and
    /// * the other Def::ops() (only included, if Def::is_set()) in this order.
    ///@{
    Defs deps() const noexcept;
    const Def* dep(size_t i) const noexcept { return deps()[i]; }
    size_t num_deps() const noexcept { return deps().size(); }
    ///@}
 
    /// @name has_dep
    /// Checks whether one Def::deps() contains specific elements defined in Dep.
    /// This works up to the next *mutable*.
    /// For example, consider the Tuple `tup`: `(?, lam (x: Nat) = y)`:
    /// ```
    /// bool has_hole = tup->has_dep(Dep::Hole); // true;
    /// bool has_mut   = tup->has_dep(Dep::Mut); // true;
    /// bool has_var   = tup->has_dep(Dep::Var); // false - y is contained in another mutable;
    /// ```
    ///@{
    bool has_dep() const { return dep_ != 0; }
    bool has_dep(Dep d) const { return has_dep(unsigned(d)); }
    bool has_dep(unsigned u) const { return dep_ & u; }
    ///@}
 
    /// @name proj
    /// @anchor proj
    /// Splits this Def via Extract%s or directly accessing the Def::ops in the case of Sigma%s or Arr%ays.
    /// ```
    /// std::array<const Def*, 2> ab = def->projs<2>();
    /// std::array<u64, 2>        xy = def->projs<2>([](auto def) { return Lit::as(def); });
    /// auto [a, b]                  = def->projs<2>();
    /// auto [x, y]                  = def->projs<2>([](auto def) { return Lit::as(def); });
    /// Vector<const Def*> projs1    = def->projs(); // "projs1" has def->num_projs() many elements
    /// Vector<const Def*> projs2    = def->projs(n);// "projs2" has n elements - asserts if incorrect
    /// // same as above but applies Lit::as<nat_t>(def) to each element
    /// Vector<const Lit*> lits1     = def->projs(   [](auto def) { return Lit::as(def); });
    /// Vector<const Lit*> lits2     = def->projs(n, [](auto def) { return Lit::as(def); });
    /// ```
    ///@{
 
    /// Yields Def::as_lit_arity(), if it is in fact a Lit, or `1` otherwise.
    nat_t num_projs() const { return isa_lit_arity().value_or(1); }
    nat_t num_tprojs() const; ///< As above but yields 1, if Flags::scalarize_threshold is exceeded.
 
    /// Similar to World::extract while assuming an arity of @p a, but also works on Sigma%s and Arr%ays.
    const Def* proj(nat_t a, nat_t i) const;
    const Def* proj(nat_t i) const { return proj(num_projs(), i); }   ///< As above but takes Def::num_projs as arity.
    const Def* tproj(nat_t i) const { return proj(num_tprojs(), i); } ///< As above but takes Def::num_tprojs.
 
    /// Splits this Def via Def::proj%ections into an Array (if `A == std::dynamic_extent`) or `std::array` (otherwise).
    /// Applies @p f to each element.
    template<nat_t A = std::dynamic_extent, class F> auto projs(F f) const {
        using R = std::decay_t<decltype(f(this))>;
        if constexpr (A == std::dynamic_extent) {
            return projs(num_projs(), f);
        } else {
            assert(A == as_lit_arity());
            std::array<R, A> array;
            for (nat_t i = 0; i != A; ++i) array[i] = f(proj(A, i));
            return array;
        }
    }
 
    template<class F> auto tprojs(F f) const { return projs(num_tprojs(), f); }
 
    template<class F> auto projs(nat_t a, F f) const {
        using R = std::decay_t<decltype(f(this))>;
        return Vector<R>(a, [&](nat_t i) { return f(proj(a, i)); });
    }
    template<nat_t A = std::dynamic_extent> auto projs() const {
        return projs<A>([](const Def* def) { return def; });
    }
    auto tprojs() const {
        return tprojs([](const Def* def) { return def; });
    }
    auto projs(nat_t a) const {
        return projs(a, [](const Def* def) { return def; });
    }
    ///@}
 
    /// @name var
    /// @anchor var
    /// Retrieve Var for *mutables*.
    /// @see @ref proj
    ///@{
    MIM_PROJ(var, )
    /// Not necessarily a Var: E.g., if the return type is `[]`, this will yield `()`.
    const Def* var();
    /// Only returns not `nullptr`, if Var of this mutable has ever been created.
    const Var* has_var() { return var_; }
    /// As above if `this` is a *mutable*.
    const Var* has_var() const {
        if (auto mut = isa_mut()) return mut->has_var();
        return nullptr;
    }
    ///@}
 
    /// @name Free Vars and Muts
    /// * local_muts() / local_vars() are cached and hash-consed.
    /// * free_vars() are computed on demand and cached in mutables.
    ///   They will be transitively invalidated by following users(), if a mutable is mutated.
    ///@{
 
    /// Mutables reachable by following *immutable* deps(); `mut->local_muts()` is by definition the set `{ mut }`.
    Muts local_muts() const;
 
    /// Var%s reachable by following *immutable* deps().
    /// @note `var->local_vars()` is by definition the set `{ var }`.
    Vars local_vars() const;
 
    /// Compute a global solution by transitively following *mutables* as well.
    Vars free_vars() const;
    Vars free_vars();
    Muts users() { return muts_; } ///< Set of mutables where this mutable is locally referenced.
    bool is_open() const;          ///< Has free_vars()?
    bool is_closed() const;        ///< Has no free_vars()?
    ///@}
 
    /// @name external
    ///@{
    bool is_external() const { return external_; }
    void make_external();
    void make_internal();
    void transfer_external(Def* to) { make_internal(), to->make_external(); }
    ///@}
 
    /// @name Casts
    /// @see @ref cast_builtin
    ///@{
    // clang-format off
    template<class T = Def> const T* isa_imm() const { return isa_mut<T, true>(); }
    template<class T = Def> const T*  as_imm() const { return  as_mut<T, true>(); }
    template<class T = Def, class R> const T* isa_imm(R (T::*f)() const) const { return isa_mut<T, R, true>(f); }
    // clang-format on
 
    /// If `this` is *mut*able, it will cast `const`ness away and perform a `dynamic_cast` to @p T.
    template<class T = Def, bool invert = false> T* isa_mut() const {
        if constexpr (std::is_same<T, Def>::value)
            return mut_ ^ invert ? const_cast<Def*>(this) : nullptr;
        else
            return mut_ ^ invert ? const_cast<Def*>(this)->template isa<T>() : nullptr;
    }
 
    /// Asserts that `this` is a *mutable*, casts `const`ness away and performs a `static_cast` to @p T.
    template<class T = Def, bool invert = false> T* as_mut() const {
        assert(mut_ ^ invert);
        if constexpr (std::is_same<T, Def>::value)
            return const_cast<Def*>(this);
        else
            return const_cast<Def*>(this)->template as<T>();
    }
    ///@}
 
    /// @name Dbg Getters
    ///@{
    Dbg dbg() const { return dbg_; }
    Loc loc() const { return dbg_.loc(); }
    Sym sym() const { return dbg_.sym(); }
    std::string unique_name() const; ///< name + "_" + Def::gid
    ///@}
 
    /// @name Dbg Setters
    /// Every subclass `S` of Def has the same setters that return `S*`/`const S*` via the mixin Setters.
    ///@{
    // clang-format off
    template<bool Ow = false> const Def* set(Loc l) const { if (Ow || !dbg_.loc()) dbg_.set(l); return this; }
    template<bool Ow = false>       Def* set(Loc l)       { if (Ow || !dbg_.loc()) dbg_.set(l); return this; }
    template<bool Ow = false> const Def* set(Sym s) const { if (Ow || !dbg_.sym()) dbg_.set(s); return this; }
    template<bool Ow = false>       Def* set(Sym s)       { if (Ow || !dbg_.sym()) dbg_.set(s); return this; }
    template<bool Ow = false> const Def* set(       std::string s) const { set(sym(std::move(s))); return this; }
    template<bool Ow = false>       Def* set(       std::string s)       { set(sym(std::move(s))); return this; }
    template<bool Ow = false> const Def* set(Loc l, Sym s        ) const { set(l); set(s); return this; }
    template<bool Ow = false>       Def* set(Loc l, Sym s        )       { set(l); set(s); return this; }
    template<bool Ow = false> const Def* set(Loc l, std::string s) const { set(l); set(sym(std::move(s))); return this; }
    template<bool Ow = false>       Def* set(Loc l, std::string s)       { set(l); set(sym(std::move(s))); return this; }
    template<bool Ow = false> const Def* set(Dbg d) const { set(d.loc(), d.sym()); return this; }
    template<bool Ow = false>       Def* set(Dbg d)       { set(d.loc(), d.sym()); return this; }
    // clang-format on
    ///@}
 
    /// @name debug_prefix/suffix
    /// Prepends/Appends a prefix/suffix to Def::name - but only in `Debug` build.
    ///@{
#ifndef NDEBUG
    const Def* debug_prefix(std::string) const;
    const Def* debug_suffix(std::string) const;
#else
    const Def* debug_prefix(std::string) const { return this; }
    const Def* debug_suffix(std::string) const { return this; }
#endif
    ///@}
 
    /// @name Rebuild
    ///@{
    Def* stub(World& w, const Def* type) { return stub_(w, type)->set(dbg()); }
    Def* stub(const Def* type) { return stub(world(), type); }
 
    /// Def::rebuild%s this Def while using @p new_op as substitute for its @p i'th Def::op
    const Def* rebuild(World& w, const Def* type, Defs ops) const {
        assert(isa_imm());
        return rebuild_(w, type, ops)->set(dbg());
    }
    const Def* rebuild(const Def* type, Defs ops) const { return rebuild(world(), type, ops); }
 
    /// Tries to make an immutable from a mutable.
    /// This usually works if the mutable isn't recursive and its var isn't used.
    virtual const Def* immutabilize() { return nullptr; }
    bool is_immutabilizable();
 
    const Def* refine(size_t i, const Def* new_op) const;
 
    /// Rewrites Def::ops by substituting `this` mutable's Var with @p arg.
    DefVec reduce(const Def* arg) const;
    DefVec reduce(const Def* arg);
    /// As above but only rewrites `this->op(i)`.
    const Def* reduce(size_t i, const Def* arg) const;
    ///@}
 
    /// @name Type Checking
    ///@{
    virtual const Def* check(size_t, const Def* def) { return def; }
    virtual const Def* check() { return type(); }
    const Def* zonk() const;
    ///@}
 
    /// @name dump
    ///@{
    void dump() const;
    void dump(int max) const;
    void write(int max) const;
    void write(int max, const char* file) const;
    std::ostream& stream(std::ostream&, int max) const;
    ///@}
 
    /// @name dot
    /// Dumps DOT to @p os while obeying maximum recursion depth of @p max.
    /// If @p types is `true`, Def::type() dependencies will be followed as well.
    ///@{
    void dot(std::ostream& os, uint32_t max = 0xFFFFFF, bool types = false) const;
    /// Same as above but write to @p file or `std::cout` if @p file is `nullptr`.
    void dot(const char* file = nullptr, uint32_t max = 0xFFFFFF, bool types = false) const;
    void dot(const std::string& file, uint32_t max = 0xFFFFFF, bool types = false) const {
        return dot(file.c_str(), max, types);
    }
    ///@}
 
protected:
    /// @name Wrappers for World::sym
    /// These are here to have Def::set%ters inline without including `mim/world.h`.
    ///@{
    Sym sym(const char*) const;
    Sym sym(std::string_view) const;
    Sym sym(std::string) const;
    ///@}
 
private:
    virtual Def* stub_(World&, const Def*) { fe::unreachable(); }
    virtual const Def* rebuild_(World& w, const Def* type, Defs ops) const = 0;
 
    Vars free_vars(bool&, bool&, uint32_t run);
    void invalidate();
    Def* unset(size_t i);
    const Def** ops_ptr() const {
        return reinterpret_cast<const Def**>(reinterpret_cast<char*>(const_cast<Def*>(this + 1)));
    }
    bool equal(const Def* other) const;
 
protected:
    mutable Dbg dbg_;
    union {
        NormalizeFn normalizer_; ///< Axiom only: Axiom%s use this member to store their normalizer.
        const Axiom* axiom_;     ///< App only: Curried App%s of Axiom%s use this member to propagate the Axiom.
        const Var* var_;         ///< Mutable only: Var of a mutable.
        mutable World* world_;
    };
    flags_t flags_;
    u8 curry_ = 0;
    u8 trip_  = 0;
 
private:
    Node node_;
    bool mut_      : 1;
    bool external_ : 1;
    unsigned dep_  : 6;
    u32 mark_ = 0;
#ifndef NDEBUG
    size_t curr_op_ = 0;
#endif
    u32 gid_;
    u32 num_ops_;
    size_t hash_;
    Vars vars_; // Mutable: local vars; Immutable: free vars.
    Muts muts_; // Immutable: local_muts; Mutable: users;
    mutable u32 tid_ = 0;
    const Def* type_;
 
    template<class D, size_t N> friend class Sets;
    friend class World;
    friend void swap(World&, World&) noexcept;
    friend std::ostream& operator<<(std::ostream&, const Def*);
};
 
/// @name DefDef
///@{
using DefDef = std::tuple<const Def*, const Def*>;
 
struct DefDefHash {
    size_t operator()(DefDef pair) const {
        if constexpr (sizeof(size_t) == 8)
            return hash((u64(std::get<0>(pair)->gid()) << 32_u64) | u64(std::get<1>(pair)->gid()));
        else
            return hash_combine(hash_begin(std::get<0>(pair)->gid()), std::get<1>(pair)->gid());
    }
};
 
struct DefDefEq {
    bool operator()(DefDef p1, DefDef p2) const { return p1 == p2; }
};
 
template<class To> using DefDefMap = absl::flat_hash_map<DefDef, To, DefDefHash, DefDefEq>;
using DefDefSet                    = absl::flat_hash_set<DefDef, DefDefHash, DefDefEq>;
///@}
 
class Var : public Def, public Setters<Var> {
private:
    Var(const Def* type, Def* mut)
        : Def(Node, type, Defs{mut}, 0) {}
 
public:
    using Setters<Var>::set;
 
    /// @name ops
    ///@{
    Def* mut() const { return op(0)->as_mut(); }
    ///@}
 
    static constexpr auto Node = mim::Node::Var;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
class Univ : public Def, public Setters<Univ> {
public:
    using Setters<Univ>::set;
    static constexpr auto Node = mim::Node::Univ;
 
private:
    Univ(World& world)
        : Def(&world, Node, nullptr, Defs{}, 0) {}
 
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
class UMax : public Def, public Setters<UMax> {
public:
    using Setters<UMax>::set;
    static constexpr auto Node = mim::Node::UMax;
 
private:
    UMax(World&, Defs ops);
 
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
class UInc : public Def, public Setters<UInc> {
private:
    UInc(const Def* op, level_t offset)
        : Def(Node, op->type()->as<Univ>(), {op}, offset) {}
 
public:
    using Setters<UInc>::set;
 
    /// @name ops
    ///@{
    const Def* op() const { return Def::op(0); }
    level_t offset() const { return flags(); }
    ///@}
 
    static constexpr auto Node = mim::Node::UInc;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
class Type : public Def, public Setters<Type> {
private:
    Type(const Def* level)
        : Def(Node, nullptr, {level}, 0) {}
 
public:
    using Setters<Type>::set;
 
    /// @name ops
    ///@{
    const Def* level() const { return op(0); }
    ///@}
 
    static constexpr auto Node = mim::Node::Type;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
class Lit : public Def, public Setters<Lit> {
private:
    Lit(const Def* type, flags_t val)
        : Def(Node, type, Defs{}, val) {}
 
public:
    using Setters<Lit>::set;
 
    /// @name Get actual Constant
    ///@{
    template<class T = flags_t> T get() const {
        static_assert(sizeof(T) <= 8);
        return bitcast<T>(flags_);
    }
    ///@}
 
    using Def::as;
    using Def::isa;
 
    /// @name Casts
    ///@{
    /// @see @ref cast_lit
    template<class T = nat_t> static std::optional<T> isa(const Def* def) {
        if (!def) return {};
        if (auto lit = def->isa<Lit>()) return lit->get<T>();
        return {};
    }
    template<class T = nat_t> static T as(const Def* def) { return def->as<Lit>()->get<T>(); }
    ///@}
 
    static constexpr auto Node = mim::Node::Lit;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
class Nat : public Def, public Setters<Nat> {
public:
    using Setters<Nat>::set;
    static constexpr auto Node = mim::Node::Nat;
 
private:
    Nat(World& world);
 
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
/// A built-in constant of type `Nat -> *`.
class Idx : public Def, public Setters<Idx> {
private:
    Idx(const Def* type)
        : Def(Node, type, Defs{}, 0) {}
 
public:
    using Setters<Idx>::set;
    using Def::as;
    using Def::isa;
 
    /// @name isa
    ///@{
 
    /// Checks if @p def is a `Idx s` and returns `s` or `nullptr` otherwise.
    static const Def* isa(const Def* def);
    static const Def* as(const Def* def) {
        auto res = isa(def);
        assert(res);
        return res;
    }
    static std::optional<nat_t> isa_lit(const Def* def);
    static nat_t as_lit(const Def* def) {
        auto res = isa_lit(def);
        assert(res.has_value());
        return *res;
    }
    ///@}
 
    /// @name Convert between Idx::isa and bitwidth and vice versa
    ///@{
    // clang-format off
    static constexpr nat_t bitwidth2size(nat_t n) { assert(n != 0); return n == 64 ? 0 : (1_n << n); }
    static constexpr nat_t size2bitwidth(nat_t n) { return n == 0 ? 64 : std::bit_width(n - 1_n); }
    // clang-format on
    static std::optional<nat_t> size2bitwidth(const Def* size);
    ///@}
 
    static constexpr auto Node = mim::Node::Idx;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
class Proxy : public Def, public Setters<Proxy> {
private:
    Proxy(const Def* type, Defs ops, u32 pass, u32 tag)
        : Def(Node, type, ops, (u64(pass) << 32_u64) | u64(tag)) {}
 
public:
    using Setters<Proxy>::set;
 
    /// @name Getters
    ///@{
    u32 pass() const { return u32(flags() >> 32_u64); } ///< IPass::index within PassMan.
    u32 tag() const { return u32(flags()); }
    ///@}
 
    static constexpr auto Node = mim::Node::Proxy;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const override;
 
    friend class World;
};
 
/// @deprecated A global variable in the data segment.
/// A Global may be mutable or immutable.
/// @deprecated Will be removed.
class Global : public Def, public Setters<Global> {
private:
    Global(const Def* type, bool is_mutable)
        : Def(Node, type, 1, is_mutable) {}
 
public:
    using Setters<Global>::set;
 
    /// @name ops
    ///@{
    const Def* init() const { return op(0); }
    void set(const Def* init) { Def::set(0, init); }
    ///@}
 
    /// @name type
    ///@{
    const App* type() const;
    const Def* alloced_type() const;
    ///@}
 
    /// @name Getters
    ///@{
    bool is_mutable() const { return flags(); }
    ///@}
 
    /// @name Rebuild
    ///@{
    Global* stub(const Def* type) { return stub_(world(), type)->set(dbg()); }
    ///@}
 
    static constexpr auto Node = mim::Node::Global;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const override;
    Global* stub_(World&, const Def*) override;
 
    friend class World;
};
 
} // namespace mim