def.h

Go to the documentation of this file.

#pragma once
 
#include <optional>
#include <span>
 
#include <fe/assert.h>
#include <fe/cast.h>
#include <fe/enum.h>
 
#include "mim/config.h"
 
#include "mim/util/dbg.h"
#include "mim/util/sets.h"
#include "mim/util/util.h"
#include "mim/util/vector.h"
 
// clang-format off
#define MIM_NODE(m)                                                                                                \
    m(Lit,    Judge::Intro) /* keep this first - causes Lit to appear left in Def::less/Def::greater*/             \
    m(Axm,    Judge::Intro)                                                                                        \
    m(Var,    Judge::Intro)                                                                                        \
    m(Global, Judge::Intro)                                                                                        \
    m(Proxy,  Judge::Intro)                                                                                        \
    m(Hole,   Judge::Hole )                                                                                        \
    m(Type,   Judge::Meta ) m(Univ,  Judge::Meta ) m(UMax,    Judge::Meta) m(UInc,   (Judge::Meta               )) \
    m(Pi,     Judge::Form ) m(Lam,   Judge::Intro) m(App,     Judge::Elim)                                         \
    m(Sigma,  Judge::Form ) m(Tuple, Judge::Intro) m(Extract, Judge::Elim) m(Insert, (Judge::Intro | Judge::Elim)) \
    m(Arr,    Judge::Form ) m(Pack,  Judge::Intro)                                                                 \
    m(Join,   Judge::Form ) m(Inj,   Judge::Intro) m(Match,   Judge::Elim) m(Top,    (Judge::Intro              )) \
    m(Meet,   Judge::Form ) m(Merge, Judge::Intro) m(Split,   Judge::Elim) m(Bot,    (Judge::Intro              )) \
    m(Reform, Judge::Form ) m(Rule,  Judge::Intro)                                                                 \
    m(Uniq,   Judge::Form )                                                                                        \
    m(Nat,    Judge::Form )                                                                                        \
    m(Idx,    Judge::Intro)
 
#define MIM_IMM_NODE(m)                                                                                            \
    m(Lit)                                                                                                         \
    m(Axm)                                                                                                         \
    m(Var)                                                                                                         \
    m(Proxy)                                                                                                       \
    m(Type)  m(Univ)  m(UMax)    m(UInc)                                                                           \
    m(Pi)    m(Lam)   m(App)                                                                                       \
    m(Sigma) m(Tuple) m(Extract) m(Insert)                                                                         \
    m(Arr)   m(Pack)                                                                                               \
    m(Join)  m(Inj)   m(Match)   m(Top)                                                                            \
    m(Meet)  m(Merge) m(Split)   m(Bot)                                                                            \
    m(Rule)                                                                                                        \
    m(Uniq)                                                                                                        \
    m(Nat)                                                                                                         \
    m(Idx)
 
#define MIM_MUT_NODE(m)                                                                                            \
    m(Global)                                                                                                      \
    m(Hole)                                                                                                        \
    m(Pi)    m(Lam)                                                                                                \
    m(Sigma)                                                                                                       \
    m(Arr)   m(Pack)                                                                                               \
    m(Rule)
// clang-format on
 
namespace mim {
 
class App;
class Axm;
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, _) node,
    MIM_NODE(CODE)
#undef CODE
};
 
#define CODE(node, _) +size_t(1)
static constexpr size_t Num_Nodes = size_t(0) MIM_NODE(CODE);
#undef CODE
 
/// 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
};
 
/// [Judgement](https://ncatlab.org/nlab/show/judgment).
enum class Mut {
    // clang-format off
    Mut = 1 << 0, ///< Node may be mutable.
    Imm = 1 << 1, ///< Node may be immmutable.
    // clang-format on
};
///@}
 
} // namespace mim
 
#ifndef DOXYGEN
// clang-format off
template<> struct fe::is_bit_enum<mim::Dep>   : std::true_type {};
template<> struct fe::is_bit_enum<mim::Judge> : std::true_type {};
template<> struct fe::is_bit_enum<mim::Mut>   : std::true_type {};
// clang-format on
#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 // D is only needed to make the resolution `D::template set` lazy
#ifdef _MSC_VER
    __declspec(empty_bases)
#endif
        Setters {
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              | Inj               | Match             |
/// | Meet              | Merge             | Split             |
/// | Reform            | Rule              |                   |
/// | Nat               | Lit               |                   |
/// | Idx               | Lit               |                   |
/// In addition there is:
/// * Var: A variable. Currently the following Def%s may be binders:
///     * Pi, Lam, Sigma, Arr, Pack
/// * Axm: To introduce new entities.
/// * Proxy: Used for intermediate values during optimizations.
/// * Hole: A metavariable filled in by the type inference (always mutable as 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;
    virtual const Def* arity() const;
    ///@}
 
    /// @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:
    /// If Def::is_set() is ...
    ///     * `false`, [set](@ref Def::set) the [operands](@ref Def::ops) from left to right.
    ///     * `true`, Def::unset() the operands first and then start over:
    ///       ```
    ///       mut->unset()->set({a, b, c});
    ///       ```
    ///
    /// MimIR assumes that a mutable is *final*, when its last operand is set.
    /// Then, Def::check() will be invoked.
    ///@{
    bool is_set() const;            ///< Yields `true` if empty or the last op is set.
    Def* set(size_t i, const Def*); ///< Successively set from left to right.
    Def* set(Defs ops);             ///< Set @p ops all at once (no Def::unset necessary beforehand).
    Def* unset();                   ///< Unsets all Def::ops; works even, if not set at all or only partially set.
 
    /// Update type.
    /// @warning Only make type-preserving updates such as removing Hole%s.
    /// Do this even before updating all other ops()!.
    Def* set_type(const Def*);
    ///@}
 
    /// @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::arity(), if it is a Lit, or `1` otherwise.
    nat_t num_projs() const;
    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 {
            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 noexcept { return external_; }
    void make_external();
    void make_internal();
    void transfer_external(Def* to);
    ///@}
 
    /// @name Casts
    /// @see @ref cast_builtin
    ///@{
    bool is_mutable() const noexcept { return mut_; }
 
    // 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>(); }
    // clang-format on
 
    /// If `this` is *mutable*, 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<Ow>(sym(std::move(s))); return this; }
    template<bool Ow = false>       Def* set(       std::string s)       { set<Ow>(sym(std::move(s))); return this; }
    template<bool Ow = false> const Def* set(Loc l, Sym s        ) const { set<Ow>(l); set<Ow>(s); return this; }
    template<bool Ow = false>       Def* set(Loc l, Sym s        )       { set<Ow>(l); set<Ow>(s); return this; }
    template<bool Ow = false> const Def* set(Loc l, std::string s) const { set<Ow>(l); set<Ow>(sym(std::move(s))); return this; }
    template<bool Ow = false>       Def* set(Loc l, std::string s)       { set<Ow>(l); set<Ow>(sym(std::move(s))); return this; }
    template<bool Ow = false> const Def* set(Dbg d) const { set<Ow>(d.loc(), d.sym()); return this; }
    template<bool Ow = false>       Def* set(Dbg d)       { set<Ow>(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;
 
    /// @see World::reduce
    template<size_t N = std::dynamic_extent>
    constexpr auto reduce(const Def* arg) const {
        return reduce_(arg).span<N>();
    }
 
    /// First Def::op that needs to be dealt with during reduction; e.g. for a Pi we don't reduce the Pi::dom.
    /// @see World::reduce
    virtual constexpr size_t reduction_offset() const noexcept { return size_t(-1); }
    ///@}
 
    /// @name Type Checking
    ///@{
 
    /// Checks whether the `i`th operand can be set to `def`.
    /// The method returns a possibly updated version of `def` (e.g. where Hole%s have been resolved).
    /// This is the actual `def` that will be set as the `i`th operand.
    virtual const Def* check([[maybe_unused]] size_t i, const Def* def) { return def; }
 
    /// After all Def::ops have ben Def::set, this method will be invoked to check the type of this mutable.
    /// The method returns a possibly updated version of its type (e.g. where Hole%s have been resolved).
    /// If different from Def::type, it will update its Def::type to a Def::zonk%ed version of that.
    virtual const Def* check() { return type(); }
 
    /// If Hole%s have been filled, reconstruct the program without them.
    /// Only gues up to but excluding other mutables.
    /// @see https://stackoverflow.com/questions/31889048/what-does-the-ghc-source-mean-by-zonk
    const Def* zonk() const;
 
    /// If *mutable, zonk%s all ops and tries to immutabilize it; otherwise just zonk.
    const Def* zonk_mut() const;
    ///@}
 
    /// zonk%s all @p defs and retuns a new DefVec.
    static DefVec zonk(Defs defs);
 
    /// @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 Syntactic Comparison
    ///
    enum class Cmp {
        L, ///< Less
        G, ///< Greater
        E, ///< Equal
        U, ///< Unknown
    };
    [[nodiscard]] static Cmp cmp(const Def* a, const Def* b);
    [[nodiscard]] static bool less(const Def* a, const Def* b);
    [[nodiscard]] static bool greater(const Def* a, const Def* b);
 
    /// @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:
    Defs reduce_(const Def* arg) const;
    virtual Def* stub_(World&, const Def*) { fe::unreachable(); }
    virtual const Def* rebuild_(World& w, const Def* type, Defs ops) const = 0;
 
    template<bool init>
    Vars free_vars(bool&, uint32_t);
    void invalidate();
    const Def** ops_ptr() const {
        return reinterpret_cast<const Def**>(reinterpret_cast<char*>(const_cast<Def*>(this + 1)));
    }
    bool equal(const Def* other) const;
 
    template<Cmp>
    [[nodiscard]] static bool cmp_(const Def* a, const Def* b);
 
protected:
    mutable Dbg dbg_;
    union {
        NormalizeFn normalizer_; ///< Axm only: Axm%s use this member to store their normalizer.
        const Axm* axm_;         ///< App only: Curried App%s of Axm%s use this member to propagate the Axm.
        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*);
};
 
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;
    static constexpr size_t Num_Ops = 1;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    friend class World;
};
 
class Univ : public Def, public Setters<Univ> {
public:
    using Setters<Univ>::set;
    static constexpr auto Node      = mim::Node::Univ;
    static constexpr size_t Num_Ops = 0;
 
private:
    Univ(World& world)
        : Def(&world, Node, nullptr, Defs{}, 0) {}
 
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    friend class World;
};
 
class UMax : public Def, public Setters<UMax> {
public:
    using Setters<UMax>::set;
    static constexpr auto Node      = mim::Node::UMax;
    static constexpr size_t Num_Ops = std::dynamic_extent;
 
    enum Sort { Univ, Kind, Type, Term };
 
private:
    UMax(World&, Defs ops);
 
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    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;
    static constexpr size_t Num_Ops = 1;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    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;
    static constexpr size_t Num_Ops = 1;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    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;
    static constexpr size_t Num_Ops = 0;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    friend class World;
};
 
class Nat : public Def, public Setters<Nat> {
public:
    using Setters<Nat>::set;
    static constexpr auto Node      = mim::Node::Nat;
    static constexpr size_t Num_Ops = 0;
 
private:
    Nat(World& world);
 
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    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;
    static constexpr size_t Num_Ops = 0;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    friend class World;
};
 
class Proxy : public Def, public Setters<Proxy> {
private:
    Proxy(const Def* type, u32 pass, u32 tag, Defs ops)
        : 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;
    static constexpr size_t Num_Ops = std::dynamic_extent;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const final;
 
    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;
    static constexpr size_t Num_Ops = 1;
 
private:
    const Def* rebuild_(World&, const Def*, Defs) const final;
    Global* stub_(World&, const Def*) final;
 
    friend class World;
};
 
} // namespace mim