Simplify.cpp

// This file is part of the Luau programming language and is licensed under MIT License; see LICENSE.txt for details

#include "Luau/Simplify.h"

#include "Luau/BuiltinDefinitions.h"
#include "Luau/Clone.h"
#include "Luau/Common.h"
#include "Luau/DenseHash.h"
#include "Luau/RecursionCounter.h"
#include "Luau/Set.h"
#include "Luau/Type.h"
#include "Luau/TypeArena.h"
#include "Luau/TypeIds.h"
#include "Luau/TypePairHash.h"
#include "Luau/TypeUtils.h"

#include <algorithm>

LUAU_FASTINT(LuauTypeReductionRecursionLimit)
LUAU_FASTFLAG(LuauSolverV2)
LUAU_DYNAMIC_FASTINTVARIABLE(LuauSimplificationComplexityLimit, 8)
LUAU_DYNAMIC_FASTINTVARIABLE(LuauTypeSimplificationIterationLimit, 128)
LUAU_FASTFLAGVARIABLE(LuauRelateHandlesCoincidentTables)

namespace Luau
{

using SimplifierSeenSet = Set<std::pair<TypeId, TypeId>, TypePairHash>;

struct TypeSimplifier
{
    NotNull<BuiltinTypes> builtinTypes;
    NotNull<TypeArena> arena;

    DenseHashSet<TypeId> blockedTypes{nullptr};

    int recursionDepth = 0;

    TypeId mkNegation(TypeId ty) const;

    TypeId intersectFromParts(TypeIds parts);

    TypeId intersectUnionWithType(TypeId left, TypeId right);

    TypeId intersectUnions(TypeId left, TypeId right);

    TypeId intersectNegatedUnion(TypeId left, TypeId right);

    TypeId intersectTypeWithNegation(TypeId left, TypeId right);

    TypeId intersectNegations(TypeId left, TypeId right);

    TypeId intersectIntersectionWithType(TypeId left, TypeId right);

    // Attempt to intersect the two types.  Does not recurse.  Does not handle
    // unions, intersections, or negations.
    std::optional<TypeId> basicIntersect(TypeId left, TypeId right);

    std::optional<TypeId> basicIntersectWithTruthy(TypeId target) const;

    std::optional<TypeId> basicIntersectWithFalsy(TypeId target) const;

    TypeId intersect(TypeId left, TypeId right);

    TypeId union_(TypeId left, TypeId right);

    TypeId simplify(TypeId ty);

    TypeId simplify(TypeId ty, DenseHashSet<TypeId>& seen);

    std::optional<TypeId> intersectOne(TypeId target, TypeId discriminant) const;

    std::optional<TypeId> subtractOne(TypeId target, TypeId discriminant) const;

    std::optional<Property> intersectProperty(const Property& target, const Property& discriminant, DenseHashSet<TypeId>& seen) const;

    std::optional<TypeId> intersectWithSimpleDiscriminant(TypeId target, TypeId discriminant, DenseHashSet<TypeId>& seen) const;

    std::optional<TypeId> intersectWithSimpleDiscriminant(TypeId target, TypeId discriminant) const;
};

// Match the exact type false|nil
static bool isFalsyType_DEPRECATED(TypeId ty)
{
    ty = follow(ty);
    const UnionType* ut = get<UnionType>(ty);
    if (!ut)
        return false;

    bool hasFalse = false;
    bool hasNil = false;

    auto it = begin(ut);
    if (it == end(ut))
        return false;

    TypeId t = follow(*it);

    if (auto pt = get<PrimitiveType>(t); pt && pt->type == PrimitiveType::NilType)
        hasNil = true;
    else if (auto st = get<SingletonType>(t); st && st->variant == BooleanSingleton{false})
        hasFalse = true;
    else
        return false;

    ++it;
    if (it == end(ut))
        return false;

    t = follow(*it);

    if (auto pt = get<PrimitiveType>(t); pt && pt->type == PrimitiveType::NilType)
        hasNil = true;
    else if (auto st = get<SingletonType>(t); st && st->variant == BooleanSingleton{false})
        hasFalse = true;
    else
        return false;

    ++it;
    if (it != end(ut))
        return false;

    return hasFalse && hasNil;
}

// Match the exact type ~(false|nil)
bool isTruthyType_DEPRECATED(TypeId ty)
{
    ty = follow(ty);

    const NegationType* nt = get<NegationType>(ty);
    if (!nt)
        return false;

    return isFalsyType_DEPRECATED(nt->ty);
}

Relation flip(Relation rel)
{
    switch (rel)
    {
    case Relation::Subset:
        return Relation::Superset;
    case Relation::Superset:
        return Relation::Subset;
    default:
        return rel;
    }
}

// FIXME: I'm not completely certain that this function is theoretically reasonable.
Relation combine(Relation a, Relation b)
{
    switch (a)
    {
    case Relation::Disjoint:
        switch (b)
        {
        case Relation::Disjoint:
            return Relation::Disjoint;
        case Relation::Coincident:
            return Relation::Superset;
        case Relation::Intersects:
            return Relation::Intersects;
        case Relation::Subset:
            return Relation::Intersects;
        case Relation::Superset:
            return Relation::Intersects;
        }
        break;
    case Relation::Coincident:
        switch (b)
        {
        case Relation::Disjoint:
            return Relation::Coincident;
        case Relation::Coincident:
            return Relation::Coincident;
        case Relation::Intersects:
            return Relation::Superset;
        case Relation::Subset:
            return Relation::Coincident;
        case Relation::Superset:
            return Relation::Intersects;
        }
        break;
    case Relation::Superset:
        switch (b)
        {
        case Relation::Disjoint:
            return Relation::Superset;
        case Relation::Coincident:
            return Relation::Superset;
        case Relation::Intersects:
            return Relation::Intersects;
        case Relation::Subset:
            return Relation::Intersects;
        case Relation::Superset:
            return Relation::Superset;
        }
        break;
    case Relation::Subset:
        switch (b)
        {
        case Relation::Disjoint:
            return Relation::Subset;
        case Relation::Coincident:
            return Relation::Coincident;
        case Relation::Intersects:
            return Relation::Intersects;
        case Relation::Subset:
            return Relation::Subset;
        case Relation::Superset:
            return Relation::Intersects;
        }
        break;
    case Relation::Intersects:
        switch (b)
        {
        case Relation::Disjoint:
            return Relation::Intersects;
        case Relation::Coincident:
            return Relation::Superset;
        case Relation::Intersects:
            return Relation::Intersects;
        case Relation::Subset:
            return Relation::Intersects;
        case Relation::Superset:
            return Relation::Intersects;
        }
        break;
    }

    LUAU_UNREACHABLE();
    return Relation::Intersects;
}

// Given A & B, what is A & ~B?
Relation invert(Relation r)
{
    switch (r)
    {
    case Relation::Disjoint:
        return Relation::Subset;
    case Relation::Coincident:
        return Relation::Disjoint;
    case Relation::Intersects:
        return Relation::Intersects;
    case Relation::Subset:
        return Relation::Disjoint;
    case Relation::Superset:
        return Relation::Intersects;
    }

    LUAU_UNREACHABLE();
    return Relation::Intersects;
}

static bool isTypeVariable(TypeId ty)
{
    return get<FreeType>(ty) || get<GenericType>(ty) || get<BlockedType>(ty) || get<PendingExpansionType>(ty);
}

Relation relate(TypeId left, TypeId right, SimplifierSeenSet& seen);

Relation relateTableToExternType(const TableType* table, const ExternType* cls, SimplifierSeenSet& seen)
{
    // If either the table or the extern type have an indexer, just bail.
    // There's rapidly diminishing returns on doing something smart for
    // indexers compared to refining exact members.
    if (table->indexer || cls->indexer)
        return Relation::Intersects;

    for (auto& [name, prop] : table->props)
    {
        if (auto propInExternType = lookupExternTypeProp(cls, name))
        {
            LUAU_ASSERT(prop.readTy && propInExternType->readTy);
            // For all examples, consider:
            //
            //  declare extern type Foobar with
            //      prop: string | number
            //  end
            //
            switch (relate(*prop.readTy, *propInExternType->readTy, seen))
            {
            case Relation::Disjoint:
                // Consider `{ read prop: boolean }` and `Foobar`, these types are
                // disjoint as `_.prop` would be `never.
                return Relation::Disjoint;
            case Relation::Coincident:
                // Consider `{ read prop: string | number }` and `Foobar`, we don't really
                // learn anything about these types.
                break;
            case Relation::Intersects:
                // Consider `{ read prop: string | boolean }` and `Foobar`, these types
                // intersect (imagine a `Foobar` initialized with `prop = "foo"`).
                return Relation::Intersects;
            case Relation::Subset:
                // Consider `{ read prop: string }` and `Foobar`: we should _roughly_
                // consider this the same as intersecting.
                return Relation::Intersects;
            case Relation::Superset:
                // This is the only mildly interesting case, consider
                // `{ read prop: string | number | boolean }` and `Foobar`.
                // We can _probably_ consider `Foobar` the subset here.
                break;
            }
        }
    }

    // If all the properties of the table were either coincident or
    // supersets of the extern property, then we claim that the table
    // is a superset.
    return Relation::Superset;
}

/**
 * @return The relationship between the single property on the right and its corresponding property
 * in the left table.
 */
Relation relateTableToProp(const TableType* leftTable, const std::string& propName, const Property& rightProp, SimplifierSeenSet& seen)
{
    // If the left table does not have the property at all,
    // assume an intersection.
    auto leftProp = leftTable->props.find(propName);
    if (leftProp == leftTable->props.end())
        return Relation::Intersects;

    if (leftProp->second.isShared() && rightProp.isShared())
    {
        switch (relate(*leftProp->second.readTy, *rightProp.readTy, seen))
        {
        case Relation::Disjoint:
            // The two read properties are disjoint, so the tables are disjoint, e.g.:
            //
            //  { y: string, x: number? } & { read x: string }
            //
            return Relation::Disjoint;
        case Relation::Coincident:
            return Relation::Coincident;
        // For _all_ other cases, two shared properties indicate a non-empty intersection.
        case Relation::Subset:
            // If the left property is a subset, intersection implies a widening of
            // write type of the left property, as in:
            //
            //  { y: string, x: number } & { x: number? }
            //
            return Relation::Intersects;
        case Relation::Superset:
            // If the left property is a superset, intersection implies a narrowing
            // of the read type of the left property, as in:
            //
            //  { y: string, x: number? } & { x: number }
            //
            return Relation::Intersects;
        case Relation::Intersects:
            // Intersection applies both of the above cases: a widened write type and
            // a narrowed read type:
            //
            //  { y: string, x: number? } & { x: string? }
            //
            return Relation::Intersects;
        default:
            // And for good measure, default to intersection.
            return Relation::Intersects;
        }
    }

    // Otherwise we want to hard match on the case of:
    //
    //  { ..., x: T } & { read x: U }
    //
    // ... or ...
    //
    //  { ..., read x: T } & { read x: U }
    //
    // We will use the relation between T and U here.
    if (!leftProp->second.readTy || !rightProp.isReadOnly())
        return Relation::Intersects;

    switch (relate(*leftProp->second.readTy, *rightProp.readTy, seen))
    {
    case Relation::Disjoint:
        // The two read properties are disjoint, so the tables are disjoint, e.g.:
        //
        //  { y: string, x: number? } & { read x: string }
        //
        return Relation::Disjoint;
    case Relation::Coincident:
        // If the two read types are coincident, then the left property is a
        // subset if it also has a write part.
        return leftProp->second.writeTy ? Relation::Subset : Relation::Coincident;
    case Relation::Subset:
        // If the left table's property is a subset of the right property, then
        // the left table is a subset, as in:
        //
        //  { y: string, x: number } & { read x: number? } => these tables intersect
        return Relation::Subset;
    case Relation::Superset:
        // If the left table's property is a superset of the right property, then
        // the two tables intersect, as in:
        //
        //  { y: string, x: number? } & { read x: number }
        //
        return Relation::Intersects;
    case Relation::Intersects:
        // If the left table's property intersects with the right property, then
        // the two tables intersect, as in:
        //
        //  { y: string, x: number? } & { read x: string? } => these tables intersect
        return Relation::Intersects;
    default:
        // And for good measure, default to intersection.
        return Relation::Intersects;
    }
}

Relation relateTables(const TableType* leftTable, const TableType* rightTable, SimplifierSeenSet& seen)
{
    // FIXME CLI-189216: As noted in the body this is not complete.
    if (leftTable->state != TableState::Sealed || rightTable->state != TableState::Sealed)
        return Relation::Intersects;

    if (rightTable->props.size() == 1 && !rightTable->indexer)
    {
        auto it = rightTable->props.begin();
        auto res = relateTableToProp(leftTable, it->first, it->second, seen);
        // If the single property is coincident with the member in the left table, then
        // by width subtyping the left table is a subset.
        if (res == Relation::Coincident)
            return Relation::Subset;
        return res;
    }

    if (leftTable->props.size() == 1 && !leftTable->indexer)
    {
        auto it = leftTable->props.begin();
        auto res = flip(relateTableToProp(rightTable, it->first, it->second, seen));
        // If the single property is coincident with the member in the right table, then
        // by width subtyping the right table is a subset (so we return superset).
        if (res == Relation::Coincident)
            return Relation::Superset;
        return res;
    }

    // This can potentially not account for something like
    //
    //  { x: number, y: number } & { x: number, y: number, z: number }
    //
    // ... where we _ought_ to say superset.
    if (leftTable->props.size() != rightTable->props.size() || leftTable->indexer.has_value() != rightTable->indexer.has_value())
        return Relation::Intersects;

    bool hasSubset = false;

    for (const auto& [rightName, rightProp] : rightTable->props)
    {
        switch (relateTableToProp(leftTable, rightName, rightProp, seen))
        {
        case Relation::Disjoint:
            return Relation::Disjoint;
        case Relation::Superset:
        case Relation::Intersects:
            // We're being _very_ conservative here. We could update this in the future to
            // account for a case like:
            //
            //  (T & { x: number }) & (T & { read x: number? })
            //
            // ... by running this loop twice.
            return Relation::Intersects;
        case Relation::Subset:
            hasSubset = true;
            break;
        case Relation::Coincident:
            break;
        }
    }

    if (!leftTable->indexer)
    {
        LUAU_ASSERT(!rightTable->indexer);
        return hasSubset ? Relation::Subset : Relation::Coincident;
    }

    if (relate(leftTable->indexer->indexType, rightTable->indexer->indexType, seen) != Relation::Coincident)
        return Relation::Intersects;

    if (relate(leftTable->indexer->indexType, rightTable->indexer->indexType, seen) != Relation::Coincident)
        return Relation::Intersects;

    return hasSubset ? Relation::Subset : Relation::Coincident;
}

Relation relateTables_DEPRECATED(TypeId left, TypeId right, SimplifierSeenSet& seen)
{
    NotNull<const TableType> leftTable{get<TableType>(left)};
    NotNull<const TableType> rightTable{get<TableType>(right)};
    LUAU_ASSERT(1 == rightTable->props.size());
    // Disjoint props have nothing in common
    // t1 with props p1's cannot appear in t2 and t2 with props p2's cannot appear in t1
    bool foundPropFromLeftInRight = std::any_of(
        begin(leftTable->props),
        end(leftTable->props),
        [&](auto prop)
        {
            return rightTable->props.count(prop.first) > 0;
        }
    );
    bool foundPropFromRightInLeft = std::any_of(
        begin(rightTable->props),
        end(rightTable->props),
        [&](auto prop)
        {
            return leftTable->props.count(prop.first) > 0;
        }
    );

    if (!foundPropFromLeftInRight && !foundPropFromRightInLeft && leftTable->props.size() >= 1 && rightTable->props.size() >= 1)
        return Relation::Intersects;

    const auto [propName, rightProp] = *begin(rightTable->props);

    auto it = leftTable->props.find(propName);
    if (it == leftTable->props.end())
    {
        // Every table lacking a property is a supertype of a table having that
        // property but the reverse is not true.
        return Relation::Superset;
    }

    const Property leftProp = it->second;

    if (!leftProp.isShared() || !rightProp.isShared())
        return Relation::Intersects;

    Relation r = relate(*leftProp.readTy, *rightProp.readTy, seen);
    if (r == Relation::Coincident && 1 != leftTable->props.size())
    {
        // eg {tag: "cat", prop: string} & {tag: "cat"}
        return Relation::Subset;
    }
    else
        return r;
}

// A cheap and approximate subtype test
Relation relate(TypeId left, TypeId right, SimplifierSeenSet& seen)
{
    // TODO nice to have: Relate functions of equal argument and return arity

    left = follow(left);
    right = follow(right);

    if (left == right)
        return Relation::Coincident;

    std::pair<TypeId, TypeId> typePair{left, right};
    if (!seen.insert(typePair))
    {
        // TODO: is this right at all?
        // The thinking here is that this is a cycle if we get here, and therefore its coincident.
        return Relation::Coincident;
    }

    if (get<UnknownType>(left))
    {
        if (get<AnyType>(right))
            return Relation::Subset;

        if (get<UnknownType>(right))
            return Relation::Coincident;

        if (get<ErrorType>(right))
            return Relation::Disjoint;

        return Relation::Superset;
    }

    if (get<UnknownType>(right))
        return flip(relate(right, left, seen));

    if (get<AnyType>(left))
    {
        if (get<AnyType>(right))
            return Relation::Coincident;

        return Relation::Superset;
    }

    if (get<AnyType>(right))
        return flip(relate(right, left, seen));

    // Type variables
    // * FreeType
    // * GenericType
    // * BlockedType
    // * PendingExpansionType

    // Tops and bottoms
    // * ErrorType
    // * AnyType
    // * NeverType
    // * UnknownType

    // Concrete
    // * PrimitiveType
    // * SingletonType
    // * FunctionType
    // * TableType
    // * MetatableType
    // * ExternType
    // * UnionType
    // * IntersectionType
    // * NegationType

    if (isTypeVariable(left) || isTypeVariable(right))
        return Relation::Intersects;

    // if either type is a type function, we cannot know if they'll be related.
    if (get<TypeFunctionInstanceType>(left) || get<TypeFunctionInstanceType>(right))
        return Relation::Intersects;

    if (get<ErrorType>(left))
    {
        if (get<ErrorType>(right))
            return Relation::Coincident;
        else if (get<AnyType>(right))
            return Relation::Subset;

        return Relation::Disjoint;
    }
    else if (get<ErrorType>(right))
        return flip(relate(right, left, seen));

    if (get<NeverType>(left))
    {
        if (get<NeverType>(right))
            return Relation::Coincident;

        return Relation::Subset;
    }
    else if (get<NeverType>(right))
        return flip(relate(right, left, seen));

    if (get<IntersectionType>(left))
        return Relation::Intersects;
    else if (get<IntersectionType>(right))
        return Relation::Intersects;

    if (auto ut = get<UnionType>(left))
    {
        for (TypeId part : ut)
        {
            Relation r = relate(part, right, seen);
            if (r == Relation::Superset || r == Relation::Coincident)
                return Relation::Superset;
        }
        return Relation::Intersects;
    }
    else if (auto ut = get<UnionType>(right))
    {
        std::vector<Relation> opts;
        for (TypeId part : ut)
        {
            Relation r = relate(left, part, seen);

            if (r == Relation::Subset || r == Relation::Coincident)
                return Relation::Subset;
        }
        return Relation::Intersects;
    }

    if (auto rnt = get<NegationType>(right))
    {
        Relation a = relate(left, rnt->ty, seen);
        switch (a)
        {
        case Relation::Coincident:
            // number & ~number
            return Relation::Disjoint;
        case Relation::Disjoint:
            if (get<NegationType>(left))
            {
                // ~number & ~string
                return Relation::Intersects;
            }
            else
            {
                // number & ~string
                return Relation::Subset;
            }
        case Relation::Intersects:
            // ~(false?) & ~boolean
            return Relation::Intersects;
        case Relation::Subset:
            // "hello" & ~string
            return Relation::Disjoint;
        case Relation::Superset:
            // ~function & ~(false?)  -> ~function
            // boolean & ~(false?)    -> true
            // string & ~"hello"      -> string & ~"hello"
            return Relation::Intersects;
        }
    }
    else if (get<NegationType>(left))
        return flip(relate(right, left, seen));

    if (auto lp = get<PrimitiveType>(left))
    {
        if (auto rp = get<PrimitiveType>(right))
        {
            if (lp->type == rp->type)
                return Relation::Coincident;

            return Relation::Disjoint;
        }

        if (auto rs = get<SingletonType>(right))
        {
            if (lp->type == PrimitiveType::String && rs->variant.get_if<StringSingleton>())
                return Relation::Superset;

            if (lp->type == PrimitiveType::Boolean && rs->variant.get_if<BooleanSingleton>())
                return Relation::Superset;

            return Relation::Disjoint;
        }

        if (lp->type == PrimitiveType::Function)
        {
            if (get<FunctionType>(right))
                return Relation::Superset;

            return Relation::Disjoint;
        }
        if (lp->type == PrimitiveType::Table)
        {
            if (get<TableType>(right))
                return Relation::Superset;

            return Relation::Disjoint;
        }

        if (get<FunctionType>(right) || get<TableType>(right) || get<MetatableType>(right) || get<ExternType>(right))
            return Relation::Disjoint;
    }

    if (auto ls = get<SingletonType>(left))
    {
        if (get<FunctionType>(right) || get<TableType>(right) || get<MetatableType>(right) || get<ExternType>(right))
            return Relation::Disjoint;

        if (get<PrimitiveType>(right))
            return flip(relate(right, left, seen));

        if (auto rs = get<SingletonType>(right))
        {
            if (ls->variant == rs->variant)
                return Relation::Coincident;

            return Relation::Disjoint;
        }
    }

    if (get<FunctionType>(left))
    {
        if (auto rp = get<PrimitiveType>(right))
        {
            if (rp->type == PrimitiveType::Function)
                return Relation::Subset;

            return Relation::Disjoint;
        }

        return Relation::Intersects;
    }

    if (auto lt = get<TableType>(left))
    {
        if (auto rp = get<PrimitiveType>(right))
        {
            if (rp->type == PrimitiveType::Table)
                return Relation::Subset;

            return Relation::Disjoint;
        }

        if (auto rt = get<TableType>(right))
        {
            if (FFlag::LuauRelateHandlesCoincidentTables)
            {
                return relateTables(lt, rt, seen);
            }
            else
            {
                // TODO PROBABLY indexers and metatables.
                if (1 == rt->props.size())
                {
                    Relation r = relateTables_DEPRECATED(left, right, seen);
                    /*
                     * A reduction of these intersections is certainly possible, but
                     * it would require minting new table types. Also, I don't think
                     * it's super likely for this to arise from a refinement.
                     *
                     * Time will tell!
                     *
                     * ex we simplify this
                     *     {tag: string} & {tag: "cat"}
                     * but not this
                     *     {tag: string, prop: number} & {tag: "cat"}
                     */
                    if (lt->props.size() > 1 && r == Relation::Superset)
                        return Relation::Intersects;

                    return r;
                }

                if (1 == lt->props.size())
                    return flip(relate(right, left, seen));

                return Relation::Intersects;
            }
        }

        if (auto re = get<ExternType>(right))
            return relateTableToExternType(lt, re, seen);

        // TODO metatables

        return Relation::Disjoint;
    }

    if (auto ct = get<ExternType>(left))
    {
        if (auto rct = get<ExternType>(right))
        {
            if (isSubclass(ct, rct))
                return Relation::Subset;

            if (isSubclass(rct, ct))
                return Relation::Superset;

            return Relation::Disjoint;
        }

        if (auto tbl = get<TableType>(right))
            return flip(relateTableToExternType(tbl, ct, seen));

        return Relation::Disjoint;
    }

    return Relation::Intersects;
}

// A cheap and approximate subtype test
Relation relate(TypeId left, TypeId right)
{
    SimplifierSeenSet seen{{}};
    return relate(left, right, seen);
}

TypeId TypeSimplifier::mkNegation(TypeId ty) const
{
    TypeId result = nullptr;

    if (ty == builtinTypes->truthyType)
        result = builtinTypes->falsyType;
    else if (ty == builtinTypes->falsyType)
        result = builtinTypes->truthyType;
    else if (auto ntv = get<NegationType>(ty))
        result = follow(ntv->ty);
    else
        result = arena->addType(NegationType{ty});

    return result;
}

namespace
{

enum class Inhabited
{
    Yes,
    No
};

Inhabited intersectOneWithIntersection(TypeSimplifier& simplifier, TypeIds& source, TypeIds& dest, TypeId candidate)
{
    if (dest.count(candidate) > 0)
        return Inhabited::Yes;

    if (auto itv = get<IntersectionType>(candidate))
    {
        for (auto subPart : itv)
        {
            if (intersectOneWithIntersection(simplifier, source, dest, subPart) == Inhabited::No)
                return Inhabited::No;
        }

        return Inhabited::Yes;
    }

    if (source.empty())
    {
        dest.insert(candidate);
        return Inhabited::Yes;
    }

    for (TypeId ty : source)
    {
        // All examples are presented in `candidate & ty` format.
        switch (relate(candidate, ty))
        {
        case Relation::Disjoint:
            // If the candidate and a member of the intersection are
            // disjoint, then the entire intersection is uninhabited, for
            // example:
            //
            //  boolean & string
            return Inhabited::No;
        case Relation::Subset:
            // If the candidate is a _subset_ of the member of the
            // intersection, then replace this entry if we don't already
            // have the candidate in the set, e.g.:
            //
            //  true & boolean
            dest.insert(candidate);
            break;
        case Relation::Coincident:
        case Relation::Superset:
            // If the candidate and a member of the intersection are
            // coincident, or the incoming part is a superset, then do
            // nothing, e.g.:
            //
            //  boolean & true
            //  boolean & boolean
            dest.insert(ty);
            break;
        case Relation::Intersects:
        {
            // If the candidate and a member of the intersection may
            // intersect, then attempt to replace the member with
            // a simpler type, e.g.:
            //
            //  boolean & ~(false?)
            if (std::optional<TypeId> simplified = simplifier.basicIntersect(candidate, ty))
                dest.insert(*simplified);
            else
            {
                dest.insert(candidate);
                dest.insert(ty);
            }
            break;
        }
        }
    }

    return Inhabited::Yes;
}
} // namespace

TypeId TypeSimplifier::intersectFromParts(TypeIds parts)
{
    if (parts.size() == 0)
        return builtinTypes->unknownType;

    if (parts.size() == 1)
        return *parts.begin();

    TypeIds source;
    TypeIds dest;

    source.reserve(parts.size());
    dest.reserve(parts.size());

    for (TypeId part : parts)
    {
        // We use the candidate, part, to construct a new intersection by
        // intersecting every element in source against part, and then
        // inserting it into dest.
        if (intersectOneWithIntersection(*this, source, dest, part) == Inhabited::No)
            return builtinTypes->neverType;

        // At this point, source will contain some intersection, and dest will contain
        // the intersection we want to retain for the next interation.

        // We swap the two, so that we can use `source` as the basis for the next iteration.
        std::swap(source, dest);

        // Then clear the `dest` without reallocating the underlying hashtable, to avoid
        // allocating.
        dest.clearWithoutRealloc();
    }

    IntersectionBuilder ib(arena, builtinTypes);
    for (auto ty : source)
        ib.add(ty);

    return ib.build();
}

TypeId TypeSimplifier::intersectUnionWithType(TypeId left, TypeId right)
{
    const UnionType* leftUnion = get<UnionType>(left);
    LUAU_ASSERT(leftUnion);

    bool changed = false;
    size_t maxSize = DFInt::LuauSimplificationComplexityLimit;

    if (leftUnion->options.size() > maxSize)
        return addIntersection(arena, builtinTypes, {left, right});

    UnionBuilder ub(arena, builtinTypes);
    ub.reserve(leftUnion->options.size());

    for (TypeId part : leftUnion)
    {
        TypeId simplified = intersect(right, part);
        changed |= simplified != part;

        if (get<NeverType>(simplified))
        {
            changed = true;
            continue;
        }

        ub.add(simplified);

        // Initial combination size check could not predict nested union iteration
        if (ub.size() > maxSize)
            return addIntersection(arena, builtinTypes, {left, right});
    }

    if (!changed)
        return left;

    return ub.build();
}

TypeId TypeSimplifier::intersectUnions(TypeId left, TypeId right)
{
    const UnionType* leftUnion = get<UnionType>(left);
    LUAU_ASSERT(leftUnion);

    const UnionType* rightUnion = get<UnionType>(right);
    LUAU_ASSERT(rightUnion);

    std::set<TypeId> newParts;

    // Combinatorial blowup moment!!

    // combination size
    size_t optionSize = (int)leftUnion->options.size() * rightUnion->options.size();
    size_t maxSize = DFInt::LuauSimplificationComplexityLimit;

    if (optionSize > maxSize)
        return arena->addType(IntersectionType{{left, right}});

    UnionBuilder ub{arena, builtinTypes};
    for (TypeId leftPart : leftUnion)
    {
        for (TypeId rightPart : rightUnion)
        {
            TypeId simplified = intersect(leftPart, rightPart);

            ub.add(simplified);

            // Initial combination size check could not predict nested union iteration
            if (ub.size() > maxSize)
                return addIntersection(arena, builtinTypes, {left, right});
        }
    }

    return ub.build();
}

TypeId TypeSimplifier::intersectNegatedUnion(TypeId left, TypeId right)
{
    // ~(A | B) & C
    // (~A & C) & (~B & C)

    const NegationType* leftNegation = get<NegationType>(left);
    LUAU_ASSERT(leftNegation);

    TypeId negatedTy = follow(leftNegation->ty);

    const UnionType* negatedUnion = get<UnionType>(negatedTy);
    LUAU_ASSERT(negatedUnion);

    bool changed = false;
    TypeIds newParts;

    for (TypeId part : negatedUnion)
    {
        Relation r = relate(part, right);
        switch (r)
        {
        case Relation::Disjoint:
            // If A is disjoint from B, then ~A & B is just B.
            //
            // ~(false?) & true
            // (~false & true) & (~nil & true)
            // true & true
            newParts.insert(right);
            break;
        case Relation::Coincident:
        // If A is coincident with or a superset of B, then ~A & B is never.
        //
        // ~(false?) & false
        // (~false & false) & (~nil & false)
        // never & false
        //
        // fallthrough
        case Relation::Superset:
            // If A is a superset of B, then ~A & B is never.
            //
            // ~(boolean | nil) & true
            // (~boolean & true) & (~boolean & nil)
            // never & nil
            return builtinTypes->neverType;
        case Relation::Subset:
        case Relation::Intersects:
            // If A is a subset of B, then ~A & B is a bit more complicated.  We need to think harder.
            //
            // ~(false?) & boolean
            // (~false & boolean) & (~nil & boolean)
            // true & boolean
            TypeId simplified = intersectTypeWithNegation(mkNegation(part), right);
            changed |= simplified != right;
            if (get<NeverType>(simplified))
                changed = true;
            else
                newParts.insert(simplified);
            break;
        }
    }

    if (!changed)
        return right;

    return intersectFromParts(std::move(newParts));
}

std::optional<TypeId> TypeSimplifier::basicIntersectWithTruthy(TypeId target) const
{
    target = follow(target);

    if (isApproximatelyTruthyType(target))
        return target;

    if (isApproximatelyFalsyType(target))
        return builtinTypes->neverType;

    if (is<UnknownType>(target))
        return builtinTypes->truthyType;

    if (is<AnyType>(target))
        // any = *error-type* | unknown, so truthy & any = *error-type* | truthy
        return arena->addType(UnionType{{builtinTypes->truthyType, builtinTypes->errorType}});

    if (is<NeverType, ErrorType>(target))
        return target;

    if (is<FunctionType, TableType, MetatableType, ExternType>(target))
        return target;

    if (auto pt = get<PrimitiveType>(target))
    {
        switch (pt->type)
        {
        case PrimitiveType::NilType:
            return builtinTypes->neverType;
        case PrimitiveType::Boolean:
            return builtinTypes->trueType;
        default:
            return target;
        }
    }

    if (auto st = get<SingletonType>(target))
        return st->variant == BooleanSingleton{false} ? builtinTypes->neverType : target;

    return std::nullopt;
}

std::optional<TypeId> TypeSimplifier::basicIntersectWithFalsy(TypeId target) const
{
    target = follow(target);

    if (isApproximatelyTruthyType(target))
        return builtinTypes->neverType;

    if (isApproximatelyFalsyType(target))
        return target;

    if (is<NeverType, ErrorType>(target))
        return target;

    if (is<AnyType>(target))
        // any = *error-type* | unknown, so falsy & any = *error-type* | falsy
        return arena->addType(UnionType{{builtinTypes->falsyType, builtinTypes->errorType}});

    if (is<UnknownType>(target))
        return builtinTypes->falsyType;

    if (is<FunctionType, TableType, MetatableType, ExternType>(target))
        return builtinTypes->neverType;

    if (auto pt = get<PrimitiveType>(target))
    {
        switch (pt->type)
        {
        case PrimitiveType::NilType:
            return builtinTypes->nilType;
        case PrimitiveType::Boolean:
            return builtinTypes->falseType;
        default:
            return builtinTypes->neverType;
        }
    }

    if (auto st = get<SingletonType>(target))
        return st->variant == BooleanSingleton{false} ? builtinTypes->falseType : builtinTypes->neverType;

    return std::nullopt;
}

TypeId TypeSimplifier::intersectTypeWithNegation(TypeId left, TypeId right)
{
    const NegationType* leftNegation = get<NegationType>(left);
    LUAU_ASSERT(leftNegation);

    TypeId negatedTy = follow(leftNegation->ty);

    if (negatedTy == right)
        return builtinTypes->neverType;

    if (auto ut = get<UnionType>(negatedTy))
    {
        // ~(A | B) & C
        // (~A & C) & (~B & C)
        bool changed = false;
        TypeIds newParts;

        for (TypeId part : ut)
        {
            Relation r = relate(part, right);
            switch (r)
            {
            case Relation::Coincident:
            // ~(false?) & nil
            // (~false & nil) & (~nil & nil)
            // nil & never
            //
            // fallthrough
            case Relation::Superset:
                // ~(boolean | string) & true
                // (~boolean & true) & (~boolean & string)
                // never & string

                return builtinTypes->neverType;

            case Relation::Disjoint:
                // ~nil & boolean
                newParts.insert(right);
                break;

            case Relation::Subset:
            // ~false & boolean
            // fallthrough
            case Relation::Intersects:
                // FIXME: The mkNegation here is pretty unfortunate.
                // Memoizing this will probably be important.
                changed = true;
                newParts.insert(right);
                newParts.insert(mkNegation(part));
            }
        }

        if (!changed)
            return right;

        return intersectFromParts(std::move(newParts));
    }

    if (auto rightUnion = get<UnionType>(right))
    {
        // ~A & (B | C)
        bool changed = false;
        std::set<TypeId> newParts;

        for (TypeId part : rightUnion)
        {
            Relation r = relate(negatedTy, part);
            switch (r)
            {
            case Relation::Coincident:
                changed = true;
                continue;
            case Relation::Disjoint:
                newParts.insert(part);
                break;
            case Relation::Superset:
                changed = true;
                continue;
            case Relation::Subset:
            // fallthrough
            case Relation::Intersects:
                changed = true;
                newParts.insert(arena->addType(IntersectionType{{left, part}}));
            }
        }

        if (!changed)
            return right;
        else if (0 == newParts.size())
            return builtinTypes->neverType;
        else if (1 == newParts.size())
            return *begin(newParts);
        else
            return arena->addType(UnionType{std::vector<TypeId>{begin(newParts), end(newParts)}});
    }

    if (auto pt = get<PrimitiveType>(right); pt && pt->type == PrimitiveType::Boolean)
    {
        if (auto st = get<SingletonType>(negatedTy))
        {
            if (st->variant == BooleanSingleton{true})
                return builtinTypes->falseType;
            else if (st->variant == BooleanSingleton{false})
                return builtinTypes->trueType;
            else
                // boolean & ~"hello"
                return builtinTypes->booleanType;
        }
    }

    Relation r = relate(negatedTy, right);

    switch (r)
    {
    case Relation::Disjoint:
        // ~boolean & string
        return right;
    case Relation::Coincident:
    // ~string & string
    // fallthrough
    case Relation::Superset:
        // ~string & "hello"
        return builtinTypes->neverType;
    case Relation::Subset:
    // ~string & unknown
    // ~"hello" & string
    // fallthrough
    case Relation::Intersects:
        // ~("hello" | boolean) & string
        // fallthrough
    default:
        return arena->addType(IntersectionType{{left, right}});
    }
}

TypeId TypeSimplifier::intersectNegations(TypeId left, TypeId right)
{
    const NegationType* leftNegation = get<NegationType>(left);
    LUAU_ASSERT(leftNegation);

    if (get<UnionType>(follow(leftNegation->ty)))
        return intersectNegatedUnion(left, right);

    const NegationType* rightNegation = get<NegationType>(right);
    LUAU_ASSERT(rightNegation);

    if (get<UnionType>(follow(rightNegation->ty)))
        return intersectNegatedUnion(right, left);

    Relation r = relate(leftNegation->ty, rightNegation->ty);

    switch (r)
    {
    case Relation::Coincident:
        // ~true & ~true
        return left;
    case Relation::Subset:
        // ~true & ~boolean
        return right;
    case Relation::Superset:
        // ~boolean & ~true
        return left;
    case Relation::Intersects:
    case Relation::Disjoint:
    default:
        // ~boolean & ~string
        return arena->addType(IntersectionType{{left, right}});
    }
}

TypeId TypeSimplifier::intersectIntersectionWithType(TypeId left, TypeId right)
{
    const IntersectionType* leftIntersection = get<IntersectionType>(left);
    LUAU_ASSERT(leftIntersection);

    if (leftIntersection->parts.size() > (size_t)DFInt::LuauSimplificationComplexityLimit)
        return addIntersection(arena, builtinTypes, {left, right});

    bool changed = false;
    TypeIds newParts;

    for (TypeId part : leftIntersection)
    {
        Relation r = relate(part, right);
        switch (r)
        {
        case Relation::Disjoint:
            return builtinTypes->neverType;
        case Relation::Coincident:
            newParts.insert(part);
            continue;
        case Relation::Subset:
            newParts.insert(part);
            continue;
        case Relation::Superset:
            newParts.insert(right);
            changed = true;
            continue;
        default:
            newParts.insert(part);
            newParts.insert(right);
            changed = true;
            continue;
        }
    }

    // It is sometimes the case that an intersection operation will result in
    // clipping a free type from the result.
    //
    // eg (number & 'a) & string --> never
    //
    // We want to only report the free types that are part of the result.
    for (TypeId part : newParts)
    {
        if (isTypeVariable(part))
            blockedTypes.insert(part);
    }

    if (!changed)
        return left;

    return intersectFromParts(std::move(newParts));
}

std::optional<TypeId> TypeSimplifier::basicIntersect(TypeId left, TypeId right)
{
    left = follow(left);
    right = follow(right);

    if (get<AnyType>(left) && get<ErrorType>(right))
        return right;
    if (get<AnyType>(right) && get<ErrorType>(left))
        return left;
    if (get<AnyType>(left))
        return arena->addType(UnionType{{right, builtinTypes->errorType}});
    if (get<AnyType>(right))
        return arena->addType(UnionType{{left, builtinTypes->errorType}});
    if (get<UnknownType>(left))
        return right;
    if (get<UnknownType>(right))
        return left;
    if (get<NeverType>(left))
        return left;
    if (get<NeverType>(right))
        return right;

    if (auto pt = get<PrimitiveType>(left); pt && pt->type == PrimitiveType::Boolean)
    {
        if (auto st = get<SingletonType>(right); st && st->variant.get_if<BooleanSingleton>())
            return right;
        if (auto nt = get<NegationType>(right))
        {
            if (auto st = get<SingletonType>(follow(nt->ty)); st && st->variant.get_if<BooleanSingleton>())
            {
                if (st->variant == BooleanSingleton{true})
                    return builtinTypes->falseType;
                else
                    return builtinTypes->trueType;
            }
        }
    }
    else if (auto pt = get<PrimitiveType>(right); pt && pt->type == PrimitiveType::Boolean)
    {
        if (auto st = get<SingletonType>(left); st && st->variant.get_if<BooleanSingleton>())
            return left;
        if (auto nt = get<NegationType>(left))
        {
            if (auto st = get<SingletonType>(follow(nt->ty)); st && st->variant.get_if<BooleanSingleton>())
            {
                if (st->variant == BooleanSingleton{true})
                    return builtinTypes->falseType;
                else
                    return builtinTypes->trueType;
            }
        }
    }

    if (const auto [lt, rt] = get2<TableType, TableType>(left, right); lt && rt)
    {
        if (1 == lt->props.size())
        {
            const auto [propName, leftProp] = *begin(lt->props);
            const bool leftPropIsRefinable = leftProp.isShared() || leftProp.isReadOnly();

            auto it = rt->props.find(propName);
            if (it != rt->props.end() && leftPropIsRefinable && it->second.isShared())
            {
                Relation r = relate(*leftProp.readTy, *it->second.readTy);

                switch (r)
                {
                case Relation::Disjoint:
                    return builtinTypes->neverType;
                case Relation::Superset:
                case Relation::Coincident:
                    return right;
                case Relation::Subset:
                    if (1 == rt->props.size() && leftProp.isShared())
                        return left;
                    break;
                default:
                    break;
                }
            }
        }
        else if (1 == rt->props.size())
            return basicIntersect(right, left);

        // If two tables have disjoint properties and indexers, we can combine them.
        if (!lt->indexer && !rt->indexer && lt->state == TableState::Sealed && rt->state == TableState::Sealed)
        {
            if (rt->props.empty())
                return left;

            bool areDisjoint = true;
            for (const auto& [name, leftProp] : lt->props)
            {
                if (rt->props.count(name))
                {
                    areDisjoint = false;
                    break;
                }
            }

            if (areDisjoint)
            {
                TableType merged{TableState::Sealed, TypeLevel{}, lt->scope};
                merged.props = lt->props;

                for (const auto& [name, rightProp] : rt->props)
                    merged.props[name] = rightProp;

                return arena->addType(std::move(merged));
            }
        }

        return std::nullopt;
    }

    if (isApproximatelyTruthyType(left))
        if (auto res = basicIntersectWithTruthy(right))
            return res;

    if (isApproximatelyTruthyType(right))
        if (auto res = basicIntersectWithTruthy(left))
            return res;

    if (isApproximatelyFalsyType(left))
        if (auto res = basicIntersectWithFalsy(right))
            return res;

    if (isApproximatelyFalsyType(right))
        if (auto res = basicIntersectWithFalsy(left))
            return res;


    Relation relation = relate(left, right);
    if (left == right || Relation::Coincident == relation)
        return left;

    if (relation == Relation::Disjoint)
        return builtinTypes->neverType;
    else if (relation == Relation::Subset)
        return left;
    else if (relation == Relation::Superset)
        return right;

    return std::nullopt;
}

TypeId TypeSimplifier::intersect(TypeId left, TypeId right)
{
    RecursionLimiter rl("TypeSimplifier::intersect", &recursionDepth, 15);

    left = simplify(left);
    right = simplify(right);

    if (left == right)
        return left;

    if (get<AnyType>(left) && get<ErrorType>(right))
        return right;
    if (get<AnyType>(right) && get<ErrorType>(left))
        return left;
    if (get<UnknownType>(left) && !get<ErrorType>(right))
        return right;
    if (get<UnknownType>(right) && !get<ErrorType>(left))
        return left;
    if (get<AnyType>(left) && get<UnionType>(right))
        return union_(builtinTypes->errorType, right);
    if (get<UnionType>(left) && get<AnyType>(right))
        return union_(builtinTypes->errorType, left);
    if (get<AnyType>(left))
        return arena->addType(UnionType{{right, builtinTypes->errorType}});
    if (get<AnyType>(right))
        return arena->addType(UnionType{{left, builtinTypes->errorType}});
    if (get<UnknownType>(left))
        return right;
    if (get<UnknownType>(right))
        return left;
    if (get<NeverType>(left))
        return left;
    if (get<NeverType>(right))
        return right;

    if (auto lf = get<FreeType>(left))
    {
        Relation r = relate(lf->upperBound, right);
        if (r == Relation::Subset || r == Relation::Coincident)
            return left;
    }
    else if (auto rf = get<FreeType>(right))
    {
        Relation r = relate(left, rf->upperBound);
        if (r == Relation::Superset || r == Relation::Coincident)
            return right;
    }

    if (isTypeVariable(left))
    {
        blockedTypes.insert(left);
        return addIntersection(arena, builtinTypes, {left, right});
    }

    if (isTypeVariable(right))
    {
        blockedTypes.insert(right);
        return addIntersection(arena, builtinTypes, {left, right});
    }

    if (get<UnionType>(left))
    {
        if (get<UnionType>(right))
            return intersectUnions(left, right);
        else
            return intersectUnionWithType(left, right);
    }
    else if (get<UnionType>(right))
        return intersectUnionWithType(right, left);

    if (get<IntersectionType>(left))
        return intersectIntersectionWithType(left, right);
    else if (get<IntersectionType>(right))
        return intersectIntersectionWithType(right, left);

    if (get<NegationType>(left))
    {
        if (get<NegationType>(right))
            return intersectNegations(left, right);
        else
            return intersectTypeWithNegation(left, right);
    }
    else if (get<NegationType>(right))
        return intersectTypeWithNegation(right, left);

    std::optional<TypeId> res = basicIntersect(left, right);
    if (res)
        return *res;
    else
        return arena->addType(IntersectionType{{left, right}});
}

TypeId TypeSimplifier::union_(TypeId left, TypeId right)
{
    RecursionLimiter rl("TypeSimplifier::union", &recursionDepth, 15);

    left = simplify(left);
    right = simplify(right);

    if (get<NeverType>(left))
        return right;
    if (get<NeverType>(right))
        return left;

    if (auto leftUnion = get<UnionType>(left))
    {
        bool changed = false;

        UnionBuilder ub(arena, builtinTypes);
        ub.reserve(leftUnion->options.size());
        for (TypeId part : leftUnion)
        {
            if (get<NeverType>(part))
            {
                changed = true;
                continue;
            }

            Relation r = relate(part, right);
            switch (r)
            {
            case Relation::Coincident:
            case Relation::Superset:
                return left;
            case Relation::Subset:
                ub.add(right);
                changed = true;
                break;
            default:
                ub.add(part);
                ub.add(right);
                changed = true;
                break;
            }
        }

        if (!changed)
            return left;

        // If the left-side is changed but has no parts, then the left-side union is uninhabited.
        if (ub.size() == 0)
            return right;

        return ub.build();
    }
    else if (get<UnionType>(right))
        return union_(right, left);

    Relation r = relate(left, right);
    if (left == right || r == Relation::Coincident || r == Relation::Superset)
        return left;

    if (r == Relation::Subset)
        return right;

    if (auto as = get<SingletonType>(left))
    {
        if (auto abs = as->variant.get_if<BooleanSingleton>())
        {
            if (auto bs = get<SingletonType>(right))
            {
                if (auto bbs = bs->variant.get_if<BooleanSingleton>())
                {
                    if (abs->value != bbs->value)
                        return builtinTypes->booleanType;
                }
            }
        }
    }

    if (const auto [lt, rt] = get2<TableType, TableType>(left, right); lt && rt)
    {
        if (1 == lt->props.size() && 1 == rt->props.size())
        {
            const auto [propName, leftProp] = *begin(lt->props);
            const auto [rightPropName, rightProp] = *begin(rt->props);

            if (rightPropName != propName)
                return arena->addType(UnionType{{left, right}});

            // Consider:
            //
            //  { prop: number? } | { prop: string? }
            //
            // Even though these two tables share a property, we cannot
            // simplify this type any further, otherwise we can, say,
            // launder a `{ prop: number? }` into a `{ prop: string? }`
            // and then write a string to it.
            //
            // We also elect to not simplify unsealed tables.
            if (!leftProp.isReadOnly() || !rightProp.isReadOnly() || lt->state != TableState::Sealed || rt->state != TableState::Sealed)
                return arena->addType(UnionType{{left, right}});

            // At this point, we have two read-only singleton tables, e.g.:
            //
            //  { read prop: number? } | { read prop: string? }
            //
            // We can relate these two properties and produce a simplified
            // version, with some special cases.

            switch (relate(*leftProp.readTy, *rightProp.readTy))
            {
            case Relation::Coincident:
            case Relation::Superset:
                // The left property is a superset (or coincident) of the
                // right, for example:
                //
                //  { read prop: number? } | { read prop: number }
                //
                return left;
            case Relation::Subset:
                // The left property is a subset of the right, for example:
                //
                //  { read prop: nil } | { read prop: false? }
                //
                return right;
            case Relation::Disjoint:
            case Relation::Intersects:
                // If we are disjoint *or* there's some overlap, then
                // we can create a new read-only singleton table with
                // a single property.
                //
                // We probably could do something quicker here for disjoint,
                // given that the union should just mint a new union type
                // anyhow.
                TableType result;
                result.state = TableState::Sealed;
                result.props[propName] = Property::readonly(union_(*leftProp.readTy, *rightProp.readTy));
                return arena->addType(std::move(result));
            }
        }
    }

    return arena->addType(UnionType{{left, right}});
}

TypeId TypeSimplifier::simplify(TypeId ty)
{
    DenseHashSet<TypeId> seen{nullptr};
    return simplify(ty, seen);
}

TypeId TypeSimplifier::simplify(TypeId ty, DenseHashSet<TypeId>& seen)
{
    RecursionLimiter limiter("TypeSimplifier::simplify", &recursionDepth, 60);

    ty = follow(ty);

    if (seen.find(ty))
        return ty;
    seen.insert(ty);

    if (auto nt = get<NegationType>(ty))
    {
        TypeId negatedTy = follow(nt->ty);
        if (get<AnyType>(negatedTy))
            return arena->addType(UnionType{{builtinTypes->neverType, builtinTypes->errorType}});
        else if (get<UnknownType>(negatedTy))
            return builtinTypes->neverType;
        else if (get<NeverType>(negatedTy))
            return builtinTypes->unknownType;
        if (auto nnt = get<NegationType>(negatedTy))
            return simplify(nnt->ty, seen);
    }

    // Promote {x: never} to never
    if (auto tt = get<TableType>(ty))
    {
        if (1 == tt->props.size())
        {
            if (std::optional<TypeId> readTy = begin(tt->props)->second.readTy)
            {
                TypeId propTy = simplify(*readTy, seen);
                if (get<NeverType>(propTy))
                    return builtinTypes->neverType;
            }
        }
    }

    return ty;
}

namespace
{

bool isSimpleDiscriminant(TypeId ty, DenseHashSet<TypeId>& seen)
{
    ty = follow(ty);
    // If we *ever* see a recursive type, bail right away, clearly that is
    // not simple.
    if (seen.contains(ty))
        return false;
    seen.insert(ty);

    // NOTE: We could probably support `{}` as a simple discriminant.
    if (auto ttv = get<TableType>(ty); ttv && ttv->props.size() == 1 && !ttv->indexer)
    {
        auto prop = begin(ttv->props)->second;
        return (!prop.readTy || isSimpleDiscriminant(*prop.readTy, seen)) && (!prop.writeTy || isSimpleDiscriminant(*prop.writeTy, seen));
    }

    if (auto nt = get<NegationType>(ty))
        return isSimpleDiscriminant(nt->ty, seen);

    return is<PrimitiveType, SingletonType, ExternType>(ty) || isApproximatelyTruthyType(ty) || isApproximatelyFalsyType(ty);
}

/**
 * There are some types that are "simple", and thus easy to intersect against:
 * - The "truthy" (`~(false?)`) and "falsy" (`false?`) types are simple.
 * - Primitive types, singleton types, and extern types are simple
 * - Table types are simple if they have no indexer, and have a single property
 *   who's read and write types are also simple.
 * - Cyclic types are never simple.
 */
bool isSimpleDiscriminant(TypeId ty)
{
    DenseHashSet<TypeId> seenSet{nullptr};
    return isSimpleDiscriminant(ty, seenSet);
}

} // namespace

std::optional<TypeId> TypeSimplifier::intersectOne(TypeId target, TypeId discriminant) const
{
    switch (relate(target, discriminant))
    {
    case Relation::Disjoint: // No A is a B or vice versa
        return builtinTypes->neverType;
    case Relation::Subset:     // Every A is in B
    case Relation::Coincident: // Every A is in B and vice versa
        return target;
    case Relation::Superset: // Every B is in A
        return discriminant;
    case Relation::Intersects:
    default:
        // Some As are in B and some Bs are in A.  ex (number | string) <-> (string | boolean).
        return std::nullopt;
    }
}

std::optional<TypeId> TypeSimplifier::subtractOne(TypeId target, TypeId discriminant) const
{
    target = follow(target);
    discriminant = follow(discriminant);

    if (auto nt = get<NegationType>(discriminant))
        return intersectOne(target, nt->ty);

    switch (relate(target, discriminant))
    {
    case Relation::Disjoint: // A v B is empty => A - B is equivalent to A
        return target;
    case Relation::Subset:     // A v B is A => A - B is empty
    case Relation::Coincident: // Same as above: A == B so A - B = {}
        return builtinTypes->neverType;
    case Relation::Superset:
    case Relation::Intersects:
    default:
        return std::nullopt;
    }
}

std::optional<Property> TypeSimplifier::intersectProperty(const Property& target, const Property& discriminant, DenseHashSet<TypeId>& seen) const
{
    // NOTE: I invite the reader to refactor the below code as a fun coding
    // exercise. It looks ugly to me, but I don't think we can make it
    // any cleaner.

    Property prop;
    prop.deprecated = target.deprecated || discriminant.deprecated;

    // We're trying to follow the following rules for both read and write types:
    // * If the type is present on both properties, intersect it, and return
    //   `std::nullopt` if we fail.
    // * If the type only exists on one property or the other, take that.

    if (target.readTy && discriminant.readTy)
    {
        prop.readTy = intersectWithSimpleDiscriminant(*target.readTy, *discriminant.readTy, seen);
        if (!prop.readTy)
            return std::nullopt;
    }
    else if (target.readTy && !discriminant.readTy)
        prop.readTy = target.readTy;
    else if (!target.readTy && discriminant.readTy)
        prop.readTy = discriminant.readTy;

    if (target.writeTy && discriminant.writeTy)
    {
        prop.writeTy = intersectWithSimpleDiscriminant(*target.writeTy, *discriminant.writeTy, seen);
        if (!prop.writeTy)
            return std::nullopt;
    }
    else if (target.writeTy && !discriminant.writeTy)
        prop.writeTy = target.writeTy;
    else if (!target.writeTy && discriminant.writeTy)
        prop.writeTy = discriminant.writeTy;

    return {prop};
}

std::optional<TypeId> TypeSimplifier::intersectWithSimpleDiscriminant(TypeId target, TypeId discriminant, DenseHashSet<TypeId>& seen) const
{
    if (seen.contains(target))
        return std::nullopt;

    target = follow(target);
    discriminant = follow(discriminant);

    if (auto ut = get<UnionType>(target))
    {
        seen.insert(target);
        TypeIds options;
        for (TypeId option : ut)
        {
            auto result = intersectWithSimpleDiscriminant(option, discriminant, seen);

            if (!result)
                return std::nullopt;

            if (is<UnknownType>(result))
                return builtinTypes->unknownType;

            if (!is<NeverType>(*result))
                options.insert(*result);
        }
        if (options.empty())
            return builtinTypes->neverType;
        if (options.size() == 1)
            return *options.begin();
        return arena->addType(UnionType{options.take()});
    }

    if (auto it = get<IntersectionType>(target))
    {
        seen.insert(target);
        TypeIds parts;
        for (TypeId part : it)
        {
            auto result = intersectWithSimpleDiscriminant(part, discriminant, seen);
            if (!result)
                return std::nullopt;

            if (is<NeverType>(*result))
                return builtinTypes->neverType;

            if (auto subIntersection = get<IntersectionType>(*result))
            {
                for (TypeId subOption : subIntersection)
                {
                    if (is<NeverType>(subOption))
                        return builtinTypes->neverType;
                    if (!is<UnknownType>(result))
                        parts.insert(*result);
                }
            }
            else if (!is<UnknownType>(*result))
                parts.insert(*result);
        }
        if (parts.empty())
            return builtinTypes->unknownType;
        if (parts.size() == 1)
            return *parts.begin();
        return arena->addType(IntersectionType{parts.take()});
    }

    if (auto ttv = get<TableType>(target))
    {
        if (auto discTtv = get<TableType>(discriminant))
        {
            // The precondition of this function is that `discriminant` is
            // simple, so if it's a table it *must* be a sealed table with
            // a single property and no indexer.
            LUAU_ASSERT(discTtv->props.size() == 1 && !discTtv->indexer);
            const auto discProp = begin(discTtv->props);
            if (auto tyProp = ttv->props.find(discProp->first); tyProp != ttv->props.end())
            {
                auto property = intersectProperty(tyProp->second, discProp->second, seen);
                if (!property)
                    return std::nullopt;
                if (property->readTy && is<NeverType>(follow(property->readTy)))
                    return builtinTypes->neverType;
                if (property->writeTy && is<NeverType>(follow(property->writeTy)))
                    return builtinTypes->neverType;

                // If the property we get back is pointer identical to the
                // original property, return the underlying property as an
                // optimization.
                if (tyProp->second.readTy == property->readTy && tyProp->second.writeTy == property->writeTy)
                    return target;

                CloneState cs{builtinTypes};
                TypeId result = shallowClone(target, *arena, cs, /* clonePersistentTypes */ true);
                auto resultTtv = getMutable<TableType>(result);
                LUAU_ASSERT(resultTtv);
                resultTtv->props[tyProp->first] = *property;
                // Shallow cloning clears out scopes, so let's put back the
                // scope from the original type.
                resultTtv->scope = ttv->scope;
                return result;
            }

            CloneState cs{builtinTypes};
            TypeId result = shallowClone(target, *arena, cs, /* clonePersistentTypes */ true);
            // Shallow cloning clears out scopes, so let's put back the
            // scope from the original type.
            auto resultTtv = getMutable<TableType>(result);
            LUAU_ASSERT(resultTtv);
            resultTtv->props.emplace(discProp->first, discProp->second);
            resultTtv->scope = ttv->scope;
            return result;
        }

        // At this point, we're doing something like:
        //
        //  { ... } & ~nil
        //
        // Which can be handled via fallthrough.
    }

    // FIXME: We could probably return to this.
    if (is<FreeType, GenericType, BlockedType, PendingExpansionType, TypeFunctionInstanceType>(target))
        return std::nullopt;

    if (isApproximatelyTruthyType(discriminant))
        return basicIntersectWithTruthy(target);

    if (isApproximatelyTruthyType(target))
        return basicIntersectWithTruthy(discriminant);

    if (isApproximatelyFalsyType(discriminant))
        return basicIntersectWithFalsy(target);

    if (isApproximatelyFalsyType(target))
        return basicIntersectWithFalsy(discriminant);

    if (is<AnyType>(target))
        return arena->addType(UnionType{{builtinTypes->errorType, discriminant}});

    if (is<ErrorType>(target))
        return builtinTypes->errorType;

    if (auto nty = get<NegationType>(discriminant))
        return subtractOne(target, nty->ty);

    return intersectOne(target, discriminant);
}

std::optional<TypeId> TypeSimplifier::intersectWithSimpleDiscriminant(TypeId target, TypeId discriminant) const
{
    DenseHashSet<TypeId> seenSet{nullptr};
    return intersectWithSimpleDiscriminant(target, discriminant, seenSet);
}

SimplifyResult simplifyIntersection(NotNull<BuiltinTypes> builtinTypes, NotNull<TypeArena> arena, TypeId left, TypeId right)
{
    TypeSimplifier s{builtinTypes, arena};

    // fprintf(stderr, "Intersect %s and %s ...\n", toString(left).c_str(), toString(right).c_str());

    TypeId res = s.intersect(left, right);

    // fprintf(stderr, "Intersect %s and %s -> %s\n", toString(left).c_str(), toString(right).c_str(), toString(res).c_str());

    return SimplifyResult{res, std::move(s.blockedTypes)};
}

SimplifyResult simplifyIntersection(NotNull<BuiltinTypes> builtinTypes, NotNull<TypeArena> arena, TypeIds parts)
{
    TypeSimplifier s{builtinTypes, arena};

    TypeId res = s.intersectFromParts(std::move(parts));

    return SimplifyResult{res, std::move(s.blockedTypes)};
}

SimplifyResult simplifyUnion(NotNull<BuiltinTypes> builtinTypes, NotNull<TypeArena> arena, TypeId left, TypeId right)
{
    TypeSimplifier s{builtinTypes, arena};

    TypeId res = s.union_(left, right);

    // fprintf(stderr, "Union %s and %s -> %s\n", toString(left).c_str(), toString(right).c_str(), toString(res).c_str());

    return SimplifyResult{res, std::move(s.blockedTypes)};
}


std::optional<TypeId> intersectWithSimpleDiscriminant(
    NotNull<BuiltinTypes> builtinTypes,
    NotNull<TypeArena> arena,
    TypeId target,
    TypeId discriminant
)
{
    if (!isSimpleDiscriminant(discriminant))
    {
        if (isSimpleDiscriminant(target))
            return intersectWithSimpleDiscriminant(builtinTypes, arena, discriminant, target);
        return std::nullopt;
    }

    TypeSimplifier s{builtinTypes, arena};

    return s.intersectWithSimpleDiscriminant(target, discriminant);
}


} // namespace Luau