extension PureXML.Schema {
    /// XSD 1.0 "Particle Valid (Restriction)" for the common cases: a complex
    /// type derived by restriction must have a content model that accepts a
    /// subset of its base's. Covers element-vs-element name matching,
    /// element-vs-wildcard admission, wildcard narrowing, occurrence-range
    /// subsumption, and the group recursions (sequence/all order-preserving with
    /// emptiable skips, choice mapping, sequence-into-choice, sequence-into-all).
    /// The spec's effective-total-range arithmetic for MapAndSum is approximated
    /// by per-particle checks, a documented simplification.
    enum ParticleRestriction {
        /// The reason a restriction's content is not a valid subset of its
        /// base's, or nil when it is. `types` resolves an element's
        /// ``ElementType/typeReference`` so element-vs-element pairs can check that
        /// the restricting type derives from the base's (NameAndTypeOK).
        static func violation(restricted: ContentType, base: ContentType, types: [String: ElementType], derivation: [String: TypeDerivation]) -> String? {
            switch (restricted, base) {
            case (.empty, .empty), (.simpleContent, _), (_, .simpleContent):
                nil
            case let (.empty, .elementOnly(baseParticle)), let (.empty, .mixed(baseParticle)):
                emptiable(baseParticle) ? nil : "the base content is required, so the restriction cannot be EMPTY"
            case (.mixed, .elementOnly):
                "a restriction cannot add mixed content to an element-only base"
            case let (.elementOnly(restrictedParticle), .empty):
                // A content-free particle (an empty group, or one whose members all
                // have maxOccurs=0) accepts only the empty sequence, so it is a valid
                // restriction of EMPTY; only a particle that can contribute a child is
                // adding content.
                contentFree(restrictedParticle) ? nil : "the base content is EMPTY, so the restriction cannot add content"
            case (.mixed, .empty):
                "the base content is EMPTY, so the restriction cannot add content"
            case let (.elementOnly(restrictedParticle), .elementOnly(baseParticle)),
                 let (.elementOnly(restrictedParticle), .mixed(baseParticle)),
                 let (.mixed(restrictedParticle), .mixed(baseParticle)):
                // The W3C restriction algorithm is defined over the normalized content
                // model (pointless particles eliminated): a `maxOccurs=0` member
                // removed, a `{1,1}` single-member group unwrapped, a nested
                // same-compositor `{1,1}` group spliced. Without this, e.g. a base
                // `sequence(any{2,3})` is not seen as the wildcard it equals.
                valid(restrictedParticle.normalized(), baseParticle.normalized(), types, derivation)
                    ? nil
                    : "the restricted content model is not a subset of the base's"
            }
        }

        /// Whether `restricted` accepts a subset of `base` (the pairwise check).
        ///
        /// Occurrence subsumption (`rangeSubsumed`) is applied only to the
        /// same-kind pairings (element/element, wildcard/wildcard, group/group) and
        /// element/wildcard, where the two particles' own occurrence ranges are
        /// directly comparable. For the cross-kind pairings (element/group and
        /// group/element via RecurseAsIfGroup; group/wildcard via
        /// NSRecurseCheckCardinality) the occurrence is matched *inside* the case,
        /// against a member or as an effective total range, never the derived
        /// particle's own occurrence against the base group/wildcard's. Applying the
        /// outer check there over-rejects: e.g. `sequence(e1{2,3})` restricting
        /// `sequence(e1{1,3},...)` once the pointless single-member group is
        /// normalized to the bare `e1{2,3}` (`3 > 1` against the base sequence).
        static func valid(_ restricted: Particle, _ base: Particle, _ types: [String: ElementType], _ derivation: [String: TypeDerivation]) -> Bool {
            // A restriction that can never contribute a child (its own `maxOccurs=0`,
            // or a group whose every member is content-free) accepts only the empty
            // sequence, so it is a valid restriction of any emptiable base. Checked
            // before the structural cases, which would otherwise recurse into members
            // that never occur and wrongly reject (e.g. a `maxOccurs="0"` sequence
            // whose members have their own non-trivial occurrences).
            if contentFree(restricted) {
                return emptiable(base)
            }
            switch (restricted.term, base.term) {
            case let (.element(restrictedName, _, restrictedTypeName, _), .element(baseName, _, baseTypeName, _)):
                return rangeSubsumed(restricted, base)
                    && sameName(restrictedName, baseName)
                    && elementTypeRestrictionOK(restrictedTypeName, baseTypeName, types, derivation)
            case let (.element(name, _, _, _), .wildcard(wildcard)):
                return rangeSubsumed(restricted, base) && wildcard.admits(name)
            case let (.wildcard(restrictedWildcard), .wildcard(baseWildcard)):
                return rangeSubsumed(restricted, base) && narrows(restrictedWildcard, baseWildcard)
            case let (.element, .group(baseGroup)):
                // RecurseAsIfGroup: the lone element, as a one-particle sequence,
                // against the base group. The element's occurrence is matched against
                // a base member inside `groupValid`, not against the base group's own
                // occurrence, so no outer `rangeSubsumed` here.
                return groupValid(Group(compositor: .sequence, particles: [restricted]), baseGroup, types, derivation)
            case let (.group, .element(baseName, _, baseTypeName, _)):
                // The mirror of RecurseAsIfGroup, by the same cardinality argument as
                // group/wildcard: a base element accepts only its own name, its own
                // occurrence count of times. The derived group is a valid restriction
                // iff every leaf it can contain is that element (same name, and a type
                // that derives from the base's) AND the group's effective total
                // occurrence range is within the base element's. Comparing the derived
                // group's *own* occurrence to the element's would be wrong (a derived
                // `sequence(a,a)` emits two a's at occurrence {1,1}); the effective
                // range is the count that must be bounded.
                return leavesRestrict(restricted, toElement: baseName, baseTypeName, types, derivation)
                    && rangeWithinWildcard(
                        derivedMin: restricted.effectiveOccurrenceMin(),
                        derivedMax: restricted.effectiveOccurrenceMax(),
                        wildcardMin: base.minOccurs,
                        wildcardMax: base.maxOccurs,
                    )
            case let (.group(restrictedGroup), .group(baseGroup)) where restrictedGroup.compositor == .sequence && baseGroup.compositor == .choice:
                // MapAndSum (Sequence:Choice, XSD 1.0 §3.9.6): each derived particle is
                // a valid restriction of some base branch, AND the derived sequence's
                // occurrence count times its number of particles is within the base
                // choice's occurrence range. The outer occurrence subsumption used by
                // the same-compositor cases is wrong here (it would compare a derived
                // `sequence{2,4}` directly to a base `choice{3,9}`); the count-product
                // is the spec's measure, deliberately stricter than a pure subset.
                let members = restrictedGroup.particles.filter { $0.maxOccurs != 0 }
                let branches = baseGroup.particles.filter { $0.maxOccurs != 0 }
                return rangeWithinWildcard(
                    derivedMin: restricted.minOccurs * members.count,
                    derivedMax: restricted.maxOccurs.map { $0 * members.count },
                    wildcardMin: base.minOccurs,
                    wildcardMax: base.maxOccurs,
                )
                    && members.allSatisfy { member in branches.contains { valid(member, $0, types, derivation) } }
            case let (.group(restrictedGroup), .group(baseGroup)):
                // A content-free derived group was already handled above; here the
                // derived group can contribute content, so its outer occurrence must be
                // subsumed and its members must map onto the base's.
                return rangeSubsumed(restricted, base) && groupValid(restrictedGroup, baseGroup, types, derivation)
            case let (.group, .wildcard(wildcard)):
                // NSRecurseCheckCardinality: every leaf the group can contain is
                // admitted by the wildcard (an element by name, a nested wildcard by
                // narrowing), AND the group's effective total occurrence range is
                // within the wildcard's occurrence range.
                return leavesAdmitted(restricted, by: wildcard)
                    && rangeWithinWildcard(
                        derivedMin: restricted.effectiveOccurrenceMin(),
                        derivedMax: restricted.effectiveOccurrenceMax(),
                        wildcardMin: base.minOccurs,
                        wildcardMax: base.maxOccurs,
                    )
            default:
                return false
            }
        }

        private static func groupValid(_ restricted: Group, _ base: Group, _ types: [String: ElementType], _ derivation: [String: TypeDerivation]) -> Bool {
            // A particle that can never occur (maxOccurs=0) contributes nothing to the
            // language, so it is removed before mapping (a "pointless particle"). Left
            // in, it would wrongly consume a base particle and starve the particles
            // after it (e.g. `(e1 maxOccurs=0, e2)` restricting `(any)`).
            let restrictedParticles = restricted.particles.filter { $0.maxOccurs != 0 }
            let baseParticles = base.particles.filter { $0.maxOccurs != 0 }
            switch (restricted.compositor, base.compositor) {
            case (.sequence, .sequence), (.all, .all):
                return recurse(restrictedParticles, baseParticles, skippedMustBeEmptiable: true, types, derivation)
            case (.choice, .choice):
                // RecurseLax: an order-preserving mapping (W3C requires it; a choice
                // `(b|a)` is NOT a valid restriction of `(a|b)`), but skipped base
                // branches need not be emptiable. The base's substitution-group
                // expansion must therefore be in document order for this to be right.
                return recurse(restrictedParticles, baseParticles, skippedMustBeEmptiable: false, types, derivation)
            case (.sequence, .choice):
                // RecurseAsIfGroup for an element against a base choice arrives here as a
                // one-particle synthetic sequence: each derived particle fits some base
                // branch. (A genuine multi-particle sequence restricting a choice is
                // handled by MapAndSum at the particle level, where the occurrence
                // count-product is available.)
                return restrictedParticles.allSatisfy { particle in
                    baseParticles.contains { valid(particle, $0, types, derivation) }
                }
            case (.sequence, .all):
                // RecurseUnordered: map onto distinct base particles in any order.
                return recurseUnordered(restrictedParticles, baseParticles, types, derivation)
            default:
                return false
            }
        }

        /// Whether a particle can never contribute a child element or wildcard: it
        /// has `maxOccurs=0`, or is a group all of whose members are content-free
        /// (an empty group included). Its only legal content is the empty sequence.
        static func contentFree(_ particle: Particle) -> Bool {
            if particle.maxOccurs == 0 { return true }
            switch particle.term {
            case .element, .wildcard:
                return false
            case let .group(group):
                return group.particles.allSatisfy(contentFree)
            }
        }

        /// Order-preserving mapping of the restricted particles onto the base's.
        /// A skipped base particle must be emptiable for sequence/all (Recurse);
        /// for choice (RecurseLax) skipping is free.
        private static func recurse(
            _ restricted: [Particle],
            _ base: [Particle],
            skippedMustBeEmptiable: Bool,
            _ types: [String: ElementType],
            _ derivation: [String: TypeDerivation],
        ) -> Bool {
            var baseIndex = 0
            for particle in restricted {
                var matched = false
                while baseIndex < base.count {
                    let candidate = base[baseIndex]
                    baseIndex += 1
                    if valid(particle, candidate, types, derivation) {
                        matched = true
                        break
                    }
                    if skippedMustBeEmptiable, !emptiable(candidate) { return false }
                }
                if !matched { return false }
            }
            if skippedMustBeEmptiable {
                while baseIndex < base.count {
                    if !emptiable(base[baseIndex]) { return false }
                    baseIndex += 1
                }
            }
            return true
        }

        /// Any-order mapping onto distinct base particles; unmapped base
        /// particles must be emptiable.
        private static func recurseUnordered(_ restricted: [Particle], _ base: [Particle], _ types: [String: ElementType], _ derivation: [String: TypeDerivation]) -> Bool {
            var used = [Bool](repeating: false, count: base.count)
            for particle in restricted {
                guard let index = base.indices.first(where: { !used[$0] && valid(particle, base[$0], types, derivation) }) else {
                    return false
                }
                used[index] = true
            }
            return base.indices.allSatisfy { used[$0] || emptiable(base[$0]) }
        }

        /// Occurrence-range subsumption: the restriction may not occur less often
        /// than the base requires, nor more often than the base allows.
        private static func rangeSubsumed(_ restricted: Particle, _ base: Particle) -> Bool {
            guard restricted.minOccurs >= base.minOccurs else { return false }
            guard let baseMax = base.maxOccurs else { return true }
            guard let restrictedMax = restricted.maxOccurs else { return false }
            return restrictedMax <= baseMax
        }

        /// Whether a particle can match no children at all.
        static func emptiable(_ particle: Particle) -> Bool {
            if particle.minOccurs == 0 { return true }
            switch particle.term {
            case .element, .wildcard:
                return false
            case let .group(group):
                switch group.compositor {
                case .sequence, .all:
                    return group.particles.allSatisfy(emptiable)
                case .choice:
                    return group.particles.contains(where: emptiable)
                }
            }
        }

        /// Whether the restricted wildcard admits no namespace the base refuses.
        private static func narrows(_ restricted: Wildcard, _ base: Wildcard) -> Bool {
            switch (restricted.namespace, base.namespace) {
            case (_, .any):
                true
            case (.other, .other):
                restricted.targetNamespace == base.targetNamespace
            case let (.enumerated(narrow), .enumerated(wide)):
                narrow.isSubset(of: wide)
            case let (.enumerated(narrow), .other):
                !narrow.contains("") && !narrow.contains(base.targetNamespace ?? "")
            default:
                false
            }
        }

        private static func sameName(_ lhs: PureXML.Model.QualifiedName, _ rhs: PureXML.Model.QualifiedName) -> Bool {
            lhs.localName == rhs.localName && (lhs.namespaceURI ?? "") == (rhs.namespaceURI ?? "")
        }

        /// Whether every leaf a particle can contain is admitted by `wildcard`: an
        /// element by name, a nested wildcard by narrowing.
        private static func leavesAdmitted(_ particle: Particle, by wildcard: Wildcard) -> Bool {
            switch particle.term {
            case let .element(name, _, _, _): wildcard.admits(name)
            case let .wildcard(inner): narrows(inner, wildcard)
            case let .group(group): group.particles.allSatisfy { leavesAdmitted($0, by: wildcard) }
            }
        }

        /// Whether every leaf a particle can contain is the named base element (same
        /// name, and a type that derives from the base element's). A wildcard leaf is
        /// rejected: it can match names other than the base element's, so it is not a
        /// subset. Used when the base content model is a single element.
        private static func leavesRestrict(
            _ particle: Particle,
            toElement baseName: PureXML.Model.QualifiedName,
            _ baseTypeName: String?,
            _ types: [String: ElementType],
            _ derivation: [String: TypeDerivation],
        ) -> Bool {
            switch particle.term {
            case let .element(name, _, typeName, _):
                sameName(name, baseName) && elementTypeRestrictionOK(typeName, baseTypeName, types, derivation)
            case .wildcard:
                false
            case let .group(group):
                group.particles.allSatisfy { leavesRestrict($0, toElement: baseName, baseTypeName, types, derivation) }
            }
        }

        /// Whether a derived effective total range fits within a base wildcard's
        /// occurrence range: at least the wildcard's minimum, at most its maximum.
        private static func rangeWithinWildcard(derivedMin: Int, derivedMax: Int?, wildcardMin: Int, wildcardMax: Int?) -> Bool {
            guard derivedMin >= wildcardMin else { return false }
            guard let wildcardMax else { return true }
            guard let derivedMax else { return false }
            return derivedMax <= wildcardMax
        }
    }
}