diff --git a/ql/ql/src/queries/performance/CyclicJoin.ql b/ql/ql/src/queries/performance/CyclicJoin.ql
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+/**
+ * @name Cyclic join
+ * @description Finds non-recursive predicate bodies and query `where` clauses whose
+ *              join graph contains an irreducible cycle (a triangle or a chordless
+ *              cycle of length 4, 5, 6, 7, 8, 9 or 10). Such alpha-cyclic conjunctive joins cannot
+ *              be evaluated optimally by any binary join plan and are the canonical case
+ *              that worst-case-optimal join algorithms (e.g. Leapfrog Triejoin) accelerate.
+ * @id ql/cyclic-join
+ * @tags performance
+ *       join-order
+ */
+
+import ql
+import codeql_ql.ast.internal.AstNodeNumbering
+
+/**
+ * A scope that gives rise to a conjunctive join: a predicate body or a query's
+ * `from`/`where`. Atoms within the same scope are (positively) conjoined.
+ */
+class JoinScope extends AstNode {
+  JoinScope() {
+    this instanceof Predicate or
+    this instanceof Select
+  }
+}
+
+/** Gets the nearest enclosing predicate or select of `n`. */
+JoinScope getScope(AstNode n) {
+  result = n.getParent+() and
+  not exists(JoinScope closer |
+    closer = n.getParent+() and
+    result = closer.getParent+()
+  )
+}
+
+/** Holds if `f` combines sub-formulas, i.e. it is not a single join atom. */
+predicate isConnective(Formula f) {
+  f instanceof Conjunction or
+  f instanceof Disjunction or
+  f instanceof Negation or
+  f instanceof Quantifier or
+  f instanceof IfFormula or
+  f instanceof Implication or
+  f instanceof HigherOrderFormula
+}
+
+/**
+ * An atomic formula: a leaf formula sitting in formula position inside a
+ * conjunctive context (its parent is a connective, a `select`, or a predicate
+ * body). Atoms nested inside an expression/another atom, or inside a `not`,
+ * are excluded because they do not contribute positive join edges.
+ */
+class Atom extends Formula {
+  Atom() {
+    not isConnective(this) and
+    exists(AstNode p | p = this.getParent() |
+      isConnective(p) or
+      p instanceof Select or
+      p instanceof Predicate
+    ) and
+    not this.getParent+() instanceof Negation
+  }
+}
+
+/** Holds if atom `a` references variable `v`. */
+predicate touches(Atom a, VarDef v) {
+  exists(VarAccess va | va = a.getAChild*() and va.getDeclaration() = v)
+}
+
+/** Holds if atom `a` uses a transitive-closure (`+`/`*`) call, i.e. is recursive. */
+predicate isRecursiveAtom(Atom a) {
+  exists(Call c | c = a.getAChild*() and c.isClosure(_))
+}
+
+/**
+ * Holds if, within scope `s`, there is a (non-recursive) atom joining the two
+ * distinct variables `u` and `v` -- an edge of the primal (join) graph.
+ */
+predicate primalEdge(JoinScope s, VarDef u, VarDef v) {
+  u != v and
+  exists(Atom a |
+    getScope(a) = s and
+    not isRecursiveAtom(a) and
+    touches(a, u) and
+    touches(a, v)
+  )
+}
+
+/**
+ * Holds if scope `s` contains an irreducible triangle over the distinct
+ * variables `x < y < z`. "Irreducible" means no single atom covers all three
+ * variables, so the three edges necessarily come from three different atoms and
+ * the join is genuinely alpha-cyclic (not GYO-reducible).
+ */
+predicate triangle(JoinScope s, VarDef x, VarDef y, VarDef z) {
+  getPreOrderId(x) < getPreOrderId(y) and
+  getPreOrderId(y) < getPreOrderId(z) and
+  primalEdge(s, x, y) and
+  primalEdge(s, y, z) and
+  primalEdge(s, x, z) and
+  // conformality: no atom covers the whole clique (which would make it acyclic)
+  not exists(Atom cover |
+    getScope(cover) = s and
+    touches(cover, x) and
+    touches(cover, y) and
+    touches(cover, z)
+  )
+}
+
+/**
+ * Holds if scope `s` contains a chordless 4-cycle `p0 - p1 - p2 - p3 - p0` over
+ * four distinct variables. Chordlessness (no diagonal edge) guarantees the cycle
+ * is irreducible/alpha-cyclic and that no atom covers it. `p0` is the minimum and
+ * `p1 < p3` to report each such cycle once.
+ */
+predicate square(JoinScope s, VarDef p0, VarDef p1, VarDef p2, VarDef p3) {
+  getPreOrderId(p0) < getPreOrderId(p1) and
+  getPreOrderId(p0) < getPreOrderId(p2) and
+  getPreOrderId(p0) < getPreOrderId(p3) and
+  getPreOrderId(p1) < getPreOrderId(p3) and
+  p1 != p2 and
+  p2 != p3 and
+  // the four sides
+  primalEdge(s, p0, p1) and
+  primalEdge(s, p1, p2) and
+  primalEdge(s, p2, p3) and
+  primalEdge(s, p3, p0) and
+  // the two diagonals must be absent (chordless)
+  not primalEdge(s, p0, p2) and
+  not primalEdge(s, p1, p3)
+}
+
+/**
+ * Holds if scope `s` contains a chordless 5-cycle `p0 - p1 - p2 - p3 - p4 - p0`
+ * over five distinct variables. `p0` is the minimum and `p1 < p4` orients the
+ * cycle so each one is reported once; all five chords are required absent, which
+ * also rules out any covering atom (irreducible/alpha-cyclic).
+ */
+predicate pentagon(JoinScope s, VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4) {
+  getPreOrderId(p0) < getPreOrderId(p1) and
+  getPreOrderId(p0) < getPreOrderId(p2) and
+  getPreOrderId(p0) < getPreOrderId(p3) and
+  getPreOrderId(p0) < getPreOrderId(p4) and
+  getPreOrderId(p1) < getPreOrderId(p4) and
+  p1 != p3 and
+  p2 != p4 and
+  // the five sides
+  primalEdge(s, p0, p1) and
+  primalEdge(s, p1, p2) and
+  primalEdge(s, p2, p3) and
+  primalEdge(s, p3, p4) and
+  primalEdge(s, p4, p0) and
+  // all five chords must be absent (chordless)
+  not primalEdge(s, p0, p2) and
+  not primalEdge(s, p0, p3) and
+  not primalEdge(s, p1, p3) and
+  not primalEdge(s, p1, p4) and
+  not primalEdge(s, p2, p4)
+}
+
+/**
+ * Holds if scope `s` contains a chordless 6-cycle
+ * `p0 - p1 - p2 - p3 - p4 - p5 - p0` over six distinct variables. `p0` is the
+ * minimum and `p1 < p5` orients the cycle so each one is reported once; all nine
+ * chords are required absent, which also rules out any covering atom.
+ */
+predicate hexagon(JoinScope s, VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5) {
+  getPreOrderId(p0) < getPreOrderId(p1) and
+  getPreOrderId(p0) < getPreOrderId(p2) and
+  getPreOrderId(p0) < getPreOrderId(p3) and
+  getPreOrderId(p0) < getPreOrderId(p4) and
+  getPreOrderId(p0) < getPreOrderId(p5) and
+  getPreOrderId(p1) < getPreOrderId(p5) and
+  p1 != p3 and
+  p1 != p4 and
+  p2 != p4 and
+  p2 != p5 and
+  p3 != p5 and
+  // the six sides
+  primalEdge(s, p0, p1) and
+  primalEdge(s, p1, p2) and
+  primalEdge(s, p2, p3) and
+  primalEdge(s, p3, p4) and
+  primalEdge(s, p4, p5) and
+  primalEdge(s, p5, p0) and
+  // all nine chords must be absent (chordless)
+  not primalEdge(s, p0, p2) and
+  not primalEdge(s, p0, p3) and
+  not primalEdge(s, p0, p4) and
+  not primalEdge(s, p1, p3) and
+  not primalEdge(s, p1, p4) and
+  not primalEdge(s, p1, p5) and
+  not primalEdge(s, p2, p4) and
+  not primalEdge(s, p2, p5) and
+  not primalEdge(s, p3, p5)
+}
+
+/**
+ * Holds if scope `s` contains a chordless 7-cycle
+ * `p0 - p1 - p2 - p3 - p4 - p5 - p6 - p0` over seven distinct variables. `p0` is
+ * the minimum and `p1 < p6` orients the cycle so each one is reported once; all 14
+ * chords are required absent, which also rules out any covering atom.
+ */
+predicate heptagon(
+  JoinScope s, VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6
+) {
+  getPreOrderId(p0) < getPreOrderId(p1) and
+  getPreOrderId(p0) < getPreOrderId(p2) and
+  getPreOrderId(p0) < getPreOrderId(p3) and
+  getPreOrderId(p0) < getPreOrderId(p4) and
+  getPreOrderId(p0) < getPreOrderId(p5) and
+  getPreOrderId(p0) < getPreOrderId(p6) and
+  getPreOrderId(p1) < getPreOrderId(p6) and
+  p1 != p3 and
+  p1 != p4 and
+  p1 != p5 and
+  p2 != p4 and
+  p2 != p5 and
+  p2 != p6 and
+  p3 != p5 and
+  p3 != p6 and
+  p4 != p6 and
+  // the seven sides
+  primalEdge(s, p0, p1) and
+  primalEdge(s, p1, p2) and
+  primalEdge(s, p2, p3) and
+  primalEdge(s, p3, p4) and
+  primalEdge(s, p4, p5) and
+  primalEdge(s, p5, p6) and
+  primalEdge(s, p6, p0) and
+  // all 14 chords must be absent (chordless)
+  not primalEdge(s, p0, p2) and
+  not primalEdge(s, p0, p3) and
+  not primalEdge(s, p0, p4) and
+  not primalEdge(s, p0, p5) and
+  not primalEdge(s, p1, p3) and
+  not primalEdge(s, p1, p4) and
+  not primalEdge(s, p1, p5) and
+  not primalEdge(s, p1, p6) and
+  not primalEdge(s, p2, p4) and
+  not primalEdge(s, p2, p5) and
+  not primalEdge(s, p2, p6) and
+  not primalEdge(s, p3, p5) and
+  not primalEdge(s, p3, p6) and
+  not primalEdge(s, p4, p6)
+}
+
+/**
+ * Holds if scope `s` contains a chordless 8-cycle
+ * `p0 - p1 - p2 - p3 - p4 - p5 - p6 - p7 - p0` over eight distinct variables. `p0`
+ * is the minimum and `p1 < p7` orients the cycle so each one is reported once; all
+ * 20 chords are required absent, which also rules out any covering atom.
+ */
+predicate octagon(
+  JoinScope s, VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6,
+  VarDef p7
+) {
+  getPreOrderId(p0) < getPreOrderId(p1) and
+  getPreOrderId(p0) < getPreOrderId(p2) and
+  getPreOrderId(p0) < getPreOrderId(p3) and
+  getPreOrderId(p0) < getPreOrderId(p4) and
+  getPreOrderId(p0) < getPreOrderId(p5) and
+  getPreOrderId(p0) < getPreOrderId(p6) and
+  getPreOrderId(p0) < getPreOrderId(p7) and
+  getPreOrderId(p1) < getPreOrderId(p7) and
+  p1 != p3 and
+  p1 != p4 and
+  p1 != p5 and
+  p1 != p6 and
+  p2 != p4 and
+  p2 != p5 and
+  p2 != p6 and
+  p2 != p7 and
+  p3 != p5 and
+  p3 != p6 and
+  p3 != p7 and
+  p4 != p6 and
+  p4 != p7 and
+  p5 != p7 and
+  // the eight sides
+  primalEdge(s, p0, p1) and
+  primalEdge(s, p1, p2) and
+  primalEdge(s, p2, p3) and
+  primalEdge(s, p3, p4) and
+  primalEdge(s, p4, p5) and
+  primalEdge(s, p5, p6) and
+  primalEdge(s, p6, p7) and
+  primalEdge(s, p7, p0) and
+  // all 20 chords must be absent (chordless)
+  not primalEdge(s, p0, p2) and
+  not primalEdge(s, p0, p3) and
+  not primalEdge(s, p0, p4) and
+  not primalEdge(s, p0, p5) and
+  not primalEdge(s, p0, p6) and
+  not primalEdge(s, p1, p3) and
+  not primalEdge(s, p1, p4) and
+  not primalEdge(s, p1, p5) and
+  not primalEdge(s, p1, p6) and
+  not primalEdge(s, p1, p7) and
+  not primalEdge(s, p2, p4) and
+  not primalEdge(s, p2, p5) and
+  not primalEdge(s, p2, p6) and
+  not primalEdge(s, p2, p7) and
+  not primalEdge(s, p3, p5) and
+  not primalEdge(s, p3, p6) and
+  not primalEdge(s, p3, p7) and
+  not primalEdge(s, p4, p6) and
+  not primalEdge(s, p4, p7) and
+  not primalEdge(s, p5, p7)
+}
+
+/**
+ * Holds if scope `s` contains a chordless 9-cycle
+ * `p0 - p1 - p2 - p3 - p4 - p5 - p6 - p7 - p8 - p0` over nine distinct variables.
+ * `p0` is the minimum and `p1 < p8` orients the cycle so each one is reported once;
+ * all 27 chords are required absent, which also rules out any covering atom.
+ */
+predicate enneagon(
+  JoinScope s, VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6,
+  VarDef p7, VarDef p8
+) {
+  getPreOrderId(p0) < getPreOrderId(p1) and
+  getPreOrderId(p0) < getPreOrderId(p2) and
+  getPreOrderId(p0) < getPreOrderId(p3) and
+  getPreOrderId(p0) < getPreOrderId(p4) and
+  getPreOrderId(p0) < getPreOrderId(p5) and
+  getPreOrderId(p0) < getPreOrderId(p6) and
+  getPreOrderId(p0) < getPreOrderId(p7) and
+  getPreOrderId(p0) < getPreOrderId(p8) and
+  getPreOrderId(p1) < getPreOrderId(p8) and
+  p1 != p3 and
+  p1 != p4 and
+  p1 != p5 and
+  p1 != p6 and
+  p1 != p7 and
+  p2 != p4 and
+  p2 != p5 and
+  p2 != p6 and
+  p2 != p7 and
+  p2 != p8 and
+  p3 != p5 and
+  p3 != p6 and
+  p3 != p7 and
+  p3 != p8 and
+  p4 != p6 and
+  p4 != p7 and
+  p4 != p8 and
+  p5 != p7 and
+  p5 != p8 and
+  p6 != p8 and
+  // the nine sides
+  primalEdge(s, p0, p1) and
+  primalEdge(s, p1, p2) and
+  primalEdge(s, p2, p3) and
+  primalEdge(s, p3, p4) and
+  primalEdge(s, p4, p5) and
+  primalEdge(s, p5, p6) and
+  primalEdge(s, p6, p7) and
+  primalEdge(s, p7, p8) and
+  primalEdge(s, p8, p0) and
+  // all 27 chords must be absent (chordless)
+  not primalEdge(s, p0, p2) and
+  not primalEdge(s, p0, p3) and
+  not primalEdge(s, p0, p4) and
+  not primalEdge(s, p0, p5) and
+  not primalEdge(s, p0, p6) and
+  not primalEdge(s, p0, p7) and
+  not primalEdge(s, p1, p3) and
+  not primalEdge(s, p1, p4) and
+  not primalEdge(s, p1, p5) and
+  not primalEdge(s, p1, p6) and
+  not primalEdge(s, p1, p7) and
+  not primalEdge(s, p1, p8) and
+  not primalEdge(s, p2, p4) and
+  not primalEdge(s, p2, p5) and
+  not primalEdge(s, p2, p6) and
+  not primalEdge(s, p2, p7) and
+  not primalEdge(s, p2, p8) and
+  not primalEdge(s, p3, p5) and
+  not primalEdge(s, p3, p6) and
+  not primalEdge(s, p3, p7) and
+  not primalEdge(s, p3, p8) and
+  not primalEdge(s, p4, p6) and
+  not primalEdge(s, p4, p7) and
+  not primalEdge(s, p4, p8) and
+  not primalEdge(s, p5, p7) and
+  not primalEdge(s, p5, p8) and
+  not primalEdge(s, p6, p8)
+}
+
+/**
+ * Holds if scope `s` contains a chordless 10-cycle
+ * `p0 - p1 - p2 - p3 - p4 - p5 - p6 - p7 - p8 - p9 - p0` over ten distinct
+ * variables. `p0` is the minimum and `p1 < p9` orients the cycle so each one is
+ * reported once; all 35 chords are required absent, which also rules out any
+ * covering atom.
+ */
+predicate decagon(
+  JoinScope s, VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6,
+  VarDef p7, VarDef p8, VarDef p9
+) {
+  getPreOrderId(p0) < getPreOrderId(p1) and
+  getPreOrderId(p0) < getPreOrderId(p2) and
+  getPreOrderId(p0) < getPreOrderId(p3) and
+  getPreOrderId(p0) < getPreOrderId(p4) and
+  getPreOrderId(p0) < getPreOrderId(p5) and
+  getPreOrderId(p0) < getPreOrderId(p6) and
+  getPreOrderId(p0) < getPreOrderId(p7) and
+  getPreOrderId(p0) < getPreOrderId(p8) and
+  getPreOrderId(p0) < getPreOrderId(p9) and
+  getPreOrderId(p1) < getPreOrderId(p9) and
+  p1 != p3 and
+  p1 != p4 and
+  p1 != p5 and
+  p1 != p6 and
+  p1 != p7 and
+  p1 != p8 and
+  p2 != p4 and
+  p2 != p5 and
+  p2 != p6 and
+  p2 != p7 and
+  p2 != p8 and
+  p2 != p9 and
+  p3 != p5 and
+  p3 != p6 and
+  p3 != p7 and
+  p3 != p8 and
+  p3 != p9 and
+  p4 != p6 and
+  p4 != p7 and
+  p4 != p8 and
+  p4 != p9 and
+  p5 != p7 and
+  p5 != p8 and
+  p5 != p9 and
+  p6 != p8 and
+  p6 != p9 and
+  p7 != p9 and
+  // the ten sides
+  primalEdge(s, p0, p1) and
+  primalEdge(s, p1, p2) and
+  primalEdge(s, p2, p3) and
+  primalEdge(s, p3, p4) and
+  primalEdge(s, p4, p5) and
+  primalEdge(s, p5, p6) and
+  primalEdge(s, p6, p7) and
+  primalEdge(s, p7, p8) and
+  primalEdge(s, p8, p9) and
+  primalEdge(s, p9, p0) and
+  // all 35 chords must be absent (chordless)
+  not primalEdge(s, p0, p2) and
+  not primalEdge(s, p0, p3) and
+  not primalEdge(s, p0, p4) and
+  not primalEdge(s, p0, p5) and
+  not primalEdge(s, p0, p6) and
+  not primalEdge(s, p0, p7) and
+  not primalEdge(s, p0, p8) and
+  not primalEdge(s, p1, p3) and
+  not primalEdge(s, p1, p4) and
+  not primalEdge(s, p1, p5) and
+  not primalEdge(s, p1, p6) and
+  not primalEdge(s, p1, p7) and
+  not primalEdge(s, p1, p8) and
+  not primalEdge(s, p1, p9) and
+  not primalEdge(s, p2, p4) and
+  not primalEdge(s, p2, p5) and
+  not primalEdge(s, p2, p6) and
+  not primalEdge(s, p2, p7) and
+  not primalEdge(s, p2, p8) and
+  not primalEdge(s, p2, p9) and
+  not primalEdge(s, p3, p5) and
+  not primalEdge(s, p3, p6) and
+  not primalEdge(s, p3, p7) and
+  not primalEdge(s, p3, p8) and
+  not primalEdge(s, p3, p9) and
+  not primalEdge(s, p4, p6) and
+  not primalEdge(s, p4, p7) and
+  not primalEdge(s, p4, p8) and
+  not primalEdge(s, p4, p9) and
+  not primalEdge(s, p5, p7) and
+  not primalEdge(s, p5, p8) and
+  not primalEdge(s, p5, p9) and
+  not primalEdge(s, p6, p8) and
+  not primalEdge(s, p6, p9) and
+  not primalEdge(s, p7, p9)
+}
+
+/** Gets a printable name for `v`. */
+string varName(VarDef v) { result = v.getName() }
+
+/**
+ * Holds if scope `s` contains an irreducible cyclic join of length `len` over
+ * the variables described by `vars`.
+ */
+predicate cyclicJoin(JoinScope s, int len, string vars) {
+  exists(VarDef x, VarDef y, VarDef z |
+    triangle(s, x, y, z) and
+    len = 3 and
+    vars = varName(x) + ", " + varName(y) + ", " + varName(z)
+  )
+  or
+  exists(VarDef p0, VarDef p1, VarDef p2, VarDef p3 |
+    square(s, p0, p1, p2, p3) and
+    len = 4 and
+    vars = varName(p0) + ", " + varName(p1) + ", " + varName(p2) + ", " + varName(p3)
+  )
+  or
+  exists(VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4 |
+    pentagon(s, p0, p1, p2, p3, p4) and
+    len = 5 and
+    vars =
+      varName(p0) + ", " + varName(p1) + ", " + varName(p2) + ", " + varName(p3) + ", " +
+        varName(p4)
+  )
+  or
+  exists(VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5 |
+    hexagon(s, p0, p1, p2, p3, p4, p5) and
+    len = 6 and
+    vars =
+      varName(p0) + ", " + varName(p1) + ", " + varName(p2) + ", " + varName(p3) + ", " +
+        varName(p4) + ", " + varName(p5)
+  )
+  or
+  exists(VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6 |
+    heptagon(s, p0, p1, p2, p3, p4, p5, p6) and
+    len = 7 and
+    vars =
+      varName(p0) + ", " + varName(p1) + ", " + varName(p2) + ", " + varName(p3) + ", " +
+        varName(p4) + ", " + varName(p5) + ", " + varName(p6)
+  )
+  or
+  exists(
+    VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6, VarDef p7
+  |
+    octagon(s, p0, p1, p2, p3, p4, p5, p6, p7) and
+    len = 8 and
+    vars =
+      varName(p0) + ", " + varName(p1) + ", " + varName(p2) + ", " + varName(p3) + ", " +
+        varName(p4) + ", " + varName(p5) + ", " + varName(p6) + ", " + varName(p7)
+  )
+  or
+  exists(
+    VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6, VarDef p7,
+    VarDef p8
+  |
+    enneagon(s, p0, p1, p2, p3, p4, p5, p6, p7, p8) and
+    len = 9 and
+    vars =
+      varName(p0) + ", " + varName(p1) + ", " + varName(p2) + ", " + varName(p3) + ", " +
+        varName(p4) + ", " + varName(p5) + ", " + varName(p6) + ", " + varName(p7) + ", " +
+        varName(p8)
+  )
+  or
+  exists(
+    VarDef p0, VarDef p1, VarDef p2, VarDef p3, VarDef p4, VarDef p5, VarDef p6, VarDef p7,
+    VarDef p8, VarDef p9
+  |
+    decagon(s, p0, p1, p2, p3, p4, p5, p6, p7, p8, p9) and
+    len = 10 and
+    vars =
+      varName(p0) + ", " + varName(p1) + ", " + varName(p2) + ", " + varName(p3) + ", " +
+        varName(p4) + ", " + varName(p5) + ", " + varName(p6) + ", " + varName(p7) + ", " +
+        varName(p8) + ", " + varName(p9)
+  )
+}
+
+from JoinScope s, int len, string vars
+where cyclicJoin(s, len, vars)
+select s,
+  "This " + s.getAPrimaryQlClass() + " contains a non-recursive cyclic (" + len.toString() +
+    "-cycle) join over variables: " + vars + "."