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The Greendlinger lemma says that a nonempty freely reduced null-homotopic combinatorial loop in a complex satisfying the metric small cancellation condition $C^{\prime }(1/6)$ contains a consecutive segment of a cell boundary longer than half that boundary. In a reduced disc diagram without boundary spurs this is supplied by a boundary disc diagram shell, or by the single-cell case. The long segment can be replaced by the strictly shorter complementary segment.

The useful precise diagram version also allows degenerate diagrams. A reduced disc diagram is either a single vertex, a single closed cell, a disc diagram ladder, or has at least three disc diagram spurs or shells whose inner paths consist of at most three pieces. This is the Greendlinger ladder theorem. For such a shell, writing the perimeter as $QP$, where $Q$ is the exterior arc and $P$ is the union of its interior pieces, the strict $C^{\prime }(1/6)$ inequalities giveA ladder is a chain of cells and possibly connecting edges, with the two boundary paths running along its opposite sides. In a nontrivial ladder with at least two blocks its two ends are shells or spurs. The single-cell case also has the long exterior arc conclusion when viewed against an appropriate boundary decomposition.

A tree diagram with no two-cells should not be described as containing a shell: it has a spur unless it is a point. Retaining the spur and ladder alternatives is essential for the quasiconvexity argument below.

A piece is a nontrivial combinatorial path that occurs in two essentially distinct positions on oriented cell attaching paths. These may be different cells or distinct admissible occurrences on one attaching path; occurrences identified by the defining attaching-path symmetry are not counted twice. In a symmetrized group presentation, it is a common initial segment of distinct relator words. Internal arcs of a reduced disc diagram are pieces, because otherwise the neighboring cells form a cancellable pair.

Let $X$ be a finite combinatorial 2-complex satisfying $C^{\prime }(1/6)$, and let $Y$ be a connected CW subcomplex of its universal cover with no missing shells. If the largest cell perimeter is $L$, then $Y^{(1)}$ is $(L/2)$-quasiconvex in the universal cover's unit-edge 1-skeleton. Bound an ambient metric geodesic and an intrinsic metric geodesic of $Y^{(1)}$ by a reduced disc diagram. A shell on the ambient side would shorten a metric geodesic. A shell on the intrinsic side belongs to $Y$ and would shorten its intrinsic metric geodesic. Spurs are likewise excluded from side interiors. The Greendlinger ladder theorem forces a ladder between the two marked corners. Each cell gives a path of length at most half its perimeter from either side to the other.