Delimiting diagrams

We introduce the unifying notion of delimiting diagram. Hitherto unrelated results such as: Minimality of the internal needed strategy for orthogonal first-order term rewriting systems, maximality of the limit strategy for orthogonal higher-order pattern rewrite systems (with maximality of the strategy F∞ for the λ-calculus as a special case), and uniform normalisation of balanced weak Church–Rosser abstract rewriting systems, all are seen to follow from the property that any pair of diverging steps can be completed into a delimiting diagram. Apart from yielding simple uniform proofs of those results, the same methodology yields a proof of maximality of the strategy F∞ for the λx −-calculus. As far was we know, this is the first time that a strategy has been proven maximal for a λ-calculus with explicit substitutions. Throughout notation is from and unqualified references are to [5]. Let→,_ be ARSs (Def. 8.2.2). Definition. _ delimits → if for all b← a _ c, for some k, l, d, b _ d← c and n+k ≤ m+ l. Local delimitation, obtained by restricting n,m to 1, does not imply delimitation as witnessed by a← b c→ d with _ =→, the problem being (Fig. 1.3) that tiling with locally delimiting diagrams may not terminate. If it does, the completed diagram is delimiting, cf. the Extension Lemma (EL) below. First we prove the Progress Lemma (PL). These will be our tools. Lemma (Progress). If _ locally delimits →, and b← a _ c, then d← c for some d. Proof. By induction on n. In case n ≤ m, setting d = c works, and in case m = 0, d = b does. Supposing n > m ≥ 1 (see the figure below), there exist b and c such that b ← b ← a _ c _ c. By local delimitation, there are n ≤ m, d such that b _ ′ d ← ′ c. By the IH for LD = local delimitation c′