Calculating Church-Rosser Proofs in Kleene Algebra
ReIMICS '01 Revised Papers from the 6th International Conference and 1st Workshop of COST Action 274 TARSKI on Relational Methods in Computer Science
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We prove Church-Rosser statements in Kleene algebra in the spirit of the point-free style of functional programming. Proofs of Church-Rosser theorems are simple and general, using only algebraic properties of the regular operations. They are fixed point-based, induction-free and often amenable to automata, hence suited to mechanization. In the strip lemma of the $\lambda$-calculus, the term and algebra part are cleanly separated. Relating Kleene algebras with allegories explains Church-Rosser diagrams categorially. Using allegoric techniques, we also prove algebraic variants of Newman''s lemma and Church-Rosser statements modulo an equivalence relation, in which the well-foundedness assumptions are derived. Here, point-wise techniques are introduced via tabulations, but still the construction avoids the invention of complex induction orderings and measures. Our results can be read as a case study for the possibilities and limitations of point-free allegorial formal methods.