On the confluence of lambda-calculus with conditional rewriting. F. Blanqui, C. Riba and C. Kirchner. Journal version of FOSSACS'06. TCS 411(37), 43 pages, 2010.
The confluence of untyped lambda-calculus with unconditional rewriting is now well understood. In this paper, we investigate the confluence of lambda-calculus with conditional rewriting and provide general results in two directions.
First, when conditional rules are algebraic. This extends results of Müller and Dougherty for unconditional rewriting. Two cases are considered, whether beta-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing beta-reduction to weakly normalizing beta-reduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve.
Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules.