Chiba et al. (2006) proposed a framework of program transformation of term rewriting systems by developed templates. Contrast to the previous framework of program transformation by templates based on lambda calculus, this framework provides a method to verify the correctness of transformation automatically. Tupling (Bird, 1980) is a well-known technique to eliminate redundant recursive calls for improving efficiency of programs. In Chiba et al.'s framework, however, one can not use tuple symbols to construct developed templates. Thus their framework is not capable of tupling transformations. In this paper, we propose a more flexible notion of templates so that a wider variety of transformations, including tupling transformations, can be handled.

Recursive programs are often easier to read and write than iterative ones, but their execution frequently requires large numbers of procedure calls and stack operations. This causes problems in program optimization related to inline coding and the locality of data references. In addition to these problems, defining programs recursively sometimes leads to repetitive execution of similar computations, causing programs to have exponential time complexity. As a result, recursion removal methods, which transform a given recursive program to an iterative one without using the stack and increasing the amount of computation time, have been studied since the 1970s. In 1998, our group proposed a recursion removal method for a linear recursive program. In this paper, we extend the method to deal with non-linear recursive programs with one descent function (RPODs), which are programs of the form f(x) = if p(x) then b(x) else a(c(x),f(d(x)),f(d2(x)),...,f(dn(x))). First, we define the cumulative function for an RPOD. Next, based on the new cumulative function, several transformation techniques for RPODs are shown. These include a unified method of deriving logarithmic-order iterative programs or loop-free programs. Finally, the relationships between our method and various tupling strategies are discussed.