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We propose a practical method for defining and proving properties of general (i.e., not necessarily structural) recursive functions in proof assistants based on type theory. The idea is to define the graph of the intended function as an inductive relation, and to prove that the relation actually represents a function, which is by construction the function that we are trying to define. Then, we generate induction principles for proving other arbitrary properties of the function. The approach has been experimented in the Coq proof assistant, but should work in like-minded proof asistants as well. It allows for functions with mutual recursive calls, nested recursive calls, and works also for the standard encoding of partial functions using total functions over a dependent type that restricts the original function's domain. We present simple examples and report on a larger case study (sets of integers represented as ordered lists of intervals) that we have conducted in the context of certified static analyses.
Vlad Rusu
Vlad.Rusu@irisa.fr
@TechReport{pichardie05a,
Author = {Pichardie, D. and Rusu, V.},
Title = {Defining and Reasoning About General Recursive Functions in Type Theory: a Practical Method},
Number = {1766},
Institution = {Irisa},
Month = {November},
Year = {2005}
}
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