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Higher-order rewriting is a framework in which one can write higher-order programs and study their properties. One such property is termination: the situation that for all inputs, the program eventually halts its execution and produces an output. Several tools have been developed to check whether higher-order rewriting systems are terminating. However, developing such tools is difficult and can be error-prone. In this paper, we present a way of certifying termination proofs of higher-order term rewriting systems. We formalize a specific method, namely the polynomial interpretation method, that is used to prove termination. In addition, we give a program that turns the output of Wanda, a termination analysis tool for higher-order rewriting systems, into a Coq script, so that we can check whether the output is a valid proof of termination.

Time complexity in rewriting is naturally understood as the number of steps needed to reduce terms to normal forms. Establishing complexity bounds to this measure is a well-known problem in the rewriting community. A vast majority of techniques to find such bounds consist of modifying termination proofs in order to recover complexity information. This has been done for instance with semantic interpretations, recursive path orders, and dependency pairs. In this paper, we follow the same program by tailoring tuple interpretations to deal with innermost complexity analysis. A tuple interpretation interprets terms as tuples holding upper bounds to the cost of reduction and size of normal forms. In contrast with the full rewriting setting, the strongly monotonic requirement for cost components is dropped when reductions are innermost. This weakened requirement on cost tuples allows us to prove the innermost version of the compatibility result: if all rules in a term rewriting system can be strictly oriented, then the innermost rewrite relation is well-founded. We establish the necessary conditions for which tuple interpretations guarantee polynomial bounds to the runtime of compatible systems and describe a search procedure for such interpretations.

We define nominal equational problems of the form $$\exists \overline{W} \forall \overline{Y} : P$$ ∃ W ¯ ∀ Y ¯ : P , where $$P$$ P consists of conjunctions and disjunctions of equations $$s\approx _\alpha t$$ s ≈ α t , freshness constraints $$a\#t$$ a # t and their negations: $$s \not \approx _\alpha t$$ s ≉ α t and "Equation missing", where $$a$$ a is an atom and $$s, t$$ s , t nominal terms. We give a general definition of solution and a set of simplification rules to compute solutions in the nominal ground term algebra. For the latter, we define notions of solved form from which solutions can be easily extracted and show that the simplification rules are sound, preserving, and complete. With a particular strategy for rule application, the simplification process terminates and thus specifies an algorithm to solve nominal equational problems. These results generalise previous results obtained by Comon and Lescanne for first-order languages to languages with binding operators. In particular, we show that the problem of deciding the validity of a first-order equational formula in a language with binding operators (i.e., validity modulo $$\alpha $$ α -equality) is decidable.