details

property	value
status	complete
benchmark	27.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n033.star.cs.uiowa.edu
space	Der95

run statistics

property	value
solver	Wanda
configuration	FirstOrder
runtime (wallclock)	0.0223751068115 seconds
cpu usage	0.019597988
max memory	1396736.0

stage attributes

key	value
output-size	2136
starexec-result	YES

output

/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. f : [o] --> o g : [o * o] --> o h : [o * o] --> o h(f(X), Y) => f(g(X, Y)) g(X, Y) => h(X, Y) As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95]. Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows: f : [v] --> v g : [v * i] --> v h : [v * i] --> v We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): h(f(X), Y) >? f(g(X, Y)) g(X, Y) >? h(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: f = \y0.2 + y0 g = \y0y1.2 + y1 + 2y0 h = \y0y1.1 + y1 + 2y0 Using this interpretation, the requirements translate to: [[h(f(_x0), _x1)]] = 5 + x1 + 2x0 > 4 + x1 + 2x0 = [[f(g(_x0, _x1))]] [[g(_x0, _x1)]] = 2 + x1 + 2x0 > 1 + x1 + 2x0 = [[h(_x0, _x1)]] We can thus remove the following rules: h(f(X), Y) => f(g(X, Y)) g(X, Y) => h(X, Y) All rules were succesfully removed. Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold. +++ Citations +++ [FuhGieParSchSwi11] C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski. Proving Termination by Dependency Pairs and Inductive Theorem Proving. In volume 47(2) of Journal of Automated Reasoning. 133--160, 2011. [Gra95] B. Gramlich. Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems. In volume 24(1-2) of Fundamentae Informaticae. 3--23, 1995. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. popout

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