details

property	value
status	complete
benchmark	AotoYamada_05__002.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n140.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.206383 seconds
cpu usage	0.206709
user time	0.182001
system time	0.024708
max virtual memory	113188.0
max residence set size	6672.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: cons : [b * c] --> c false : [] --> a filter : [b -> a * c] --> c filtersub : [a * b -> a * c] --> c nil : [] --> c true : [] --> a Rules: filter(f, nil) => nil filter(f, cons(x, y)) => filtersub(f x, f, cons(x, y)) filtersub(true, f, cons(x, y)) => cons(x, filter(f, y)) filtersub(false, f, cons(x, y)) => filter(f, y) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]). We thus obtain the following dependency pair problem (P_0, R_0, computable, formative): Dependency Pairs P_0: 0] filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y)) 1] filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y) 2] filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y) Rules R_0: filter(F, nil) => nil filter(F, cons(X, Y)) => filtersub(F X, F, cons(X, Y)) filtersub(true, F, cons(X, Y)) => cons(X, filter(F, Y)) filtersub(false, F, cons(X, Y)) => filter(F, Y) Thus, the original system is terminating if (P_0, R_0, computable, formative) is finite. We consider the dependency pair problem (P_0, R_0, computable, formative). We apply the subterm criterion with the following projection function: nu(filter#) = 2 nu(filtersub#) = 3 Thus, we can orient the dependency pairs as follows: nu(filter#(F, cons(X, Y))) = cons(X, Y) = cons(X, Y) = nu(filtersub#(F X, F, cons(X, Y))) nu(filtersub#(true, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) nu(filtersub#(false, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_0, R_0, computable, f) by (P_1, R_0, computable, f), where P_1 contains: filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y)) Thus, the original system is terminating if (P_1, R_0, computable, formative) is finite. We consider the dependency pair problem (P_1, R_0, computable, formative). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: * 0 : This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. +++ Citations +++ [FuhKop19] C. Fuhs, and C. Kop. A static higher-order dependency pair framework. In Proceedings of ESOP 2019, 2019. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. [KusIsoSakBla09] K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui. Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems. In volume 92(10) of IEICE Transactions on Information and Systems. 2007--2015, 2009. popout

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all output return to Higher Order Rewriting Union Beta