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Higher Order Rewriting Union Beta pair #487093783
details
property
value
status
complete
benchmark
kop12lmcs2.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n143.star.cs.uiowa.edu
space
Kop_13
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.151472 seconds
cpu usage
0.151556
user time
0.135799
system time
0.015757
max virtual memory
113188.0
max residence set size
6160.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: cons : [nat -> nat * funlist] --> funlist false : [] --> bool head : [funlist] --> nat -> nat if : [bool * nat -> string * nat -> string] --> nat -> string nil : [] --> funlist s : [nat] --> nat tail : [funlist] --> funlist test : [nat -> nat] --> bool true : [] --> bool Rules: if(true, f, g) => f if(false, f, g) => g test(/\x.s(x)) => true head(cons(f, x)) => f tail(cons(f, x)) => x 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]). In order to do so, we start by eta-expanding the system, which gives: if(true, F, G, X) => F X if(false, F, G, X) => G X test(/\x.s(x)) => true head(cons(F, X), Y) => F Y tail(cons(F, X)) => X We thus obtain the following dependency pair problem (P_0, R_0, computable, formative): Dependency Pairs P_0: Rules R_0: if(true, F, G, X) => F X if(false, F, G, X) => G X test(/\x.s(x)) => true head(cons(F, X), Y) => F Y tail(cons(F, X)) => X 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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 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.
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