details

property	value
status	complete
benchmark	kripke.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	Hamana_17

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.061863 seconds
cpu usage	0.059015
user time	0.036919
system time	0.022096
max virtual memory	113188.0
max residence set size	5324.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: abs : [] --> (a -> b) -> Arrab app : [] --> Arrab -> a -> b box : [] --> a -> Boxa unbox : [] --> Boxa -> a Rules: app (abs (/\x.f x)) y => f y abs (/\x.app y x) => y unbox (box x) => x box (unbox x) => x Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: Alphabet: abs : [a -> b] --> Arrab app : [Arrab * a] --> b box : [a] --> Boxa unbox : [Boxa] --> a ~AP1 : [a -> b * a] --> b Rules: app(abs(/\x.~AP1(F, x)), X) => ~AP1(F, X) abs(/\x.app(X, x)) => X unbox(box(X)) => X box(unbox(X)) => X app(abs(/\x.app(X, x)), Y) => app(X, Y) ~AP1(F, X) => F X Symbol ~AP1 is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. Additionally, we can remove some (now-)redundant rules. This gives: Alphabet: abs : [a -> b] --> Arrab app : [Arrab * a] --> b box : [a] --> Boxa unbox : [Boxa] --> a Rules: app(abs(/\x.X(x)), Y) => X(Y) abs(/\x.app(X, x)) => X unbox(box(X)) => X box(unbox(X)) => X We observe that the rules contain a first-order subset: unbox(box(X)) => X box(unbox(X)) => X Moreover, the system is finitely branching. Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS. According to the external first-order termination prover, this system is indeed Ce-terminating: || Input TRS: || 1: unbox(box(PeRCenTX)) -> PeRCenTX || 2: box(unbox(PeRCenTX)) -> PeRCenTX || 3: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX || 4: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY || Number of strict rules: 4 || Direct POLO(bPol) ... removes: 4 1 3 2 || TIlDePAIR w: x1 + 2 * x2 + 1 || box w: 2 * x1 + 1 || unbox w: 2 * x1 + 1 || Number of strict rules: 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, all): Dependency Pairs P_0: Rules R_0: app(abs(/\x.X(x)), Y) => X(Y) abs(/\x.app(X, x)) => X unbox(box(X)) => X box(unbox(X)) => X Thus, the original system is terminating if (P_0, R_0, computable, all) is finite. We consider the dependency pair problem (P_0, R_0, computable, all). 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. popout

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