details

property	value
status	complete
benchmark	lambda_prod.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n142.star.cs.uiowa.edu
space	Hamana_17

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.0703641 seconds
cpu usage	0.066885
user time	0.043487
system time	0.023398
max virtual memory	113188.0
max residence set size	4956.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: app : [] --> arrab -> a -> b lam : [] --> (a -> b) -> arrab pair : [] --> a -> b -> prodab pia : [] --> prodab -> a pib : [] --> prodab -> b Rules: app (lam (/\x.f x)) y => f y lam (/\x.app y x) => y pia (pair x y) => x pib (pair x y) => y pair (pia x) (pib 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: app : [arrab * a] --> b lam : [a -> b] --> arrab pair : [a * b] --> prodab pia : [prodab] --> a pib : [prodab] --> b ~AP1 : [a -> b * a] --> b Rules: app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) lam(/\x.app(X, x)) => X pia(pair(X, Y)) => X pib(pair(X, Y)) => Y pair(pia(X), pib(X)) => X app(lam(/\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: app : [arrab * a] --> b lam : [a -> b] --> arrab pair : [a * b] --> prodab pia : [prodab] --> a pib : [prodab] --> b Rules: app(lam(/\x.X(x)), Y) => X(Y) lam(/\x.app(X, x)) => X pia(pair(X, Y)) => X pib(pair(X, Y)) => Y pair(pia(X), pib(X)) => X We observe that the rules contain a first-order subset: pia(pair(X, Y)) => X pib(pair(X, Y)) => Y pair(pia(X), pib(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: pia(pair(PeRCenTX,PeRCenTY)) -> PeRCenTX || 2: pib(pair(PeRCenTX,PeRCenTY)) -> PeRCenTY || 3: pair(pia(PeRCenTX),pib(PeRCenTX)) -> PeRCenTX || 4: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTX || 5: TIlDePAIR(PeRCenTX,PeRCenTY) -> PeRCenTY || Number of strict rules: 5 || Direct POLO(bPol) ... removes: 4 1 3 5 2 || TIlDePAIR w: 2 * x1 + 2 * x2 + 1 || pair w: 2 * x1 + 2 * x2 + 1 || pib w: 2 * x1 + 1 || pia 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(lam(/\x.X(x)), Y) => X(Y) lam(/\x.app(X, x)) => X pia(pair(X, Y)) => X pib(pair(X, Y)) => Y pair(pia(X), pib(X)) => X Thus, the original system is terminating if (P_0, R_0, computable, all) is finite. popout

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all output return to Higher Order Rewriting Union Beta