/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem 1: 

(VAR x)
(RULES
f(0) -> s(0)
f(s(0)) -> s(0)
f(s(s(x))) -> f(f(s(x)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(s(x))) -> F(f(s(x)))
 F(s(s(x))) -> F(s(x))
-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(s(x))) -> F(f(s(x)))
 F(s(s(x))) -> F(s(x))
-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(s(x))) -> F(f(s(x)))
 F(s(s(x))) -> F(s(x))
->->-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 F(s(s(x))) -> F(f(s(x)))
 F(s(s(x))) -> F(s(x))
-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))
-> Usable rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = X + 2
[0] = 0
[s](X) = 2.X + 2
[F](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(s(x))) -> F(s(x))
-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(s(x))) -> F(s(x))
->->-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))

Problem 1: 

Subterm Processor:
-> Pairs:
 F(s(s(x))) -> F(s(x))
-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))
->Projection:
 pi(F) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0) -> s(0)
 f(s(0)) -> s(0)
 f(s(s(x))) -> f(f(s(x)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.