/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
half(0) -> 0
half(s(s(x:S))) -> s(half(x:S))
log(s(0)) -> 0
log(s(s(x:S))) -> s(log(s(half(x:S))))
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
 LOG(s(s(x:S))) -> HALF(x:S)
 LOG(s(s(x:S))) -> LOG(s(half(x:S)))
-> Rules:
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 log(s(0)) -> 0
 log(s(s(x:S))) -> s(log(s(half(x:S))))

Problem 1: 

SCC Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
 LOG(s(s(x:S))) -> HALF(x:S)
 LOG(s(s(x:S))) -> LOG(s(half(x:S)))
-> Rules:
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 log(s(0)) -> 0
 log(s(s(x:S))) -> s(log(s(half(x:S))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
->->-> Rules:
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 log(s(0)) -> 0
 log(s(s(x:S))) -> s(log(s(half(x:S))))
->->Cycle:
->->-> Pairs:
 LOG(s(s(x:S))) -> LOG(s(half(x:S)))
->->-> Rules:
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 log(s(0)) -> 0
 log(s(s(x:S))) -> s(log(s(half(x:S))))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 log(s(0)) -> 0
 log(s(s(x:S))) -> s(log(s(half(x:S))))
->Projection:
 pi(HALF) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 LOG(s(s(x:S))) -> LOG(s(half(x:S)))
-> Rules:
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 log(s(0)) -> 0
 log(s(s(x:S))) -> s(log(s(half(x:S))))
-> Usable rules:
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[half](X) = 2.X
[log](X) = 0
[0] = 2
[fSNonEmpty] = 0
[s](X) = 2.X + 2
[HALF](X) = 0
[LOG](X) = X

Problem 1.2: 

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

The problem is finite.