/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 v_NonEmpty:S x:S)
(RULES
fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
p(s(0)) -> 0
p(s(s(x:S))) -> s(p(s(x:S)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(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:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
 FAC(s(x:S)) -> P(s(x:S))
 P(s(s(x:S))) -> P(s(x:S))
-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
 FAC(s(x:S)) -> P(s(x:S))
 P(s(s(x:S))) -> P(s(x:S))
-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 P(s(s(x:S))) -> P(s(x:S))
->->-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->->Cycle:
->->-> Pairs:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
->->-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 P(s(s(x:S))) -> P(s(x:S))
-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->Projection:
 pi(P) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 FAC(s(x:S)) -> FAC(p(s(x:S)))
-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
-> Usable rules:
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[fac](X) = 0
[p](X) = 1/2.X
[*](X1,X2) = 0
[0] = 1
[fSNonEmpty] = 0
[s](X) = 2.X.X + 1
[FAC](X) = 1/2.X.X
[P](X) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 fac(s(x:S)) -> *(fac(p(s(x:S))),s(x:S))
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.