/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 x y z)
(RULES
f(0,0,z) -> s(z)
f(0,s(0),s(0)) -> s(s(0))
f(0,s(0),s(s(z))) -> f(0,s(0),z)
f(0,s(s(y)),s(0)) -> f(0,y,s(0))
f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))
f(0,y,0) -> s(y)
f(s(0),y,z) -> f(0,s(y),s(z))
f(s(x),0,s(z)) -> f(x,s(0),z)
f(s(x),s(y),0) -> f(x,y,s(0))
f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
f(x,0,0) -> s(x)
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(0,s(s(y)),s(0)) -> F(0,y,s(0))
-> Rules:
 f(0,0,z) -> s(z)
 f(0,s(0),s(0)) -> s(s(0))
 f(0,s(0),s(s(z))) -> f(0,s(0),z)
 f(0,s(s(y)),s(0)) -> f(0,y,s(0))
 f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))
 f(0,y,0) -> s(y)
 f(s(0),y,z) -> f(0,s(y),s(z))
 f(s(x),0,s(z)) -> f(x,s(0),z)
 f(s(x),s(y),0) -> f(x,y,s(0))
 f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
 f(x,0,0) -> s(x)
->Projection:
 pi(F) = 2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 F(0,s(0),s(s(z))) -> F(0,s(0),z)
-> Rules:
 f(0,0,z) -> s(z)
 f(0,s(0),s(0)) -> s(s(0))
 f(0,s(0),s(s(z))) -> f(0,s(0),z)
 f(0,s(s(y)),s(0)) -> f(0,y,s(0))
 f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))
 f(0,y,0) -> s(y)
 f(s(0),y,z) -> f(0,s(y),s(z))
 f(s(x),0,s(z)) -> f(x,s(0),z)
 f(s(x),s(y),0) -> f(x,y,s(0))
 f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
 f(x,0,0) -> s(x)
->Projection:
 pi(F) = 3

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 F(0,s(s(y)),s(s(z))) -> F(0,s(s(y)),s(z))
 F(0,s(s(y)),s(s(z))) -> F(0,y,f(0,s(s(y)),s(z)))
-> Rules:
 f(0,0,z) -> s(z)
 f(0,s(0),s(0)) -> s(s(0))
 f(0,s(0),s(s(z))) -> f(0,s(0),z)
 f(0,s(s(y)),s(0)) -> f(0,y,s(0))
 f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))
 f(0,y,0) -> s(y)
 f(s(0),y,z) -> f(0,s(y),s(z))
 f(s(x),0,s(z)) -> f(x,s(0),z)
 f(s(x),s(y),0) -> f(x,y,s(0))
 f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
 f(x,0,0) -> s(x)
->Projection:
 pi(F) = 2

Problem 1.3: 

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

Problem 1.3: 

Subterm Processor:
-> Pairs:
 F(0,s(s(y)),s(s(z))) -> F(0,s(s(y)),s(z))
-> Rules:
 f(0,0,z) -> s(z)
 f(0,s(0),s(0)) -> s(s(0))
 f(0,s(0),s(s(z))) -> f(0,s(0),z)
 f(0,s(s(y)),s(0)) -> f(0,y,s(0))
 f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))
 f(0,y,0) -> s(y)
 f(s(0),y,z) -> f(0,s(y),s(z))
 f(s(x),0,s(z)) -> f(x,s(0),z)
 f(s(x),s(y),0) -> f(x,y,s(0))
 f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
 f(x,0,0) -> s(x)
->Projection:
 pi(F) = 3

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.4: 

Subterm Processor:
-> Pairs:
 F(s(x),s(y),s(z)) -> F(s(x),s(y),z)
-> Rules:
 f(0,0,z) -> s(z)
 f(0,s(0),s(0)) -> s(s(0))
 f(0,s(0),s(s(z))) -> f(0,s(0),z)
 f(0,s(s(y)),s(0)) -> f(0,y,s(0))
 f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))
 f(0,y,0) -> s(y)
 f(s(0),y,z) -> f(0,s(y),s(z))
 f(s(x),0,s(z)) -> f(x,s(0),z)
 f(s(x),s(y),0) -> f(x,y,s(0))
 f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
 f(x,0,0) -> s(x)
->Projection:
 pi(F) = 3

Problem 1.4: 

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

The problem is finite.