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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 F(x,f(s(s(y)),f(z,w))) -> F(s(x),f(y,f(s(z),w)))
 F(x,f(s(s(y)),f(z,w))) -> F(s(z),w)
 F(x,f(s(s(y)),f(z,w))) -> F(y,f(s(z),w))
 F(x,f(s(s(y)),nil)) -> F(s(x),f(y,f(s(0),nil)))
-> Rules:
 L(f(s(s(y)),f(z,w))) -> L(f(s(0),f(y,f(s(z),w))))
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
-> Usable rules:
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = 2.X1 + X2
[0] = 0
[nil] = 1
[s](X) = X + 2
[F](X1,X2) = X1 + X2

Problem 1.1: 

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

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 F(x,f(s(s(y)),f(z,w))) -> F(s(z),w)
 F(x,f(s(s(y)),f(z,w))) -> F(y,f(s(z),w))
 F(x,f(s(s(y)),nil)) -> F(s(x),f(y,f(s(0),nil)))
-> Rules:
 L(f(s(s(y)),f(z,w))) -> L(f(s(0),f(y,f(s(z),w))))
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
-> Usable rules:
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = 2.X1 + X2
[0] = 0
[nil] = 2
[s](X) = X + 1
[F](X1,X2) = 2.X1 + X2

Problem 1.1: 

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

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 F(x,f(s(s(y)),f(z,w))) -> F(y,f(s(z),w))
 F(x,f(s(s(y)),nil)) -> F(s(x),f(y,f(s(0),nil)))
-> Rules:
 L(f(s(s(y)),f(z,w))) -> L(f(s(0),f(y,f(s(z),w))))
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
-> Usable rules:
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = 2.X1 + X2
[0] = 0
[nil] = 2
[s](X) = X + 2
[F](X1,X2) = X2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 L#(f(s(s(y)),f(z,w))) -> L#(f(s(0),f(y,f(s(z),w))))
-> Rules:
 L(f(s(s(y)),f(z,w))) -> L(f(s(0),f(y,f(s(z),w))))
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
-> Usable rules:
 f(x,f(s(s(y)),f(z,w))) -> f(s(x),f(y,f(s(z),w)))
 f(x,f(s(s(y)),nil)) -> f(s(x),f(y,f(s(0),nil)))
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = X1.X2 + X1 + 1/2.X2 + 1/2
[0] = 0
[nil] = 2
[s](X) = 2.X + 1/2
[L#](X) = X

Problem 1.2: 

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

The problem is finite.