/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 x1)
(RULES
P(p(x1)) -> x1
P(x1) -> Q(Q(p(x1)))
Q(p(q(x1))) -> q(p(Q(x1)))
Q(q(x1)) -> x1
p(P(x1)) -> x1
p(Q(Q(x1))) -> Q(Q(p(x1)))
p(p(x1)) -> q(q(x1))
q(Q(x1)) -> x1
q(q(p(x1))) -> p(q(q(x1)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 P#(x1) -> Q#(Q(p(x1)))
 P#(x1) -> Q#(p(x1))
 P#(x1) -> P(x1)
 Q#(p(q(x1))) -> Q#(x1)
 Q#(p(q(x1))) -> P(Q(x1))
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

SCC Processor:
-> Pairs:
 P#(x1) -> Q#(Q(p(x1)))
 P#(x1) -> Q#(p(x1))
 P#(x1) -> P(x1)
 Q#(p(q(x1))) -> Q#(x1)
 Q#(p(q(x1))) -> P(Q(x1))
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q#(x1)
 Q#(p(q(x1))) -> P(Q(x1))
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q#(x1)
 Q#(p(q(x1))) -> P(Q(x1))
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = X
[Q](X) = 2.X + 2
[p](X) = 2.X + 2
[q](X) = 2.X + 2
[Q#](X) = 2.X + 2
[P](X) = 2.X + 2
[Q](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> P(Q(x1))
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> P(Q(x1))
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> P(Q(x1))
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = X
[Q](X) = X
[p](X) = 2.X + 2
[q](X) = 2.X + 2
[Q#](X) = X
[P](X) = 2.X + 2
[Q](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> Q#(p(x1))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = X
[Q](X) = 2.X + 2
[p](X) = 2.X + 2
[q](X) = 2.X + 2
[Q#](X) = 2.X + 2
[P](X) = 2.X + 2
[Q](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(Q(Q(x1))) -> P(x1)
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = 2.X
[Q](X) = 2.X + 2
[p](X) = 2.X + 2
[q](X) = 2.X + 2
[Q#](X) = 2.X + 2
[P](X) = 2.X + 2
[Q](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(p(x1)) -> Q(q(x1))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = X
[Q](X) = 2.X + 2
[p](X) = 2.X + 2
[q](X) = X
[Q#](X) = 2.X + 2
[P](X) = 2.X + 2
[Q](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 P(p(x1)) -> Q(x1)
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = 2.X
[Q](X) = X + 2
[p](X) = X + 2
[q](X) = X + 2
[Q#](X) = X + 1
[P](X) = X + 1
[Q](X) = X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(q(x1))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = 2.X
[Q](X) = 2.X + 2
[p](X) = 2.X + 2
[q](X) = 2.X + 2
[Q#](X) = 2.X + 2
[P](X) = 2.X + 2
[Q](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(x1)
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
 Q(q(p(x1))) -> Q(x1)
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = 2.X
[Q](X) = 2.X + 2
[p](X) = 2.X + 2
[q](X) = 2.X + 2
[Q#](X) = 2.X + 2
[P](X) = 2.X + 2
[Q](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
->->-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 Q#(p(q(x1))) -> Q(p(Q(x1)))
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
-> Usable rules:
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[P](X) = X
[Q](X) = 2.X + 1
[p](X) = 2.X
[q](X) = 1/2.X
[Q#](X) = 2.X + 2
[P](X) = 2.X + 1/2
[Q](X) = 1/2.X + 1/2

Problem 1: 

SCC Processor:
-> Pairs:
 P(Q(Q(x1))) -> Q#(Q(p(x1)))
 Q(q(p(x1))) -> P(q(q(x1)))
-> Rules:
 P(p(x1)) -> x1
 P(x1) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 q(Q(x1)) -> x1
 q(q(p(x1))) -> p(q(q(x1)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.