/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 x1)
(RULES
0(2(0(x1))) -> 1(0(1(x1)))
0(2(1(x1))) -> 1(0(2(x1)))
0(2(R(x1))) -> 1(0(1(R(x1))))
1(2(0(x1))) -> 2(0(1(x1)))
1(2(1(x1))) -> 2(0(2(x1)))
1(2(R(x1))) -> 2(0(1(R(x1))))
L(2(0(x1))) -> L(1(0(1(x1))))
L(2(1(x1))) -> L(1(0(2(x1))))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 0#(2(0(x1))) -> 0#(1(x1))
 0#(2(0(x1))) -> 1#(0(1(x1)))
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
 L#(2(0(x1))) -> 0#(1(x1))
 L#(2(0(x1))) -> 1#(0(1(x1)))
 L#(2(0(x1))) -> 1#(x1)
 L#(2(0(x1))) -> L#(1(0(1(x1))))
 L#(2(1(x1))) -> 0#(2(x1))
 L#(2(1(x1))) -> 1#(0(2(x1)))
 L#(2(1(x1))) -> L#(1(0(2(x1))))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))

Problem 1: 

SCC Processor:
-> Pairs:
 0#(2(0(x1))) -> 0#(1(x1))
 0#(2(0(x1))) -> 1#(0(1(x1)))
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
 L#(2(0(x1))) -> 0#(1(x1))
 L#(2(0(x1))) -> 1#(0(1(x1)))
 L#(2(0(x1))) -> 1#(x1)
 L#(2(0(x1))) -> L#(1(0(1(x1))))
 L#(2(1(x1))) -> 0#(2(x1))
 L#(2(1(x1))) -> 1#(0(2(x1)))
 L#(2(1(x1))) -> L#(1(0(2(x1))))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(0(x1))) -> 0#(1(x1))
 0#(2(0(x1))) -> 1#(0(1(x1)))
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->->Cycle:
->->-> Pairs:
 L#(2(0(x1))) -> L#(1(0(1(x1))))
 L#(2(1(x1))) -> L#(1(0(2(x1))))
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(0(x1))) -> 0#(1(x1))
 0#(2(0(x1))) -> 1#(0(1(x1)))
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
-> Usable rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[1](X) = X + 1
[2](X) = X + 2
[R](X) = 2.X + 1
[0#](X) = 2.X + 2
[1#](X) = 2.X + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 0#(2(0(x1))) -> 1#(0(1(x1)))
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(0(x1))) -> 1#(0(1(x1)))
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(0(x1))) -> 1#(0(1(x1)))
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
-> Usable rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[1](X) = X + 1
[2](X) = X + 2
[R](X) = X + 2
[0#](X) = 2.X + 2
[1#](X) = 2.X + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(0(x1))) -> 1#(x1)
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
-> Usable rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[1](X) = X + 1
[2](X) = X + 2
[R](X) = X + 1
[0#](X) = X + 1
[1#](X) = X + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(1(x1))) -> 0#(2(x1))
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
-> Usable rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[1](X) = X + 1
[2](X) = X + 2
[R](X) = 2.X + 2
[0#](X) = 2.X
[1#](X) = 2.X + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(1(x1))) -> 1#(0(2(x1)))
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
-> Usable rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[1](X) = X + 1
[2](X) = X + 2
[R](X) = 2.X + 1
[0#](X) = 2.X + 2
[1#](X) = 2.X + 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 1#(2(0(x1))) -> 0#(1(x1))
 1#(2(0(x1))) -> 1#(x1)
 1#(2(1(x1))) -> 0#(2(x1))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1#(2(0(x1))) -> 1#(x1)
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))

Problem 1.1: 

Subterm Processor:
-> Pairs:
 1#(2(0(x1))) -> 1#(x1)
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Projection:
 pi(1#) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 L#(2(0(x1))) -> L#(1(0(1(x1))))
 L#(2(1(x1))) -> L#(1(0(2(x1))))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
-> Usable rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[1](X) = 2.X
[2](X) = 2.X + 2
[R](X) = 2
[L#](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 L#(2(1(x1))) -> L#(1(0(2(x1))))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 L#(2(1(x1))) -> L#(1(0(2(x1))))
->->-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 L#(2(1(x1))) -> L#(1(0(2(x1))))
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
-> Usable rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[1](X) = 2.X
[2](X) = 2.X + 2
[R](X) = 2.X + 2
[L#](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(2(0(x1))) -> 1(0(1(x1)))
 0(2(1(x1))) -> 1(0(2(x1)))
 0(2(R(x1))) -> 1(0(1(R(x1))))
 1(2(0(x1))) -> 2(0(1(x1)))
 1(2(1(x1))) -> 2(0(2(x1)))
 1(2(R(x1))) -> 2(0(1(R(x1))))
 L(2(0(x1))) -> L(1(0(1(x1))))
 L(2(1(x1))) -> L(1(0(2(x1))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.