YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
0(2(0(x1:S))) -> 1(0(1(x1:S)))
0(2(1(x1:S))) -> 1(0(2(x1:S)))
0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
1(2(0(x1:S))) -> 2(0(1(x1:S)))
1(2(1(x1:S))) -> 2(0(2(x1:S)))
1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
L(2(0(x1:S))) -> L(1(0(1(x1:S))))
L(2(1(x1:S))) -> L(1(0(2(x1:S))))
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(0(x1:S))) -> 0#(1(x1:S))
 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
 0#(2(0(x1:S))) -> 1#(x1:S)
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
-> Usable rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
->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:S))) -> 1#(0(1(x1:S)))
 0#(2(0(x1:S))) -> 1#(x1:S)
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
 0#(2(0(x1:S))) -> 1#(x1:S)
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
->->-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
 0#(2(0(x1:S))) -> 1#(x1:S)
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
-> Usable rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
->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:S))) -> 1#(x1:S)
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(0(x1:S))) -> 1#(x1:S)
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
->->-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(0(x1:S))) -> 1#(x1:S)
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
-> Usable rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
->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) = X
[1#](X) = X

Problem 1.1: 

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

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(1(x1:S))) -> 0#(2(x1:S))
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
-> Usable rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
->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 + 2
[1#](X) = 2.X + 1

Problem 1.1: 

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

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
 1#(2(0(x1:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
-> Usable rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
->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:S))) -> 0#(1(x1:S))
 1#(2(0(x1:S))) -> 1#(x1:S)
 1#(2(1(x1:S))) -> 0#(2(x1:S))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1#(2(0(x1:S))) -> 1#(x1:S)
->->-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 L#(2(0(x1:S))) -> L#(1(0(1(x1:S))))
 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
-> Usable rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
->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:S))) -> L#(1(0(2(x1:S))))
-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
->->-> Rules:
 0(2(0(x1:S))) -> 1(0(1(x1:S)))
 0(2(1(x1:S))) -> 1(0(2(x1:S)))
 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
 1(2(0(x1:S))) -> 2(0(1(x1:S)))
 1(2(1(x1:S))) -> 2(0(2(x1:S)))
 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
 L(2(1(x1:S))) -> L(1(0(2(x1:S))))

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.