YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
0(q1(x1:S)) -> q2(1(x1:S))
0(q2(x1:S)) -> 0(q0(x1:S))
1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
1(q2(x1:S)) -> q2(1(x1:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 0#(q1(x1:S)) -> 1#(x1:S)
 1#(q0(0(x1:S))) -> 0#(0(q1(x1:S)))
 1#(q0(0(x1:S))) -> 0#(q1(x1:S))
 1#(q0(1(x1:S))) -> 0#(1(q1(x1:S)))
 1#(q0(1(x1:S))) -> 1#(q1(x1:S))
 1#(q1(0(x1:S))) -> 0#(q1(x1:S))
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q1(x1:S)) -> 1#(x1:S)
 1#(q0(0(x1:S))) -> 0#(0(q1(x1:S)))
 1#(q0(0(x1:S))) -> 0#(q1(x1:S))
 1#(q0(1(x1:S))) -> 0#(1(q1(x1:S)))
 1#(q0(1(x1:S))) -> 1#(q1(x1:S))
 1#(q1(0(x1:S))) -> 0#(q1(x1:S))
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q1(x1:S)) -> 1#(x1:S)
 1#(q0(0(x1:S))) -> 0#(0(q1(x1:S)))
 1#(q0(0(x1:S))) -> 0#(q1(x1:S))
 1#(q0(1(x1:S))) -> 0#(1(q1(x1:S)))
 1#(q0(1(x1:S))) -> 1#(q1(x1:S))
 1#(q1(0(x1:S))) -> 0#(q1(x1:S))
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
->->-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q1(x1:S)) -> 1#(x1:S)
 1#(q0(0(x1:S))) -> 0#(0(q1(x1:S)))
 1#(q0(0(x1:S))) -> 0#(q1(x1:S))
 1#(q0(1(x1:S))) -> 0#(1(q1(x1:S)))
 1#(q0(1(x1:S))) -> 1#(q1(x1:S))
 1#(q1(0(x1:S))) -> 0#(q1(x1:S))
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
-> Usable rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[1](X) = 2.X + 1
[q0](X) = 2.X + 2
[q1](X) = 2.X + 1
[q2](X) = 2.X + 2
[0#](X) = X + 2
[1#](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 1#(q0(0(x1:S))) -> 0#(0(q1(x1:S)))
 1#(q0(0(x1:S))) -> 0#(q1(x1:S))
 1#(q0(1(x1:S))) -> 0#(1(q1(x1:S)))
 1#(q0(1(x1:S))) -> 1#(q1(x1:S))
 1#(q1(0(x1:S))) -> 0#(q1(x1:S))
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1#(q0(1(x1:S))) -> 1#(q1(x1:S))
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
->->-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1#(q0(1(x1:S))) -> 1#(q1(x1:S))
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
-> Usable rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 1
[1](X) = 2.X
[q0](X) = 2.X + 1
[q1](X) = 2.X
[q2](X) = 2.X + 1
[1#](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
->->-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
-> Usable rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[1](X) = 2.X
[q0](X) = 2.X + 2
[q1](X) = 2.X
[q2](X) = 2.X + 2
[1#](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
->->-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S)))
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
-> Usable rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[1](X) = 2.X + 2
[q0](X) = X + 1
[q1](X) = 2.X + 1
[q2](X) = X + 1
[1#](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
->->-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->->Cycle:
->->-> Pairs:
 1#(q2(x1:S)) -> 1#(x1:S)
->->-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 1#(q1(1(x1:S))) -> 1#(q1(x1:S))
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1](X) = 2.X + 2
[q1](X) = 2.X
[1#](X) = 2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 1#(q2(x1:S)) -> 1#(x1:S)
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Projection:
 pi(1#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(q1(x1:S)) -> q2(1(x1:S))
 0(q2(x1:S)) -> 0(q0(x1:S))
 1(q0(0(x1:S))) -> 0(0(q1(x1:S)))
 1(q0(1(x1:S))) -> 0(1(q1(x1:S)))
 1(q1(0(x1:S))) -> 1(0(q1(x1:S)))
 1(q1(1(x1:S))) -> 1(1(q1(x1:S)))
 1(q2(x1:S)) -> q2(1(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.