YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
0(q2(0(x1:S))) -> q2(0(0(x1:S)))
0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
q0(0(x1:S)) -> 0'(q1(x1:S))
q0(1'(x1:S)) -> 1'(q3(x1:S))
q1(0(x1:S)) -> 0(q1(x1:S))
q1(1'(x1:S)) -> 1'(q1(x1:S))
q2(0'(x1:S)) -> 0'(q0(x1:S))
q3(1'(x1:S)) -> 1'(q3(x1:S))
q3(b(x1:S)) -> b(q4(x1:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 0#(q1(1(x1:S))) -> 0#(1'(x1:S))
 0#(q1(1(x1:S))) -> 1'#(x1:S)
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q1(1(x1:S))) -> 0#(1'(x1:S))
 0#(q1(1(x1:S))) -> 1'#(x1:S)
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q1(1(x1:S))) -> 0#(1'(x1:S))
 0#(q1(1(x1:S))) -> 1'#(x1:S)
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q1(1(x1:S))) -> 0#(1'(x1:S))
 0#(q1(1(x1:S))) -> 1'#(x1:S)
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 1
[1](X) = 2.X + 2
[b](X) = 2.X + 1
[q4](X) = 2.X + 1
[0#](X) = 2.X + 2
[0'#](X) = 2.X + 2
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q1(1(x1:S))) -> 1'#(x1:S)
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q1(1(x1:S))) -> 1'#(x1:S)
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q1(1(x1:S))) -> 1'#(x1:S)
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[0'](X) = 2
[1'](X) = 1
[q0](X) = 2.X + 2
[q1](X) = 2
[q2](X) = X
[q3](X) = 2.X + 2
[1](X) = X
[b](X) = 2.X + 2
[q4](X) = 2.X
[0#](X) = 2
[0'#](X) = 2
[1'#](X) = 1
[Q0](X) = 2
[Q1](X) = 2
[Q2](X) = X
[Q3](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q1(1(x1:S))) -> Q2(0(1'(x1:S)))
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 2
[1](X) = 2.X + 2
[b](X) = X + 2
[q4](X) = 2.X + 2
[0#](X) = 2.X + 1
[0'#](X) = X + 2
[1'#](X) = 2.X + 1
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q2(0(x1:S))) -> 0#(0(x1:S))
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 2
[1](X) = 2.X + 2
[b](X) = X + 1
[q4](X) = 2
[0#](X) = 2.X + 2
[0'#](X) = 2.X + 2
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q2(0(x1:S))) -> Q2(0(0(x1:S)))
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X
[1](X) = 2.X + 2
[b](X) = 2.X
[q4](X) = 2.X
[0#](X) = 2.X + 2
[0'#](X) = 2.X + 1
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 1
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 1
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q2(1'(x1:S))) -> 0#(1'(x1:S))
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 1
[0'](X) = X
[1'](X) = 2.X
[q0](X) = X
[q1](X) = 2.X
[q2](X) = 2.X + 2
[q3](X) = 0
[1](X) = 2.X + 2
[b](X) = 2.X
[q4](X) = 0
[0#](X) = 2.X + 1
[0'#](X) = 2.X + 1
[1'#](X) = 2.X + 1
[Q0](X) = 2.X + 1
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 1
[Q3](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(q2(1'(x1:S))) -> Q2(0(1'(x1:S)))
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = X
[1](X) = 2.X + 2
[b](X) = 2
[q4](X) = 2.X + 1
[0#](X) = 2.X + 2
[0'#](X) = 2.X
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X
[Q3](X) = X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> 0#(q1(x1:S))
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0'#(q1(1(x1:S))) -> 0'#(1'(x1:S))
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 1
[1](X) = 2.X + 2
[b](X) = X + 1
[q4](X) = X + 2
[0'#](X) = 2.X + 2
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0'#(q1(1(x1:S))) -> 1'#(x1:S)
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 2
[1](X) = 2.X + 2
[b](X) = 2
[q4](X) = X + 2
[0'#](X) = 2.X + 2
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0'#(q1(1(x1:S))) -> Q2(0'(1'(x1:S)))
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 1
[1'](X) = 2.X + 2
[q0](X) = 2.X + 1
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[q4](X) = 1
[0'#](X) = X + 2
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0'#(q2(0(x1:S))) -> 0'#(0(x1:S))
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = X
[1](X) = 2.X + 2
[b](X) = X + 1
[q4](X) = X
[0'#](X) = X + 2
[1'#](X) = X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0'#(q2(0(x1:S))) -> Q2(0'(0(x1:S)))
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X
[1'](X) = 2.X + 2
[q0](X) = 2.X
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 1
[1](X) = 2.X + 2
[b](X) = 1
[q4](X) = 2.X + 2
[0'#](X) = 2.X + 2
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0'#(q2(1'(x1:S))) -> 0'#(1'(x1:S))
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = X
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 1
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[q4](X) = X + 1
[0'#](X) = X
[1'#](X) = 2.X + 2
[Q0](X) = 2.X
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0'#(q2(1'(x1:S))) -> Q2(0'(1'(x1:S)))
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 1
[1'](X) = 2.X + 2
[q0](X) = 2.X + 1
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X
[1](X) = 2.X + 2
[b](X) = X + 2
[q4](X) = 2.X + 2
[0'#](X) = X + 2
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 1
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> 0'#(q1(x1:S))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> 0'#(q0(x1:S))
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1'#(q1(1(x1:S))) -> 1'#(1'(x1:S))
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 1
[1'](X) = 2.X + 2
[q0](X) = 2.X + 1
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 1
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[q4](X) = X + 1
[1'#](X) = 2.X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1'#(q1(1(x1:S))) -> 1'#(x1:S)
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 2
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[q4](X) = 2.X + 2
[1'#](X) = X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = X + 1
[Q3](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1'#(q1(1(x1:S))) -> Q2(1'(1'(x1:S)))
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X
[1](X) = 2.X + 2
[b](X) = X + 2
[q4](X) = 2.X + 2
[1'#](X) = 2.X + 1
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1'#(q2(0(x1:S))) -> 1'#(0(x1:S))
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = X
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 1
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[q4](X) = 2.X
[1'#](X) = X
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 2
[Q3](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1'#(q2(0(x1:S))) -> Q2(1'(0(x1:S)))
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2
[1'](X) = 1
[q0](X) = X + 2
[q1](X) = 2.X + 2
[q2](X) = X
[q3](X) = 2.X + 2
[1](X) = 2.X + 2
[b](X) = 2
[q4](X) = X + 2
[1'#](X) = 2
[Q0](X) = 2
[Q1](X) = 2
[Q2](X) = X
[Q3](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1'#(q2(1'(x1:S))) -> 1'#(1'(x1:S))
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X + 2
[1'](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 2
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[q4](X) = 2.X
[1'#](X) = 2.X + 1
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X + 1
[Q3](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 1'#(q2(1'(x1:S))) -> Q2(1'(1'(x1:S)))
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
-> Usable rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[0'](X) = 2.X
[1'](X) = 1
[q0](X) = X
[q1](X) = X
[q2](X) = X
[q3](X) = 2
[1](X) = X + 2
[b](X) = 0
[q4](X) = X + 2
[1'#](X) = 2
[Q0](X) = 2.X
[Q1](X) = 2
[Q2](X) = X
[Q3](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q0(0(x1:S)) -> Q1(x1:S)
 Q0(1'(x1:S)) -> 1'#(q3(x1:S))
 Q0(1'(x1:S)) -> Q3(x1:S)
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> 1'#(q1(x1:S))
 Q1(1'(x1:S)) -> Q1(x1:S)
 Q2(0'(x1:S)) -> Q0(x1:S)
 Q3(1'(x1:S)) -> 1'#(q3(x1:S))
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q3(1'(x1:S)) -> Q3(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->->Cycle:
->->-> Pairs:
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> Q1(x1:S)
->->-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 Q3(1'(x1:S)) -> Q3(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Projection:
 pi(Q3) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 Q1(0(x1:S)) -> Q1(x1:S)
 Q1(1'(x1:S)) -> Q1(x1:S)
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Projection:
 pi(Q1) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(q1(1(x1:S))) -> q2(0(1'(x1:S)))
 0(q2(0(x1:S))) -> q2(0(0(x1:S)))
 0(q2(1'(x1:S))) -> q2(0(1'(x1:S)))
 0'(q1(1(x1:S))) -> q2(0'(1'(x1:S)))
 0'(q2(0(x1:S))) -> q2(0'(0(x1:S)))
 0'(q2(1'(x1:S))) -> q2(0'(1'(x1:S)))
 1'(q1(1(x1:S))) -> q2(1'(1'(x1:S)))
 1'(q2(0(x1:S))) -> q2(1'(0(x1:S)))
 1'(q2(1'(x1:S))) -> q2(1'(1'(x1:S)))
 q0(0(x1:S)) -> 0'(q1(x1:S))
 q0(1'(x1:S)) -> 1'(q3(x1:S))
 q1(0(x1:S)) -> 0(q1(x1:S))
 q1(1'(x1:S)) -> 1'(q1(x1:S))
 q2(0'(x1:S)) -> 0'(q0(x1:S))
 q3(1'(x1:S)) -> 1'(q3(x1:S))
 q3(b(x1:S)) -> b(q4(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.