/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem 1: 

(VAR x1)
(RULES
a(q1(b(x1))) -> q2(a(y(x1)))
a(q2(a(x1))) -> q2(a(a(x1)))
a(q2(y(x1))) -> q2(a(y(x1)))
q0(a(x1)) -> x(q1(x1))
q0(y(x1)) -> y(q3(x1))
q1(a(x1)) -> a(q1(x1))
q1(y(x1)) -> y(q1(x1))
q2(x(x1)) -> x(q0(x1))
q3(y(x1)) -> y(q3(x1))
q3(bl(x1)) -> bl(q4(x1))
y(q1(b(x1))) -> q2(y(y(x1)))
y(q2(a(x1))) -> q2(y(a(x1)))
y(q2(y(x1))) -> q2(y(y(x1)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(q1(b(x1))) -> A(y(x1))
 A(q1(b(x1))) -> Q2(a(y(x1)))
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

SCC Processor:
-> Pairs:
 A(q1(b(x1))) -> A(y(x1))
 A(q1(b(x1))) -> Q2(a(y(x1)))
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(q1(b(x1))) -> A(y(x1))
 A(q1(b(x1))) -> Q2(a(y(x1)))
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(q1(b(x1))) -> A(y(x1))
 A(q1(b(x1))) -> Q2(a(y(x1)))
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 1
[y](X) = 2.X + 2
[b](X) = 2.X + 2
[bl](X) = 2.X + 1
[q4](X) = 1
[x](X) = 2.X + 2
[A](X) = 2.X + 2
[Q0](X) = X + 2
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = 2.X + 2
[Y](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A(q1(b(x1))) -> Q2(a(y(x1)))
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(q1(b(x1))) -> Q2(a(y(x1)))
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(q1(b(x1))) -> Q2(a(y(x1)))
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 1
[q0](X) = 2.X + 2
[q1](X) = 2.X
[q2](X) = 2.X + 2
[q3](X) = 2.X
[y](X) = 2.X
[b](X) = 2.X + 2
[bl](X) = X + 2
[q4](X) = X + 2
[x](X) = 2.X + 2
[A](X) = X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X
[Q2](X) = X
[Q3](X) = 2.X + 1
[Y](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(q1(b(x1))) -> Y(x1)
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 2
[y](X) = 2.X + 2
[b](X) = 2.X + 2
[bl](X) = X + 2
[q4](X) = 2.X + 2
[x](X) = 2.X + 2
[A](X) = X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = 2.X + 2
[Y](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(q2(a(x1))) -> A(a(x1))
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[q0](X) = X + 2
[q1](X) = 2.X
[q2](X) = 2.X + 2
[q3](X) = X
[y](X) = 2.X + 1
[b](X) = 2.X + 2
[bl](X) = 2.X + 2
[q4](X) = X
[x](X) = X + 1
[A](X) = 2.X + 1
[Q0](X) = X + 2
[Q1](X) = 2.X + 2
[Q2](X) = 2.X
[Q3](X) = 2.X
[Y](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(q2(a(x1))) -> Q2(a(a(x1)))
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(q2(y(x1))) -> A(y(x1))
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[q0](X) = 2.X + 2
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = 2.X + 2
[y](X) = 2.X + 2
[b](X) = 2.X + 2
[bl](X) = 2
[q4](X) = 2.X + 2
[x](X) = 2.X + 2
[A](X) = X + 2
[Q0](X) = 2.X + 2
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = 2.X + 2
[Y](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(q2(y(x1))) -> Q2(a(y(x1)))
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[q0](X) = 2.X + 1
[q1](X) = 2.X + 2
[q2](X) = 2.X + 2
[q3](X) = X
[y](X) = 2.X + 2
[b](X) = 2.X + 2
[bl](X) = 1
[q4](X) = 2
[x](X) = X + 2
[A](X) = 2.X + 2
[Q0](X) = X + 1
[Q1](X) = 2.X + 2
[Q2](X) = X + 2
[Q3](X) = X + 2
[Y](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> A(q1(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q0(a(x1)) -> Q1(x1)
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
 Q1(y(x1)) -> Y(q1(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q0(y(x1)) -> Q3(x1)
 Q0(y(x1)) -> Y(q3(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->->Cycle:
->->-> Pairs:
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

SCC Processor:
-> Pairs:
 Q0(y(x1)) -> Y(q3(x1))
 Q2(x(x1)) -> Q0(x1)
 Q3(y(x1)) -> Q3(x1)
 Q3(y(x1)) -> Y(q3(x1))
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Q0(y(x1)) -> Y(q3(x1))
 Q2(x(x1)) -> Q0(x1)
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->->Cycle:
->->-> Pairs:
 Q3(y(x1)) -> Q3(x1)
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))


The problem is decomposed in 2 subproblems.

Problem 1.1.1: 

Reduction Pair Processor:
-> Pairs:
 Q0(y(x1)) -> Y(q3(x1))
 Q2(x(x1)) -> Q0(x1)
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2
[q0](X) = 2.X + 1
[q1](X) = 2
[q2](X) = X
[q3](X) = 2
[y](X) = 0
[b](X) = 1
[bl](X) = 2
[q4](X) = 2.X + 2
[x](X) = 1
[Q0](X) = 2
[Q2](X) = 2.X + 1
[Y](X) = 1

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 Q2(x(x1)) -> Q0(x1)
 Y(q1(b(x1))) -> Q2(y(y(x1)))
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Q2(y(a(x1)))
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Q2(y(y(x1)))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1.1.1: 

Reduction Pair Processor:
-> Pairs:
 Y(q1(b(x1))) -> Y(y(x1))
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
-> Usable rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2.X + 2
[q0](X) = 2.X
[q1](X) = 2.X
[q2](X) = 2.X + 2
[q3](X) = 2.X
[y](X) = 2.X
[b](X) = 2.X + 2
[bl](X) = X + 2
[q4](X) = 2.X + 1
[x](X) = 0
[Y](X) = 2.X

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Y(y(x1))
->->-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))

Problem 1.1.1: 

Subterm Processor:
-> Pairs:
 Y(q1(b(x1))) -> Y(x1)
 Y(q2(a(x1))) -> Y(a(x1))
 Y(q2(y(x1))) -> Y(y(x1))
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Projection:
 pi(Y) = 1

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.1.2: 

Subterm Processor:
-> Pairs:
 Q3(y(x1)) -> Q3(x1)
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Projection:
 pi(Q3) = 1

Problem 1.1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 Q1(a(x1)) -> Q1(x1)
 Q1(y(x1)) -> Q1(x1)
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Projection:
 pi(Q1) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(q1(b(x1))) -> q2(a(y(x1)))
 a(q2(a(x1))) -> q2(a(a(x1)))
 a(q2(y(x1))) -> q2(a(y(x1)))
 q0(a(x1)) -> x(q1(x1))
 q0(y(x1)) -> y(q3(x1))
 q1(a(x1)) -> a(q1(x1))
 q1(y(x1)) -> y(q1(x1))
 q2(x(x1)) -> x(q0(x1))
 q3(y(x1)) -> y(q3(x1))
 q3(bl(x1)) -> bl(q4(x1))
 y(q1(b(x1))) -> q2(y(y(x1)))
 y(q2(a(x1))) -> q2(y(a(x1)))
 y(q2(y(x1))) -> q2(y(y(x1)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.