YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
a(a(l2(x1:S))) -> l2(a(a(x1:S)))
a(a(x1:S)) -> x1:S
a(l1(x1:S)) -> l1(a(a(a(x1:S))))
b(l1(x1:S)) -> b(r2(x1:S))
b(l2(x1:S)) -> b(r1(x1:S))
r1(a(x1:S)) -> a(a(a(r1(x1:S))))
r1(b(x1:S)) -> l1(b(x1:S))
r2(a(x1:S)) -> a(a(a(r2(x1:S))))
r2(b(x1:S)) -> l2(a(b(x1:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(a(l2(x1:S))) -> A(a(x1:S))
 A(a(l2(x1:S))) -> A(x1:S)
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
 B(l1(x1:S)) -> B(r2(x1:S))
 B(l1(x1:S)) -> R2(x1:S)
 B(l2(x1:S)) -> B(r1(x1:S))
 B(l2(x1:S)) -> R1(x1:S)
 R1(a(x1:S)) -> A(a(a(r1(x1:S))))
 R1(a(x1:S)) -> A(a(r1(x1:S)))
 R1(a(x1:S)) -> A(r1(x1:S))
 R1(a(x1:S)) -> R1(x1:S)
 R2(a(x1:S)) -> A(a(a(r2(x1:S))))
 R2(a(x1:S)) -> A(a(r2(x1:S)))
 R2(a(x1:S)) -> A(r2(x1:S))
 R2(a(x1:S)) -> R2(x1:S)
 R2(b(x1:S)) -> A(b(x1:S))
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(l2(x1:S))) -> A(a(x1:S))
 A(a(l2(x1:S))) -> A(x1:S)
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
 B(l1(x1:S)) -> B(r2(x1:S))
 B(l1(x1:S)) -> R2(x1:S)
 B(l2(x1:S)) -> B(r1(x1:S))
 B(l2(x1:S)) -> R1(x1:S)
 R1(a(x1:S)) -> A(a(a(r1(x1:S))))
 R1(a(x1:S)) -> A(a(r1(x1:S)))
 R1(a(x1:S)) -> A(r1(x1:S))
 R1(a(x1:S)) -> R1(x1:S)
 R2(a(x1:S)) -> A(a(a(r2(x1:S))))
 R2(a(x1:S)) -> A(a(r2(x1:S)))
 R2(a(x1:S)) -> A(r2(x1:S))
 R2(a(x1:S)) -> R2(x1:S)
 R2(b(x1:S)) -> A(b(x1:S))
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(l2(x1:S))) -> A(a(x1:S))
 A(a(l2(x1:S))) -> A(x1:S)
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->->Cycle:
->->-> Pairs:
 R1(a(x1:S)) -> R1(x1:S)
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->->Cycle:
->->-> Pairs:
 R2(a(x1:S)) -> R2(x1:S)
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->->Cycle:
->->-> Pairs:
 B(l1(x1:S)) -> B(r2(x1:S))
 B(l2(x1:S)) -> B(r1(x1:S))
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 A(a(l2(x1:S))) -> A(a(x1:S))
 A(a(l2(x1:S))) -> A(x1:S)
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
-> Usable rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[l1](X) = 2.X
[l2](X) = 2.X + 2
[A](X) = 2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 A(a(l2(x1:S))) -> A(x1:S)
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(l2(x1:S))) -> A(x1:S)
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 A(a(l2(x1:S))) -> A(x1:S)
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
-> Usable rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[l1](X) = 2.X + 1
[l2](X) = 2.X + 2
[A](X) = 2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 A(l1(x1:S)) -> A(a(a(x1:S)))
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
-> Usable rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[l1](X) = 2.X + 2
[l2](X) = 0
[A](X) = 2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 A(l1(x1:S)) -> A(a(x1:S))
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
-> Usable rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[l1](X) = X + 2
[l2](X) = 0
[A](X) = 2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(l1(x1:S)) -> A(x1:S)
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))

Problem 1.1: 

Subterm Processor:
-> Pairs:
 A(l1(x1:S)) -> A(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Projection:
 pi(A) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 R1(a(x1:S)) -> R1(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Projection:
 pi(R1) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 R2(a(x1:S)) -> R2(x1:S)
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Projection:
 pi(R2) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 B(l1(x1:S)) -> B(r2(x1:S))
 B(l2(x1:S)) -> B(r1(x1:S))
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
-> Usable rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[a](X) = [0 1;1 0].X
[b](X) = [1 0;1 0].X + [0;1]
[r1](X) = [0 1;1 0].X + [1;1]
[r2](X) = [0 1;1 0].X
[l1](X) = [0 1;1 0].X + [1;1]
[l2](X) = [0 1;0 0].X + [1;0]
[B](X) = [1 0;1 0].X

Problem 1.4: 

SCC Processor:
-> Pairs:
 B(l2(x1:S)) -> B(r1(x1:S))
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B(l2(x1:S)) -> B(r1(x1:S))
->->-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 B(l2(x1:S)) -> B(r1(x1:S))
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
-> Usable rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[b](X) = 0
[r1](X) = 2.X
[r2](X) = 2
[l1](X) = 2.X
[l2](X) = 2.X + 2
[B](X) = 2.X

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(l2(x1:S))) -> l2(a(a(x1:S)))
 a(a(x1:S)) -> x1:S
 a(l1(x1:S)) -> l1(a(a(a(x1:S))))
 b(l1(x1:S)) -> b(r2(x1:S))
 b(l2(x1:S)) -> b(r1(x1:S))
 r1(a(x1:S)) -> a(a(a(r1(x1:S))))
 r1(b(x1:S)) -> l1(b(x1:S))
 r2(a(x1:S)) -> a(a(a(r2(x1:S))))
 r2(b(x1:S)) -> l2(a(b(x1:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.