YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
b(a(a(b(x1:S)))) -> b(a(b(x1:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(b(a(x1:S))) -> A(a(b(b(a(a(x1:S))))))
 A(b(a(x1:S))) -> A(a(x1:S))
 A(b(a(x1:S))) -> A(b(b(a(a(x1:S)))))
 A(b(a(x1:S))) -> B(a(a(x1:S)))
 A(b(a(x1:S))) -> B(b(a(a(x1:S))))
 B(a(a(b(x1:S)))) -> B(a(b(x1:S)))
-> Rules:
 a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
 b(a(a(b(x1:S)))) -> b(a(b(x1:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 A(b(a(x1:S))) -> A(a(b(b(a(a(x1:S))))))
 A(b(a(x1:S))) -> A(a(x1:S))
 A(b(a(x1:S))) -> A(b(b(a(a(x1:S)))))
 A(b(a(x1:S))) -> B(a(a(x1:S)))
 A(b(a(x1:S))) -> B(b(a(a(x1:S))))
 B(a(a(b(x1:S)))) -> B(a(b(x1:S)))
-> Rules:
 a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
 b(a(a(b(x1:S)))) -> b(a(b(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B(a(a(b(x1:S)))) -> B(a(b(x1:S)))
->->-> Rules:
 a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
 b(a(a(b(x1:S)))) -> b(a(b(x1:S)))
->->Cycle:
->->-> Pairs:
 A(b(a(x1:S))) -> A(a(b(b(a(a(x1:S))))))
 A(b(a(x1:S))) -> A(a(x1:S))
 A(b(a(x1:S))) -> A(b(b(a(a(x1:S)))))
->->-> Rules:
 a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
 b(a(a(b(x1:S)))) -> b(a(b(x1:S)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 B(a(a(b(x1:S)))) -> B(a(b(x1:S)))
-> Rules:
 a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
 b(a(a(b(x1:S)))) -> b(a(b(x1:S)))
->Projection:
 pi(B) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 A(b(a(x1:S))) -> A(b(b(a(a(x1:S)))))
-> Rules:
 a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
 b(a(a(b(x1:S)))) -> b(a(b(x1:S)))
-> Usable rules:
 a(b(a(x1:S))) -> a(a(b(b(a(a(x1:S))))))
 b(a(a(b(x1:S)))) -> b(a(b(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = 2
[b](X) = 1/2.X
[A](X) = X

Problem 1.2: 

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

The problem is finite.