YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x1:S)) -> B(b(b(x1:S)))
 A(a(x1:S)) -> B(b(x1:S))
 A(a(x1:S)) -> B(x1:S)
 B(b(b(b(b(x1:S))))) -> A(a(a(x1:S)))
 B(b(b(b(b(x1:S))))) -> A(a(x1:S))
 B(b(b(b(b(x1:S))))) -> A(x1:S)
-> Rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(b(b(b(b(x1:S))))) -> a(a(a(x1:S)))
-> Usable rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(b(b(b(b(x1:S))))) -> a(a(a(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = 3.X + 1/2
[b](X) = 2.X + 1/4
[A](X) = 3.X + 2
[B](X) = 2.X + 4/3

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x1:S)) -> B(b(x1:S))
 A(a(x1:S)) -> B(x1:S)
 B(b(b(b(b(x1:S))))) -> A(a(a(x1:S)))
 B(b(b(b(b(x1:S))))) -> A(a(x1:S))
 B(b(b(b(b(x1:S))))) -> A(x1:S)
-> Rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(b(b(b(b(x1:S))))) -> a(a(a(x1:S)))
-> Usable rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(b(b(b(b(x1:S))))) -> a(a(a(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = 3.X + 4/3
[b](X) = 2.X + 2/3
[A](X) = 2/3.X + 3/4
[B](X) = 3/4.X + 1/3

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x1:S)) -> B(x1:S)
 B(b(b(b(b(x1:S))))) -> A(a(a(x1:S)))
 B(b(b(b(b(x1:S))))) -> A(a(x1:S))
 B(b(b(b(b(x1:S))))) -> A(x1:S)
-> Rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(b(b(b(b(x1:S))))) -> a(a(a(x1:S)))
-> Usable rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(b(b(b(b(x1:S))))) -> a(a(a(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = X + 1/2
[b](X) = X + 1/3
[A](X) = 4.X + 1/2
[B](X) = 4.X + 3/2

Problem 1: 

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

The problem is finite.