YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Narrowing Processor:
-> Pairs:
 A(a(x1:S)) -> A(b(b(b(x1:S))))
 A(b(b(c(x1:S)))) -> A(a(a(b(x1:S))))
 A(b(b(c(x1:S)))) -> A(a(b(x1:S)))
 A(b(b(c(x1:S)))) -> A(b(x1:S))
-> Rules:
 a(a(x1:S)) -> a(b(b(b(x1:S))))
 a(b(b(c(x1:S)))) -> a(a(a(b(x1:S))))
 b(a(x1:S)) -> b(b(c(x1:S)))
->Narrowed Pairs:
->->Original Pair:
 A(a(x1:S)) -> A(b(b(b(x1:S))))
->-> Narrowed pairs:
 A(a(a(x1:S))) -> A(b(b(b(b(c(x1:S))))))

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.