YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.