YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.