YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.