YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.