YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.