YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x1:S)) -> B(b(b(x1:S)))
 A(a(x1:S)) -> B(b(x1:S))
 A(a(x1:S)) -> B(x1:S)
 B(c(d(x1:S))) -> A(x1:S)
 B(c(x1:S)) -> B(x1:S)
-> Rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(c(d(x1:S))) -> a(x1:S)
 b(c(x1:S)) -> c(b(x1:S))
 b(x1:S) -> c(c(d(x1:S)))
 c(x1:S) -> d(d(d(x1:S)))
-> Usable rules:
 a(a(x1:S)) -> b(b(b(x1:S)))
 b(c(d(x1:S))) -> a(x1:S)
 b(c(x1:S)) -> c(b(x1:S))
 b(x1:S) -> c(c(d(x1:S)))
 c(x1:S) -> d(d(d(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
[d](X) = X
[A](X) = 3.X + 2
[B](X) = 3.X + 1/2

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.