YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.