YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(d(x1:S)) -> B(d(x1:S))
 A(x1:S) -> B(b(x1:S))
 A(x1:S) -> B(x1:S)
 B(d(b(x1:S))) -> A(d(x1:S))
 B(d(b(x1:S))) -> D(x1:S)
 D(f(x1:S)) -> B(d(x1:S))
 D(f(x1:S)) -> D(x1:S)
-> Rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
-> Usable rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = X + 3
[b](X) = X + 2/3
[d](X) = 4.X + 3
[c](X) = 0
[f](X) = X + 1/3
[A](X) = 1/2.X + 1/3
[B](X) = 1/2.X
[D](X) = 2.X + 4/3

Problem 1: 

SCC Processor:
-> Pairs:
 A(x1:S) -> B(b(x1:S))
 A(x1:S) -> B(x1:S)
 B(d(b(x1:S))) -> A(d(x1:S))
 B(d(b(x1:S))) -> D(x1:S)
 D(f(x1:S)) -> B(d(x1:S))
 D(f(x1:S)) -> D(x1:S)
-> Rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(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(d(x1:S))
 B(d(b(x1:S))) -> D(x1:S)
 D(f(x1:S)) -> B(d(x1:S))
 D(f(x1:S)) -> D(x1:S)
->->-> Rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(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(d(x1:S))
 B(d(b(x1:S))) -> D(x1:S)
 D(f(x1:S)) -> B(d(x1:S))
 D(f(x1:S)) -> D(x1:S)
-> Rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
-> Usable rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = X + 3
[b](X) = X + 2/3
[d](X) = 4.X + 1/4
[c](X) = 0
[f](X) = X + 1/4
[A](X) = 1/3.X + 3/2
[B](X) = 1/3.X + 1
[D](X) = 4/3.X + 4/3

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(x1:S) -> B(x1:S)
 B(d(b(x1:S))) -> A(d(x1:S))
 B(d(b(x1:S))) -> D(x1:S)
 D(f(x1:S)) -> B(d(x1:S))
 D(f(x1:S)) -> D(x1:S)
-> Rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
-> Usable rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = X + 3
[b](X) = X + 2/3
[d](X) = 4.X + 3/2
[c](X) = 0
[f](X) = X + 1/4
[A](X) = 1/2.X + 4/3
[B](X) = 1/2.X + 3/4
[D](X) = 2.X + 4/3

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 B(d(b(x1:S))) -> D(x1:S)
 D(f(x1:S)) -> B(d(x1:S))
 D(f(x1:S)) -> D(x1:S)
-> Rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
-> Usable rules:
 a(d(x1:S)) -> d(b(c(b(d(x1:S)))))
 a(x1:S) -> b(b(f(b(b(x1:S)))))
 b(d(b(x1:S))) -> a(d(x1:S))
 d(f(x1:S)) -> b(d(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = 1/2.X + 4/3
[b](X) = 3/4.X + 1/3
[d](X) = 4.X
[c](X) = 0
[f](X) = 4/3.X + 1/2
[B](X) = 1/3.X + 4
[D](X) = X + 4

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.