YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.