YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(b(x1:S)) -> R(x1:S)
 D(a(x1:S)) -> A(a(d(x1:S)))
 D(a(x1:S)) -> A(d(x1:S))
 D(a(x1:S)) -> D(x1:S)
 D(x1:S) -> A(x1:S)
 R(a(x1:S)) -> D(r(x1:S))
 R(a(x1:S)) -> R(x1:S)
 R(x1:S) -> D(x1:S)
-> Rules:
 a(b(x1:S)) -> b(r(x1:S))
 d(a(x1:S)) -> a(a(d(x1:S)))
 d(x1:S) -> a(x1:S)
 r(a(x1:S)) -> d(r(x1:S))
 r(x1:S) -> d(x1:S)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.