YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.