YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 L#(a(a(x1:S))) -> L#(a(b(c(x1:S))))
 L#(a(a(x1:S))) -> B(c(x1:S))
 L#(a(a(x1:S))) -> C(x1:S)
 B(a(a(x1:S))) -> B(c(x1:S))
 B(a(a(x1:S))) -> C(x1:S)
 C(b(x1:S)) -> B(a(x1:S))
 C(a(x1:S)) -> C(x1:S)
-> Rules:
 L(a(a(x1:S))) -> L(a(b(c(x1:S))))
 b(a(a(x1:S))) -> a(b(c(x1:S)))
 c(b(x1:S)) -> b(a(x1:S))
 c(R(x1:S)) -> b(a(R(x1:S)))
 c(a(x1:S)) -> a(c(x1:S))

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.