YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(c(x1:S)) -> A(x1:S)
 A(c(x1:S)) -> C(a(x1:S))
 A(l(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> L(a(x1:S))
 C(a(r(x1:S))) -> A(x1:S)
 L(r(a(x1:S))) -> A(l(c(c(r(x1:S)))))
-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))

Problem 1: 

SCC Processor:
-> Pairs:
 A(c(x1:S)) -> A(x1:S)
 A(c(x1:S)) -> C(a(x1:S))
 A(l(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> L(a(x1:S))
 C(a(r(x1:S))) -> A(x1:S)
 L(r(a(x1:S))) -> A(l(c(c(r(x1:S)))))
-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(c(x1:S)) -> A(x1:S)
 A(c(x1:S)) -> C(a(x1:S))
 A(l(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> L(a(x1:S))
 C(a(r(x1:S))) -> A(x1:S)
 L(r(a(x1:S))) -> A(l(c(c(r(x1:S)))))
->->-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 A(c(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> L(a(x1:S))
 C(a(r(x1:S))) -> A(x1:S)
 L(r(a(x1:S))) -> A(l(c(c(r(x1:S)))))
-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(c(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> L(a(x1:S))
 L(r(a(x1:S))) -> A(l(c(c(r(x1:S)))))
->->-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 A(c(x1:S)) -> A(x1:S)
 A(l(x1:S)) -> L(a(x1:S))
 L(r(a(x1:S))) -> A(l(c(c(r(x1:S)))))
-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(l(x1:S)) -> L(a(x1:S))
 L(r(a(x1:S))) -> A(l(c(c(r(x1:S)))))
->->-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))
->->Cycle:
->->-> Pairs:
 A(c(x1:S)) -> A(x1:S)
->->-> Rules:
 a(c(x1:S)) -> c(a(x1:S))
 a(l(x1:S)) -> l(a(x1:S))
 c(a(r(x1:S))) -> r(a(x1:S))
 l(r(a(x1:S))) -> a(l(c(c(r(x1:S)))))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.