YES

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
b(u(x:S)) -> a(e(x:S))
c(u(x:S)) -> b(x:S)
d(u(x:S)) -> c(x:S)
d(x:S) -> e(u(x:S))
v(e(x:S)) -> x:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 C(u(x:S)) -> B(x:S)
 D(u(x:S)) -> C(x:S)
-> Rules:
 b(u(x:S)) -> a(e(x:S))
 c(u(x:S)) -> b(x:S)
 d(u(x:S)) -> c(x:S)
 d(x:S) -> e(u(x:S))
 v(e(x:S)) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 C(u(x:S)) -> B(x:S)
 D(u(x:S)) -> C(x:S)
-> Rules:
 b(u(x:S)) -> a(e(x:S))
 c(u(x:S)) -> b(x:S)
 d(u(x:S)) -> c(x:S)
 d(x:S) -> e(u(x:S))
 v(e(x:S)) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.