YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
f(s(X:S),X:S) -> f(X:S,a(X:S))
f(X:S,c(X:S)) -> f(s(X:S),X:S)
f(X:S,X:S) -> c(X:S)
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(s(X:S),X:S) -> f(X:S,a(X:S))
 f(X:S,c(X:S)) -> f(s(X:S),X:S)
 f(X:S,X:S) -> c(X:S)
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(X:S),X:S) -> F(X:S,a(X:S))
 F(X:S,c(X:S)) -> F(s(X:S),X:S)
-> Rules:
 f(s(X:S),X:S) -> f(X:S,a(X:S))
 f(X:S,c(X:S)) -> f(s(X:S),X:S)
 f(X:S,X:S) -> c(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(X:S),X:S) -> F(X:S,a(X:S))
 F(X:S,c(X:S)) -> F(s(X:S),X:S)
-> Rules:
 f(s(X:S),X:S) -> f(X:S,a(X:S))
 f(X:S,c(X:S)) -> f(s(X:S),X:S)
 f(X:S,X:S) -> c(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.