YES

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
a(d,0) -> 0
a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
a(f,0) -> a(s,0)
a(p,a(s,x:S)) -> x:S
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> 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:
 A(d,a(s,x:S)) -> A(d,a(p,a(s,x:S)))
 A(d,a(s,x:S)) -> A(p,a(s,x:S))
 A(d,a(s,x:S)) -> A(s,a(d,a(p,a(s,x:S))))
 A(d,a(s,x:S)) -> A(s,a(s,a(d,a(p,a(s,x:S)))))
 A(f,a(s,x:S)) -> A(d,a(f,a(p,a(s,x:S))))
 A(f,a(s,x:S)) -> A(f,a(p,a(s,x:S)))
 A(f,a(s,x:S)) -> A(p,a(s,x:S))
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 A(d,a(s,x:S)) -> A(d,a(p,a(s,x:S)))
 A(d,a(s,x:S)) -> A(p,a(s,x:S))
 A(d,a(s,x:S)) -> A(s,a(d,a(p,a(s,x:S))))
 A(d,a(s,x:S)) -> A(s,a(s,a(d,a(p,a(s,x:S)))))
 A(f,a(s,x:S)) -> A(d,a(f,a(p,a(s,x:S))))
 A(f,a(s,x:S)) -> A(f,a(p,a(s,x:S)))
 A(f,a(s,x:S)) -> A(p,a(s,x:S))
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(d,a(s,x:S)) -> A(d,a(p,a(s,x:S)))
->->-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->->Cycle:
->->-> Pairs:
 A(f,a(s,x:S)) -> A(f,a(p,a(s,x:S)))
->->-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Narrowing Processor:
-> Pairs:
 A(d,a(s,x:S)) -> A(d,a(p,a(s,x:S)))
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Narrowed Pairs:
->->Original Pair:
 A(d,a(s,x:S)) -> A(d,a(p,a(s,x:S)))
->-> Narrowed pairs:
 A(d,a(s,x:S)) -> A(d,x:S)

Problem 1.1: 

SCC Processor:
-> Pairs:
 A(d,a(s,x:S)) -> A(d,x:S)
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(d,a(s,x:S)) -> A(d,x:S)
->->-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S

Problem 1.1: 

Subterm Processor:
-> Pairs:
 A(d,a(s,x:S)) -> A(d,x:S)
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Projection:
 pi(A) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Narrowing Processor:
-> Pairs:
 A(f,a(s,x:S)) -> A(f,a(p,a(s,x:S)))
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Narrowed Pairs:
->->Original Pair:
 A(f,a(s,x:S)) -> A(f,a(p,a(s,x:S)))
->-> Narrowed pairs:
 A(f,a(s,x:S)) -> A(f,x:S)

Problem 1.2: 

SCC Processor:
-> Pairs:
 A(f,a(s,x:S)) -> A(f,x:S)
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(f,a(s,x:S)) -> A(f,x:S)
->->-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S

Problem 1.2: 

Subterm Processor:
-> Pairs:
 A(f,a(s,x:S)) -> A(f,x:S)
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Projection:
 pi(A) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(d,a(s,x:S)) -> a(s,a(s,a(d,a(p,a(s,x:S)))))
 a(d,0) -> 0
 a(f,a(s,x:S)) -> a(d,a(f,a(p,a(s,x:S))))
 a(f,0) -> a(s,0)
 a(p,a(s,x:S)) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.