YES

Problem 1: 

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

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(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:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
 F(s(x:S),y:S) -> -#(s(x:S),y:S)
 F(s(x:S),y:S) -> -#(y:S,s(x:S))
 F(s(x:S),y:S) -> F(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 F(s(x:S),y:S) -> P(-(s(x:S),y:S))
 F(s(x:S),y:S) -> P(-(y:S,s(x:S)))
 F(x:S,s(y:S)) -> -#(s(y:S),x:S)
 F(x:S,s(y:S)) -> -#(x:S,s(y:S))
 F(x:S,s(y:S)) -> F(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 F(x:S,s(y:S)) -> P(-(s(y:S),x:S))
 F(x:S,s(y:S)) -> P(-(x:S,s(y:S)))
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(s(x:S)) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
 F(s(x:S),y:S) -> -#(s(x:S),y:S)
 F(s(x:S),y:S) -> -#(y:S,s(x:S))
 F(s(x:S),y:S) -> F(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 F(s(x:S),y:S) -> P(-(s(x:S),y:S))
 F(s(x:S),y:S) -> P(-(y:S,s(x:S)))
 F(x:S,s(y:S)) -> -#(s(y:S),x:S)
 F(x:S,s(y:S)) -> -#(x:S,s(y:S))
 F(x:S,s(y:S)) -> F(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 F(x:S,s(y:S)) -> P(-(s(y:S),x:S))
 F(x:S,s(y:S)) -> P(-(x:S,s(y:S)))
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(s(x:S)) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
->->-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(s(x:S)) -> x:S
->->Cycle:
->->-> Pairs:
 F(s(x:S),y:S) -> F(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 F(x:S,s(y:S)) -> F(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
->->-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(s(x:S)) -> x:S


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(s(x:S)) -> x:S
->Projection:
 pi(-#) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(s(x:S),y:S) -> F(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 F(x:S,s(y:S)) -> F(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(s(x:S)) -> x:S
-> Usable rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 p(s(x:S)) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = X1 + 1/2.X2
[f](X1,X2) = 0
[p](X) = 1/2.X
[0] = 0
[fSNonEmpty] = 0
[s](X) = 2.X + 2
[-#](X1,X2) = 0
[F](X1,X2) = 2.X1 + 2.X2
[P](X) = 0

Problem 1.2: 

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

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(x:S,s(y:S)) -> F(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 f(s(x:S),y:S) -> f(p(-(s(x:S),y:S)),p(-(y:S,s(x:S))))
 f(x:S,s(y:S)) -> f(p(-(x:S,s(y:S))),p(-(s(y:S),x:S)))
 p(s(x:S)) -> x:S
-> Usable rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 p(s(x:S)) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = X1 + 1/2
[f](X1,X2) = 0
[p](X) = 1/2.X + 1/2
[0] = 0
[fSNonEmpty] = 0
[s](X) = 2.X + 2
[-#](X1,X2) = 0
[F](X1,X2) = 1/2.X2
[P](X) = 0

Problem 1.2: 

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

The problem is finite.