YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(f(x:S,a),y:S) -> F(x:S,y:S)
 F(f(x:S,a),y:S) -> F(y:S,f(x:S,y:S))
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))

Problem 1: 

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

Problem 1: 

Forward Instantiation Processor:
-> Pairs:
 F(f(x:S,a),y:S) -> F(x:S,y:S)
 F(f(x:S,a),y:S) -> F(y:S,f(x:S,y:S))
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
->Instantiated Pairs:
->->Original Pair:
 F(f(x:S,a),y:S) -> F(x:S,y:S)
->-> Instantiated pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
->->Original Pair:
 F(f(x:S,a),y:S) -> F(y:S,f(x:S,y:S))
->-> Instantiated pairs:
 F(f(x:S,a),f(x3:S,a)) -> F(f(x3:S,a),f(x:S,f(x3:S,a)))
 F(f(x:S,a),f(x5:S,a)) -> F(f(x5:S,a),f(x:S,f(x5:S,a)))

Problem 1: 

SCC Processor:
-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
 F(f(x:S,a),f(x3:S,a)) -> F(f(x3:S,a),f(x:S,f(x3:S,a)))
 F(f(x:S,a),f(x5:S,a)) -> F(f(x5:S,a),f(x:S,f(x5:S,a)))
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
 F(f(x:S,a),f(x3:S,a)) -> F(f(x3:S,a),f(x:S,f(x3:S,a)))
 F(f(x:S,a),f(x5:S,a)) -> F(f(x5:S,a),f(x:S,f(x5:S,a)))
->->-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
 F(f(x:S,a),f(x3:S,a)) -> F(f(x3:S,a),f(x:S,f(x3:S,a)))
 F(f(x:S,a),f(x5:S,a)) -> F(f(x5:S,a),f(x:S,f(x5:S,a)))
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
-> Usable rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = 1/2.X2
[a] = 2
[F](X1,X2) = X2

Problem 1: 

SCC Processor:
-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
 F(f(x:S,a),f(x5:S,a)) -> F(f(x5:S,a),f(x:S,f(x5:S,a)))
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
 F(f(x:S,a),f(x5:S,a)) -> F(f(x5:S,a),f(x:S,f(x5:S,a)))
->->-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
 F(f(x:S,a),f(x5:S,a)) -> F(f(x5:S,a),f(x:S,f(x5:S,a)))
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
-> Usable rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X1,X2) = 1/2.X2
[a] = 2
[F](X1,X2) = X2

Problem 1: 

SCC Processor:
-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
->->-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))

Problem 1: 

Subterm Processor:
-> Pairs:
 F(f(f(x3:S,a),a),y:S) -> F(f(x3:S,a),y:S)
 F(f(f(x5:S,a),a),y:S) -> F(f(x5:S,a),y:S)
-> Rules:
 f(f(x:S,a),y:S) -> f(y:S,f(x:S,y:S))
->Projection:
 pi(F) = 1

Problem 1: 

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

The problem is finite.