YES

Problem 1: 

(VAR v_NonEmpty:S x0:S x1:S x2:S x3:S)
(RULES
p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(b(x1:S))),p(a(a(x0:S)),x3:S))
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(x0:S)),x3:S)
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(b(x1:S))),p(a(a(x0:S)),x3:S))
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(x0:S)),x3:S)
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(b(x1:S))),p(a(a(x0:S)),x3:S))
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(x0:S)),x3:S)
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->->-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(b(x1:S))),p(a(a(x0:S)),x3:S))
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(x0:S)),x3:S)
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Usable rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[p](X1,X2) = X1 + X2 + 2
[a](X) = 2.X
[b](X) = 0
[P](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(x0:S)),x3:S)
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(x0:S)),x3:S)
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->->-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(a(a(x0:S)),x3:S)
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Usable rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[p](X1,X2) = 2.X1 + X2 + 2
[a](X) = 2.X
[b](X) = 0
[P](X1,X2) = 2.X1 + X2

Problem 1: 

SCC Processor:
-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->->-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 P(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> P(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
-> Usable rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[p](X1,X2) = 1/2.X1 + 2/3.X2
[a](X) = 3.X + 3/2
[b](X) = 0
[P](X1,X2) = 4.X1 + 4.X2

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 p(a(a(x0:S)),p(x1:S,p(a(x2:S),x3:S))) -> p(x2:S,p(a(a(b(x1:S))),p(a(a(x0:S)),x3:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.