YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
a(p(x1:S)) -> p(a(A(x1:S)))
a(A(x1:S)) -> A(a(x1:S))
p(A(A(x1:S))) -> a(p(x1:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 a#(p(x1:S)) -> a#(A(x1:S))
 a#(p(x1:S)) -> P(a(A(x1:S)))
 a#(A(x1:S)) -> a#(x1:S)
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))

Problem 1: 

SCC Processor:
-> Pairs:
 a#(p(x1:S)) -> a#(A(x1:S))
 a#(p(x1:S)) -> P(a(A(x1:S)))
 a#(A(x1:S)) -> a#(x1:S)
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 a#(p(x1:S)) -> a#(A(x1:S))
 a#(p(x1:S)) -> P(a(A(x1:S)))
 a#(A(x1:S)) -> a#(x1:S)
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
->->-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 a#(p(x1:S)) -> a#(A(x1:S))
 a#(p(x1:S)) -> P(a(A(x1:S)))
 a#(A(x1:S)) -> a#(x1:S)
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
-> Usable rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[p](X) = X + 1
[A](X) = X
[a#](X) = X + 1
[P](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 a#(A(x1:S)) -> a#(x1:S)
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 a#(A(x1:S)) -> a#(x1:S)
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
->->-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 a#(A(x1:S)) -> a#(x1:S)
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
-> Usable rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[p](X) = 1
[A](X) = 2.X + 2
[a#](X) = 2.X
[P](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
->->-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 P(A(A(x1:S))) -> a#(p(x1:S))
 P(A(A(x1:S))) -> P(x1:S)
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
-> Usable rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X + 1
[p](X) = 1/2.X + 1
[A](X) = X + 1
[a#](X) = 2.X + 1/2
[P](X) = X + 1/2

Problem 1: 

SCC Processor:
-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 P(A(A(x1:S))) -> a#(p(x1:S))
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 P(A(A(x1:S))) -> a#(p(x1:S))
->->-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 a#(p(x1:S)) -> P(a(A(x1:S)))
 P(A(A(x1:S))) -> a#(p(x1:S))
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
-> Usable rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[a](X) = X + 2/3
[p](X) = 1/4.X + 1
[A](X) = X + 2
[a#](X) = 4/3.X + 3
[P](X) = 1/3.X + 3

Problem 1: 

SCC Processor:
-> Pairs:
 P(A(A(x1:S))) -> a#(p(x1:S))
-> Rules:
 a(p(x1:S)) -> p(a(A(x1:S)))
 a(A(x1:S)) -> A(a(x1:S))
 p(A(A(x1:S))) -> a(p(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.