YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
P(p(x1:S)) -> x1:S
R(r(x1:S)) -> x1:S
R(x1:S) -> r(x1:S)
p(P(x1:S)) -> x1:S
r(P(P(x1:S))) -> P(P(r(x1:S)))
r(R(x1:S)) -> x1:S
r(p(x1:S)) -> p(p(r(P(x1:S))))
r(r(x1:S)) -> x1:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 R#(x1:S) -> r#(x1:S)
 r#(P(P(x1:S))) -> P#(P(r(x1:S)))
 r#(P(P(x1:S))) -> P#(r(x1:S))
 r#(P(P(x1:S))) -> r#(x1:S)
 r#(p(x1:S)) -> P#(x1:S)
 r#(p(x1:S)) -> p#(p(r(P(x1:S))))
 r#(p(x1:S)) -> p#(r(P(x1:S)))
 r#(p(x1:S)) -> r#(P(x1:S))
-> Rules:
 P(p(x1:S)) -> x1:S
 R(r(x1:S)) -> x1:S
 R(x1:S) -> r(x1:S)
 p(P(x1:S)) -> x1:S
 r(P(P(x1:S))) -> P(P(r(x1:S)))
 r(R(x1:S)) -> x1:S
 r(p(x1:S)) -> p(p(r(P(x1:S))))
 r(r(x1:S)) -> x1:S

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 P(p(x1:S)) -> x1:S
 R(r(x1:S)) -> x1:S
 R(x1:S) -> r(x1:S)
 p(P(x1:S)) -> x1:S
 r(P(P(x1:S))) -> P(P(r(x1:S)))
 r(R(x1:S)) -> x1:S
 r(p(x1:S)) -> p(p(r(P(x1:S))))
 r(r(x1:S)) -> x1:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.