YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
0(x1:S) -> 1(x1:S)
4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 4#(5(4(5(x1:S)))) -> 4#(4(5(5(x1:S))))
 4#(5(4(5(x1:S)))) -> 4#(5(5(x1:S)))
 4#(5(4(5(x1:S)))) -> 5#(5(x1:S))
-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)

Problem 1: 

SCC Processor:
-> Pairs:
 4#(5(4(5(x1:S)))) -> 4#(4(5(5(x1:S))))
 4#(5(4(5(x1:S)))) -> 4#(5(5(x1:S)))
 4#(5(4(5(x1:S)))) -> 5#(5(x1:S))
-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 4#(5(4(5(x1:S)))) -> 4#(4(5(5(x1:S))))
 4#(5(4(5(x1:S)))) -> 4#(5(5(x1:S)))
->->-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 4#(5(4(5(x1:S)))) -> 4#(4(5(5(x1:S))))
 4#(5(4(5(x1:S)))) -> 4#(5(5(x1:S)))
-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
-> Usable rules:
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[4](X) = X + 2
[5](X) = 2.X + 2
[2](X) = X + 1
[4#](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 4#(5(4(5(x1:S)))) -> 4#(5(5(x1:S)))
-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 4#(5(4(5(x1:S)))) -> 4#(5(5(x1:S)))
->->-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 4#(5(4(5(x1:S)))) -> 4#(5(5(x1:S)))
-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
-> Usable rules:
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[4](X) = 2.X + 2
[5](X) = 2.X
[2](X) = 2
[4#](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(x1:S) -> 1(x1:S)
 4(5(4(5(x1:S)))) -> 4(4(5(5(x1:S))))
 5(5(5(5(5(5(4(4(4(4(4(4(x1:S)))))))))))) -> 2(x1:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.