YES

Problem 1: 

(VAR v_NonEmpty:S x1:S x2:S x3:S x4:S x5:S)
(RULES
f(0,0,0,0,0) -> 0
f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(0,0,0,0,s(x5:S)) -> F(x5:S,x5:S,x5:S,x5:S,x5:S)
 F(0,0,0,s(x4:S),x5:S) -> F(x4:S,x4:S,x4:S,x4:S,x5:S)
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)

Problem 1: 

SCC Processor:
-> Pairs:
 F(0,0,0,0,s(x5:S)) -> F(x5:S,x5:S,x5:S,x5:S,x5:S)
 F(0,0,0,s(x4:S),x5:S) -> F(x4:S,x4:S,x4:S,x4:S,x5:S)
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,0,0,0,s(x5:S)) -> F(x5:S,x5:S,x5:S,x5:S,x5:S)
 F(0,0,0,s(x4:S),x5:S) -> F(x4:S,x4:S,x4:S,x4:S,x5:S)
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
->->-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 F(0,0,0,0,s(x5:S)) -> F(x5:S,x5:S,x5:S,x5:S,x5:S)
 F(0,0,0,s(x4:S),x5:S) -> F(x4:S,x4:S,x4:S,x4:S,x5:S)
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Projection:
 pi(F) = 5

Problem 1: 

SCC Processor:
-> Pairs:
 F(0,0,0,s(x4:S),x5:S) -> F(x4:S,x4:S,x4:S,x4:S,x5:S)
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,0,0,s(x4:S),x5:S) -> F(x4:S,x4:S,x4:S,x4:S,x5:S)
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
->->-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 F(0,0,0,s(x4:S),x5:S) -> F(x4:S,x4:S,x4:S,x4:S,x5:S)
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Projection:
 pi(F) = 4

Problem 1: 

SCC Processor:
-> Pairs:
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
->->-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 F(0,0,s(x3:S),x4:S,x5:S) -> F(x3:S,x3:S,x3:S,x4:S,x5:S)
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Projection:
 pi(F) = 3

Problem 1: 

SCC Processor:
-> Pairs:
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
->->-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 F(0,s(x2:S),x3:S,x4:S,x5:S) -> F(x2:S,x2:S,x3:S,x4:S,x5:S)
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Projection:
 pi(F) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
->->-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 F(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> F(x1:S,x2:S,x3:S,x4:S,x5:S)
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Projection:
 pi(F) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0,0,0,0,0) -> 0
 f(0,0,0,0,s(x5:S)) -> f(x5:S,x5:S,x5:S,x5:S,x5:S)
 f(0,0,0,s(x4:S),x5:S) -> f(x4:S,x4:S,x4:S,x4:S,x5:S)
 f(0,0,s(x3:S),x4:S,x5:S) -> f(x3:S,x3:S,x3:S,x4:S,x5:S)
 f(0,s(x2:S),x3:S,x4:S,x5:S) -> f(x2:S,x2:S,x3:S,x4:S,x5:S)
 f(s(x1:S),x2:S,x3:S,x4:S,x5:S) -> f(x1:S,x2:S,x3:S,x4:S,x5:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.